Wandering interface method - Revision history

Nice and Tidy at 14:07, 11 February 2010

2010-02-11T14:07:35Z

← Older revision		Revision as of 16:07, 11 February 2010
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	The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P({\mathcal A})</math>.		The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P({\mathcal A})</math>.

	This goal is achieved in practice by employing a [[canonical ensemble]] where the [[~~Boundary~~ conditions \|box shape]] as described by the surface area,		This goal is achieved in practice by employing a [[canonical ensemble]] where the [[Periodic boundary conditions \|box shape]] as described by the surface area,
	<math>{\mathcal A}</math> is considered as an extra ensemble variable.		<math>{\mathcal A}</math> is considered as an extra ensemble variable.
	During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual [[Metropolis Monte Carlo]] scheme. This provides a [[Markov chain \|Markovian random walk]] of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:		During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual [[Metropolis Monte Carlo]] scheme. This provides a [[Markov chain \|Markovian random walk]] of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:

Nice and Tidy: Changed reference to Cite format.

2009-05-11T10:13:38Z

Changed reference to Cite format.

← Older revision		Revision as of 12:13, 11 May 2009
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	The '''wandering interface method''' (WIM) is a general, versatile and robust procedure for the calculation of surface [[Helmholtz energy function \|free energies]]. It can be employed to calculate liquid-vapor [[surface tension]] and solid-fluid interfacial tension alike. It does not require the explicit evaluation of the [[Virial equation of state \| virial]] and may be employed for both continuous and discontinuous [[Idealised models \|model potentials]].		The '''wandering interface method''' (WIM) is a general, versatile and robust procedure for the calculation of surface [[Helmholtz energy function \|free energies]]
			<ref>[http://link.aps.org/doi/10.1103/PhysRevE.75.061609 L.G. MacDowell and P. Bryk "Direct Calculation of Interfacial Tensions from Computer Simulation: Results for Freely Jointed Tangent Hard Sphere Chains", Physical Review E '''75''' 061609 (2007)]</ref>.
			It can be employed to calculate liquid-vapor [[surface tension]] and solid-fluid interfacial tension alike. It does not require the explicit evaluation of the [[Virial equation of state \| virial]] and may be employed for both continuous and discontinuous [[Idealised models \|model potentials]].

	The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P({\mathcal A})</math>.		The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P({\mathcal A})</math>.
Line 7:		Line 9:
	During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual [[Metropolis Monte Carlo]] scheme. This provides a [[Markov chain \|Markovian random walk]] of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:		During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual [[Metropolis Monte Carlo]] scheme. This provides a [[Markov chain \|Markovian random walk]] of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:
	:<math> k_BT\ln P({\mathcal A}) \propto -\gamma {\mathcal A} </math>		:<math> k_BT\ln P({\mathcal A}) \propto -\gamma {\mathcal A} </math>
	Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P({\mathcal A})</math> in a logarithmic plot.		Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P({\mathcal A})</math> on a logarithmic plot.

	Since the system will tend to stretch continuously so as to achieve a vanishing surface area, one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the [[microstate]] probability distribution is given by:		Since the system will tend to stretch continuously so as to achieve a vanishing surface area, one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the [[microstate]] probability distribution is given by:
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	Several other choices are possible and have been tested since then.		Several other choices are possible and have been tested since then.


	==References==		==References==
	~~#[http:~~//link.aps.org/doi/10.1103/PhysRevE.75.061609 L.G. MacDowell and P. Bryk "Direct Calculation of Interfacial Tensions from Computer Simulation: Results for Freely Jointed Tangent Hard Sphere Chains", Physical Review E '''75''' 061609 (2007)]		<references/>
	[[category: Computer simulation techniques]]		[[category: Computer simulation techniques]]

Noe: slight change in notation to keep consistene with other pages

2009-05-11T09:45:15Z

slight change in notation to keep consistene with other pages

← Older revision		Revision as of 11:45, 11 May 2009
Line 1:		Line 1:
	The '''wandering interface method''' (WIM) is a general, versatile and robust procedure for the calculation of surface [[Helmholtz energy function \|free energies]]. It can be employed to calculate liquid-vapor [[surface tension]] and solid-fluid interfacial tension alike. It does not require the explicit evaluation of the [[Virial equation of state \| virial]] and may be employed for both continuous and discontinuous [[Idealised models \|model potentials]].		The '''wandering interface method''' (WIM) is a general, versatile and robust procedure for the calculation of surface [[Helmholtz energy function \|free energies]]. It can be employed to calculate liquid-vapor [[surface tension]] and solid-fluid interfacial tension alike. It does not require the explicit evaluation of the [[Virial equation of state \| virial]] and may be employed for both continuous and discontinuous [[Idealised models \|model potentials]].

	The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>.		The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P({\mathcal A})</math>.

