In the previous chapters, the theory of pure liquid metals has been developed in some depth. The philosophy has been, in essence, to regard a pure liquid metal as a collection of suitably screened ions, interacting via effective ion-ion pair potentials. Recognizing that these pair potentials have features characteristic of liquid metals because the effective ion-ion interactions are mediated by the (almost totally degenerate) assembly of conduction electrons, nevertheless the treatment then is, essentially, that of a onecomponent liquid. In a more basic, fully first-principles treatment of a pure liquid metal, it is really to be viewed as a two-component system: ions, say in liquid Na, Na+, and electrons, e−.

In the present chapter, before turning to this approach to a pure liquid metal, some aspects of the theory of the previous chapters will be generalized to apply to binary liquid-metal alloys, such as Na-K or Na-Cs. Really, such alloys are three-component systems (compare Appendix 14.6), but in this chapter, following the philosophy of the earlier chapters, they will be treated as two-component systems. Thus, in liquid Na-K, the short-range order in the binary liquid metal alloy will be described by three partial pair correlation functions, namely, gNa-Na(r), gk-k(r) and gNa-K(r). The corresponding partial structure factors are SNa-Na(k), and so on. Various linear combinations of these structure factors are often very helpful, as emphasized especially in the work of Bhatia and Thornton (1970).

Of course, such a structural description has to be given for each concentration of the alloy.

In this chapter, an introduction will first be given to some progress in characterizing critical behaviour in terms of critical exponents. This will then be illustrated by calculations based on model equations of state. These will, in fact, give a route to the calculation of the liquid-vapour coexistence curves of liquid metals. Then some other nonuniversal properties, and especially the critical constants of the fluid alkalis, will be considered in relation to simple plasma models. Reference will also be made to spinodal curves.

In Chapter 6, when dealing with freezing theory, the order parameters were referred to; these were, in fact, the Fourier components of the periodic density in the crystalline phase. Again, in the treatment of the liquid-vapour critical behaviour, the concept of the order parameter is basic: Let us take this as a starting point.

Concept of order parameter

In the liquid-gas system, to be treated in detail below, the transition is from the high-temperature (or low-pressure) gas phase to the liquid phase, which has, as discussed quantitatively in the previous chapters, marked shortrange order that is absent in the gaseous phase. Further lowering of temperature or increase of pressure normally produces a further transition to the solid phase, having crystalline long-range order (see Chapter 16). A typical phase diagram is shown in Figure 9.1. Having used the variables T and P, the third variable V is dependent, of course, and could have been used to distinguish the phases.

The use of pair force laws to predict liquid structure is exemplified by the calculations of Swamy (1986) on liquid Na and Al. Using the hypernetted chain ((5.5) with E → c), and also the method of Machin-Woodhead and Chihara (MWC): see Chihara (1984; and other references there), Swamy has explored the results of using various oscillatory potentials to assess the applicability of these integral equations for such force laws. The results thereby obtained have been compared with molecular-dynamics simulation. His studies indicate that the HNC equation underestimates the main peak in S(k). The MWC method seems to give good results for S(k) according to Swamy's studies for shorter-range potentials, but when oscillatory effects are included, it is also deficient near the first peak of S(k).

While difficulties remain with integral-equation treatments of liquid structure for such direct calculations of S(k) from a given potential, it has been known for a long time that the inverse problem of extracting φ(r) from an experimentally determined S(k) (Johnson and March, 1963) is a much more stringent probe of an approximate integral equation. This point has been made quite clear in the work of Levesque, Weis and Reatto (1985; see also Reatto, 1988), which is summarized in Section 5.4.

In this appendix, procedures will be described by which the inversion of liquid structure data can yield pair potentials when these are constrained by requirements imposed by the form of pseudopotential theory, already discussed in Chapters 3 and 4.

Many complications in the metal physics description of binary alloys have been referred to as arising from the concentration dependence of the force fields. Thus a simpler approach seems called for in discussing the metallic binary liquid-glass transition.

