The extrapolation of phase equilibrium curves of mixtures in the isobaric-isothermal Gibbs ensemble

MOLECULAR PHYSICS 2002 VOL 100 NO 21 3429plusmn3441

The extrapolation of phase equilibrium curves of mixtures in theisobaricplusmnisothermal Gibbs ensemble

TAMAAcirc S KRISTOAcirc F JAAcirc NOS LISZI and DEZSOO BODA

Department of Physical Chemistry University of VeszpreAcirc m H-8201 VeszpreAcirc mPO Box 158 Hungary

(Received 20 March 2002 revised version accepted 14 May 2002)

A powerful extrapolation scheme is proposed to determine the vapourplusmnliquid and liquidplusmnliquid equilibrium curves of mixtures by performing a single isothermalplusmnisobaric Gibbs ensem-ble Monte Carlo (GEMC) simulation The coexistence curves for the mole fraction and thedensity are extrapolated as functions of the temperature and the pressure by second-orderTaylor series The coe cients of the Taylor series which are the temperature and pressurederivatives of these quantities along the coexistence curves can be calculated from the dataproduced by a single GEMC simulation on the basis of macructuation formulas We show that theapplication of a PadeAcirc approximant considerably widens the temperature and pressure rangewhere the extrapolation is accurate Using Lennard-Jones mixtures as test systems we showthat the technique is able to produce quite accurate equilibrium curves at regxed temperature inthe function of the pressure and vice versa The procedure yields good results not only forvapourplusmnliquid but also for liquidplusmnliquid coexistence curves The calculation of the vapourpressure curves at a regxed composition of the liquid side is straightforward with the method

1 Introduction

In our previous paper we suggested an idea to extra-

polate the vapourplusmnliquid equilibrium (VLE) curves ofpure macruids by performing a single isothermal Gibbs

ensemble Monte Carlo (GEMC) simulation [1] Thebasic idea followed from our series expansion methods

[2plusmn7] which were originated from the NpT plus test par-

ticle method of Fischer and co-workers [8 9] In these

methods separate NpT [2] NVT [3] adiabatic [4] orgrand canonical (GC) [5plusmn7] simulations were performed

in the two phases Taylor series expansions were con-structed to extrapolate the various functions expressing

the relationships between the various intensive par-

ameters in the given ensemble (the middothellippT dagger function inthe NpT ensemble the phellipmiddotT dagger function in the GC

ensemble etc where T is the temperature p is the press-

ure middot is the chemical potential N is the number ofparticles and V is the volume) (Phase equilibrium can

be determined trivially if one knows the equation of

state [10plusmn12] of the given system The above functionsgiving middotplusmnpplusmnT relationships are sometimes termed

`chemical potential equations of statersquo) The lower andhigher order coe cients of the series were calculated

from simple ensemble averages and macructuation for-

mulas respectively which were obtained from the

simulation in a straightforward way The VLE curves

can be determined if we have the above functions in

both phases It is important to emphasize that these

methods are able to yield the VLE data not only at

the point where the simulations are actually performed

but also for a certain temperature rangeThis attractive capability together with the advanta-

geous features of the GEMC technique leads us to the

extrapolation scheme introduced in [1] The GEMC

technique is the most widely used method to determine

phase coexistence of molecular macruids elegant easy to

program and produces the phase equilibrium data

directly [13 14] Our basic idea was to extrapolate theVLE curves directly instead of extrapolating the various

one-phase thermodynamic functions like in the case of

the series expansion methods We have shown that

similar to the series expansion methods the VLE

curves can be extrapolated by Taylor series and the

coe cients can be obtained through macructuation for-

mulas [15 16] from the GEMC simulation Moreoverwe have pointed out that the application of a PadeAcirc

approximant signiregcantly improves the convergence of

the extrapolation series The procedure unireges the

advantageous features of the GEMC technique and

the series expansion methods The method resembles the

GibbsplusmnDuhem integration introduced by Kofke [17 18]

for pure macruids but yields the dp=dT ˆ centh=T centv deriva-tive directly and takes into account higher order terms

Molecular Physics ISSN 0026plusmn8976 printISSN 1362plusmn3028 online 2002 Taylor amp Francis Ltdhttpwwwtandfcoukjournals

DOI 1010800026897021015864 1

Author for correspondence e-mail bodaalmos veinhu

We call this an èxtrapolation schemersquo because wewant to emphasize that the basic simulation method isstill the GEMC technique and the information which isnecessary to build the extrapolation functions can begained from the GEMC simulation without a majormodiregcation of the original code Basically the heartof the technique lies in the important feature of the mol-ecular simulations in which they sample the phase spacenot only in the state point where they are performed butin the vicinity of this point In this way they are able toobtain information from one simulation for a neigbour-hood of the simulation state point This capabilitystands behind our series expansion methods where theinformation comes from various thermodynamic deriva-tives and the macructuations corresponding to them How-ever this capability is in close connetion with othersophisticated simulation techniques like the histogramreweighting [19plusmn24] and the thermodynamic scaling [25plusmn28] methods

The phase equilibrium simulation of mixtures is moreproblematic than that of pure macruids because the systemhas more degrees of freedom The application of theGEMC technique [14] for mixtures is straightforwardalthough in some cases not without di culties Never-theless this is the most widely used method to determinephase coexistence in mixtures but only in one equilib-rium point (the term èquilibriumrsquo will mean phase equi-librium henceforth if not stated otherwise) The GibbsplusmnDuhem integration method has been extended to mix-tures [29] There was partial success in extending theNpT plus test particle method to mixtures [30plusmn33] Theprocedure as the GEMC technique is able to producethe equilibrium point at one hellipT pdagger point but not for ahellipT pdagger domain since the calculation of the mole fractionderivatives proved to be too di cult Recently Vrabecand Hasse have developed a version of this methodusing a pseudo GC simulation on the vapour side [34]So far our GC technique [7] is the only method whichutilizes the full capabilities of the series expansionmethods to determine phase equilibrium of mixturesThis technique is able to produce the VLE curves in acertain T or p range but we encountered di culties inapplying it to liquidplusmnliquid equilibria (LLE)

In this paper we extend our extrapolation scheme [1]for the case of mixtures using constant temperature andpressure GEMC simulations We show that the equilib-rium mole fractions and densities can be extrapolatedover the hellipT pdagger plane with Taylor series andor a PadeAcircapproximant We test the method on various Lennard-Jones (LJ) mixtures applying it for both vapourplusmnliquid(VL) and liquidplusmnliquid (LL) equilibria It will be shownthat the technique is applicable and e cient in the caseof mixtures and provides a promising tool to studyphase coexistence in many component systems

2 MethodThe GEMC simulation is performed for a binary mix-

ture at the basic point hellipT 0 p0dagger namely at regxed tempera-ture T 0 and pressure p0 Let us express the second-order Taylor series for the equilibrium curve of a generalconreggurational quantity f around the basic pointhellipT 0 p0dagger as a function of the temperature and the press-ure As a matter of fact it would be more proper to talkabout an equilibrium surface f hellipshy pdagger but we will use theterm èquilibrium curversquo since in practice the phasecoexistence is studied by regxing T or p and using theother as a variable For f any of the intensive quantitiescan be substituted nevertheless in this paper we willconsider the equilibrium curves for the number density

raquo ˆ N=V and the mole fraction of component Ax ˆ xA ˆ NA=N Of course their equilibrium valuecan be taken in any of the two phases For the phaseswe will refer to them using the roman numbers I and IIThe method will be proved to be applicable to both VLand LL equilibria

