Exact solution of the Percus Yevick integral equation for hard spheres: Difference between revisions
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Carl McBride (talk | contribs) m (New page: The exact solution for the Percus-Yevick integral equation for hard spheres was derived by M. S. Wertheim in 1963 \cite{PRL_1963_10_000321} (See also \cite{JMP_1964_05_00643}) (and for mix...) |
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(and for mixtures by in Lebowitz 1964 \cite{PR_1964_133_00A895}). | (and for mixtures by in Lebowitz 1964 \cite{PR_1964_133_00A895}). | ||
The direct correlation function is given by (\cite{PRL_1963_10_000321} Eq. 6) | The direct correlation function is given by (\cite{PRL_1963_10_000321} 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} | <math>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}</math> | ||
where | where | ||
\eta = \frac{1}{6} \pi R^3 \rho | <math>\eta = \frac{1}{6} \pi R^3 \rho</math> | ||
and $R$ is the hard sphere diameter.\\ | and $R$ is the hard sphere diameter.\\ | ||
The equation of state is (\cite{PRL_1963_10_000321} Eq. 7) | The equation of state is (\cite{PRL_1963_10_000321} Eq. 7) | ||
\beta P \rho = \frac{(1+\eta+\eta^2)}{(1-\eta)^3} | <math>\beta P \rho = \frac{(1+\eta+\eta^2)}{(1-\eta)^3}</math> | ||
Everett Thiele (1963 \cite{JCP_1963_39_00474}) also studied this system, | Everett Thiele (1963 \cite{JCP_1963_39_00474}) also studied this system, | ||
resulting in (Eq. 23) | resulting in (Eq. 23) | ||
h_0(r) = ar+ br^2 + cr^4 | <math>h_0(r) = ar+ br^2 + cr^4</math> | ||
where (Eq. 24) | where (Eq. 24) | ||
a = \frac{(2x+1)^2}{(x-1)^4} | <math>a = \frac{(2x+1)^2}{(x-1)^4}</math> | ||
and | and | ||
b= - \frac{12x + 12x^2 + 3x^3}{2(x-1)^4} | <math>b= - \frac{12x + 12x^2 + 3x^3}{2(x-1)^4}</math> | ||
and | and | ||
c= \frac{x(2x+1)^2}{2(x-1)^4} | <math>c= \frac{x(2x+1)^2}{2(x-1)^4}</math> | ||
and where $x=\rho/4$.\\ | and where $x=\rho/4$.\\ | ||
The pressure via the pressure route (Eq.s 32 and 33) is | The pressure via the pressure route (Eq.s 32 and 33) is | ||
P=nkT\frac{(1+2x+3x^2)}{(1-x)^2} | <math>P=nkT\frac{(1+2x+3x^2)}{(1-x)^2}</math> | ||
and the compressibility route is | and the compressibility route is | ||
P=nkT\frac{(1+x+x^2)}{(1-x)^3} | <math>P=nkT\frac{(1+x+x^2)}{(1-x)^3}</math> | ||
==References== |
Revision as of 13:20, 23 February 2007
The exact solution for the Percus-Yevick integral equation for hard spheres was derived by M. S. Wertheim in 1963 \cite{PRL_1963_10_000321} (See also \cite{JMP_1964_05_00643}) (and for mixtures by in Lebowitz 1964 \cite{PR_1964_133_00A895}). The direct correlation function is given by (\cite{PRL_1963_10_000321} Eq. 6)
where
and $R$ is the hard sphere diameter.\\ The equation of state is (\cite{PRL_1963_10_000321} Eq. 7)
Everett Thiele (1963 \cite{JCP_1963_39_00474}) also studied this system, resulting in (Eq. 23)
where (Eq. 24)
and
and
and where $x=\rho/4$.\\ The pressure via the pressure route (Eq.s 32 and 33) is
and the compressibility route is