Scaling analysis of narrow necks in curvature models of fluid lipid-bilayer vesicles

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Abstract
Under appropriate conditions fluid lipid-bilayer vesicles in aqueous solution take the form of two (or more) compact shapes connected by a narrow neck (or necks). We study the limit (termed ‘‘vesiculation’’) in which the neck radius a approaches zero. On the basis of elastic equations, derived originally by Deuling and Helfrich [J. Phys. (Paris) 37, 1335 (1976)] for a bending-energy model (the spontaneous-curvature model), we show analytically that, at vesiculation, the local curvatures of the two regions joined by the neck satisfy a simple, universal ‘‘kissing’’ (osculation) condition. Furthermore, for points near but not at the vesiculation limit, a is small but nonzero and there is characteristic scaling behavior. For example, in the surface tension (σ) and pressure (p) variables, the vesiculation boundary is a line in the (σ,p) plane, and the quantity a lna scales linearly with the distance (Δσ,Δp) from the boundary. These relations have been observed numerically, but no analytic discussion has previously appeared in the literature. Results for the spontaneous-curvature model generalize easily to other (more physical) bending-energy models.