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Small Volume Fraction Limit of the Diblock Copolymer Problem: II. Diffuse-Interface Functional

Rustum Choksi, Mark A. Peletier

arXiv:1004.4271v1math.APmath.FAnlin.AOnlin.PS

TL;DR

The small-volume regime raises how the energy behaves as η shrinks, since η^4/ε determines whether constant or localized states have lower energy. The paper studies this behavior through functionals defined over Dirac point masses and establishes convergence bounds for the diffuse-interface problem.

  • Problem

    The small-volume regime raises how Eε,η behaves as η shrinks, since η^4/ε determines whether constant or localized states have lower energy.

    Source: §3 · p. 8

  • Method

    The analysis studies small-η behavior through functionals defined over Dirac point masses and proves lower-bound, compactness, and upper-bound results as ε and η tend to zero.

    Source: §3 · pp. 7–8

  • Results

    Bounded-energy sequences have subsequences converging weakly to limits with countable support, and the theorem supplies recovery sequences and constrained-minimizer results under additional bounds.

    Source: §3 · p. 8

  • Takeaways & Limitations

    The asymptotic description represents small-volume behavior with functionals defined over Dirac point masses.

    Source: §3 · p. 7

    In three dimensions, the recovery construction does not give optimal energy-closeness scaling, and its ε bounds are stronger than the ideal condition because the local minimizer behavior is unknown.

    Source: §3 · p. 9

Abstract

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We present the second of two articles on the small volume fraction limit of a nonlocal Cahn-Hilliard functional introduced to model microphase separation of diblock copolymers. After having established the results for the sharp-interface version of the functional (arXiv:0907.2224), we consider here the full diffuse-interface functional and address the limit in which epsilon and the volume fraction tend to zero but the number of minority phases (called particles) remains O(1). Using the language of Gamma-convergence, we focus on two levels of this convergence, and derive first- and second-order effective energies, whose energy landscapes are simpler and more transparent. These limiting energies are only finite on weighted sums of delta functions, corresponding to the concentration of mass into `point particles'. At the highest level, the effective energy is entirely local and contains information about the size of each particle but no information about their spatial distribution. At the next level we encounter a Coulomb-like interaction between the particles, which is responsible for the pattern formation. We present the results in three dimensions and comment on their two-dimensional analogues.

1 Introduction

The paper studies the nonlocal Cahn–Hilliard functional in a limit where the minority-phase volume fraction and interface parameter vanish while the number of particles remains O(1). Its first- and second-order effective energies are defined on weighted Dirac masses: the first describes individual particle masses, while the second includes interactions between particles.

1.1 The Functional

  • The functional balances gradient penalization, phase separation, and a nonlocal term favoring rapid oscillation.For sufficiently large γ, the nonlocal term may favor intermediate-scale variation and patterns with an intrinsic length scale.Source: §1.1 · p. 2
  • The small-volume regime keeps the number of minority-phase particles O(1) as ε and the mass fraction f tend to zero together.The paper focuses on three dimensions and comments on analogous two-dimensional results.Source: §1.1 · p. 2

1.2 The Spherical Phase in Diblock Copolymers

  • The order parameter represents relative monomer density, and the mass constraint fixes the relative abundance of the A-region.The two preferred states are u = 0 and u = 1; small inclusions of one phase in the other are called particles.Source: §1.2 · p. 3
  • The analysis uses Gamma-convergence to derive first- and second-order effective energies for limits concentrated on weighted Dirac point masses.The diffuse-interface parameter and minority-phase fraction tend to zero with a suitable scaling.Source: §1.2 · p. 4
  • The first-order effective energy is a sum of local particle energies depending on each particle’s mass, but not on particle positions.The next-order energy contains Coulomb-like interactions expected to enforce a periodic array.Source: §1.2 · p. 4

2 Some definitions and notation

The section establishes notation for the flat torus, measures, and the Green’s function used in the analysis. It also identifies total variation with perimeter for characteristic functions of finite-perimeter sets.

2 Some definitions and notation

  • The flat torus Td is treated as an additive group, and weak-* convergence of Radon measures is tested against continuous functions.The same convergence notation is used for functions when they are regarded as measures.Source: §2 · p. 5
  • The Green’s function GTd is introduced as the solution of the stated equation for −∆ on the torus.The text gives local descriptions in three and two dimensions.Source: §2 · p. 5
  • The nonlocal double integral can use u rather than its zero-average version because GTd has zero average.The integral pairs u(x)u(y) with GTd(x − y).Source: §2 · p. 6
  • For a characteristic function of a finite-perimeter set, total variation on Td is its perimeter.The passage defines total variation through the measure of the gradient.Source: §2 · pp. 5–6

3 Rescalings and Statements of the Results

The section rescales the diffuse-interface problem and states its first- and second-order limits as the parameters vanish. The limits describe point-particle configurations, with compactness and recovery requiring conditions that relate the interface width to particle size.

