Computation of phase equilibria

2010-08-19T08:09:42Z

Roberta:

Thermodynamic equilibrium implies, for two phases <math> \alpha </math> and <math> \beta </math>:
* equal [[temperature]]s: <math> T_{\alpha} = T_{\beta} </math>
* equal [[pressure]]s: <math> p_{\alpha} = p_{\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]] ==
Simulations can be carried out using either the [[Monte Carlo]] or the [[molecular dynamics]] technique.
Assuming that one has some knowledge on the [[phase diagrams | 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 'first guess' method =====

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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 to see 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 'first guess' method =====
It can be useful to take into account classical thermodynamics to improve the previous analysis. This is
because is is not unusual have large uncertainties in the results for the properties.
The basic idea is to use [[thermodynamic consistency]] requirements to improve the analysis.
== Methodology in the [[Isothermal-isobaric ensemble|NpT]] ensemble ==
=== Low temperature: <math> \left. T << T_c \right. </math> ===
For temperatures well below the [[critical points | critical point]], provided that the calculation of the chemical potential
of the liquid phase using [[Widom test-particle method]] gives precise results, the following strategy can be used to obtain a 'quick' result:
#Perform an <math> NpT </math> simulation of the liquid phase at zero pressure, i.e. <math> p \simeq 0 </math>
#Arrive at an initial estimate, <math> \mu^{(1)} </math> for the coexistence value of the chemical potential by computing, in the liquid phase: <math> \left. \mu^{(1)} = \mu_l (N,T,p=0) \right. </math>
#Make a first estimate of the coexistence pressure, <math> p^{(1)} </math>, by computing, either via simulation or via the [[Virial coefficients of model systems |virial coefficients]] of the gas phase, the pressure at which the gas phase fulfils: <math> \left. \mu_g(N,T, p^{(1)} ) = \mu^{(1)} \right. </math>
#Refine the results, if required, by performing a simulation of the liquid phase at <math> \left. p^{(1)} \right. </math>, or use estimates of <math> \left( \partial V / \partial p \right)_{N,T,p=0} </math> (from the initial simulation) and the gas equation of state data to correct the initial estimates of pressure and chemical potential at coexistence.
Note that this method works only if the liquid phase remains metastable at zero pressure.
=== Weak first order transitions ===
There are situations where other strategies based on [[Isothermal-isobaric ensemble|NpT]] ensemble simulations can be used.
Similar approaches have also been applied in the [[Monte Carlo in the grand-canonical ensemble|Grand Canonical]] ensemble.
In practice, the ''free energy barrier'' between the two phases has to be low enough to allow the simulated system
to cross from one phase to the other when it is being simulated. Somehow, the situation is now the opposite to that
described in Section 2.1.
Taking into the account the classical partition function in the [[isothermal-isobaric ensemble|NpT]] ensemble it can be written:

: <math> \left. P(V|N,p,T) \propto \exp \left[ - \frac{ p V}{k_B T} \right] \exp \left[ - \frac{A(N,V,T)}{k_B T} \right] \right. </math> ,

where <math> \left. P(V|N,p,T) \right. </math> stands for the probability of the volume <math> \left. V \right. </math> at given
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>:
*<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:

:<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,
and <math> \left. {\mathcal A} \right. </math> is the
surface area, which depends on the thermodynamic
variables <math> \left. (N,V,T) \right. </math> and the geometry of the [[Periodic boundary conditions |simulation box]].
For small values of the surface tension, small system sizes and good simulation algorithms it could be possible for
pressures close to <math> p_{eq} </math> to sample in a simulation the whole region of densities between
vapour and liquid densities. If such is the case the phase equilibria conditions can be computed
by reweighing techniques applied on the volume histograms of the simulation.
==== Simple reweighing of the volume probability distribution ====
Suppose that a precise <math> \left. NpT \right. </math> simulation has been carried out at pressure <math> \left. p_0 \right. </math>. From that simulation a volume probability distribution, <math> \left. P_0(V|N,p_0,T) \right. </math>
has been computed. It is possible to use this function to estimate the distributions of volume for pressure values
close to <math> \left. p_0 \right. </math>;

: <math> \left. P(V|N,p,T) \propto \exp \left[ - \frac{ (p-p_0) V }{k_B T } \right] P_0(V|N,p_0,T) \right. </math>

