Dipola o igin o he gas-liquid coexis ence o he ha d-co e 1:1 elec oly e model
J. M. Rome o-En ique,1,*L. F. Rull,1and A. Z. Panagio opoulos2
1Depa amen o de Fı
´sica A o
´mica, Molecula y Nuclea , A
´ eadeFı
´sica Teo
´ ica, Uni e sidad de Se illa,
Apa ado Co eos 1065, 41080 Se illa, Spain
2Depa men o Chemical Enginee ing, P ince on Uni e si y, P ince on, New Je sey 08544
共Recei ed 26 June 2002; published 9 Oc obe 2002兲
We p esen a sys ema ic s udy o he e ec o he ion pai ing on he gas-liquid phase ansi ion o ha d-co e
1:1 elec oly e models. We s udy a class o dipola dime models ha depend on a pa ame e Rc, he maximum
sepa a ion be ween he ions ha compose he dime . This pa ame e can a y om
␴
⫾ ha co esponds o he
igh ly e he ed dipola dime model o Rc→⬁ ha co esponds o he S illinge -Lo e desc ip ion o he ee
ion sys em. The coexis ence cu e and c i ical poin pa ame e s a e ob ained as a unc ion o Rcby g and-
canonical Mon e Ca lo echniques. Ou esul s show ha his dependence is smoo h bu nonmono onic and
con e ges asymp o ically owa ds he ee ion case o ela i ely small alues o Rc. This ac allows us o
desc ibe he gas-liquid ansi ion in he ee ion model as a ansi ion be ween wo dime ized luid phases. The
ole o he unpai ed ions can be conside ed as a pe u ba ion o his pic u e.
DOI: 10.1103/PhysRe E.66.041204 PACS numbe 共s兲: 64.70.Fx, 05.10.Ln, 05.70.Jk
I. INTRODUCTION
In ecen yea s he e has been an inc easing in e es in
phase ansi ions be ween luid phases o di e en elec oly e
concen a ions in ionic solu ions. Two di e en egimes ha e
been iden i ied expe imen ally 关1兴. The ‘‘sol ophobic’’ e-
gime occu s o la ge sol en dielec ic cons an s ha e ec-
i ely u n o Coulombic in e ac ions. Consequen ly, sol o-
phobic phase ansi ions a e p ima ily d i en by un a o able
in e ac ions be ween solu e and sol en . This beha io is
well desc ibed by he usual heo y o nonelec oly e solu-
ions wi h sho - ange in e ac ions, and clea ly leads o Ising-
like c i ical beha io . By con as , in he ‘‘Coulombic’’ e-
gime he sol en has a low dielec ic cons an and
elec os a ic in e ac ions be ween he solu e ions d i e he
phase sepa a ion. In his case, he phase diag ams a e qui e
asymme ic, and appa en ly mean- ield c i ical beha io has
been claimed 关2–4兴, al hough Ising-mean- ield c osso e has
also been seen wi hin a na owe ange o empe a u es
a ound he c i ical han in nonionic luids 关5–8兴. Some o
hese expe imen al s udies sugges ha he e exis s a new
cha ac e is ic leng h in hese sys ems ha compe es wi h he
co ela ion leng h o densi y luc ua ions 关7,8兴.
Elec oly e sys ems in he Coulomb egime a e o en
modeled as cha ged ha d sphe es embedded in a uni o m
dielec ic con inuum 共p imi i e models兲. Mos s udied is he
‘‘ es ic ed p imi i e model’’ 共RPM兲, in which he ions a e o
equal size. A apo -liquid phase ansi ion a e y low em-
pe a u es and densi ies was p edic ed heo e ically 25 yea s
ago 关9兴and by ea ly compu e simula ion s udies 关10兴. Im-
p o emen o compu e simula ion echniques has allowed an
inc easingly p ecise de e mina ion o he coexis ence pa am-
e e s 关11–16兴. The e ha e also been ecen s udies o p imi-
i e models wi h asymme y in size 关17–19兴and cha ge
关19,20兴. Ve y ecen esul s sugges ha he c i ical beha io
o he RPM belongs o he Ising uni e sali y class 关16兴.
