Development of a Novel Visualization and Measurement Apparatus for the PVT Behaviours of

Development of a Novel Visualization and Measurement Apparatus for the PVT Behaviours of Polymer/Gas Solutions

by

Yao Gai Gary Li

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Department of Mechanical and Industrial Engineering University of Toronto

© Copyright by Yao Gai Gary Li 2008

University of Toronto Mechanical and Industrial Engineering Department

ii

Development of a Novel Visualization and Measurement Apparatus for the

PVT Behaviours of Polymer/Gas Solutions

Yao Gai Gary Li

Degree of Doctor of Philosophy 2008 Graduate Department of Mechanical and Industrial Engineering

University of Toronto

ABSTRACT

The Pressure-Volume-Temperature (PVT) for polymer/gas solutions is an important

fundamental property of which accurate data measurement has not been reported until recently.

The diffusivity, solubility, and surface tension are critical physical properties of polymer/gas

solutions in understanding and controlling polymer processing such as, foaming, blending, and

extracting reaction. However, the determination of these properties relies on accurate PVT data

as a prerequisite. Due to the difficulties involved in measuring the specific volume while

maintaining a sufficiently high pressure and temperature to achieve a single-phase polymer/gas

solution, accurate PVT data or volume swelling measurement of polymer/gas solutions is not yet

available. In this research, a new methodology was proposed and developed for direct

measuring the PVT properties of polymer melts saturated with high-pressure gas at elevated

temperatures. The ultimate goal is to develop and construct an apparatus that would provide

more accurate fundamental properties through PVT measurement to the foaming industry,

which is heavily involved with polymer/gas mixtures.


iii

In memory of my grandfather Liang who passed away during my Ph.D. program

your love, support and encouragement will always be

in my heart and my soul

To those who are affected

in the catastrophic earthquake in my home province

at the extremely difficult time our hearts unite

May God Bless Us and Our Nation

To my great parents

give me the chance to live for their endless love and support

during good and bad times I owe it with all of me


iv

Acknowledgements

I would like to express my sincere gratitude to my supervisor Professor Chul B. Park for

providing guidance and encouragement throughout my research. I would like also to thank my

Ph. D. thesis committee: Prof. Yu Sun, Prof. Hani Naguib Prof. Craig Simmons and the external

examiner Prof. Musa Kamal.

My gratitude is extended to the Department of Mechanical and Industrial Engineering at

the University of Toronto for providing the University of Toronto Open Fellowships. Especially

I like to thank Brenda Fung from the Graduate Office for her help during my graduate study

years. I also like to thank Sheila Baker from the purchasing office for making all the purchasing

an easy job. Also my thanks go to Teresa Wang and Geoffrey Chow. I like to also thank Oscar

Del Rio for his support in dealing with computer problems.

I also wish to acknowledge the professional technical support from Jeff Sansome, Dave

Eisdaile, Mike Smith, Len Rooseman, and Thai Do in the Machine Tool Laboratory.

I would like to thank my colleagues in the Microcellular Plastics Manufacturing

Laboratory for their support and friendship. With their help and encouragement, the Ph.D.

became more enjoyable. So my special thanks go to Dr. Jin Wang, Dr. Donglai Xu, Dr.

Gangjian Guo, Dr. Guangming Li, Dr. Hongbo Li, Dr. Patrick Lee, Dr. Qingping Guo, Dr.

Zhenjing Zhu, Dr. Takashi Kuboki, Dr. Kyungmin Lee, Dr. Balasubramanian Maridass, Dr.

Bhuwnesh Kumar, Dr. Kelvin Lee, Dr. Xia Liao, Dr. Wenli Zhu, Lilac Cuiling Wang, Sue

Chang, Kelly Jinjin Zhang, Nan Chen, Wang Jing, Sunny Leung, Anson Wong, Raymond Chu,

Jeff Qingfeng Wu, Richard Lee, John Lee, Peter Jung, Ryan Kim, Mohammed Hassan,

Mohammed Serry, Johnny Park, Kanghong Lee, Esther Lee. Their support is so valuable in

helping me to finish the thesis.


v

My students Fedir Gul, Satbinder Dhesy and Yu Shuang Zhu also contributed a lot to

this research project.

I also want to thank Dr. Dean Peachy, Melissa Miller and their son Daniel of my home

stay family. They have become my Canadian family while I am far away from home. They did

not just provide me a place to live, but also give me the parental care, support and help at the

crucial first year when I tried to get acquainted to this new environment.

A Chinese proverb says “At home one relies on one’s parents, and outside on one’s

friends”. I had felt this tremendously from the understanding, support, and encouragement I

have received from my friends over those years. Although we may be far apart from some of

you at this moment, but the friendship would last forever. Could not really name each every of

them here, but just a few, they are Uncle David Xue, Auntie Huang, Uncle Hao’s family, Auntie

Kong’s family, Emmanuel Chen’s family, Dr. Hongbo Li’s family, Ross and Nancy Hayes,

Jeannette Zhang, Yvonne Feng, Audrey Lin, Alfred Sum, Jean Hsu, Xuan Yang, Tara Wu.

A special tribute needs to be paid to April Binnie for proof reading my thesis.

Finally I like to take this chance to express my deepest gratitude to my family, my

relatives and friends in China. Their endless love, continuing support, tremendous amount of

encouragement and patience made it all possible for me to completed school journey today.


vi

Table of Contents

Abstract …………………………………………………………………………………………ii

Dedication ………………………………………………………………………………………iii

Acknowledgements …………………………………………………………………………….iv

Table of Contents ……………………………………………………………………………....vi

List of Figures ………………………………………………………………………………….xi

Nomenclature ……………………………………………………………………………….....xv

Chapter 1. Introduction ................................................................................................................ 1

1.1 Preamble ........................................................................................................................... 1 1.2 Thermoplastic Foams ....................................................................................................... 2 1.3 Plastic Foaming Process ................................................................................................... 4

1.3.1 Introduction ..................................................................................................... 4 1.3.2 Batch Process ................................................................................................... 5 1.3.3 Continuous Process .......................................................................................... 6

1.4 PVT Measurement ............................................................................................................ 9 1.5 Role of PVT in Solubility and Surface Tension ............................................................. 10 1.6 Motivation and Scope/ Objective ................................................................................... 13 1.7 Thesis Contribution ........................................................................................................ 15 1.8 Thesis Organization ........................................................................................................ 16

Chapter 2. Literature Review ..................................................................................................... 19

2.1 Background on PVT Property Measurement .................................................................. 19 2.1.1 PVT Measurement for Pure Polymer ............................................................ 19 2.1.2 PVT Measurement for Polymer/Gas Solutions ............................................. 21

2.2 Background on Polymer/Gas Solutions Property ........................................................... 27 2.2.1 Solubility and Diffusivity .............................................................................. 27 2.2.2 Nucleation ...................................................................................................... 35 2.2.3 Surface Tension ............................................................................................. 37

2.2.3.1 Spinning Drop Method ...............................................................41 2.2.3.2 Drop Shape Techniques .............................................................42

2.3 Background on 3-D Object Reconstruction from 2-D Images ....................................... 45 2.3.1 Introduction ................................................................................................... 45 2.3.2 Active 3-D Reconstruction Method ............................................................... 45


vii

2.3.3 Passive 3-D Reconstruction Method ............................................................. 46 2.3.3.1 Multi-view Stereo Method .........................................................47 2.3.3.2 Model-based Multi-view Stereo Method ...................................48 2.3.3.3 Volumetric Modeling Method ...................................................49 2.3.3.3.1 Shape from Silhouettes ...........................................................50 2.3.3.3.2 Shape from Photoconsistency .................................................55

2.4 Summary ......................................................................................................................... 56 Chapter 3. Theoretical Background .......................................................................................... 58

3.1 Theoretical Background on Equation of State ................................................................ 58 3.1.1 Introduction ................................................................................................... 58 3.1.2 Equation of State (EOS) ................................................................................ 59 3.1.3 Equations of State for Polymers and Polymer/Gas Solutions ....................... 60

3.1.3.1 Introduction ................................................................................60 3.1.3.2 Sanchez-Lacombe (SL) EOS .....................................................61 3.1.3.3 Simha-Somcynsky (SS) EOS .....................................................62 3.1.3.4 Statistical Associating Fluid Theory (SAFT) .............................63

3.2 Models of PVT Measurement ......................................................................................... 64 3.3 Solubility Measurement Using MSB .............................................................................. 66 3.4 Viscoelastic Models of Polymers ................................................................................... 68

3.4.1 Maxwell Equation .......................................................................................... 69 3.4.2 Boltzmann’s Superposition Principle ............................................................ 70 3.4.3 Doi-Edwards Model and Entanglement ........................................................ 70

3.5 Surface Tension Measurement ....................................................................................... 71 3.6 Summary ......................................................................................................................... 74

Chapter 4. Design and Construction of Novel Apparatus for PVT measurement ................ 75

4.1 Conceptual Design of PVT Measurement Apparatus ..................................................... 75 4.1.1 Introduction ................................................................................................... 75 4.1.2 Analysis of the PVT Apparatus for Polymer/Gas Solutions ......................... 77 4.1.3 Detailed Analysis and Decomposition of FRs and DPs ................................ 78

4.1.3.1 Hardware Attribute ....................................................................78 4.1.3.2 Future Decomposition of FR1 and DP1 (Second Level) ...........83 4.1.3.3 Software Attribute ......................................................................86

4.2 Detailed Design and Construction of the PVT Apparatus .............................................. 89 4.2.1 Overview of the Apparatus ............................................................................ 89 4.2.2 High Pressure and Temperature Visualization Chamber .............................. 90 4.2.3 Charged Couple Device (CCD) Camera and Optical Lens ........................... 93


viii

4.2.4 XY Stage and Precision Control .................................................................... 94 4.2.5 Rotational Device .......................................................................................... 94 4.2.6 Image Processing Terminal ........................................................................... 97 4.2.7 Light Source, Noise and Vibration Control ................................................... 97

4.3 Theory Background and Algorithms Construction ........................................................ 99 4.3.1 Edge Detection and Volume Integration for Axisymmetric Drop Shape .... 101 4.3.2 Algorithms for 3D Volumetric Calculation of Asymmetric Drop Shape .... 104

4.3.3.1 Rotation Device and Degree of Asymmetry ............................105 4.3.3.2 Modeling of Degree of Asymmetry: Radial Asymmetry .........106 4.3.3.3 Determination of the Reference Radius ...................................107 4.3.3.4 Definition of Degree of Asymmetry (DOA) ............................109 4.3.3.5 Volume Integration of Asymmetry Drop Shape ......................110

4.4 Integration of Hardware and Software GUI Interface .................................................. 113 4.4.1 XY Stage Control Software ......................................................................... 113 4.4.2 GUI Construction for Image Capture and Rotation ..................................... 114

4.5 Summary ....................................................................................................................... 115 Chapter 5. Validation of the Proposed Design ........................................................................ 117

5.1 Introduction .................................................................................................................. 117 5.2 Empirical Verification of Error Reduction from Image Reconstruction ...................... 117 5.3 Empirical Verification Using Tait Equation for Pure Polymer PVT ............................ 119

5.3.1 Experimental Procedure and Materials ........................................................ 119 5.3.2 Density Measurement for Pure Axisymmetric Linear and Branched PP .... 120 5.3.3 Density Measurement for Pure PS685D with Asymmetry .......................... 123

5.4 Summary ....................................................................................................................... 128 Chapter 6. Measurement of the PVT Data for Polymer/Gas Solutions ................................ 130

6.1 Introduction .................................................................................................................. 130 6.2 PVT Measurement with Axisymmetry for Linear/Branched PP/CO2 Solutions .......... 130

6.2.1 Experimental Procedure .............................................................................. 130 6.2.2 Experimental Setup ...................................................................................... 132 6.2.3 Experimental Materials ................................................................................ 132 6.2.4 Volume Swelling of Linear/Branched PP/CO2 ........................................... 132 6.2.5 Effect of Temperature and Pressure on PVT of PP/CO2 Solutions ............. 136

6.2.5.1 Effect of Temperature on Volume Swelling ............................136 6.2.5.2 Effect of Pressure on Volume Swelling ...................................136

6.2.6 Effect of Branch Structure on PVT of PP/CO2 Solutions ........................... 137 6.3 Measurement of PVT Data for PS and PS/Gas Solutions ............................................ 141


ix

6.3.1 Measurement of PVT Data with Asymmetry for PS and PS/CO2 ............... 141 6.3.1.1 Introduction ..............................................................................141 6.3.1.2 Experiment Procedure ..............................................................142 6.3.1.3 Experimental Material ..............................................................143

6.3.2 The Temperature Effect on the Degree of Asymmetry ............................... 143 6.3.3 Temperature and Pressure Effect of PVT of PS/CO2 .................................. 145 6.3.4 Measurement of PVT for PS/HFC-152a Solution ....................................... 146

6.3.4.1 Introduction ..............................................................................146 6.3.4.2 Experimental Procedure and Material ......................................148 6.3.4.3 Temperature and Pressure Effect of PVT for PS/HFC-152a ...149

6.4 Validity of Equation of States Using Experimental Results ......................................... 150 6.4.1 Volume Swelling From EOS Predictions .................................................... 150 6.4.2 Polymer Chain Entanglement Hypothesis ................................................... 154

6.4.2.1 Introduction ..............................................................................154 6.4.2.2 Small Amplitude Oscillatory Shear Method ............................156

6.4.3 Experimental Procedure and Materials ........................................................ 157 6.4.3.1 Sample Preparation ..................................................................158 6.4.3.2 Gap Zeroing .............................................................................159

6.4.4 Chain Entanglement of Linear and Branched PP ........................................ 160 6.4.4.1 Strain Sweep and Frequency Test ............................................160 6.4.4.2 Chain Entanglement of Linear and Branched PP .....................166

6.5 Determination of Accurate Solubility Based on the PVT Data .................................... 167 6.5.1 Corrected PP/CO2 Solubility from PVT Measurement ............................... 167 6.5.2 Specific Volume of Linear/Branched PP/CO2 ............................................. 174

6.6 Summary ....................................................................................................................... 178 Chapter 7. Measurement of Surface Tension for Polymer/Gas Solutions ........................... 181

7.1 Introduction .................................................................................................................. 181 7.2 Experimental Materials ................................................................................................. 184 7.3 Measurement of Surface Tension for Linear/Branch PP/Gas Solutions ...................... 184

7.3.1 Surface Tension for Linear/Branched PP/CO2 Solutions ............................ 184 7.3.2 Temperature and Pressure Effect on Surface Tension ................................. 188 7.3.3 Branch Effect on Surface Tension between Linear and Branched PP......... 192

7.4 Summary ....................................................................................................................... 195 Chapter 8. Conclusions and Future Work .............................................................................. 197

8.1 Conclusions .................................................................................................................. 197 8.2 Future work ................................................................................................................... 201


x

REFERENCES…..………………………………………………………………………….. 203 Appendix 1…………………………………………………………………………………… 233

Appendix 2...…………………………………………………………………………………. 243

Appendix 3...…………………………………………………………………………………. 256

Appendix 4...…………………………………………………………………………………. 259

Appendix 5...…………………………………………………………………………………. 260

Appendix 6...…………………………………………………………………………………. 261

Appendix 7...…………………………………………………………………………………. 262

Appendix 8...…………………………………………………………………………………. 264


xi

List of Figures

Figure 1-1 Overview of Microcellular Foaming Process ................................................................ 5

Figure 1-2 Overview of a Batch Foaming Process .......................................................................... 5

Figure 1-3 Schematic of a Single Foaming Extruder ...................................................................... 6

Figure 1-4 MSB Solubility Apparatus Schematic and Working Principle .................................... 11

Figure 1-5 Building Blocks of Foaming Research and Industry ................................................... 14

Figure 2-1 Schematic of Bellow Type Dilatometer ....................................................................... 20

Figure 2-2 Schematic of LVDT Dilatometer ................................................................................. 24

Figure 2-3 Schematic of Modified MSB for Polymer/Gas PVT Measurement ............................ 26

Figure 2-4 Schematic of Critical Nucleation Radius and Activation Energy ................................ 36

Figure 2-5 Surface Tension and Gravity of Small Particle at Liquid/Particle Interface* ............. 38

Figure 2-6 3-D Object Reconstruction Scheme ............................................................................. 51

Figure 2-7 Visual Hull Construction from Volume Intersection ................................................... 52

Figure 3-1 Generalized Maxwell Model Represented by Spring-Damper Systems...................... 69

Figure 3-2 Sessile Drop Coordinate System Definition ................................................................ 73

Figure 4-1 Axiomatic Design Mapping Process from FRs to DPs ................................................ 76

Figure 4-2 PVT System Schematic ............................................................................................... 78

Figure 4-3 CAD Model of Chamber Body with Rotational Device .............................................. 92

Figure 4-4 CAD Model of the High T and P Visualization Chamber ........................................... 92

Figure 4-5 Location of Rotational Device and High T and P Rotary Seal .................................... 96


xii

Figure 4-6 Actual PVT Apparatus Setup ....................................................................................... 99

Figure 4-7 Methodology Flowchart for Axisymmetry Drop ....................................................... 100

Figure 4-8 Image Reconstruction Schematic ............................................................................... 101

Figure 4-9 Detected Sessile Drop Edge and Volume Integration over Vertical Span................. 102

Figure 4-10 Axisymmetric Drop Side and Top View Schematic ................................................ 103

Figure 4-11 Volume Determination at ith Level of Axisymmetric Drop Shape .......................... 103

Figure 4-12 Methodology Flowchart for Asymmetry Drop ........................................................ 106

Figure 4-13 Schematic of Reference Radius Definition .............................................................. 108

Figure 4-14 Radius Asymmetry at ith Level of Asymmetric Drop Shape ................................... 110

Figure 4-15 Asymmetric Circular Area after Nθ Rotations at ith Level ....................................... 111

Figure 4-16 Area Determination of the Circular Section within Angle θ ................................... 111

Figure 4-17 Volume Determination at ith Level of Asymmetric Drop Shape ............................. 112

Figure 4-18 XY Stage Software Control GUI ............................................................................. 113

Figure 4-19 GUI Interface of the Image Capture and Rotation ................................................... 114

Figure 5-1 Accuracy Improvement from Sphere Image Reconstruction ..................................... 118

Figure 5-2 Pure Linear PP Measured and Tait Calculated Densities .......................................... 122

Figure 5-3 Pure Branched PP Measured and Tait Calculated Densities ..................................... 123

Figure 5-4 Actual Sessile Drop Profile and Level Selection at 150oC ........................................ 124

Figure 5-5 Profile Top View at 150oC ......................................................................................... 124




xiii

Figure 5-8 Profile Top View at 200oC from Cooling .................................................................. 126

Figure 5-9 Profile Top View at 150oC from Cooling .................................................................. 126

Figure 5-10 Density Measurement from Asymmetry Drop Shape and Tait Equation ................ 127

Figure 6-1 Linear PP/CO2 Swelling vs. a) Temperatures and b) Pressures................................. 134

Figure 6-2 Branched PP/CO2 Swelling vs. a) Temperatures and b) Pressures ............................ 135

Figure 6-3 Linear and Branched PP/CO2 Swelling at 180oC ...................................................... 138



Figure 6-6 Schematic of Semicrystalline Polymer Structure* .................................................... 141

Figure 6-7 Degree of Asymmetry of Pure PS at Temperature Rising and Cooling Process ....... 144

Figure 6-8 PS/CO2 Volume Swelling vs. a) Temperatures and b) Pressures .............................. 146

Figure 6-9 PS/HFC-152a Volume Swelling at 150oC and 190oC ............................................... 150

Figure 6-10 Linear PP/CO2 and EOS Swelling at 180oC ............................................................ 151



Figure 6-13 Branched PP/CO2 and EOS Swelling at 180oC ....................................................... 152



Figure 6-16 Schematic of SAOS Experiment Wave ................................................................... 157

Figure 6-17 ARES Rheometer ..................................................................................................... 158

Figure 6-18 Linear PP Strain Sweep Test.................................................................................... 161

Figure 6-19 Branched PP Strain Sweep Test ............................................................................... 161


xiv

Figure 6-20 Linear PP 180oC Frequency Sweep Test ................................................................. 162



Figure 6-23 Branched 180oC Frequency Sweep Test .................................................................. 164

Figure 6-24 Branched PP 200oC Frequency Sweep Test ............................................................ 165

Figure 6-25 Branched PP 220oC Frequency Sweep Test ............................................................ 165

Figure 6-26 Linear and Branched PP Chain Entanglement Density at T above Tm .................... 166

Figure 6-27 Corrected Linear PP/CO2 Solubility at 180oC ......................................................... 170



Figure 6-30 Corrected Solubility of Branched PP/CO2 at 180oC ................................................ 172



Figure 6-33 Specific Volume of LPP/CO2 and BPP/CO2 at 180oC ............................................ 176



Figure 7-1 Surface Tension of Linear PP/CO2 at Various Temperatures and Pressures ............. 188

Figure 7-2 Equilibrium Surface Tension of PS/CO2 Solution ..................................................... 189

Figure 7-3 Surface Tension of Branched PP/CO2 at Various Temperatures and Pressures ........ 191

Figure 7-4 Surface Tension of Linear and Branched PP/CO2 at 180oC ...................................... 192




xv

Nomenclature

A = Helmholtz free energy (J) Ai = asymmetric circular shape area at ith layer of the drop profile (cm2) Aθi = area of each section on the circular shape within angle θi (cm2) [A] = design matrix

resA = residual Helmholtz free energy (J/mol) segmentA = segment Helmholtz energy per mole of molecules (J/mol)

chainA = Helmholtz energy increment due to bonding per mole of molecules (J/mol)

nassociatioA = Helmholtz energy due to associating per mole of molecules (J/mol)

2α = cross section of the penetrate molecule area (m2) 0B = constant 0β = strain amplitude

c = capillary constant C = solution concentration (mol/cm3) c = the number of external degrees of freedom per chain

sC = solubility of gas in the polymer (g-gas/g-polymer) D = diffusion coefficient (m2/s)

DPs = design parameters DE = cohesive energy density (J/mol*m3)

asymmetryE = radial degree of asymmetry (m2) FRs = function requirements

mFΔ = free energy of mixing (J) g = gravity (m/s2)

iG = relaxation modulus (Pa)

NG = plateau modulus (Pa) ( )ω'G 'G = storage modulus (Pa)

''G = loss modulus (Pa) ∗Δ homG = Gibbs free energy (J) h = thickness of the sample sheet (cm)

sHΔ = molar heat of sorption (J) H = enthalpy (J) H0 = Henry’s law constant

homJ = homogenous nucleation rate (per unit volume of polymer) (#/m3.s)

hetJ = heterogeneous nucleation rate (per unit area of nucleating agent)( #/m2.s)

K = Boltzmann’s constant ≈1.380658x10-23 (J/K)

12k = binary interaction parameter dimensionless L = final sample length (cm)


xvi

0L = initial sample length (cm) M = Molecular weight

tM = mass uptake at time t (g)

∞M = equilibrium mass uptake after an infinite time (g) eM = entanglement molecular weight (g) masssamplem = initial sample mass (g)

spherem = sphere ball mass (g) mixturem = polymer/gas solution mass (g)

Mrotation = Number of rotation integer number N = number of points integer number

Nlayer = number of layer along drop profile integer number AN = Avogadro’s constant ≈6.0221367x1023 molecules/mol

Nφ = number of small circular section within angle φ P* = characteristic pressure (MPa) P~ = reduced pressure dimensionless

bubbleP = bubble pressure (Pa) sysP = solution system pressure (Pa)

PΔ = pressure difference across the gas and polymer/gas interface (Pa)

sP = saturation pressure (Pa) platep = perimeter of the plate (m) R = gas constant (J/K*mol) R1 = first principle radii of curvature (m) R2 = second principle radii of curvature (m) R0 = radius of curvature at the origin (m)

crR = critical nuclei radius (m) 0,θjr = drop radius at any angle (m)

tippedestalr = radius of sessile pedestal tip (mm) S = entropy (J) s = number of segments per chain dimensionless wS = volume swelling ratio dimensionless

T = Temperature (K or oC) T~ = reduced temperature dimensionless T* = characteristic temperature (K or oC) teq = equilibrium time (s) U = system internal energy (J)

fυ = total specific free-volume accessible for diffusion (cm3/g)

pυ = specific volume of the polymer (cm3/g)

0υ = specific occupied volume of the polymer chains (cm3/g) v = velocity of propagation (m/s)

V* = characteristic volume (cm3/g) 0v = constant


xvii

1v = constant VΔ = change of sample volume (cm3) 0V = initial sample volume (cm3) BV = total volume of sample holder (cm3) PV = polymer sample volume (cm3) sV = swellon volume (cm3)

V(T,P teq) = equilibrium polymer/gas solution volume (cm3) V(T,P,tini) = initial volume of pure polymer (cm3)

volumesampleinitalV = initial sample volume (cm3) taitV = Tait calculated volume (cm3) θ,jV = volume at any angle (cm3)

sphereV = sphere ball volume (cm3) PPbranchedυ = specific volume of branched PP (cm3/g)

PPlinearυ = specific volume of linear PP (cm3/g) W = free barrier energy for nucleation (J) ( )TPW , = balance readout at P and T (g)

gW = gas dissolution in polymer (g) ( )TW ,0 = balance readout at vacuum and T (g) ω = rotational velocities (radian/s) ψ = phase angle (degree) x = interaction parameter dimensionless

apparentX = gas apparent solubility (g-gas/g-polymer) correctedX = corrected solubility (g-gas/g-polymer)

segZ = compressibility factor from segment chainZ = compressibility factor from chain

nassociatioZ = compressibility factor from association gφ = volume fraction of gas dimensionless

pφ = volume fraction of polymer dimensionless

gn = molar fraction of gas

pn = molar fraction of polymer lgγ = surface tension between gas and polymer/gas interface (mJ/m2)λ = jump length (m) θ = contact angle (degree) λ = wavelength (m) Ω = number of states accessible to the system dimensionless ∗ρ = characteristic density (g/cm3)

ρ~ = reduced density dimensionless sphereρ = sphere ball density (g/cm3)

gasρ = gas density (g/cm3)

iτ = relaxation time (s)


xviii

iσ = stress (Pa)

iη = viscosity (Pa.s) •

λ = shear deformation rate (1/s) ( 0,1θ+− jNx , 0,θjx ) = coordinate on the profile at any angle

σ = stress amplitude (Pascal) ( )ωη∗ = complex viscosity (Pa.s)

mixturep,ρ = polymer/gas solution density (g/cm3)


1

Chapter 1. Introduction

1.1 Preamble

The consumption of foam products in today’s technology continues to grow at

exponential rates worldwide. A simple and most general definition of foam can be described as

a substance that is formed by trapping many gas bubbles in a liquid or solid. Foamed polymers

are used in a variety of applications in the modern world because of their unique properties

compared with unfoamed products. Some of those properties include weight reduction, superior

insulating abilities, buoyancy, and energy dissipation, convenience and comfort features.

Polymeric foams have a wide range of applications from furniture, bedding, insulation,

packaging, appliances, sports applications, shock and sound absorbers, to the transportation and

automotive industry. Some specific examples of using polymeric foams include food trays,

insulation boards used for building construction, automobile seat, bumpers fascia, headliners

and sports helmets, etc.

There are two major classes of polymeric foams: thermoplastic and thermoset foams. The

thermoplastic foams can be recycled and reprocessed, but the thermoset foams are not

recyclable and are intractable due to their heavily cross-linked chain structure and their non

thermo-reversibility[1]. Polymeric foams can also be further classified as flexible, semi-flexible,

rigid or semi-rigid depending on the composition, cell morphology and other physical and

thermal properties[2]. Polymeric foams are produced by a variety of processes, depending on the

applications. Several known foam processing techniques include extrusion, injection molding,

compression molding, blow molding, rotational molding and thermoforming. With the

increasing demand of the foam market in the past decades, a vast amount of effort has also been


2

invested on the fundamental study of polymeric foams. Furthermore, studies were conducted

related to the foaming process in order to be able to improve and optimize the current

technology for developing innovative materials which could achieve better performance and

regulatory compliance.

1.2 Thermoplastic Foams

Plastic foams or cellular plastics are normally referred to as cellular, expanded or sponge

plastics. They generally consist of two phases, a solid polymer matrix and gaseous phase [2].The

gas phase typically exists in the polymer matrix as dispersed voids or cells throughout the

material. This cellular structure is formed by introducing either physical blowing agents (PBAs)

such as volatile liquids, inert gases that can be quickly released upon thermodynamic instability

or chemical blowing agents (CBAs) that can generate gases during the plastic foam process.

The cellular structure of foams or cellular materials offers many advantages over traditional

materials and non-cellular polymers. As previously mentioned, there are two types of foams:

thermoplastics and thermoset. Due to their thermo-reversible nature, thermoplastics are the

preferred class of material for processing purposes. The production of thermoplastic and

thermosetting materials has contributed appreciably to the development of the polymeric foam

industry, which itself dates back to the first half of 20th century. The production of the first

synthetic plastic foam dates back a few decades. In 1931, Swedish inventors C. G. Munters and

J. G. Tandberg invented the extrusion process for foaming polystyrene (PS) and their patent was

filed in 1935 in United States[3,4].

Commercial production of Styrofoam®, developed by Dow Chemical Co. using a

continuous extrusion process, began in 1943. The process of utilizing expandable polystyrene


3

granules containing a solvent blowing agent for the production of polystyrene foam was

developed by Badische Anilin and Soda Fabrik (BASF) of Germany in the 1950’s [5]. In the

1930s both cellular polyvinyl chloride (PVC) and urethane foams were reported to be in

production in Germany [6]. Polyethylene foams were first introduced in a DuPont patent, U.S.

Pat. 2,256,483, in 1942. In 1958, Dow first commercialized a highly expanded polyethylene

foam product prepared by the extrusion process [7]. Polypropylene (PP) foams were introduced

only a couple of decades ago. In the years following their discovery, PP became one of today’s

most widely used polymeric materials because of their relatively high service temperature and

good abrasion resistance. For instance, polypropylene foams are widely used in the automobile

industry in areas such as, instrument panels, door trims, and quarter panels. In particular, the use

of molded PP foam beads as energy absorbers in car bumpers was one of the greatest

innovations in foam use [2]. High melt strength polypropylene (HMS PP), first presented by

Himont [8], with branched long chains, demonstrated a wider processing window and a much

better foaming structure over the linear conventional PP [9,10]. The world market of PP has

grown from approximately 1.5 million tons in the 1970’s, to over 25 million tons in the year

2000 [11].

There are many other types of foams besides those aforementioned. These include foams

based on butadiene-styrene, neoprene, Poly (Acrylonitrile, Butadiene, Styrene) (ABS), cellulose

and many others. However, during the last decade, efforts have been made to achieve

thermoplastic foams with finer cell sizes and more uniform cell distribution because of their

better mechanical and thermal properties[12-18]. All in all, the steady growth of polymeric foam

consumption in the last several decades is strong evidence of the important role foams play in

our society.


4

1.3 Plastic Foaming Process

1.3.1 Introduction

Foaming is a unique technology and, as stated previously, almost every single type of

polymer material or resin could be used to make foam products. Having the right foaming

technique is crucial for foam product manufacturing. Thermoplastic foams are produced by the

creation of a dispersed gaseous phase through a polymer melt and gas expansion process. The

gaseous phase can be generated by the separation of a dissolved gas, vaporization of a volatile

liquid, or through the release of gas from a chemical reaction by adding chemical blowing

agents.

The added compound in the polymer melt to release gas is called a blowing agent (BA).

There are two types of blowing agents depending on their roles in gas generation: 1) physical

blowing agents (PBAs), and 2) chemical blowing agents (CBAs). PBAs are mostly liquids or

supercritical fluids, such as CO2, and N2. They do not undergo chemical transformation and will

be released in a gaseous phase upon thermodynamic instability, i.e. reduction in pressure or

increase in temperature. CBAs, on the other hand, are the compounds or mixtures that are added

into a polymer melt to release gas as result of chemical reactions. CBAs were not used in any of

the experiments involved in this research study. The PBA used in the experiments is CO2. There

exist two distinctive foaming processes, namely, batch foaming and continuous foaming. For

both batch and continuous foaming processes, there are three important stages involved: 1)

polymer/gas solution formation, 2) cell nucleation, and 3) cell growth. Figure 1-1 is an

illustration of the 3 stages of the microcelluar foaming process.


5

+gas

polymer

Two Systems Two Phase

Polymer/Gas Mixture Single Phase

Polymer/Gas Solution Thermodynamic

Instability Micocellular Structure

gas injection mixing & diffusion diffusion nucleation cell growth

Figure 1-1 Overview of Microcellular Foaming Process 1.3.2 Batch Process

Batch foaming process is the simplest foaming method. It consists of a single stage or

multi-stages. The following figure shows a simple schematic of a batch foaming process.

pressurized cavity

pressure vessel o-ringsample

inert gas

control

glycerin bath

sample

FOAMINGSATURATION

Figure 1-2 Overview of a Batch Foaming Process

Batch foaming confines the polymer resins or pellets in an enclosed pressure vessel

when the vessel is heated and pressurized. Sometimes this is described as the polymer pellets

being impregnated with a blowing agent in an autoclave. In batch foaming, time is required for

the blowing agents to saturate the polymer pellets. Typically it takes several hours to several

days for a polymer sheet to become saturated. The amount of time required depends on the

polymer dimension, the diffusivity and the saturation temperature. The impregnated polymer

pellets are expanded or foamed upon an abrupt discharge of the pellets into the atmospheric


6

pressure or by heating up the unexpanded parts using a steam or oil bath, as shown on the right

of Figure 1-2. The batch process is primitive, but it comprises the three important characteristics

of the foaming process: 1) polymer/gas solution forms upon saturation of the polymer pellets; 2)

cell nucleation and 3) cell growth. The cell nucleation and cell growth happen together

following the dramatic change in thermodynamic instability brought by the abrupt change in

pressure or temperature.

1.3.3 Continuous Process

The continuous process is the expanded version of the batch foaming process that

realizes each of those three characteristics at each individual stage of the foaming in

chronological order. The aforementioned extrusion and injection moldings are typical

continuous foaming processes. During the foaming process, different mechanisms and

parameters such as, temperature, pressure and shear, etc. can occur and sometimes overlap in

different stages. The whole foaming process is a very complicated kinetic procedure in which a

large amount of research has been dedicated to explore and study its mechanisms, the effect of

each parameter, as well as their combined effects. Using a very simple schematic of a typical

single extruder shown in Figure 1-3, the continuous process can be clearly and better illustrated.

polymer/gas solution formation

plastication of polymer nucleation

hopper

plasticating screw nucleation device

diffusion enhancing device

high pressure gas

cell growth

cooling unit

Figure 1-3 Schematic of a Single Foaming Extruder


7

The first stage is the polymer/gas solution formation stage or the stage of gas dissolution.

In this stage, the blowing agents such as, CO2 or N2 are injected under high pressure into the

mixing section of the extruder. The blowing agents are well mixed with the molten polymer at

relatively high temperatures, usually above the polymer melting point Tm for semi-crystalline

polymers or glass transition temperature Tg for amorphous polymers. The solution formation is

governed by gas diffusion into the polymer. The diffusion processes are relatively slow and the

cycle time is relatively long. Few techniques have been used to increase the diffusion process,

such as temperature increase or pressure increase. Increasing the temperature could expedite the

diffusivity and also increasing the gas pressure could also boost the diffusion rate by increasing

the local gas concentration. Since the diffusion governs how much gas could dissolve into the

polymer matrix, the solubility limit is the overriding parameter in this stage. The amount of gas

dissolved into the polymer is a controllable processing parameter that affects melt solution

homogenization, foaming dynamics, and stability. The Flory-Huggins equation (1-1) is a

mathematical mean to determine how much gas can be dissolved in the polymer [19].

( )pgppggm xnnnKTF φφφ ++=Δ lnln (1-1)

Where mFΔ is free energy of mixing, K is the Boltzmann’s constant, T is the absolute

temperature in Kelvin, gφ is the volume fraction of gas, pφ is the volume fraction of polymer.

gn and pn are molar fraction of gas and polymer, x is the interaction parameter to take into

account the energy of the interdispersing polymer and gas molecules.

The second stage is the cell nucleation. This stage subjects the polymer/gas solution

from stage one, which forms the polymer/gas single solution, to a thermodynamic instability in

order to nucleate bubbles. This thermodynamic instability is achieved by rapidly lowering the

gas solubility in the polymer via either a temperature increase and/or a pressure drop. Upon the


8

introduction of thermodynamic instability, the single phase solution system then looks for a state

of lower free energy by clustering gas molecules in the form of cell nuclei. The formation of cell

nuclei provides a relatively small mean free distance for the gas molecules in solution to diffuse

through. This cell nucleation process happens when the polymer/gas melt is decompressed at the

nucleation device shown in Figure 1-4.

There are two types of cell nucleation: homogenous and heterogeneous nucleation.

Homogenous nucleation happens throughout the material and the heterogeneous nucleation

happens at high-energy regions such as phase boundaries. From thermodynamic law, the critical

bubble radius is expressed in equation (1-2) as follows:

( )PPRbcr −

∗= γ2 (1-2)

Where crR is the critical nuclei radius, γ is the surface tension between the gas and polymer/gas

interface, Pb is the bubble pressure and P is the surrounding or polymer/gas solution pressure.

According to the homogeneous nucleation theory [20], the Gibbs free energy for the

formation of a nucleus with critical size is given by

32hom

3

16γ

π

PG

Δ=Δ ∗ (1-3)

The PΔ is the pressure difference across the gas and polymer/gas interface, γ is again the

surface tension. The bubble nucleation process is a critical stage in plastic foam production

since it governs the cell morphology and in return determines the properties of plastic foams.

The nucleation process will be described in detail in Chapter 2.

The last stage of the continuous foaming process is the cell growth. Once the cell nuclei

are formed, they tend to grow if the size is bigger than the critical bubble radius crR . The cell

continues to grow and the polymer density is reduced as the gas that is initially dissolved in the


9

polymer matrix which surrounds the nuclei, diffuse into the nucleated cells. Cell growth is also

important in that it determines the final foam density as well as cell morphology. The growth of

cells is limited by the diffusion rate, the stiffness of the viscoelastic polymer/gas solution, the

availability of gas inside the solution, and the time allowed for cells to grow.

After the cell growth stage, the temperatures of polymer melt decreases as it leaves the

nucleation device or die exit.

1.4 PVT Measurement

Thermoplastic foamed products have become very popular in recent years.

Thermoplastic foams exhibit unique mechanical and thermal properties including cushioning,

impact resistance, insulation and buoyancy. They are distinguishable from their unfoamed

counterparts by virtue of their cellular structure. The unique cellular or microcellular structure is

achieved through the expansion of a blowing agent dissolved in a molten thermoplastic resin by

a batch or continuous foam process.

When gas dissolves into a molten polymer during the polymer/gas formation stage, the

polymer swells (dilates) due to gas sorption. Upon the introduction of thermodynamic instability

such as, temperature increase or dramatic pressure decrease in the polymer/gas single solution,

the cell will nucleate homogenously or heterogeneously and grow from the dissolved gas to

eventually form the cellular foam. Therefore, the pressure-volume-temperature (PVT)

relationship for polymeric materials is a subject of great importance and interest to polymer

scientists and engineers, particularly from a process design standpoint. The PVT

(pressure-volume-temperature) properties of a polymer/gas solution can be established by

measuring the amount of polymer swelling as a function of the concentration of dissolved gas at

certain temperatures and pressure levels. . Information on the PVT properties is crucial for the


10

successful processing of plastic foams because both the cell nucleation and cell growth

behaviors are strongly dependent on these properties. The importance of the PVT property,

especially the PVT of polymer/gas solutions towards solubility and surface tensions, will be

described more clearly in the next section when the role of PVT in the solubility and surface

tension is depicted.

The PVT for pure polymer has been extensively studied in the past decade through

empirical and theoretical methods. A comprehensive PVT database for a variety of pure

polymers has already been established[21-24]. The study of PVT on polymer/gas solutions

(mixtures or binary system) has been the focus for many researchers in the past decade as well.

Using the gravimetric method seems to have been the most popular method for conducting the

study, but most of the experimental conditions were not at elevated temperatures and the results

were not extensive. Hence, the conclusion here is that the PVT properties of polymer/gas

solutions at high temperatures and pressures have not been extensively investigated in the

molten state, due to the difficulties of maintaining the high pressure and high temperature

required in the measurement.

1.5 Role of PVT in Solubility and Surface Tension

In continuous foaming, supercritical fluids used as blowing agents such as CO2 or N2 are

always the most important aspects of the study because how they react with the polymer matrix

determines the quality of the final foam products. Mentioned earlier, the formation of the

polymer/gas solution is part of the foaming process. How much blowing agents are indeed

needed to be able to form a homogenous polymer/gas single solution depends on information

provided on gas solubility. The most popular methods used to determine solubility is the


11

gravimetric method using the magnetic suspension balance (MSB) system and theoretical EOS

calculations. The following figure shows a schematic of a MSB and its working principle.

High pressure gasVacuum

F(0,T)microbalance microbalance

F(P,T)

Polymer Sample

Solubility

Uncompensated Solubility

Volume Swelling Compensation

Buoyancy Effect

( ) ( ) ( )sPBgasg

VVVTWTPWW +++−= ρ,0,

High pressure gasVacuum

F(0,T)microbalance microbalance

F(P,T)

Polymer Sample

Solubility

Uncompensated Solubility

Volume Swelling Compensation

Buoyancy Effect

( ) ( ) ( )sPBgasg

VVVTWTPWW +++−= ρ,0,

Figure 1-4 MSB Solubility Apparatus Schematic and Working Principle

As seen from the above figure, the volume VB (volume of the coupling device plus the

volume sample container), volume of polymer sample Vp, and the swelling volume Vs of the

polymer account for the buoyancy effect and need to be considered in order to obtain reliable

solubility data. The first two volumes could be determined relatively easily, but the bottleneck

issue is how to determine the real volume dilation Vs of polymer due to gas dissolution. MSB

could monitor the weight change accurately but not the volumetric changes. Therefore, due to

the lack of the volume swelling information because of the difficulties in measuring direct

volume swelling, the MSB solubility study has to rely on theoretical equations of state (EOS) in

order to make the volume swelling predictions. Although the EOS have been proven to be

reliable when they are used to analyze the property of pure substances and mixtures, it has not

been thoroughly verified when they are used for swelling prediction in solubility calculations.

One of the reasons is that due to the complexity of the molecular reactions among polymer/gas


12

matrices under various high temperature and pressure conditions, the EOS that is based on pure

thermodynamic theories could not properly model those molecular reactions and behaviours

even with proposed assumptions. Therefore, the role of obtaining accurate volume swelling

information, i.e. the PVT data when forming the polymer/gas single solution is important not

just to facilitate better solubility measurement but also to enable a good polymer/gas single

solution formation. Naturally, the question is how PVT plays an important role in surface

tension measurement application.

During the second stage of the foaming process, the critical nucleation radius, the free

energy barrier, and the nucleation rate are physical factors that would definitely influence the

outcome of the nucleation stage macroscopically. When examining the equations (1-2, 1-3, 2-9,

2-10, 2-11, 2-12) that used to quantify those factors, it is found that one of the most crucial

parameters is the surface tension, or interfacial tension (γ ) between the gas bubble and the

surrounding polymer/gas mixture. A simple definition from a molecular point of view is that the

surface tension is the intermolecular force that contracts the surface. As is self-explanatory from

those equations, the surface tension parameters do play a significant role in determining the free

energy barrier, critical cell radius, cell nucleation rate, which would all eventually determine the

cell nucleation density and the cell morphology.

There have been intensive studies and literature conducted on the surface tension.

Among them, the sessile drop method based on the Axial-symmetric Drop Shape Analysis

(ADSA) was used, which is one of the most popular and efficient methods to obtain the surface

tension data. The ADSA method has been used by many researchers to study surface tension

of not only polymer materials, but also other substances. Since supercritical CO2 is a promising

solvent for application in polymer blending and foaming, intensive surface tension studies of

polymer/CO2 solutions have also been carried out from various literatures.[25-27] In the next two


13

chapters, when the detailed mathematical formulas are unveiled, we will discover not

surprisingly, that the density information of the polymer/gas mixture is one of the key input

parameters for surface tension determination. As for polymer/gas mixtures studies, the current

method of determining the density is either using the solubility from MSB combining with the

EOS volume swelling prediction or using the EOS calculation alone. This suggests that accurate

surface tension information also relies on accurate PVT information due to the requirement for

accurate polymer/gas densities information.

In conclusion, in the two most critical stages of the foaming process, it is a strong

demonstration that the solubility of gas in polymer and surface tension (or the interfacial tension)

between the gas and polymer/gas interface have dominating effects on the solution formation

and cell nucleation stages. More importantly, without the accurate PVT information of

polymer/gas solution, both solubility and surface tension would not be obtained accurately.

Therefore the method of how to obtain the accurate PVT data is the most crucial

behind-the-scenes factor in facilitating more reliable solubility and surface tension

measurements. Additionally, properties such as diffusivity of the gas in the melt, the shear and

extensional viscosities of polymer/gas solutions are also strongly influenced by the PVT data of

the respective polymer/gas solution.

1.6 Motivation and Scope/ Objective

The polymer foaming industry is divided into three major building blocks in Figure 1-6:

1) fundamental study, 2) simulation research, and 3) polymer foaming processing. The

fundamental and simulation studies serve as the sources of new and innovative knowledge for

supporting and improving the know-how and technology in real foaming process. These

fundamental properties could also be directly utilized in the foaming process without going


14

through the simulation stage. Any improvement in fundamental and simulation would lead to

potential success to the polymer foaming industry overall. As shown in the following diagram,

the PVT properties, solubility and surface tension of polymer and polymer /gas mixtures along

with others belong to the fundamental section.

Fundamental

MaterialsRheologySolubility/DiffusivityPVTSurface TensionCrystallization

Simulation

Experimental SimulationComputer Simulation

Processing

ExtrusionInjection moldingRotomoldingCompression Molding

Figure 1-5 Building Blocks of Foaming Research and Industry

Precise solubility measurement using the Magnetic Suspension Balance (MSB) method

requires the true volume swelling (PVT) data of the molten polymer to account for the buoyancy

effect. The validity of the Sanchez and Lacombe (SL) and Simha and Somcynsky (SS) EOS

used to theoretically predict the volume dilation would also be important and necessary from the

accurately measured PVT information. In order to improve the design of the nucleation device

to achieve the best cell morphology and to have better foam properties, the surface tension is

influential on the nucleation stages of the foaming process. Precise surface tension measurement

using the ADSA method also requires accurate measured PVT data for determining the density

of the polymer/gas solution. Therefore, developing a new experimental apparatus that would

measure PVT information is a significant contribution. It will assist not only in obtaining more

accurate solubility and surface tension data, etc. but in providing more accurate fundamental

data for improvement in simulation studies. Thereby it will improve the foaming

experimentation which would eventually bring more innovative improvement to the foaming


15

industry. In addition, having accurately measured PVT information will help to verify the EOS

predictions on volume swelling. Those serve as the biggest motivation of this thesis.

The ultimate objective is to develop and construct a novel apparatus for the first time to

accurately measure the PVT properties of various polymer/gas solutions over a wide range of

elevated temperatures and pressures. Consequently, a database for the PVT properties of various

polymer/gas solutions, such as PP/CO2, PS/CO2, PP/N2 will be built upon the measurement

results for the first time.

1.7 Thesis Contribution

For the first time in academic study of polymers and the polymer foaming industry, the

design and construction of a novel apparatus that can measure the PVT property of polymer/gas

solutions at elevated temperatures and pressures is proposed and realized in this research. The

major contributions of this research are broken down as follows:

1. Successfully designed and constructed a novel apparatus to experimentally measure the

PVT property of polymer/gas under elevated temperatures and pressure for the first time

2. Conducted experiments on the measurement of PVT for pure polymer (PP and PS)

3. Conducted experiments on the measurement of PVT for polymer/gas solutions (PP/CO2,

PS/CO2)

4. Obtained accurate CO2 solubility data in PP measured from MSB based on the measured

PVT data for PP/CO2 solution

5. Obtained accurate CO2 and PP/CO2 solution surface tension data based on the measured

PVT data for PP/CO2 solution


16

6. Successfully designed and constructed a rotational device to study the degree of

asymmetry of polymer with high viscosity and melt strength

7. Studied the degree of asymmetry of PS drop through PS density measurement

8. Hypothesized the effects of the polymer chain entanglement density on polymer swelling

9. Measured polymer chain entanglement to support the hypothesis

10. Obtained the best comparison between the measured PVT data and the EOS predicted

values

1.8 Thesis Organization

In order to detail each of the above contributions, the thesis is divided into eight chapters.