Using this reweighing procedure we can estimate the value of <math> \left. p_{eq} \right. </math>; that is: the value
of pressure for which the volume distribution presents two peaks of equal height.
The analysis of the form of the distributions at the equilibrium conditions for different system sizes (i.e.
in the current case for different values of <math> \left. N \right. </math>) can be useful to distinguish between
continuous and first order [[phase transitions]].
=== See also ===
[[Surface tension]]
== Van der Waals loops in the [[canonical ensemble|canonical ]] ensemble ==
It is possible to compute the liquid-vapour equilibrium without explicit calculations of the chemical potential (or the pressure)
by performing a number of simulations sampling appropriately the vapour, liquid, and intermediate regions.
As an example, consider a simple fluid at a given subcritical temperature (<math> \left. T < T_c \right. </math>). We can perform a number of simulations for a given number of particles, <math> \left. N \right. </math> and different densities:

: <math> \rho_i = i \times \Delta \rho; \; \; \; \; i=1, 2, \cdots, m </math>

In these simulations, we can compute the pressure (or the chemical potential) and fit the result to an appropriate equation.
With such an ''equation of state'' the phase equilibria can be estimated.
If two phase equilibria exists, a ''loop'' in the representation of <math> \left. p = p (\rho) \right. </math> (or <math> \left. \mu = \mu (\rho) \right. </math>)
should appear.
* Computing <math> \left. \mu(\rho) \right. </math> from the equation of state given as <math> \left. p(\rho) \right. </math>:
For fixed temperature and number of particles:

: <math> \left. d A = - p d V \right. </math>
: <math> \left. d A = \frac{ p N }{\rho^2} d \rho ; \; \; \; \; d a = \frac{ p }{\rho^2} d \rho \right. </math>

where <math> \left. A \right. </math> is the [[Helmholtz energy function]] and
<math> \left. a = A/N \right. </math>, integrating:

:<math> \left. a(\rho) = a(\rho_0) + \int_{\rho_0}^{\rho} \frac{ p(\rho') }{(\rho')^2} d \rho' \right. </math>

On the other hand

:<math> \left. A = - p V + \mu N ; \; \; \; a = - p/ \rho + \mu \right. </math>

therefore:

: <math> \left. \mu(\rho) - p(\rho)/\rho = \mu(\rho_0) - p(\rho_0)/\rho_0 + \int_{\rho_0}^{\rho} \frac{ p(\rho') }{(\rho')^2} d \rho' \right. </math>
: <math> \left. \mu(\rho) = \mu(\rho_0) + p(\rho)/\rho - p(\rho_0)/\rho_0 + \int_{\rho_0}^{\rho} \frac{ p(\rho') }{(\rho')^2} d \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>.
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 ====
Some precautions should be taken if this procedure is used:
* The precision of the simulation results in the two phase region will be poor (so, large simulations are required to have a good estimation of the equation of state)
* 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==
An example using the [[Lennard-Jones model]],
*[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)]
and its application to [[water]]
*[http://dx.doi.org/10.1063/1.2183308 Ramón García Fernández, José L. F. Abascal, and Carlos Vega "The melting point of ice Ih for common water models calculated from direct coexistence of the solid-liquid interface", Journal of Chemical Physics '''124''' 144506 (2006)]
== 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 Ref. 1).
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
box with its own [[periodic boundary conditions]].
This separation of the two phases into different boxes is in order to suppress any influence due to [[interface | interfacial]] effects.
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
the two phases.
If the overall conditions are of phase separation, it is expected that two phases will appear in
different simulation boxes.
====External links====
*[http://kea.princeton.edu/jerring/gibbs/ Gibbs ensemble Monte Carlo code] on the [http://kea.princeton.edu/ Panagiotopoulos Group Homepage]
[[category: Monte Carlo]]
[[category: Computer simulation techniques]]

== Mixtures ==
=== Symmetric mixtures ===
Examples of symmetric [[mixtures]] can be found both in lattice of continuous model. The [[Ising Models | Ising model]] can be
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.
== See also==
*[[Gibbs-Duhem integration]]
==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]]

Law of corresponding states

2010-08-19T08:05:39Z

Roberta:

The '''law of corresponding states''' is an empirical law which encapsulates the finding that the [[equations of state]] for many real gases are remarkably similar when they are expressed in terms of reduced [[temperature]]s (<math>T_r = T/T_c</math>), [[pressure]]s, (<math>p_r = p/p_c</math>) and volumes (<math>V_r = V/V_c</math>), where the subscript <math>c</math> represents the value of the property at the [[Critical points|critical point]]. This law was first described by [[Johannes Diderik van der Waals]] in his 1873 thesis, and forms the subject of a paper by him in 1913 <ref>[http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=1493&view=image&startrow=1 Johannes Diderik van der Waals "The law of corresponding states for different substances", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''15 II''' pp. 971-981 (1913)]</ref>
==Assumptions==

