Saddle-splay constant

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... 

When discussing bulk energies, this term is usually omitted. For curved surfaces of liquid crystals, e.g., in the description of lyotropic and membrane liquid crystals, however, this term becomes important." The surface term, K24, is usually important in case of saddle-splay (potato chips type) deformations and is called saddle-splay constant. 

I J l K k k Lo M m N Electric current (A) Nucleation rate (s 1 m 3) Length of one chain link (m) in a polymer chain Spring constant (Nm-1), equilibrium constant (e.g. mol/L) Bending rigidity (J) Saddle-splay modulus (J) Thickness of a polymer brush (m) Molar mass (kg mol-1), torque in Ch. 11 (Nm) Mass (kg), molecular mass (kg per molecule) Number of molecules (dimensionless or mol), number of segments in a linear polymer chain... 

The result, Eq. (43), can also be used to calculate the elastic constants of interfaces in ternary diblock-copolymer systems [100]. The saddle-splay modulus is found to be always positive, which favors the formation of ordered bicontinuous structures, as observed experimentally [9] and theoretically [77,80] in diblock-copolymer systems. In contrast, molecular models for diblock-copolymer monolayers [68,69], which are applicable to the strong-segregation limit, always give a negative value of k. This result can be understood intuitively [68], as the volume of a saddle-shaped film of constant thickness is smaller than... 

It is often taken as constant in the absence of further information. K is the Gaussian curvature (the product of the two principal curvatures) and is the 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 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 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... 

The structure of the BPIII phase actually resembles the L3 (so-called sponge) phase of lyotropic liquid crystals/ and the smectic blue phases observed and studied recently. Both the sponge and smectic blue phases are optically isotropic, and the smectic blue are optically active as well. Theoretical arguments show that the reason for the defect structure is the negative value of tire saddle-splay elastic constant, K24, which makes the defects energetically favorable. A sketch of the sponge phase and of the smectic blue phase is shown in Figure 6.24. 

We note tiiat bent-core smectic liquid crystals are also good candidates to form optically isotropic SmBlue type phases due to tiieir bent-shape, which may favor negative saddle-splay elastic constant. In fact, there are a number of experimental observations of optically isotropic and optically active smectic phases. ... 

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... 

The constant K4 is rarely mentioned in the literature. In many instances, the saddle-splay term does not usually influence the equilibrium equations because of an application of the divergence theorem, as mentioned previously. However, recent investigations on possible experiments for measuring K4 have been briefly commented upon by Virga [273, p.l23]. 

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