Gibbs-Duhem integration

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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

Consider two thermodynamic phases: and , at thermodynamic equilibrium at certain conditions. The thermodynamic equilibrium implies:

  • Equal temperature in both phases: , i.e. thermal equilbirum.
  • Equal pressure in both phases , i.e. mechanical equilbrium.
  • Equal chemical potentials for the components , i.e. material equilibrium.

In addition if we are dealing with a statistical mechanics model, with certain parameters that we can represent as , the model should be the same in both phases.

Example: phase equilibria of one-component system

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 we wil have for any phase:

Taking into account that is the Gibbs free energy per particle

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:

whrere for any porperty we can define: (i.e. the difference between the values of the property in the phases). Taking a path with, for instance constante , 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) runs to estimate the values of for both

phases at given values of .

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

References

  1. 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)
  2. 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)