Gibbs-Duhem integration: Difference between revisions

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The so-called '''Gibbs-Duhem integration''' refers  to a number of methods that couple
== History ==
molecular [[Computer simulation techniques |simulation techniques]]  with [[Thermodynamic relations |thermodynamic equations]] in order to draw
The so-called Gibbs-Duhem Integration referes to a number of methods that couple
[[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 (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. Thermodynamic equilibrium implies:
The thermodynamic equilibrium implies:


* Equal temperature in both phases: <math> T = T_{a} = T_{b} </math>, i.e. thermal equilbirum.
* 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 equilbrium.
* Equal [[pressure]] in both phases <math> p = p_{a} = p_{b} </math>, i.e. mechanical equilibrium.
* Equal chemical potentials 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 if we are dealing with a statistical mechanics model, with certain parameters that we can represent as <math> \lambda </math> , the
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.


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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 we wil have for any phase:
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|Gibbs free energy]] per particle
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 \frac{1}{N} \left[ \frac {\partial (\beta \mu )}{\partial \lambda }\right]_{\beta,\beta p} </math>
: <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>


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).
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 constante <math> \beta </math>, the coexistence line will  follow the trajectory produced by the solution of the
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:
differential equation:


<math> d(\beta p) = - \frac{ \Delta \bar{L} }{\Delta \bar{V} } d \lambda. </math>
:<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]]).


TO BE CONTINUED
* 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 ==
#[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)]
<references/>
#[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) ]
'''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 12:02, 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 , at thermodynamic equilibrium at certain conditions. Thermodynamic equilibrium implies:

  • Equal temperature in both phases: , i.e. thermal equilibrium.
  • Equal pressure in both phases , i.e. mechanical equilibrium.
  • Equal chemical potentials for the components , 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 two phases of the systems are at equilibrium, this implies:

Given the thermal equilibrium we can also write:

where

  • , where is the Boltzmann constant

When a differential change of the conditions is performed one will have, for any phase:

Taking into account that is the Gibbs energy function per particle

where:

  • is the internal energy (sometimes written as ).
  • is the volume
  • is the number of particles

are the mean values of the energy and volume for a system of 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:

Again, let us suppose that we have a phase coexistence at a point given by and that we want to modify slightly the conditions. In order to keep the system at the coexistence conditions:

Therefore, to keep the system on the coexistence conditions, the changes in the variables are constrained to fulfill:

where for any property we can define: (i.e. the difference between the values of the property in the phases). Taking a path with, for instance constant , the coexistence line will follow the trajectory produced by the solution of the differential equation:

(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 .

  • A procedure to solve numerically the differential equation (Eq.1)

Peculiarities of the method (Warnings)[edit]

  • 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]

Related reading