It is worthwhile to use the reciprocal temperatureshy ˆ 1=kT instead of the temperature (where k is theBoltzmann constant) because of the more compactform of the formulas Let us deregne the zeroth regrst-and second-order terms of the series expansion respect-ively by

f0 ˆ f hellipshy 0 p0dagger hellip1dagger

f1 ˆ f1hellipshy pdagger

ˆ fshy

sup3 acute

0

hellipshy iexcl shy 0dagger Dagger fp

sup3 acute

0

hellipp iexcl p0dagger hellip2dagger

and


ˆ 1

2

2fshy 2

Aacute

0

hellipshy iexcl shy 0dagger2 Dagger 2fshy p

Aacute

0

hellipshy iexcl shy 0daggerhellipp iexcl p0dagger

Dagger 1

2

2fp2

Aacute

0

hellipp iexcl p0dagger2 hellip3dagger

Here the subscript 0 means that the quantities in ques-tion have to be taken at the basic point hellipshy 0 p0dagger whilethe subscripts 1 and 2 refer to regrst- and second-orderterms in shy and p The regrst-order and second-orderTaylor series expansions of the equilibrium curves ofquantity f around hellipshy 0 p0dagger are expressed as

f T1 ˆ f0 Dagger f1 hellip4dagger

and

f T2 ˆ f0 Dagger f1 Dagger f2 hellip5dagger

3430 T KristoAcirc f et al

The superscripts T1 and T2 mean that f hellipshy pdagger is extra-polated by the regrst- and second-order Taylor seriesexpansions respectively as a function of shy and p Inequation (1) f0 is the equilibrium value of f at thebasic point and is obtained directly from the GEMCsimulation as an ensemble average f0 ˆ h f i Inequation (2) the regrst derivatives of f are the derivativestaken along the phase coexistence curve (in the litera-ture these sorts of derivatives are usually denoted by thesubscript frac14) and can be expressed as macructuation for-mulas

fshy

sup3 acute

0

ˆ h f ishy

sup3 acute

0

ˆ h f ihHi iexcl h fHi hellip6dagger

and

fp

sup3 acute

0

ˆ h f ip

sup3 acute

0

ˆ shy 0 h f ihV i iexcl h fV ipermil Š hellip7dagger

where H ˆ U Dagger p0V is the total enthalpy V is the totalvolume and U is the total conreggurational energy ofthe system By `totalrsquo we mean that the total quantityof the system is the sum of the corresponding quantitiesin the two phases eg H ˆ HI Dagger HII The bracketsdenote Gibbs ensemble averages

In equation (3) the second derivatives can beobtained by di erentiating equations (6) and (7) oncemore

2fshy 2

Aacute

0

ˆ h f ishy

sup3 acute

0

hHi Dagger h f i hHishy

sup3 acute

0

iexcl h fHishy

sup3 acute

0

hellip8dagger

2fshy p

Aacute

0

ˆ

shyshy h f ihV i iexcl h fV ipermil Š

sup3 acute

0

ˆ h f ihV i iexcl h fV ipermil Š

Dagger shy 0

h f ishy

sup3 acute

0

hV i Dagger h f i hV ishy

sup3 acute

0

micro

iexcl h fV ishy

sup3 acute

0

para hellip9dagger

and

2fp2

Aacute

0

ˆ shy 0

h f ip

sup3 acute

0

hV i Dagger h f i hV ip

sup3 acute

0

iexcl h f V ip

sup3 acute

0

micro para

hellip10dagger

Substituting the corresponding macructuation formulas inequations (6) and (7) for the derivatives on the right-hand side (RHS) of equations (8)plusmn(10) we obtain thefollowing higher order macructuation formulas containingtriple correlations

2fshy 2

Aacute

0

ˆ 2h f ihHi2 iexcl h f ihH2i iexcl 2h fHihHi Dagger h fH2i

hellip11dagger

2fshy p

Aacute

0

ˆ h f ihV i iexcl h fV ipermil Š Dagger shy 0 2h f ihHihV ipermil

iexcl h fHihV i iexcl h fV ihHi iexcl h f ihHV i Dagger h fHV iŠ

hellip12dagger

and

2fp2

Aacute

0

ˆ shy 20 2h f ihV i2 iexcl h f ihV 2i iexcl 2h fV ihV i Dagger h fV 2ih i

hellip13dagger

Note that in equations (6)plusmn(13) all derivatives andensemble averages have to be taken at the basic pointhellipshy 0 p0dagger The partial derivative =shy means that thepressure is kept constant and vice versa We also empha-size that the ensemble averages of f H V and theirdouble and triple products which are necessary to con-struct the above macructuation formulas are yielded by theGEMC simulation without a major modiregcation of theoriginal code Therefore by performing a single GEMCsimulation we can calculate everything to express theTaylor series to extrapolate the phase coexistence curvesof a thermodynamic quantity f over a certain domain ofthe hellipshy pdagger plane around the basic point hellipshy 0 p0dagger

Nevertheless as we will show in section 3 the second-order Taylor series is insu cient to extrapolate the equi-librium curves over a wide range of the variables shy andp The calculation of the higher order terms (containingquaternary correlations) although theoretically poss-ible practically it is not Even the macructuation formulascorresponding to the second derivatives (equations (11)plusmn(13)) are obtained with a large statistical uncertaintyFortunately we have the useful tool of the PadeAcirc approx-imation [3536] to improve the extrapolation The PadeAcircapproximants were designed to improve the convergenceof series whose terms are known only up to n orderThey reproduce the expansion up to n order and theyprovide an estimate of the remainder of the series Forexample a PadeAcirc approximant was successfully used inthe the free energy expansion in the perturbation theoryof polar macruids proposed by Stell et al [37] The second-order series expansion of the equilibrium curve of thequantity f can be replaced by

f P ˆ f0 Dagger f11 iexcl hellip f2=f1dagger hellip14dagger

where the superscript P means that equation (20) is the01 PadeAcirc approximant of f T2

The extrapolation of phase equilibrium curves of mixtures 3431

The PadeAcirc approximant diverges when f2=f1 ˆ 1 As wewill show in the next section by discussing the results forvarious LJ mixtures the PadeAcirc approximation curvesbehave badly if f2=f1 is close to 1 or greater than 1 Inthe range 0 lt f2=f1 lt 1 sometimes the PadeAcirc approxi-mant brings an improvement over the second-orderseries but this improvement is rarely considerable Iff2=f1 is close to 0 and positive then the PadeAcirc approxi-mant does not diverge but the second-order series aregood enough so replacing them with the PadeAcirc approx-imant is unnecessary In general it is worthwhile tochoose f T2 over the PadeAcirc approximant when f2=f1 gt 0In contrast if f2=f1 lt 0 the PadeAcirc approximant signireg-cantly improves the convergence of the second-orderseries f T2 Therefore we propose the following criteriafor choosing between the two routes

f hellipshy pdagger ˆf T2 if f2=f1 gt 0

f P if f2=f1 lt 0

(hellip15dagger

It is important to note that equation (15) is not a uni-versal recipe The user of the technique has to be crea-tive in its use and for other systems conclusionsdi erent from those we drew for LJ mixtures studiedin this work might be drawn Let us cite from the textbookNumerical Recipes [35] `PadeAcirc approximation has theuncanny knack of picking the function you had in mindfrom among all the possibilities Except when it does notThis is the downside of the PadeAcirc approximation it isuncontrolled There is in general no way to tell howaccurate it is or how far out in x it can usefully extendIt is a powerful but in the end still mysterious techniquersquoAs we will see in the next section despite its `downsidersquothe PadeAcirc approximation proves to be quite useful Notethat this problem with the divergence of the PadeAcirc approxi-mation did not occur in our study for pure macruids [1]

In our previous procedure for pure macruids [1] we haveshown that in the case of the pressure and the chemicalpotential the regrst temperature derivatives can beexpressed using simple ensemble averages throughClapeyron-like equations (equations (9) and (13) in [1])and we need to use macructuation formulas only in thesecond derivatives Similarly in the case of mixtureswe have the possibility to express the regrst derivativesof the chemical potential(s) in the following way

shy middoti

shyˆ hellip1 iexcl xI

i daggerhII iexcl hellip1 iexcl xIIi daggerhI

xIIi iexcl xI

ihellip16dagger

and

shy middoti

pˆ shy

hellip1 iexcl xIi daggervII iexcl hellip1 iexcl xII

i daggervI

xIIi iexcl xI

i hellip17dagger

where h ˆ H=N and v ˆ V =N are the one-particleenthalpy and one-particle volume respectively and i

refers to species A or B By di erentiating the aboveequations and using the macructuation formulas(equations (6) and (7)) for the regrst derivatives of xI

i xII

i hI hII vI and vII the second derivatives of shy middoti andthus the extrapolation curves of the equilibrium chemi-cal potential can be expressed Nevertheless we rarelyregnd data for the equilibrium chemical potential in simu-lation studies of phase equilibria of mixtures thereforewe do not pursue this question in this work

