The Frank Constants

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 

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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... 

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 =... 

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. 

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... 

Here are the Frank constants, n is a unit vector in the direction of the director (n and -n will be assumed equivalent), and the permittivity tensor of an NLC at the frequency of the light field is... 

The temperature dependence of the threshold power is determined by the temperature dependence of the Frank constant and of the permittivity. Bearing in mind the fact that the Frank constant K varies with temperature as the square of the order parameter while the anisotropy of the permittivity (Ref. 12), we can separate the temperature-dependent part in expression (12)forPFr intheformPpr° As the temperature... 

M. Bradshaw, E. Raynes, J. Bunning, and T. E. Faber, The Frank constants of some nematic liquid crystals,/ Phys. Fr., 46,1513 (1985],... 

Here we used the projection operator 3 = 3ij — riiiij, and is the Levi-Civita symbol. In the nematodynamics the Frank constants Ki (i = 1, 2, 3) should be adiabatic ones however, since e-p is the homogeneous quadratic form of V n,- the adiabatic and isothermal constants are the same [12]. In this section, we shall consider the case where the external electric and magnetic fields are absent. By virtue of local equilibrium, we have the following thermodynamic equations in laboratory frame ... 

The constants iFi=i-3 are the Frank elastic constants and correspond to the three fundamental deformations of the director field splay, twist and bend. Then-magnitude is of the order of 10 N, which is a fairly small value, meaning that on a length scale of several hundred microns (a typical size of a bulk sample), the nematic liquid crystal can easily deform or be deformed. It should be added that the Frank elastic constants depend on the order in the liquid crystal, being approximately proportional to 5. Consequently, the Frank constants Ki are temperature dependent and increase with decreasing temperature. 

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