The following paragraphs walk through each individual chapter briefly.

Chapter 2 presents a literature review related to the study of this thesis, namely, the PVT

measurement, solubility, diffusivity, nucleation and surface tension studies. Since the

methodology proposed is closely associated with image processing, it is important and

necessary to present the current works that has been done in computer vision and image

processing on 3-D object reconstruction from 2-D images as part of the literature survey. The

literature gathered in this chapter on the study of PVT measurement for pure polymer and

polymer/gas solutions reviews the methodology used in the research and shows the current

progress and status. The law of thermodynamic, thermodynamic equilibrium and the equation of

states (EOS) are elaborated on because they are heavily utilized in the solubility measurement.

Chapter 3 focuses on the theoretical background study. The three most important aspects

that closely relate in with this research are: 1) solubility and equations of state (EOS) from


17

thermodynamic phase-equilibrium, 2) theories on PVT study and measurement, and 3) study of

surface tension in polymer processing.

Chapter 4 first describes the conceptual design of the PVT system. The axiomatic design

approach is used and provides a scientific and systematic basis for analyzing the design problem.

The details of the analysis are depicted in this chapter. Meanwhile, the design of every

component from both the hardware and software aspects of the system is also described in detail.

The last section of this chapter focuses on the algorithm development, including image

reconstruction, edge detection and volume integration. The definition of degree of asymmetry is

defined mathematically as well.

Chapter 5 emphasizes the validity of the PVT system using both theoretical and

empirical approaches. The volume measurement of precision stainless sphere demonstrates the

error reduction using the PVT system. Comparison between the measured pure polymer density

and the Tait equation calculated density is the main focus of this chapter A couple of different

polymer materials, such as PP and PS were used as the case study.

Chapter 6 turns the focus of the study to the PVT property of polymer/gas solutions. The

first section of this chapter shows the PVT study for polymer/gas system based on axisymmetry

assumption of the polymer/gas sample under high temperatures and pressures. The polymer

resins used for this study are linear polypropylene (PP) and branched polypropylene. The

blowing agent used is CO2. The experimental procedures are described for conducting the

measurement. The results of the volume swelling for linear and branched PP/CO2 solutions are

summarized. The calculation of the specific volume of PP/CO2 solution is also illustrated. The

effects of the temperature and pressure as well as the branching structure on the PVT properties

are explained in depth.


18

The second focus of Chapter 6 is to address the issue of asymmetry if the polymer has

high melt strength and viscosity at relatively high temperature and does not easily form a

axisymmetric profile. The polymer resin used in this study is polystyrene (PS) and the blowing

agent is CO2. The rotational device is employed to capture the asymmetry and also to

compensate the asymmetry effect on PVT measurement by rotating the sample at a given angle

for one revolution. The effects of temperature and pressure as well as the asymmetry on the final

PVT for pure PS and PS/CO2 are illustrated. The comparison between the measured PVT data

with those predicted from EOS is also examined in details. The conclusion and comments are

made based on the direct comparison between the predicted and measured volume swelling. The

last section of this chapter shows the steps for determining more accurate solubility using the

measured PVT data.

Chapter 7 addresses the issue of the importance of accurate PVT measurement in the

surface tension study. Experimental procedures are reviewed for the surface tension method.

The materials used for the surface tension study are the same as described in Chapter 6. The

surface tension results are calculated based on the density information from PVT experiments.

The effects of temperature and pressure on the surface tension of PP/CO2 solution are depicted

in detail.

Chapter 8 provides a summary and suggested future work for this research.


19

Chapter 2. Literature Review

2.1 Background on PVT Property Measurement

2.1.1 PVT Measurement for Pure Polymer

The relationship between pressure, volume and temperature ( PVT ) for polymeric

materials is a subject of importance to polymer scientists and engineers, particularly from a

process design standpoint. Pressure-volume-temperature (PVT) information for a variety of pure

polymers at temperatures above the melting point Tm of crystalline polymers or the glass

transition temperature Tg for amorphous polymers has been established through both empirical

and theoretical studies. Zoller et al. [21,22,24], Rodgers[28] and Sato et al.[29] have used the

bellow-type dilatometer to measure the density of a pure polymer, in which the polymer sample

and liquid mercury were confined inside the dilator. Figure 2-1 shows the simple schematic of

the bellow type dilatometer. Zollar et al. have described the detailed measurement procedures

and used this method to build up a PVT database for various pure polymers. This type of

dilatometer, however, is not capable of measuring the PVT properties of polymer/gas solutions.


20

Electric Heaters

LVDT CoreLVDT Coil

Bellows

Pressure Fluid Inlet

Plastic Sample

Mercury

Electric Heaters

LVDT CoreLVDT Coil

Bellows

Pressure Fluid Inlet

Plastic Sample

Mercury

Figure 2-1 Schematic of Bellow Type Dilatometer

Numerous theoretical equations of state for polymer liquids have been developed[30-45]

over the past couple of decades. The equations derived from thermodynamic principles can be

used to predict the properties of polymer blends and solutions. It seems that nearly all equations

of state for polymer liquids provide a reasonably good fit to PVT data, especially at low

pressures. However, there is a major difference between the various equations in their abilities

to fit PVT data over a wide range of pressure and to predict thermodynamic properties of

polymer blends and solutions, particularly phase separation behavior. Most equations require an

empirical interaction parameter, determined from at least one experimental data point, in order

to describe accurately the phase behavior.

The modified cell model of Dee and Walsh[31,32], the Simha-Somcynsky (SS) hole

theory[43,44], the Prigogine cell model[39,40], and the semiempirical model of Hartmann and

Haque[34], were all found to provide good fits of polymer liquid PVT data over the full range of

experimental pressures. The Flory-Orwoll-Vrij[33] and the Sanchez-Lacombe (SL) lattice-fluid

equations of states [41,42]are both significantly less accurate over the wider pressure range.

Perhaps the most common empirical representation of polymeric PVT data is that of the Tait


21

equation[28,46]. It is, in fact, not a true equation of state, but rather an isothermal compressibility

model. For different polymer resins, the format of the Tait equation is in similar format but the

coefficients of the equation are different depending on the specific material grade. The apparatus

to measure the PVT of pure polymer and fit with Tait eqauion is commercially available and the

most popular one is the Gnomix PVT apparatus developed and upgraded based on the

bellow-type dilatometer principle.

It has been reported that the pendent drop or sessile drop method based on axisymmetric

drop shape analysis (ADSA)[26,27,47-57]commonly used for performing surface tension

measurements, was used for measuring the density of pure polymer melts under ambient

pressure. Wulf et al.[57]used the ADSA method to determine the surface tension and the

density of pure polystyrene (PS) at ambient pressure and temperature up to 236oC. The major

limitation of this method is associated with the inaccuracy of boundary determination and the

strict axisymmetric drops.

2.1.2 PVT Measurement for Polymer/Gas Solutions In polymeric foaming process, for example, supercritial CO2 (scCO2) is being utilized as

an environmentally friendly foaming agent replacing the conventional chlorofluorocarbons

(CHFC), hydrofluorocarbon, and hydrocarbon foaming agents. When polymer and CO2 form

single phase solution system, CO2 gas dissolves into a molten polymer, and the polymer swells

(or dilates) due to gas sorption. The amount of polymer swelling or dilation is characterized by

its PVT properties, which can be obtained by measuring the equilibrium state volume of a

polymer/gas solution at any specific temperature and pressure. In polymer foaming applications,

cell nucleation and growth are governed by physical properties such as solubility[58,59],

diffusivity[60],and surface tension[26,27,61]. However, the determination of these properties relies


22

on the PVT data, i.e., the polymer swelling caused by gas dissolution. For example, currently,

the most commonly used technique for gas solubility measurement in polymer melts is the use

of a magnetic suspension balance system[62-65] in which the accurate determination of gas

solubility depends on the buoyancy correction of the equilibrium polymer/gas solution volume,

i.e., the swelling of the polymer due to the dissolved gas. Therefore, knowing their PVT

properties (the polymer swelling by dissolved gas) is critical for understanding and controlling

foam processing. Unfortunately, due to the high pressures and high temperatures involved in the

measurement, there have been very limited studies on the PVT properties of polymer/gas

solutions.

Attempts have been made to measure polymer swelling under high pressure gas

conditions[66-85]. Visual observation and induction gauges are the most popular scheme among

those studies in which the length change in one or more dimensions is measured in the presence

of high pressure CO2 gas.

Bonavoglia et al.[66] measured the swelling of poly(methyl methacrylate) (PMMA),

poly(tetrafluoroethylene), and poly(vinylidenefluoride) in CO2 (5–23 MPa) at temperatures in

the range of 313–353 K. Ender [68] used the coil of a linear variable differential transformer

(LVDT) to measure the volume swelling of various elastomers under high pressures in CO2 at

relatively low temperatures. Shenoy et al.[83] also used a LVDT to evaluate CO2 polymer

plasticization. Their results show that at 22oC under CO2 pressure,

polystyrene-block-polybutadiene-block-polystyrene (SBS) elastomer undergoes compression

due to hydrostatic pressure. However, sample expansion occurs upon depressurization. At 45oC,

SBS undergoes swelling of 0.7% due to CO2 plasticization, while no post-pressurization

expansion is observed. The contrasting result is explained by change in PS domain mobility and


23

discontinuity in the density-pressure relationship. The general concept of using the LVDT to

detect the dimension change to measure the volumetric change is illustrated in Figure 2-2.

Ender[68] has used the coil of a linear variable differential transformer (LVDT) to

measure the volume swelling of various elastomers under high pressures in CO2 at relatively

low temperatures. Foster et al.[69]used a cylinder-piston-type method in a compressibility

chamber to acquire PVT data by multiplying the cross-sectional area of the piston with the

linear displacement of the piston. Nikitin et al.[70] used a visual observation cell to study the

swelling of PS/ CO2 solutions at 335 K. They observed the diffusion front of CO2 in PS and

calculated a diffusion coefficient. Rajendran et al.[73] and Bonavoglia et al.[66] also employed a

direct visual observation method to measure polymer swelling in CO2. Royer et al.[74]observed

CO2 induced swellings of three poly (dimethylsiloxane) (PDMS) samples with different

molecular weights in a view cell and found that the molecular weight did not affect the

magnitude of swelling at 303, 323, and 343 K under pressures ranging from 0 to 27.6 MPa.

Wissinger et al.[75] measured the swelling of poly(methyl methacrylate) (PMMA),

polycarbonate (PC), and polystyrene (PS) in contact with CO2 at temperature from 306 to 338 K

and pressure up to 10 MPa. They used a cathetometer shown in Figure 2-2 for measuring the

length of the thin polymer films in a high-pressure view cell and identified two distinct types of

swelling and sorption isotherms. One is characterized by swelling and sorption that began to

level off at elevated pressures. The other is swelling and sorption that continued to increase with

CO2 pressure.

Zhang et al. [76] measured CO2 sorption and swelling of PMMA, poly(vinylpyridine),

polyisoprene, and three block copolymers in the presence of CO2 at 308 K and at a pressure up

to 10 MPa. They also used a cathetometer to directly measure the dimension of polymer samples

in a high-pressure view cell.


24

Some researchers proposed non-direct measurement schemes, which combine both

gravimetric and volumetric methods. Using so-called combined gravimetric-dynamic method,

Keller et al.[77] studied the swelling of PC at temperature, 293 K, under pressures ranging from 0

to 6 MPa. They combined the gravimetric method with a dynamic method where the inertia of

the polymer sample was determined by slow oscillations of a rotational pendulum or by the

relaxation motion of a floating rotator.

The dynamic method showed the same order of magnitude of polymer swelling as

Wissinger and Paulatitis [75]showed for the PC/ CO2 system. Using a pressure decay apparatus

with a vibrating-wire force sensor, Hilic et al.[78] conducted simultaneous measurement of the

solubility of nitrogen (N2) at pressures up to 70 MPa and CO2 at pressures up to 45 MPa in PS

and the associated polymer swelling. The vibrating wire sensor acted as a balance to weigh the

polymer sample. They measured the swelling of PS with a precision of 0.5% at three isotherms

from 313 to 353 K, which were below the glass transition temperature of PS and observed a

significant change in both solubility and swelling at temperatures between 363 and 383 K.

Gotthardt et al.[81] studied the volume changes caused by the sorption of H2O, Ar, N2,

CO2, CH4 and Acetone in Bisphenol-A polycarbonate and of CO2 in different substituted

polycarbonates using a dilatometer at room temperature.

gas vacuum

pressure transducer

Δl

high pressure window

polymer sample

gas vacuum

pressure transducer

Δl

high pressure window

polymer sample

Figure 2-2 Schematic of LVDT Dilatometer


25

Holck et al. [85]used a gravimetric sorption balance and a dilatometer based on capacitance

sensor to measure the CO2 sorption induced dilation in polysulfone and compared it with

experimental and molecular modeling results at 308 K and pressures up to 50 bar.

The most proposed methods of measuring polymer swelling require a specific shape or

state of the polymeric material. There are few papers which report swelling measurements of

molten polymers or polymeric solutions at temperatures higher than the glass transition

temperature.

Unfortunately, most of approaches are only valid for isotropic swelling and the polymer

have been pre-treated carefully to release internal stresses and ensure the isotropic behaviour.

Furthermore, the warpage of the samples limits its application and results in an inaccurate length

measurement. In general, most of the methods described above operate at fairly low

temperatures, ranging from 25°C to 100°C which are below glass transition temperature of

polymer in order to keep an appropriate shape of the sample.

Park et al. [71,79] measured the PVT properties of polymer/CO2 solutions using a foaming

extruder and a positive-displacement gear pump mounted on the extruder. The major drawback

of this method is the required (and limited) extrusion processing conditions for measuring the

PVT data.

Funami et al.[86] developed a new method of directly measuring the densities of two

polymer melt-CO2 single-phase solutions, poly(ethylene glycol) (PEG)-CO2 and polyethylene

(PE)-CO2 at high pressure and temperature using magnetic suspension balance (MSB) seen in

the following Figure 2-3.


26

Figure 2-3 Schematic of Modified MSB for Polymer/Gas PVT Measurement

A thin disc-shaped platinum plate was submerged in the polymer-CO2 single-phase

solution in the MSB high-pressure cell. The weight of the plate was measured while keeping

temperature and CO2 pressure in the sorption cell at a specified level. Since the buoyancy

force exerted on the plate by the polymer/CO2 solution reduced the apparent weight of the plate,

the density of the polymer/CO2 mixture could be calculated by subtracting the true weight of the

plate from its measured weight. However, this density measuring method has some limitations

on applicable polymers. When the plate is moving up and down in the polymer melt during the

position changeover operation, a dragging force is generated. Therefore, the readout could not

be guaranteed when the viscosity of polymer melt is high.

An recent attempt to calculate the surface tension and density simultaneously using

surface tension ADSA-P method[87,88] to study the PS/N2 solution at pressure up to 500 psi and

200oC. The limitation is the ADSA method cannot handle asymmetry PS drop shape due to its

high viscosity and elasticity even at low temperature. The results is not accurate at low

temperature range and plus the operating pressure is low.


27

2.2 Background on Polymer/Gas Solutions Property

2.2.1 Solubility and Diffusivity As was explained in the continuous foaming process, the solution formation of the

polymer and blowing agents is the first very critical stage, which would determine the ultimate

properties of the foam products. When a blowing agent such as CO2 is dissolved into polymer,

several physical properties of polymer such as glass transition temperature, melting temperature,

surface tension, and viscosity are changed depending on solubility of CO2 into the polymer.

Knowing the precise solubility of the blowing agent would be crucial to determine the solubility

pressure or exactly how much gas are required for achieving the polymer/gas single phase

solution formation. Therefore a thorough understanding about the interaction between the

physical blowing agents and polymer melts would be useful not only in the selection and

development of suitable blowing agents but also for optimization of the plastics foam

production process.

The solubility of gas in polymer melts is defined as the maximum amount of gas that can

dissolve into the polymer melts at a specific temperature and pressure. When a gas permeates

through a polymer membrane, several processes are involved: 1) the gas is absorbed at the

entering interface, 2) the gas dissolves at the interface to establish a quick equilibrium and 3) the

dissolved penetrate molecules diffuse through the membrane. The mechanism of permeation

involves both solution and diffusion. Microscopic and macroscopic diffusion are the two

approaches to study the diffusion.

Microscopic diffusion includes two most important models: the Meares’ Molecular

model and the Free-volume model. In the molecular model, the molecules penetrate through a

polymer matrix and it is usually assumed that microcavities of different sizes are formed and

destroyed continuously in the polymer. Small, dissolved molecules trapped in these locations are


28

able to move into the direction of a driving force by the cooperative motion of adjacent polymer

chains[89]. Meares’s Molecular model is expressed in equation (2-1) as follows:

AD NE λα

π24

= (2-1)

where DE is the cohesive energy density that requires disrupting neighbouring polymer chains,

λ is the jump length, 2α is the cross section of the penetrate molecule, and AN is the

Avogadro’s constant. Meares’s model deals with prediction of the energy required for a

diffusion jump and it does not predict beforehand the value of diffusion coefficient. More

complex models[90-92] theoretically describe the overcoming of attractive forces between the

chains and the energy required for bending of polymer chains.

Another molecular model is the free-volume model proposed by Cohen and Turnbull[93]

and developed by Fujita[94]. The free-volume model describes the absolute velocity of the

penetrating molecule in the polymer matrix and it relates the diffusion coefficient to the free

volume available in the system. The equation is:

⎟⎟⎠

⎞⎜⎜⎝

⎛ −∗=

f

BADυ

exp (2-2)

0υυυ −= pf (2-3)

The D in the equation is diffusion coefficient, fυ is the total specific free-volume accessible

for diffusion, pυ is the specific volume of the polymer and 0υ is the specific occupied volume

of the polymer chains. Other forms of the free-volume model[95] are also available.

The diffusion coefficient D is a macroscopic property of a polymer-penetrate system that

does not give detailed or particular information about molecular motions of polymer chains

during the diffusion process. The concentration of penetrate within the gas/polymer interface is


29

not known unless the solubility of the gas in the polymer has been determined. In simple cases,

it is customary to assume that the gas/polymer system follows Henry’s law[96] which gives the

general relationship between the concentrationsof penetrate and the pressure on polymer:

ss

s PRTHHC ⎟

⎠⎞

⎜⎝⎛ Δ−= exp0 (2-4)

where sC is solubility of gas in the polymer, H0 is the constant of Henry’s law, R is the gas

constant, T is the temperature in Kelvin, sHΔ is the molar heat of sorption in Joule and sP is

the saturation pressure in Pascal. The molar heat of sorption can be a negative or positive value

depending on the polymer-gas system.

The fundamental equation of macroscopic diffusion was derived by Fick in 1855. These

are Fick’s first and second law. The mathematical theory is based on the hypothesis that the rate

of transfer of diffusing substance through unit area of a section is proportional to the

concentration gradient measured normal to the section. Fick’s first law relates the diffusion flux

to the concentration difference and the diffusion coefficient is assumed to be constant across the

membrane.

adxdCDj =−= (2-5)

where a is a constant, and if the flux and concentration gradient can be accurately determined,

then the diffusion coefficient can be evaluated directly.

Fick’s second law considers the mass balance:

2

2

dx

CdDdxdj

dtdc

−=−= (2-6)


30

Equation (2-6) assumes one-direction diffusion. In many polymer systems, however, the

diffusion coefficient D depends markedly on the concentration and D varies from point to point,

in which case equation (2-6) becomes:

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

==xCD

xdxdj

dtdc

(2-7)

Diffusion and sorption are the direct causes of the volume swelling when gas is dissolved into

the polymer matrix. There are a few models available to study the sorption behaviour of rubbery

and glassy polymers.

Crank and Park and other researchers used the kinetic study of the sorption and

desorption of gases and vapours in polymers as a means for determining the diffusion

coefficient. In the batch process, a sheet of polymer sample is saturated with the blowing agent

by placing it in a high-pressure chamber connected to the blowing agent reservoir. At this high

pressure, the blowing agent will continuously diffuse into the polymer matrix until the solubility

limit is reached. Although the solubility limit theoretically occurs at time infinity, the

instantaneous concentration of the blowing agent in the polymer can still be obtained using the

following equation[97]:

( )

( )∑∞

=∞ ⎥⎥⎦

⎤

⎢⎢⎣

⎡ +−

+−=

02

22

2212exp

12

181m

th

tmD

mMM π

π (2-8)

where D is diffusivity, tM is the mass uptake at time t and similarly ∞M is the equilibrium

mass uptake after an infinite time at which the equilibrium of the system is reached; t is the

elapsed time and h is the thickness of the sample sheet.

The amount of mass uptake eventually levels off at ∞M in the absorption process, and

the solubility limit can be calculated by dividing the mass uptake ( ∞M ) by the mass of the


31

polymer sample. On the other hand, the solubility limit of gas dissolved into the polymer

depends on the system pressure and temperature and can be estimated by Henry's law[96].

At the processing pressure and temperature, the estimation of the solubility of CO2 in

some polymers has been given in the literature[17,98]. There are many methods and techniques

available to determine the solubility. These methods include: 1) pressure decay method[99-102], 2)

piezoelectric sorption method[103-106], 3) Gas-liquid chromatography method[107-109], and 4)

Gravimetric method[104,110-115]. The following paragraphs briefly describe each of these

measurement methods.

The pressure decay method was first proposed by Newitt and Weale [102] in 1948 for the

experimental measurement of gas solubility in polystyrene melt. The principle used is that in a

confined sorption chamber at constant temperature with known volume, when the gas is

absorbed into the polymer matrix, the pressure of the chamber will decrease accordingly.

Therefore, by measuring the pressure drop due to gas sorption, the solubility can be determined.

The apparatus built based on the pressure decay method is called the Durrill’s apparatus

[99] With the pressure decay experimental approach, Lundberg et al. measured the solubility of

nitrogen and methane in polyethylene[101], solubility of methane in polystyrene[116], and

solubilities of methane in polyisobutylene[100]. Newitt et al. investigated the solubility of

hydrogen and nitrogen in polystyrene[102]; Durrill et al. systematically determined the solubility

of nitrogen (N2), helium(He), carbon dioxide (CO2), and argon(Ar) in polyethylene,

polyisobutylene, polypropylene, polystyrene and polymethylmethacrylate [99]. Sato et al. used

the pressure decay method to study the solubility of carbon dioxide and nitrogen in molten

polypropylene (PP) and high-density polyethylene (HDPE). The solubilities were measured at

temperatures 433.2, 453.2, and 473.2 K and pressures up to 17 MPa. The solubility increased

almost linearly with pressure. While the solubility of carbon dioxide decreased with increasing


32

temperature, that of nitrogen increased in the temperature range examined. The solubility of

nitrogen in glassy polystyrene (PS) was measured at 313.2, 333.2, and 353.2 K and pressures up

to 17 MPa.

In 1975, Bonner et al. creatively devised a novel piezoelectric sorption technique to

determine gas solubility in polymers at high temperature and high pressure conditions [103-105].

The principle of the piezoelectric sorption method is based on the fact that the relevant

frequency of piezoelectric crystal increases with increase in mass loaded on the surface of the

crystal. Therefore the relevant frequency of piezoelectric crystal can be applied to determine

solubility of gas dissolved in a polymer coating layer on the surface of the crystal[104]. With the

piezoelectric sorption method, Bonner et al. measured the solubility of nitrogen and ethylene in

low-density polyethylene[104,105]. The drawback of the piezoelectric sorption method is that it

can not be used to measure the solubility at high temperature due to the polymer sample’s

viscosity change at high temperature.

Gas-liquid chromatography was first applied by Prausnitz et al. to determine the gas

solubility in polymer at high temperature in 1976[107,108]. Based on the gas-liquid

chromatography, the solubility of ethylene, n-butane, vinylacetate, n-hexane, benzene, toluene,

and n-octane in low-density polyethylene were successfully measured[108]. In addition, the

solubility of methyl ethyl ketone, acetone, isopropyl alcohol, vinyl acetate, sulfur dioxide,

methyl chloride, ethane, ethylene, and carbon dioxide in polyethylene and in ethylene-vinyl

acetate copolymer were studied[107]. Gas-liquid chromatography is a simple and rapid method

for measuring solubility of volatile compounds in polymer melts. But the main problem of this

method is that the pressure is limited to ambient pressure range.

The volumetric and gravimetric methods have been widely used to measure not only the

diffusivity but also the solubility of CO2 in polymer[65,66,70,73-76,99,117,118]. The gravimetric method


33

was applied to study the sorption of gas in polymer melt. Based on the gravimetric method,

Bonner etc. studied the sorption of benzene in polyisoprene and polyisobutylene at 80°C; the

sorption of cyclohexane in polyisobutylene at 100°C; and the sorption of cyclohexane in

ethylenevinyl acetate copolymer (EVA), isooctane in EVA, isooctane in poly(vinyl acetate) at

110°C [119].

Baird investigated the kinetics sorption of n-pentane in polystyrene at 30°C[109]. The

solubility of vinyl chloride monomer (VCM) in PVC powders has been studied by Berens etc. at

temperatures from 30 to 110 °C[111]. Berens et al. also investigated the solubility of nitrogen,

carbon dioxide, vinylchloride, methanol, acetone, n-butane, and benzene in PVC at temperatures

below Tg[112]. Kamiya et al. precisely determined the solubility of gases in polymers at high

pressure. The solubilities of N2 and CO2 in LDPE, CO2 in polycarbonate (PC); and N2, CH4,

C2H6, and CO2 in polysulfone and PS were measured as a function of pressure up to 50 atm[114].

They carefully measured the polymer swelling and exactly determined the solubility of N2, Ar,

CO2 in poly(vinyl benzoate) at 25-65 oC and up to 5Mpa[115].

Typically a quartz spring or a microbalance with precision was employed to measure the

in situ weight gain of a polymer sample as gas dissolved into it. Recently, a new apparatus based

on the gravimetric concept called magnetic suspension balance (MSB) became widely used in

measuring gas solubility in a variety of polymers at relatively high temperature and pressure. In

this research, the gravimetric method called magnetic suspension balance (MSB) will be the

apparatus utilized for the solubility measurement. Figure 1-4 from Chapter 1 shows a schematic

and a picture of the MSB apparatus. Sato et al. [62,64,120,121], Oshima et al.[86,117,122,123] and Li et

al.[58,60,124-126] have carried out intensive studies on blowing agents, such as CO2, N2 and butane

solubility in a variety of polymers or polymer blends, such as PP, PS, and PVC using the MSB

apparatus. The MSB and equation of state will be discussed in greater detail in the theoretical


34

background section in Chapter 3. Although the gravimetric method is accurate in measuring the

solubility at moderate high temperature and pressure, a correction for buoyancy is necessary due

to the volume swelling or dilation of the polymer sample from the gas dissolution. In general,

the method needs correction of the volume swelling predicted using the equation of state (EOS)

for polymer. Detailed theoretical background regarding EOS will be discussed in Chapter 3 as

well since EOS is intensively used to predict the volume swelling of polymer/gas solutions in

order to help account for the buoyancy effect in the solubility measurement using the MSB

method.

There have been several studies on CO2 solubility and diffusivity in polymer and

associated change in polymer property[123,127-132]. Recently, Tomasko et al. made a

comprehensive review on CO2 solubility and diffusivity in polymer, the effects of CO2

dissolution on the polymer property, and their applications.

As seen from the equations, diffusivity determines how fast the gas can diffuse into the

polymer melts and it is a dynamic process while, on the other hand, solubility determines how

much gas can be dissolved into polymer melts at a saturated equilibrium state. In this research,

since our focus is on the PVT property of the polymer/gas solution at equilibrium states, which

means the polymer is completely saturated with blowing agents, solubility information is the

most important aspect that will be dealt with. As mentioned previously, one of the contributions

of this thesis is to obtain more accurate volume swelling from PVT measurement so that the

buoyancy effect caused by the polymer volume dilation due to gas dissolution could be more

precisely accounted for. Therefore, more accurate solubility can be determined from the MSB

method.


35

2.2.2 Nucleation

As discussed previously in the section on foam processing, cell nucleation is the second

critical stage of the foaming fabrication process. Over the last few decades, vast number of

studies already been devoted to the study of cell nucleation. From the classical thermodynamic

law[133], when a thermodynamically equilibrium system is subject to any degree of instability,

there exists a critical size in the saturated system for bubbles to be nucleated. The following

equation determines the critical nuclei size:

systembubble

critical PPR

−=

γ2 (2-9)

This is known as the Laplace equation, and it represents the mechanical equilibrium between the

critical nucleus and its surroundings. γ is the surface tension at the phase interface.

For a metastable system, if the size of the nuclei is bigger than the critical size, the

bubble will start to grow, then the equilibrium would be interrupted and the system becomes

unstable. The relationship between the free energy barrier and the critical radius resulting from

thermodynamic instability is shown in the figure 2-4. The X axis represents the radius of the

nuclei and the Y axis represents the free energy barrier or the activation energy. The activation

energy is the energy needed to enable the bubbles to be nucleated at the critical radius so that the

nucleated bubble can keep growing once its radius is at or over this critical bubble radius,

otherwise the bubble will collapse. The activation energy or the energy barrier is at the

maximum when the bubble reaches the critical radius. When the bubble is growing beyond this

critical radius, the energy required is dramatically decreased.


36

criticalhetR criticalRhom

Heterogeneous Nucleation Homogeneous Nucleation

Nucleus Radius R

actW

Figure 2-4 Schematic of Critical Nucleation Radius and Activation Energy

Nucleation of bubbles in the polymer can be described using the classical nucleation

theory[134-136], that was originally developed for a single-component system where the second

phase is created by evaporation of the liquid when superheated. The theory was then extended

by Blander and Katz[134] to a diffusion system where one component is volatile and forms

bubbles. There are two types of nucleation: homogenous nucleation and heterogeneous

nucleation.

The rate can also be determined through the classic nucleation theory. For homogenous

nucleation, the nucleation rate is articulated using the following equation:

( ) ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−=

2

3

hom3

16exp2

systembubble PPkTmNJ πγ

πγ

(2-10)

Bubble nucleation is heterogeneous when it is initiated at some preferred sites by mixing

the polymer with an additive, or the nucleation sites could have impurities inside the polymer

matrix. In general, nucleation tends to occur at the boundary of the matrix and additive rather

than inside the polymer matrix as with homogenous nucleation.

For heterogeneous nucleation, the nucleation rate is determined as:


37

( ) ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−=

2

33/2

316exp2

systembubblehet

PPkTmQNJ πγ

πγ

(2-11)

Combined with the second law of thermodynamics and Gibbs’ free energy theory, the

initial energy needs to be overcame or the so-called free barrier energy for nucleation is derived

for both homogenous and heterogeneous systems as follows:

( )23

316

systembubble PPW

−=

πγ (2-12)

The above equations are a strong demonstration that surface tension serves as one of the

important parameters in determining every aspect of the nucleation process, from the critical

bubble radius, free energy barrier to nucleation rate. The surface tension between the gaseous

phase and the polymer/gas solution do play a significant role during the polymer process where

the cell size and cell density are important issues in controlling the final morphology of the

foamed products.

This would then lead to the next section of our discussion, which will reveal another

strong reason why this research is important and viable not only for academia but also for the

foaming industry.

2.2.3 Surface Tension

Surface tension is an intermolecular force caused by attraction and propulsion of

molecules at the interface between two mediums. Figure2-5 vividly illustrated the force balance

between gravity (fw) and surface tension (fs) of a small particle on a liquid surface. The unit for

Surface Tension used to be “dyne/cm”, but has been standardized to “mN/m” as a SI unit.

Numerous techniques have been developed for the measurement of interfacial properties. Wu[137]

gives a complete review on different techniques to measure the surface tension of polymer melts.


38

Some of the major methods including the Wilhelmy plate technique[138-143]the drop weight

method[139-142], the oscillating jet method [139,142] the capillary wave method[139,142], the spinning

drop method[139-142,144-146], and drop shape techniques [139-143,147-160] are discussed briefly in this

section. Among those methods, the spinning drop method, drop shape techniques, i.e. the

pendant drop and sessile which are based on the analysis of the equilibrium shape of a drop in a

forced field, are often applied in the surface tension measure for polymer melts and polymer/gas

solutions.

Figure 2-5 Surface Tension and Gravity of Small Particle at Liquid/Particle Interface* *Source: http://en.wikipedia.org/wiki/Surface_tension

As the name implies, the Wilhelmy plate technique uses a detection plate for surface

tension measurement. The plate used is usually made of platinum or glass. When the bottom of a

vertically-oriented detection plate makes contact with a liquid surface, the liquid wets the plate

surface upward and a meniscus is created. At this moment, the surface area of the liquid is

expanded and surface tension tends to contract the surface area as a counteraction, and immerse

the plate downward. This method determines surface tension by measuring the force bringing

the plate downward via a counter balance. The measured downward force F directed to the plate

is related directly to the liquid surface tension [161]. The relation between the force and the

surface tension is expressed as follows:


39

θργ

cosplatelv p

gVF Δ+= (2-13)

where platep is the perimeter of the plate, V is the volume of the displaced liquid, ρΔ is the

density difference between the liquid and air, and g is gravity. In general, if only the

measurement of surface tension is desired, then the plate is roughened to produce a zero contact

angle (complete wetting). This approach has a relatively simple and a high degree of accuracy

(results are typically given with an error of approximately 0.2%), but a major disadvantage of

the Wilhelmy plate technique is the requirement of a relatively large amount of liquid. The use

of a large reservoir can also make it difficult to maintain a high degree of purity, which is of

critical importance to all surface tension measurements since the introduction of impurities,

even small quantities, can dramatically affect the interfacial properties[162]. Moreover, this

method is rarely applied for surface tension of polymer melts or polymer/gas solution because of

their high viscosity.

Another similar method to the Wilhelmy plate method is the Du Nouy ring method

[139,140,143]. In this method, a circular loop of wire is used in place of a platinum plate. However,

while the Wilhelmy Plate method keeps the plate bottom immersed in the liquid surface, this

method requires lowering the liquid surface to detect the peak force upon removal of the ring

from the liquid surface. The disadvantages are: 1) the shape of the thin wire ring material

changes, which affects the measurement results, 2) the ring is less accurate when dealing with

higher viscosity materials, and 3) variations in surface tension over time cannot be measured

due to the mechanics of removing the ring from the surface[143].

The drop weight method was proposed in the 1900s. Since the weight of the drop falling

off the capillary correlates with the interfacial tension, the surface tension is then determined

through the following equation [163-165]:


40

frgV

πργ

2Δ

= (2-14)

where V is the drop volume, r is the radius of the capillary and f is an empirical factor tabulated

as a function of cRr / ( cR , is a characteristic dimension defined as 3/1V )[163,164,166]. There are

some drawbacks to this method. First, although the technique itself is simple, the measurement

of the drop weight is very sensitive to vibration. Second, the vibrations of the apparatus can

cause premature separation of the drop from the end of the capillary before the drop reaches the

critical size. Last, since adsorption may occur when conducting the surface tension

measurements of multi-component solutions, the results may not reflect equilibrium saturation

of the solutes at the interface.

The oscillating jet method uses the oscillating jet generated by forcing the liquid through

an elliptical orifice with properties of standing waves[139,142,167]. In the absence of viscosity and

compressibility, the surface tension is related to measurable physical properties including the

wavelength of the oscillations, liquid density, mean radius of the orifice (i.e., average of the

minimum and maximum radii), and flow rate. In this method, the wavelengths are measured by

passing parallel light waves perpendicular to the jet stream[142]. Bohr[168] applied the method to a

real liquid by taking into account the influence of the velocity profile of the jet on surface

tension. The oscillating jet method has been used to study the surface tension of surfactant

solutions[139]. This method also gives reasonably accurate values of surface tension even if the

jet velocity profile is not included in the calculations [142]. A major problem associated with this

method is the cost of equipment because of the high degree of accuracy required in measuring

wavelengths.

If a deep body of liquid is perturbed by a vibrator, the surface of the liquid will oscillate

where the wavelength of the surface waves depends on the liquid surface tension and gravity[139].


41

Such waves are called capillary-gravitational waves, as theoretically formulated by Kelvin[169].

The proper equation is as follows:

ρλπγ

πλ 2

22 +=

gv (2-15)

where v is the velocity of propagation, λ is the wavelength, g is acceleration due to gravity, γ

is surface tension, and ρ is the liquid density. It is clear that the propagation velocity is

determined by gravity for long waves and by surface tension for short waves (i.e., called

capillary waves) [142].

The theory of capillary waves is more complicated for viscous liquids, especially for

surfactant solutions with viscoelastic surface properties[142]. Similar to the oscillating jet method,

the theory of oscillations at a flat interface is based on the analysis of the Navier-Stokes

hydrodynamic equations and the boundary conditions at an interface. The details can be found

in References[170-173]. Again, similar to the oscillating jet method, the major problem associated

with the capillary method is the cost of equipment. In general, this method is more complex

from both theoretical and experimental aspects. Another disadvantage of this method is the large

amount of liquid required for each experiment.

Only a handful of methods are developed that could be used to measure the interfacial

tension of polymer melts or polymer/gas solutions, due to difficulties of the high temperature

involved and high viscosities of polymer melts. Those measurement methods are elaborated in

the following sections.

2.2.3.1 Spinning Drop Method

In this technique, a drop of liquid (or a bubble) is suspended in a denser liquid, and both

the drop and the surrounding liquid are contained in a horizontal tube spun about its longitudinal


42

axis[142]. As a result of spinning, gravity has little effect on the shape of the drop. At low

rotational velocities (ω ), the drop (bubble) has an ellipsoidal shape, but when ω is sufficiently

large, it becomes cylindrical. Under the latter condition, the interfacial tension is calculated from

the following equation[174]:

23

41 ρωγ Δ= r (2-16)

where r is the radius of the cylindrical drop, and ρΔ is the density difference between the drop

and the surrounding liquid. One of the advantages of the spinning drop method is its

applicability to determine surface tension of highly viscous liquids when many traditional

methods are unsuitable. For instance, this method is appropriate for polymer melts with a

viscosity of 300-500 Pa*s [142]. In these experiments, a solid polymer is initially placed in the

tube that is heated to the melting temperature of the polymer while spinning in an oven with a

control window.

2.2.3.2 Drop Shape Techniques

Drop shape methods have been developed to determine the liquid-vapor or liquid-liquid

interfacial tensions and the contact angle from the shape of a pendant drop, sessile drop, or

captive bubble. In essence, the shape of a drop is determined by a combination of surface

tension and gravity effects. Surface tension tends to make a drop spherical whereas gravity tends

to elongate a pendant drop or flatten a sessile drop. When gravitational and surface tensional

effects are comparable, then, in principle, one can determine the surface tension from an

analysis of the shape of the drop. There are several advantages using the drop shape method: 1)

it only requires small amounts of the liquid, 2) it is easy to handle, and 3) it can be used in many

difficult experimental conditions such as studies of temperature or pressure dependence of


43

liquid-fluid interfacial tensions. Drop shape methods have been applied to materials ranging

from organic liquids to molten metals and from pure solvents to concentrated solutions.

Also, since the profile of the drop may be recorded as digital images, it is possible to

study interfacial tensions in dynamic systems, where the properties are time dependent.

In the drop shape method the balance between surface tension and external forces, such as

gravity, is reflected mathematically in the so-called Laplace equation of capillarity.

Currently, the pendant drop or sessile drop is the most commonly used drop shape

method to measure the interfacial tension for polymer melts and is also most promising for the

challenging cases such as high pressure and high temperature application. Harrison et al.

measured interfacial tensions for PS oligomer (Mw=1850)/CO2[175] at 45 °C up to 310 bar where

γ decreases from 37.4 dyn/cm at 1bar to 1.5dyn/cm at 310 bar. Recently the sessile drop method

based on the Axisymmetric Drop Shape Analysis (ADSA) has been used intensively to study the

interfacial tension of polymer/gas solutions and has become one of the most popular and

efficient methods to obtain the surface tension data.

Rotenberg [54,176] developed the powerful technique, Axisyrnmetric Drop Shape Analysis

(ADSA), which fits the measured profile to a Laplacian curve using a nonlinear procedure. The

simplicity and accuracy of the ADSA method was further improved as Cheng [56,177]

implemented image processing techniques to detect the edge of the drop automatically. He

incorporated an automated edge detection technique into the ADSA program that improved

considerably the accuracy of the results and the efficiency of the "first generation" of the ADSA

technique (i.e. Rotenberg ADSA). del Rio [178,179] developed a second generation of ADSA to

overcome the deficiencies of the numerical schemes of the first generation using more efficient

algorithms. He used the curvature at the apex rather than the radius of curvature and the angle of

vertical alignment as optimization parameters.


44

The latest third generation of ADSA developed by Mina et al.[180] would overcome some

inconsistent results from the previous two generations of ADSA. The detailed development of

this third generation ADSA can be found in Mina’s Ph.d. thesis[180].

Since supercritical CO2 is a promising solvent for application in polymer blending and

foaming, intensive surface tension studies of polymer/CO2 solutions have also been carried out

as documented in various literature. Huyk et al.[26,27,181] studied the surface tension of polymer

melts in supercritical carbon dioxide using Axisymmetric Drop Shape Analysis-Profile

(ADSA-P) method. Huyk et al.[26] investigated the surface tension of polystyrene (PS) melts in

supercritical carbon dioxide using the ADSA-P method at temperature ranges from 180oC to

210oC with pressure up to 4000 psi. The relationship between surface tension and density is

described by the empirical Macleod equation. They also studied the effects of molecular weight

distribution on surface tension using PS/CO2 [181] .

Again when closely examined, the equations for determining surface tension have been

closely associated with the density information of both phases. To be specific, the density

difference of the two interacting phases is one of the key variables of those formulas. This will

be seen implicitly in Chapter 3, when the theoretical background is illustrated. When the

formula is used to calculate the surface tension of polymer/gas solutions, the determination of

polymer/gas solution density once again becomes dependent on either pure equation of state

(EOS) calculations based on laws of thermodynamics or the corrected MSB solubility

measurement with the EOS predicted volume swelling. Although a more detailed theoretical

background will be described in Chapter 3, it is clear now that there is the need for obtaining

accurate density information for surface tension measurement. This is another one of the

contributions mentioned in the first Chapter. As for now, a more complete picture is painted for

the motives behind this research.


45

2.3 Background on 3-D Object Reconstruction from 2-D Images

2.3.1 Introduction Reconstruction of a complex three-dimensional (3-D) rigid object from its two-dimensional

(2-D) images is a challenging computer vision problem under general imaging conditions. This

problem has sparked recent interest in the computer vision community[182-186] as a result of new

applications in telepresence, virtual reality and other graphics-oriented problems that require

realistic textured object models. Without prior information about the imaging environment such

as camera geometry, lighting conditions, object and background surface properties, etc., it

becomes very difficult to gain knowledge on the 3-D structures of captured rigid objects. There

are mainly two approaches for capturing the 3D shape of a real object: active and passive

methods. The active method can be grouped into two categories: contact and non-contact

scanners.

2.3.2 Active 3-D Reconstruction Method Contact 3D scanners probe the subject through physical touch. A coordinate measuring

machine (CMM) is an example of a contact 3D scanner. It is mostly used in manufacturing and

can be very precise. The disadvantage of CMM is that it requires contact with the object to

perform the scanning. The other disadvantage of CMM is that they are relatively slow compared

to the other scanning methods. Physically moving the arm on which the probe is mounted can be

very slow and the fastest CMM can only operate on a few hundred hertz. Laser range scanners

and encoded light projecting systems are non-contact systems that use active triangulation to

acquire precise 3D data[187]. 3D laser range scanners use laser light to probe the environment.

This technique is called triangulation because the laser dot, camera and laser emitter form a

triangle.


46

Structured light 3D scanners project a pattern of light on the subject and look at the

deformation of the pattern on the subject. The pattern may be one dimensional or two

dimensional. The advantage of structured light 3D scanners is speed. Instead of scanning one

point at a time, structured light scanners can scan multiple points or the entire field of view at

once. This reduces or eliminates the problem of distortion from motion. Some existing systems

are capable of scanning moving objects in real-time.

Recently, Song Zhang and Peisen Huang[188] developed a real-time scanner using the

digital fringe projection and phase-shifting technique (a varied structured light method). The

system is able to capture, reconstruct, and render the high-density details of the dynamically

deformable objects (such as facial expressions) at 40 frames per second.

2.3.3 Passive 3-D Reconstruction Method

Compared to active scanners, passive methods work in an ordinary environment with

simple devices and the mathematics of differential geometry. A digital RGB camera from

different viewpoints captures the images of the target object while the target sits and rotates on a

turntable or as it uses multiple cameras to capture the images of the target at different

viewpoints. The 3D information is then extracted from the sequence of 2D (color) images by

using the following major techniques: 1) multi-view-stereo method 2) model based multi-view

stereo method and 3) volumetric modeling method. The volumetric modeling methods were

then later divided into two different methods and they are Shape from Silhouettes and Shape

from photoconsistency.


47

2.3.3.1 Multi-view Stereo Method Early passive 3-D object model reconstruction attempts have been based on image matching.

This class of methods includes multi-view stereos which recovers 3-D geometry by using

correspondences across images, and by applying surface fitting and triangulation over the

estimated 3-D clouds of points[189-191]. The pairwise disparity estimation allows computing

image to image correspondences between adjacent rectified image pairs, and independent depth

estimates for each camera viewpoint. Takeo Kanade et al.[192] used a 4-camera system capable of

video rate image acquisition. The four cameras were used in a converging configuration for

more effective use of the camera view spaces. In addition, to recover dense stereo range data

from each set of images, they projected a varying sinusoidal pattern onto the scene to enhance

local intensity discriminability at each pixel and facilitate matches. This resulted in a

multi-baseline stereo system with active illumination. They also described in detail the

implementation of the depth recovery algorithm which involved the preprocessing stage of

image rectification.

In stereo processing, a short baseline means that the estimated distance will be less

precise due to narrow triangulation. For more precise distance estimation, a longer baseline is

desired. But with a longer baseline, however, a larger disparity range must be searched in order

to find a match. As a result, matching is more difficult, and there is a greater possibility of a

false match. Therefore, there is a tradeoff between precision and accuracy in the matching

process. T. Nakahara et al. [190] presents a stereo matching method that uses multiple stereo pairs

with various baselines generated by a lateral displacement of a camera to obtain precise distance

estimates without suffering from ambiguity. Matching is performed simply by computing the

sum of squared-difference (SSD) values. An advantage of this method is that it can eliminate

false matches and increase precision without any search or sequential filtering.


48

Francis Schmitt and Carlos Hernández Esteban[193] presented another method which

combines both the shape from the silhouette technique with the multi-stereo carving technique

for the reconstruction of a 3D real object from a sequence of high-definition images. Their

method is fast and accurate for the estimation of the carving depth at each vertex of the 3D mesh.

The quality of the final textured 3D reconstruction models given in their study validated the

method.

There are disadvantages of using the multi-view stereo method. In order to find effective

correspondences, views must be close together and correspondences must be found in the

images. Resultant partial 3-D clouds of points for different views must be triangulated and fused

into a single consistent model. In case of sparse correspondence, a parameterized surface model

has to be fit to represent the sparse 3-D points for surface modeling. The explicit handling of

occlusion between views is not addressed.

2.3.3.2 Model-based Multi-view Stereo Method

In order to avoid multi-view stereo drawbacks, the model-based multi-view stereo

approach is proposed by J. Malik et al.[194]. Model-based techniques differ from traditional

stereos in that they measure how the actual scene of the images deviates from an initially given

approximate 3-D model. Model-based approaches reduce the correspondence problem for views,

which are relatively apart. Occlusion problems are also addressed by the use of an initial model

and when the fusion of view-based partial surfaces are eliminated. J. Malik et al.[194] presented

an approach for creating realistic synthetic views of existing architectural scenes from a sparse

set of still photographs using the model-based multi-view stereo techniques. The approach

combines both geometry-based and image-based modeling, plus rendering techniques and

consists of two components. The first component is an easy-to-use photogrammetric modeling


49

system which facilitates the recovery of a basic geometric model of the photographed scene.

