# Gibbs-Duhem integration

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]

- A good initial point must be known to start the procedure (See
^{[3]}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

## References[edit]

- ↑ 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)

**Related reading**

- 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)