A rigorous derivation of mean-field models describing 2D micro phase separation

We study the free boundary problem describing the micro phase separation of diblock copolymer melts in the regime that one component has small volume fraction $$\rho $$ρ such that the micro phase separation results in an ensemble of small disks of one component. We consider the two dimensional case in this paper, whereas the three dimensional case was already considered in Niethammer and Oshita (Calc Var PDE 39:273–305, 2010). Starting from the free boundary problem restricted to disks we rigorously derive the heterogeneous mean-field equations on a time scale of the order of $${\mathcal {R}}^{3}\ln (1/\rho )$$R3ln(1/ρ), where $${\mathcal {R}}$$R is the mean radius of disks. On this time scale, the evolution is dominated by coarsening and stabilization of the radii of the disks, whereas migration of disks becomes only relevant on a larger time scale.


Introduction
Diblock copolymer molecules consist of subchains of two different type of monomers, say A-and B-monomers. The different type of subchains tend to segregate, and hence the phase separation take place. However since the subchains are chemically bonded, the two subchains mix on a macroscopic scale, while on a molecular scale, A-and B-subchains still segregate and the micro-domains are formed. This is called micro phase separation. For more physical background on this phenomenon we refer to [2,8].
In the strong segregation regime, energetically favorable configurations have been characterized in the Ohta-Kawasaki theory [18] by minimizers of an energy functional, which is in the two dimensional case of the form Communicated Here (0, L) 2 ⊂ R 2 is the domain covered by the copolymers, ⊂ [0, L) 2 denotes the region covered by, say, A-monomers, ρ = | | L 2 ∈ (0, 1) the average density, σ ∈ R + = (0, ∞) is a parameter related to the polymerization index, χ is the characteristic function of , and H 1 denotes one dimensional Hausdorff measure.
The first term in the energy prefers large blocks of monomers, the second favors a very fine mixture. Competition between these terms leads to minimizers of E which represent micro phase separation.
Starting with the pioneering work [15], where the Ohta-Kawasaki theory is formulated on a bounded domain as a singularly perturbed problem and the limiting sharp interface problem is identified, there has been a large body of analytical work. Minimizers of the energy functionals have been characterized in [1,3,4,20], the existence/stability of stationary solutions has been investigated in [16,17,19,21] and a time dependent model has been considered in [7,9]. The mean field models in the three dimensional case have been derived in [6,10,12].
We consider the gradient flow of the energy, which is a standard way to set up a model for the evolution of the copolymer configuration that decreases energy and preserves the volume fraction. Then the evolution equation becomes the following extension of the Mullins-Sekerka evolution for phase separation in binary alloys [11]. The normal velocity v of the where [∇w · n] denotes the jump of the normal component of the gradient of the potential across the interface. Here n denotes the outer normal to and The potential w is for each time determined via − w = 0 i n (0, L) 2 where κ is the curvature of ∂ . We are interested in the case that the volume of (t) is preserved in time and can thus impose Neumann or periodic boundary conditions for w on ∂(0, L) 2 . In what follows we will consider a periodic setting and hence always require that the potential w is (0, L) 2 -periodic. For local well-posedness of this evolution see [5]. The evolution defined by (2)-(4) has a formal interpretation as a gradient flow of the energy (1) on a Riemannian manifold. Indeed, consider the manifold of subsets of the 2dimensional flat torus T of length L with fixed volume, that is M := { ⊂ T ; | | = L 2 ρ}, whose tangent space T M at an element ∈ M is described by all kinematically admissible normal velocities of ∂ , that is, The Riemannian structure is given by the following metric tensor on the tangent space: where w α : T → R (α = 1, 2) solves for v α ∈ T M (α = 1, 2). The gradient flow of the energy (1) is now the dynamical system where at each time the velocity is the element of the tangent space in the direction of steepest descent of the energy. In other words, v is such that for allṽ ∈ T (t) M. Choosingṽ = v we immediately obtain the energy estimate associated with each gradient flow, which is In what follows we consider the micro phase separation in the two dimensional case in the regime where the fraction of A-monomers is much smaller than the one of B-monomers. In this case A-phase consists of an ensemble of many small approximately circular particles. We reduce the evolution to the gradient flow on circular particles.
For that purpose we define the submanifold N ⊂ M of all sets which are the union of disjoint balls where N is the number and i = 1, . . . , N an enumeration of the particles with centers in the torus T. Since the normal velocity v satisfies v = d R i dt + d X i dt · n on ∂ B R i (X i ), the tangent space can be identified with the hyperplane such that V i describes the rate of change of the radius of particle i and ξ i the rate of change of its center. We use the abbreviation The metric tensor is then given by For the following it will be convenient to split the metric tensor into the radial and shift part respectively. For any {V i , ξ i } i , we write where u and φ are harmonic in-and outside the particles and where We consider the energy We obtain the differentials of the energies in the direction of a tangent vectorZ Herew : T → R is a function w α satisfying (9) for Z α =Z. The integration by parts yields From now on we consider an arrangement of particles as described above which evolves according to the gradient flow equation. This means that for any t ≥ 0, it holds for (11) for allZ ∈ T Y N . SinceZ is an arbitrary element of the tangent space we conclude from (11) that w satisfies 1 and for all i such that R i > 0, with a Lagrange parameter λ(t) that ensures volume conservation. Equations (12) and (13) are the analogue of (4) in the restricted setting. Our aim is to identify the evolution in the limit of vanishing volume fraction of particles. More precisely, we consider a sequence of systems characterized by the parameter in the limit ε → 0. Here d is defined by and R by Then 1 d 2 denotes the number density of particles, and π R 2 the average volume of particles. Here and throughout this paper we use the abbreviation i = i:R i >0 .
Our main result informally says that when L ∼ L sc , with on the time scale of order R 3 ln(1/ρ), the number density of particles with radius r and center x, denoted by ν = ν(t, r , x) (suitably normalized), satisfies where ψ = ψ(t, x) satisfies for each t that in the limit ε → 0. Here σ is also suitably normalized.
We remark that on the other hand, in the case that L L sc , that is, in the very dilute case, one obtains a homogeneous version where ψ is constant in space, and is replaced by λ(t). More precisely that the number density of particles with radius r , denoted by ν(t, r ) (suitably normalized), satisfies