The modeling system is effective and robust because it exploits the constraints that are

characteristic of architectural scenes.

The second component is a model-based stereo algorithm, which recovers how the real

scene deviates from the basic model. By making use of the model, the stereo approach can

robustly recover accurate depth from image pairs with large baselines. Consequently, this

approach can model large architectural environments with far fewer photographs than current

image-based modeling approaches.

The view-dependent texture mapping method is also presented to better simulate

geometric detail on basic models. This approach can recover models for use in either

geometry-based or image-based rendering systems. The results were demonstrated when it was

used to create realistic renderings of architectural scenes from viewpoints far from the original

photographs.

However, model-based multi-view stereo approaches need an initial 3-D geometric

model which brings additional user interaction.

2.3.3.3 Volumetric Modeling Method

In order to avoid the disadvantages of multi-view stereos and model-based stereos, the

3-D scene space solution or the volumetric modeling method has been proposed[195-197] . This

method represents the volume of the object by making occupancy decisions about whether a

volume element i.e. voxel, contains object volume in 3-D scenes. The volumetric modeling

method allows views to be captured apart from each other, which may cause problems in other

methods. More importantly, this method eliminates correspondence problems. On the other hand,

the final model evolves from an initial volumetric scene representation in which the object of


50

interest lies. In this context, there is no need to fuse an initial model and resultant 3-D object

data which is the case in multi-view and model-based stereo systems. The occlusion problem is

also addressed by using regular tessellation of the initial bounding volume as voxels. However,

volumetric modeling of object spaces depends on the calibration of the cameras for each image,

and on an initial bounding volume of the object space.

As mentioned in the introduction, the volumetric scene modeling approaches can be

further grouped into two separate approaches: 1) shape from silhouettes and 2) shape from

photoconsistency.

2.3.3.3.1 Shape from Silhouettes

Shape from silhouettes technique deals with the reconstruction of 3-D solid models by

volume intersections from a set of silhouette images. Silhouette-based 3-D reconstructions have

been well known in literature and in practice. It constructs the approximated visual hull of the

object from a finite image sequence. One of the first complete 3-D modeling systems based on

silhouette images is the so-called Hannover system by Niem et al.[198-200] Many concepts and

techniques of the current systems are based on the ones explained and implemented in this

system. Matsumoto et al.[201] improved the silhouette technique by introducing the new concept

of voting in the implementation of the volume intersection.

For practical 3-D reconstruction solutions, the problem can be simplified by using

controlled imaging environments. In such an environment, the camera makes a controlled

motion around the object, and the background surface and lighting are selected to reduce the

specularities on the acquired image. The camera has to be calibrated in such a setup to obtain the

internal and external parameters defining the physical properties of the camera and also the

camera imaging positions with respect to the rotary table turn angles. Niem et al.[198-200] and


51

Matsumoto et al.[201] used a setup consisting of a rotary table (turn-table) with a fixed camera to

obtain a controlled camera motion around the object. Altalay et al.[202] has described a complete,

end-to-end system that explains the steps of using the silhouette method. The flowchart of the

silhouette-based method for 3-D reconstruction is shown in the following Figure.

Silhouette Extraction

Silhouette based Volume Intersection

3‐D Model Fine Tuning using

Photoconstistency

Appearance Reconstruction

Image Acquisition

Camera CalibrationBounding Cube Estimation

Figure 2-6 3-D Object Reconstruction Scheme There are basically six major steps of using shapes from silhouettes after image acquisition: 1)

calibration of a turn-table and extraction of the rotation axis based on vision geometry 2)

silhouette extraction 3) visual hull generation 4) voxelization 5) 3-D model fine tuning using

photoconsistency 6) appearance reconstruction

The first step is the calibration of the turn-table and the extraction of the rotation axis.

This is essential in achieving a high quality of 3-D reconstruction. One major source of

calibration error is the inaccuracy in the 3-D to 2-D mappings of the n known points. A

multi-image camera calibration approach proposed by Lavest et al.[203] is used to estimate the

precise 3-D coordinates of the geometrical calibration target when the intrinsic and extrinsic

camera parameters were represented by the unknown vector.


52

The second step is to extract the silhouette of the real object from the camera images. In

controlled environments, this segmentation of the real object against the background can be

facilitated by using a background of uniform colour.

In the third step, a bounding pyramid volume is constructed using the focal point of the

camera and the silhouette. The convex hull of this bounding volume is formed by the lines of

sight from the camera focal point through different contour points of the object silhouette shown

in the following schematic. The coarse object volume or the convex hull of the object can be

computed to obtain the bounding box of the object when the camera parameters and the object

silhouettes on the images are known.

Figure 2-7 Visual Hull Construction from Volume Intersection

The fourth step is voxelization. The bounding box of the object is then discretized into

small cubes or voxels. Assuming that each side of the box is divided into n voxels, a voxel space

V containing n3 voxels is generated. Subsequently, each cube in the voxel space is projected

onto the images by using the related camera parameters. If the projected cube region on any

selected image is totally not contained by the silhouette region, it is removed from the voxel

space. Otherwise, it is kept in the object voxel space.

The next step is to augment the silhouette-based reconstruction with photometric information

existing in the images to minimize the problem of excess voxels from the visual hull

approximation due to the concavities on the object and the insufficient camera viewing points.


53

In the approximated visual hull of the object obtained from finite sets of images, each

surface voxel has a set of images from which the voxel can be seen. If it is assumed that a

surface voxel is placed in the correct depth, meaning that it is on the real object surface, then the

stereo theory states that its projections on the images from which it can be seen, must be the

corresponding regions. In other words voxels having correct depth placement in the

approximated visual hull have photoconsistent projections on the images from which voxels are

seen without occlusion. The idea, theory and algorithm of using photoconsistency and carving in

volumetric voxel spaces is first introduced by Seitz et al. [195] and have recently been improved

by several researchers.[195,197,204-206] The approaches using photoconsistency to generate 3-D

models is generally called voxel coloring. Starting from an initial set of opaque voxels, voxel

coloring algorithm carves the voxels that are not photoconsistent as it goes through the opaque

voxels. The algorithm stops when all the opaque voxels are color consistent.

Furthermore, carving the voxels of the model resulting from silhouette and the use of

multiple view-based stereo has also been described. but the details and the experimental results

are not given. The system described by Gibson et al.[207] requires no pre-calibration of the

camera and uses full perspective geometry. The authors claim that self-calibration is used.

However, there is no detailed information given in the paper related to this subject. Fitzgibbon

et al. [208] describes the projective geometry of single axis rotation (rotation similar to that

discussed in our work) and gives its automatic and optimal estimation from an image sequence.

However, it is shown that 3-D structures and cameras can be calibrated up to an overall

two-parameter ambiguity. The two-parameter reconstruction ambiguity is removed by

specifying the camera aspect ratio and parallel scene lines. Apart from the use of uncalibrated

multiple views, the system is very similar to that of Hannover. In addition, the work does not

indicate any further refinement to remove the disadvantages of volume intersection method.


54

A study by Mendonça et al.[209] described a method for motion estimation of an object

rotating around a fixed axis. Based on this information, 3-D positions of the points on the

contours of the objects were found by the triangulation and epipolar parameterization. The

method is relatively simple particularly compared to that proposed by Fitzgibbon et al.[208] in the

sense that it only needs a few number of correspondences. One point which is not very clear in

the study of Mendonça et al.[209] is that for the reconstruction of an object model, some means of

interpolation would be required since only contour information is used.

The last step is the texture mapping and visualization of the reconstructed object in 3-D.

Volumetric voxel representation should be transformed to a representation by triangular patches.

A simple and practical technique, called the Marching Cubes Algorithm[210,211] is generally used

in computer graphics to do such a conversion. Marching Cubes Algorithm uses the isolevel

information of the vertices of the voxel in order to interpolate an isosurface passing through the

voxel. Having constructed the geometry of the object, it is needed to recover the appearance, in

other words, the texture of the surface should be determined. Texture mapping is a well-known

technique that is used to achieve a high degree of realism in virtual reality applications. In

image-based 3-D model reconstructions, the texture of the model is extracted from the images of

the object. This increases the realism of the reconstructed model considerably[212].

As mentioned previously, the main drawback of the silhouette-based volume intersection

approach is the additional excess volume coming from an insufficient number of viewing

positions and from the concavities on the object to be modeled. It is impossible to engrave the

additional volume filling the concavities, even if the infinite number of images is used in the

process. In order to carve excess voxels from the inferred visual hull, which is approximated by

silhouette based volume intersection, an algorithm based on photoconsistency is developed.


55

2.3.3.3.2 Shape from Photoconsistency

Rather than using binary silhouette images, shapes from the photoconsistency technique

employs additional colored photometric information.[195] This improves reconstruction results at

the excess voxels existing in the approximated visual hull. Shapes from photoconsistency

methods use the color consistency constraints to distinguish object surface voxels from other

voxels. Color constraints for a voxel states that a surface voxel has color consistent projection

regions on the images from which it is seen. Use of shape from the photoconsistency approach

avoids point correspondence difficulties. 3-D reconstructions based on photoconsistency require

camera parameters for each used view and a model for the object surface reflectance.

Furthermore, a very important issue in these methods is the criterion for consistency

checks. Most of the consistency criteria require a threshold or input from the user. In addition,

voxel visibility problems have to be well addressed since the color consistency checks for a

voxel requires the set of images from which the voxel is visible. Efficiency in the visibility test

for every voxel is essential. In order to easily maintain visibility information for voxels, voxel

coloring algorithm proposed by Seitz and Dyer [195] restricts the position of cameras so that no

object point is contained within the convex hull of the camera centers. However, there are

several extensions to the initial voxel coloring algorithm for the visibility problem[196,197,213-215].

Instead of using these extensions and alternatives requiring additional space and complex

updates as carving progress, or using multi sweeps along the coordinate axes, traversing a ray

from the voxel center through the camera center is proposed to check visibility. It is also

possible to find the maximum photoconsistent voxels on the ray and this eliminates threshold

checks while carving the voxel from the initial set. This algorithm is mostly inspired from the

work of Matsumoto et al.[205]


56

In recent years, there has been a significant interest in 3-D reconstruction from

uncalibrated views. These auto-calibration techniques have been the object of a tremendous

amount of work[216-220] and as such effective methods to derive the epipolar geometry and the

trifocal tensor from point correspondences, have been devised[208,221]. However, most of these

methods assume that it is possible to run an interest operator such as a corner detector [208,220]to

extract from one of the images a sufficiently large number of points that can then be reliably

matched in the other images. When using images exhibiting too little texture, such interest

points are not reliable. It has been shown that projective, affine and Euclidean reconstructions

can be obtained from uncalibrated views[216,222-225]. However, these methods are sensitive to

noise and initialization and generally, the reconstruction results are valid up to a scale factor.

2.4 Summary

This chapter helps to walk through most of the literature reviews associated with the

most relevant topics of this research. The methods for pure polymer PVT measurement are

described. The methods and apparatus used to measure the PVT of polymer/gas systems are also

explained and the difficulties associated with them in measuring polymer/gas solutions PVT at

high temperature and pressure are addressed. The topic of solubility and diffusivity is also

touched upon since solubility is very important processing parameter in the polymer foaming

industry. Knowing how to accurately measure gas solubility inside a polymer melt would help

not only in achieving better products, but also in process and equipment design in terms of

knowing the right amount gas required at any specific temperature and pressure. Nucleation is

also an important issue in polymer process. The rate of nucleation would determine the cell size

and cell density that could ultimately affect cell morphology. To a large extent, the cell


57

morphology is the key determining factor of the foam properties, such as mechanical, acoustic,

and insulation properties. The development of surface tension and methods of measuring it are

also addressed in detail. Since surface tension is one of the important factors in cell nucleation

and the determination of surface tension requires density information of the polymer/gas

solution as an input, the importance of getting accurate density information through PVT

measure for surface tension is also self-explanatory. The up-to-date works on 3-D object

reconstruction based on 2-D images in the content of computer vision and image processing are

also investigated. The setup of our proposed methodology is similar in the way that 2-D images

are used for obtaining 3-D information, i.e. volumetric value. Due to the nature of our

experiment the approach proposed in this research is unique and the simplest design. Details

regarding the design of the system would be elaborated in Chapter 4. In summary this chapter

not only serves as a background survey, but also gives the reasons and the motives behind this

research.


58

Chapter 3. Theoretical Background

3.1 Theoretical Background on Equation of State

3.1.1 Introduction

The behaviours of any natural system can be described using thermodynamic law, which

includes postulates and equations. According to the classical thermodynamics, the following

three basic terms need to be introduced into a system: 1) Enthalpy (H), 2) Helmholtz free energy

(A), and 3) Gibbs free energy (G). Their relationships[226,227] are described as follows:

pVUH += (3-1)

TSUA −= (3-2)

TSHG −= (3-3) Where U is the system internal energy; S is the entropy; p is system pressure; and T is the

temperature of system. According to the Boltzmann equation (3-4) the entropy ( S ) of the

system could be determined as long as the Ω (the number of states accessible to the system) is

known, where K is Boltzmann's constant.

( )Ω= lnKS (3-4)

If a thermodynamic model is available and can be applied to calculate all the above state

properties (H, S, A, G) of a system, we can easily get the following equation to describe the

pressure-volume-temperature (PVT) relation of the system, which is the so-called equation of

state (EOS):

TVAP ⎟⎠⎞

⎜⎝⎛∂∂

−= (3-5)


59

The ultimate target of thermodynamics is to describe the macroscopic properties of

systems at equilibrium condition. For example, thermodynamic models are used for calculations

of the enthalpies, heat capacities, and phase equilibria. Regarding equilibrium systems,

statistical thermodynamics provides a link between the microscopic world of atoms/molecules

and the macroscopic world of bulk properties. Statistical thermodynamics provides means to

calculate fundamental thermodynamic quantities or macroscopic properties from knowledge of

the quantum states available to a system comprising of particles, molecules or components. The

challenge of statistical thermodynamic is how to properly describe the small particle in a

molecular or sub-molecular level [226].

3.1.2 Equation of State (EOS)

Equation of state (EOS) is a mathematical relation between volume, pressure,

temperature, and composition[228,229]. Originally, equations of state were developed and used

mainly for pure components. They were first applied to non-polar mixtures in 1970s[230,231].

Subsequently, EOS was developed rapidly for the calculation of phase equilibrium in non-polar

and polar mixtures. The Van der Waals EOS was the first equation to predict vapour-liquid

coexistence. Many equations of state have been proposed, but most of them are categorized into

two classes: empirical (or semi-empirical) and theoretical.

Equations of state can be used over a wide range of temperature and pressure and they

can be applied to mixtures of diverse components, ranging from light gases to heavy liquids.

They can be used to calculate vapour-liquid, liquid-liquid and supercritical fluid-phase

equilibrium. The calculation of phase equilibrium has been discussed intensively in the literature

recently and this literature is reviewed in detail[232]. For the polymer system, a variety of EOSs

were also proposed for the correlation of polymer PVT behavior. The Flory EOS[33,233,234], the


60

Simha-Somcynsky EOS[43,44,235-237], and the Sanchez-Lacombe EOS[41,42,238-240] are the most

widely used EOS for polymer system. In this research, SL and SS are the two equations of state

which would be used to assist in solubility studies and will be explained more in the following

sections.

3.1.3 Equations of State for Polymers and Polymer/Gas Solutions

3.1.3.1 Introduction

The details of the derivation of SL, SS and SAFT EOS are not repeated here since those

works have already been well established in the literature by many others. The basic

thermodynamic guidelines upon which those EOS are built are stated briefly. The need to

interpret and correlate the properties of polymer liquids and their solutions or polymer/solvent

solution is the initiation of many thermodynamic models.

The majority of thermodynamic theories for polymer liquids are roughly based on four

models or theories: 1) cell models, 2) lattice-fluid models, 3) hole models, and 4) perturbation

theories. Those inlcude the cell model of Prigogine[40], the modified cell theory of Dee and

Walsh[31,32], lattice theory of Flory[233] and Huggins [241], the lattic fluid model of Sanchez and

Lacombe[41,42,239,240] and Panayiotou and Vera [37,242], the non-equilibrium lattic fluid (NEFL)

model of Doghieri and Sarti[243], the group-contribution lattice fluid model of parekh and

Danner[244], and the lattice hole theory of Simha and Somcynsky[44,235].

More recently, the statistical associating fluid thoery (SAFT) modified based on

Wertheim’s thermodynamic perturbation theory (TPT)[245-248] was developed by Chapman et

al.[249,250] Later Huang and Radosz[251,252] modified the SAFT model. The criteria for the first

three models are the mathematical formalism used in accounting for the compressibility and

thermal expansion of the system. Cell models restrict volumetric changes in the system to


61

changes in cell volume, that is, the space surrounding a polymer segment placed on a lattice

framework. In contrast, lattice-fluid models allow empty sites or lattice vacancies while the cell

volume is assumed constant. Hole models make allowances for both cell expansion and lattice

vacancies[28]. The main assumption of perturbation theories is that the residual (difference to

ideal-gas state) part of the Helmholtz energy of a system Ares can be written as the sum of

different terms whereas the main contribution is described by the Helmholtz energy of a chosen

reference system[253]. Most of the theoretical models use mixing rules to describe the mixtures

without changing the structure of the model and also use the reduced variables scaled by

characteristic parameters. Zoller[254] and Rodgers[28] have worked on the PVT behavor of

polymers and compared the predictions using various models. The details of those EOS, except

for SL and SS EOS, are omitted in this research because they are not used in solubility

measurement. SL and SS are the EOS used with solubility measurement in this study and is

explained more mathematically in the following sections.

3.1.3.2 Sanchez-Lacombe (SL) EOS

The Sanchez-Lacombe (SL) EOS is the most widely used thermodynamic model to

describe the PVT property of polymer/gas solutions. As previously mentioned, SL EOS is

derived based on lattice fluid theory which allows for vacancies in the lattice and assumes the

polymer has a flexible liquid structure. The reduced form of SL EOS is through equation

(3-6):

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+−−−= ρρρ ~11~1ln~~~ 2

rTP (3-6)


62

In the above relations, ρ~,~,~ PT and V~ are reduced parameters. They are calculated from the

characteristic reducing parameters P*, T*, V* and ∗ρ as follows:

∗∗∗∗ ==== VVVPPPTTTVV /~;/~;/~;/~/1 ρ ;∗∗

∗=

ρTR

PMr (3-7)

r is the fraction of occupied lattice sites by a molecule composed of r segments or “mers” and M

is the molecular weight. Each molecule in this theory is characterized by P*, T*, ∗ρ that are

used as the normalization constants for the reduced parameters. These characteristic parameters

are typically fit to vapour pressure or liquid density data and are tabulated for other polymers.

Typical gas and polymer characteristic parameters could be found in the literature [91-93].

Mixtures are handled using volume fraction based mixing rules with an adjustable binary

interaction parameter 12k .

3.1.3.3 Simha-Somcynsky (SS) EOS

Hole theory was proposed by Simha and his co-workers[43,44,235,236] in order to improve

on the cell model for the liquid state. Hole theory considers that each lattice of a site can

accommodate either a small molecule or a chain segment. Also as in lattice fluid theory, hole

theory adopts an improvement of the cell model for the liquid state by the introduction of

vacancies in the lattice [44], which describes the major part of the thermal expansion. Changes in

cell volume itself, which have a non-negligible influence on the thermodynamic properties, are

also allowed. In the Simha-Somcynsky hole mode, the “square well” approximation to the cell

potential is used and non-nearest neighbour contributions to the lattice energy are included. The

resulting coupled equations of state must be solved simultaneously with an expression that


63

minimizes the partition function with respect to the fraction of occupied sites. The reduced

equation for SS EOS is:

TQyQTVp ~

)2045.1011.1(2)1(~/~~22

1 −+−= −η (3-8)

( )22 033.3409.2~613/1)1ln(1

3QQ

Ty

yy

ss

cs

−+−−

=⎥⎦

⎤⎢⎣

⎡ −+

−⎟⎠⎞

⎜⎝⎛

ηη

(3-9)

Again ρ~,~,~ PT and V~ are reduced parameters in the above relations. They are calculated from

the characteristic reducing parameters P*, T*, V* and ∗ρ . Small s is the number of segments

per chain, c is the number of external degrees of freedom per chain, and

1/ 6 1/ 31/( ) , 2Q yV and yQη −= =% are dimensionless quantities.

3.1.3.4 Statistical Associating Fluid Theory (SAFT)

Although the statistical associating fluid theory (SAFT) model is not the focus of this

research, due to its rising popularity, there is a need to explain this equation of state model in

greater detail. The equation of state based on Statistical Associating Fluid Theory (SAFT)

describes the molecules which move freely in continuous space. SAFT EOS was extended based

on Wertherim’s Thermodynamic Perturbation theory[245-248] from Chapman et al.[249,250] Huang

and Radosz developed the modified SAFT equation of state. The SAFT EOS accounts for

hard-sphere repulsive forces, dispersion forces, chain formation and association. It could be

represented as a sum of four Helmholtz function terms:

nassociatiochainsegmentres AAAA ++= (3-10) or it could be also in the form of the compressibility factor:

nassociatiochainsegment ZZZZ +++=1 (3-11)


64

The SAFT model describes liquid molecules as equal-sized spherical segments interacting

according to a square-well potential. The detail derivation of the above equations can be found

from studies of Huang and Radosz.

3.2 Models of PVT Measurement

Several models or equations of state (EOS) have been proposed to describe the PVT

behavior of polymeric liquids based on thermodynamic law. Those equations of state for

polymer are either empirical or theoretical. The most widely used empirical equation is the Tait

equation[46].

The Tait equation is an isothermal compressibility fitting model rather than a true

equation of state. It reliably calculates the specific volume of pure polymer at different

temperatures and pressures using PVT data measured from a bellow type dilatometer.

Tait proposed an empirical model of average isothermal compressibility for fresh and sea water

as follows:

( ) ( ) ( ) ( )( ) ⎥

⎦

⎤⎢⎣

⎡+−+

=PTB

TPCPTBTPvTPv ,, 0 (3-12)

Where 0P is the initial pressure and ( )TPv ,0 , ( )TB and ( )TC are functions of temperature

only. Wohl [255] took the similar concept and rewrote the compressibility definition to derive the

following equation:

( )( ) ( )

( ) ⎥⎦

⎤⎢⎣

⎡+

=⎥⎦⎤

⎢⎣⎡

∂∂

−PTB

TCP

TPvTPv T

,,

1

0

(3-13)

After performing integration on the above equation, the following can be derived:


65

( ) ( ) ( ) ( ) ⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

−+−=

0

00 1ln1,,

PTBPP

TCTPvTPv (3-14)

By letting 00 =P the equation becomes:

( ) ( ) ( ) ( ) ⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+−=

TBPTCTvTPv 1ln1,0, (3-15)

Where the zero-pressure isotherm is:

( ) TvvTv 10,0 += (3-16)

And the ( )TB is also known as:

( ) TBeBTB 10

−= (3-17)

where 010 , Bandvv are constants. They are determined based on the empirical fitted data for

each different material grade. The ( )TC is a universal constant and is independent of

temperature. The value is 0.0894. Therefore the following equation is the finalized general Tait

equation form:

( ) ( ) ( ) ⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+−=

TBPTvTPv 1ln0894.01,0, (3-18)

The equation has various forms and coefficients depending on the grade of the polymer sample

used.

Empirical studies have been carried out to examine the PVT relationship for polymer/gas

solution empirically besides using EOS to calculate the PVT relation. Most of the background

for polymer/gas PVT measurement were already explained in previous literature section.

Foster et al.[69] used a cylinder-piston-type method in a compressibility chamber to

acquire PVT data by multiplying the cross-sectional area of the piston with the linear

displacement of the piston. Ender[68] has used the coil of a linear variable differential

transformer (LVDT) to measure the volume swelling of various elastomers under high pressures


66

in CO2 at relatively low temperatures. Wissinger et al.[75] presented a method of measuring

polymer swelling by determining the changes in length of a thin polymer film. And the change

in volume is related to its changes in length using the following relation:

13

00−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

ΔLL

VV (3-19)

Unfortunately, their approach is only valid for isotropic swelling. Furthermore, the

warpage of the samples limits its application and results in an inaccurate length measurement. In

general, most of the methods described above operate at fairly low temperatures, ranging from

25°C to 100°C. Park et al. [71] measured the PVT properties of polymer/CO2 solutions using a

foaming extruder and a positive-displacement gear pump mounted on the extruder. The major

drawback of this method is the required (and limited) extrusion processing conditions for

measuring the PVT data.

3.3 Solubility Measurement Using MSB

Having talked in Chapter 2 about what has been done in measuring the solubility data of

gas in the polymer matrix, the focus now is on how solubility is measured and calculated using

magnetic suspension balance (MSB), since one of the motives of this research is to facilitate a

better MSB solubility measurement by using PVT apparatus. Seen from the previous MSB

schematic, the polymer sample was placed in a container inside the sorption chamber before the

experiment. The chamber was then sealed, evacuated, and preheated to the experimental

temperature. The read-out from the balance at the vacuum condition (P=0) and this specific

temperature (T) was denoted as ( )TW ,0 . The chosen blowing agents, such as CO2, N2 or butane

were compressed and introduced into the sorption chamber. The pressure was maintained by a

syringe pump. A Jalabo TD-6 heating circulator was utilized to control the temperature of the


67

sorption chamber with precision. The mass of the sample will increase during the sorption

period untill the equilibrium state is reached or in other words, the sample will be saturated with

the surrounding gas reservoir at that pressure. When the equilibrium state is reached, the sample

mass will not change unless a high pressure is introduced by having more compressed gas enter

the sorption chamber. At that particular pressure (P) and temperature (T), the weight read-out of

the balance is denoted as ( )TPW , . Therefore the amount of gas that dissolved into the polymer

at each pressure and temperature condition is denotes as gW and is calculated through the

following equation:

( ) ( ) ( )sPBgasg VVVTWTPWW +++−= ρ,0, (3-20)

Where gasρ is the density of the gas and can be measured in situ by the function of MSB; BV

is the total volume of the sample holder and measuring load coupling devices shown in Figure

1-5. This BV volume is determined using the buoyancy method with the high-pressure gas; and

PV is the volume of the pure polymer sample at pressure P and temperature T. This volume can

be determined from the Tait equation of pure polymer with the weight of the polymer sample.

sV is the swellon volume of the polymer/gas mixture due to the gas dissolution. If this swellon

volume term is being ignored, then the reading of the balance will be the so-called “apparent”

solubility of the polymer/gas solution.

In order to get accurate solubility data, volume swelling has to be taken into account. But

with the lack of direct measurement data, the only method is to utilize those equations of state

(EOS) to predict the swelling information. This is one of the reasons why EOS is important and

has been detailed in the beginning of this chapter. Among the various EOS, the most popular

ones are the SL, SS and SAFT EOS. The SL and SS EOS are also going to be the focus of this

research in the latter chapter when the swellings from direct measurement from the newly


68

developed PVT apparatus are compared with those predictions. By taking into account the true

volume swelling sV from the direct measurement, the equation can then be used to determine

the true solubility directly without relying on the predicted values from EOS.

3.4 Viscoelastic Models of Polymers

Since the hypothesis made in the later section from the research results is strongly

related with the viscoelasticity property of polymer material, it is necessary to elaborate on the

study of polymer’s viscoelastic characteristic. The polymer materials are non-Newtonian fluid in

nature. Newtonian fluid is a fluid that does not pose elastic properties and has no memory of

past deformations. The study that needs to be done with Newtonian fluid is the measurement of

its deformation rate, rather than its strain. Since the polymer material is non-Newtonian fluid

and is both elastic and viscous in nature the so-called viscoelasticity is one of the most important

rheological properties of polymer melt. This uniqueness of the elastic and viscous nature of

polymer determines many of the properties of polymer melt in experiments and industrial

processing. The following example describes the distinctive difference between elasticity and

viscoelasticity. Rubber is different from polymer material in that the chains in the former are

cross-linked by chemical means. A rubber material can be regarded as macroscopic

three-dimensional network formed of strands of polymer chain with both ends as cross-links.

Because the cross-links in rubber keep the chain strands from moving away from their relative

positions over a distance larger than the strand size itself, the chains do not flow with the

deformation and are able to recover their conformations once the deformation is released. In

contrast to this behaviour, an un-cross-linked polymer exhibits the elastic response only at its

linear region for momentary deformations. But under a prolonged deformation, it cannot recover


69

its original shape because the molecular chain has flowed. Therefore, for a polymeric melt, the

elasticity and viscosity are both considered. The slow flow property of the polymer is

characterized by its viscosity. There are two different approaches to model the viscoelasticity of

polymeric fluids[256]: 1) models based on the polymer chain dynamics within the linear region,

such as Maxwell equation and Boltzmann’s superposition principle, and 2) models based on the

molecular theory such as Elastic Dumbbell Model, the Rouse Model and the Doi-Edwards

Model.

3.4.1 Maxwell Equation

An easy way to describe the Maxwell equation is to use a simple spring and damper to

represent the elastic and viscous natures of the viscoelasticity of polymeric fluids.

F

……

.

damper spring

F

……

.

damper spring

Figure 3-1 Generalized Maxwell Model Represented by Spring-Damper Systems

Figure 3-1 shows a simplified and generalized Maxwell model represented by spring and

damper combinations. The general Maxwell equation is as follows:

( ) ( )∑=i

i tt σσ (3-21)

with

( ) ( )•

−=+ λτη

στ

σi

ii

ii tt

dtd 1

(3-22)


70

and

i

ii G

ητ −= (3-23)

Where iτ is the relaxation time of the viscoelastic system, iσ is the stress, iη is the viscosity,

•

λ is the shear deformation rate and iG is the relaxation modulus.

3.4.2 Boltzmann’s Superposition Principle

Boltzmann’s superposition principle assumes that if the stress at the present time t is

caused by a step strain at an earlier time t’, the stress is linearly proportional to the strain, and

the proportionality or the modulus decreases with the separation of the time, t-t’. The modulus, a

decaying function of t-t’, is denoted by G(t-t’). Boltzmann’s superposition principle says that

when a system has been applied with small step strains at different times, t1, t2, ..etc…before the

present time t, then all those stresses caused by the individual step strains are independent of

each other, and the total stress at t could be simply expressed as the summation of all those

stresses:

( ) ( ) ( )∑ Δ−−=i

ii tttGt λσ (3-24)

3.4.3 Doi-Edwards Model and Entanglement

The theories developed based on molecular interaction are the most complicated area in

polymer rheology. Since it is not the main focus of this research, mathematical rigour is not

attempted in this introductory treatment for the Elastic Dumbbell Model and the Rouse Model.

Since the effect of the polymer chain entanglement density on the volume swelling hypothesis is

postulated in a later section based on research findings, the Doi-Edwards Model developed from


71

polymer chain entanglement is worth taking a brief look at. Chain entanglement is a kind of

intermolecular interaction that occurs in a concentrated long-chain region of polymer melt.

Unlike other types of intermolecular interactions, it does not involve any energetic change and

does not give rise to a change in the electron density, hence chain entanglement cannot be

observed by microscopes or static scattering or absorption spectroscopies. The Doi-Edwards

Model could be represented in a simple form as follows:

N

e GRTM ρ

= (3-25)

eM is the entanglement molecular weight and NG is the plateau modulus[257-259].

Chain entanglement mainly affects chain motions. Another strong effect of chain

entanglement is the observation of a clear plateau in the linear relaxation modulus

( )tG [257,259,260] and storage modulus spectrum ( )ω'G [260]. Those terms could become clear in

the later sections when the rotational rheometer is used to study the polymer chain entanglement

density.

3.5 Surface Tension Measurement

As described in the earlier background and literature review, many different techniques

and apparatus have been developed to study and measure surface tension. Out of all those

methods, the drop shape method becomes the most used because of numerous advantages. For

instance, drop shape method only requires a small amount of sample to conduct an experiments

in comparison with the Wilhelmy plate technique. Also the drop shape method is easy to handle

and can be used in difficult experimental conditions such as in studies of high temperature and

pressure effects on surface tension. In addition, this method can be applied to a variety of


72

materials, ranging from organic liquids to molten metals and from pure solvents to concentrated

solutions. In our case, this method is utilized for polymer/gas solutions. Also, as pointed out by

Hoofar et al. [180], the drop shape method could be used to study surface tension in a dynamic

system by recording digital images where the properties are time dependent. So in this section, a

brief theoretical background based on what has been introduced previously will be expanded on

to illustrate the mathematics behind the drop shape method.

Traditionally, the Young-Laplace capillary equation, as shown in equation (3-26) is the

basis of the surface tension measurement. The equation actually shows a balance between

surface tension and external forces, such as gravity, in mathematical form. It also shows the

relationship between the radii curvature and the pressure difference across a curved interface.

⎛ ⎞⎜ ⎟⎝ ⎠

lg1 2

1 1Δ P = γ +R R

(3-26)

where γlg is the surface tension, R1 and R2 represent the two principle radii of curvature, and ΔP

is the pressure difference across the interface. In the absence of any other external forces

except gravity, it can be expressed as a linear function of elevation:

gzPP ρΔ+Δ=Δ 0 (3-27)

Where 0PΔ is the pressure difference at a reference plane and z is the vertical coordinate of the

drop measured from the reference plane. From the above two equations, it is evident that, if the

shape information of the drop is given, then surface tension determination would be possible.

On the other hand, if the surface tension is known, the shape of the drop could also be

determined. From Figure 3-2, the Young-Laplace equation can be expressed as a set of the

following first-order differential equations:

dx = cosφds

(3-28)


73

dz = sinφds

(3-29)

dφ sinφ= 2 + z -ds x

χ (3-30)

χ, the shape parameter is defined as:

20 lg= ΔρgR /γχ (3-31)

where R0 is the radius of curvature at the origin of the (x, z) coordinate system and Δρ is the

density difference between the polymer-gas solution and gas phases. The overall method to

determine γlg is outlined as follows: 1) Obtain the (x, z) coordinates of the boundary profile of a

sessile drop in equilibrium with the blowing agent using the captured images from the PVT

visualization system; 2) Starting from an initial estimate of the γlg and the empirically derived

parameters, a sessile drop profile is also calculated from Equations (3-28) to (3-31) [261]; 3) This

theoretically determined sessile drop profile will be compared with the ones measured from the

images taken by the PVT visualization system. The former will be updated in an iterative

process until the best fit between the two is obtained; 4) From the updated sessile drop profile,

the surface tension value will be determined using the Young-Laplace equation. The detailed

numerical method can be found in Ref. [261]

Figure 3-2 Sessile Drop Coordinate System Definition


74

3.6 Summary

The main purpose of this chapter is to build a strong and in-depth background on the

theories of the issues that will be used or addressed in this research. Starting from the

thermodynamic laws to describe single component or mixture systems, the postulates and

relations are briefly introduced to lay the ground work for deriving the thermodynamic

equations of state (EOS). Equations of state (EOS) are intensively used to describe and

determine the property of pure substances as well as mixtures. SS EOS and SL EOS are the ones

used in conjunction with MSB solubility measurement through providing predicted volume

swelling to compensate the buoyancy effect. Therefore, much more attention is paid to closely

looking at their mathematical formulations.

Following the EOS, the literature of PVT measurement for pure polymer and

polymer/gas mixtures is briefly iterated. The brief derivation of the Tait equation shows that the

Tait equation has a general format for different types of polymer materials and that it is the

coefficients that distinguish among polymers. The pros and cons of the methodologies used for

polymer/gas PVT measurement are depicted. The major drawback of those methods is their low

operating temperature where there is not polymer/gas single solution formation. In this research

the experimental temperature is usually above the melting point (Tm) of crystalline polymer,

such as PP or glassy transition temperature (Tg) of amorphous polymer, such as PS. A brief

introduction on the modeling of this unique viscoelasticity nature of polymer or polymer/gas

solution provides a foundation for the later study on the polymer chain entanglement effect on

the volume swelling. Lastly, usage of the drop shape method to measure the surface tension is

addressed mathematically so that it is easier to physically see the determining factor in surface

tension measurement.


75

Chapter 4. Design and Construction of Novel Apparatus for PVT measurement

4.1 Conceptual Design of PVT Measurement Apparatus

4.1.1 Introduction

The familiar phrase, “problem solving is what engineers do” is quite true and

non-deniable, in that the essence of it explains that engineers need to use the right technique for

problem solving. Therefore, for our design problem, the well-known Axiomatic design method

is applied and used to generate feasible design solutions.

The Axiomatic design method was developed by Suh[262] at the Massachusetts Institute

of Technology (MIT) as a systematic framework for guiding engineering designs based on two

axioms. The design process itself is actually a mapping process from functional domain to

physical domain. The things that need to be achieved in the design are defined as function

requirements (FRs) and the corresponding parameters that could meet the function requirement

are defined as design parameters (DPs). Under the context of the FRs and DPs, design is also

defined as the creation of a synthesized solution to satisfy perceived needs through the mapping

process between the FRs and DPs. Figure 4-1 shows the general mapping relationship.


76

…....

FRs

……

1

2

3

4

5

FRs

……

1

2

3

4

5

Mapping

Functional Space Physical Space

…....

FRs

……

1

2

3

4

5

FRs

……

1

2

3

4

5

Mapping

Functional Space Physical Space

Figure 4-1 Axiomatic Design Mapping Process from FRs to DPs

Many researchers have been using this method to successfully achieve their ultimate objectives

in their research.There are two axioms that govern a good design. The following lists the two

axioms

Axiom 1 The Independence Axiom

Maintain the independence of functional requirement Axiom 2 The Information Axiom

Minimize the information content Axiom 1 deals with the relationship between functions and physical variables, and

Axiom 2 deals with the complexity of the design. Once those FRs are defined and the DPs are

identified, the mapping process between the functional domain and the physical domain can be

represented by a design equation as

[ ] DPsAFRs = (4-1)

where FRs and DPs are vectors and [A] is the so-called design matrix. The elements of

the matrix represent the degree of coupling between iFR and jDP . If the coefficient of the

design matrix A is small or near zero, then the function requirement of FRs correspond to a

weak or independent magnitude relationship with DPs, which implies that the changes of that


77

particular DP could not alter that FR or vice versa. If there is strong relationship between FRs

and DPs, the coefficient A is denoted by some non-zero value, mostly 1 or just a symbol. In

order to satisfy the independent axiom A, the design matrix A has to be a diagonal or triangular

matrix. A design with a diagonal matrix is an uncoupled design and with a triangular matrix it is

a decoupled design. Uncoupled or decoupled designs have a set of DPs that meet the specific

FRs and their relationship also meets the independence axiom. On the other hand, coupled

designs are not satisfactory because they fail to meet the requirement of Axiom 1 and needs to

be reconsidered in picking the right DPs.

In this research, the primary goal is to develop a new system that could measure the PVT

information for polymer/gas solutions at high temperatures and pressures; hence the axiomatic

design method was chosen to help create a conceptual design of the PVT measurement system.

4.1.2 Analysis of the PVT Apparatus for Polymer/Gas Solutions

Our proposed new method adopts the pendent/sessile drop technique and combines it

with a visualization system that allows for the tracking of the swelling of a polymer melt due to

gas dissolution over a wide range of elevated temperatures and pressures.

The overall design of the system consists of two major functional attributes: the software

attribute and the hardware attribute. The software attribute is the implementation of the function

modules that perform individual tasks as well as the integration of these modules. The

individual tasks are as follows: image capture, image reconstruction, and volume integration.

The hardware attribute is the actual physical components that are either delicately machined

components or are standard parts purchased off the shelf, to be able to realize the functions

required from the software attribute when integrated together. Such components comprise of a


78

camera system, high temperature and pressure visualization chamber, light source, and image

processing terminal, etc. Figure 1 shows a schematic of the overall PVT system.

Image Processor

Temperature Controller

CCD Camera

Rotational Droplet Rod

High T and P Cell

High Pressure Supply

Gas Source

Backlighting

Vibration Free Table (VFT)

XY Precision Stage Motor

Figure 4-2 PVT System Schematic Those two major functional attributes are then integrated to construct the complete

apparatus. The software algorithm, such as the camera movement and image capture, etc. that

are the next level FRs within the software attribute, were developed and implemented in the

actual system to facilitate the functions such as stage movement and camera triggering for image

capturing, etc. through the apparatus.

4.1.3 Detailed Analysis and Decomposition of FRs and DPs

4.1.3.1 Hardware Attribute

As mentioned, hardware attributes are those experimental components that work only

when integrated with software functional modules. The process of defining the major functional

requirements (FRs) is explained in the following paragraphs.

Since the design adopts the concept of the pendent/sessile drop technique with

visualization, the apparatus must consist of a customized high-temperature and pressure


79

visualization chamber, a charged couple device (CCD) camera, and a computer terminal

installed with the image grab card for the CCD camera. In order to be able to provide a high

temperature and pressure environment, the chamber system itself also needs to have electrical

heater(s), temperature control system, pressure supply system, as well as a light source with a

diffuser for the illumination of the CCD camera. In order to improve the accuracy of the

calculation of the image processing, an XY lead ballscrew stage with a nanometer resolution is

needed so that the camera can move to capture multiple images for the purpose of getting higher

resolution through image reconstruction. Also in order to eliminate most of the unexpected

external noise (vibration mainly), the whole system setup would need a vibration free table as a

base so that the system could enjoy a vibration free environment as much as possible during the

experiment. One additional defined function requirement is to have the system capable of

rotating the pendent/sessile droplet in any desired angle to study the effect of asymmetry drop

shape, as well as to obtain reliable results when the drop shape is not axisymmetric. This

rotation would allow a 360 revolution view of the polymer sample so that a complete profile can

be captured on camera to obtain more accurate volume information for some polymers, i.e.

polystreyene (PS) with higher melt strength and viscosity which can not form axisymetric

shapes as easily compared with others. As such, the solution to this requirement is to have a step

motor as one of the hardware components to provide the rotation.

Therefore, from the above description, the specific functional requirements (FRs) for

developing a comprehensive PVT measurement system that is able to achieve the ultimate

objectives can be summarized as follows:

FR1 = Visualization

FR2 = Image capture

FR3 = Multiple image capture


80

FR4 = Sample rotation

FR5 = Vibration noise reduction

FR6 = Data storage/processing

Also with the brief descriptions from each FRs, in order to fulfill the above FRs, the

following design parameters (DPs) are proposed and summarized respectively:

DP1 = Steel chamber with sapphire window

DP2 = Charged Couple Device (CCD) camera

DP3 = Precision XY stage

DP4 = Step motor

DP5 = Vibration free table

DP6 = Computer terminal

With all FRs and DPs identified, the design equation for the hardware of the system can be

written in the following matrix:

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

6

5

4

3

2

1

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

6

5

4

3

2

1

DPDPDPDPDPDP

AAAAAA

AAAAAA

AAAAAA

AAAAAA

AAAAAA

AAAAAA

FRFRFRFRFRFR

(4-2)

The diagonal elements Aii of the design matrix can be noted using symbol "x", simply

because each jDP is chosen directly to accomplish the corresponding iFR . An examination of

all the non-diagonal elements of the matrix is required in order to determine the effects of each

jDP on the other iFR .


81

The steel chamber with sapphire window only provides the visualization capability to the

system and will not affect other functional requirements, such as image capture, image rotation,

etc. Therefore the steel chamber with sapphire window has no relationship with other FRs and

the remaining coefficients other than A11 on that row are denoted as “0” based on the coefficient

definition of the design matrix. Similarly, the CCD camera would not affect other FRs except

the image capture; therefore all the coefficients, such as A21, A23, A24 and A25 in that row, except

A22, are all zeros.

The purpose of multiple image capture is to increase the resolution of the reconstructed

image from those captured ones. By increasing the resolution, the number of pixels would

increase and the pixel size would reduce which is beneficial for volume integration. This image

reconstruction concept will be elaborated more in a later section. In order to accomplish this task,

the camera needs to be mounted on a stage that has precision movement. Therefore this XY

stage with micrometer movement and nanometer error resolution would fulfill the requirement.

In addition, the capture of multiple images is also strongly associated with the CCD camera. If

the camera is not working, then this multiple image capture functional requirement would not be

realized even with a functional precision stage. Therefore the A32 and A33 should be non-zero.

Other than A32 and A33, all other coefficients terms should be zero since the XY stage does not

have a strong relationship with other FRs.

The sample rotation mechanism is used to capture the asymmetry of some highly viscous

polymers, such as PS, that cannot easily form nice axisymmetric dorm drop shapes at desired

temperatures. An ideal axisymmetric dorm drop shape would be the best for image analysis to

achieve the best accurate volume integration results. The step motor is used to couple with the

sample droplet so that the sample can rotate inside the chamber. It is clear to see that the use of a


82

rotational motor is pretty much independent of other FRs. As a result, the coefficients other than

A33, should all be zeros.

Noise reduction and control is another key aspect of the system design, which helps to

eliminate any external factor that can affect the accuracy of experimental results. The

installation of a vibration free table would certainly help to damp out any external noise that

could propagate to the chamber system as well as keep the whole apparatus setup level. So the

coefficient A55 is definitely a non-zero and denoted as big X. The installation of the vibration

control device would not affect image storage and processing but does have some relationship

with other FRs. For instance, if the vibration is so severe and without the vibration free device,

those FRs, such as image capture, sample rotation, etc, could not be carried out properly. On the

other hand, the chance of having detrimental vibration noise is not likely, so the relationship of

the vibration free table with other FRs is there, but not strong and it is being recognized and

denoted using small x.

The last major function requirement is data storage and processing. This functional

requirement could be easily met by using a computer with sufficient storage space. For data

processing, the relevant tools or software required will be properly installed on the computer

terminal. It may appear as though the computer terminal will only serve as the storage and

processing station that has nothing to do with other FRs, but it indeed will affect the image

capture. Without the storage and processing ability, the captured images would have nowhere to

go and would not be able to be processed. In this case, the coefficients of A62, A66 should not be

zeros. Other coefficients on this row are then zeros. From the above detail analysis, the matrix

(4-2) can be reformatted as:


83

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

6

5

4

3

2

1

6

5

4

3

2

1

0000000000000000000

DPDPDPDPDPDP

XXXXXXxxxx

XXXX

XXX

FRFRFRFRFRFR

(4-3)

Equation 4-3 is a lower triangular matrix, which implies that the functional requirements

can be achieved if the design parameters are implemented in the proper sequence. . Furthermore,

the design in this parental level is an uncoupled design. The first Axiom is satisfied by having a

decoupled or uncoupled design.

In order to meet the requirement of Axiom 2, it is necessary to decompose these FRs and

DPs to lower level hierarchies to try to minimize the design information.

4.1.3.2 Future Decomposition of FR1 and DP1 (Second Level)

As stated in the previous section, in order to minimize the design information of the

parent level, it is sometimes good practice to future decompose some FRs and DPs into sub

levels. In this case, one more level of decomposition of FR1 and DP2 would need to be added

because there are more functional requirements identified on the visualization chamber system.

In another words, there are more design details that need to be added onto the visualization

chamber to achieve other types of functional requirements. In addition, the corresponding DPs

are also required in response to those second-level FRs from the chamber system. The following

sections depict the detailed analysis of the decomposition and formulation of the second level

design matrix.

Firstly, the chamber body needs to be maintained at a high temperature and pressure

reservoir. Therefore, a heating device and pressure supply equipment need to be attached to the


84

chamber. Secondly, due to the introduction of high pressure gas inside the chamber, a proper

sealing solution is required at the visualization window, as well as at the connecting surface of

the chamber with other component(s) while maintaining a high pressure level. In summary,

those additional functional requirements for the chamber would be identified as:

FR11 = High temperature

FR12 = High pressure

FR13 = Sealing at high temperature and pressure

In response to the second-level FR11, the electric heaters and temperature controller

system are needed. The heaters are inserted into the grooves that are machined out of the

chamber body. Those heaters are connected to a temperature controller, which can actively

control and maintain the set temperature of the chamber system using PID control. In response

to the high pressure functional requirement FR12, the gas injection and release port are needed

so that the chamber can be pressurized by allowing high pressure gases to be injected. Last,

since the pressurized chamber could have leakage at some of the openings, such as the place the

sapphire windows are housed, proper high temperature and pressure seals and seal groove are

required to be used. In summary, those additional design parameters DPs are listed as below:

DP11 = Electric heater and temperature control system

DP12 = High pressure supply unit

DP13 = High temperature and pressure seal

The design equation then can be represented as:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

13

12

11

333231

232221

131211

13

12

11

DPDPDP

AAA

AAA

AAA

FRFRFR

(4-4)


85

Together with the temperature controller, the electric heaters that are inserted into the

groves into the chamber body would provide the heat capacity to ensure the chamber

temperature is maintained at desired levels. The pressure supply unit through the gas injection

port into the chamber would provide pressurized gas into the chamber cavity. Those two

functional requirements are independent of each other; in that neither one could have an affect

on each other. Therefore, A12, A13 and A21 are zeros. The rest of the coefficients of the first two

rows are non-zeros. If the seal happens to be defected or worn out after being used for many

experiments, then the high pressure gas supply alone would not have the chamber maintained at

high pressure. Hence A23 is not zero. On the other hand, the choice of high temperature and

pressure seal would always be providing proper seal and heat resistance no matter if the heat or

the gas supply is working or not. Therefore A31 and A32 are zeros and A33 is non-zero.

