Order Parameter, Phase Transition, and Free Energies

Order Parameter, Phase Transition, and Free Energies... 

ORDER PARAMETER, PHASE TRANSITION, AND FREE ENERGIES... 

Salje (1985) interpreted overlapping (displacive plus Al-Si substitutional) phase transitions in albite in the light of Landau theory (see section 2.8.1), assigning two distinct order parameters Q n and to displacive and substitutional disorder and expanding the excess Gibbs free energy of transition in the appropriate Landau form ... 

At the N-I transition, the free energy is the same for both isotropic and nematic phases with different order parameters, 0 and Sc. Thus the N-I transition temperature is obtained as... 

Fig. 6.5 (a) Temperature dependence of the order parametru in the Landau-de Gennes model (B) and (C) are coefficients of the expansion. Tjvj Tc is experimental value of the isotropic—nematic phase transition temperature corresponding to equality of free energy densities for the two phases, (b) Experimental dependence of the order parameter for 5CB and the characteristic temperature points Tc, Tc and Tc defined in accordance with the model of panel (a)... 

The free energy can be used to determine the specific heat and the temperature dependence of the specific heat can tell us something about the possible order-disorder phase transitions in magnetic systems. A is an extensive property and hence depends on the system size. It is here conveniently written as a product of the system size N) and a system-size independent parameter which can be considered as the free energy per site. 

Depiction of a first order transition in terms of the variation of the Landau free energy density with the order parameter at several reduced temperatures. For details see text. Reproduced from N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Copyright (c) July 21, 1992 reprinted with permission of Westview Press of the Perseus books Group. 

The order parameter thus characterizes the transition, and the Landau free energy expansion in this order parameter, and in eventual secondary order parameters coupled to the first, has to be invariant under the symmetry operations of the disordered phase, at the same time as the order parameter itself should describe the order in the condensed phase as closely as possible. In addition to having a magnitude (zero for T> Tc, nonzero for T< T ), it should have the same symmetry as that phase. Further requirements of a good order parameter are that it should correctly predict the order of the transition, and that it should be as simple as possible. As an example, the tensori-al property of the nematic order parameter... 

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