The result
In this section, we will introduce suitably rescaled variables, state the precise assumption on our initial particle arrangement, and present the statement of our main result. We assume from now on that L = L sc for the ease of presentation, and we will rescale the spatial variables by L sc such that L sc = L = 1 and hence d = ε, R = ε exp(−1/ε 2 ) =: α ε .
Notice that ρ = πα 2 ε ε −2 and ln(1/ρ) ∼ ε −2 . We introduceR i ,t,V i ,ξ ,ŵ,σ andμ via From now on we only deal with the rescaled quantities and drop the hats in the notation. We denote the joint distribution of particle centers and radii at a given time t by ν ε t ∈ (C 0 p ) * , which is given by where C 0 p stands for the space of continuous functions on R + × T with compact support contained in R + × T. Here T denotes the unit flat torus, and R + = (0, ∞). Note that since ζ(r , x) = 0 for r = 0, particles which have vanished do not enter the distribution. The natural space for ν ε t and its limit ν t is the space (C 0 p ) * of Borel measures on R + × T. We are now going to make the assumptions on our initial particle arrangement precise. Notice first, that in view of (15) and (16) we have It follows immediately, that that is the surface energy of the initial particle arrangement is finite. Furthermore it is natural to assume that initially the nonlocal energy is uniformly bounded in ε, that is where C is independent of ε and where μ ε (0, We will see later [cf. (84)], that the nonlocal energy controls i ε 2 R 4 i . Hence, finiteness of the nonlocal energy initially also implies i ε 2 R 4 i (0) ≤ C. For our analysis we need a little more than this. We need a certain tightness assumption which ensures, that not too much mass is contained in very large particles as ε → 0. More precisely, we assume that Finally, we assume that initially particles are well separated in the sense that we assume that there is γ > 0, such that In accordance with the notation in (22) we will use in the following the abbreviation Otherwise the domain of integration is specified.
The natural space for potentials of diffusion fields is H 1 (T). Furthermore we will denote by • H 1 (T) the subspace of H 1 (T) of functions with mean value zero. We can now state our main result which informally says that ν ε t converges as ε → 0 to a weak solution of (18)- (19).
Theorem 2.1 Let T > 0 be given and assume that the assumptions in Sect. 2 are satisfied. Then there exists a subsequence, again denoted by ε → 0, and a weakly continuous map (18) and (19) hold in the following weak sense for all ζ ∈ H 1 (T) and almost all t ∈ (0, T ).
The proof of Theorem 2.1 goes similarly to the approach for the three dimensional case in [12]. However in contrast to the three dimensional case we need to estimate 1/R i term in the proof that the tightness property is preserved in time (see Lemmas 3.9 and 3.10) since the Lagrange multiplier diverges when particles disappear.

