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«1. Introduction. Nucleation and growth processes arise in a variety of natural and technological applications (cf. [12] and the references therein), ...»

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∗, †, AND ‡


Abstract. The aim of this paper is to derive a general theory for the averaging of heterogeneous processes with stochastic nucleation and deterministic growth. We start by generalizing the classical Johnson-Mehl-Avrami-Kolmogorov theory based on the causal cone to hetereogeneous growth situations. Moreover, we relate the computation of the causal cone to a Hopf-Lax formula for Hamilton-Jacobi equations describing the growth of grains. As an outcome of the approach we obtain formulae for the expected values of geometric densities describing the growth processes, in particular we generalize the standard Avrami-Kolmogorov relations for the degree of crystallinity.

By relating the computation of expected values to mesoscale averaging, we obtain a suitable description of the process at the mesoscale. We show how the variance of these mesoscale averages can be estimated in terms of quotients of the typical length on the micro- and on the mesoscale.

Moreover, we discuss the efficient computation of the mesoscale averages in the typical case when the nucleation and growth rates are obtained from mesoscopic fields (such as e.g. temperature).

Finally, we give a short outlook to possible extension such as polycrystalline growth, which turns out to be rather straight-forward when starting from our general framework.

Keywords: Nucleation, Growth, Multiscale Models, Averaging, Hamilton-Jacobi equations.

Subject Classification (MSC 2000): 49L25, 60D05, 82C26, 92C15

1. Introduction. Nucleation and growth processes arise in a variety of natural and technological applications (cf. [12] and the references therein), such as e.g.

solidification and phase-transition of materials (cf. e.g. [49]), semiconductor crystal growth (cf. [37]), biomineralization (cf. e.g. [48]), DNA replication (cf. e.g. [29]).

The mathematical modelling of such processes can, roughly speaking, be divided into

two parts:

• Models focussing on the geometric growth of objects, such as a part of theory of free boundary problems (cf. e.g. [1, 19, 45] and [44] as a collection of references), often completely disregarding nucleation phenomena or even the presence of multiple objects (e.g. crystals). Usually, such models are moving boundary problems with a law for the growth of a phase boundary in normal direction.

• Models focussing on the kinetics of nucleation, often completely disregarding the geometric aspect of the growth processes. Usually such models are meaneld or rate equations, often without spatial dependence (cf. e.g. [2, 4, 10, 26, 28, 32]).

The aim of this paper is to bridge between these two type of models, the microscopic front growth and the macroscopic average of many nuclei, by introducing mesoscale models that locally average the microscopic models in presence of a large number of grains. The special way of averaging allows to describe systems with a very high number of grains (for which it is impossible to simulate the growth of every single grain), but still provides information about local averages for geometric quantities such as contact interface densities. The starting point of averaging procedures are the global spatial averages derived by Kolmogorov [32], Avrami [2], and Johnson ∗ Institut f¨ r Industriemathematik, Johannes Kepler Universit¨t, Altenbergerstr. 69, A 4040 Linz, u a Austria. e-mail: martin.burger@jku.at. Supported by the Austrian Science Foundation under project SFB F 13/08.

† Dipartimento di Matematica, Universit` di Milano, Via C. Saldini 50, I-20133 Milano, Italy.

a e-mail: vincenzo.capasso@unimi.it ‡ Istituto Nazionale di Fisica Nucleare, and Dipartimento di Matematica, Universit` di Milano, a Via C. Saldini 50, I-20133 Milano, Italy. e-mail: livio.pizzocchero@mat.unimi.it and Mehl [31] for constant growth rates and simple nucleation laws. These global averages have later been extended to time-dependence of the nucleation and growth process and certain other effects (cf. [11, 22, 23, 35]). First steps towards heterogeneous nucleation and growth processes have been made in [40], who derived a system of local rate equations by formal arguments, whose solution is actually close to the local averages we shall derive in this paper. The computation of local averages has been applied for the first time with partly formal arguments by the authors in the context of polymer crystallization (cf. [7, 8, 9, 34]). In this paper we shall derive such a local averaging approach in a mathematically rigorous way for an important and rather general class of nucleation and growth models and derive new error estimates for the averaged quantities.

The setup of this paper is the following: we shall consider a nucleation process in time and space, which is a stochastic Poisson process with rate α = α(x, t). This nucleation process generates a sequence of random variables Xk ∈ Rd and Tk ∈ R+ describing the spatial location and time of the k-th nucleation event. The k-th nucleus shall be represented by the set Θk (t). Moreover, we assume that the growth of a nuclei occurs with a nonnegative normal velocity G(x, t), i.e., the velocity of boundary points is determined by

–  –  –

where n is the unit outer normal. We shall consider the growth from a spherical nucleus from an infinitesimal radius R → 0.

Without further notice we shall assume that α and G are bounded and continuous functions on Rd × [0, T ] with

–  –  –

Moreover, we assume that G Lipschitz-continuous with respect to the spatial variable x.

As mentioned above, the case of particular interest is a three-scale situation in the growth process with respect to space, i.e., there exist

• A macroscale corresponding to a length L, in which the whole process takes place.

• A microscale corresponding to the length := G0 T related to typical grain sizes obtained in the interval [0, T ]

• A mesoscale corresponding to a length λ such that λ L, which marks the finest resolution for the description that is of practical importance.

