Laplace pressure as a surface stress in fluid vesicles

TL;DR

This paper demonstrates that Laplace pressure in fluid vesicles can be represented as a divergence of a surface stress, leading to a conserved effective stress that simplifies understanding of vesicle equilibrium shapes.

Contribution

It introduces a novel formulation expressing Laplace pressure as a divergence of a position-dependent surface stress, revealing new geometrical insights into vesicle equilibrium shapes.

Findings

Effective surface stress is conserved in equilibrium.

In cylindrical geometry, the cross-section is a planar Euler elastic curve.

Only spherical geometries have vanishing effective stress in higher dimensions.

Abstract

Consider a surface, enclosing a fixed volume, described by a free-energy depending only on the local geometry; for example, the Canham-Helfrich energy quadratic in the mean curvature describes a fluid membrane. The stress at any point on the surface is determined completely by geometry. In equilibrium, its divergence is proportional to the Laplace pressure, normal to the surface, maintaining the constraint on the volume. It is shown that this source itself can be expressed as the divergence of a position-dependent surface stress. As a consequence, the equilibrium can be described in terms of a conserved `effective' surface stress. Various non-trivial geometrical consequences of this identification are explored. In a cylindrical geometry, the cross-section can be viewed as a closed planar Euler elastic curve. With respect to an appropriate centre the effective stress itself vanishes; this provides a remarkably simple relationship between the curvature and the position along the loop. In two or higher dimensions, it is shown that the only geometry consistent with the vanishing of the effective stress is spherical. It is argued that the appropriate generalization of the loop result will involve `null' stresses.