F om a heo e ical poin o iew, di e en app oaches
ha e been used in o de o explain he apo -liquid ansi ion
o he p imi i e models. In eg al equa ions such as he mean
sphe ical app oxima ion 关21,22兴, as well as he Debye-
Hu
¨ckel heo y 关21兴and Poisson-Bol zmann app oaches 关23兴
ha e succes ully been applied o i . Fo he RPM, he mos
succes ul heo ies a e he pai ing heo ies ha conside he
ionic luid as a mix u e o bound pai s and ee ions in
chemical equilib ium 关21,24兴, in he spi i o he Bje um’s
ideas 关25兴. Howe e , in all hese heo ies he ansi ion is
d i en by he ee ions, e en when in same cases, as in he
Debye-Hu
¨ckel-Bje um app oach, he associa ed pai s a e
he dominan species. Analy ical 关26兴and compu e simula-
ion 关27兴 esul s, on he o he hand, show ha he s uc u e o
he apo phase is domina ed by neu al clus e s, mos ly
dime s and e ame s. Compu e simula ions ha e also dem-
ons a ed ha he phase en elope o he RPM esembles ha
o he cha ged ha d dumbbell model 关28,17兴. This esul has
been con i med by heo e ical s udies 关29,30兴. The ques ion
ha he p esen wo k examines in de ail is he in luence o
ionic associa ion on he apo -liquid ansi ion o p imi i e
model elec oly es. In con as o an ea lie s udy 关17兴, which
only conside ed igh ly bound dime s, he e we examine a
ange o models wi h a ying alues o Rc, he maximum
sepa a ion be ween ions in a dime . Rc→⬁co esponds o
he S illinge -Lo e desc ip ion 关31兴o he ee ion sys em.
The s uc u e o his pape is as ollows. In Sec. II, we
examine he mic oscopic s uc u e o he coexis ing phases
o he ionic luid and compa e he co ela ion unc ions o
hose o a igh ly e he ed dime model. In Sec. III, an exac
chemical ep esen a ion o he ionic luid as a mix u e o
associa ed pai s and ee ions is in oduced. The ole o he
associa ed pai s on he phase coexis ence is s udied in de ail
in Sec. IV and he pape closes wi h discussion and conclu-
sions.
*Co esponding au ho . Elec onic add ess: [email p o ec ed]
PHYSICAL REVIEW E 66, 041204 共2002兲
1063-651X/2002/66共4兲/041204共10兲/$20.00 ©2002 The Ame ican Physical Socie y66 041204-1
II. MICROSCOPIC STRUCTURE OF THE IONIC AND
TETHERED DIMER SYSTEMS AT COEXISTENCE
In his sec ion we analyze he ole o pai ing on gas-liquid
coexis ence o p imi i e model ionic luids. We ha e s udied
by compu e simula ion an 1:1 size-asymme ic p imi i e
model, in which he ions a e modeled as ha d sphe es o
diame e s
␴
⫹and
␴
⫺, ca ying cha ges ⫹qand ⫺q, e-
spec i ely, embedded on a dielec ic con inuum o dielec ic
cons an D(D⫽1 o he acuum兲. The in e ac ion po en ial
be ween wo ions sepa a ed by a dis ance ij is gi en by
uij共 ij兲⫽uhs共 ij兲⫹qiqj
D ij,共1兲
whe e uhs( ij) is he ha d-co e po en ial ha akes he alue
⫹⬁i ij⬍1
2(
␴
i⫹
␴
j) and 0 o he wise.