The equation (4-4) could then be rewritten as:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

13

12

11

13

12

11

000

00

DPDPDP

XXX

X

FRFRFR

(4-5)

The newly rewritten matrix equation (4-5) proves to be an upper triangular matrix. It

implies that the second level design is also a decoupled design with all the design elements

added onto the body chamber. Each of the functional requirements could be achieved without

interfering with the others. In conclusion, this is a sound design solution.

With those proposed decoupled designs, the big picture for constructing the hardware

part of the apparatus is now becoming clear. The CCD camera, with its optical lens that is

synchronized with the computer image-processing terminal, is mounted on a high-resolution XY

stage driven by a precision step motor providing precise x and y directional movements. The

visualization chamber, where the pendent/sessile droplet sample is housed, is located between


86

the light source and the CCD optical lens. The chamber is maintained at user-defined

temperatures and pressures through precision PID temperature controls and syringe pump

pressure controls. Another step motor (stepper) coupled with the pendent or sessile droplet

would rotate the sample if required to capture the asymmetry drop shape of high viscous

polymer/gas solution and to obtain more accurate results. The diffuser attached to the light

removes the noise from the light, making it more uniform when it shines on the pendent or

sessile drop, and the vibration free table damps out the external vibration noise so that the

system that sits on it and would not be affected. A similar approach is taken to finish the

design process for the software part of the PVT apparatus in the following sections.

4.1.3.3 Software Attribute

Having chosen the Axiomatic design technique to successfully finish the hardware

component design, the same design concepts and logic could be used in the design of the

software of the system. The design of software is easier compared with hardware design and it is

mostly involved with algorithm designs in order to achieve certain functionality. The

implementation of the algorithm is going to be addressed in a later section.

In the PVT measurement system, the image of the pendent or sessile drops is first

captured and saved. The most important step during the image analysis is the acquisition and

determination of the drop boundary based on the pixel intensity values of the greyscale image.

Normally, the image contains the image data in the form of digital picture element or pixels. The

value of each pixel is called the intensity or grey level (in the black-and-white case). There are

256 grey levels to represent an image, with 0 representing black and 255 representing white.

The intensities of pixels at the boundary layers gradually vary and fade out into the

neighbouring black or white background (in our case, the sessile drop is a black or white


87

background (in our case, the sessile drop is black object and the background is white). The

errors in image processing are mostly involved in the assessment of the exact boundary

determination. Either using the high precision/resolution CCD camera or enlarging the drop by

an optical system will reduce the pixel size and increase the number of pixel points, which in

turn reduces the relative errors involved in the boundary determination.

Thus, instead of acquiring a small, single image, it is proposed to take multiple images

for PVT measurements. This is the FR3 in the hardware attribute section. The detaied procedure

is: the sessile or pendant droplet sample is first magnified at the maximum magnification of the

CCD camera; then, sequential images are taken locally along the drop boundary to cover the

whole drop. Therefore, these captured individual images need to be reconstructed into one

single image during post-image analysis. The drop profile of the reconstructed images is

required to be identified as boundary coordinates of the pixel unit. Then, based on those pixel

coordinates, numerical integration algorithm is needed to compute the volume based on the drop

profile. Lastly, the calibration for converting the pixel unit into metric length is required so that

the pixel volume information can be represented in terms of the metric dimension. Hence, the

functional requirements FRs for the software development are:

FR1 = Image reconstruction

FR2 = Image profile recognition

FR3 = Volume integration

FR4 = Volume conversion

In response to the first FR1, an image stitching software or a simple algorithm could be

utilized for reconstructing those individual images into one image. For image profile recognition,

the precise subpixel edge detection algorithm is then employed to identify the edge boundary

coordinates in terms of the pixel unit. The Canny edge detector is used and it is proven to be the


88

best subpixel edge detector.[56,263-267] Polynomial curve fitting and Simpson’s integration

techniques are used to develop the algorithms to compute the polymer-gas drop volume. Finally

a calibration procedure performed before the experiment would be able to obtain the conversion

factor between the pixel and metric unit. Therefore, the design parameters (DPs) to satisfy the

FRs can be chosen as follows:

DP1 = Image stitching algorithm

DP2 = Edge detection algorithm

DP3 = Numerical integration algorithm

DP4 = Calibration technique

The design equation can be represented in the following matrix:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

4

3

2

1

44434241

34333231

24232221

14131211

4

3

2

1

DPDPDPDP

AAAA

AAAA

AAAA

AAAA

FRFRFRFR

(4-6)

Since the calibration process is commenced usually before the experiment, it is desired

to switch their FR4 and DP4 to the front to be FR1 and DP1. The sequence of other FRs and

DPs remained. After the swap, the calibration DP1 is then independent of all other FRs. Hence

the coefficient of the first row should be all zeros, except A11. The edge detection algorithm

(DP3) can not be finished unless the image reconstruction (DP2) is carried out successfully. The

edge detection and image reconstruction has no relationship with the calibration (DP1),

therefore the coefficients in the middle two rows are all zeros except A22, A32 and A33. For the last

DP4, the numerical integration for volume would not be possible until all of the above are

realized. Hence, it strongly depends on the successful completion of all the others functional


89

requirements and that its coefficients are all non-zeros. A perfect lower triangular matrix is

generated from the software design and it is written as below:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

4

3

2

1

4

3

2

1

000000

DPDPDPDP

XXXXXXX

XXX

FRFRFRFR

(4-7)

This low triangular matrix again illustrates that the proposed design for the software

construction is a perfect design with the right sequence of order as shown in 4-6 and 4-7. The

sequence of the design parameter is the actual experimental sequence or step in post-image

processing, so it makes more sense now to explain why it is a perfect design with triangular

matrix. For instance, the image profile detection can not be done until the images are captured

and reconstructed. Similarly, only after the profile of the images is detected and being

represented in terms of pixel coordinates, can the use of the integration technique that calculates

the drop volume that is based on the pixel boundary profile, actually proceed.

4.2 Detailed Design and Construction of the PVT Apparatus

4.2.1 Overview of the Apparatus

Based on the conceptual design of the PVT measurement apparatus, the design of the

detailed system components is carried out in this section. The overall system consists of four

major components: 1) high temperature and pressure visualization chamber, 2) CCD camera, 3)

precision XY stage, 4) rotational device and 5) light source.

The high temperature and pressure chamber is the actual housing where the sample is

placed. Since there is no off-the-shelf equipment available, the high temperature and pressure

chamber has to be designed from scratch and each of its components are machined from the

departmental machine shop with precision.


90

The high-pressurized gas (blowing agent) is injected into the chamber to maintain the

desired pressure level. This requires the use of high pressure syringe pump that could pump the

gas into high pressure levels before delivering into the high temperature and pressure chamber.

The electrical cartridge heaters and temperature controller are chosen for the chamber to elevate

and maintain the desired chamber temperature.

The CCD camera provides high-resolution image capturing capability for taking the

sessile drop of polymer/gas solution pictures at high temperatures and pressures. From those

captured images, we are able to use image-processing tools to track the volume changing

information at the equilibrium state of the polymer/gas solution.

The precision XY stage is to facilitate the move of the camera to take multiple images

for reconstruction. The camera is carefully mounted on the stage and the stage is controlled

through the control software via sending the pulse signal to the step motors to control and

implement the X and Y direction movement. The rotation device gives the system the ability to

accommodate some polymer/gas with an asymmetrical drop shape.

The light source provides the necessary illumination into the chamber cell so that the

camera could capture the image. The light source needs to be uniform and stable to get rid of

any possible noise, therefore, a lighting system with uniformity and stability control is chosen

for the system construction.

4.2.2 High Pressure and Temperature Visualization Chamber

The high pressure and temperature visualisation chamber is the major component of the

PVT apparatus. It is the place where the sessile drop sample is housed. The design of the

chamber system is complex and delicate. Many functions need to be realized in this small

chamber while maintaining a high degree of freedom for easy operation. The detail functional


91

requirement and corresponding design parameters have already been depicted in the previous

section. Firstly, the chamber needs to have a visualization window so that the light can go

through and at the same time the camera can take images. Secondly, the chamber needs to be

designed so that there is gas injection and an exit channel so that the high pressure gas can be

injected to maintain high chamber pressure, as well as for releasing the high pressure gas upon

completion of the experiment.

In addition to the gas injection and release, the thickness of the chamber would also be

considered as a key factor as the safety concerns when handling with high pressure gas, the

safety factor of 6 is chosen to make sure the chamber material itself, plus its fasten bolts, would

be able to bear such high pressures of up to 6000psi pressure load. Thirdly, the chamber should

have a place to insert the cartridge electrical heaters so that the system can be heated up to the

desired temperature. How to have proper seal when dealing with high pressure gas is always the

important issue when designing and making the system. Any leakage (bigger than 0.001ml)

from the system while running experiment is unacceptable. Therefore it is critical to choose the

proper seal for the chamber. The place needs to be sealed are the visualization window opening,

the place where the rotational motor shaft goes through as well as at the junction place where

the chamber meets other parts. Costume made rotary seal is used for the rotational motor shaft.

Moreover, there should be a house for locating the rotational motor right underneath the

chamber so that the motor shaft can be perfectly aligned and coupled with the sessile droplet.

Lastly, there need to have a cooling system using either water or air that can be attached

to the system to cool the sections where the rotary seal and two bearings are located as shown in

Figure 4-3. This is to prevent the build up of the local heat conducted from the chamber body so

that the durability and performance of the seal and bearing can be maintained over a long

duration of time. Figure 4-3 shows the CAD assembled chamber on the left, and on the right is


92

the cross-sectional view of the whole chamber system to give a big picture of where all the

added detail designs are located in the chamber.

Visualization Chamber

Rotary Seal

Shaft Bearing Cushion

Base Support

Rotating Motor

Shaft Bearing House

Visualization Chamber

Rotary Seal

Shaft Bearing Cushion

Base Support

Rotating Motor

Shaft Bearing House

Figure 4-3 CAD Model of Chamber Body with Rotational Device

Figure 4-4 shows more features of the chamber parts with the sapphire window, such as the

window seal groove, cartridge heater holes, cap screw holes, etc. The four evenly distributed

heaters would ensure the thermal uniformity of the chamber body so to keep stable drop profile.

34 5

6

1: Cap screw holes 2: Cartridge heater holes 3: Sapphire window

4: Thread fitting 5: Chamber Space 6: Static Seal

1

2

34 5

6

34 5

6

1: Cap screw holes 2: Cartridge heater holes 3: Sapphire window

4: Thread fitting 5: Chamber Space 6: Static Seal

1

2

Figure 4-4 CAD Model of the High T and P Visualization Chamber


93

4.2.3 Charged Couple Device (CCD) Camera and Optical Lens

The camera is a vital component of the PVT apparatus. There would not be any

information captured for the purpose of image processing if the camera were not available.

Picking a high resolution camera would ensure the good quality of the images captured. The

Pulnix TM-4100CL monochrome progressive scan CCD camera with dual-tap output and a

frame rate of 15 fps at full 2048x2048 resolution is used for the purpose of image acquisition.

The TM-4100CL camera features the latest Kodak KAI-4021 CCD imager for the best image

quality and sensitivity. This camera is used in many different applications, ranging from

machine vision, intelligent transportation systems to high-definition graphics, etc.

The CCD camera provides an analog video signal that is digitized by a frame grabber

installed in a host computer. The digitized image contains the image data in the form of digital

picture elements or pixels. The value of each pixel is called the intensity or grey level (in the

black-and-white case). The most commonly used equipment utilizes 640 x 480 pixels and 256

grey levels to represent an image where 0 and 255 represent black and white, respectively. Thus,

a digitized image is mathematically represented by an array of real numbers from 0 to 255.

In order to receive a magnified image, a set of lenses is chosen to be assembled with the

CCD camera to form the complete imaging acquisition system. This set of macro lens system

from Schneider comprises of an Apo-Componon 8/40 optical lens, a Unifoc 12 helical mount, a

50 mm extension tube and a C-Mount camera adaptor. When the aperture of the lens and the

focal length are adjusted each time, the magnification of the image is changed as well. Having

the biggest magnification is theoretically sound for our purpose, but the image quality is also

very important. Therefore, before the experiment, the aperture of the lens and the light intensity

are adjusted so that the best quality of image and at the same time, the maximum magnification,

can be attained.


94

4.2.4 XY Stage and Precision Control

An XYRB™ crossed roller table from the Danaher Motion Company is used as the

precision XY stage to facilitate the CCD camera movement. The crossed roller stage is a low

profile stage designed for a wide range of applications. Common uses include factory

automation, microelectronics assembly, and laser machining. The stage uses 3mm crossed roller

ways and provides high stiffness. Precision ground preloaded ball screws allow for a higher

speed than a lead-screw driven stage. A specialized monolithic center couples the upper and

lower axes, making them stage stiff and compact. Both motors are mounted at the center section,

so that they move with the lower axis. The two precision step motors provide two axial

freedoms, X and Y, respectively. The total linear travel span of the motor stage in X and Y

direction is 2 inches. The step motor has 400 steps per revolution with 8 micrometer position

accuracy for its encoder. The total load capacity is 75 Kg, which is rigid enough to carry the

weight of the CCD camera without significantly affecting its performance. The stage supports 3

axes controls and high accuracy of positioning control. The controller and amplifier are

provided from the manufacturer. The amplifier provides the power to the motor stage. The

controller could control three axes: X, Y and Z. The first two are used for the XY stage stepper

and the last one, the Z, is used for controlling the step motor coupled with the sessile droplet for

rotational purposes. PID control is utilized to achieve accurate motor positioning control with

minimal amount of error due to repetitive movements.

4.2.5 Rotational Device It has been assumed that the shape for some polymer/gas sessile drops is axisymmetrical or

has axisymmetry, which means the profile of the drop is axisymmetrical in one revolution. In

other words, it is assumed that the PP/gas sessile drop would form a near-perfect parabolic or


95

dorm shape that would give us the best drop profile for performing the volume integration. This

assumption holds since PP has low melt strength and viscosity when it is at a temperature well

above its melting point. In addition, the plasticization effect of dissolved gas at high

temperatures would future reduce its melt viscosity and help the polymer melt to flow easier.

But this assumption does not hold for all polymers. For amorphous polymers like polystyrene

(PS), it has high melt strength and viscosity and it does not have melting point Tm, instead, it has

the so-called glassy transition temperature Tg. The PS polymer matrix behaves like rubber and

the melt strength remains relatively high at higher temperatures. The processing temperature for

PS is usually around 200oC, which posts the problem for PVT experimental studies of the

PS/gas melt at relatively low temperature with its non-symmetric drop shape using the same

technique for PP material. Therefore, the motive for creating the rotational device is to be able

to capture and study the degree of asymmetry and its effect on the PVT measurement with

respect to the temperature and pressure.

The purpose of the rotation device is to capture the asymmetry of the sessile drop shape

for polymers with higher melt strength, such as PS used in this study. In addition, the rotational

device would also help to compensate the effect of asymmetric drop shape on the PVT

measurement via capturing the drop profiles at different angels. The rotational device comprises

of a step motor, an amplifier, a power supply, two ball bearings, and a coupling device. The

coupling device is used to connect the motor shaft with the sessile droplet so that when the

motor rotates, the sessile droplet could also rotate simultaneously at the same pace as the motor.

Therefore, through carefully programmed motor move steps, the precise rotational angle could

be realized on the sample droplet. The ball bearing is housed as shown in figure 4-6 at the two

far ends of the motor shaft to keep it aligned while rotating. The stepper has 500 steps per

revolution, which means that 500 pulses are sent to the motor to trigger the 1 revolution (360o)


96

rotation of the motor shaft. The motor is connected to the XY stage controller as its third axis.

The desired rotational angle is set in the GUI program when the rotation is needed. The rotation

angle is usually as an integer number, such as 5 or 18 degrees.

Another key feature involved with the rotation device is the rotary seal. As seen from the

following figure, the motor shaft is coupled with the sessile droplet inside the chamber and

without a proper seal the gas could leak through the shaft and the pressure would not be

maintained inside the chamber. The solution to this problem is the use of the high temperature

and pressure rotary seal custom designed and produced for this particular application from

Trelleborg Seal Inc. The rotary seal is made out of special Teflon Perfluoro Rubber (FFKM)

material that can handle temperatures of up to 260oC and 5000psi pressure. The rotational

device including the motor and the location of the rotary seal is shown in the section view of the

following Figure 4-5. The sessile droplet where the polymer sample pellet sits on top is also

shown in this figure. This figure also provides a clear picture of where the sessile drop locates in

the chamber during the real experimental environment.

Rotational Device

High T and P Rotary Seal


Rotational Device


Ball Bearings

Coupling Device

Sessile Droplet

Rotational Device



Rotational Device


Ball Bearings

Coupling Device

Sessile Droplet

Figure 4-5 Location of Rotational Device and High T and P Rotary Seal


97

4.2.6 Image Processing Terminal

Computers have become one of the most important parts of people’s daily lives and even

more so for those in academia. Computers are used to serve as image storage and processing

terminals in our PVT system. The images captured from the experimental are stored in the

computer memory. The image processing software, such as the edge detector and image

stitching tool are installed. The developed volume integration algorithm and some other

necessity that was needed in the post image processing are all installed on the computer terminal.

Without this powerful computer device, the time-consuming calculations of dealing with large

amounts of images and pixel points would not even be possible.

4.2.7 Light Source, Noise and Vibration Control

Light source is used to provide illumination to the CCD camera so that the image can be

captured. A white light source composed of a wide range of wavelengths, white light forms

chromatic effects such as rainbow and chromatic aberration that can cause blurring and loss of

clearness of the image of the drop at the edge[268,269]. These chromatic effects can be reduced

using light with a narrow range of wavelengths. Therefore, a band-pass filter (i.e., a filter that

transmits wavelengths between the two cut-off wavelengths of the filter) can be used to pass

only a narrow band of the visible wavelengths. The choice of an optical filter involves a trade

off between the intensity and the bandwidth of light. In other words, the wider the bandwidth,

the higher the light intensity. An appropriate filter is expected to reduce the effects of both

chromatic aberration and the rainbow effect, while maintaining sufficient intensity for the

illumination.

The results from surface tension measurements[180] show that the use of any optical filter

(red, green, or blue) reduces the discrepancy between the surface tension values obtained for


98

large and small drops. This may be explained by the fact that filters reduce the chromatic effects

caused by white light, so they improve the quality of the image. In other words, chromatic

effects make the image of the drop blurry at the interface (i.e. the edge). Thus, the edge of the

image deviates from the real physical edge; hence, the error increases. The results indicate that

the use of the blue filter (with a wavelength range of 400-500 nm) reduces the discrepancy

between the surface tension values obtained for large and small drops more than the other filters

examined. Thus, the blue filter is used in this research. In addition to the blue filter, a light

source from SCHOTT with uniformity and intensity control was purchased to be used as the

system light source to achieve the best image quality. The controller of the light source could

not only vary the intensity of the light but could also to control the uniformity of the light shone

on the sample.

For the purpose of eliminating any possible external mechanical noise, a vibration free

table is used as the base for the whole experimental setup. The table is supported by air at a

constant pressure level. In other words, the table is acting as a damping system so that any

vibration noise caused by external sources reaching the table would be damped out from the

air-floating table. Therefore the effect of the vibration on the PVT system sitting on the table

would be minimized. Another important usage of this vibration free table is to keep the whole

PVT apparatus levelled through its adjustable air cushions. The following Figure 4-6 shows a

real picture of the constructed PVT apparatus based on the proposed design methodology. The

detail components mentioned before such as the CCD camera, the lens, the blue filter, the water

cooling and the vibration free table, etc. are all shown in this figure.


99

8

12

3

4

5

6

7

119

10

Figure 4-6 Actual PVT Apparatus Setup

The high temperature and pressure chamber(4), water cooling system (6) and rotational

device (5) are sat on a millimetre resolution rail at the centre of the vibration free table (3). The

CCD camera (2) and the XY stage (1) are in line with the chamber to the right; The polarizer

diffuser (10), blue filter (9) and light source (8) are in line with the chamber to the left.

4.3 Theory Background and Algorithms Construction

The overall methodology and experimental procedure are elucidated in two different

flow charts. The first flow chart in Figure 4-7 shows the case of dealing with the axisymmetry

drop shape and the second flow chart in Figure 4-12 is used for the asymmetry drop shape case.

The second flow chart is illustrated in a later section. However, for the time being, it is assumed

that the polymer pendent/sessile drop is axisymmetric, which means that the droplet profile is

axisymmetric with respect to its vertical centerline. At the beginning of each experiment, the

camera parameters, such as the working distance relative to the drop sample, the image contrast

1 XY Precision stage 2 CCD camera 3 Vibration free table 4 Visualization chamber 5 Rotational motor 6 Water cooling 7 Syringe pump 8 Pressure transducer 9 Blue filter 10 Polarizer diffuser


100

through optical lens, and so on, are adjusted for maximum zoom while ensuring the best image

quality at the boundary. Calibration is then carried out to determine the pixel size in x and y

orientation, with respect to the XY stage movement. The calibration determines the size of the

pixel in terms of the metric length based on the movement of the stage since the unit step length

of the stage is known. This is a crucial step since it not only helps to correct some possible

optical distortions but also provides the conversion factor between the image pixel size and the

real drop’s metric dimension for volume integration.

Axialsymmetry

Drop shape

Image Recon. &

Edge Detection

Volume Swelling &

PVT Determination

Ultimate Result

Axialsymmetry

Drop shape

Image Recon. &

Edge Detection

Volume Swelling &

PVT Determination

Ultimate Result

Figure 4-7 Methodology Flowchart for Axisymmetry Drop

The number of images to cover the whole pendant/sessile drop is calculated based on the

magnification of the optical system and the resolution of the CCD camera. For a given number,

the step-wise movements of the XY stage in both x and y directions are determined. Those x and

y increments are programmed and stored into the motion controller to guide the step motor to

move the XY stage and camera with precision. The images are captured when the camera pauses

briefly in between each new x and y movement. The series of images are then combined and

reconstructed to form a complete drop image based on the movements using the image


101

processing tool. Figure 4-8 show an illustrative schematic of the image reconstruction using

pendent drop as an example. The algorithm for image reconstruction is shown in the Appendix1.

Figure 4-8 Image Reconstruction Schematic

4.3.1 Edge Detection and Volume Integration for Axisymmetric Drop Shape

After the image reconstruction is completed, the precise edge detection algorithm is used

to detect the drop boundary and generate the edge pixel coordinates. The Canny edge detector is

used to extract the boundary pixel information into coordinates. There are many other edge

detection algorithms available besides the Canny edge detector, such as the Sobel edge detection.

However, Canny’s edge detector is the best-known subpixel edge detector even for poor quality

images. The Canny edge detector defines the detection and localization criteria of a class of

edges based on the first derivative. The detected edge information is in the form of coordinates

based on the sessile drop boundary pixel information. The output of such an operator will have

high magnitudes at the pixels where the gray levels change rapidly, i.e., at the edge of the drop.

After this stage, the edge is a discrete curve in which the minimum distance between the two

adjacent points is equal to the size of a pixel. To achieve higher precision, the edge is

smoothened using a cubic spline technique to provide subpixel resolution. The boundary points

can be represented as (xi,xj) and a simple f(x) notation can be used to represent the drop profile

after spline fit through all the points.

Monitor Virtual


102

21

.

.

.

.

.

.

.

NN-1N-2

Sessile Drop

Boundary

2N-1, 2N . . . . . . . . . . . . . . .0,1,2

xj

xi

)( ixf

21

.

.

.

.

.

.

.

NN-1N-2

Sessile Drop

Boundary

2N-1, 2N . . . . . . . . . . . . . . .0,1,2

xj

xi

)( ixf

Figure 4-9 Detected Sessile Drop Edge and Volume Integration over Vertical Span

Among the numerical integration methods, composite Simpson’s 1/3 method which has

superior and finer segments, is selected[270] to compute the final volume of the pure polymer or

polymer-gas solution drop. Simpson's rule is a Newton-Cotes formula for approximating the

integral of a function using quadratic polynomials (i.e., parabolic arcs instead of the straight line

segments used in the trapezoidal rule). Simpson's rule can be derived by integrating a third-order

Lagrange interpolating polynomial fit to the function at three equally spaced points. Compared

with differential geometry approach dealing with Euclidean space, working with numerical

integration in Cartesian coordinate is more straightforward and conventional in this study.

General Simpson’s 1/3 rule has the following form:

( ) ( ) ( ) ( )∫ ∑∑ ⎥⎦

⎤⎢⎣

⎡+++==

−

=

−

=

b

a n

n

jj

n

ii xfxfxfxfhdxxfI

2

6,4,2

1

5,3,10 )(*4*4**

31

(4-8)

where ( )

nxx

nabh n )( 1−

=−

=

For the volume integration of the axisymmetric drop shape, Simpson’s rule is used to

compute the volume of the drop as the volume summation of every small frustum of a cone at


103

infinitesimal small height along the splined drop profile shown in Figure 4-9. Since the drop

shape is axisymmetric, the radius at any layer ii is equal respect to its centerline in side and top

views as schematically shown in Figure 4-10.

riR −liR −

riR −liR −( )iRfi

riR −liR −

riR −liR −( )iRfi

Figure 4-10 Axisymmetric Drop Side and Top View Schematic

Figure 4-11 schematically illustrate how the volume is computed at any given level i

along the drop profile.

iR

1+iRhΔ

( )iRf i

iR

1+iRhΔ

( )iRf

iR

1+iRhΔ

( )iRf i

Figure 4-11 Volume Determination at ith Level of Axisymmetric Drop Shape

The unit volume shown in the above Figure 4-10 is determined as:

( )211

2 ****31

++ ++Δ= iiiii RRRRhV π (4-9)

where ( ) ( )[ ]ii RfRfh −=Δ +1

With the summation of all the unit volume along the spline drop profile, the total volume

is expressed as:

( ) hRRRRdhhAVlayer

layer

N

iiiiiNtotal Δ++== ∑∫

=++∞→

***31lim)(

1

211

2π (4-10)


104


If the notations from Figure 4-9 are used, an alternative expression based on equation

4-10 can be used as follow for the total drop volume:

( )∑∑∫=

++=

∞→++Δ==

N

jjijijiji

N

iN

total RRRRhdhhAV1

21,1,,

2,

1

***31lim)( π (4-11)

where

( )( )

( ) Nxfxfh

xxR

xxR

N

jijiNji

jijiNji

/)()(

2/

2/

1

1,1,121,

,,12,

−=Δ

−=

−=

++−++

−+

The i= 1,2,……2N and j = 1, 2, ….N. 2N is the total number of the pixel points. The

algorithm is also attached in Appendix 1.

The volume swelling ratio in this study was defined by comparing the final equilibrium volume

with the initial volume as follows:

( )( )

( )PPsample

tPT

tPT

tPTw m

V

V

VS eq

ini

eq

υ*,,

,,

,,== (4-12)

where V(T,P teq) is the measured equilibrium polymer/gas solution volume at temperature T,

pressure P, and equilibrium time teq. V(T,P,tini) is the volume of the neat PP sample at temperature

T, pressure P. PPυ is the specific volume of the pure polymer sample which can be calculated

using Tait equation (5-1), (5-2) and (5-3) in Chapter 5. Unlike the generalized Tait equation

(3-18), those Tait equations are derived specifically for the polymer resins used in this research.

4.3.2 Algorithms for 3D Volumetric Calculation of Asymmetric Drop Shape

In the case of asymmetric sessile drop, the rotational device is utilized to compensate the

effect of the asymmetric drop shape on the final volume integration through rotating the sessile


105

drop at small angle. The degree of asymmetry of each drop is defined and analyzed for each

experiment. The volume integration of the asymmetric drop shape is slightly different than the

aforementioned axisymmetric scenario in how to obtain the cross sectional area of the drop in

radial direction. The detail algorithm development of defining the degree of asymmetry as well

as the volume integration is elucidated in the following sections and the algorithm is included in

Appendix 1.

4.3.3.1 Rotation Device and Degree of Asymmetry

The purpose of having the rotational device as part of the PVT system has been

emphasized previously. It is one of the key design features that allows the PVT system to not

only capture the asymmetry drop shape of polymer/gas with high viscosity and melt strength,

but also to compensate the effect of the asymmetry drop shape in volume determination. The

details of establishing the definition of degree of asymmetry and the development of the

mathematical algorithm are described in the following sections. As mentioned earlier, the

methodology is slightly different in dealing with asymmetric drops. Figure 4-12 shows the

methodology flowchart for an asymmetry drop shape case.


106

Asymmetry

Drop shape

Rotational Device

Determine Degree of Asymmetry

Rotation Done?

Volume Swelling &

PVT Determination

Asymmetry

Drop shape

Rotational Device

Determine Degree of Asymmetry

Rotation Done?

Volume Swelling &

PVT Determination

Figure 4-12 Methodology Flowchart for Asymmetry Drop

4.3.3.2 Modeling of Degree of Asymmetry: Radial Asymmetry

It is, then, very important to be able to define a term that could describe degree of

asymmetry or the skewness for the study. The approach for defining the degree of asymmetry is

radial asymmetry of the sessile drop with respect to a reference radius.

The concept of using the radius to define the degree of asymmetry is to directly

determine how the radius of any asymmetric drop shape differs from the reference radius. The

reference radius is assumed to be axisymmetric or near axisymmetric and it is picked where the

shape of the polymer sessile drop is approximately at near symmetry. The polymer used in

particular is the PS. The experiments to determine the effect of an asymmetry drop shape are

carried out at various temperatures in vacuum conditions. The vacuum conditions would allow

for the study of the effect of asymmetry of the drop shape on the pure polymer PVT

measurement, but would also allow for the study of the temperature effect on the asymmetry.


107

During the actual experiment, the radius of the PS sessile drop at the 250oC, the 200oC

and the 150oC during the temperature cooling round are chosen to be the reference points. It is

believed that at those temperatures during the cooling the PS drop has better shape because as

the temperature reached 250oC, the PS polymer chain should have already been relaxed and the

shape should have become much more axisymmetric compared with the same temperatures

during the initial heating process. Therefore, the temperatures in the cooling process are the

temperatures that will maintain better axisymmetric shape and will be used as the reference

temperatures. This is the direct method that characterizes the differences in radius in order to

quantitatively define the degree of asymmetry. The detailed mathematical modeling is described

in the following sections.

4.3.3.3 Determination of the Reference Radius

The experimental temperatures are carefully planned and set to help conduct the

experiment. For instance, 150oC, 200oC and 250oC are the three temperatures that are needed to

be used in the experiment. At the start of the experiment, the temperature is raised (temperature

increasing ramp) from 150oC to 200oC and finally to 250oC. About 15 to 20 minutes is allocated

at each temperature to ensure equilibrium state is reached. The temperature is then decreased

(temperature decreasing ramp) in a reverse fashion from 250oC to 200oC and 150oC at the end.

The time maintained at each temperature during the cooling round is the same as the

temperature rising. Since high temperature helps get more axisymmetric drop shape, it is

believed that the drop shape at the temperature during the cooling round is much closer to the

ideal symmetry. Therefore the radii of the drop along the drop profile at the temperatures in the

increasing ramp are used as the reference radius. TGA test (Appendix 7) is carried out to make

sure the PS sample is not degraded or decomposed at 250oC


108

Since the drop is also rotated at different angles, the average radius value obtained from

the radii at those different angles at the same level on the drop profile is defined and used as the

overall reference radius of the drop for that temperature.

Figure 4-12 shows a schematic of how the reference radius is picked at each angle for

asymmetry drops. The left is the side view of a drop and the right is the top view of a drop at

angle 0θ . At each rotation angle, the profile of the sessile drop is determined through a subpixel

edge detector and the edge information is stored in pixel coordinates. Along the drop profile at

angle 0θ , the radius is defined as

( ) 2/000 ,,1, θθθ jjNj xxr −= +− (4-13)

The same mathematical is repeated for the drop profile at other rotated angles, such as 1θ . Then

those radii at each of the same levels but at different angles, are summed and averaged as the

average reference radius for that drop profile at that particular level.

0,1 θ+− jNx0,θjx 0,θjr

1θ

0θ

1−Mθ0

0,1 θ+− jNx0,θjx 0,θjr

1θ

0θ

1−Mθ0

Figure 4-13 Schematic of Reference Radius Definition As shown in the above Figure 4-13, the reference average radius is determined through the

following equation:


109

( ) 2/*

1,,1

11, ijijN

rotationM

i

N

jrotationij xx

MNr θθθ −= +−

==∑∑ (4-14)

where θπΔ

=rotationM

where j is integer from 1, 2, 3. ……2N, where 2N is the number of points and N is the

number of levels from the drop profile. Angle Δθ is the specified amount of rotating angle and it

is a user-defined value. Mrotation is the number of rotating of the axisymmetric drop shape based

on the value of Δθ which is always chosen based on the motor steps per revolution, It is

preferred to have an integer number for Δθ so that it can be evenly divided by 2π. In the

experiments carried out in this study, the step motor has 500 steps for each revolution. In order

to have integer number of turn at each rotation, when the rotation angle is 18oC, the steps the

motor turns is 50steps/rotation which is a reasonable setting. Therefore, the angle is chosen at

18o apart every rotation starting from 0o. With 11 rotations, the whole drop profile would be

covered completely.

4.3.3.4 Definition of Degree of Asymmetry (DOA)

Then the degree of asymmetry is calculated for all the layers by comparing the radius of

from asymmetric profile with the reference radius at corresponding layers. Figure 4-14 shows

schematically how to obtain the radius from asymmetric drop profile at ith level.

Then this is repeated for all the profiles at different rotating angles. Finally, the total

error, denoted as E, from all the layers and rotating angles are summed and averaged. This

average values is then normalized with respect to the surface area ( )2* tippedestalrπ of the stainless

steel pedestal since the pedestal surface radius tippedestalr is a constant value. Then this


110

normalized Easymmetry is just a number without unit and it is then defined as the degree of

asymmetry for the sample at that particular temperature.

( )2

1,,

12 **1 ∑∑

=

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

∗=

N

jijij

M

irotationtippedestalasymmetry rR

MNrE

rotation

θθπ (4-15)

where M rotation is the number of rotations. This newly defined error term asymmetryE could be

used to as an index or definition of the Degree of Asymmetry (DOA) of the PS sessile drop.

From the magnitude of the DOA, it could then serves as an indicator to how the shape of the PS

sessile drop is close to the ideal symmetry case.

4.3.3.5 Volume Integration of Asymmetry Drop Shape

When the sessile drop is non-symmetric, the integration used in the axisymmetry case is

not applicable. But the edge detection and spline technique are still applicable in finding out the

boundary information. As described in the asymmetry definition, since the shape is not

axisymmetric, at angle 1θ , the radius at each level along the profile is not uniform. Figure 4-14

shows the radius at ith level of a asymmetric drop profile..

1θ

i1,θriR −1,θliR −

1,θriR −

1,θliR −

( )iRf

1θ

i1,θriR −1,θliR −

1,θriR −

1,θliR −

( )iRf

Figure 4-14 Radius Asymmetry at ith Level of Asymmetric Drop Shape

After number of rotations, the radii at each angle from ith level form an asymmetry

area. Figure 4-15 shows schematically the formation of this asymmetric circular area.

θN


111

1θ

i

1θ

1,θriR −1,θliR − 1,θriR −

1,θliR −

2θ3θ4θ

Nθ

2,θriR −

2,θliR −

3,θliR − 4,θliR −

NliR θ,−NriR θ,−

( )iRf

( )jR θ

1θ

i

1θ

1,θriR −1,θliR − 1,θriR −

1,θliR −

2θ3θ4θ

Nθ

2,θriR −

2,θliR −

3,θliR − 4,θliR −

NliR θ,−NriR θ,−

( )iRf

( )jR θ

Figure 4-15 Asymmetric Circular Area after Nθ Rotations at ith Level

Spline technique is then applied to the circumference of this asymmetric area and

Simpson integration is used to accurately obtain the areas of each of the small circular section

bounded within each rotation angle θ. The detail area integration mathematically of the area is

illustrated graphically in Figure 4-16 as follow:

1θ

1,θriR −

NliR θ,−Nθ

1,θriR −

NliR θ,−

iRφ

1+iRφ

φΔ

1θ

1,θriR −

NliR θ,−Nθ

1,θriR −

NliR θ,−

iRφ

1+iRφ

φΔ

Figure 4-16 Area Determination of the Circular Section within Angle θ From the spline, each of the small θ i circular section is divided into number of even smaller

sections with angle Δφ. The area within each Δφ is determined as:

21

2360 ⎟⎟⎠

⎞⎜⎜⎝

⎛ +Δ= +ii

i

RRA φφφ

φπ (4-16)


112

Therefore the area of θ i section is the summation of all those Aφ i and it expressed as:

( )2

1

121 2360

lim360 ∑∫

=

+

∞→ ⎟⎟⎠

⎞⎜⎜⎝

⎛ +Δ==

φφφ

φθ

φπφφπN

i

iiN

iiRR

dRA (4-17)

After obtain the areas within each θ i angle, the total area of this ith level asymmetric circular

section is the summation of all the Aθ i areas as:

( )∑

=−−−−−− ++++++=

θ

θθθθθθ

N

ilirilrlri AAAAAAA

12211 L

(4-18)

As shown in the following Figure 4-17, the area at ith level is then integrated along the

splined profile to calculate the volume of hΔ at ith level.

iA

1+iA( )iRfhΔi

iA

1+iA( )iRfhΔi

iA

1+iA( )iRfhΔi

Figure 4-17 Volume Determination at ith Level of Asymmetric Drop Shape This unit volume is computed as:

( ) hAAAAV iiiii Δ+∗+= ++ *31

11 (4-19)


Similarly, the volume of the asymmetric sessile drop is the volume summation of those

unit volume with infinitesimal small height along the splined drop profile. The total volume is

expressed as:

( )∑∫=

++∞→

++∗Δ∗==layerN

iiiii

layerNtotal AAAAhdhhAV

111 *

3lim)( π

(4-20)


113

Where ( ) ( )[ ]ii RfRfh −=Δ +1 One more note here is that there may be different approaches for volume estimation for

the asymmetric drop and the algorithm used in this research takes into the consideration of the

asymmetric drop shape and rigorously analyzes the asymmetric cross sections at different levels

for area and volume integration. It yields fairly accurate results based on the pure PS density

measurement.

4.4 Integration of Hardware and Software GUI Interface

4.4.1 XY Stage Control Software The XY stage is controlled through software from Galil Motion Co. The software

provides a graphic user interface (GUI) shown in the following Figure 4-18 for basic control of

the motion system.

Figure 4-18 XY Stage Software Control GUI

The commands are listed in a manual and the control of the system can be achieved

either through entering the command into the interface or via a script written with all the basic


114

commands. A simple script is written to initialize the XY stage at the beginning of each

experiment. For instance, the speed of the motor shaft rotation that determines the axis moving

speed when driven by the motor, and the maximum moving range of each axis, etc. would be

included in this script. When the script file is downloaded into the controller, the motor would

follow the setting when it drives the XY stage. During the calibration process, the interface is

used and those basic commands are entered manually to complete the calibration process.

4.4.2 GUI Construction for Image Capture and Rotation

In order to improve the efficiency and ability for easy operation during the

experimentation, the process of setting up the camera movement steps, rotation angles and the

image storage are automated through the development of the graphic user interface using Visual

Basics language. Figure 4-19 shows a screen shot of the GUI interface. The detailed

development algorithm is attached in Appendix 2 for reference.

12

3

4

5

6

Figure 4-19 GUI Interface of the Image Capture and Rotation

1 Monitor window 2 x & y movement input 3 Rotation input

4 Time duration input 5 Data storage directory input 6 Action buttons


115

The GUI allows users to enter the number of images needed to be captured by providing

the step movement for the XY stage so that the camera will move by following the exact

command. It also allows the user to enter the rotational angle of the sample droplet when

dealing with asymmetry measurement. Also, the time between each capture can also be

specified via the interface. Once all of the inputs are correctly entered, pressing the start button

would then start the experimental process and the images would be automatically saved in the

computer terminal.

4.5 Summary

The first half of the chapter described the design process using Axiomatic design

technique in detail. Each of the apparatus components was also thoroughly described. Although

the design process utilized the Axiomatic design tool, the actual implementation of the

components was an iterative process. When problems were anticipated during the apparatus

constructing, the designs were then re-evaluated for modifications until the problems or issues

were resolved.

There are some criterions for designing or choosing specific components. The selection

criteria are mostly dependent on system improvement, trial and error, etc. For instance, the

original CCD camera has low resolution about 640x480. When the image reconstruction proved

to be pros to error reduction, then a higher resolution CCD camera (2048 x 2048) is acquired to

achieve better image resolution even for a single image. The light source with variable intensity

is picked so that we could have various intensities under different conditions. The optimal light

intensity is selected after trial and errors when the best image quality is achieved.


116

The second half of chapter was devoted to describe the algorithm development for

volume integration of both axisymmetric and asymmetric sessile drop cases. In the case of

asymmetric sessile drop, rotational device was used to facilitate the algorithm development. The

definition of degree of asymmetry was also defined and detail mathematics to calculate the

degree of asymmetry was developed and presented.


117

Chapter 5. Validation of the Proposed Design

5.1 Introduction

There are several ways to verify the validity of our proposed design as well as the accuracy

and repeatability of our constructed PVT measurement apparatus based on our design

methodology. These methods are the precision stainless sphere ball volume measurements to

verify the significance of error reduction in using image reconstructions and empirical approach

of using the PVT system to measure the densities of pure polymers and compare them with Tait

calculated densities to verify the system accuracy and repeatability in dealing with both

axisymmetric and asymmetric sessile drops.

5.2 Empirical Verification of Error Reduction from Image Reconstruction

In order to demonstrate and verify the improvement in accuracy using the concept of

multiple image reconstruction, an absolute volume determination experiment is carried out using

a known volume precision sphere. The precision sphere was placed on the tip of a custom made

needle rod through a magnetic force inside the visualization chamber with maximum camera

magnification and optimal image quality. The temperature of the chamber was at room

temperature and no gas was injected to pressurize the chamber. Three separate experimental

case studies were carried out for the reconstruction. The first case captured one image, the

second case captured 6 images, and the third case took 9 images. The mass and density of the

reference stainless spheres were measured using a density measurement device (ASTM


118

D792-00) and the actual volume of the sphere was determined by using the water displacement

method (sphere

spheresphere

mV

ρ= ) as the “absolute” volume for error analysis.

The error analysis is carried out by comparing the measured volume using the proposed

approach with the “absolute” volume. It is clearly shown from Figure 5-1 the accuracy is

dramatically improved as the number of images used for reconstruction increases. For a constant

camera resolution and field of view (FOV), the image will have a smaller pixel size if the

camera’s magnification is increased. The accuracy will be improved as the pixel size decreases

continuously and the number of images increases during edge detection and image

reconstruction. The error was about 2.8% in the single sphere image experiment. However, the

error decreased dramatically to 0.776% for 6 images and to 0.361% for 9 images. This

demonstrates that our proposed methodology gives clearer edge information in terms of the

pixel size. It is believed that the accuracy will be further improved with accurate calibration,

higher resolution camera.

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

2.5

3.0

Erro

r (%

)

Number of Image

Single Image 6 Image Reconstruction 9 Image Reconstruction

Figure 5-1 Accuracy Improvement from Sphere Image Reconstruction


119

But for dynamic measurement in observing the volume change overtime, a single image

may have to be used because the volume of polymer specimen is changing continuously while

the camera moves around and captures the images. The average time needed for taking 6 images

and 9 images were roughly 20 seconds and 31 seconds, respectively.

5.3 Empirical Verification Using Tait Equation for Pure Polymer PVT

Several models or equations of state (EOS) have been proposed to describe the PVT

behavior of polymeric liquids based on thermodynamic law. Those equations of state for

polymer are either empirical or theoretical. The most widely used empirical equation is the Tait

equation in describing the pure polymer properties. The Tait equation is an isothermal

compressibility fitting model rather than a true equation of state. It reliably calculates the

specific volume of pure polymer at different temperatures and pressures using PVT data

measured from a bellow type dilatometer to obtain required fitting parameters. The detailed Tait

equation derivation has been described in Chapter 2; the different forms of Tait equation were

used for different polymer materials to calculate the density of the pure polymers at different

temperatures. At the same time, the pure polymer densities were also determined through the

PVT system. The density results from Tait and from the experiment measurement will be

compared.

5.3.1 Experimental Procedure and Materials

Pure polymer materials are used to serve the purpose of verifying the accuracy and

reliability of the PVT system. The pure polymer materials are linear and branched PP, PS 685D.

For PP and PS the procedures of conducting the pure PVT experiment are almost the same. A

simplified procedure is as follows:


120

1) Select polymer pellet and measure the sample weight using precision balance

2) Prepare the sessile drop sample using vacuum oven

3) Conduct the PVT experiment at specific temperature at vacuum condition

4) Calculate the density of pure polymer at each temperature using the PVT measured

volume and mass information

5) Determine the pure polymer density using Tait equation at each set temperature and

vacuum

6) Compare the densities values obtained from experiment in step 4 with the ones

calculated from Tait in step 5.

5.3.2 Density Measurement for Pure Axisymmetric Linear and Branched PP

Following the simplified procedure, the pellet is picked from the virgin resin and the

weight of the pellet is picked such that it is between 8 to 12 mg due to the diameter limitation of

the sessile pedestal. The weight of the pellet is measured using the microbalance from MSB

system. The single pellet is then carefully placed in the middle of the heated pedestal with

tweezers. The pedestal with sample pellet is then placed in the vacuum oven at the temperature

near melting point for semi-crystalline polymer or the glass transition temperature for

amorphous polymer for about an hour to get rid of possible air pocket or moisture inside the

sample upon melting. Then the sessile sample is naturally cooled in the vacuum condition to

room temperature afterwards. The sessile drop with the best shape is picked for the experiment.

In addition, the size of the sessile drop depends on the size of the sessile pedestal. The size of

the pedestal can be changed if a bigger size is necessary.

Specific care was taken to make sure to get the best axisymmetric drop profile and at the

same time to make sure the sample did not stay in the oven for too long to prevent degradation.


121

Again, the TGA tests (Appendix 7) show that the neither the temperature nor the duration of the

experiment time would cause degradation of the sample.

Density measurements for neat PP were conducted in vacuum at five different

temperatures of 180°C, 190°C, 200°C, 210°C, and 220°C, which are all above their melting

points. The vacuum pump helped to achieve the vacuum reservoir inside the visualization

chamber. After approximately half an hour in the vacuum at each of these temperatures, the

images were taken. The density experiments were carried out three times at each temperature to

ensure repeatability, and the average values were reported as the final values.

On the other hand, the Tait equations for branched PP and linear PP resins used in this

research have the following forms, respectively[124]

PP

PPbranched+

++

=8

2

9

6

10*146.1

T10*1.221

10*773.5

10*485.6υ (5-1)

PP

PPlinear+

++

=7

2

9

6

10*86.9

T10*1.06

10*45.6

10*46.7υ (5-2)

The specific volume υ has units of m3/kg, pressure P is in Pascal and temperature T is in

degrees Celsius.

Figure 5-2 shows the comparison between the measured PP densities and the known data

derived from Tait equation for linear PP at those five different temperatures.


