Exact solution of the Percus Yevick integral equation for hard spheres: Difference between revisions

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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)
\begin{equation}
 
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>
\end{equation}
 
where
where
\begin{equation}
 
\eta = \frac{1}{6} \pi R^3 \rho
<math>\eta = \frac{1}{6} \pi R^3 \rho</math>
\end{equation}
 
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)
\begin{equation}
 
\beta P \rho = \frac{(1+\eta+\eta^2)}{(1-\eta)^3}  
<math>\beta P \rho = \frac{(1+\eta+\eta^2)}{(1-\eta)^3}</math>
\end{equation}
 
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)
\begin{equation}
 
h_0(r) = ar+ br^2 + cr^4  
<math>h_0(r) = ar+ br^2 + cr^4</math>
\end{equation}
 
where (Eq. 24)
where (Eq. 24)
\begin{equation}
 
a = \frac{(2x+1)^2}{(x-1)^4}
<math>a = \frac{(2x+1)^2}{(x-1)^4}</math>
\end{equation}
 
and
and
\begin{equation}
 
b= - \frac{12x + 12x^2 + 3x^3}{2(x-1)^4}
<math>b= - \frac{12x + 12x^2 + 3x^3}{2(x-1)^4}</math>
\end{equation}
 
and
and
\begin{equation}
 
c= \frac{x(2x+1)^2}{2(x-1)^4}
<math>c= \frac{x(2x+1)^2}{2(x-1)^4}</math>
\end{equation}
 
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
\begin{equation}
 
P=nkT\frac{(1+2x+3x^2)}{(1-x)^2}
<math>P=nkT\frac{(1+2x+3x^2)}{(1-x)^2}</math>
\end{equation}
 
and the compressibility route is
and the compressibility route is
\begin{equation}
 
P=nkT\frac{(1+x+x^2)}{(1-x)^3}
<math>P=nkT\frac{(1+x+x^2)}{(1-x)^3}</math>
\end{equation}
 
==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

References

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