The size asymme y is cha ac e ized by he pa ame e ␭,
de ined as
␭⫽
␴
⫺
␴
⫹
.共2兲
Mon e Ca lo simula ions in he neu al g and-canonical
ensemble we e pe o med, cha ac e ized by a empe a u e T
and he con igu a ional chemical po en ial o a pai o un-
like ions
␮
⫽2
␮
eal⫺3kBTln(⌳⫹⌳⫺), wi h
␮
eal⬅(
␮
⫹
⫹
␮
⫺)/2 and ⌳⫾being he he mal de B oglie wa eleng hs
o he ionic species,
⌳⫾⫽
冑
h2
2
␲
m⫾kBT.共3兲
As usual, cubic boxes o leng h Lunde pe iodic bounda y
condi ions a e used. The long- anged cha ac e o he Cou-
lombic in e ac ions is handled by he use he Ewald summa-
ion echnique wi h conduc ing bounda y condi ions, wi h
518 Fou ie -space wa e ec o s and eal-space damping pa-
ame e
␬
⫽5. The ela i e e o due o he in ini e sums
unca ion in he elec os a ic ene gy is less han 10⫺5 o
andom con igu a ions in small sys ems 关32兴, and his choice
has been also alida ed by di ec simula ions o he RPM
关15兴. In o de o speed up he simula ions, he basic s eps o
he simula ions 共inse ion and dele ions o pai s o unlike
ions兲a e biased ollowing Re . 关11兴. Mo eo e , a ine-
disc e iza ion app oxima ion is used: he posi ions a ailable
o each ion a e he si es o a simple cubic la ice o spacing a.
This me hodology has succes ully been applied o he RPM
关14兴, o 1:1 size-asymme ic p imi i e models 关17兴, and o
z:1 size-asymme ic p imi i e models 关14兴, and allows a
speedup ela i e o he con inuum calcula ions o a ac o o
100 o small sys ems. The esul s a e almos indis inguish-
able om he con inuum ones o a disc e iza ion pa ame e
␨
⬅
␴
⫾/a⫽10 关14,17,19兴, whe e
␴
⫾⫽1
2(
␴
⫹⫹
␴
⫺) is he
unlike-ion collision diame e . This alue o he disc e iza ion
pa ame e (
␨
⫽10) was used in he p esen s udy. His og am
eweigh ing echniques 关33兴and mixed- ield ini e-size scal-
ing me hods 关34兴we e used o ob ain he apo -liquid en e-
lopes and he e ec i e c i ical poin s, espec i ely. Fo he
sake o compa ison, we ha e also s udied igh ly e he ed
dipola dime sys ems 关17,28兴, consis ing o Npai s o a
posi i e and a nega i e ion es ic ed o emain a sepa a ions
␴
dsa is ying
␴
⫾⭐
␴
d⭐1.02
␴
⫾. Simula ion de ails and
some p elimina y esul s o a ange o alues o ␭a e p e-
sen ed in Re . 关17兴.
The unlike-ion collision diame e
␴
⫾p o ides he basic
leng h scale app op ia e o de ining bo h he educed em-
pe a u e and educed densi y ia
T*⫽kBTD
␴
⫾/q2and
␳
*⫽2N
␴
⫾
3,/L3共4兲
whe e N⫹⫽N⫺⬅Nis he pa icle numbe o each ionic spe-
cies 关11,17,22兴. The educed simula ion box leng h is de ined
simila ly ia L*⫽L/
␴
⫾, and he educed ene gies and
chemical po en ial as U*⫽UD
␴
⫾/q2and
␮
*
⫽
␮
D
␴
⫾/q2.
The alue o he asymme y pa ame e conside ed in his
pape is ␭⫽3. Fo ha case, he gas-liquid coexis ence o
he ionic luid shows a shi in bo h empe a u e and densi y
wi h espec o he e he ed dime luid 共see Fig. 1兲, which is
a gene al ea u e when compa ing ionic and e he ed dime
sys ems 关17兴. On he o he hand, he asymme y is no high
so as o a o la ge chainlike neu al clus e s, as occu s o
bigge alues o ␭关17,18兴. These ea u es quali y his case o
be a ypical example o mode a ely asymme ic 1:1 elec o-
ly es, including he RPM.
Figu e 1 makes he simila i y be ween he phase diag am
o he ionic and he e he ed dime luids clea . I also sug-
ges s in an indi ec way ha he pai ing plays a decisi e ole
on he gas-liquid coexis ence in he ionic luid. In o de o
cla i y such a ole, we ha e s udied he ion-ion adial dis i-
bu ion unc ions gij( ) co esponding o he ionic sys ems,
and he co esponding ones o he e he ed dime sys ems.
