Tribological and mechanical characterization of thin polymer …techniques was developed to assess...
Transcript of Tribological and mechanical characterization of thin polymer …techniques was developed to assess...
Tribological and Mechanical Characterization of
Thin Polymer Films
A Dissertation Presented
by
Qian Sheng
to
The Department of Mechanical and Industrial Engineering
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in the field of
Mechanical Engineering
Northeastern University
Boston, Massachusetts
July 2013
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Abstract
Mechanical and tribological characteristics of polymer films are crucial to their
successful implementation as thin protective coatings. Hot filament chemical vapor
deposition (HFCVD) is a relatively new technique which enables deposition of various
polymers on a variety of surfaces. In this dissertation a toolbox of experimental
techniques was developed to assess the quality of thin polymer films. These include
assessment of friction, durability and interfacial adhesion. While this work is primarily
focused on polytetrafluoroethylene (PTFE), assessments of perfluoroalkoxy (PFA) and
poly(trivinyltrimethylcyclotrisiloxane) (Poly(V3D3)) were also carried out.
Frictional and durability characteristics of thin PTFE films deposited on aluminum
substrates were investigated by using a ball-on-disk and ball-on-plate configurations,
respectively. PFA and Poly(V3D3) were deposited on glass and were likewise tested. The
effects of normal force, sliding speed and surface roughness on the coefficient of friction
(COF) and durability were quantitatively examined by the analysis of their variance
(ANOVA). The results show that native surface roughness of the substrate has the most
significant effects on the COF and durability; and, that the smooth interface is dominated
by adhesion. Experiments indicated that PTFE-thin film durability can be optimized if it
is deposited on a mid-range, (Ra ~ 0.5 μm) surface roughness.
In order to assess the interfacial adhesion properties of thin PTFE films, a micro-
indentation based technique was developed. Experimentally, the technique involved
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monitoring the peel-off radius of the PTFE as subjected to different levels of loads by a
conical micro-indenter. The mechanics of PTFE indentation was simulated by the finite
element method (FEM) to obtain quantitative measurements of interfacial fracture
toughness out of the experimental measurements. It was determined that a material
penetration model was necessary to model the indenter-PTFE interaction. A finite
element model utilizing continuum damage mechanics (CDM) was implemented in
ABAQUS/Explicit to simulate the penetration of the bulk PTFE. Effects of different
damage/failure parameters were investigated systematically and compared to
experimental results to determine the adequate bulk fracture energy and damage initiation
strain levels. The bonding in the PTFE-glass interface was modeled by a cohesive zone
mechanics (CZM) model. A normalized relationship between the delamination radius, the
indentation load, coating thickness, and the material properties of the coating was
developed based on the finite element results. Fracture toughness of the PTFE-glass
interface was determined quantitatively. Values reported in this work include a relatively
more comprehensive treatment of the energy spent in penetrating the thin film, and thus
are somewhat lower than those reported in the literature. Both the experimental and the
theoretical aspects of the work also show that it is more difficult to delaminate thinner
coatings.
The results of this work will be useful in assessing the effectiveness of different
deposition techniques, in choosing optimal coating thickness, and in preparing optimal
surface roughness for improved durability.
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Curriculum Vitae
The author was born on July 5, 1982 in Nanjing, Jiangsu Province, in the People’s
Republic of China. He studied at Nanjing University of Aeronautics and Astronautics, in
Nanjing, China and received BS and MS degrees from the Department of Mechanical and
Electrical Engineering in July 2004 and April 2007, respectively. He received
scholarships during 2000-2006 and was awarded the honor of excellent graduate student
in 2007. After graduating in 2007, he worked as a mechanical engineer in the Powertrain
Business Unit of Comau, which focuses on production systems in FIAT Corporation, in
Shanghai, China.
He started his Ph.D. studies in the Department of Mechanical and Industrial Engineering
of Northeastern University (Boston, MA, the United States), in the fall of 2008. He
worked as a teaching and research assistant at the Applied (Bio) Mechanics and
Tribology Laboratory and instructed the labs of mechanics of materials and finite element
method for five years. His research topic involved experimental investigation and
numerical modeling for the tribology and mechanics of thin polymer films. His
dissertation advisor was Professor Sinan Müftü.
He is a student member of the American Society of Mechanical Engineers (ASME) and
the Adhesion Society.
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Acknowledgements
I thank Professor Sinan Müftü, my dissertation advisor, for his help and guidance on my
research work throughout the last five years. I thank Professor Kai-Tak Wan for fruitful
discussions related to this work and his permission to perform nano-indentation tests in
his laboratory at Northeastern University. I thank Professor Andrew Gouldstone for his
helpful suggestions related to this work and his great help during my teaching assistant
work. I also acknowledge the support from the Department of Mechanical and Industrial
Engineering for providing me the teaching assistantship.
I am grateful for the support and contributions of Aleksandr J. White at GVD
Corporation (Cambridge, MA) to the experimental work of this dissertation. I thank
Michael A. Karnath for starting the initial phase of this work and the journal publication
on Tribology Transactions. I also acknowledge the help from Dr. Guangxu Li, Dr. Jiayi
Shi, and Mr. Michael Robitaille for showing me the use of nano-indenter and helpful
discussions.
I thank all of my colleagues in the Applied (Bio) Mechanics and Tribology Laboratory
for helpful suggestions on my research, and for the friendship throughout my doctoral
studies.
I acknowledge the help and guidance of my teachers at different stages of my education
in China.
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Finally, I thank my parents for their love, support and education. I remember my
grandparents with love for their perseverance and integrity. I thank my wife, Qingkun Liu,
for her love, support and patience.
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Table of Contents
Abstract ................................................................................................................................ i
Curriculum Vitae ............................................................................................................... iii
Acknowledgements ............................................................................................................ iv
Table of Contents ............................................................................................................... vi
List of Figures ..................................................................................................................... x
List of Tables .................................................................................................................. xvii
Nomenclature .................................................................................................................... xx
1 Introduction ............................................................................................................... 1
2 Mechanics Review for Thin-Film Polymer Tribology .............................................. 4
2.1 Indentation of an Elastic Half-Space .................................................................... 4
2.1.1 Point Load on an Elastic Half-Space .......................................................... 4
2.1.2 Hertzian Contact ......................................................................................... 7
2.2 Indentation of a Layered Elastic Half-Space........................................................ 9
2.3 Indentation on an Elastic-Plastic Layered Half-Space ....................................... 14
2.4 Indentation on Soft Thin-Film Material ............................................................. 15
2.5 Material Properties Investigation by Indentation ............................................... 20
2.6 Indentation-Induced Interfacial Delamination ................................................... 25
2.6.1 Definitions of Adhesion Energy and Interfacial Fracture Toughness ...... 25
2.6.2 Mathematical Descriptions of Interfacial Delamination ........................... 27
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2.6.3 A Relationship between Non-dimensional Delamination Radius and
Indentation Load ..................................................................................................... 37
2.6.4 Film Thickness Effect on Interfacial Delamination .................................. 39
2.7 Bulk Material Damage/Failure Criteria in Finite Element Analysis .................. 41
2.7.1 Introduction of Bulk Material Damage/Failure ........................................ 41
2.7.2 Implementation of Bulk Material Damage/Failure Criteria in Finite
Element Analysis ..................................................................................................... 42
2.8 Cohesive Zone Model in Finite Element Analysis ............................................ 45
2.8.1 Description of Cohesive Zone Model ....................................................... 45
2.8.2 Implementation of Cohesive Zone Model in ABAQUS ........................... 48
3 Review of PTFE Material Properties ...................................................................... 55
3.1 Molecular Structure of PTFE ............................................................................. 55
3.2 Mechanical Properties of PTFE ......................................................................... 59
3.2.1 Young’s Modulus and Yield Stress .......................................................... 59
3.2.2 Viscoelastic and Plastic Properties of PTFE ............................................. 63
3.2.3 Frictional Characteristics .......................................................................... 66
3.2.4 Wear Characteristics ................................................................................. 76
3.3 Summary ............................................................................................................ 78
4 Friction and Durability of Thin PTFE Films on Rough Aluminum Substrates ...... 80
4.1 Introduction ........................................................................................................ 80
4.2 Materials and Methods ....................................................................................... 81
4.3 Results ................................................................................................................ 83
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4.3.1 COF ........................................................................................................... 86
4.3.2 Durability .................................................................................................. 92
4.4 Discussion .......................................................................................................... 96
4.5 Summary and Conclusion ................................................................................ 103
5 Frictional Characteristics of Thin PFA and Silicone Films on Glass Substrates .. 105
5.1 Introduction ...................................................................................................... 105
5.2 Materials and Methods ..................................................................................... 108
5.3 Results .............................................................................................................. 108
5.3.1 Friction Characteristics of Thin PFA Films ............................................ 108
5.3.2 Friction Characteristics of Thin Silicone Films ...................................... 112
5.4 Discussion ........................................................................................................ 115
5.5 Summary and Conclusion ................................................................................ 117
6 Simulation of Material Damage during Indentation of a Soft Polymer ................ 119
6.1 Axi-Symmetric Finite Element Analysis of Thin-Film Indentation ................ 119
6.1.1 Materials and Methods ............................................................................ 119
6.1.2 Results and Discussions .......................................................................... 122
6.2 3D Finite Element Model of Indentation of PTFE Thin-Film by Using a
Material Damage Approach ....................................................................................... 125
6.2.1 Materials and Methods ............................................................................ 125
6.2.2 Results and Discussions .......................................................................... 129
6.3 Summary and Conclusions ............................................................................... 135
7 Interfacial Delamination of PTFE Thin Films ...................................................... 137
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7.1 Introduction ...................................................................................................... 137
7.2 Materials and Methods ..................................................................................... 138
7.3 Results and Discussion ..................................................................................... 146
7.4 Summary and Conclusion ................................................................................ 155
8 Summary, Conclusions and Future Work ............................................................. 157
Bibliography ................................................................................................................... 162
Appendix 1 ABAQUS/Explicit Verification by Modeling 2D Interface of Indentation 170
Appendix 2 Material Properties of Fused Silica Using Nano-Indentation ..................... 172
Appendix 3 Mechanical Properties of Thin Polymer Films Using Indentation.............. 175
Appendix 4 Material Properties of Glass Substrates ...................................................... 179
Appendix 5 Green’s Function ......................................................................................... 180
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List of Figures
Figure 2.1. Concentrated point load on a 3D elastic half space. ........................................ 6
Figure 2.2. Hertzian contact on an elastic half-space (a << R). Note that a is the contact
radius at the interface of spherical indenter and flat surface, δH is the indentation
displacement, R is the indenter radius, p0 is the maximum contact pressure. ..................... 9
Figure 2.3. Spherical indenter on a layered elastic half-space. Note that 2b is the contact
diameter at the interface of indenter and thin film; t is film thickness; ur(r, z) and uz(r, z)
are the horizontal and vertical displacement, respectively. .............................................. 11
Figure 2.4. Stresses on an element of thin-film coating................................................... 11
Figure 2.5. Schematic of indentation on PTFE thin-film coating. ................................... 15
Figure 2.6. Schematic of delamination for three types of indentation cases (adapted from
Ritter et al. [13]). Note that type I indentation is elastic deformation under the indenter;
type II is plastic deformation under the indenter; type III is penetration of coating by
indenter. ............................................................................................................................ 19
Figure 2.7. A schematic representation of load, W, as a function of displacement, δ, for
an indentation test [16]. Note that Wmax is the peak load; δmax is the displacement
corresponded at the peak load; δf is the final depth of the contact impression after
unloading; S is the initial unloading stiffness. .................................................................. 22
Figure 2.8. A section view of an indentation showing parameters in the analysis
(Adapted from [16]). Note that δc is the vertical distance along which contact is made; δs
xi
is the displacement between the initial surface and the surface at the perimeter of the
contact; δf is the final depth of the residual hardness impression. .................................... 23
Figure 2.9. Schematic of driving force/resistance curves (Adapted from Anderson [30]).
........................................................................................................................................... 27
Figure 2.10. The schematic representation of interfacial delamination by conical indenter.
Note that 2b is the contact diameter at the interface of indenter and coating; 2c is the
delamination diameter at the interface of coating and substrate; δ is indentation
displacement; Ψ is half conical angle; W is indentation load; t is film thickness. ............ 28
Figure 2.11. Schematic representation of a delaminated, residually stressed film (adapted
from Marshall and Evans [31]). Note that 2c is delamination diameter, σR is the residual
stress, t is film thickness, ΔR is film expansion radius. ..................................................... 29
Figure 2.12. Schematic representation of a stress-free film with indentation-induced
delamination (adapted from Marshall and Evans [31]). Note that σ0 is the indentation
stress, Δ0 is film expansion radius related to indentation volume, V0. .............................. 33
Figure 2.13. Schematic of annular-plate model for delamination (adapted from Rosenfeld
et al. [32]). ......................................................................................................................... 35
Figure 2.14. Material damage initiation and failure [39]. Note that (a) typical ductile
material; (b) elastic and perfectly plastic material. ........................................................... 42
Figure 2.15. Examples of the traction-separation relationship for the cohesive zone
model, for (a) constant traction; (b) trapezoidal traction; (c) bilinear traction. ................ 47
Figure 2.16. Schematic representation of the bilinear traction-separation law
implemented in ABAQUS. ............................................................................................... 51
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Figure 3.1. Schematic representation of PTFE and PE molecular structures [54]. Note
that (a) The zigzag backbone of a PTFE molecular segment; (b) A PTFE molecular chain;
(c) A PE molecular chain. ................................................................................................. 56
Figure 3.2. Crystalline structure of bulk PTFE (Makinson and Tabor [55]). Note that (a)
crystalline block or ‘band’; (b) crystalline slices or ‘striae’ after sliding; (c) hexagonal
array of chains within the slices. ....................................................................................... 59
Figure 3.3. Comparison between experimental data and predicted behavior in uniaxial
tension at different strain-rate (T = 20° C, strain-rates, : 1.2×10-3
/s and 2.3×10-4
/s) [69].
........................................................................................................................................... 65
Figure 3.4. Comparison between predicted and experimental stress relaxation results [69].
........................................................................................................................................... 66
Figure 3.5. Schematic of interfacial and bulk regimes of friction [71]. Note that W is the
normal force, Ffric is the friction force, t1 is the thickness of bulk region, t2 is the thickness
of interfacial region. .......................................................................................................... 68
Figure 3.6. The bulk deformations due to plastic flow and viscoelastic losses [71]. ...... 68
Figure 3.7. The rolling friction of a rigid sphere on bulk PTFE and the quantity, E-1/3
tan(δ), as a function of temperature [71]. .......................................................................... 69
Figure 3.8. Relations between the cohesive energy density (CED) of the polymers and
friction coefficient, wear rate for similar polymer-polymer combinations [77]. .............. 74
Figure 3.9. Frictional and wear characteristics of PTFE with respect to cohesive energy
density (CED). (a) Relations between friction coefficient and the difference in CED for
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dissimilar polymer-PTFE combinations; (b) Relations between the wear rate of polymer
pin and the difference in CED for dissimilar polymer-PTFE combinations [77]. ............ 75
Figure 4.1. Configurations of PTFE frictional and durability tests. ................................ 82
Figure 4.2. Surface roughness, Ra as a function of grit size, g and mean particle diameter,
dp, respectively. ................................................................................................................. 85
Figure 4.3. The COF of aluminum substrate without PTFE thin films. .......................... 86
Figure 4.4. The COF of PTFE on aluminum substrates with different roughness. (a) Ra =
1.28 μm and 2.34 μm; (b) Ra = 0.01 μm and 0.57 μm. ....................................................... 89
Figure 4.5. Durability characteristics of 1 μm PTFE on roughened and polished
aluminum substrates. (a) sliding distance to failure; (b) The COF. .................................. 94
Figure 4.6. The COF comparisons between the prediction by Equation (4.6) and
experiment measurement. ................................................................................................. 99
Figure 4.7. Worn surfaces of PTFE thin films on aluminum substrate with different
roughness Ra (normal force is 5 N, sliding speed is 0.42 mm/s). .................................... 101
Figure 4.8. COF histories on aluminum substrates with different surface roughness. Note
that (a) - (d) were tested on 1 µm PTFE films deposited on aluminum substrate; (e) was
tested on aluminum substrate without PTFE films. ........................................................ 102
Figure 5.1. Schematic representation of PTFE and PFA molecule formulae. ............... 107
Figure 5.2. Schematic representation of poly(V3D3) molecular structure. Note that the
hexagonal units show the intact siloxane rings, acting as cross-linking moieties for
backbone chains [4]. ....................................................................................................... 107
Figure 5.3. The COF as a function of PFA film thickness............................................. 111
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Figure 5.4. The COF as a function of silicone film thickness. ...................................... 114
Figure 5.5. Microscopic observations of 1 μm PFA worn surfaces (sliding speed: 1 mm/s).
......................................................................................................................................... 116
Figure 5.6. Microscopic observations of 1 μm silicone worn surfaces (sliding speed: 1
mm/s). .............................................................................................................................. 117
Figure 6.1. The geometry and mesh configuration of 2D axi-symmetric finite element
model............................................................................................................................... 121
Figure 6.2. Convergence studies of element number in thickness direction of PTFE films
(Hp = 30 MPa). ................................................................................................................ 122
Figure 6.3. Comparison of the experimental and the calculated data to FEA results. ... 124
Figure 6.4. von-Mises stress distribution of indentation at different normal force of (a)
1.5 N and (b) 15 N calculated by finite element analysis................................................ 124
Figure 6.5. The geometry and mesh configuration of 3D finite element model. ........... 128
Figure 6.6. Comparison of the measured and calculated contact width, 2b, to FEA
prediction for 10 μm PTFE films using shear damage model. ........................................ 130
Figure 6.7. Comparison of the measured and calculated contact width, 2b, to FEA result
for 10 μm PTFE films by ductile damage model. ........................................................... 131
Figure 6.8. Effects of equivalent plastic strain, 0pl , (0.1, 0.5, 5% are not shown on the
graph for clarity of illustration) on the contact width, 2b, for Γf = 20 J/m2. ................... 132
Figure 6.9. Effects of bulk fracture toughness, Γf, (40, 500 J/m2 are not shown on the
graph for clarity of illustration) on the contact width, 2b, for0
pl = 1%.......................... 133
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Figure 6.10. von-Mises stress distribution of PTFE thin films at steady state
configurations. Note that a is the contact radius at the interface of ball indenter and
substrate; b is the contact radius at the interface of ball indenter and PTFE films. ........ 134
Figure 6.11. von-Mises stress distribution of the glass substrate at steady state
configurations. ................................................................................................................ 135
Figure 7.1. The geometry and mesh configuration of 3D finite element model for the
delamination simulation. ................................................................................................. 142
Figure 7.2. Convergence studies for 1, 5 and 10 μm PTFE delamination simulations (W =
1 N, Γi = 100 mJ/m2, Young’s modulus E = 3 GPa, yield stress σY = 35 MPa for PTFE
material properties). ........................................................................................................ 144
Figure 7.3. 10 μm PTFE delamination contours predicted by finite element simulation
(Young’s modulus, E = 3 GPa, hardness, σY = 35 MPa). ............................................... 145
Figure 7.4. The coordinates of nodes with interfacial delamination. ............................. 146
Figure 7.5. Thickness effects on the interfacial delamination of PTFE thin films (normal
force: 0.5 N). Note that 2c represents the delamination diameter at the interface. ......... 148
Figure 7.6. Load effects on the interfacial delamination of PTFE thin films (film
thickness: 3 μm). ............................................................................................................. 149
Figure 7.7. The predictions of delamination diameter, 2c, as a function of film thickness
and material properties. ................................................................................................... 152
Figure 7.8. The curve fitting of non-dimensional delamination radius, C , and indentation
force, F , from finite element simulation. ...................................................................... 154
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Figure 7.9. The comparison of 10 μm PTFE finite element simulation to Rosenfeld et al’s
formulations with respect to different material properties. ............................................. 155
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List of Tables
Table 2.1. Geometric constant, ε, for different indenters [17, 22]. .................................. 24
Table 3.1. Characteristics of the helical conformation of PTFE [51]. ............................. 58
Table 3.2. Tensile properties of bulk and free-standing films at room temperature [58].
Note that free-standing film is tested by different authors. .............................................. 62
Table 3.3. Mechanical properties of bulk PTFE in the tension test [62] (strain rate, =
5×10-3
s-1
). ......................................................................................................................... 62
Table 3.4. Mechanical properties of bulk PTFE in the compression test (strain rate, =
10-3
s-1
). Note that failure strain and stress were not reported in this compression test [63].
