Excerpt from the Proceedings of the COMSOL Users Conference 2006 Paris
Numerical Simulation with “Comsol Multiphysics” of Crystallization Kinetics of Semi-Crystalline Polymer during Cooling: Application to Injection Moulding Process. N. Brahmia*, 1, P. Bourgin 1, M. Boutaous*, 1, D. Garcia2 1 Laboratoire de Recherche Pluridisciplinaire en Plasturgie « LR2P ». Site de plasturgie de l’INSA de Lyon BP 807-01108 Bellignat. 2Pôle Européen de Plasturgie. 2, rue P.M. Curie -01100 Bellignat.
[email protected] 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, the polymer 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.
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 often the 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 using Comsol multiphysics software.
2. Experimental section In this study, the polymer used is an injection moulding grade of a semicrystalline polymer which is polypropylene. Before starting simulation, several characterizations of the material such as, viscosity, thermal conductivity, heat capacity and specific volume must be determined.
Keywords: Injection moulding, semicrystalline polymer, crystallization kinetics, thermal properties, heat transfer
2.1 Thermal properties
1. Introduction 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 (α). This parameter, as shown by Fulchiron [1], is defined by:
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 of rheological 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
α =
Xc X∞
(1)
Where X c 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 α: (2) C p (α , T ) = α C p sc (T ) + (1 − α ) C p a (T )
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Excerpt from the Proceedings of the COMSOL Users Conference 2006 Paris
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 polymer was 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 for Cpsc and Cpa: C p a ( T ) = 3.4656 T + 1774.2
(3)
C ps c (T ) = 8.3619 T + 1329.2 With T in degree °C and Cp in J.kg-1.K-1.
(4)
Figure (2) shows that the thermal conductivity of amorphous and semicrystalline phase, are temperature dependent and a linear fitting gives: (6) λ sc (W / m . K ) = 10 -4 T + 0.1868
λ a (W / m .K ) = 10 -4 T + 0.1394 Where T in °C and λ 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: (8) V (α , T , P ) = α V sc (T ) + (1 − α ) V a (T )
8000
Where Va
7000
Cp (J/kg.K)
is
the specific volume of the
amorphous sate of the material and Vsc is the specific 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: (9) V a = 3 × 10 −3 T + 1 .2108
6000 5000 4000 3000 2000 1000 20
40
60
80
100 120 140 160 Temperature (°C)
Figure 1: Evolution temperature.
of
heat
180
200
220
Vs c = 2.6 ×10 −3 T + 1.1055 Where T in °C and V in m3.kg-1
240
(10)
capacity versus 1,5 P = 40MP P = 80MP P = 120MP P = 160MP
1,45
The thermal conductivity is also described by a mixing rule between solid state and liquid state as follows: (5) λ (T , α ) = α λ sc (T ) + λ a (T )
Thermal conductivity (W/m K)
(7)
1,4
Vsp(cm³/g)
1,35 1,3 1,25 1,2
0,24
1,15
0,22
1,05
1,1
1 0,2
50
100
150 Temperature (°C)
200
250
Figure3: PVT diagram of PP.
0,18
0,16
2.2 Crystallization Kinetic
0,14 20
40
60
80
Crystallization is the process whereby an ordered structure is produced from a disordered phase, usually a melt or solution. It represents a mix between nucleation and growth. When a semicrystalline polymer melt is cooled down into its crystallization temperature range, crystallization starts around discrete points called “nuclei”, and the crystal grow around nuclei to form what is referred to as “spherulites”. Once
100 120 140 160 180 200 220 240 260 Temperature (°C)
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].
