Saddle-splay modulus

What are typical values for the parameters k and kl The bending rigidity of surfactant films is typically of the order of 1-20 kBT at room temperature. Factors that reduce k are short alkyl chains, cosurfactants, double-chain surfactants with unequal chains, and cis-unsaturated bonds. For the saddle-splay modulus only few measurements have been done. It tends to be negative with an amount much smaller than the bending rigidity for the same system. 

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 results are in line with those in table 1.6. The increase of fcj with hydrocarbon chain length of the surfactant is well established and agrees with expectation. Negative values of the saddle-splay modulus k mean that the oil tends to form droplets in the water. 

Kozlovsky, Y., Efrat, A., Siegel, D.A., Kozlov, M.M. Stalk phase formation Effects on dehydration and saddle splay modulus. Biophys. J. 2004, 87, 2508-21. 

Safran, S.A. (1991) Saddle-splay modulus and the stability of spherical microemulsions. Phys. Rev. A, 43, 2903-2904. 

Figure 11.9 Bar bending modulus k (a) and bar saddle-splay modulus i< (b) for the systems H20(D20)/NaCI as well as H20/NaCI-n-decane/triolein-CioE4 as a function of 3. With increasing p the bending modulus of the amphiphilic film k decreases, while the saddle-splay modulus i< becomes less negative.

This form for the free energy per unit area was discussed by Helfirich and states that the mean curvature which minimizes the free energy has a value Co, termed the spontaneous curvature of the membrane. The energy cost of deviating from the spontaneous curvature is the bending or curvature modulus, k. The parameter k, known as the saddle-splay modulus, measures the energy cost of saddlelike deformations. 

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... 

This expression can also be obtained from a calculation of the free energies of spherical and cylindrical oil droplets in water (or vice versa) [93,98]. This calculation can also be used to extract the saddle-splay modulus k of Eq. (41), with the result... 

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... 

But what about the K24 term The so-called saddle-splay modulus K24 is important only for particular situations, in which a distortion has a two- or three-dimensional structures such as nematic droplets in the isotropic solutions [8] or blue phases [9]. The free energy term including modulus K24 is a so-called divergence term because it has a form of divn to the first degree. Hence, if one performs the... 

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... 

Cq is the membrane intrinsic curvature, that is, the curvature of the membrane with zero deformation (e.g., a single layer of cylindrical molecules should have an intrinsic curvature of zero, but cone-shaped molecules will pack two dimensionally to have a curvature with no energy cost, k is the bending modulus, and is known as the saddle-splay modulus. As we can see in Figure 3.10, it is quite possible to have a curved membrane with a mean curvature of zero in the case of saddle-like deformations. Therefore, it becomes clear that to describe the energy cost of a saddle deformation the extra terms are necessary. 

Here R and / 2 are the local principal radii of curvature, i/o = 1 /Ro is the spontaneous curvature, k is the bending modulus, and k is the saddle splay modulus. The half-sum of the local curvatures is called the mean curvature H ... 

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