Cluster distribution in the association description of square-well fluids

An association description of a complex fluid is often helpful by offering a systematic approach to the complicated interactions in such a fluid that determine its thermodynamic behavior. Here, we use the simple example of a square-well fluid in one and three dimensions to identify and illustrate a number of important issues in the design of equations of state based on association equilibria of geometrical clusters. Our main focus is on the cluster distribution, i.e., on the densities of monomers, dimers, trimers etc. at a given temperature and overall density. In one dimension (1D), the cluster distribution can be obtained exactly, as we show here; in three dimensions (3D), an exact solution does not seem to be feasible. For this reason, we compare the exact results in 1D with data obtained from computer simulations, with the predictions of the association theory formulated by Fantoni et al., and with results of an approach based on the virial expansion that we proposed in an earlier work. These three methods can be applied in three dimensions as well, and we carry over the lessons learned in 1D to the 3D case to identify strengths and shortcomings of the theoretical approaches. Our findings emphasize the importance of accurate activity coefficients, which are available in 1D, but have to be approximated in 3D. For this purpose, the size and the shape of the exclusion volumes of clusters have to be known. A phenomenon that seems to have been neglected so far in this context is the effective shrinking of clusters as the overall density is increased; this effect becomes particularly important at high densities.