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Numerical Simulation with Comsol Multiphysics of Crystallization

Kinetics of Semi-Crystalline Polymer during Cooling: Application to

Injection Moulding Process.

N. Brahmia*, 1

, P. Bourgin1, M. Boutaous

*, 1, D. Garcia

2

1Laboratoire de Recherche Pluridisciplinaire en Plasturgie LR2P . Site de plasturgie de lINSA de Lyon

BP 807-01108 Bellignat. 2Ple Europen de Plasturgie. 2, rue P.M. Curie -01100 Bellignat.nadia.brahmia@insa-lyon.fr

Abstract: Thermoplastic polymers are

frequently used in several industries. They

represent the most important group of polymeric

materials. During injection moulding process,

the cooling phase including solidification is oftenthe most significant part of a manufacturing

cycle. For semicrystalline polymers, where

different crystal structures appear, the thermo

mechanical histories play a very important role

in the estimation of the final parts properties.

During the cooling stage, the behavior of these

polymers is very difficult to model because of

the complication of the crystallization process

which is coupled to heat transfer and material

properties change. The aim of this work is the

study of the coupling phenomena between the

crystallization and the thermal kinetics in the

polymer injection moulding process, by usingComsol multiphysics software.

Keywords: Injection moulding, semicrystalline

polymer, crystallization kinetics, thermal

properties, heat transfer

1. Introduction

Injection moulding is one of the most widely

used processes in the polymer processing

industry. The computer simulation of the process

is an essential tool to predict the final properties

of the part. For this calculation the knowledge ofrheological and thermomechanical material

characteristics is of great importance. To

understand the crystallization process of

semicrystalline thermoplastics, during the

injection moulding process, it is necessary to

take into account the kinetics of crystallization

and to consider the temperature and

crystallization dependency of all the

thermophysical properties of the material. All

these variables parameters make the simulation

more complicated. Generally, the simulations of

injection process, consider isothermal or isobar

case. In this work we have used Comsol

multiphysics software to study the coupling

between heat transfer and crystallization kinetics

during the solidification of a semicrystalline

polymer, such as polypropylene. First, thepolymer studied was precisely characterized and

the crystallization kinetics was defined, using

isothermal and no-isothermal DSC experiments.

The results are compared to some data find in the

literature, and then the influence of several

parameters are computed and commented.

2. Experimental section

In this study, the polymer used is an injection

moulding grade of a semicrystalline polymer

which is polypropylene. Before startingsimulation, several characterizations of the

material such as, viscosity, thermal conductivity,

heat capacity and specific volume must be

determined.

2.1 Thermal properties

To study heat transfer during crystallization

of a polymer, it is necessary to take into account

these thermal properties as a function of

temperature and relative crystallinity (). Thisparameter, as shown byFulchiron [1], is defined

by:

=X

Xc (1)

WhereXc is the crystallinity depending on time,

temperature and pressure and X, the

crystallinity at the end of the solidification.The heat capacity is described using simple

mixing rule between the solid state and the liquid

state values weighted by the relative crystallinity

:

)()1()(),( TCTCTC apscpp += (2)

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Where Cpsc and Cpa are the heat capacity of the

solid state and the liquid state, respectively.

The specific heat Cp in each phase of the polymerwas obtained by using calorimetric techniques

(DSC), as shown in figure (1).

For the polymer used in this work, Le Bot[2]

proposed a linear temperature dependency versus

temperature forCpsc and Cpa:

1774.2+T3.4656)( =TC ap (3)

1329.2+T8.3619)( =TC cps (4)

With T in degree C and Cp in J.kg-1.K-1.

1000

2000

3000

4000

5000

6000

7000

8000

20 40 60 80 100 120 140 160 180 200 220 240

Temperature (C)

Cp(J/kg.K

)

Figure 1: Evolution of heat capacity versus

temperature.

