Universal amplitudes of the Casimir-like interactions between four types of rods in fluid membranes

TL;DR

This paper analytically calculates the universal Casimir-like fluctuation-induced interactions between four types of rods on fluid membranes, revealing attractive potentials, conditions for repulsion, and the potential for rods to bend and contact due to these forces.

Contribution

It introduces a classification of rods based on membrane constraints and derives universal Casimir amplitudes for their interactions, including conditions for attraction and repulsion.

Findings

All interactions are attractive for stiff constraints and proportional to L/d.

Two types of rods are equivalent, leading to six universal amplitudes.

At d/L≈0.2, the attraction reaches thermal energy kT.

Abstract

The fluctuation-induced, Casimir-like interaction between two parallel rods of length L adsorbed on a fluid membrane is calculated analytically at short separations d<<L. The rods are modeled as constraints imposed on the membrane curvature along a straight line. This allows to define four types of rods, according to whether the membrane can twist along the rod and/or curve across it. For stiff constraints, all the interaction potentials between the different types of rods are attractive and proportional to L/d. Two of the four types of rods are then equivalent, which yields six universal Casimir amplitudes. Repulsion can occur between different rods for soft constraints. Numerical results obtained for all ranges of d/L show that the attraction potential reaches kT for d/L\simeq0.2. At separations smaller than d_c \approx L(L/l_p)^(1/3), where l_p is the rod persistence length, two rods with fixed ends will bend toward each other and finally come into contact because of the Casimir interaction.