Crystal Growth and Impingement in Polymer Melts

Crystal Growth and Impingement in Polymer Melts Martin Burger

1. Introduction Crystallization of polymeric materials is a solidification process in strong interaction with heat conduction. Both of the basic mechanisms involved in the solidification from a melt, namely the nucleation and growth of crystals, are strongly influenced by the temperature and its variation. Vice versa, the latent heat is rather large for polymeric materials, so that it causes a considerable change in the heat transfer process. The interaction of solidification and heat conduction is a well-known phenomenon in models of phase-change, and involves important problems in the theory of free and moving boundary problems. In typical examples of moving boundary problems for the heat equation such as the well-studied Stefan problem (cf. [18]), the unknown boundary is assumed to be an isotherm, which leads in a mathematical model to a homogeneous Dirichlet condition for the (appropriately scaled) temperature. For polymers, the situation is different, since the phase change does not take place at a fixed temperature (or with kinetic undercooling close to this temperature), but in a rather large temperature range between the thermal melting point Tm and the glass transition temperature Tg . The nucleation of new crystals can be modeled as a heterogeneous Poisson process in space and time, with rate α = α(x, t). For many important materials such as isotactic Polypropylene (i-pp), a model for the nucleation rate in depen∂ ˜ dence of the temperature u is given by α(x, t) = ∂t N (u(x, t)) (cf. [6]), with some ˜ material function N , which varies only in the interval (Tg , Tm ), i.e., no crystals nucleate if the temperature is below the glass transition temperature or above the melting point. Usually it is assumed that new crystals nucleate as small spherical objects with initial radius R0 determined from classical nucleation theory. The nucleation process has been well investigated with respect to its stochastic properties 1991 Mathematics Subject Classification. Primary: 35R35, Secondary: 80A22. Key words and phrases. Crystal Growth, Moving Boundaries, Level Set Method. Partial support is acknowledged to the Austrian National Science Foundation FWF under grant SFB F 013/08. The author thanks Prof. Vincenzo Capasso, Dr. Alessandra Micheletti and Dr. Claudia Salani (University Milano) for useful and stimulating discussions.

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Figure 1. Impingement of polymer crystals (right) and a final tesselation from an experiment with isotactic Polypropylene. (cf. [9, 20]), its averaging (cf. [5, 8]), its numerical simulation (cf. [19]), as well as the determination of the nucleation rate from indirect measurements (cf. [2, 7]). SN The growth of the crystalline phase Θ = i=1 Θi (Θi (t) being the single crystals), is determined via a nonlinear Gibbs-Thompson relation for the velocity VN and the temperature u, (1.1)

VN = G(u) =

1 ∂u [ ] L ∂n

on Γ := ∂Θ − ∂Ω,

∂. where G is a given material function, L is the latent heat, and [ ∂n ] denotes the jump of the normal derivative on the boundary. For a certain temperature interval (e.g. for u ≥ 80◦ C in the case of isotactic Polypropylene) the function G is nonincreasing, so that the relation can be rewritten as u = G−1 (VN ), which is a nonlinear version of the typical condition used for kinetic undercooling. Surprisingly, the funtion G usually reaches a maximum at some temperature (dependent on the material) and is non-decreasing below it, which is confirmed by some experiments (cf. [21, 25]). The experimental background for this growth model in a nonisothermal situation was provided by Schulze and Naujeck [26], who considered the growth of a crystal in an environment with strong temperature gradients and found that the obtained shapes coincided with the ones predicted by a minimal-time principle. In [8], the growth model was reformulated in arbitrary spatial dimension as (1.2) (cf. [28] for an overview of different formulations of crystal growth models ). Motivated by the physics of the problem we shall assume that G is a smooth function with compact support, which is nonnegative and globally bounded. The heat transfer in the material is described by the heat equation in Ω−∂Θ, in general with parameters (such as conductivity, density, and capacity) dependent on the phase. For simplicity we restrict our attention to the case of all parameters being constant, which yields, together with the second identity in (1.2), the

Crystal Growth and Impingement in Polymer Melts

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nonlinear parabolic equation ∂u (1.2) c − k∆u = LδΓ G(u) in Ω × (0, T ), ∂t where δΓ denotes the Dirac delta distribution on Γ. This equation is supplemented by appropriate initial conditions at time t = 0, i.e., (1.3)

u = u0

in Ω × {0},

and by boundary conditions on ∂Ω × (0, T ), for simplicity we restrict our attention to Dirichlet conditions of the form (1.4)

u = uD

on ∂Ω × (0, T ).

