Pair correlation function and density profile of a liquid in contact with rigid wall

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Abstract
The numerical solution of the first two equations of the Born-Green-Yvon-Bogolyubov hierarchy of equations is reported for a liquid in the vicinity of an ideal rigid wall. The superposition approximation is applied as a closure in order to truncate the hierarchy. The pair correlation function for this geometry depends upon the distance between the particles, distance of the particles from the wall and orientation of the pair with respect to the wall. The dependence upon the first two variables was taken explicitly into account while the dependence upon the angle defining the orientation was overcome by two trigonometric terms of polar angle. The system of integral equations was solved for two points in the phase space of the Lennard-Jones system: ρ0σ3 = 0·65, kT/ε = 3·67 and ρ0σ3 = 0.45, kT/ε = 2·93. The results show that the pair correlation function deviates appreciably from the isotropic liquid bulk behaviour. The resulting density profile is rather close to the one obtained with the homogeneous and isotropic pair correlation function which means that a simple ansatz for the pair correlation function leads to fortuitous cancellation of errors in the calculations of the density profile.