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  .
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
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)
- A good initial point must be known to start the procedure (See  and computation of phase equilibria).
- 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
- 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)
- 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)
- 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)
- 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)
- Gerassimos Orkoulas "Communication: Tracing phase boundaries via molecular simulation: An alternative to the Gibbs–Duhem integration method", Journal of Chemical Physics 133 111104 (2010)