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Excerpt from the Proceedings of the COMSOL Users Conference 2006 Paris
respectively. Tg is the glass transition temperature of the material. Differentiation of equation (12) leads to differential form of Nakamura crystallization kinetics, which is more useful in modeling:
all the spherulites meet their neighbors, the crystallization process is complete. The Avrami equation [4] has been widely used to describe polymer crystallization kinetics under isothermal conditions in the form: (11) α ( t ) = 1 − exp ( − k t n ) 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 is constant, Nakamura & al [5] developed the following equation from Avrami theory [4]:
t
0
n −1 ∂α = n K ( T ) (1 − α ) [− ln (1 − α )) ] n ∂t
To determine K(T), different DSC measurements of isothermal and no isothermal crystallization were carried out, following a measurement protocol used previously by Koscher [7,8] and Juo [9] and. By fitting the experimental results using equations (13) and (14), the parameters which permitted to determine K (T), were defined.
n
α ( t ) = 1 − exp − ∫ K ( T ) dt
(12 )
Where K (T) is the non-isothermal crystallization rate constant. K ( T ) = [k ( T ) ] = ln ( 2 ) 1 n
1 n
1 t1 2
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 system is 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).
(13)
Where t 1 is the crystallization half time. 2
1 t1 2
indicates the overall rate of isothermal
crystallization. By assuming that the number of nucleation sites is independent of temperature and all sites are activated at the same time, Hoffman & al [6] developed the following expression to describe the overall rate of crystallization as a function of temperature: 1 t1 2
1 = t1 2
−U * exp exp R (T − T ∞ ) 0
−Kg T ∆T
(15 )
Figure 4: Model Geometry
(14 )
The general energy equation has been used to describe the heat transfer in the system: ∂T ∂ ∂T + Q (16) = λ (α , T ) ∂ t ∂y ∂ y Where Q is a term source defined by: ∂ α (17) Q = ρ (α , T ) ∆ H ∂ t ∆H is the total enthalpy released during crystallization process. The crystallization kinetics is given by Nakamura equation as follows: n −1 ∂α (18) = n K ( T ) (1 − α ) ( − ln( 1 − α )) n ∂t The Nakamura constant K (T) as proposed by Koscher [7, 8] is given by:
ρ (α , T ) C p (α , T )
Where T is the crystallization temperature, and is a factor that includes all the terms which 1 t1 2 0
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, ∆ T = T f − T is the super cooling. According to Hoffman & al [6] the parameter U* and T∞ can be assigned “universal” values 6284 J/mol.K and Tg -30K
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Excerpt from the Proceedings of the COMSOL Users Conference 2006 Paris
1
4 K (T ) = π N 0 (T ) 3 G0 3
1 0,9
Kg U* . exp − × exp − T (T − T ) R ( T − T∞ ) f
Relative crystallinity
(19)
The polymer tested by Koscher [7, 8] is an injection grade of isotactic polypropylene (Eltex HV 252). The values of different parameters in K(T), are listed as follows:
3 . 10 T + 2124 10 . 68 T + 1451
ρ a (α, T) ρ sc (α, T)
1 (1 . 138 + 6 . 773 × 10
−5
T + 0 . 189
− 4 . 96 × 10
−4
T + 0 . 31
1 (1 . 077 + 4 . 225 × 10
−4
0,4
0 150
140
130 120 Temperature (°C)
110
100
90
Figure 5: Evolution of relative crystallinity versus temperature: Comparison between Comsol multiphysics and Matlab results for constant cooling rate (20°C/min).
For the same time step, the evolution of relative crystallinity was calculated for a cooling rate of 20°C/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 constant values 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).
T)
−4
0,5
0,1
*
− 6 . 25 × 10
0,6
0,2
G0 = 2.83 × 10 2 , Kg = 5.5 × 10 5 (K ), U = 6284 J/mol.K, Tf = 210°C, T∞ = -30°C, ∆H =90 × 103 J/kg and n =3. The material properties are defined by the general equations: (2), (5) and (8). For amorphous and semicrystalline states the functions are listed in table 1: Cpa (α, T) Cp sc (α, T) λ a (α, T) λsc (α, T)
0,7
0,3
(20)
N 0 ( T ) = exp ( 0 . 156 × ( T f − T ) + 15 . 1 ) 2
Simulation _Comsol Simulation_Matlab
0,8
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 the polymer, and decreases linearly: T = T0 − Vref × t , where T0 represent the initial
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 the experimental results.
temperature of the system and Vref is the cooling rate value. The Nakamura 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.
