Curvature Elasticity of Fluid Membranes

In general, one must consider the chemical potential of a molecule at the interface and in the solution. The equality of the two chemical potentials is the criterion for equilibrium and hence determines the area per molecule on the interface. When the amount of interface is fixed, as in the case of a single water-oil interface, this equality fixes 27 (see the problems at the end of Chapter 2). However, when the amount of interface can vary to minimize the free energy, 27 is determined by minimizing the interfacial free energy per molecule the chemical potential then determines the number of interfaces that exist in the system as well as the (small) volume fraction of surfactant that is not incorporated in these interfaces the properties of each interface are determined to a first approximation by the minimization of the local free energy of the film. 

In addition to being characterized by the area per amphiphile, the interfacial membrane is also characterized by its thickness. A, which can also change under deformations of the film. For simplicity, we assume that the equation of state of the flat membrane determines the thickness as a function of the area 

First consider a locally flat, isolated interface. Saturation occurs when the interfacial free energy achieves a minimum  

We now consider a curved interface with radii of curvature k and K2 (see Chapter 1). The mean curvature is 

We write an expansion of the free energy per molecule, /, for small curvatures (the actual small parameter is the product of the curvature and the membrane thickness) up to second order in k, K2. As explained in Chapter 1, the two invariants of the surface to this order in curvature are the mean and Gaussian curvatures since the free energy of a fluid membrane must be invariant under rotations of the coordinate system, / is a function of H, and K to the order we consider. Thus, 

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Deuling H, Helfrich W (1976) The curvature elasticity of fluid membranes a catalogue of vesicle shapes. J Phys (France) 37 1335-1345... 

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. 

Since the solubility of lipids in water is very low, the number of lipid molecules in a membrane is essentially constant over typical experimental time scales. Also, the osmotic pressure generated by a small number of ions or macromolecules in solution, which cannot penetrate the lipid bUayer, keeps the internal volume essentially constant. The shape of fluid vesicles [176] is therefore determined by the competition of the curvature elasticity of the membrane, and the constraints of constant volume V and constant surface area S. In the simplest case of vanishing spontaneous curvature, the curvature elasticity is given by (98). In this case, the vesicle shape in the absence of thermal fluctuations depends on a single dimensionless parameter, the reduced volume V = V/Vo, where Vb = (47t/3)1 o nd Ro = (5/4 r) are the volume and radius of a sphere of the same surface area S, respectively. The calculated vesicle shapes are shown in Fig. 23. There are three phases. For reduced volumes not too far from the sphere, elongated prolate shapes are stable. In a small range of reduced volumes of V e [0.592,0.651], oblate discocyte shapes have the lowest curvature energy. Finally, at very low reduced volumes, cup-like stomatocyte shapes are found. 

The dynamical behavior of fluid vesicles in simple shear flow has been stodied experimentally [190-193], theoretically [194-201], numerically with the boundary-integral technique [202,203] or the phase-field method [203,204], and with meso-scale solvents [37,180,205]. The vesicle shape is now determined by the competition of the curvature elasticity of the membrane, the constraints of constant volume V and constant surface area S, and the external hydrodynamic forces. 

The situation is also very different with regard to curvature elasticity when we compare 2-dimensional membranes with giant 1-dimensional micelles. The most general expression for the curvature elasticity of a 2-dimensional fluid film is [5.9]... 

These scaling relations are an immediate consequence of the scale invariance noted for the curvature elasticity (5.9) of fluid membranes. An isotropic dilation of factor A transforms Ci and to Ci/A and C2/A, respectively, whilst dA transforms to A dA. The whole thing leaves d ei unchanged. 

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. 

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 ... 

This potential force occurs in microstructured fluids like microemulsions, in cubic phases, in vesicle suspensions and in lamellar phases, anywhere where an elastic or fluid boundary exists. Real spontaneous fluctuations in curvature exist, and in liposomes they can be visualised in video-enhtuiced microscopy [59]. Such membrane fluctuations have been invoked as a mechanism to account for the existence of oil- or water-swollen lamellar phases. Depending on the natural mean curvature of the monolayers boimding an oil region - set by a mixture of surfactant and alcohol at zero -these swollen periodic phases can have oil regions up to 5000A thick With large fluctuations the monolayers are supposed to be stabilised by steric hindrance. Such fluctuations and consequent steric hindrance play some role in these systems and in a complete theory of microemulsion formation. 

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... 

When aggregation produces very extended objects (giant 1-dimensional micelles or 2-dimensional fluid membranes), the degrees of freedom associated with their curvature elasticity play an essential role in determining phase properties. The situation is very different depending on how many dimensions the object has. 

Below, we first introduce the most general mechanical description of the surface moments (torques) exerted on the boundary between two fluid phases. Then, we consider the thermodynamics of a curved interface (membrane) in terms of the work of flexural deformation. Next, we specify the bending rheology by means of the model of Helfiich [202]. Finally, we review the available expressions for the contributions of the electrostatic, steric, and van der Waals interactions to the interfacial bending moment and curvature elastic moduli. These expressions relate the interfacial flexural properties to the properties of the adsorbed surfactant molecules. 

The flow of many red blood cells in wider capillaries has also been investigated by several simulation techniques. Discrete fluid-particle simulations - an extension of DPD - in combination with bulk-elastic discocyte cells (in contrast to the membrane elasticity of real red blood cells) have been employed to investigate the dynamical clustering of red blood cells in capillary vessels [223,224], An immersed finite-element model - a combination of the immersed boundary method for the solvent hydrodynamics [225] and a finite-element method to describe the membrane elasticity - has been developed to study red blood cell aggregation [226]. Finally, it has been demonstrated that the LB method for the solvent in combination with a triangulated mesh model with curvature and shear elasticity for the membrane can be used efficiently to simulate RBC suspensions in wider capillaries [189]. 

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