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Theoretical Methods for Mineral Sciences Research

The theory programme in Mineral Sciences is closely linked to both the experimental and computational programmes. Much of our theory work has its origins in the study of phase transitions.

Displacive phase transitions in framework structures and Rigid Unit Modes

cristobalite_phases.jpg
The two phases of cristobalite at ambient pressure. The phase transition involves rotations of the tetrahedra, as seen in this projection of the structures of the two phases. The high-temperature cubic phase has many low-energy rigid unit modes, which allow the tetrahedra to have disordered orientations to avoid formation of the linear Si–O–Si bonds seen in this average structure.

Many materials can be described as frameworks of groups of atoms that are themselves linked together as relatively stuff polyhedra. For example, the crystalline and amorphous phases of silica contain SiO4 tetrahedra that are linked to each other by sharing corner oxygen atoms. We have developed a programme of work to understand the origin of displacive phase transitions in framework structures. The underlying idea is that the polyhedra are much stiffer than the forces involved in rotating two tetrahedra around a shared vertex. This allows us to develop models in which we focus on deformations in which the network of polyhedra can distort without the polyhedra themselves distorting. These deformations will correspond to low-energy phonon modes as well as potential static deformations associated with phase transitions. Calculations of the low-frequency modes give the set of potential soft modes for displacive phase transitions. This theoretical approach has also been used to understand the nature of high-temperature phases, and has been applied to an understanding of negative thermal expansion, and to understanding the structural basis for zeolite catalysis.

Contact: Martin Dove

Elasticity and phase transitions

landau_equations.jpg
Representative Landau free energy function with coupling between all elastic degrees of freedom and the order parameter Q. Minimisation gives equations for the variation of temperature of the elastic constants through their coupling with the order parameter.

Landau theory gives us a tool with which we can write the free energy of a system that undergoes a phase transition in terms of a fundamental parameter that describes the structural changes associated with the phase transition, the so-called order parameter. The free energy is usually expanded as a series in the order parameter, with the temperature-dependence usually being assigned to the quadratic term (for good theoretical reasons). This expansion of the free energy can be extended to include other quantities, such as elastic strains. We have developed a programme of work to incorporate strain as a general variable (noting that strain is a second-rank tensor), and from the formalism it is possible to obtain the dependence of the elastic constants as functions of temperature or pressure in the vicinity of a structural phase transition.

Contact: Michael Carpenter

Kinetics of cation-ordering phase transitions

Phase transitions that involve cation ordering are often limited by kinetic constraints. We have developed methods for understanding the kinetics of cation-ordering phase transitions using Ginzberg-Landau theory, in which the time-dependence of the free energy is written in a form similar to that for the equilibrium properties. This enables the time-dependence of the order parameter to be calculated directly.

Contact: Ekhard Salje

Domains and microstructure

When a material undergoes a structural phase transition, it will inevitably form domains which have the same distorted structure but with different permutations of axes. For example, in a cubic-tetragonal ferroelectric phase transition, the polarisation can point along any of the ±x, ±y or ±z directions, and in practice all 6 possibilities will form as domains in an initial single crystal. We have studied the formation of domains using simple models that combine local ordering with strain deformations. More recently we have been studying the transport of cations along domain walls.

 

Contact: Ekhard Salje, Mark Calleja or William Lee

Needle domains in CaTiO3 perovskite. The origin of the needle shapes has been studied using theory and computational tools.


Experimental observation of domains in CaTiO3 perovskite arising from a displacive phase transition. The needle shapes have been studied using analytical methods developed within the group.


quartz_domain_wall.jpg
Domain wall in quartz generated by molecular dynamics simulations.


Surface relaxations

We have been studying some simple models of relaxation at surfaces. It has been shown that the relaxation propagates into the bulk of the crystal in several different ways to a depth of several layers. More recently we have investigated the overlap of surface relaxations in thin crystals and shown that this overlap should lead to an interaction between the opposite surfaces of the crystals.

Contact: William Lee or Ekhard Salje


Mineral Sciences

 

Last updated on 11-Jun-09 14:43

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