Undulated bilayer interfaces in the planar functionalized Cahn-Hilliard equation

Keith Promislow; Qiliang Wu

doi:10.3934/dcdss.2022035

Article Contents

2022, Volume 15, Issue 9: 2633-2656. Doi: 10.3934/dcdss.2022035

This issue Previous Article Large-amplitude modulation of periodic traveling waves Next Article The amplitude system for a Simultaneous short-wave Turing and long-wave Hopf instability

Undulated bilayer interfaces in the planar functionalized Cahn-Hilliard equation

Keith Promislow^1, and
Qiliang Wu^2, ,

1.
Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824, USA
2.
Department of Mathematics, Ohio University, Morton Hall 321, 1 Ohio University, Athens OH 45701, USA

^* Corresponding author: Qiliang Wu
^* Corresponding author: Qiliang Wu

Received: April 2021

Revised: January 2022

Early access: February 28, 2022

Published online: February 28, 2022

K. Promislow acknowledges support from NSF-DMS grant 1813203. Q. Wu acknowledges support from NSF-DMS grant 1815079.

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Abstract

Experiments with diblock co-polymer melts display undulated bilayers that emanate from defects such as triple junctions and endcaps, [8]. 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.

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Mathematics Subject Classification: Primary: 35B36, 35Q92, 37L10.

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Figure 1. Cryo-TEM images of blends of amphiphilic diblock polymer in water. A mixture of diblocks with hydrophobic/hydrophillic chain lengths of 170/110 and 46/58, respectively. This mixture produces visible undulations behind the endcaps, see the red boxes outlining the structures labeled '2' and '3'. Reprinted with permission from Figures 7 & 8 of [8]

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Figure 2. The operator $ \mathbb{L}( \varepsilon) $ admits two purely imaginary eigenvalues (blue crosses) with algebraic multiplicity 2 when $ \varepsilon = 0. $ Given $ \varepsilon>0 $, for $ \alpha_0>0 $ they remain purely imaginary while for $ \alpha_0<0 $ they split into four geometrically simple modes (red crosses)

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