Phase equilibria and critical behavior of square‐well fluids of variable width by Gibbs ensemble Monte Carlo simulation
Abstract
The vapor–liquid phase equilibria of square†well systems with hard†sphere diameters σ, well†depths ε, and ranges λ=1.25, 1.375, 1.5, 1.75, and 2 are determined by Monte Carlo simulation. The two bulk phases in coexistence are simulated simultaneously using the Gibbs ensemble technique. Vapor–liquid coexistence curves are obtained for a series of reduced temperatures between about Tr=T/Tc=0.8 and 1, where Tc is the critical temperature. The radial pair distribution functions g(r) of the two phases are calculated during the simulation, and the results extrapolated to give the appropriate contact values g(σ), g(λσ−), and g(λσ+). These are used to calculate the vapor†pressure curves of each system and to test for equality of pressure in the coexisting vapor and liquid phases. The critical points of the square†well fluids are determined by analyzing the density†temperature coexistence data using the first term of a Wegner expansion. The dependence of the reduced critical temperature T* c=kTc/ε, pressure P* c=Pcσ3/ε, number density Ï * c=Ï cσ3, and compressibility factor Z=P/(Ï kT), on the potential range λ, is established. These results are compared with existing data obtained from perturbation theories. The shapes of the coexistence curves and the approach to criticality are described in terms of an apparent critical exponent β. The curves for the square†well systems with λ=1.25, 1.375, 1.5, and 1.75 are very nearly cubic in shape corresponding to near†universal values of β (β≊0.325). This is not the case for the system with a longer potential range; when λ=2, the coexistence curve is closer to quadratic in shape with a near†classical value of β (β≊0.5). These results seem to confirm the view that the departure of β from a mean†field or classical value for temperatures well below critical is unrelated to long†range, near†critical fluctuations.
Full text
Phase equilib ia and c i ical beha io o squa e-well luids o a iable wid h
by Gibbs ensemble Mon e Ca lo simula ion
Lou des Vega, En ique de Miguel, and Luis F. Ruil
Depa amen o de isica A bmica Molecula y Nuclea , Uni e sidad de Se iUa, Ap do. 1065,
Se illa 41080, Spain
Geo ge Jackson and Ian A. McLu e
Depa men
o
Chemis y, Uni e si y
o
She ieeld, She ield, S3 7HF, Uni ed Kingdom
(Recei ed 20 May 199 1; accep ed 11 Oc obe 199 1)
The apo -liquid phase equilib ia o squa e-well sys ems wi h ha d-sphe e diame e s o, well-
dep hs E, and anges il = 1.25, 1.375, 1.5, 1.75, and 2 a e de e mined by Mon e Ca lo
simula ion. The wo bulk phases in coexis ence a e simula ed simul aneously using he Gibbs
ensemble echnique. Vapo -liquid coexis ence cu es a e ob ained o a se ies o educed
empe a u es be ween abou
T,.
=
T/T,
= 0.8 and 1, whe e
T,
is he c i ical empe a u e. The
adial pai dis ibu ion unc ions g( ) o he wo phases a e calcula ed du ing he simula ion,
and he esul s ex apola ed o gi e he app op ia e con ac alues g(a), g(/Za- ), and
g(;la -I- ). These a e used o calcula e he apo -p essu e cu es o each sys em and o es o
equali y o p essu e in he coexis ing apo and liquid phases. The c i ical poin s o he squa e-
well luids a e de e mined by analyzing he densi y- empe a u e coexis ence da a using he i s
e m o a Wegne expansion. The dependence o he educed c i ical empe a u e
T = kT,/q
p essu e
P
=
P,o-‘/E,
numbe densi y p: = pE d, and comp essibili y ac o Z =
P /(pkT),
on
he po en ial ange il, is es ablished. These esul s a e compa ed wi h exis ing da a ob ained
om pe u ba ion heo ies. The shapes o he coexis ence cu es and he app oach o c i icali y
a e desc ibed in e ms o an appa en c i ical exponen 8. The cu es o he squa e-well
sys ems wi h il = 1.25, 1.375, 1.5, and 1.75 a e e y nea ly cubic in shape co esponding o
nea -uni e sal alues o p ( lzO.325). This is no he case o he sys em wi h a longe
po en ial ange; when ;1 = 2, he coexis ence cu e is close o quad a ic in shape wi h a nea -
classical alue o p (PzO.5). These esul s seem o con i m he iew ha he depa u e o l
om a mean- ield o classical alue o empe a u es well below c i ical is un ela ed o long-
ange, nea -c i ical luc ua ions.
I. INTRODUCTION
Sys ems o pa icles in e ac ing ia he squa e-well po-
en ial ha e been ex ensi ely s udied by s a is ical mechani-
cal me hods. Such in e es is in a iably due o he ac ha
he squa e-well po en ial is he simples model which in-
cludes he p esence o a ac i e and epulsi e o ces. The
po en ial ene gy u( ) o a pai o squa e-well pa icles sepa-
a ed by a dis ance
is gi en by
i
+ 03, i <q
u( ) = -E,
i a< </2o; (1)
0, i &la,
whe e u is he ha d-sphe e diame e o he pa icle, il is he
educed ange o he po en ial well, and E is i s dep h. The
model is o subs an ial heo e ical impo ance in s udies o
sys ems wi h a a ying po en ial ange since i can ep esen
h ee limi ing cases, namely, he ha d-sphe e, he sho -
ange s icky-sphe e, and he long- ange an de Waals
luids.
The e has been a se ies o s udies in which in eg al equa-
ions we e sol ed nume ically o he s uc u al and he mo-
dynamic p ope ies o squa e-well sys ems. These ha e in-
cluded solu ions o he Pe cus-Ye ick (PY),‘-’
hype ne ed chain (HNC) ,‘*’ and mean-sphe ical app oxi-
ma ions (MSA) ‘ 9 o he Oms ein-Zemike (OZ) in eg al
equa ion, and o he Y on-Bo n-G een (YBG) equa-
ion’s2’ based on he supe posi ion app oxima ion. In he
con ex o he p esen s udy, i is wo h discussing he esul s
o he YBG heo y in mo e de ail. Nume ical calcula ions
o squa e-well sys ems wi h a po en ial ange o il = 1.85
ha e shown ha he YBG equa ion appea s o ha e a egion
o long- ange solu ions cha ac e ized by nea -uni e sal al-
ues o he c i ical exponen s y = 1.24, /3 = 0.33, and S = 4.4
o he comp essibili y on he c i ical isocho e, he coexis-
ence cu e, and he c i ical iso he m, espec i ely.‘7-2’ This
is in di ec con as wi h he esul s o PY heo y in which
he c i ical exponen s seem o be classical.22 Mo e accu a e
s udies sugges ha he YBG equa ion does no exhibi a ue
c i ical poin .23-26 I is his con used s a e o a ai s which, in
pa , mo i a ed he p esen s udy.
