Fully Consistent Finite-Strain Landau Theory for High-Pressure Phase Transitions

Landau theory (LT) is an indispensable cornerstone in the thermodynamic description of phase transitions. As with structural transitions, most applications require one to consistently take into account the role of strain. If temperature drives the transition, the relevant strains are, as a rule, small enough to be treated as infinitesimal, and therefore one can get away with linearized elasticity theory. However, for transitions driven by high pressure, strains may become so large that it is absolutely mandatory to treat them as finite and deal with the nonlinear nature of the accompanying elastic energy. In this paper, we explain how to set up and apply what is, in fact, the only possible consistent Landau theory of high-pressure phase transitions that systematically allows us to take these geometrical and physical nonlinearities into account. We also show how to incorporate available information on the pressure dependence of elastic constants taken from experiment or simulation. We apply our new theory to the example of the high-pressure cubic-tetragonal phase transition in strontium titanate, a model perovskite that has played a central role in the development of the theory of structural phase transitions. Armed with pressure-dependent elastic constants calculated by density-functional theory, we give an accurate description of recent high-precision experimental data and predict a number of elastic transition anomalies accessible to experiments.

Popular Summary

High-pressure phase transitions in crystals and minerals occur in diverse fields such as astrophysics, seismology and geology, chemistry, and nanotechnology. Landau’s thermodynamic theory of structural phase transitions has been enormously successful at ambient pressures. However, a concise and fully consistent extension to high pressures—where finite strain has to be taken into account—has been missing up until now. We provide an extension of Landau’s theory using a careful combination of concepts from nonlinear elasticity theory and information supplied by experiments and electronic density-functional theory.

Precise knowledge of the elastic properties of earth-forming materials and their anomalies at pressure- and temperature-induced phase transitions is integral to understanding the physical properties of Earth (e.g., interpreting seismic signals from earthquakes). In theories based on the use of infinitesimal strain measures, neglecting the pressure dependence of the elastic constants and Landau coupling coefficients results in errors that can quickly reach 100%. The power of our new theory, in which we replace infinitesimal strain measures with the Lagrangian strain tensor, is illustrated by a practical application to recently published experimental data on the cubic-to-tetragonal high-pressure transition (10 GPa) at room temperature in the perovskite strontium titanate, one of the most studied archetypal model systems in the field of structural phase transitions.

We expect that our theory will be valuable for understanding phase-transition anomalies as a function of pressure and temperature and for predicting phase diagrams of earth- and planetary-forming materials.

Figure 1

$P$ dependence of Lagrangian strain components ${\eta }_1(P)$ and ${\eta }_3(P)$ (solid colored lines) as obtained from a fit of our theory to the experimental data of Ref. [23] (black data points). Insets: Corresponding results of the infinitesimal approach using the parameters of the left column of Table III of Ref. [23].

Figure 2

Effective $P$ dependence of Landau coefficients $A_R(P)\equiv A_R[\widehat{\mathit{X}}]$, $B_R(P)\equiv B_R[\widehat{\mathit{X}}]$, and $C_R(P)\equiv C_R[\widehat{\mathit{X}}]$. Note the change of sign of the coefficient $B_R(P)$ near $P=P_c$, which implies the tendency of the transition toward becoming tricritical or even weakly first order under pressure. Inset: $P$ dependence of parameters $d_{ii}(P)\equiv d_{ii}[\widehat{\mathit{X}}]$.

Figure 3

Solid colored lines: Pressure dependencies of elastic constants $C_{11}(P)$, $C_{12}(P)$, $C_{13}(P)$, and $C_{33}(P)$ as determined from combining our present theory with the approach of Ref. [33]. Dotted black lines: Background elastic constants.

Figure 4

Main plot: Difference $\mathrm{\Delta }\overline{\widehat{\epsilon }}(P)={\overline{\hat{\epsilon }}}_3-{\overline{\hat{\epsilon }}}_3$ of spontaneous strain components as obtained from experiment vs our fit (blue data points and function). For comparison, the dotted red line shows our result for the squared order parameter ${\overline{Q}}^2(P)$ after a suitable rescaling. Left upper inset: Experimental data for the squared soft-mode frequency compared to a suitable rescaled fit result for the inverse order-parameter susceptibility ${\chi }^{-1}(P)$ as obtained from our fit. Right lower inset: $\mathrm{\Delta }\overline{\widehat{\epsilon }}$ parametrized as a function of the squared equilibrium order parameter ${\overline{Q}}^2$. The dotted line corresponds to an assumed proportionality between $\mathrm{\Delta }\overline{\widehat{\epsilon }}(P)$ and ${\overline{Q}}^2(P)$ as it would follow from a naive application of LT.