Pair distribution function multicomponent systems

We start with detailed definitions of the singlet and the pair distribution functions. We then introduce the pair correlation function, a function which is the cornerstone in any molecular theory of liquids. Some of the salient features of these functions are illustrated both for one- and for multicomponent systems. Also, we introduce the concepts of the generalized molecular distribution functions. These were found useful in the application of the mixture model approach to liquid water and aqueous solutions. 

Comparing structural information from X-ray and neutron diffraction provides a very valuable way to validate MD simulation results of glasses. In some simple systems, the partial pair distribution function or partial structure factors of all atom pairs can be determined experimentally and they provide excellent validations for simulated structures. However, as the composition becomes more complicated and more elements included, larger number of pair contributions will complicate the comparison and the validation becomes more and more difficult in multicomponent glass systems. For example, for binary oxides, e.g. sodium silicate, there are six partial pair distribution functions, but for a four component systems, for example the bioactive glass composition, there are a total of fifteen partials contributions. The overlap between partial contributions makes it very challenging to assign the peaks and to determine the quality of comparison and hence the validation of the simulated structure models. 

Equation 68 is equivalent to Eq. 64 for a one-component system described by a pair potential. Extension to the multicomponent case requires a summation over all pairs of the integrals of the potentials with the unperturbed distribution functions i.e.,... 

Most of the properties of the molecular distribution functions discussed in Chapter 5 hold for mixtures as well. In this section, we dwell upon some new features that are specific to multicomponent systems. We shall be mainly interested in the properties of the various pair correlation functions in a mixture of two components with spherical molecules. 

In this chapter we shall briefly summarize a method due chiefly to McMiixan and Mayer [1945] and to Kirkwood and Buff [1951] which expresses the thermodjmamic properties of a multicomponent system in terms of integrals of the radial distribution function of the different types of molecular pairs present in the mixture. )... 

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