We finally note that (1.3) has an equivalent enthalpy formulation of the form ∂e − k∆u = 0, e = cu − Lχ, ∂t with χ being the indicator function of Ω. In the following we shall discuss the mathematical analysis and numerical simulation of the initial-boundary value problem (1.3), (1.4), (1.5), coupled to the growth law (1.2). We assume that the nucleation events {(Xj , Tj )}j=1,...,N , representing the origin and time of nucleations, as well as the initial shape of a nucleated crystal (a small spherulite), are given a-priori, analysis and simulation of the stochatic aspects of nucleation shall be discussed elsewhere. (1.5)

2. Mathematical Analysis In the following we shall be concerned with the mathematical analysis of (1.2) coupled to the heat transfer problem (1.3), (1.4), (1.5). The analysis of this problem is challenging in the single crystal case already, where only local-in-time existence and uniqueness could be proven so far by Friedman and Velazquez [11] using fixedpoint arguments and regularity (of the temperature and the free boundary) in a H¨older class C 2+α , α > 0. In presence of multiple crystals, the mathematical analysis of the crystal growth via (1.2) leads to additional difficulties: 1. Because of the non-equilibrium condition at the interface, there is no weak formulation of the problem (such as for the two-phase Stefan problem, cf. [18]) that would circumvent the problem of dealing with the interface explicitely. 2. Even before impingement the growth of a fixed crystal is influenced by all other crystals via changes in the temperature, i.e., we have to deal with a nonlocal front growth problem. 3. One cannot expect much regularity of the free boundary Γ even if all parameters and initial shapes are analytic. This can be seen from a simple case with G ≡ 1 and two spherulitic crystals. Before impingement, the shape Γ(t) is analytic, but the impingement creates a non-Lipschitz

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Martin Burger singularity. For each time after the impingement event, the shape Γ(t) is only locally Lipschitz. Thus, the best type of regularity we can expect is ”locally Lipschitz almost everywere”.

A detailed analysis can be carried out in the spatially one-dimensional case, where regularity of the free boundary plays no role since all crystals are compact intervals and Γ(t) consists of a finite number of points. Global existence and uniqueness for this case has been obtained recently (cf. [3]); here we only state the main result: Theorem 1. Let Ω ⊂ R1 be a bounded interval, let T ∈ R+ be arbitrary and let (Xj , Tj ) ∈ Ω × [0, T ], j = 1, . . . , M be given nucleation events. Then the coupled problem (1.2), (1.3), (1.4), (1.5) has a unique weak solution u ∈ L∞ (0, T ; C 0,1 (Ω))

and

Γ ∈ L∞ ([0, T ]; KM ),

where KM is the set of all subsets of Ω with cardinality less or equal to 2M equipped with the Hausdorff metric. Moreover, for each time interval [S1 , S2 ] that does not contain a nucleation or impingement event, we have u ∈ C(S1 , S2 ; C 0,1 (Ω))

and

Γ ∈ C 1 ([0, T ]; Km ).

In multi-dimensional situations, a promising approach is the representation of the growing crystalline phase via the level set approach, i.e., (2.1)

Θ(t) = { x ∈ Ω | φ(x, t) ≤ 0 },

where φ is a continuous functions. A weak formulation of the evolution of Ω(t) and Γ(t) = ∂Ω(t) via (1.2) is provided by computing a viscosity solution of the firstorder Hamilton-Jacobi equation (cf. e.g. [10, 17] for details on viscosity solutions) (2.2)

