2019/12/19 by Stan Alama, Lia Bronsard, Alama, Stan +5
Computer Science · Materials Science · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Block Copolymer Self-Assembly #FOS: Mathematics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1912.08971
openalex publication_date 2019/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a two-dimensional ternary inhibitory system derived as a sharp-interface limit of the Nakazawa-Ohta density functional theory of triblock copolymers. This free energy functional combines an interface energy favoring micro-domain growth with a Coulomb-type long range interaction energy which prevents micro-domains from unlimited spreading. Here we consider a limit in which two species are vanishingly small, but interactions are correspondingly large to maintain a nontrivial limit. In this limit two energy levels are distinguished: the highest order limit encodes information on the geometry of local structures as a two-component isoperimetric problem, while the second level describes the spatial distribution of components in global minimizers. We provide a sharp rigorous derivation of the asymptotic limit, both for minimizers and in the context of Gamma-convergence. Geometrical descriptions of limit configurations are derived; among other results, we will show that, quite unexpectedly, coexistence of single and double bubbles can arise. The main difficulties are hidden in the optimal solution of two-component isoperimetric problem: compared to binary systems, not only it lacks an explicit formula, but, more crucially, it can be neither concave nor convex on parts of its domain.