Saddle bending elasticity

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. 

Let be the stretching elasticity of the membrane, and let k, k, and Cq be the bending elasticity, the saddle splay bending elasticity, and the spontaneous curvature of the membrane [5,6]. Later on, attention is restricted to a quasispherical flaccid fluctuating vesicle. Only the case of fluctuations with magnitudes much less than the radius of the vesicle are considered. Let the area of the tension-free flat membrane of the vesicle be Sg, and its volume (assumed constant in time) be If Choose a laboratory frame of reference XYZ with origin O inside the vesicle. Let 0, q> be the... 

The A" and K22, terms, which vanish identically in the apolar nematic and cholesteric phases, are not considered here. As discussed above, the volume integrals of the free energy terms containing the splay-bend elastic constant and the saddle-splay elastic constant K24 can be transformed into integrals over the nematic surface... 

Bending elasticity moduli can be measured, or at least estimated experimentally. The spontaneous curvature is close to the reciprocal radius of spherical micelles in equilibrium with solubilizate, and therefore is rather straightforward to measure. The bending modulus is more difficult to measure experimentally.Both k and k are usually presented in units of thermal energy, ksT. Typical values of the bending modulus vary from 0.1 ( flexible monolayers) to 10 k T ( rigid monolayers). There is still very little data concerning the experimental value of the saddle splay... 

We have already mentioned in the Introduction that microemulsions containing long-chain amphiphiles can be described by interfacial models. These models are based on the curvature elasticity of the amphiphilic monolayer and thus contain as material parameters the bending rigidity and the saddle-splay modulus. These parameters have to be calculated from a more microscopic model. A somewhat similar problem occurs in the... 

The three (positive) elastic constants Kn (splay), K22 (twist), and K33 (bend) are associated to the three principal deformations. In the surface term, fs is the contribution of the two anchorings, k is the unit vector normal to the surface and directed outward, K13 is the splay-bend constant, and K24 is the saddle-splay constant. The two last surface terms play only for thin films the mere existence of the splay-bend constant K13 is a matter of debate. In the framework of Landau-de Gennes analysis, = K33 and the elastic... 

The phase diagram of such curved surfaces in the amphiphilic system has been studied by Huse and Liebler [8] on the basis of the elastic energy of the surface. The phase behavior is determined by the balance of the surface tension, the bending rigidity and the saddle-splay modulus of the curved interface and the IPMS structure appears when the saddle-splay modulus increased to a some critical value. Mathematically, more than 30 species of curved surfaces have been reported as IPMS [9] and, experimentally, several types of IPMS structures have been found in cubic phases of lipid-water systems [10]. Concerning the formation mechanism of the cubic network, Ranpon and Charvolin [6] found the epitaxial relationships between reticular planes of these three phases in the... 

The strain tensor must conform to the symmetry of the liquid crystal phase, and as a result, for nonpolar, nonchiral uniaxial phases there are ten nonzero components of kij, of which four are independent ( i i, 22> A 33 and 24)- These material constants are known as torsional elastic constants for splay (k, 1), twist ( 22) bend ( 33) and saddle-splay ( 24) terms in 24 do not contribute to the free energy for configurations in which the director is constant within a plane, or parallel to a plane. The simplest torsional strains considered for liquid crystals are one dimensional, and so neglect of 24 is reasonable, but for more complex director configurations and at surfaces, k24 can contribute to the free energy [7]. In particular 24 is important for curved interfaces of liquid crystals, and so must be included in the description of lyotropic and membrane liquid crystals [8]. Evaluation of Eq. (16) making the stated assumptions, leads to [9] ... 

The effect of surface elastic constants on the nematic director configurations is of basic interest for the elastic theory of liquid crystals and plays a critical role in those device applications where the nematic is confined to a curved geometry. The saddle-splay surface elastic constant, K24, and the splay-bend surface elastic constant, K13, defied measurement for more than sixty years, since the pioneering work of Oseen, who made the first steps toward the elastic theory of liquid crystals. 

Measurements of the saddle-splay surface elastic constant, K24, and the splay-bend surface elastic constant, K13, were first introduced by Oseen [1] in 1933 from a phenomenological viewpoint, and later by Nehring and Saupe [2] from a molecular standpoint. These constants tend to be neglected in conventional elastic continuum treatments for fixed boundary conditions because they do not ent the Euler-Lagrange equation for bulk equilibrium. Experimental determination of tiie two surface elastic constants is undoubtedly a difficult task, since their effects are hard to discriminate from those of ordinary sur ce anchoring [3]. 

H e Ki, K2 and K3 correspond to the splay, twist, and bend bulk elastic constants, respectively. Further, the terms K24 and K13, known as the saddle-splay surface elastic constant and mixed splay-bend surface elastic c( istant, respectively, are called surface elastic constants because they enter EQN (3) as divergences of a volume integral, which are converted to surface integrals via Green s theorem. It should be noted that the second derivative of the K24 term in EQN (3) is apparent however, this is... 

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