Exact solution of the Percus Yevick integral equation for hard spheres

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The exact solution for the Percus Yevick integral equation for the hard sphere model was derived by M. S. Wertheim in 1963 Ref. 1 (See also Ref. 2) (and for mixtures by in Lebowitz 1964 Ref. 3). The direct correlation function is given by (Ref. 1 Eq. 6)

C(r/R) = - \frac{(1+2\eta)^2 - 6\eta(1+ \frac{1}{2} \eta)^2(r/R) + \eta(1+2\eta)^2\frac{(r/R)^3}{2}}{(1-\eta)^4}

where

\eta = \frac{1}{6} \pi R^3 \rho

and R is the hard sphere diameter. The equation of state is (Ref. 1 Eq. 7)

\beta P \rho = \frac{(1+\eta+\eta^2)}{(1-\eta)^3}

Everett Thiele (1963 Ref. 4}) also studied this system, resulting in (Eq. 23)

\left.h_0(r)\right. = ar+ br^2 + cr^4

where (Eq. 24)

a = \frac{(2x+1)^2}{(x-1)^4}

and

b= - \frac{12x + 12x^2 + 3x^3}{2(x-1)^4}

and

c= \frac{x(2x+1)^2}{2(x-1)^4}

and where x = ρ / 4. The pressure via the pressure route (Eq.s 32 and 33) is

P=nkT\frac{(1+2x+3x^2)}{(1-x)^2}

and the compressibility route is

P=nkT\frac{(1+x+x^2)}{(1-x)^3}

[edit] References

  1. M. S. Wertheim "Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres", Physical Review Letters 10 321 - 323 (1963)
  2. M. S. Wertheim "Analytic Solution of the Percus-Yevick Equation", Journal of Mathematical Physics, 5 pp. 643-651 (1964)
  3. J. L. Lebowitz, "Exact Solution of Generalized Percus-Yevick Equation for a Mixture of Hard Spheres", Physical Review 133 pp. A895 - A899 (1964)
  4. Everett Thiele "Equation of State for Hard Spheres", Journal of Chemical Physics, 39 pp. 474-479 (1963)
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