Simple Integral Equation and Density Functional Study of a Hard Sphere Fluid in a Pore Formed by Two Hard Walls D. Henderson* Department of Chemistry and Biochemistry, Brigham Young UniVersity, ProVo, Utah 84602 S. Sokolowski Department for the Modelling of Physico-Chemical Processes, Faculty of Chemistry, MCS UniVersity, 20031 Lublin, Poland D. T. Wasan Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Illinois 60616 ReceiVed: October 24, 1997; In Final Form: January 21, 1998 The structure of a hard sphere fluid in a pore formed by two parallel hard walls is studied as a function of the separation of the walls using the singlet theory of Henderson et al. and Lozada-Cassou with the Percus- Yevick (PY) and hypernetted chain (HNC) closures, which were studied previously, and a modified version of the Verlet (MV) closure. In addition, the density functional formalism of Tarazona et al. (TME) is employed. As noted earlier, the singlet theory yields poor results when the PY and HNC closures are used but yields quite good results when the MV closure is used. The TME approach also yields good results. The MV and singlet theory results are comparable in accuracy with those obtained from the second-order PY (PY2) theory but are much less demanding computationally. Recently, we 1 studied a hard sphere fluid in a pore formed by two parallel hard walls, separated by a distance H using the singlet theory of Lozada-Cassou 2 with the Percus-Yevick (PY) and hypernetted chain (HNC) closures. This approach is a singlet level theory in the sense that the coordinates of only one hard sphere are considered; however, a total of three particles (one fluid particle and two giant, or surface, particles) are considered. The Lozada-Cassou theory is an extension of singlet theory of Henderson et al. 3 (HAB). It was found that the singlet theory gave poor results with these closures. This is not surprising since it is well-known that the HAB approach can be inaccurate when these closures are used. In contrast, good agreement with the simulations of Wertheim et al. 4 were obtained when a second-order Percus-Yevick (PY2) theory was used. The PY2 theory is a second-order theory in the sense that the coordinates of two hard spheres are considered. However, a total of four particles are considered. As a result, the calculations are difficult. Some time ago, it was shown 5 that good results both for a hard sphere fluid and for a hard sphere fluid near a single wall, i.e., the HAB approach, were obtained with a modified Verlet 6 (MV) closure. This point has been made again in a recent paper. 7 Thus, one might hope that the MV closure might yield good results for a narrow pore. One aim of this work is to examine this hypothesis. In addition, we apply the density functional (DF) approach of Tarazona et al. (TME), 8,9 with the Carnahan-Starling free energy for hard spheres, to this system. This has been done previously; however, we include these results for comparison. We consider a fluid of hard spheres of diameter d between two parallel hard walls, separated by the distance H. Thus, H -d is the separation between the two planes of closest approach of the hard sphere fluid. The original version of the Ornstein- Zernike (OZ) equation, useful for bulk fluids, is well-known. HAB observed that the OZ formalism can be applied to inhomogeneous fluids. The HAB equation is where h and c are the total and direct correlation functions for the fluid-pore pair, F is the density of the bulk fluid, and c B (t) is the direct correlation function of the bulk fluid, which is presumed to be known. Lozada-Cassou made the important observation that eq 1 can be applied to a pore as well as to a fluid near a single wall. The PY and HNC theories result from the combination of eq 1 and the PY closure, or the HNC closure, In the above equations, γ(r) ) h(r) - c(r), and y(r) is the cavity or background function. The function C(s) is obtained using some approximation for the bulk fluid. We use the PY result because it is quite accurate and is analytic so that the integration in eq 2 can be performed analytically. Further details about the original OZ equation, closures, and the extensions discussed above can be found in a recent book on inhomogeneous fluids. 10 The results of the singlet theory with the PY and HNC closures are displayed in Figures 1-3. The density profile is h(z 1 ) - c(z 1 ) )F ∫ -∞ ∞ h(z 2 ) C(|z 1 - z 2 |)dz 2 (1) C(s) ) 2π ∫ s ∞ tc B (t)dt (2) γ(r) ) y(r) - 1 (3) γ(r) ) ln y(r) (4) 3009 J. Phys. Chem. B 1998, 102, 3009-3011 S1089-5647(97)03444-5 CCC: $15.00 © 1998 American Chemical Society Published on Web 03/31/1998