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

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

Convergence of phase-field approximations to the Gibbs-Thomson law

We prove the convergence of phase-field approximations of the Gibbs-Thomson law. This establishes a relation between the first variation of the Van-der-Waals-Cahn-Hilliard energy and the first variation of the area functional. We allow for folding of diffuse interfaces in the limit and the occurrence of higher-multiplicities of the limit energy measures. We show that the multiplicity does not affect the Gibbs-Thomson law and that the mean curvature vanishes where diffuse interfaces have collided. We apply our results to prove the convergence of stationary points of the Cahn-Hilliard equation to constant mean curvature surfaces and the convergence of stationary points of an energy functional that was proposed by Ohta-Kawasaki as a model for micro-phase separation in block-copolymers.Comment: 25 page

Sharp-interface problem of the Ohta-Kawasaki model for symmetric diblock copolymers

The Ohta-Kawasaki model for diblock-copolymers is well known to the scientific community of diffuse-interface methods. To accurately capture the long-time evolution of the moving interfaces, we present a derivation of the corresponding sharp-interface limit using matched asymptotic expansions, and show that the limiting process leads to a Hele-Shaw type moving interface problem. The numerical treatment of the sharp-interface limit is more complicated due to the stiffness of the equations. To address this problem, we present a boundary integral formulation corresponding to a sharp interface limit of the Ohta-Kawasaki model. Starting with the governing equations defined on separate phase domains, we develop boundary integral equations valid for multi-connected domains in a 2D plane. For numerical simplicity we assume our problem is driven by a uniform Dirichlet condition on a circular far-field boundary. The integral formulation of the problem involves both double- and single-layer potentials due to the modified boundary condition. In particular, our formulation allows one to compute the nonlinear dynamics of a non-equilibrium system and pattern formation of an equilibrating system. Numerical tests on an evolving slightly perturbed circular interface (separating the two phases) are in excellent agreement with the linear analysis, demonstrating that the method is stable, efficient and spectrally accurate in space.Comment: 34 pages, 10 figure

Phase separation on biological membranes

We provide a detailed mathematical analysis of a model for lipid raft formation in cell membranes which was recently proposed by Garcke, Rätz, Röger and the author. Lipid rafts are domains of a specific molecule composition (mostly saturated lipids and cholesterols) in biological membranes. In principle, the proposed model is a phase-field model describing phase separation between saturated and unsaturated lipids. Additionally, the model is based the assumption that active transport processes of cholesterols into and out of the membrane influence the phase separation within the membrane, due to a high affinity between cholesterols and saturated lipids. As such, the model takes the form of an extended Cahn-Hilliard equation which contains additional terms to account for the cholesterol transport. We prove results on the existence and regularity of solutions, their long-time behaviour, and on the existence of stationary solutions. Moreover, we investigate three different asymptotic regimes. The first two are connected to model parameters: We study the case of large cytosolic diffusion and investigate the effect of a infinitely large affinity between cholesterols and saturated lipids. The third is a detailed analysis of the sharp-interface limit of the phase-field model. The first case leads to the reduction of coupled bulk-surface equations in the lipid raft model to a system of surface equations with non-local contributions. Subsequently, we recover the well-known Ohta-Kawasaki equations as the limit for infinitely large affinity between cholesterols and saturated lipids. We prove the convergence of solutions of the lipid raft model to weak solutions of the sharp-interface limit in the sense of varifolds. Finally, we directly prove the existence of weak solutions to the sharp-interface limit

The mean curvature at the first singular time of the mean curvature flow

Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)$, for $t\in [0,T)$. We prove that the mean curvature blows up at the first singular time $T$ if all singularities are of type I. In the case $n = 2$, regardless of the type of a possibly forming singularity, we show that at the first singular time the mean curvature necessarily blows up provided that either the Multiplicity One Conjecture holds or the Gaussian density is less than two. We also establish and give several applications of a local regularity theorem which is a parabolic analogue of Choi-Schoen estimate for minimal submanifolds