Computation of phase equilibria: Difference between revisions
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Thermodynamic equilibrium implies, for two phases <math> \alpha </math> and <math> \beta </math>: | |||
* Equal [[temperature]]: <math> T_{\alpha} = T_{\beta} </math> | |||
* Equal [[pressure]]: <math> p_{\alpha} = p_{\beta} </math> | |||
* Equal [[chemical potential]]: <math> \mu_{\alpha} = \mu_{\beta} </math> | |||
The computation of phase equilibria using computer simulation can follow a number of different strategies. | |||
=== Independent simulations for each phase at fixed <math> T </math> in the [[canonical ensemble]] === | === Independent simulations for each phase at fixed <math> T </math> in the [[canonical ensemble]] === | ||
Simulations can be carried out either using [[Monte Carlo]] or [[Molecular dynamics]] techniques. | |||
Assuming that one has some knowledge on the phase diagram of the system, one can try the following recipe: | |||
- Fix a temperature and a number of particles | - Fix a temperature and a number of particles | ||
- Perform a | - Perform a limited number of simulations in the low density region (where the gas phase density is expected to be) | ||
- Perform a | - Perform a limited number of simulations in the moderate to high density region (where the liquid phase should appear) | ||
- In these simulations we can compute for each density (at fixed | - In these simulations we can compute for each density (at fixed temperature) the values of the pressure and the | ||
chemical potentials (for instance using the [[Widom test-particle method]]) | chemical potentials (for instance using the [[Widom test-particle method]]) | ||
==== A quick (and dirty?) method ==== | |||
Using the previously obtained results the following somewhat unsophisticated procedure can be used to obtain a first inspection of the possible phase equilibrium. | |||
Fit the simulation results for each branch by using appropriate functional forms: | |||
:<math> \left. \mu_{v}(\rho) \right. ; p_v(\rho);\mu_l(\rho); p_l(\rho) </math> | |||
Use the fits to build, for each phase, a table with three entries: <math> \rho, p, \mu </math>, then plot for both tables | |||
<math> \mu </math> as a function of <math> p </math> and check if the two lines intersect. The crossing point | |||
provides (to within statistical uncertainty, the errors due to [[finite size effects]], etc.) the coexistence conditions. | |||
Use the fits to build for each phase a table with three entries: <math> \rho, p, \mu </math>, | |||
<math> \mu </math> as a function of <math> p </math> and check if the two lines | |||
==== Improving the dirty method ==== | ==== Improving the dirty method ==== | ||
It can be useful to take into account classical thermodynamics to improve the previous analysis. This can be useful | |||
because is is not unusual have large uncertainties in the results of the properties. | |||
because | The basic idea is to use [[thermodynamic consistency]] requirements to improve the analysis. | ||
=== Methodology in the [[Isothermal-isobaric ensemble|NpT]] ensemble === | |||
The basic idea is to use | |||
=== | |||
=== Van der Waals loops, in the [[canonical ensemble|canonical ]] ensemble === | === Van der Waals loops, in the [[canonical ensemble|canonical ]] ensemble === | ||
=== Direct simulation of the two phase system in the [[Canonical ensemble]] === | === Direct simulation of the two phase system in the [[Canonical ensemble]] === | ||
== Mixtures == | == Mixtures == | ||
=== Symmetric mixtures === | |||
=== | ==References== | ||
[[category: computer simulation techniques]] | |||
Revision as of 18:16, 22 September 2007
Thermodynamic equilibrium implies, for two phases and :
- Equal temperature:
- Equal pressure:
- Equal chemical potential:
The computation of phase equilibria using computer simulation can follow a number of different strategies.
Independent simulations for each phase at fixed in the canonical ensemble
Simulations can be carried out either using Monte Carlo or Molecular dynamics techniques. Assuming that one has some knowledge on the phase diagram of the system, one can try the following recipe:
- Fix a temperature and a number of particles
- Perform a limited number of simulations in the low density region (where the gas phase density is expected to be)
- Perform a limited number of simulations in the moderate to high density region (where the liquid phase should appear)
- In these simulations we can compute for each density (at fixed temperature) the values of the pressure and the chemical potentials (for instance using the Widom test-particle method)
A quick (and dirty?) method
Using the previously obtained results the following somewhat unsophisticated procedure can be used to obtain a first inspection of the possible phase equilibrium.
Fit the simulation results for each branch by using appropriate functional forms:
Use the fits to build, for each phase, a table with three entries: , then plot for both tables as a function of and check if the two lines intersect. The crossing point provides (to within statistical uncertainty, the errors due to finite size effects, etc.) the coexistence conditions.
Improving the dirty method
It can be useful to take into account classical thermodynamics to improve the previous analysis. This can be useful because is is not unusual have large uncertainties in the results of the properties. The basic idea is to use thermodynamic consistency requirements to improve the analysis.