Gibbs-Duhem integration: Difference between revisions
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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 | |||
The so-called Gibbs-Duhem | [[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> | ||
molecular simulation techniques with thermodynamic equations in order to draw | <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>. | ||
phase coexistence lines. | |||
The method was proposed by Kofke ( | |||
== 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: | ||
* Equal temperature in both phases: <math> T = T_{a} = T_{b} </math>, i.e. thermal | * 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 | * Equal [[pressure]] in both phases <math> p = p_{a} = p_{b} </math>, i.e. mechanical equilibrium. | ||
* Equal chemical | * Equal [[chemical potential]]s for the components <math> \mu_i = \mu_{ia} = \mu_{ib} </math>, i.e. ''material'' equilibrium. | ||
In addition if | 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. | ||
== Example: phase equilibria of one- | == Example: phase equilibria of one-component system == | ||
Notice: The derivation that follows is just a particular route to perform the integration | Notice: The derivation that follows is just a particular route to perform the integration | ||
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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 | 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 + | ||
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</math> | </math> | ||
Taking into account that <math> \mu </math> is the [[Gibbs energy function | Taking into account that <math> \mu </math> is the [[Gibbs energy function]] per particle | ||
: <math> d \left( \beta\mu \right) = \frac{E}{N} d \beta + \frac{ V }{N } d (\beta p) + | : <math> d \left( \beta\mu \right) = \frac{E}{N} d \beta + \frac{ V }{N } d (\beta p) + | ||
\left[ \frac{ \partial (\beta \mu) }{\partial \lambda} \right]_{\beta,\beta p} d \lambda. | \left[ \frac{ \partial (\beta \mu) }{\partial \lambda} \right]_{\beta,\beta p} d \lambda. | ||
</math> | </math> | ||
where: | |||
* <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. N \right. </math> is the number of particles | |||
<math> \left. \right. E, V </math> are the mean values of the energy and volume for a system of <math> \left. N \right. </math> particles | |||
in the isothermal-isobaric ensemble | |||
Let us use a bar to design quantities divided by the number of particles: e.g. <math> \bar{E} = E/N; \bar{V} = V/N </math>; | Let us use a bar to design quantities divided by the number of particles: e.g. <math> \bar{E} = E/N; \bar{V} = V/N </math>; | ||
and taking into account the definition: | and taking into account the definition: | ||
: <math> \bar{L} \equiv | : <math> \bar{L} \equiv \left[ \frac {\partial (\beta \mu )}{\partial \lambda }\right]_{\beta,\beta p} </math> | ||
Again, let us suppose that we have a phase coexistence at a point given by <math>\left[ \beta_0, (\beta p)_0, \lambda_0 \right]</math> and that | Again, let us suppose that we have a phase coexistence at a point given by <math>\left[ \beta_0, (\beta p)_0, \lambda_0 \right]</math> and that | ||
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constrained to fulfill: | constrained to fulfill: | ||
<math> \left( \Delta \bar{E} \right) d \beta + \left( \Delta \bar{V} \right) d (\beta p) + \left(\Delta \bar{L} \right) d \lambda = 0 </math> | :<math> \left( \Delta \bar{E} \right) d \beta + \left( \Delta \bar{V} \right) d (\beta p) + \left(\Delta \bar{L} \right) d \lambda = 0 </math> | ||
where for any property <math> X </math> we can define: <math> \Delta X \equiv X_a - X_b </math> (i.e. the difference between the values of the property in the phases). | |||
Taking a path with, for instance constant <math> \beta </math>, the coexistence line will follow the trajectory produced by the solution of the | |||
differential equation: | |||
:<math> d(\beta p) = - \frac{ \Delta \bar{L} }{\Delta \bar{V} } d \lambda. </math> (Eq. 1) | |||
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 [[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>. | |||
* A procedure to solve numerically the differential equation (Eq.1) | |||
== Peculiarities of the method (Warnings) == | |||
* 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 | |||
* Care must be taken to reduce (and estimate) possible departures from the correct coexistence lines | |||
== References == | == References == | ||
<references/> | |||
'''Related reading''' | |||
*[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]] | |||
Latest revision as of 13:02, 28 September 2010
The so-called Gibbs-Duhem integration refers to a number of methods that couple molecular simulation techniques with thermodynamic equations in order to draw phase coexistence lines. The original method was proposed by David Kofke [1] [2].
