(258f) On the Use of Scaled Particle Theory to Determine a Broad Range of Hard-Sphere Fluid Properties with High Accuracy

The thermophysical and structural properties of hard-sphere fluids are important for various theoretical descriptions of liquids, glasses and complex fluids. For example, the reversible work of forming a cavity (a spherical region devoid of particle centers) in a hard-sphere fluid is used in several solvation theories of liquids, in which one attempts to predict the free energy cost of adding a solute molecule to a chosen solvent. In addition, since the depletion, or entropic, potential of a colloid particle at a certain position is equal to the difference in the free energy between that position and a reference position far away from another particle or surface, certain methods developed to predict the depletion potential require an accurate estimate of the work of cavity formation. Furthermore, various models of liquid behavior require knowledge of other hard-sphere fluid properties, such as the pressure, the surface (or boundary) tension and/or excess surface adsorption of the hard-sphere fluid near a cavity surface or another hard wall, as well as important structural information, including the nearest-neighbor probability density distribution and the pair correlation function for two hard spheres at or near contact.

Many theories for predicting most of the above mentioned properties of hard-sphere fluids stem from the Scaled Particle Theory (SPT) of Reiss, Frisch and Lebowitz (Reiss et al., 1959, J. Chem. Phys., 31, 369). SPT predicts the work of cavity formation in the hard-sphere fluid exactly for cavities smaller than a hard-sphere solvent particle and develops an interpolation function that spans from small cavity radii to the limit of macroscopic size. The interpolation is based on a number of exact conditions imposed by geometry and thermodynamics and yields an equation of state (EOS). By suitable manipulation, this interpolation function also provides information about, say, the surface tension, excess surface adsorption, and the pair correlation function near contact. Unfortunately, no current version of SPT can produce all of these properties, including the EOS, with high accuracy. The original form of SPT provides a very good estimate of the surface tension, though its pressure predictions are relatively poor at high packing fractions. In contrast, one recent version of SPT produces an accurate EOS at all packing fractions, yet fails to predict the surface tension with the same accuracy. Consequently, ongoing effort has been directed to improve current forms of SPT such that it is able to reproduce all properties of interest with relative accuracy.

First, we present a simple method for computing a broad range of important hard-sphere properties within the framework of SPT by adapting the interpolation method to an arbitrary EOS. In addition to incorporating several exact conditions of SPT, the interpolation utilizes both the pressure and chemical potential of the hard sphere fluid as provided by a particular EOS, such as the Carnhan-Starling, and so is now thermodynamically consistent (which is in contrast to several previous versions of SPT adapted to a given EOS that neglected to enforce thermodynamic consistency). Our new interpolation method provides very accurate predictions of both the work of cavity formation and hard sphere interfacial properties when any accurate hard sphere equation of state is used. Second, we also revisit the recently proposed six-condition version of SPT (Heying and Corti, 2004, J. Phys. Chem. B, 108, 19756) and similarly modify its chosen interpolation function to produce, now self-consistently, a comparably accurate EOS along with accurate predictions of the surface tension. Both modified versions of SPT highlight the importance of an appropriately chosen SPT interpolation function. Unlike what may have been suggested by previous forms of SPT, we show that a more complex interpolation function is needed to describe both small and macroscopic cavities. Due to its reliance upon geometric and physical arguments, a careful examination of the finer details of SPT should lead to an improved understanding of the thermodynamics of hard-sphere fluids.