Pe u ba ion heo y has o en been used o calcula e he
he modynamics p ope ies o squa e-well luids.27-33 In
his pape we compa e he c i ical cons an s ob ained om
he second-o de heo y3’ o sys ems wi h a ange o alues
o ;1 wi h es ima es om compu e simula ion. A pe u ba-
ion heo y o luids which uses he squa e-well luid as he
e e ence has also been de eloped by de1 Rio
e a1.34V35
They
ha e ex ended he app oach o deal wi h squa e-well luids
2296 J. Chem. Phys. 96 (3), 1 Feb ua y 1992 0021-9606/92/032296-i 0$06.00 0 1992 Ame ican Ins i u e o Physics
o a iable well wid h.3c39 The heo y gi es easonable
ag eemen wi h exis ing simula ion da a o sho and long
anges o he po en ial and mos o he luid ange.
In addi ion o heo y, a numbe o Mon e Ca lo simula-
ion s udies o squa e-well luids ha e been unde -
aken.4043p8*30 The i s molecula dynamics simula ions o
squa e-well sys ems we e pe o med by Alde
e al.*
By and
la ge, only sys ems wi h a po en ial ange o II = 1.5 we e
examined al hough some esul s a e a ailable o o he po-
en ial anges ( 1.125 < ;I < 2) .30,43 Many empe a u es and
densi ies in he luid s a e ha e been in es iga ed, bu only
ske ches o he sys ems’ phase equilib ia ha e been simula -
ed. Di ec simula ion echniques can be applied o he s udy
o phase equilib ia. 45+46 The simples me hod in ol es simu-
la ing he wo coexis ing phases sepa a ed by an in e ace in
a single simula ion box. Mos o he simula ion s udies ocus-
ing on he p ope ies o he in e acial egion ha e been e-
iewed by Rowlinson and Widom.47 Fo su icien ly la ge
sys em sizes, he me hods can be used o s udy bulk phase
coexis ence. Chapela e
aL4’
ha e used he molecula dy-
namics me hod o s udy he phase equilib ia o he squa e-
well sys em wi h /z = 1.5 by con ining he luid in a box be-
ween pa allel ha d walls ha a e pe pendicula o he z
di ec ion. Ini ially he luid is homogeneous and is gi en a
densi y close o he c i ical alue, bu a as spinodal decom-
posi ion occu s and a slab o liquid condenses a he cen e o
he box wi h apo on ei he side o i . Mo e ecen ly, Bena-
ides
e a1.39
ha e pe o med a simila s udy o he squa e-
well luid wi h/z = 3. Howe e , hese ypes o di ec simula-
ion s udies o phase equilib ia possess signi ican d awbacks
because he me hod is es ic ed o a ilm o liquid con ined
be ween pa allel pla es. Unless he size o he sys em is e y
la ge he con inemen causes he coexis ence p ope ies o
he luid o be di e en om he bulk coexis ence p ope ies
o in e es . In ac , he me hod b eaks down al oge he o
empe a u es close o he c i ical poin . The e ec o con in-
ing a luid be ween pa allel pla es is discussed in Sec. IV wi h
e e ence o he squa e-well luid wi h /z = 1.5.
A mo e ecen app oach which is used o di ec ly simu-
la e phase equilib ia is he so-called Gibbs ensemble Mon e
Ca lo echnique. 49*50 In his me hod he wo coexis ing ho-
mogeneous phases in he modynamic equilib ium a e simu-
la ed in sepa a e boxes wi hou he p esence o an in e ace.
The simula ion in ol es h ee dis inc Mon e Ca lo mo es:
phase space is sampled andomly in each box by mo ing he
molecules ollowing he usual Me opolis Mon e Ca lo
scheme; a andom change in olume is made so ha an in-
c ease in olume o one box co esponds o a dec ease in
olume in he o he , and ice e sa; inally, molecules a e
in e changed be ween he wo boxes. Since he in e ace is
no included in Gibbs ensemble simula ions, he coexis ence
p ope ies a e expec ed o be a good ep esen a ion o bulk
beha io , and in con as o he o me me hod, he con ine-
men p oblem does no a ise. I has also been sugges ed51.52
ha because he me hod allows o luc ua ions in he
numbe s o pa icles and olumes o he subsys ems, he
Gibbs ensemble me hod leads o be e es ima es o he c i i-
cal cons an s o he in ini e sys em and a close p oximi y o
he c i ical poin can be achie ed.
Gibbs ensemble Mon e Ca lo simula ions ha e now
been pe o med o a numbe o po en ial models, including
pu e Lenna d-Jones luids and mix u es,49’50*53954 he sym-
me ical nonaddi i e ha d sphe e sys em,55 he ha d-co e
wo-Yukawa luid,56 he Lenna d-Jones luid wi h a quad-
upole in e ac ion, ” he nonsphe ical Gay-Be ne luid,58
squa e-well dia omics,59 and mo e ealis ic models o al-
kanes and wa e .60 6’
In his wo k we p esen he i s esul s o Gibbs ensem-
ble Mon e Ca lo simula ions o he apo -liquid phase
equilib ia o squa e-well luids wi h po en ial anges o
il = 1.25, 1.375, 1.5, 1.75, and 2. The c i ical cons an s o
he squa e-well sys ems a e ob ained by analyzing he da a
using he i s e m o a Wegne expansion. The pu pose o
he s udy is wo old: i s , o p o ide accu a e apo -liquid
coexis ence da a o he squa e-well sys ems o compa ison
wi h mean- ield and pe u ba ion heo ies; second, o de e -
mine he dependence o he c i ical cons an s on ;1 and o
analyze he o e all shape o he coexis ence cu es in e ms
o an appa en c i ical exponen 0. In iew o he success o
he Gibbs me hod in es ima ing he c i ical poin , he ap-
p oach o he c i ical poin is examined wi h pa icula a -
en ion placed on he shape o wo-phase en elope o he
a ious anges o he po en ial. De ails o he simula ions
and he da a analysis a e discussed in Se s. II and III, and he
esul s a e p esen ed in Sec. IV.
II. COMPUTER SIMULATIONS
In Gibbs ensemble Mon e Ca lo simula ions he coexis -
ing apo and liquid phases a e moni o ed simul aneously as
sepa a e subsys ems wi hou he p esence o an in e ace.
