Decorated phases in triblock copolymers: zeroth- and first-order analysis
Stanley Alama, Lia Bronsard, Xinyang Lu, Chong Wang
TL;DR
The work analyzes a two-dimensional inhibitory ternary system modeling triblock copolymers, deriving a zeroth-order sharp-interface energy $E_L$ via $\Gamma$-convergence that combines interfacial perimeter with a nonlocal Green-function interaction. At first order, it identifies a lens-shaped minority phase that lies on the boundary between two dominant backgrounds, yielding a limiting energy $E_0$ governed by lens energies $e_0(m)=c_1\sqrt{m}+c_2m^2+c_3m$ and discrete mass placements. The results show that minority masses become finitely many and equal in optimal configurations, with the optimal count $N^*=\!M(\tfrac{c_1}{2c_2})^{-2/3}$, and supply a precise lens-shaped morphology $\Lambda^*$ that minimizes perimeter-diameter trade-offs. The theory extends to negative interaction coefficients under a smallness constraint and provides recovery sequences built from balls and lenses, contributing to the mathematical understanding of pattern formation in triblock copolymers.
Abstract
We study a two-dimensional inhibitory ternary system characterized by a free energy functional which combines an interface short-range interaction energy promoting micro-domain growth with a Coulomb-type long-range interaction energy which prevents micro-domains from unlimited spreading. Here we consider a scenario in which two species are dominant and one species is vanishingly small. In this scenario two energy levels are distinguished: the zeroth-order energy encodes information on the optimal arrangement of the dominant constituents, while the first-order energy gives the shape of the vanishing constituent. This first-order energy also shows that, for any optimal configuration, the vanishing phase must lie on the boundary between the two dominant constituents and form lens clusters also known as vesica piscis.