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 .

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 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

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)