........................................................................................................................................... 63
Table 4.1. Surface roughness parameter of Ra and σ2 for aluminum substrates. Note that
the mean particle diameter dp for each grit size is adapted from Orvis et al. [91]. ........... 84
Table 4.2. Experimental data for frictional characteristics of 1 μm PTFE coating
deposited on glass substrates. ........................................................................................... 90
Table 4.3. ANOVA test for frictional tests of 1 μm PTFE coating on aluminum substrates
(The response is COF). ..................................................................................................... 91
Table 4.4. Two-way ANOVA test for PTFE coatings on Ra = 2.34 μm aluminum
substrates. .......................................................................................................................... 91
Table 4.5. Two-way ANOVA test for PTFE coating on Ra = 1.28 μm aluminum
substrates. .......................................................................................................................... 91
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Table 4.6. Two-way ANOVA test for PTFE coatings on Ra = 0.57 μm aluminum
substrates. .......................................................................................................................... 91
Table 4.7. Two-way ANOVA test for PTFE coatings on Ra = 0.01 μm aluminum
substrates. .......................................................................................................................... 92
Table 4.8. Experimental data for durability characteristics of 1 μm PTFE coating
deposited on aluminum substrates. Note that ave. represents average value and std.
represents standard deviations. ......................................................................................... 95
Table 4.9. ANOVA test for durability tests of 1 μm PTFE coating on aluminum
substrates (The response is sliding distance to failure). .................................................... 95
Table 4.10. ANOVA test for durability tests of 1 μm PTFE coating on aluminum
substrates (The response is COF). .................................................................................... 96
Table 4.11. The determination of parameters in Equation (4.6) by using curve-fitting. .. 98
Table 5.1. Three-way ANOVA for the COF of PFA thin films.................................... 111
Table 5.2. Two-way ANOVA for the COF of 0.3 μm PFA films. ................................. 111
Table 5.3. Two-way ANOVA for the COF of 1 μm PFA films. .................................... 112
Table 5.4. Two-way ANOVA for the COF of 5 μm PFA films. .................................... 112
Table 5.5. Three-way ANOVA analysis for the COF of silicone films. ........................ 115
Table 5.6. Two-way ANOVA analysis for the COF of 0.3 μm silicone films. .............. 115
Table 5.7. Two-way ANOVA analysis for the COF of 1 μm silicone films. ................. 115
Table 6.1. Material properties in 2D axisymmetric finite element analysis [3, 62]. ...... 120
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Table 6.2. Material properties used in 3D finite element simulation using material
damage model. Note that 0
pl is the equivalent plastic strain for damage initiation, Γf is the
fracture energy for damage evolution. ............................................................................ 126
Table 7.1. Material properties in 3D FEA delamination model. Note: Diamond and glass
properties were taken from Oliver and Pharr [16]; The PTFE properties were measured
by using nano-indentation, except for Poisson’s ratio, ν, reported in Karnath et al. [3]. 141
Table 7.2. Mesh sizes and aspect ratios for modeling all thick PTFE films. ................. 141
Table 7.3. The number and type of elements implemented in delamination model. ..... 144
Table 7.4. Experimentally measured delamination diameter, 2c, for different film
thickness, t, and normal force, W. Note that Ave. 2c is the average delamination diameter;
Std. Dev. 2c is its standard deviation. ............................................................................. 150
xx
Nomenclature
Chapter 2
W Normal force
Wmax Peak normal force
σrr, σθθ, σzz Normal stress components
rr , , zz Averaged normal stresses in the r, θ and z- direction
r , , z Averaged strains in the r, θ and z-direction
ρ Distance in cylindrical coordinates
γrz Shear strain
τrz, τrθ, τθz Shear stress components
Fr, Fz Body forces in the r- and z-direction
ur(r, z), uz(r, z) Displacements in r-, z-direction
r, θ, z Cylindrical coordinate axes
G Shear modulus
E* The composite Young’s modulus
E, Ei Young’s modulus of the materials in contact
ν, νi Poisson’s ratio of the materials in contact
Er Reduced Young’s modulus
a, 2a Contact radius, diameter between the indenter and substrate
b, 2b Contact radius, diameter between the indenter and thin-film layer
R, Ri Radius of the contact bodies
δH Approach distance in the substrate
δ Total indentation approach
p0 Maximum contact pressure
τi (i = 1, 2) Interfacial shear stress
t Film thickness
xxi
f(r) A function describing the indenter profile
Hp Hardness of PTFE
H Hardness of materials
Hc, Hs Hardness of coating, substrate
FH Hertzian contact force
σY Yield stress
k, λ, ω, ς Coefficients in Ritter’s equations
K1, I1, K1’, I1’ Modified Bessel functions of the second kind and their derivatives
a0, b0 Half the indenter diagonal in the substrate and coating, respectively
S Initial unloading stiffness
β A correction parameter for indenter geometry
Ap Projected area of contact
δmax Maximum indentation displacement at peak load
δf Final depth of contact impression
δs Deflection of the surface
δc Contact depth
ε Geometric constant for indenter
F(δc) Area/shape function with different indenter geometry
Ci (i =1, …, 8) Constants in the shape/area function
Wa Adhesion energy
Γi Interfacial fracture toughness, fracture energy release rate
Γc Experimental measurement of interfacial fracture toughness
γf, γs, γfs Surface energies of film, substrate and interface
Γp Plastic dissipation energy
Γfric Energy loss due to friction
ΓR Fracture resistance
U, UR, UR’ Total strain energy of the system
Af Crack area
σ0, σb Stress induced by indentation
xxii
σR Residual stress
σc Critical stress of buckling
Uc, Us Energy stored in the film and remaining system, respectively
UB Energy difference between buckled and unbuckled plates
c, 2c Delamination radius
Ψ Half conical angle
ΔR, Δ0 Expansion radius of the plate
B A constant in the derivation of interfacial fracture toughness
α The slope of buckling load versus displacement
V0 Indentation volume
κ Constant related to material properties
ξ Constant related to indenter geometry
W , C Non-dimensional force and radius
res Non-dimensional residual stress
b Bergers vector
tc Critical film thickness for delamination
k a constant used in the plate theory
D Damage variable
0
pl Equivalent plastic strain for damage initiation
pl
f Equivalent plastic strain for material failure
0
pl Equivalent plastic strain rate
εpl
Equivalent plastic strain variable
pl
fu Equivalent plastic displacement for material failure
upl
Equivalent plastic displacement variable
Γf Bulk fracture energy
L Characteristic length of finite element mesh
η Stress state parameter
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σi (i = 1, 2, 3) Principal stress
σeqv von-Mises stress
ξs Shear stress ratio
τmax Maximum shear stress
ks Material parameter
p Pressure stress
K0 Initial elastic stiffness
σc0, τc0 Maximum cohesive strengths in normal and shear directions
σm Magnitude of effective traction vector
σm0 Critical traction magnitude for damage initiation
δ0, δc Critical separation for damage initiation and failure, respectively
δmax Maximum separation
δm Magnitude of effective displacement
δn, δs Displacements in normal and shear directions
δmc, δm0 Critical effective displacement for failure and damage initiation
ΓI, ΓII Interfacial fracture toughness of mode-I and –II, respectively
WI, WII Work done mode-I and –II tractions, respectively
φ Phase angle
Γm Interfacial fracture toughness in mixed-mode
α0 Power coefficient in energy-based damage evolution criterion
Chapter 3, 4 and 5
Strain rate
T Temperature
μ The coefficient of friction
μr The rolling coefficient of friction
t1 Thickness of bulk region
t2 Thickness of interfacial region
xxiv
W Normal force
Ffric Friction force
R Indenter radius
Ar Real contact area
p Flow stress
τ Interfacial Shear strength
τ0 A constant of adhesion friction
α Pressure coefficient
tan(δ) Loss tangent of material
Tg Glass-transition temperature
E, E*, Ei Young’s modulus
ν, νi Poisson’s ratio
Ra Average surface roughness
σ2 Variance of surface roughness
g Grit size
dp Mean particle diameter
v sliding speed
c1, c2, c3, c4 Constants for speed effects on COF
p-value null-hypothesis parameter of ANOVA
Chapter 6, 7 and 8
tr Residual thickness of film
a, 2a Contact radius, diameter at the interface of ball indenter and
substrate
b, 2b Contact radius, diameter at the interface of ball indenter and PTFE
coating
c, 2c Delamination radius, diameter at the interface
xxv
E, ν Young’s modulus and Poisson’s ratio
σY Yield stress of PTFE coating
Hp, H Film hardness
W Normal force
Γf Bulk fracture energy
Γi Evaluation of interfacial fracture toughness
Γc Interfacial fracture toughness determined by experiment
K0 Initial elastic stiffness
t Film thickness
α Coefficient of linear interpolation
0
pl Equivalent plastic strain for damage initiation
L0, L1, L2, h Dimensions of finite element mesh
NL0, NL1, Nh, Nt Number of elements in finite element mesh
r Average radius of delamination in simulation
ri Delamination radius in simulation
s Standard deviation of delamination radius
F , C Non-dimensional force and delamination radius, respectively
p, m, n, q Coefficients for non-dimensional force and radius
Appendices
R Indenter radius
E, ν Young’s modulus, Poisson’s ratio
G Shear modulus
H Hardness of materials
b Contact radius of ball indenter and substrate
τ0 Interfacial shear stress
S Initial unloading stiffness
δs Deflection of surface
xxvi
δc Contact depth
ε Correction factor for indenter geometry
δmax Maximum indentation displacement
ΔW, Δδ Variations of indentation load and displacement, respectively
Ap Projected contact area
g(x, s) Green’s function
v(x), p(x) Displacement and distributed force as a function of coordinates
1
1 Introduction
Polymers are a large class of naturally occurring or artificially synthesized, covalently
bonded materials ranging from natural biopolymers, such as DNA, RNA and proteins, to
plastics such as synthetic rubber and silicone [1]. Recently, as more functional polymeric
materials are widely applied in many areas, efforts have been made to investigate
material properties and characteristics of polymers by using theoretical and experimental
methodologies.
Gleason’s group in MIT developed two thin-film polymer coating technologies, including
PTFE (Polytetrafluoroethylene) by using Hot Filament-Chemical Vapor Deposition
(HFCVD) and Poly(V3D3) (poly(trivinyltrimethylcyclotri-siloxane)) by using Initiated
Chemical Vapor Deposition (iCVD) [2, 3]. These polymers have shown extraordinary
characteristics in terms of material properties. In particular, low coefficient of friction
(COF), chemical inertness, and low dielectric constant of PTFE allow its application in
micromechanical devices and integrated circuits [3]. Poly (V3D3) can be applied to
encapsulate implantable microelectronics and be used to protect circuit boards from
environmental effects due to its biocompatibility as well as dielectric and insulating
properties [4, 5].
However, tribological and mechanical properties of these two thin-film coatings are not
clearly understood, which would otherwise significantly influence their performance in
practice. This motivates this dissertation. The general goal of this work is to study the
2
mechanism of friction and wear of polymer coatings and their adhesion properties from
the perspective of contact and fracture mechanics. In particular, this work started by
conducting friction and durability (wear) tests of thin-film PTFE coatings on smooth and
roughened aluminum substrates. Frictional characteristics of thin PFA and Poly (V3D3)
films on glass substrates are also studied. The analysis of variance (ANOVA) is
introduced to quantitatively examine the relative contributions of the effects, including
normal force, sliding speed, and surface roughness on COF and durability.
Another major concern related to successful performance of thin polymer coatings is their
adhesion properties. A micro-indentation based technique was used to characterize
adhesion of thin polymer films. In this method interfacial delamination is induced and the
delamination diameter is compared to a model of interfacial fracture. A 3D finite element
model was developed to investigate the delamination radius as a function of indentation
force, film thickness, material properties and interfacial fracture toughness. In practice,
the interfacial fracture toughness evaluated by indentation tests, is a system parameter,
which consists of contributions from the adhesion energy at the film-substrate interface,
and the plastic dissipation underneath the indenter during the indentation.
Chapter 2 introduces a literature survey related to this work, including indentation
mechanics of a half-space and a layered half-space with elastic or elastic-plastic
properties, mathematical description of interfacial delamination, bulk material
damage/failure criteria and cohesive zone model (CZM) implemented in finite element
analysis. In Chapter 3 we present an overview of PTFE material properties mostly in bulk
3
form and thin-film form. In Chapter 4 the friction and durability characteristics of thin-
film PTFE coatings on rough aluminum substrates are discussed. Effects of the multiple
variables are analyzed based on the statistical variation of the measurements. In Chapter 5
we present the friction characteristics of thin silicone and PFA films deposited on glass
substrates. In Chapter 6 finite element simulation for soft polymer coatings using bulk
damage/failure criteria is presented and compared to the previously developed close-form
formulations. In Chapter 7 experimental investigation and numerical simulation of
interfacial delamination are presented and a relation between non-dimensional
indentation force and delamination radius is developed to evaluate the interfacial fracture
toughness. Chapter 8 gives a summary and conclusions, and recommendations for future
research. Appendices 1-5 contain verification of ABAQUS/Explicit for the indentation
simulation, material properties of thin-film polymer coatings and glass substrates using
nano-indentation and Green’s function.
4
2 Mechanics Review for Thin-Film Polymer
Tribology
In this chapter, mechanics literature related to this dissertation is reviewed. This includes
the mechanics of indentation of a half-space and that of a layered half-space with elastic
or elastic-plastic material properties. The indentation methodology and its mathematical
descriptions are reviewed to help evaluate material properties. The load-interfacial
delamination radius relationships in thin-film indentation are reviewed. The bulk material
damage and cohesive zone models, used in finite element simulations, are presented.
2.1 Indentation of an Elastic Half-Space
2.1.1 Point Load on an Elastic Half-Space
Mechanics of an elastic half-space subjected to a concentrated load, W, as shown in
Figure 2.1, can be modeled by using the equations of equilibrium expressed in the
cylindrical coordinate system. For an axis-symmetric case, the equations of equilibrium
are given as follows,
10r rrrr rz
rFr r z r
(2.1)
1 10zrz zz
rz zFr r z r
(2.2)
5
where σrr, σθθ, σzz are the normal stress components and τrz, τrθ, τθz are the shear stress
components, Fr, Fz are the body forces in unit of load per unit volume. The coordinate
axes r, θ and z are defined in Figure 2.1. The normal and shear stress components σzz and
τrz on the surface of the elastic half-space (z = 0), away the concentrated load are
negligible, and expressed as follows,
,0 0zz r , r (2.3)
0 0,rz r , r (2.4)
The deformation of an elastic half-space due to a concentrated force, W, located at the
origin (Figure 2.1) can be expressed by the following displacement components in the r-
and z-directions [6],
3, 1 2
4r
W rz zu r z
G r
(2.5)
2
3
2 1,
4z
W zu r z
G
(2.6)
where 2 2r z , G is the shear modulus, ν is the Poisson’s ratio. The corresponding
normal and shear stress variations in the half-space are expressed as follows [7],
2
2 2 5
1 31 2
2rr
W z zr
r r
(2.7)
6
2 2 3
11 2
2
W z z
r r
(2.8)
3
5
3
2zz
W z
(2.9)
2
5
3
2rz
W rz
(2.10)
Figure 2.1. Concentrated point load on a 3D elastic half space.
7
2.1.2 Hertzian Contact
Hertz contact theory deals with the contact of two elastic spheres. In what follows friction
and adhesion at the interface are neglected. When two spheres are pressed against each
other a circular contact region with radius, a, develops. It is assumed that a is much
smaller than the radius of curvature of the two contacting bodies. Figure 2.2
schematically depicts the contact of a rigid sphere of radius, R, with an elastic half-space.
In this contact scenario, the normal stress component, σzz, and the shear stress
components, τrz and τθz, are zero outside of the contact region. Therefore, the boundary
conditions are given as follows,
0 0( , )z r and 0 0,rzτ r , r a (2.11)
,0 0zz r , r a (2.12)
The z-direction displacement at the contact interface (r ≤ a) is expressed as follows,
2 2
2 2 1,0 1
2 2H H Hz
r ru r R R r R R
R R
(2.13)
where δH is the approach distance of the spherical indenter.
Using Equation (2.6) and the Green’s function, shown in Appendix 5, the deflection of
the elastic half space under the spherical indenter is found as follows,
8
2 21 1
,0 ( , )z
S
Wu r p s dsd
E r E
2
2 20
*
12
4
πpνa r
E a
for r ≤ a (2.14)
where p0 is the maximum contact pressure, E* is the composite Young’s modulus,
defined as follows,
2 2
1 2
*
1 2
1 11
E E E
(2.15)
where Ei, νi (i = 1, 2) are Young’s modulus and Poisson’s ratio of the materials.
By combining Equations (2.13), (2.14) and (2.15), the contact radius, a, is found as
follows,
1/3
*
3
4
WRa
E
(2.16)
Similarly the contact approach δH is found as,
1/3
2/3
* *2
3 9
4 16H
WW
E a E R
(2.17)
For two contacting elastic spheres, R can be expressed as follows,
1 2
1 1 1
R R R (2.18)
9
where Ri (i = 1, 2) are the radii of contacting bodies.
Figure 2.2. Hertzian contact on an elastic half-space (a << R). Note that a is the contact
radius at the interface of spherical indenter and flat surface, δH is the indentation
displacement, R is the indenter radius, p0 is the maximum contact pressure.
2.2 Indentation of a Layered Elastic Half-Space
Figure 2.3 shows a schematic of indentation of a thin elastic film on a rigid substrate.
Matthewson presents the governing equilibrium equation from which analytical
expressions of stress and strain in the film are obtained [8]. He assumed that the stress
can be averaged in the thickness direction. The stress acting on a small volume of the
10
thin-film coating is illustrated in Figure 2.4. The interface between the film and the
substrate is assumed to be frictionless, where the shear stress τ0 is zero. The equilibrium
equation in r-direction is shown as follows,
1 0rrrrd
dr r t
(2.19)
where rr , are the averaged radial and circumferential stresses, τ1 is the shear stress
acting at the interface of coating-indenter interface, and t is the film thickness.
In the contact region (r < b), the shear stress at z = t is zero and the average strain in the z-
direction, z , is determined by the indenter geometry. Thus, the boundary conditions are
expressed as follows,
0 , , 0rzr t G r t (2.20)
z
f r
t (2.21)
where f(r) is a function describing the indenter profile.
11
Figure 2.3. Spherical indenter on a layered elastic half-space. Note that 2b is the contact
diameter at the interface of indenter and thin film; t is film thickness; ur(r, z) and uz(r, z)
are the horizontal and vertical displacement, respectively.
Figure 2.4. Stresses on an element of thin-film coating.
Outside the contact region (r > b), the average stress in the z-direction, zz , is zero,
therefore, the boundary condition is shown as follows,
12
2
2 01 2
zz r z z
GG
(2.22)
The interfacial shear and normal stress as a function of contact radius were presented for
both spherical and conical indenters. The results indicate the stress distribution is quite
sensitive to the Poisson’s ratio of the coating and the interfacial shear stress for nearly
incompressible materials is largest. The normalized interfacial shear stress distribution
given by Matthewson [8] was reproduced by using our finite element method, and
presented in the Appendix 1.
Chadwick [9] discussed axisymmetric indentation of frictionless spherical indenter on an
incompressible elastic layer by using the Wiener-Hopf integral equations. The governing
equilibrium equations are the same as Equations (2.1) and (2.2). The well-bonded and
slippery interfaces between a thin film and a substrate were considered. In particular, for
both cases, the boundary conditions described in Figure 2.3 are expressed as follows,
2
,0 ,2
z
ru r r b
R (2.23)
,0 0,zz r r b (2.24)
,0 0,rz r r b (2.25)
Note that the contact radius at the indenter-thin film interface is represented by b while
the contact radius at the indenter-substrate (elastic half-space) is represented by a for ease
13
of identification. Additionally, along a well-bonded interface the radial and normal
displacement components are zero,
, 0ru r t , , 0zu r t (2.26)
On the other hand, along a slippery interface the shear stress is zero,
, 0rz r t (2.27)
For these conditions the indentation force, W, as a function of the film thickness t is
shown to be,
2 3
3
2
3
ERW
t
for bonded interface (2.28)
t
ERW
3
2 2 for slippery interface (2.29)
where E is the Young’s modulus of the film, t is the film thickness, and R is the indenter
radius. Equations (2.28) and (2.29) show that the indentation force, W, on bonded layers
is inversely proportional to the cubic of film thickness while only to the film thickness on
slippery layers. This also indicates a larger force is required to equally indent a bonded
layer compared to a slipping layer.
Chen and Engel [10] proposed a general numerical method to analyze the contact stress
of one or two parallel elastic layers bonded to a homogeneous half-space. In particular,
the boundary value problems for both flat and parabolic punches were solved using
14
integral least square approach and Reissner energy method. The results showed that the
normal stress between a layer and flat punch is tensile when the layers are thinner.
However, the normal stress at the interface becomes compressive when no separation is
between the layer and punch.
2.3 Indentation on an Elastic-Plastic Layered Half-Space
Kral et al. analyzed repeated indentations of an elastic-plastic, layered medium by a rigid
sphere using finite element analysis [11]. They found that with full plasticity the contact
pressure develops a high pressure peak at the contact edge, instead of a relatively uniform
contact pressure on a homogeneous half-space. A significant tensile radial stress develops
near the contact edge under the maximum load and increases for thinner, stiffer, and
harder layers and by increasing strain hardening of the layer as well as the substrate. A
tensile hoop stress, formed at the surface near the contact edge, decreases with increasing
strain hardening. Both tensile radial and hoop stresses were thought to be responsible for
ring and radial cracks of the layer interface.
Using the same finite element model, Kral et al. reported the mechanics of the substrate
[12]. In particular, they showed that large tensile, radial and hoop stresses develop in the
film-substrate interface under the maximum load and increases with layer stiffness,
hardness and the strain-hardening exponents. The interfacial shear stress and maximum
von-Mises stress developed in the substrate depend only on strain-hardening exponent in
fully plastic deformation. In addition, the stresses at the interface remain predominantly
15
compressive for both loading and unloading cycles as the tensile hoop stress arises as a
band surrounding the plastic zone in the substrate and prevents the expansion of the
plastic zone.
2.4 Indentation on Soft Thin-Film Material
Karnath et al. developed closed-form formulae to model the indentation of plastically
flowing PTFE layer, schematically shown in Figure 2.5, deposited on the glass substrate
by using Hertz contact theory [3].
Figure 2.5. Schematic of indentation on PTFE thin-film coating.
16
In the case of purely elastic contact, the contact of a spherical indenter and the flat
surface is described by Hertzian contact equations, where the contact radius, a is
evaluated by Equation (2.16). The approach, δH, of the two surfaces is given by using
Equation (2.17).
The indentation of a thin PTFE film of thickness, t, deposited over a substrate by a
spherical indenter can be split into two processes. When the indentation load is below a
critical value, the ball makes contact with the PTFE film only; and the external force, W,
is balanced by the restoring force from the plastically deforming PTFE layer. As the load
is increased, the ball and the glass substrate eventually make contact; and the external
force, W, is balanced by the combined effects of the deforming PTFE film and glass.
These relationships can be expressed as follows,
2
2 2
,
,
p
H p
H πb δ tW
δ tF H π b a
(2.30)
where Hp is the hardness of the PTFE and FH is the Hertzian contact force at the ball
substrate (glass) interface. In addition, the relation of FH to δH is shown as follows,
*2/12/3
3
4ERF HH (2.31)
where R is the radius of ball indenter, E* is the composite Young’s modulus of two
contacting bodies, given in Equation (2.15).
The geometric relationship,
17
RRRb 2222 (2.32)
can be used to describe the width of the indentation on the top of the PTFE film as a
function of the indenter approach distance, δ. Note that when δ is greater than t, the glass
substrate deforms by δH and the following relationship prevails δ = δH + t. By combining
Equations (2.30) – (2.32), a relationship between the external force, W, and the approach
distance of the indenter, δ, is obtained as follows,
1/2 3/2
2 , ( )
( )4 / 3 2 ,
p
H p
H π R δ δ δ tW
δ tR E δ H π R δ δ R δ t
(2.33)
where 3p YH σ and σY is the yield stress of the PTFE film.
Ritter et al. classified the indentation on thin polymer coatings as three different modes
when the coating underneath the indenter has elastic or plastic deformation and the
polymer coating was entirely penetrated, which are schematically shown in Figure 2.6
[13]. In the first mode which the coating was elastically deformed, the equilibrium
equations, Equation (2.19), in terms of normal stresses and strains averaged through the
coating thickness was used to derive the indentation load, W, as a function of contact
radius, b. The mathematical expression was shown as follows,
4 2 42 2
1
6 142
1 2 6 1 2 3 8 2 2
tG tb kb b b GbW b I G b
R t Rt Rt
(2.34)
The coefficients, k, λ, ω, ς, are determined as follows:
18
1/2
3 1 2
2 1k
1 1
1 1 1 1
/ / / 1 6 / 2
/ / / / 1 2
b t K b t K b t t R
kK b t I kb t I kb t K b t
1
1
1 64
4 / 2 1 2
t kbkI
K b t R t
1/2
6 1
4
where 1 /K b t , 1 /I kb t , 1 /K b t , 1 /I kb t are the modified Bessel functions of the
second kind and their derivatives respectively; t is the film thickness, G is the shear
modulus, R is the indenter radius, b is the contact radius at the indenter-coating interface.
