Frank constant

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. 

Note 3 The names of Oseen, Zocher, and Frank are associated with the development of the theory for the elastic behaviour of nematics and so the elastic constants may also be described as the Oseen-Zocher-Frank constants, although the term Frank constants is frequently used. 

Here V n and V x n are the divergence and the curl of n. The three contributions to Wd are associated with the three independent modes of distortion splay, twist, and bend, depicted in Fig, 10-6. Terms of higher order than quadratic in Vn are only required if spatial distortions become severe. The Frank constants K, K2, and are of the order u /a,... 

For liquid-crystal-display applications, the most important viscosity is the twist viscosity yi values of yi for many different liquid crystals can be found in Yun (1973). The ratio where Ki is one of the Frank constants, has units of length squared per unit time... 

In the simplest case, where all three Frank constants are equal, this reduces to... 

A gradient theory that predicts the Frank constants was proposed by Nehring and Saupe (1972), Priest (1973), and Straley (1973). There have been many modifications and... 

Very few data exist for the viscosities or Frank constants of discotic nematics—that is, nematics composed of disc-Uke particles or molecules (Chandrasekhar 1992). One can estimate values of the Leslie viscosities from the Kuzuu-Doi equations (10-20) by setting the aspect ratio p equal to the ratio of the thickness to the diameter of the particles thus /j — 0 for highly anisotropic disks. This implies that R(p) —1, and Eq. (10-20b) implies that the viscosity o 2 is large and positive for discoidal nematics, while it is negative for ordinary nematics composed of prolate molecules or particles. If, as expected, is much smaller in magnitude than 0 2. the director (which is orthogonal to the disks) will tend... 

Here is a typical Leslie viscosity, AT is a Frank constant, V is a flow velocity, /i is a length scale of the flow geometry, such as the tube diameter in Poiseuille flow, and ftn is the average shear rate. The Ericksen number is the ratio of the flow-induced viscous stress 6Ye.fi = /h to the Frank stress K/h. The appropriate Leslie viscosity or Frank constant... 

At low enough shear rates, polymeric nematics ought to obey the same Leslie-Ericksen continuum theory that describes so well the behavior of small-molecule nematics. The main difference is that polymers have a much higher molecular aspect ratio than do small molecules, which leads to greater inequalities in the the numerical values of the various viscosities and Frank constants and to much higher viscosities. 

Because of the difficulty with which polymeric nematic monodomains are prepared, there are few measurements of Leslie viscosities and Frank constants for LCPs reported in the literature. The most complete data sets are for PBG solutions, reported by Lee and Meyer (1990), who dissolved the polymer in a mixed solvent of 18% dioxane and 82% dichloromethane with a few percent added dimethylformamide. Some of these data, measured by light scattering and by the response of the nematic director to an applied magnetic field, are shown in Figs. 11-19 and 11-20 and in Table 11-1. While the twist constant has a value of around K2 0.6 x 10 dyn, which is believed to be roughly independent of concentration and molecular weight, the splay and bend constants ATj and K3 are sensitive to concentration and molecular weight. 

X C2. Although these ratios are not consistent with data for PBG solutions, they are consistent with measurements of the Frank constants of suspensions of the tobacco mosaic virus (Lee and Meyer 1986), which is an extremely rigid particle. Thus, discrepancies between theory and measurements for PBG are most likely due to the partial flexibility of PBG molecules. 

A Landau-de Gennes theory for the Frank constants of long semiflexible worm-Uke chains at low order parameter S gives (Shimada et al. 1988) K = K7, = 0K2 =... 

Figure 11.21 Predictions of the dimensionless Frank constants for hard rods as functions of reduced concentration nL"dv/A, using a numerical solution of the Onsager equation. At the concentration v = v, the abscissa has the value n L Ju/4 = 4. (From Lee and Meyer, reprinted with permission from J. Chem. Phys. 84 3443, Copyright 1986, American Institute of Physics.)...

The Frank constants for thermotropic polymers are rarely measured. Zheng-Min and Kleman (1984) report values of K 30 x 10 dyn, K7 3 x 10 dyn for... 

The Frank constant thus diverges as T- T and 17 (7) tends to zero in the same limit. 

The renormalized Frank constant approaches a universal value as... 

Near the A-N transition, the present theory leads to the same conclusions as de Gennes s model discussed in 5.5.2 the divergent contributions to and in the nematic phase are given by (5.5.18) and (5.5.19). Near the C N transition, all three Frank constants diverge ... 

Very few quantitative measurements of the physical properties have been reported. The Frank constants for splay and bend have been determined using the Freedericksz method. Interestingly the values are of the same order as for nematics of rod-like molecules. The twist constant has not yet been measured. The faet that the diamagnetic anisotropy is negative makes it somewhat more difficult to measure these constants by... 

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