Fluctuation-Induced Interactions between Rods on a Membrane

Abstract

We consider the interaction between two rods embedded in a fluctuating surface. The modification of fluctuations by the rods leads to an attractive long-range interaction between them. We consider fluctuations governed by either surface tension (films) or bending rigidity (membranes). In both cases the interaction falls off with the separation of the rods as $1/R^4$. The orientational part of the interaction is proportional to $\cos^2\left[ θ_1+θ_2 \right]$ in the former case, and to $\cos^2\left[ 2\left(θ_1+θ_2\right) \right]$ in the latter, where $θ_1$ and $θ_2$ are angles between the rods and the line joining them. These interactions are somewhat reminiscent of dipolar forces and will tend to align collections of such rods into chains.