Pair correlation function, uniform fluids

A quantity of central importance in the study of uniform liquids is the pair correlation function, g r), which is the probability (relative to an ideal gas) of finding a particle at position r given that there is a particle at the origin. All other structural and thermodynamic properties can be obtained from a knowledge of g r). The calculation of g r) for various fluids is one of the long-standing problems in liquid state theory, and several accurate approaches exist. These theories can also be used to obtain the density profile of a fluid at a surface. 

Here, we report some basic results that are necessary for further developments in this presentation. The merging process of a test particle is based on the concept of cavity function (first adopted to interpret the pair correlation function of a hard-sphere system [75]), and on the potential distribution theorem (PDT) used to determine the excess chemical potential of uniform and nonuniform fluids [73, 74]. The obtaining of the PDT is done with the test-particle method for nonuniform systems assuming that the presence of a test particle is equivalent to placing the fluid in an external field [36]. 

The choice of the weighting function w depends on the version of density functional theory used. For highly inhomogeneous confined fluids, a smoothed or nonlocal density approximation is introduced, in which the weighting function is chosen to give a good description of the hard sphere direct pair correlation function for the uniform fluid over a... 

All of the approximate DFTs of melting assume that the properties of the uniform (liquid) phase are well characterized, and, in particular, that the free energy and pair correlation function of the liquid are known. The crystalline phase is viewed as a highly inhomogeneous fluid with a spatially varying density p(r) having the symmetry of the crystalline... 

Readers might have seen a similar picture from a result of a molecular dynamics simulation. If you look carefully, there are places where molecules are densely crowded, while in some places molecules are scarce. If one takes the average of the number of molecules over the entire volume of the container, N/V = p, it is of course a constant, and it does not include useful information with respect to the structure and dynamics of the fluid. However, if one takes a product of densities at two different places r and r, and takes a thermal average over configurations, namely, u r)u r )) = p(r,r ), it then contains ample information about the structure and dynamics of liquids. The quantity is called density-density pair correlation function . When the fluid is uniform, the quantity can be expressed by a function of only the distance between the two places, such that p(r, F) p( r — r ). 

We shall see from the Omstein-Zemike theory of the pair-correlation function h(r) in a uniform fluid, that near the critical point... 

Now p is r-dependent, rather than z-dependent as it was in (9.2), and the relevant solution is that whidi for large r is asymptotic to the mean density p of the homogeneous fluid udiik the uniform p, hne is ft( T). We have the mean local density p(r) related to the pair-correlation function h(r) by... 

When the smoothed or nonlocal density approximation (or NL-DFT model) is used, the weighting function is chosen so that the hard-sphere direct pair-correlation function is well described for the uniform fluid over a wide range of densities. One example of such a weighting function is the model proposed by Tarazona [69], which uses the Percus-Yevick theory for approximating the correlation function over a wide range of density. In this case, the weighting function is expanded as a power series of the smoothed density. The use of a smoothed density in NL-DFT provides an oscillating density profile expected of a fluid adjacent to a sohd surface, the existence of which is corroborated by molecular simulation results [17,18]. 

The interpolation between the low and high density limits, which is inherent to this variational approach, leads in a very natural way to the scaled particle theory for the structure and thermodynamics of isotropic fluids of hard particles. This unifies, for the first time the Percus Yevick theory, which is based on diagram expansions, and the scaled particle theory of Reiss, Frisch and Lebowitz, and, at the same time yields the analytical expressions of the dcf conformal to those of the hard spheres. It provides an unified derivation of the most comprehensive analytic description available of the hard sphere thermodynamics and pair distribution functions as given by the Percus Yevick and scaled particle theories, and yields simple explicit expressions for the higher direct direct correlation functions of the uniform fluid. 

Once the problem was so dearly recognized then a solution, at least in principle, was to hand. Instead of aamniing a random or uniform distribution of molecules we introduce distribution or correlation functions into our expressions for the mean interactions of molecules at two positions in the fluid. These functions are measures of the conditional probability of the occurrence of pairs (or larger groups) of molecules at specified points. Their calculation is one of the principal aims of modem theories liquids. However there are still many problems, particularly those connected with phase transitions, which we cannot solve expliddy in terms of dosed expressions for these distribution functions, and we often have recourse to mean-field approximations even today. 

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