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

TL;DR

A boundary integral formulation corresponding to a sharp interface limit of the Ohta-Kawasaki model for diblock-copolymers allows one to compute the nonlinear dynamics of a non-equilibrium system and pattern formation of an equilibrating system.

Abstract

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.

Figures (10)

• Figure 1: A schematic diagram of Ohta-Kawasaki problem. The interior domain $\Omega^-$ is the disjoint union of three connected and bounded regions $\Omega^{-}_1, \Omega^{-}_2$ and $\Omega^{-}_3.$ The boundary of $\Omega^-$ consists of $\Gamma=\partial \Omega^-_1 \cup \partial \Omega^-_2 \cup \partial \Omega^-_3.$ The outer region $\Omega^+$ is bounded and surrounds $\Omega^-$.
• Figure 2: Comparison of results from the nonlinear simulation and the linear analysis for $R(t)$ and $\delta(t)$ against time. We choose $\sigma=0.47, R_{\infty}=10, N=1024$, and $\Delta t = 2\times 10^{-3}$ to obtain the match between the two setups and the simulation are stopped when the linear analysis results starts to over-predict the nonlinear results at $t_{\rm end}=1.0$.
• Figure 3: Time evolution of the interface
• Figure 4: Demonstration of spectral accuracy and second-order convergence in time of the nonlinear simulation.
• Figure 5: Time evolution of 4 elliptic regions with semi-axes $a=1.5$ and $b=1.0$. The other parameters are $R_{\infty}=4$ and surface tension $\sigma=0.47$. The system enters equilibrium at $t_{eq}=8.75.$ Centroids of the domains D1, D2, D3, and D4 are at $\left(2,0\right)$, $\left(0,2\right)$, $\left(-2,0\right)$, and $\left(0,-2\right)$ at $t=0$, respectively.