Gibbs-Duhem integration - Revision history

Carl McBride: Changed references to Cite format

2010-09-28T11:02:37Z

Changed references to Cite format

← Older revision		Revision as of 13:02, 28 September 2010
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	The so-called '''Gibbs-Duhem integration''' refers to a number of methods that couple		The so-called '''Gibbs-Duhem integration''' refers to a number of methods that couple
	molecular [[Computer simulation techniques \|simulation techniques]] with [[Thermodynamic relations \|thermodynamic equations]] in order to draw		molecular [[Computer simulation techniques \|simulation techniques]] with [[Thermodynamic relations \|thermodynamic equations]] in order to draw
	[[Computation of phase equilibria \| phase coexistence]] lines. The original method was proposed by David Kofke (~~Refs~~. 1 ~~and 2~~).		[[Computation of phase equilibria \| phase coexistence]] lines. The original method was proposed by David Kofke <ref>[http://dx.doi.org/10.1080/00268979300100881 David A. Kofke, "Gibbs-Duhem integration: a new method for direct evaluation of phase coexistence by molecular simulation", Molecular Physics '''78''' pp 1331 - 1336 (1993)]</ref>
			<ref>[http://dx.doi.org/10.1063/1.465023 David A. Kofke, "Direct evaluation of phase coexistence by molecular simulation via integration along the saturation line", Journal of Chemical Physics '''98''' pp. 4149-4162 (1993)]</ref>.

	== Basic Features ==		== Basic Features ==
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	== Peculiarities of the method (Warnings) ==		== Peculiarities of the method (Warnings) ==

	* A good initial point must be known to start the procedure (See ~~Ref~~. 3 and the ~~entry:~~ [[computation of phase equilibria]]).		* A good initial point must be known to start the procedure (See <ref>[http://dx.doi.org/10.1063/1.2137705 A. van 't Hof, S. W. de Leeuw, and C. J. Peters "Computing the starting state for Gibbs-Duhem integration", Journal of Chemical Physics '''124''' 054905 (2006)]</ref> and [[computation of phase equilibria]]).

	* The ''integrand'' of the differential equation is computed with some numerical uncertainty		* The ''integrand'' of the differential equation is computed with some numerical uncertainty
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	== References ==		== References ==
	~~#[http://dx.doi.org~~/10.1080/00268979300100881 David A. Kofke, "Gibbs-Duhem integration: a new method for direct evaluation of phase coexistence by molecular simulation", Molecular Physics '''78''' pp 1331 - 1336 (1993)]		<references/>
	~~#[http://dx.doi.org/10.1063/1.465023 David A. Kofke, "Direct evaluation of phase coexistence by molecular simulation via integration along the saturation line", Journal of Chemical Physics~~ '''98''' ~~pp. 4149-4162 (1993)]~~		'''Related reading'''
	#[http://dx.doi.org/10.1063/1.~~2137705~~ A. van 't Hof, S. W. de Leeuw~~, and C. J. Peters~~ "~~Computing the starting state for~~ Gibbs-Duhem integration", Journal of Chemical Physics '''124''' ~~054905~~ (2006)]		*[http://dx.doi.org/10.1063/1.2137706 A. van 't Hof, C. J. Peters, and S. W. de Leeuw "An advanced Gibbs-Duhem integration method: Theory and applications", Journal of Chemical Physics '''124''' 054906 (2006)]
	#[http://dx.doi.org/10.1063/1.~~2137706 A. van 't Hof, C. J. Peters, and S. W. de Leeuw~~ "An ~~advanced Gibbs-Duhem~~ integration method~~: Theory and applications~~", Journal of Chemical Physics '''~~124~~''' ~~054906~~ (~~2006~~)]		*[http://dx.doi.org/10.1063/1.3486090 Gerassimos Orkoulas "Communication: Tracing phase boundaries via molecular simulation: An alternative to the Gibbs–Duhem integration method", Journal of Chemical Physics '''133''' 111104 (2010)]

	[[category: computer simulation techniques]]		[[category: computer simulation techniques]]

