Martynov Sarkisov: Difference between revisions
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'''Martynov''' and '''Sarkisov''' proposed | '''Martynov''' and '''Sarkisov''' proposed | ||
an expansion of the | an expansion of the [[bridge function]] in terms of basis functions: | ||
:<math>B(\rho, T, r)= - \sum_{i=1}^\infty A_i (\rho,T) \phi^i (\rho, T, r)</math> | :<math>B(\rho, T, r)= - \sum_{i=1}^\infty A_i (\rho,T) \phi^i (\rho, T, r)</math> | ||
| Line 6: | Line 6: | ||
where <math>\phi</math> is the chosen basis function and <math>A_i</math> are the coefficients determined from | where <math>\phi</math> is the chosen basis function and <math>A_i</math> are the coefficients determined from | ||
[[thermodynamic consistency]] conditions. | [[thermodynamic consistency]] conditions. | ||
The Martynov-Sarkisov closure is based on | The Martynov-Sarkisov [[Closure relations | closure]] is based on | ||
the expansion of the | the expansion of the bridge function in powers of the [[thermal potential]]. | ||
The closure in terms of the bridge function (Eq. 16 of <ref>[http://dx.doi.org/10.1080/00268978300102111 G. A. Martynov; G. N. Sarkisov "Exact equations and the theory of liquids. V", Molecular Physics '''49''' 1495-1504 (1983)]</ref>), for [[hard sphere model | hard sphere]]s, is | |||
:<math>B[\omega(r)]= - A_2 \omega(r_{12})^2 = \sqrt{(1+2\gamma(r))}-\gamma(r) -1</math> | :<math>B[\omega(r)]= - A_2 \omega(r_{12})^2 = \sqrt{(1+2\gamma(r))}-\gamma(r) -1</math> | ||
where <math>\omega(r)</math> is the thermal potential and <math>A_2=1/2</math>. (This closure formed the basis for the | where <math>\omega(r)</math> is the thermal potential and <math>A_2=1/2</math>. (This closure formed the basis for the | ||
[[Ballone-Pastore-Galli- | [[Ballone-Pastore-Galli-Gazzillo]] closure for hard sphere mixtures). | ||
Charpentier and Jaske ( | Charpentier and Jaske <ref>[http://dx.doi.org/10.1063/1.1332808 I. Charpentier and N. Jakse "Exact numerical derivatives of the pair-correlation function of simple liquids using the tangent linear method", Journal of Chemical Physics '''114''' pp. 2284-2292 (2001)]</ref> have | ||
observed that the value of <math>A_2</math> differs drastically from 0.5 for temperatures | observed that the value of <math>A_2</math> differs drastically from 0.5 for temperatures | ||
greater than <math>T^*\approx 2.74</math>, thus the Martynov-Sarkisov closure is deficient in the supercritical domain. | greater than <math>T^*\approx 2.74</math>, thus the Martynov-Sarkisov closure is deficient in the supercritical domain. | ||
==References== | ==References== | ||
<references/> | |||
'''Related reading''' | |||
doi | *[http://dx.doi.org/10.1063/1.466138 G. Sarkisov and D. Tikhonov "Martynov–Sarkisov integral equation for the simple fluids" Journal of Chemical Physics '''99''' pp. 3926-3932 (1993)] | ||
*[http://dx.doi.org/10.1063/1.478276 G. A. Martynov, G. N. Sarkisov and A. G. Vompe "New closure for the Ornstein–Zernike equation" Journal of Chemical Physics '''110''' pp. 3961-3969 (1999)] | |||
*[http://dx.doi.org/10.1063/1.1365107 Gari Sarkisov "Approximate integral equation theory for classical fluids", Journal of Chemical Physics '''114''' pp. 9496-9505 (2001)] | |||
[[Category: Integral equations]] | |||
Latest revision as of 13:38, 11 November 2009
Martynov and Sarkisov proposed an expansion of the bridge function in terms of basis functions:
where is the chosen basis function and are the coefficients determined from thermodynamic consistency conditions. The Martynov-Sarkisov closure is based on the expansion of the bridge function in powers of the thermal potential.
The closure in terms of the bridge function (Eq. 16 of [1]), for hard spheres, is
![{\displaystyle B[\omega (r)]=-A_{2}\omega (r_{12})^{2}={\sqrt {(1+2\gamma (r))}}-\gamma (r)-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36cd24f0b254c56f94c7bb57e712e81f07721a2b)
where is the thermal potential and . (This closure formed the basis for the Ballone-Pastore-Galli-Gazzillo closure for hard sphere mixtures). Charpentier and Jaske [2] have observed that the value of differs drastically from 0.5 for temperatures greater than , thus the Martynov-Sarkisov closure is deficient in the supercritical domain.
References[edit]
- ↑ G. A. Martynov; G. N. Sarkisov "Exact equations and the theory of liquids. V", Molecular Physics 49 1495-1504 (1983)
- ↑ I. Charpentier and N. Jakse "Exact numerical derivatives of the pair-correlation function of simple liquids using the tangent linear method", Journal of Chemical Physics 114 pp. 2284-2292 (2001)
Related reading
- G. Sarkisov and D. Tikhonov "Martynov–Sarkisov integral equation for the simple fluids" Journal of Chemical Physics 99 pp. 3926-3932 (1993)
- G. A. Martynov, G. N. Sarkisov and A. G. Vompe "New closure for the Ornstein–Zernike equation" Journal of Chemical Physics 110 pp. 3961-3969 (1999)
- Gari Sarkisov "Approximate integral equation theory for classical fluids", Journal of Chemical Physics 114 pp. 9496-9505 (2001)