Stresses in lipid membranes

TL;DR

This paper derives the stress tensor and conservation laws for lipid membranes modeled by the Helfrich Hamiltonian, providing insights into their equilibrium shapes and fluid properties using Noether's theorem.

Contribution

It introduces a systematic derivation of the stress and torque tensors for lipid membranes and connects conservation laws to physical forces and torques.

Findings

Derived the shape equation as a normal projection of stress conservation.

Identified the stress tensor and torque tensor for lipid membranes.

Obtained the first integral of the shape equation for axially symmetric membranes.

Abstract

The stresses in a closed lipid membrane described by the Helfrich hamiltonian, quadratic in the extrinsic curvature, are identified using Noether's theorem. Three equations describe the conservation of the stress tensor: the normal projection is identified as the shape equation describing equilibrium configurations; the tangential projections are consistency conditions on the stresses which capture the fluid character of such membranes. The corresponding torque tensor is also identified. The use of the stress tensor as a basis for perturbation theory is discussed. The conservation laws are cast in terms of the forces and torques on closed curves. As an application, the first integral of the shape equation for axially symmetric configurations is derived by examining the forces which are balanced along circles of constant latitude.