141.108.6.68: /* Example: phase equilibria of one-component system */

2010-03-01T16:07:45Z

Example: phase equilibria of one-component system

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	where		where
	* <math> \beta := 1/k_B T </math>, where <math> k_B </math> is the [[Boltzmann constant]]		* <math> \beta := 1/k_B T </math>, where <math> k_B </math> is the [[Boltzmann constant]]
	When a differential change of the conditions is performed one will, ~~have~~ for any phase:		When a differential change of the conditions is performed one will have, for any phase:

	: <math> d \left( \beta\mu \right) = \left[ \frac{ \partial (\beta \mu) }{\partial \beta} \right]_{\beta p,\lambda} d \beta +		: <math> d \left( \beta\mu \right) = \left[ \frac{ \partial (\beta \mu) }{\partial \beta} \right]_{\beta p,\lambda} d \beta +

Carl McBride: /* Example: phase equilibria of one-component system */ Added internal link.

2008-02-18T17:08:42Z

Example: phase equilibria of one-component system: Added internal link.

← Older revision		Revision as of 19:08, 18 February 2008
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	The Gibbs-Duhem integration technique, for this example, will be a numerical procedure covering the following tasks:		The Gibbs-Duhem integration technique, for this example, will be a numerical procedure covering the following tasks:

	* Computer simulation (for instance using [[Metropolis Monte Carlo]] in the NpT ensemble) runs to estimate the values of <math> \bar{L}, \bar{V} </math> for both		* Computer simulation (for instance using [[Metropolis Monte Carlo]] in the [[Isothermal-isobaric ensemble \|NpT ensemble]]) runs to estimate the values of <math> \bar{L}, \bar{V} </math> for both
	phases at given values of <math> [\beta, \beta p, \lambda ] </math>.		phases at given values of <math> [\beta, \beta p, \lambda ] </math>.

Nice and Tidy at 11:31, 15 February 2008

2008-02-15T11:31:57Z

← Older revision		Revision as of 13:31, 15 February 2008
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	~~== History ==~~		The so-called '''Gibbs-Duhem integration''' refers to a number of methods that couple
	The so-called Gibbs-Duhem ~~Integration~~ refers to a number of methods that couple		molecular [[Computer simulation techniques \|simulation techniques]] with [[Thermodynamic relations \|thermodynamic equations]] in order to draw
	molecular simulation techniques with thermodynamic equations in order to draw		[[Computation of phase equilibria \| phase coexistence]] lines. The original method was proposed by David Kofke (Refs. 1 and 2).
	[[Computation of phase equilibria \| phase coexistence]] lines. The method was proposed by Kofke (~~Ref~~ 1-2).

	== Basic Features ==		== Basic Features ==

	Consider two thermodynamic phases: <math> a </math> and <math> b </math>, at thermodynamic equilibrium at certain conditions.		Consider two thermodynamic phases: <math> a </math> and <math> b </math>, at thermodynamic equilibrium at certain conditions. Thermodynamic equilibrium implies:
	~~The thermodynamic~~ equilibrium implies:

	* Equal [[temperature]] in both phases: <math> T = T_{a} = T_{b} </math>, i.e. thermal equilibrium.		* Equal [[temperature]] in both phases: <math> T = T_{a} = T_{b} </math>, i.e. thermal equilibrium.
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	* Equal [[chemical potential]]s for the components <math> \mu_i = \mu_{ia} = \mu_{ib} </math>, i.e. ''material'' equilibrium.		* Equal [[chemical potential]]s for the components <math> \mu_i = \mu_{ia} = \mu_{ib} </math>, i.e. ''material'' equilibrium.

	In addition if ~~we are~~ dealing with a statistical ~~mechanics~~ model, ~~with~~ certain parameters that we can ~~represent~~ as <math> \lambda </math> , the		In addition, if one is dealing with a statistical mechanical [[models \|model]], having certain parameters that can be represented as <math> \lambda </math>, then the
	model should be the same in both phases.		model should be the same in both phases.