He e, we p esen a b ie summa y o he Gibbs ensemble
algo i hm o Panagio opoulos49*50 as applied o squa e-well
luids wi h a iable ange. Fo a clea e and mo e de ailed
desc ip ion o he echnique, he eade is e e ed o he
o iginal pape s. A ho ough s a is ical mechanical de i a-
ion o he algo i hm has been p esen ed by Smi e
a1.,51 52
and he me hod has also been desc ibed in a couple o ecen
e iews.46*62
A he s a o he simula ion N, = 256 squa e-well pa -
icles ep esen ing one o he coexis ing phases a e placed on
a ace-cen ed-cubic ( ee) la ice in a cubic box 1 o olume
V, wi h he usual pe iodic bounda y condi ions (PBC) and
minimum image con en ion (MIC) .45 Simila ly, N, = 256
pa icles o he o he phase a e placed in an equi alen box 2
o olume V2. One box ep esen s he apo phase and he
o he he liquid phase so ha he o al numbe o pa icles
unde s udy is N = N, + N2 = 5 12. Du ing he cou se o
he simula ion, he olumes and he numbe s o pa icles o
he wo subsys ems a e allowed o a y in such a way ha he
empe a u e
T,
he o al olume V= V, + V, and he o al
numbe o pa icles N = N, + N2 emain cons an . Since
he simula ion ensu es ha he p essu es
P,
and
P2,
he
chemical po en ials p, and p2, and he empe a u e a e
equal, andom luc ua ions will o ce he subsys ems in o
egions o phase space ep esen ing he wo coexis ing
phases.
A simula ion cycle in he Gibbs ensemble Mon e Ca lo
Vega e &: Phase equilib ia o he squa e-well luid 2297
J. Chem. Phys., Vol. 96, No. 3, 1 Feb ua y 1992
2298 Vega
e a/.:
Phase equilib ia o he squa e-well luid
me hod comp ises h ee dis inc ypes o mo es: N, and N2
ial pa icle displacemen s wi hin each box, one ial ol-
ume change, and a numbe o ial pa icle in e changes
which depend on he densi y o he sys em. In he case o
displacemen mo es, he pa icles a e chosen and displaced
andomly wi hin he boxes ollowing he well es ablished
canonical NVT Me opolis scheme.63 The accep ance p ob-
abili y o each subsys em is adjus ed o -40% by a ying
he maximum ex en o pa icle displacemen s. A e he N,
and N2 ial displacemen s in each box, a andom change in
olume is a emp ed. The olume changes o he wo boxes
a e coupled, wi h he o e all olume V emaining cons an
in o de o sa is y he condi ion o equali y o p essu es in he
wo egions. The accep ance p obabili y o he olume
change is essen ially gi en by he p esc ip ion o Wooda o
simula ions in he iso he mal-isoba ic
NPT
ensemble. The
maximum allowable olume displacemen is adjus ed o gi e
a-40% accep ance a e o he new ial con igu a ions.
The las s age o he Gibbs ensemble me hod in ol es pa i-
cle in e changes be ween he wo subsys ems in o de o en-
su e equali y o chemical po en ial in he wo phases. This is
achie ed by c ea ing a pa icle a a andom posi ion in one
subsys em and annihila ing a andomly chosen pa icle in
he o he egion. The c ea ion/annihila ion mo es a e emi-
niscen o Mon e Ca lo simula ions in he g and canonical
,U VT ensemble. The numbe o in e change mo es ha a e
a emp ed is adjus ed so ha - l%-5% o he o al numbe
o pa icles a e in e changed pe cycle; ypically 250-3000
in e change a emp s a e equi ed o he squa e-well luid
in he densi y ange o in e es .
model, he p essu e canno be ob ained di ec ly om simula-
ion. The i ial equa ion can be used, howe e , i he adial
pai -dis ibu ion unc ion g( ) is known o , mo e speci ical-
ly, i i s con ac alues g(a), g(/Za- ), and g(/Za+ ) co e-
sponding o he discon inui ies o he po en ial a e known.
By de e mining he dis ances be ween pai s o pa icles in
each subsys em once e e y cycle, his og ams a e cons uc ed
which when no malized gi e g( ) o he wo coexis ing
phases. The con ac alues a e ob ained by ex apola ing he
da a o g( ) .& The equa ion o s a e can be ob ained om
he p essu e equa ion’
P
-= 1 +p-p+%m --A3[gbw) -g(Ao+)lI,
PkT
(2)
whe ep =
N/Vis
he numbe densi y. Hence, i he con ac
alues o g( ) a e known o a gi en numbe densi y, he
comp essibili y ac o Z and he p essu e o he squa e-well
luid can be calcula ed om Eq. (2). The p essu es o he
coexis ing apo and liquid phases a e de e mined in his
way o all o he sys ems s udied. The unce ain ies in he
alues o he p essu es a e di icul o calcula e, bu can be
es ima ed om he e o s in he mean densi ies o each phase
and he e o s incu ed when g( ) is ex apola ed o gi e he
con ac alues.
The simula ion o phase equilib ia o a gi en s a e poin
equi es a o al o 2-7
X
10’ such cycles. A e equilib a ing
he sys em o 1-2X IO5 cycles, a u he l-5
X
lo5 cycles a e
pe o med o accumula e he a e ages o he p ope ies o
he wo coexis ing phases. The long uns a e equi ed o he
highe -densi y con igu a ions. Ve le neighbo 1is P a e
used o each subsys em o speed up he calcula ion o pai
in e ac ions, he lis s being ecalcula ed e e y cycle. The
mean numbe s o pa icles N, and N, , olumes V, and I’,,
and ene gies
E,
and
E,
o he wo coexis ing phases 1 and 2
a e ob ained as con igu a ional a e ages o e he accumula-
ion s age o he simula ion. In his way he coexis ence a-
po and liquid densi ies o he squa e-well sys em a a ixed
empe a u e a e de e mined. The e o s in he a e age p op-
e ies o in e es a e es ima ed by calcula ing he s anda d
de ia ions o blocks o 25 cycles. The ini ial con igu a ions
used in he s udies a low empe a u e a e ee la ices. The
inal equilib ium con igu a ions ob ained om he i s un
a e hen used as he s a ing poin o simula ions a a highe
empe a u e. By epea ing he simula ions o a se ies o
empe a u es and po en ial anges, he apo -liquid coexis-
ence cu es a e de e mined be ween he c i ical empe a-
u e and - 80% o i s alue o sys ems wi hal = 1.25, 1.375,
1.5, 1.75, and 2.
In he ollowing sec ion we discuss how he c i ical con-
s an s a e ob ained by analyzing he phase equilib ium da a
and how he app oach o he c i ical poin can be desc ibed
in e ms o a Wegne expansion wi h he co esponding c i i-
cal exponen s. The esul s o he Gibbs-ensemble simula-
ions o he phase-equilib ia and c i ical beha io o he
squa e-well sys ems a e p esen ed in Sec. IV.
III. CRITICAL POINT
An ad an age o he Gibbs ensemble Mon e Ca lo ech-
nique o e o he compu e simula ion me hods is ha a clos-
e p oximi y o he c i ical poin can be achie ed. By a ca e-
ul analysis o he su ace con ibu ions o he ee ene gy
densi y, Smi
e al.