Basic Features[edit]
Consider two thermodynamic phases: and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b } , at thermodynamic equilibrium at certain conditions. Thermodynamic equilibrium implies:
- Equal temperature in both phases: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T = T_{a} = T_{b} } , i.e. thermal equilibrium.
- Equal pressure in both phases Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = p_{a} = p_{b} } , i.e. mechanical equilibrium.
- Equal chemical potentials for the components Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_i = \mu_{ia} = \mu_{ib} } , i.e. material equilibrium.
In addition, if one is dealing with a statistical mechanical model, having certain parameters that can be represented as , then the model should be the same in both phases.
Example: phase equilibria of one-component system[edit]
Notice: The derivation that follows is just a particular route to perform the integration
- Consider that at given conditions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T , p, \lambda } two phases of the systems are at equilibrium, this implies:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_{a} \left( T, p, \lambda \right) = \mu_{b} \left( T, p, \lambda \right) }
Given the thermal equilibrium we can also write:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta \mu_{a} \left( \beta, \beta p, \lambda \right) = \beta \mu_{b} \left( \beta, \beta p, \lambda \right) }
where
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta := 1/k_B T } , where is the Boltzmann constant
When a differential change of the conditions is performed one will have, for any phase:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \left( \beta\mu \right) = \left[ \frac{ \partial (\beta \mu) }{\partial \beta} \right]_{\beta p,\lambda} d \beta + \left[ \frac{ \partial (\beta \mu) }{\partial (\beta p)} \right]_{\beta,\lambda} d (\beta p) + \left[ \frac{ \partial (\beta \mu) }{\partial \lambda} \right]_{\beta,\beta p} d \lambda. }
Taking into account that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu } is the Gibbs energy function per particle
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \left( \beta\mu \right) = \frac{E}{N} d \beta + \frac{ V }{N } d (\beta p) + \left[ \frac{ \partial (\beta \mu) }{\partial \lambda} \right]_{\beta,\beta p} d \lambda. }
where:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. E \right. } is the internal energy (sometimes written as ).
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. V \right. } is the volume
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. N \right. } is the number of particles
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \right. E, V } are the mean values of the energy and volume for a system of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. N \right. } particles in the isothermal-isobaric ensemble
Let us use a bar to design quantities divided by the number of particles: e.g. ; and taking into account the definition:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{L} \equiv \left[ \frac {\partial (\beta \mu )}{\partial \lambda }\right]_{\beta,\beta p} }
Again, let us suppose that we have a phase coexistence at a point given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ \beta_0, (\beta p)_0, \lambda_0 \right]} and that we want to modify slightly the conditions. In order to keep the system at the coexistence conditions:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \left[ \beta \mu_{a} - \beta \mu_b \right] = 0 }
Therefore, to keep the system on the coexistence conditions, the changes in the variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta, (\beta p), \lambda } are constrained to fulfill:
where for any property Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X } we can define: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta X \equiv X_a - X_b } (i.e. the difference between the values of the property in the phases). Taking a path with, for instance constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta } , the coexistence line will follow the trajectory produced by the solution of the differential equation:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(\beta p) = - \frac{ \Delta \bar{L} }{\Delta \bar{V} } d \lambda. } (Eq. 1)
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 for both
phases at given values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\beta, \beta p, \lambda ] } .
- A procedure to solve numerically the differential equation (Eq.1)
Peculiarities of the method (Warnings)[edit]
- A good initial point must be known to start the procedure (See [3] and computation of phase equilibria).
- The integrand of the differential equation is computed with some numerical uncertainty
- Care must be taken to reduce (and estimate) possible departures from the correct coexistence lines
References[edit]
- ↑ 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)
- ↑ 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)
- ↑ 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)
Related reading
- 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)
- Gerassimos Orkoulas "Communication: Tracing phase boundaries via molecular simulation: An alternative to the Gibbs–Duhem integration method", Journal of Chemical Physics 133 111104 (2010)