122

180 190 200 210 2200.710

0.715

0.720

0.725

0.730

0.735

0.740

0.745

0.750

Dens

ity (g

/cm

3 )

Temperature (oC)

This work Tait Equation from Gnomix Measurement

Figure 5-2 Pure Linear PP Measured and Tait Calculated Densities

The results show an excellent agreement between the Tait calculated data and the

experimentally obtained values. Figure 5-3 shows the same comparison between the Tait

calculated density and the measured density of pure branched PP at those four temperatures.

Since repeated experiments were carried out, the error bars throughout this thesis show the

standard errors from those experiments. Again, the obtained results illustrate a very good

agreement between experimental results and theoretical calculations. Providing the Tait

equation for pure polymer PVT calculations has been proven to be very accurate and reliable,

these pure polymer density experiments validate our method of approach as well as the

experimental set-up for measuring the volume of a polymer /CO2 solution.


123

190 200 210 2200.70

0.71

0.72

0.73

0.74

0.75

0.76

0.77

0.78

0.79

0.80

0.81

0.82

Dens

ity (g

/cm

3 )

Temperature (oC)

Experiment Measured Density Tait Calculated Density

Figure 5-3 Pure Branched PP Measured and Tait Calculated Densities 5.3.3 Density Measurement for Pure PS685D with Asymmetry

In addition to the pure PP density measurement, in the study of the shape parameter,

particularly the asymmetry shape effect of the drop on the PVT measurement, the pure PS

density is also measured at various temperatures. The main purpose of the asymmetry study is

to demonstrate that from the use of rotational device, the effect of asymmetry on the density

calculation could be compensated so that accurate volume can still be obtained despite the

asymmetric drop shape.

The PS resins are PS 685D supplied from Dow Chemical and its Tait equation has the

following form:

( ) ( )[ ] ( )( ) ⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−∗

∗+−−+=

37.376exp*1ln0894.0137.3769724.0,

TDBCPTATPv (5-3)

where A, B, C, D are constant coefficients and their values are A= 5.996*10-4, C= 0.0689476


124

, B=1.79784*103, and D=-4.761*10-3, respectively. Temperature T is in Kelvin, and pressure is

in psi. The PS Tait equation 5-3 has more resemblance with the general Tait equation described

in Chapter 2.

The Figure 5-4 shows the asymmetry drop profile from an actual experiment sessile

drop.

0 200 400 600 800 1000 1200 1400 1600 1800 2000-2200

-2000

-1800

-1600

-1400

-1200

-1000

-800

-600

Level 1Level 2Level 3

0 200 400 600 800 1000 1200 1400 1600 1800 2000-2200

-2000

-1800

-1600

-1400

-1200

-1000

-800

-600

Level 1Level 2Level 3

Figure 5-4 Actual Sessile Drop Profile and Level Selection at 150oC

In order to show how the drop shape looks like from top view, the radii at randomly

chose three levels from the drop profile at each of the experimental temperature at all the

rotational angles are plotted from Figure 5-5 to 5-9.

‐400

‐300

‐200

‐100

0

100

200

300

400

‐400 ‐300 ‐200 ‐100 0 100 200 300 400

Top View at 150oC

Level 1

Level 2

Level 3

Figure 5-5 Profile Top View at 150oC


125

‐400

‐300

‐200

‐100

0

100

200

300

400

‐400 ‐200 0 200 400

Top View at 200oC

Level 1

Level 2

Level 3


‐400

‐300

‐200

‐100

0

100

200

300

400

‐400 ‐200 0 200 400

Top View at 250oC

Level 1

Level 2

Level 3


Figure 5-5 to 5-7 showed the drop radius profile at those three layers at 150oC, 200oC

and 250oC respectively during the temperature rising round. Figure 5-8 and 5-9 showed the

radius profile from 200oC and 150oC during the temperature cooling round. The figures clearly

showed the actual circumferential profile of the asymmetry drop at three different levels of the

drop. Figure 5-4 showed the worst asymmetry shape at the temperature 150oC and Figure 5-6

showed more circular shape at 250oC. At temperatures during the cooling process, Figure 5-7


126

and 5-8 showed that the drop shape still remained fairly good circular shape when temperature

dropped.

‐400

‐300

‐200

‐100

0

100

200

300

400

‐400 ‐200 0 200 400

Top View at 200oC

Level 1

Level 2

Level 3

Figure 5-8 Profile Top View at 200oC from Cooling

‐400

‐300

‐200

‐100

0

100

200

300

400

‐400 ‐200 0 200 400

Top View at 150oC

Level 1

Level 2

Level 3

Figure 5-9 Profile Top View at 150oC from Cooling

The volumes of the PS sessile drop were determined and the densities were calculated at

from all the rotated angles during the temperature rising round. Figure 5-10 showed the density

results comparison with Tait equation. The results indicated that the measured densities from

asymmetry sessile drop agreed very well with Tait equation values at all three temperatures.

Therefore the conclusion drawn here is that PS sessile drop is not axisymmetric in nature at low


127

temperature, but with the help of rotational device, the effect of asymmetry drop shape is

compensated and the volume calculation is still reliable and accurate.

125 150 175 200 225 250 2750.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15 Measured Density Tait Density

Den

sity

(g/c

m3 )

Temperature (0C)

Figure 5-10 Density Measurement from Asymmetry Drop Shape and Tait Equation

In other words, with the ability to rotate the sample droplet, accurate density information

can still be obtained for polymer melts or polymer/gas solutions with asymmetric drop shape

regardless the temperature effect on the asymmetry nature of the drop profile. The effect of

temperature on asymmetry is shown in Chapter 6.


128

5.4 Summary

The emphasis of this chapter is on rigorous verifications of the validity of the proposed

image reconstruction method in error reduction as well as the accuracy and repeatability of

using this newly constructed system to measure the PVT information of pure polymer and

polymer/gas solutions at high temperatures and pressures.

A precision stainless sphere ball was used to demonstrate the error reduction from image

reconstruction. Three sets of images were taken for reconstruction purposes demonstrated that

the accuracy improved significantly when more images were taken for reconstruction. The

second part of the chapter focused on the pure polymer density measurement to demonstrate the

accuracy and repeatability of the system in measuring volume. Three pure polymer materials,

namely, linear PP, branched PP and PS 685D were used in the experiments at different

temperatures and in vacuum conditions. The measured densities were compared with densities

calculated from the Tait equation. It was found that the empirically measured density matched

very well with the theoretically calculated value; therefore, the accuracy, the reliability of using

this newly developed system to conduct experiments has been confirmed.

During the experiment with pressurized supercritical blowing agent (CO2), it is not

feasible to measure the initial volume at each pressure level using inert gas, such as Helium for

the same sample pellet unless different samples are used to carry out the experiment separately

at each pressure level at one temperature. This would prolong the experiment time and

complicate the experimental procedure. In addition, the inert gas molecular may still dissolve

into the polymer sample which would affect the measurement outcome. Since the verification

proves the system accuracy, using Tait equation to compute the initial sample volume would

reduce the experiment complexity but also provide a precise initial sample volume. Since the

pure PP PVT measurement at high temperature and vacuum conditions already verified the


129

system reliability, it is not that significantly important to measure the specific gravity of PP at

room temperature.


130

Chapter 6. Measurement of the PVT Data for Polymer/Gas Solutions

6.1 Introduction

In a polymer/gas solution systems, when gas dissolves into a molten polymer the

polymer swells (or dilates) due to gas sorption. The amount of polymer swelling or dilation is

characterized by its PVT properties, which can be obtained by measuring the equilibrium state

volume of a polymer/gas solution at any specific temperature and pressure. The ultimate goal of

this research is to develop a new method for direct PVT property measurements of polymer/gas

solutions. The method utilizes the pendent/sessile droplet technique and combines it with a high

temperature and pressure visualization chamber, where the polymer swelling caused by high

pressure gas, can be monitored. The detailed development of this PVT apparatus has been

discussed in the previous chapter. The system enables the assessment not only of the dynamic

change of the polymer/gas solution volume with respect to time, but also, more importantly, the

final volume at equilibrium state. The overall methodology is illustrated in a flow chart seen in

Figure 4-7.

6.2 PVT Measurement with Axisymmetry for Linear/Branched PP/CO2

Solutions

6.2.1 Experimental Procedure The overall methodology and experimental procedure are elucidated in a flow chart as

shown in Figure 4-7. It is assumed that the polymer pendent/sessile drop is axisymmetric to


131

begin with, which means the droplet profile is axisymmetric with respect to its vertical

centerline.

At the beginning of each experiment, the camera parameters, such as the working

distance relative to the drop sample, the image contrast through optical lens, and so on, are

adjusted for maximum zoom while ensuring the best image quality at the boundary.

The calibration is then carried out to determine the pixel size in the x and y orientation,

with respect to the XY stage movement. The calibration determines the size of the pixel in terms

of the metric length, which is based on the movement of the stage, since the unit step length of

the stage is known. This is a crucial step since it not only provides the conversion factor

between the image pixel size and the real drop’s metric dimension at the image reconstruction,

but it also helps to correct some possible optical distortions. The detail calibration technique is

summarized in Appendix 3.

The number of images needed to cover the whole pendant/sessile drop is calculated

based on the magnification of the optical system and the resolution of the CCD camera. For a

given number, the step-wise movements of the XY stage in both x and y directions are

determined. Those x and y increments are programmed and stored into the motion controller to

guide the step motor to move the XY stage and camera with precision. The images are captured

when the camera pauses briefly in between each of the new x and y movements. The series of

images is then combined and reconstructed to form a complete drop image based on the

movements using an image processing tool.

After the image reconstruction is completed, the precise Canny edge detection algorithm

is used to detect the drop boundary and generate the edge pixel coordinates. The Canny edge,

which is based on the first derivative and localization criteria of a class of edges, is the most

successful subpixel edge detector. The boundary profile coordinates are identified. To improve


132

the accuracy of the drop profile coordinates, repeated points, i.e., pixels with the same

grey-levels at one location, are averaged into distinctive data points (xi, xj) using the smoothing

spline technique.

Finally the Simpson’s 1/3 rule is applied for integration to compute the final volume of

the polymer-gas solution drop. Among the numerical integration methods, composite Simpson’s

1/3 method has superior and finer segments over the Trapezodial rule. A detailed mathematical

formulation is described in Chapter 4.

6.2.2 Experimental Setup

In Chapter 4, the system components are introduced one by one in detail. The PVT

apparatus is the assembly of those components together. In brief, the experimental apparatus

consists of the following components: a high-pressure chamber with a sapphire visualization

window; a 2024 x 2024 resolution JAI Pulnix TM4100 CL camera with a control software (Easy

Grab); Schneider 4/80 lens and extension tubes; a temperature controller (Omega CN132) with

thermocouple (Omega RTD); four cartridge heaters; an automatic high-precision XY stage with

Galil motion controller and control board; a manual 1” XYZ stage to adjust the position of the

light source; a syringe pump connected to the gas tank; and a backlight source with a light

equalizer/diffuser.

6.2.3 Experimental Materials

Linear polypropylene (Borealis MD 55) and Branched PP (Borealis HMS WB130) with

Tm = 162oC ~ 165°C, CO2 (99.99% purity, BOC Coleman grade) were used

6.2.4 Volume Swelling of Linear/Branched PP/CO2

First of all, the absorption test is carried out using MSB system to see how long it take

the pellet sample to be saturated and reach equilibrium at all pressure levels at the lowest


133

temperature 180oC for both linear and branched PP. It is found out that approximately 15 to 20

minutes would be the time for a resin pellet to reach equilibrium under pressurized supercritical

CO2 reservoir. The results are included in the thesis in Appendix 8.

The density measurement for PP/CO2 solutions were also conducted at 180°C, 200°C,

and 220°C under high pressure CO2. At each temperature, the pressure started from 1000 psi

and went up to 4500 psi, at 500 psi increments. At each pressure level, the PP/CO2 solution was

maintained for 35 to 45 minutes to ensure the equilibrium of the system was reached.

Equilibrium was considered to have been achieved when the total volume of the polymer/gas

solution no longer changed.

When a polymer melt is exposed to a high pressure gas, two competing mechanisms will

affect the specific volume. On the one hand, the hydraulic pressure will decrease the specific

volume of compressible polymer/gas solutions. On the other hand, the dissolved gas under high

pressure would cause the polymer to swell and increase the specific volume. The latter is

typically higher than the hydraulic pressure effect. This means that the presence of gas evidently

enhances the overall activity of the polymer/gas system[271] and thus, creates more free volume

for the CO2 molecule to penetrate into after the gas fills the existing free volume. The increased

specific volume (i.e., increased free volume) causes an increase in the solubility and diffusivity

[124]. The dissolved CO2 causes a plasticization effect to reduce the viscosity of the polymer/gas

mixtures and to increase the chain mobility [132,272]. Also this increased specific volume

decreases the surface tension of polymer[273]. In summary, despite the hydraulic compression

effect, the high-pressure gas increases the specific volume and thereby affects the solubility,

diffusivity, viscosity and surface tension. All of these fundamental parameters are critical in

determining the foaming behaviours, and it should be emphasized that the swelling caused by

gas dissolution governs all of these parameters.


134

180 200 2200.981.001.021.041.061.081.101.121.141.161.181.201.221.24

Swel

ling

Rat

io

Temperature (oC)

1000psi 1500psi 2000psi 2500psi 3000psi 3500psi 4000psi 4500psi

a) at Temperatures

500 1000 1500 2000 2500 3000 3500 4000 4500

1.00

1.04

1.08

1.12

1.16

1.20

1.24

Swel

ling

Rat

io

Pressure (psi)

1800C Linear PP/CO2

2000C Linear PP/CO2

2200C Linear PP/CO2

b) Pressures

Figure 6-1 Linear PP/CO2 Swelling vs. a) Temperatures and b) Pressures


135

180 200 2200.981.001.021.041.061.081.101.121.141.161.181.201.221.24

Volu

me

Swel

ling

Rat

io

Temperature (oC)

1000psi 1500psi 2000psi 2500psi 3000psi 3500psi 4000psi 4500psi

a) Temperatures

500 1000 1500 2000 2500 3000 3500 4000 4500

1.00

1.04

1.08

1.12

1.16

1.20

1.24

Swel

ling

Rat

io

Pressure (psi)

1800C Branched PP/CO2



b) Pressures

Figure 6-2 Branched PP/CO2 Swelling vs. a) Temperatures and b) Pressures

As depicted from equation (4-12), the swelling ratio is the quotient of the total PP/CO2

mixture volume at an equilibrium state over the initial pure polymer volume. The final volume

is measured experimentally and the initial pure polymer volume is determined as the product of


136

the pre-weighted mass of the pure polymer sample and its density is calculated using the Tait

equation. Figures 6-1 and 6-2 show the volume swelling of linear PP/CO2 and branched PP/CO2

solutions at all the temperatures and pressures, respectively.

6.2.5 Effect of Temperature and Pressure on PVT of PP/CO2 Solutions

6.2.5.1 Effect of Temperature on Volume Swelling

When the pressure is constant, higher temperatures result in less volume swelling in the

PP/CO2 solution. This is true for both linear PP/CO2 and branched PP/CO2 solutions. It is clearly

seen from the above Figure 6-1 a) and 6-2 a) that the volume swelling of both PP/CO2 solutions

decreases when the temperature increases. As the temperature increases, the polymer chains

become softer and more relaxed. As a result, the free volume increases and the specific volume

increases. On the other hand, the solubility of CO2 in PP is known to decrease as the

temperature increases. This means that the diffusion of CO2 out of the polymer matrix increases

more at a higher temperature compared to the increased free volume of polymer. So, despite the

increased free volume in the PP matrix, the PP/CO2 solution will lose CO2 at higher

temperatures. As the pressure climbed higher, the volume swelling was governed by the gas

content, and the sensitivity of the volume swelling with respect to the temperature became

smaller.

6.2.5.2 Effect of Pressure on Volume Swelling

As discussed, the increasing solubility of CO2 at a higher pressure caused more CO2 to

dissolve into the PP matrix, inducing more volume swelling. It is seen from the results that when

pressure increases at each individual temperature, the volume swelling ratio increases for both

linear and branched PP/CO2 mixtures. Both Figures 6-1 b) and 6-2 b) show that the measured

volume swelling ratio of the linear and branched PP/CO2 solution increased as the pressure


137

increased at 180°C, 200°C, and 220°C. At a fixed temperature, when the pressure increased, the

volume of the polymer/gas mixture also increased, so did the volume swelling ratio. Since the

solubility of CO2 increases with an increase in the pressure as mentioned above [124], more CO2

dissolves into the PP matrix, which would induce more volume swelling.

However, as the pressure went higher, it was observed that the equilibrium volume and

volume swelling ratio will eventually approach a plateau region. In other words, the rate of

increase in volume swelling becomes smaller and the swelling graph starts to level off as the

pressure keeps increasing. This implies that the volume will not expand indefinitely as the

pressure goes up at a constant temperature. Wissinger et al. [75]also reported similar behaviours

of volume swelling and sorption that began to level off and reached limiting values for PC/CO2

and PMMA/CO2 systems at low temperatures of about 35oC and 32.7oC, respectively, when

pressure was in the range of 70 atm to 100 atm.

On the one hand, after passing some pressure levels, such as after 3000psi, the hydraulic

effect from the higher pressure may start to exhibit a stronger volume compression effect than it

does at relatively low pressure levels. If this is the case, then it will counteract or prevent more

volume swelling due to gas dissolution at very high pressures. On other hand, the chain

structures of both linear and branched PP would also be part of the reason in restricting itself

from expanding indefinitely through absorbing more gas at high pressures. The long chain

structure effect and the possible implications of chain entanglement on volume swelling will be

analyzed in later sections.

6.2.6 Effect of Branch Structure on PVT of PP/CO2 Solutions

As the effects of pressure and temperature on the volume swelling have been explained,

it is now appropriate to analyze the polymer itself to study how its long chain structure,


138

especially the branched long chain structure in branched PP, would affect the volume swelling.

This section is going to focus more on the branching effect on the PVT rather than other aspects

of structure effect, such as chain entanglement and chain entanglement density. The latter will

be dealt with in a separate section in detail. The swelling between linear and branched PP at

each temperature were shown separately. Figure 6-3 shows the comparisons of the volume

swelling between linear and branched PP/CO2 at 180oC.

500 1000 1500 2000 2500 3000 3500 4000 45001.00

1.04

1.08

1.12

1.16

1.20

1.24

Swel

ling

Rat

io

Pressure (psi)

1800C Linear PP/CO2


Figure 6-3 Linear and Branched PP/CO2 Swelling at 180oC

Figure 6-4 shows the comparisons of the volume swelling between linear and branched

PP/CO2 at 200oC and, lastly, Figure 6-5 shows the comparisons of the volume swelling between

linear and branched PP/CO2 at 220oC. The volume swelling data was tabulated with error in

Appendix 6.


139

500 1000 1500 2000 2500 3000 3500 4000 4500

1.00

1.04

1.08

1.12

1.16

1.20

Volu

me

Swel

ling

Pressure (psi)

2000C Linear PP/CO2



500 1000 1500 2000 2500 3000 3500 4000 4500

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

1.18

Volu

me

Swel

ling

Pressure (psi)

2200C Linear PP/CO2



Of all the results, one immediate conclusion could be quickly reached is that: in general,

the linear PP/CO2 mixture has a higher volume swelling over the branched PP/CO2 mixture due

to the branched chain structure and possible high polymer chain entanglement density of


140

branched PP. When long chained structures of linear PP are being branched, the movement of

the branched chain structure or chain mobility is greatly restricted. The melt viscosity, as well as

the melt strength of the branched PP is significantly enhanced over the linear PP. The

improvement of the melt strength is good in preventing the cell collapse in the foaming process

because cell walls have higher strength and rigidity and could withstand the force exerted from

cell growth. This will in turn help to maintain the cell structure and eventually the foam

morphology. Thereby, the mechanical properties of branched PP foam products are proven to be

better than those foamed from linear PP[9].

As shown in Figure 6-3, at 180˚C branched and linear PP/CO2 have similar volume

swelling at pressure levels of up to 2500psi. But when the pressure goes higher than 2500 psi,

branched PP starts to have less swelling. In other temperatures, linear PP/CO2 always has higher

swelling over branched PP/CO2 at all pressures. This implies that the branching structure would

be effective in preventing volume dilation at relatively high temperatures and pressures.

When the pressure kept increasing, the branched PP/CO2 had a much smaller volume

swelling compared to linear PP/CO2. Since the long-chain branched PP from Borealis is

produced by the Daploy process, which is characterized by an after-treatment of the native

polypropylene granules out of the polymer synthesis at temperatures well below the melt

temperature of polypropylene[274,275], mircogels were formed and found in very small amounts

of the final product (usually below 1wt% concentration). The effect of the branch would involve

the promotion of strain hardening that increases the melt strength, which could then result in

zero-shear viscosity increase and reduction in fluidity.

At higher temperatures of 200oC and 220oC, the resistances, such as increased melt

strength and strain hardening generated from branched structures, were counter-balanced to


141

some extent by the increase of temperature and pressure, the linear PP still has significant higher

volume swelling than branched PP at all pressures.

6.3 Measurement of PVT Data for PS and PS/Gas Solutions

6.3.1 Measurement of PVT Data with Asymmetry for PS and PS/CO2 6.3.1.1 Introduction

Polymer has been categorized into two different types according to their matrix structure,

namely, amorphous and semi-crystalline. When ordinary small molecules pack themselves in

regular three-dimensional arrays, they can then be considered as crystalline. When polymer is

amorphous, the molecules lack position order on the molecules. Semicrystalline polymer

contains both crystalline and amorphous regions in their polymer matrix as seen in the following

two dimensional schematic.

Crystalline region

Amorphous region

Figure 6-6 Schematic of Semicrystalline Polymer Structure* *source: http://chem.chem.rochester.edu/

Semicrystalline polymers have true melting temperatures (Tm) at which the ordered

regions break up and become disordered. In contrast, the amorphous regions soften over a

relatively wide temperature range (always lower than Tm) known as the glass transition (Tg).

Fully amorphous polymers do not exhibit Tm and when polymers are all above Tg they become


142

liquids. Typical semicrystalline polymers are like polyethylene (PE) and polypropylene (PP) and

conventional amorphous polymers are polystyrene (PS) and poly(methyl methacrylate)

(PMMA).

PS is used in this study to emphasize the effect of the high melt strength on the

formation of the sessile drop shape, and thereby the effect on the final results of PVT

measurement. In order to easily capture the effect of asymmetry, pure PS density experiments

are carried out. The experiments were carried out. The experiment was carefully constructed so

that the rotational device was utilized to rotate the droplet at very small angle for one revolution

to capture the drop shape. The results from those each individual angle will be compared and

analyzed.

6.3.1.2 Experiment Procedure

At each of the temperatures, enough time was allocated for the sample to reach

equilibrium and the chamber was vacuumed. The sessile droplet was rotated at every 18oC. The

reason for picking this angle was to make the number of the rotation and the motor rotation step

integer numbers with 500 steps per revolution from the step motor. A single image of the drop

was captured at every rotated angle. In the cooling round, the temperatures were cooled from

250oC to 200oC and finally to 150oC.

At each of the temperatures, there were 11 pictures taken since the 11 rotations were at

18o for each rotation. In this temperature setting, the asymmetry shape effect would be

dramatically differentiated from the two 150oC and 200oC temperatures since the they retained a

better axisymmetric shape from high temperature of 250oC during the cooling and since

presumably the shape would be closer to symmetry at high temperatures, which was shown to


143

be true in the experimental results. Three sets of experiments were carried out to repeat the

measurement three times and the results from every experiment were analyzed and compared.

After conducting the pure PS density experiment to investigate how the asymmetry

could effect the experimental results, another three sets of sample were prepared at 250oC to

conduct the PS/ CO2 volume swelling experiment with CO2 as blowing agent. For the PVT

experiment, the same three temperatures, 150oC, 200oC to 250oC, were used and the pressures

started from 1000 psi to 4000 psi at 1000 psi increments.

6.3.1.3 Experimental Material

Polystyrene (PS685D) supplied from Dow Chemcial Company and CO2 (99.99% purity,

BOC Coleman grade) were used in this study.

6.3.2 The Temperature Effect on the Degree of Asymmetry

As has been defined and modeled in Chapter 3, there are two approaches to define the

degree of asymmetry: radial asymmetry and volumetric asymmetry. Figure 6-7 shows the plot of

the degree of asymmetry vs. temperatures. This degree of asymmetry is based on the radius

symmetry defined in Chapter 4. See equation 4-17. It is very obvious to observe that when the

temperature increases from 150oC to 250oC, the degree of asymmetry decreases dramatically

from the first experiment and the third experiment.

It is also observed that the degree of asymmetry is reduced in the first experiment during

the temperature rising process. For example, in the second experiment, the Easymmetry value

dropped from 1 at 150oC to 0.5 at 200oC and eventually reached almost 0 at 250oC. From the

definition of Easymmetry the bigger values implies that the radius of the drop at that temperature

deviates more from the reference radius, which is more axisymmetric In other words, the bigger

the Easymmetry value, the more the drop shape will have asymmetry. For instance, the Easymmetry


144

value for the first experiment is around 2 at 150oC and 1.7 at 200oC. This means the shape of the

sessile drop from the first experiment is more skewed to start with than those in the second and

third experiment. On the contrary, since the third experiment started with a better drop shape, its

Easymmetry value is much smaller even at the low temperatures. Those results indicate that the

increase in temperature helps the high viscous PS polymer chain structure to relax more and

form more symmetry shapes, which is desired in accurate volume calculation.

On the other hand, during the temperature cooling process, the degree of asymmetry

almost remains unchanged by going from 250oC to 150oC. From Figure 6-7, the Easymmetry values

are pretty much the same (close to 0) for all three temperatures. The results illustrate that the

higher the temperature, the better the sessile drop shape is to close near the axisymmetric dorm

shape and the better the drop shape forms at high temperatures, which could be retained even as

the temperature dropped, which means the polymer chain contraction due to the temperature

drop would not significantly affect the overall shape.

140 160 180 200 220 240 260-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Nor

mal

ized

Deg

ree

of A

sym

met

ry E

asym

met

ry

Temperature (oC)

Experiment 1 Experiment 2 Experiment 3

Figure 6-7 Degree of Asymmetry of Pure PS at Temperature Rising and Cooling Process


145

6.3.3 Temperature and Pressure Effect of PVT of PS/CO2

Three PS sessile drop samples were prepared at 250oC to help form a more axisymmetric

drop shape. In order to prevent sample degradation due to the long exposure at high temperature,

the PS sessile droplet stayed at 250oC for only 15 minutes. The following Figure 6-8 shows the

swelling ratio of PS/CO2 solutions at all three temperatures and four pressures. The results are

very similar to the case of PP/CO2 swelling at high temperatures and pressures. When the

pressure is constant, higher temperatures also result in less volume swelling in the PS/CO2

solution. It is clearly seen from the above Figure 6-8 a), that the volume swelling decreases

when the temperature increases at all pressure levels. As the temperature increases, the polymer

chains become softer and more relaxed. As a result, the free volume increases and the specific

volume increase. On the other hand, the solubility of CO2 in PS decreases as the temperature

increases. The volume swelling significantly drops at 250oC compared to the drop from 150oC

to 200oC. This means that the diffusion of CO2 out of the polymer matrix increases more at a

higher temperature compared to the increased free volume of polymer. So the PS/CO2 solution

loses more CO2 at higher temperatures.

150 200 2501.00

1.02

1.04

1.06

1.08

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1.14

1.16

1.18

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1000psi 2000psi 3000psi 4000psi

a) Temperatures


146

1000 2000 3000 40001.00

1.05

1.10

1.15

1.20

1.25

Swel

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Rat

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Pressure (psi)

150oC PS/CO2

200oC PS/CO2

250oC PS/CO2

b) Pressures

Figure 6-8 PS/CO2 Volume Swelling vs. a) Temperatures and b) Pressures

The increasing solubility of CO2 at a higher pressure caused more CO2 to dissolve into

the PS matrix, inducing more volume swelling. It is seen from Figure 6-8 b) that when pressure

increases at each individual temperature, the volume swelling ratio increases. But as the

pressure goes higher, we observe that the equilibrium volume and the rate of volume swelling

will eventually slow down which is also observed in the PP/CO2 case. This implies that the

volume will not expand indefinitely in PS/CO2 case as the pressure goes up at a constant

temperature. This trend is obvious at 250oC temperature.

6.3.4 Measurement of PVT for PS/HFC-152a Solution 6.3.4.1 Introduction

A variety of physical blowing agents (PBAs), such as chlorofluorocarbon (CFCs),

Hydrochlorofluorocarbons (HCFCs), Hydrofluorocarbons (HFC), butane, or pentane are

currently used in the plastic foam industry for low-density foam processing because of their low

diffusivity and high solubility [123]. The drawback for CFCs is their environmental hazards. The


147

depletion of the ozone layer associated with their usage led to the Montreal Protocol in

1987[276,277]. Under this Protocol, the production of CFCs has been banned in Europe and United

States since 1995. The most commonly used HCFCs for foam blowing agents (i.e.,HCFC-141b,

HCFC-22 and HCFC-142b) will be phased out beginning January 1st 2010. In addition, there

is a significant challenge for the plastic foam manufactures those countries who have ratified the

Kyoto Protocol [276,277] and have committed to reduce their emissions of carbon dioxide and five

other greenhouse gases, i.e., water vapor, carbon dioxide, methane, nitrous oxide, and ozone..

Therefore, the issue to find the next generation of environmental substances which can

be used as a replacement for CFCs and HCFCs in foam manufacturing is becoming imminent.

Currently, extensive research has been devoted to develop new blowing agents for foam

production [132,271,278]. The potential blowing agent replacement candidates will be HFCs (134a,

152a, or experimental HFCs), hydrocarbons (HCs) and inert gases such as carbon dioxide (CO2)

and nitrogen (N2). Among these conceivable surrogated gases, HFCs, i.e., HFC 134a and HFC

152a, offer superior thermal insulation capabilities and are desirable candidates for the

replacement of CFCs and HCFCs[279-281]. Nevertheless, the low solubility and low diffusivity

associated with HFC 134a [280,282] has made the foaming processes challenging despite its

reasonably good R-value. In order to obtain low-density foams, it is necessary to employ a high

system pressure to increase the HFC 134a content that could be dissolved. Previous research has

demonstrated that the use of high HFC 134a content would lead to foams exhibiting a rather

poor morphology. Gendron et al.[283] indicated that foaming polystyrene (PS) with HFC-134a

content above 7.5 wt.% would result in large voids, which were in the order of a few millimeters,

due to the inhomogeneous dissolution of the HFC-134a. On the other hand, it is known that

HFC-152a has a higher solubility and diffusivity. However, there are serious concerns regarding

the storage and long term insulation performance of the end-product due to its flammability and


148

fast diffusion coefficient at room temperature. Moreover, it is believed that the rapid diffusion of

HFC-152a will significantly decrease the amount of gas that will remain in the foam over time

and deteriorate the thermal insulation efficiency of the product over its lifetime.

6.3.4.2 Experimental Procedure and Material The volume swelling of PS685D was carried out at two separate termperatures. They

were 150oC and 190oC, respectively. The blowing agent used in this study is HFC-152a

(Formacel®

Z-2) provided by Dupont. The vacuum pump helps to achieve the vacuum reservoir

inside the chamber at the very beginning of each experiment. The initial temperature is set at

150oC. Once the temperature goes beyond the glassy transition point, the solidified sessile

drop sample starts re-melting back to the sessile drop shape and the HFC-152a gas is then

injected into the chamber to reach the first pressure level which is 200 psi. The pressure goes up

at 200 psi increaments each time until it reaches 800 psi for 150oC cases. For 190oC cases, the

same pressure levels, namely, 200 psi, 400 psi and 800 psi were studied. The camera started

taking the images. The whole camera motion repeated every 10 minutes up to 60 minutes to

ensure that the polymer melt reaches equilibrium state at the 150oC. After finished at 150oC,

the pressure was increased to 400 psi and the previous steps were repeated until the experiment

finished at 800psi. Then a new set of experiments were conducted at 1900C with similar

procedures being used at 1500C. The whole experimentation was completed when all of the

temperature levels were conducted. Image edge detection and volume integration techniques are

utilized to generate the volume information of the polymer sample at these temperatures.

The volume swelling ratios at those pressures were plotted. Also the SS and SL

predicted volume swelling for the same materials at each corresponding pressure were also

plotted on the same graph for comparison purposes. Figures 6-10 and 6-11 show that the total


149

volume of the PS685D/HFC-152a solution expands as the pressure increases each time. The

density of the HFC-152a increases in the confined chamber space as the pressure increases,

hence, more HFC-152a molecular will penetrate into the PS685D polymer matrix causing more

dilation until it reaches the saturation point. In this study, instead of using EOS to predict the

volume swelling, we used our PVT apparatus to measure the volume swelling experimentally.

6.3.4.3 Temperature and Pressure Effect of PVT for PS/HFC-152a The volume of the PS/ HFC-152a increases as the pressure increases. For the fixed

temperature, as the pressure increases, the volume of the polymer/gas mixture increases, as does

the volume swelling ratio. Since the solubility of HFC-152a increases[284] as the pressure goes

up, there is more HFC-152a dissolving into the PS matrix to induce more volume swelling. It is

seen from Figure 6-9 that as the pressure goes up, the volume of the polymer/gas mixture

increases significantly. From the experiments, it was observed that the time for the mixture to

reach a stabilized state is less when the pressure is higher because of the P increased plasticizing

effect. In order not to have the hydraulic effect, the pressure would have to be the same

throughout the volume swelling calculation according to our earlier definition at each pressure

level. As previously mentioned, the Tait equation is ultilized here to compute the initial volume

of the polymer/gas mixture as the base for computing volume swelling ratio.


150

0 100 200 300 400 500 600 700 8000.94

0.96

0.98

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

Volu

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Swel

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Pressure

PS/HFC-152a at 150oC PS/HFC-152a at 190oC

Figure 6-9 PS/HFC-152a Volume Swelling at 150oC and 190oC

For the same pressure level, PS/ HFC-152a volume swelling decreases as the temperature

increases. This could be seen clearly from those above figures. For example, at the 800 psi

pressure level, 1500C PS/ HFC-152a has the higher volume swelling. At higher temperatures,

although the liquid-polymer is softer and more relaxed, the solubility of HFC-152a decreases as

the temperature increases[284] and the overall polymer/gas volume decreases, thereby, the

volume swelling drops.

6.4 Validity of Equation of States Using Experimental Results

6.4.1 Volume Swelling From EOS Predictions

As mentioned earlier, the volume swellings have been often predicted using EOS based

on thermodynamic principles. The materials used in the experiment were the same and the

temperature and pressure parameter settings were similar to the ones used in the EOS for the

predictions. The following figures show all of the comparisons between experimentally

measured swelling ratios with SL and SS EOS predictions at different temperatures for both

linear PP/CO2 and branched PP/CO2 solutions.


151

Figure 6-10, Figure 6-11 and Figure 6-12 show the comparisons for linear PP/CO2 with

SL and SS EOS predictions at 180oC, 200oC and 220oC, respectively.

500 1000 1500 2000 2500 3000 3500 4000 4500 50001.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

1.60

Swel

ling

Rat

io

Pressure (psi)

180oC SL Swelling 180oC SS Swelling 180oC PVT Exp. Swelling

Figure 6-10 Linear PP/CO2 and EOS Swelling at 180oC

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

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1.40

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200oC SL Swelling 200oC SL Swelling 200oC PVT Exp. Swelling



152

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.95

1.00

1.05

1.10

1.15

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1.25

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1.40

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1.50

1.55

Volu

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Pressure (Psi)



The following Figure 6-13, Figure 6-14 and Figure 6-15 show the comparisons for

branched PP/CO2 with SL and SS EOS predictions at 180oC, 200oC and 220oC, respectively.

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

Volu

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Swel

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Pressure (Psi)


Figure 6-13 Branched PP/CO2 and EOS Swelling at 180oC


153

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

1.00

1.05

1.10

1.15

1.20

1.25

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Pressure (psi)



500 1000 1500 2000 2500 3000 3500 4000 4500 50000.95

1.00

1.05

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1.25

1.30

1.35

1.40

1.45

Swel

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Pressure (psi)



As shown in all of the results, at all three temperatures, EOS predictions show increasing

volume swelling as the pressure increases. The SL EOS show a concave upward increasing

trend and the SS EOS show a more linear increase. The experimental volume swelling results

also show the increase of volume swelling as pressure increases. However, the rate of increase,

i.e., the slope of the volume swelling curve, decreases as the pressure increases and starts to


154

level off as the pressure goes up to a certain high value at all three temperatures. Disregard the

discrepancies between the SL and SS EOS predictions, for our results show that SL and SS EOS

overestimate the swelling ratio of polymer/gas solutions. The SS EOS predictions are somewhat

close to the experimental measured results. In other words, the SS EOS predictions are better

than SL EOS predictions.

Although those EOS do provide a means to help obtain the volume swelling when a

direct experiment measurement is not available. Due to the limitation of using purely

thermodynamic laws to model and predict the real time molecular movement and interactions,

the discrepancies can not be avoided when compared with the accurately measured values.

Since these EOS have been mainly developed for neat polymers and blends, some modifications

would be required for accommodating the polymer/gas mixture behaviours. It would be to

improve their fitting parameters based on the experimental results so that they could make better

predictions. Further experiments and theoretical work need to be conducted to make these

results fit well with EOS.

6.4.2 Polymer Chain Entanglement Hypothesis

6.4.2.1 Introduction It seems that there is an upper limit for polymer/gas volume swelling, based on the

obtained volume swelling results in the previous figures. It is then hypothesized that due to the

polymer chain entanglement, the volume swelling of polymer/gas (linear PP/CO2 and branched

PP/CO2), solutions would eventually reach a plateau at some high pressure ranges even greater

than 4500psi, which is the highest pressure tested in the experiment. The chain entanglement

would have similar effects on the volume swelling regardless of the polymer chain structure.


155

By comparing the experimental volume swelling to the EOS predicted values, it is seen

that the experimentally measured volume swelling would not increase indefinitely with the

increase of the pressure. The experimental volume swelling results also show the increase of

volume swelling as the pressure increases, but the rate of increase, i.e., the slope of the volume

swelling curve, decreases as the pressure increases and starts to level off as the pressure goes up

to a certain high value at all three temperatures.

The same phenomena were also observed from the branched PP/CO2 volume swelling. It

is, then, hypothesized that the polymer chain entanglement would eventually be the determining

factor as postulated previously in the study of volume swelling of linear PP/CO2 mixture.

Wissinger et al. reported similar behaviors of volume swelling and sorption that began to level

off and reached limiting values for PC/CO2 and PMMA/CO2 systems at low temperatures of

about 35oC and 32.7oC, respectively, when the pressure was in the range of 70 atm to 100 atm.

Besides the possible increased hydraulic effect at higher pressures, it is contemplated

that due to the polymer chain entanglement, the polymer matrix would not be stretched

indefinitely unless the internal force is strong enough to break the chain entanglement. In other

words, there is an upper limit for the volume of polymer to expand because of the constraint

posed by the polymer chain entanglement. An entanglement is visualized as the looping of a

portion of one polymer chain backbone about another polymer chain and the entanglement

density is the number of entanglement junctions per unit volume and is calculated through

Equation (6-9). The entanglement junctions confine the motion of chain molecules under

applied stress as transient ‘cross-link’ point. Chain entanglement is an important feature of

polymers, which is one of the factors controlling rheological, viscoelastic, solid mechanical and

adhesive properties of polymers.


156

The effects of the entanglement on viscosity and elasticity[285], intrinsic brittleness and

toughness[286], and deformation mechanism[287,288] have been reported. It will be important to

investigate how the entanglement limits the swelling of polymer melts. The simple approach is

to obtain and compare the chain entanglement densities of linear and branched PP at

temperatures above their melting point. The polymer chain entanglement density could be

measured at zero-shear viscosity of the polymer melt using the dynamic rheometer analysis.

In our PVT experiment, since the gas dissolution process is fairly steady

macroscopically, there is no significant shear applied onto the polymer matrix. Therefore, it is

feasible to use a rotational rheometer to conduct the zero-shear viscosity measurement. At very

low frequencies, such as 10-4 Hz, it is assumed that there is almost no shear applied to the

polymer melt using the rotational rheometer and that the viscosity measured at this point would

be considered as the zero-shear viscosity. As the frequency increases, the polymer chain

undergoes stress release and the polymer long chain molecules then expand and relax over time

and the dense polymer chain entanglement will then be greatly released. Thus, the viscosity, and

the resistance to flow the polymer melt, would be dramatically reduced. This is the so-called

shear thinning effect.

6.4.2.2 Small Amplitude Oscillatory Shear Method

The small amplitude oscillatory shear (SAOS) is the method generally used to determine

the LVE properties of molten polymers. This type of test can be done in either strain- or

stress-controlled mode. The features of this SAOS experiment is shown in the following Figure

6-16[289].


157

Figure 6-16 Schematic of SAOS Experiment Wave

For SAOS in strain-controlled mode, a sinusoidal strain is imposed on the sample:

( ) ( )tt ωββ sin0= (6-1)

where 0β is the strain amplitude, and ω is the frequency. If 0β is sufficiently small, the

resulting stress is also sinusoidal with a phase angle ψ :

( ) ( )ψωσσ += tt sin0 (6-2) whereσ is the stress amplitude. The amplitude ratio is defined as

00 / βσ≡dG (6-3)

The storage modulus 'G and loss modulus ''G are:

( )ψcos'dGG = (6-4)

( )ψsin''dGG = (6-5)

( ) ωψωη /cos/''dGG == (6-6)

( ) ωψωη /sin/''''dGG == (6-7)

The absolute value of the complex viscosity is:

( ) ( )[ ] ( )[ ]2''2' ωηωηωη +=∗ (6-8)

6.4.3 Experimental Procedure and Materials

Rotational rheometers are widely used to measure the rheological properties due to their

simplicity and easy operation. The Advanced Rheometric Expansion System (ARES) rotational


158

rheometer is utilized to measure the complex viscosity (Eta’), the storage shear modulus (G’)

and the loss modulus (G”). There are two tests conducted in order to obtain that data. The first

test is the dynamic stain sweep to determine the linear regime since the maximum strain for

linear viscoelastic behaviours varies with frequency, temperature and polymer. The second test

is the dynamic frequency sweep test where the selected frequency range is applied to the sample

above the melting point. Figure 6-17 shows a picture of the rheometer used in this study.

Figure 6-17 ARES Rheometer

The material used is PS 685D from Dow Chemical. The key procedures of using the

rhometer are described in the following and the step procedure is attached in Appendix 4.

6.4.3.1 Sample Preparation

Since the parallel disk is used in the rheometer as the sample platform, the disk-shape

sample is therefore needed to be prepared prior the measurement. The most convenient method

of sample-disk preparation is compression molding. The compression molding method

minimizes the strain and thermal histories of the polymer sample under the high temperatures

above the melting point. The big round disk sample is prepared and then cut into small pieces to


159

fit between those parallel disks of the rheometer. The linear and branched PP samples were

molded at 165oC using a stainless steel mold plate with a diameter of 100 mm and a 2.0 mm

thickness. The thickness of the mold is important because the sample must be slightly thicker

than the rheometer gap, but thin enough to avoid a large normal force during the sample loading.

6.4.3.2 Gap Zeroing

Variations in temperature and removal of the fixture can change the zero position of the

upper plate. Therefore, the gap must be corrected for each experiment and at each temperature.

Both the fixtures and the instrument frame will expand until they reach thermal equilibrium, and

this affects the gap. To avoid problems arising from non-equilibrium thermal behaviours, the

chamber is also preheated to the experimental temperature for several hours prior to zero the gap.

The ARES require gas flow to heat up the system, and air is used instead of nitrogen to reduce

operating costs.

Initial zeroing is carried out by setting the gap to zero after lowering the upper fixture up

until it touches the bottom plate. The upper fixture plate is first lowered manually to a position

very close to the bottom plate. Then the “zero” button in the GUI interface is pressed. The

fixture automatically moves to touch the bottom plate. Touching is confirmed by monitoring the

normal force, as well as the gap value displayed on the GUI interface. The software from ARES

will confirm when the gap between the two plates becomes zero from its online monitor

software. A sample must be loaded carefully to avoid air bubbles between the sample and the

lower plate. After placing the sample on the plate, it is pushed down in the center and a spatula

is used to sweep the bubbles from the center to the rim. Before setting the desired gap, the edge

of the sample is trimmed with the gap set slightly larger than its final value. The step-by-step

procedure for sample loading and gap zeroing is as follows:


160

There were a total of 6 experiments that were carried out to do the measurements since we had

to run each material with three different temperatures, namely, 180oC, 200oC and 220oC. Those

temperature settings followed exactly with our PVT experiment so that we could really correlate

the polymer chain entanglement density with our postulated hypothesis.

6.4.4 Chain Entanglement of Linear and Branched PP 6.4.4.1 Strain Sweep and Frequency Test

There were two tests carried out to obtain the data. The first test was the dynamic stain

sweep to determine the linear regime since the maximum strain for linear viscoelastic

behaviours varies with frequency, temperature and polymer. Figures 6-18 and 6-19 show the

strain sweep test results for both linear and branched PP polymers. An important piece of

information extracted from those results was the maximum strain of those materials. The

technique for checking the values is to look for the time where the G’ and G’’ start to change. It

is easy to figure out from the graphs that at the beginning, the values of G’ and G’’ were steady

while the strain value kept increasing. At the time when the strain value passed 10, big changes

started to show for both G’ and G’’. Therefore, the value 10 was the maximum strain picked for

both of the materials as being the strain input for the frequency sweep test.


161

10-2 10-1 100 101 102104

105

100

101

Strain [%]

G' (

)

[Pa]

G" (

)

[Pa]

tan_delta ()

[ ]

LPP Strainsweep

Figure 6-18 Linear PP Strain Sweep Test

10-1 100 101 102104

105

10-1

100

101

Strain [%]

G' (

)

[Pa]

G" (

)

[Pa]

tan_delta ()

[ ]

BPP Strainsweep

Figure 6-19 Branched PP Strain Sweep Test


162

The second test was the dynamic frequency sweep test, where the selected frequency

range was applied to the sample above melting point. The frequency test result plots are shown

from Figure 6-20 to Figure 6-25. The first three figures are the results for linear PP.

10-4 10-3 10-2 10-1 100 101 102100

101

102

103

104

105

106

102

103

104

105

Freq [Hz]

G' (

)

[Pa]

G" (

)

[Pa] Eta* (

) [Pa-s]

LPP_180oC_frequencysweep

Figure 6-20 Linear PP 180oC Frequency Sweep Test


163

10-3 10-2 10-1 100 101 102100

101

102

103

104

105

106

102

103

104

105

Freq [Hz]

G' (

)

[Pa]

G" (

)

[Pa] Eta* (

) [Pa-s]

LPP_200oC_frequencysweep


10-4 10-3 10-2 10-1 100 101 102

100

101

102

103

104

105

106

102

103

104

105

Freq [Hz]

G' (

)

[Pa]

G" (

)

[Pa] Eta* (

) [Pa-s]

LPP_220oC_requencysweep


The following three are the frequency sweep test for branched PP at all three

temperatures.


164

10-4 10-3 10-2 10-1 100 101 102101

102

103

104

105

102

103

104

105

Freq [Hz]

G' (

)

[Pa]

G" (

)

[Pa] Eta* (

) [Pa-s]

BPP_180oC_frequencysweep

Figure 6-23 Branched 180oC Frequency Sweep Test

As expected, the results from those plots showed that the complex viscosity of both the

linear and branched PP had decreased as the frequency increased. Duo to the limitation of the

frequency range of the rheometer and the time required to run extremely low frequencies, the

frequencies were chosen to start from 5x10-3 instead of from 10-4. From the graphs it is easy to

see that the complex viscosity tended to have plateaus at really low frequencies. This was

especially shown in the linear PP frequency test. At those plateaus, we could extract the value

for the corresponding G’ values from the graph.