The paper is organized as follows: In Section 2 we shall introduce the causal cone, which describes the set of nucleation events leading to coverage of a point by the grains, and relate it to Hopf-Lax formulae for the solution of Hamilton-Jacobi equations modeling the growth. In Section 3 we introduce the stochastic model of the nucleation process and compute expected values of some random geometric measures such as the phase function and nucleation numbers. The efficient computation of approximations to these expected values in typical multiscale situations is discussed in Section 4. Section 5 is devoted to the estimation of variances of these random variables and perform a mesoscale averaging, which allows to bound the variances in terms of the relative scale λ. Finally, we discuss some extensions such as crystalline growth in Section 6.

l λ λ L Fig. 1.1. Sketch of macro- (L), meso- (λ), and microscale ( ) in a nucleation and growth process.

2. Causal Cone and Hopf-Lax Formula. In this section we shall define the growth model in detail and verify its well-posedness. Moreover, we establish the connection between the total phase and the ”freely grown grains”, i.e., the objects obtained for a single nucleation. Finally, we introduce the causal cone, which we relate to Hopf-Lax-type formula backward in time.

2.1. The Growth Model. We start with a weak formulation of the growth model provided by the level set method. For this sake we assume that the nucleation times Tk and the nucleation locations Xk are given and fixed (i.e., we investigate single realizations of the stochastic nucleation process), such that 0 = T1 ≤ T2 ≤...

Then we consider the following growth model: The function φ is determined as the unique continuous viscosity solution of

–  –  –

In order to prove the existence of the limit, we employ a Hopf-Lax formula, which

yields a quasi-explicit formula for the solution of (2.1):

Proposition 2.1. Let φR be the unique viscosity solution of (2.1) in a time interval (Tk−1, Tk ). Then

–  –  –

for all t ∈ (Tk−1, Tk ).

Proof. The result follows from a standard Hopf-Lax formula (cf. [24]).

A consequence of the Hopf-Lax formula for the viscosity solution is a representation formula for the phase ΘR :

Lemma 2.2.

For t ∈ [Tm, Tm+1 ), the phase ΘR (t) is determined by

–  –  –

Vice versa, if x is an element of the set on the right-hand side of this inclusion, then ξ(Tk ) ∈ BR (Xk ) for any R and therefore x ∈ ∩R0 ΘR (t), which implies equality of the sets.

2.2. The Freely Grown grains. Given the nucleation events (Xk, Tk ), we can also define the evolution of a freely grown single grain Ωk (t), which nucleates in Xk at time Tk and then grows with normal velocity G on ∂Ωk (t). Using the same technique as in the previous section, we may conclude that Ωk (t) = { x ∈ Rd | ∃ξ ∈ W 1,∞ ([Tk, t]) : ξ(Tk ) = Xk, ξ(t) = x, ˙ |ξ(s)| ≤ G(ξ(s), s), s ∈ (Tk, t) } (2.6) for t ≥ Tk and Ωk (t) = ∅ for t Tk.

It seems obvious that the total phase Θ(t) is the union of the freely grown grains, but this statement is not true for general growth laws like mean curvature motion.

In our case, this statement holds and is a simple consequence of the representation formulae for Θ(t) and Ωk (t).

Corollary 2.4. The equality Θ(t) = k Ωk (t) holds for all t ∈ R+.

Subject to the Hopf-Lax formula, Corollary 2.4 is a rather simple result, but it clarifies a discussion going on the literature on modelling crystallization processes for a long time (cf. [23, 47]), whether it makes sense to use freely grown grains (which do not correspond to physical objects in general) or not. Our result just states that from a rigorous mathematical viewpoint, it is equivalent to take the union of the freely grown or the union of the real grains to obtain the phase. Of course, we cannot expect this result to be true for more general growth laws, like curvature-dependent velocity.

We will discuss such cases and their difficulties in Section 6.

For the freely grown grains, we can somehow revert time, i.e., derive a condition whether x ∈ Ωk (t) for fixed x ∈ Rd and t ∈ R+.

Proposition 2.5. For (x, t) ∈ Rd × R+, the following two statements are equivalent:

(i) x ∈ Ωk (t).

(ii) There exists η ∈ W 1,∞ ([0, t − Tk ]) such that η(t − Tk ) = Xk, η(0) = x, |η(s)| ≤ G(ξ(s), t − s), s ∈ (0, t − Tk ).

˙ Proof. The assertion follows immediately from the above Hopf-Lax-formula and a transformation of the time variable from s to t − s, and the use of the new variable η(s) := ξ(t − s).

2.3. The Causal Cone. So far, we have taken a Lagrangian approach and looked at the evolution of the grain away from the location they nucleated. Alternatively, we can use an Eulerian approach, i.e., fix a time t and a location a spatial x, and investigate under which condition the point x will be covered by the phase Θ(t) at time t. This investigation is simplified significantly by the results of the previous section, which allow to look at the freely grown grains.

Looking at the possible nucleation events for which x ∈ Θ(t) leads in a natural way to the causal cone defined by

–  –  –

Due to the change of the time direction, we may consider the causal cone as the space-time region covered by a grain growing backward in time with the given growth rate G.

The representation of the causal cone via the Hopf-Lax formula allows some immediate conclusions on its geometric structure:

Proposition 2.6. The causal cone C(x, t) can be decomposed in the form

–  –  –

2.4. Arrival Times and Nucleation Events. Due to the above reasoning we can define a ”maximal” nucleation time S at y leading to coverage of x at time t, i.e.,

–  –  –

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