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 referes to a number of methods that couple
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 equilbirum.
* 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 equilbrium.
* 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 we wil have for any phase:
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>


whrere for any porperty <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).
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 constante <math> \beta </math>, the coexistence line will  follow the trajectory produced by the solution of the
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: and , at thermodynamic equilibrium at certain conditions. The thermodynamic equilibrium implies:

  • Equal temperature in both phases: , i.e. thermal equilibrium.
  • Equal pressure in both phases , i.e. mechanical equilibrium.
  • Equal chemical potentials for the components , i.e. material equilibrium.

In addition if we are dealing with a statistical mechanics model, with certain parameters that we can represent as , 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 two phases of the systems are at equilibrium, this implies:

Given the thermal equilibrium we can also write:

where

  • , where is the Boltzmann constant

When a differential change of the conditions is performed one will, have for any phase:

Taking into account that is the Gibbs free energy per particle

where:

  • is the internal energy
  • is the volume
  • is the number of particles

are the mean values of the energy and volume for a system of particles in the isothermal-isobaric ensemble

Let us use a bar to design quantities divided by the number of particles: e.g. ; and taking into account the definition:

Again, let us suppose that we have a phase coexistence at a point given by and that we want to modify slightly the conditions. In order to keep the system at the coexistence conditions:

Therefore, to keep the system on the coexistence conditions, the changes in the variables are constrained to fulfill:

where for any property we can define: (i.e. the difference between the values of the property in the phases). Taking a path with, for instance constant , the coexistence line will follow the trajectory produced by the solution of the differential equation:

(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 for both

phases at given values of .

  • 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

  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)