Reference hyper-netted chain - Revision history

Carl McBride: Changed Ref.s

2008-04-15T14:25:57Z

Changed Ref.s

← Older revision		Revision as of 16:25, 15 April 2008
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	~~Note: the~~ reference-~~HNC~~ (RHNC) is sometimes referred to as the modified-HNC (MHNC) Ref.1		The '''reference hyper-netted chain''' (RHNC) is sometimes referred to as the modified-HNC (MHNC) (Ref. 1)

	Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality':		Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality':
	"...the bridge functions.. ..constitute the same family of curves, irrespective of the assumed pair potential"		"...the bridge functions.. ..constitute the same family of curves, irrespective of the assumed pair potential"
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	These are the conditions that will determine the optimum values of <math>\sigma</math> and <math>\epsilon</math>		These are the conditions that will determine the optimum values of <math>\sigma</math> and <math>\epsilon</math>
	that minimise the free energy.		that minimise the free energy.
	The RHNC satisfies the 'Hiroike' termodynamic relation test Ref. 7 and 8, i.e.		The RHNC satisfies the 'Hiroike' termodynamic relation test Ref. 7 and 9, i.e.
	:<math>\left.\frac{\partial U}{\partial V}\right\vert_{T} = T \left.\frac{\partial p}{\partial T}\right\vert_{V} -p</math>		:<math>\left.\frac{\partial U}{\partial V}\right\vert_{T} = T \left.\frac{\partial p}{\partial T}\right\vert_{V} -p</math>
	For highly asymmetric mixtures see Ref. 8.		For highly asymmetric mixtures see Ref. 9.

	==References==		==References==

Carl McBride: /* References */ Added a reference

2008-04-15T14:22:56Z

References: Added a reference

← Older revision		Revision as of 16:22, 15 April 2008
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	Note: the reference-HNC (RHNC) is sometimes referred to as the modified-HNC (MHNC) Ref.1		Note: the reference-HNC (RHNC) is sometimes referred to as the modified-HNC (MHNC) Ref.1

	Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality':		Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality':
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	#[http://dx.doi.org/10.1103/PhysRevA.28.2374 F. Lado, S. M. Foiles and N. W. Ashcroft, "Solutions of the reference-hypernetted-chain equation with minimized free energy", Physical Review A '''28''' 2374 - 2379 (1983)]		#[http://dx.doi.org/10.1103/PhysRevA.28.2374 F. Lado, S. M. Foiles and N. W. Ashcroft, "Solutions of the reference-hypernetted-chain equation with minimized free energy", Physical Review A '''28''' 2374 - 2379 (1983)]
	#[http://dx.doi.org/10.1143/JPSJ.12.326 Kazuo Hiroike "Radial Distribution Function of Fluids I", Journal of the Physical Society of Japan '''12''' pp. 326-334 (1957)]		#[http://dx.doi.org/10.1143/JPSJ.12.326 Kazuo Hiroike "Radial Distribution Function of Fluids I", Journal of the Physical Society of Japan '''12''' pp. 326-334 (1957)]
			#[http://dx.doi.org/10.1143/JPSJ.12.864 Kazuo Hiroike "Radial Distribution Function of Fluids II", Journal of the Physical Society of Japan '''12''' pp. pp. 864-873 (1957)]
	#[http://dx.doi.org/10.1063/1.2102891 S. Amokrane, A. Ayadim, and J. G. Malherbe "Structure of highly asymmetric hard-sphere mixtures: An efficient closure of the Ornstein-Zernike equations", Journal of Chemical Physics, '''123''' 174508 (2005)]		#[http://dx.doi.org/10.1063/1.2102891 S. Amokrane, A. Ayadim, and J. G. Malherbe "Structure of highly asymmetric hard-sphere mixtures: An efficient closure of the Ornstein-Zernike equations", Journal of Chemical Physics, '''123''' 174508 (2005)]

Carl McBride: RHNC moved to Reference hyper-netted chain

2007-10-25T13:08:01Z

RHNC moved to Reference hyper-netted chain

← Older revision	Revision as of 15:08, 25 October 2007
(No difference)

Nice and Tidy at 13:22, 21 June 2007

2007-06-21T13:22:44Z

Nice and Tidy at 15:52, 28 February 2007

2007-02-28T15:52:08Z

← Older revision		Revision as of 17:52, 28 February 2007
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	Note: the reference-HNC (RHNC) is sometimes referred to as the modified-HNC (MHNC) Ref.1		Note: the reference-HNC (RHNC) is sometimes referred to as the modified-HNC (MHNC) Ref.1

	Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality':		Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality':
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	The RHNC satisfies the 'Hiroike' termodynamic relation test Ref. 7 and 8, i.e.		The RHNC satisfies the 'Hiroike' termodynamic relation test Ref. 7 and 8, i.e.
	:<math>\left.\frac{\partial U}{\partial V}\right\vert_{T} = T \left.\frac{\partial p}{\partial T}\right\vert_{V} -p</math>		:<math>\left.\frac{\partial U}{\partial V}\right\vert_{T} = T \left.\frac{\partial p}{\partial T}\right\vert_{V} -p</math>
	For highly asymmetric mixtures see Ref. 9.		For highly asymmetric mixtures see Ref. 8.

	==References==		==References==

Nice and Tidy: /* References */

2007-02-28T15:49:20Z

References

← Older revision		Revision as of 17:49, 28 February 2007
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	#[http://dx.doi.org/10.1103/PhysRevA.28.2374 F. Lado, S. M. Foiles and N. W. Ashcroft, "Solutions of the reference-hypernetted-chain equation with minimized free energy", Physical Review A '''28''' 2374 - 2379 (1983)]		#[http://dx.doi.org/10.1103/PhysRevA.28.2374 F. Lado, S. M. Foiles and N. W. Ashcroft, "Solutions of the reference-hypernetted-chain equation with minimized free energy", Physical Review A '''28''' 2374 - 2379 (1983)]
	#[http://dx.doi.org/10.1143/JPSJ.12.326 Kazuo Hiroike "Radial Distribution Function of Fluids I", Journal of the Physical Society of Japan '''12''' pp. 326-334 (1957)]		#[http://dx.doi.org/10.1143/JPSJ.12.326 Kazuo Hiroike "Radial Distribution Function of Fluids I", Journal of the Physical Society of Japan '''12''' pp. 326-334 (1957)]
	#[http://dx.doi.org/ ~~JCP_2005_123_174508~~]		#[http://dx.doi.org/10.1063/1.2102891 S. Amokrane, A. Ayadim, and J. G. Malherbe "Structure of highly asymmetric hard-sphere mixtures: An efficient closure of the Ornstein-Zernike equations", Journal of Chemical Physics, '''123''' 174508 (2005)]



	[[Category: Integral equations]]		[[Category: Integral equations]]

Nice and Tidy: /* References */

2007-02-28T15:47:06Z

References

← Older revision		Revision as of 17:47, 28 February 2007
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	#[http://dx.doi.org/10.1016/0375-9601(82)90207-9 F. Lado "A local thermodynamic criterion for the reference-hypernetted chain equation", Physics Letters A '''89''' pp. 196-198 (1982)]		#[http://dx.doi.org/10.1016/0375-9601(82)90207-9 F. Lado "A local thermodynamic criterion for the reference-hypernetted chain equation", Physics Letters A '''89''' pp. 196-198 (1982)]
	#[http://dx.doi.org/10.1103/PhysRevA.28.2374 F. Lado, S. M. Foiles and N. W. Ashcroft, "Solutions of the reference-hypernetted-chain equation with minimized free energy", Physical Review A '''28''' 2374 - 2379 (1983)]		#[http://dx.doi.org/10.1103/PhysRevA.28.2374 F. Lado, S. M. Foiles and N. W. Ashcroft, "Solutions of the reference-hypernetted-chain equation with minimized free energy", Physical Review A '''28''' 2374 - 2379 (1983)]
	#[http://dx.doi.org/ ~~JPSJ_1957_12_00326~~]		#[http://dx.doi.org/10.1143/JPSJ.12.326 Kazuo Hiroike "Radial Distribution Function of Fluids I", Journal of the Physical Society of Japan '''12''' pp. 326-334 (1957)]
	#[http://dx.doi.org/ JCP_2005_123_174508]		#[http://dx.doi.org/ JCP_2005_123_174508]


	[[Category: Integral equations]]		[[Category: Integral equations]]

