A Study of Pair Correlation Functions Using Classical DFT

At the heart of physics of fluids are particle distribution functions. If all of these functions of a fluid are known, the state can be fully described. With a universal theory of particle distribution functions, physics of fluids is done. Of particular interest is the radial distribution function (rdf), which is related to the second particle distribution function, because the average excess internal energy, pressure and isothermal compressibility naturally follow from it. Here the function is obtained for `soft' particles by calculating the density profile around a particle fixed in the origin, acting as an external potential. This is called the test-particle method. In order to theoretically describe `soft' particles, the short ranged (repulsive) forces and the long ranged (usually attractive) forces of the interaction potential are separated. The repulsive forces are calculated from a weighted density theory (FMT) and the long ranged forces are added as a `small' perturbation. The FMT has been proven to be accurate, but in order to describe the perturbation well we need to know how particles correlate for larger inter-particle separations in inhomogeneous systems. In particular for inhomogeneous systems it is difficult to say something about this but we can distinguish approximations for high and low densities. The result is a 1D non-linear integral equation with squares, cubes, fractions and logarithms in the weighted form of the unknown for the reference system and a perturbation which is linear in the unknown. The results are satisfactory for high and low densities, but for the intermediate range the results are less satisfactory. For a range of high densities we obtain a measure for the short-ranged correlations by fitting with MD results.