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Gibbs-Duhem integration

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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:  a and  b  , at thermodynamic equilibrium at certain conditions. Thermodynamic equilibrium implies:

  • Equal temperature in both phases:  T = T_{a} = T_{b} , i.e. thermal equilibrium.
  • Equal pressure in both phases  p = p_{a} = p_{b} , i.e. mechanical equilibrium.
  • Equal chemical potentials for the components  \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  \lambda , 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  T , p, \lambda two phases of the systems are at equilibrium, this implies:
 \mu_{a} \left( T, p, \lambda \right) = \mu_{b} \left( T, p, \lambda \right)

Given the thermal equilibrium we can also write:

 \beta \mu_{a} \left( \beta, \beta p, \lambda \right) = \beta \mu_{b} \left( \beta, \beta p, \lambda \right)


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

 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  \mu is the Gibbs energy function per particle

 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. V \right. is the volume
  •  \left. N \right. is the number of particles

 \left. \right. E, V are the mean values of the energy and volume for a system of  \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.  \bar{E} = E/N; \bar{V} = V/N ; and taking into account the definition:

 \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 \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:

 d \left[ \beta \mu_{a} - \beta \mu_b \right] = 0

Therefore, to keep the system on the coexistence conditions, the changes in the variables  \beta, (\beta p), \lambda are constrained to fulfill:

  \left( \Delta  \bar{E} \right) d  \beta + \left( \Delta \bar{V} \right) d (\beta p) + \left(\Delta  \bar{L} \right) d \lambda = 0

where for any property  X we can define:  \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  \beta , the coexistence line will follow the trajectory produced by the solution of the differential equation:

 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:

phases at given values of  [\beta, \beta p,  \lambda ] .

  • 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


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