Bending Elasticity of Fluid Membranes

Bending elasticity is a long-standing concept of continuum mechanics and has been used mostly to deal with solid rods and plates. More recently, it has been applied to fluid membranes, especially the lipid bilayers of giant vesicles, to understand their equilibrium shapes and shape fluctuations. For continuum theory to be applicable, the membranes should be reasonably smooth or, in other words, not fluctuate too much. 

Strictly speaking, fluid membranes such as amphiphilic monolayers and bilayers are continua only in the two lateral dimensions, whereas in the normal direction their extension is molecular. The simplest model of them is a mathematical surface without any thickness. However, thin as it is, the third dimension of the membranes determines their elastic properties of stretching and bending. 

The bending elasticity of fluid membranes is closely related to the director field elasticity of liquid crystals. Of the three elastic deformations in nematics, which are splay, bend, and twist, only splay remains as it does in the case of smectics. In fact, a membrane is like an isolated smectic layer and this is why membrane curvature is sometimes expressed in terms of splay and saddle splay. 

The following is a brief introduction to fluid membrane bending elasticity. The emphasis is on some basic ideas and not on particular models or applications. The theory of vesicle shapes is treated in Chapter 7. A subject to be included in the following is non-Hookean bending elasticity, i.e. energy terms of higher than 

Ciant Vefiick s Edited by P. L. Luisi and P. Waldc 2000 John Wiley Sons Ltd. 

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Helfrich, Wolfgang, Bending Elasticity of Fluid Membranes, 6, 51 see also Klosgen, Beate, 6, 243 see also Thimmel, Johannes, 6, 253. 

The contributions of the normal bending elasticity prevent the formation of permanent highly curved pieces of fluid membranes. The typical length scale of cubic structures for example is around 250 A. Also, no stable unilamellar vesicles are found below a radius of about 20 nm [29] because the high curvature requires a lot of energy and results in high membrane tension and an affinity for fusion to end up with a bigger, and less curved, new vesicle. 

The theoretical approach to describe all these effects was formulated long ago [7], at the time purely on the basis of Hookean deformations that eontribute to the bending elastic energy density of fluid membranes ... 

Liposome is a closed vesicle with the lipid-bilayer membrane, which takes a variety of shapes such as biconcave discocytes, cup-shaped stomatocytes and prolate and oblate ellipsoids, depending on the temperature and the osmotic conditions [1]. Among various attempts made to explain these shapes [2,3], Helfrich has discussed the bending elastic energy [4] of fluid membranes formed by lipids as... 

A wide variety of shape transformations of fluid membranes has been extensively studied theoretically in the past two decades using a bending elasticity model proposed by Canham and Helfrich [1]. The model has succeeded in explaining equilibrium shapes of the erythrocyte. However, much attention has recently been paid to shape deformations induced by internal degrees of freedom of membranes. For example, the bending elasticity model cannot explain the deformation from the biconcave shape of the erythrocyte to the crenated one (echinocytosis) [2, 3]. It is pointed out [3] that a local asymmetry in the composition between two halves of the bilayer plays an important role in the crenated shape. It has been observed [4] that a lateral phase separation occurs on an artificial two-component membrane where domains prefer local curvatures depending on the composition. In order to study the shape deformation accompanied by the intramembrane phase separation, we consider a two-component membrane as the simplest case of real biomembranes composed of several kinds of amphiphiles. 

Soft membranes attracted the attention of physicists in recent years because of novel properties related to their nonplanarity. Typically, thermal energies are sufficent to produce marked deviations from the planar state. In the case of fluid membranes, these fluctuations depend primarily on their bending elasticity. The energy of bending per unit area, g, is usually expressed by a quadratic form in the principal curvatures, C and C2 which are splays in the language of liquid crystals ... 

E. A. Evans and W. Rawicz, "Entropy-driven tension and bending elasticity in condensed-fluid membranes," Phys. Rev. Lett., 64, 2094-7 (1990) E. A. Evans, "Entropy-driven tension in vesicle membranes and unbinding of adherent vesicles," Langmuir, 7, 1900-8 (1991). 

In the simplest model for analyzing the shape of elastic vesicles with thin fluid membranes, the bending energy is made proportional to the integrated curvatures over the closed membrane surface... 

The theory of plate bending shows that the elasticities of bending and stretching are closely related because both derive from the bulk elasticity of the plate [2]. In principle, this theory can be adapted to fluid membranes so that it accounts for their elasticities of lateral and transverse dilations and of molecular tilt. The elastic modulus of lateral shear is made zero to obtain fluidity. 

Fluid membranes are molecularly thin two-dimensional objects in three-dimensional space. Their extreme deformability distinguishes them from liquid crystals and other three-dimensional systems. Moreover, their fluidity and complex internal structure sets them apart from polymers. The intervention of higher order bending elasticity and its consequences for membrane shape and fluctuations seem to be related to these characteristic properties. 

In the molten state, molecules may difluse within their monolayer therein the membrane resembles a two-dimensional liquid. Fluid membranes are soft and therefore easily deformable, whereas solid membranes are stiff and rather break on tearing than to adjust into a new shape. Nevertheless bending of fluid bilayers requires energy. The final equilibrium shape of an object that is confined by a fluid membrane is given by the minima of the bending elastic energy [8,9]. 

Fig. 32 Snapshots of vesicles in capillary flow, with bending rigidity K/k T = 20 and capillary radius / cap = 1-4/fo- a Fluid vesicle with discoidal shape at the mean fluid velocity v T/ffcap =41, both in side and top views, b Elastic vesicle (RBC model) with parachute shape at t m r/Rcap — 218 (with shear modulus nRl/ksT = 110). The blue arrows represent the velocity field of the solvent, c Elastic vesicle with shpper-like shape at v r/Rcap = 80 (with iiRl/k T = 110). The inside and outside of the membrane are depicted in red and green, respectively. The upper front quarter of the vesicle in (b) and the front half of the vesicle in (c) are removed to allow for a look into the interior, the black circles indicate the lines where the membrane has been cut in this procedure. Thick black lines indicate the walls of the cylindrical capillary. From [187]...

The elastic membrane model assumes that the cell is a thin-walled sphere filled with incompressible fluid. Because the wall is thin, it may be treated as a mechanical membrane. It can be presumed that the wall cannot support out-of-plane shear stresses or bending moments. This situation is described as plane stress, as the only non-zero stresses are in the plane of the cell wall. Furthermore, the stresses can be expressed as... 

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