On the Phase Diagram for Microphase Separation of Diblock Copolymers: An Approach via a Nonlocal Cahn–Hilliard Functional

Abstract

We consider analytical and numerical aspects of the phase diagram for microphase separation of diblock copolymers. Our approach is variational and is based upon a density functional theory which entails minimization of a nonlocal Cahn–Hilliard functional. Based upon two parameters which characterize the phase diagram, we give a preliminary analysis of the phase plane. That is, we divide the plane into regions wherein a combination of analysis and numerics is used to describe minimizers. In particular we identify a regime wherein the uniform (disordered state) is the unique global minimizer; a regime wherein the constant state is linearly unstable and where numerical simulations are currently the only tool for characterizing the phase geometry; and a regime of small volume fraction wherein we conjecture that small well-separated approximately spherical objects are the unique global minimizer. For this last regime, we present an asymptotic analysis from the point of view of the energetics which will be complemented by rigorous $\Gamma$-convergence results to appear in a subsequent article. For all regimes, we present numerical simulations to support and expand on our findings.