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| The so-called '''Gibbs-Duhem integration''' refers to a number of methods that couple | | CURRENTLY THIS ARTICLE IS UNDER CONSTRUCTION |
| molecular [[Computer simulation techniques |simulation techniques]] with [[Thermodynamic relations |thermodynamic equations]] in order to draw | | == History == |
| [[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>
| | The so-called Gibbs-Duhem Integration referes to a number of methods that couple |
| <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>.
| | molecular simulation techniques with thermodynamic equations in order to draw |
| | phase coexistence lines. |
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| | The method was proposed by Kofke (Ref 1-2). |
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| == Basic Features == | | == Basic Features == |
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| Consider two thermodynamic phases: <math> a </math> and <math> b </math>, at thermodynamic equilibrium at certain conditions. Thermodynamic equilibrium implies: | | Consider two thermodynamic phases: <math> a </math> and <math> b </math>, at thermodynamic equilibrium at certain conditions. |
| | The thermodynamic equilibrium implies: |
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| * 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 equilbirum. |
| * 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 equilbrium. |
| * Equal [[chemical potential]]s for the components <math> \mu_i = \mu_{ia} = \mu_{ib} </math>, i.e. ''material'' equilibrium. | | * Equal chemical potentials for the components <math> \mu_i = \mu_{ia} = \mu_{ib} </math>, i.e. ''material'' equilibrium. |
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| 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 | | 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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| == Example: phase equilibria of one-component system == | | == Example: phase equilibria of one-compoment system == |
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| 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 one will have, for any phase: | | When a differential change of the conditions is performed we wil have for any phase: |
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| : <math> d \left( \beta\mu \right) = \left[ \frac{ \partial (\beta \mu) }{\partial \beta} \right]_{\beta p,\lambda} d \beta +
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| \left[ \frac{ \partial (\beta \mu) }{\partial (\beta p)} \right]_{\beta,\lambda} d (\beta p) +
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| \left[ \frac{ \partial (\beta \mu) }{\partial \lambda} \right]_{\beta,\beta p} d \lambda.
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| </math>
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| Taking into account that <math> \mu </math> is the [[Gibbs energy function]] per particle
| | : <math> d \mu = \left( \frac{ \partial \mu }{\partial T} \right)_{p,\lambda} d T + |
| : <math> d \left( \beta\mu \right) = \frac{E}{N} d \beta + \frac{ V }{N } d (\beta p) +
| | \left( \frac{ \partial \mu }{\partial p} \right)_{T,\lambda} d p + |
| \left[ \frac{ \partial (\beta \mu) }{\partial \lambda} \right]_{\beta,\beta p} d \lambda. | | \left( \frac{ \partial \mu }{\partial \lambda} \right)_{T,p} d \lambda. |
| </math> | | </math> |
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| where:
| | Taking into account that <math> \mu </math> is the [[Gibbs energy function|Gibbs free energy]] per particle: |
| * <math> \left. E \right. </math> is the [[internal energy]] (sometimes written as <math>U</math>).
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| * <math> \left. V \right. </math> is the volume
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| * <math> \left. N \right. </math> is the number of particles
| | TO BE CONTINUED .. soon |
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| <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
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| in the isothermal-isobaric ensemble
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| 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>;
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| and taking into account the definition:
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| : <math> \bar{L} \equiv \left[ \frac {\partial (\beta \mu )}{\partial \lambda }\right]_{\beta,\beta p} </math>
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| 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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| we want to modify slightly the conditions. In order to keep the system at the coexistence conditions:
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| : <math> d \left[ \beta \mu_{a} - \beta \mu_b \right] = 0 </math>
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| Therefore, to keep the system on the coexistence conditions, the changes in the variables <math> \beta, (\beta p), \lambda </math> are
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| constrained to fulfill:
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| :<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>
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| 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).
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| Taking a path with, for instance constant <math> \beta </math>, the coexistence line will follow the trajectory produced by the solution of the
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| differential equation:
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| :<math> d(\beta p) = - \frac{ \Delta \bar{L} }{\Delta \bar{V} } d \lambda. </math> (Eq. 1)
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| The Gibbs-Duhem integration technique, for this example, will be a numerical procedure covering the following tasks:
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| * 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
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| phases at given values of <math> [\beta, \beta p, \lambda ] </math>.
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| * A procedure to solve numerically the differential equation (Eq.1)
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| == Peculiarities of the method (Warnings) ==
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| * 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]]).
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| * The ''integrand'' of the differential equation is computed with some numerical uncertainty
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| * Care must be taken to reduce (and estimate) possible departures from the correct coexistence lines
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| == References == | | == 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, Mol. Phys. '''78''' , pp 1331 - 1336 (1993)] |
| '''Related reading'''
| | #[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, J. Chem. Phys. '''98''' ,pp. 4149-4162 (1993) ] |
| *[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)]
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| *[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)]
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| [[category: computer simulation techniques]]
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