Instability and Periodic Deformation in Bilayer Membranes Induced by Freezing

The instability and periodic deformation of bilayer membranes during freezing processes are studied as a function of the difference of the shape energy between the high and the low temperature membrane states. It is shown that there exists a threshold stability condition, bellow which a planar configuration will be deformed. Among the deformed shapes, the periodic curved square textures are shown being one kind of the solutions of the associated shape equation. In consistency with recent expe rimental observations, the optimal ratio of period and amplitude for such a texture is found to be approximately equal to (2)^{1/2}\pi.Comment: 8 pages in Latex form, 1 Postscript figure. To be appear in Mod. Phys. Lett. B. 199

Diffusion of a Deformable Body in a random Flow

We consider a deformable body immersed in an incompressible liquid that is randomly stirred. Sticking to physical situations in which the body departs only slightly from its spherical shape, we calculate the diffusion constant of the body. We give explicitly the dependence of the diffusion constant on the velocity correlations in the liquid and on the size of the body. We emphasize the particular case in which the random velocity field follows from thermal agitation.Comment: 9 pages, 2 figures, late

Parameterization invariance and shape equations of elastic axisymmetric vesicles

The issue of different parameterizations of the axisymmetric vesicle shape addressed by Hu Jian-Guo and Ou-Yang Zhong-Can [ Phys.Rev. E {\bf 47} (1993) 461 ] is reassesed, especially as it transpires through the corresponding Euler - Lagrange equations of the associated elastic energy functional. It is argued that for regular, smooth contours of vesicles with spherical topology, different parameterizations of the surface are equivalent and that the corresponding Euler - Lagrange equations are in essence the same. If, however, one allows for discontinuous (higher) derivatives of the contour line at the pole, the differently parameterized Euler - Lagrange equations cease to be equivalent and describe different physical problems. It nevertheless appears to be true that the elastic energy corresponding to smooth contours remains a global minimum.Comment: 10 pages, latex, one figure include

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

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.Comment: 11 pages, 11 figures. Phys. Rev. E (to appear in Sept. 1999

Gravity-Induced Shape Transformations of Vesicles

We theoretically study the behavior of vesicles filled with a liquid of higher density than the surrounding medium, a technique frequently used in experiments. In the presence of gravity, these vesicles sink to the bottom of the container, and eventually adhere even on non - attractive substrates. The strong size-dependence of the gravitational energy makes large parts of the phase diagram accessible to experiments even for small density differences. For relatively large volume, non-axisymmetric bound shapes are explicitly calculated and shown to be stable. Osmotic deflation of such a vesicle leads back to axisymmetric shapes, and, finally, to a collapsed state of the vesicle.Comment: 11 pages, RevTeX, 3 Postscript figures uuencode

Willmore minimizers with prescribed isoperimetric ratio

Motivated by a simple model for elastic cell membranes, we minimize the Willmore functional among two-dimensional spheres embedded in R^3 with prescribed isoperimetric ratio

Area-Constrained Planar Elastica

We determine the equilibria of a rigid loop in the plane, subject to the constraints of fixed length and fixed enclosed area. Rigidity is characterized by an energy functional quadratic in the curvature of the loop. We find that the area constraint gives rise to equilibria with remarkable geometrical properties: not only can the Euler-Lagrange equation be integrated to provide a quadrature for the curvature but, in addition, the embedding itself can be expressed as a local function of the curvature. The configuration space is shown to be essentially one-dimensional, with surprisingly rich structure. Distinct branches of integer-indexed equilibria exhibit self-intersections and bifurcations -- a gallery of plots is provided to highlight these findings. Perturbations connecting equilibria are shown to satisfy a first order ODE which is readily solved. We also obtain analytical expressions for the energy as a function of the area in some limiting regimes.Comment: 23 pages, several figures. Version 2: New title. Changes in the introduction, addition of a new section with conclusions. Figure 14 corrected and one reference added. Version to appear in PR

Well-posedness of Hydrodynamics on the Moving Elastic Surface

The dynamics of a membrane is a coupled system comprising a moving elastic surface and an incompressible membrane fluid. We will consider a reduced elastic surface model, which involves the evolution equations of the moving surface, the dynamic equations of the two-dimensional fluid, and the incompressible equation, all of which operate within a curved geometry. In this paper, we prove the local existence and uniqueness of the solution to the reduced elastic surface model by reformulating the model into a new system in the isothermal coordinates. One major difficulty is that of constructing an appropriate iterative scheme such that the limit system is consistent with the original system.Comment: The introduction is rewritte

Phase ordering and shape deformation of two-phase membranes

Within a coupled-field Ginzburg-Landau model we study analytically phase separation and accompanying shape deformation on a two-phase elastic membrane in simple geometries such as cylinders, spheres and tori. Using an exact periodic domain wall solution we solve for the shape and phase ordering field, and estimate the degree of deformation of the membrane. The results are pertinent to a preferential phase separation in regions of differing curvature on a variety of vesicles.Comment: 4 pages, submitted to PR

Impermeability effects in three-dimensional vesicles

We analyse the effects that the impermeability constraint induces on the equilibrium shapes of a three-dimensional vesicle hosting a rigid inclusion. A given alteration of the inclusion and/or vesicle parameters leads to shape modifications of different orders of magnitude, when applied to permeable or impermeable vesicles. Moreover, the enclosed-volume constraint wrecks the uniqueness of stationary equilibrium shapes, and gives rise to pear-shaped or stomatocyte-like vesicles.Comment: 16 pages, 7 figure