” ha e shown ha apo -liquid coexis-
ence canno be obse ed close o bu below he c i ical
poin . Since Me opolis Mon e Ca lo simula ions o a ca-
nonical ensemble o e es ima e he alue o he c i ical em-
pe a u e, he Gibbs ensemble leads o a be e es ima e o i s
alue. They explain his unexpec ed obse a ion in e ms o
luc ua ions in he olumes and numbe s o pa icles o he
wo subsys ems allowed in Gibbs ensemble simula ions.
The p essu es o he coexis ing phases ha e also been
calcula ed o each one o hese sys ems in o de o de e -
mine he co esponding apo -p essu e cu es and o e i y
he equali y o p essu e o gi en coexis ence poin s. In he
case o a discon inuous po en ial such as he squa e-well
In a s udy o pa icula in e es , he a ailable da a o a
numbe o sys ems exhibi ing apo -liquid and liquid-liquid
phase equilib ia ha e been e-examined.67 The app oach o
he c i ical poin was desc ibed in e ms o an e ec i e c i i-
cal exponen and a Wegne expansion o accoun o he
co ec ion o scaling ou side he ange o he pu e powe law
beha io . I was sugges ed ha he mean- ield o classical
c i ical exponen s ne e accu a ely desc ibe he shape o a
apo -liquid coexis ence cu e. We shall ques ion his con-
clusion o he coexis ence da a o he squa e-well sys ems
wi h ;1 = 2 and 3 in Sec. IV. The s udy also showed ha he
depa u e o he e ec i e c i ical exponen s om hei classi-
cal alues o e a wide ange o educed empe a u es well
J. Chem. Phys., Vol. 96, No. 3, 1 Feb ua y 1992
below he c i ical poin is un ela ed o long- ange, nea -c i i-
cal luc ua ions. This obse a ion was con i med o he
Gibbs ensemble simula ion da a o he Lenna d-Jones sys-
em49 in which long- ange luc ua ions could no be ep e-
sen ed since he maximum sys em size in es iga ed was
N = 500, and i also ies in wi h he conclusions o Smi e
al.”
The o e all consequence o his, as a as simula ions in
he Gibbs ensemble a e conce ned, is ha al hough he co -
ela ion leng hs o luc ua ions a e limi ed by he ini e size
o he box, he e ec i e c i ical exponen has a alue close o
he uni e sal alue ob ained by he eno maliza ion g oup
(RG) heo y o in ini e co ela ion leng hs.
The e ec i e c i ical exponen was i s de ined and
used by Ve scha el 6* in 1896 as a sensi i e measu e o he
shape o a coexis ence cu e. Fo a apo o densi y pU in
coexis ence wi h a liquid o densi y p,, i is de ined as
Be =
JMp, -p )
alnI ’
whe e = 1 -
T/T,
and
T,
is he c i ical empe a u e. In
he limi o small alues o co esponding o empe a u es
jus below he c i ical poin ,
(p, -p ) =WIS9 (4)
whe e B, is he leading ampli ude e m. The uni e sal alue
de e mined om RG heo y o p = 0.325 implies a cubic
shape o he coexis ence cu e, whils he classical mean-
ield alue o p = 0.5 ep esen s a quad a ic coexis ence
cu e.
(3)
RG heo y also allows co ec ions o scaling o be calcu-
la ed ou side his asymp o ic c i ical egion. Wegne 69’70
showed ha away om he c i ical poin , Eq. (4) can be
w i en as an expansion o he o m
(pi-p,) =BoI la+B,I IP+A’+B*I Is+2A’+..., (5)
whe e A, is one o he so-called gap exponen s aking he RG
alue o A, = 0.5 o he apo -liquid sys ems o in e es ,
and Bj a e he co ec ion ampli udes o coe icien s. Equa-
ion (5) can be used wi h apo -liquid coexis ence da a o
es ima e he sys em’s c i ical empe a u e
T,
and c i ical ex-
ponen i. In o de o es ima e he c i ical densi y pc, an
equa ion o he “diame e s”
(pu
+
p,
)/2 o he coexis ence
cu e mus be used,”
The anomaly in he diame e o he coexis ence cu e cha -
ac e ized by C, is weak and di icul o obse e. By combining
Eqs. (5) and (6) an equa ion o he densi y on each b anch
can be ob ained,
p* =pc+C,I I~+C21 I+C31 I~+A1+~..
*~(BoI I~+B,I Is+A~+B21 IP+2A~+...).
(7)
He e,
p
_ and
p +
ep esen he apo and liquid phase den-
si ies, espec i ely.
O iginally, i was hoped ha mos o he e ms in he
abo e exp ession would be used o analyze he da a ob ained
in his simula ion s udy; howe e , since he esul s ob ained
Vega
&al.:
Phase equilib ia o he squa e-well luid 2299
we e o insu icien p ecision o a e y accu a e da a analy-
sis, some o he e ms in he exp ession we e neglec ed. Mo e
speci ically, we excluded he e ms C, and C, in he exp es-
sion o he diame e , and all he highe e ms Bi o i > 0 in
he Wegne expansion. In ou s udy, empe a u es mode -
a ely close o he c i ical poin we e in es iga ed co espond-
ing o 0 < I I < 0.2, and he gap exponen e ms which de-
sc ibe beha io a om he c i ical poin we e expec ed o
be small compa ed wi h he leading e ms. The e o s in he
Gibbs ensemble coexis ence densi ies we e oo la ge o de e -
mine whe he o no he ex ended scaling co ec ions had o
be included. Equa ion (7) hen simpli ies o
P* =pc +C2l l *PoI ~ (8)
in which we a e e ec i ely assuming ec ilinea diame e s
and using only he leading ampli ude e m (4) o he Wegne
expansion. This exp ession allows us o i he coexis ence
da a and ob ain es ima es o
pc, T,,
p, and he ampli ude
e ms C, and B, . The appa en c i ical exponen ob ained in
his way is expec ed o be e y simila o he e ec i e c i ical
exponen de ined by Eq. (3) and we shall use i o desc ibe
he o e all shape o he coexis ence cu es ob ained o he
squa e-well luids wi h di e en alues o ;1.
In o de o es ima e he c i ical p essu e
PC,
he apo -
p essu e cu e o he luid mus be de e mined. By i ing he
apo -p essu e da a ob ained om he simula ion o an
equa ion o he Clausius-Clapey on” o m,
he alue o
PC
co esponding o he alue o
T,
ob ained
om Eq. (8) is calcula ed.
The c i ical cons an s o he squa e-well sys ems we e
es ima ed in his way by i ing he Gibbs ensemble coexis-
ence da a o Eqs. (8) and (9) using a nonlinea leas -
squa es p ocedu e. 66 In he ollowing sec ion we epo e-
sul s o he c i ical poin s and exponen s ob ained o he
squa e-well luids wi h a ying il, and we e-examine he
exis ing da a o he Lenna d-Jones, ha d-co e wo-
Yukawa, and Gay-Beme po en ial models.
