Editing Computation of phase equilibria
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Thermodynamic equilibrium implies, for two phases <math> \alpha </math> and <math> \beta </math>: | Thermodynamic equilibrium implies, for two phases <math> \alpha </math> and <math> \beta </math>: | ||
* equal [[temperature]]s: <math> T_{\alpha} = T_{\beta} </math> | * equal [[temperature]]s: <math> T_{\alpha} = T_{\beta} </math> | ||
* equal [[pressure]]s: <math> p_{\alpha} = p_{\beta} </math> | * equal [[pressure]]s: <math> p_{\alpha} = p_{\beta} </math> | ||
* equal [[chemical potential]]s: <math> \mu_{\alpha} = \mu_{\beta} </math> | * equal [[chemical potential]]s: <math> \mu_{\alpha} = \mu_{\beta} </math> | ||
The computation of phase equilibria using computer simulation can follow a number of different strategies. Here we will focus mainly | |||
on [[first-order transitions]] in fluid phases, usually [[Gas-liquid phase transitions |liquid-vapour]] equilibria. | |||
== Independent simulations for each phase at fixed temperature in the [[canonical ensemble]] == | == Independent simulations for each phase at fixed temperature in the [[canonical ensemble]] == | ||
Simulations can be carried out using either the [[Monte Carlo]] or the [[molecular dynamics]] technique. | Simulations can be carried out using either the [[Monte Carlo]] or the [[molecular dynamics]] technique. | ||
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conditions <math> \left. N,P,T \right. </math>. Let <math> p_{eq} </math> the pressure at which the phase transition occurs. In such a | conditions <math> \left. N,P,T \right. </math>. Let <math> p_{eq} </math> the pressure at which the phase transition occurs. In such a | ||
case the following scenario is expected for <math> \left. P(V|N,p,T) \right. </math>: | case the following scenario is expected for <math> \left. P(V|N,p,T) \right. </math>: | ||
*<math> \left. P(V|N,p_{eq},T) \right. </math> has two maxima, corresponding to the liquid and vapor pure phases, with <math> \left. P(V_v|N,p_{eq},T) = P(V_l|N,p_{eq},T) = P_{v/l} \right. </math> | *<math> \left. P(V|N,p_{eq},T) \right. </math> has two maxima, corresponding to the liquid and vapor pure phases, with | ||
: <math> \left. P(V_v|N,p_{eq},T) = P(V_l|N,p_{eq},T) = P_{v/l} \right. </math> | |||
*The probability of a given intermediate volume at <math> \left. p_{eq} \right. </math> can be estimated (from macroscopic arguments) as: | *The probability of a given intermediate volume at <math> \left. p_{eq} \right. </math> can be estimated (from macroscopic arguments) as: | ||
:<math> \left. P(V|N,p_{eq},T) \simeq P_{v/l} \times \exp \left[ - \frac{ \gamma(T) \mathcal A }{k_B T } \right] \right. </math>, | : <math> \left. P(V|N,p_{eq},T) \simeq P_{v/l} \times \exp \left[ - \frac{ \gamma(T) \mathcal A }{k_B T } \right] \right. </math>, | ||
where <math> \left. \gamma(T) \right. </math> is the [[surface tension]] of the vapor-liquid interface, | where <math> \left. \gamma(T) \right. </math> is the [[surface tension]] of the vapor-liquid interface, | ||
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A similar procedure can be built up to compute <math> \left. p(\rho) \right. </math> | A similar procedure can be built up to compute <math> \left. p(\rho) \right. </math> | ||
from <math> \left. \mu(\rho) \right. </math>. | from <math> \left. \mu(\rho) \right. </math>. | ||
Once <math> \left. p(\rho) \right. </math> and <math> \left. \mu(\rho) \right. </math> are known it is straightforward to compute the coexistence point. | Once <math> \left. p(\rho) \right. </math> and <math> \left. \mu(\rho) \right. </math> are known it is straightforward to compute the coexistence point. | ||
==== Practical details ==== | ==== Practical details ==== | ||
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* The simulation results in the two phase region will depend dramatically on the system size (calculations with different number of particles become convenient to check the quality of the phase equilibria results) | * The simulation results in the two phase region will depend dramatically on the system size (calculations with different number of particles become convenient to check the quality of the phase equilibria results) | ||
== Direct simulation of the two phase system== | == Direct simulation of the two phase system== | ||
#[http://dx.doi.org/10.1063/1.1474581 James R. Morris and Xueyu Song "The melting lines of model systems calculated from coexistence simulations", Journal of Chemical Physics '''116''' 9352 (2002)] | |||
== Gibbs ensemble Monte Carlo for one component systems== | == Gibbs ensemble Monte Carlo for one component systems== | ||
The [[Gibbs ensemble Monte Carlo]] method is often considered as a smart variation of the standard canonical ensemble procedure (See | The [[Gibbs ensemble Monte Carlo]] method is often considered as a 'smart' variation of the standard canonical ensemble procedure (See Ref. 1). | ||
The simulation is, therefore, carried out at constant volume, temperature and number of particles. | The simulation is, therefore, carried out at constant volume, temperature and number of particles. | ||
The whole system is divided into two non-interacting parts, each one has its own simulation | The whole system is divided into two non-interacting parts, each one has its own simulation | ||
box with its own [[periodic boundary conditions]]. | box with its own [[boundary conditions |periodic boundary conditions]]. | ||
This separation of the two phases into different boxes is in order to suppress any influence due to | This separation of the two phases into different boxes is in order to suppress any influence due to interracial effects. | ||
The two subsystems can interchange volume and particles. The rules for these interchanges are | The two subsystems can interchange volume and particles. The rules for these interchanges are | ||
built up so as to guarantee conditions of both chemical and mechanical equilibrium between | built up so as to guarantee conditions of both chemical and mechanical equilibrium between | ||
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[[category: Monte Carlo]] | [[category: Monte Carlo]] | ||
[[category: Computer simulation techniques]] | [[category: Computer simulation techniques]] | ||
== Mixtures == | == Mixtures == | ||
=== Symmetric mixtures === | === Symmetric mixtures === | ||
Examples of symmetric [[mixtures]] can be found both in lattice of continuous model. The [[Ising Models | Examples of symmetric [[mixtures]] can be found both in lattice of continuous model. The [[Ising Models]] can be | ||
viewed as mixture of two different chemical species which de-mix at low | viewed as mixture of two different chemical species which de-mix at low | ||
temperature. The symmetry in the interactions can be exploited to simplify the calculation of phase diagrams. | temperature. The symmetry in the interactions can be exploited to simplify the calculation of phase diagrams. | ||
== See also== | == See also== | ||
*[[Gibbs-Duhem integration]] | *[[Gibbs-Duhem integration]] | ||
==References== | ==References== | ||
#[http://dx.doi.org/10.1080/00268978700101491 Athanassios Panagiotopoulos "Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble", Molecular Physics '''61''' pp. 813-826 (1987)] | |||
[[category: computer simulation techniques]] | [[category: computer simulation techniques]] |