	This goal is achieved in practice by employing a [[canonical ensemble]] where the [[Boundary conditions \|box shape]] as described by the surface area,		This goal is achieved in practice by employing a [[canonical ensemble]] where the [[Boundary conditions \|box shape]] as described by the surface area,
	<math>A</math> is considered as an extra ensemble variable.		<math>{\mathcal A}</math> is considered as an extra ensemble variable.
	During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual [[Metropolis Monte Carlo]] scheme. This provides a [[Markov chain \|Markovian random walk]] of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:		During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual [[Metropolis Monte Carlo]] scheme. This provides a [[Markov chain \|Markovian random walk]] of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:
	:<math> k_BT\ln P(A) \propto -\gamma A </math>		:<math> k_BT\ln P({\mathcal A}) \propto -\gamma {\mathcal A} </math>
	Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P(A)</math> in a logarithmic plot.		Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P({\mathcal A})</math> in a logarithmic plot.

	Since the system will tend to stretch continuously so as to achieve a vanishing surface area, one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the [[microstate]] probability distribution is given by:		Since the system will tend to stretch continuously so as to achieve a vanishing surface area, one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the [[microstate]] probability distribution is given by:
	:<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.		:<math> P(U,{\mathcal A}) \propto \exp(-\beta U + W({\mathcal A}) )</math>.
	where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval. In the original implementation of WIM, the following biasing function was chosen:		where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval. In the original implementation of WIM, the following biasing function was chosen:
	:<math>		:<math>
	\exp(W(A)) =		\exp(W({\mathcal A})) =
	\left \{		\left \{
	\begin{array}{cc}		\begin{array}{cc}
	0 & A < A_{min} \\		0 & {\mathcal A} < {\mathcal A}_{min} \\
	1 & A_{min} < A < A_{max} \\		1 & {\mathcal A}_{min} < {\mathcal A} < {\mathcal A}_{max} \\
	0 & A > A_{max}		0 & {\mathcal A} > {\mathcal A}_{max}
	\end{array}		\end{array}
	\right .		\right .

Nice and Tidy: Slight tidy.

2009-05-11T09:13:20Z

Slight tidy.

Nice and Tidy: Wandering Interface Method moved to Wandering interface method: Lower-case name space.

2009-05-11T09:04:18Z

Wandering Interface Method moved to Wandering interface method: Lower-case name space.

← Older revision	Revision as of 11:04, 11 May 2009
(No difference)

Lgmac at 21:59, 10 May 2009

2009-05-10T21:59:37Z

← Older revision		Revision as of 23:59, 10 May 2009
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	Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P(A)</math> in a logarithmic plot.		Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P(A)</math> in a logarithmic plot.

	Since the system will tend to stretch continuously so as to achieve a vanishing surface area, ~~in practice~~ one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the microstate probability distribution is given by:		Since the system will tend to stretch continuously so as to achieve a vanishing surface area, one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the microstate probability distribution is given by:
	:<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.		:<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.
	where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval. In the original implementation of WIM, the following biasing function was chosen:		where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval. In the original implementation of WIM, the following biasing function was chosen:

Lgmac at 21:53, 10 May 2009

2009-05-10T21:53:52Z

← Older revision		Revision as of 23:53, 10 May 2009
Line 1:		Line 1:
	The Wandering Interface Method (WIM) is a general, versatile and robust procedure for the calculation of surface free energies. It can be employed to calculate liquid-vapor surface ~~tensions~~ and solid-fluid interfacial ~~tensions~~ alike. It does not require the explicit evaluation of the virial and may be employed for both continuous and discontinuous model potentials.		The Wandering Interface Method (WIM) is a general, versatile and robust procedure for the calculation of surface free energies. It can be employed to calculate liquid-vapor [[surface tension]] and solid-fluid interfacial tension alike. It does not require the explicit evaluation of the virial and may be employed for both continuous and discontinuous model potentials.

	The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>.		The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>.

	This goal is achieved in practice by employing a canonical ensemble where the box shape as described by the surface area,		This goal is achieved in practice by employing a [[canonical ensemble]] where the box shape as described by the surface area,
	<math>A</math> is considered as an extra ensemble variable.		<math>A</math> is considered as an extra ensemble variable.
	During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual metropolis scheme. This provides a markovian random walk of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:		During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual [[metropolis Monte Carlo]] scheme. This provides a markovian random walk of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:
	:<math> k_BT\ln P(A) \propto -\gamma A </math>		:<math> k_BT\ln P(A) \propto -\gamma A </math>
	Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P(A)</math> in a logarithmic plot.		Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P(A)</math> in a logarithmic plot.

Lgmac at 21:41, 10 May 2009

2009-05-10T21:41:28Z

← Older revision		Revision as of 23:41, 10 May 2009
Line 1:		Line 1:
	The Wandering Interface Method (WIM) is a general, versatile and robust procedure for the calculation of surface free energies. It can be employed to calculate liquid-vapor surface tensions and solid-fluid interfacial tensions alike.		The Wandering Interface Method (WIM) is a general, versatile and robust procedure for the calculation of surface free energies. It can be employed to calculate liquid-vapor surface tensions and solid-fluid interfacial tensions alike. It does not require the explicit evaluation of the virial and may be employed for both continuous and discontinuous model potentials.

	The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>.		The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>.