IV. RESULTS AND DISCUSSION
The phase equilib ia a e de e mined using he Gibbs en-
semble Mon e Ca lo echnique desc ibed in Sec. II o
squa e-well luids wi h po en ial anges cha ac e ized by
il = 1.25, 1.375, 1.5, 1.75, and 2. The esul ing apo -liquid
coexis ence cu es a e shown in Figs. l-6, he apo p es-
su e cu es a e shown in Fig. 7, and he ;1 dependence o he
c i ical empe a u e, p essu e, densi y, comp essibili y ac-
o , and exponen a e gi en in Table VI.
In he ollowing discussion, i is use ul o educe he
empe a u e and ene gy wi h espec o he squa e-well
dep heas
T*
= kT/eandE * =
E/E,
hep essu eisw i en
in e ms o E and he ha d-sphe e diame e c o he pa icles,
i.e.,
P
* =
P~/E,
he densi y is educed wi h espec o (T as
p* =pd,
and h e comp essibili y ac o is gi en by
Z=
P/(pkT)
= P*/(p*T*).Thesubsc ip conanyo he
a iables deno es he c i ical poin alues.
The apo -liquid coexis ence cu es o he squa e-well
J. Chem. Phys., Vol. 96, No. 3,1 Feb ua y 1992
2300 Vega
e a/.:
Phase equilib ia o he squa e-well luid
T”
0.72
0.70
0.66
0.66
0.64 I
0.0
FIG. 1. The empe a u e-densi y apo -liquid coexis ence cu e o a
squa e-well luid wi h a po en ial ange o /2 = 1.25. The densi ies o coexis -
ing apo p: and liquid p: phases (squa es) and diame e s (p: + p:)/Z
( iangles) a e ob ained om Gibbs ensemble Mon e Ca lo simula ions; he
e o ba s ep esen one s anda d de ia ion. The solid cu e and line o e -
ilinea diame e s a e ob ained by i ing Eq. (8) o he simula ion da a. The
es ima ed c i ical poin (ci cle) is also shown.
luid wi h ;1 = 1.25, 1.375, 1.5, and 1.75 a e shown in Figs.
l-4, espec i ely, as
T
-
p
p ojec ions o he
PVT
su aces.
The da a poin s ep esen he esul s o Gibbs ensemble
Mon e Ca lo simula ions, and he con inuous solid cu es
a e ob ained by a leas -squa es i o Eq. (8) o he simula-
ion da a. The esul s o he Gibbs ensemble simula ions o
he densi ies, ene gies, and p essu es o he coexis ing apo
and liquid phases a e summa ized in Tables I-IV, and he
alues o he c i ical cons an s a e gi en in Table VI. The
alues o p as lis ed in Table VI indica e coexis ence cu es
which a e close o cubic han quad a ic in shape as expec ed
om he RG heo y alue o p = 0.325. The appa en c i i-
cal exponen p = 0.34 0.02 ound o il = 1.75 is signi i-
can ly la ge han hose o he sys ems wi h a sho e po en-
ial ange, bu he o e all shape o he coexis ence cu e is
s ill nea -cubic.
Also shown in Fig. 3 as solid iangles is he apo -
liquid coexis ence simula ion da a ob ained by Chapela
e
a1.48
They used he molecula dynamics me hod o s udy he
T*
0.1 0.2 0.3 0.4 0.5 0.6 0.7 c
P*
FIG. 2. The empe a u e-densi y apo -liquid coexis ence cu e o a FIG. 4. The empe a u e-densi y apo -liquid coexis ence cu e o a
squa e-well luid wi h a po en ial ange o L = 1.375. See he cap ion o Fig. squa e-well luid wi h a po en ial ange o /2 = 1.75. See he cap ion o Fig. 1
1 o mo e de ails. o mo e de ails.
1.25-
1.20.
1.15.
T* 1.10-
1.05.
l.OO-
0.95 i-
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6
P*
FIG. 3. The empe a u e-densi y apo -liquid coexis ence cu e o a
squa e-well luid wi h a po en ial ange o A= 1.5. See he cap ion o Fig. 1
o mo e de ails. Also shown on his igu e as solid iangles a e he co e-
sponding da a ob ained om molecula dynamics simula ions (Re . 48).
phase equilib ia o he sys em wi h 2 = 1.5 by con ining he
luid in a box be ween pa allel ha d walls. I is clea ha he
p esence o he in e ace and he con ining walls shi s he
phase beha io away o he bulk coexis ence alues. Be-
cause o he la ge c i ical luc ua ions p esen in such simula-
ions, he me hod should no be used o p edic phase equili-
b ia o empe a u es close o he c i ical poin
(0.85 <
T/T,
< 1.0). The Gibbs ensemble Mon e Ca lo
echnique allows a close p oximi y o he c i ical poin o be
achie ed and p o ides an accu a e es ima e o bulk phase
equilib ia.
The coexis ence cu e ob ained o he sys em wi h
/z = 2 om simula ions in he Gibbs ensemble a e shown in
Fig. 5 (also see Tables V and VI). In con as o he esul s
o he sho e po en ial anges, he c i ical exponen
l= 0.53 0.11 ound o his sys em indica es a classical
c i ical beha io wi h a quad a ic shape o he coexis ence
cu e. Also shown in he igu e as a dashed cu e is he
heo e ical phase equilib ia de e mined by using he accu-
1.85
1.60
1.75
1.70
T*
1.65
-I
0.1 0.2 0.3 0.4 0.5 0.6
J. Chem. Phys., Vol. 96, No. 3,i Feb ua y 1992
Vega e a/.: Phase equilib ia o he squa e-well luid 2301
TABLE I. Vapo -liquid coexis ence da a om Gibbs ensemble Mon e Ca lo simula ions o N = 512 squa e-
well molecules wi hapo en ial angeo A = 1.25. Thedensi iesp*, ene gies E *, andp essu esP* o hecoexis -
ing apo and liquid phases a e labeled u and I, espec i ely. The e o s in p* and E * ep esen one s anda d
de ia ion o e blocks o 25 cycles, and he e o in P* is es ima ed om he e o s in he densi y and he
ex apola ed con ac alues o g( ).
T* P: P: ES W P p:
0.66 0.062 0.001 0.823 0.010 - 0.63 0.07 - 4.36 0.07 0.027 0.002 0.025 0.008
0.68 0.060 0.003 0.792 0.018 - 0.55 0.06 - 4.16 O.ll 0.033 0.002 0.033 0.013
0.70 0.085 0.007 0.761 0.014 - 0.74 0.11 - 3.97 0.07 0.037 0.005 0.045 0.016
0.72 0.103 0.006 0.708 0.014 - 0.83 0.12 - 3.69 0.07 0.049 0.006 0.054 0.015
0.73 0.127 0.011 0.684 kO.022 - 0.99 0.12 - 3.57 0.11 0.046 0.012 0.059 0.022
0.75 0.161 0.016 0.597kO.028 - 1.17kO.14 - 3.17kO.12 0.067*0.016 0.079 i-0.025
0.76 0.208 0.021 0.516 0.039 - 1.46 0.14 - 2.84 . 0.16 0.072 0.026 0.098 0.034
TABLE II. Vapo -liquid coexis ence da a om Gibbs ensemble Mon e Ca lo simula ions o N = 512 squa e-
well molecules wi h a po en ial ange o A = 1.375. See Table I o de ails.
