Computation of phase equilibria - Revision history

Carl McBride: /* See also */ Added a couple of internal links

2011-05-06T11:12:17Z

See also: Added a couple of internal links

← Older revision		Revision as of 13:12, 6 May 2011
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	temperature. The symmetry in the interactions can be exploited to simplify the calculation of phase diagrams.		temperature. The symmetry in the interactions can be exploited to simplify the calculation of phase diagrams.
	== See also==		== See also==
			*[[Constrained cell method]]
	*[[Gibbs-Duhem integration]]		*[[Gibbs-Duhem integration]]
			*[[Phase switch Monte Carlo]]

	==References==		==References==
	<references/>		<references/>
	[[category: computer simulation techniques]]		[[category: computer simulation techniques]]

Carl McBride: /* Direct simulation of the two phase system */ resized image

2011-03-03T13:59:37Z

Direct simulation of the two phase system: resized image

← Older revision		Revision as of 15:59, 3 March 2011
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	* The simulation results in the two phase region will depend dramatically on the system size (calculations with different number of particles become convenient to check the quality of the phase equilibria results)		* The simulation results in the two phase region will depend dramatically on the system size (calculations with different number of particles become convenient to check the quality of the phase equilibria results)
	== Direct simulation of the two phase system==		== Direct simulation of the two phase system==
	[[Image:direct_coexistence.png\|~~thumb~~\|right]]		[[Image:direct_coexistence.png\|300px\|right]]
	Direct simulation of the two phase system was first implemented by Abraham <ref>[http://dx.doi.org/10.1103/PhysRevB.23.6145 Farid F. Abraham "Two-dimensional melting, solid-state stability, and the Kosterlitz-Thouless-Feynman criterion", Physical Review B '''23''' pp. 6145-6148 (1981)]</ref>.		Direct simulation of the two phase system was first implemented by Abraham <ref>[http://dx.doi.org/10.1103/PhysRevB.23.6145 Farid F. Abraham "Two-dimensional melting, solid-state stability, and the Kosterlitz-Thouless-Feynman criterion", Physical Review B '''23''' pp. 6145-6148 (1981)]</ref>.
	It has since been applied to the [[Lennard-Jones model]] <ref>[http://dx.doi.org/10.1063/1.1474581 James R. Morris and Xueyu Song "The melting lines of model systems calculated from coexistence simulations", Journal of Chemical Physics '''116''' 9352 (2002)]</ref>		It has since been applied to the [[Lennard-Jones model]] <ref>[http://dx.doi.org/10.1063/1.1474581 James R. Morris and Xueyu Song "The melting lines of model systems calculated from coexistence simulations", Journal of Chemical Physics '''116''' 9352 (2002)]</ref>