Rescaling

  • Balancing the nonlocal self-interaction with the other terms under an O(1) particle-count assumption gives the scaling γ ∼ 1/η^3.The stated balance uses the leading H^-1 contribution of order 1/η.Source: §3 · p. 6

Effective energies

  • E0 is finite on weighted sums of distinct Dirac masses, and its constrained minimum is determined by the local energy of particle masses.The next-order functional is introduced to capture the energy beyond this first-order minimum.Source: §3 · p. 7

Theorem 3.1

  • Bounded energy yields a subsequential limit with countable support; bounded next-order energy additionally forces a global minimizer of E0 under the mass constraint.The lower-bound result assumes ε_n = o(η_n^(4+ζ)) for some ζ > 0; the upper bound uses ε_n ≤ C1(η_n), with a stronger C2 condition for next-order recovery at minimizers.Source: §3 · p. 8

Parameter conditions

  • The sharp-interface approximation requires ε ≪ η, while forcing the diffuse-interface limit onto point masses requires the stronger ε ≪ η^4.The latter condition makes the energy of the constant state diverge relative to localized states; the lower-bound proof uses the still stronger ε_n = o(η_n^(4+ζ)).Source: §3 · pp. 8–9
  • In three dimensions, the upper-bound construction uses stronger ε-to-η conditions than the ideal ε_n = o(η_n); the authors say the two-dimensional analysis permits a weaker condition, up to a logarithmic correction.The three-dimensional construction uses a Modica–Mortola profile that does not give optimal energy closeness.Source: §3 · p. 9

4 Previous results for the sharp interface limit

The earlier sharp-interface analysis establishes first- and second-order Γ-limits on point-particle configurations. At second order, bounded energy imposes both global minimality of the first-order limit and restrictions on the particle masses.

Sharp-interface Γ-limits

  • The first-order limit has countable support for bounded energy, with a corresponding lower-bound statement.The second-order theorem also supplies an upper-bound recovery sequence for finite E0 and F0.Source: §4 · p. 9
  • For bounded second-order energy, the limit is a global minimizer of E0 under the mass constraint and is a weighted sum of Dirac masses with weights in M.The result also gives a lower bound for F0.Source: §4 · p. 9

Conditions on particle masses

  • Second-order boundedness requires the particle masses to satisfy the minimizing condition in (4.6), which in turn implies only finitely many nonzero masses.The compactness condition also requires each mass to have a minimizer for the local problem defining e0(mi).Source: §4 · p. 10

5 Proof of Theorem 3.1

The proof transfers the diffuse-interface problem to the previously analyzed sharp-interface limit. It establishes lower bounds by truncation and constructs upper-bound recovery sequences by smoothing sharp-interface configurations with a quantified interface profile.

Lower bound and compactness

  • The lower-bound proof truncates diffuse-interface sequences to sharp-interface sequences with the same limit and a small energy difference.It uses ε_n = o(η_n^(4+ζ)) to control the truncation and interfacial-energy error.Source: §5 · pp. 10, 13

Upper bound

  • For the upper bound, the proof modifies a sharp-interface recovery sequence near its boundary using a quantified Modica–Mortola profile construction.The construction invokes the Otto–Viehmann lemma [24]; the cited profile convergence is linked to [19, 5].Source: §5 · pp. 10–11

Lower bound and compactness

  • Applying the sharp-interface theorem after truncation gives a subsequential limit with countable support and transfers the lower-bound inequality to the diffuse-interface sequence.The argument first establishes closeness of the constructed sequences in L2 and H^-1.Source: §5 · pp. 12–13

Upper bound

  • The diffuse-interface recovery sequence is chosen with a scale-dependent approximation, and ε_n is constrained so the approximation errors vanish for first- and next-order bounds.The proof chooses separate functions C1(η_n) and C2(η_n) to enforce the respective error estimates.Source: §5 · pp. 13–15
  • For next-order recovery, the target must minimize E0 under the mass constraint, and the construction uses a finer approximation scale than for first-order recovery.The sequence uses α = η_n^2 and chooses ε_n to control the resulting energy error.Source: §5 · pp. 15–16

6 The local structure of minimizers and the variational problem that defines e0

The section studies the local variational problem defining e0 and conjectures how its minimizers change as mass increases. It relates the problem to classical results on competing shape preferences and discusses what is known about minimizers.

