CONTINUUM LIMITS OF PARTICLES INTERACTING VIA DIFFUSION

We consider a two-phase system mainly in three dimensions and we examine the coarsening of the spatial distribution, driven by the reduction of interface energy and limited by diffusion as described by the quasistatic Stefan free boundary problem. Under the appropriate scaling we pass rigorously to the limit by taking into account the motion of the centers and the deformation of the spherical shape. We distinguish between two different cases and we derive the classical mean-field model and another continuum limit corresponding to critical density which can be related to a continuity equation obtained recently by Niethammer and Otto. So, the theory of Lifshitz, Slyozov, and Wagner is improved by taking into account the geometry of the spatial distribution.


Introduction
Ostwald ripening is a very general phenomenon occurring in liquids, solids, and on solid surfaces.Basically, it takes place in the late stages of first-order phase transitions, when larger droplets, grains, crystallites, or islands grow at the expense of smaller ones via evaporation, diffusion, and condensation.In general, the kinetics of a first-order phase transition are characterized by a first stage where small droplets of a new phase are created out of the old phase, for example, solid formation in an undercooled liquid.The first stage, called nucleation, yields a large number of small particles.During the next stage, the nuclei grow rapidly at the expense of the old phase.When the phase regions are formed, the mass of the new phase is close to equilibrium and the amount of undercooling is small, but large surface area is present.In the next stage, the configuration of phase regions is coarsened, and the geometric shape of the phase regions becomes simpler and simpler, eventually tending to regions of minimum surface area with given volume.The driving force of this process comes from the need to decrease interfacial energy with increasing the average size.There have been considerable efforts in finding a theory which describes Ostwald ripening, and the Mullins-Sekerka model is a prominent candidate.In the present work we focus on this model.The Mullins-Sekerka model is a nonlocal evolution law in which the normal velocity of a propagating interface depends on the jump across the interface of the normal derivative of a function which is harmonic on either side and which equals the mean curvature on the propagating interface.The model was first introduced by Mullins and Sekerka [6] to study solidification and liquidation of materials and it has the following form where Ω is a bounded, smooth, simply connected domain; u is the chemical potential, H is the mean curvature of Γ(t); V is the normal velocity taken positive for a shrinking sphere; [[∂u/∂n]] = (∂u e /∂n e ) + (∂u i /∂n i ) is the jump of the derivative of u in the normal direction to Γ where u e , u i are the restrictions of u on the exterior Ω e and the interior Ω i of Γ(t) in Ω and n e , n i are the unit exterior normal to Ω e , Ω i ; and Γ = N i=1 Γ i is the union of the boundaries of the N particles.The Mullins-Sekerka model arises as a singular limit for the Cahn-Hilliard equation, see [1,10].Solutions to the Mullins-Sekerka model evolve in such a way that the volume of the region enclosed by Γ(t) is preserved, while the area of Γ(t) shrinks.This is a consequence of the following simple computations provided that the interfaces exist and are smooth.Let A(t) denote the area of Γ(t) and Vol(t) the volume of the enclosed region Ω i (t).Then one can compute (1.2) A single sphere or multiple spheres of the same radius are equilibria for the evolution.
In [2,4] an initial configuration of a finite number N of almost spherical particles was considered.It was shown that to each particle we can associate a center ξ ∈ R 3 , a radius R ∈ R + , and a shape function r(•) ∈ C 3+α (S 2 ) such that with r i satisfying the orthogonality conditions where •, • is the Euclidean inner product, e j ∈ R 3 , j = 1,2,3, is the standard unit vector, and S 2 is the unit sphere.We derive equations for the radii and the centers which have the Nicholas D. Alikakos et al. 217 following expressions: ) where g i and φ i are precisely estimated higher-order terms, and γ is the smooth part of a Green's function capturing the effect of the boundary.Equations (1.5) without γ were derived formally by Weins and Cahn [12] as a correction to the classical Lifshitz-Slyozov-Wagner (LSW) kinetic law that is given by the principal term in (1.5).Ignoring g i and φ i , (1.5) and (1.6) form a closed system for the radii and the centers.The purpose of the present paper is the rigorous continuum limit of (1.5), (1.6) under two different scalings and under no extra assumptions.The resulting continuity equations determine the evolution of the distribution function n(R,ξ,t) which gives the particle density of size R, at the location ξ, and at time t.They have the following form: with subcritical density and critical density where K(t) is determined by the conservation of volume.The subcritical case corresponds to the classical LSW model.The critical versus the noncritical can be understood also as follows. Consider and examine the validity of the approximate solution u(x) (In a very recent preprint, Niethammer and Velázquez [9] have obtained a remarkable estimate for the effective potential of a single particle in the supercritical case by taking into account the screening effect of the surrounding particles.In our setting of bounded domain and hence bounded volume, the limit in the supercritical case does not exist.In the infinite volume case, it does as has been shown in [9]).Clearly u satisfies the equation.Evaluating it on ∂B i , we see that the approximate solution u(x) is estimated off by Thus β = 2/3 represents the boundary of the range of validity of this approximation.Notice that the volume fraction is 3β (cf.Section 2) and tends always to zero as → 0. So, the measure of the ξ's in the limit is always zero.However, if β = 2/3, the electrostatic capacity is nonzero.This was noted in [7] and can be deduced from the remark above.It was noticed in [2], under the appropriate scaling (cf.Section 2), that the esti- holds.Thus β > 2/3 guarantees that this sum goes to zero as → 0. This coincides with the condition that the relative capacity tends to zero as → 0. The case β = 2/3 is critical and leads to a qualitatively different continuum limit.The subcritical case was first derived rigorously by Niethammer [7] for an averaged version of (1.1) that allows the restriction of the evolution to the class of spherical particles.Also [7] assumes immobile centers arranged on a regular periodic lattice on R 3 .For the rigorous justification of these hypotheses, see [3,11].The critical case represents a significant correction to the classical LSW model which takes into account the geometry of the distribution in space.A closely related continuity equation to (1.10) was derived recently by Niethammer and Otto [8] where the uniformity in space in particular makes a difference in the form of the equations.The structure of the paper is as follows.In Section 2, we introduce our scaling and we present the three fundamental estimates which will allow us to pass to the limit: the monopole approximation, the boundary estimate, and the dipole estimate concerning the estimation of the centers.In Section 3, utilizing an appropriate representation of the measure, we obtain two distinct limits corresponding to the subcritical and critical density.In passing to the limit in Section 3, we follow closely [7,8].