Carl McBride: /* Direct simulation of the two phase system */ Added an image

2011-03-03T13:54:55Z

Direct simulation of the two phase system: Added an image

← Older revision		Revision as of 15:54, 3 March 2011
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	* The simulation results in the two phase region will depend dramatically on the system size (calculations with different number of particles become convenient to check the quality of the phase equilibria results)		* The simulation results in the two phase region will depend dramatically on the system size (calculations with different number of particles become convenient to check the quality of the phase equilibria results)
	== Direct simulation of the two phase system==		== Direct simulation of the two phase system==
			[[Image:direct_coexistence.png\|thumb\|right]]
	Direct simulation of the two phase system was first implemented by Abraham <ref>[http://dx.doi.org/10.1103/PhysRevB.23.6145 Farid F. Abraham "Two-dimensional melting, solid-state stability, and the Kosterlitz-Thouless-Feynman criterion", Physical Review B '''23''' pp. 6145-6148 (1981)]</ref>.		Direct simulation of the two phase system was first implemented by Abraham <ref>[http://dx.doi.org/10.1103/PhysRevB.23.6145 Farid F. Abraham "Two-dimensional melting, solid-state stability, and the Kosterlitz-Thouless-Feynman criterion", Physical Review B '''23''' pp. 6145-6148 (1981)]</ref>.
	It has since been applied to the [[Lennard-Jones model]] <ref>[http://dx.doi.org/10.1063/1.1474581 James R. Morris and Xueyu Song "The melting lines of model systems calculated from coexistence simulations", Journal of Chemical Physics '''116''' 9352 (2002)]</ref>		It has since been applied to the [[Lennard-Jones model]] <ref>[http://dx.doi.org/10.1063/1.1474581 James R. Morris and Xueyu Song "The melting lines of model systems calculated from coexistence simulations", Journal of Chemical Physics '''116''' 9352 (2002)]</ref>
	and [[water]]		and [[water]]
	<ref>[http://dx.doi.org/10.1063/1.2183308 Ramón García Fernández, José L. F. Abascal, and Carlos Vega "The melting point of ice Ih for common water models calculated from direct coexistence of the solid-liquid interface", Journal of Chemical Physics '''124''' 144506 (2006)]</ref>.		<ref>[http://dx.doi.org/10.1063/1.2183308 Ramón García Fernández, José L. F. Abascal, and Carlos Vega "The melting point of ice Ih for common water models calculated from direct coexistence of the solid-liquid interface", Journal of Chemical Physics '''124''' 144506 (2006)]</ref>.

	== Gibbs ensemble Monte Carlo for one component systems==		== Gibbs ensemble Monte Carlo for one component systems==
	The [[Gibbs ensemble Monte Carlo]] method is often considered as a smart variation of the standard canonical ensemble procedure (See <ref>[http://dx.doi.org/10.1080/00268978700101491 Athanassios Panagiotopoulos "Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble", Molecular Physics '''61''' pp. 813-826 (1987)]</ref>).		The [[Gibbs ensemble Monte Carlo]] method is often considered as a smart variation of the standard canonical ensemble procedure (See <ref>[http://dx.doi.org/10.1080/00268978700101491 Athanassios Panagiotopoulos "Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble", Molecular Physics '''61''' pp. 813-826 (1987)]</ref>).

Carl McBride at 10:49, 3 March 2011

2011-03-03T10:49:20Z

← Older revision		Revision as of 12:49, 3 March 2011
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			The '''computation of phase equilibria''' using [[Computer simulation techniques \|computer simulation]] can follow a number of different strategies. Here we will focus mainly on [[first-order transitions]] in fluid phases, usually [[Gas-liquid phase transitions \|liquid-vapour]] equilibria.
	Thermodynamic equilibrium implies, for two phases <math> \alpha </math> and <math> \beta </math>:		Thermodynamic equilibrium implies, for two phases <math> \alpha </math> and <math> \beta </math>:
	* equal [[temperature]]s: <math> T_{\alpha} = T_{\beta} </math>		* equal [[temperature]]s: <math> T_{\alpha} = T_{\beta} </math>
	* equal [[pressure]]s: <math> p_{\alpha} = p_{\beta} </math>		* equal [[pressure]]s: <math> p_{\alpha} = p_{\beta} </math>
	* equal [[chemical potential]]s: <math> \mu_{\alpha} = \mu_{\beta} </math>		* equal [[chemical potential]]s: <math> \mu_{\alpha} = \mu_{\beta} </math>
	~~The computation of phase equilibria using computer simulation can follow a number of different strategies. Here we will focus mainly~~
	~~on [[first-order transitions]] in fluid phases, usually [[Gas-liquid phase transitions \|liquid-vapour]] equilibria.~~
	== Independent simulations for each phase at fixed temperature in the [[canonical ensemble]] ==		== Independent simulations for each phase at fixed temperature in the [[canonical ensemble]] ==
	Simulations can be carried out using either the [[Monte Carlo]] or the [[molecular dynamics]] technique.		Simulations can be carried out using either the [[Monte Carlo]] or the [[molecular dynamics]] technique.