165

10-4 10-3 10-2 10-1 100 101 102101

102

103

104

105

102

103

104

105

Freq [Hz]

G' (

)

[Pa]

G" (

)

[Pa] Eta* (

) [Pa-s]


Figure 6-24 Branched PP 200oC Frequency Sweep Test

10-4 10-3 10-2 10-1 100 101 102

101

102

103

104

105

102

103

104

105

Freq [Hz]

G' (

)

[Pa]

G" (

)

[Pa] Eta* (

) [Pa-s]


Figure 6-25 Branched PP 220oC Frequency Sweep Test


166

6.4.4.2 Chain Entanglement of Linear and Branched PP

Liu et al. [288] has studied the deformation mechanism of polyphenylquinoxaline (PPQ-E)

film using theoretical prediction and microscopic observation. The entanglement density,

estimated from the plateau modulus on the dynamic mechanical property curves, was used to

predict the deformation mechanism of PPQ-E. One of the commonly used equations to calculate

the chain entanglement density is as following[290,291]:

RT

GNV

nA

e0= (6-9)

Where NA is Avogadro constant, R is the gas constant and nG0 is the plateau modulus of the

melt at temperature T. From the measurement results, the nG0 is the G’ modulus from the

rheometer results at corresponding plateau of the complex viscosity. The polymer chain

entanglement densities of both linear and branched pp are plotted in Figure 6-26.

170 180 190 200 210 220 2301E20

1E21

1E22

1E23 Linear PP Branched PP

Cha

in E

ntan

glem

ent D

ensi

ty (m

-3)

Temperature (oC)

Figure 6-26 Linear and Branched PP Chain Entanglement Density at T above Tm


167

Figure 6-26 showed that both linear PP and branched PP has high chain entanglement

density although the entanglement density of linear PP is one magnitude smaller than that of

branched PP. When temperature increases, the chain entanglement density decreases for both

linear and branched PP. At each fixed temperature, since the chain entanglement density of

linear is smaller than branched PP, the volume swelling of linear PP should be higher than

branched PP which is indeed the results obtained. Due to the high chain entanglement density of

both linear and branched PP, there may be an upper limit for the ultimate expandability of the

polymer chains. In another word, the existence of such a highly entangled polymer chains in

both linear and branched PP may prevent the polymer chains being stretched or deformed

indefinitely under high shear. During the PVT experiment, the only possible shear can be

generated is from the volume swelling due to gas dissolution which is so small and can not

overcome the highly entangled chains. Therefore it is hypothesis at some high pressure ranges,

with the effect from high chain entanglement the volume swelling is going to reach plateau. It is

also predicted the swelling of branched PP will reach the plateau first compared with linear PP

because of higher chain entanglement density.

6.5 Determination of Accurate Solubility Based on the PVT Data

6.5.1 Corrected PP/CO2 Solubility from PVT Measurement

How the magnetic suspension balance (MSB) is used to measure the solubility of gas

inside the polymer and how the EOS has been used to help the measurement in providing

predicted volume swelling, have been introduced in Chapter 3. In this section, using the

experimentally measured volume swelling to help obtain more accurate solubility data is

elaborated. The corrected solubility is also going to be compared with those corrected from the


168

EOS. To begin, a brief derivation is needed on how to derive the solubility correction from the

MSB apparatus.

The read-out from the balance at the vacuum condition (P=0) and this specific

temperature (T) was denoted as ( )TW ,0 . When the equilibrium state is reached inside the

sorption chamber with the presence of pressurized blowing agents, the sample mass will not

change unless a high pressure is introduced by having more compress gas go into the sorption

chamber. At that particular pressure (P) and temperature (T), the weight read-out of the balance

is noted as ( )TPW , . Therefore, the amount of gas that is dissolved into the polymer at each

pressure and temperature condition is denoted as gW and is calculated through the following

equation:

( ) ( ) ( )sPBgasg VVVTWTPWW +++−= ρ,0, (6-10)

Where gasρ is the density of the gas and can be measured in situ by the function of MSB; BV

is the volume of the sample holder, including the sample container and the measuring load

coupling devices shown in the Figure 1-5. This volume is determined using the buoyancy

method with the high-pressure gas; and PV is the volume of the pure polymer sample at

pressure P and temperature T. This volume can be determined from the PVT equation of pure

polymer plus the weight of the polymer sample. The sV is the swollen volume of the

polymer/gas mixture due to the gas dissolution. If this swollen volume term becomes ignored,

then the reading of the balance will be the so-called “apparent” solubility of the polymer/gas

solution. Next, the measured weight gain is defined as the apparent weight gain through the

following equation:

( ) ( ) ( )PBgasapparentg VVTWTPWW ++−= ρ,0,, (6-11)


169

The apparent solubility, apparentX can be determined from the above equation as:

masssampleapparentgapparent mWX /,= (6-12)

In order to get an accurate solubility data, volume swelling has to be taken into account. After

obtaining more accurate and true volume swelling from the direct measurement using our PVT

apparatus, the apparent solubility can be corrected as true or accurate corrected solubility as:

masssample

sgasapparentcorrected m

VXX

∗+=

ρ (6-13)

From the definition of the volume swelling in equation (4-11), the swollen volume sV which

occurs due to gas dissolution, can be interpreted as the initial sample volume multiple the

swelling ratio:

( )1* −= wvolumesampleinitals SVV (6-14)

Combine the above equation 6-13 and 6-14, and the corrected solubility could be simplified into

the following equation:

( ) purepwgasapparentcorrected SXX ,1 υρ ∗−∗+= (6-15)

where purep,υ is the specific volume of the pure polymer sample and can be calculated

through the Tait equation. With all of the mathematics in hand, the corrected solubility is

determined from the measured volume swelling, as well as from the apparent solubility obtained

from the MSB experiment. Since the temperature and pressure settings are almost the same for

both MSB solubility and PVT measurement, the swelling ratio measured from the PVT

apparatus were directly used in the MSB measurement without major data correlation.

In each of the following figures, the apparent solubility is shown as a black square

symbol. SL and SS EOS corrected solubility are shown as purple circles and blue triangle

respectively. The experimentally corrected solubility based on equation 6-15 is represented as a

red triangle. For linear PP/CO2 solutions, the result plots are shown from Figure 6-27 to Figure


170

6-29. At pressures below 3000psi, the corrected solubility using experimental swelling is very

close to the SS EOS corrected solubility. When the pressure increases higher to above 3000 psi,

the corrected solubility from the experimental swelling becomes smaller due to small volume

swelling at high pressures, which contributes to a lesser buoyancy effect. This is more clearly

illustrated at higher temperatures, such as 200oC and 220oC. Very similar to the volume swelling

comparison between the SL and SS EOS prediction, the SL correction shows a concave inward

increase of solubility as the pressure increases at each temperature and the SS correction shows

a more linear trend of the solubility when the pressure kept increasing. The experimental

corrected solubility follows the same trend as the measured volume swelling, in that it shows the

signs of levelling off at high pressures and it also indicates that the solubility would likely reach

a plateau when the pressure increases indefinitely. In other words, it implies that when the “CO2

absorbing” capacity of the polymer matrix is reached or the polymer matrix is totally saturated

with CO2 , there is no place for the polymer matrix to stretch and the solubility limit would then

be reached at some high pressure levels.

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.0250.0500.0750.1000.1250.1500.1750.2000.2250.2500.2750.3000.3250.3500.3750.4000.4250.450

Solu

bilit

y (g

-gas

/g-p

olym

er)

Pressure (psi)

180oC Apparent Solubility 180oC Exp. Corrected Solubility 180oC SS Corrected Solubility 180oC SL Corrected Solubility

Figure 6-27 Corrected Linear PP/CO2 Solubility at 180oC


171

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.0250.0500.0750.1000.1250.1500.1750.2000.2250.2500.2750.3000.3250.3500.3750.4000.4250.450

Solu

bilit

y(g-

gas/

g-po

lym

er)

Pressure (psi)



500 1000 1500 2000 2500 3000 3500 4000 4500 50000.020.040.060.080.100.120.140.160.180.200.220.240.260.280.300.320.340.360.380.400.42

Solu

bilit

y(g-

gas/

g-po

lym

er)

Pressure (psi)



For the branched PP/CO2 solutions case, a similar conclusion can be drawn based on the

results shown in Figure 6-30, Figure 6-31 and Figure 6-32. Both SL and SS corrected solubility

showed concave inward and linear increase trends, but the experimental corrected solubility

showed the trend of reaching a plateau if the pressure kept increasing. Again, in the case of

200oC and 220oC at pressures below 3000psi, the corrected solubility using experimental


172

swelling was very close to the SS EOS corrected solubility. When the pressure increased higher

to above 3000 psi, the corrected solubility from the experimental swelling became smaller due

to volume swelling limitation at high pressures.

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.025

0.050

0.075

0.100

0.125

0.150

0.175

0.200

0.225

0.250

0.275

0.300

0.325

Solu

bilit

y (g

-gas

/g-p

olym

er)

Pressure (psi)


Figure 6-30 Corrected Solubility of Branched PP/CO2 at 180oC

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.025

0.050

0.075

0.100

0.125

0.150

0.175

0.200

0.225

0.250

0.275

0.300

0.325

Solu

bilit

y (g

-gas

/g-p

olym

er)

Pressure (psi)




173

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.025

0.050

0.075

0.100

0.125

0.150

0.175

0.200

0.225

0.250

0.275

0.300

0.325

Solu

bilit

y (g

-gas

/g-p

olym

er)

Pressure (psi)



In summary, the SL EOS corrected solubility has a bigger discrepancy with the SS EOS

and experimentally corrected solubility. The SS EOS corrected solubility is similar to

experimental corrected solubility only at pressures below 3000psi. Due to the polymer chain

structure and chain polymer entanglement, the results have shown a limited volume swelling at

higher pressures and the hypothesis postulated previously states that there is an upper cap for the

volume to expand with the gas dissolution. Therefore, this would be reflected in the final

solubility correction. Not surprisingly, the experimentally corrected solubility indicates that

there indeed exists an upper limit for gas solubility as well. The conclusion is clearly illustrated

in all of the figures. Despite the discrepancy between the SL, SS EOS corrected solubility and

experimental corrections; the directly measured swelling data from the PVT apparatus enabled

us to not rely on the EOS prediction while obtaining accurate solubility data.


174

6.5.2 Specific Volume of Linear/Branched PP/CO2

With the accurately measured volume swelling of the PP/CO2 solution that is due to the

CO2 dissolution, the buoyancy effect from the solubility measurement using MSB could be

compensated more precisely to obtain more accurate solubility information of the CO2 in the PP

matrix. This is done in the previous section using the measured swelling to get the true

corrected solubility information. This is also the reason why this section in particular is placed

right after the apparent solubility is corrected, since the determination of the mixture specific

volume required the final solubility of the polymer to be known.

With the amount of the CO2 gas dissolved in the PP matrix that were determined, the

equation used in determined the corrected solubility could be used agai to help derive the

equations to calculate the density information for the PP/CO2 solution at high temperatures and

pressures. The definition of the volume swelling ratio is again reiterated here as:

( )

( )

( )

PPsample

tPT

tPT

tPTw m

V

V

VSRatioSwellingVolume eq

ini

eq

υ*,,

,,

,,== (6-16)

Therefore, the swollen volume which occurs due to gas dissolution, can be expressed as the

difference between the initial sample volume and the final equilibrium polymer/gas solution

volume shown in the following equation (6-2):

( ) ( )inieqs tPTVtPTVV ,,,, −= (6-17) where V(T,P,teq) is the measured equilibrium polymer/gas solution volume at temperature T,

pressure P, and equilibrium time teq. The V(T,P,tini) is the volume of the neat PP sample

calculated at temperature T, pressure P using the Tait equation. The measured volume of

polymer/gas solution is:

( ) mixturepmixtureeq mtPTV ,,, υ∗= (6-18)


175

The initial sample volume is calculated as the following:

( ) purepsampleini mtPTV ,,, υ∗= (6-19)

In addition, the mixture mass of the polymer/gas solution is the summation of the initial polymer

sample mass and the dissolved gas amount:

2cosamplemixture mmm += (6-20)

2com is the dissolved CO2 inside the polymer matrix and samplem is the initial polymer sample

mass. The final corrected solubility from MSB is defined as:

sample

co

mm

s 2= (6-21)

Substitute equation (6-6) into (6-3), the final measured volume is expressed as:

( ) ( ) mixturepsampleeq smtPTV ,1,, υ∗+= (6-22) The initial volume calculated from the Tait density and initial sample mass is already expressed

in equation (6-4). Therefore, equation (6-2) can be rewritten to describe swollen volume due to

gas dissolution as:

( )[ ]purepmixturepsamples smV ,,1 υυ −∗+= (6-23) And the specific volume of the mixture could be obtained by rearranging the above equation

(6-9) as:

( )sS purepw

mixturep +

∗=

1,

,

υυ (6-24)

The s in the above equations is the same term as X corrected in equation (6-15). Therefore a

simple substitution would help in getting the final formula for obtaining the equilibrium

polymer/gas mixture specific volume as:

( )[ ]purepwgasapparent

purepwmixturep SX

S

,

,, 11 υρ

υυ

∗−++

∗= (6-25)


176

In equation (6-25), apparentX and gasρ are directly measured from the MSB apparatus, wS is the

volume swelling ratio obtained directly from the PVT apparatus. The purep,υ is the specific

volume of a pure polymer sample that can be calculated reliably from the Tait equation.

Therefore, currently the prediction of volume swelling from those EOS is not needed to for

solubility measurement using MSB. The only thing needed is the volume swelling information

for the PVT measurement. All other relevant information, such as the corrected solubility of gas

in the polymer, the density of the polymer/gas solution, can be obtained through equations 6-15

and 6-25 respectively.

Figure 6-33, Figure 6-34 and Figure 6-35 showed the specific volume of linear and

branched PP/CO2 solution verse CO2 gas pressure at each individual temperature.

500 1000 1500 2000 2500 3000 3500 4000 4500 50001.22

1.24

1.26

1.28

1.30

1.32

1.34

1.36

1.38

180oC Linear PP/CO2

180oC Branched PP/CO2

Spec

ific

Volu

me

(cm

3 /g)

Pressure (psi)

Figure 6-33 Specific Volume of LPP/CO2 and BPP/CO2 at 180oC


177

500 1000 1500 2000 2500 3000 3500 4000 4500 50001.22

1.24

1.26

1.28

1.30

1.32

1.34

1.36

1.38 200oC Linear PP/CO2


Spec

ific

Volu

me

(cm

3 /g)

Pressure (psi)


500 1000 1500 2000 2500 3000 3500 4000 4500 50001.22

1.24

1.26

1.28

1.30

1.32

1.34

1.36

1.38 220oC Linear PP/CO2


Spec

ific

Volu

me

(cm

3 /g)

Pressure(psi)


All three figures show that linear PP/CO2 have a high specific volume compared with

branched PP/CO2. This is expected because of higher volume swelling ratios observed in the

former compared with the latter. For the same amount of polymer sample, even the CO2 has

higher solubility in linear PP than in branched PP, the increase of the volume outweighs the

mass increase due to the dissolution of the CO2 gas; Therefore, the specific volume is higher


178

for linear PP/CO2 solutions. At 180oC and 200oC, both linear and branched PP/CO2 saw a slight

increase of specific volume at low-pressure ranges, from 1000 psi to 2000 psi. The specific

volume then decreased steadily after, from 2500 psi and onwards. This can be explained by

two factors: 1) the decrease of the volume swelling rate as the pressure increases, 2) the increase

of the CO2 density as the pressure increases

When the pressure keeps increasing above 2500psi, the increase of volume swelling is

getting smaller. The apparent solubility and the CO2 density increase as the pressure increases.

From equation 6-25, with an increase in the value of denominator and a decrease from the

numerator, the net result is the decrease mixture specific volume. In another word, the increase

in the mixture weight due to heavier CO2 dissolution is higher than the increase in the mixture

volume due to an already slowdown volume swelling, which resulted in a decrease in mixture

specific volume.

In addition, as mentioned in an earlier section, at higher pressures, the hydraulic effect

could also be one of the contributing factors.

6.6 Summary

Chapter 6 mostly focused on the study of PVT properties, such as volume swelling and

the corrected solubility of both linear and PP/CO2 solutions at high temperatures and pressures.

The temperatures were at 180oC, 200oC and 220oC and the pressure started from 1000psi to

4500psi. The results showed that the volume swelling increased as the pressure increased. On

the other hand, the increase in temperature resulted in a decrease in volume swelling for both

linear and branched PP/CO2 solutions. Some of the linear PP/CO2 swelling results and detailed

discussions are already published[292].


179

Due to the long chain branching (LCB) effect, the branched PP showed significantly less

volume swelling in all temperatures and pressures. It was discovered that at each fixed

temperature, the rate of volume swelling decreased as the pressure increased and there existed a

plateau or upper limit for the volume swelling of linear PP and branched PP/CO2 solutions. This

phenomenon then lead to the hypothesis made in this study that states that due to the polymer

chain structure and chain entanglement density, there was an upper limit for polymer/gas

volume swelling at high pressures. In the rheometer test conducted for linear and branched PP at

180oC, 200oC and 220oC, the results showed that branched PP had much higher chain

entanglement density than linear PP.

The volume swellings from the measurements were also compared with SL and SS EOS

prediction values for both linear and branched PP. It showed that both SL and SS predicted

continuously increasing volume swelling as the pressure increased at constant temperatures. The

discrepancy between SS EOS predictions with measured values was the smallest.

The corrected solubility data from both experiments and EOS were also plotted. The

results also indicated that the solubility tended to reach some plateaus as the pressure increased

for both linear and branched PP cases.

Experiments were carried out to investigate the effect of shape asymmetry of amorphous

PS at three different temperatures and vacuum conditions. The results demonstrated strongly

that as the temperature increased, the drop shape becomes more asymmetric and the results were

more accurate. In addition, the results from Figure 6-8 showed that the rotational device helped

to compensate or eliminate the effect of asymmetry, which would facilitate accurate PVT

measurement of asymmetric polymer drop sample. The PS/CO2 PVT showed that as the

temperature increased, the PS/CO2 had less swelling, but the swelling increased as the pressure

increased.


180

Due to the need for exploring more environmentally-friendly gas blowing agents, the

HFC-152a was used to study its PVT behaviour with PS solutions at high temperatures but with

relatively low pressures of up to only 800psi. At low pressures, the swelling of PS/HFC-152a

showed concave increasing trend as the pressure increased.


181

Chapter 7. Measurement of Surface Tension for Polymer/Gas Solutions

7.1 Introduction

The surface (interfacial) tension of polymer melts is an important thermodynamic

parameter and it plays a key role in many polymer processing applications such as foaming,

blending, coating and wetting. The effort to understand and control processes involving these

applications would benefit from a better understanding of events at the molecular level.

However, the high viscosity and the limited thermal stability of polymer melts as well as the

difficulties in carrying out experimental studies due to high temperature and pressure

requirement, accurate and reliable experimental data are often not available.

Supercritical fluids are well established as a processing solvent in various polymer

applications such as foaming, blending, modification, composites formation, particle production

and polymerization. Supercritical carbon dioxide (SF-CO2) is becoming an increasingly

interesting foaming agent in the production of microcellular polymer foams.[131,293] In order to

form polymeric foam, bubbles must first nucleate and grow within the molten or plasticized

material. The initial nucleation is induced by a change in thermodynamic instability either

through temperature increase or pressure drop. During this process, a second phase is

generated from a metastable polymer/gas homogeneous mixture. According to the classical

nucleation theory[136], properties such as diffusivity (D), gas concentration (c), surface tension

(γ), temperature (T), and degree of supersaturation are the parameters controlling the nucleation

rate, N:


182

⎥⎦

⎤⎢⎣

⎡Δ

−= 2

3

316exp

PkTcfN πγ

(7-1)

Particularly, relationships between surface tension and foaming are of great interest, as foaming

is strongly influenced by the surface tension of polymer. The energy barrier of the cell

nucleation can be lowered by lowering the surface tension. The free energy barrier equation is

brought back here again as follows:

( )23

316

systembuble PPW

−=

πγ (7-2)

As a polymer-gas mixture has a lower surface tension than that of the pure polymer, the

Gibbs free energy for the formation of a nucleus will be reduced in cubic power of the surface

tension, and in turn will cause the nucleation rate to increase exponentially. Therefore,

understanding and controlling the surface tension of polymer/gas solution is important for

optimizing the foaming process. However, although some data about the surface tension in pure

polymer at elevated pressures have been reported, only a little fundamental knowledge on the

surface tension of the polymer/gas solution is available because of the experimental difficulties

in performing measurements on highly viscous polymer liquids at high pressure and

temperature.

Jaeger et al.[291] investigated the interfacial properties of high viscous polystyrene (PS) in

supercritical CO2 at pressure up to 25 MPa. Dimitrov et al.[290] found the interfacial tension of

poly(ethylene glycol)nonylphenyl ether (PEG-NPE)-CO2 to be a linear function of CO2 density

up to 800 kg/m3. Enders et al.[294] used Cahn-Hilliard theory in combination with equations of

state (the original statistical associating fluid theory (SAFT), the perturbed-chain statistical

associating fluid theory (PC-SAFT) or the Sanchez-Lacombe (SL) lattice theory) to describe the


183

temperature and pressure dependence of the interfacial properties between PS-gas mixture and

the pure gas phase. However, it was not possible to predict the interfacial tension in

quantitative agreement with experimental data. The theoretical framework needs further

improvement with respect to the polydispersity of the polymer.

Park et al.[25-27] and Li et al.[61] recently used Axisymmetric Drop Shape Analysis (ADSA)

to determine the surface tension of polymer melt in supercritical fluids, such as supercritical

CO2 and supercritical nitrogen (N2). The ASDA approach relies on a numerical integration of

the Laplace equation of capillarity.[295] During this procedure, the density difference between

polymer-supercritical fluids mixtures and the supercritical fluids was an input parameter.[296]

One of the most important requirements for the accurate calculation of the interfacial tension of

binary systems is the correct calculation of the densities of the coexisting phases. As

mentioned in previous chapters, Sanchez-Lacombe (SL) and Simha-Somcynsky (SS) equation

of state (EOS) were normally applied to estimate the pressure-volume-temperature (PVT) data

of the polymer-supercritical fluids mixtures, which gave the density data. Nevertheless, the

experimental data for the density of polymer-supercritical fluid systems are rather rare.

Funami et al.[86] developed a new method of directly measuring the densities of two

polymer melt-CO2 single-phase solutions, poly(ethylene glycol) (PEG)-CO2 and polyethylene

(PE)-CO2 at high pressure and temperature using a magnetic suspension balance (MSB). A

thin disc-shaped platinum plate was submerged in the polymer-CO2 single-phase solution in the

MSB high-pressure cell. The weight of the plate was measured while keeping temperature and

CO2 pressure in the sorption cell at a specified level. Since the buoyancy force exerted on the

plate by the polymer-CO2 solution reduced the apparent weight of the plate, the density of the

polymer-CO2 mixture could be calculated by subtracting the true weight of the plate from its

measured weight. However, this density measuring method has some limitations on applicable


184

polymers. When the plate is moving up and down in the polymer melt during the position

changeover operation, a dragging force is generated. Therefore, the readout can not be

guaranteed when the viscosity of polymer melt is high.

To determine the density of polymer-gas, the solubility of gas in polymer and the

volume of the polymer-gas mixture are required. The volumetric and gravimetric methods

have been widely used to measure the solubility of gas in polymer and polymer blends as

described in Chapter 2 and 3. However, neither the volumetric nor the gravimetric method

alone can generate the accurate solubility data because the gas dissolution in polymer causes

volume swelling in polymer. The correction of volume swelling can be obtained either by

prediction using EOS for polymer, such as Flory EOS,[234] SL EOS, and SS EOS or direct

experimental measurement using the PVT apparatus developed in this research.

7.2 Experimental Materials

Linear PP (DM 55 Borealis, Mn=105,473), branched PP (Daploy WB130HMS Borealis,

Mn=122,395) and CO2 (Coleman grade, 99.99% purity, BOC Canada) were used in this study.

7.3 Measurement of Surface Tension for Linear/Branch PP/Gas Solutions

7.3.1 Surface Tension for Linear/Branched PP/CO2 Solutions

The technique of Axisymmetric Drop Shape Analysis Profile (ADSA-P) was used to

determine the surface tension from the image captured in PVT measurement. Surface tensions

were obtained by fitting the shape and dimensions of the axisymmetric menisci acquired to the

theoretical drop profile according to the Laplace equation of capillarity.


185

⎟⎟⎠

⎞⎜⎜⎝

⎛+=Δ

21

11RR

P γ (7-3)

where ΔP is the pressure difference across the curved interface; γ is the surface (interfacial)

tension; R1 and R2 are the principal radii of curvature of the drop. The value of surface tension

was generated as a fitting parameter after a least-squares algorithm was employed to minimize

the difference between experimental drop profiles and theoretical ones.

Within the ADSA formalism, the density was originally introduced through the capillary

constant:

( )

γρ gc ⋅Δ

= (7-4)

where c is the capillary constant; Δρ is the density difference between liquid/fluid interface; g is

the acceleration due to gravity. The determination of c through ADSA would yield the surface

tension while Δρ is a readily available input parameter.

Once the mass of CO2 dissolved in polymer and the volume of the polymer-gas mixture

are known, the density of polymer-gas mixture can be calculated from its volume and mass (sum

of the initial mass of polymer and the mass of dissolved CO2). Then, Δρ is calculated by the

following equation:

gasmixturep ρρρ −=Δ , (7-5)

where ρmixture is the density of polymer-gas solution.

The temperature and pressure dependences of the specific volume for the polymer melt

can be described by the Tait equation:

( ) ( ) ( )[ ]TBPCTVTPV +−= 1ln1,0, (7-6)


186

The Tait equations for the specific linear PP and branched PPused in this study were

derived based on the measured PVT behaviour using the bellow type dilatometer[22-24,254] are as

follows, respectively[124]:

10001086.91006.1

1045.61046.7

7

2

9

6

, ×⎟⎟⎠

⎞⎜⎜⎝

⎛+×××

++×

×=

PT

Ppurepυ (7-7)

100010146.110221.1

10773.510485.6

8

2

9

6

, ×⎟⎟⎠

⎞⎜⎜⎝

⎛+×××

++×

×=

PT

Ppurepυ (7-8)

where the specific volume υp,pure is in (cm3/g), the temperature T is in (oC) and pressure P is in

(Pa).

The solubility of CO2 in PP melts was measured gravimetrically using a MSB

(Rubotherm GmbH). When CO2 dissolves in polymer, it swells polymer. Because the

buoyancy caused by the swelling changes during the solubility measurement, to perform

accurate measurement, the weight of dissolved CO2 has to be measured with consideration of

the buoyancy effect:

( ) ( ) ( ) ( )( )spBCOCO VVVPTTWPTWPTW +++−= ,0,,,22

ρ (7-9)

where the balance readout at temperature, T, and pressure, P was recorded as W(T, P), and

vacuum (P ≈ 0) was recorded as W(T, 0). The ρCO2 is the density of gas and can be measured

in situ by the function of MSB. VB is the volume of the sample holder; and VP is the volume of

the pure polymer sample at pressure P and temperature T (without gas), which can be calculated

by evaluating the pure polymer’s PVT equations. VS is the swollen volume of the polymer/gas

mixture due to the gas dissolution. Without considering the swelling effect, the solubility

measured from the read-out of the MSB is denoted as apparent solubility apparentX . This apparent

solubility, apparentX can be determined from the above equation as:


187

masssampleapparentgapparent mWX /,= (7-10)

In order to get an accurate solubility data, the volume swelling has to be taken into

account. After obtaining more accurate and true volume swelling from direct measurement

using our PVT apparatus, the apparent solubility can be corrected as true or accurate corrected

solubility as:

masssample

sgasapparentcorrected m

VXX

∗+=

ρ (7-11)

From the definition of the volume swelling in equation (4-15) the swellon volume sV can be

interpreted as the initial sample volume multiply by the swelling ratio:

( )1* −= wvolumesampleinitals SVV (7-12)

Combine the above equation 7-11 and 7-12, the corrected solubility could be simplified into the

following equation:

( ) purepwgasapparentcorrected SXX ,1 υρ ∗−∗+= (7-13)

where purep,υ is the specific volume of pure polymer sample and can be calculated through Tait

equation. With all the mathematics in hand, the corrected solubility is determined from the

measured volume swelling as well as the apparent solubility obtained from the MSB

experiment.

Thus, the corrected CO2 solubility in polymer (WCO2) with buoyancy effect

compensation will be obtained from equation 8. As the mass and volume of polymer-gas

mixture are known, the density of polymer-gas solution phase, ρp,mixture, can be obtained from

purepw

corrected

mixturep

mixturepmixturep S

XVW

,,

,, υ

ρ∗

== (7-14)


188

7.3.2 Temperature and Pressure Effect on Surface Tension

Figure7-1 shows the equilibrium surface tension results of linear PP/CO2 system at all

three experimental temperatures and the pressures. First of all, the results show that at each

fixed temperature, the surface tension decreases when the pressure increases. When the pressure

is fixed, the surface tension decreases as the temperature increases. But this conclusion is true

only for pressures up to 2000psi. When the pressure went beyond 2000psi, the surface tension

dependence on temperature dropped dramatically. Those results are similar to what has been

reported in the study of PS/CO2 from Park et al. [25-27]. From the Figure 7-2, they observed that,

in the PS/CO2 system at temperature between 170oC and 210oC, the dependence of surface

tension on temperature becomes less with increasing pressure. When the pressure values reach

above 2000 psi, the dependence diminishes. And it was concluded that increasing temperature is

effective at reducing surface tension only when moderate pressure is applied during a polymer

process from their study. But they did not report the dependence of surface tension on

temperature at higher pressure beyond 2500 psi since their experiments were run at a maximum

pressure of 2500 psi.

180 200 2202

4

6

8

10

12

14

16

18Pressure (psi)

1000 1500 2000 2500 3000 3500 4000 4500

Surf

ace

Tens

ion

(mJ/

m2 )

Temperature (oC)

Figure 7-1 Surface Tension of Linear PP/CO2 at Various Temperatures and Pressures


189

From our experimental results, at 2500 psi, the surface tensions across all three

temperatures are similar. When the pressure was increased to 4500psi, the surface tension values

slightly increased when the temperature increased.

Figure 7-2 Equilibrium Surface Tension of PS/CO2 Solution

Figure 7-3 shows the surface tension of branched PP/CO2 at all three temperatures and

pressures up to 4500psi. It further confirms our observation from the linear PP/CO2 case. It is

clearly shown that the surface tension dependence on temperature as in the branched PP/CO2

case follows the exactly same trend as it does in the linear PP/CO2 case. Before the pressure

reaches 2000 psi, the surface tension drops when the temperature increases. The dependence of

surface tension on temperatures diminishes at pressure beyond 2000 psi. On the contrary, the

surface tensions slightly increase at higher pressures when the temperature increases.

When the pressure increases, the surface tension decreases because a higher pressure

would mean that more CO2 gas dissolves into the polymer matrix and the solubility increases.

When gas is dissolved into the polymer melt, the free volume is occupied with gas molecules at

the beginning. When more gas penetrates into the polymer chain, more free volume is created


190

accompanied by a plasticizing effect. This leads to an increase of molecular mobility and results

in a decrease of the melt viscosity. This increase in molecular mobility and decrease of the melt

viscosity would lower the surface tension of the polymer/gas mixture system. Moreover, this

increase of solubility due to an increasing in pressure would decrease the free energy of the

polymer/gas solution; hence the surface tension will decrease. Li et al.[61] used the density

gradients theory to model and predict the surface tension relation with pressure for PS/CO2 and

PP/CO2 solutions and it showed that the surface tension decreases dramatically as pressure

increases.

Park et al. [25-27] also modeled the temperature and pressure dependence of surface

tension using the second order linear regression model. It explained that when the pressure

increases, the largest factor causing the surface tension decrease is the internal energy. At higher

pressure, there are fewer holes present in the polymer matrix due to the high gas solubility. With

the removal of the holes, the free energy of the system is then decreased and, therefore, the

surface tension would drop as well. As for the reasons behind the surface tension’s

dependence on the temperature, it is explained that the increase in temperature would reduce the

segregation between molecular constituents; hence, the surface tension drops with temperature

at moderate pressure.


191

180 200 220

4

6

8

10

12

14

16

18

Pressure (psi) 1000 1500 2000 2500 3000 3500 4000 4500

Surf

ace

Tens

ion

(mJ/

m2 )

Temperature(oC)

Figure 7-3 Surface Tension of Branched PP/CO2 at Various Temperatures and Pressures

There are two competing forces between the effect of T and the effect of gas content

dissolved in the PS. First, as the T increases, the surface tension decreases as the inter-molecular

distance increases. But as the T increases, the solubility decreases because the solubility

decreases. It is known that the surface tension decreases as the amount of gas dissolved in the

polymer increases. So this lowered gas content (with an increase in T) increases the surface

tension. So as the T increases, these two mechanisms are competing with each other. At low P,

the effect of dissolved gas is not dominant. But as the P increases, the dissolved gas content

increases. So the dissolved gas content dominates the surface tension. The break point for

PP-CO2 combination is: 20.6 MPa (3000psi). Above this point, the gas content governs the

surface tension and below this point, the T governs the surface tension.

Usually contact angle is also determined when surface tension is studied. But contact

angle is not measured in this research and the contact angle will be different when different

materials were used as the pedestal. Also the contact angle is also various when different grade

pellets are used since the sample pellets are from the manufacturing and it is possible the dirty

particle or impurity may affect the drop shape but the information is not provided precisely.


192

TGA tests were carried out to see whether there were any degradation and the results shown that

the degradation is minimal. The TGA test results are going to be included in the Appendix 7 as

well.

7.3.3 Branch Effect on Surface Tension between Linear and Branched PP

It is believed that the branching of the PP molecules reduces the rate of the nucleation

process of the blowing agent because of an increase in the surface tension with branching. Since

the branched PP has larger elastic components, such as high melt elasticity and polymer chain

entanglement density, it will require a larger amount of work to create a new surface and thus a

higher surface tension. The following figures show the surface tension comparisons of linear

PP/CO2 and branched PP/CO2 at each temperature.

500 1000 1500 2000 2500 3000 3500 4000 4500 50002

4

6

8

10

12

14

16

18

20 BPP/CO2 at 180oC

LPP/CO2 at 180oC

Surf

ace

Tens

ion

(mJ/

m2 )

Pressure (psi)

Figure 7-4 Surface Tension of Linear and Branched PP/CO2 at 180oC


193

500 1000 1500 2000 2500 3000 3500 4000 4500 50002

4

6

8

10

12

14

16

18

20 BPP/CO2 at 200oC)

LPP/CO2 at 200oC)

Surf

ace

tens

ion

(mJ/

m2 )

Pressure (psi)


500 1000 1500 2000 2500 3000 3500 4000 4500 50002

4

6

8

10

12

14

16

18 BPP/CO2 at 220oC

LPP/CO2 at 220oC

Surf

ace

Tens

ion(

mJ/

m2 )

Pressure (psi)



194

From all three figures at three various temperatures, the surface tension decreases as the

pressure increases for both linear and branched PP/CO2 solutions. This is consistent with the

results seen in the previous sections as well as those in the literature aforementioned.

Those results also clearly show that the branched PP/CO2 has higher surface tension than

linear PP/CO2. Long chain branching (LCB) in branched PP polymer enhances polymer chain

entanglements and reduces polymer hydrodynamic volume. The LCB structure also greatly

improved the polymer elasticity, as evident from the increases in storage modulus, relaxation

modulus and die swell[297]. In comparison to linear polymer melts, long chain branched polymer

melts show enhanced strain-hardening due to the fact that while the backbone of the branched

macromolecule is stretched by deformation, the side chains are compressed.

In addition, the long branches of PP induce more chain entanglements in melt state,

contributing to enhancing the melt strength over linear PP[298,299]. It is also seen that the

branching structures affect the gas solubility and the swelling. Due to the increased melt

strength and viscosity of branching structure as well as the reduction of gas solubility and

volume swelling, the surface tension of branched PP is also increased over linear PP. However,

the surface tension of branched PP/CO2 and linear PP/CO2 starts to converge at higher pressure

levels from 2500 psi to 4500 psi as shown in all three figures. This means the effect of

branching on surface tension is diminished from the increase of the pressure. When the pressure

increases to higher levels, it helps to get more gas dissolved into the polymer. The plasticization

effect due to more gas dissolution into the polymer matrix would reduce the melt strength and

viscosity, which would counteract the branching effect on the surface tension. When the

pressure is high enough, its effect dominates the branching effect, therefore the surface tension

of branched PP/CO2 will decrease and the surface tension for both linear and branched PP/CO2

converges at high pressure.


195

7.4 Summary

The focus of this chapter is to combine the surface tension measurement technique with

the determined density information from PVT to obtain more accurate surface tension

measurements of polymer/gas solutions. The solution systems used in the surface tension study

are the same as those used in the PVT study from Chapter 6. They were linear PP/CO2 and

branched PP/CO2 solutions.

The effects of temperature, pressure and branching structure on the surface of PP/CO2

solutions were shown in detail based on the experimental results. It was found that the surface

tension will decrease as the temperature increases. But this decreasing dependence of surface

tension on temperature only hold at pressured up to 2000 psi. This observation from the

experimental results is similar to the one reported from Park et al. [27,181] when they saw that the

dependence of surface tension of PS/CO2 solution on temperature diminishes at a pressure of

2500 psi. Pressure also affects the surface tension through solubility. The solubility of CO2

increases as the pressure increases, the free energy of the system would drop when more gas is

dissolved into the polymer matrix to occupy the holes and therefore the surface tension also

decreases as a result of the pressure increase.

The branching structure also has impact on surface tension. Branching structure would

enhance the polymer melt strength, strain-hardening and melt viscosity, which make the

polymer chains become less mobile than linear polymer. The restriction on the chain mobility

due to branching effect will cause the gas solubility in branched polymer to decrease; thereby

the surface tension of branched PP/CO2 solution is higher than linear PP/CO2. However when

the pressure increases to higher levels, the branching effect on the surface tension become

diminishing. This indicates that the higher the pressure, the more gas is dissolved into the

polymer. The plasticization effect of having more gas dissolved into the polymer matrix would


196

reduce the melt strength and the viscosity; therefore, the surface tension will decrease as the

pressure increases. The decrease in surface tension due to the increase in pressure is higher than

the increase in surface tension from effect of branching at higher pressure, therefore the results

show that the surface tension of linear and branched PP/CO2 started to converge as pressure

increases beyond 2500psi.


197

Chapter 8. Conclusions and Future Work

8.1 Conclusions

A novel apparatus has been designed and constructed based on the need for more

accurate determination of the PVT property of polymer/gas solutions at elevated temperatures

and pressures.

During thermoplastic foaming, the polymer/gas solution formation, cell nucleation and

cell growth, are the most important stages. It has been shown that the gas solubility in a polymer

melt would determine the proper polymer/gas formation. The surface tension at both the gas and

polymer/gas interfaces would strongly affect the behaviour of cell nucleation and cell growth

which are crucial factors in determining the properties of final foam products. However,

accurate determination of both solubility and surface tensions need reliable and precise PVT

information. SL and SS EOS have been used quite extensively in predicting the volume swelling

and in calculating the density of the polymer/gas mixtures to facilitate better measurement of the

solubility and surface tensions. However, with the limitation of using pure theories, the urgency

for developing an experimental apparatus to obtain the polymer/gas solutions’ PVT information

is imminent.

This research study, motivated by these strong needs, used the Axiomatic design

methodology to propose, verify and construct a unique experimental apparatus which could

enable the accurate measurement of polymer, polymer/gas solutions PVT at high temperatures

and pressures. The design and construction process of the system were depicted in detail.

The error reduction from image reconstruction was verified using the system to measure

the volume of a stainless sphere ball. The results showed that, when the number of images taken


198

increases, the error in volume calculation drops significantly due to the pixel size reduction and

the boost in increasing pixel number. The densities of pure linear PP, branched PP and PS were

measured using the PVT apparatus at high temperatures but under vacuum conditions. Their

densities in the same condition were also calculated from the Tait equation. The density results

obtained from two different means showed excellent agreement. This proved that the PVT

system is accurate, reliable and repeatable in conducting the PVT experiments for pure polymers

as well as polymer/gas solutions.

The PVT properties of linear and branched PP/CO2 solutions were intensively studied at

high temperatures and pressures. The temperatures were at 180oC, 200oC and 220oC and the

pressure ranged from from 1000psi to 4500psi. The results showed that the volume swelling

increased as the pressure increased. On the other hand, the increase in temperature resulted in a

decrease in volume swelling for both linear and branched PP/CO2 solutions.

Due to the long chain branching (LCB) effect, the branched PP showed significantly less

volume swelling in all temperatures and pressures. The LCB helped increase the melt strength

and viscosity but restricted the chain mobility, and thereby it was less easy for the gas molecules

to penetrate into the branched structure of branched PP matrix.

It was found that, at each fixed temperature, the rate of volume swelling decreased as

pressure increased and there exists a plateau or upper limit for the volume swelling of linear PP

and branched PP/CO2 solutions. This finding indicated that due to the long chain structure and

chain entanglement, the polymer could not physically stretch or dilate indefinitely from

dissolution of the gas as pressure increased. This phenomenon then led to the hypothesis made

in this study, which dictates that due to the polymer chain structure and chain entanglement

density, there was an upper limit for polymer/gas volume swelling at high pressures. The

polymer chain entanglement density experiments were then carried out using the rotational


199

rheometer to further study the chain entanglement property of linear and branched PP. It was

discovered that branched PP had a much higher chain entanglement density than linear PP. The

chain entanglement density decreased as temperature increased. For the same polymer at high

temperatures, the low entanglement density resulted in less chain agglomerations. This made it

easier for the gas molecules to diffuse out of the matrix at a high temperatures, leading to a drop

in the solubility of the gas and a decrease in volume swelling. At the same temperature, linear

PP has less chain entanglement density and therefore has less chain entanglement that would

have less restriction on the chain mobile and allowed more gas molecules to diffuse into the

polymer matrix as the pressure increased.

As mentioned in the Chapter 1, one of the motivations of this research is to help

facilitate more accurate solubility measurements as well as verify the validity of EOS prediction

volume swelling. The volume swellings from the measurements were then compared with SL

and SS EOS prediction values for both linear and branched PP. It showed that both SL and SS

predicted continuously increasing volume swelling as the pressure increased at constant

temperature. SL showed a concave upward increase and SS showed a more linear increase. SS

predictions were the closest EOS predictions to the experimental measurement values. The

discrepancy between the EOS prediction and the measurement values may be due to the

inability of predicting the real molecular dynamics and behaviours using pure thermodynamic

laws. Nonetheless, the PVT measurement provided the means to obtain the volume

measurement directly and gave room for improvement in making those two EOS model

predictions better fit with experimental data.

The corrected solubility data from both the experiments and EOS were also plotted. The

results again showed similar trends as in the swelling comparison. The SS EOS corrected

solubility was the closet to the experimental corrected solubility. The results also indicated that


200

the solubility tended to reach some plateau as the pressure increased for both linear and

branched PP cases.

Another important topic regarding the shape asymmetry effect on the PVT measurement

was investigated thoroughly. The experiments were carried out using PS at three different

temperatures and in a vacuum condition. The results demonstrated strongly that, as the

temperature increased, the drop shape becomes more asymmetric. The results also demonstrated

that the rotational device helped to compensate the effect of asymmetric drop shape on PVT

measurement. Therefore, the rotational device enable the system to measure the PVT data

accurately even with asymmetric polymer drop. The PVT information for PS/CO2 was also

investigated. The study showed that as the temperature increased, the PS/CO2 had less swelling

but the swelling increased as the pressure increased.

Due to the need of exploring more environmentally friendly gas blowing agents, the

HFC-152a was used to study its PVT behaviour with PS solutions at high temperatures, but at

relatively low pressures of up to only 800 psi. At low pressures, the swelling of PS/HFC-152a

showed a concave increasing trend as the pressure increased. However, the temperature effect

on the volume swelling was the same and the volume swelling was also less at high

temperatures.

Finally, the surface tensions of both PP/CO2 solutions were investigated using the

experimental determined density data. At constant T, as P increases, the surface tension drops

for both linear and branched PP/CO2 solutions. There is clear competition between the

temperature effect and the gas content effect on surface tension. The gas content was the

dominate factor at pressure up to 2500psi. This observation agreed with what other researchers

discovered for PS/CO2 solutions. At pressures from 2500 psi to 3500 psi, the dependence of the

surface tension on the temperature was minimal. After the pressure surpassed 3500 psi, the


201

surface tension saw a slight increase due to decrease in solubility caused by the temperature

increased. Due to the branching effect, the surface tensions of branched PP/CO2 solutions were

higher than linear PP/CO2 only at relatively low pressures of up to 2500 psi. When the pressure

increased beyond 2500psi, the surface tension between the linear and branched PP/CO2

solutions at each temperature became very close to each other, this indicated that the increase in

pressure helps to diminish the branching effect over those two long chain polymers.

8.2 Future work

Although this research has successfully developed a unique and sound system to measure

the PVT of polymer/gas solutions accurately for the first time, there are still improvement works

that needs to be carried out to take full advantage of this system. The system improvement

would allow for potential system components modification or upgrade. For example, if the size

of the sessile drop needs to be changed, then the pedestal needs to be machined to accommodate

the change.

The following suggestions are made for the direction of future research using this PVT

measurement apparatus.

1. Conduct intensive PVT measurements on various polymer/gas solution combinations.

2. Establish a rigorous and complete PVT database for variety of polymer/gas solutions.

3. To improve the technique of fabricating a better droplet tip with a sharp edge so that the

boundary between the sample and tip can be easily distinguished.

4. This study simply compared the volume swelling results from experimental and the EOS

prediction. Discrepancies were found but they were not fully investigated, therefore it is


202

necessary and worthwhile to carry out more in-depth study on how to improve the

modelling of the EOS so that better predictions can be made.

5. Surface tension property studies were only carried out using PP/CO2 solutions. In order

to obtain more accurate surface tension values and to better understand the role of

surface tension in cell nucleation and growth, more experiments need to be carried out

using variety of polymer/gas solutions

6. The asymmetry of other polymer/gas drop shapes is worth investigating. A better index

or universal constant can be established to describe the drop shape asymmetry in general.

7. Lastly, the experiments carried out in this study were confined to the usage of only the

sessile drop. Since the design of the PVT apparatus is also capable of dealing with

pendent droplets, it would be important to conduct all the above experiments using the

pendent drops to further investigate the difference between those two techniques.


203

REFERENCES [1] Khemani, K. C., "Polymeric Foams : Science and Technology", American Chemical

Society,Washington D.C. 239 (1997).

[2] Frisch, K. C., Klempner, D., "Handbook of Polymeric Foams and Foam Technology",

Hanser, New York (1991).

[3] Munters, C. G., Pat. 2,023,204 U.S. (1935).

[4] Ruskin, B. L., Lake Publishing, Libertyville, Ill. (1965).

[5] Kennedy, R. N.,"Handbook of Foamed Plastics", Lake Publishing, Libertyville, Ill. (1965).

[6] Todd, W. D.,Lake Publishing, Libertyville, Ill. XIV. (1965).

[7] Suh, K. W. P., Park, C. B., Maurer, M. J., Tusim, M. H., De Genova, R., Broos, R., Sophiea,

D. P., "Lightweight Cellular Plastics", Advanced Materials, 12, 1779-1789 (2000).

[8] Lee, S. T., "Foam Extrusion: Principles and Practice", Technomic Publishing Company Inc.,

Lancaster, PA, (2000).

[9] Naguib, H. E., Park, C. B.,"A Study of Melt Fracture of Linear and Branched

Propylene/Butane Solutions with Foaming Additives", Annual Technical Conference -

Society of Plastics Engineers, 1025-1031 (2000).

[10] Naguib, H. E., Park, C. B., Panzer, U., Reichelt, N., "Strategies for Achieving Ultra

Low-density Polypropylene Foams", Polymer Engineering and Science, 42 (7), 1481-1492

(2002).

[11] Maier, C., Calafut, T., "Polypropylene: The Definitive User's Guide and Data Book",

Norwitch, New York (1998).

[12] Baldwin, D. F. S., N. P. Shimbo, M. MD, "Gas Dissolution and Crystallization in

Microcellular Thermoplastic Polyesters", Cellular Polymers, 38, 109-128 (1992).


204

[13] Collias, D. I., Baird, D. G., "Tensile Toughness of Microcellular Foams of Polystyrene,

Styrene-Acrylonitrile Copolymer, and Polycarbonate, and the Effect of Dissolved Gas on

the Tensile Toughness of the Same Polymer Matrices and Microcellular Foams", J. Polym.

