Gibbs-Duhem integration: Difference between revisions
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CURRENTLY THIS ARTICLE IS UNDER CONSTRUCTION | CURRENTLY THIS ARTICLE IS UNDER CONSTRUCTION | ||
== History == | == History == | ||
The so-called 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 | molecular simulation techniques with thermodynamic equations in order to draw | ||
phase coexistence lines. | phase coexistence lines. | ||
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The thermodynamic equilibrium implies: | The thermodynamic equilibrium implies: | ||
* Equal temperature in both phases: <math> T = T_{a} = T_{b} </math>, i.e. thermal | * Equal temperature in both phases: <math> T = T_{a} = T_{b} </math>, i.e. thermal equilibrium. | ||
* Equal pressure in both phases <math> p = p_{a} = p_{b} </math>, i.e. mechanical | * Equal pressure in both phases <math> p = p_{a} = p_{b} </math>, i.e. mechanical equilibrium. | ||
* Equal chemical potentials for the components <math> \mu_i = \mu_{ia} = \mu_{ib} </math>, i.e. ''material'' equilibrium. | * Equal chemical potentials for the components <math> \mu_i = \mu_{ia} = \mu_{ib} </math>, i.e. ''material'' equilibrium. | ||
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where | where | ||
* <math> \beta = 1/k_B T </math>, where <math> k_B </math> is the [[Boltzmann constant]] | * <math> \beta = 1/k_B T </math>, where <math> k_B </math> is the [[Boltzmann constant]] | ||
When a differential change of the conditions is performed | When a differential change of the conditions is performed one will, have for any phase: | ||
: <math> d \left( \beta\mu \right) = \left[ \frac{ \partial (\beta \mu) }{\partial \beta} \right]_{\beta p,\lambda} d \beta + | : <math> d \left( \beta\mu \right) = \left[ \frac{ \partial (\beta \mu) }{\partial \beta} \right]_{\beta p,\lambda} d \beta + | ||
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constrained to fulfill: | constrained to fulfill: | ||
<math> \left( \Delta \bar{E} \right) d \beta + \left( \Delta \bar{V} \right) d (\beta p) + \left(\Delta \bar{L} \right) d \lambda = 0 </math> | :<math> \left( \Delta \bar{E} \right) d \beta + \left( \Delta \bar{V} \right) d (\beta p) + \left(\Delta \bar{L} \right) d \lambda = 0 </math> | ||
where for any property <math> X </math> we can define: <math> \Delta X \equiv X_a - X_b </math> (i.e. the difference between the values of the property in the phases). | |||
Taking a path with, for instance | Taking a path with, for instance constant <math> \beta </math>, the coexistence line will follow the trajectory produced by the solution of the | ||
differential equation: | differential equation: | ||
<math> d(\beta p) = - \frac{ \Delta \bar{L} }{\Delta \bar{V} } d \lambda. </math> (Eq. 1) | :<math> d(\beta p) = - \frac{ \Delta \bar{L} }{\Delta \bar{V} } d \lambda. </math> (Eq. 1) | ||
The Gibbs-Duhem integration technique, for this example, will be a numerical procedure covering the following tasks: | The Gibbs-Duhem integration technique, for this example, will be a numerical procedure covering the following tasks: | ||
Revision as of 18:18, 2 March 2007
CURRENTLY THIS ARTICLE IS UNDER CONSTRUCTION
History
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 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: , i.e. thermal equilibrium.
- 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 equilibrium.
- 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:
- 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_{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 one will, 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
- 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) = \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:
- 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. E \right. } is the internal energy
- 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. V \right. } is the volume
- 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. N \right. } is the number of particles
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. \right. E, V } are the mean values of the energy and volume for a system 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 \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. 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 \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 }
where for any property 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: (i.e. the difference between the values of the property in the phases). Taking a path with, for instance constant 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. } (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 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}, \bar{V} } for both
phases at given values 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 [\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
- 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
- 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)
- 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)