Fo his pu pose, we ha e conside ed he gas and liquid
s a es a coexis ence o a empe a u e T⬇0.95Tc, wi h Tc
being he co esponding c i ical empe a u e. The ion-ion a-
dial dis ibu ion unc ions o he ionic sys ems, o T*
⫽0.0425 and he coexis ing densi ies
␳
*⫽0.0076(15) and
FIG. 1. Gas-liquid phase diag am o he ionic sys em 共dia-
monds兲and he e he ed dime luid 共ci cles兲wi h a e he leng h
equal o 1.02
␴
⫾. The size asymme y pa ame e is se ␭⫽3. The
c i ical pa ame e s ha e been ob ained om he ex apola ion o
L*→⬁o he esul s o L*⫽12,15, and 18 while he subc i ical
coexis ence cu es we e ob ained using L*⫽15.
ROMERO-ENRIQUE, RULL, AND PANAGIOTOPOULOS PHYSICAL REVIEW E 66, 041204 共2002兲
041204-2
␳
*⫽0.162(3) a e plo ed in Figs. 2 and 3, espec i ely. In
bo h cases, he unlike-ion adial dis ibu ion unc ion be-
comes e y la ge close o con ac , indica ing he associa ion
in bound pai s o unlike ions. Mo eo e , he like-ion adial
dis ibu ion unc ions show maxima a ound *⬅
␴
⫾⫽1.5
共in he case o g⫺⫺ , his peak coincides wi h he con ac
alue兲, and he e is a seconda y maximum in g⫹⫺ o *
⬇2.5. These obse a ions allow us o conclude ha he e is
high co ela ion be ween pai s o associa ed unlike ions. We
ecall ha he ange o densi ies in which he gas-liquid co-
exis ence occu s p e en s packing e ec s, so he s uc u e is
comple ely gi en by he Coulombic in e ac ions.
The ion-ion adial dis ibu ion unc ions o he e he ed
dime luid a e plo ed in Fig. 4 o he gas b anch 关T*
⫽0.0405,
␳
*⫽0.0119(15)] and in Fig. 5 o he liquid
b anch 关T*⫽0.0405,
␳
*⫽0.206(2)]. The compa ison be-
ween he ionic luid and e he ed dime luid mic oscopic
s uc u es con i ms he quali a i e simila i y be ween bo h
sys ems. A u he es o his simila i y is ound in he com-
pa ison o he neu al clus e popula ions in he gas b anch.
We use Gillan’s de ini ion o a clus e 关26兴. Two ions iand j
a e di ec ly bound when he dis ance be ween hem is less
han Rij
c. This condi ion de ines ma hema ically an equi a-
lence ela ionship, and he equi alences classes in which he
ions g oup a e he clus e s. As he in e ac ion be ween like
ions is epulsi e, he dependence o he clus e de ini ion on
R⫹⫹
co R⫺⫺
cshould be e y weak 共i hey a e aken smalle
han he mean dis ance be ween wo like ions兲. On he o he
hand, he clus e de ini ion is no e y sensi i e o he alue
o R⫹⫺
ci ha alue lies be ween he i s minimum o g⫹⫺
and he mean dis ance be ween agg ega es. In his wo k we
ha e used R⫹⫹
c⫽R⫺⫺
c⫽R⫹⫺
c⫽1.5
␴
⫾. As expec ed, he mi-
c oscopic s uc u e o he gas phase is domina ed by he
N-ion neu al clus e s 关17,27兴. Thei ac ions N o he ionic
and e he ed ion sys ems a he gas phase in coexis ence o
T⫽0.95Tca e qui e simila 共see Fig. 6兲, al hough he e h-
e ed dime sys ems ha e sligh ly highe ac ions o neu al
clus e s han do ionic sys ems, specially o la ge N, in ag ee-
men wi h he esul s p e iously epo ed 关17兴.
Despi e he simila i ies ound be ween he ionic luid and
he e he ed dime luid, he e a e some di e ences ha can
a ionalize he quan i a i e di e ences be ween hei phase
diag ams. The unlike-ion adial dis ibu ion unc ions o he
e he ed dime luid di e quali a i ely om he ionic coun-
e pa s close o he con ac alue, since in he o me he e is
a jump a he maximum allowed sepa a ion o wo ions o a
FIG. 2. Ion-ion adial dis ibu ion unc ions co esponding o he
ionic sys em a T*⫽0.0425 and
␳
*⫽0.0076(15) 共 he gas phase兲.
The inse shows he beha io o g⫹⫺( *) a dis ances close o he
con ac alue.
