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 (Refs. 1 and 2).

Basic Features

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

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:

$\beta \mu _{a}\left(\beta ,\beta p,\lambda \right)=\beta \mu _{b}\left(\beta ,\beta p,\lambda \right)$

where

• $\beta :=1/k_{B}T$, where $k_{B}$ is the Boltzmann constant

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

$d\left(\beta \mu \right)={\frac {E}{N}}d\beta +{\frac {V}{N}}d(\beta p)+\left[{\frac {\partial (\beta \mu )}{\partial \lambda }}\right]_{\beta ,\beta p}d\lambda .$

where:

• $\left.E\right.$ is the internal energy (sometimes written as $U$).

• $\left.V\right.$ 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:

• Computer simulation (for instance using Metropolis Monte Carlo in the NpT ensemble) runs to estimate the values of ${\bar {L}},{\bar {V}}$ for both

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

• 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 Ref. 3 and the entry: 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

1. 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)
2. 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)
3. 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)
4. 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)