Editing Gibbs-Duhem integration
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The so-called | CURRENTLY THIS ARTICLE IS UNDER CONSTRUCTION | ||
molecular | == History == | ||
The so-called Gibbs-Duhem Integration referes to a number of methods that couple | |||
molecular simulation techniques with thermodynamic equations in order to draw | |||
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 | * Equal temperature in both phases: <math> T = T_{a} = T_{b} </math>, i.e. thermal equilbirum. | ||
* Equal | * Equal pressure in both phases <math> p = p_{a} = p_{b} </math>, i.e. mechanical equilbrium. | ||
* Equal | * Equal chemical potentials 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 | When a differential change of the conditions is performed we wil 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]] per particle | Taking into account that <math> \mu </math> is the [[Gibbs energy function|Gibbs free energy]] 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> | ||
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 \left[ \frac {\partial (\beta \mu )}{\partial \lambda }\right]_{\beta,\beta p} </math> | : <math> \bar{L} \equiv \frac{1}{N} \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> | |||
whrere for any porperty <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 | Taking a path with, for instance constante <math> \beta </math>, the coexistence line will follow the trajectory produced by the solution of the | ||
differential equation: | 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: | 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 | * Computer simulation (for instance using [[Metropolis Monte Carlo]]) 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>. | ||
* A procedure to solve numerically the differential equation (Eq.1) | * A procedure to solve numerically the differential equation (Eq.1) | ||
== Peculiarities of the method | == Peculiarities of the method == | ||
== 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)] | |||
#[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) ] | |||