The field theory of symmetrical layered electrolytic systems and the thermal Casimireffect

Abstract

We present a general extension of a field-theoretic approach developed in earlierpapers to the calculation of the free energy of symmetrically layered electrolyticsystems which is based on the sine–Gordon field theory for the Coulomb gas. Themethod is to construct the partition function in terms of the Feynman evolutionkernel in the Euclidean time variable associated with the coordinate normal to thesurfaces defining the layered structure. The theory is applicable to cylindricalsystems and its development is motivated by the possibility that a static van derWaals or thermal Casimir force could provide an attractive force stabilizing adielectric tube formed from a lipid bilayer, an example of which is provided by thet-tubules occurring in certain muscle cells. In this context, we apply the theoryto the calculation of the thermal Casimir effect for a dielectric tube of radiusR andthickness δ formed from such a membrane in water. In a grand canonical approachwe find that the leading contribution to the Casimir energy behaves like−kBTLκC/R which gives rise to an attractive force which tends to contract the tube radius. We find thatκC∼0.3 for the case of typical lipid membrane t-tubules. We conclude that except in the case of avery soft membrane this force is insufficient to stabilize such tubes against thebending stress which tends to increase the radius. We briefly discuss the role ofthe lipid membrane reservoir implicit in the approach and whether its nature inbiological systems may possibly lead to a stabilizing mechanism for such lipid tubes.