In our previous paper [1] for pure macruids we havegiven a method which provides an opportunity partlyto check the inner consistency of the simulation partlyto give an estimation for the extrapolation range of thevarious approximations A similar procedure could beused in the case of mixtures on the basis of equations (16)and (17) The shy and the p derivatives of the chemicalpotential can be calculated in a hellipshy pdagger point (not necess-arily the basic point) in two di erent ways (1) by di er-entiating the extrapolation function ( f T2 or f P) of shy middotior (2) by extrapolating the quantities on the right-handside of equations (16) and (17) (xI

i vII) If the tworoutes give results for the derivatives of shy middoti that do notdi er from each other more than a prescribed value theextrapolation can be accepted as satisfactory Neverthe-less on the basis of our experience in the case of puremacruids [1] it seems more straightforward to estimate theextrapolation range in a visual way by comparing theextrapolation curves obtained from two adjacent basicpoints (whether they coincide or not see reggure 3 insection 31) In addition the formulas are too complexin the case of mixtures therefore we do not givean extensive study to this question We just state theexistence of the possibility

3 Results and discussionWe test the above extrapolation scheme for various

LJ mixtures for which we found phase equilibrium datain the literature We study both VL and LL equilibria atboth regxed temperature and regxed pressure In table 1 wecollected the various systems whose phase coexistenceproperties are considered in this work Table 1 alsoshows the values of the temperature and the pressurefor which our GEMC simulations have been performed(we denote by `regxedrsquo which one is considered to be con-stant during the evaluation of the data) the referencesfrom which the potential parameters and the data forcomparison have been taken and the numbers of regguresin which the equilibrium results are presented for thegiven casey The results for the LJ mixtures are pre-


y We do not tabulate the direct results of our GEMC simu-lations only the extrapolated phase coexistence curvesNevertheless the authors will be glad to send the data uponrequest bodaalmosveinhu or kristoftalmosveinhu

sented in reduced units where the potential parameters

of component A were used to reduce T ˆ kT =degA is the

reduced temperature shy ˆ 1=T ˆ degA=kT is the reduced

reciprocal temperature p ˆ pfrac143A=degA is the reduced

pressure and raquo ˆ Nfrac143A=V ˆ raquofrac143

A is the reduced density

We briemacry summarize the particulars of our GEMCsimulations (more detailed descriptions of the technique

can be found in books [38 39] and in reviews [40 41])

The isothermalplusmnisobaric GEMC calculations were per-

formed for the LJ mixtures using N ˆ 512 particles The

simulations were started either from a face-centred cubic

lattice conregguration or from an output conregguration of

a previous run An equilibration period of 20 000plusmn50 000cycles was followed by a production period of 300 000plusmn

400 000 cycles Each cycle consisted of N attempted par-

ticle displacements in the two subsystems one

uncoupled volume change of the subsystems and about

40plusmn600 attempted particle transfers (depending on the

densities) between the subsystems The type of move

to be performed and the particle for the displacementand transfer steps were randomly selected The maxi-

mum changes (displacement of a selected molecule and

volume change) were adjusted to obtain where possible

a 40plusmn50 acceptance rate for the attempted move The

number of attempted exchanges was chosen to obtain 2

or 3 successful particle transfers per cycleThe interactions were truncated at a spherical cut-o

distance equal to half of the box length of the respective

simulation cell Long-range corrections for the LJ inter-

actions were estimated by assuming that the pair corre-

lation functions are unity beyond the cut-o radius In

the simulations all the quantities necessary to build up

the macructuation formulas were also accumulated Esti-mates for the statistical errors were made by dividing

the whole runs into about 40plusmn50 blocks and calculating

the standard deviation of the block averages

31 Vapourplusmnliquid equilibrium at constant temperatureThe e ciency of our procedure in extrapolating VLE

curves has been tested on system I whose potential par-ameters have been taken from the paper of Vrabec et al

[31] who studied this system by the NpT plus test par-ticle method [30] They published VLE results for thereduced temperatures T ˆ 075 and 1 We have per-

formed GEMC simulations for these temperatures atthe reduced pressures p ˆ 002 005 and 0065 (forT ˆ 075) and at p ˆ 006 and 008 (for T ˆ 1)

The direct (not extrapolated) VLE results of these andsome additional GEMC simulations are represented byopen squares in the reggures We expressed the various

extrapolation equations as functions of the pressureand displayed them (together with reference data) in

the reggures in the following way

first-order Taylor series f T1 dotted line

second-order Taylor series f T2 dashed line

Pade approximant f P dot-dashed line

f T2 or f P on the basis ofequation (15) solid line with error bars

results in basic points open squaresliterature data reglled circles

The meaning of the types of the lines and symbols is thesame in every reggure if we do not state otherwise The

error bars of the solid curves are calculated from theerror propagation law in the following way


Table 1 The potential parameters and state points for the various LJ mixtures whose phase equilibria is studied inthis work Also shown are the numbers of reggures where the equilibrium curves of the given case are presentedcompared to literature results taken from the works of Vrabec et al [31 45] van Leeuwen et al [42] and Guo et al[44]

System

I II III

degB=degA 05 07298 075

degAB=degA 07071 05980 06062frac14B=frac14A 1 09398 095

frac14AB=frac14A 1 09699 0975

Equilib VL LL LLT 075 regxed 1 regxed 075 0875 1 0889 regxed 0825 09

p 002 005 0065 006 008 006 regxed 012 0125 regxed

Reference [31] [42 45] [44]

Figures 1 3 4 2 3 5 6 7 8 9 10 11

macrf hellipshy pdagger ˆX

i

f hellipshy pdaggerf hellipidagger

0

shyshyshyshyshy

shyshyshyshyshymacrf hellipidagger0 hellip18dagger

where macrf hellipshy pdagger is the error of the extrapolated equilib-rium value calculated either from f T2 or from f P fortemperature shy and pressure p f hellipidagger

0 is one of the shy orp derivatives of f at hellipshy 0 p0dagger and macrf hellipidagger

0 is the statistical

error of f hellipidagger0 calculated from the block average method

Figures 1 and 2 show the VLE curves for T ˆ 075 and

1 respectively for the mole fraction of component AThe results obtained from various basic simulations per-

formed at various pressures are shown in di erent insetsfor the sake of clarity The data for lower mole fractions

belong to the vapour phaseThe conclusions we make at the discussion of these

reggures will more or less be valid in all cases It can beseen from the reggures that the regrst-order series (dotted

lines) appropriately reproduce the tangents of the

curves nevertheless they are insu cient to extrapolatethe curves for a wider range of pressure For the behav-

iour of the second-order and the PadeAcirc extrapolationsthe vapour (left) side of the upper inset of reggure 2

(T ˆ 1 and p ˆ 008) is a typical example The

second-order curves (dashed lines) are able to extrapo-late for a wider pressure range but they diverge for

higher jp iexcl p0j values if f2=f1 lt 0 (p gt p0 in reggure 2)In this domain the PadeAcirc approximant works much

better than the second-order Taylor series In the otherdomain (p lt p0 in reggure 2) the PadeAcirc approximant

diverges and the second-order Taylor series yields thebetter extrapolation Sometimes we encounter the f2 ˆ f1


0 02 04 06 08 1x

0

002

004

p

002

004

006

008

p

004

006

008

p

T=075

p 0 =0065

p 0 =005

p 0 =002

Figure 1 Vapourplusmnliquid coexistence curves of system I forthe mole fraction of component A as functions of thepressure obtained from the regrst-order (dotted lines) andthe second-order (dashed lines) Taylor series as well asfrom the PadeAcirc approximants of the second-order Taylorseries (dot-dashed lines) at regxed temperature T ˆ 075The solid lines with error bars represent the curvesobtained from the criterion given by equation (15) Thereglled circles are the NpT +TP data of Vrabec et al [31]and the open squares are our GEMC simulation resultsobtained for various basic points The di erent insets ofthe reggure contain the results obtained from the GEMCsimulations performed for the di erent basic pressuresp0 ˆ 002 005 and 0065