In the second mode, which the coating beneath the spherical indenter is plastically
deformed, the indentation force, W, is balanced with the coating hardness. The load, W,
as a function of contact radius, b, is shown as follows [14],
2
cW H b (2.35)
where Hc is the hardness of coating.
19
Figure 2.6. Schematic of delamination for three types of indentation cases (adapted from
Ritter et al. [13]). Note that type I indentation is elastic deformation under the indenter;
type II is plastic deformation under the indenter; type III is penetration of coating by
indenter.
When the coating was completely penetrated, typically by a Vickers indenter (pyramidal
geometry), the indentation force, W, is balanced with the hardness from the substrate as
well as coating, which can be expressed as follows [13, 15],
20
2 2 2
0 0 02 2c sW H b a H a (2.36)
where Hs is the hardness of substrate, a0, b0 are half the indenter diagonal in the substrate
and the coating, respectively.
2.5 Material Properties Investigation by Indentation
In general, micro and nano indentation can be used to determine mechanical properties
(Young’s modulus, E, and the hardness, H) of materials which are difficult to test by
other methods. Therefore, indentation is especially useful for evaluating material
properties of thin films in micro- or nano-scale [16, 17]. ASTM E2546 standard gives the
specifications of instrumented indentation testing. In particular, according to this standard
the test sample thickness should be at least ten times greater than the indentation depth
and six times greater than the indentation radius to avoid the substrate effect or residual
stress concentration [18]. The reason behind the rule of 10% indentation approach with
respect to film thickness is that the plastic zone associated with the indentation is entirely
contained within the film and negligible elastic deformation of the substrate contributes
to the evaluations of material properties [19].
It is found that displacements recovered during first unloading are not elastic in a range
of materials, for example, polymers, which lead to inaccurate evaluation of elastic
properties. It is reported that a nose phenomenon at the loading-unloading peak is
detected as the creeping effect in a wide range of polymers [20]. The ways to minimize
creep effects are to include hold periods of peak load in loading sequence or to use
21
unloading curves obtained after several cycles of loading [16]. Additionally, when a very
thin film (less than 1 μm) is indented, thermal drift, which is brought by changes in the
dimension of the contact (indenter, specimen) from thermal expansion or contraction due
to temperature changes, becomes a significant error source to affect the determination of
material properties [19]. In order to minimize the effects of thermal drift, the indentation
test is typically performed in an insulated enclosure, or in thermal equilibrium,
established by waiting a sufficient period.
Figure 2.7 shows a typical loading-unloading graph of indentation. The material
properties are determined based on the unloading curve. In particular, the initial
unloading stiffness, S, is evaluated as follows,
2r p
WS E A
(2.37)
where β is a correction parameter, Er is the reduced modulus, Ap is the projected area of
contact. The value of correction parameter, β, depends on the shape of the indenter,
which varies from 1.012, 1.034 to 1.067 for Vicker’s, Berkovich and conical indenters,
respectively [21]. The reduced modulus, Er, is defined as follows,
2 21 11 s i
r s iE E E
(2.38)
where Es, Ei are Young’s moduli of the specimen and indenter, respectively, and νs, νi are
the Poisson’s ratios of the specimen and the indenter, respectively. Thus, the Young’s
22
modulus of the specimen, Es, shown in Equation (2.38), is expressed as follows with
respect to Er and Ei,
12
211
1i
s s
r i
EE E
(2.39)
Figure 2.7. A schematic representation of load, W, as a function of displacement, δ, for
an indentation test [16]. Note that Wmax is the peak load; δmax is the displacement
corresponded at the peak load; δf is the final depth of the contact impression after
unloading; S is the initial unloading stiffness.
23
The hardness of specimen, H, is evaluated as follows,
max
p
WH
A (2.40)
A relationship between the projected area of indentation as a function of the tip
displacement was obtained for a variety of indenters. Figure 2.8 schematically shows a
section view of indentation, where the parameters in the analysis are specified. The total
indentation displacement, δ, is expressed in terms of the contact depth, δc, as follows:
s c (2.41)
Figure 2.8. A section view of an indentation showing parameters in the analysis
(Adapted from [16]). Note that δc is the vertical distance along which contact is made; δs
is the displacement between the initial surface and the surface at the perimeter of the
contact; δf is the final depth of the residual hardness impression.
24
Table 2.1. Geometric constant, ε, for different indenters [17, 22].
indenter geometry conical Spherical Berkovich Flat
geometric constant, ε 0.72 0.75 0.75 1.00
The deflection of the surface at the contact perimeter, δs, for different indenters, was
investigated by Sneddon [22], Doerner and Nix [23], and Oliver and Pharr [17]. It is
expressed in terms of the maximum indentation load, Wmax, and the unloading stiffness, S,
as follows,
maxs
W
S
(2.42)
where ε is a geometric constant, given in Table 2.1 for different indenter types.
The relation of the projected contact area, Ap, and the contact depth, δc, which is
expressed as the area or shape function, F(δc), was established experimentally. Pharr [24]
and Sakharova et al. [25] presented the area functions, F(δc), for a range of indenters with
different geometries. In particular, the projected contact area, Ap, as a function of contact
depth, δc, for a geometrically-perfect Berkovich indenter is shown as follows [16],
224.5p c cA F (2.43)
However, the imperfection of the tip can lead to slightly different relations. Oliver and
Pharr presented the relationship of contact area, Ap, and contact depth, δc, to characterize
other indenters which deviate from the Berkovich indenter as follows [17],
25
2 1 1/2 1/4 1/128
1 2 3 824.5p c c c c c cA F C C C C (2.44)
where C1 - C8 are constants. The projected contact area, Ap, is critical for evaluating the
initial unloading stiffness, S, and specimen hardness, Hs.
2.6 Indentation-Induced Interfacial Delamination
2.6.1 Definitions of Adhesion Energy and Interfacial Fracture
Toughness
Volinsky et al. [26] elaborate on the definitions of adhesion energy and interfacial
fracture toughness. In particular, the adhesion energy, or the true work of adhesion at the
interface, Wa, is the amount of energy required to create new surfaces by breaking the
bonds of materials, which is mathematically expressed as follows,
a f s fsW (2.45)
where γf, γs are the surface energies of the film and substrate, respectively, γfs is the
energy of the interface . The parameter, Wa, is an intrinsic property of the film-substrate
interface, which depends on different types of bonding, and it is a constant for a given
film-substrate interface.
Interfacial fracture toughness, Γi, or the practical work of adhesion, is the amount of
energy to delaminate thin films from the substrate. Ideally, the adhesion energy, Wa, is
assumed to be equal to interfacial fracture toughness, Γi, by Griffith fracture theory.
26
However, the process of thin film delamination from the substrate usually involves
plastic deformation, which makes it difficult to extract the adhesion energy from the total
energy measured. In general, the interfacial fracture toughness, Γi, can be quantified as
follows [26, 27],
i a p fricW (2.46)
where Γp is the energy spent in plastic deformation of the film and the substrate, Γfric is
the energy loss due to friction. In fact, Γp and Γfric are functions of the adhesion energy,
Wa, in many cases.
In fracture mechanics, the strain energy release rate, or the crack driving force is used as
a measure of the interfacial fracture toughness, Γi [28, 29]. In a controlled indentation test,
the interfacial fracture toughness, Γi, is determined by obtaining the equilibrium state
with the interfacial fracture resistance, ΓR, which is mathematically shown as follows,
Ri R
f
U
A
(2.47)
where UR is the total strain energy of the system, Af is the crack area.
In particular, when Γi > ΓR, unstable crack growth takes place; when Γi ≤ ΓR, the crack
growth is stable. In the experiments reported in Chapter 7, we measure the interfacial
fracture toughness, Γc, which aims to differentiate the theoretical evaluation of Γi, at
initiation of crack growth instead of obtaining the shape of the R-curve (resistance curve),
which indicates the material resistance to crack extension [30]. Figure 2.9 shows a rising
27
Figure 2.9. Schematic of driving force/resistance curves (Adapted from Anderson [30]).
R-curve for determining the interfacial fracture toughness, Γc. R-curve can be different
for a variety of materials and structures.
Next section describes a general method to theoretically model indentation-induced
delamination by evaluating the strain energy of the system.
2.6.2 Mathematical Descriptions of Interfacial Delamination
Indentation-induced delamination, schematically shown in Figure 2.10, can be used as a
controlled adhesion test to measure the adhesive property of a thin film which is affected
by its fracture resistance and strength.
Marshall and Evans [31] modeled the delamination process by assuming that the thin film
behaves as a clamped plate. They calculated the energy release rate, Γi, from changes in
28
the strain energy of the system, which includes the effects of indentation and the residual
stresses. First, they investigated the delamination of a thin film with biaxial residual
stress, σR. The total strain energy, UR, includes the energy in the film above the crack, Uc,
as well as in the remaining system, Us.
R c sU U U (2.48)
Figure 2.10. The schematic representation of interfacial delamination by conical
indenter. Note that 2b is the contact diameter at the interface of indenter and coating; 2c
is the delamination diameter at the interface of coating and substrate; δ is indentation
displacement; Ψ is half conical angle; W is indentation load; t is film thickness.
29
Figure 2.11. Schematic representation of a delaminated, residually stressed film (adapted
from Marshall and Evans [31]). Note that 2c is delamination diameter, σR is the residual
stress, t is film thickness, ΔR is film expansion radius.
In order to evaluate the energy, Uc, a section of film around the perimeter of the crack
was assumed to be subjected to the residual stress, σR, as shown in Figure 2.11.
The expansion radius of the plate, ΔR, due to the residual stress, σR, can be evaluated as
follows,
1 f R
R
f
c
E
(2.49)
where Ef, νf are the Young’s modulus and the Poisson’s ratio of the film, c is the
delamination radius. The increase of strain energy in the film, Uc, can be calculated as
follows,
30
2 21 f R
c R R
f
t cU ct
E
(2.50)
The total strain energy, UR, in the system becomes,
2 21 f R
R s
f
t cU U
E
(2.51)
Since UR is independent of crack length for an unbuckled plate, UR, is shown as follows:
1 f
R
f
t BU
E
(2.52)
where B is a constant. Thus, the remaining strain energy, Us, can be expressed by
combining Equations (2.51) and (2.52) as follows,
2 21 f R
s
f
t B cU
E
(2.53)
When the edge stress, σR, is greater than a critical value, σc, the film buckles. The
buckling stress for a clamped circular plate was shown as follows [31],
2
2
14.68
12 1c f
tE
c
(2.54)
The difference in strain energy between buckled and unbuckled plates can be expressed
as follows,
31
2 21 1f R c
B
f
t cU
E
(2.55)
where α represents the slope of buckling load versus edge displacement after buckling. It
can be expressed in terms of the Poisson’s ratio, ν, as follows [27],
1
11 0.902 1
(2.56)
Buckling of plate reduces the total strain energy. Thus the total strain energy of the
system, RU , becomes
R R BU U U (2.57)
Based on the definition, the energy release rate, Γi, for the buckled plate R c is
evaluated as follows,
2 21 11
2
f R cRi
f
tdU
c dc E
(2.58)
The second case by Marshall and Evans considers the delamination that occurs at the
interface of a stress-free film as shown in Figure 2.12. The deformation within the plastic
zone around the contact was found by considering volume conservation and radial
displacement mode. The expansion radius Δ0 in terms of indentation volume V0 can be
shown as follows,
32
0 02V ct (2.59)
The stress induced by the indentation at the edge of the plate, σ0, is evaluated as follows,
0 0
0 21 2 1
f f
f f
E E V
v c c t
(2.60)
The total strain energy, UR, when the film is not buckled ( 0 c ), includes the strain
energy, Uc, in the film caused by the stress of indentation, σ0, and the residual strain
energy, Us, stored in the remaining system, as shown in Figure 2.12. The mathematical
expression is shown as follows,
2
2 22 2
0011
2
ff
R c s
f f
t B ct cU U U
E E
(2.61)
where B is constant. The energy release rate, Γi, is calculated as follows,
2 2
011
2 2
fRi
f
tdU
c dc E
(2.62)
When the film is buckled ( 0 c ), the difference of strain energy due to buckling can
be expressed as follows,
2 2
01 1f c
B
f
t cU
E
(2.63)
33
Figure 2.12. Schematic representation of a stress-free film with indentation-induced
delamination (adapted from Marshall and Evans [31]). Note that σ0 is the indentation
stress, Δ0 is film expansion radius related to indentation volume, V0.
The total strain energy, RU , is shown as follows,
R R BU U U (2.64)
Thus the energy release rate, Γi, for buckled plate is shown as follows,
22
0 0
11 1
2f c
i
f
t
E
34
22
0 0
1 11
2 1 0.902 1f c
f
t
E
(2.65)
In the third case, delamination of the interface is analyzed with the combined effects of
residual stress and indentation stress [31]. The total strain energy, U, can be evaluated
with a combination of Equations (2.53), (2.61) and (2.63), where the edge stress is
changed as the sum of residual stress, σR, and indentation stress, σ0, as follows,
2
20 2
0
2 2 2 2
0
11
2
1
ff
i B s R
f
R c R
B ctU U U U c
E
c B c
(2.66)
The derivations of energy release rate, Γi, presented above are based on the assumption of
mixed-mode, where the mode-II predominates the crack tip [29].
Rosenfeld et al. investigated the energy release rate, Γi, for an epoxy by neglecting the
residual stress and buckling [32]. Figure 2.13 shows the interfacial delamination by
conical indenter, which was schematically shown in Figure 2.10, was modeled as an
annular plate. The outside surface of the plate, which adheres to the remaining portion of
coating, was constrained and a fixed pressure, σb (r = b), was applied on the inside
surface. The radial and circumferential stresses, σrr and σθθ, as a function of radius are
shown as follows,
2 2
2 2
1 /
1 /rr b
c r
c b
(2.67)
35
2 2
2 2
1 /
1 /b
c r
c b
(2.68)
where1
1
f
f
v
, νf is the Poisson’s ratio of the epoxy coating.
The total strain energy, UR, of the plate is evaluated by the following integral,
2 2 2c
R rr f rrb
f
tU rdr
E
(2.69)
where t is the coating thickness, Ef is the Young’s modulus of epoxy. The energy release
rate, Γi, can be evaluated based on the strain energy, U, as follows:
22 2
2
2 11 1
2 1 / 1
f bRi
f f f
tdU
c dc E c b
(2.70)
Figure 2.13. Schematic of annular-plate model for delamination (adapted from Rosenfeld
et al. [32]).
36
In order to derive the energy release rate, Γi, as a function of indentation force, W, and the
contact radius, b, hardness of the material, H, is considered. This gives the following
relationship,
22
WH
b (2.71)
Note that the indenter used is a pyramidal indenter with diagonal 2b. For σb, the Tresca
yield criterion gives the following relationship,
b Y H (2.72)
where σY is the yield stress of the film.
For a polymer, the relationship between H and σY is given as follows [32, 33],
2.25 YH (2.73)
By combining Equations (2.72) and (2.73), the stress, σb, is shown as follows,
1.25 0.556b Y H (2.74)
Thus, Equation (2.70) for evaluation of Γi can be rewritten by combing Equations (2.73-
2.74):
22 2
22 1 0.556 2
1 1f
i f f
f
H t Hc
E W
22 2 20.627 1
1 2 1f
f f
f
H t Hc
E W
(2.75)
Jayachandran et al. [34] simulated the indentation of a PMMA coating by using 2D
axisymmetric finite element model and evaluated adhesive shear stress by using
37
Matthewson formulations [14], where the critical contact radius, b, for delamination was
estimated. However, the interfacial shear stress was underestimated without modeling the
interface damage.
2.6.3 A Relationship between Non-dimensional Delamination Radius
and Indentation Load
In order to characterize the interfacial delamination, Marshall and Evans developed the
non-dimensional parameters for indentation load and delamination radius, which
otherwise depend on film material properties, interface fracture toughness and indenter
geometry [28]. The following is a summary of their work. In particular, the non-
dimensional force W is expressed as follows,
2
0
3
12 11
2 1 14.68
VW
t
(2.76)
where V0 is indentation volume which depends on the indenter geometry.
For a pyramidal indenter, the relationship between the projected contact area, Ap, and the
indentation volume, V0, can be shown to be as follows,
2/3
03 2
cotp
VA
(2.77)
where Ψ is the half apex angle of the indenter.
38
Using Equation (2.40), the film hardness, H can be evaluated in terms of the projected
area, Ap, as follows,
2/3
2/3
0
cot
3 2p
WWH
A V
(2.78)
The non-dimensional force W can be calculated by using Equations (2.76) and (2.78) as
follows,
3/2 3/2
3 3
1cot 1 1
7.342
W WW
H t H t
(2.79)
where
cot 1
0.031cot 17.34 2
is a constant that depends on the indenter
geometry and the plate boundary conditions.
The non-dimensional delamination radius, C , is expressed as follows,
2
51
iC cE t
(2.80)
Non-dimensional force,W , given in Equation (2.79), was related to the delamination
radius, C , as follows [28],
22 1
2
1 / 2 1 1
1 res
W W
C
(2.81)
39
where res is the normalized residual deposition stress, defined as follows,
21 1
res R
i
t
E
(2.82)
For the conical indenter, used in micro-indentation, the indentation volume, V0, with
respect to the ratio of normal load, W, to the hardness, H, can be shown to be as follows,
3/2
0
cot
3
WV
H
(2.83)
For this case, the coefficient, ξ, in Equation (2.79) becomes,
3/2
1cot0.024cot 1
7.34
(2.84)
2.6.4 Film Thickness Effect on Interfacial Delamination
Li et al. [33], Sheng et al. [35] and Ritter et al. [15] found that thinner polymer films were
difficult to delaminate at the interface, showing film thickness dependence on the
delamination. Moody et al. showed that the interfacial fracture toughness of epoxy
decreases with decreasing film thickness [36]. The thickness dependence was attributed
to insufficient amount of elastic strain energy in the thinner films for the initiation and
propagation of delamination [27].
Volinsky et al. [37] discussed the contribution of plastic energy dissipation, Γp, in a
ductile thin film, mostly for metals, to the interfacial fracture toughness, Γi, as described
40
in Equation (2.46). A plastic strip model for estimating plastic energy dissipation at the
interfacial crack tip was shown as follows,
2
ln 1Yp
f
tt
E b
(2.85)
where t is film thickness, σY is film yield stress, b is the magnitude of the Burgers vector.
Equation (2.85) shows the thickness dependence on the plastic energy dissipation, where
this model assumes extension of plastic zone through the entire film thickness.
Evans and Hutchinson [29] derived a relation of critical film thickness for delamination,
where the delamination radius and film thickness are sensitive to variations in the
residual stress and the adherence. The equation is shown as follows,
2
0 12
f
c
f
t ckE
(2.86)
where σ0 is the stress induced by the indentation, c is delamination radius, k is a constant
used in the plate theory.
41
2.7 Bulk Material Damage/Failure Criteria in Finite Element Analysis
2.7.1 Introduction of Bulk Material Damage/Failure
Material damage and failure take place often and have been a critical issue in engineering.
Two of the most known accidents with respect to material failure were brittle fracture of
Liberty ships during the World War II, and the explosion of the Challenger Space Shuttle
due to malfunction of an O-ring seal [30]. Continuum Damage Mechanics (CDM), as a
relatively new branch of solid mechanics, is used to investigate damage and failure
characteristics for different materials using numerical methods and boundary integral
equation [38].
Two major mechanisms are used to describe material damage and failure of brittle and
ductile materials. In particular, brittle damage is characterized in the form of cleavage of
crystallographic planes with negligible inelastic deformation, typically observed in
polycrystalline metals at low temperature. Ductile damage behaves quite differently,
where large plastic deformation occurs around crystalline defects and causes localized
necking regions [38].
42
2.7.2 Implementation of Bulk Material Damage/Failure Criteria in
Finite Element Analysis
In continuum damage mechanics, a damage variable, D, is used to indicate the damage
initiation as well as material failure, where the value of D is in a range of 0 - 1. Figure
2.14 schematically shows mechanical characteristics of ductile materials and materials
with the assumption of elastic-perfectly plastic behavior. In particular, damage initiates at
point c (D = 0), where the equivalent plastic strain, 0pl , denotes initiation point of damage.
When the damage variable, D, equals 1, where the plastic energy dissipation decays to
point d, damage accumulation leads to material failure with a complete loss of load-
carrying capability.
(a) (b)
Figure 2.14. Material damage initiation and failure [39]. Note that (a) typical ductile
material; (b) elastic and perfectly plastic material.
43
A literature review shows a number of macroscopic or micromechanical material
properties, such as density, cross-section area, Young’s modulus, yield stress, have been
chosen as phenomenological parameters to characterize material deterioration. For
example, Rabotnov [38] selected the reduction of cross-section area caused by micro-
cracking as the measure of damage initiation. Lemaitre [40] replaced a true stress in the
material constitutive law by an effective stress as damage variable to describe isotropic
ductile damage in metals.
Two primary damage/failure criteria used in ABAQUS include ductile and shear damage.
In particular, the ductile damage criterion is used to predict the initiation of damage
formed by growth and coalescence of voids at local plastic deformations. This
phenomenological model describes the relationship of the equivalent plastic strain, plD , at
the onset of damage, as a function of stress state parameter, η, as well as equivalent strain
rate, pl . The damage variable, D, for ductile damage, is calculated as follows,
,
pl
plplD
dD
(2.87)
1 2 3
2 2 21 2 3 1 2 2 3 1 3
(2.88)
where σi (i = 1, 2, 3) are the principal stresses [41].
The shear criterion is often used to characterize the damage caused by shear band
localization [41]. This model describes the relationship of the equivalent plastic strain,
44
plS , at the onset of damage, as a function of shear stress ratio, ξs, and plastic strain rate,
pl . The damage variable, D, is expressed as follows,
,
pl
plplsS
dD
(2.89)
max
eqv s
s
k p
(2.90)
2 2 2
1 2 2 3 3 1
1
2eqv (2.91)
where τmax is the maximum shear stress, ks is a material parameter, p is pressure stress and
σeqv is the von-Mises stress, σi (i = 1, 2, 3) are the principal stresses.
Material is assumed to degrade linearly after initiation of damage, for numerical
implementation, schematically shown in Figure 2.14(b). A stress-displacement response
was proposed for damage evolution. In order to prevent mesh dependence on strain
localization plastic displacement was defined. An equivalent plastic displacement, upl
,
independent on the characteristic length, L, of the finite element mesh, was introduced to
evaluate the bulk fracture energy, Γf, during the damage evolution. The bulk fracture
toughness, Γf, schematically represented as the green-color region in Figure 2.14, is
expressed as follows [39],
0 0
pl plf f
pl
upl pl pl pl
f L d du
(2.92)
45
pl plu L (2.93)
where σpl
is the stress with plastic deformation, pl
f ,pl
fu are the plastic strain, and
displacement, respectively, at the point of material failure, L is the characteristic length of
finite element mesh, related to the coordinates of integration points among different types
of elements.
As the fracture energy is dissipated in a linear form, the fracture toughness, Γf, as shown
in Equation (2.92), can be expressed as follows,
1
2
pl
f Y fu (2.94)
where σY is the yield stress at the damage initiation.
