Obtaining effective pair potentials in colloidal monolayers using a thermodynamically consistent inversion scheme

The structure and stability of colloidal monolayers depends crucially on the effective pair interaction potential u(r) between colloidal particles. In this study, we construct a novel method for extracting u(r) from the two-dimensional (2D) radial distribution function g(r) of dense colloidal monolayers. The method is based on the Ornstein-Zernike relation and the HMSA closure first proposed by Zerah and Hansen (Zerah, G.; Hansen, J.-P. Self-consistent integral equations for fluid pair distribution functions: Another attempt. J. Chem. Phys. 1986, 84(4), 2336-2343). The HMSA closure contains a single fitting parameter which is determined by requiring thermodynamic consistency between the virial and compressibility equations of state. The accuracy of the HMSA inversion scheme is compared to a 2D predictor corrector scheme based on hard-disk fluids (HDPC) previously proposed by us (Law, A. D.; Buzza, D. M. A. Determination of interaction potentials of colloidal monolayers from the inversion of pair correlation functions: A two-dimensional predictor-corrector method. J. Chem. Phys. 2009, 131, 094704) and the conventional "one-step" inversion methods of HNC and Percus-Yevick (PY). The accuracy of all these schemes is tested against Monte Carlo simulation data for g(r) from monolayers interacting via a range of commonly encountered potentials, including both purely repulsive potentials and potentials containing an attractive well. For all the potentials studied, we find that the accuracy of the HMSA and HDPC schemes is superior to HNC and PY, especially as we go to higher densities. The HDPC and HMSA schemes are particularly accurate for hard-core and soft-core fluids, respectively, at high density and are therefore complementary to each other. Finally, we find that, even in the presence of experimentally realistic levels of noise in the input g(r) data, both HMSA and HDPC schemes are able to faithfully extract the salient features of the underlying interaction potentials. Both these schemes therefore provide convenient and accurate methods for extracting u(r) from experimental g(r) data for general 2D monolayers.