Cohen and Turnbull (1959) [see also Turnbull and Cohen (1961) and Cohen and Grest (1979)] proposed a free-volume model in order to examine the thermodynamic and diffusive behaviour in the vicinity of the glass transition. In this model [see Li, Moore, and Wang (1988a, b)]:

1. An atom in the supercooled liquid or glass, for the most part, vibrates in a cage formed by its surrounding atoms, and

2. The atom inside the cage may escape to a void and diffuse from its original position, when it gains sufficient activation energy to overcome the barrier between its cage and the void.

The void referred to is defined as having a free volume greater than an atomic volume and is adjacent to the cage.

Point (1) has been established to be valid from a computer simulation of the static and dynamic properties for a Lennard-Jones (LJ) system (Kimura and Yonezawa, 1983). However, the same computer study implies that point (2) may not be actually applicable to the atomic mean square displacements.

Because of the fundamental differences between a LJ system and a metal, Li, Moore, and Wang (1988a,b) have made similar computer studies of metallic binary systems, which can become metallic glasses by a rapid quench from the melt not only in computer experiments but also in the laboratory (in which LJ systems such as argon never become glassy).

In Chapter 5, the statistical mechanics of homogeneous bulk liquids was treated, mainly within a pair potential framework. The generalization of this theory will next be set up to deal with inhomogeneous systems. One important application will be to the study of the liquid-vapour interface, developed in Chapter 12. However, the immediate application will be to the theory of freezing of liquid metals.

Specifically, the focal point of this chapter is the idea that correlation functions in a bulk liquid near to its freezing point already contain valuable information pertaining to the properties of its solid near melting. An example that can be cited here is the theory of the liquid-solid transition due to Kirkwood and Monroe (1941). This theory was based on the so-called hierarchy of statistical mechanical equations for the various order distribution functions (see, for example, Hill, 1956), generalized to apply to inhomogeneous systems (e.g., a solid with a periodic rather than constant single-particle density). The Kirkwood-Monroe theory certainly requires as basic knowledge the pair potential φ(r) and the homogeneous pair function g(r) discussed at length in the preceding chapters.

Single-particle density related to direct correlation function

Lovett and Buff (1980) have revived interest in the question of whether classical statistical-mechanical equations, such as the first member of the Born-Green-Yvon hierarchy, which connects the singlet density ρ(r) and the liquid pair correlation function g(r) can admit more than one solution for ρ for a given g (see Kirkwood and Monroe, 1941). Actually, these workers focused on the equation relating ρ and the Ornstein-Zernike direct correlation function c(r) of a liquid.

This book is about the theory of liquid metals. The interplay between electronic and ionic structure is a major feature of such systems. This should occasion no suprise, as even a pure liquid metal is a two-component system: positive ions and conduction electrons. Therefore, as in a binary liquid mixture such as argon and krypton, where three partial structure factors SArAr, SKrKr, and SArKr are required to describe the short-range atomic order, so in liquid metal Na, for instance, one needs SNa+Na+, SNa+e, and See for a structural characterization.

For a very fundamental treatment, the preceding description would be the correct starting point to treat liquid metal Na. Indeed, the theory of liquid metals has been developed in this manner. However, it is still true that, for many important purposes, a simpler picture suffices. Thus, in the chapter following this outline, attention will be focused on the ion-ion structure factor, which will simply be written as S(k); k = 4π sin θ/λ with 2θ the angle of scattering of X rays or neutrons and λ the wavelength of the radiation. It will be emphasized that it is indeed S(k) that is measured in suitable neutron-scattering experiments.

Then, in the following chapter, the use of this knowledge of structure will be considered in relation to the thermodynamics of liquid metals. Following this, electron screening of ions will be treated with the theme stressed above, the interplay between electronic and ionic structure, leading to a treatment of effective interionic forces. This theory will then be confronted with an approach based on the so-called inverse problem—namely, that of extracting an effective ion-ion interaction from the measured structure factor S(k).