Proof of Theorem 2.1
We can deduce Theorem 2.1 by the homogenization of Rayleigh Principle (see Theorem 3.11). This will be obtained from the homogenization of metric tensor (Lemmas 3.5, 3.6) and the limit of the differential of the energy (Lemma 3.8). Also we need some a-priori estimates, which are given by a series of lemmas. The proof of Lemmas in this section will be given in Sect. 4. For readers convenience we will not abbreviate the arguments.

Gradient flow structure
In rescaled variables the submanifold N turns into and the tangent space We will always denote by The notationZ ε will be used for an arbitrary element of the tangent space. Furthermore we use the abbreviation We define the energy in rescaled variables as χ ∪B i − π and T μ ε dx = 0, and the metric tensor Notice that the potentials are only determined up to additive constants. In what follows we fix this constant by requiring that Tũ Equations (12) and (13) for the direction of steepest descent, turn into for some λ ε (t) ∈ R and for all i such that R i > 0.
Here and in what follows we abbreviate, with some abuse of notations, for a disk B R (X ) the perimeter by |∂ B R | and its area by |B R |. Now the energy estimate (8) reads for all t 1 > 0. Finally, the Rayleigh principle says that Z ε satisfies for allZ ε ∈ T Y N and nonnegative β ∈ C ∞ ([0, T ]).

A priori estimates and weak limits
It follows from definitions (15), (16) and the facts that volume of particles is conserved and the number can only decrease, that On the other hand the uniform bound on the energy in (34) implies the following.
. As will be shown in Sect. 4, we have which yields in particular Bounds (39) and (40) yield a weak Hölder regularity in t of {ν ε t } t : Using Arzela-Ascoli's Theorem, (36) and (42) imply that there exists a weakly continuous family {ν t } t of nonnegative Borel measures on R + × T such that for a subsequence for ζ in a countable subset of C 0 p ∩ C ∞ . Again by (36), we see that we can extend the locally uniform convergence in (43) to all ζ ∈ C 0 p . Obviously, the bound (36) is conserved and due to (37) and (38) we have The uniform control of the signed Borel measures {r 2 dρ ε t dt} ε and {r 2 dψ ε t dt} ε on R + × T×[0, T ] implied by (40) ensures the weak convergence, where the limits can be regarded as bounded linear functionals on L 2 (r 2 dν t dt) and L 2 (r 2 dν t dt) 2 respectively, and hence by such that for a subsequence for all nonnegative β ∈ C 0 ([0, T ]). Thus the limit of (39) is
We are going to show below that for any T > 0, particles do not collide on a time interval [0, T ] for sufficiently small ε. More precisely Lemma 3.3 For any T > 0 we can find ε 0 > 0 such that for all ε ∈ (0, ε 0 ]. Thus it follows that the marginal of ν t with respect to x has a bounded Lebesgue density. Hence it follows from (46) that the functional Hence K (t, ·) ∈ H 1 (T) is uniquely determined via (29) up to additive constants.
In order to prove Lemma 3.3 we show the following.
Lemma 3.4 (slow motion of the particle centers) As long as (51) is satisfied, we have

Homogenization of the metric tensor
We identify the -limit for the metric tensor and provide the necessary results to pass to the limit in the metric tensor. The following is a lower semicontinuity result.

Lemma 3.5 (lower semicontinuity) For all nonnegative
where for almost all t the function u(t, ·) ∈ Furthermore we show that for any tangent vector of the limit manifold (ṽ,ξ), there exists an approximating sequence along which the metric tensor is continuous. Lemma 3.6 (construction) For anyṽ ∈ L 2 (r 2 dν t dt) with ṽ r 2 dν t = 0 for almost all t and anyξ ∈ L 2 (r 2 dν t dt) 3 there exists a sequenceZ ε withZ ε ∈ T Y ε N ε such thatZ ε weakly converges to (ṽ,ξ), and Note that the contribution from the drift term and the radial part do not interact in the limit ε → 0.

