Spheres and Prolate and Oblate Ellipsoids from an Analytical Solution of Spontaneous Curvature Fluid Membrane Model

TL;DR

This paper analytically explores solutions to the Helfrich spontaneous curvature membrane model, revealing a variety of shapes including spheres, ellipsoids, and biconcave forms, with the biconcave shape having minimal curvature energy.

Contribution

It provides an analytical classification of membrane shapes derived from the Helfrich model, including new insights into the biconcave shape and other geometries.

Findings

Identifies a family of shapes including spheres, ellipsoids, and biconcave forms.

Shows the circular biconcave shape has the lowest local curvature energy among closed shapes.

Describes self-intersecting shapes like inverted biconcave and nodoidlike cylinders.

Abstract

An analytic solution for Helfrich spontaneous curvature membrane model (H. Naito, M.Okuda and Ou-Yang Zhong-Can, Phys. Rev. E {\bf 48}, 2304 (1993); {\bf 54}, 2816 (1996)), which has a conspicuous feature of representing the circular biconcave shape, is studied. Results show that the solution in fact describes a family of shapes, which can be classified as: i) the flat plane (trivial case), ii) the sphere, iii) the prolate ellipsoid, iv) the capped cylinder, v) the oblate ellipsoid, vi) the circular biconcave shape, vii) the self-intersecting inverted circular biconcave shape, and viii) the self-intersecting nodoidlike cylinder. Among the closed shapes (ii)-(vii), a circular biconcave shape is the one with the minimum of local curvature energy.