Nice and Tidy: /* References */

2007-02-28T15:42:33Z

References

← Older revision		Revision as of 17:42, 28 February 2007
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	#[http://dx.doi.org/10.1080/00268978700100441 Anatol Malijevský and Stanislav Labík "The bridge function for hard spheres", Molecular Physics, '''60''' pp. 663-669 (1987)]		#[http://dx.doi.org/10.1080/00268978700100441 Anatol Malijevský and Stanislav Labík "The bridge function for hard spheres", Molecular Physics, '''60''' pp. 663-669 (1987)]
	#[http://dx.doi.org/10.1080/00268978900101181 Stanislav Labík and Anatol Malijevský "Bridge function for hard spheres in high density and overlap regions", Molecular Physics, '''67''' pp. 431-438 (1989)]		#[http://dx.doi.org/10.1080/00268978900101181 Stanislav Labík and Anatol Malijevský "Bridge function for hard spheres in high density and overlap regions", Molecular Physics, '''67''' pp. 431-438 (1989)]
	#[http://dx.doi.org/ ~~PLA_1982_89_0196~~]		#[http://dx.doi.org/10.1016/0375-9601(82)90207-9 F. Lado "A local thermodynamic criterion for the reference-hypernetted chain equation", Physics Letters A '''89''' pp. 196-198 (1982)]
	#[http://dx.doi.org/10.1103/PhysRevA.28.2374 F. Lado, S. M. Foiles and N. W. Ashcroft, "Solutions of the reference-hypernetted-chain equation with minimized free energy", Physical Review A '''28''' 2374 - 2379 (1983)]		#[http://dx.doi.org/10.1103/PhysRevA.28.2374 F. Lado, S. M. Foiles and N. W. Ashcroft, "Solutions of the reference-hypernetted-chain equation with minimized free energy", Physical Review A '''28''' 2374 - 2379 (1983)]
	#[http://dx.doi.org/ JPSJ_1957_12_00326]		#[http://dx.doi.org/ JPSJ_1957_12_00326]

Nice and Tidy: /* References */

2007-02-28T15:30:57Z

References

← Older revision		Revision as of 17:30, 28 February 2007
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	#[http://dx.doi.org/10.1103/PhysRevA.8.2548 F. Lado "Perturbation Correction for the Free Energy and Structure of Simple Fluids", Physical Review A '''8''' 2548 - 2552 (1973)]		#[http://dx.doi.org/10.1103/PhysRevA.8.2548 F. Lado "Perturbation Correction for the Free Energy and Structure of Simple Fluids", Physical Review A '''8''' 2548 - 2552 (1973)]
	#[http://dx.doi.org/10.1080/00268978700100441 Anatol Malijevský and Stanislav Labík "The bridge function for hard spheres", Molecular Physics, '''60''' pp. 663-669 (1987)]		#[http://dx.doi.org/10.1080/00268978700100441 Anatol Malijevský and Stanislav Labík "The bridge function for hard spheres", Molecular Physics, '''60''' pp. 663-669 (1987)]
	#[http://dx.doi.org/ ~~MP_1989_67_0431~~]		#[http://dx.doi.org/10.1080/00268978900101181 Stanislav Labík and Anatol Malijevský "Bridge function for hard spheres in high density and overlap regions", Molecular Physics, '''67''' pp. 431-438 (1989)]
	#[http://dx.doi.org/ PLA_1982_89_0196]		#[http://dx.doi.org/ PLA_1982_89_0196]
	#[http://dx.doi.org/10.1103/PhysRevA.28.2374 F. Lado, S. M. Foiles and N. W. Ashcroft, "Solutions of the reference-hypernetted-chain equation with minimized free energy", Physical Review A '''28''' 2374 - 2379 (1983)]		#[http://dx.doi.org/10.1103/PhysRevA.28.2374 F. Lado, S. M. Foiles and N. W. Ashcroft, "Solutions of the reference-hypernetted-chain equation with minimized free energy", Physical Review A '''28''' 2374 - 2379 (1983)]

Nice and Tidy: /* References */

2007-02-28T15:28:30Z

References

← Older revision		Revision as of 17:28, 28 February 2007
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	Note: the reference-HNC (RHNC) is sometimes referred to as the modified-HNC (MHNC) Ref.1		Note: the reference-HNC (RHNC) is sometimes referred to as the modified-HNC (MHNC) Ref.1

	Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality':		Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality':
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	#[http://dx.doi.org/10.1103/PhysRevA.20.1208 Yaakov Rosenfeld and N. W. Ashcroft "Theory of simple classical fluids: Universality in the short-range structure", Physical Review A '''20''' pp. 1208 - 1235 (1979)]		#[http://dx.doi.org/10.1103/PhysRevA.20.1208 Yaakov Rosenfeld and N. W. Ashcroft "Theory of simple classical fluids: Universality in the short-range structure", Physical Review A '''20''' pp. 1208 - 1235 (1979)]
	#[http://dx.doi.org/10.1103/PhysRevA.8.2548 F. Lado "Perturbation Correction for the Free Energy and Structure of Simple Fluids", Physical Review A '''8''' 2548 - 2552 (1973)]		#[http://dx.doi.org/10.1103/PhysRevA.8.2548 F. Lado "Perturbation Correction for the Free Energy and Structure of Simple Fluids", Physical Review A '''8''' 2548 - 2552 (1973)]
	#[http://dx.doi.org/ ~~MP_1987_60_0663~~]		#[http://dx.doi.org/10.1080/00268978700100441 Anatol Malijevský and Stanislav Labík "The bridge function for hard spheres", Molecular Physics, '''60''' pp. 663-669 (1987)]
	#[http://dx.doi.org/ MP_1989_67_0431]		#[http://dx.doi.org/ MP_1989_67_0431]
	#[http://dx.doi.org/ PLA_1982_89_0196]		#[http://dx.doi.org/ PLA_1982_89_0196]

← Older revision		Revision as of 15:22, 21 June 2007
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	Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality':		Rosenfeld and Ashcroft (1979) (Ref. 1) proposed the `anzatz of universality':
	"...the bridge functions.. ..constitute the same family of curves, irrespective of the assumed pair potential"		"...the bridge functions.. ..constitute the same family of curves, irrespective of the assumed pair potential"
	The basis of the method is to solve the modified HNC equation		The basis of the method is to solve the modified [[HNC]] equation
	(with inclusion of the one-parameter bridge ~~functions~~		(with inclusion of the one-parameter [[bridge function]]s
	appropriate to hard spheres), and determine the only free parameter <math>\eta</math>		appropriate to [[hard sphere model \| hard spheres]]), and determine the only free parameter <math>\eta</math>
	(related to the hard-sphere diameter) by requiring thermodynamic consistency.		(related to the hard-sphere diameter) by requiring [[thermodynamic consistency]].
	Fred Lado (Ref. 2) and Rosenfeld and Ashcroft (1979) (Ref. 3)		Fred Lado (Ref. 2) and Rosenfeld and Ashcroft (1979) (Ref. 3)
	noticed that the ~~OZ equation~~ can always be written in the form		noticed that the [[Ornstein-Zernike relation]] can always be written in the form
	:<math>\gamma_{12} = \rho \int_V (h_{13} - \gamma_{13}) h_{23} ~{\rm d}(3)</math>		:<math>\gamma_{12} = \rho \int_V (h_{13} - \gamma_{13}) h_{23} ~{\rm d}(3)</math>
	In view of this a hybrid solution between the [[HNC\| hyper-netted chain]]		In view of this a hybrid solution between the [[HNC\| hyper-netted chain]]
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	(See also Ref.s 6 and 7)		(See also Ref.s 6 and 7)
	The RHNC closure is given by (Eq. 17 Ref. 7)		The RHNC closure is given by (Eq. 17 Ref. 7)
	:<math> c\left(r\right) = h(r) - \ln [g(r) e^{\beta \~~phi~~(r)}] + B_0(r) </math>		:<math> c\left(r\right) = h(r) - \ln [g(r) e^{\beta \Phi(r)}] + B_0(r) </math>
	along with the constraint (Eq. 18 Ref. 7)		along with the constraint (Eq. 18 Ref. 7)
	:<math>\rho \int [g(r) - g_0(r)] \delta B_0(r) dr_3 = 0 </math>		:<math>\rho \int [g(r) - g_0(r)] \delta B_0(r) dr_3 = 0 </math>
	Incorporating a reference potential <math>\~~phi_0~~(r)= \~~phi_0~~(r;\sigma,\epsilon)</math>		where <math>\Phi(r)</math> is the [[intermolecular pair potential]].
			Incorporating a reference potential <math>\Phi_0(r)= \Phi_0(r;\sigma,\epsilon)</math>
	this equation becomes (Eqs. 19a and 19b in Ref. 7)		this equation becomes (Eqs. 19a and 19b in Ref. 7)
	:<math>\rho \int [g(r) - g_0(r)] \sigma \frac{\partial B_0(r)}{\partial \sigma}dr_3 = 0 </math>		:<math>\rho \int [g(r) - g_0(r)] \sigma \frac{\partial B_0(r)}{\partial \sigma}dr_3 = 0 </math>