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We present a general extension of a field-theoretic approach developed inearlier papers to the calculation of the free energy of symmetrically layeredelectrolytic systems which is based on the Sine-Gordon field theory for theCoulomb gas. The method is to construct the partition function in terms of theFeynman evolution kernel in the Euclidean time variable associated with thecoordinate normal to the surfaces defining the layered structure. The theory isapplicable to cylindrical systems and its development is motivated by thepossibility that a static van der Waals or thermal Casimir force could providean attractive force stabilising a dielectric tube formed from a lipid bilayer,an example of which are t-tubules occurring in certain muscle cells. In thiscontext, we apply the theory to the calculation of the thermal Casimir effectfor a dielectric tube of radius $R$ and thickness $\delta$ formed from such amembrane in water. In a grand canonical approach we find that the leadingcontribution to the Casimir energy behaves like $-k_BTL\kappa_C/R$ which givesrise to an attractive force which tends to contract the tube radius. We findthat $\kappa_C \sim 0.3$ for the case of typical lipid membrane t-tubules. Weconclude that except in the case of a very soft membrane this force isinsufficient to stabilise such tubes against the bending stress which tend toincrease the radius. We briefly discuss the role of lipid membrane reservoirimplicit in the approach and whether its nature in biological systems maypossibly lead to a stabilising mechanism for such lipid tubes.Comment: 28 pages, 2 figures, LaTe

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