Difference between revisions of "Mean spherical approximation"

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:<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>{\rm g}_{ij}(r) \equiv h_{ij}(r) 1=0 ~~~~~~~~ r < \sigma_{ij} = (\sigma_i   \sigma_j)/2</math>
  
  
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:<math>g(r) = \frac{c(r) + \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}</math>
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:<math>g(r) = \frac{c(r)   \beta \Phi_2(r)}{1-e^{\beta \Phi_1(r)}}</math>
  
  
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:<math>B(r) \approx B^{\rm MSA}(s) = \ln (1+s)-s</math>
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:<math>B(r) \approx B^{\rm MSA}(s) = \ln (1 s)-s</math>
  
  

Revision as of 11:59, 4 July 2007

The Lebowitz and Percus mean spherical approximation (MSA) (1966) (Ref. 1) closure relation is given by


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


In the Blum and Høye mean spherical approximation for mixtures (Refs 2 and 3) the closure is given by


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


and

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 Ref. 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-Anderson division of the Lennard-Jones potential. By introducing the definition (Eq. 10 Ref. 4)


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


one can arrive at (Eq. 11 in Ref. 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

See Ref. 5.

References

  1. J. L. Lebowitz and J. K. Percus "Mean Spherical Model for Lattice Gases with Extended Hard Cores and Continuum Fluids", Physical Review 144 pp. 251 - 258 (1966)
  2. L. Blum and J. S. Høye "Solution of the Ornstein-Zernike equation with Yukawa closure for a mixture", Journal of Statistical Physics, 19 pp. 317-324 (1978)
  3. Lesser Blum "Solution of the Ornstein-Zernike equation for a mixture of hard ions and Yukawa closure" Journal of Statistical Physics, 22 pp. 661-672 (1980)
  4. Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics 103 pp. 2625-2633 (1995)
  5. Andrés Santos "Thermodynamic consistency between the energy and virial routes in the mean spherical approximation for soft potentials" Journal of Chemical Physics 126 116101 (2007)