Nematic energy

It is worth remarking for future reference that when w is the nematic energy Wp in (2.50) we have... 

Recall that the saddle-splay term in the nematic energy Wp in equation (2.49) is given by... 

The nematic energy becomes wp = 0 when n is a constant unit vector and, in the absence of body forces, G = 0. The equilibrium equations (2.143) are obviously satisfied when n = n by setting A = 0, obtained by taking the scalar product of the equilibrium equations with no. 

In order to simplify the presentation the one-constant approximation will be assumed for the nematic energy density given by equation (2.67). To analyse axial disclinations we consider configurations in which the director referred to Cartesian axes takes the form... 

To simplify matters we again make the one-constant approximation to the nematic energy density wp in equation (2.64) or its equivalent form in (2.67). The total energy over a unit length in y is then... 

Following Leslie [175], a rate of work hypothesis will now be invoked. This assumes that the rate at which forces and moments do work on a volume of nematic will be absorbed into changes in the nematic energy wp (cf. equations (2.49) and (2.50)) or the kinetic energy, or will be lost by means of viscous dissipation. The rate of work postulate is taken to be... 

We now turn to the balance of angular momentum equations (4.120). To first order in the approximations introduced above at (5.515) and (5.517), the nematic energy density (2.50) satisfies (cf. equation (4.131))... 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

Using this order parameter, the free energy in the nematic phase close to a transition to the smectic phase can be shown to be given by [20, 88, 89, 91]... 

C and I account for gradients of the smectic order parameter the fifth tenn also allows for director fluctuations, n. The tenn is the elastic free-energy density of the nematic phase, given by equation (02.2.9). In the smectic... 

Fig. 16. Gibbs energy-temperature diagram if FCC and ECC are present in the system. Ai-isotropic (undeformed) melt, A2-deformed melt (nematic phase) points 1 and 4 - melting temperatures of FCC and ECC under unconstrained conditions (transition into isotropic melt) points V and 2 -melting temperatures of FCC and ECC under isometric conditions (transition into nematic phase), point 3 - melting temperature of nematic phase (transition into isotropic melt but not completely randomized)...

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