Mean spherical approximation - Revision history

Carl McBride: Updated internal link + tidy + Cite references format

2012-02-16T12:07:41Z

Updated internal link + tidy + Cite references format

← Older revision		Revision as of 14:07, 16 February 2012
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	The '''~~Lebowitz and Percus~~''' ~~mean spherical approximation~~ (MSA) ~~(1966) (Ref. 1)~~ [[Closure relations \| closure relation]] is given by		The '''mean spherical approximation''' (MSA) [[Closure relations \| closure relation]] of Lebowitz and Percus is given by <ref>[http://dx.doi.org/10.1103/PhysRev.144.251 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)]</ref>:


	:<math>c(r) = -\beta \omega(r), ~~~~ r>\sigma.</math>		:<math>c(r) = -\beta \omega(r), ~ ~ ~ ~ r>\sigma.</math>


	In the '''Blum and Høye''' mean spherical approximation for mixtures (~~Refs 2 and 3~~) the closure ~~is given by~~		In the '''Blum and Høye''' mean spherical approximation for [[mixtures]] the closure is given by <ref>[http://dx.doi.org/10.1007/BF01011750 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)]</ref>
			<ref>[http://dx.doi.org/10.1007/BF01013935 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)]</ref>:


	:<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>


	and		and

	:<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 <math>h_{ij}(r)</math> and <math>c_{ij}(r)</math> are the total and the direct correlation ~~functions~~ for two spherical		where <math>h_{ij}(r)</math> and <math>c_{ij}(r)</math> are the [[Total correlation function \|total]] and the [[direct correlation function]]s for two spherical
	molecules of ''i'' and ''j'' species, <math>\sigma_i</math> is the diameter of ~~'''~~i'' species of molecule.		molecules of <math>i</math> and <math>j</math> species, <math>\sigma_i</math> is the diameter of <math>i</math> species of molecule.
	Duh and Haymet (Eq. 9 ~~Ref~~. 4) write the MSA approximation as		Duh and Haymet (Eq. 9 in <ref name="Duh and Haymet">[http://dx.doi.org/10.1063/1.470724 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)]</ref>) write the MSA approximation as


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	where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the		where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the
	[[Weeks-Chandler-~~Anderson~~ perturbation theory \| Weeks-Chandler-~~Anderson~~ division]]		[[Weeks-Chandler-Andersen perturbation theory \| Weeks-Chandler-Andersen division]]
	of the [[Lennard-Jones model \| Lennard-Jones]] potential.		of the [[Lennard-Jones model \| Lennard-Jones]] potential.
	By introducing the definition (Eq. 10 ~~Ref. 4~~)		By introducing the definition (Eq. 10 in <ref name="Duh and Haymet"> </ref>)


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	one can arrive at (Eq. 11 in ~~Ref. 4~~)		one can arrive at (Eq. 11 in <ref name="Duh and Haymet"> </ref>)


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	==Thermodynamic consistency==		==Thermodynamic consistency==
			<ref>[http://dx.doi.org/10.1063/1.2712181 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)]</ref>
			==References==
			<references/>

	~~See Ref. 5.~~

	~~==References==~~
	#[http://dx.doi.org/10.1103/PhysRev.144.251 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)]
	#[http://dx.doi.org/10.1007/BF01011750 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)]
	#[http://dx.doi.org/10.1007/BF01013935 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)]
	#[http://dx.doi.org/10.1063/1.470724 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)]
	#[http://dx.doi.org/10.1063/1.2712181 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)]
	[[Category:Integral equations]]		[[Category:Integral equations]]

Carl McBride: Reverted edits by 83.149.19.6 (Talk); changed back to last version by Carl McBride

2007-07-04T12:38:02Z

Reverted edits by 83.149.19.6 (Talk); changed back to last version by Carl McBride

← Older revision		Revision as of 14:38, 4 July 2007
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	:<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>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>


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	:<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>

83.149.19.6 at 10:59, 4 July 2007

2007-07-04T10:59:47Z

← Older revision		Revision as of 12:59, 4 July 2007
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	:<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>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>


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	:<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>

Carl McBride at 09:18, 4 July 2007

2007-07-04T09:18:10Z

← Older revision		Revision as of 11:18, 4 July 2007
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	~~The~~ '''Blum and Høye''' mean spherical approximation ~~(MSA) (1978-1980)~~ (Refs 2 and 3) closure is given by		In the '''Blum and Høye''' mean spherical approximation for mixtures (Refs 2 and 3) the closure is given by

Carl McBride: Reverted edits by 72.3.237.81 (Talk); changed back to last version by Carl McBride

2007-07-04T09:17:12Z

Reverted edits by 72.3.237.81 (Talk); changed back to last version by Carl McBride

← Older revision		Revision as of 11:17, 4 July 2007
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	:<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>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>


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	:<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>

72.3.237.81 at 00:04, 4 July 2007

2007-07-04T00:04:59Z

← Older revision		Revision as of 02:04, 4 July 2007
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	:<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>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>


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	:<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>

Carl McBride at 17:07, 26 June 2007

2007-06-26T17:07:38Z

← Older revision		Revision as of 19:07, 26 June 2007
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	The '''Lebowitz and Percus''' mean spherical approximation (MSA) (1966) (Ref. 1) closure is given by		The '''Lebowitz and Percus''' mean spherical approximation (MSA) (1966) (Ref. 1) [[Closure relations \| closure relation]] is given by

Carl McBride at 14:17, 20 June 2007

2007-06-20T14:17:38Z

← Older revision		Revision as of 16:17, 20 June 2007
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	where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the [[~~WCA~~ division]] of the [[Lennard-Jones model \| Lennard-Jones]] potential.		where <math>\Phi_1</math> and <math>\Phi_2</math> comes from the
			[[Weeks-Chandler-Anderson perturbation theory \| Weeks-Chandler-Anderson division]]
			of the [[Lennard-Jones model \| Lennard-Jones]] potential.
	By introducing the definition (Eq. 10 Ref. 4)		By introducing the definition (Eq. 10 Ref. 4)

Carl McBride at 14:56, 22 March 2007

2007-03-22T14:56:29Z

← Older revision		Revision as of 16:56, 22 March 2007
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	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>.

			==Thermodynamic consistency==

			See Ref. 5.

	==References==		==References==

Carl McBride: /* References */

2007-03-22T14:55:47Z

References

← Older revision		Revision as of 16:55, 22 March 2007
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	#[http://dx.doi.org/10.1007/BF01013935 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)]		#[http://dx.doi.org/10.1007/BF01013935 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)]
	#[http://dx.doi.org/10.1063/1.470724 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)]		#[http://dx.doi.org/10.1063/1.470724 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)]
			#[http://dx.doi.org/10.1063/1.2712181 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)]
	[[Category:Integral equations]]		[[Category:Integral equations]]