Mean spherical approximation

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The mean spherical approximation (MSA) closure relation of Lebowitz and Percus is given by [1]:

c(r) = -\beta \omega(r), ~ ~ ~ ~ r>\sigma.

In the Blum and Høye mean spherical approximation for mixtures the closure is given by [2] [3]:

{\rm g}_{ij}(r) \equiv h_{ij}(r) +1=0 ~ ~ ~ ~ ~ ~ ~ ~ r < \sigma_{ij} = (\sigma_i + \sigma_j)/2


c_{ij}(r)= \sum_{n=1} \frac{K_{ij}^{(n)}}{r}e^{-z_nr} ~ ~ ~ ~  ~ ~ \sigma_{ij} < r

where h_{ij}(r) and c_{ij}(r) are the total and the direct correlation functions for two spherical molecules of i and j species, \sigma_i is the diameter of i species of molecule. Duh and Haymet (Eq. 9 in [4]) write the MSA approximation as

g(r) = \frac{c(r) + \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}

where \Phi_1 and \Phi_2 comes from the Weeks-Chandler-Andersen division of the Lennard-Jones potential. By introducing the definition (Eq. 10 in [4])

\left.s(r)\right. = h(r) -c(r) -\beta \Phi_2 (r)

one can arrive at (Eq. 11 in [4])

B(r) \approx B^{\rm MSA}(s) = \ln (1+s)-s

The Percus Yevick approximation may be recovered from the above equation by setting \Phi_2=0.

Thermodynamic consistency[edit]