02 04 06 08 1x

0

005

01

p

p 0 =008

p 0 =006

005

01

015

p

T=10

Figure 2 Same as in reggure 1 but the regxed temperature isT ˆ 1 The di erent insets of the reggure contain theresults obtained from the GEMC simulations performedfor the di erent basic pressures p0 ˆ 06 and 08 Themeaning of the lines and symbols is the same as in reggure 1

case in the range of pressure studied These cases pro-duce the diverging behaviour of the dot-dashed curves

like in the upper inset of reggure 2 As a summary we can

conclude that the solid lines yielded by the scheme givenin equation (15) produce the best extrapolation curves

The error bars become large for large jp iexcl p0j as amatter of fact comparing the solid curves to the refer-

ence data the error bars seem to be overestimatedIn reggure 3 we illustrate the capabilities of our tech-

nique The solid curves are the same as in reggures 1 and 2

but without error bars The reglled symbols are theselected data of Vrabec et al [31] while the open sym-

bols are the results of our grand canonical procedure

proposed earlier [7] This procedure belongs to theseries expansion methods and similarly to the present

technique is able to produce the equilibrium curves for

a certain temperature andor pressure range The detailsof the method is given elsewhere [7] It is important to

emphasize that the grand canonical data seen in reggure 3have been obtained by performing only 8 pairs of simu-

lations (8 in liquid and 8 in vapour phase) The extra-

polation obtained from the present method seems towork even better It cannot be overemphasized that

the curves shown in reggure 3 have been obtained byperforming only 5 () GEMC simulations The agree-

ment with our grand canonical data is better than the

agreement with the results of Vrabec et al [31]In reggures 4 and 5 one can see the VLE curves for the

reduced coexistence densities for T ˆ 075 and 1 re-

spectively Similar conclusions can be drawn as in thecase for the mole fraction (reggures 1 and 2)

32 Vapourplusmnliquid equilibrium at constant pressureFigures 6 and 7 show the VLE results at the regxed

pressure p ˆ 006 using temperature as a variableThe basic GEMC simulation for T ˆ 10 is the sameas that used in the regxed temperature case (see the lowerinsets of reggures 2 and 5) Similarly the simulation forT ˆ 075 is also the same as that used previously (upperinsets of reggures 1 and 4) but here we had to extrapolatenot only as a function of the temperature but also forthe pressure from the basic pressure of the simulationp0 ˆ 0065 to the regxed pressure of the present calcula-tions p ˆ 006 An additional simulation was per-formed for the intermediate temperature T 0 ˆ 0875

Initially the behaviour of the extrapolation curves inreggures 6 and 7 somehow seems worse than the behav-iour of the curves in reggures 1plusmn5 This would imply thatthe extrapolation as a function of the pressure worksbetter than as a function of the temperature Neverthe-less we would like to draw attention to the di erentscales of the y axis in reggures 1plusmn5 and reggures 6 and 7The temperature range in reduced units in reggures 6 and7 (07plusmn105) is much wider than the reduced pressurerange considered in reggures 1plusmn5 (0plusmn014) Taking theextrapolation curves in reggures 6 and 7 over a tempera-ture range comparable to the pressure ranges in reggures 1-5 (less than 01 in reduced units) around the basic points


0 02 04 06 08 1x

0

003

006

009

012

015

p

T 0 =075

T 0 =10

Figure 3 Same as in reggures 1and 2 for temperatures T ˆ075 and 1 Only the solidcurves are shown withouterror bars The reglled circlesand squares are the NpT +TPdata of Vrabec et al [31] whilethe open squares are dataobtained from our GC method[7]

(open squares) one can regnd that the extrapolation withrespect to T is satisfactory in this domain Moreoveras we will see below the case is reversed for the LLE theextrapolation as a function of T seems better than withrespect to p So we believe that we cannot make thestatement that any of the p and the T extrapolationsworks better for a similar range of the variable inreduced units

At the end of discussing our VLE results we wouldrepeatedly like to point out that all the extrapolationcurves seen in reggures 1plusmn7 were obtained by performingonly 6 GEMC simulations

33 Liquidplusmnliquid equilibrium at constant temperatureIn this and the subsequent section we demonstrate

that our method is equally useful in the case of LLE

The original GEMC technique was proved to be applic-able for LLE [42plusmn44] nevertheless it was not trivial thatthis extrapolation scheme works well when both phasesare dense liquids Any simulation method encountersdi culties in systems with high density since particle

exchange or particle insertion become problematic inthis domain For instance we were unable to obtainsatisfactory results for LLE using our GC method [7]Therefore the simulator has to deal with simulationsfor high density systems with extra care In light ofthese considerations it was a pleasant surprise that

our extrapolation procedure gives quite good resultsfor LLE too

To study pressure dependence at a regxed temperaturewe chose the potential parameters from the work of van

Leeuwen et al [42] (system II) In this case a deviationfrom the LorentzplusmnBerthelot combining rule is appliedthe unlike-pair energy parameter is decreased

degAB ˆ 07hellipdegAdegBdagger1=2 In this way the attraction of likespecies is stronger than that of unlike species and LLimmiscibility occurs According to the GEMC simula-tions of van Leeuwen et al this system shows a LL

phase separation at T ˆ 0889 above p sup1 0084 witha three-phase (one vapour and two liquids) line at thispressure


0 01 02 r (V)

0

002

004

p

p 0 =002

0

002

004

006

p

p 0 =005

T=075

004

006

008

p

06 075 09 r (L)

p 0 =0065

Figure 4 Same as in reggure 1 but the VL coexisting densitiesare presented as functions of the pressure at regxed tem-perature T ˆ 075

0 01 02 r (V)

0

005

01

p

005

01

015

p

p 0 =008

p 0 =006

T=10

05 06 07 r (L)


Figures 8 and 9 show the coexistence mole fractions

and densities respectively as functions of the pressure

at T ˆ 0889 (In some cases the various curves wouldbe indistinguishable from each other and in these cases

only the solid lines are shown) Our basic simulation wasperformed for p0 ˆ 012 At this pressure our results

seem to deviate from the data of van Leeuwen et al in

phase L1 Nevertheless the data of van Leeuwen et al

seem somewhat too uncertain as they have quite largeerror bars Probably this is partly due to the fact that

the authors performed relatively short simulations

(which nonetheless were long enough to locate the

LL phase separation) and partly due to the large macructua-

tions in the pressure (the constant pressure GEMC

simulations were unstable therefore the authors used

the constant volume ensemble where the pressure is

required to be calculated)Because of the uncertainty of the data of van Leeuwen

et al we used the data [45] provided by Dr Jadran

Vrabec obtained by their novel Grand Equilibrium

method [34] Their results are shown by open triangles

in reggures 8 and 9 It is seen that their data are in goodagreement with our extrapolation curves

Moreover our procedure is able to extrapolate to

even lower pressures properly At lower pressures our


0 02 04 06 08 1x

07

08 T

T 0 =075

08

09

1

T T 0 =0875

09

1

T

p=006

T 0 =10

Figure 6 Vapourplusmnliquid coexistence curves for the molefraction of component A as functions of the temperatureat regxed pressure p ˆ 006 The di erent insets of thereggure contain the results obtained from the GEMC simu-lations performed for the di erent basic temperaturesT 0 ˆ 075 0875 and 1 The meaning of the lines andsymbols is the same as in reggure 1

005 01 r (V)

07

08 T

08

09

1

T

T 0 =0875

T 0 =075

09

1

T T 0 =10

p=006

07 08 r (L)

Figure 7 Same as in reggure 6 but the VL coexisting densitiesare presented as functions of the temperature at regxedpressure p ˆ 006

constant pressure GEMC simulations were also

unstable therefore we performed a constant volume

simulation that is displayed by open squares in

reggures 8 and 9 As is seen in these reggures the agreement

between the extrapolation results and the constantvolume GEMC data for lower pressures is satisfactory

