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: 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 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


  • 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


Related reading