T*
P: P: E: E: p: p:
0.860 0.092 * 0.004 0.722 0.009 - 1.13 0.11 - 4.67 0.07 0.019 0.005 0.022 * 0.010
0.880 0.072 & 0.004 0.687 0.008 - 0.78 0.06 - 4.43 0.05 0.027 0.004 0.029 . 0.015
0.900 0.104 0.008
0.681 0.016 - 1.08 0.15 - 4.38 0.10 0.042 0.010 0.038 0.018
0.920 0.109 0.004 0.641&-0.011 -l.lO*O.lO -4.ll 0.09 0.048&0.011 0.051+0.016
0.930 0.110 0.003 0.620 * 0.013 - 1.11 0.10 - 3.99 0.09 0.055 . 0.008 0.059 + 0.019
0.940 0.150~0.011 0.625 0.021 - 1.40 0.16 -4.00*0.13 0.062&0.016 0.068*0.024
0.945 0.142 & 0.012 0.608 0.021 - 1.34 0.17 - 3.90 0.13 0.064 0.016 0.078 0.023
0.950 0.151 *0.015 0.593 kO.032 - 1.46kO.17 -3.81 j10.18 0.083 kO.015 0.077 0.031
0.960 0.171 0.024 0.551 kO.054 - 1.54kO.19 - 3.58kO.28 0.092*0.021 0.085&0.034
0.970 0.202*0.048 0.476kO.137 - 1.76*0.33 - 3.17kO.77 0.096 0.033 0.101 kO.045
TABLE III. Vapo -liquid coexis ence da a om Gibbs ensemble Mon e Ca lo simula ions o N = 5 12 squa e-
well molecules wi h a po en ial ange o /z = 1.5. See Table I o de ails.
T* P: P: E: ET p: p:
1.00 0.038 0.001 0.659 0.006 - 0.55 0.02 - 5.18 0.05 0.031 0.001 0.031 0.009
1.05 0.053 0.002 0.632 0.007 - 0.72 0.04 - 4.96 * 0.06 0.042 0.002 0.044 0.012
1.08 0.051 0.004 0.599 0.015 - 0.66 0.08 - 4.63 0.11 0.043 0.003 0.052 0.017
1.10 0.058 * 0.003 0.579 0.017 - 0.68 0.08 - 4.42 0.12 0.047 0.003 0.056 & 0.021
1.12 0.078 0.004 0.567 0.018 - 0.95 0.08 - 4.45 0.13 0.069 0.005 0.063 0.024
1.15 0.092 0.009 0.535 0.022 - 1.12 0.11 - 4.23 0.14 0.074 0.009 0.078 0.028
1.18 0.133 . 0.010 0.503 0.022 - 1.52 0.12 - 4.01 0.14 0.090 + 0.014 0.087 0.035
1.20 0.147 & 0.017 0.448 0.040 - 1.62 0.17 - 3.66 0.24 0.111 0.020 0.096 0.041
TABLE IV. Vapo -liquid coexis ence da a om Gibbs ensemble Mon e Ca lo simula ions o N = 5 12 squa e-
well molecules wi h a po en ial ange o /z = 1.75. See Table I o de ails.
T* P: P: E: E: p: p:
1.55 0.048 0.004 0.537 0.013 - 0.86 + 0.11 - 6.14 + 0.13 0.056 . 0.004 0.058 0.017
1.57 0.053 0.005 0.529 0.014 - 0.93 0.12 - 6.05 + 0.15 0.051 o : 0.007 0.062 . 0.024
1.60 0.059 0.004 0.514 0.018 - 1.04 0.11 - 5.89 0.18 0.060 ~ 0.007 0.064 0.027
1.62 0.062~0.004 0.503 kO.018 - 1.06-j=O.l1 - 5.78kO.18 0.070~0.006 0.075 *0.031
1.65 0.071 * 0.005 0.483 0.019 - 1.18 0.12 - 5.57 0.18 0.071 0.010 0.086 0.033
1.68 0.104 k 0.005 0.489 * 0.008 - 1.73 0.09 - 5.62 0.09 0.095 0.016 0.105 0.028
1.70 0.106~0.007 0.474-&0.010 - 1.73kO.12 -5.46&0.10 0.112~0.018 0.111 kO.029
1.72 0.116 0.004 0.457 0.012 - 1.86 0.09 - 5.30 0.11 0.128 0.018 0.126 0.032
1.73 0.118~0.008 0.445*0.013 -1.87 0.12 -5.17&0.13 0.131*0.020 0.132*0.035
J. Chem. Phys., Vol. 96, No. 3, 1 Feb ua y 1992
2302 Vega
e a/:
Phase equilib ia o he squa e-well luid
2.7
2.6
T*
2.5
2.4
2.3
(
FIG. 5. The empe a u e-densi y apo -liquid coexis ence cu e o a
squa e-well luid wi h a po en ial ange o /2 = 2. See he cap ion o Fig. 1 o
mo e de ails. The dashed cu e is ob ained om VDW mean- ield heo y
(Re . 39).
a e ha d-sphe e equa ion o s a e o Ca nahan and S a -
ling’* oge he wi h a mean- ield a ac i e e m; his is es-
sen ially he an de Waals (VDW) equa ion o s a e o he
squa e-well luid o de1 Rio and co-wo ke s.3~39 The VDW
app oxima ion unde es ima es he c i ical empe a u e a
T:
= 2.64 and o e es ima es he c i ical densi y a
p: = 0.249. Al hough he mean- ield coexis ence cu e is in
poo ag eemen wi h he simula ion da a close o he c i ical
empe a u e i p o ides a easonable desc ip ion o he a-
po -liquid equilib ia o empe a u es a enough om he
c i ical poin (
T
* < 2.4). This is no unexpec ed o a squa e-
well luid wi h long- ange in e ac ions such as ;1 = 2 since in
he limi o in ini e ange he squa e-well po en ial is well
desc ibed by he VDW app oxima ion. The alidi y o his
app oxima ion o he coexis ence cu e o he sys em wi h
;1= 3 has also been no ed in a sepa a e s udy by Bena ides
e
a1.39
The disc epancy be ween he VDW heo y and simula-
ion is mo e ma ked o he sys ems wi h he smalle alues
o R. By using highe -o de pe u ba ion heo ies (see o
example Re s. 36-39), a be e ep esen a ion o he apo -
liquid coexis ence cu es o sho - ange squa e-well luids
can be achie ed.
