Free energy Frank

The Helfrich-Prost model was extended in a pair of papers by Ou-Yang and Liu.181182 These authors draw an explicit analogy between tilted chiral lipid bilayers and cholesteric liquid crystals. The main significance of this analogy is that the two-dimensional membrane elastic constants of Eq. (5) can be interpreted in terms of the three-dimensional Frank constants of a liquid crystal. In particular, the kHp term that favors membrane twist in Eq. (5) corresponds to the term in the Frank free energy that favors a helical pitch in a cholesteric liquid crystal. Consistent with this analogy, the authors point out that the typical radius of lipid tubules and helical ribbons is similar to the typical pitch of cholesteric liquid crystals. In addition, they use the three-dimensional liquid crystal approach to derive the structure of helical ribbons in mathematical detail. Their results are consistent with the three conclusions from the Helfrich-Prost model outlined above. 

Fgi is the elastic Frank free energy which describes the slowly varying spatial distortions of the director the free energy density f i is a function of the elastic modes of deformation of a nematic liquid crystal and is given by [19,23]... 

Topologically, it turns out that the helical structure of the cholesteric cannot be deformed continuously to produce a cubic lattice without creating defects. Thus BP I and BP II are unique examples in nature of a regular three-dimensional lattice composed of disclination lines. Possible unit cells of such a disclination network, arrived at by minimizing the Oseen-Frank free energy, are shown in fig. 4.8.3. The tubes in the diagram represent disclination lines, whose cores are supposed to consist of isotropic (liquid) material. Precisely which of these configurations represents the true situation is a matter for further study. 

In other words, both twist and bend distortions are absent, leaving only the splay term in the Oseen-Frank free energy expression (3.3.7). It is seen from fig. 5.3.1, that by merely bending or corrugating the layers a splay deformation can be readily achieved without affecting the layer thickness. 

To discuss the critical behaviour of the twist and bend elastic constants in the nematic phase, we observe that the Frank free energy expression should include the contribution due to smectic short-range order ... 

Our task is to find an analytical expression for 9(z) at different fields. The scheme is as follows. First we shall write a proper integral equation for the free energy. Then, following the variational procedure discussed in Section 8.3, we compose the Euler equation corresponding to the free energy minimum and solve this differential equation for 9(z). To simphfy the problem we use the one-constant approximation Kii= K22 = Ks3= K.ln our geometry, and only one derivative, namely the bend term with dnjdz, is essential in the Frank free energy form (8.15) ... 

If we intend to calculate precisely the threshold field for the two-dimensional distortion we should write the Frank free energy with the director compcments (12.34) and the field term (Ea/47t)(En) and then make minimization of the free energy with respect to the two variables cp and 9 [18]. For a qualitative estimation of the threshold we prefer to follow the simple arguments by Helfrich [17]. We consider a one-dimensional (in layer plane xy) periodic distortion of a cholesteric... 

This term comes from calculating the usual Frank free energy (i.e., with bend, twist, and splay, and equal elastic constants) outside a disclination core of radius Rc but inside a cutoff radius size R. ... 

In the early 1970s Nehring and Saupe proposed that one should add a term linear in second gradients of director to the Frank free energy, namely... 

As shown by Frank and Evans 41 , solutions of apolar substances in water are characterized by a large entropy of mixing, leading to a high positive free energy of dissolving. 

If the two representations are equivalent then Eqs. (3.79) and (3.80) describe how A s and B s must be transformed in terms of a s and /Ts. (These identities are performed explicitly by Sanchez and Di Marzio, [49]. Frank and Tosi [105] further show that if a s and /Ts are chosen to satisfy detailed balance conditions, that is equilibrium behaviour, then the occupation numbers of the two representations are only equivalent if the nv s are in an equilibrium distribution within each stage. This is likely to be true if there is a high fold free energy barrier at the end of each stem deposition, and thus will probably be a good representation for most polymers. In particular, the rate constant for the deposition of the first stem, A0 must contain the high fold free energy term, i.e. ... 

From a consideration of the electrostatic free energy alone it is not immediately obvious how the arrangement of ions in the solution as a whole is related to the moving polarized central ion and its structured ionic cloud. It is reasonable to think that the induced multipoles impose restrictions on the mixing of the ions, so that the energy and entropy of configurations described by the structured ionic cloud are lower than in the DH model, as envisaged earlier by Frank and Thompson (16). Such considerations do not lead directly to predictions of the... 

Blandamer and Symons (57) assumed in their work that the free energies of hydration of Rb+(g) and Cl (g) were identical, since the two ions have crystal radii of the same magnitude. Jain (58) has also applied the radii to calculate absolute free energies of ionic hydration. His treatment is in category (ii) and is based on the model developed by Frank and coworkers (59). The equation used is similar to that employed by Stokes (60) and utilizes calculated van der Waals radii of the ions as well as the crystal radii. To obtain the best agreement it was necessary to assume that the effective dielectric constant in water is 2.7. 

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