FIG. 3. Same as in Fig. 2 in he liquid b anch: T*⫽0.0425 and
␳
*⫽0.162(3).
FIG. 4. Ion-ion adial dis ibu ion unc ions co esponding o he
e he ed dime sys em a T*⫽0.0405 and
␳
*⫽0.0119(15) 共 he gas
phase兲. The inse shows he beha io o g⫹⫺( *) a dis ances close
o he con ac alue.
FIG. 5. Same as in Fig. 4 in he liquid b anch: T*⫽0.0405 and
␳
*⫽0.206(2).
DIPOLAR ORIGIN OF THE GAS-LIQUID . . . PHYSICAL REVIEW E 66, 041204 共2002兲
041204-3
dime , while he ionic g⫹⫺ akes smoo hly highe alues on
ha ange o alues o *. Consequen ly, he condi ion on
he con inemen o he ions ha compose a e he ed dime
mus be elaxed in o de o e ine he pai ing concep in he
ionic luids. In he ollowing sec ion we will in oduce an
app op ia e amewo k o such a goal.
III. CHEMICAL PICTURE OF IONIC FLUIDS
We conside an ionic luid (q⫹⫽⫺q⫺⬅q) con ained in a
olume Vand in equilib ium wi h a ese oi a a empe a-
u e Tand chemical po en ials
␮
⫹and
␮
⫺. Fo simplici y,
he neu al g and-canonical ensemble, in which only neu al
con igu a ions a e allowed, will be conside ed. This en-
semble has been shown o be equi alen o he usual g and-
canonical ensemble in he he modynamic limi 关35兴. The
neu al g and-canonical pa i ion unc ion can be w i en as
⌶⫽兺
Ni⫽0
⬁
冉
e
␤␮
⌳⫹
3
冊
Ni
冉
e
␤␮
⌳⫺
3
冊
Ni
ZU共Ni,Ni兲,共5兲
whe e Niis he numbe o ions o each species,
␤
⫽1/(kBT),
␮
⬅(
␮
⫹⫹
␮
⫺)/2, ⌳⫹and ⌳⫺a e he he mal de
B oglie wa eleng hs co esponding o each species, and
ZU(Ni,Ni) is he canonical con igu a ional pa i ion unc ion
co esponding o a ixed numbe o ions a he empe a u e T
and enclosed in he olume V,
ZU共Ni,Ni兲⫽1
共Ni!兲2
冕
V2Nid 1
⫹d 1
⫺...d Ni
⫹d Ni
⫺e⫺
␤
U共6兲
wi h U⬅Uphys⫽Uhc⫹兺i⬍jqiqj/(D ij) he o al po en ial
ene gy and Uhc he ha d-co e con ibu ion, equal o ⫹⬁i
wo pa icles o e lap, and 0 o he wise 共o he empe ed po-
en ials can be used, bu he main esul s o his sec ion will
emain unchanged兲. I he luid pa icles a e s ongly associ-
a ed in o bound (⫹,⫺) pai s, as occu s in he low-
empe a u e and low-densi y egion in which he apo -
liquid ansi ion occu s o he ionic luids, he ‘‘physical’’
ep esen a ion desc ibed abo e will be inadequa e, and i can
be eplaced by a ‘‘chemical’’ pic u e, in which he luid is
composed o associa ed pai s and ee ions. Fi s , a ule ha
unequi ocally iden i ies bound pai s o each con igu a ion
o he ionic luid 共up o a se o null measu e in he con igu-
a ional space兲is needed. Once such a ule is de ined, we can
w i e he canonical con igu a ional pa i ion unc ion as
ZU共Ni,Ni兲⫽兺
Np⫽0
Ni
ZU
*共N ,N ,Np兲,共7兲
whe e Npis he numbe o bound pai s, N ⫽Ni⫺Npis he
numbe o ee ions o each species, and ZU
*(N ,N ,Np)is
he canonical con igu a ional pa i ion unc ion co espond-
ing o a sys em o Npassocia ed pai s and N ee ions o
each species,
ZU
*共N ,N ,Np兲⫽1
Np!
1
共N !兲2
冕
V2(Np⫹N )d共1兲...
