Fluctuations and Liquid Crystal Phase Transitions

The calorimetric measurements provide information only on the static thermal quantities H and Cp. The dynamic thermal quantities such as the thermal diffusivity D and thermal conductivity K, which are linked to the dynamics of the fluctuations near phase transitions, are substantially more difficult to measure. Only recently have high-resolution techniques with sufficiently small thermal gradients been used for the study of liquid crystal phase transitions. 

Many liquid crystal phase transitions involve broken continuous symmetries in real space and their interactions on a molecular scale are short range [1]. As a result, fluctuations have long been known to be an important feature of liquid crystal phase transitions even weakly first order (discontinuous) ones. Compared to major advances in our understanding of fluctuation controlled second-order (continuous) phase transitions, relatively little is known about fluctuation phenomena (critical phenomena) at first-order phase transitions such as the nematic-isotropic transition. 

In the vicinity of fluctuation dominated phase transitions, the temperature dependence of thermodynamic parameters such as the specific heat at constant pressure, Cp=e , and the order parameter, y/ e, are all related to through a free energy density giving rise to scaling relations. For example a=2-vdmdp=(d-2) v/2 [2]. Despite the variety of their continuous broken symmetries, most liquid crystal phase transitions are expected to fall in the 3D-XY (helium) universality class with, a=-0.01, V-0.67 and y8-0.33. 

Here we give an overview of fluctuation dominated thermotropic liquid crystal phase transitions with a few hints of emerging aspects. From this perspective, the situation may be fairly summarized by noting that while analogies to phase transition models in spin-space (e.g. XY model with two components for the order parameter or Ising model with one) or momentum space (superconductivity) are a powerful tool to predict qualitative behavior for fluctuation dominated, real space, high temperature liquid crystal phase transitions, there is a significant gap between several quantitative (and qualitative) expectations and experimental measurements. 

Although originally intended as a theory of second-order phase transitions, the Landau theory can easily be generalized to include first-order phase transitions. de Gennes was the first to successfully apply Landau s theory to the first-order liquid-crystal phase transitions. It is the purpose of the present chapter to develop this Landau-de Gennes theory of liquid-crystal phase transitions and to discuss and illustrate its use. In the following sections, the derivation and discussion of the basic equations will be followed by application of the theory to the calculation of thermodynamic properties and fluctuation phenomena of liquid-crystal phase transitions, and by a description of some of the theory s more novel predictions and their experimental verifications. 

The simplest type of liquid-crystal phase is the nematic phase (see Fig. 23.9b) TBBA undergoes a transition from liquid to nematic at 237°C. In a nematic liquid crystal, the molecules display a preferred orientation in a particular direction, but their centers are distributed at random, as they would be in an ordinary liquid. Although liquid-crystal phases are characterized by a net orientation of molecules over large distances, not all the molecules point in exactly the same direction. There are fluctuations in the orientation of each molecule, and only on average do the molecules have a greater probability of pointing in a particular direction. 

Zhang Z, Zuckermann M J and Mouritsen O G 1993 Phase transition and director fluctuations in the 3-dimensional Lebwohl-Lasher model of liquid crystals/Mo/. Phys. 80 1195-221... 

In a smectic A liquid crystal one can easily define two directions the normal to the layers p and an average over the molecular axes, the director, h. In the standard formulation of smectic A hydrodynamics these two directions are parallel by construction. Only in the vicinity of phase transitions (either the nematic-smectic A or smectic A-smectic C ) has it been shown that director fluctuations are of physical interest [33, 44, 45], Nevertheless h and p differ significantly in their interaction with an applied shear flow. 

This approximate expression, using the Maier-Saupe theory for S2 and 54 and taking R(p) 1, agrees reasonably well with measurements of X for a variety of liquid crystals (see Fig. 10-10), as long as there is no transition to a smectic phase near the temperature range considered. When a smectic-A phase is nearby, as is the case for 8CB, then smecticlike fluctuations of the nematic state can significantly reduce A. For 8CB, for example, A drops to around 0.3-0.4 when T — 34°C (Kneppe et al. 1981 Mather et al. 1995), which is around 0.7°C above the transition to the smectic-A phase. 

In a chiral smectic (Sc ) phase, the tilt angle is the same within a layer, but the tilt direction processes and traces a helical path through a stack of layers (Figure 43). It has been demonstrated that when such a helix is completely unwound, as in a surface stabilized ferroelectric liquid crystal cell, then changing the tilt of the molecules fi om +0 to —0 by alternating the direction of an applied field results in a substantial electro-optic effect, which has the features of veiy fast switching (%1 - lOps), high contrast and bistability [87]. The smectic A phase of chiral molecules may also exhibit an electro-optic effect, this arises due to molecular tilt fluctuations which transition is approached, which are combined with a... 

Kauzmann did not accept this implication, however. He argued instead that in the temperature interval between 7 and Tk, the probability of crystallization was increasing so that before the isoentropic point could be reached, the time scale for crystallization would become the same as that for configurational relaxation in the amorphous phase, as discussed in Section II above. This would precipitate a first-order phase transition by the spontaneous growth of fluctuations in the appropriate direction. Such an event would necessarily terminate the liquid-state metastable free-energy surface and make the apparent entropy-crossing problem metaphysical and, Kauzmann therefore reasoned, of no consequence. 

---