Eng. Sci., 35 (14), 1167-1177 (1995).

[14] Collias, D. I., Baird, D. G., Borggreve, R. J. M., "Impact Toughening of Polycarbonate by

Microcellular Foaming", Polymer, 35 (18), 3978-3983 (1994).

[15] Collias, D. I. B., D. G., "Impact Behavior of Microcellular Foams of Polystyrene and

Styrene-Acrylonitrile Copolymer, and Single-Edge-Notched Tensile Toughness of

Microcellular Foams of Polystyrene, Styrene-Acrylonitrile Copolymer, and

Polycarbonate", J. Polym. Eng. Sci., 35 (14), 1178-1183 (1995).

[16] Daniel F. Baldwin, C. B. P., Nam P. Suh, "Microcellular Sheet Extrusion System Process

Design Models for Shaping and Cell Growth Control ", Polym. Eng. Sci., 38 (4), 535 - 705

(1998).

[17] Park, C. B., Baldwin, D. F., Suh, N. P., "Effect of the Pressure Drop Rate on Cell

Nucleation in Continuous Processing of Microcellular Polymers", Polym. Eng. Sci., 35,

432-440 (1995).

[18] Seeler, K. A., Kumar, V., "Tension-Tension Fatigue of Microcellular Polycarbonate: Initial

Results", Journal of Reinforced Plastics and Composites, 12 (3), 359-376 (1993).

[19] Lee, S. T., Ramesh, N. S., "Polymeric Foams Mechanisms and Materials", CRC Press,

Washington, D.C. 336 (2004).

[20] Suh, C., "The Nucleation of Microcellular Thermoplastic Foam With Additives: Part I:

Theoretical Considerations", J. Polym. Eng. Sci., 27 (7), 485-492 (1987).

[21] Zoller, P., "The Pressure-Volume-Temperature Properties of Three Well-Characterized

Low-Density Polyethylenes", J. Appl. Polym. Sci., 23 (4), 1051-1056 (1979).


205

[22] Zoller, P., Bolli, P., Pahud, V., Ackermann, H., "Apparatus for Measuring

Pressure-volume-temperature Relationships of Polymers to 350oC and 2200 kg/cm2",

Review of Scientific Instruments, 47 (8), 948-952 (1976).

[23] Zoller, P., Hoehn, H. H., "Pressure-volume-temperature Properties of Blends of

Poly(2,6-dimethyl-1,4-phenylene ether) with Polystyrene", Journal of Polymer Science,

Polymer Physics Edition, 20 (8), 1385-1397 (1982).

[24] Zoller, P., Walsh, D., "Standard Pressure-Volume-Temperature Data for Polymers",

Technomic Publishing, Lancaster, Pennsylvania (1995).

[25] Park, H., Park, C. B., Tzoganakis, C., Chen, P., "Effect of Molecular Weight on the Surface

Tension of Polystyrene Melt in Supercritical Nitrogen", Ind. Eng. Chem. Res., 46 (11),

3849-3851 (2007).

[26] Park, H., Park, C. B., Tzoganakis, C., Tan, K. H., Chen, P., "Surface Tension Measurement

of Polystyrene Melts in Supercritical Carbon Dioxide", Industrial and Engineering

Chemistry Research, 45 (5), 1650-1658 (2006).

[27] Park, H., Thompson, R. B., Lanson, N., Tzoganakis, C., Park, C. B., Chen, P., "Effect of

Temperature and Pressure on Surface Tension of Polystyrene in Supercritical Carbon

Dioxide", Journal of Physical Chemistry B, 111, 3859-3868 (2007).

[28] Rodgers, P. A., "Pressure-volume-temperature Relationships for Polymeric :Liquids: A

Review of Equations of State and Their Characteristic Parameters for 56 Polymers",

Journal of Applied Polymer Science, 48 (6), 1061-1080 (1993).

[29] Sato, Y., Yamasaki, Y., Takishima, S., Masuoka, H., "Precise Measurement of the PVT of

Polypropylene and Polycarbonate up to 330oC and 200 MPa", Journal of Applied Polymer

Science, 66 (1), 141-150 (1997).


206

[30] Benedetto, A. T. D., "Molecular Properties of Amorphous High Polymers. I. A Cell Theory

for Amorphous High Polymers", J. Polym. Sci., 1 (11), 3459-3466 (1963).

[31] Dee, G. T., Walsh, D. J., "Equations of State for Polymer Liquids ", Macromolecules, 21

(3), 811-815 (1988).

[32] Dee, G. T., Walsh, D. J., "A Dodified Cell Model Equation of State for Polymer Liquids",

Macromolecules, 21, 815-817 (1988).

[33] Flory, P. J., Orwoll, R. A., Vrij, A., "Statistical Thermodynamics of Chain Molecule

Liquids. I. An Equation of State for Normal Paraffin Hydrocarbons", J. Am. Chem. Soc.,

86 (17), 3507-3514 (1964).

[34] Hartmann, B., Haque, M. A., "Equation of State for Polymer Liquids", J. Appl. Polym. Sci.,

30, 1553 (1985).

[35] Lennard-Jones, J. E., Devonshire, A. F., "Critical Phenomena in Gases-I", Proc. R. Soc.

London Ser. A. Mathematica and Physics Science, 163 (912), 53-71 (1937).

[36] Nies, E., Stroeks, A., "A Modified Hole Theory of Polymeric Fluids. 1. Equation of State

of Pure Components", Macromolecules, 23 (18), 4088-4092 (1990).

[37] Panayiotou, C., Vera, J. H., "An Improved Lattice-Fluid Equation of State for Pure

Component Polymeric Fluids", J. Polym. Eng. Sci., 22 (6), 345-348 (1982).

[38] Paul, D. R., Benedetto, A. T. D., J. Polym. Sci., C16, 1269 (1967).

[39] Prigogine, I., Bellemans, A., Mathot, V., "The Molecular Theory of Solutions",

North-Holland, Amsterdam (1957).

[40] Prigogine, I., Trappeniers, N., Mathot, V., "Statistical Thermodynamics of r-mers and r-mer

Solutions", Disc. Faraday Sci., 15, 93 (1953).

[41] Sanchez, I. C., Lacombe, R. H., "An Elementary Molecular Theory of Classical Fluids.

Pure Fluids", J. Phys. Chem., 80 (21), 2352-2362 (1976).


207

[42] Sanchez, I. C., Lacombe, R. H., J. Polym. Sci. Polym. Lett. Ed., 15, 71 (1977).

[43] Simha, R., "Configurational Thermodynamics of the Liquid and Glassy Polymeric States",

Macromolecules, 10 (5), 1025-1030 (1977).

[44] Simha, R., Somcynsky, T., "On the Statistical Thermodynamics of Spherical and Chain

Molecule Fluids", Macromolecules, 2 (4), 342-350 (1969).

[45] Spencer, R. S., Gilmore, G. D., "Equation of State for Polystyrene ", J. Appl. Phys., 20 (6),

502-506 (1949).

[46] Tait, P. G. in Phys. Chem., London (1888).

[47] Kwok, D. Y., Cheung, L. K., Park, C. B., Neumann, A. W., "Study on the Surface Tensions

of Polymer Melts Using Axisymmetic Drop Shape Analysis", Polymer Engineering and

Science, 38 (5), 757-764 (1998).

[48] Kwok, D. Y., Chiefalo, P., Khorshiddoust, B., S. Lahooti, Cabrerizo-Vilchez, M. A., Rio, O.

d., Neumann, A. W., In Book "Determination of Ultralow Interfacial Tension by

Axisymmetric Drop Shape Analysis", ACS Series, Washington, D.C. 24. (1995).

[49] Kwok, D. Y., Hui, W., Lin, R., Neumann, A. W., "Liquid-fluid Interfacial Tensions

Measured by Axisymmetric Drop Shape Analysis: Comparison between the Pattern of

Interfacial Tensions of Liquid-Liquid and Solid-Liquid Systems", Langmuir, 11,

2669-2673 (1995).

[50] Kwok, D. Y., Lin, R., Mui, M., Neumann, A. W., "Low Rate Dynamic and Static Contact

Angles and the Determination of Solid Surface Tensions", Colloids Surf. A: Physicochem.

Eng. Aspects, 116, 1851-1859 (1996).

[51] Kwok, D. Y., Tadros, B., Deol, H., Vollhardt, D., R. Miller, Cabrerizo-Vilchez, M. A.,

Neumann, A. W., "Axisymmetric Drop Shape Analysis as a Film Balance: Rate


208

Dependence of the Collapse Pressure and Molecular Area of Close Packing of

1-octadecanol", Langmuir, 12, 1851-1859 (1996).

[52] Lahooti, S., Rio, O. I. d., Cheng, P., Neumann, A. W., In Book "Axisymmetric Drop Shape

Analysis (ADSA)", Marcel Dekker, New York, 10. (1996).

[53] Li, J., Miller, R., Wüstneck, R., Möhwald, H., Neumann, A. W., "Use of Pendent Drop

Technique as a Film Balance at Liquid/liquid Interfaces", Colloids Surf. A :

Physicochemical and Engineering Aspects, 96 (3), 295-299 (1995).

[54] Rotenberg, Y., Boruvka, L., Neumann, A. W., "Determination of Surface Tension and

Contact Angle from the Shapes of Axisymmetric Fluid Interfaces", J. Colloid Interface

Sci., 93 (1), 169-183 (1983).

[55] Liu, D., Li, H., Noon, M. S., Tomasko, D. L., "CO2-induced PMMA Swelling and Multiple

Thermodynamic Property Analysis Using Sanchez-Lacombe EOS", Macromolecules, 38

(10), 4416-4424 (2005).

[56] Chen, P., Li, D., Voruvka, L., Rotenberg, Y., Newmann, A. W., "Automation of

Axisymmetric Drop Shape Analysis for Measurements of Interfacial Tensions and Contact

Angles", Colloids and Surfaces, 43 (2), 151-167 (1990).

[57] Wulf, M., Michel, S., Grundke, K., Rio, O. I. d., Kwok, D. Y., Neumann, A. W.,

"Simultaneous Determination of Surface Tension and Density of Polymer Melts Using

Axisymmetric Drop Shape Analysis1", J. of Colloid and Interface Science, 210 (1),

172–181 (1999).

[58] Li, H. B., Li, G., Leung, S., Park, C. B. in SPE, ANTEC (2005).

[59] Sato, Y., Yurugi, M., Fujiwara, K., Takishima, S., Masuoka, H., "Solubilities of Carbon

Dioxide and Nitrogen in Polystyrene under High Temperature and Pressure", Fluid Phase

Equilibria, 125 (1-2), 129-138 (1996).


209

[60] Li, G., Li, H. B., Turng, L. S., Gong, S., Zhang, C., "Measurement of Gas Solubility and

Diffusivity in Polylactide", Fluid Phase Equilibria, 246 (1-2), 158-166 (2006).

[61] Li, H. B., Lee, J., Tomasko, D. L., "Effect of Carbon Dioxide on the Interfacial Tension of

Polymer Melts", Ind. Eng. Chem. Res., 43 (2), 509-514 (2004).

[62] Sato, Y., Takikawa, T., Sorakubo, A., Takishima, S., Masuoka, H., Imaizumi, M.,

"Solubility and Diffusion Coefficient of Carbon Dioxide in Biodegradable Polymers",

Industrial and Engineering Chemistry Research, 39 (12), 4813-4819 (2000).

[63] Sato, Y., Takikawa, T., Takishima, S., Masuoka, H., J. Supercrit. Fluids, 19, 187–198

(2001).

[64] Sato, Y., Takikawa, T., Yamane, M., Takishima, S., Masuoka, H., "Solubility of Carbon

Dioxide in PPO and PPO/PS Blends", Fluid Phase Equilibria, 194-197, 847-858 (2002).

[65] Wong, B., Zhang, Z., Handa, Y. P., "High-Precision Gravimetric Technique for

Determining the Solubility and Diffusivity of Gases in Polymers", J. Polym. Sci. Part B:

Polym. Phys., 36 (12), 2025–2032 (1998).

[66] Bonavoglia, B., Storti, G., Morbidelli, M., Rajendran, A., Mazzotti, M., "Sorption and

Swelling of Semicrystalline Polymers in Supercritical CO2", J Polym Sci Part B: Polymer

Phys 44 (11), 1531-1546 (2006).

[67] Buckley, D. J., Berger, M., Poller, D., "The Swelling of Polymer Systems in Solvents. I.

Method for Obtaining Complete Swelling-Time Curves", J. Polym. Sci., 56 (163),

163-174 (1962).

[68] Ender, D. H., "Elastomeric Seals", Chemtech, 16, 52-56 (1986).

[69] Foster, G. N., Waldman, N., Grisly, R., "Pressure-volume-temperature Behavior of

Polypropylene Polymer Engineering & Science", Polym. Eng. Sci., 6 (2), 131-134 (1996).


210

[70] Nikitin, L. N., Gallyamov, M. O., Vinokur, R. A., Nikolaec, A. Y., Said-Galiyev, E. E.,

Khokhlov, A. R., Jespersen, H. T., Schaumburg, K., "Swelling and Impregnation of

Polystyrene Using Supercritical Carbon Dioxide", J. Supercritical Fluids, 26 (3), 263-273

(2003).

[71] Park, C. B., Park, S. S., Ladin, D., Naguib, H. E., "On-line Measurement of the PVT

Properties of Polymer Melts Using a Gear Pump", Advances in Polymer Technology, 23

(4), 316-327 (2004).

[72] Park, S. S., Park, C. B., Ladin, D., Naguib, H. E., Tzoganakis, C., J. Manuf. Sci. Eng., 124,

86 (2002).

[73] Rajendran, A., Bonavoglia, B., Forrer, N., Storti, G., Mazzotti, M., Morbidelli, M.,

"Simultaneous Measurement of Swelling and Sorption in a Supercritical CO2-Poly(methyl

methacrylate) System", Ind. Eng. Chem. Res., 44 (8), 2549-2560 (2005).

[74] Royer, J. R., DeSimone, J. M., Khan, S. A., "Carbon Dioxide-Induced Swelling of

Poly(dimethylsiloxane)", Macromolecules, 32 (26), 8965-8973 (1999).

[75] Wissinger, R. G., Paulaitis, M. E., "Swelling and Sorption in Polymer-CO2 Mixtures at

Elevated Pressures", Journal of Polymer Science, Part B (Polymer Physics), 25 (12),

2497-2510 (1987).

[76] Zhang, Y., Gangwani, K. K., Lemert, R. M., "Sorption and Swelling of Block Copolymers

in the Presence of Supercritical Fluid Carbon Dioxide", J Supercritical Fluids, 11 (1-2),

115-134 (1997).

[77] Keller, J. U., Rave, H., Staudt, R., "Measurement of Gas Absorption in a Swelling

Polymeric Material by a Combined Gravimetric-Dynamic Method", Macromol Chem

Phys 200 (10), 2269-2275 (1999).


211

[78] Hilic, S., Boyer, S. A. E., Padua, A. A. H., Grolier, J. P. E., "Simultaneous Measurement of

the Solubility of Nitrogen and Carbon Dioxide in Polystyrene and of the Associated

Polymer Swelling", J Polym Sci Part B: Polym Phys, 39, 2063 (2001).

[79] Park, S. S., Park, C. B., Ladin, D., Naguib, H. E., Tzoganakis, C., "Development of a

Dilatometer for Measurement of the PVT Properties of a Polymer/CO2 Solution Using a

Foaming Extruder and a Gear Pump", Journal of Manufacturing Science and Engineering,

Transactions of the ASME, 124 (1), 86-91 (2002).

[80] Hirose, T., Mizoguchi, K., Kamiya, Y., "Dilation of Polyethylene by Sorption of Carbon

Dioxide", Journal of Polymer Science Part B: Polymer Physics, 24 (9), 2107-2115 (1986).

[81] Gotthardt, P., Gruger, A., Brion, H. G., Plaetschke, R., Kirchheim, R., "Volume Change of

Glassy Polymers by Sorption of Small Molecules and Its Relation to the Intermolecular

Space", Macromolecules, 30, 8058-8065 (1997).

[82] Kamiya, Y., Naito, Y., Terada, K., Mizoguchi, K., "Volumetric Properties and Interaction

Parameters of Dissolved Gases in Poly(dimethylsiloxane) and Polyethylene",

Macromolecules, 33, 3111-3119 (2000).

[83] Shenoy, S., Woerdeman, D., Sebra, R., Garach-Domech, A., Wynne, K. J., "Quantifying

Polymer Swelling Employing a Linear Variable Differential Transformer: CO2 Effects on

SBS Triblock Copolymer", Macromol. Rapid Commun., 23, 1130–1133 ( 2002).

[84] Wind, J. D., Sirard, S. M., Paul, D. R., Green, P. F., Johnston, K. P., Koros, W. J., "Carbon

Dioxide-Induced Plasticization of Polyimide Membranes: Pseudo-Equilibrium

Relationships of Diffusion, Sorption, and Swelling", Macromolecules, 36 (17), 6433-6441

(2003).


212

[85] Holck, O., Siegert, M. R., Heuchel, M., Bohning, M., "CO2 Sorption Induced Dilation in

Polysulfone: Comparative Analysis of Experimental and Molecular Modeling Results",

Macromolecules, 39 (26), 9590-9604 (2006).

[86] Funami, E., Taki, K., Ohshima, M., "Density Measurement of Polymer/CO2 Single-phase

Solution at High Temperature and Pressure Using a Gravimetric Method", Journal of

Applied Polymer Science, 105 (5), 3060-3068 (2007).

[87] Park, H.,"Surface Tension Measurement of Polystyrenes in Supercritical Fluids",

University of Waterloo, Waterloo, Ph.D. Thesis (2007).

[88] Park, H., Park, C. B., Tzoganakis, C., Tan, K.-H., Chen., P., "Simultaneous Determination

of the Surface Tension and Density of Polystyrene in Supercritical Nitrogen", Ind. & Eng.

Chem. Res., 47 (13)2008).

[89] Frisch, H. L., "Diffusion of Small Molecules in Polymers", CRC Critical Reviews in Solid

States and Material Science, 11 (2)1983).

[90] Brandt, W. W., "Model Calculation of the Temperature Dependence of Small Molecule

Diffusion in High Polymers", J. Phys. Chem., 63 (7), 1080-1085 (1959).

[91] DiBenedetto, A. T., "Molecular Properties of Amorphous High Polymers, 1. A Cell Theory

for Amorphous High Polymers", J. Polym. Sci. Part A: General Papers, 1 (11), 3459 -

3476 (1963).

[92] Pace, R. J., Datyner, A., "Statistical Mechanical Model for Diffusion of Simple Pentrants in

Polymers. I. Theory", J. Polym. Sci., Polym. Phys. Ed., 17 (3), 437-451 (1979).

[93] Cohen, M. H., Turnbull, D., "Free-Volume Model of the Amorphous Glass Transition", J.

Chem. Phys., 34 (1), 120-125 (1961).

[94] Fujita, H., "Organic vapors above the glass transition temperature, in :diffusion in polymer",

Acad. Press., New York (1968).


213

[95] Vrentas, J. S., Duda, J. L., "Diffusion in Polymer-Solvent Systems, 1. Reexamination of the

Free-Volume Theory", J. Polym. Phys., 15 (3), 403 - 416 (1977).

[96] Krevelen, D. W. V., "Properties of Polymers", Elsevier Scientific Publishers Company,

(1990).

[97] Crank, J., Park, G. S., "Diffusion in Polvmers", Academic Press Inc.,New York (1968).

[98] Park, C. B., Suh, N. P., "Filamentary Extrusion of Microcellular Polymers Using a Rapid

Decompressive Element", Polym. Eng. Sci., 36, 34 (1996).

[99] Durrill, P. L., Griskey, R. G., "Diffusion and Solution of Gases in Thermally Softened or

Molten Polymers: Part I. Development of Technique and Determination of Data", AICHE

J., 12 (6), 1147-1151 (1966).

[100] Lundberg, J. L., Mooney, E. J., Rogers, C. E., Journal of Polymer Science, Polymer

Physics Edition, 7, 947-962 (1969).

[101] Lundberg, J. L., Wilk, M. B., Huyett, M. J., "Solubilties and Diffusivities of N2 in PE", J.

Appl. Phys., 31 (6), 1131-1132 (1960).

[102] Newitt, D. M., Weale, K. E., "Solution and Diffusion of Gases in Polystyrene at High

Pressures", Journal of the Chemical Society, 1541-1549 (1948).

[103] Bonner, D. C., "Solubility of Supercritical Gases in Polymers-A Review", Polym Eng Sci,

17 (2), 65-72 (1977).

[104] Bonner, D. C., Cheng, Y., "A New Method for Determination of Equilibrium Sorption of

Gases by Polymers at Elevated Temperatures and Pressures", J .Polym. Sci. Polym. Lett.

Ed., 13 (2), 259 (1975).

[105] Cheng, Y. L., Bonner, D. C., "Solubility of Ethylene in Liquid, Low-Density Polyethylene

to 69 Atmospheres", Journal of Polymer Science, Polymer Physics Edition, 15 (4),

593-603 (1977).


214

[106] Cheng, Y. L., Bonner, D. C., "Solubility of Nitrogen and Ethylene in Molten,

Low-Density Polyethylene to 69 Atmospheres", Journal of Polymer Science, Polymer

Physics Edition, 16 (2), 319-333 (1978).

[107] Liu, D. D., Prausnitz, J. M., "Solubilities of Gases and Volatile Liquids in Polyethylene

and in Ethylene-Vinyl Acetate Copolymers in the Region 125-225o C", Industrial &

Engineering Chemistry Fundamentals, 15 (4), 330-335 (1976).

[108] Maloney, D. P., Prausnitz, J. M., "Solubilities of Ethylene and Other Organic Solutes in

Liquid, Low-Density Polyethylene in the Region 124o to 300oC", AICHE J., 22 (1), 74-82

(1976).

[109] Maloney, D. P., Prausnitz, J. M., "Solubility of Ethylene in Liquid, Low-Density

Polyethylene at Industrial-Separation Pressures", Industrial & Engineering Chemistry

Process Design and Development 15 (1), 216-220 (1976).

[110] Baird, B. R., Hopfenberg, H. B., Stannett, V. T., "The Effect of Molecular Weight and

Orientation on the Sorption of nPentane by Glassy Polystyrene", Polym. Eng. Sci. , 11 (4),

274-283 (1971).

[111] Berens, A. R., "The Solubility of Vinyl Chloride in Poly(Viny1 Chloride)", Angewandte

Makromolekulare Chemie, 47 (1), 97-110 (1975).

[112] Berens, A. R., "Gravimetric and Volumetric Study of the Sorption of Gases and Vapors in

Poly(Viny1 Chloride) Powders", J. Polym. Eng. Sci., 20 (1), 95-101 (1980).

[113] Fechter, J. M. H., Hopfenberg, H. B., Koros, W. J., "Characterization of Glassy State

Relaxations by Low Pressure Carbon Dioxide Sorption in Poly( MethylMethacrylate)", J.

Polym. Eng. Sci., 21 (14), 925-929 (1981).

[114] Kamiya, Y., Hirose, T., Mizoguchi, K., Naito, Y., "Gas sorption in Poly(vinyl benzoate)",

J. Polym. Sci.,: Part B 24 (7), 1525-1539 (1986).


215

[115] Kamiya, Y., Mizoguchi, K., Naito, Y., Hirose, T., J. Polym. Sci. Polym. Phys. Ed., 24,

535-547 (1986).

[116] Lundberg, J. L., Wilk, M. B., Huyett, M. J., "Estimation of Diffusivities and Solubilities

from Sorption Studies", Journal of Polymer Science 57 (165), 275-299 (1962).

[117] Areerat, S., Funami, E., Hayata, Y., Nakagawa, D., Ohshima, M., "Measurement and

Prediction of Diffusion Coefficients of Supercritical CO2 in Molten Polymers", Polymer

Engineering and Science, 44 (10), 1915-1924 (2004).

[118] Sato, Y., Fujiwara, K., Takikawa, T., Sumarno, Takishima, S., Masuoka, H., "Solubilities

and Diffusion Coefficients of Carbon Dioxide and Nitrogen in Polypropylene,

High-density Polyethylene, and Polystyrene under High Pressures and Temperatures",

Fluid Phase Equilibria, 162 (1-2), 261-276 (1999).

[119] Bonner, D. C., Prausnitz, J. M., "Thermodynamic Properties of Some Concentrated

Polymer Solutions", Journal of Polymer Science, Polymer Physics Edition, 12 (1), 51-73

(1974).

[120] Kleinrahm, R., W.Wagner, "Measurement and Correlation of the Equilibrium Liquid and

Vapor Densities and the Vapor-Pressure Along the Coexistence Curve of Methane", J.

Chem. Thermodyn., 18, 739–760 (1986).

[121] Sato, Y., Takikawa, T., Takishima, S., Masuoka, H., "Solubilities and Diffusion

Coefficients of Carbon Dioxide in Poly(vinyl acetate) and Polystyrene", J. Supercrit.

Fluids, 19 (2), 187–198 (2000).

[122] Areerat, S., Hayata, Y., Katsumoto, R., Kegasawa, T., Egami, H., Ohshima, M.,

"Solubility of Carbon Dioxide in Polyethylene/titanium Dioxide Composite under High

Pressure and Temperature", Journal of Applied Polymer Science, 86 (2), 282-288 (2002).


216

[123] Areerat, S., Nagata, T., Ohsima, M., "Measurement and Prediction of LDPE/CO2 Solution

Viscosity", Polymer Engineering and Science, 42 (11), 2234-2245 (2002).

[124] Li, G., Wang, J., Park, C. B., Simha, R., "Measurement of Gas Solubility in

Linear/Branched PP Melts", Journal of Polymer Science Part B-Polymer Physics, 45 (17),

2497-2508 (2007).

[125] Li, G., Gunkel, F., Wang, J., Park, C. B., Altstädt, V., "Solubility Measurements of N2

and CO2 in Polypropylene and Ethene/Octene Copolymer", Journal of Applied Polymer

Science, 103 (5), 2945-2953 (2007).

[126] Li, G., Li, H., Wang, J., Park, C. B., "Investigating the Solubility of CO2 in

Polypropylene Using Various EOS Models", Cellular Polymers, 25 (4), 237-248 (2006).

[127] Condo, P. D., Paul, D. R., Johnston, K. P., "Glass Transitions of Polymers with

Compressed Fluid Diluents: Type I1 and I11 Behavior", Macromolecules, 27 (2), 365-371

(1994).

[128] Gehardt, L. J., Manke, C. W., Gulari, E., "Rheology of Polydimethylsiloxane Swollen

With Supercritical Carbon Dioxide", J Polym Sci Part B: Polym Phys, 35, 523 (1997).

[129] Handa, Y. P., Kruus, P., O'Neill, M., "High-pressure calorimetric study of plasticization of

poly(methyl methacrylate) by methane, ethylene, and carbon dioxide", J Polym Sci Part B:

Polym Phys. Chem., 34, 2635-2639 (1996).

[130] Kwang, C., Manke, C. W., Gulari, E., "Rheology of Molten Polystyrene with Dissolved

Supercritical and Near-Critical Gases", J. Polym. Sci., Part B: Polym. Phys. Chem., 37,

2771 (1999).

[131] Tomasko, D. L., Li, H. B., Liu, D. H., Han, X. M., Wingert, M. J., Lee, L. J., Koelling, K.

W., "A Review of CO2 Applications in the Processing of Polymers", Ind. Eng. Chem. Res.,

42 (25), 6431-6456 (2003).


217

[132] Lee, M., Park, C. B., Tzoganakis, C., "Measurements and Modeling of PS/Supercritical

CO2 Solution Viscosities", J. Polym. Eng. Sci., 39 (1), 99-109 (1999).

[133] Ward, C. A., Levart, E., "Conditions for Stability of Bubble Nuclei in Solid Surfaces

Contacting a Liquid-Gas Solution", J. Appl. Phys., 56 (2), 491-501 (1984).

[134] Blander, M., Katz, J. L., "Bubble Nucleation in Liquids", AlchE J., 21 (5), 833-848

(1975).

[135] Gibbs, J. W., "Thermodynamics", Dover, New York (1961).

[136] Colton, Suh, "The Nucleation of Microcellular Thermoplastic Foam With Additives: Part

I: Theoretical Considerations", J. Polym. Eng. Sci., 27 (7), 485-492 (1987).

[137] Wu, S., "Polymer Interface and Adhesion", Marcel Dekker, New York (1982).

[138] Whilhelmy, L., Ann. Phys., 119, 177 (1863).

[139] Adamson, A. W., "Physical Chemistry of Surfaces", John Wiley & Sons Inc.,New York

(1990).

[140] Drelich, J., Fang, C., White, C. L., In Book "Measurement of Interfacial Tension in

Fluid-Fluid Systems", Marcel Dekker, Inc. (2002).

[141] Hartland, S., "Surface and Interfacial Tension: Measurement, Theory, and Applications",

Marcel Dekker, (2004).

[142] Rusanov, A. I., Prokhorov, V. A., "Interfacial Tensiometry", Elsevier, Amsterdam

(1996).

[143] Spelt, J. K., Vargha-Butler, E. I., In Book "Contact Angle and Liquid Surface Tension

Measurements: General Procedure and Techniques", Marcel Deklter Inc., New York,

Chapter 8. (1996).


218

[144] Elmendorp, J. J., Vos, G. D., "Measurement of Interfacial Tensions of Molten Polymer

Systems by Means of the Spinning Drop Method", Polymer Engineering and Science, 26,

415-417 (1986).

[145] Joseph, D. D., et al., "A Spinning Drop Tensioextensometer", J. Rheol., 36 (4), 621-662

(1992).

[146] Petersen, R. C. M., W., D., Smith, R. D., "The Formation of Polymer Fibers from the

Rapid Expansion of Supercritical Fluid Solutions", Polymer Engineering and Science, 27

(22), 1693-1697 (1987).

[147] Anastasiadis, S. H., et al., "The Determination of Polymer Interfacial Tension by Drop

Image Processing: Comparison of Theory and Experiment for the Pair, Poly(dimethyl

siloxane)/polybutadiene", Polymer Engineering and Science, 26 (20), 1410-1418 (1986).

[148] Anastasiadis, S. H., et al., "The Determination of Interfacial Tension by Video Image

Processing of Pendant Fluid Drops", Colloid and Interface Science, 119 (1), 55-66 (1987).

[149] Anastasiadis, S. H. G., Irena, K., Jeffrey, T., "Interfacial Tension of Immiscible Polymer

Blends: Temperature and Molecular Weight Dependence", Macromolecules, 21,

2980-2987 (1988).

[150] Arashiro, E. Y., Demarquette, N. R., "Influence of Temperature, Molecular Weight, and

Polydispersity of Polystyrene on Interfacial Tension Between Low-Density Polyethylene

and Polystyrene", Journal of Applied Polymer Science, 74, 2423-2431 (1999).

[151] Demarqutte, N. R., Kamal, M. R., "Interfacial Tension in Polymer Melts. I: An Improved

Pendant Drop Appratus", Polymer Engineering and Science, 34 (24), 1823-1833 (1994).

[152] Kamal, M. R., Lai-Fook, R., Demarquette, N. R., "Interfacial in Polymer Melts Part II:

Effects of Temperature and Molecular Weight on Interfacial Tension", Polymer

Engineering and Science, 34 (24), 1834-1839 (1994).


219

[153] Nose, T., et al., "Interfacial Tension of Demixed Polymer Solutions Near Critical Solution

Temperature", Contemporary Topics in Polymer Science, 4, 789-805 (1984).

[154] Roe, R.-J., "Interfacial Tension between Polymer Liquids", Journal of Colloid and

Interface Science, 31 (2), 228-235 (1969).

[155] Runke, T., Song, B., Springer, J., "Surface- and Interfacial Tensions of Liquid Crystalline

Polymers", Berichte der Bunsen-Gesellschaft, 98 (3), 508-511 (1994).

[156] Shinozaki, K., et al., "Interfacial Tension of Demixed Polymer Solutions Near the Critical

Temperature: Polystyrene + Methylcyclohexane", Polymer, 23 (5), 728-734 (1982).

[157] Shinozaki, K., Abe, M., Nose, T., "Interfacial Tension of Demixed Polymer Solutions

Over a Wide Range of Reduced Temperature", Polymer, 23 (5), 722-727 (1982).

[158] Shinozaki, K., Saito, Y., Nose, T., "Interfacial Tension Between Demixed Symmetrical

Polymer Solutions: Polystyrene-poly(dimethylsiloxane)- Propylbenzene System", Polymer,

23 (13), 1937-1943 (1982).

[159] Wu, S., "Surface and Interfacial Tensions of Polymer Melts. I. Polyethylene,

Polyisobutylene, and Poly(vinyl acetate)", Journal of Colloid and Interface Science, 31 (2),

153-161 (1969).

[160] Wu, S., "Surface and interfacial tensions of polymer melts. II. Poly(methyl methacrylate),

poly(n-butyl methacrylate), and polystyrene", Journal of Physical Chemistry, 74 (3),

632-638 (1970).

[161] Neumann, A. W., Good, R. J., "Techniques of Measuring Contact Angles", Plenum Press,

New York (1979).

[162] Furlong, D. N., Hartland, S., "Wall Effect in the Determination of Surface Tension using a

Wilhelmy Plate", J. Colloid Interface Sci., 71 (2), 301-315 (1979).


220

[163] Boucher, E. A., Evans, M. J. B.,"Pendent Drop Profiles and Related Capillary

Phenomena",Series A, Mathematical and Physical Sciences, Proceedings of the Royal

Society of London, London, 349-374 (1975).

[164] Boucher, E. A., Kent, H. J., "Capillary Phenomena Equilibrium and Stability of Pendent

Drops", J. Colloid Interface Science, 67 (1), 10-15 (1978).

[165] Tate, T., Phil. Mag., 27, 176 (1864).

[166] Harkins, W. D., Brown, F. E., "The Determination of Surface Tension (Free Surface

Energy), and the Weight of Falling Drop: the Surface Tension of Water and Benzene by

the Capiliary Height Method", J. Am. Chem. Soc., 41 (4), 499-524 (1919).

[167] Dukhin, S. S., Kretzschmar, G., Miller, R., "Dynamics of Adsorption at Liquid Interfaces:

Theory, Experiment, Application", Elsevier, Amsterdam (1995).

[168] Bohr, N., "Determination of the Surface-Tension of Water by the Method of Jet

Vibration", Phil. Trans. Roy. Soc. Ser. A, 209, 281-317 (1909).

[169] Kelvin, L., "Hydrokinetic Solutions and Observations", Philos. Mag., 42, 362-377 (1871).

[170] Hansen, R. S., Ahmad, J., "Progress in Surface and Membrane Science", Academic Press,

New York (1971).

[171] Lucassen-Reynders, E. H., Lucassen, J., "Properties of Capillary Waves", Adv. Colloid

Interface Sci., 2, 347 (1969).

[172] Mann, J. A., " Plenum Press, New York (1984).

[173] Nosltov, B. A., Kochurova., N. N., " Leningrad State Univ.,Leningrad (1985).

[174] Vonnegut, B., Rev. Sci. Inst., 13 1942).

[175] Bashforth, F., Adams, J. C., "An Attempt to Test the Theory of Capillary Action",

Cambridge (1892).


221

[176] Rotenberg, Y., "The Determination of the Shape of Non-Axisyrnmetric Drops and the

Calculation of Surface Tension, Contact Angle, Surface Area and Volume of

Axisymmetric Drops", University of Toronto, Toronto, Ph.D. Thesis (1983).

[177] Cheng, P., "Automation of Axisyrnmetric Drop Shape Analysis Using Digital Image

Processing", University of Toronto, Toronto, Ph.D. Thesis (1990).

[178] Rio, O. I. d.,"On the Generalization of Axisymmetric Drop Shape Analysis", University of

Toronto, Toronto, M.A.Sc. Thesis (1993).

[179] Rio, O. I. d., Neumann, A. W., "Axisymmetric Drop Shape Analysis: Computational

Methods for the Measurement of Interfacial Properties from the Shape and Dimensions of

Pendant and Sessile Drops", J. Colloid Interface Sci., 196 (2), 136-147 (1997).

[180] Hoorfar, M.,"Development of a Third Generation of Axisymmetric Drop Shape Analysis

(ADSA)", University of Toronto, Toronto, Ph.D. Thesis (2006).

[181] Park, H., Park, C. B., Tzoganakis, C., Chen, P., Ind. Eng. Chem. Res., 46, 3849-3851

(2007).

[182] Tomasi, C., Kanade, T., "Shape and Motion from Image Streams under Orthography: A

Factorization Method", Int. J. of Computer Vision, 9 (2), 137–154 (1992).

[183] Kanade, T., Narayanan, P. J., Rander., P. W., "Virtualized Reality: Concepts and Early

Results", Proc. IEEE Workshop on Representation of Visual Scenes, 69–76 (1995).

[184] S. Moezzi, A. K., Kuramura, D. Y., Jain, R., "Reality Modeling and Visualization from

Multiple Video Sequences", IEEE Computer Graphics and Applications, 16 (6), 58–63

(1996).

[185] Beardsley, P., Torr, P., Zisserman, A., "3D Model Acquisition from Extended Image

Sequences", Proc. European Conf. on Computer Vision, 683–695 (1996).


222

[186] Robert, L., "Realistic Scene Models from Image Sequences", Proc. Imagina 97 Conf,

8–13 (1997).

[187] Mayer, R., "Scientific Canadian: Invention and Innovation From Canada's National

Research Council", Raincoast Books, Vancouver (1999).

[188] Zhang, S., Huang, P., "High-Resolution, Real-time 3-D Shape Measurement", Optical

Engineering, 45 (12), 123601-123609 (2006).

[189] Zheng, J. Y., "Acquiring 3-D Models from a Sequence of Contours", IEEE Trans. Pattern

Anal. Machine Intell, 16, 163–178 (1994).

[190] Kanade, T., Okutomi, M., Nakahara, T., "A Multiple-Baseline Method", Proc. DARPA

Image Understanding Workshop, 409–426 (1992).

[191] Kanade, T., Kang, S. B., Webb, J., Zitnick, C. L., "A Multibaseline Stereo System with

Active Illumination and Real-time Image Acquisition", Proc. Int. Conf. Computer Vision,

88–93 (1995).

[192] Kang, S. B., Webb, J. A., Zitnick, C. L., Kanade, T., "A Multibaseline Stereo System with

Active Illumination and Real-time Image Acquisition", Proc. Int. Conf. Computer Vision,

88–93 (1995).

[193] Esteban, C. H. a., Schmitt, F., "Multi-Stereo 3D Object Reconstruction", Proc. Int. Symp.

3-D Data Processing Visualization Transmission,, 159–166 (2002).

[194] Debevec, P. E., Taylor, C. J., Malik, J., "Modeling and Rendering Architecture from

Photographs: A Hybrid Geometry and Image-based Approach", Proc. Computer Graphics,

Annu. Conf. Series, 11–20 (1996).

[195] Seitz, S. M., Dyer, C. R., "Photorealistic Scene Reconstruction by Voxel Coloring", Int. J.

Comput. Vis., 35 (2), 151-173 (1999).


223

[196] K. N. Kutulakos and S. M. Seitz, , v., no. 3, pp. 1999–218, 2000., "A theory of shape by

space carving", Int. J. Comput. Vis., 38 (3), 199–218 (2000).

[197] Culbertson, W. B., Malzbender, T., Slabaugh, G., "Generalized Voxel Coloring", Proc. Int.

Conf. Computer Vision., 100–115 (1999).

[198] Niem, W., "Robust and Fast Modelling of 3D Natural Objects from Multiple Views",

SPIE Proceedings Image and Video Processing II, 2182, 388-397 (1994).

[199] Niem, W., "Automatic Reconstruction of 3D Objects Using a Mobile Camera", Image and

Vision Computing, 17 (2), 125-134 (1999).

[200] Niem, W., Buschmann, R.,"Automatic Modeling of 3-D Natural Objects From Multiple

Views", European Workshop on Combined Real and Synthetic Image Processing for

Broadcast and Video Production Hamburg, Germany, (1994).

[201] Matsumoto, Y., Terasaki, H., Sugimoto, K., Arakawa, T., "A Portable Three-Dimensional

Digitizer", Proc. Int. Conf. Recent Advances 3-D Digital Imaging Modeling, 7 (1997).

[202] Mülayim, A. Y., Yılmaz, U., Atalay, V., "Silhouette-Based 3-D Model Reconstruction

From Multiple Images", IEEE Transactions on Systems, Man and Cybernetics--Part B:

Cybernetics, 33 (4), 582-591 (2003).

[203] Lavest, J. M., Viala, M., Dhome, M., "Do We Really Need an Accurate Calibration

Pattern to Achieve a Reliable Camera Calibration?", Proc. Eur. Conf. Computer Vision,

158–174 (1998).

[204] Kutulakos, K. N., Seitz, S. M., "A Theory of Shape by Space Carving", Int. J. Comput.

Vis., 38 (3), 199–218 (2000).

[205] Matsumoto, Y., Fujimura, K., Kitamura, T., "Shape-from-silhouette/stereo and Its

Application to 3-D Digitizer", Proc. Discrete. Geometry Computer Imagery Conf.,

177–188 (1999).


224

[206] Zhang, L., Seitz, S. M., "Image-Based Multiresolution Modeling by Surface Deformation",

Robotics Institute Carnegie Mellon University, CMU-RI-TR-00-07 (2000).

[207] Gibson, D. P., Campbell, N. W., Thomas, B. T., "The Generation of 3-D Models Without

Camera Calibration", Proc. Computer Graphics Interactive Techniques, 146–149 (1998).

[208] Fitzgibbon, A. W., Cross, G., Zisserman, A., "Automatic 3-D Model Construction for

Turn-table Sequences", Proc. 3D Structure from Multiple Images of Large-Scale

Environments, European Workshop SMILE '98, 155-170 (1998).

[209] Mendonça, P. R. S., Wong, K.-Y. K., Cipolla, R., "Camera Pose Estimation and

Reconstruction from Image Profiles Under Circular Motion", Proc. Eur. Conf. Computer

Vision, 864–877 (2000).

[210] Lorensen, W. E., Cline, H. E., "Marching Cubes: A high Resolution 3-D Surface

Reconstruction Algorithm", ACM Comput. Graph., 21 (4), 163-169 (1987).

[211] Montani, C., Scateni, R., Scopigno, R., "Discretized Marching Cubes", Visualization,

281-287 (1994).

[212] Weinhaus, F. M., Devarajan, V., "Texture Mapping 3-D Models of Realworld Scenes",

ACM Comput. Surv., 29 (4), 325–365 (1997).

[213] Max, N., "Hierarchical Rendering of Trees from Precomputed Multi-layer Z-buffers",

Proc. Eurographics Rendering Workshop, 165–174 (1996).

[214] Shade, J., Gortler, S., He, L., Szelisky, R.,"Layered Depth Images", Proc. 25th Int. Conf.

Computer Graphics Interactive Techniques, 231–242 (1998).

[215] Eisert, P., Steinbach, E., Girod, B.,"Multi-hypothesis, Volumetric Reconstruction of 3-D

Objects from Multiple Calibrated Camera Views", Proc. Int. Conf. Acoustics Speech

Signal Processing, 3509–3512 (1999).


225

[216] Faugeras, O. D.,"What Can Be Seen in Three Dimensions with an Uncalibrated Stereo

Rig?", Proc. Eur. Conf. Computer Vision, 563–578 (1992).

[217] Hartley, R. I.,"Estimation of Relative Camera Positions for Uncalibrated Cameras", Proc.

Eur. Conf. Computer Vision, 579–587 (1992).

[218] Luong, Q. T., Viéville, T., "Canonical Representations for the Geometries of Multiple

Projective Views", Comput. Vis. Image Understand., 64 (2), 193–229 (1996).

[219] Triggs, B., "Linear Projective Reconstruction from Matching Tensors", Image Vis.

Comput., 15 (8), 617–625 (1997).

[220] Pollefeys, M., Koch, R., Gool, L. V., "Self-Calibration and Metric Reconstruction in Spite

of Varying and Unknown Internal Camera Parameters", Int. J. Comput. Vis., 32 (1), 7-25

(1999).

[221] Z. Zhang, R. D., Faugeras, O., Luong, Q. T., "A Robust Technique for Matching Two

Uncalibrated Images Through the Recovery of the Unknown Epipolar Geometry", J. Artif.

Intell., 78, 87-119 (1995).

[222] Koch, R., Pollefeys, M., Gool, L. V.,"Multi Viewpoint Stereo from Uncalibrated Video

Sequences", Proc. Eur. Conf. Computer Vision, 55-71 (1998).

[223] Hartley, R. I., Zisserman, A., "Multiple View Geometry in Computer Vision", Cambridge

Univ. Press,Cambridge, UK (2000).

[224] Maybank, S. J., Faugeras, O. D., "A Theory of Self-Calibration of a Moving Camera", Int.

J. Comput. Vis., 8 (2), 123-151 (1992).

[225] Fusiello, A., "Uncalibrated Euclidean Reconstruction: A Review", Image Vis. Comput., 18

(67), 555-563 (2000).

[226] Moore, W., "Physical Chemistry", Prentice-Hall, Inc. Englewood Cliffs New Jersey

(1972).


226

[227] Alberty, R. A., "Physical Chemistry", John Wiley & Sons,New York (1987).

[228] Wang, W., Liu, X., Zhong, C., Twu, C. H., Coon, J. E., "Simplified Hole Theory Equation

of State for Liquid Polymers and Solvents and Their Solutions", Ind. Eng. Chem. Res., 36

(6), 2390-2398 (1997).

[229] Twu, C. H., Bluck, D., Cunningham, J. R., Coon, J. E., "A Cubic Equation of State with a

New Alpha Function and a New Mixing Rule", Fluid Phase Equilibia, 69, 33-50 (1991).

[230] Peng, D. Y., Robinson, D. B., "A New Two-Constant Equation of State", Ind. Eng. Chem.

Fundam., 15 (1976).

[231] Soave, G., "Equilibrium Constant from a Modified Redlich-Kwong Equation of State",

Journal of Chemical Engineering Science, 27 (6), 1197-1203 (1972).

[232] Wei, Y. S., Sadus, R. J., "Equations of State for the Calculation of Fluid-Phase Equilibria",

AlchE J., 46 (1), 169-196 (2000).

[233] Flory, P. J., "Thermodynamics of High-Polymer Solutions", J. Chem. Phys., 10, 51

(1942).

[234] Flory, P. J., Orwoll, R. A., Vrij, A., "Statistical Thermodynamics of Chain Molecule

Liquids. II. Liquid Mixtures of Normal Paraffin Hydrocarbons", J. Am. Chem. Soc., 86

(17), 3515-3520 (1964).

[235] Simha, R.,"Transitions, Relaxations, and Thermodynamics in the Glassy State",Polym.

Eng. Sci. (USA), Miami Beach, FL, USA, 82-86 (1980).

[236] Simha, R., Jain, R. K., "Statistical Thermodyanmics of Blends : Equation of State and

Phase Relations", Journal of Polymer Engineering and Science, 24 (17), 1284-1290

(1984).

[237] Jain, R. K., Simha, R., "Statistical Thermodynamics of Multicomponent Fluids. 2.

Equation of State and Phase Relations", Macromolecules, 17, 2663-2668 (1984).


227

[238] Sanchez, I. C., Cho, J., "A Universal Equation of State for Polymer Liquids", Polymer, 36

(15), 2929-2939 (1995).

[239] Sanchez, I. C., Lacombe, R. H., "Statistical Thermodynamics of Polymer Solutions",

Macromolecules, 11 (6), 1145-1156 (1978).

[240] Sanchez, I. C., Rodgers, P. A., "Solubility of Gases in Polymers", Pure & Appl. Chem., 62

(11), 2107-2114 (1990).

[241] Huggins, M. L., "Thermodynamic Properties of Solutions of Long-chain Compounds",

Annals of New York Academy Science 43 l(1942).

[242] Panayiotou, C., Vera, J. H., "Statistical Thermodynamics of r-mer Fluids and Their

Mixtures", Polym. J., 14 (9), 681-694 (1982).

[243] Doghieri, F., Sarti, G. C., "Nonequilibrium Lattice Fluids: A Predictive Model for the

Solubility in Glassy Polymers", Macromolecules, 29 (24), 7885 - 7896 (1996).

