Editing Gibbs-Duhem integration
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The so-called | == History == | ||
molecular | The so-called Gibbs-Duhem Integration refers to a number of methods that couple | ||
[[Computation of phase equilibria | phase coexistence]] lines. The | molecular simulation techniques with thermodynamic equations in order to draw | ||
[[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. | ||
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 | In addition if we are dealing with a statistical mechanics model, with certain parameters that we can represent as <math> \lambda </math> , the | ||
model should be the same in both phases. | model should be the same in both phases. | ||
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where | where | ||
* <math> \beta | * <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 | 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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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 | * 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 | ||
phases at given values of <math> [\beta, \beta p, \lambda ] </math>. | phases at given values of <math> [\beta, \beta p, \lambda ] </math>. | ||
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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 | * 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 | ||
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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)] | |||
''' | #[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)] | ||
#[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)] | |||