34 Liquidplusmnliquid equilibrium at constant pressureTo investigate temperature dependence at constant

pressure a system studied by Guo et al [44] waschosen (system III) This system is very similar tosystem II used by van Leeuwen et al [42] The resultsfor the equilibrium mole fractions and densities forthe constant pressure p ˆ 0125 can be seen inreggures 10 and 11 respectively The behaviour of ourextrapolation curves around the basic temperaturesT 0 ˆ 0825 and 09 is extremely good There is a slightdeviation from the data of Guo et al [44] for highertemperatures To check whether our results are consis-tent at high temperature we performed an additionalsimulation for p0 ˆ 0935 (denoted by open squares)This point supports our extrapolation results obtained


01 013 016 x (L1)

0075

01

0125

p

T=0889

09 092 x (L2)

Figure 8 Liquidplusmnliquid coexistence curves of system II forthe mole fraction of component A as functions of thepressure at regxed temperature T ˆ 0889 The meaningof the lines is the same as in reggure 1 The reglled circlesare the GEMC data of van Leeuwen et al [42] while theopen triangles are the data of Vrabec [45] obtained by theGrand Equilibrium method [34] The open squares atp ˆ 012 are the results of our constant pressureGEMC simulations performed at the basic point whilethe open squares at p ˆ 00822 are the results of ourconstant volume GEMC simulations

055 06 065 r (L1)

0075

01

0125

p

T=0889

075 076 r (L2)

Figure 9 Same as in reggure 8 but the LL coexisting densitiesare presented as functions of the pressure at regxed tem-perature T ˆ 0889 The meaning of the lines and sym-bols is the same as in reggures 1 and 8 respectively

0 01 02 x (L1)

08

085

09

095

T

08

085

09

095

T

08 09 1x (L2)

T 0 =09

T 0 =0825

p=0125

Figure 10 Liquidplusmnliquid coexistence curves of system III forthe mole fraction of component A as functions of thetemperature at regxed pressure p ˆ 0125 The meaningof the lines is the same as in reggure 1 The reglled circlesare the GEMC data of Guo et al [44] the open squaresare our GEMC simulation results obtained for variousbasic points The di erent insets of the reggure containthe results obtained from the GEMC simulations per-formed for the di erent basic temperatures T 0 ˆ 0825and 09

from the T 0 ˆ 09 basic simulation against the data ofGuo et al

35 Vapourplusmnliquid equilibrium at regxed liquidcomposition

The vapourplusmnliquid equilibrium when the mole frac-tion of the liquid is constant is important from a prac-tical point of view This is the case when the size of theliquid phase is appropriately large the evaporation of asmall portion of the liquid does not change its composi-tion We are interested in the question how does thevapour pressure change with the temperature whenthe mole fraction of the liquid phase is kept at a pre-scribed value We realize that our extrapolation schemeyields a function for the liquid side equilibrium molefraction xhellipshy pdagger By regxing xL and solving the equationxL ˆ xhellipshy pdagger we can calculate the phellipT dagger curve corre-sponding to the prescribed value of xL Figure 12shows the curves obtained from the extrapolationequations given by equation (15) for xL ˆ 08 and 09For comparison the data of Vrabec et al [31] whereavailable are also shown The vapour pressure curvefor the pure LJ macruid (xL ˆ 1) is also plotted using the

correlation equation of Vrabec and co-workers [9](dashed line)

It is seen that the behaviour of our curves is satisfac-tory Adding some of the more volatile component B tothe pure liquid of component A at a given temperaturethe vapour pressure increases

36 Computer time requirementTo study the computer time requirement we have

plotted some extrapolated equilibrium mole fractiondata as functions of the simulation time in reggure 13All results refer to the case shown in the middle insetof reggure 1 The temperature is T ˆ 075 the pressure ofthe basic simulation is p ˆ 005 We extrapolated fromthis pressure to a higher pressure p ˆ 0065 (reglledcircles) and to a lower pressure p ˆ 0035 (open circles)on the basis of equation (15) (solid lines with error barsin reggure 1) The results of GEMC simulations per-formed at these pressures are also shown as horizontallines (dashed for p ˆ 0065 and dot-dashed forp ˆ 0035 these data are shown as open squares inreggure 1) The CPU time on the abscissa is calculatedconsidering a 533 MHz Pentium processor PC

It can be seen that the extrapolated mole fraction datapractically do not change after 25 h and they agree withthe reference data within the error bars The 50 h simu-lations were performed just from exaggarated precau-tion This 25 h is 4plusmn5 times longer simulation timethan that one usually uses in a GEMC simulation


05 06 07 08 r

08

085

09

095

T

08

085

09

095

T

p=0125

T 0 =0825

T 0 =09

Figure 11 Same as in reggure 10 but the LL coexisting den-sities are presented as functions of the temperature atregxed pressure p ˆ 0125 The meaning of the lines andsymbols is the same as in reggures 1 and 10 respectively

07 08 09 1 T

0

003

006

009

p

x L =08

x L =09

x L =1

Figure 12 The vapour pressure as a function of the tempera-ture at regxed liquid mole fractions The solid lines are ourextrapolation curves The meaning of the symbols is thesame as in reggure 1 The dashed line is the vapour pressurecurve of the pure LJ macruid calculated from the correlationequation given by Lotreg et al [9]

Nevertheless we believe that the obtained equilibriumcurves are worth the extra computation time

A detailed discussion on the comparison of this pro-cedure with other methods can be found in our previouspaper [1]

4 ConclusionsWe have proposed an e cient extrapolation tech-

nique to determine the VL and LL equilibrium curvesof binary mixtures by performing a single GEMCsimulation The coe cients of the extrapolation seriescan be calculated from the data yielded by the GEMCsimulation To obtain these data it is unnecessary tosubstantially modify the original GEMC code Onemerely has to accumulate not only the usual physicalquantities but their double and triple products fromwhich the macructuation formulas are built The evaluation

of the data produced by the simulation can be per-formed separately and it is not a di cult task Webelieve that the results are worth the extra e ort

The authors thank Dr Jadran Vrabec for providingreference data for reggures 8 and 9

References[1] Boda D KristoAcirc f T Liszi J and Szalai I 2002

Molec Phys 100 1989[2] Boda D Liszi J and Szalai I 1995 Chem Phys

Lett 235 140[3] Szalai I Liszi J and Boda D 1995 Chem Phys

Lett 246 214[4] KristoAcirc f T and Liszi J 1997 Z Phys Chem 199 61[5] Boda D Liszi J and Szalai I 1996 Chem Phys

Lett 256 474[6] Boda D Chan K Y and Szalai I 1997 Molec

Phys 92 1067[7] Boda D KristoAcirc f T Liszi J and Szalai I 2001

Molec Phys 99 2011[8] MoEgrave ller D and Fischer J 1990 Molec Phys 69

463[9] Lotfi A Vrabec J and Fischer J 1992 Molec

Phys 76 1319[10] Johnson J K Zollweg J A and Gubbins K E

1993 Molec Phys 78 591[11] Mecke M MuEgrave ller A Winkelmann J Vrabec J

Fischer J Span R and Wagner W 1996 Int JThermophys 17 391

[12] Szalai I Kronome G and LukaAcirc cs T 1997 J chemSoc Faraday Trans 93 3737

[13] Panagiotopoulos A Z 1987 Molec Phys 61 813[14] Panagiotopoulos A Z Quirke N Stapleton M

and Tildesley D J 1988 Molec Phys 63 527[15] Escobedo F A 2000 J chem Phys 113 8444[16] Escobedo F A 1998 J chem Phys 108 8761[17] Kofke D A 1993 Molec Phys 78 1331[18] Kofke D A 1993 J chem Phys 98 4149[19] McDonald I R and Singer K 1967 Discuss

Faraday Soc 43 40[20] Ferrenberg A M and Swendsen R H 1988 Phys

Rev Lett 61 2635[21] Kiyohara K Gubbins K E and Panagiotopoulos

A Z 1997 J chem Phys 106 3338[22] Kiyohara K Gubbins K E and Panagiotopoulos

A Z 1998 Molec Phys 94 803[23] Conrad P B and de Pablo J J 1998 Fluid Phase

Equilib 150 51[24] Potoff J J and Panagiotopoulos A Z 1998 J

chem Phys 109 10914[25] Valleau J P 1991 J comput Phys 96 193[26] Valleau J P 1993 J chem Phys 99 4718[27] Graham I S and Valleau J P 1990 J chem Phys

