Spontaneous Curvature Induced Stretching-Bending Mode Coupling in Membranes

E I Kats

doi:10.3390/physics3020025

Abstract

In this paper, a simple example to illustrate what is basically known from the Gauss’ times interplay between geometry and mechanics in thin shells is presented. Specifically, the eigen-mode spectrum in spontaneously curved (i.e., up-down asymmetric) extensible polymerized or elastic membranes is studied. It is found that in the spontaneously curved crystalline membrane, the flexural mode is coupled to the acoustic longitudinal mode, even in the harmonic approximation. If the coupling (proportional to the membrane spontaneous curvature) is strong enough, the coupled modes dispersions acquire the imaginary part, i.e., effective damping. The damping is not related to the entropy production (dissipation); it comes from the redistribution of the energy between the modes. The curvature-induced mode coupling makes the flexural mode more rigid, and the acoustic mode becomes softer. As it concerns the transverse acoustical mode, it remains uncoupled in the harmonic approximation, keeping its standard dispersion law. We anticipate that the basic ideas inspiring this study can be applied to a large variety of interesting systems, ranging from still fashionable graphene films, both in the freely suspended and on a substrate states, to the not yet fully understood lipid membranes in the so-called gel and rippled phases.

Highlights

It is my pleasure and honor to present my work in the Special Issue of journal of Physics, dedicated to the 70th birthday of Prof
From Equation (6), one can see that the transverse acoustic mode, which is decoupled from flexural mode in the harmonic approximation, has the standard dispersion law: ωt2 = μq2
The damping is not related to the entropy production, since there is no any dissipative term in the action (3)

Summary

Introduction

It is my pleasure and honor to present my work in the Special Issue of journal of Physics, dedicated to the 70th birthday of Prof. The paper I am presenting ( from the realm of soft matter physics) is a small token of my gratitude and respect to Misha Tribelsky. In many realistic and experimentally relevant situations, the “up-down” symmetry is spontaneously broken [3,5] It can be nonuniformly broken over the membrane surface, e.g., due to asymmetric adsorption of different molecular species. In such a situation, the up-down non-symmetric free energy in the harmonic approximation can be written in terms of the scalar out-of-plane displacement f and the 2D vector u of the in-plane displacements: [2,3,5].

Basic Derivation

Eigen-Modes in Spontaneously Curved Membranes

Findings

Outlook and Conclusions