# Membrane buckling and the determination of Gaussian curvature modulus

## TL;DR

This paper investigates membrane buckling under stress to develop a new method for measuring the Gaussian curvature modulus, a key parameter in membrane shape transformations, which has been difficult to determine experimentally.

## Contribution

It introduces a novel approach using buckling behavior of membranes under stress to accurately determine the Gaussian curvature modulus.

## Key findings

- Rectangular membrane buckling occurs continuously.
- Circular membrane buckling can be discontinuous depending on boundary conditions.
- Stress-strain analysis of buckled circular membranes helps measure Gaussian curvature modulus.

## Abstract

Biological membranes are able to exhibit various morphology due to the fluidity of the lipid molecules within the monolayers. The shape transformation of membranes has been well described by the classical Helfrich theory, which consists only a few phenomenological parameters, including the mean and the Gaussian curvature modulus. Though various methods have been proposed to measure the mean curvature modulus, determination of the Gaussian curvature modulus remains difficult both in experiments and in simulation. In this paper we study the buckling process of a rectangular membrane and a circular membrane subject to compressive stresses and under different boundary conditions. We find that the buckling of a rectangular membrane takes place continuously, while the buckling of a circular membrane can be discontinous depending on the boundary conditions. Furthermore, our results show that the stress-strain relationship of a buckled circular membrane can be used to effectively determine the Gaussian curvature modulus.