	This goal is achieved in practice by ~~considering~~ a canonical ensemble~~, with the usual constant N, V, T variables,~~ where the box shape as described by the surface area,		This goal is achieved in practice by employing a canonical ensemble where the box shape as described by the surface area,
	<math>A</math> is considered as an extra ensemble variable.		<math>A</math> is considered as an extra ensemble variable.
	During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual metropolis scheme. This provides a markovian random walk of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:		During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual metropolis scheme. This provides a markovian random walk of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:

Lgmac at 21:39, 10 May 2009

2009-05-10T21:39:21Z

← Older revision		Revision as of 23:39, 10 May 2009
Line 11:		Line 11:
	Since the system will tend to stretch continuously so as to achieve a vanishing surface area, in practice one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the microstate probability distribution is given by:		Since the system will tend to stretch continuously so as to achieve a vanishing surface area, in practice one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the microstate probability distribution is given by:
	:<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.		:<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.
	where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval.		where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval. In the original implementation of WIM, the following biasing function was chosen:

	:<math>		:<math>
	exp(W(A)) =		\exp(W(A)) =
	\left \{		\left \{
	\begin{array}{cc}		\begin{array}{cc}
Line 23:		Line 22:
	\right .		\right .
	</math>		</math>

			Several other choices are possible and have been tested since then.


	==References==		==References==
	#[http://link.aps.org/doi/10.1103/PhysRevE.75.061609 L.G. MacDowell and P. Bryk "Direct Calculation of Interfacial Tensions from Computer Simulation: Results for Freely Jointed Tangent Hard Sphere Chains", Physical Review E '''75''' 061609 (2007)]		#[http://link.aps.org/doi/10.1103/PhysRevE.75.061609 L.G. MacDowell and P. Bryk "Direct Calculation of Interfacial Tensions from Computer Simulation: Results for Freely Jointed Tangent Hard Sphere Chains", Physical Review E '''75''' 061609 (2007)]

Lgmac at 21:37, 10 May 2009

2009-05-10T21:37:52Z

@@ Line 12: / Line 12: @@
 :<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.
 where <math>W</math> is a  biasing function chosen so as to constraint the fluctuations within some convenient interval.
 ==References==
 #[http://link.aps.org/doi/10.1103/PhysRevE.75.061609  L.G. MacDowell and P. Bryk "Direct Calculation of Interfacial Tensions from Computer Simulation: Results for Freely Jointed Tangent Hard Sphere Chains", Physical Review E '''75''' 061609 (2007)]

← Older revision		Revision as of 11:13, 11 May 2009
Line 1:		Line 1:
	The ~~Wandering Interface Method~~ (WIM) is a general, versatile and robust procedure for the calculation of surface free energies. It can be employed to calculate liquid-vapor [[surface tension]] and solid-fluid interfacial tension alike. It does not require the explicit evaluation of the virial and may be employed for both continuous and discontinuous model potentials.		The '''wandering interface method''' (WIM) is a general, versatile and robust procedure for the calculation of surface [[Helmholtz energy function \|free energies]]. It can be employed to calculate liquid-vapor [[surface tension]] and solid-fluid interfacial tension alike. It does not require the explicit evaluation of the [[Virial equation of state \| virial]] and may be employed for both continuous and discontinuous [[Idealised models \|model potentials]].

	The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>.		The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>.

	This goal is achieved in practice by employing a [[canonical ensemble]] where the box shape as described by the surface area,		This goal is achieved in practice by employing a [[canonical ensemble]] where the [[Boundary conditions \|box shape]] as described by the surface area,
	<math>A</math> is considered as an extra ensemble variable.		<math>A</math> is considered as an extra ensemble variable.
	During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual [[~~metropolis~~ Monte Carlo]] scheme. This provides a ~~markovian~~ random walk of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:		During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual [[Metropolis Monte Carlo]] scheme. This provides a [[Markov chain \|Markovian random walk]] of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:
	:<math> k_BT\ln P(A) \propto -\gamma A </math>		:<math> k_BT\ln P(A) \propto -\gamma A </math>
	Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P(A)</math> in a logarithmic plot.		Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P(A)</math> in a logarithmic plot.

	Since the system will tend to stretch continuously so as to achieve a vanishing surface area, one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the microstate probability distribution is given by:		Since the system will tend to stretch continuously so as to achieve a vanishing surface area, one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the [[microstate]] probability distribution is given by:
	:<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.		:<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.
	where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval. In the original implementation of WIM, the following biasing function was chosen:		where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval. In the original implementation of WIM, the following biasing function was chosen:
Line 28:		Line 28:
	==References==		==References==
	#[http://link.aps.org/doi/10.1103/PhysRevE.75.061609 L.G. MacDowell and P. Bryk "Direct Calculation of Interfacial Tensions from Computer Simulation: Results for Freely Jointed Tangent Hard Sphere Chains", Physical Review E '''75''' 061609 (2007)]		#[http://link.aps.org/doi/10.1103/PhysRevE.75.061609 L.G. MacDowell and P. Bryk "Direct Calculation of Interfacial Tensions from Computer Simulation: Results for Freely Jointed Tangent Hard Sphere Chains", Physical Review E '''75''' 061609 (2007)]
			[[category: Computer simulation techniques]]