SIMULATION OF POLYMER CRYSTAL GROWTH WITH VARIOUS MORPHOLOGIES USING A PHASE-FIELD MODEL

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1 SIMULATION OF POLYMER CRYSTAL GROWTH WITH VARIOUS MORPHOLOGIES USING A PHASE-FIELD MODEL M. Asle Zaeem,*, S. Nouranian, Mark F. Horstemeyer Department of Materials Science and Engineering Missouri University of Science and Technology, Rolla, MO 65409, USA Center for Advanced Vehicular Systems Mississippi State University, Starkville, MS ABSTRACT A finite element-based phase-field model was developed to simulate crystal growth in semi-crystalline polymers with various crystal morphologies. The original Kobayashi s phase-field model for solidification of pure materials was adopted to account for polymer crystallization. Evolution of a nonconserved phase-field variable was considered to track the interface between the melt and the crystalline phases. A local free energy density was used to account for the meta-stable states in polymer solidification. The developed model was successfully applied for simulation of two and three dimensional, single- and polycrystalline morphologies (hexagonal and spherulitic) in isotactic polypropylene (ipp). These morphologies were compared based on different super-cooling conditions and interface anisotropy. The unique aspect of this work is that the employed model is capable of simulating multiple arbitrarily oriented crystals and has no limitations with respect to the crystal morphology. The results show significant thermal effects on the shape and growth rate of ipp crystals.. INTRODUCTION Microstructure evolution during solidification and crystallization has significant effect on macroscale properties of materials [, ]. The type of microstructure formed during crystal growth depends on thermo-physico-chemical factors such as temperature distribution, presence of impurities, chemical composition or concentrations of phases in multi-component materials, etc. Recently, the phase-field approach has been applied to polymer crystallization, where the morphology of multifaceted single polymer crystals [3-6] and complex hierarchical polymer crystal structures, such as spherulites [7-], have been simulated after modifying the original phase-field equations of metallic alloy solidification []. In this study, a phase-field finite-element numerical simulation of crystal growth is performed for a representative semi-crystalline polymer system, i.e., isotactic polypropylene (ipp). This work builds on previous work by Kyu et al. [3], Mehta et al. [4, 7], and Xu et al. [8] and is part of a larger initiative to predict the morphology of nanocomposites with semi-crystalline polymer matrices. Furthermore, the three dimensional (3D) representation of the polymer crystal growth has been attempted for the first time. The employed model is capable of simulating multiple arbitrarily oriented crystals with no * Correspondence to: zaeem@mst.edu.

2 limitations on the crystal morphology. This is crucial for the accurate prediction of crystal structure evolution and the build-up of spherulite boundaries in polymers.. THEORETICAL DESCRIPTION The original Kobayashi s model for solidification of pure materials [] was modified to include the meta-stable states associated with imperfect polymer crystallization [3]. A non-conserved phase-field variable φ ( r, t) was considered, which is a continuous function in time ( t ) and space ( r ) dimensions, and takes a value of zero in the melt and one in the crystalline region. The total free energy of the system of melt and solidifying crystals F( φ, m( T )) includes a local free energy density f local ( φ, m( T )) representing the energies from the melt and crystalline phases, and a nonlocal free energy density representing the energies from interfacial regions. The latter is called interface gradient energy density f (φ) : grad F( φ, m) = f cryst ( φ, m) dv = { f local ( φ, m) + f grad ( φ)} dv, () V V where m is a function of temperature T. Similar to Xu et al. [5], the local free energy density of Harrowell-Oxtoby [3] was adopted to account for the meta-stable states in polymer solidification and spatio-temporal development of imperfect semi-crystalline morphologies. In this work, the local free energy density is f local φ = 4 η η ηm m η η 3 φ ( φ, m) W φ( φ η)( φ + m) dφ = W φ + φ () 0 W is a coefficient representing the energy barrier for nucleation (formation energy of the crystalline phase), which is a function of crystallization temperature T c. The phase-field variable at a stable 0 solidification phase is η = T m / T, where 0 m T m is the equilibrium melting temperature and T m is the melting temperature at a specific crystallization temperature. η is equal to one in Kobayashi s model, which results in a stable phase at φ = [6, ]. Similar to [], the relationship m( T ) = α tan ( γ ( Tm T )) was used in our model, where α and γ are constants satisfying π η 0 m ( T ) <. The term ( m ) in Equation can be related to an unstable energy barrier for crystallization [5]. The nonlocal gradient energy density in Equation is f grad ( φ) = κ ( φ). (3) where κ is the gradient energy coefficient. κ can be related to the surface energy σ through where κ = εσ,