The basic estimates
We consider three characteristic length scales in the problem: the size of the particle, the distortion from the spherical shape, and the distance between particles.There are two equivalent normalizations that we can adopt (i) the size of particle = O(1), Λ = size of the domain = O(1/ ), and the distance between particles = O(1/ β ), 0 < β ≤ 1, where size/distance = β ; (ii) the size of particle = O( ), Λ = size of the domain = O (1), and the distance between particles = O( η ), η < 1, where size/distance = 1−η .The relation between η and β is β = 1 − η.For β = 1, η = 0, we have the corresponding finite case.We find it convenient to work with scaling (ii).Under the first scaling, the extinction time is O(1) while under the second scaling, the extinction of the particle happens at O( 3 ).Although we adopt the second scaling where the size of the particle is O( ), we rescale time by setting τ = 3 t.In this new setting, the extinction time is of the order of T = O (1).
In what follows we denote by N the number of active particles, particles for which R i (t) > 0. We denote by N Ω the initial number of particles where according to our scaling ( The R, ξ equations after rescaling time take the form ) 3) The error terms were derived in [4] and they are estimated here under the scaling hypotheses, size/distance = 1−η .From the structure of (2.2), (2.3), we observe that in order to pass to the limit, the estimation of the terms 3  h =i ∂γ/∂ξ h is essential.
that the new particles we keep adding are further and further away.Similar considerations are needed for the other sums as well.However, the summability of this series determines the subcritical and the critical density.
We consider Ω ⊂ R 3 a bounded smooth domain.We take the particles initially at t = 0 satisfying c 1 η ≤ d(ξ i ,ξ j ) ≤ c 2 η .Let ξ i be in the center of a sphere that is contained in Ω as shown in Figure 2.1, and we sum over the shell.
The number of particles in the infinitesimal shell is estimated by πr 2 ∆r/d 3 and the strength of the shell is (2.4) So, we obtain the estimate (2.5) Analogously, we have the estimate (2.6) So, by utilizing (2.5), (2.6), and the scaling, we have In the case where η < 1/3, 1−3η → 0, we obtain the mean field model.When η = 1/3, 1−3η is not negligible any more and we have the critical density.The ξ equations in both the subcritical and critical cases tend to 0 as → 0.