The variational problem defining e0

  • e0(m) minimizes a two-term energy at fixed mass, whose terms favor different shapes; balls minimize the first term but are worst for the second.The section frames the local structure problem through this competition.Source: §6 · p. 16

Connection to Poincaré’s problem

  • The shape problem recalls Poincaré’s fluid-body problem: rearrangement results establish balls as unique minimizers for the potential problem.The section cites Poincaré, the Riesz Rearrangement Inequality, Burchard, and Lieb.Source: §6 · p. 16

Conjectured minimizers

  • For m ≤ m*, the conjectured minimizer is a single ball; for m > m*, it ceases to exist, with m* ≈ 22.066.The conjectured threshold is the positive zero of f(m) − 2f(m/2).Source: §6 · p. 17
  • For m > m*, separating two equal balls lowers the limiting energy because the H−1 interaction penalizes objects at finite distance.The separated sequence has limiting energy 2f(m/2).Source: §6 · p. 17
  • Above a further threshold m** > m*, three equal balls are conjectured to form a minimizing sequence, with analogous sequences for larger masses.For three balls the limiting value is 3f(m/3).Source: §6 · p. 17

Known minimizer results

  • Prior sharp-interface results show bounded-energy sequences converge to weighted delta functions whose weights correspond to masses admitting minimizers of e0, but give no explicit range for those weights.An unpublished second-variation calculation indicates that a ball remains stable up to mc ≈ 62.83, beyond m*.Source: §6 · p. 17

7 Analogous results in two dimensions

In two dimensions, the H−1 norm is critical, so the small-volume scaling and limiting energies differ from the three-dimensional case. The section gives compactness and recovery results, characterizes optimal masses, and identifies the leading and next-order energy terms.

Scaling and dimensional differences

  • The two-dimensional H−1 norm is critical, requiring ε to be slaved to η with exponent 3 rather than the three-dimensional exponent 4.After rescaling with v = u/η^2, γ is slaved to η.Source: §7 · pp. 17, 20

First-order energy

  • The explicit two-dimensional first-order energy has a dominant H−1 contribution whose logarithmic scaling makes that term additive.Source: §7 · p. 18

Second-order energy

  • The next term in the expansion is recovered as the limit of the rescaled two-dimensional energy.Source: §7 · p. 18

Optimal mass sequences

  • Optimal mass sequences contain finitely many nonzero terms, all equal; if one mass is below 2^-2/3π, it is the only nonzero term.Source: §7 · p. 19

Limiting weights

  • Bounded second-order energy imposes minimality and compactness conditions on limiting weights, with the minimality condition requiring an optimal sequence for the total mass.Source: §7 · p. 19

Compactness and lower bound

  • The lower-bound result gives countable support for the limit under the stated scaling, and bounded second-order energy makes that limit a constrained global minimizer.Source: §7 · p. 19

Recovery sequence

  • The two-dimensional upper-bound scaling is less restrictive because first-order minimizers are circular droplets, which can be regularized using a one-dimensional optimal profile.The profile approximates a Heaviside function and yields explicit dependence of C0 on η.Source: §7 · p. 20

Estimate condition

  • For the stated two-dimensional estimate, the constant C0 diverges as the bound on the Laplacian approaches 4π.Source: §7 · p. 20

8 Discussion, dynamics, and related work

Together with the sharp-interface study, this paper analyzes the small-volume regime and highlights open questions about three-dimensional local particle structure. The discussion distinguishes the problem’s focus on minimizer support from related work and connects the limiting energies to particle dynamics and pattern formation.

Open problems

  • The authors identify three-dimensional local structure as a priority for rigorously assessing how the nonlocal term affects particle shape.They note that many open problems remain in this area.Source: §8 · p. 22

Relation to Ginzburg-Landau theory

  • Unlike Ginzburg-Landau analyses of gauge-invariant vorticity, this asymptotic study concerns the support of minimizers.Source: §8 · p. 22

Related results

  • Ren and Wei prove existence and stability of sphere-like solutions and locate their centers near minimizers of a delta-measure effective energy.Their techniques are local rather than global.Source: §8 · p. 22
  • For the sharp-interface problem in two dimensions, Muratov proves global minimizers are nearly identical small circular droplets separated by large distances, without determining their placement [20].Source: §8 · p. 23

Dynamics

  • Particle dynamics separate material exchange and migration: small particles exchange mass and migrate, while migration is associated with self-organized patterns and a Coulomb-like interaction at the next energy level.The discussion links the position and radius gradient-flow structures in to the first- and second-order limit functionals; rigorous dynamics remain an open question, with approaches in,, and.Source: §8 · p. 23
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