Carl McBride: Changed references to Cite format

2011-03-03T10:43:28Z

Changed references to Cite format

← Older revision		Revision as of 12:43, 3 March 2011
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	* The simulation results in the two phase region will depend dramatically on the system size (calculations with different number of particles become convenient to check the quality of the phase equilibria results)		* The simulation results in the two phase region will depend dramatically on the system size (calculations with different number of particles become convenient to check the quality of the phase equilibria results)
	== Direct simulation of the two phase system==		== Direct simulation of the two phase system==
	~~An example using~~ the [[Lennard-Jones model]],		Direct simulation of the two phase system was first implemented by Abraham <ref>[http://dx.doi.org/10.1103/PhysRevB.23.6145 Farid F. Abraham "Two-dimensional melting, solid-state stability, and the Kosterlitz-Thouless-Feynman criterion", Physical Review B '''23''' pp. 6145-6148 (1981)]</ref>.
	*[http://dx.doi.org/10.1063/1.1474581 James R. Morris and Xueyu Song "The melting lines of model systems calculated from coexistence simulations", Journal of Chemical Physics '''116''' 9352 (2002)]		It has since been applied to the [[Lennard-Jones model]] <ref>[http://dx.doi.org/10.1063/1.1474581 James R. Morris and Xueyu Song "The melting lines of model systems calculated from coexistence simulations", Journal of Chemical Physics '''116''' 9352 (2002)]</ref>
	and ~~its application to~~ [[water]]		and [[water]]
	*[http://dx.doi.org/10.1063/1.2183308 Ramón García Fernández, José L. F. Abascal, and Carlos Vega "The melting point of ice Ih for common water models calculated from direct coexistence of the solid-liquid interface", Journal of Chemical Physics '''124''' 144506 (2006)]		<ref>[http://dx.doi.org/10.1063/1.2183308 Ramón García Fernández, José L. F. Abascal, and Carlos Vega "The melting point of ice Ih for common water models calculated from direct coexistence of the solid-liquid interface", Journal of Chemical Physics '''124''' 144506 (2006)]</ref>.
	== Gibbs ensemble Monte Carlo for one component systems==		== Gibbs ensemble Monte Carlo for one component systems==
	The [[Gibbs ensemble Monte Carlo]] method is often considered as a smart variation of the standard canonical ensemble procedure (See ~~Ref~~. 1).		The [[Gibbs ensemble Monte Carlo]] method is often considered as a smart variation of the standard canonical ensemble procedure (See <ref>[http://dx.doi.org/10.1080/00268978700101491 Athanassios Panagiotopoulos "Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble", Molecular Physics '''61''' pp. 813-826 (1987)]</ref>).
	The simulation is, therefore, carried out at constant volume, temperature and number of particles.		The simulation is, therefore, carried out at constant volume, temperature and number of particles.
	The whole system is divided into two non-interacting parts, each one has its own simulation		The whole system is divided into two non-interacting parts, each one has its own simulation
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	*[[Gibbs-Duhem integration]]		*[[Gibbs-Duhem integration]]
	==References==		==References==
	~~#[http:~~//dx.doi.org/10.1080/00268978700101491 Athanassios Panagiotopoulos "Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble", Molecular Physics '''61''' pp. 813-826 (1987)]		<references/>
	[[category: computer simulation techniques]]		[[category: computer simulation techniques]]

Carl McBride: Reverted edits by Roberta (talk) to last revision by 161.111.20.32

2010-08-20T09:40:28Z

Reverted edits by Roberta (talk) to last revision by 161.111.20.32

← Older revision		Revision as of 11:40, 20 August 2010
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	# In these simulations we can compute for each density (at fixed temperature) the values of the pressure and the chemical potentials (for instance using the [[Widom test-particle method]])		# In these simulations we can compute for each density (at fixed temperature) the values of the pressure and the chemical potentials (for instance using the [[Widom test-particle method]])
	===== A quick 'first guess' method =====		===== A quick 'first guess' method =====