The limit of the differential of the energy
We identify the limit of the differential of the energy. To that aim we identify the limit of the potentials μ ε so as to prove the convergence for the nonlocal part of the energy.
Note that Here, as also in the homogenization of the metric tensor, the key idea is that the potentials can be represented as a sum of monopoles, which represent the self-interaction of particles, plus a slowly varying field, which represents the interaction between different particles. We set l = γ ε, where γ > 0 is as in (51). We write μ ε = iμ i +μ ε with it holds that The slowly varying fieldμ ε converges strongly to K and this enables us to pass to the limit in the differential of the energy.

Tightness
In order to prove Lemma 3.8 we first need to show that the tightness property (26) is preserved in time so that no mass is lost at infinity in the limit ε → 0.

Lemma 3.9 (tightness) For any t > 0 we have
This lemma is crucial to our proof. The proof is much more difficult than three dimensional case. In fact, the main idea of the proof is to show by asymptotics that, at least in some average sense, where u is the Lagrange multiplier that ensures the volume conservation. In contrast to the three dimensional case, the Lagrange multiplier may diverge. However we can still show the following a-priori estimate, and thus, at least on average, V i ≤ 0, if R i is sufficiently large, and no mass can escape to infinity as ε → 0, from which one deduces Lemma 3.9. for all ε ∈ (0, ε 1 ].

Homogenization of Rayleigh principle
The main task that remains to be done now is to determine the equation for the velocity function v. It will be characterized as the minimizer in the Rayleigh principle. Thus our task is to characterize the limits of Z ε that satisfy (35). Notice that we can use Lemma 3.8 with Z ε −Z ε where Z ε is the direction of steepest descent andZ ε is as in Lemma 3.6. The main result from which one deduces Theorem 2.1 is the following.
The Euler-Lagrange equation for (62) becomes r 2 v = r (u − σ K − λ) − 1 − σ r 3 with λ being a Lagrange multiplier that ensures the constraint r v dν t = 0. Setting ψ(t, x) ≡ u − σ K − λ, we can then derive (28)-(29). The proof is basically straightforward and goes similarly to the one in Chapter 6 of [13]. We omit the details here.
Notice that in the formulation (62) we need that v−ṽ has compact support in the r -variable. This is due to the fact that we cannot guarantee that the term r v dν t which appears in the differential of the surface energy is well-defined.

Proof of Lemmas
Proof of Lemma 3. 4 We set l = γ ε, where γ is as in (51). Notice that due to (51) and For further use we collect some properties of φ i . It is easily checked that φ i is continuous in T, harmonic in B l (X i )\∂ B i and satisfies Furthermore, φ i = 0 on ∂ B l (X i ) and Integration by parts yields Together with (66), (67) and (33) we find We estimate the last term on the right hand side of (68) via We can write μ ε = i μ i +μ ε with Notice that μ i is continuous in T, and satisfies − μ i = ε 2 These imply thatμ ε , and hence ∇μ ε also are harmonic in ∪ i B l (X i ). Then, due to the properties of μ i and the mean value theorem, we have It is easily checked that u i is continuous in T, harmonic in B l (X i )\∂ B i and satisfies Then from a similar argument as above, we see that Thus, in summary we find This completes the proof of Lemma.