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Excerpt from the Proceedings of the COMSOL Users Conference 2006 Paris
- In order to control the thermal kinetics in the polymer, we have assumed a linear decrease of the temperature at the interface metal/polymer as follows: T = T0 − VCooling × t 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 147°C, 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 6: Comparison at constant cooling rate of relative crystallinity, between Comsol multiphysics results, Koscher measurements and Koscher calculations [8]
Figure 7: Comparison of relative crystallinity for isothermal crystallization, between Comsol multiphysics results, experimental measurements, and Koscher calculations [8]
4.3 Simulation of effective cooling of the polymer plate: coupling of the heat and the crystallization equations
Figure 8: Temperature evolution at different positions in the polymer plate, (Vcooling = 6°C/min)
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.
Figure (9) presents the evolution of relative crystallinity as function of time at different locations in the plate. We notice that the polymer in the skin layer crystallizes more quickly then in the core layer. The total crystallization time is about 180 sec.
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Excerpt from the Proceedings of the COMSOL Users Conference 2006 Paris
two different thermal kinetics: VCooling = 12 °C/min and VCooling = 30°C/min.
1,1 x=0 x = 2 mm x = 3.5 mm x = 4.8 mm
1
0,8
1,1 1
0,7
0,9
0,6
0,8
0,5
Relative Crystallinity
Relative Crystllinity
0,9
0,4 0,3 0,2 0,1 0 300
400
500
600 700 Time (s)
800
900
1000
0,6 0,5 0,4 0,3 0,2 0,1
Vcooling = 12°C/min
0
Vcooling = 30°C/min
-0,1
Figure 9: Evolution of relative crystallinity at different position of the plate, for Vcooling = 6°C/min.
4.3.1
0,7
70
80
90
100 110 Temperature (°C)
120
130
140
Figure 11: Effect of cooling rate on crystallization at polymer core layer.
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.
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). 220 200 180 160 T em p era tu re (°C )
210 Vcooling = 12°C/min V cooling= 30°C/min
190
Temperature (°C)
170 150 130 110
140 120 100 80 T0 = 180°C T0 = 190°C T0 = 200°C T0 = 208 °C
60 40
90
20
70
0 0
50
100
200
300
400
500
Time (s)
30
Figure 12: Effect of initial temperature temperature profile at polymer core.
10 0
100
200
300
400 500 Time (s)
600
700
800
900
Figure 10: Evolution of temperature at core polymer for two different cooling rates
on
It’s 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 doesn’t changes.
The cooling time is reduced of about (242 sec) when we increase the VCooling of about 18°C/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
Figure (13) shows, the evolution of relative crystallinity at the interface (mould/polymer) and at polymer core.
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Excerpt from the Proceedings of the COMSOL Users Conference 2006 Paris
numerical simulations, with the coupling between the heat transfer equation and crystallization equation, are easily solved and interesting 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. .
1,1 1
Relative Crystallinity
0,9 0,8 0,7 0,6 0,5 0,4 0,3
T0 = 180°C, x = 0 T0 = 208°C, x = 0 T0 = 180°C, x = 4.5mm T0 = 208°C, x = 4.5mm
0,2 0,1 0
6. References
-0,1 60
70
80
90 100 Temperature (°C)
110
120
130
1. R. Fulchiron, E. Koscher, G. Poutot, D. Delaunay, G. Régnier, 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 Injectés: Application à la Prédiction des Retraits, thèse 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, N°2 (1990). 4. M. Avrami, Granulation, Phase Change and Microstructure: Kinetics of Phase Change, Journal of Chemistry and Physics, Vol. 9, pp177184 (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 Polypropylène : Aspects Cinétiques et Morphologiques, thèse de doctorat, université ‘Claude Bernard’ Lyon 1, 2002. 9. J. Juo, Numerical Simulation of StressInduced Crystallization of Injection Molded Semicrystalline Thermoplastics, PHD Degree, Faculty of New Jersey Institute of Technology, (2001).
Figure 13: Effect of initial temperature on relative crystallinity at polymer core layer for Vcooling = 30°C/min.
We note that crystallization begins approximately 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 onedimensional 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 at different 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
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