Codimension One Minimizers of Highly Amphiphilic Mixtures

TL;DR

This paper introduces a modified FCH functional to model highly amphiphilic systems, analyzing the behavior of minimizers near codimension one interfaces and establishing convergence and energy bounds for sequences with bounded energy.

Contribution

It develops a new functional form for modeling highly amphiphilic mixtures and provides rigorous analysis of minimizers and their convergence properties near interfaces.

Findings

Bounded energy sequences converge to bilayer profiles.

Sequences with limited tangential variation satisfy lim inf inequality.

Construction of sequences that violate lim inf inequality depending on parameters.

Abstract

We present a modified form of the Functionalized Cahn Hilliard (FCH) functional which models highly amphiphilic systems in solvent. A molecule is highly amphiphilic if the energy of a molecule isolated within the bulk solvent molecule is prohibitively high. For such systems once the amphiphilic molecules assemble into a structure it is very rare for a molecule to exchange back into the bulk. The highly amphiphilic FCH functional has a well with limited smoothness and admits compactly supported critical points. In the limit of molecular length epsilon approaches 0, we consider sequences with bounded energy whose support resides within an epsilon-neighborhood of a fixed codimension one interface. We show that the FCH energy is uniformly bounded below, independent of epsilon >0, and identify assumptions on tangential variation of sequences that guarantee the existence of subsequences that converge to a weak solution of a rescaled bilayer profile equation, and show that sequences with limited tangential variation enjoy a lim inf inequality. For fixed codimension one interfaces we construct bounded energy sequences which converge to the bilayer profile and others with larger tangential variation which do not converge to the bilayer profile but whose limiting energy can violate the lim inf inequality, depending upon the energy parameters.