Mean spherical approximation: Difference between revisions
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:<math>c(r) = -\beta \omega(r), ~~~~ r>\sigma.</math> | :<math>c(r) = -\beta \omega(r), ~~~~ r>\sigma.</math> | ||
The | The '''Blum and Hoye''' mean spherical approximation (MSA) (1978-1980) (Refs 2 and 3) closure is given by | ||
closure is given by | |||
:<math>{\rm g}_{ij}(r) \equiv h_{ij}(r) +1=0 ~~~~~~~~ r < \sigma_{ij} = (\sigma_i + \sigma_j)/2</math> | :<math>{\rm g}_{ij}(r) \equiv h_{ij}(r) +1=0 ~~~~~~~~ r < \sigma_{ij} = (\sigma_i + \sigma_j)/2</math> | ||
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:<math>c_{ij}(r)= \sum_{n=1} \frac{K_{ij}^{(n)}}{r}e^{-z_nr} ~~~~~~ \sigma_{ij} < r</math> | :<math>c_{ij}(r)= \sum_{n=1} \frac{K_{ij}^{(n)}}{r}e^{-z_nr} ~~~~~~ \sigma_{ij} < r</math> | ||
where | 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 | molecules of ''i'' and ''j'' species, <math>\sigma_i</math> is the diameter of '''i'' species of molecule.\ | ||
Duh and Haymet (Eq. 9 \cite{JCP_1995_103_02625}) write the MSA approximation as | Duh and Haymet (Eq. 9 \cite{JCP_1995_103_02625}) write the MSA approximation as | ||
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==References== | ==References== | ||
#[PR_1966_144_000251] | #[PR_1966_144_000251] | ||
#[JSP_1978_19_0317_nolotengoSpringer] | |||
#[JSP_1980_22_0661_nolotengoSpringer] | |||
Revision as of 13:10, 23 February 2007
The Lebowitz and Percus mean spherical approximation (MSA) (1966) (Ref. 1) closure is given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c(r) = -\beta \omega(r), ~~~~ r>\sigma.}
The Blum and Hoye mean spherical approximation (MSA) (1978-1980) (Refs 2 and 3) closure is given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm g}_{ij}(r) \equiv h_{ij}(r) +1=0 ~~~~~~~~ r < \sigma_{ij} = (\sigma_i + \sigma_j)/2}
and
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ij}(r)= \sum_{n=1} \frac{K_{ij}^{(n)}}{r}e^{-z_nr} ~~~~~~ \sigma_{ij} < r}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{ij}(r)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ij}(r)} are the total and the direct correlation functions for two spherical molecules of i and j species, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i} is the diameter of 'i species of molecule.\ Duh and Haymet (Eq. 9 \cite{JCP_1995_103_02625}) write the MSA approximation as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(r) = \frac{c(r) + \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}}
where $\Phi_1$ and $\Phi_2$ comes from the WCA division of the LJ potential.\\ By introducing the definition (Eq. 10 \cite{JCP_1995_103_02625})
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(r) = h(r) -c(r) -\beta \Phi_2 (r)}
one can arrive at (Eq. 11 \cite{JCP_1995_103_02625})
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(r) \approx B^{\rm MSA}(s) = \ln (1+s)-s}
The Percus Yevick approximation may be recovered from the above equation by setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_2=0} .
References
- [PR_1966_144_000251]
- [JSP_1978_19_0317_nolotengoSpringer]
- [JSP_1980_22_0661_nolotengoSpringer]