Effect of Thermal Undulations on the Bending Elasticity and Spontaneous Curvature of Fluid Membranes

Membranes are an important subject of study in physical chemistry and biology. They can be considered as material surfaces with a surface energy depending on the curvature tensor. Usually, mathematical models developed in the literature consider the dependence of surface energy only on mean curvature with an added linear term for Gauss curvature. Therefore, for closed surfaces the Gauss curvature term can be eliminated because of the Gauss-Bonnet theorem. In [18], the depende...

The Helfrich energy is commonly used to model the elastic bending energy of lipid bilayers in membrane mechanics. The governing differential equations for certain geometric characteristics of the shape of the membrane can be obtained by applying variational methods (minimization principles) to the Helfrich energy functional and are well-studied in the axisymmetric framework. However, the Helfrich energy functional and the resulting differential equations involve a number of p...

We consider a dilute solution of infinitely rigid rods near a curved, perfectly repulsive surface and study the contribution of the rod depletion layer to the bending elastic constants of membranes. We find that a spontaneous curvature state can be induced by exposure of BOTH sides of the membrane to a rod solution. A similar result applies for rigid disks with a diameter equal to the rod's length. We also study the confinement of rods in spherical and cylindrical repulsive s...

Recent theoretical advances in elasticity of membranes following Helfrich's famous spontaneous curvature model are summarized in this review. The governing equations describing equilibrium configurations of lipid vesicles, lipid membranes with free edges, and chiral lipid membranes are presented. Several analytic solutions to these equations and their corresponding configurations are demonstrated.

We use molecular dynamics to study the vibrations of a thermally fluctuating two-dimensional elastic membrane clamped at both ends. We directly extract the eigenmodes from resonant peaks in the frequency domain of the time-dependent height and measure the dependence of the corresponding eigenfrequencies on the microscopic bending rigidity of the membrane, taking care also of the subtle role of thermal contraction in generating a tension when the projected area is fixed. At fi...

We consider the propagation of flexural waves across a nearly flat, thin membrane, whose stress-free state is curved. The stress-free configuration is specified by a quenched height field, whose Fourier components are drawn from a Gaussian distribution with power law variance. Gaussian curvature couples the in-plane stretching to out-of-plane bending. Integrating out the faster stretching modes yields a wave equation for undulations in the presence of an effective random pote...

The quantitative understanding of membranes is still rooted in work performed in the 1970s by Helfrich and others, concerning amphiphilic bilayers. However, most biological membranes contain a wide variety of nonamphiphilic molecules too. Drawing analogy with the physics of nematic/non-nematic mixtures, we present a dynamical (out of equilibrium) description of such multicomponent membranes. The approach combines nematohydrodynamics in the linear regime and a proper use of (d...

The Gaussian (saddle splay) rigidity of fluid membranes controls their equilibrium topology but is notoriously difficult to measure. In lipid mixtures, typical of living cells, linear interfaces separate liquid ordered (LO) from liquid disordered (LD) bilayer phases at subcritical temperatures. Here we consider such membranes supported by curved supports that thereby control the membrane curvatures. We show how spectral analysis of the fluctuations of the LO-LD interface prov...

In multicomponent membranes, internal scalar fields may couple to membrane curvature, thus renormalizing the membrane elastic constants and destabilizing the flat membranes. Here, a general elasticity theory of membranes is considered that employs a quartic curvature expansion. The shape of the membrane and its deformation energy near a long rod-like inclusion are studied analytically. In the limit where one can neglect the end-effects, the nonlinear response of the membrane ...

Comparison of a few simple models of fluid and solid membranes illustrates how shear stresses can arise from a bending energy through a coupling between curvature and surface stresses, a feature incidental to the fluid or solid nature of the material. In particular, it is shown how a fluid-like Helfrich bending energy contributes shear stresses, while a related solid-like energy, the correct continuum limit of a Seung-Nelson discrete bending term, produces a stress tensor wit...