94 7894[28] Kiyohara K Spyriouni T Gubbins K E and

Panagiotopoulos A Z 1996 Molec Phys 89 965[29] Mehta M and Kofke D A 1994 Chem Eng Sci

49 2633[30] Vrabec J and Fischer J 1995 Molec Phys 85 781[31] Vrabec J Lotfi A and Fischer J 1995 Fluid Phase

Equilib 112 173[32] Vrabec J and Fischer J 1996 AIChE J 43 212


10 20 30 40 Simulation time h

006

009

012

x(V)

05

07

09

x(L)

T=075

Figure 13 The equilibrium mole fraction as a function of thesimulation time The vapour (bottom inset) and liquid(top inset) side mole fractions are shown at the tempera-ture T ˆ 075 for the pressures p ˆ 0065 (reglled circles)and p ˆ 0035 (open circles) as extrapolated from thebasic simulation performed at T 0 ˆ 075 and atp0 ˆ 005 The symbols with error bars are the resultsobtained from extrapolation while the lines are the resultsobtained directly from GEMC simulations performed atthe pressures p ˆ 0065 (dashed lines) and p ˆ 0035(dot-dashed lines) The CPU time refers to a 533 MHzPentium processor PC

[33] Kronome G Szalai I Wendland M and FischerJ 2000 J molec Liq 85 237

[34] Vrabec J and Hasse H 2002 Molec Phys (in thepress)

[35] Press W H Teukolsky S A Vetterling W Tand Flannery B P 1994 Numerical Recipes inFortran The Art of Scientiregc Computing 2nd Edn(Cambridge Cambridge University Press)

[36] Graves-Morris P R 1979 PadeAcirc Approximation andits Applications Lecture Notes in Mathematics Vol 765edited by L Wuytack (Berlin Springer-Verlag)

[37] Gray C G and Gubbins K E 1984 Theory ofMolecular Fluids Vol 1 Fundamentals (OxfordClarendon Press)

[38] Frenkel D and Smit B 1996 UnderstandingMolecular Simulations (San Diego Academic Press)

[39] Sadus R J 1999 Molecular Simulation of FluidsTheory Algorithms and Object-orientation (AmsterdamElsevier)

[40] Panagiotopoulos A Z 1992 Fluid Phase Equilib 7697

[41] Baus M Rull L F and Ryckaert J P 1995Observation Prediction and Simulation of PhaseTransitions in Complex Fluids (Dordrecht Kluwer)

[42] van Leeuwen M E Peters C J de Swaan AronsJ and Panagiotopoulos A Z 1991 Fluid PhaseEquilib 66 57

[43] Georgoulaki A M Ntorous I V Tassios D Pand Panagiotopoulos A Z 1994 Fluid PhaseEquilib 100 153

[44] Guo M Li Y and Lu J 1994 Fluid Phase Equilib98 129

[45] Vrabec J 2002 private communication


We call this an èxtrapolation schemersquo because wewant to emphasize that the basic simulation method isstill the GEMC technique and the information which isnecessary to build the extrapolation functions can begained from the GEMC simulation without a majormodiregcation of the original code Basically the heartof the technique lies in the important feature of the mol-ecular simulations in which they sample the phase spacenot only in the state point where they are performed butin the vicinity of this point In this way they are able toobtain information from one simulation for a neigbour-hood of the simulation state point This capabilitystands behind our series expansion methods where theinformation comes from various thermodynamic deriva-tives and the macructuations corresponding to them How-ever this capability is in close connetion with othersophisticated simulation techniques like the histogramreweighting [19plusmn24] and the thermodynamic scaling [25plusmn28] methods

The phase equilibrium simulation of mixtures is moreproblematic than that of pure macruids because the systemhas more degrees of freedom The application of theGEMC technique [14] for mixtures is straightforwardalthough in some cases not without di culties Never-theless this is the most widely used method to determinephase coexistence in mixtures but only in one equilib-rium point (the term èquilibriumrsquo will mean phase equi-librium henceforth if not stated otherwise) The GibbsplusmnDuhem integration method has been extended to mix-tures [29] There was partial success in extending theNpT plus test particle method to mixtures [30plusmn33] Theprocedure as the GEMC technique is able to producethe equilibrium point at one hellipT pdagger point but not for ahellipT pdagger domain since the calculation of the mole fractionderivatives proved to be too di cult Recently Vrabecand Hasse have developed a version of this methodusing a pseudo GC simulation on the vapour side [34]So far our GC technique [7] is the only method whichutilizes the full capabilities of the series expansionmethods to determine phase equilibrium of mixturesThis technique is able to produce the VLE curves in acertain T or p range but we encountered di culties inapplying it to liquidplusmnliquid equilibria (LLE)

In this paper we extend our extrapolation scheme [1]for the case of mixtures using constant temperature andpressure GEMC simulations We show that the equilib-rium mole fractions and densities can be extrapolatedover the hellipT pdagger plane with Taylor series andor a PadeAcircapproximant We test the method on various Lennard-Jones (LJ) mixtures applying it for both vapourplusmnliquid(VL) and liquidplusmnliquid (LL) equilibria It will be shownthat the technique is applicable and e cient in the caseof mixtures and provides a promising tool to studyphase coexistence in many component systems

2 MethodThe GEMC simulation is performed for a binary mix-

ture at the basic point hellipT 0 p0dagger namely at regxed tempera-ture T 0 and pressure p0 Let us express the second-order Taylor series for the equilibrium curve of a generalconreggurational quantity f around the basic pointhellipT 0 p0dagger as a function of the temperature and the press-ure As a matter of fact it would be more proper to talkabout an equilibrium surface f hellipshy pdagger but we will use theterm èquilibrium curversquo since in practice the phasecoexistence is studied by regxing T or p and using theother as a variable For f any of the intensive quantitiescan be substituted nevertheless in this paper we willconsider the equilibrium curves for the number density

raquo ˆ N=V and the mole fraction of component Ax ˆ xA ˆ NA=N Of course their equilibrium valuecan be taken in any of the two phases For the phaseswe will refer to them using the roman numbers I and IIThe method will be proved to be applicable to both VLand LL equilibria

It is worthwhile to use the reciprocal temperatureshy ˆ 1=kT instead of the temperature (where k is theBoltzmann constant) because of the more compactform of the formulas Let us deregne the zeroth regrst-and second-order terms of the series expansion respect-ively by

f0 ˆ f hellipshy 0 p0dagger hellip1dagger


ˆ fshy

sup3 acute

0

hellipshy iexcl shy 0dagger Dagger fp

sup3 acute

0

hellipp iexcl p0dagger hellip2dagger

and


ˆ 1

2

2fshy 2

Aacute

0

hellipshy iexcl shy 0dagger2 Dagger 2fshy p

Aacute

0

hellipshy iexcl shy 0daggerhellipp iexcl p0dagger

Dagger 1

2

2fp2

Aacute

0

hellipp iexcl p0dagger2 hellip3dagger

Here the subscript 0 means that the quantities in ques-tion have to be taken at the basic point hellipshy 0 p0dagger whilethe subscripts 1 and 2 refer to regrst- and second-orderterms in shy and p The regrst-order and second-orderTaylor series expansions of the equilibrium curves ofquantity f around hellipshy 0 p0dagger are expressed as

f T1 ˆ f0 Dagger f1 hellip4dagger

and

f T2 ˆ f0 Dagger f1 Dagger f2 hellip5dagger



fshy

sup3 acute

0

ˆ h f ishy

sup3 acute

0


and

fp

sup3 acute

0

ˆ h f ip

sup3 acute

0




2fshy 2

Aacute

0

ˆ h f ishy

sup3 acute

0


sup3 acute

0

iexcl h fHishy

sup3 acute

0

hellip8dagger

2fshy p

Aacute

0

ˆ


sup3 acute

0


Dagger shy 0

h f ishy

sup3 acute

0


sup3 acute

0

micro

iexcl h fV ishy

sup3 acute

0

para hellip9dagger

and

2fp2

Aacute

0

ˆ shy 0

h f ip

sup3 acute

0


sup3 acute

0

iexcl h f V ip

sup3 acute

0

micro para

hellip10dagger


2fshy 2

Aacute

0


hellip11dagger

2fshy p

Aacute

0



hellip12dagger

and

2fp2

Aacute

0


hellip13dagger








f P if f2=f1 lt 0

(hellip15dagger



shy middoti



xIIi iexcl xI

ihellip16dagger

and

shy middoti

pˆ shy


i daggervI

xIIi iexcl xI

i hellip17dagger



i xII
























































System

I II III

degB=degA 05 07298 075


frac14AB=frac14A 1 09699 0975


p 002 005 0065 006 008 006 regxed 012 0125 regxed

Reference [31] [42 45] [44]