2.8 Cohesive Zone Model in Finite Element Analysis
2.8.1 Description of Cohesive Zone Model
Cohesive zone model (CZM), which behaves like a non-linear material described by a
traction-separation law, is developed to simulate nucleation and propagation of interfacial
delamination [42]. This non-linear fracture mechanics approach was developed based on
the analysis of plastic zone of a crack tip. Dugdale presented a plastic strip model to
analyze the stress intensity factor ahead of a crack tip, where a constant closure stress
equal to yield stress was applied, by superimposing elastic plane-stress solutions [43].
Barenblatt generalized the plastic strip model and pointed out the stress in the cohesive
46
zone ahead of the crack is supposed to be a function of the separation instead of a
constant yield stress [44]. Cottrell [45] put forward the concept of crack-bridging as a
unifying theory for fracture at various length scales, from atomic bond breaking to large-
scale bridging in fibers of composite materials. A bridging law which relates the surface
tractions in the bridging zone, or cohesive zone to the relative separation displacements
was established to describe the fracture mechanisms of plastic zone at a crack front. Bao
and Suo reviewed the crack-bridging law and emphasized the implications of strength
and resistance toughness for ceramic matrix composite [46].
The bridging law, or traction-separation relation in the cohesive zone model depends on
the material as well as its associated fracture mechanism [42]. In particular, the fracture
mechanism of metals is characterized as large plastic deformation and void nucleation,
growth and coalescence ahead of a crack tip, while the fracture takes place by breaking
atomic bonds in ideally brittle materials, which yields distinctly different traction-
separation relations. The traction-separation relationship for elastic-plastic fracture is
derived from micromechanics models or determined experimentally, while the relation
for brittle fracture can be obtained from an inter-atomic bond potential.
47
(a) (b) (c)
Figure 2.15. Examples of the traction-separation relationship for the cohesive zone
model, for (a) constant traction; (b) trapezoidal traction; (c) bilinear traction.
Some relatively simple traction-separation relationships are defined in analytical and
numerical analysis, as shown in Figure 2.15. In particular, Figure 2.15(a) shows a
constant traction in the bridging zone, which was assumed in the Dugdale model.
Trapezoidal and bilinear relationships are often used in practice. Camanho et al. [47]
predicted the interfacial delamination of composite materials with mixed-mode loading
using bilinear traction-separation relation and gave comparable results with experimental
testing. Sorensen and Jacobsen [48] determined the cohesive law with trapezoidal
traction-separation relations by using J-integral approach and characterized large-scale
failure of carbon fiber-epoxy composites.
Among traction-separation relations, shown in Figure 2.15, two parameters of the
maximum stress, σc0, and the critical displacement, δc, are critical to characterize the
bridging law. The interfacial fracture toughness, Γi, which represents the area of traction-
separation relation, can be evaluated as [46],
48
00
c
i c cd
(2.95)
The strength, σc0, of an inorganic material, is roughly one tenth of its Young’s modulus (~
1010
N/m2), and the critical separation, δc, is on the order of lattice spacing (~ 10
-10 m)
[46]. An order-of-magnitude estimate for interfacial fracture toughness, Γi, can reach ~ 1
J/m2, which is close to the surface energy per unit area of a solid. The strength, σc0, for an
elastic-plastic fracture in metals can reach the yield stress (~ 108 N/m
2) and δc is in the
order of 10-6
m, which gives rise to a fracture energy up to 10 J/m2 and much higher than
the surface energy [42].
2.8.2 Implementation of Cohesive Zone Model in ABAQUS
The cohesive zone model is appropriate to model adhesion and delamination of interfaces
between two dissimilar materials, where the material properties are elastic or elastic-
plastic [49]. In ABAQUS, surface-based cohesive behavior or conventional cohesive
elements by specifying the traction-separation law can be used to model the interfacial
delamination [39]. However, surface-based interface is suitable for simulating a wide
range of cohesive interaction, where the interface thickness is negligibly small. If the
interface adhesive layer has a finite thickness and available macroscopic properties, such
as stiffness and strength, cohesive elements will be a better option for simulations.
Several traction-separation laws are schematically shown in Figure 2.15. A bilinear
traction-separation law as well as its implementation in ABAQUS is presented next as
49
this relation is adopted to investigate interfacial delamination of thin polymer film
through this dissertation.
Figure 2.16 schematically shows the parameters required to define cohesive interface in
mode-I, including initial elastic stiffness, K0, maximum cohesive strength, σc0, and a
critical separation, δc. The interface initially opens elastically with initial elastic stiffness,
K0, until the interface stress equals the maximum cohesive strength, σc0, which indicates
the initiation of damage. A damage variable, D, which is in the 0-1 range, is used to
quantify the interface status and is mathematically expressed using a damage evolution
rule,
max 0
max 0
c
c
D
(2.96)
00
0
c
K
(2.97)
where δ0 is the critical separation for damage initiation and δmax is the maximum
separation during the entire loading history. When the interface is partially delaminated,
with 0 < D < 1, the opening stress related to displacement can be expressed as follows,
01σ D K δ (2.98)
Equation (2.97) applies for both loading and unloading cases. In particular, during
loading (D = 0), the opening stress, σ, linearly increases with displacement, δ. During
unloading, Equation (2.98) can be rewritten as follows,
50
0
1σ
DK δ
(2.99)
Equation (2.99) indicates the damage variable, D, increases by increasing the opening
displacement, δ.
Equation (2.96) and (2.97) substituted into Equation (2.98), the stress at the cohesive
interface is shown as follows,
max 0 max
0 0
0 max 0 max 0
1c c
c c
c c
(2.100)
Since δmax = δ during loading, Equation (2.100) becomes,
0
0
cc
c
δ δσ σ
δ δ
(2.101)
During loading, the maximum separation, δmax, equals the separation, δ, where the
damage parameter, D, increases with increasing the opening displacement. During the
unloading, the maximum separation, δmax, and damage parameter, D, remain constant.
When c , the damage parameter, D, equals 1 and the interface loses its load-carrying
capability (σ = 0).
51
Figure 2.16. Schematic representation of the bilinear traction-separation law
implemented in ABAQUS.
The traction-separation laws of specifying shear and tear modes (mode-II and mode-III)
can be defined separately, with a set of similar parameters as mode-I discussed above.
Under a mixed-mode condition (mode-I and-II), a few damage initiation criteria are
available in ABAQUS, including the maximum stress separation criterion and the
quadratic stress separation criterion. In this work, we adopted the maximum stress
criterion, given as follows,
0 0
max , 1c c
(2.102)
where τ is the interfacial shear stress in either of two shear directions, in the case
of tension and 0 otherwise.
52
The mixed-mode damage initiation criterion shows the critical magnitude of the traction
vector depends on the ratio of the shear to normal traction, defined by a phase angle φ of
mode mix. In particular, φ is expressed as follows,
tan
(2.103)
Note that the Macauley bracket assumes that compression does not cause damage, which
makes Equation (2.103) be valid mathematically.
The magnitude of the effective traction vector, σm, is shown as follows,
2 2
cos sinm
(2.104)
By substituting Equation (2.104) into Equation (2.102), the critical traction magnitude for
damage initiation, σm0, is expressed as follows,
0 00 max ,
cos sin
c cm
(2.105)
In order to characterize damage evolution of the parameter, D, by a combination of
normal and shear deformation across the interface, an effective displacement, δm, is
defined as follows,
2 2
m n s (2.106)
53
where δn and δs are displacements in normal and shear directions, respectively. The
damage parameter, D, in the mixed-mode is evaluated by using the evolution law, shown
in Equation (2.96):
max 0
max 0
mc m
mc m
D
(2.107)
where δmc is the critical effective displacement, depending on the mode mix as well as the
damage evolution criterion. The energy-based damage evolution criterion was adopted
and shown as follows,
0 0
1I II
I II
W W
(2.108)
where α0 is power coefficient, ΓI, ΓII are the interfacial fracture toughness under mode-I, -
II, respectively. WI, WII are the work done by the traction in normal and shear modes,
where were mathematically shown as follows,
0
n
IW d
(2.109)
0
s
IIW d
(2.110)
Also, δm can be expressed in terms of separations, δn, δs as follows,
cos sin
n sm
(2.111)
54
Combining Equation (2.111) and (2.107), the damage parameter, D, is shown as follows,
0 0
0 0
cos sinmc n m mc s m
mc m n mc m s
D
(2.112)
Since the area under bilinear traction-separation law shows the work done by the
tractions, Equations (2.109) and (2.110) can be rewritten as follows,
2
0
1cos
2I m mcW (2.113)
2
0
1sin
2II m mcW (2.114)
The total interfacial fracture toughness in mixed-mode, Γm, can be shown as follows,
0
1
2m I II m mcW W (2.115)
Therefore, the strength of interface with mixed-mode depends on normal and shear
stresses, and the interfacial fracture toughness depends on the toughness in mode-I and -II.
In ABAQUS, five parameters are required to define cohesive interface in mixed-mode,
including the initial stiffness, K0, normal and shear stresses, σc0, τc0, and interfacial
fracture toughness in normal and shear directions, ΓI and ΓII.
55
3 Review of PTFE Material Properties
In this chapter, material properties of PTFE are reviewed. In particular, the literature
related to the molecular structure and mechanical properties, frictional and wear
characteristics in bulk and thin-film forms are surveyed.
3.1 Molecular Structure of PTFE
PTFE (Polytetrafluoroethylene) is an important engineering material with low friction
coefficient and dielectric constant, which is biocompatible, chemically inert and stable
under relatively high temperatures [50, 51]. Because of these characteristics it is widely
used as a solid lubricant, protective film, and an electrical insulator [52].
A number of mechanical properties of PTFE closely depend on its molecular structure,
which consists of 20,000-200,000 repeating units of tetrafluoroethylene (- CF2 - CF2 -)n
[53]. Bunn and Howells [54] investigated the molecular structure of PTFE at different
temperatures by X-ray diffraction. They found that the fluorocarbon molecules of PTFE
have a helical chain that twists at every 13-atom repeating unit. Figure 3.1(a) shows the
schematic of the twisted zigzag backbone in one PTFE segment. Figure 3.1(b) is the
illustration that depicts the fluorocarbon molecules assembled around the backbone of
PTFE segment, while Figure 3.1(c) depicts the molecular structure of PE (polyethylene) –
another polymer composed of hydrocarbon molecules. Compared to the molecular
56
structure of the PE, the smooth profile of the PTFE molecule is one of the main reasons
behind the low coefficient of friction (COF) of PTFE [54].
(a) (b) (c)
Figure 3.1. Schematic representation of PTFE and PE molecular structures [54]. Note
that (a) The zigzag backbone of a PTFE molecular segment; (b) A PTFE molecular chain;
(c) A PE molecular chain.
57
PTFE undergoes three types of thermally activated structural transformations at 19, 30,
and 150° C respectively. Below 19° C, the molecular chain is a 136 helix, i.e. every 13 –
CF2- units presented in six twists form an 180° rotation [51]. This is considered to be the
most stable phase [54]. At 19° C, a first-order crystal transformation occurs where a 157
helical conformation in a hexagonal cell is formed. Above 30° C, the 157 helical
configuration remains, but the conformation disorder increases. Above 150° C, the
conformation disorder of the helix chain significantly increases, which is detrimental to
maintaining the smooth profile of PTFE molecular chain.
It has been reported that the COF of PTFE varies with temperature, and this variation has
been attributed to the different helical conformations of PTFE at different phases [51]. In
particular, the COF values for PTFE below 19° C are somewhat greater than that between
19° 30° C [52]. As mentioned above, a 136 helix conformation exists below 19° C,
while another helix conformation, 157, is found above 19° C. Table 3.1 shows the
residues per turn. This table shows that the density of helical conformation of 157, is
slightly lower than that of 136, which has a very small effect on the PTFE chain
conformation [51]. However, the major difference of inter-chain distance between helical
conformations has significant impact on the conformational disorder where large inter-
chain distance allows the helix to have more room available to untwist or develop
disorders [51]. Bunn and Howells found that the spacing between molecular chains for
different helix conformations and found them to be 5.62 Å below 20° C, and 5.66 Å
above 20° C [54].
58
Makinson and Tabor [55] proposed a model of crystalline structure of bulk PTFE shown
in Figure 3.2. Bulk PTFE is composed of a large number of individual units called
“crystalline blocks” or “bands” with the dimensions of 10 to 100 μm long and 0.2 to 1 μm
wide. Figure 3.2(a) and (b) illustrate that one individual crystalline block is formed by
many small crystalline slices or “striae” which are typically about 200 Å thick. Figure
3.2(c) depicts each individual slice separated by disordered regions consisting of highly
oriented sheets of molecules. This crystalline model indicates that shear takes place more
easily within the amorphous regions between individual slices which makes slip occur
more frequently along the backbone of molecular chain. Recently, Sawyer et al. [56, 57]
investigated the influence of orientation of PTFE molecular structure on the COF values.
They found low friction forces and low barriers to interfacial slip, as the sliding direction
was parallel to PTFE molecular chain’s backbone. In contrast, when the sliding direction
was perpendicular to the PTFE chain backbone, they found high friction forces and high
wear, as molecular reorientation and chain scission happened.
Table 3.1. Characteristics of the helical conformation of PTFE [51].
136 157
Pitch (Å) 2.813 2.786
Unit twist (°) 166.2 168
Rise per residue (Å) 1.298 1.30
Residues per turn 2.167 2.143
59
Figure 3.2. Crystalline structure of bulk PTFE (Makinson and Tabor [55]). Note that (a)
crystalline block or ‘band’; (b) crystalline slices or ‘striae’ after sliding; (c) hexagonal
array of chains within the slices.
3.2 Mechanical Properties of PTFE
In order to thoroughly understand mechanical properties of PTFE, we review the
literature and summarize the findings of the previous investigations. This review includes
the mechanical properties of bulk and thin-film PTFE (modulus of elasticity, hardness,
viscoelasticity and plasticity) and frictional and wear characteristics. This knowledge will
shed little light on our research about PTFE thin-film coatings.
3.2.1 Young’s Modulus and Yield Stress
Wang et al. [58] investigated the thickness dependence of Young’s modulus and hardness
of on-wafer, ultrathin PTFE by using the Dynamic Contact Module (DCM) technique.
Experimental results showed that both Young’s modulus and hardness of PTFE thin film
strongly depend on the film thickness for films thinner than 500 nm. On the other hand
for thicker PTFE films, the Young’s modulus and hardness of the coatings were found to
60
be independent of the thickness, with the reported values of 2.3 GPa and 58 MPa,
respectively. The authors reported that there were significant differences in
thermophysical and mechanical properties of the thin-film polymer and the bulk polymer,
because the molecular structure of ultrathin (less than 100 nm thick) and thin (100 – 1000
nm thick) polymer films reorganize and give rise to a significantly different structure
compared to that of the bulk polymer [58]. Additionally, the Young’s modulus of bulk
and free-standing film measured by other methods is found to be 1/6 of the values
obtained by using nano-indentation, indicating the strength of polymers has a high
pressure dependence.
Table 3.2 gives the tensile properties of the bulk sample and free-standing films at room
temperature [58]. Lucas et al. [59] also measured Young’s modulus and hardness of
PTFE thin-films (500 – 1500 nm thick) on silicon substrate and reported 1 GPa for
Young’s modulus and 30 – 55 MPa for hardness. They found that the material properties
measured were independent of film thickness, which was consistent with the findings by
Wang et al [58]. However, the difference of Young’s modulus at the same indentation
depth of 500 nm reported by these two experiments was possibly caused by different
coating fabrication procedures and indentation heads applied.
Wang et al. [60] further assumed that the higher Young’s modulus and hardness of the
thinner films could be attributed to the first thin layer crystallite, as this thin layer of
PTFE was very dense and well attached to the substrate surface. This explanation was
built on the finding by Jones et al. [61] that polymer films less than 100 nm thick had
61
almost identical molecular properties to the same materials in bulk volume. Wang et al.
[60] also observed by SEM that the lamellar structure of the PTFE films develops
differently at various depth levels; the lamellae were nucleated directly on the surface of
the first ultrathin layer of PTFE crystals and grew into three dimensions. This thickness
dependence of morphology was thought to be critical to the mechanical properties of
PTFE. Since there are always a few of amorphous regions left between the PTFE
lamellae, the thin-film PTFE coating exhibits a lower modulus and hardness at its
amorphous state [60].
Rae and Brown [62] explored mechanical properties of bulk PTFE by using a Hopkinson
bar in tension tests. The tensile tests for samples of Dupont 7A and 7C Teflon (PTFE)
were conducted at a certain range of strain-rates 2 × 10-4
– 0.1 s-1
and temperatures -50 to
23° C. It was found that the tensile mechanical properties of PTFE, such as Young’s
modulus and yield stress1, are affected by strain-rate and temperature, but only to a
limited extent by crystallinity. From the stress-strain diagrams tested at different
temperatures and strain-rates, it was found that bulk PTFE behaves like elastic-plastic
materials until the strain reaches over 50%.
1 Rae and Brown reported that the Young’s modulus was determined from the initial tangent modulus and
the yield stress was calculated with a 2% offset [62].
62
Table 3.2. Tensile properties of bulk and free-standing films at room temperature [58].
Note that free-standing film is tested by different authors.
Materials Young’s Modulus (GPa) Yield Strength (MPa)
Bulk sample 0.41 9.0
15 mm free-standing film 0.4 10.3
15 mm free-standing film 0.4 17
Rae and Dattelbaum [63] also investigated mechanical properties of bulk PTFE by using
compression tests. Same as the tension tests, samples of Dupont 7A and 7C Teflon (PTFE)
were tested in compression at strain-rates in range of 10-4
– 1 s-1
and temperatures in
range of -198 to 200° C. Also, mechanical properties of PTFE were investigated at both
large and small strains in terms of its ductile properties. Experimental results showed the
mechanical properties are significantly affected by strain-rate and temperature.
Additionally, it was found that the Poisson’s ratio at small strains in tension was roughly
0.36, which was different from 0.46 obtained in compression tests.
Table 3.3. Mechanical properties of bulk PTFE in the tension test [62] (strain rate, =
5×10-3
s-1
).
Temperature
(˚ C)
Young’s Modulus
(GPa)
Yield Stress
(MPa)
Failure
Strain
Failure Stress
(MPa)
0 1.29 16.43 1.04 114.29
23 0.79 11.43 1.43 132.14
50 0.51 5.72 1.64 139.29
100 0.29 3.57 1.68 103.57
63
Table 3.4. Mechanical properties of bulk PTFE in the compression test (strain rate, =
10-3
s-1
). Note that failure strain and stress were not reported in this compression test [63].
Temperature
(˚ C)
Young’s Modulus
(GPa)
Yield Stress
(MPa)
0 0.82 16.38
26 0.43 8.57
50 0.38 7.53
100 0.21 4.29
3.2.2 Viscoelastic and Plastic Properties of PTFE
PTFE exhibits viscoelastic properties since it was found that its COF decreases with
slower sliding speeds, higher normal force, and high temperatures, which was believed to
be associated with the occurrence of relaxation between molecular chains in the
amorphous region [52]. Also, its viscoelastic characteristics can be explained by the
peculiar molecular structure of PTFE. In particular, its smooth molecular profile gives
rise to low COF and such low value is obtained once a transfer film with an oriented
molecular structure is created during the sliding. It is worth noting that remarkable
increase in the COF was found when the PTFE was irradiated to produce cross-linked
chains [52].
PTFE also exhibits temperature-dependent creep behavior. PTFE can have either
amorphous or semi-crystalline structure, where the degree of crystallinity and the size
and distribution of the crystallites have a large effect on the mechanical properties [64].
At temperatures well below the glass-transition, long molecular chains are rigid, resulting
64
in brittle behavior; however, at high temperature, backbone bonds rotate and allow
molecules to partially disentangle and move relative to one another [64].
Steijn [65, 66] examined the effects of viscoelasticity on the frictional behavior of PTFE,
and explored the effects of time, temperature and environment on PTFE in pure-sliding
tests. He concluded that the sliding behavior was influenced by the time lapse between
sliding experiments, sliding speed and thermal history of the sliding components.
Khan and Zhang [67] proposed a finite deformation viscoelasto-plastic constitutive
relation, which combined standard linear viscoelastic model and Khan, Huang, and Liang
(KHL)’s viscoplastic model [68] to characterize the strain-rate hardening, creep and
relaxation behavior of PTFE. They used the standard spring-dashpot model to represent
viscoelastic behavior and KHL model to describe viscoplastic behavior. Additionally, the
relaxation of PTFE only depends on initial strain with no effect on succeeding material
behavior; but, creep is influenced by both viscoelastic and plastic deformation, and has an
effect on subsequent material response.
Bergström and Hilbert [69] developed a new constitutive model, referred to as the dual
network fluoropolymer (DNF) model for predicting the time and temperature-dependent
mechanical behavior of PTFE. This model overcame the deficiency of Khan and Zhang’s
constitutive model [67], which can only predict the characteristics of fluoropolymers at
isothermal conditions. The DNF model incorporates experimental characteristics by
decomposing the material behavior of PTFE into a viscoplastic response and a
viscoelastic response. The viscoelastic response is further decomposed into the response
65
of two molecular networks acting in parallel: the first network holds the equilibrium
(long term) of the viscoelastic response and the second network the time-dependent (short
term) deviation from the viscoelastic equilibrium state. They modeled the mechanical
behavior of PTFE, in uniaxial tension and subsequent relaxation by finite element
analysis and found that numerical prediction matched well with experimental results, as
shown in Figure 3.3 and 3.4.
Figure 3.3. Comparison between experimental data and predicted behavior in uniaxial
tension at different strain-rate (T = 20° C, strain-rates, : 1.2×10-3
/s and 2.3×10-4
/s) [69].
66
Figure 3.4. Comparison between predicted and experimental stress relaxation results
[69].
3.2.3 Frictional Characteristics
The frictional characteristics of bulk PTFE was initially investigated by Makinson and
Tabor [55], who showed that the polymer is transferred to the slider surface. The
experiments on both glass and PTFE surfaces showed two separate friction regimes that
depend on sliding speed and temperature. Microscope observations showed a very thin
film was drawn over the surface, where the molecules inside the film were oriented with
the molecular chains parallel to the sliding direction. The low friction values during the
sliding tests were possibly caused by the fact that the shearing of the slices within the
crystal made the disordered regions slip at very small shear stresses.
Pooley and Tabor [70] investigated the relationship between the frictional behavior of
PTFE and its molecular structure. They observed that the coefficient of friction fell to a
low value (µ < 0.1) at the beginning of sliding, once the slider acquired a very thin
67
transfer film of PTFE with preferred orientation. However, the bulk properties of PTFE
were thought to be responsible for the high static friction, because within the bulk of the
PTFE lumpy transfer of PTFE was observed. This was attributed to strong interfacial
adhesion and shearing. They concluded that the frictional characteristics of the polymer
were closely related to the rigidity and smooth profile of the molecules instead of the
degree of crystallinity or the crystalline texture of the polymers.