⫻d共Np兲d 1
⫹d 1
⫺...d N
⫹d N
⫺e⫺
␤
U*,
共8兲
whe e (i)⬅
兵
⫹, ⫺
其
ia e he posi ions o he ions ha com-
pose he associa ed pai i, and i
⫹(⫺)co esponds o he co-
o dina es o a ⫹(⫺) ee ion i, espec i ely. The po en ial
ene gy U*⬅Uchem does no coincide, in gene al, wi h he
physical po en ial ene gy Uphys, since di e en con igu a-
ions o he ‘‘chemical’’ mix u e can be compa ible wi h a
gi en ionic con igu a ion.
Subs i u ing Eq. 共7兲in o Eq. 共5兲and ea anging he e-
sul ing exp ession, he neu al g and-canonical pa i ion
unc ion can be w i en as
⌶⫽兺
N ⫽0
⬁
兺
Np⫽0
⬁
冉
e
␤␮
⌳⫹
3
冊
N
冉
e
␤␮
⌳⫺
3
冊
N
冉
e
␤␮
p
⌳p
6
冊
Np
ZU
*共N ,N ,Np兲
共9兲
wi h
␮
p⬅2
␮
and ⌳p⫽
冑
⌳⫹⌳⫺. These las de ini ions co -
espond o he chemical equilib ium condi ions be ween he
ee ions and he associa ed pai s 关21,36兴. We ema k ha
his de i a ion is independen o he c i e ion chosen o de-
ine he pai s. Howe e , he choice mus be such ha
ma ches he mic oscopic s uc u e o he luid. As we ha e
seen in he p eceding Sec ion, he apo s uc u e is domi-
na ed by neu al agg ega es, mos ly dime s and e ame s.
This ac is consis en wi h he Bje um ideas o pai ing 关25兴.
Consequen ly, a dis ance-based c i e ion seems o be he
mos app op ia e: wo unlike ions ha a e close han Rc(Rc
being a sui able cu o dis ance兲a e conside ed as an associ-
a ed pai . I is easy o see ha such a c i e ion does no de ine
uniquely he associa ed pai s when he popula ion o e am-
e s and highe -o de neu al clus e s is no negligible, as i
can be seen om Fig. 7, since he e a e di e en a ange-
men s o associa ed pai s ha a e compa ible wi h he same
ionic con igu a ion. Hence, a mo e sys ema ic c i e ion is
needed in o de o be able o de ine associa ed pai s o al-
mos e e y ionic con igu a ion, while keeping he in ui i e
FIG. 6. F ac ion No N-ion neu al clus e s. The squa es co -
espond o he ionic luid a T*⫽0.0425 and
␳
*⫽0.0076(15), and
he iangles o he e he ed dime luid a T*⫽0.0405 and
␳
*
⫽0.0119(15).
ROMERO-ENRIQUE, RULL, AND PANAGIOTOPOULOS PHYSICAL REVIEW E 66, 041204 共2002兲
041204-4
de ini ion o an associa ed pai as encompassing unlike ions
ha a e a he closes dis ances. We use a sui able modi ica-
ion o he S illinge -Lo e pai de ini ion on a gi en ionic
con igu a ion 关31兴. In his p esc ip ion, all he dis ances be-
ween wo unlike ions a e compu ed, and hen he i s pai is
de ined as he wo unlike ions a he closes dis ance. This
s ep is epea ed, aking in o accoun only ions ha emain
unpai ed om p e ious s eps o he e alua ion o (⫹,⫺)
dis ances. We s op when he minimum dis ance be ween wo
unlike ions is g ea e han Rc, conside ing he emaining
ions as ee ions 共no ee ions we e conside ed in Re . 关31兴,
and consequen ly Rc⬅⬁ in ha case兲. This p o ocol p o ides
a unique con igu a ion o associa ed pai s and ee ions o
almos each ionic con igu a ion, since he me hod is ambigu-
ous only in a subse o he ionic con igu a ions in which a
leas wo (⫹,⫺) dis ances a e equal, and he measu e o
such a se is null in he con igu a ional space.
Ob iously, di e en pai ing p esc ip ions can be used.