Figures 1 3 4 2 3 5 6 7 8 9 10 11


i


0

shyshyshyshyshy




















0 02 04 06 08 1x

0

002

004

p

002

004

006

008

p

004

006

008

p

T=075

p 0 =0065

p 0 =005

p 0 =002


02 04 06 08 1x

0

005

01

p

p 0 =008

p 0 =006

005

01

015

p

T=10

























0 02 04 06 08 1x

0

003

006

009

012

015

p

T 0 =075

T 0 =10














0 01 02 r (V)

0

002

004

p

p 0 =002

0

002

004

006

p

p 0 =005

T=075

004

006

008

p

06 075 09 r (L)

p 0 =0065


0 01 02 r (V)

0

005

01

p

005

01

015

p

p 0 =008

p 0 =006

T=10

05 06 07 r (L)























0 02 04 06 08 1x

07

08 T

T 0 =075

08

09

1

T T 0 =0875

09

1

T

p=006

T 0 =10


005 01 r (V)

07

08 T

08

09

1

T

T 0 =0875

T 0 =075

09

1

T T 0 =10

p=006

07 08 r (L)










01 013 016 x (L1)

0075

01

0125

p

T=0889

09 092 x (L2)


055 06 065 r (L1)

0075

01

0125

p

T=0889

075 076 r (L2)


0 01 02 x (L1)

08

085

09

095

T

08

085

09

095

T

08 09 1x (L2)

T 0 =09

T 0 =0825

p=0125











05 06 07 08 r

08

085

09

095

T

08

085

09

095

T

p=0125

T 0 =0825

T 0 =09


07 08 09 1 T

0

003

006

009

p

x L =08

x L =09

x L =1


































006

009

012

x(V)

05

07

09

x(L)

T=075

















fshy

sup3 acute

0

ˆ h f ishy

sup3 acute

0


and

fp

sup3 acute

0

ˆ h f ip

sup3 acute

0




2fshy 2

Aacute

0

ˆ h f ishy

sup3 acute

0


sup3 acute

0

iexcl h fHishy

sup3 acute

0

hellip8dagger

2fshy p

Aacute

0

ˆ


sup3 acute

0


Dagger shy 0

h f ishy

sup3 acute

0


sup3 acute

0

micro

iexcl h fV ishy

sup3 acute

0

para hellip9dagger

and

2fp2

Aacute

0

ˆ shy 0

h f ip

sup3 acute

0


sup3 acute

0

iexcl h f V ip

sup3 acute

0

micro para

hellip10dagger


2fshy 2

Aacute

0


hellip11dagger

2fshy p

Aacute

0



hellip12dagger

and

2fp2

Aacute

0


hellip13dagger








f P if f2=f1 lt 0

(hellip15dagger



shy middoti



xIIi iexcl xI

ihellip16dagger

and

shy middoti

pˆ shy


i daggervI

xIIi iexcl xI

i hellip17dagger



i xII
























































System

I II III

degB=degA 05 07298 075


frac14AB=frac14A 1 09699 0975


p 002 005 0065 006 008 006 regxed 012 0125 regxed

Reference [31] [42 45] [44]

Figures 1 3 4 2 3 5 6 7 8 9 10 11


i


0

shyshyshyshyshy




















0 02 04 06 08 1x

0

002

004

p

002

004

006

008

p

004

006

008

p

T=075

p 0 =0065

p 0 =005

p 0 =002


02 04 06 08 1x

0

005

01

p

p 0 =008

p 0 =006

005

01

015

p

T=10

























0 02 04 06 08 1x

0

003

006

009

012

015

p

T 0 =075

T 0 =10














0 01 02 r (V)

0

002

004

p

p 0 =002

0

002

004

006

p

p 0 =005

T=075

004

006

008

p

06 075 09 r (L)

p 0 =0065


0 01 02 r (V)

0

005

01

p

005

01

015

p

p 0 =008

p 0 =006

T=10

05 06 07 r (L)























0 02 04 06 08 1x

07

08 T

T 0 =075

08

09

1

T T 0 =0875

09

1

T

p=006

T 0 =10


005 01 r (V)

07

08 T

08

09

1

T

T 0 =0875

T 0 =075

09

1

T T 0 =10

p=006

07 08 r (L)










01 013 016 x (L1)

0075

01

0125

p

T=0889

09 092 x (L2)


055 06 065 r (L1)

0075

01

0125

p

T=0889

075 076 r (L2)


0 01 02 x (L1)

08

085

09

095

T

08

085

09

095

T

08 09 1x (L2)

T 0 =09

T 0 =0825

p=0125











05 06 07 08 r

08

085

09

095

T

08

085

09

095

T

p=0125

T 0 =0825

T 0 =09


07 08 09 1 T

0

003

006

009

p

x L =08

x L =09

x L =1


































006

009

012

x(V)

05

07

09

x(L)

T=075


















f P if f2=f1 lt 0

(hellip15dagger



shy middoti



xIIi iexcl xI

ihellip16dagger

and

shy middoti

pˆ shy


i daggervI

xIIi iexcl xI

i hellip17dagger



i xII
























































System

I II III

degB=degA 05 07298 075


frac14AB=frac14A 1 09699 0975


p 002 005 0065 006 008 006 regxed 012 0125 regxed

Reference [31] [42 45] [44]

Figures 1 3 4 2 3 5 6 7 8 9 10 11


i


0

shyshyshyshyshy




















0 02 04 06 08 1x

0

002

004

p

002

004

006

008

p

004

006

008

p

T=075

p 0 =0065

p 0 =005

p 0 =002


02 04 06 08 1x

0

005

01

p

p 0 =008

p 0 =006

005

01

015

p

T=10

























0 02 04 06 08 1x

0

003

006

009

012

015

p

T 0 =075

T 0 =10














0 01 02 r (V)

0

002

004

p

p 0 =002

0

002

004

006

p

p 0 =005

T=075

004

006

008

p

06 075 09 r (L)

p 0 =0065


0 01 02 r (V)

0

005

01

p

005

01

015

p

p 0 =008

p 0 =006

T=10

05 06 07 r (L)























0 02 04 06 08 1x

07

08 T

T 0 =075

08

09

1

T T 0 =0875

09

1

T

p=006

T 0 =10


005 01 r (V)

07

08 T

08

09

1

T

T 0 =0875

T 0 =075

09

1

T T 0 =10

p=006

07 08 r (L)










01 013 016 x (L1)

0075

01

0125

p

T=0889

09 092 x (L2)


055 06 065 r (L1)

0075

01

0125

p

T=0889

075 076 r (L2)


0 01 02 x (L1)

08

085

09

095

T

08

085

09

095

T

08 09 1x (L2)

T 0 =09

T 0 =0825

p=0125











05 06 07 08 r

08

085

09

095

T

08

085

09

095

T

p=0125

T 0 =0825

T 0 =09


07 08 09 1 T

0

003

006

009

p

x L =08

x L =09

x L =1


































006

009

012

x(V)

05

07

09

x(L)

T=075
































































System

I II III

degB=degA 05 07298 075


frac14AB=frac14A 1 09699 0975


p 002 005 0065 006 008 006 regxed 012 0125 regxed

Reference [31] [42 45] [44]