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Pitzer <ref>[http://dx.doi.org/10.1063/1.1750496 Kenneth S. Pitzer "Corresponding States for Perfect Liquids", Journal of Chemical Physics '''7''' pp. 583-590 (1939)]</ref> produced a list of assumptions in order for the law of corresponding states to apply. This list was later modified by Guggenheim <ref name="Guggenheim"> </ref>. These are:
#There is negligible difference between [[Fermi–Dirac statistics]] and [[Bose–Einstein statistics]] for the system (i.e. the system behaves classically).
#The effect of quantisation of the translational [[Degree of freedom |degrees of freedom]] is negligible (i.e. the system behaves classically).
#The molecules are spherically symmetrical, either actually or by virtue of rapid and free rotation.
#The intramolecular degrees of freedom are assumed to be completely independent of the volume per molecule.
#The potential energy will be taken as a function only of the various intermolecular distances.
#The [[Intermolecular pair potential | potential energy for a pair of molecules]] can be written as <math>A\Phi (r/r_0)</math> where <math>r</math> is the intermolecular distance, and <math>A</math> and <math>r_0</math> are characteristic constants, and <math>\Phi</math> is a universal function.
==Examples==
For [[argon]], [[krypton]], [[nitrogen]], [[oxygen]], [[carbon dioxide]] and [[methane]] one has <ref name="Guggenheim"> [http://dx.doi.org/10.1063/1.1724033 E. A. Guggenheim "The Principle of Corresponding States", Journal of Chemical Physics '''13''' pp. 253-261 (1945)]</ref>

:<math>\frac{p_cV_c}{RT_c}\approx 0.292</math>

(for pressure measured in atmospheres, and volume in cm<sup>3</sup>mole<sup>-1</sup>)

For [[neon]], [[argon]], and [[oxygen]] one has <ref name="Guggenheim"> </ref>

:<math>\frac{T_B}{T_c} \approx 2.7</math>

where <math>T_B</math> is the [[Boyle temperature]].

For [[neon]], [[argon]], [[krypton]],and [[xenon]] one has <ref name="Guggenheim"> </ref>

:<math>\frac{T_{tp}}{T_c} \approx 0.555</math>

where <math>T_{tp}</math> is the [[triple point]].
==Acentric factor==
The acentric factor <ref>[http://dx.doi.org/10.1021/ja01618a002 Kenneth S. Pitzer, David Z. Lippmann, R. F. Curl Jr., Charles M. Huggins, Donald E. Petersen "The Volumetric and Thermodynamic Properties of Fluids. II. Compressibility Factor, Vapor Pressure and Entropy of Vaporization", Journal of the American Chemical Society '''77''' pp. 3433-3440 (1955)]</ref> is defined in terms of the [[vapour pressure]] at <math>T_r = 0.7 </math>.
It has been shown that a number of substances have the behavior if they share the same acentric factor.
==Colloids==
The law of corresponding states has been extended to suspensions of spherical [[colloids]] that interact via a [[Intermolecular pair potential | pair potential]]
by Noro and Frenkel
<ref>[http://dx.doi.org/10.1063/1.1288684 Massimo G. Noro and Daan Frenkel "Extended corresponding-states behavior for particles with variable range attractions", Journal of Chemical Physics '''113''' 2941-2944 (2000)]</ref>.
==References==
<references/>
'''Related material'''
*[http://dx.doi.org/10.1016/S0031-8914(38)80037-9 J. de Boer and A. Michels "Contribution to the quantum-mechanical theory of the equation of state and the law of corresponding states. Determination of the law of force of helium", Physica '''5''' pp. 945-957 (1938)]
*[http://dx.doi.org/10.1063/1.3072156 Patrick Grosfils and James F. Lutsko "Dependence of the liquid-vapor surface tension on the range of interaction: A test of the law of corresponding states", Journal of Chemical Physics '''130''' 054703 (2009)]
*Hong Wei Xiang "The Corresponding-States Principle and its Practice", Elsevier Science (2005) ISBN 0-444-52062-7
[[category: equations of state]]

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Computation of phase equilibria

Law of corresponding states