The thermal conductivity is also described by a

mixing rule between solid state and liquid state

as follows:

)()(),( TTT asc += (5)

0,14

0,16

0,18

0,2

0,22

0,24

20 40 60 80 100 120 140 160 180 200 220 240 260

Temperature (C)

Thermalconductivity(W/mK)

Figure 2. Thermal conductivity versus temperature.

The thermal conductivity was measured by an

apparatus based on the transient line source

method, which is ideally suited to polymers

because measurements can be made quickly,

before the onset of thermal degradation and no

sample preparation is required [3].

Figure (2) shows that the thermal conductivity of

amorphous and semicrystalline phase, are

temperature dependent and a linear fitting gives:

0.1868+T10)./( -4=KmWsc (6)

0.1394+T10)./( -4=KmWa (7)

Where T in C and in W.m-1.K-1

The density is the inverse of specific volume

which is deduced from PVT diagrams: figure (3).

It was evaluated as a weighted mean value of the

pure amorphous and crystalline densities

according to:

)()1()(),,( TVTVPTV asc += (8)

WhereaV is the specific volume of the

amorphous sate of the material and Vsc is thespecific volume of the solid state.

The PVT experimental results were obtained

using a piston-type dilatometer (GNOMIX).

A linear fit for specific volume for each phase is

obtained as:

2108.1103 3 += TVa (9)

1.1055+T102.6 3=csV (10)

Where T in C and V in m3.kg-1

1

1,05

1,1

1,15

1,2

1,25

1,3

1,35

1,4

1,45

1,5

50 100 150 200 250

Temperature (C)

Vsp(cm/g)

P = 40MP

P = 80MP

P = 120MP

P = 160MP

Figure3: PVT diagram of PP.

2.2 Crystallization Kinetic

Crystallization is the process whereby an

ordered structure is produced from a disorderedphase, usually a melt or solution. It represents a

mix between nucleation and growth. When asemicrystalline polymer melt is cooled down into

its crystallization temperature range,crystallization starts around discrete points callednuclei, and the crystal grow around nuclei to

form what is referred to as spherulites. Once

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all the spherulites meet their neighbors, thecrystallization process is complete.

TheAvrami equation [4] has been widely used to

describe polymer crystallization kinetics under

isothermal conditions in the form:

)(exp1)( ntkt = (11)

Where n is the Avrami index, k is the Avrami

isothermal crystallization rate constant.

For non isothermal, isokinetic conditions, and

assuming that the number of activated nuclei isconstant, Nakamura & al [5] developed the

following equation fromAvrami theory [4]:

)12()(exp1)(

0

nt

dtTKt

=

WhereK(T) is the non-isothermal crystallizationrate constant.

[ ]

==

2

1

11 1)2(ln)()(

tTkTK nn

(13)

Where

2

1t is the crystallization half time.

2

1

1

t

indicates the overall rate of isothermal

crystallization. By assuming that the number of

nucleation sites is independent of temperatureand all sites are activated at the same time,Hoffman & al [6] developed the following

expression to describe the overall rate ofcrystallization as a function of temperature:

)14(exp)(

exp11 *

02

1

2

1

=

TT

K

TTR

U

tt

g

Where T is the crystallization temperature, and

021

1

t

is a factor that includes all the terms which

are independent of temperature. R is the

universal gas constant, U* is the activation

energy of the transport of crystallizing units

across the phase boundary. T is the temperature

under which such as transport ceases. Kg is the

spherulite growth rate, TTT f = is the super

cooling. According to Hoffman & al [6] the

parameter U*

and T can be assigneduniversal values 6284 J/mol.K and Tg -30K

respectively. Tg is the glass transitiontemperature of the material.

Differentiation of equation (12) leads todifferential form of Nakamura crystallization

kinetics, which is more useful in modeling:

[ ] )15())1(ln)1()(1

n

n

TKnt

=

To determineK(T), different DSC measurements

of isothermal and no isothermal crystallization

were carried out, following a measurement

protocol used previously byKoscher [7,8] and

Juo [9] and. By fitting the experimental resultsusing equations (13) and (14), the parameters

which permitted to determine K (T), were

defined.

