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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Be o e we conclude, i is impo an o poin ou ha he
appa en c i ical exponen s es ima ed he e only gi e he gen-
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