Studies in the density functional theory of freezing: relative stability of FCC & BCC structures

Abstract

In this chapter we have developed a minimisation technique which has been demonstrated to yield excellent results for realistic systems. The technique is particularly useful and necessary for non?close?packed structures where non?Gaussian corrections are large. We have used it in a variety of situations (to be discussed in subsequent chapters) where accurate comparisons of free energies are important. In conclusion, we have demonstrated that the density functional theory incorporating three?body effects in a simple manner can explain the observed BCC, FCC and liquid phase diagram of charge?stabilised colloidal suspensions. This calculation lends further support to the conjecture put forward in the previous chapter, namely, that although three?body correlations are important in stabilising the BCC lattice, their relevance to the FCC–BCC relative stability question may, in fact, be minimal since they can perhaps be modelled as a set of universal numbers or, at worst, a universal function. We predict non?Gaussian corrections to the solid density to be significant for the BCC lattice which should be verifiable experimentally. Charge?stabilised colloidal systems offer us a model system where the length and time scales involved are such that various aspects involving phase transformations can be studied at a level of detail which is impossible in atomic systems. The methods of density functional theory can therefore be tested thoroughly by comparing theoretical predictions with real laboratory experiments. In the future we hope to study, among other things, laser?induced freezing, BCC–FCC interfaces, lattice defects, shear melting and phase transformations in colloidal suspensions consisting of two or more species differing in their charge or in their particle diameters. In this chapter, we have considered an FCC–BCC interface within a simple density functional scheme. Our results predict the interfacial width to be small and the interfacial energy to be low in the scale of typical interfacial energies for realistic solids. Several issues of considerable physical interest have, however, not been addressed in our greatly simplified theory. We shall comment on them briefly here, mentioning specific calculations which we hope to undertake in the future to investigate these issues. The formation of one phase in another occurs by the production of an interface. The dynamics of nucleation therefore depends crucially on the energetics of the interface in question. Interfaces between various orientations of the FCC and the BCC lattices are expected to have very different interfacial energies. Further, the energy of an interface is expected to depend on its position within the unit cell and is expected to have periodic modulations reflecting the periodicity of the underlying lattice (analogous to the Peierls–Nabarro barrier for dislocation motion; see Rajlakshmi et al. 1988) which will have important consequences for the motion of these interfaces during the growth of one phase within another. For example, the close