Bifurcation of solutions of Ornstein–Zernike equation in the gas–liquid coexistence region of simple fluids

Abstract

One objective of this paper is to investigate bifurcation of solutions of Ornstein–Zernike equation in liquid state theory for the gas–liquid coexistence region of simple fluids. The analysis uses a discrete form of the Ornstein–Zernike equation in real space, together with a closure relation which provides nearly self consistent thermodynamic properties of fluids governed by interaction potentials like the Lennard-Jones potential. Accurate asymptotic forms of correlation functions are incorporated in the real space algorithm so as to mitigate the influence of cutoff length used in the discrete approximations. The global error reduction of this algorithm is similar to that of Simpson’s rule. It is found that there is no spinodal on the vapor side on a sub-critical isotherm. However, there is a density at which compressibility vanishes, but this lies on a nonphysical branch. There are fold bifurcation points on the vapor and liquid sides together with a ‘no-real-solution-region’ in between. This is followed by a region of negative compressibility extending up to the spinodal on the liquid side. Importantly, it is found that in regions where compressibility is non-positive, there are infinite number of solutions to OZE which violate an integral constraint that is needed for applications to thermodynamics. The emerging picture is somewhat similar to that in the case of the hypernetted-chain-closure. It is also found that complex solutions connect the physical solutions that exist outside the ‘no-real-solution-region’ as well as in region of negative compressibility. The second objective of the paper, in the light of bifurcation scenario, is to evolve an interpretation of the coupling-parameter expansion of correlation functions pertaining to the region of negative compressibility. It is shown that the expansion provides an accurate description of results of the restricted ensembles, implied in mean field theories and simulation methods. Bifurcation of singlet density to an in-homogeneous distribution at a spinodal is also established using a simple equation, which uses the direct correlation function. Thus the paper re-establishes the fact that restriction to a homogeneous singlet density is the root cause of all inconsistencies found in the gas–liquid coexistence region.