Ferroelastic phase transitions: structure and microstructure.

TLDR: In this article, the derivation of the Landau potential G = ½A θS [coth(θS/T) − coth(αS/TC)]Q2 + ¼BQ4 + … is derived as a solution of the general ϕ4 model.

Abstract: Landau-type theories describe the observed behaviour of phase transitions in ferroelastic and co-elastic minerals and materials with a high degree of accuracy. In this review, the derivation of the Landau potential G = ½AθS [coth(θS/T) − coth(θS/TC)]Q2 + ¼BQ4 + … is derived as a solution of the general ϕ4 model. The coupling between the order parameter and spontaneous strain of a phase transition brings the behaviour of many phase transitions to the mean-field limit, even when the atomistic mechanism of the transition is spin-like. Strain coupling is also a common mechanism for the coupling between multiple order parameters in a single system. As well as changes on the crystal structure scale, phase transitions modify the microstructure of materials, leading to anomalous mesoscopic features at domain boundaries. The mesostructure of a domain wall is studied experimentally using X-ray diffraction, and interpreted theoretically using Ginzburg–Landau theory. One important consequence of twin meso­structures is their modified transport properties relative to the bulk. Domain wall motion also provides a mechanism for superelastic behaviour in ferroelastics. At surfaces, the relaxations that occur can be described in terms of order parameters and Landau theory. This leads to an exponential profile of surface relaxations. This in turn leads to an exponential interaction energy between surfaces, which can, if large enough, destabilize symmetrical morphologies in favour of a platelet morphology. Surface relaxations may also affect the behaviour of twin walls as they intersect surfaces, since the surface relaxation may lead to an incompatibility of the two domains at the surface, generating large strains at the relaxation. Landau theory may also be extended to describe the kinetics of phase transitions. Time-dependent Landau theory may be used to describe the kinetics of order–disorder phase transitions in which the order parameter is homogeneous. However, the time-dependent Landau theory equations also have microstructural solutions, explaining the formation of microstructures such as tweed.