In o de o e i y he adequacy o he p inciple o co e-
sponding s a es o he squa e-well luids wi h di e en al-
ues o & he coexis ence cu es o Figs. l-5 ha e been eplo -
ed in e ms o he educed a iables
T,
=
T/T,
and
p = p/p= in Fig. 6. I is clea ha he p inciple holds mode -
a ely well o sys ems wi h small alues o il; he coexis ence
cu es o II = 1.25, 1.375, 1.5, and 1.75 a e shown as he
solid cu es wi h p og essi ely la ge wid hs. I b eaks
down, howe e , o he longe ange po en ial wi h /z = 2
(do ed cu e) co esponding o a mo e poin ed cu e in he
p oximi y o he c i ical poin . This sugges s ha he p inci-
ple o co esponding s a es can desc ibe he sepa a e beha-
io s o sys ems wi h small and la ge A bu canno simul a-
neously desc ibe bo h. Also shown as he dashed cu e in
Fig. 6 is he apo -liquid equilib ia ob ained om mean-
ield heo y (c . Fig. 5). The o e all shapes o he coexis-
ence cu es change om he cubic cha ac e o he sys em
wi h ;1= 1.25 o he quad a ic cha ac e o he sys em wi h
A = 2.
The c i ical p essu es o he squa e-well luids can be
ob ained om he simula ion esul s by i ing Eq. (9) o he
apo -p essu e da a. The loga i hm o he apo p essu e is
plo ed as a unc ion o he ecip ocal empe a u e o he
sys ems wi h;1 = 1.25, 1.375, 1.5, 1.75, and 2 in Fig. 7. He e,
he p essu es o he apo (squa es) and liquid (solid
squa es) a e shown; he p essu es o he coexis ing phases
calcula ed om he Gibbs ensemble simula ions a e close
bu no exac ly equal o each o he . The s aigh lines shown
on he igu e ep esen a leas -squa es i o Eq. (9) o he
da a and om hese he c i ical p essu es can be es ima ed.
The esul s ob ained o he sys ems wi h di e en alues o
;I a e summa ized in Table VI. Fo he sys em wi h R = 2,
close p oximi y o he c i ical poin could no be simula ed
and he e o in he es ima ed alue o he c i ical p essu e is
expec ed o be qui e la ge. In o de o check he alue o
c i ical p essu e ob ained o his sys em, an addi ional NVT
Mon e Ca lo simula ion was pe o med o
T
= 2.764 and
p = 0.225 co esponding o he c i ical poin . A c i ical
p essu e o P, = 0.210 0.006 was ob ained by de e min-
ing he con ac alues o he adial dis ibu ion unc ion; his
esul is in good ag eemen wi h he alue o
P
= 0.197 & 0.026 ob ained om he Clausius-Clapey on
plo (Fig. 7).
The alues o he c i ical cons an s es ima ed om he
simula ion da a a e summa ized in Table VI, and a compa i-
TABLE V. Vapo -liquid coexis ence da a om Gibbs ensemble Mon e Ca lo simula ions o N = 512 squa e-
well molecules wi h a po en ial ange o 1 = 2. See Table I o de ails.
T* P: P: E: E: p: p
2.35 0.085 0.005 0.571 0.007 - 1.90 0.08 - 9.42 0.11 0.110 0.010 0.120 0.019
2.45 0.106 & 0.005 0.521 0.008 - 2.32 0.05 - 8.62 0.14 0.128 0.016 0.135 0.024
2.50 0.108 0.010 0.489 0.008 - 2.31 0.10 - 8.12 0.13 0.141 0.018 0.132 0.025
2.52 0.107 0.010 0.472 0.019 - 2.21 0.15 - 7.87 0.18
0.144 0.017 * 0.150 0.032 &
2.53 0.107 0.008 0.466 0.011 - 2.29 0.09 - 7.77 0.17
0.147 0.015 0.149 0.029
2.54 0.098 0.011 0.447 0.014 - 2.00 0.21 - 7.49 0.22 0.146 0.016 0.155 0.038
2.56 0.106 0.010 0.457 0.017 - 2.16 & 0.16 - 7.63 0.27
0.150 0.017 0.160 0.041
2.57 0.110~0.010 0.431 * o.014 -2.31 kO.11 -7.23*0.20 0.153*0.020 0.158~0.040
2.58 0.120 -&0.008 0.428 kO.011 -2.51 * o.09 - 7.19 0.17 0.156 0.025 0.162~0.&0
J. Chem. Phys., Vol. 96, No. 3,1 Feb ua y 1992
Vega
&al.:
Phase equilib ia o he squa e-well luid 2303
TABLE VI. The c i ical empe a u e Ty, p essu e P , densi y p:, comp essibili y ac o Z,, and exponen p es ima ed om he Gibbs ensemble Mon e
Ca lo (MC) da a o squa e-well luids wi h a iable po en ial anges R. The i ed alues o 4, C, , A, and Ba e also gi en. The e o s a e es ima ed om he
espec i e e o s in he densi ies o he coexis ing apou and liquid phases. Also shown in he able a e he co esponding alues ob ained om second-o de
pe u ba ion heo y (PT) (Re . 30).
R T: p:
PF ZC B BO G A B
1.250 MC 0.764 0.004 0.081 0.015 0.370 0.023
0.29 0.07
0.28 0.04 1.35 0.13 0.56 0.16 4.95 - 5.70
PT 0.913 0.133 0.34 0.43
1.375 MC 0.974 0.010 0.105 0.023 0.355 0.045
0.30 0.11
0.25 0.07 1.10 0.17 0.40 0.48 9.99 - 11.93
PT 1.11 0.148 0.34 0.39
I.500 MC 1.219*0.08 0.108 0.016 0.299*0.023 0.30 0.07 0.30 0.02 1.04 0.04 0.27 0.13 3.67 - 7.19
PT 1.35 0.153 0.31 0.36
1.750 MC 1.811 0.013 0.179 0.020 0.284 0.009 0.35 0.05 0.34 0.02 0.95 0.01 0.04 0.03 5.79 - 13.60
PT 2.04 0.196 0.25 0.38
2.ooO MC 2.764 0.023 0.197 0.026 0.225 0.018 0.32 0.07 j, 0.53 0.11 1.31 0.12 0.72 0.08 1.50 - 8.63
PT 2.88 0.255 0.24 0.37
son is made wi h he co esponding esul s ob ained om
second-o de he modynamic pe u ba ion heo y.30 The
pe u ba ion heo y p o ides only a easonable desc ip ion
o he simula ion alues wi h imp o ing ag eemen as /2 is
inc eased. Ca ley 33 has used adial dis ibu ion unc ions
calcula ed om in eg al equa ion heo y oge he wi h i s -
o de pe u ba ion heo y o sol e o phase coexis ence in
he sys em wi h R = 1.5, and has ob ained c i ical poin al-
ues o
T
= 1.35 and &’ = 0.30 0.023. The alue o he
c i ical densi y compa es a o ably wi h he Gibbs ensemble
es ima e o p = 0.299 0.023, bu he esul ob ained o
he c i ical empe a u e is conside ably highe han he sim-
ula ion alue o
TT =
1.219 0.008.