	~~[http://www.uk.bestessays.com/ essay writing service]~~

	Using the previously obtained results the following, somewhat unsophisticated, procedure can be used to obtain a first inspection of the possible phase equilibrium:		Using the previously obtained results the following, somewhat unsophisticated, procedure can be used to obtain a first inspection of the possible phase equilibrium:
	#Fit the simulation results for each branch by using appropriate functional forms: <math> \left. \mu_{v}(\rho) \right. ; p_v(\rho);\mu_l(\rho); p_l(\rho) </math>		#Fit the simulation results for each branch by using appropriate functional forms: <math> \left. \mu_{v}(\rho) \right. ; p_v(\rho);\mu_l(\rho); p_l(\rho) </math>

Roberta at 08:09, 19 August 2010

2010-08-19T08:09:42Z

← Older revision		Revision as of 10:09, 19 August 2010
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	# In these simulations we can compute for each density (at fixed temperature) the values of the pressure and the chemical potentials (for instance using the [[Widom test-particle method]])		# In these simulations we can compute for each density (at fixed temperature) the values of the pressure and the chemical potentials (for instance using the [[Widom test-particle method]])
	===== A quick 'first guess' method =====		===== A quick 'first guess' method =====

			[http://www.uk.bestessays.com/ essay writing service]

	Using the previously obtained results the following, somewhat unsophisticated, procedure can be used to obtain a first inspection of the possible phase equilibrium:		Using the previously obtained results the following, somewhat unsophisticated, procedure can be used to obtain a first inspection of the possible phase equilibrium:
	#Fit the simulation results for each branch by using appropriate functional forms: <math> \left. \mu_{v}(\rho) \right. ; p_v(\rho);\mu_l(\rho); p_l(\rho) </math>		#Fit the simulation results for each branch by using appropriate functional forms: <math> \left. \mu_{v}(\rho) \right. ; p_v(\rho);\mu_l(\rho); p_l(\rho) </math>

161.111.20.32: /* Gibbs ensemble Monte Carlo for one component systems */

2010-08-06T17:11:24Z

Gibbs ensemble Monte Carlo for one component systems

← Older revision		Revision as of 19:11, 6 August 2010
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	*[http://dx.doi.org/10.1063/1.2183308 Ramón García Fernández, José L. F. Abascal, and Carlos Vega "The melting point of ice Ih for common water models calculated from direct coexistence of the solid-liquid interface", Journal of Chemical Physics '''124''' 144506 (2006)]		*[http://dx.doi.org/10.1063/1.2183308 Ramón García Fernández, José L. F. Abascal, and Carlos Vega "The melting point of ice Ih for common water models calculated from direct coexistence of the solid-liquid interface", Journal of Chemical Physics '''124''' 144506 (2006)]
	== Gibbs ensemble Monte Carlo for one component systems==		== Gibbs ensemble Monte Carlo for one component systems==
	The [[Gibbs ensemble Monte Carlo]] method is often considered as a 'smart' variation of the standard canonical ensemble procedure (See Ref. 1).		The [[Gibbs ensemble Monte Carlo]] method is often considered as a smart variation of the standard canonical ensemble procedure (See Ref. 1).
	The simulation is, therefore, carried out at constant volume, temperature and number of particles.		The simulation is, therefore, carried out at constant volume, temperature and number of particles.
	The whole system is divided into two non-interacting parts, each one has its own simulation		The whole system is divided into two non-interacting parts, each one has its own simulation

Nice and Tidy: /* Gibbs ensemble Monte Carlo for one component systems */

2010-02-11T14:05:50Z

Gibbs ensemble Monte Carlo for one component systems

← Older revision		Revision as of 16:05, 11 February 2010
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	The simulation is, therefore, carried out at constant volume, temperature and number of particles.		The simulation is, therefore, carried out at constant volume, temperature and number of particles.
	The whole system is divided into two non-interacting parts, each one has its own simulation		The whole system is divided into two non-interacting parts, each one has its own simulation
	box with its own [[~~boundary conditions \|~~periodic boundary conditions]].		box with its own [[periodic boundary conditions]].
	This separation of the two phases into different boxes is in order to suppress any influence due to [[interface \| interfacial]] effects.		This separation of the two phases into different boxes is in order to suppress any influence due to [[interface \| interfacial]] effects.
	The two subsystems can interchange volume and particles. The rules for these interchanges are		The two subsystems can interchange volume and particles. The rules for these interchanges are
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	[[category: Monte Carlo]]		[[category: Monte Carlo]]
	[[category: Computer simulation techniques]]		[[category: Computer simulation techniques]]

