Undulated Bilayer Interfaces in the Planar Functionalized Cahn-Hilliard Equation

TL;DR

This paper models undulated bilayer structures observed in diblock copolymer melts using a functionalized Cahn-Hilliard equation with localized inhomogeneities, revealing long-range oscillatory decay of perturbations.

Contribution

It introduces a mathematical framework to mimic defects in bilayers and analyzes the decay scale of undulations in the functionalized Cahn-Hilliard model.

Findings

Undulated bilayer solutions exhibit oscillatory width perturbations.

Decay length of perturbations scales as (;\varepsilon^{-1/2};).

Perturbations decay on a longer scale than the bilayer width.

Abstract

Experiments with diblock co-polymer melts display undulated bilayers that emanate from defects such as triple junctions and endcaps, \cite{batesjain_2004}. Undulated bilayers are characterized by oscillatory perturbations of the bilayer width, which decay on a spatial length scale that is long compared to the bilayer width. We mimic defects within the functionalized Cahn-Hillard free energy by introducing spatially localized inhomogeneities within its parameters. For length parameter $\varepsilon\ll1$, we show that this induces undulated bilayer solutions whose width perturbations decay on an $O\!\left(\varepsilon^{-1/2}\right)$ inner length scale that is long in comparison to the $O(1)$ scale that characterizes the bilayer width.