Figures 1 3 4 2 3 5 6 7 8 9 10 11


i


0

shyshyshyshyshy




















0 02 04 06 08 1x

0

002

004

p

002

004

006

008

p

004

006

008

p

T=075

p 0 =0065

p 0 =005

p 0 =002


02 04 06 08 1x

0

005

01

p

p 0 =008

p 0 =006

005

01

015

p

T=10

























0 02 04 06 08 1x

0

003

006

009

012

015

p

T 0 =075

T 0 =10














0 01 02 r (V)

0

002

004

p

p 0 =002

0

002

004

006

p

p 0 =005

T=075

004

006

008

p

06 075 09 r (L)

p 0 =0065


0 01 02 r (V)

0

005

01

p

005

01

015

p

p 0 =008

p 0 =006

T=10

05 06 07 r (L)























0 02 04 06 08 1x

07

08 T

T 0 =075

08

09

1

T T 0 =0875

09

1

T

p=006

T 0 =10


005 01 r (V)

07

08 T

08

09

1

T

T 0 =0875

T 0 =075

09

1

T T 0 =10

p=006

07 08 r (L)










01 013 016 x (L1)

0075

01

0125

p

T=0889

09 092 x (L2)


055 06 065 r (L1)

0075

01

0125

p

T=0889

075 076 r (L2)


0 01 02 x (L1)

08

085

09

095

T

08

085

09

095

T

08 09 1x (L2)

T 0 =09

T 0 =0825

p=0125











05 06 07 08 r

08

085

09

095

T

08

085

09

095

T

p=0125

T 0 =0825

T 0 =09


07 08 09 1 T

0

003

006

009

p

x L =08

x L =09

x L =1


































006

009

012

x(V)

05

07

09

x(L)

T=075

















i


0

shyshyshyshyshy




















0 02 04 06 08 1x

0

002

004

p

002

004

006

008

p

004

006

008

p

T=075

p 0 =0065

p 0 =005

p 0 =002


02 04 06 08 1x

0

005

01

p

p 0 =008

p 0 =006

005

01

015

p

T=10

























0 02 04 06 08 1x

0

003

006

009

012

015

p

T 0 =075

T 0 =10














0 01 02 r (V)

0

002

004

p

p 0 =002

0

002

004

006

p

p 0 =005

T=075

004

006

008

p

06 075 09 r (L)

p 0 =0065


0 01 02 r (V)

0

005

01

p

005

01

015

p

p 0 =008

p 0 =006

T=10

05 06 07 r (L)























0 02 04 06 08 1x

07

08 T

T 0 =075

08

09

1

T T 0 =0875

09

1

T

p=006

T 0 =10


005 01 r (V)

07

08 T

08

09

1

T

T 0 =0875

T 0 =075

09

1

T T 0 =10

p=006

07 08 r (L)










01 013 016 x (L1)

0075

01

0125

p

T=0889

09 092 x (L2)


055 06 065 r (L1)

0075

01

0125

p

T=0889

075 076 r (L2)


0 01 02 x (L1)

08

085

09

095

T

08

085

09

095

T

08 09 1x (L2)

T 0 =09

T 0 =0825

p=0125











05 06 07 08 r

08

085

09

095

T

08

085

09

095

T

p=0125

T 0 =0825

T 0 =09


07 08 09 1 T

0

003

006

009

p

x L =08

x L =09

x L =1


































006

009

012

x(V)

05

07

09

x(L)

T=075







































0 02 04 06 08 1x

0

003

006

009

012

015

p

T 0 =075

T 0 =10














0 01 02 r (V)

0

002

004

p

p 0 =002

0

002

004

006

p

p 0 =005

T=075

004

006

008

p

06 075 09 r (L)

p 0 =0065


0 01 02 r (V)

0

005

01

p

005

01

015

p

p 0 =008

p 0 =006

T=10

05 06 07 r (L)























0 02 04 06 08 1x

07

08 T

T 0 =075

08

09

1

T T 0 =0875

09

1

T

p=006

T 0 =10


005 01 r (V)

07

08 T

08

09

1

T

T 0 =0875

T 0 =075

09

1

T T 0 =10

p=006

07 08 r (L)










01 013 016 x (L1)

0075

01

0125

p

T=0889

09 092 x (L2)


055 06 065 r (L1)

0075

01

0125

p

T=0889

075 076 r (L2)


0 01 02 x (L1)

08

085

09

095

T

08

085

09

095

T

08 09 1x (L2)

T 0 =09

T 0 =0825

p=0125











05 06 07 08 r

08

085

09

095

T

08

085

09

095

T

p=0125

T 0 =0825

T 0 =09


07 08 09 1 T

0

003

006

009

p

x L =08

x L =09

x L =1


































006

009

012

x(V)

05

07

09

x(L)

T=075




























0 01 02 r (V)

0

002

004

p

p 0 =002

0

002

004

006

p

p 0 =005

T=075

004

006

008

p

06 075 09 r (L)

p 0 =0065


0 01 02 r (V)

0

005

01

p

005

01

015

p

p 0 =008

p 0 =006

T=10

05 06 07 r (L)























0 02 04 06 08 1x

07

08 T

T 0 =075

08

09

1

T T 0 =0875

09

1

T

p=006

T 0 =10


005 01 r (V)

07

08 T

08

09

1

T

T 0 =0875

T 0 =075

09

1

T T 0 =10

p=006

07 08 r (L)










01 013 016 x (L1)

0075

01

0125

p

T=0889

09 092 x (L2)


055 06 065 r (L1)

0075

01

0125

p

T=0889

075 076 r (L2)


0 01 02 x (L1)

08

085

09

095

T

08

085

09

095

T

08 09 1x (L2)

T 0 =09

T 0 =0825

p=0125











05 06 07 08 r

08

085

09

095

T

08

085

09

095

T

p=0125

T 0 =0825

T 0 =09


07 08 09 1 T

0

003

006

009

p

x L =08

x L =09

x L =1


































006

009

012

x(V)

05

07

09

x(L)

T=075





































0 02 04 06 08 1x

07

08 T

T 0 =075

08

09

1

T T 0 =0875

09

1

T

p=006

T 0 =10


005 01 r (V)

07

08 T

08

09

1

T

T 0 =0875

T 0 =075

09

1

T T 0 =10

p=006

07 08 r (L)










01 013 016 x (L1)

0075

01

0125

p

T=0889

09 092 x (L2)


055 06 065 r (L1)

0075

01

0125

p

T=0889

075 076 r (L2)


0 01 02 x (L1)

08

085

09

095

T

08

085

09

095

T

08 09 1x (L2)

T 0 =09

T 0 =0825

p=0125











05 06 07 08 r

08

085

09

095

T

08

085

09

095

T

p=0125

T 0 =0825

T 0 =09


07 08 09 1 T

0

003

006

009

p

x L =08

x L =09

x L =1


































006

009

012

x(V)

05

07

09

x(L)

T=075
























01 013 016 x (L1)

0075

01

0125

p

T=0889

09 092 x (L2)


055 06 065 r (L1)

0075

01

0125

p

T=0889

075 076 r (L2)


0 01 02 x (L1)

08

085

09

095

T

08

085

09

095

T

08 09 1x (L2)

T 0 =09

T 0 =0825

p=0125











05 06 07 08 r

08

085

09

095

T

08

085

09

095

T

p=0125

T 0 =0825

T 0 =09


07 08 09 1 T

0

003

006

009

p

x L =08

x L =09

x L =1


































006

009

012

x(V)

05

07

09

x(L)

T=075

























05 06 07 08 r

08

085

09

095

T

08

085

09

095

T

p=0125

T 0 =0825

T 0 =09


07 08 09 1 T

0

003

006

009

p

x L =08

x L =09

x L =1


































006

009

012

x(V)

05

07

09

x(L)

T=075
















































006

009

012

x(V)

05

07

09

x(L)

T=075
















The extrapolation of phase equilibrium curves of mixtures in the isobaric-isothermal Gibbs ensemble

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Transcript of The extrapolation of phase equilibrium curves of mixtures in the isobaric-isothermal Gibbs ensemble