Proof of Lemma 3.3 In Sect. 3.3 we will see that
T 0 T |∇φ ε | 2 + |∇u ε | 2 dx dt ≤ C (cf. (81)). Furthermore, due to (34), we have T |∇μ ε | 2 dx ≤ C. Then the statement follows from Lemma 3.4 and Proof of Lemma 3. 5 We can prove in the same way as in [13] since (51) is satisfied. Hence, it remains to show that and Step 1: Monopoles. Our goal is to construct a good approximation of φ ε which is based on cutting off the single monopole solutions. To that aim we define φ i as in (63) with γ is as in (51) and thus the balls {B l (X i )} i are disjoint. Using (64) and (65), we find Step 2: A lower bound. We will show that for any given δ > 0 we have if ε is sufficiently small. Indeed, due to the fact that On the other hand From the corresponding Euler-Lagrange equation we see that the minimizerψ is orthogonal to all divergence-free function and hence a gradient. We find that ψ = ∇φ i and thus (77) follows from (76).
Step 3: Approximation of φ ε . Withφ we have for any k > 0. In fact, forφ ε := φ ε −φ ε it follows from the definitions and (64) that for all ζ ∈ • H 1 (T). We now define L ε ∈ (H 1 (T)) * via We observe that due to − ∂ B l (X i ) ξ i · n d S = 0 we can write We can then estimate where the last estimate follows from Trace Theorem and Poincaré's inequality. This gives for any k > 0, which proves (79).
Step 4: Bounds on the individual terms.
Step 5: Lower semicontinuity. Since we have now established (81), we obtain the existence of the weak limit ξ as explained before in Sect. 3.2, and hence the assertion (74) follows from (49) and (77).
Step 6: The mixed term vanishes in the limit.
Proof of Lemma 3. 6 We need only to show that for givenξ as above we can findφ ε such that since the corresponding result forũ ε can be proved in the same way as in [13]. The proof is in fact quite similar to the proof of Lemma 3.5, since there the minimization property of {ξ i } i is not used in the construction, which is henceforth quite general.
Step 1: Construction for smoothξ . We first assume thatξ is smooth. Then we defineξ i := ξ(X i ) and construct φ i as in (63) for ξ i =ξ i , and a correspondingφ ε as in (78). The property (76) implies as ε → 0. Here we used (43) and thatξ has compact support. Furthermore, we obtain also exactly as in the proof of Lemma 3.5 thatφ ε :=φ ε −φ ε converges to zero strongly in L 2 ((0, T ); H 1 (T)), and as a consequence the mixed term vanishes in the limit.
In order to finish the proof of the lemma we have to show that we can approximatẽ ξ ∈ L 2 (r 2 dν t dt) 3 by smooth functionsξ n with compact support such that T 0 β |ξ n | 2 r 2 dν t dt → T 0 β |ξ | 2 r 2 dν t dt as n → ∞. But this follows from a density argument. Proof of Lemma 3. 1 We first prove (38). Set l = γ ε, where γ is as in (51). Due to the fact that On the other hand, for each i, where the infimum is taken over all ψ ∈ (L 2 (B l (X i ))) 2 which satisfy . We see that the minimizerψ is orthogonal to all divergence-free vector-valued functions.
Proof of Lemma 3. 10 We can prove this in a similar way as in [14, section 3.4] by making use of (32) and Lemma A.1 although we are using Neumann boundary condition on ∂ B i instead of Dirichlet boundary condition.
2e−1 by 0 < R i ≤ 1/ε and |B l (X i )|/|A i | ≤ 2 for small ε with l = γ ε and we see that Now we estimate the second term of the right hand side of (85).
It follows from (85) and (86) that We estimate the L 2 norm of v ε . Using we see that We estimate the second term of the right hand side of (90).
Note that due to Then it follows from (90) and (91) that Then by (88), Thus we obtain from (87),

Proof of Lemma 3.9
Step 1: An expression for V i . Similarly to φ i in the proof of Lemma 3.4 we introduce a suitable test function v i . Here it is the capacity potential of B i with respect to B l (X i ), with l = γ ε. This gives that With this definition we also have Due to (32) it follows that Using that μ ε = i μ i +μ ε with μ i as in (70) we obtain Similarly using that φ ε = i φ i +φ ε with φ i as in (63) we obtain It follows that Step 2: A bound on ∂ B l (X i ) u ε + φ ε − 2σ μ ε d S.

(98)
Step 3: A bound for λ ε (t). We go back to (96) to estimate λ ε (t). For that purpose we multiply (96) by ε 2 and sum over i. We find First notice that due to i ln l α ε R i ≤ C due to (84). As a consequence, we also have 1 i a ε ≤ C. Taking m = 0 in (98), we find  (101) Step 4: Completion of Proof. Now we go back to (96), multiply with ε 2 R 2 i and sum, but only over i such that R i ≥ M. We find (102) Here we used a ε ε 2 = 1 1+ε 2 ln γ R i = 1 + O(ε) as ε → 0 for i such that R i ≥ M, ε R i ≤ 1 and M > γ .