3 σ ( θ, θ 0 ) = δ cos[ j ( θ θ 0 )] ; φ y θ = tan φ x. σ ( θ,θ 0 ) is a function accounting for the anisotropy of the surface tension, in which θ is the growth angle and θ0 is the preferential orientation with respect to the horizontal coordinate. ε is the reference surface energy and δ represents the strength of surface anisotropy. When j is equal to four, the system represents a four-fold symmetric crystal structure, similar to microstructures resulted from the solidification of cubic metallic alloys. When j is equal to six, a six-fold symmetric crystal structure is obtained similar to the microstructure resulted from the solidification of hexagonal metallic alloys, e.g., snowflakes structures. The evolution equation of non-conserved order parameter is (4) φ( r, t) δ F = M, (5) δφ( r, t) where M is the mobility coefficient. Using Equations -5 and after some manipulation, the evolution equation of φ becomes φ = M ε φ σσ + ε x y φ η σσ + ε ( σ φ) Wφ( φ η)( φ + m), (6) y x where σ = σ θ. The two-dimensional transient differential equation governing the heat transfer within the calculation domain is given by T φ = α T + K, (7) where α = k ( ρc p ) is the thermal diffusivity, k is the thermal conductivity, ρ is the density, C p is the specific heat, K = L C p, and L is the latent heat of solidification. For computational convenience, we use non-dimensional groups identified by overbars. Nondimensional groups enhance versatility in accounting for different polymer systems. Non-dimensional time and space dimensions are: t = t τ, x = x X, and y = y X. In this research, τ = 0 6 s and 6 X =.5 0 m are the scaling factors for time and space, respectively. For a specific polymer, τ and X can be related to the diffusion coefficient and radius of gyration of a polymer chain [5]. Scaled temperature is T = T T ) ( T T ). Considering dimensionless thermal diffusivity and latent heat as ( c m c α = ατ X and K K ( T m Tc ) =, respectively, Equation 7 becomes T φ = α T + K, (8) 3

4 3. RESULTS AND DISCUSSION For crystallization from a single nucleus, a square domain with a side length of 50 µ m (0 nondimensional length scale) was considered and a circular nucleus with a diameter of µ m (0.4 nondimensional length scale) was built in the center of the square domain. Initial conditions for the nucleus were T = Tc and φ =, and initial conditions for the rest of the domain were T = Tm and φ = 0. Polymer crystallization can progress at different temperatures and the crystallization patterns depend on the crystallization temperature and the super-cooling. Based on the non-dimensional scheme presented before K, so increasing the value of K decreases the super-cooling and brings the system ( T m T c ) closer to isothermal crystallization. In all simulation results, γ = 0 and ε = Figure shows the two-dimensional (D) representation of the evolution of a single spherulite from the ipp melt as a function of time at a fixed non-dimensional latent heat ( K =.5) and nondimensional thermal diffusivity ( α = 0. 8). The surface anisotropy (δ ) is taken to be zero. As can be seen, a spherulitic structure (circular pattern in D) with a well-defined circular perimeter (boundary) is formed. t =0.0 s t =0. s t =0.4 s t =0.8 s Figure - D representations of a single spherulite growth of isotactic polypropylene crystal. K =.5, α = 0. 8, and δ = 0. Decreasing the super-cooling by increasing the value of K (isothermal crystallization) reduces the crystal growth rate and results in a loosely defined spherulite boundary (Figure ). 4

5 K =.5 K =.5 K =.75 K = Figure - D representations of the growth of a single ipp spherulite versus super-cooling K. α = 0. 8, t =0.8 s, and δ = 0. ( T m T c ) If the strength of the surface anisotropy (δ ) is increased, a hexagonal crystal structure with a sixfold symmetry results. This morphology is shown in Figures 3 and 4 for different super-cooling conditions and times. As can be seen in Figure 3, the well-defined hexagonal boundary disappears with increasing K (approaching the isothermal crystallization). K =.75 K = K =.5 Figure 3 - D representations of the growth of a single ipp crystal with a six-folds symmetry versus super-cooling K. α = 0. 8, t =0.8 s, and δ = ( T m T c ) 5