Briscoe and Tabor [71] investigated the role of mechanical properties on friction of
polymers. They separated the friction force into two components, the adhesion
component and the ploughing component. Figure 3.5 shows a schematic from their work,
where the thin interfacial region has thickness t2. The energy is dissipated in processes
which were believed to resemble plastic shear or fracture in interfacial layers of the
polymer or at the original interface. However, once a large amount of material has been
removed by the rigid slider, marked by the thickness t1, the ploughing component of
friction force arises due to bulk deformation of the material. Figure 3.6 shows that the
plastic losses or viscoelastic losses could occur due to energy dissipation in the bulk
deformation region. In the case of adhesion component, the frictional force is the product
of shear stress and the real contact area. However, practically, it is difficult to measure
the real contact area, therefore evaluations often have to rely on mathematical modeling,
combined with an understanding of the bulk deformation of the solid.
68
Figure 3.5. Schematic of interfacial and bulk regimes of friction [71]. Note that W is the
normal force, Ffric is the friction force, t1 is the thickness of bulk region, t2 is the thickness
of interfacial region.
Figure 3.6. The bulk deformations due to plastic flow and viscoelastic losses [71].
69
Figure 3.7. The rolling friction of a rigid sphere on bulk PTFE and the quantity, E-1/3
tan(δ), as a function of temperature [71].
For the case of a sphere with radius, R, indenting a flat at low loads, the real contact area,
Ar, can be evaluated based on the elastic deformation at the contact interface as follows:
2/3
*
3
4r
WRA
E
(3.1)
where W is the normal force, E* is the composite modulus, mathematically expressed in
Equation (2.15).
As the normal load is increased permanent plastic deformation takes place in the material.
The real contact area, Ar, for a fully developed subsurface plastic region can be shown as
follows:
r
WA
p (3.2)
70
where p is the flow stress, or mean contact pressure of the bulk polymer.
The shear strength τ of high density thin-film polymers was investigated as a function of
applied mean pressure, p, and reported as follows [71]:
0 p (3.3)
where α is the pressure coefficient, τ0 is a parameter, typically a factor of ten lower than
the shear strength of the bulk polymer.
The COF value, µ, can then be evaluated by combining Equations (3.2) and (3.3) as
follows:
0 0fric r
r
F pA
W p A p p
(3.4)
Briscoe [72] assumed that the bulk contact pressure, p, is approximately equal to the
hardness of the polymer at heavy loads. Thus, the COF, μ, at high loads can be estimated
to α since the constant τ0 is rather small compared to the hardness of the polymer, p. He
also found that the COF, µ, calculated in Equation (3.4) matched well with the
experimental measurement of PTFE sliding test on glass, which verified the frictional
work dissipated within a thin surface layer. However, at low loads the COF, μ,
significantly depends upon the surface topography and Equation (3.4) was not applicable
[71].
71
As for the ploughing component of friction, Figure 3.6 illustrates the two situations of
simple plastic flow and viscoelastic grooving occurring at the interface. These are closely
related to the bulk properties of the polymers. The COF, µ, due to the energy loss of
viscoelastic grooving was given as follows [71]:
1/3 1/31/3 2/3 21 tan
2W R E
(3.5)
where W is the normal load, R is indenter radius, E, ν are Young’s modulus and Poisson’s
ratio, respectively, tan(δ) is the loss tangent of the material. Figure 3.8 shows the loss,
1/3
tanE
, as a function of temperature and that the variation of the rolling COF, μr,
with temperature behaves in the same way. The good agreement between theory and
experiment indicated that mechanical properties of the polymer largely govern the
frictional losses.
More recently, Myshkin et al. [73] investigated the effects of load, sliding speed and
temperature on the friction of thin PTFE films. They found that the COF, µ, decreases
with increasing the load in a range of moderate normal loads 0.02 – 1 N, but remains
constant with the load in the range of 10 – 100 N. They also report that the speed-
independent friction was only within a limited range of speed (0.1-10 mm/s) for PTFE.
However, for higher speed values the friction force depends only slightly on the speed.
The sliding speed has a pronounced effect on friction near the glass-transition
temperature, Tg, while friction hardly depends on the sliding speed at lower temperatures
since the segments of main molecular chain were frozen [74]. The trend that the COF, μ
72
increases by increasing the sliding speed was found in several polymer films, including
Polytetrafluoroethylene (PTFE), Polypropylene (PP), high density polyethylene (HDPE),
low density polyethylene (LDPE), Polystyrene (PS), which were investigated by Briscoe
and Tabor [75, 76]. However, Polymethylmethacrylate (PMMA) shows distinctly
different speed-dependence effect on the COF because viscoelastic retardation in
compression was believed as a significant factor in controlling the shear strength below
the glass transition temperature while the strain rate in shear predominated in other
polymers [76]. As for the effects of temperature, the heat induced during sliding results
from the deformation of material in the actual contact spots and the breakdown of
adhesion bonds. This gives rise to the correlation of COF with hardness and shear
strength. It was believed that adhesion is the basic mechanism of friction of polymers at
the highly elastic state over smooth surfaces or when the polymer was heated around the
glass-transition temperature.
Jia et al. [77] investigated the relations between cohesive energy density2 (CED) and
tribological properties in sliding of two polymers. They found that for similar polymer
combinations the friction coefficient is higher when the CED has a high value, as shown
in Figure 3.8. However, Figure 3.9 indicates that for two dissimilar mated polymers the
2 The ratio of cohesive energy over molar volume for monomer unit or atomic group in unit of (J/cm
3),
which provides a criterion to measure the strength of secondary bonds in a polymer or between two
polymers [75].
73
lower friction coefficient is associated with the absolute value of the difference in the
CED values, i.e. larger difference of CED of the two mated polymers gives rise to smaller
coefficient of friction. Thus, they concluded that the sliding friction properties of a
polymer-polymer pair are significantly influenced by the adhesion between two polymers
in contact. Additionally, they examined the wear behavior of various polymer-polymer
combinations with the relations of cohesive energy density (CED) [77]. They found that
the wear rate for similar polymer-polymer combination is decreased with increasing CED,
as shown in Figure 3.8. In Figure 3.9(b), it was observed that the wear rate of dissimilar
polymer-PTFE combinations appeared not to be closely associated with the CED
difference.
Wieleba [78] studied the effects of a number of steel counterface roughness parameters,
on the COF of PTFE composites. He found that the shape of the asperities has the most
significant impacts on the COF, while the height of asperities has the most significant
effects on wear of the PTFE composites.
The draw direction, which easily forms the orientation of the molecular chains of linear
polymer, with respect to friction-induced direction is one of important factors that affect
the tribological properties of PTFE [79]. Liu et al. investigated the dependence of COF
on the orientation of molecular chains of drawn bulk PTFE by using a pin-on-disc tester
[80]. They performed the sliding tests of the drawn PTFE along three different sliding
directions, including parallel with the draw direction, transverse to the draw direction and
perpendicular to the draw direction. They observed that the COF depends on sliding
74
direction; friction-induced orientation occurs on the worn surfaces; and, the transfer films
with high crystallinity are formed on the counterparts.
Figure 3.8. Relations between the cohesive energy density (CED) of the polymers and
friction coefficient, wear rate for similar polymer-polymer combinations [77].
75
(a)
(b)
Figure 3.9. Frictional and wear characteristics of PTFE with respect to cohesive energy
density (CED). (a) Relations between friction coefficient and the difference in CED for
dissimilar polymer-PTFE combinations; (b) Relations between the wear rate of polymer
pin and the difference in CED for dissimilar polymer-PTFE combinations [77].
76
3.2.4 Wear Characteristics
It is known that PTFE has much higher wear rate on the order of 10-7
cm/cm than other
crystalline polymers such as polyethylene, polypropylene, whose wear rates are on the
order of 10-9
cm/cm [81]. Tanaka et al. experimentally investigated the effects of heat
treatment, sliding speed and temperature on wear properties of PTFE, and found that the
wear rate is affected by the width of the bands 3 in the fine structure instead of the
crystallinity. Also, the effects of speed and temperature on the wear rate reflected the
viscoelastic nature of shear deformation at the amorphous region between crystalline
slices [81]. Hollander and Lancaster [82] investigated the effects of the topography of
metal counterface on the wear properties of the polymers and found that the wear rates of
polymers vary inversely with the average radius of curvature of the asperities, R, of metal
counterfaces. Thus, experimental results indicated that the topography of the counterface
for any polymer-metal combination is the predominant factor in determining the
magnitude of the wear rates of polymers. This is not only true for the initial stage of
sliding between fresh surfaces, but also at the later stages where the topography forms by
the sliding process itself. Additionally, the formation of transfer films was found on a
counterface during repeated sliding and believed to play significant roles in reducing the
3 Band: the structure of PTFE appeared as long bands with striations perpendicular to the length of the
bands. This structure is remarkably contrasted to the spherulitic structure of other crystalline polymers [79].
77
localized stresses and increasing the real contact area, which eventually has effects on
wear and the reduction in wear rate.
Steijn [83] discussed the characteristics of polymer wear by considering phenomena
occurring at the interface, including the formation of transfer films, metal pick-up and
surface melting. He focused on the polymer behavior from the response of plastics when
subjected to rubbing and rolling, and therefore separated the primary wear mechanisms
into adhesive and abrasive wear. The experiment results indicated that the wear
characteristics of polymers depend on the behavior of response to the dynamics of the
wear system, including a transfer film formation, surface melting, and elastic or
viscoelastic deformation at the contact region. Also, the wear properties of polymers were
varied and multifaceted since the material properties of polymers deviated from different
elastic moduli and melting points.
Lhymn [84] investigated the tribological failure sequence of PTFE and carbon-fiber-
reinforced PTFE on stainless steel plate by microstructural observation and wear rate
measurement. The wear rates of PTFE indicated that a material removal process by a
decohesional flaking mechanism occurs in adhesive wear, which produced extensive
heating at the contact area.
Bahadur [85] studied the development of transfer layers between polymer-polymer as
well as polymer-metal interfaces during sliding, and their role in mechanism of wear. He
found that the wear rate of polymer-metal pair was strongly influenced by the cohesion of
78
transfer film, adhesion of transfer film to the counterface, and protection of rubbing
polymer surface from metal asperities by the transfer film.
Myshkin et al. [73] discussed the common types of wear on polymers, including abrasion,
adhesion and fatigue. They found that PTFE is susceptible to friction transfer, which the
transfer of material from one surface to another occurrs due to localized bonding between
contacting solid surfaces, when rubbing against both metals and polymers. It was
discovered that PTFE is transferred in the form of flakes of very small size at the initial
stage of friction, which changed the roughness of both surfaces in contact. The roughness
of polymer surface underwent large deviations during unsteady wear until the steady
wear was reached, while metal surface roughness was modified due to transfer of PTFE.
However, he concluded that the effects of transfer film on the wear characteristics may
not be significant when small particles of soft polymer material with micrometer size
were transferred onto the hard mating surface, such as metals.
3.3 Summary
The literature survey presented the material properties of PTFE in bulk and thin film
forms. The effects of testing parameters on the frictional and wear characteristics of
PTFE were also investigated and shown as follows:
The COF depends on molecular structure, normal force, sliding speed,
temperature, surface roughness of substrate, adhesion and ploughing at the contact
region, sliding direction with respect to molecular chain orientation.
79
The wear rate depends on normal force, sliding speed, temperature, topography of
substrate, film transfer formation, elastic or viscoelastic deformation at the contact
region.
80
4 Friction and Durability of Thin PTFE
Films on Rough Aluminum Substrates
In this chapter, an experimental evaluation of the friction and durability characteristics of
thin PTFE films, deposited on aluminum substrates with different surface roughness is
presented.
4.1 Introduction
Polytetrafluoroethylene (PTFE) is a chemically inert engineering polymer with low
coefficient of friction (COF), and low dielectric constant [3]. In general PTFE is used as a
low friction coating in thrust bearings [86], and also as a load bearing surface in
applications that require low frictional resistance [87]. PTFE has also been used in thin
film form in micromechanical devices due to its low friction and low surface energy, and
in medical devices due to its chemical and thermal stability [88]. Recently, PTFE-carbon
nanotube (CNT) composites have attracted attention due to their improved wear
resistance [89]. Methods of thin-film PTFE deposition include spin- and dip-coating,
chemical vapor deposition (CVD), and hot-filament chemical vapor deposition (HFCVD)
[2]. The latter deposition technique allows thin-film PTFE to be deposited on almost any
type of substrate, including temperature sensitive materials [90].
81
A survey of the literature related to the frictional and wear characteristics of bulk PTFE
has been presented in Chapter 3.
4.2 Materials and Methods
Friction and durability characteristics of 1 μm thin, PTFE films deposited on aluminum
substrates by HFCVD technique were investigated. In particular, the effects of the normal
load, the sliding speed and the surface roughness were investigated by using a universal
micro-tribotester (UMT-2; CETR, Campbell, CA). The friction tests were conducted by
using the ball-on-plate configuration, schematically shown in Figure 4.1(a), where the
normal force and sliding speed were the independently controlled variables and the
tangential force was measured. A sliding distance of 25-mm was used in the tests. The
coefficient of friction (COF) was defined as the ratio of the tangential force to the normal
force. The reported COF values were chosen from the steady-state regime of the tests,
and each parameter combination was repeated at least five times. In the durability tests,
the ball-on-disk configuration of the instrument was used to monitor the COF of the
interface. Figure 4.1(b) shows such a configuration, where the substrate is attached to a
rotary table. The tests were stopped when the dynamic COF reached the solid-on-solid
friction value. The radius of the test track on aluminum substrates was set to 4 mm. Each
test was repeated at least three times for each testing parameter. The number of tests was
increased when the standard deviation was large.
82
(a) Ball-on-plate configuration for friction test
(b) Ball-on-disk configuration for durability test
Figure 4.1. Configurations of PTFE frictional and durability tests.
Steel (Rockwell hardness of C60-67), spherical balls with the diameter of 6.35 mm were
used as indenters. All of the balls were cleaned with hand soap before the tests, in order
to eliminate the effects of smeared materials. 5052 aluminum was used as the substrate.
83
All the tests were performed in the laboratory environment, where the temperature was in
a range of 20.1 to 27.3° C and the relative humidity (RH) was between 12% to 38%. The
majority of tests were conducted at 25° C and 30% RH.
Thin PTFE films (1 μm thick) were deposited on roughened aluminum plates by using the
HFCVD technique [3]. The surface roughness of the 5052 aluminum substrates were
modified by polishing and by bead blasting with 80, 120, 150, 180, 220 and 320-grit
silica particles. After the tests were completed, the wear tracks were characterized by
using an optical microscope (Meiji, ML 8500).
4.3 Results
The correlation between the various techniques used in this work to adjust the surface
roughness and the average surface roughness (Ra) and the variance of surface roughness
(σ2) are presented in Table 4.1. Figure 4.2 shows the relation between the Ra of the
surface and the size of the silica particles, quantified by units of grit, g, and by mean
particle diameter, dp. Figure 4.2 shows that the following relationships can be established
between Ra versus g, and Ra versus dp,
0.70745.017aR g (4.1-a)
0.6401exp 0.0067a pR d (4.1-b)
Note that the R2 values for these relationships are 0.9122 and 0.9812, respectively.
84
In order to establish a baseline solid-on-solid COF at the aluminum-steel interface,
friction tests were conducted on uncoated roughened aluminum plates. Figure 4.3 shows
the measured COF values as a function of normal load and surface roughness. It is seen
that the COF at the aluminum-steel interface is greater than 0.6. This value of COF was
set as the determinant of failure of the PTFE coating for friction and durability tests of
PTFE coated on substrates.
Table 4.1. Surface roughness parameter of Ra and σ2 for aluminum substrates. Note that
the mean particle diameter dp for each grit size is adapted from Orvis et al. [91].
Aluminum
substrates
Ra
(μm)
Std. Dev.
(μm)
σ2
(μm2)
Std. Dev.
(μm)
dp
(μm)
80-grit 2.340 0.620 9.820 5.430 190
120-grit 1.285 0.035 1.680 0.042 115
150-grit 1.280 0.230 2.640 0.900 92
180-grit 1.110 0.099 1.405 0.120 82
220-grit 0.995 0.007 1.270 0 68
320-grit 0.820 0.014 1.030 0.014 36
Polished 0.010 0.003 0.0002 0.0001
85
(a)
(b)
Figure 4.2. Surface roughness, Ra as a function of grit size, g and mean particle diameter,
dp, respectively.
86
Figure 4.3. The COF of aluminum substrate without PTFE thin films.
4.3.1 COF
The effects of sliding speed and normal force on the COF between the steel, spherical
slider and the PTFE coated aluminum plates are shown in Figure 4.4 and in Table 4.2.
Statistical analysis of the result is presented in Table 4.3-4.7. In addition to the results
presented in Figure 4.4, the effects of sliding speed and normal force on the COF were
also observed in the durability tests. Those cases are presented later in this chapter. Here,
three sliding speed values of 0.1, 1 and 5 mm/s, and four normal load values of 2.5, 5, 10
and 15 N were used. The Ra values of the surface roughness of the plates were 0.01, 0.57,
1.28 and 2.34 µm. Figure 4.4(a) shows the test results for Ra = 1.28 and 2.34 µm. It is
seen that the COF increases with sliding speed in the speed range of 0.1-5 mm/s.
Similarly a trend of increasing COF with increasing normal load in the load range of 2.5-
87
15 N is observed. It also appears that the COF decreases somewhat when the normal load
becomes 15 N. Figure 4.4(b) gives the COF values for the same parameters, but for the
two smoother surfaces with Ra = 0.57 and 0.01 µm. In the case of Ra = 0.57 µm, the COF
is clearly reduced with increasing normal load. A clear increase in COF is seen when the
sliding speed is increased from 0.1 mm/s to the higher values, but no clear change is seen
between 1 and 5 mm/s. In the case of the smoothest surface it is seen that the COF is
nearly independent of the load but depends on the sliding speed.
Results presented in Figure 4.4 show a non-linear relationship between the COF and the
independent variables; normal force, sliding speed and surface roughness. The average
COF values and its corresponding standard deviations for all the variables in this test are
given in Table 4.2. In order to assess the significance of the functional dependencies
stated above, analysis of variance (ANOVA) of the experimental data was carried out.
The null hypotheses tested were that load/speed/roughness has no significance on the
COF. In addition, the effects of the interactions between the variables were also tested. A
three way ANOVA was conducted with the degrees of freedom as shown in Table 4.3.
For each variable the p-value is smaller than 0.05, which indicates that the null
hypothesis is rejected, and that the normal load, the sliding speed, and the surface
roughness, and all three of their combinations have significant effects on COF. By
normalizing the sum of the square (SS) it is possible to obtain the relative contributions
of each factor when the entire data set (Table 4.2) is analyzed as a whole. This shows that
96.36% of the COF is due to surface roughness effects. Sliding speed contributes 1.97%
88
to COF, whereas the contribution of normal load (0.05%) is lower than the error (0.33%).
In order to investigate the effects of normal load and sliding speed on the COF for surface
roughness value, two-factor ANOVA tests were conducted, as shown in Table 4.4 - 4.7.
This analysis shows that sliding speed and normal load and their combinations have
significant effects (p < 0.05) on the COF, except for the case of Ra = 0.57 µm, where the
interaction effects of load and speed are rejected by the ANOVA. The analysis shows that
for each of the rough substrates (Ra = 0.57, 1.28 and 2.34 µm) the sliding speed has more
significance on the COF as compared to normal load. In particular, for the polished
substrate (Ra = 0.01 µm) the sliding speed contributes 97.6% to COF whereas the
contribution of the normal load is negligible.
89
(a) Ra = 1.28 μm and 2.34 μm
(b) Ra = 0.01 μm and 0.57 μm
Figure 4.4. The COF of PTFE on aluminum substrates with different roughness. (a) Ra =
1.28 μm and 2.34 μm; (b) Ra = 0.01 μm and 0.57 μm.
90
Table 4.2. Experimental data for frictional characteristics of 1 μm PTFE coating
deposited on glass substrates.
Normal force
(N)
Sliding speed
(mm/s)
Ra (μm) of
substrates COF (ave.)
COF
(std.dev.)
2.5 0.1 2.34 0.2927 0.0144
5 0.1 2.34 0.3122 0.0059
10 0.1 2.34 0.3202 0.0128
15 0.1 2.34 0.3253 0.0036
2.5 1 2.34 0.3141 0.0128
5 1 2.34 0.3405 0.0136
10 1 2.34 0.3573 0.0017
15 1 2.34 0.3541 0.0069
2.5 5 2.34 0.3356 0.0182
5 5 2.34 0.3452 0.0084
10 5 2.34 0.3540 0.0077
15 5 2.34 0.3358 0.0088
2.5 0.1 1.28 0.2579 0.0083
5 0.1 1.28 0.2513 0.0090
10 0.1 1.28 0.2654 0.0052
15 0.1 1.28 0.2513 0.0023
2.5 1 1.28 0.2696 0.0124
5 1 1.28 0.2729 0.0060
10 1 1.28 0.2862 0.0059
15 1 1.28 0.2868 0.0030
2.5 5 1.28 0.2815 0.0050
5 5 1.28 0.2818 0.0055
10 5 1.28 0.2994 0.0053
15 5 1.28 0.2871 0.0117
2.5 0.1 0.57 0.1579 0.0025
5 0.1 0.57 0.1469 0.0033
10 0.1 0.57 0.1343 0.0046
15 0.1 0.57 0.1318 0.0074
2.5 1 0.57 0.1810 0.0043
5 1 0.57 0.1682 0.0041
10 1 0.57 0.1608 0.0055
15 1 0.57 0.1630 0.0042
2.5 5 0.57 0.1778 0.0061
5 5 0.57 0.1670 0.0029
10 5 0.57 0.1552 0.0047
15 5 0.57 0.1508 0.0023
2.5 0.1 0.01 0.0275 0.0014
5 0.1 0.01 0.0329 0.0006
10 0.1 0.01 0.0377 0.0005
15 0.1 0.01 0.0422 0.0003
2.5 1 0.01 0.0527 0.0017
5 1 0.01 0.0500 0.0008
10 1 0.01 0.0542 0.0006
15 1 0.01 0.0579 0.0006
2.5 5 0.01 0.0971 0.0009
5 5 0.01 0.0995 0.0062
10 5 0.01 0.0987 0.0019
15 5 0.01 0.0954 0.0008
91
Table 4.3. ANOVA test for frictional tests of 1 μm PTFE coating on aluminum substrates
(The response is COF).
Source SS df MS F p Fcri %TTS
Load 0.00135 3 0.00045 9.42 p < 0.05 2.65 0.05
Speed 0.05339 2 0.0267 559.92 p < 0.05 3.04 1.97
Roughness 2.61615 3 0.87205 18290.25 p < 0.05 2.65 96.36
Load × Speed 0.00164 6 0.00027 5.74 p < 0.05 2.14 0.06
Load × Roughness 0.01406 9 0.00156 32.76 p < 0.05 1.92 0.52
Speed × Roughness 0.01692 6 0.00282 59.14 p < 0.05 2.14 0.62
Load×Speed×Roughness 0.00238 18 0.00013 2.77 p < 0.05 1.62 0.09
Error 0.00915 192 0.00005 0.33
Total 2.71504 239
Table 4.4. Two-way ANOVA test for PTFE coatings on Ra = 2.34 μm aluminum
substrates.
