Thermodynamic Properties of Liquid Crystal Phase Transitions

In this section we use Landau-de Gennes theory to calculate the thermodynamic properties of liquid-crystal phase transitions in terms of the phenomenological parameters a, B, C, and Tc. In particular, we will derive expressions for Tc, the transition temperature S c, the equilibrium order parameter value in the low temperature phase at the transition and As, the transition entropy per molecule. We will also calculate the temperature dependence of S (the equilibrium value of S) for T close to Tc, and h, the height of the free-energy barrier between S = 0 and S = S c at T = Tc. 

To calculate the thermodynamic properties listed above, it is sufficient to consider a spatially uniform system in which the order parameter value is spatially invariant. This means that the spatial derivative terms in the Landau free-energy density, which are important for the calculation of fluctuation phenomena as shown in the next section, can be neglected for the present purpose. Therefore, from Eq. [42] we get 

From Z(T) one can calculate the free energy density of the system ffHT)  

Thermodynamic properties of the system can be derived directly from F (T). 

In Appendix A it is shown that, if Z T) is in the form of Eq. [46], then F (T) is equal to the value of the absolute minimum of the Landau free energy density fF at temperature T. Moreover, the (temperature-dependent) equilibrium value of the order parameter, S , is equal to the value of S which minimizes at each T. Therefore, in order to obtain F (T), we differentiate fF with respect to S and set the result equal to zero for S = S  

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