∂φ + V |∇φ| = 0, ∂t

in Rd × (0, T ),

where V is extension of the velocity VN to the whole space Rd and the initial value φ(., 0) is a uniformly continuous function satisfying (2.1) for t = 0. This approach, originally introduced for numerical purposes by Osher and Sethian [23], does not rely on any parametrization and allows to treat topological changes in an automatic way. We refer to the monographs by Osher and Fedkiw [22] and by Sethian [27] for an overview of the level set method and its applications. Nucleation events can be included by restarting the level set evolution at each nucleation time with a new level set function including the nucleated object, and therefore we shall restrict our attention to a time interval (0, T ) without nucleation in the following, assuming that some initial shape Θ(0) is given. If we assume that a temperature u ∈ C(0, T ; C 0,1 (Ω)) is given and if the domain Ω has a sufficiently smooth boundary, we can find some extension u ˜ on RN × (0, T ) such that the (natural) extension velocity defined by V = G(˜ u) is bounded, nonnegative, and Lipschitz continuous. For such an evolution we can collect the following results:

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• Well-posedness: For each uniformly continuous initial value, the level set equation (2.2) has a uniformly continuous unique viscosity solution (cf. [10]). • Stable dependence: The solutions of the level set equation depend in a Lipschitz-continuous way (in the norm of C(0, T ; C 0,1 (Ω))) on the velocity V (cf. [12]). • Non-fattening: If we start with a level set function such that Γ(0) = { x ∈ Ω | φ(x, t) = 0 } has zero Lebesgue measure, then, for all t > 0, the set Γ(t) coincides with the zero level set of φ(., t), which has zero Lebesgue measure (cf. [1]). A recent result by Ley [14] shows that even an estimate of the form |∇φ| ≥ η > 0 holds in the viscosity sense in a neighborhood of Γ(t). • Partial regularity: If we assume in addition that V ≥ ρ in Ω × (0, T ) for some ρ ∈ R+ (which is the case in the temperature range of interest), then for almost every t ∈ (0, T ), the set Γ(t) is locally Lipschitz almost everywhere with respect to the Hd−1 -measure (cf. [4]). Hence, by the level set approach we obtain a well-posed evolution, which yields exactly the maximal regularity we expected. An interesting open problem for future research is the coupling of the level set evolution with the heat transfer model (1.3)-(1.5). A promising approach to global existence seems to be the use of a relaxation model as for the Stefan problem (cf. e.g. [29]) and (or) some smoothing of the form χ² = f ² (φ), with f ² approximating  if s < 0  1 [0, 1] if s = 0 . f (s) =  0 if s > 0 For the smoothed problem, the map φ 7→ u is continuous and compact, which allows, together with the above stable dependence results for the solutions of the level set equation, to use a fixed point argument to deduce the existence of a solution. Moreover, analogous a-priori estimates as in [29] can be obtained for the temperature u in the norm of L1 (0, T ; W 2,1 (Ω)), which might allow to deduce existence by similar weak-star convergence techniques as in [29].

3. Numerical Simulation The numerical simulation of crystal growth consists of two main tasks, namely the numerical solution of the level set equation (2.2) and of the heat transfer problem (1.3)-(1.5). In addition, we have to implement the coupling of these problems via the velocity V = G(u) and the sourve term in (1.3). For solving first-order Hamilton-Jacobi equations like (2.2), several schemes have been proposed over the last two decades, most of them using finite differences. A frequently used class of solvers are essentially nonoscillatory schemes (cf. [24]), originally developed for conservation laws. These type of schemes allows high order approximation of solution, can be implemented efficiently, and is explicit in time

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(like the majority of schemes for conservation laws and Hamilton-Jacobi equations). As any other explicit scheme, there is a stability bound on the time step τ in dependence of the spatial grid size h, namely the so-called CFL-condition (3.1)