Source SS df MS F p Fcri %TTS
Load 0.00751 3 0.0025 22.1437 p < 0.05 2.80 27.75
Speed 0.01159 2 0.0058 51.2621 p < 0.05 3.19 42.83
Interaction 0.002536 6 0.0004 3.7411 0.00392 2.29 9.38
Error 0.005424 48 0.0001 20.05
Total 0.027052 59
Table 4.5. Two-way ANOVA test for PTFE coating on Ra = 1.28 μm aluminum
substrates.
Source SS df MS F p Fcri %TTS
Load 0.002129 3 0.0007 13.4469 p < 0.05 2.7981 13.58
Speed 0.010219 2 0.0051 96.8102 p < 0.05 3.1907 65.18
Interaction 0.000796 6 0.0001 2.5141 0.03388 2.2946 5.08
Error 0.002533 48 0.0001 16.16
Total 0.015678 59
Table 4.6. Two-way ANOVA test for PTFE coatings on Ra = 0.57 μm aluminum
substrates.
Source SS df MS F p Fcri %TTS
Load 0.00543 3 0.0018 87.3099 p < 0.05 2.7981 39.19
Speed 0.00722 2 0.0036 174.1259 p < 0.05 3.1907 52.11
Interaction 0.00021 6 3.497E-5 1.6868 0.14469 2.2946 1.51
Error 0.000995 48 2.073E-5 7.18
Total 0.013855 59
92
Table 4.7. Two-way ANOVA test for PTFE coatings on Ra = 0.01 μm aluminum
substrates.
Source SS df MS F p Fcri %TTS
Load 0.000337 3 0.000112 26.7897 p < 0.05 2.7981 0.80
Speed 0.041286 2 0.020643 4921.982 p < 0.05 3.1907 97.60
Interaction 0.000477 6 7.953E-5 18.962 p < 0.05 2.2946 1.13
Error 0.000201 48 4.194E-6 0.48
Total 0.042302 59
4.3.2 Durability
Durability of the 1 μm thick PTFE film was tested for a smaller set of variables due to the
long duration of such tests. Results are presented in Figure 4.5 and Table 4.8. Normal
force values of 2.5 and 5 N, and sliding speed values of 0.42 mm/s and 4.2 mm/s were
used on three rough (Ra = 0.57, 1.28, and 2.34 μm) and one smooth (Ra = 0.01 μm)
substrate. The sliding distance to failure was monitored during these tests, where failure
was assumed to take place when COF became greater than 0.6. Figure 4.5(a) shows that
sliding distance to failure has a clear dependence on the surface roughness of the
substrate. The smoothest surface, which has the lowest COF value, fails within 1.5 m of
sliding. The case of Ra = 0.57 μm survived for several hundred meters of sliding, whereas
the case of Ra = 2.34 μm failed within a few meters of sliding. Moreover, the sliding
distance to failure, and hence durability increased with increasing speed and decreasing
normal force. These differences were more apparent for the smoother surface (Ra = 0.57
and 1.28 μm), whereas all cases failed fast on the rough surface.
93
The COF values corresponding to the durability distances are plotted in Figure 4.5(b).
Interestingly, the comparison of the results indicates that low friction value to be related
to shorter durability. Tables 4.9 - 4.10 give analysis of variance (ANOVA) results for
durability and COF. The analysis shows that the normal load, sliding speed, surface
roughness and their combinations have significant effects on the durability and COF (p <
0.05), except for the combined effects of load and surface roughness, and load, speed and
roughness for the case of COF. The analysis also indicates that the roughness of the
substrate has the most significant effect on the durability, as compared to the normal load
and the sliding speed.
94
(a)
(b)
Figure 4.5. Durability characteristics of 1 μm PTFE on roughened and polished
aluminum substrates. (a) sliding distance to failure; (b) The COF.
95
Table 4.8. Experimental data for durability characteristics of 1 μm PTFE coating
deposited on aluminum substrates. Note that ave. represents average value and std.
represents standard deviations.
Normal force
(N)
Sliding
velocity
(mm/s)
Ra (μm) of
aluminum
substrates
Sliding
distance
- ave. (m)
Sliding
distance
- std. (m)
COF
(ave.)
COF
(std.)
2.5 4.19 2.34 0.94 0.15 0.251 0.021
2.5 0.42 2.34 28.24 13.46 0.206 0.024
2.5 4.19 1.28 225.89 55.82 0.220 0.036
2.5 0.42 1.28 117.51 5.22 0.214 0.003
2.5 4.19 0.57 906.16 252.49 0.193 0.014
2.5 0.42 0.57 252.53 151.48 0.142 0.009
2.5 4.19 0.01 0.75 0.06 0.069 0.002
2.5 0.42 0.01 1.14 0.56 0.050 0.002
5 4.19 2.34 0.71 0.07 0.197 0.003
5 0.42 2.34 19.90 32.82 0.142 0.017
5 4.19 1.28 153.26 43.42 0.148 0.019
5 0.42 1.28 89.81 43.58 0.143 0.012
5 4.19 0.57 364.56 89.19 0.137 0.004
5 0.42 0.57 212.22 52.94 0.104 0.007
5 4.19 0.01 0.62 0.17 0.068 0.002
5 0.42 0.01 1.14 0.56 0.051 0.002
Table 4.9. ANOVA test for durability tests of 1 μm PTFE coating on aluminum
substrates (The response is sliding distance to failure).
Source SS df MS F p Fcri %TTS
Load 89380.0 1 89380 13.54 0.0009 4.15 3.39
Speed 162328.2 1 162328.2 24.59 p < 0.05 4.15 6.16
Roughness 1460745.8 3 486915.3 73.75 p < 0.05 2.90 55.44
Load × Speed 54390.5 1 54390.5 8.24 0.0072 4.15 2.06
Load × Roughness 172180.6 3 57393.5 8.69 0.0002 2.90 6.54
Speed × Roughness 348633.0 3 116211 17.6 p < 0.05 2.90 13.23
Load×Speed×Roughness 135644.6 3 45214.9 6.85 0.0011 2.90 5.15
Error 211282.8 32 6602.6 8.02
Total 2634585.5 47
96
Table 4.10. ANOVA test for durability tests of 1 μm PTFE coating on aluminum
substrates (The response is COF).
Source SS df MS F p Fcri %TTS
Load 0.02551 1 0.02551 126.27 p < 0.05 4.15 12.62
Speed 0.01114 1 0.01114 55.13 p < 0.05 4.15 5.51
Roughness 0.14436 3 0.04812 238.22 p < 0.05 2.90 71.40
Load × Speed 0.00014 1 0.00014 0.68 0.4163 4.15 0.07
Load × Roughness 0.00942 3 0.00314 15.55 p < 0.05 2.90 4.66
Speed × Roughness 0.00505 3 0.00168 8.33 0.0003 2.90 2.50
Load×Speed×Roughness 0.00011 3 0.00004 0.18 0.9101 2.90 0.05
Error 0.00646 32 0.0002 3.20
Total 0.20218 47
4.4 Discussion
Makinson and Tabor [55], and Briscoe and Tabor [71] found that the COF of the bulk
PTFE increases by increasing the sliding speed or decreasing the normal force and the
temperature. Karnath et al. reported similar observations for the thin-film PTFE [3].
Experimental evidence also shows film transfer has significant effects on the COF [55].
In this work, similar speed and load dependencies on the COF were observed.
Briscoe and Tabor [71] observed that the COF of polymer films, deposited on smooth
glass substrates, are dominated by their intrinsic shear property. In this adhesion
dominated mode, the COF, μ, for polymers was found to depend on the shear strength, τ,
as follows,
0 0fric r
r
F τ αp ττ Aμ α
W p A p p
(4.2)
97
where Ffric is the friction force, W is the normal force, Ar is the real contact area, τ is the
shear strength which, for polymers, depends on a temperature-dependent constant value
τ0 and a pressure-dependent value αp with α as the pressure coefficient. The average
contact pressure, p, can be evaluated as follows,
r
Wp
A (4.3)
2/3
*
3
4r
WRA
E
(4.4)
where2 21 2
1 2
1 11
E EE
, Ei, νi (i = 1, 2) are the Young’s modulus and Poisson’s ratio of
the two contacting bodies, respectively, and R is the radius of indenter. By combining
Equations (4.2) – (4.4), the following expression is obtained,
2/3
0
1/3
3
4
R
W E
(4.5)
For PTFE films, the constants in Equation (4.5) have been reported as τ0 = 1×106 Pa, α =
0.08 [76]. By considering the effects of sliding speed on the COF, the following
relationship is suggested [75],
2/3
01 2 3 41/3 *
3ln exp
4
Rc v c c v c
W E
(4.6)
98
where v is the sliding speed and c1, c2, c3 and c4 are empirical constants. These constants
are determined by using the experimental results on smooth aluminum substrate (Ra =
0.01 μm), and given in Table 4.11.
The COF as a function of normal force, W, is shown in Figure 4.6. For the thin film PTFE
on smooth substrate the friction is dominated by adhesion effects and depends on sliding
speed, v; as well as the intrinsic constants τ0 and α are given by Equation (4.6).
On the other hand, the COF values on rough aluminum substrates are greater than that on
smooth substrates as shown in Figure 4.4. This difference is attributed to the solid-solid
contacts between the substrate and the indenter.
Table 4.11. The determination of parameters in Equation (4.6) by using curve-fitting.
2.5 N 5 N 10 N 15 N
c1 0.349 0.501 0.320 0.485
c2 1.023 1.002 1.062 1.063
c3 0.158 0.165 0.172 0.137
c4 -0.785 -0.811 -0.749 -0.635
99
Figure 4.6. The COF comparisons between the prediction by Equation (4.6) and
experiment measurement.
The durability tests presented here are different than the traditional wear tests, where the
PTFE films are severely removed from the substrates. Figure 4.7 shows microscopic
observations of the substrates after failure for the case of 5 N normal force and 0.42 mm/s
sliding speed. Figure 4.8 shows typical COF histories for different aluminum substrates
during the durability tests. The plots indicate that the PTFE film stabilizes the interfacial
COF to 0.06, on the mirror-polished substrate (Ra = 0.01 μm) while on the rougher
substrates, much larger variations are seen. Figure 4.8(e) shows the frictional test of the
steel ball on the uncoated aluminum substrate. Comparison of Figure 4.8(a) and 4.8(e)
indicates that the large variations in the COF could be due to solid-solid contact.
Interestingly, low COF on the smooth substrate yields lower durability. However, the
100
rough surfaces with medium roughness (Ra = 0.57 and 1.28 μm) give much longer
durability. It is possible that the rough interfaces act as a good reservoir for the storage of
PTFE debris and significantly reduce the wear rate to allow the interface last longer.
Various degrees of mechanical interlocking effects appear on the rough surfaces, which
gradually yield shorter durability. These results can be used to find an optimal roughness
for the PTFE-aluminum interface for improved durability.
The wear mechanism of PTFE is determined by a combination of adhesive and abrasive
wear modes. Adhesive wear is usually associated with conditions in which the asperities
in contact are sheared and the fragments are adhered and removed repeatedly between
two contacting surfaces [92]. Abrasive wear often takes place when a rough hard surface
slides against a softer surface, and the interface is damaged by ploughing and fracture
[92]. Lhymn showed experimentally that different wear mechanisms have significant
effects on the removal volume of PTFE [84]. In particular, the wear depends on the
frictional heat generated at the contact region, proportional to the normal force. Heat
generation induces a very large amount of material removal due to decohesional flaking
[84].
The prediction of COF using the modified Briscoe and Tabor equation, shown in Figure
4.6, and the COF history in Figure 4.8(a) indicate that adhesion governs the sliding and
durability of the PTFE film in the case of smooth aluminum substrate. The durability
tests on the rough substrates (Figure 4.7b-d) show a distinctly different wear mode, which
is attributed to an abrasive process.
101
(a) Ra = 0.01 μm (b) Ra = 0.57 μm
(c) Ra = 1.28 μm (d) Ra = 2.34 μm
Figure 4.7. Worn surfaces of PTFE thin films on aluminum substrate with different
roughness Ra (normal force is 5 N, sliding speed is 0.42 mm/s).
102
(a) Ra = 0.01 µm (b) Ra = 0.57 µm
(c) Ra = 1.28 µm (d) Ra = 2.34 µm
(e) Ra = 0.57 µm
Figure 4.8. COF histories on aluminum substrates with different surface roughness. Note
that (a) - (d) were tested on 1 µm PTFE films deposited on aluminum substrate; (e) was
tested on aluminum substrate without PTFE films.
103
4.5 Summary and Conclusion
The goal of this work was to characterize the friction and durability characteristics of thin
PTFE films on roughened aluminum substrates. The following conclusions are reached
based on this work:
The experiments showed that the COF and durability of PTFE film depend on
normal force, sliding speed and surface roughness of substrate.
In particular, the COF increases by increasing the surface roughness, Ra, and the
sliding speed. The durability improves by increasing sliding speed or decreasing
the normal force, but has a non-linear relationship with surface roughness, Ra.
ANOVA analysis indicated that the surface roughness of substrate has the most
significant effects on the COF and durability. The sliding speed contributes more
to the COF and durability as compared to the normal force.
The COF histories of durability tests on different aluminum substrates showed
that the PTFE film well lubricates the smooth surface and stabilizes the COF to
0.06, while it does not lubricate the rough surfaces well, indicating solid-solid
contacts.
The modified equation based on Briscoe and Tabor model predicted the COF on
smooth substrate well, indicating that the adhesion governs the sliding and
durability process.
The durability on rough surfaces was much likely to be associated with abrasive
wear mode.
104
The test results gave guidelines for designing an interface of PTFE film and rough
substrate for improved durability.
105
5 Frictional Characteristics of Thin PFA
and Silicone Films on Glass Substrates
In this chapter, we report on the characterization of the frictional properties of thin PFA
and poly(V3D3) films as a function of film thickness, normal force and sliding speed. A
brief introduction is given to frictional properties of these two polymers. Analysis of
variance (ANOVA) was used to seek correlations between the variables.
5.1 Introduction
PFA (Perfluoroalkoxy) is a polymer in the fluoropolymer family with a molecular
structure similar to PTFE, as shown in Figure 5.1. Physical properties of this polymer,
including chemical stability, high temperature resistance, low COF and anti-stiction are
also similar to PTFE [93]. These extraordinary properties allow it to be widely used as
plastic hardware in laboratory environment, as well as tubing in applications involving
highly corrosive environment. The dielectric properties of PFA are superior to that of
PTFE; for example, the dielectric strength of 100 μm thick PFA-film is four times higher
than the similar PTFE-film [93]. Processing of bulk PFA is easier than PTFE in typical
methods for thermoplastics, including extrusion, injection and moulding, as PFA does not
demonstrate high viscosity at high temperatures.
106
The second thin polymer film that we examined is poly(trivinyltrimethylcyclotrisiloxane),
poly(V3D3), developed by GVD corporation (Cambridge, MA), by using initial chemical
vapor deposition (iCVD)4, a relatively new technique [4]. The polymerization process
includes trivinyl-trimethyl-cyclotrisiloxane (V3D3) treated as a monomer and it is
initiated with tertbutyl peroxide (TBP). The all-dry deposition process generates a highly
cross-linked matrix material, shown in Figure 5.2. Poly(V3D3) coating exhibits high
adhesive strength to silicon substrates and is insoluble in both polar and nonpolar
solvents. Extraordinary dielectric properties were also observed compared to other
polymer materials, for example, parylene-C, and its non-cytotoxic properties allow
potential applications in medical devices [5].
The Young’s modulus and hardness of PFA are reported as 586 MPa and 60 MPa,
respectively [94]. The material properties of poly(V3D3) is not clear. The friction and
durability characteristics of theses polymers have not been thoroughly reported from the
review of literature, which motivates the experimental investigations in this work.
4 iCVD: this technique uses a radical generating initiator species, combined with the monomer and fed to
the CVD reactor. When both species pass over a resistively heated filament array, the temperature is high
enough for the weak initiator to break bonds while the chemical structure of monomer remains unaffected.
107
(a) PTFE (b) PFA
Figure 5.1. Schematic representation of PTFE and PFA molecule formulae.
Figure 5.2. Schematic representation of poly(V3D3) molecular structure. Note that the
hexagonal units show the intact siloxane rings, acting as cross-linking moieties for
backbone chains [4].
108
5.2 Materials and Methods
Friction characteristics of PFA and silicone films were tested by using the method
presented in Chapter 4. The steady-state COF at the interface of thin polymer films,
deposited on glass substrates were monitored. Three thick PFA films (0.3, 1, 5 μm) and
two thick silicone films (0.3, 1 μm) were tested. The normal forces were in the range of
2.5 - 15 N, and the sliding speeds during the test were between 0.01 - 1 mm/s. Each
parameter was measured at least three times.
5.3 Results
5.3.1 Friction Characteristics of Thin PFA Films
The COF for all three thick PFA films (0.3, 1, 5 μm) are presented in Figure 5.3. In
general, the highest sliding speed gives the highest COF values on 0.3 and 5 μm PFA
films while the sliding speed of 0.01 and 0.1 mm/s give comparably lower COF. For the 1
µm films the COF values are comparable for all speeds. The medium sliding speed, 0.1
mm/s gives lowest COF at a low load range of 2.5 – 5 N while 0.01 and 1 mm/s gives
higher COF at a low load range. In order to assess the significance of the normal load, the
sliding speed and the film thickness on the COF, analysis of the variance (ANOVA) of
the experimental data was performed. The null hypotheses of the analyses were
load/speed/film thickness have no significance on the COF. Additionally, the interaction
effects between the variables for COF were also tested. Table 5.1 shows the results of the
three-way ANOVA analysis with the specified degrees of freedom. The p-values for all
109
variables were found to be less than 0.05, indicating that the load, the sliding speed and
the film thickness and the interactions of these variables having significant effects on the
COF. The relative contribution of each variable was obtained by normalizing the sum of
squares (SS). This showed that the sliding speed and the film thickness contribute 26.67%
and 33.22% to the COF, respectively. These contributions are much greater than the
contribution from the normal force (4.23%). In order to evaluate the effects of normal
force and sliding speed on the COF for each separate film, two-way ANOVA analysis
were carried out, as presented in Tables 5.2-5.4. These results indicate that sliding speed
has the most significant effect on the COF as compared to the normal force and their
interactions for the 0.3 and 5 μm PFA films. For the 1 μm PFA films, the interaction of
sliding speed and normal force has the most significance on the COF, and the sliding
speed contributes more to COF as compared to normal force.
110
(a) 0.3 μm
(b) 1 μm
111
(c) 5 μm
Figure 5.3. The COF as a function of PFA film thickness.
Table 5.1. Three-way ANOVA for the COF of PFA thin films.
Source SS df MS F p Fcri %TTS
Load 0.0078 3 0.0026 33.28 p < 0.05 2.6895 4.23
Speed 0.0490 2 0.0245 314.51 p < 0.05 3.0812 26.67
Thickness 0.0610 2 0.0305 391.76 p < 0.05 3.0812 33.22
Load × Speed 0.0039 6 0.0007 8.36 p < 0.05 2.1845 2.13
Load × Thickness 0.0041 6 0.0007 8.75 p < 0.05 2.1845 2.23
Speed × Thickness 0.0309 4 0.0077 99.00 p < 0.05 2.4566 16.79
Load×Speed×Thickness 0.0215 12 0.0018 22.98 p < 0.05 1.8437 11.69
Error 0.0056 72 0.0001 3.05
Total 0.1838 107
Table 5.2. Two-way ANOVA for the COF of 0.3 μm PFA films.
Source SS df MS F p Fcri %TTS
Load 0.0017 3 0.0006 10.36 0.00015 3.0088 7.54
Speed 0.0129 2 0.0065 118.83 p < 0.05 3.4028 57.67
Interaction 0.0065 6 0.0011 19.89 p < 0.05 2.5082 28.96
Error 0.0013 24 5.429E-5 5.82
Total 0.0224 35
112
Table 5.3. Two-way ANOVA for the COF of 1 μm PFA films.
Source SS df MS F p Fcri %TTS
Load 0.0009 3 0.0003 12.60 p < 0.05 3.0088 9.86
Speed 0.0027 2 0.0014 56.74 p < 0.05 3.4028 29.60
Interaction 0.0050 6 0.0008 34.67 p < 0.05 2.5082 54.27
Error 0.0006 24 2.423E-5 6.26
Total 0.0093 35
Table 5.4. Two-way ANOVA for the COF of 5 μm PFA films.
Source SS df MS F p Fcri %TTS
Load 0.0093 3 0.0031 19.88 p < 0.05 3.0088 10.51
Speed 0.0642 2 0.0321 206.77 p < 0.05 3.4028 70.51
Interaction 0.0139 6 0.0023 14.89 p < 0.05 2.5082 15.23
Error 0.0037 24 0.0002 4.09
Total 0.0911 35
5.3.2 Friction Characteristics of Thin Silicone Films
The COF values for 0.3 and 1 μm thick silicone films are presented in Figure 5.4. The test
conditions, load and sliding speed, were the same as in the friction tests of the PFA films.
Note that 5 µm thick film was not available for these tests. In general, Figure 5.4
indicates that the COF increases by decreasing the sliding speed. This trend is opposite to
the one observed for the PTFE friction tests [3]. It is also seen that the COF decreases by
increasing the normal force, and eventually stabilizes. This trend is similar to that
exhibited by PTFE coatings. A film thickness dependence similar to PTFE is also
observed.
In order to evaluate the functional dependencies of the tested variables, ANOVA was
performed on the experimental data. Results of the three-way ANOVA test is shown in
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Table 5.5, where the null hypothesis was stated as the load/speed/film thickness as well
as the interaction of these variables have no significance on the COF. The p-values for
the load, speed and thickness indicate the null-hypothesis is rejected, which shows that
load/speed/film thickness have significant effects on the COF. In particular, the film
thickness and sliding speed contribute 39.01% and 16.68% to the COF, respectively.
These are much greater than the other effects. In addition, the interactions of load and
speed, and thickness and speed also have significant effects on COF, whereas the
interactions of load and thickness (p > 0.05) do not.
Two-way ANOVA analysis was also carried out for each separate silicone film, in order
to investigate the effects of normal force and sliding speed on the COF. Tables 5.6 - 5.7
show the ANOVA analysis results, indicating that the sliding speed contributes more to
the COF as compared to normal force.
114
(a) 0.3 μm
(b) 1 μm
Figure 5.4. The COF as a function of silicone film thickness.
115
Table 5.5. Three-way ANOVA analysis for the COF of silicone films.
