We aim to study the behavior of polymer crystallization during cooling stage in injection molding more accurately, the multiscale model and multiscale algorithm proposed in our previous work (Ruan et al., 2012) have been extended to the 3D polymer crystallization case. Our multiscale model incorporates two distinct length scales: a coarse grid for the heat diffusion and a fine grid for the crystal morphology evolution (nucleation, growth, and impingement). Our multiscale algorithm couples the different methods on different length scales, namely, the finite volume method (FVM) on the coarse grid and the pixel coloring method on the fine grid. By using these multiscale model and multiscale algorithm, simulations for 3D polymer crystallization are carried out. Macroscopic variables, for example, temperature, relative crystallinity, as well as the microscopic structural characters, for example, crystal morphology development, and mean size of spherulites, are investigated at various cooling conditions. We also show the importance of coupling heat transfer with crystallization as well as 3D numerical studies.
Nowadays, semicrystalline polymers play an important role in industry due to their advantages of enhanced mechanical properties, ease of manufacturing, and so forth [
Polymer crystallization is a typical multiscale processing: the molecular chains fold together and form ordered regions called lamellae, which compose larger spheroidal structures named spherulites [
To date, a number of numerical investigations have appeared on spherulitic crystallization in polymer melts during cooling. Charbon and Swaminarayan [
In the numerical study of 3D polymer crystallization, Raabe [
The objective of this article is to extend the multiscale model and the multiscale computational method proposed in our pervious work [
During the cooling stage, the process of crystallization is coupled with the heat transfer which makes the modeling more difficult and complex. On the one hand, it is well known that the kinetic parameters of nucleation and growth in microscopic scale are strongly dependent on the temperature or processing conditions. On the other hand, the crystallization is an exothermic process which releases the heat and affects the thermal field in macroscopic level.
Thermomechanical histories are of great importance in the estimate of the final product properties. The corresponding simplified energy equation is [
In the accurate modeling, it is important to consider the thermal properties as a function of temperature and relative crystallinity. The thermal capacity, thermal conductivity can be described by the “mixing rule” of the solid state and the liquid state values to get [
Crystallization is a mechanism of phase change in semicrystalline polymeric materials [
Polymer nucleation is an important factor which affects the final morphology. Usually, the nucleation may vary material from material. In the numerical study, one may adopt an empirical nucleation relation which is derived from fitting the experiment data. Here, we use the following relation of nucleation density [
Growth rate is another important factor which affects the development of morphology. Here, we adopt the Hoffman-Lauritzen theory [
With the growth of spherulites, it is inevitable to meet with the “impingement.” Impingement is happed in spherulites which contact with their neighbors or the walls. Different spherulites impinge to form grain boundaries. With the help of grain boundaries, it is possible to calculate the mean size of spherulites.
In our paper, we assume that the temperature field is in the macroscopic level while the crystal morphology evolution is in the microscopic level (see Figure
Schematic representation of different length scale for computation.
In our algorithm, the domain is first divided by a coarse grid. FVM with cell vertex scheme is used on this coarse grid to calculate the temperature. Then, each coarse grid is subdivided into a number of cubes to form the fine grid. The pixel coloring method is employed on this fine grid to track the development of crystal morphology. We assume the temperature on each coarse grid is the same which determines the nucleation and growth rate of spherulites through (
On the coarse grid, FVM is used to solve the energy equation. The reason why we use FVM is because this method uses the control volume which is more like the cell in the statistics for the microscopic information. Figure
Schematic of a control volume (a) 3D grid (b) 2D grid (profile of 3D grid).
On the fine grid, the pixel-coloring method [
Figure
Multiscale algorithm.
3D polymer crystallization during cooling stage is studied here. Figure
Computational geometry.
The material we considered here is iPP and the parameters used in the simulation are [
We test three meshes in this problem computation, namely, Mesh1:
Evolution of temperature and relative crystallinity at the intersection line of
Evolution of temperature and relative crystallinity during cooling at the plane of
Evolution of temperature and relative crystallinity (a) considering the latent heat (b) without considering the latent heat.
Latent heat released by the crystallization is the bridge in macro-micro-coupling. To determine the importance of the contribution of the latent heat, the results of our simulation are compared with the results of model which ignores the latent heat. Figure
Evolution of crystal morphology at the control volume of
Evolution of crystal morphology at the control volume of
Final morphology in the skin layer and in the core layer is compared in Figure
Crystal morphology at different positions (a) skin (b) core.
Mean radius of each spherulite and the distribution of spherulite size at different positions are shown in Figure
Spherulite size and distribution of spherulite size at different positions.
In order to highlight the importance of 3D simulation, we also give the comparison of the results obtained in 2D case with the 3D case. Since the problem we studied is a quasi-three dimensional problem, it can be also reduced as a two-dimensional one. Here, we consider the profile of
Figure
Evolution of temperature and relative crystallinity (2D results).
Evolution of crystal morphology at the whole computational geometry in 2D case is shown in Figure
Evolution of crystal morphology in 2D case (a)
Spherulite size and distribution of spherulite size at different positions (2D results).
The comparison in this section tells us that the macroscopic and microscopic values obtained in 2D simulation are quantitatively different from that in 3D simulation. Therefore, if the models, algorithms, and computational conditions are allowed, we should consider the 3D simulation in order to obtain the more precise results.
Effects of cooling rate and initial temperature are also investigated in this 3D polymer crystallization case. Without loss of generality, we here choose the “core” control volume as our cell for showing the results.
We change the boundary conditions by varying the cooling rate as 1 K/min, 2 K/min, 5 K/min in order to study the effects of cooling rate.
Figure
Evolution of relative crystallinity and the distribution of spherulite size in core layer for different cooling rates.
Thus, we will conclude that if the designer wants to obtain the smaller size of the spherulites, he shall impose a relatively higher cooling rate boundary condition.
We varying the initial temperature as 470 K, 480 K, 490 K in order to study the effects of initial temperature.
Figure
Evolution of relative crystallinity and the distribution of spherulite size in core layer with different initial temperatures.
Thus, if the designer wants to improve efficiency, he should cool down the melt with a relative lower initial temperature.
We have extended our previous proposed multiscale model and multiscale algorithm to the simulation of 3D polymer crystallization during cooling stage. With the multiscale model and multiscale algorithm, we obtained the temperature distributions and relative crystallinity at various locations in the mold cavity, meanwhile, we also predicted the crystal morphology development and its size as well as its distributions. We have also shown the importance of coupling between the heat transfer with crystallization as well as 3D numerical studies. Results presented in this paper shows that latent heat released by crystallization plays a very important role in macro-micro-coupling which should be considered in order to predict the more accurate crystal morphology; 2D simulation is qualitatively agreement with the 3D simulation (not only for the variables predicted in macroscopic level and microscopic level, but also for the effects of cooling rate and initial temperature), however, in the view of quantitative analysis, results obtained for these two cases have some differences. Therefore, if the computational conditions are allowed, we recommend the 3D simulation.
Future work will concentrate on 3D crystallization simulation during cooling coupled with heat transfer in reinforced system as well as flow-induced crystallization (FIC) simulation in injection molding.
The financial supports provided by the Henan Scientific and Technological Research Project (no. 122102210198), Key Scientific and Technological Research Project of Department of Education of Henan Province (no. 12B110006), Youth Scientific Foundation of Henan University of Science and Technology (no. 2012QN015), and the Doctoral Foundation of Henan University of Science and Technology (no. 09001612) are fully acknowledged.