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 (Refs. 1 and 2).

1 Basic Features
2 Example: phase equilibria of one-component system
3 Peculiarities of the method (Warnings)
4 References

[edit] 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 $μ i = μ i a = μ i b$ , i.e. material equilibrium.

In addition, if one is dealing with a statistical mechanical model, having certain parameters that can be represented as $λ$ , then the model should be the same in both phases.

[edit] 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 $T, p,λ$ 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

$β: = 1 / k B T$ , 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 $μ$ 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 $β,(β p),λ$ 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 $β$ , the coexistence line will follow the trajectory produced by the solution of the differential equation: