Exact solution of the Percus Yevick integral equation for hard spheres

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 $x=\rho /4$. 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

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

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)