Pe haps he mos in e es ing esul is he change in he
shape o he coexis ence cu e as he ange is inc eased. The
sys ems wi hal = 1.25,1.375,1.5, and 1.75 ha eanea -cubic
shape wi h appa en c i ical exponen s close o he uni e sal
alue o /3 = 0.325. A small inc ease in he c i ical exponen
is, howe e , appa en . These esul s a e in good ag eemen
wi h he alue o p = 0.33 ob ained o he sys em wi h
R = 1.85 om nume ical s udies using he YBG in eg al
equa ion 17-*’ al hough a mo e accu a e analysis has sugges -
ed ha he equa ion does no exhibi a ue c i ical
poin .23-26 Fo he sys em wi h R = 2 he shape o he coexis-
ence cu e is nea ly quad a ic co esponding o he classical
mean- ield alue o p = 0.5. In o de o ensu e ha his e-
sul is no due he small sys em size o N = 5 12, a la ge
sys em o N= 1000 was also examined; he esul ing da a
we e e y simila o ha o he smalle sys em. In he case o
he squa e-well luid wi h il = 3 s udied by Bena ides e
a1.,39 he appa e n c i ical exponen es ima ed om he sim-
ula ion da a has inc eased op = 0.77 0.15. I hei da a is
analyzed using Eq. (8) he bes - i ed pa ame e s a e
T~=11.68~0.15,p~=0.181 0.010,~=0.77 0.15,
Be = 1.32 + 0.06, and C2 = 0.43 0.03. I mus be no ed,
howe e , ha since he empe a u es in es iga ed o his
sys em a e a om he c i ical alue (0.5 <
T/T, < 0.8))
i
is di icul o es ima e he c i ical cons an s and exponen s.
Exis ing Gibbs ensemble simula ion da a o he phase
equilib ia o he Lenna d-Jones,49 ’o ha d-co e wo-
l.oM)
0 075
0.950
0.925
0.900
0.075
0.650 ~
- _^
” “”
".ZS ".SO 0,;s l.bO l.iS 1.50 1,;s 2.bo 2.i5 2.;0
PI
FIG. 6. The apo -liquid coexis ence cu es o squa e-well luids wi h
a iable po en ial ange A plo ed in e ms o he educed empe a u e
T, = T/T, and densi y p, =p/pc. The solid cu es wi h p og essi ely
la ge wid hs ep esen he sys ems wi h ,l = 1.25, 1.375, 1.5, and 1.75; he
do ed cu e ep esen s he coexis ence cu e o he sys em wi h R = 2.
Also shown as a dashed cu e is he coexis ence cu e ob ained om VDW
mean- ield heo y (Re . 39).
-1.25 2 1.75
1.5 1.375 1.25
-1.75 1.25
-2.25
IIlP *
-2.75
-3.25
-3.75
..,,I_ . ;
0.25 0.50 0.75 1 .oo 1.25 1.50
-4.25
0.25 0.50 0.75 1 .oo 1.25 1.50 5
i/ -
FIG. 7. The apo -p essu e cu es o squa e-well luids wi h a iable po-
en ial ange /2. The cu es a e labeled wi h he app op ia e alues o ,%, and
he p essu es o he coexis ing apo (solid squa es) and liquid phases
(open squa es) a e shown oge he wi h he es ima ed c i ical poin s (ci -
cles) .
J. Chem. Phys., Vol. 96, No. 3,1 Feb ua y 1992
2304 Vega
e a/:
Phase equilib ia o he squa e-well luid
Yukawa,56 and he Gay-Be ne5* luids ha e been eana-
lyzed using Eq. (8) as desc ibed in Sec. III. Bes i s o he
simula ion da a we e ob ained wi h he ollowing alues o
he pa ame e s:
T: =
1.321 0.005, p: = 0.321 0.017,
l= 0.36 0.05, B0 = 1.11 0.05, and C, = 0.20 0.07
o he Lenna d-Jones luid;
T$ =
1.294 0.009,
/.I: = 0.342 0.021, /? = 0.33 0.08, B, = 1.07 0.07,
and C, = 0.12 & 0.04 o he ha d-co e wo-Yukawa luid;
and
T,* = 0.488 0.004,
p: = 0.101 0.009,
l= 0.32 0.03, &, = 0.43 0.03, and C, = 0.16 0.05
o he Gay-Be ne luid. The es ima ed alues o he appa -
en c i ical exponen indica e ha he shapes o he coexis-
ence cu es we e nea ly cubic o hese h ee sys ems. A
conside able amoun o ca e mus be aken in de e mining
alues o he c i ical poin s and c i ical exponen s; a small
change in he c i ical exponen can make qui e a signi ican
di e ence on he es ima ed alues o he c i ical cons an s.
Fo example, in ob aining he c i ical poin o wa e om
Gibbs ensemble simula ion da a, de Pablo e a1.6’ s a e ha a
classical c i ical exponen wi h a ixed alue o /3 = 0.5 was
used. Thei esul s would ha e been qui e di e en had hey
used he co ec uni e sal alue o p = 0.325.
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e al quali a i e shape o he wo-phase en elope, and ha he
shape o some o he coexis ence cu es may change qui e
d ama ically close o he c i ical poin . Since o squa e-well
sys ems wi h la ge alues o il i is inc easingly di icul o
simula e s a e poin s close o he c i ical poin , he e o s in
he es ima ed alues o he c i ical exponen s may be consid-
e able. This ha ing been said, he beha io o he squa e-well
luid is in s a k con as wi h ha obse ed o he Lcnna d-
Jones sys em.67 The e ec i e c i ical exponen o he Len-
na dJones luid is ai ly cons an and close o ha ob ained
by he RG me hod (/3 = 0.325) o e a wide ange o em-
pe a u es well below he c i ical poin . This does no seem o
be he case o he squa e-well luids wi h long- ange in e ac-
ions in which he c i ical exponen s would ha e o dec ease
qui e d ama ically o he uni e sal alue as he c i ical poin
is app oached, assuming, o cou se, ha hese sys ems exhib-
i uni e sal c i ical beha io .
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ACKNOWLEDGMENTS
We wish o hank Fe nando de1 Rio o use ul discus-
sions and Ian L. Pegg o sugges ing he o m o Eq. (7).
This esea ch was suppo ed in pa by g an s om he Ac-
ciones In eg adas p ogzamme o he B i ish Council and
om he Conseje ia de Educaci6n y Ciencia o he Jun a de
Andalucia. Gene ous alloca ions o compu e ime a he
Cen o In o ml ico Cien i ico de Andalucia (CICA) a e
also acknowledged.
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J. Chem. Phys., Vol. 96, No. 3, 1 Feb ua y 1992