Source SS df MS F p Fcri %TTS
Load 0.0321 3 0.0107 8.15 p < 0.05 2.734 8.42
Speed 0.0637 2 0.0318 24.23 p < 0.05 3.126 16.68
Thickness 0.1489 1 0.1489 113.32 p < 0.05 3.976 39.01
Load × Speed 0.0280 6 0.0047 3.55 p < 0.05 2.229 7.33
Load × Thickness 0.0042 3 0.0014 1.06 0.3771 2.734 1.09
Speed × Thickness 0.0118 2 0.0059 4.50 0.0162 3.126 3.10
Load×Speed×Thickness 0.0300 6 0.0050 3.80 p < 0.05 2.229 7.86
Error 0.0631 48 0.0013 16.52
Total 0.3817 71
Table 5.6. Two-way ANOVA analysis for the COF of 0.3 μm silicone films.
Source SS df MS F p Fcri %TTS
Load 0.0093 3 0.0031 5.02 p < 0.05 3.009 24.20
Speed 0.0127 2 0.0064 10.30 p < 0.05 3.009 33.13
Interaction 0.0016 6 0.0003 0.42 0.86 2.508 4.08
Error 0.0148 24 0.0006 38.59
Total 0.0384 35
Table 5.7. Two-way ANOVA analysis for the COF of 1 μm silicone films.
Source SS df MS F p Fcri %TTS
Load 0.0270 3 0.009 4.48 p < 0.05 3.009 13.89
Speed 0.0627 2 0.0314 15.61 p < 0.05 3.403 32.28
Interaction 0.0564 6 0.0094 4.68 p < 0.05 2.508 29.02
Error 0.0482 24 0.0020 24.82
Total 0.1944 35
5.4 Discussion
The wear track of the 1 μm PFA and silicone films were observed under an optical
microscope after the sliding tests. Figure 5.5- 5.6 show the microscopic observations for
1 mm/s in the load range of 2.5 - 15 N. Worn PFA surfaces show clean and regular
appearance while the silicone films give a “crystalline” worn track; the piled-up material
is discontinuous at places. The distinct worn appearances of those two polymers are
116
possibly related to interfacial shear strength, which gives rise to different COF values
[76]. However, no experimental data exists in the literature for the τ0 and α values of
these materials to make a comparison. Obtaining this data was beyond the scope of this
dissertation.
(a) 2.5 N (b) 5 N
(c) 10 N (d) 15 N
Figure 5.5. Microscopic observations of 1 μm PFA worn surfaces (sliding speed: 1
mm/s).
117
(a) 2.5 N (b) 5 N
(c) 10 N (d) 15 N
Figure 5.6. Microscopic observations of 1 μm silicone worn surfaces (sliding speed: 1
mm/s).
5.5 Summary and Conclusion
We investigated the frictional characteristics of PFA and silicone thin films with respect
to the normal force, sliding speed and film thickness. The observations and conclusions
are as follows:
The COF of PFA increases by increasing the sliding speed while the COF of
silicone increases by reducing the speed.
118
The ANOVA analysis showed that sliding speed and film thickness have more
significant effects on the COF as compared to normal force. The coupling of the
load and speed is not significant.
119
6 Simulation of Material Damage during
Indentation of a Soft Polymer
The goal of this chapter is to demonstrate the use of material damage mechanics in
simulating indentation of very soft materials. To this end the experiments used to obtain
the contact width, 2b, reported by Karnath et al. [3] in COF measurements were
simulated. 2D axi-symmetric and 3D finite element analyses with material damage are
presented.
6.1 Axi-Symmetric Finite Element Analysis of Thin-Film Indentation
6.1.1 Materials and Methods
Indentation of a 10 μm thick PTFE thin-film deposited on a glass substrate by a spherical
indenter (Figure 2.5) was modeled in the axisymmetric configuration. The PTFE film and
the glass were assumed to behave in elastic/perfectly-plastic manner and the ball indenter
was modeled to be rigid. The indenter diameter was 6.25 mm. The PTFE material
properties were based on the experimental results obtained by Rae and Brown [62], and
Rae and Dattelbaum [63]. Figure 6.1 gives material properties used in this analysis.
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Table 6.1. Material properties in 2D axisymmetric finite element analysis [3, 62].
Material Young’s modulus, E (GPa) Poisson’s ration, ν Yield stress, σY (MPa)
PTFE 0.496 0.44 5 – 10
Glass 72 0.22 1000
Commercially available finite element analysis programming package,
ABAQUS/Explicit (Simulia, Providence, RI), was used. The indentation problem is
verified in Appendix 1. The coating-substrate interface was assumed to be perfectly
bonded, and the bottom surface of the substrate was fixed in all degrees of freedom. The
symmetry axis was modeled by constraining the x-component of the displacement as
shown in Figure 6.1. The interface between the indenter and the coating was given a COF
value of 0.06. 8 quadrilateral elements were placed in the thickness direction of the
coating, and adaptive remeshing was implemented to prevent excessive element
distortion. A total number of 91272 CAX4R reduced integration elements was assigned
to the PTFE film and the substrate. The surface-based contact pair algorithm was
implemented in contact simulations. This indentation force was applied on the indenter in
a ramped manner with a rise time of 0.1 second, and kept constant thereafter.
Figure 6.1(b) shows the finite element mesh used in this simulation. A convergence study
was carried out by varying the number of elements between 2 and 10 in the thickness
direction. The film hardness in the simulation is set as Hp = 30 MPa. A plot of the contact
width, 2b, with number of elements used through the thickness of the thin-film shows that
121
(a) The geometry of modeling indentation
(b) Fine meshes around the contact region in ABAQUS/Explicit
Figure 6.1. The geometry and mesh configuration of 2D axi-symmetric finite element
model.
122
Figure 6.2. Convergence studies of element number in thickness direction of PTFE films
(Hp = 30 MPa).
convergence is achieved with 4 elements, as shown in Figure 6.2. The mesh size in the
glass substrate was the same down to a depth of 150 μm, and thereafter it was gradually
increased in regions away from the contact region.
6.1.2 Results and Discussions
The indentation process was modeled by the FEM as described above. The contact width,
2b, as schematically defined in Figure 2.5 was predicted and compared to experimental
measurements and a closed-form solution given in Equations (2.32) – (2.33).
Figure 6.3 shows comparison of the finite element results, the predictions of Equations
(2.32) - (2.33), and the experimental measurements for the 10 μm thick PTFE film. The
123
PTFE hardness, Hp, was adjusted as a fitting parameter. Two sets of finite element results
are shown with the PTFE hardness values of Hp = 15 MPa and 30 MPa. The analytical
results are shown for Hp = 0 and 10 MPa. Note that the Hp values used in both of these
approaches are close to physically measured values (40 MPa) in Appendix 2. This figure
shows good agreement between the measured and the analytically calculated values.
These results indicate that the contribution of the Hertzian contact between the steel ball
and the glass substrate dominates the interfacial contact conditions. For most of the load
range, the ball directly contacts the glass substrate. The finite element analysis does not
predict the contact width, 2b, as a function of normal force, W, as well as the analytical
relationship, especially for low load situations.
Figure 6.4 shows the interfacial conditions predicted by the finite element analysis, where
the residual indentation half-width, b, and residual thickness, tr, are shown. In the case of
1.5 N normal force, b and tr are 136.8 μm and 7.95 μm, respectively. When the
indentation force is increased to 15 N, these values become 288 μm and 2 μm. The finite
element analysis predicts a micron-scale thin PTFE film to remain trapped in the ball-
substrate interface. The compliance of the interface modeled by the finite element method
is close, but not identical to the experimental conditions as seen in Figure 6.4. Thus, it is
concluded that the PTFE layer has a smaller contribution than modeled by FEM, and the
model presented through Equations (2.32) – (2.33) describes the physics of the problem
more closely.
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Figure 6.3. Comparison of the experimental and the calculated data to FEA results.
(a) (b)
Figure 6.4. von-Mises stress distribution of indentation at different normal force of (a)
1.5 N and (b) 15 N calculated by finite element analysis.
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6.2 3D Finite Element Model of Indentation of PTFE Thin-Film by
Using a Material Damage Approach
Experimental evidence and analytical formulation show that the indenter is making direct
contact with the substrate. This indicates that the PTFE film is penetrated easily. In order
to model penetration of the PTFE film by the indenter, we employed material damage
model available in ABAQUS, introduced in Section 2.7.2, to model the indentation of
this soft PTFE film.
6.2.1 Materials and Methods
The steel ball and the glass substrate were modeled as elastic perfectly-plastic materials
with properties given in Table 6.2. The PTFE thin film was modeled as an elastic/
perfectly-plastic material and the damage of the material was considered. Shear and
ductile material damage models were investigated as possible damage criteria for this soft
polymer film. Additionally, the equivalent plastic strain 0
pl (0.1 - 50%) and fracture
energy Γf (10 – 1000 J/m2) were used for defining the initiation and evolution of the bulk
material damage, respectively. Section 2.7 presents the physical interpretation of these
two damage parameters, 0
pl and Γf and their implementations in ABAQUS. It is assumed
that the facture energy of PTFE film is dissipated in a linear form during damage
evolution. Table 6.2 gives the material properties used in this 3D finite element analysis.
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Table 6.2. Material properties used in 3D finite element simulation using material
damage model. Note that 0
pl is the equivalent plastic strain for damage initiation, Γf is the
fracture energy for damage evolution.
Material E (GPa) ν σY (MPa) 0
pl
Γf (J/m2)
Steel 210 0.30 380 N/A N/A
PTFE 0.496 0.44 10 0.1 - 50 10 - 1000
Glass 72 0.22 1000 N/A N/A
Figure 6.5(a) shows the configuration used in the analysis. The film was assumed to be
bonded to the substrate. The bottom surface of substrate was constrained in all degrees of
freedom. The surfaces along the quarter-symmetric axis were fixed in x- or z-directions,
respectively. A constant pressure was applied on the top surface of ball indenter, with
which was initially ramped in 0.1 second to the final value. The element-based general
contact algorithm was implemented, which allows the contact between the indenter and
the PTFE film even after elements are removed due to material damage.
The ball indenter was partitioned into two regions, including a fine mesh region-I and
free mesh region-I. The PTFE film was partitioned as fine mesh region-II and free mesh
region-II. The fine mesh region-I subtends 6.5˚, and makes the total length of the fine
mesh region-I be equal to that of the fine mesh region-II. We find the use of comparable
element size in the fine mesh region-I and –II to be essential for convergence of these
contact simulations. In particular, the free mesh region-I in the ball indenter was meshed
with 45,487-C3D4 tetrahedral elements, and the fine mesh region-I was meshed with
48,600-C3D8R hexahedral elements. The fine mesh region-II and free mesh region-II in
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PTFE film were meshed with 80,000 and 16,000-C3D8R hexahedral elements,
respectively. The substrate was meshed with the number of 120,000 graded C3D8R
hexahedral elements with gradually increasing size away from the contact region. Figure
6.5(b) shows the mesh for 3D finite element model around the contact region. The
thickness direction in both fine mesh region-I and -II was given 4 and 8 elements for
convergence studies. Figure 6.6 indicates that 8-element in thickness direction achieves a
converged solution.
128
(a) The geometry of 3D finite element model
(b) Meshes around the contact region in ABAQUS
Figure 6.5. The geometry and mesh configuration of 3D finite element model.
129
6.2.2 Results and Discussions
Figure 6.6 shows comparison of the calculated and the measured contact width, 2b, to the
finite element results for the 10 μm thick PTFE film by using the shear damage model.
The damage initiation was assumed to start at 1% of equivalent plastic strain, and the
material fails when the fracture energy, Γf, reaches 20 J/m2. The predicted contact width,
2b, agrees with the measured and the calculated (Equations 2.32 – 2.33) results. The pile-
up of PTFE films predicted in finite element analysis possibly increases the calculation of
contact width, 2b as compared to the measured values. Difference in using 4 or 8-
elements through the film-thickness is small. Note that the error bars indicating the
standard deviation of numerical analysis, shown in Figure 6.6, are due to the rectangular
mesh configuration we adopted. The particular algorithm of collecting all data points and
evaluating the standard deviation is presented in section 7.2.
When the contact width, 2b, predicted by using shear damage model and 2D axi-
symmetric model without material failure in Figure 6.3 and 6.6 are compared, it is seen
that the thin-layer of coating that is trapped in the ball-glass interface leads to
overestimation of the material compliance. The improved prediction by using the material
damage model verifies the contribution of Hertzian contact at the ball-glass interface,
which dominates interfacial contact conditions at loads higher than 2.5 N.
Figure 6.7 shows the contact width, 2b, predicted by using the ductile material damage
criterion. Comparison of Figure 6.6 and 6.7 shows that damage models have significant
130
effects on the prediction of contact width, 2b, and the shear damage model is more
suitable for simulating PTFE damage. Piggot [95] found that the interface failure of fiber-
enforced polymers, for example, polyethylene (PE), polypropylene (PP), and poly(methyl
mechacrylate) (PMMA) was caused by high mean shear stress. The fact that shear
fracture dominates the polymer failure is consistent with the damage mechanism used in
this finite element simulation.
Figure 6.6. Comparison of the measured and calculated contact width, 2b, to FEA
prediction for 10 μm PTFE films using shear damage model.
131
Figure 6.7. Comparison of the measured and calculated contact width, 2b, to FEA result
for 10 μm PTFE films by ductile damage model.
Next, the effects of equivalent plastic strain, 0pl , and fracture energy, Γf, on the prediction
of contact width, 2b were investigated. Figure 6.8 shows the effects of equivalent plastic
strain, 0pl ( = 0.1%, 0.5%, 1%, 5%, 10%, 20%, and 50%), for Γf = 20 J/m
2. Results
indicate that the equivalent plastic strain, 0pl , has negligible effects on contact width, 2b,
in the range of 0.1 – 10% strain. Joyce performed fracture experiments on bulk PTFE to
evaluate the fracture toughness, Γf, and reported that the offset yield stress of 1% plastic
strain, at room temperature, is 10 - 15 MPa [96, 97]. In order to be consistent with this
finding of damage initiation value, we used 1% plastic strain in our simulations.
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Figure 6.8. Effects of equivalent plastic strain, 0pl , (0.1, 0.5, 5% are not shown on the
graph for clarity of illustration) on the contact width, 2b, for Γf = 20 J/m2.
Figure 6.9 shows the contact width, 2b, as a function of fracture energy, Γf. The
equivalent plastic strain is set 1% and the fracture toughness, Γf, is varied in the range of
10, 20, 30, 40, 50, 100, 500 and 1000 J/m2. The result indicates that the fracture
toughness, Γf, has significant effects on the prediction of contact width, 2b. Bulk fracture
toughness values, Γf, which are in the range of 20 - 50 J/m2, give results similar to
experiments. Note that this range is only several percent of the toughness values reported
by Joyce [96]. This work shows that PTFE is easily deformable at a high contact pressure,
resulting in low fracture toughness.
133
Figure 6.9. Effects of bulk fracture toughness, Γf, (40, 500 J/m2 are not shown on the
graph for clarity of illustration) on the contact width, 2b, for0
pl = 1%.
Steady state configurations of the thin-film are represented in Figure 6.10 for different
normal loads (1.25, 2.5, 5 and 10 N), and for 0pl = 1% and Γf = 20 J/m
2. The indentation
radius, b, and the contact radius, a, are computed based on such results, as marked on this
figure. Close inspection of the figures reveals material pile-up at the edge of indentation
radius, b, where the maximum von-Mises stress in the PTFE film is also found. The
volume of the damage material increases with load. At low load levels the PTFE film is
partially penetrated and the stresses generated in the film provide the equilibrium with
respect to the indentation force; whereas, at higher loads, the indenter makes direct
contact with the glass substrate, as hypothesized in the analytical results. The von-Mises
stress distribution of the glass substrate at steady state configurations is shown in Figure
134
6.11. Note that the maximum stress at the normal load of 10 N is lower than the specified
yield stress, 1 GPa, which indicates glass substrate elastically deformed.
Figure 6.10. von-Mises stress distribution of PTFE thin films at steady state
configurations. Note that a is the contact radius at the interface of ball indenter and
substrate; b is the contact radius at the interface of ball indenter and PTFE films.
135
Figure 6.11. von-Mises stress distribution of the glass substrate at steady state
configurations.
6.3 Summary and Conclusions
Simulations of the indentation of thin PTFE films on a glass substrate were carried out by
using the FEM with a material damage model. The material damage parameters pertinent
to PTFE were determined by comparison to experimental results.
Shear damage model was found to be more suitable to model the interfacial
contact of soft polymer films as compared to ductile damage model.
The effects of equivalent plastic strain, 0
pl , and bulk fracture toughness, Γf, as
indicators of the damage initiation and evolution, respectively, on the contact
width, 2b, were investigated.
136
The simulation results as compared to experimental measurements showed that
bulk fracture toughness, Γf, has more significant effects on the contact width, 2b
predictions as compared to the equivalent plastic strain, 0
pl .
This work showed that the equivalent plastic strain 0
pl = 1% and fracture
toughness Γf = 20 J/m2 represent the experimentally obtained results well.
137
7 Interfacial Delamination of PTFE Thin
Films
Interfacial adhesion is a major concern with respect to the performance of thin polymer
films in developing new thin film processes, such as hot filament chemical vapor
deposition (HFCVD). In this chapter, we present an experimental investigation and
numerical simulation of the interfacial fracture toughness of PTFE by micro-indentation.
7.1 Introduction
Indentation is one of the common techniques, (Figure 2.10), to evaluate the interfacial
fracture toughness and shear strength of the thin film-substrate interface. Evans and
Hutchinson derived the stress intensity factor of a circular interfacial crack by assuming
mixed-mode fracture (I and II) at the crack tip, and showed that the stress intensity varies
directly with the indentation volume and inversely with the delamination radius and film
thickness [29]. Marshall and Evans gave a generalized relationship between the
normalized load and the delamination diameter to quantitatively predict the interfacial
fracture toughness. They showed that the delamination diameter depends on the normal
load and the film thickness, and verified the results experimentally by using a ZnO film,
deposited on a silicon substrate [28]. Ritter et al. evaluated the interfacial shear strength,
based on Matthewson’s formulations for the cases when the films were elastically or
138
plastically deformed or penetrated. Delamination of a soft polymer coating was found to
be caused by the shear failure at the perimeter of the contact region [13].
In this work, micro-indentation experiments were performed to induce delamination at
the interface of a PTFE film and a glass substrate. A 3D finite element model was
developed to simulate the delamination of the PTFE-film, and the results were compared
to experimental measurements. The interfacial fracture toughness, Γc, for the system was
then determined.
7.2 Materials and Methods
HFCVD was used to deposit 1, 2, 3, 5 and 10 μm thick PTFE films on glass substrates.
Micro-indentation tests of the PTFE films were conducted by using a micro-tribotester
(UMT-2 by CETR, Campbell, CA). A Rockwell C indenter (200 μm radius and 120°
cone angle) was used in the tests. Five indentation loads (W = 0.5, 0.75, 1, 2 and 3 N)
were used. The loading rate during the tests was 0.01 mm/s and the hold period was 30
seconds. Each indentation load was repeated twenty times at different locations. An
optical microscope (MX51 by Olympus, Japan) was used to examine the interface after
the indentation, and measure the delamination diameter, 2c, as shown schematically in
Figure 2.10. Young’s modulus, E, and hardness, Hp, of PTFE films were measured by
using a nanoindenter with a Berkovich tip (Nano Bionix, UTM-150, MTS, Eden Prairie,
MN). The detailed results of nano-indentation are provided in Appendices 3-4. The yield
stress, σY, which is obtained by using the relationship Hp = 3σY, is reported in Table 7.1.
139
As shown in Chapter 6, the PTFE film is penetrated very easily by the indenter during
indentation; and, straightforward finite element simulation of the compression of the
coating under the indenter is not adequate for modeling this very soft material [3]. In
order to simulate the penetration process, 3D finite element analysis that considers
material damage under the indenter was carried out by using ABAQUS/Explicit (Simulia,
Providence, RI). Shear damage criterion was applied to the bulk of the PTFE film by
giving equivalent plastic strain, 0pl (1%), as the damage initiation, and bulk fracture
toughness, Γf (20 J/m2), as the damage evolution criteria.
For the glass-PTFE interface, bilinear traction-separation law was used to specify the
delamination, as shown schematically in Figure 2.16. In particular, the normal stress and
shear stress were given 5 MPa as delamination initiation, and the stiffness, K0 is given as
4×104 N/mm
3. Interfacial fracture toughness, Γi, was given in a range of 50 - 1000 mJ/m
2
for delamination evolution. This approach allows for simulating mode-I and -II fracture
effects in the film-substrate interface. The PTFE film and the substrate were modeled as
elastic-perfectly plastic, while the indenter was modeled as elastic, due to its relatively
high Young’s modulus. Bergström and Boyce [98] presented a modified constitutive
model for time-dependent mechanical behaviors of elastomers and soft biological tissues
under cyclic loading. Bergström and Hilbert [69] developed a Dual Network
Fluoropolymer (DNF) constitutive model to simulate the time and temperature-dependent
mechanical behavior of bulk fluoropolymers including PTFE. In this work, we did not
140
use these models for simplicity and assumed isothermal condition for thin PTFE films.
The material properties used in the finite element analysis are given in Table 7.1.
The finite element mesh density and the aspect ratio of the 3D hexahedral (C3D8R)
elements near the contact region were found to be critical for the convergence of the
predicted delamination radius. Table 7.2 shows dimensions of the fine mesh regions and
mesh sizes for modeling the delaminations of 1, 5 and 10 μm PTFE films. Figure 7.1(a)
shows the geometry of simulation and mesh configurations. In particular, the fine mesh
regions-I and –IV, on the PTFE film, and the indenter, respectively, were given
comparable mesh sizes. Fine mesh region-III, of the substrate was given comparable
mesh size, to the fine mesh region-IV, considering the eventual contact of the indenter
and the substrate. Fine mesh region-II of the PTFE film was given mesh size comparable
to the substrate, in order to predict the delamination. The mesh regions of the PTFE film
and the substrate away from the contact were gradually increased. Figure 7.1(b) shows
the finite element mesh for the 10 μm thick PTFE film. The element-based general
contact algorithm was implemented, which allows the contact between the indenter and
the PTFE film even after elements are removed due to material damage. Table 7.3 shows
the number of elements and element types implemented in the simulations.
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Table 7.1. Material properties in 3D FEA delamination model. Note: Diamond and glass
properties were taken from Oliver and Pharr [16]; The PTFE properties were measured
by using nano-indentation, except for Poisson’s ratio, ν, reported in Karnath et al. [3].
Material E (GPa) ν σY (MPa) 0pl (%) Γf (J/m
2)
Diamond 1141 0.07 N/A N/A N/A
PTFE 3 0.44 35 1 20
Glass 72 0.22 1000 N/A N/A
Table 7.2. Mesh sizes and aspect ratios for modeling all thick PTFE films.
t (μm) L0 (μm) L1 (μm) L2 (μm) h (μm) Nt NL0 NL1 Nh Ratio
1 100 300 10 10 4 150 175 40 2.67
5 200 600 10 10 8 125 150 8 2.56
10 200 600 10 10 8 75 125 8 2.13
142
(a) The geometry
(b) Elements around the contact region for 10 μm PTFE film
Figure 7.1. The geometry and mesh configuration of 3D finite element model for the
delamination simulation.