Howe e , his p o ocol p o ides an explici exp ession o
he chemical po en ial ene gy Uchem,
Uchem⫽Uphys⫹兺
i⫽1
Np
p共
兵
⫹, ⫺
其
i兲
⫹兺
1⭐i⬍j⭐Np
pp共
兵
⫹, ⫺
其
i,
兵
⫹, ⫺
其
j兲
⫹兺
i⫽1
Np兺
j⫽1
N
p⫹共
兵
⫹, ⫺
其
i, j
⫹兲
⫹兺
i⫽1
Np兺
j⫽1
N
p⫺共
兵
⫹, ⫺
其
i, j
⫺兲
⫹兺
i⫽1
N 兺
j⫽1
N
⫹⫺共 i
⫹, j
⫺兲,共10兲
whe e Uphys is he physical po en ial ene gy due o he ha d-
co e and ele os a ic in e ac ions, and he o he e ms co e-
spond o e ec i e in e ac ions o en opic o igin needed o
educe he con igu a ional space o he chemical sys em. The
e m p(
兵
⫹, ⫺
其
i) depends only on he posi ions o he ions
ha compose an associa ed pai and con ines hem o be a a
dis ance sho e han Rc,
p共
兵
⫹, ⫺
其
i兲⫽
再
0,
兩
i
⫹⫺ i
⫺
兩
⭐Rc
⫹⬁
兩
i
⫹⫺ i
⫺
兩
⬎Rc
共11兲
The pai wise po en ial ene gy pp(
兵
⫹, ⫺
其
i,
兵
⫹, ⫺
其
j) co -
esponds o a s e ic hind ance condi ion be ween wo associ-
a ed pai s 关31兴, since he dis ances be ween wo unlike ions
co esponding o di e en pai s canno be sho e han he
minimum dis ance be ween he ions ha compose each pai ,
pp共
兵
⫹, ⫺
其
i,
兵
⫹, ⫺
其
j兲⫽
再
⫹⬁
兩
i
⫿⫺ j
⫾
兩
⭐dij
0 o he wise 共12兲
wi h dij⬅min(
兩
i
⫹⫺ i
⫺
兩
,
兩
j
⫹⫺ j
⫺
兩
).
The in e ac ion be ween he associa ed pai s and he ee
ions is modi ied by he e m p⫾(
兵
⫹, ⫺
其
i, j
⫾), which p e-
en s he ee ion om being close o he unlike ion o he
associa ed pai han i s pa ne ,
p⫾共
兵
⫹, ⫺
其
i, j
⫾兲⫽
再
⫹⬁,
兩
i
⫿⫺ j
⫾
兩
⭐
兩
i
⫹⫺ i
⫺
兩
0, o he wise
共13兲
Finally, wo unlike ee ions canno be a a sho e dis-
ance han Rcdue o he e m ⫹⫺( i
⫹, j
⫺),
⫹⫺共 i
⫹, j
⫺兲⫽
再
⫹⬁,
兩
i
⫹⫺ j
⫺
兩
⭐Rc
0,
兩
i
⫹⫺ j
⫺
兩
⬎Rc
共14兲
I is no ha d o see ha a mix u e o associa ed pai s and
ee ions wi h a po en ial ene gy gi en by Eq. 共10兲is com-
ple ely equi alen o he o iginal ionic sys em. I is in e es -
ing o no e ha , excep he e m on p ha ies he ions ha
compose an associa ed pai , he o he e ms a e pai wise,
sho - anged modi ica ions o he ha d-co e condi ions, so
he po en ial ene gy o an allowed mix u e con igu a ion is
he same as in he co esponding ionic con igu a ion. Mo e-
o e , e e y con igu a ion o associa ed pai s and ee ions
ob ained om an ionic con igu a ion by he pai ing p oce-
du e desc ibed abo e ul il he cons ains induced by he
added po en ials. Con e sely, he mix u e con igu a ion is he
same as he one ob ained om he co esponding ionic con-
igu a ion by he S illinge -Lo e p o ocol.
This analysis p o ides an exac chemical ep esen a ion o
he ionic sys em as a mix u e o bound pai s and ee ions in
chemical equilib ium. The equi alence be ween ep esen a-
ions o he sys em is no only a he le el o he modynamic
p ope ies, bu also in he mic oscopic s uc u e, unlike p e-
ious s udies based on he ma ching o he chemical and he
physical ee ene gies, e.g., ia he i ial coe icien s 关37,38兴.
