The Ornstein-Zernike equation for hard spheres near a hard wall

Abstract

Hard-sphere-hard-wall density profiles are studied using the Henderson-Abraham-Barker (HAB) formulation of the Ornstein-Zernike (OZ) equation. A new method of numerically solving this equation is proposed that consists of a combination of Newton-Raphson and Picard iteration approaches. The new method is rapidly convergent and insensitive to the initial estimate. A new reference hypernetted chain (RHNC) theory for the OZ equation is proposed that uses the bridge function of a bulk fluid hard sphere reference system, and this is tested against simulation data and the results of alternative theories at reduced bulk fluid number densities up to 0·913. The proposed RHNC theory gives excellent results for the density profiles. The Percus-Yevick (PY) and hypernetted chain (HNC) theories give rather poor results. Hybrid theories, such as the PY/HNC (the PY bulk fluid closure and the HNC wall fluid closure) and the HNC/PY are even worse, as are those obtained by combining accurate bulk-fluid correlation functions with the PY and the HNC closures for the wall fluid function. The Martynov-Sarkisov (MS/MS) theory is better than PY/PY and HNC/HNC, but solutions do not exist above number density 0·675. We also provide evidence of the superiority of the RHNC theory over other alternative methods, including those based on the Born-Green-Yvon, density functional and non-uniform OZ approaches. We believe that the RHNC is the best currently available theory for the hard-sphere-hard-wall density profile.