Nice and Tidy: /* Example: phase equilibria of one-component system */

2008-02-15T11:24:18Z

Example: phase equilibria of one-component system

← Older revision		Revision as of 13:24, 15 February 2008
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	where		where
	* <math> \beta = 1/k_B T </math>, where <math> k_B </math> is the [[Boltzmann constant]]		* <math> \beta := 1/k_B T </math>, where <math> k_B </math> is the [[Boltzmann constant]]
	When a differential change of the conditions is performed one will, have for any phase:		When a differential change of the conditions is performed one will, have for any phase:

Nice and Tidy at 10:26, 13 February 2008

2008-02-13T10:26:31Z

← Older revision		Revision as of 12:26, 13 February 2008
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	#[http://dx.doi.org/10.1063/1.2137705 A. van 't Hof, S. W. de Leeuw, and C. J. Peters "Computing the starting state for Gibbs-Duhem integration", Journal of Chemical Physics '''124''' 054905 (2006)]		#[http://dx.doi.org/10.1063/1.2137705 A. van 't Hof, S. W. de Leeuw, and C. J. Peters "Computing the starting state for Gibbs-Duhem integration", Journal of Chemical Physics '''124''' 054905 (2006)]
	#[http://dx.doi.org/10.1063/1.2137706 A. van 't Hof, C. J. Peters, and S. W. de Leeuw "An advanced Gibbs-Duhem integration method: Theory and applications", Journal of Chemical Physics '''124''' 054906 (2006)]		#[http://dx.doi.org/10.1063/1.2137706 A. van 't Hof, C. J. Peters, and S. W. de Leeuw "An advanced Gibbs-Duhem integration method: Theory and applications", Journal of Chemical Physics '''124''' 054906 (2006)]
			[[category: computer simulation techniques]]

Carl McBride: /* Basic Features */ Added internal links

2008-01-29T15:43:46Z

Basic Features: Added internal links

← Older revision		Revision as of 17:43, 29 January 2008
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	The thermodynamic equilibrium implies:		The thermodynamic equilibrium implies:

	* Equal temperature in both phases: <math> T = T_{a} = T_{b} </math>, i.e. thermal equilibrium.		* Equal [[temperature]] in both phases: <math> T = T_{a} = T_{b} </math>, i.e. thermal equilibrium.
	* Equal pressure in both phases <math> p = p_{a} = p_{b} </math>, i.e. mechanical equilibrium.		* Equal [[pressure]] in both phases <math> p = p_{a} = p_{b} </math>, i.e. mechanical equilibrium.
	* Equal chemical ~~potentials~~ for the components <math> \mu_i = \mu_{ia} = \mu_{ib} </math>, i.e. ''material'' equilibrium.		* Equal [[chemical potential]]s for the components <math> \mu_i = \mu_{ia} = \mu_{ib} </math>, i.e. ''material'' equilibrium.

	In addition if we are dealing with a statistical mechanics model, with certain parameters that we can represent as <math> \lambda </math> , the		In addition if we are dealing with a statistical mechanics model, with certain parameters that we can represent as <math> \lambda </math> , the

Nice and Tidy: /* Peculiarities of the method (Warnings) */

2007-09-25T11:27:19Z

Peculiarities of the method (Warnings)

← Older revision		Revision as of 13:27, 25 September 2007
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	== Peculiarities of the method (Warnings) ==		== Peculiarities of the method (Warnings) ==

	* A good initial point must be known to start the procedure (See Ref. 3).		* A good initial point must be known to start the procedure (See Ref. 3 and the entry: [[computation of phase equilibria]]).

	* The ''integrand'' of the differential equation is computed with some numerical uncertainty		* The ''integrand'' of the differential equation is computed with some numerical uncertainty

Nice and Tidy: /* History */

2007-09-25T11:26:18Z

History

← Older revision		Revision as of 13:26, 25 September 2007
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	The so-called Gibbs-Duhem Integration refers to a number of methods that couple		The so-called Gibbs-Duhem Integration refers to a number of methods that couple
	molecular simulation techniques with thermodynamic equations in order to draw		molecular simulation techniques with thermodynamic equations in order to draw
	phase coexistence lines. The method was proposed by Kofke (Ref 1-2).		[[Computation of phase equilibria \| phase coexistence]] lines. The method was proposed by Kofke (Ref 1-2).

	== Basic Features ==		== Basic Features ==

Carl McBride: /* Example: phase equilibria of one-component system */

2007-06-07T16:31:56Z

Example: phase equilibria of one-component system

← Older revision		Revision as of 18:31, 7 June 2007
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	where:		where:
	* <math> \left. E \right. </math> is the internal energy		* <math> \left. E \right. </math> is the [[internal energy]] (sometimes written as <math>U</math>).

	* <math> \left. V \right. </math> is the volume		* <math> \left. V \right. </math> is the volume