The field theory of symmetrical layered electrolytic systems and the thermal Casimir effect

TL;DR

This paper extends a field-theoretic approach to calculate the thermal Casimir effect in layered electrolytic systems, specifically applied to lipid membrane tubes, revealing forces that influence their stability.

Contribution

It develops a generalized Sine-Gordon field theory method for layered electrolytic systems and applies it to biological lipid tubes to analyze their stability under Casimir forces.

Findings

The Casimir energy behaves like -k_BTLκ_C/R, indicating an attractive force.

The coefficient κ_C is approximately 0.3 for lipid membrane tubes.

The Casimir force alone is insufficient to stabilize the tubes against bending stress.

Abstract

We present a general extension of a field-theoretic approach developed in earlier papers to the calculation of the free energy of symmetrically layered electrolytic systems which is based on the Sine-Gordon field theory for the Coulomb gas. The method is to construct the partition function in terms of the Feynman evolution kernel in the Euclidean time variable associated with the coordinate normal to the surfaces defining the layered structure. The theory is applicable to cylindrical systems and its development is motivated by the possibility that a static van der Waals or thermal Casimir force could provide an attractive force stabilising a dielectric tube formed from a lipid bilayer, an example of which are t-tubules occurring in certain muscle cells. In this context, we apply the theory to the calculation of the thermal Casimir effect for a dielectric tube of radius $R$ and thickness $\delta$ formed from such a membrane in water. In a grand canonical approach we find that the leading contribution to the Casimir energy behaves like $-k_BTL\kappa_C/R$ which gives rise to an attractive force which tends to contract the tube radius. We find that $\kappa_C \sim 0.3$ for the case of typical lipid membrane t-tubules. We conclude that except in the case of a very soft membrane this force is insufficient to stabilise such tubes against the bending stress which tend to increase the radius. We briefly discuss the role of lipid membrane reservoir implicit in the approach and whether its nature in biological systems may possibly lead to a stabilising mechanism for such lipid tubes.