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

We shall set S(y; x, t) = 0 if (y, 0) ∈ C(x, t), since in this case a nucleation at y will / never create a grain covering x at time t.

The Hopf-Lax formulas used to derive the causal cone oﬀer the possibility to interpret maximal nucleation times from a geometric optics point of view (cf. [5]).

Note that S(y; x, t) corresponds to the time, when a front starting at x at time t and travelling (in negative time direction) with the Hamiltonian H(x, t, p) := G(x, t)|p| arrives at y. It is well-known (cf. [42]) that (ﬁxing x and t), the arrival time σ = t − S is a (positive) viscosity solution of the eikonal equation

Example. In [41], the growth of a grain with a time-homogeneous growth rate G(x, t) ≡ G2 (x) has been considered for linear functions G2 (x) = ax + b and spatial dimension d = 2 ( a ∈ R2, b ∈ R+ ). Without restriction of generality one can assume that a = (a1, 0), and by solution of a problem in the calculus of variation, analogous to [41] the causal cone is obtained as

the region union of the random grains, which is now a random closed set (RACS).

The well known theory of Choquet-Matheron [33] shows that it is possible to assign a unique probability law associated with a random closed set Ξ ⊂ Rd on the measurable space (F, σF ) of the family of closed sets in Rd endowed with the σalgebra generated by the hit-or-miss topology, by assigning its hitting function H Ξ.

The hitting function of Ξ is deﬁned as HΞ : K ∈ K → P(Ξ ∩ K = ∅).

More precisely, we deﬁne a random closed set Ξ as a random object Ξ : (Φ, A, P) → (F, σF ).

Moreover, we denote by K the family of compact sets in Rd.

In our case, using the above analysis of the growth process, it is possible to show [17] that a unique probability measure PΘ can be associated with the germgrain process Θ = Θ(t), t ∈ R+. From now on, in a canonical sense, we shall denote by P this probability measure, and by E (respectively V), expectations (respectively variances) with respect to this probability, whenever they exist as ﬁnite values.

3.1. Stochastic Geometric Measures. In the following we discuss the quantitative description of the geometric process Θ, which can be obtained in terms of mean densities of volumes, surfaces, edges, and vertices (at the respective Hausdorﬀ dimensions), based on the analysis in [14, 43]. Let Θ(t) be a d−dimensional random closed set having boundary of Hausdorﬀ dimension d − 1, with integer d.

The mean local volume density and mean local surface density, respectively, of the random closed Θ(t) at point x are deﬁned by E[Hd (Θ(t) ∩ Br (x))] ρ(x, t) := lim (3.2) Hd (Br (x)) r→0

. for any B ∈ BRd.

3.2. The Volume Density. It seems obvious that ρ will be inﬂuenced by the nature of the nucleation and growth processes. However, the probability that the point x is not covered by time t may be expressed in terms of the probability that no nucleation event occurs inside the causal cone (see also [9] ),

For small free volume density ρ∗, the ﬁrst order terms on the right-hand side dominate and thus, ρ(x, t) ≈ ρ∗ (x, t). For increasing ρ∗, there is an obvious saturation eﬀect yielding ρ(x, t) ρ∗ (x, t). This relation can also be express as an evolution equation

**for the (volume) density:**

which widely is known as (Johnson-Mehl-) Avrami-Kolmogorov formula (cf. [18, 23, 27, 38, 46, 50] and the references therein).

4.1. Diﬀerential Equations in Spatial Dimension One. For nucleation and growth processes in spatial dimension one (d = 1) one can easily show by a comparison result for ordinary diﬀerential equations that

This shows that E(x, t; s) ⊂ B(x; R+ (x, t, s)), where B(x; R) denotes the ball around x with radius R. By similar reasoning we can deduce the inclusion E(x, t; s) ⊃ B(x; R− (x, t, s)) with

Note that the constant α0 3 T is a bound for the number of nucleations in a microscale cell in the time interval [0, T ], which is usually small or of order one.

5. Variance and Mesoscale Averaging. In the following we discuss the variance of the random variable χ(x, t) := IΞ(t) (x), taking the values 0 or 1 only, and the variance of a local averaging on a mesoscale, which yields error estimates for corresponding mesoscale quantities.

5.1. Variance of the Volume Density. The volume density ρ(x, t) is the expected value of the random variable χ(x, t). Hence, χ(x, t)2 = χ(x, t) and we obtain for the local variance

Clearly, with such a pessimistic estimate the variance is not decreased, since we have not introduced information about the possible dependence or independence of the random variables χ(x, t) at diﬀerent locations. This will be done in the following section within a local averaging procedure.

5.2. Error Estimates for Mesoscale Averaging. In the following we shall consider local averages of the volume density at the mesoscale. We assume that λ = 3N with ≥ G0 T. We perform the local averaging over a cell

and therefore we have to expect that the variance grows in time like T d.

6. Extensions. In the following we consider some possible extensions of the mesoscale averaging to further situations of interest. We shall not develop a detailed theory for these cases, but only outline the major analogies and diﬀerences to the growth considered above.

6.1. Anisotropic Growth. In anisotropic growth, which appears for many materials with an underlying crystal structure such as metals or semiconductors, the form of the normal velocity G in the growth model (1.1) has to be changed to

We shall assume that H(x, t,.) is a convex function for all (x, t). In this case, the level set formulation of the growth model is given by Ω(t) = {φ(., t) ≤ 0} for φ being the viscosity solution of the Hamilton-Jacobi equation

The causal cone can be deﬁned in the same way as in the isotropic growth situation above, and using (6.1) one can also derive an analogous Hopf-Lax representation of the causal cone as

The basic ideas and results of mesosale averaging such as the Avrami-Kolmogorov formula remain unchanged in the anisotropic setting, the only diﬀerence is the slightly more complicated computation of the causal cone.

6.2. Polycrystalline Growth. A challenging example in modern semiconductor processing is the growth of polycrystalline structures on amorphous substrates (cf. e.g. [39]). In these processes, a crystalline material is deposited on an amorphous substrate and crystals nucleate randomly. Since the material is crystalline, each nuclei has a special orientation, which is a random variable in the nucleation process.

Hence, nucleation should be modeled as a Poisson process in D × R+ × S d with a rate α = α(x, t, ν) for ν ∈ S d.

The initial orientation of the nuclei determines the subsequent anisotropic growth of the crystal, i.e., the level set formulation of the growth of the j-th grain becomes Ωj (t) = {φj (., t) ≤ 0}

which is the polycrystalline equivalent of the classical Avrami-Kolmogorov formula (4.2).

Acknowledgements. Stimulating discussions on the subject of the paper are acknowledged to Alessandra Micheletti (University Milano) and on polycrystalline growth to Hajdin Ceric (Technical University Vienna) and Peter Smereka (University Michigan).

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