	== Mixtures ==		== Mixtures ==
	=== Symmetric mixtures ===		=== Symmetric mixtures ===

Carl McBride: More trivial changes.

2008-10-30T13:05:01Z

More trivial changes.

← Older revision		Revision as of 15:05, 30 October 2008
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	conditions <math> \left. N,P,T \right. </math>. Let <math> p_{eq} </math> the pressure at which the phase transition occurs. In such a		conditions <math> \left. N,P,T \right. </math>. Let <math> p_{eq} </math> the pressure at which the phase transition occurs. In such a
	case the following scenario is expected for <math> \left. P(V\|N,p,T) \right. </math>:		case the following scenario is expected for <math> \left. P(V\|N,p,T) \right. </math>:
	*<math> \left. P(V\|N,p_{eq},T) \right. </math> has two maxima, corresponding to the liquid and vapor pure phases, with		*<math> \left. P(V\|N,p_{eq},T) \right. </math> has two maxima, corresponding to the liquid and vapor pure phases, with <math> \left. P(V_v\|N,p_{eq},T) = P(V_l\|N,p_{eq},T) = P_{v/l} \right. </math>

	: <math> \left. P(V_v\|N,p_{eq},T) = P(V_l\|N,p_{eq},T) = P_{v/l} \right. </math>

	*The probability of a given intermediate volume at <math> \left. p_{eq} \right. </math> can be estimated (from macroscopic arguments) as:		*The probability of a given intermediate volume at <math> \left. p_{eq} \right. </math> can be estimated (from macroscopic arguments) as:

	: <math> \left. P(V\|N,p_{eq},T) \simeq P_{v/l} \times \exp \left[ - \frac{ \gamma(T) \mathcal A }{k_B T } \right] \right. </math>,		:<math> \left. P(V\|N,p_{eq},T) \simeq P_{v/l} \times \exp \left[ - \frac{ \gamma(T) \mathcal A }{k_B T } \right] \right. </math>,

	where <math> \left. \gamma(T) \right. </math> is the [[surface tension]] of the vapor-liquid interface,		where <math> \left. \gamma(T) \right. </math> is the [[surface tension]] of the vapor-liquid interface,
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	A similar procedure can be built up to compute <math> \left. p(\rho) \right. </math>		A similar procedure can be built up to compute <math> \left. p(\rho) \right. </math>
	from <math> \left. \mu(\rho) \right. </math>.		from <math> \left. \mu(\rho) \right. </math>.

	Once <math> \left. p(\rho) \right. </math> and <math> \left. \mu(\rho) \right. </math> are known it is straightforward to compute the coexistence point.		Once <math> \left. p(\rho) \right. </math> and <math> \left. \mu(\rho) \right. </math> are known it is straightforward to compute the coexistence point.
	==== Practical details ====		==== Practical details ====
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	== Mixtures ==		== Mixtures ==
	=== Symmetric mixtures ===		=== Symmetric mixtures ===
	Examples of symmetric [[mixtures]] can be found both in lattice of continuous model. The [[Ising Models]] can be		Examples of symmetric [[mixtures]] can be found both in lattice of continuous model. The [[Ising Models \| Ising model]] can be
	viewed as mixture of two different chemical species which de-mix at low		viewed as mixture of two different chemical species which de-mix at low
	temperature. The symmetry in the interactions can be exploited to simplify the calculation of phase diagrams.		temperature. The symmetry in the interactions can be exploited to simplify the calculation of phase diagrams.