Editing Mean spherical approximation
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The '''mean spherical approximation | The '''Lebowitz and Percus''' mean spherical approximation (MSA) (1966) (Ref. 1) closure is given by | ||
:<math>c(r) = -\beta \omega(r), ~~~~ r>\sigma.</math> | |||
The '''Blum and Hoye''' mean spherical approximation (MSA) (1978-1980) (Refs 2 and 3) closure is given by | |||
:<math>{\rm g}_{ij}(r) \equiv h_{ij}(r) +1=0 ~~~~~~~~ r < \sigma_{ij} = (\sigma_i + \sigma_j)/2</math> | |||
and | and | ||
:<math>c_{ij}(r)= \sum_{n=1} \frac{K_{ij}^{(n)}}{r}e^{-z_nr} ~ ~ ~ ~ | :<math>c_{ij}(r)= \sum_{n=1} \frac{K_{ij}^{(n)}}{r}e^{-z_nr} ~~~~~~ \sigma_{ij} < r</math> | ||
where <math>h_{ij}(r)</math> and <math>c_{ij}(r)</math> are the total and the direct correlation functions for two spherical | |||
molecules of ''i'' and ''j'' species, <math>\sigma_i</math> is the diameter of '''i'' species of molecule. | |||
Duh and Haymet (Eq. 9 Ref. 4) write the MSA approximation as | |||
:<math>g(r) = \frac{c(r) + \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}</math> | :<math>g(r) = \frac{c(r) + \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}</math> | ||
where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the [[WCA division]] of the [[Lennard-Jones]] potential. | |||
By introducing the definition (Eq. 10 \cite{JCP_1995_103_02625}) | |||
:<math>s(r) = h(r) -c(r) -\beta \Phi_2 (r)</math> | |||
:<math> | |||
one can arrive at (Eq. 11 \cite{JCP_1995_103_02625}) | |||
:<math>B(r) \approx B^{\rm MSA}(s) = \ln (1+s)-s</math> | :<math>B(r) \approx B^{\rm MSA}(s) = \ln (1+s)-s</math> | ||
The [[Percus Yevick]] approximation may be recovered from the above equation by setting <math>\Phi_2=0</math>. | The [[Percus Yevick]] approximation may be recovered from the above equation by setting <math>\Phi_2=0</math>. | ||
==References== | ==References== | ||
#[PR_1966_144_000251] | |||
#[JSP_1978_19_0317_nolotengoSpringer] | |||
#[JSP_1980_22_0661_nolotengoSpringer] | |||
[ | #[JCP_1995_103_02625] |