Cylindrical equilibrium shapes of fluid membranes

Abstract

Within the framework of the well-known curvature models, a fluid lipid bilayer membrane is regarded as a surface embedded in the three-dimensional Euclidean space whose equilibrium shapes are described in terms of its mean and Gaussian curvatures by the so-called membrane shape equation. In the present paper, all solutions to this equation determining cylindrical membrane shapes are found and presented, together with the expressions for the corresponding position vectors, in explicit analytic form. The necessary and sufficient conditions for such a surface to be closed are derived and several sufficient conditions for its directrix to be simple or self-intersecting are given.

Minimal surfaces bounded by elastic lines

“…Beyond a critical value of the surface tension (relative to the bending stiffness, as discussed earlier), the planar disc buckled into a twofold mode consistent with our linear stability analysis. The geometry of this shape has been rigorously described (Flaherty et al 1972;Arreaga et al 2002;Vassilev et al 2008), and is related to the problem of the collapse of a infinitely long cylindrical pipe under a uniform pressure difference. For kL 3 /a ≈ 643, the planar elliptical shape transitions to a twisted non-planar saddle-like shape, as experimentally observed.…”

Comment on “Shape transition of unstrained flattest single-walled carbon nanotubes under pressure” [J. Appl. Phys. 115, 044512 (2014)]

“…Let ðxðsÞ; yðsÞ; zÞ be the position vector of a tube cross section profile curve C in a certain Cartesian frame (x, y, z), such that the z-axis is coincident with the tube axis. It is established by the authors of this note [7][8][9] that the differential structure of Eq. (2) allows representation of the parametric equations of the curve C in the form…”

“…Apparently, Mu et al 1 had not noticed the available exact analytic results concerning this problem, [5][6][7][8][9] which are well recognized in the current literature, [10][11][12] and therefore, they performed the foregoing task deriving approximate expressions for the curvature and parametric equations of the tube cross section profile curve C.…”

Geometry of the anisotropic minimal surfaces

“…Here ∆ S is the Laplace-Beltrami operator on the surface S, k c andk -denote the bending and the Gaussian rigidities of the membrane, σ -the tensile stress of the membrane, Ih -the spontaneous curvature of the bilayer and p -the osmotic pressure difference between the external and internal part. Its explicit solutions are discussed in [1,4,5,6,9,10,14,16] and in [15] the reader will find a whole chapter devoted to this subject. From mathematical point of view the main difficulty in solving (2) is that it represents a nonlinear fourth order partial differential equation for the position vector x running on the surface S. A fortunate circumstance is that this differential equation can be rewritten in the form of a system of four differential equations of second order.…”