3. Mathematical modeling

Our aim is the analysis of a cooling of plate

of semicrystalline polymer like in the cooling

phase of injection moulding process. The systemis composed of a plate of polymer between two

blocks of metal. The thickness of each block is

of 30 mm, and of the plate is of 5mm. The width

of the system is of about 50 mm (Figure 4).

Figure 4: Model Geometry

The general energy equation has been used todescribe the heat transfer in the system:

)16(),(),(),( Qy

TT

yt

TTCT p +

=

Where Qis a term source defined by:

tHTQ

= ),( (17)

H is the total enthalpy released duringcrystallization process.

The crystallization kinetics is given byNakamura equation as follows:

n

n

TKnt

1

))1ln(()1()(

=

(18)

The Nakamura constant K (T) as proposed byKoscher[7, 8] is given by:

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)19()(

exp.)(

exp

)(3

4)(

*

03

1

0

=

TTT

K

TTR

U

GTNTK

f

g

The polymer tested by Koscher [7, 8] is an

injection grade of isotactic polypropylene (EltexHV 252). The values of different parameters in

K(T), are listed as follows:

)1.15)(156.0(exp)(0 += TTTN f (20)

G0 =21083.2 ,Kg=

5105.5 (K2), U

*= 6284

J/mol.K, Tf =210C, T= -30C, H =90 10

3

J/kgand n =3.

The material properties are defined by the

general equations: (2), (5) and (8). Foramorphous and semicrystalline states the

functions are listed in table 1:

Cpa (, T) 212410.3 +T

Cp sc (, T) 145168.10 +T

a (, T) 189.01025.6 5 + T

sc (, T) 31.01096.44 + T

a (, T) )10773.6138.1(14 T+

sc (, T) )10225.4077.1(14

T+

Table 1: Material properties of PP Eltex HV225

4. Results and discussion

4.1 Numerical validation of the model

The objective of this section, is to ensure that

Comsol multiphysics, solves correctly the

problem, without any numerical errors. To do,

we have opted to compare the results to those

find in the literature. Due to the lake of data

obtained in similar systems, the phenomenon

was simplified. We have assumed that the

system is cooled at constant cooling rate as in

differential scanning calorimeter (DSC). The

temperature is homogenously distributed in thepolymer, and decreases linearly:

tVTT ref = 0 , where T0 represent the initial

temperature of the system and Vref is the cooling

rate value. TheNakamura crystallization kinetics

was then solved using diffusion equation in

general module of Comsol multiphysics

software, and also by Matlab software using a

finite differences scheme to discretize the

equations.

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

90100110120130140150Temperature (C)

Relativecrystallinity

Simulation _Comsol

Simulation_Matlab

Figure 5: Evolution of relative crystallinity versustemperature: Comparison between Comsolmultiphysics and Matlab results for constant coolingrate (20C/min).

For the same time step, the evolution of relative

crystallinity was calculated for a cooling rate of20C/min. The result shows no difference

between the two calculations and this had

permitted us to validate our model numerically.

4.2. Comparison with experimental results:

The second validation of the model is made

by comparison with experimental results from

literature [8]. The system which was simplified

before, as a simple case of DSC, was studied for

constant and variable temperatures. The relative

crystallinity was calculated for several constantvalues of cooling rates and temperatures.

Comparison between our simulations results,

Koscher & al [8] experimental results and their

own calculations were carried out as shown in

figure (6) and (7).

For constant cooling rate, our results were more

representative of the experimental results,

compared to those calculated in the literature.

This simulation permitted to improve the

calculation of crystallization kinetics. In the

figure (6), we can see that the Comsol

simulations follow with a good agreement theexperimental results.