The boundary estimate.
We are also interested in estimating where γ is the smooth part of the Green's function capturing the effect of the boundary and γ satisfies (2.11) Proof.From classical elliptic theory [5] one has the desired estimates.We take ξ h on the boundary and in the center of a sphere.Then (2.15) 2.3.The estimate for the centers.We assume that ) and consider the system , (2.17) for r = O(1), globally in time.

Nicholas D. Alikakos et al. 225
Set ) where and for s h < d, (2.40) By continuous dependence, (2.18), and the structure of the ξ-equations, there is T 0 > 0 depending only on c and η, such that

.42)
We have This step is a normalization, scaling out the distance and summing over a lattice.In what follows, we are imagining the point (ξ i + ξ j )/2 in the center of a sphere that encloses Ω.

Passage to the limit
Set where dµ t is an O(1) quantity, and is defined via The Riesz representation theorem provides the existence of a unique positive Borel, and regular measure µ t ∈ (C c ) * where z ∈ C c (Ω × (0,∞)) with compact support in Ω × (0,∞).We begin with some lemmas from Niethammer and Otto [7,8].
Lemma 3.1.The following hold true: Proof.(i) From the definition of the measure, we have So, for z = 1, (ii) Again, from the definition of the measure for z = R 3 , we have and hence For the sake of simplicity in what follows, we use the same notation for a sequence and for its subsequence.
Lemma 3.2.The following hold true: 3 , uniformly in , with C independent of , and depending only on the initial distribution; As soon as we can choose a subsequence such that (ii) We know the compact embedding BV into L 1 , BV into L p .If we show that 1/R is uniformly bounded in 1/R ∈ L ∞ (0,T) BV (0,T), then we are going to have strong convergence of a subsequence of 1/R in L p .Note that We are going to show that By the interpolation fact, From the definition of measure describing initial data, we conclude that It is known that N 3η is a function of bounded variation.We would like to show that i 3η R i (t) is also a function of bounded variation.Indeed, (It is known (see [3]) that at the extinction time 1/(T − t) 2/3 .)So, the desired quotient is a function of bounded variation.
(iv) From the definition of the measure, we have By utilizing (i), (ii), we have the desired expression for R.
(v) We fix t 1 , t 2 and we calculate So, from Lemma 3.1(i), the assertion follows.
Our aim is the passage to the limit as → 0 and the derivation of the conservation law in a weak sense.Then, as → 0, the conservation law is satisfied: where where K(t) is such that (d/dt) R 3 dµ t = 0.
Proof.The following hold: .22)We distinguish between two cases: (I) the subcritical for η < 1/3 and (II) the critical for η = 1/3.(I) Subcritical case which also tends to zero utilizing the basic estimate.So, by Lemma 3.2(ii), (iii), in the limit as → 0, we obtain Hence, the expected conservation law is an immediate consequence of (3.24) and Lemma 3.2(v).
(II) Critical case.It is important to note that for η = 1/3,

.27)
This scaling fact explains why η = 1/3 is special in evaluating terms like We have (3.28) We are going to pass to the limit in (2.2), that is, In view of the arguments in (I) above, we only need to prove that as → 0, uniformly in δ, where We define the test function where θ is defined as in Figure 3.1.So, Utilizing the basic estimate, the last two terms of (3.33) in terms of δ are of the order of magnitude of o(1).So,  The last step which will complete the proof is the estimation of the double sum (1/NR) k,h,k =h (R h /|ξ h − ξ k |)(R k /R − 1) and the determination of K(t) in the limit.This will be accomplished in three steps.
Step 2. As soon as the above double sum is bounded, there exists a sequence n such that it converges to K(t), uniformly in time.
Step 3. The constant K(t) is determined via the conservation of volume formula to the limit so that the leading order reads: Utilizing the definition of the measure, we have equivalently As soon as the ξ equations for η = 1/3 tend to zero, the desired conservation law is an immediate consequence of (3.28) and Lemma 3.2(v).
Remark 3.4.Niethammer and Otto [8] obtained rigorous arguments based on homogenization for gradient flow structure under the hypotheses that the centers approximately do not move and the particles are radially symmetric.The case of the critical density is examined where the evolution for {R i } i reduces only to an evolution of the joint distribution of particle radii and particle center n(R,ξ,t).They derived the kinetic equation