Gibbs-Duhem integration

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History

The so-called Gibbs-Duhem Integration referes to a number of methods that couple molecular simulation techniques with thermodynamic equations in order to draw phase coexistence lines.

The method was proposed by Kofke (Ref 1-2).

Basic Features

Consider two thermodynamic phases: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b } , at thermodynamic equilibrium at certain conditions. The thermodynamic equilibrium implies:

• Equal temperature in both phases: $T=T_{a}=T_{b}$, i.e. thermal equilbirum.
• Equal pressure in both phases Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = p_{a} = p_{b} } , i.e. mechanical equilbrium.
• Equal chemical potentials for the components Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_i = \mu_{ia} = \mu_{ib} } , i.e. material equilibrium.

In addition if we are dealing with a statistical mechanics model, with certain parameters that we can represent as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda } , 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta \mu_{a} \left( \beta, \beta p, \lambda \right) = \beta \mu_{b} \left( \beta, \beta p, \lambda \right) }

where

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta = 1/k_B T } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B } is the Boltzmann constant

When a differential change of the conditions is performed we wil have for any phase:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu } is the Gibbs free energy per particle

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

Let us use a bar to design quantities divided by the number of particles: e.g. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{E} = E/N; \bar{V} = V/N } ; and taking into account the definition:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{L} \equiv \frac{1}{N} \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \left[ \beta \mu_{a} - \beta \mu_b \right] = 0 }

Therefore, to keep the system on the coexistence conditions, the changes in the variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta, (\beta p), \lambda } are constrained to fulfill:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( \Delta \bar{E} \right) d \beta + \left( \Delta \bar{V} \right) d (\beta p) + \left(\Delta \bar{L} \right) d \lambda = 0 }

whrere for any porperty Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X } we can define: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 constante Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta } , the coexistence line will follow the trajectory produced by the solution of the differential equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(\beta p) = - \frac{ \Delta \bar{L} }{\Delta \bar{V} } d \lambda. }

TO BE CONTINUED

References

1. David A. Kofke, Gibbs-Duhem integration: a new method for direct evaluation of phase coexistence by molecular simulation, Mol. Phys. 78 , pp 1331 - 1336 (1993)
2. David A. Kofke, Direct evaluation of phase coexistence by molecular simulation via integration along the saturation line, J. Chem. Phys. 98 ,pp. 4149-4162 (1993)