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Figure 6: Comparison at constant cooling rate ofrelative crystallinity, between Comsol multiphysics

results, Koscher measurements and Koscher

calculations [8]

Figure 7: Comparison of relative crystallinity for

isothermal crystallization, between Comsolmultiphysics results, experimental measurements, and

Koschercalculations [8]

4.3 Simulation of effective cooling of the

polymer plate: coupling of the heat and

the crystallization equations

In this section, we consider the system

presented in figure (4). Some simplifications are

assumed compared to the reel conditions in

injection moulding process.

- At initial time, the temperature of all the system(mould and polymer) is homogeneous T (t =0) =

T0.

- The cooling boundary conditions are imposed

at the interface metal/polymer. Due to the plate

dimensions, the heat transfer occurs only by

conduction in one dimensional direction. The

contact between the polymer and the mould is

assumed to be perfect, without any thermal

contact resistance.

- In order to control the thermal kinetics in the

polymer, we have assumed a linear decrease of

the temperature at the interface metal/polymer asfollows:

tVTT Cooling = 0 Where, Vcooling represents the cooling rate.

Figure (8) shows the temperature evolution in

the plate during the cooling process, at several

locations. At the middle of the plate (at x= 0),

due to the polymer thermal inertia, the

temperature diminution begins after 60 sec. The

variation which occurs between (725 and 895)

sec corresponds to the crystallization of polymer.

It is characterized by a plateau that appear at

147C, and which corresponds to the release of

latent heat of crystallization. After 920 sec the

polymer is entirely solidified. The temperature

evolution in the solid phase tends to an

asymptotic form (the system tends to be

thermally homogeneous).

Figure 8: Temperature evolution at different positionsin the polymer plate, (Vcooling = 6C/min)

Figure (9) presents the evolution of relative

crystallinity as function of time at different

locations in the plate. We notice that the polymerin the skin layer crystallizes more quickly then in

the core layer. The total crystallization time is

about 180sec.

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0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

1,1

300 400 500 600 700 800 900 1000

Time (s)

RelativeCrystllinity

x = 0

x = 2 mm

x = 3.5 mm

x = 4.8 mm

Figure 9: Evolution of relative crystallinity atdifferent position of the plate, for Vcooling = 6C/min.

4.3.1 Effect of the cooling rate

In order to study the effect of the cooling rate

on the material properties, several simulations

with different boundary conditions are made.

The effect of the thermal kinetics on the

crystallization rate can be observed on the figure

(10). As expected, we observe that the decrease

in initial temperature leads to the acceleration of

the crystallization and conversely.

10

30

50

70

90

110

130

150

170

190

210

0 100 200 300 400 500 600 700 800 900

Time (s)

Temperature(C)

Vcooling = 12C/min

V cooling= 30C/min

Figure 10: Evolution of temperature at core polymerfor two different cooling rates

The cooling time is reduced of about (242 sec)

when we increase the VCooling of about 18C/min.

Figure (11) shows the evolution of relative

crystallinity at the polymer core for different

cooling rates. It shows that the crystallization is

shifted to the lower temperatures when the

cooling rate is more important. A difference of

10 C between the corresponding temperatures

of the half-time transformation is observed, for

two different thermal kinetics: VCooling = 12

C/min and VCooling = 30C/min.

-0,1

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

1,1

70 80 90 100 110 120 130 140Temperature (C)

RelativeCrystallinity

Vcooling = 12C/min

Vcooling = 30C/min

Figure 11: Effect of cooling rate on crystallization atpolymer core layer.

4.3.2 Effect of initial temperature

To explore the polymer thermal history, we have

studied the effect of the initial temperature in the

material behavior, figure (12).

0

20

40

60

80

100

120

140

160

180

200

220

0 100 200 300 400 500Time (s)

Temperature(C)

T0 = 180C

T0 = 190C

T0 = 200C

T0 = 208 C

Figure 12: Effect of initial temperature ontemperature profile at polymer core.

Its clear from the evolution of temperature

profile in the polymer core, that initial

temperature affects drastically the cooling

kinetics. When T0 increases, the crystallization

begin with a delay, but the rate of cooling and

the temperature of crystallization doesnt

changes.

Figure (13) shows, the evolution of relative

crystallinity at the interface (mould/polymer) and

at polymer core.

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-0,1

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

1,1

60 70 80 90 100 110 120 130Temperature (C)

RelativeCrystallinity

T0 = 180C, x = 0

T0 = 208C, x = 0

T0 = 180C, x = 4.5mm

T0 = 208C, x = 4.5mm

Figure 13: Effect of initial temperature on relative

crystallinity at polymer core layer for Vcooling =30C/min.

We note that crystallization beginsapproximately at the same temperature at the

interface mould/polymer and also at the polymer

core, but it continues on more important ranges

of temperatures at the interfaces. Indeed, at the

polymer core, crystallization plateau slow down

considerably the cooling, so, the temperature

changes less quickly, whereas crystallization

continues. At the interface, temperatures

decrease quickly, and the crystallization takes a

large range of temperatures.

5. Conclusion

Comsol multiphysics software was used to

simulate the cooling of a polymer as in the

injection moulding process of a semicrystalline

polymer. To simplify the injection mould

configuration, we have considered a one-

dimensional heat transfer in a system of

polypropylene plate, initially at melt

temperature, cooled between two metallic

blocks. To take into account crystallization process of the material,Nakamura differential

equation was coupled to the equations system.

Simulations permitted us to determine profile

temperature and relative crystallinity evolution atdifferent positions of the system. The good

agreement between Matlab calculations and

Comsol multiphysics simulations, in a simplified

case, permitted us to validate our model. The

study of the effect of different parameters such

as the initial temperature of the system and the

cooling rate, on the crystallization was also

carried out. In spite of simplifyied Nakamura

crystallization model, relative crystallinity had

been well represented. The difficulties of

numerical simulations, with the coupling

between the heat transfer equation and

crystallization equation, are easily solved andinteresting results and comparisons are obtained.

The next objective will be to use a more

developed crystallization kinetics model in a 3D

simulation of the heat transfer in the injection

moulding process.

.

6. References

1. R. Fulchiron, E. Koscher, G. Poutot, D.Delaunay, G. Rgnier, Analysis of Pressure

Effect on the Crystallization Kinetics of

Polypropylene: Dilatometric Measurements and

Thermal Gradient Modelling, J. Macromol. Sci,Phys., Vol. (3-4), pp 297-314 (2001).

2. P. Le Bot, Comportement Thermique des

Semi-Cristallins Injects: Application la

Prdiction des Retraits, thse de doctorat, Epun

de Nantes, 1998.

3. H. LOBO, C. Cohen, Measurement of

Thermal Conductivity of Polymer, Polymer

Engineering and Science, Vol.30, pp 65-70, N2(1990).

4. M. Avrami, Granulation, Phase Change and

Microstructure: Kinetics of Phase Change,

Journal of Chemistry and Physics, Vol. 9, pp177-

184 (1941).5. K. Nakamura, T. Watanabe, Some Aspects of

Non-Isothermal Crystallization of Polymers: I.

Relationship between Crystallization

Temperature, Crystallinity, and Cooling

Conditions, Journal of applied Polymer, Vol. 26,

pp 1077-1091 (1972).6. J.D. Hoffman, G. T. Davis and J.I Lauritzen,

Crystalline and Non-Crystalline Solids, Treatise

on Solids, J. B. Hannay, Plenum., New York

(1976).

7. E. Koscher, R. Fulchiron, Influence of Shear

on Polypropylene Crystallization: Morphology:

Development and Kinetics, Polymer, Vol.43, pp

6931-6942 (2002).

8. E. Koscher, Effets du Cisaillements sur la

Cristallisation du Polypropylne : Aspects

Cintiques et Morphologiques, thse de doctorat,

universit Claude Bernard Lyon 1, 2002.

9. J. Juo, Numerical Simulation of Stress-

Induced Crystallization of Injection Molded

Semicrystalline Thermoplastics, PHD Degree,

Faculty of New Jersey Institute of Technology,

(2001).

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