6 K =.5; t =0.55 s K =.5; t =0.7 s K =.75; t =0.8 s Figure 4 - D representations of a single crystal growth crystal with six-folds symmetry versus supercooling K. α = 0. 8, t =0.8 s, and δ = ( T m T c ) A multi-nucleus crystallization simulation with appropriate parameters results in a complex spherulitic morphology (Figure 5) that resembles the true morphology of the isotactic polypropylene (Figure 6). The spherulites grow until they form well-defined boundaries. In this case, the nucleation density determines the size, distribution, and other attributes of the spherulites. t =0.0s t =0.06s t =0.s t =0.s Figure 5 - D representations of multiple spherulitic growth of isotactic polypropylene. K =.5, α = 0.85, and δ = 0. 6

7 Figure 6 Surface of isotactic polypropylene viewed under light microscope, where the spherulitic morphology can be seen. A 3D representation of the growth of a single spherulite is given in Figure 7. In this figure, the evolution of both phase-field variable and iso-surface representing the outer boundary of the spherulite is shown. t =0.0 s t =0.05 s t =0.0 s t =0.5 s Figure 7 - Evolution of a 3D spherulite; top row represents the phase-field variable and the bottom row shows the iso-surface. K =.5, α = 0. 85, and δ = 0. 7

8 4. SUMMARY A finite element-based phase-field model was developed based on Kobayashi s model for pure materials to simulate crystal growth in a representative semi-crystalline polymer, i.e., isotactic polypropylene (ipp). The developed model is capable of predicting various crystal morphologies in D and 3D representations. Furthermore, the model accounts for multiple arbitrarily oriented crystals. Predictions of the model agree well with the experimentally validated crystal morphologies for ipp. The results show significant thermal effects on the shape and growth rate of ipp crystals. This work is a preliminary phase in the development of a phase-field model for advanced nanocomposite material systems with semi-crystalline polymer matrices. REFERENCES [] Askeland, D. (003) The Science of Materials and Engineering. Belmont, CA: Thompson, Brooks/Cole. [] Herlach, D.M. (Editor) (004) Solidification and Crystallization. Weinheim, Germany: Wiley- VCH. [3] Kyu, T., Mehta, R., and Chiu, H.-W. (000) Spatiotemporal Growth of Faceted and Curved Single Crystals. Physical Review E 6(4): [4] Mehta, R., Keawwattana, W., and Kyu, T. (004) Growth Dynamics of Isotactic Polypropylene Single Crystals During Isothermal Crystallization from a Miscible Polymeric Solvent. Journal of Chemical Physics 0(8): [5] Xu, H., Matkar, R., and Kyu, T. (005) Phase-Field Modeling on Morphological Landscape of Isotactic Polystyrene Single Crystals. Physical Review E 7:0804. [6] Wang, D., Shi, T., Chen, J., An, L., and Jia, Y. (008) Simulated Morphological Landscape of Polymer Single Crystals by Phase Field Model. Journal of Chemical Physics 9: [7] Mehta, R. and Kyu, T. (004) Dynamics of Spherulitic Growth in Blends of Polypropylene Isomers. Journal of Polymer Science: Polymer Physics 4(5): [8] Xu, H., Keawwattana, W., and Kyu, T. (005) Effect of Thermal Transport on Spatiotemporal Emergence of Lamellar Branching Morphology During Polymer Spherulitic Growth. Journal of Chemical Physics 3:4908. [9] Gránásy, L., Pusztai, T., Tegze, G., Warren, J.A., and Douglas, J.F. (005) Growth and Form of Spherulites. Physical Review E 7:0605. [0] Gránásy, L., Pusztai, T., Börzsönyi, T., Tóth, G.I., Tegze, G., Warren, J.A., and Douglas, J.F. (006) Polycrystalline Patterns in Far-From-Equilibrium Freezing: A Phase Field Study. Philosophical Magazine 86(4): [] Asanishi, M., Takaki, T., and Tomita, Y. (007) Polymer Spherulite Growth Simulation During Crystallization by Phase-Field Method. Proceedings of AES ATEMA 007 International Conference, Montreal, Canada, August 06-0: [] Kobayashi, R. (993) Modeling and Numerical Simulations of Dendritic Crystal Growth. Physica D 63: [3] Harrowell, P.R. and Oxtoby, D.W. (987) On the Interaction between Order and a Moving Interface: Dynamical Disordering and Anisotropic Growth Rates. Journal of Chemical Physics 86(5):