Existence, Bifurcation, and Geometric Evolution of Quasi-Bilayers in the Multicomponent Functionalized Cahn-Hilliard Equation

TL;DR

This paper models multicomponent bilayer structures using a generalized Cahn-Hilliard framework, analyzing their stability, geometric evolution, and bifurcation behavior relevant to biological and synthetic membranes.

Contribution

It introduces the multicomponent functionalized Cahn-Hilliard model and derives a sharp interface energy capturing curvature effects and bifurcation mechanisms.

Findings

Derived a Canham-Helfrich energy with curvature from lipid aspect ratio.

Constructed asymmetric bilayer profiles using a billiard limit potential.

Identified layer-by-layer pearling as the main bifurcation mechanism.

Abstract

Multicomponent bilayer structures arise as the ubiquitous plasma membrane in cellular biology and as blends of amphiphilic copolymers used in electrolyte membranes, drug delivery, and emulsion stabilization within the context of synthetic chemistry. We develop the multicomponent functionalized Cahn-Hilliard (mFCH) free energy as a model which allows competition between bilayers with distinct composition and between bilayers and higher codimensional structures, such as co-dimension two filaments and co-dimension three micelles. We investigate the stability and slow geometric evolution of multicomponent bilayer interfaces within the context of gradient flows of the mFCH, addressing the impact of aspect ratio of the lipid/copolymer unit on the intrinsic curvature and the codimensional bifurcation. In particular we derive a Canham-Helfrich sharp interface energy whose intrinsic curvature arises through a Melnikov parameter associated to lipid aspect ratio. We construct asymmetric homoclinic bilayer profiles via a billiard limit potential and show that the dominant co-dimensional bifurcation mechanism is via the layer-by-layer pearling observed experimentally.