Balancing torques in membrane-mediated interactions: Exact results and numerical illustrations

TL;DR

This paper develops an exact and numerical framework for analyzing membrane-mediated interactions, focusing on torque balance and shape constraints, with explicit solutions for symmetric two-particle configurations including parallel cylinders.

Contribution

It introduces a divergence-free tensor framework for membrane torques, providing exact solutions for symmetric two-particle systems and illustrating complex nonlinear behaviors.

Findings

Exact solutions for two parallel cylinders on membranes.

Torque balance constrains membrane shape significantly.

Nonlinear behaviors emerge in symmetric particle interactions.

Abstract

Torques on interfaces can be described by a divergence-free tensor which is fully encoded in the geometry. This tensor consists of two terms, one originating in the couple of the stress, the other capturing an intrinsic contribution due to curvature. In analogy to the description of forces in terms of a stress tensor, the torque on a particle can be expressed as a line integral along any contour surrounding the particle. Interactions between particles mediated by a fluid membrane are studied within this framework. In particular, torque balance places a strong constraint on the shape of the membrane. Symmetric two-particle configurations admit simple analytical expressions which are valid in the fully nonlinear regime; in particular, the problem may be solved exactly in the case of two membrane-bound parallel cylinders. This apparently simple system provides some flavor of the remarkably subtle nonlinear behavior associated with membrane-mediated interactions.