Computation of phase equilibria

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The computation of phase equilibria using computer simulation can follow different strategies.

Liquid-vapor equilibria of one component systems

The thermodynamic equilibrium implies, for two phases $\alpha$ and $\beta$:

• Equal temperature: $T_{\alpha }=T_{\beta }$

• Equal pressure: $p_{\alpha }=p_{\beta }$

• Equal chemical potential: $\mu _{\alpha }=\mu _{\beta }$

Independent simulations for each phase at fixed $T$ in the canonical ensemble

The simulations can be carried out either using Monte Carlo or Molecular dynamics techhniques. Let us assume that we have some knowledge on the phase diagram of the system. We could try the following recipe:

- Fix a temperature and a number of particles

- Perform a few simulations in the low density region (where the gas phase density is expected to be)

- Perform a few simulations in the moderate / high density region (where the liquid phase should appear)

- In these simulations we can compute for each density (at fixed T) the values of the pressure and the chemical potentials (for instance using the Widom test-particle method)

A quick (dirty?) method

Using the results the following (unsophisticated) procedure can be used to get a first inspection on the possible phase equilbrium.

Fit the simulation results for each branch using appropriate functional forms:

$\mu _{v}(\rho );p_{v}(\rho );\mu _{l}(\rho );p_{l}(\rho )$

Use the fits to build for each phase a table with three entries: $\rho ,p,\mu$, Plot for both tables $\mu$ as a function of $p$ and check if the two lines cross. The crossing point gives (within statistical uncertainities, errors due to finite size, 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 we can have large uncertainties in the results of the properties ....

The basic idea is to use the thermodynamic consistence requirements to improve the analysis....

TO BE CONTINUED ..

Van der Waals loops, in the canonical ensemble

TO BE CONTINUED ...