Hard Spheres and Lennard-Jones Particles

This function is depicted in Fig. 5.9. Note that the force is attractive everywhere in the range a R 2a, i.e., 

This is quite a surprising result if we remember that we are dealing with hard-sphere particles. We now examine this phenomenon for a system of three almost hard spheres, two of them fixed at Ri and R2 (with R = IR2 - Ri ), and a third particle serving as a solvent. Clearly, this is the simplest solvent we can envisage besides a vacuum. The 

FIGURE 5.9. The forms of W R)/kT for hard spheres at low densities. The curves are computed for spheres of diameter CT = 1.0 and two densities, p = 0.1 and p = 0.4. Note that for 1 / 2, we have an attractive potential of average force. 

FIGURE 5.10. An almost-hard-sphere potential function. As the angle a tends to zero, one gets the hard-sphere potential U R). 

FIGURE 5.11. Two almost-hard spheres, numbered 1 and 2, at a distance of cr 2(7. A third particle, 

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Computer simulation results for S2 are somewhat sparse and involve the usual uncertainties involved in extrapolating results for a truncated T(r) used in a periodic box to untruncated T(r) in an infinite system." Nevertheless for polarizable hard-sphere and Lennard-Jones particles, it is probably safe to say that the estimates currently available from the combined use of analytic and simulation input are enough to provide a reliable guide to the p and dependence of Sj over the full fluid range of those variables. The most comprehensive studies of have been made by Stell and Rushbrooke" and by Graben, Rushbrooke, and Stell," for the hard-sphere and Lennard-Jones cases, respectively. Both these works utilize the simulation results of Alder, Weis, and Strauss," as well as exact density-expansion results, and numerical results of the Kirkwood superposition approximation... 

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