[244] Parekh, V. S., Danner, R. P., "Prediction of Polymer PVT Behaviour using the Group

Contribution Lattice-fluid EOS", Journal of Polymer Science, Part B: Polymer Physics,

33 (3), 395-402 (1995).

[245] Wertheim, M. S., "Fluids with Highly Directional Attractive Forces: I. Statistical

Thermodynamics", J. Stat. Phys. , 35 (1-2), 19-34 (1984).

[246] Wertheim, M. S., "Fluids with Highly Directional Attractive Forces: II. Thermodynamic

Perturbation Theory and Integral Equations", J. Stat. Phys., 35 (1-2), 35-47 (1984).

[247] Wertheim, M. S., "Fluids with Highly Directional Attractive Forces: III. Multiple

Attraction Sites", J. Stat. Phys., 42 (3-4), 459-476 (1986).

[248] Wertheim, M. S., "Fluids with Highly Directional Attractive Forces: IV. Equilibrium

Polymerization", J. Stat. Phys., 42 (3-4), 477-492 (1986).


228

[249] Chapman, W. G., Gubbins, K. E., Jackson, G., Radosz, M., "SAFT: Equation-of-State

Solution Model for Associating Fluids", Fluid Phase Equilibria, 52, 31-38 (1989).

[250] Chapman, W. G., Gubbins, K. G., Jackson, G., Radosz, M., "New Reference Equation of

State for Associating Liquids", Ind. Eng. Chem. Res., 29 (8), 1709-1721 (1990).

[251] Huang, S. H., Radosz, M., "Equation of State for Small, Large, Polydisperse, and

Associating Molecules", Ind. Eng. Chem. Res., 29, 2284 (1990).

[252] Huang, S. H., Radosz, M., "Equation of State for Small, Large, Polydisperse, and

Associating Molecules: Extension to Fluid Mixtures", Ind. Eng. Chem. Res., 30 (8),

1994-2005 (1991).

[253] Sadowski, G., "Thermodynamics of Polymer Systems", Macromolecular Symposia 206

(1), 333-346 (2004).

[254] Zoller, P., In Book "PVT Relationships and Equation of State of Polymers", Wiley, New

York. (1989).

[255] Wohl, A., "Investigations on the Equation of State. IV. The Compression Equation of the

Liquids. - Equation of State of strongly Consolidated Materials", Zeitschrift fur

Physikalische Chemie, 99, 234 (1921).

[256] Lin, Y.-H., "Polymer Viscoelasticity: Basics, Molecular Theories and Experiments",

World Scientific New Jersey (2003).

[257] Ferry, J. D., "Viscoelastic Properties of Polymers", Wiley,New York (1980).

[258] Graessley, W. W., "The Entanglement Concept in Polymer Rheology", Adv. Polym. Sci.,

16 (1974).

[259] Mark, H., Tobolsky, A. V., "Physical Chemistry of High Polymeric Systems",

Interscicence, New York (1950).


229

[260] Onogi, S., Masuda, T., Kitagawa, K., "Rheological Properties of Anionic Polystyrenes. I.

Dynamic Viscoelasticity of Narrow-Distribution Polystyrenes", Macromolecules, 3 (2)

(1970).

[261] Jennings, J. W., Pallas, N. R., "An Efficient Method for the Determination of Interfacial

Tension from Drop Profiles", Langmuir, 4 (4), 960-967 (1988).

[262] Suh, N. P., "The Principle of Design", Oxford, New York (1990).

[263] Canny, J., "A Computational Approach to Edge Detection", IEEE Transactions on pattern

Analysis and Machine Intelligence, 8 (6), 679-698 (1986).

[264] Davis, L. S., "A Survey of Edge Detection Techniques", Computer Graphics and Image

Processing, 4, 248-270 (1975).

[265] Green, W. B., "Digital Image Processing", Van Nostrand Reinhold, New York (1983).

[266] Pratt, W. K., "Digital Image Processing", Wiley-Interscience, New York (1978).

[267] Rosenfeld, A., Kak, A. C., "Digital Image Processing", Academic Press, New York

(1982).

[268] Guo, K. H., Uemura, T., Yang, W., "Reflection-Interference Method to Determine Droplet

Profiles", Applied Optics, 24 (16), 2655-2659 (1985).

[269] Thomas, L. P., Gratton, R., Marino, B. M., Diez, J. A., "Droplet Profiles Obtained from

the Intensity Distribution of Refraction Patterns", Applied Optics, 34 (25), 5840-5848

(1995).

[270] Chapra, S. C., "Applied Numerical Methods with MATLAB for Engineers and Scientist",

McGraw-Hill, Boston (2005).

[271] Lee., S. T., Park., C. B., Ramesh, N. S., "Polymeric Foams Science and Technolgoy",

Taylor and Francis Group Publishing, Boca Raton Florida (2007).


230

[272] Guo, Q., Park, C. B., Xu, X., Wang, J., "Relationship of Fractional Free Volumes Derived

from the Equations of State (EOS) and the Doolittle Equation", Journal of Cellular

Plastics, 43 (1), 69-82 (2007).

[273] Wong, B., Zhang, Z., Handa, Y. P., J. Polym. Sci. Part B: Polym. Phys. , 36, 2025-2032

(1998).

[274] Panzer, U.," SPO' 98 Proceedings of 8th International Business Forum on Speciality

Polyolefins, Houston, 209-242 (1998).

[275] Ratzsch, M., Panzer, U., Hesse, A., Bucka, H.," SPE ANTEC Tech. , 2071 (1999).

[276] Sato, Y., Iketani, T., Takishima, S., Masuoka, H., "Solubility of Hydrofluorocarbon

(HFC-I34a, HFC-152a) and Hydrochlorof luorocarbon (HCFC-142b) Blowing Agents in

Polystyrene", J. Polym. Eng. Sci., 40 (6), 1369-1375 (2000).

[277] Wang, M., Sato, Y., Iketani, T., Takishima, S., Masuoka, H., Watanabe, T., Fukasawa, Y.,

"Solubility of HFC-134a, HCFC-142b, Butane, and Isobutane in Low-density

Polyethylene at Temperatures from 383.15 to 473.15 K and at Pressures up to 3.4 MPa",

Fluid Phase Equilib, 232 (1-2), 1-8 (2005).

[278] Naguib, H. E., Xu, X., Park, C. B., Hesse, A., Panzer, U., Reichelt, N.,"Effects of

Blending of Branched and Linear Polypropylene Resins on the Foamability", SPE,

ANTEC, Technical Papers, 1623-1630 (2001).

[279] Li, G., Leung, S. N., Wang, J., Park, C. B.,"Solubilities of Blowing Agent Blends", AIChE

2006 Annual Meeting, (2006).

[280] Park, C. B., Behravesh, A. H., Venter, R. D., "Low-Density, Microcellular Foam

Processing in Extrusion Using CO2", J. Polym. Eng. Sci., 38 (11), 1812-1823 (1998).

[281] Suh, K. W., Park, C. B., Maurer, M. J., Tusim, M. H., Genova, R. D., Broos, R., Sophiea,

D. P., "Lightweight Cellular Plastics", Advanced Materials, 12 (23), 1779-1789 (2000).


231

[282] Klempner, D., Frisch, K. C., "Handbook of Polymeric Foams and Foam Technology",

Oxford University Press, Munich, Vienna, New York (1991).

[283] Gendron, R., Champagne, M. F., "Foaming PS with HFC-134a in Supercritical State",

SPE ANTEC, 2700-2704 (2006).

[284] Gendron, R., Huneault, M., Tatibouet, J., Vachon, C., "Foam Extrusion of Polystyrene

Blown with HFC-134a", Cell. Polym., 21 (5), 315-341 (2002).

[285] Ajji, A., Carreau, P. J., Schreiber, H. P., "Chain Entanglement and Viscoelastic Properties

of Molten Polymers", J. Polym. Sci.: Part B: Polymer Physics, 24 (9), 1983-1990 (1986).

[286] Wu, S., "Control of Intrinsic Brittleness and Toughness of Polymers and Blends by

Chemical Structure: A Review", Polym. Int., 29 (3), 229-247 (1992).

[287] Donald, A. M., M, E., "The Competition Between Shear Deformation and Crazing in

Glassy Polymers", J. Mater. Sci., 17, 1871-1879 (1982).

[288] Liu, W. J., Shen, J. S., Wang, Z., Lu, F. C., Xu, M., "the Deformation Mechanism of

Polyphenyliquinoxaline Films", Polymer, 42 (17), 7461-7464 (2001).

[289] Park, H. E.,"Effect of Pressure and Dissolved CO2 on the Rheological Property of

Polymer Melts", McGill University, Montreal, Ph.D. Thesis (2005).

[290] Dimitrov, K., Boyadzhiev, L., Tufeu, R., "Properties of Spercritical CO2 Sturated

Ply(ethylene glycol) Nonylphenyl Ether", Macromolecular Chemistry and Physics, 200

(7), 1626-1629 (1999).

[291] Jaeger, P. T., Eggers, R., Baumgartl, H., "Interfacial Properties of High Viscous Liquids

in a Supercritical Carbon Dioxide Atmosphere", J. Supercritical Fluids, 24 (3), 203-217

(2002).

[292] Li, Y. G., Park, C. B., Li, H. B., J.Wang, "Measurement of the PVT Property of PP/CO2

Solution", Fluid Phase Equilibria, 270 (1-2), 15-22 (2008).


232

[293] Cooper, A. I., "Synthesis and Processing of Polymers Using Supercritical Carbon

Dioxide", J. Mater. Chem., 10, 207-234 (2000).

[294] Enders, S., Kahl, H., Winkelmann, J., "Interfacial Properties of Polystyrene in Contact

with Carbon Dioxide", Fluid Phase Equilibria, 228, 511-522 (2005).

[295] Andreas, J. M., Hauser, E. A., W.B., T., "Boundary Tension By Pendant Drops", J. Phys.

Chem., 42, 1001-1019 (1938).

[296] Xue, A., Tzoganakis, C., Chen, P., "Measurement of Interfacial Tension in PS/LDPE

Melts Saturated with Supercritical CO2", J. Polym. Eng. Sci., 44 (1), 18-27 (2004).

[297] Yan, D., Wang, W. J., Zhu., S., "Effect of Long Chain Branching on Rhelogical Properties

of Metalocene Polyethylene", Polymer, 40 (7), 1737-1744 (1999).

[298] Lee, Y. J., Sohn, H. S., Park, S. H., "Effect of Chain Structure of Polypropylenes on the

Melt Flow Behavior", Korea-Australia Rhelogy Journal, 12, 181-186 (2000).

[299] Wagner, M. H., "The Rhelogy of Linear and Long-Chain Branched Polymer Melts",

Macromol. Symp., 236, 219-227 (2006).


233

APPENDIX 1 Image Movement Input Algorithm %-------------------------------------------------------------------------- %%%%% This program is used for the image reconstruction based on the X-Y | %%%%% stage movement and the calibration, using all the image taken | %%%%% By Hongbo Li, Gary Li | %--------------------------------------------------------------------------| %%%% define the movement distance in terms of pixel from calibration %%%% Dy=[0, 115.91, 316.59, 432.5, 432.5, 316.59, 115.91, -115.91, -316.59 -432.5, -432.5, -316.59, -115.91 ]'; Dx=-[0, 435.25, 318.603, 116.647, -116.647, -318.603, -435.25, -435.25 -318.603, -116.647, 116.647, 318.603, 435.25 ]'; %%% save the image file name based on the number sequence %%% filenames='p1.tif' 'p2.tif' 'p3.tif' 'p4.tif' 'p5.tif' 'p6.tif' 'p7.tif' ... 'p8.tif' 'p9.tif' 'p10.tif' 'p11.tif' 'p12.tif' 'p13.tif' 'p14.tif' 'p15.tif'... 'p16.tif' 'p17.tif' 'p18.tif' 'p19.tif' 'p20.tif' 'p21.tif'... 'p22.tif' 'p23.tif' 'p24.tif' 'p25.tif' 'p26.tif' 'p27.tif' 'p28.tif' 'p29.tif'... 'p30.tif' 'p31.tif' 'p32.tif' 'p33.tif' 'p34.tif' 'p35.tif' 'p36.tif' 'p37.tif'... 'p38.tif' 'p39.tif' 'p40.tif' 'p41.tif' 'p42.tif' 'p43.tif' 'p44.tif' 'p45.tif'; %%%% Important Note %%%% %%%% Vertical movement using the x directional pixel movement values %%%% Horizontal movement using the y directional pixel movement values %%%% Remember: New Program there is no sign changes;


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Image Reconstruction Algorithm %%% Edge Reconstruction %%% clear; close all; clc; B=[ ; ]; C=[ ; ; ]; %%% Load the stage step movement and all the images file %%%% %%% Movement: Dx, Dy in the unit of pixies based on calibration %%%% %%% Image Files: "bmp" format, or "Tif" format in the sequence taken %%%% Image_Movement_Input; volume_cylinder=0; DX_total=0; DY_total=0; for i=1:length(Dx) image_file = uigetfile('*.tif;*.bmp','Choose image file'); image_file=char(filenames(i)); I = imread(image_file);

figure(1); hold; imshow(I);

%%%%possible thresholding/smoothing and edge detection routine %%%%% %%%%use the canny edge detection algorithm to obtain the edge coordinates %%% %%%%threshold value and sigma can be changed%%%%

BW1 = edge(I,'canny',0.1,1); figure(2); imshow(BW1); drop_edge = bwmorph(BW1,'remove',3 );

[X Y]= find(drop_edge); X=X+Dx(i); Y=Y+Dy(i); DX_total=DX_total+Dx(i);DY_total=DY_total+Dy(i); X=X+DX_total; Y=Y+DY_total;

%%%% Put the X and Y cooridnates into a matrix %%%% A=[X,Y]; B=[B; A]; %%%% Plot the constructed image%%%%% figure(3); plot(B(:,1),B(:,2),'b.'); end ;


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Axialsymmetric Volume Integration Algorithm %--------------------------------------------------------------------------- %Combine cropping and volume calculation script this program crops droplet %image in half by finding profile maximum and spline the boundary %calculate partial volumes separately using curve fitting and integration %Script features cropping routines and edge detection and volume calculation %routines that can be used separately %For spline fit function checkxy is used for removing duplicates and noise, %checkxy MUST be in the workin directory! %--------------------------------------------------------------------------- clc; cla; clf; syms s; v_left=0; v_right=0; v_total=0; V_final=0; image_file = uigetfile('*.tif;*.bmp','Choose image file'); I = imread(image_file); %%%%possible thresholding/smoothing routine here%%%% BW1 = edge(I,'canny',0.1,1); drop_edge = bwmorph(BW1,'remove',3 ); [X,Y] = find(drop_edge); %%%% plot to check for artifacts %%%% figure(1); plot(X,Y,’b.’); %%%%Find the highest point of the drop or the apex of the drop%%%% %%%%that is the smallest value for x %%%% minX = min( X ); index = find( X==minX ); centerY = mean( Y(index) ); %%%%2048- centerY image width, modify for larger images%%%% %%%%480 - image height, modify for actual image%%%% width=2048-centerY; Image_right = imcrop(BW1, [centerY 0 width 2048]);

figure(2); imshow(Image_right);

imwrite(Image_right,'right_part.tif','tif','Compression','none','WriteMode','overwrite');

figure(3); imshow(Image_left);

imwrite(Image_left,'left_part.tif','tif','Compression','none','WriteMode','overwrite');


236

Image_left = imcrop(BW1, [0 0 centerY 2048]); %%%%calculate the right part volume%%%% right_image_file = imread('right_part.tif'); drop_edgeR = bwmorph(right_image_file,'remove',3 ); [X1,Y1] = find(drop_edgeR); figure(4); plot(X1,Y1,'ro'); %removing duplicates, function checkxy MUST be in the working directory! [xx1,yy1] = checkxy(X1,Y1,'R'); xx_right=xx1'; yy_right=yy1'; ymin=min(Y1); yyy_right=yy1-ymin; %%%% Volume Integration Calculation %%%% pp=spline(xx_right,yy_right); [B,c,L,K,D]=unmkpp(pp); x_max=max(xx_right); x_min=min(xx_right); %ymin=min(yy_right); ymin=min(Y1); NT=10000; xright=linspace(x_min,x_max,NT); yright=ppval(pp,xright); figure(5); hold; plot(xright,yright,'o'); volume=0; for i=1:length(xright)-1 if xright(i)<= 996

v_right=v_right+1/3*pi*(abs((xright(i+1)-xright(i))))*((yright(i)-ymin)^2+(yright(i)-ymin)*(yright(i+1)-ymin)+(yright(i+1)-ymin)^2);

elseif xright(i)>= 996

volume=0; end; end; %%%%left part volume calculation%%%% left_image_file = imread('left_part.tif'); drop_edgeL = bwmorph(left_image_file,'remove',3 ); [X2,Y2] = find(drop_edgeL);


237

figure(6); plot(X2,Y2,'bd'); % removing duplicates, function checkxy MUST be in the working directory! [xx2,yy2] = checkxy(X2,Y2,'L'); xx_left=xx2'; yy_left=yy2'; ymax=max(Y2); yyy_left=abs(yy2-ymax); figure(7); hold; plot(xx_left,yy_left,'rd'); volume = 0; %%%%Calculate volume integrals %%%% pp=spline(xx_left,yy_left); xmax=max(xx_left); xmin=min(xx_left); %ymax=max(yy_left); ymax=max(Y2); NT=10000; xleft=linspace(xmin,xmax,NT); yleft=ppval(pp,xleft); figure(8); plot(xleft,yleft,'*'); for i=1:length(xleft)-1 if xleft(i)<= 986

v_left=v_left+1/3*pi*(abs((xleft(i+1)-xleft(i))))*((yleft(i)-ymax)^2+abs((yleft(i)-ymax))*abs((yleft(i+1)-ymax))+(yleft(i+1)-ymax)^2);

elseif xleft(i)>= 986 volume=0; end; end; %%%%Total volume by combining the v_left and v_right together%%%% v_total=(v_right+v_left)/2; step_size = 0.5; % the XY stage step size is 0.5 um per step cal_factor = 0.187329365; % the calibration factor is pixel/step vol_conversion_factor = ((step_size/cal_factor)*10^-4)^3; v_final=v_total*vol_conversion_factor; format long date_time = fix(clock); V =[date_time, v_final]; %%%%%saving file and appending data to existing values%%%% fid = fopen('volume data.txt','a+','n'); fprintf(fid,'%12.0f %9.2f\r\n',V); fclose(fid);


238

Function “checkxy” Algorithm function [x,y] = checkxy(x,y,Ntest) %CHECKXY check the given data x = x(:).'; y = y(:).'; NT=length(x); if any(diff(x)<0), [x,index] = sort(x); y = y(index); end if (Ntest=='R') %[b0,indx]=min(y); a0=x(indx); a0=x(NT);b0=y(NT); else if (Ntest=='L') %[b0,indx]=max(y); a0=x(indx); a0=x(NT);b0=y(NT); else error('wrong input') end end %%% Note, here we must keep the highest value of Y, since it is sensitive to %%% the final volume; %%% Depending on which part of the image, the extreme value y (at the end) %%% has to be selected mults = knt2mlt(x); if any(mults) % remove repeat sites, averaging the corresponding values for j=find(diff([mults,0])<0) y(j-mults(j)) = mean(y(j-mults(j):j)); end repeats = find(mults); x(repeats) = []; y(repeats) = []; %temp = warndlg(... %'Data has been modofied', ... % 'Repeated values found ...'); %waitfor(temp) end n0=length(x); if a0==x(n0) x(n0)=a0; y(n0)=b0; end if a0 ~=x(n0) x(n0+1)=a0; y(n0+1)=b0; end lx = length(x); if lx<2 error('There should be at least two data points.') end if lx~=length(y)


239

error(['The number ',num2str(lx),' of data sites should match',... ' the number ',num2str(length(y)),' of data values.']) End


240

Asymmetric Volume Calculation Algorithm (Fortran) C THIS ALOGRITHM IS DEVELOPED TO SPLINE THE PRIMETER OF EACH C LAYER OF THE DROP AND THEN USE SIMPSON’S RULE TO INTEGRATE C TO COMPUTE THE VOLUME OF ASYMMETRIC DROP SHAPE C THIS IS TO CALCULATE THE VOLUME OF SESSILE DROP PROGRAM VOLUME CALCUATION OF A SESSILE DROP PARAMETER (IMAX=560) ! TEMP=150 C PARAMETER (IMAX=570) ! TEMP=200 C PARAMETER (IMAX=506) ! TEMP=250 PARAMETER (KMAX=100) PARAMETER (PAI=3.1415926) REAL RL(0:KMAX,0:IMAX),RR(0:KMAX,0:IMAX) REAL THETAL(0:KMAX), THETAR(0:KMAX) REAL TOT(0:IMAX) REAL AREA(0:KMAX,0:IMAX) REAL TEMP C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C DATA INPUT TEMP=150. C TEMP=200. C TEMP=250. IF (TEMP .EQ. 150.) THEN OPEN (2, FILE='E:\3\SPLINE150L.DAT',STATUS='OLD') READ (2,*) (THETAL(K),K=1,KMAX) DO I=1,IMAX READ (2,*) (RL(K,I), K=1,KMAX) ENDDO CLOSE (2) OPEN (4, FILE='E:\3\SPLINE150R.DAT',STATUS='OLD') READ (4,*) (THETAR(K),K=1,KMAX) DO I=1,IMAX READ (4,*) (RR(K,I), K=1,KMAX) ENDDO CLOSE (4) ENDIF IF (TEMP .EQ. 200.) THEN OPEN (12, FILE='E:\3\SPLINE200L.DAT',STATUS='OLD') READ (12,*) (THETAL(K),K=1,KMAX) DO I=1,IMAX READ (12,*) (RL(K,I), K=1,KMAX) ENDDO CLOSE (12) OPEN (14, FILE='E:\3\SPLINE200R.DAT',STATUS='OLD') READ (14,*) (THETAR(K),K=1,KMAX) DO I=1,IMAX READ (14,*) (RR(K,I), K=1,KMAX) ENDDO CLOSE (14)


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ENDIF IF (TEMP .EQ. 250.) THEN OPEN (22, FILE='E:\3\SPLINE250L.DAT',STATUS='OLD') READ (22,*) (THETAL(K),K=1,KMAX) DO I=1,IMAX READ (22,*) (RL(K,I), K=1,KMAX) ENDDO CLOSE (22) OPEN (24, FILE='E:\3\SPLINE250R.DAT',STATUS='OLD') READ (24,*) (THETAR(K),K=1,KMAX) DO I=1,IMAX READ (24,*) (RR(K,I), K=1,KMAX) ENDDO CLOSE (24) ENDIF DO K=1,KMAX RL(K,0)=0. RR(K,0)=0. ENDDO C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C VOLUME CALCUATION VOL=0. DO I=1,IMAX TOT(I)=0. C DO K=1,KMAX-1 C AREA(K,I)= C & PAI*((RL(K,I)+RL(K+1,I)+RL(K,I-1)+RL(K+1,I-1))/4.)**2 C & *(THETAL(K+1)-THETAL(K))/360. C & +PAI*((RR(K,I)+RR(K+1,I)+RR(K,I-1)+RR(K+1,I-1))/4.)**2 C & *(THETAR(K+1)-THETAR(K))/360. C TOT(I)=TOT(I)+AREA(K,I) C ENDDO DO K=1,KMAX-1 AREA(K,I)=1./3.*PAI*( & (( RL(K,I-1)+RL(K+1,I-1) )/2.)**2 & +(( RL(K,I-1)+RL(K+1,I-1) )/2.)*(( RL(K,I)+RL(K+1,I) )/2.) & +(( RL(K,I)+RL(K+1,I) )/2.)**2 & )*(THETAL(K+1)-THETAL(K))/360. & + & 1./3.*PAI*( & (( RR(K,I-1)+RR(K+1,I-1) )/2.)**2 & +(( RR(K,I-1)+RR(K+1,I-1) )/2.)*(( RR(K,I)+RR(K+1,I) )/2.) & +(( RR(K,I)+RR(K+1,I) )/2.)**2 & )*(THETAR(K+1)-THETAR(K))/360. TOT(I)=TOT(I)+AREA(K,I) ENDDO TOT(I)=TOT(I)*(2.6953E-4)**2


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VOL=VOL+TOT(I)*1.*(2.6803E-4) ENDDO IF (TEMP .EQ. 150) THEN ERR=(VOL-0.014416)/0.014416 ! T=150 DEGREE C WRITE(*,*) 'VOL=', VOL, ' CC' WRITE(*,*) (VOL-0.014416) WRITE(*,*) 'ERR=', ERR*100, ' %' ENDIF IF (TEMP .EQ. 200) THEN ERR=(VOL-0.014848)/0.014848 ! T=200 DEGREE C WRITE(*,*) 'VOL=', VOL, ' CC' WRITE(*,*) (VOL-0.014848) WRITE(*,*) 'ERR=', ERR*100, ' %' ENDIF IF (TEMP .EQ. 250) THEN ERR=(VOL-0.01528)/0.01528 ! T=250 DEGREE C WRITE(*,*) 'VOL=', VOL, ' CC' WRITE(*,*) (VOL-0.01528) WRITE(*,*) 'ERR=', ERR*100, ' %' ENDIF OPEN (6, FILE='AREA.DAT', STATUS='UNKNOWN') DO I=1,IMAX WRITE (6,*) I,TOT(I) ENDDO CLOSE (6) END


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APPENDIX 2 Motion Stage GUI Development Algorithm Private Response As String Private hDmc As Long Private Controller As Integer Private RC As Long Private Px() As Long Private Py() As Long Dim stoploop As Boolean Private Declare Sub Sleep Lib "kernel32" (ByVal dwMilliseconds As Long) Private Declare Function timeGetTime Lib "winmm.dll" () As Long Private Sub Btn_AutoImage_Click() Dim i As Long Dim j As Long Dim k As Long Dim h As Long Dim Duration As Single ' in hour Dim Interval As Long ' in minute Dim UnitAngle As Long Dim Sleeptime As Long 'millisecond Dim totalloop As Long Dim Kt As Long 'get current time Dim Comma As Integer Dim Item As String Dim Totalitem As Integer Dim CommandString As String Dim SaveDirectory As String Dim Output As Long Dim Imagefile As String Dim Today As String Totalitem = List_Data.ListCount Output = 1 Today = Date$ + "-" If Txt_Duration.Text = "" Then Txt_Duration.SetFocus


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End If If Txt_Interval.Text = "" Then Txt_Interval.SetFocus End If UnitAngle = 0 If chk_Rotation.Value Then If txt_RotationAngle.Text = "" Then txt_RotationAngle.SetFocus Else UnitAngle = Val(txt_RotationAngle.Text) End If End If Txt_Duration.Enabled = False Txt_Interval.Enabled = False txt_RotationAngle.Enabled = False Duration = Val(Txt_Duration.Text) * 60 'convert hour to minute Interval = Val(Txt_Interval.Text) totalloop = Duration / Interval Sleeptime = Interval * 60 ' seconds stoploop = False Pbar_Waittime.Max = Sleeptime If txt_SaveDir.Text = "" Then SaveDirectory = "D:\GaryPics\" Else SaveDirectory = txt_SaveDir.Text End If If Totalitem > 0 Then Btn_AutoImage.Enabled = False j = 1 Do While (j <= totalloop And Not stoploop) DoEvents Pbar_Waittime.Value = 0 Btn_Pause.SetFocus '=========== If chk_Rotation.Value Then


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k = 0 'set the counter for rotation angle Do While ((k * UnitAngle) <= 360) 'get data points and display them in listbox For h = 0 To Totalitem - 1 List_Data.Selected(h) = False List_Data.Refresh Next '========= lbl_RotationAngle.Caption = Str(k) + ":" + Str(k * UnitAngle) If k > 0 Then CommandString = "PRZ=" + Str((UnitAngle * 500 / 360)) CommandString = CommandString + ";" + "BGZ" + ";" + "CB0;AM;SB0" RC = DMCCommand(hDmc, CommandString, Response, ResponseLength) Call RTrim(Response) If Len(Txt_Response.Text) > 32000 Then Txt_Response.Text = Txt_Command.Text + vbCrLf + Response Else Txt_Response.Text = Txt_Response.Text + Txt_Command.Text + vbCrLf + Response End If End If '============== '360 degree = 500 pulses 'time taken for rotating shaft by unitangle is unitangle*250ms/360 for sp=2000 '============== i = 0 Do While (i < Totalitem) 'scan each point and take picture at desired spots Item = Trim$(List_Data.List(i)) Lbl_CurrentPoint.Caption = Item CommandString = "PR" + Item CommandString = CommandString + ";" + "BGXY" + ";" + "CB0;AM;SB0" 'pause for 250 millisecs Kt = timeGetTime While Abs(timeGetTime - Kt) < 250 DoEvents


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Wend RC = DMCCommand(hDmc, CommandString, Response, ResponseLength) Call RTrim(Response) If Len(Txt_Response.Text) > 32000 Then Txt_Response.Text = Txt_Command.Text + vbCrLf + Response Else Txt_Response.Text = Txt_Response.Text + Txt_Command.Text + vbCrLf + Response End If DoEvents 'select the data point that has been processed If List_Data.Selected(i) = False Then List_Data.Selected(i) = True List_Data.Refresh End If 'Check if motion is completed by checking the value of generaloutput0 RC = DMCRefreshDataRecord(hDmc, ByVal 0&) RC = DMCGetDataRecordByItemId(hDmc, DRIdGeneralOutput0, DRIdAxis1, DRTypeLONG, Output) Do While (Output = 0) DoEvents RC = DMCRefreshDataRecord(hDmc, ByVal 0&) RC = DMCGetDataRecordByItemId(hDmc, DRIdGeneralOutput0, DRIdAxis1, DRTypeLONG, Output) Sleep (5) Loop 'Save image to files when desired position is reached If chk_Rotation.Value Then Imagefile = SaveDirectory + Today + Str(j) + "-" + Str(i + 1) + "-" + Str(k) + "-" + Str(k * UnitAngle) + ".tif" End If Sbar_Main.Panels(3).Text = "Image File saved to:" + Imagefile EBW8Image1.Save Imagefile, eGrayLevelTiff Sbar_Main.Panels(2).Text = "Scanned points:" + Str(j) + ":" + Str(i + 1) + ":" + Str(k) + ":" + Str(k * UnitAngle) i = i + 1 Loop ' end of the loop for taking pictures at various points


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'// end of the code for rotation k = k + 1 Loop '// end of the loop for rotation Else 'no rotation, only points along perimeter 'get data points and display them in listbox For h = 0 To Totalitem - 1 List_Data.Selected(h) = False List_Data.Refresh Next i = 0 Do While (i < Totalitem) 'scan each point and take picture at desired spots Item = Trim$(List_Data.List(i)) Lbl_CurrentPoint.Caption = Item CommandString = "PR" + Item CommandString = CommandString + ";" + "BGXY" + ";" + "CB0;AM;SB0" 'pause for 250 millisecs Kt = timeGetTime While Abs(timeGetTime - Kt) < 250 DoEvents Wend RC = DMCCommand(hDmc, CommandString, Response, ResponseLength) Call RTrim(Response) If Len(Txt_Response.Text) > 32000 Then Txt_Response.Text = Txt_Command.Text + vbCrLf + Response Else Txt_Response.Text = Txt_Response.Text + Txt_Command.Text + vbCrLf + Response End If DoEvents 'select the data point that has been processed If List_Data.Selected(i) = False Then List_Data.Selected(i) = True List_Data.Refresh End If 'Check if motion is completed by checking the value of generaloutput0


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RC = DMCRefreshDataRecord(hDmc, ByVal 0&) RC = DMCGetDataRecordByItemId(hDmc, DRIdGeneralOutput0, DRIdAxis1, DRTypeLONG, Output) Do While (Output = 0) DoEvents RC = DMCRefreshDataRecord(hDmc, ByVal 0&) RC = DMCGetDataRecordByItemId(hDmc, DRIdGeneralOutput0, DRIdAxis1, DRTypeLONG, Output) Sleep (5) Loop Imagefile = SaveDirectory + Today + Str(j) + "-" + Str(i + 1) + ".tif" Sbar_Main.Panels(3).Text = "Image File saved to:" + Imagefile EBW8Image1.Save Imagefile, eGrayLevelTiff Sbar_Main.Panels(2).Text = "Scanned points:" + Str(j) + ":" + Str(i + 1) i = i + 1 Loop ' end of the loop for taking pictures at various points End If 'condition check for rotation j = j + 1 Kt = timeGetTime While Abs(timeGetTime - Kt) < (Sleeptime * 1000) And Not stoploop Pbar_Waittime.Value = Abs(timeGetTime - Kt) / 1000 DoEvents Wend If stoploop Then Exit Do End If Loop ' end of the loop of each interval End If Btn_AutoImage.Enabled = True Btn_OpenData.Enabled = True Txt_Duration.Enabled = True Txt_Interval.Enabled = True Call MsgBox("Done Done Done!!!", vbOKOnly, "Warning") End Sub


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Private Sub Btn_CloseData_Click() List_Data.Clear Btn_OpenData.Enabled = True Btn_CloseData.Enabled = False Btn_AutoImage.Enabled = False End Sub Private Sub Btn_Initialization_Click() EBoard1.DriverIndex = 0 '+ change EBoard1.SetParamNm "BoardTopology", "MONO" '- change ' Create a channel and associate it with the first connector on the first board ECamera1.Mpf = "CHANNEL" ECamera1.SetParamNm "DriverIndex", 0 '+ change ECamera1.SetParamNm "Connector", "M" '- change ' Choose the video standard '+ change ECamera1.SetParamNm "CamFile", "TM-4000CL_P15RG" '- change ' Choose the camera expose duration ECamera1.SetParamNm "Expose_us", 66667 ' Choose the pixel color format ECamera1.SetParamNm "ColorFormat", "Y8" ' Set the acquisition mode to snapShot ECamera1.SetParamNm "AcquisitionMode", "SNAPSHOT" ' Choose the way the first acquisition is triggered ECamera1.SetParamNm "TrigMode", "IMMEDIATE" ' Choose the triggering mode for subsequent acquisitions ECamera1.SetParamNm "NextTrigMode", "SAME" ' Choose the number of images to acquire ECamera1.SetParamNm "SeqLength_Fr", -1 ' Choose the tap configruation 'ECamera1.SetParamNm "TapConfiguration", "BASE_2T8-2X_1Y" ' Choose the tap configruation 'ECamera1.SetParamNm "TapStructure", "DUAL_B" ' Choose the image flip style 'ECamera1.SetParamNm "ImagFlipX", "OFF" ' Choose the image flip style


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'ECamera1.SetParamNm "ImagFlipY", "OFF" ' Choose the image grab window 'ECamera1.SetParamNm "GrabWindow", "NOBLACK" ' Configure the receiving EImage: ' (set image size corresponding to camera capture) ImageSizeX = ECamera1.GetParamNm("ImageSizeX") ImageSizeY = ECamera1.GetParamNm("ImageSizeY") EBW8Image1.SetSize ImageSizeX, ImageSizeY EBW8Image1.EnableScrollBars = True ' Link the Camera with the EImage: ECamera1.Cluster = EBW8Image1 ' Enable the signals SurfaceFilled and AcquisitionFailure: ECamera1.SetParamNm "SignalEnable:2", "ON" ECamera1.SetParamNm "SignalEnable:7", "ON" '============================================ ECamera1.SetParamNm "ChannelState", "ACTIVE" Lbl_Status.Caption = "" '============================================ Btn_Initialization.Enabled = False End Sub Private Sub Btn_OpenData_Click() Dim InputData As String Dim fname As String Dim b As String Dim x As Long Dim y As Long Dim i As Integer Cdlg_Main.FileName = "" Cdlg_Main.ShowOpen fname = Cdlg_Main.FileName If fname <> "" Then 'File #1 will always be used for reading in the source data file Open fname For Input As #1 b = "" i = 0


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Do Until EOF(1) i = i + 1 'Input #1, InputData ReDim Px(i - 1) ReDim Py(i - 1) Input #1, Px(i - 1), Py(i - 1) b = Str(Px(i - 1)) + "," + Str(Py(i - 1)) List_Data.AddItem (b) Loop Close #1 End If If List_Data.ListCount > 0 Then Btn_CloseData.Enabled = True Btn_OpenData.Enabled = False Btn_AutoImage.Enabled = True End If End Sub Private Sub Btn_Pause_Click() Dim CommandString As String stoploop = True CommandString = "AB" RC = DMCCommand(hDmc, tempcommand, Response, ResponseLength) Call RTrim(Response) If Len(Txt_Response.Text) > 32000 Then Txt_Response.Text = Txt_Command.Text + vbCrLf + Response Else Txt_Response.Text = Txt_Response.Text + Txt_Command.Text + vbCrLf + Response End If Txt_Duration.Enabled = True Txt_Interval.Enabled = True If chk_Rotation.Value Then txt_RotationAngle.Enabled = True End If


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End Sub Private Sub Btn_Send_Click() Dim additionalresponse As String Dim SpeedNumber As Long Response = Space(256) ResponseLength = 256 If Txt_Command.Text = "" Then Txt_Command.Text = ";" End If RC = DMCCommand(hDmc, Txt_Command.Text, Response, ResponseLength) Call RTrim(Response) If Len(Txt_Response.Text) > 32000 Then Txt_Response.Text = Txt_Command.Text + vbCrLf + Response Else Txt_Response.Text = Txt_Response.Text + Txt_Command.Text + vbCrLf + Response End If If RC = DMCERROR_BUFFERFULL Then RC = DMCGetAdditionalResponseLen(hDmc, ResponseLength) If RC = 0 Then additionalresponse = Space$(ResponseLength + 1) RC = DMCGetAdditionalResponse(hDmc, additionalresponse, ResponseLength) RTrim (additionalresponse) If Len(Txt_Response.Text) > 32000 Then Txt_Response.Text = additionalresponse Else Txt_Response.Text = Txt_Response.Text + additionalresponse End If End If End If Txt_Command.Text = "" Txt_Response.SelStart = Len(Txt_Response.Text) End Sub Private Sub Btn_Stop_Click() Dim tempcommand As String Dim additionalresponse As String


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Response = Space(256) ResponseLength = 256 'Stop the stepper motor tempcommand = "ST" RC = DMCCommand(hDmc, tempcommand, Response, ResponseLength) tempcommand = "CS" RC = DMCCommand(hDmc, tempcommand, Response, ResponseLength) Call RTrim(Response) If Len(Txt_Response.Text) > 32000 Then Txt_Response.Text = Txt_Command.Text + vbCrLf + Response Else Txt_Response.Text = Txt_Response.Text + Txt_Command.Text + vbCrLf + Response End If End Sub Private Sub chk_Rotation_Click() If txt_RotationAngle.Enabled Then txt_RotationAngle.Enabled = False txt_RotationAngle.Text = "" Else txt_RotationAngle.Enabled = True End If End Sub Private Sub ECamera1_Signal(ByVal SignalType As MULTICAMLibCtl.enumMcEventType, ByVal Obj As Object, ByVal Last As Boolean) If SignalType = eMcSigAcquisitionFailure Then Lbl_Status.Caption = "Acquisition Failure" Else If SignalType = eMcSigSurfaceFilled Then Obj.Refresh ' Retrieve the frame rate and display it Value = ECamera1.GetParamNm("FrameRate_Hz") Lbl_Status.Caption = "FrameRate_Hz " + Value End If End If End Sub Private Sub Form_Load() Dim tempcommand As String


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Dim Kt As Long '============================== ' Controller card detecting Response = Space(256) ResponseLength = 256 DiagsOn = False hDmc = -1 Controller = 1 If Command$ <> "" Then Controller = Val(Command) End If If Controller < 1 Or Controller > 15 Then Controller = 1 End If RC = DMCOpen(Controller, 0, hDmc) If RC = 0 Then RC = DMCVersion(hDmc, Response, ResponseLength) Sbar_Main.Panels(1).Text = Response RC = DMCCommand(hDmc, ";", Response, ResponseLength) 'Initialize the stepper motors tempcommand = "MT 2.5,2,2;SP 2000,2000,2000;AC 100000,100000,100000;DC 50000,50000,50000" RC = DMCCommand(hDmc, tempcommand, Response, ResponseLength) Else hDmc = -1 MsgBox "Error: could not connect to controller " + Trim$(Str$(Controller)) + ". RC = " + Trim$(Str$(RC)) Unload Frm_Main End If Me.Show '====================================== 'Initialization of Evision Board 'initialize buttons Btn_OpenData.Enabled = True Btn_CloseData.Enabled = False Btn_AutoImage.Enabled = False


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txt_RotationAngle.Enabled = False 'Btn_Initialization_Click End Sub Private Sub Form_Unload(Cancel As Integer) stoploop = True End Sub Private Sub Txt_Command_GotFocus() Btn_Send.Default = True End Sub Private Sub Txt_Command_KeyDown(KeyCode As Integer, Shift As Integer) If KeyCode = vbKeyEscape Then TabStrip.SetFocus End If End Sub Private Sub Txt_Command_KeyPress(KeyAscii As Integer) If KeyAscii >= 97 And KeyAscii <= 122 Then KeyAscii = KeyAscii - 32 End If End Sub


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APPENDIX 3 Calibration is carried out in vacuum condition at the experimental temperature prior to

each experiment. The calibration is done at all the levels across the sample horizontally in X

direction and vertically in Y direction. The following graphically illustrates how the calibration

is determined in both directions. The number of levels is determined to cover the whole drop.

Calibration in X direction

x1x2x3x4

x

y

At level 1, the edge point on the X axis at pixel coordinate 1X of the drop is moved across the

lens (or the screen) to a new location at NX _1 with the stage moved N step towards right (or

left) direction as illustrated as following two pictures

1X NX _1


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During the N step movement, the number of pixel is determined as: 1_1 XX N −

Since each step of the stage is 0.5μm, the pixel size at level 1 can be determined as:

1_1

*00005.0XX

N

N −

Similarly, the pixel size at all other levels could be determined.

Calibration in Y direction

y1 y2 y3 y4

x

yy5 y6 y7

At level 1, the edge point on the Y axis at pixel coordinate 1Y of the drop is moved across the

lens (or the screen) to a new location at NY _1 with the stage moved N step in upward (or

downward) direction as illustrated as following two pictures

During the N step movement, the number of pixel is determined as: 1_1 YY N −

1Y

NY _1


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Since each step of the stage is 0.5μm, the pixel size at level 1 can be determined as:

1_1

*00005.0YY

N

N −

Similarly, the pixel size at all the levels could be determined. Since the pixel is not a perfect

square pixel, the pixel size would be averaged in both X and Y directions to be as the calibration

factor. This calibration factor is used as the conversion factor for the volume conversion from

pixel to metric unit.

The following table shows the calibration results from one of the experiments.

Experiment_Set_4 1 step = 0.5 um 0.00005

x initial x final step number step length pixel size pixel/step 1426 1892 2500 0.125 0.000268 0.1864 867 1346 2500 0.125 0.000261 0.1916 269 743 2500 0.125 0.000264 0.1896 537 1009 2500 0.125 0.000265 0.1888

1240 1719 2500 0.125 0.000261 0.1916 425 905 2500 0.125 0.00026 0.1920

avg 0.000263 0.1900

y initial y final step number step length pixel size pixel/step

762 1235 2500 0.125 0.000264 0.1892 1055 1530 2500 0.125 0.000263 0.1900 983 1456 2500 0.125 0.000264 0.1892 493 966 2500 0.125 0.000264 0.1892 158 632 2500 0.125 0.000264 0.1896 54 529 2500 0.125 0.000263 0.1900

Averageavg 0.000264 0.1895 0.189767


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APPENDIX 4

Rheometer Parallel-disks Sample Loading Procedure

The following is a very simple procedure that describes how to make a proper sample

loading onto the two fixture plates prior to the real measurement.

1. Lower the upper plate until it touches the bottom plate

2. Set the gap to zero

3. Raise the upper plate until the gap is 15 mm

4. Load the sample on the bottom plate

5. Lower the upper plate until the gap is 4mm

6. Close the cover and start sample heat at set temperature

7. Start to lower the upper plate with 0.5 mm at a time after the temperature reaches a set

point

8. Lower the upper plate at 2 mm gap from bottom plate

9. Trim the sample flush with the edge of the disks

10. Repeat steps 6 and 7

11. Lower the upper plate at 1.5mm gap from bottom plate

12. Trim the sample flush with the edge of the disks

13. Repeat steps 6

14. Start to lower the upper plate with 1.0 mm gap distance

15. Start the rheometer experiment


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APPENDIX 5 Table Molecular Weight Information of Linear, Branched PP and PS Sample Linear DM55 PP Branched HMS WB 130 PP PS 685D

Mw (g/mol) 481,750 654,039 338,800 Mn (g/mol) 105,473 122,395 196,200

Mw/ Mn 4.57 5.34 1.73 Molecular Weight Distribution of Linear and Branched PP

2 3 4 5 6 7 8-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Wei

ght f

ract

ion,

d(w

t)/d(

log(

MW

)))

Log(MW)

Linear PP DM55

Molecular Weight Distribution for Linear PP DM55

2 3 4 5 6 7 8-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Wei

ght f

ract

ion,

d(w

t)/d(

log(

MW

)))

Log(MW)

Branched PP HMS WB130

Molecular Weight Distribution for Branched PP HMS WB130


261

APPENDIX 6

Swelling Ratio Results for Linear and Branched PP Table 1) Volume Swelling of Linear PP/CO2 at High Temperatures and Pressures Temperature(oC) 180oC

Swelling SE Error 200oC


Swelling SE Error

Pressure (psi) 1000 1.02943 0.01555 1.01633 0.0013 1.01049 0.006271500 1.06437 0.01024 1.04834 0.0017 1.04064 0.0052 2000 1.10236 0.00772 1.0807 0.0039 1.0632 0.007322500 1.13401 0.00657 1.11105 0.00456 1.09107 0.006023000 1.16766 0.00575 1.1384 0.00227 1.11745 0.004423500 1.18741 0.0065 1.1593 5.43547E-4 1.14413 0.0059 4000 1.21233 0.00572 1.17519 0.00202 1.15738 0.002 4500 1.22621 0.0091 1.18751 0.00145 1.1668 0.00456

Table 2) Volume Swelling of Branched PP/CO2 at High Temperatures and Pressures Temperature(oC) 180oC



Swelling SE Error

Pressure (psi) 1000 1.02644 0.0018 1.01083 0.00244 1.00295 0.00141 1500 1.05647 0.00111 1.03413 0.0039 1.02743 0.00158 2000 1.098 0.00468 1.06597 0.00628 1.05408 0.00279 2500 1.12964 0.00228 1.09803 0.0082 1.08004 0.00217 3000 1.15805 0.00414 1.12338 0.00619 1.10723 3.98093E-43500 1.17643 0.00373 1.14829 0.00165 1.13041 0.00137 4000 1.19597 0.00749 1.16124 0.00455 1.14743 0.0014 4500 1.2124 0.01189 1.17419 0.013 1.15405 0.00201


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APPENDIX 7

TGA Test Results

220oC

250oC

1) Heat and Hold Test for a) Linear and Branched PP at 220oC for 3 Hours and b) for PS at 250oC for 1.5 Hours


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2) Temperature Ramp Test for Linear and Branched PP

3) Temperature Ramp Test for PS 685D Virgin Sample, Sample Out of Vacuum Oven

and Sample after Experiment


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APPENDIX 8

MSB Absorption Test for Saturation Time Determination 1) Linear PP/CO2 at 180oC

0 200 400 600 800 1000 1200

13.8

14.1

14.4

14.7

Wei

ght (

g)

Time (min)

1800C Linear PP/CO2 Saturation

190 200 210 220 230 240 250 260 270 280 290 300 310 320

14.11

14.12

14.13

14.14

Wei

ght (

g)

Time (min)

1800C Linear PP/CO2 Saturation


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2) Branched PP/CO2 at 180oC

0 100 200 300 40014.0

14.2

14.4

14.6

Wei

ght (

g)

Time (min)

1800C Branched PP/CO2 Saturation

90 95 100 105 110 115 120 125 130 135 140 145 150 155 160

14.40

14.42

14.44

Wei

ght (

g)

Time (min)

1800C Branched PP/CO2 Saturation

Development of a Novel Visualization and Measurement Apparatus for the PVT Behaviours of

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