τ sup |V (x, t)| < h. x

Since the velocity V = G(u) is usually rather small for polymeric materials, this stability bound is not restrictive and the actually chosen time step is usually far below the one allowed by (3.1). Another consequence of an explicit time scheme for the level set equation is the possibility to use its result as an input for the time step in the heat equation, which should be implicit, since the stability bound for a parabolic equation (τ ∼ h2 ) is quite restrictive. Thus, having computed φ at time t + τ , we can apply the (horizontal) method of lines to the heat equation (1.3) or (1.6) and use the the values of φ to assemble the right-hand side, leading to an elliptic boundary value problem in each time step. For the spatial discretization of the arising elliptic problem, we have several choices. A first possibility is to choose a finite element grid matching the interface, which leads to high accuracy but also to an enormous numerical effort, since the grid has to be changed in each time step. More efficient are methods with non-matching grids such as the immersed interface method (cf. [15, 16]),which is a finite difference method on a rectangular grid with additional correction terms in those cells including parts of the interface, or by a method approximating the interface terms by smoothed quantities, which is frequently used in the context of level set methods (cf. [22]). For the latter, the simplest possibility consists in using the enthalpy formulation (1.6) and approximating χ by χ² = f ² (φ), with f² as in the previous section, which yields an equation of the form ∂u ∂φ (3.2) c − k∆u = −Lδ ² (φ) , ∂t ∂t ² with δ approximating the Dirac delta distribution. After time discretization, we can employ a finite element method on a mesh fixed during the evolution or at least during several time steps. The mesh generation should be fine around the initial interface and consequently also around the interface after some time steps, provided the motion is not too fast. Only if the mesh around the interface becomes to coarse after some time steps, the domain has to be remeshed. In this way, the numerical effort can be reduced significantly. For computations with a large number of crystals (see [19] for such simulations), it seems of advantage to use a fine uniform grid (fixed during the evolution), and to apply fast Poisson solvers for the solution of the elliptic problem. Finally, we show a numerical experiment with two crystals in a domain scaled to the cell [−1, 1]2 . We use material parameters in a range typical for isotactic Polypropylene (cf. [5] for numerical values), and use a temperature scaling such that L = 1 (around 70◦ unscaled). The (unscaled) initial value is a uniform temperature u0 = 124◦ C. The heat transfer parameters, which are typically small for

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polymers are chosen as c = k = 10−2 . All solvers were implemented in the software system MATLAB, for the heat equation we use a finite element discretization with n = 19968 triangles. The level set equation is solved on a uniform grid with 256 × 256 elements using a weighted ENO-scheme of fifth order in space and third order in time, with a Godunov flux, as proposed in [13]. In the first example, shown in Figure 2, we use a uniform Dirichlet value of u = 88◦ C at the boundary. In this case, one can observe a strong reheating effect due to release of latent at the crystal boundary and the free boundary is almost an isotherm of the temperature. Moreover, the absence of strong temperature gradients in the crystals causes the shape to stay close to a spherical shape. Consequently, the crystals heat each other at a single point and the contact interface propagates outward from this point. The behavior in the second example (shown in Figure 3) is different due to different boundary values of the temperature, which are very low on the left and upper part of the rectangle (around 70◦ C) and rather high at the right and lower part (around 140◦ C). This yields a temperature gradient mainly in diagonal direction, with some changes due to reheating. In particular, the temperature around the lower crystal is so high (and the value of G(u) so low) that there is almost no growth, while the second crystal grows to a large, non-spherical shape. The impingement starts from two different contact points in this case and leaves a small liquid part inside the crystalline phase for some time.

References [1] G.Barles, H.M.Soner, P.E.Souganidis, Front propagation and phase-field theory, SIAM J. Cont. Optim. 31 (1993), 439-469. [2] M.Burger, Iterative regularization of an identification problem arising in polymer crystallization, SIAM J. Numer. Anal. 39 (2001), 1029-1055. [3] M.Burger, Growth of multiple crystals in polymer melts (2001), submitted. [4] M.Burger, Growth fronts of first-order Hamilton-Jacobi equations, SFB-Report 02-8 (University Linz, 2002), and submitted. [5] M. Burger, V. Capasso, Mathematical modelling and simulation of non-isothermal crystallization of polymers, Math. Mod. Meth. Appl. Sci. 11 (2001), 1029-1054. [6] M.Burger, V.Capasso, G.Eder, Modelling crystallization of polymers in temperature fields, ZAMM 82 (2002), 51-63. [7] M.Burger, V.Capasso, H.W.Engl, Inverse problems related to crystallization of polymers, Inverse Problems 15 (1999), 155-173. [8] M.Burger, V.Capasso, C.Salani, Modelling multi-dimensional crystallization of polymers in interaction with heat transfer, Nonlinear Anal. B, Real World Appl. 3 (2002), 139-160. [9] V.Capasso, C.Salani, Stochastic birth-and-growth processes modelling crystallization of polymers in a spatially heterogenous temperature field, Nonlinear Anal. B, Real World Appl. 1 (2000), 485-498.

