3.1 Introduction
This chapter examines the techniques of computer simulation and shows how they are applied to the study of fast-ion conductors. The methods described fall into two classes: static lattice simulations, which take no explicit account of thermal vibrations, and molecular dynamics, which model the dynamical and time-dependent properties of the system. This thesis is concerned with the application of such methods to experimental problems rather than their development and so the descriptions which follow will summarise the main features of each technique and demonstrate the value of each to the study of superionics. Both classes of simulation are well established in the field of solid state studies and are the subject of several reviews in the literature (Catlow and Mackrodt, 1982; Sangster and Dixon, 1976). Each class of simulation requires the specification of interatomic potential models to represent the forces acting between ions in the system. Reliable specifications are fundamental to the accuracy of the simulation results. This important topic is discussed in the final section of this chapter.
3.2 Aims of Computer Simulation
The aim of computer simulation is to provide information on the structure and transport properties of ionic or semi-ionic materials at the microscopic level. Such detail is complementary to conventional experimental techniques such as powder neutron diffraction, conductivity and nmr and so simulation is a useful analytical tool.
Early simulation studies examined simple systems such as the alkali halides and alkaline earth fluorides and these met with considerable
success (Catlow et al., 1977 a, b). The advent of powerful supercomputers with their parallel and concurrent architectures has, however, increased the range and scope of investigation to include more complex materials. A greater level of understanding has been afforded in, for example, the behaviour of superionic, non-cubic materials (Wolf et al., 1984 a,b; Wolf and Catlow, 1984), silicates (Catlow and Parker, 1985), zeolites (Hope, 1985) and ceramics (Lewis and Catlow, 1985). A growing area of interest is surface simulation for which associated techniques have been developed (Tasker, 1979; Colbourn, 1986) and these have been extended to examine grain boundary interfaces (Duffy and Tasker, 1983).
The basis of the modelling technique is the specification of reliable interatomic potentials from which the following properties may be calculated:
i)Crystal structure - in terms of unit-cell coordinates and cell dimensions.
These calculations may be used in conjunction with diffraction data to analyse complex structures. Alternatively, they may be used as a predictive tool when probing a system under extreme conditions of temperature or pressure.
ii)Crystal properties, such as elastic, dielectric, piezoelectric constants
and lattice dynamics.
\\\) Defect properties, which hold the key to transport properties of solids
and are of primary interest in this thesis. The static simulations characterise the nature of the defect species responsible for the transport process. This is achieved by calculating energies of formation and migration and the binding energies of defects and impurity defect species. Molecular dynamics treats thermal motion explicitly and is instrumental in identifying the microscopic mechanisms of ion transport. It is also useful for the analysis of complex and high temperature structures.
The work on LaF3 presented in Chapters 5 and 6 is principally concerned with (iii), although aspects of the first two points will also be presented.
The following sections describe the simulation methods used in this thesis. Each of the methods is based on the classical Born model approach to the solid, that is, the specification of short-range potential functions acting between point ion charges.
3.3 S tatic Lattice Sim ulations
3.3.1 Perfect Lattice
The lattice energy, Ul, of a perfect crystal may be written as:
where the summations refer to all ion pairs, i and j, such that i * j, and all trios i, j, k, where i * j * k. r^ equals the separation between all pairs of ions, qj is the charge on species i, ^ describes the short-range interaction between ion pairs and is the three body potential between atom i and two others, j and k.
The first term on the right hand side of equation 3.1 is the long-range Coulombic or electrostatic energy which provides the dominant contribution to the lattice energy, being around 90% for ionic materials. The short-range interactions represent the forces between adjacent ions. They are described by simple analytical functions containing a repulsive term due to charge cloud overlap and an attractive term consisting of dispersive and any covalent interactions. The short-range term in equation 3.1 has also been extended to include three-body, angular-dependent interactions (Leslie, 1985). For simulation work, pair potential models are generally acceptable for ionic materials, although three body terms become large in covalent systems (Sim, 1988; Sanders et al, 1984). These higher order potentials were not used in this thesis.
3.3.2 Summation Methods
The Coulombic, r 1 sums of equation 3.1, are only slowly converging in real space and cannot be truncated without leading to serious artefacts (Adams, 1983). They do, however, become rapidly convergent when transformed into reciprocal space according to the method of Ewald (1921).
The physical basis for the Ewald method is the replacement of each point ion by a Gaussian charge distribution at every lattice site. Considering the lattice to be composed of sub-lattices of opposite charge, the electrostatic potential experienced by any one reference ion in the presence of all other ions is given by,
*F = ^ + ¥ 2 (3.2).
The potential 'F-j arises from a Gaussian charge distribution of the same size and sign as the ions replaced. However, according to the definition of the Madelung constant, the charge distribution on the reference site is not considered to contribute to either of the potentials 'F-j o r x¥ 2- Potential *Fi may, therefore, be written as,
(3.3)
where x¥ a is the potential of a continuous series of Gaussian distributions and ^ t h a t of a single distribution at the reference site. Potential ¥ 2, in equation 3.2, corresponds to a lattice of point charges with an additional, but equally opposite, Gaussian charge distribution superimposed on the first.
The speed of convergence of both ^ and *F2 depends on the width of the Gaussian distribution, r|, although when taken together the total potential 'F is independent of rj. The functional forms for 'F-i and *F2 show that 'F-j converges rapidly in reciprocal space for large rj whilst *F2 converges faster
46
in real space for small tv An optimum value for rj ensures fast convergence of both parts.
Catlow and Norgett (1976) calculated the optimum value of rj by minimising the number of terms in each series, Nt, with respect to rj to give,
where s is the number of ions per unit cell and Vc is the unit cell volume.
Returning to the expression for lattice energy, equation 3.1, the short- range interaction can be summed directly in real space. The convergence is such that a cut-off distance may be supplied beyond which the interactions are negligible.
3.3.3 Energy Minimisation
To calculate equilibrium properties of a crystal structure it is first necessary to minimise the lattice energy with respect to structural parameters. This can be carried out in two ways: firstly, to constant volume where minimisation is with respect to atomic coordinates only and the cell dimensions are fixed; secondly, to constant pressure where all structural parameters including the cell dimensions may be allowed to vary. In either case, for a lattice containing s ions per unit cell, the lattice energy is first expanded about any configuration r to second order in internal and bulk strains (Born and Huang, 1954). The energy at the new configuration i' is then,
3
(3.4)
The total strain vector,