Simple and exact expressions have been found for the compressibility and pressure equations of state predicted by the Percus—Yevick equation for hard spheres. The equations of state are in good agreement with the machine calculations of Wainwright and Alder, and Wood, Parker, and Jacobson.

Equation of State for Hard Spheres

The pair distribution function for a classical fluid in thermal equilibrium is found to be more closely approximated by the Percus and Yevick (Phys. Rev., 110: 1(1958)) approximation than by the Bogoliubov-Born-Green- Kirkwood-Yvon (B.B.G.K.Y.) approximation or the hypernettedchain approximation. It is noted that the reason for this finding lies in the fact that the Percus and Yevick approximation chooses a quite advantageous function for making a Taylor expansion of the appropriate equation for a grand canonical ensemble. (T.F.H.)

Approximation Methods in Classical Statistical Mechanics

The entropic depletion force, in colloids, arises when large particles are placed in a solution of smaller ones, and sterically constrained to avoid them. In this paper, we consider a system of two parallel plates suspended in a semidilute solution of long thin rods of length L and diameter D. By numerically solving an integral equation, which is exact in the “Onsager limit” (D≪L), we obtain the depletion force between the plates. The second integral of this determines (via the Derjaguin approximation) the depletion potential between two large hard spheres of radius R, immersed in a solution of hard rods (satisfying D≪L≪R). The results for this potential are compared with our previous second order perturbation treatment [Y. Mao, M. E. Cates, and H. N. W. Lekkerkerker, Phys. Rev. Lett. 75, 4548 (1995)], as well as with newly computed third order perturbation results. There is good agreement at low and intermediate densities (which validates our numerical procedures for the integral equation) but the gradual failure of the perturbative treatments is revealed as the density increases. From the numerical results, we conclude that for typical colloidal sphere/rod mixtures, the repulsive barrier in the depletion interaction is likely to be less than the thermal energy kBT throughout the semidilute concentration range of the rods. This reverses our previous conclusion, based on second order perturbation theory, that this barrier could easily lead to kinetic stabilization of the mixture. However, it is possible that terms of order D/L (neglected here) could further modify the results. Within the present theory, the main modification to the well-known (attractive) first order interaction, as the density is raised, is by changing the depth and range of the primary attractive well.

Theory of the depletion force due to rodlike polymers

Numerical solutions of the Percus–Yevick equation for the Lennard‐Jones (6–12) potential are presented largely for high densities and the liquid region in a form particularly useful to workers who have access only to small computers which cannot be used for solving the PY equation. The solutions, given in a form suitable for calculating either the radial distribution functions or direct correlation functions, are deposited with the National Auxiliary Publications Service (Document No. NAPS‐00720). A table of densities and temperatures at which the PY solutions are available is given. Energies and pressures, including the compressibility pressure for temperatures below the critical, are also tabulated and comparisons are made with molecular dynamics results. Solutions for the hard‐sphere potential are also presented completing the tables of Throop and Bearman [J. Chem. Phys. 43, 2408 (1965)] for large intermolecular separations.

Numerical Solutions of the Percus–Yevick Equation for the Lennard‐Jones (6–12) and Hard‐Sphere Potentials

An efficient method of solving the Percus‐Yevick and related equations is described. The method is applied to a Lennard‐Jones fluid, and the solutions obtained are discussed. It is shown that the Percus‐Yevick equation predicts a phase change with critical density close to 0.27 and with a critical temperature which is dependent upon the range at which the Lennard‐Jones potential is truncated. At the phase change the compressibility becomes infinite although the virial equation of state does not show this behavior. Outside the critical region the PY equation is at least two‐valued for all densities in the range (0, 0.6).

Percus‐Yevick Equation Applied to a Lennard‐Jones Fluid

The radial distribution function for a classical fluid of particles interacting with the Lennard‐Jones potential has been computed by solving the Percus‐Yevick integral equation numerically. The solutions and the quantities, p/nkT and E/NkT, are compared with those obtained by Wood and Parker using Monte Carlo techniques. The radial distribution functions agree to better than 15% beyond the first points where they are unity while the thermodynamic quantities differ by, at most, 3% from the range of Monte Carlo values for cases where the system is believed to be in a liquid state. The quantity K=−(1/V)(∂V/∂p)N,T, has also_been computed.

Solutions to the Percus‐Yevick Equation

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Solutions to the Percus‐Yevick Equation