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Figure 2. Color plot of the temperature and shape of the growing crystals in the first example: at the initial stage (t = 1.09, t = 4.72), short time before (t = 12.9), at (t = 15.1) and after impingement (t = 15.8,t = 19.1).

Crystal Growth and Impingement in Polymer Melts

Figure 3. Color plot of the temperature and shape of the growing crystals in the second example: at the initial stage (t = 19.7, t = 34.7), short time before (t = 88.7), at (t = 93.8) and after impingement (t = 97.3, t = 100).

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[10] M.G.Crandall, P.L.Lions, On existence and uniqueness of solutions of HamiltonJacobi equations, Nonlinear Anal. 10 (1986), 353-370. [11] A.Friedman, J.L.Velazquez, A free boundary problem associated with crystallization of polymers in a temperature field, Indiana Univ. Math. Journal 50 (2001), 1609-1650. [12] E.R.Jakobsen, K.H.Karlsen, Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations, CAM Preprint 01-03 (Department of Mathematics, UCLA, 2001). [13] G.S.Jiang, D.Peng, Weighted ENO-schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput. 21 (2000), 2126-2143. [14] O.Ley, Lower bound gradient estimates for first-order Hamilton-Jacobi equations and applications to the regularity of propagating fronts, Adv. Differ. Equ. 6 (2001), 547576. [15] Z.Li, A fast iterative algorithm for elliptic interface problems, SIAM J. Numer. Anal. 35 (1998), 230–254. [16] Z.Li, Y.Shen, A numerical method for solving heat equations involving interfaces, Electron. J. Diff. Eqns., C 3 (1999), 100-108. [17] P.L.Lions, Generalized Solutions of Hamilton-Jacobi Equations (Pitman, Boston, London, Melbourne, 1982). [18] A.M.Meirmanov, The Stefan Problem (De Gruyter, Berlin, 1992). [19] A.Micheletti, M.Burger, Stochastic and deterministic simulation of nonisothermal crystallization of polymers, J.Math.Chem. 30 (2001), 169-193. [20] A.Micheletti, V.Capasso, The stochastic geometry of polymer crystallization processes, Stoch. Anal. Appl. 15 (1997), 355-373. [21] B.Monasse, J.M.Haudin, Thermal dependence of nucleation and growth rate in polypropylene by non-isothermal calorimetry, Colloid Polym. Sci. 264 (1986), 117-122. [22] S.J.Osher, R.P.Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, Berlin, Heidelberg, New York, 2002). [23] S.J.Osher, J.A.Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations, J. Comp. Phys. 79 (1988), 12–49. [24] S.J.Osher, C.W.Shu, High-order essentially nonoscillatory schemes for HamiltonJacobi equations, SIAM J. Numer. Anal. 28 (1991), 907-922. [25] E.Ratajski, H.Janeschitz-Kriegl, How to determine high growth speeds in polymer crystallization, Colloid Polym. Sci. 274 (1996), 938 - 951. [26] G.E.W.Schulze, T.R.Naujeck, A growing 2D spherulite and calculus of variations, Colloid Polym. Science 269 (1991), 689-703. [27] J.A.Sethian, Level Set Methods and Fast Marching Methods (Cambridge University Press, 2nd ed., Cambridge, 1999). [28] J.E.Taylor, J.W.Cahn, C.A.Handwerker, Geometric models of crystal growth, Acta metall. mater. 40 (1992), 1443-1472. [29] A.Visintin, Models of phase-relaxation, Diff. Int. Equations 14 (2001), 1469-1486

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