143
A constant pressure was applied on the top of the conical indenter. The bottom of
substrate was constrained in all degrees of freedom. The surfaces along the quarter-
symmetric axis were fixed in x- or z-directions, respectively.
Convergence of the finite element model was tested by using different mesh sizes.
Figure 7.2 shows the convergence studies for 1, 5 and 10 μm thick PTFE delamination
diameters for normal force, W = 1 N, and interfacial fracture toughness, Γi = 100 mJ/m2.
The number of elements is in the range of 4 - 10. This plot indicates that use of 8-
elements in the thickness direction achieves the converged results for the 5 and 10 μm
thin films, and 4-elements for the 1 μm thin film. The error bars shown in Figure 7.2 are
due to the use of a rectangular finite element mesh in an axisymmetric problem.
Figure 7.3 shows the delamination contours predicted by the finite element method for W
= 1 N, E = 3 GPa, σY = 35 MPa. The interfacial fracture toughness, Γi, is given in the
range of 50 – 200 mJ/m2. In ABAQUS, the parameter, CSDMG, monitors interfacial
delamination, with CSDMG = 1 indicating the interface is completely delaminated, and
CSDMG = 0 indicating no delamination.
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Figure 7.2. Convergence studies for 1, 5 and 10 μm PTFE delamination simulations (W =
1 N, Γi = 100 mJ/m2, Young’s modulus E = 3 GPa, yield stress σY = 35 MPa for PTFE
material properties).
Table 7.3. The number and type of elements implemented in delamination model.
Film
thickness
# of elements in film
and substrate
# of elements in
indenter
# of elements in
fine mesh region-I
# of elements in fine
mesh region-IV
1 μm 380,208 (C3D8R) 44,309 (C3D8R, C3D4) 90,000 (C3D8R) 32,000 (C3D8R)
5 μm 397,953 (C3D8R) 1,352 (C3D8R, C3D4) 80,000 (C3D8R) 352 (C3D8R)
10 μm 309,422 (C3D8R) 1,292 (C3D8R, C3D4) 45,000 (C3D8R) 480 (C3D8R)
In order to evaluate the delamination radius, the coordinates of the nodes ,i iC x z with
interfacial delamination as shown in Figure 7.4 are found by using a Python script. Since
the parameter, CSDMG, gives node-based values including the coordinates, the
coordinates, ,i iC x z are regrouped into (n+1) small data sets based on the x-coordinate
in range of (0 - xn). Within each small set, the maximum z-coordinate is searched and
145
each node on the delamination contour is denoted as, max10,C z , 1 max 2,C x z , …,
max,n nC x z . The radius of each node on the delamination boundary was then calculated
as ri (i = 0, 1, …, n) with respect to the origin. Then the averaged radius, r , and its
standard deviation, s, were evaluated as follows,
0
1
1
n
i
i
r rn
(7.1)
2
0
1 n
i
i
s r rn
(7.2)
(a) 1 N, 50 mJ/m
2 (b) 1 N, 100 mJ/m
2
(c) 1 N, 150 mJ/m
2 (d) 1 N, 200 mJ/m
2
Figure 7.3. 10 μm PTFE delamination contours predicted by finite element simulation
(Young’s modulus, E = 3 GPa, hardness, σY = 35 MPa).
146
Figure 7.4. The coordinates of nodes with interfacial delamination.
7.3 Results and Discussion
Figure 7.5 shows optical micrographs of the indented surfaces of five different PTFE
films for W = 0.5 N. No interfacial delamination was observed on the 1 and 2 μm thick
films, in Figure 7.5(a) and (b). On the other hand, typical delaminated interfaces for 5 and
10 μm thick films are seen in Figure 7.5(d) and (e). Figure 7.6 shows the load effects on
the delamination diameter, 2c, of the 3 μm thick film, where the interfacial delamination
was clearly seen at 0.75, 2 and 3 N, as shown in Figure 7.6(b), (d) and (e). These
observations show that delamination takes place for film thickness values of 3 μm and
higher, and indicate the thickness-dependence.
As the delamination takes place more readily on 5 and 10 μm films, we measured the
delamination diameter, 2c, at each indentation force for five repeats. Table 7.4 shows the
147
averaged 2c and its standard deviation. The Student’s T-test was used to test if the film
thickness has significance on the delamination diameter, 2c. The null-hypothesis was that
the delamination diameter, 2c, is not a function of film thickness. The p-value (0.32 >
0.05) indicates that the null hypothesis cannot be rejected and thus there is no significant
difference for 2c between 5 and 10 μm PTFE films.
148
(a) 1 μm (b) 2 μm
(c) 3 μm (d) 5 μm
(e) 10 μm
Figure 7.5. Thickness effects on the interfacial delamination of PTFE thin films (normal
force: 0.5 N). Note that 2c represents the delamination diameter at the interface.
2c
2c
149
(a) 0.5 N (b) 0.75 N
(c) 1 N (d) 2 N
(e) 3 N
Figure 7.6. Load effects on the interfacial delamination of PTFE thin films (film
thickness: 3 μm).
In order to develop a general understanding of the effects of interfacial fracture toughness,
Γi, and the other material properties, interfacial delamination was modeled for 1, 5 and 10
150
μm films by using a range of Γi (50 – 1000 mJ/m2), E (0.5, 3 GPa), Hp (30, 105 MPa) and
t (1, 5, 10 μm) values. Figure 7.7(a) - (c) show the computed variation of the delamination
diameter, 2c, with different PTFE material properties, for 1, 5, and 10 μm thick films,
respectively. The delamination diameter, 2c, increases with increasing normal force, W,
but decreases with increasing interfacial fracture toughness, Γi. For the large Γi values,
the gradient of 2c with respect to Γi is lower. Figure 7.7(a) - (c) also shows that for fixed
Γi and W values, a larger delamination diameter develops under the more compliant
coating.
Table 7.4. Experimentally measured delamination diameter, 2c, for different film
thickness, t, and normal force, W. Note that Ave. 2c is the average delamination diameter;
Std. Dev. 2c is its standard deviation.
t (μm) load, W (N) Ave. 2c (μm) Std. Dev. 2c (μm)
5 0.5 179.41 7.07
5 0.75 177.49 12.11
5 1 187.17 8.93
5 2 201.68 12.30
5 3 181.99 7.02
10 0.5 146.86 13.65
10 0.75 177.00 7.75
10 1 187.27 6.95
10 2 203.27 7.94
10 3 204.49 4.93
151
(a) 1 μm PTFE film
(b) 5 μm PTFE film
152
(c) 10 μm PTFE film
Figure 7.7. The predictions of delamination diameter, 2c, as a function of film thickness
and material properties.
The results presented in Figure 7.8 are normalized by defining the following non-
dimensional force, F , and delamination radius, C ,
2
nW
FHt
(7.3)
i
mp qc E
Ct Et H
(7.4)
where W is the indentation force, E is the Young’s modulus, H is the hardness of the film,
t is the film thickness, c is the delamination radius, Γi is the interfacial fracture toughness.
With these definitions, the relationship between the variables is established as follows,
C F (7.5)
153
where the coefficients α = 1.5 × 10-3
, n = 0.299, p = 0.953, m = 0.381 and q = -0.915 are
found by curve fitting as shown in Figure 7.8. Note that the R2 value for the curve fit is
0.99. The complete relationship between non-dimensional variables is shown as follows,
3
0.3810.953 0.915 0.299
21.5 10ic E W
t Et H Ht
(7.6)
The interfacial fracture toughness, Γc, can be obtained from this equation as follows,
1/0.3810.915 0.299 0.953
20.0015c
E W cEt
H Ht t
(7.7)
By using the experimentally measured values, the interfacial fracture toughness, Γc, for 5
and 10 μm PTFE films are determined by using Equation (7.7). In particular, Γc for the 5
and 10 μm thick coatings are found as 196 ± 112 mJ/m2 and 721 ± 231 mJ/m
2,
respectively. Figure 7.8 shows the comparison of the normalized experimental values and
the numerical simulations, where 1 μm PTFE simulation results are not shown. We also
used Rosenfeld et al.’s method [32], described in Equation (2.74), to calculate the
interfacial fracture toughness, Γc. The values of Γc for 5, 10 μm PTFE films were found as
1190 ± 471 mJ/m2 and 2434 ± 703 mJ/m
2, respectively, which are much higher than the
values evaluated using Equation (7.7). The energy spent in the penetration of PTFE film
was not considered in the Rosenfeld et al.’s model, which results in the overestimation of
the interfacial fracture toughness.
154
The load-dependence of the delamination radius, c, is also investigated. Equation (7.6)
gives the relation of delamination radius, c, and normal force, W, as follows,
0.772 1.35991/0.953 0.3137
0.3998 1.274
i
t Ec W
H
(7.8)
Rosenfeld et al.’s method shown in Equation (2.74) indicates that 0.5c W , which is
reasonably close to Equation (7.8). Figure 7.9 shows the comparison of our finite
element simulation to Rosenfeld et al.’s formulation for 10 μm PTFE delamination.
Figure 7.8. The curve fitting of non-dimensional delamination radius, C , and indentation
force, F , from finite element simulation.
155
Figure 7.9. The comparison of 10 μm PTFE finite element simulation to Rosenfeld et al’s
formulations with respect to different material properties.
7.4 Summary and Conclusion
Adhesion properties of PTFE thin films deposited on glass were investigated. Micro-
indentation tests were used to induce the interfacial delamination of PTFE thin films on
glass substrate. Following observations and conclusions are made:
The experiments showed that the delamination diameter, 2c, depends on the film
thickness and the normal force. The critical thickness for the occurrence of
delamination was found to be 3 μm.
The entire indentation process was simulated by using the finite element method,
with material damage model and bilinear traction-separation law. The numerical
156
results indicated that the delamination diameter, 2c, depends on film thickness,
material properties, and indentation force.
The occurrence of delamination depends on the strain energy stored in the film
during the indentation, indicating thinner films are less likely to delaminate.
Predicted interfacial fracture toughness, Γc, for 5 and 10 μm PTFE films are less
than those predicted by Rosenfeld et al.’s formulation. The difference is due to the
energy spent during penetration to cause PTFE film damage/failure.
157
8 Summary, Conclusions and Future Work
In this dissertation, we investigated the frictional and durability characteristics of thin-
film PTFE coatings deposited on rough and smooth aluminum substrates. We also
studied the frictional characteristics of thin PFA and Poly(V3D3) films on glass substrates.
The analysis of variance (ANOVA) was used to quantitatively analyze the effects of
normal force, sliding speed and surface roughness on the COF and durability.
Material damage/failure approach was introduced to numerical simulation of indentation
on a soft polymer coating because 2D-axisymmetric numerical model without using
continuum damage mechanics underestimated the contact width, compared to
experimental measurement. Different damage criteria were used and the effects of
equivalent plastic strain and fracture toughness on the prediction of contact width were
studied.
Interfacial delamination of thin PTFE films was also investigated. Micro-indentation with
a Rockwell C indenter was performed on films with different thickness and delamination
diameters were measured by using optical microscope. Numerical model using shear
damage/failure criterion and cohesive zone model was developed to predict the
delamination diameter. A relationship of non-dimensional indentation force and
delamination radius was found by curve-fitting and was used to evaluate the interfacial
fracture toughness.
158
The conclusions are summarized as follows,
Frictional and durability characteristics of thin-film PTFE coatings
It was observed that the COF and durability depend on normal force, sliding speed
and surface roughness from the substrate. In particular, the COF increases by
increasing the surface roughness, Ra, and the sliding speed. Load-dependence of COF
is observed on rough surfaces, but is negligible on smooth substrates. The durability
is improved by increasing the sliding speed or decreasing the normal force, but has a
nonlinear relationship with surface roughness, Ra. The ANOVA results indicated that
surface roughness of substrate has the most significant effects on the COF and
durability. The sliding speed contributes more to the COF and durability as compared
to the normal force.
We also observed that the interface of PTFE and smooth aluminum substrates is well
lubricated in friction and durability tests. A modified equation based on Briscoe and
Tabor friction model [75, 76], was developed to predict the COF on the smooth
surface and had good agreements with the experiments. It indicated that the adhesion
governs the sliding and durability process on smooth substrates.
Frictional characteristics of thin-film PFA and Poly(V3D3) coatings
We observed that the COF of PFA increases by increasing the sliding speed while
that of silicone increases by reducing the speed. The ANOVA showed that the sliding
159
speed and the film thickness have greater significance on the COF as compared to the
normal force.
Mechanics simulation of material damage for soft polymer coatings
The simulation results, as compared to the experiment, showed that shear
damage/failure model was suitable to characterize the interfacial contact of soft
polymers on rigid substrates.
We investigated the effects of equivalent plastic strain and bulk fracture toughness as
indicators of the damage initiation and evolution, respectively, on the prediction of
contact width. The results showed that the bulk fracture toughness has more
significant effects on the contact width as compared to the equivalent plastic strain.
We also found that the equivalent plastic strain 0
pl = 1% and fracture toughness Γf =
20 J/m2 predict the experimentally obtained results well. It showed that indentation
simulation can be used to evaluate the bulk fracture toughness for thin-film polymers,
which are difficult to determine using typical experimental methodologies.
We also observed that thin-film polymer coating at a high contact pressure has less
resistance to be deformed, indicated by simulation results.
Interfacial delamination of thin-film PTFE coatings
We performed the micro-indentation on thin PTFE films to induce interfacial
delamination. The experiments showed that the delamination diameter depends on the
160
film thickness and the normal force. In our work, the critical thickness for the
occurrence of delamination was found to be 3 μm. Thinner films were observed to
delaminate less likely because the occurrence of delamination depends on the strain
energy stored in the film during the indentation.
We employed the finite element method, using material damage/failure approach and
cohesive zone model, to simulate the delamination process. The results showed that
the prediction of delamination diameter is a function of film thickness, material
properties, indentation load and interfacial fracture toughness. The relationship of
non-dimensional indentation force and radius was developed to evaluate the
interfacial fracture toughness of 5 and 10 μm PTFE coatings. The evaluations of
interfacial fracture toughness were found to be much smaller as compared to those
values using Rosenfeld et al.’s equations. The difference is due to the energy spent
during penetration to cause PTFE film damage/failure, which is not considered in the
mathematical description of Rosenfeld et al.’s model.
Future Work
In the future, a simulation of scratch on rough surface may be modeled to investigate
the effects of surface roughness on the COF of soft polymer coatings. More refined
nonlinear material properties may be included in the finite element model to study the
effects of creep response on the prediction of contact radius and adhesion properties.
The experiments related to the evaluation of non-linear material properties may be
proposed, including drop-ball and peel tests. The frictional and durability tests on
161
other thin polymer films may be performed by considering different indenter and
substrate materials, which probably expands the overview and understandings to the
mechanisms of polymer tribology in thin-film form.
162
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Appendix 1 ABAQUS/Explicit Verification by
Modeling 2D Interface of Indentation
In order to verify if ABAQUS/Explicit is applicable to the analysis of indentation
interface, a 2D-axisymmetric finite element model as shown schematically in Figure 6.1
was used. An analysis of this configuration was given by analytical functions by
Matthewson [8]. The same mesh and boundary conditions were employed. The coating
and the substrate were modeled as elastic. In particular, the Young’s modulus and the
Poisson’s ratio of substrate were given as 72 GPa, and 0.22, respectively. A variety of the
Poisson’s ratios and thus shear moduli were assigned to the coating to investigate the
variations of interfacial shear stress as a function of contact radius. Table A1.1 shows
material properties and contact radius, b, evaluated in the finite element analysis.
Figure A1.1 shows the normalized interfacial shear stress variation in the radial direction
as predicted by finite element analysis and Matthewson’s close-form formulation [8].
Results predicted by the finite element method indicate a similar trend to analytical
solution. The deviations are attributed to i) finite element mesh, which affects the
evaluation of contact radius, and ii) the assumption of averaged stress in Matthewson’s
model.
171
Table A1.1 Material properties of glass using nano-indentation.
Indenter radius,
R (mm)
Poisson’s
ratio, ν
Young’s modulus, E
(MPa)
Shear modulus, G
(MPa)
Contact radius, b
(mm)
3.125 0.3 5000 1923 0.131
3.125 0.4 5000 1786 0.112
3.125 0.45 5000 1724 0.095
Figure A1.1 The comparisons of interfacial shear strength between finite element
analysis and Matthewson’s solution [8].
172
Appendix 2 Material Properties of Fused Silica
Using Nano-Indentation
The sample of fused silica is selected as the calibration of nano-indentation. The
experimental method and instrument were presented in Section 7.2. Figure A2.1 shows
the load-displacement graphs of fused silica for two test repeats. In order to compare to
material properties investigated by Oliver and Pharr [16], the maximum indentation load
was set as 120 mN. The material properties are determined as follows. The initial
unloading stiffness, S, is evaluated using Equation (2.37), shown as follows,
36
9
Δ 120.325 100.249 10
Δ 1021.969 540.031 10
WS
δ
(N/m) (A2.1)
The deflection of the surface, δs, is calculated using Equation (2.42) and shown as
follows,
39max
6
120.325 100.72 347.93 10
0.249 10s
Wδ ε
S
(m) (A2.2)
The contact depth, δc, is evaluated using Equation (2.41),
9 9
max 1021.969 347.93 10 674.04 10c sδ δ δ (m) (A2.3)
The contact area, Ap, for Berkovich indenter, was estimated by using Equation (2.43) and
shown as follows,
173
Figure A2.1 Indentation test on fused silica.
2
2 9 1224.5 24.5 674.04 10 11.13 10p cA δ (m2) (A2.4)
Thus, the hardness of fused silica, Hs, is calculated using Equation (2.40),
3
max
12
120.325 1010.81
11.13 10s
p
WH
A
(GPa) (A2.5)
The reduced modulus, Er, is evaluated based on Equation (2.37),
610
12
0.249 106.397 10 63.97
2 2 1.034 11.13 10r
p
S π πE
β A
(GPa) (A2.6)
174
The Young’s modulus of fused silica, Es, is evaluated using Equation (2.39),
1 12 2
2 21 1 0.071 1
1 1 0.18 65.663.97 1141
i
s s
r i
νE ν
E E
(GPa) (A2.7)
where Young’s modulus of indenter, Ei = 1141 GPa, the Poisson’s ratio, νi = 0.07 [16].
The Young’s modulus obtained by Oliver and Pharr is 69.3 GPa with the standard
deviation of 0.39 GPa.
175
Appendix 3 Mechanical Properties of Thin
Polymer Films Using Indentation
Nano-indentation is used to characterize mechanical properties of polymers in thin-film
and bulk forms. The experimental method and background are presented in Section 2.5
and 7.2, respectively. Figures A3.1 - A3.4 show the load-displacement graphs for 5, 10
µm PTFE films, bulk PTFE and 5 µm silicone film, respectively. The Young’s modulus
and the hardness were evaluated by using Oliver and Pharr’s theory [16]. Table A3.1
shows the material properties of polymers, including PTFE and silicone, in thin-film and
bulk forms, respectively. It indicates that thin-film PTFE coatings have greater Young’s
modulus and hardness than bulk PTFE, but lower than 5 µm silicone films. The student’s
T-test was performed to test if the material properties are as a function of film thickness
among 5 and 10 µm PTFE films. The null-hypothesis is that the material properties of
PTFE are not a function of film thickness. The p-values shown in Table A3.2 indicate
that film thickness has no significance on the material properties between 5 and 10 µm
PTFE films.
Lucas et al. [59] reported that the material properties of PTFE film on silicon wafer are
independent of the film thickness (0.5 – 15 µm). Wang et al. [58] examined the thickness
effects on material properties of ultra-thin PTFE films (48.1 – 1141 nm), spin coated on
the silicon wafers. Thickness-dependence effect is seen on thinner films (< 500 nm).
When the thickness is greater than 500 nm, the test results indicate no thickness-
176
dependence on material properties. In particular, Young’s modulus, E = 2.3 GPa, and
hardness, H = 58 MPa. Wang et al. [58] also found that PTFE in thin-film form exhibits
higher modulus compared to bulk sample and free-standing film. It is possible that
polymers have a high pressure dependence of the strength, which gives rise to higher
modulus in compression than in tension. The molecular organizations in ultra-thin and
thin polymer films are different from those in bulk forms, resulting in significance in
thermophysical and mechanical properties [60].
Figure A3.1 The load-displacement graph for 5 µm PTFE film.
177
Figure A3.2 The load-displacement graph for 10 µm PTFE film.
Figure A3.3 The load-displacement graph for bulk PTFE (thickness, t = 1.5 inches).
178
Figure A3.4 The load-displacement graph for 5 µm silicone film.
Table A3.1 Material properties of polymers in thin-film and bulk forms.
5 µm PTFE 10 µm PTFE bulk PTFE 5 µm silicone
Young’s modulus (GPa) 2.71 2.80 0.85 3.55
Std. Dev. (GPa) 0.26 0.17 0.07 0.26
Hardness (GPa) 0.11 0.10 0.04 0.23
Std. Dev. (GPa) 0.01 0.01 0.01 0.02
Table A3.2 The student’s T-test of material properties for 5 and 10 µm PTFE films.
Young’s modulus, E Hardness, H
p-value 0.517 0.095
179
Appendix 4 Material Properties of Glass
Substrates
Material properties of glass substrate, where thin polymer films are supposed to be
deposited by using HFCVD, are also examined. Table A4.1 shows the Young’s modulus
and the hardness of glass with ten test repeats. The evaluations of Young’s modulus and
hardness are given as 62.59 ± 7.83 GPa, and 9.45 ± 1.85 GPa, respectively.
Table A4.1 Material properties of glass using nano-indentation.
Test # W (mN) δmax (nm) E (GPa) H (GPa)
1 4.99 208.23 54.34 9.26
2 4.99 215.34 56.66 7.60
3 4.99 200.77 72.71 7.92
4 4.99 197.91 62.12 9.87
5 4.99 213.71 49.56 9.35
6 4.99 207.84 60.21 8.26
7 4.99 188.11 73.57 10.16
8 4.99 195.38 67.44 9.49
9 4.99 203.91 63.59 8.45
10 4.99 183.28 64.66 14.13
Average 62.59 9.45
Std.Dev. 7.83 1.85
180
Appendix 5 Green’s Function
A Green’s function g(x,s) represents the effect/displacement at a position x due to a force
of unit magnitude acting at s, which is schematically shown in Figure A5.1. Note that a
Green’s function is symmetrical, i.e. g(x,s) = g(s,x). The displacement v(x) for any
distributed force, p(x), applied in the range l1 ≤ x ≤ l2, can be obtained by using an
influence (Green’s) function as follows [7],
2
1
,l
lp xv x g x s dx (2.11)
Figure A5.1 Schematic of the displacement for distributed force, p(x) using the Green’s
function [7].