Integral equations and the pressure at the liquid solid interface

The relationship between the pressure in a fluid and the density-functional which controls the density profile of a fluid confined between two walls is examined. As a result, the conditions which must be fulfilled by an approximate density functional to yield a bulk pressure P, a normal pressure, P W at the interface between a hard wall and the fluid, and a fluid density, ¤üW, adjacent to the wall which satisfy the exact relations P W=P=k B T ¤üW, are established. The density-functional which yields the HNC closure for a hard-sphere fluid near a hard wall is modified so that the modified functional yields the Carnahan-Starling bulk pressure and hence fulfils the necessary conditions. The density profile to which this gives rise is compared with the results of computer simulation for various bulk densities when the fluid is bounded by two hard walls separated by a distance equal to 8 diameters of the hard sphere. The agreement is found to be very good even at a bulk reduced density of 0·91.