In hose s udies he e ec i e po en ials gi en by Eqs. 共11兲
and 共14兲could be guessed, bu he o he e ms 共 ha a ise
om ma ching hi d- and highe -o de i ial coe icien s兲
FIG. 7. Di e en ways o pai ing ions using he c i e ion ha
wo unlike ions a e pai ed i he dis ance be ween hem is sho e
han he cu o Rc. The ca ion 4 can be pai ed wi h ei he he anion
1 o he anion 2. In he i s case he ions 2 and 3 emain ee, and
in he la e case he ions 1 and 3 cons i u e ano he associa ed pai .
DIPOLAR ORIGIN OF THE GAS-LIQUID . . . PHYSICAL REVIEW E 66, 041204 共2002兲
041204-5

we e no clea ly iden i ied. Fu he mo e, hey ha e no been
used a all in pai ing heo ies o elec oly es. We mus s ess
he impo ance o he e ec i e new e m gi en by Eq. 共12兲in
he po en ial ene gy, which educes he con igu a ion space
a ailable o he associa ed pai s. This e ec is specially im-
po an o high alues o Rc. An ex eme case ha illus-
a es he e ec o missing he s e ic hind ance e m is he
loosely e he ed dime luid, in which he ions ha compose
he pai can be a any ela i e dis ance g ea e han
␴
⫾.In
his case, he canonical pa i ion unc ion o he dime luid is
Np!ZU(Np,Np), whe e Npis he numbe o dime s,
ZU(Np,Np) is gi en by Eq. 共6兲and he Np! ac o is he
numbe o di e en ways o pai ing he unlike ions in each
ionic con igu a ion and lead o di e en dime con igu a-
ions. As a consequence, in he he modynamic limi he
Helmhol z ee ene gy pe pa icle do he loosely e he ed
dime luid di e ges as
d
kBT⬃⫺lnNp⫹
冉
2 i
kBT⫹1
冊
,共15兲
whe e iis he ionic Helmhol z ee ene gy pe ion, which is
well beha ed in he case o neu al sys ems 关35兴.
The associa ion deg ee o he ionic luid in he coexis -
ence egion has been s udied by compu e simula ion in he
amewo k o he p e ious analysis. In he usual g and-
canonical simula ions, we ha e iden i ied he associa ed pai s
by he S illinge -Lo e ule wi h Rc⫽
冑
3L/2, i.e., all he
ions a e associa ed, and he minimum image con en ion is
used in o de o calcula e he dis ances be ween unlike ions.
The p obabili y dis ibu ions p( ) o a pai ha ing an in e -
nal sepa a ion dis ance be ween ions is plo ed in Fig. 8 o
he gas and liquid phases a T*⫽0.0443. Su p isingly, bo h
dis ibu ions a e p ac ically iden ical, despi e he ac ha he
co esponding densi ies a e qui e di e en . The dis ibu ions
show a e y p onounced peak o ⫽
␴
⫾, and a local maxi-
mum a ound ⫽2.5
␴
⫾, app oxima ely whe e g⫹⫺ p esen s
he second local maximum. Fo la ge alues o , he dis i-
bu ions p( ) decay o ze o. These esul s con i m he s ong
associa ion in he ionic luid a low empe a u es, and ha he
s uc u e is weakly a ec ed by a ia ions in densi y, a leas
in he ange in which he gas-liquid coexis ence occu s. Fo
alues o close o he con ac alue, he dis ibu ion is well
desc ibed by he nonin e ac ing pai luid p obabili y dis i-
bu ion, gi en by he ollowing exp ession:
pideal共 兲⬇
2exp
冉
q2
DkBT
冊
冕
␴
⫾
Rcy2exp
冉
q2
DkBTy
冊
dy
⬃q2 2
DkBT
␴
⫾
4exp
冉
q2
DkBT
␴
⫾
冋
␴
⫾
⫺1
册
冊
共16兲
alid o T*Ⰶ1 and an in e nal cu o RcⰇ
␴
⫾. Howe e ,
o ⲏ1.5
␴
⫾,p( ) de ia es om he ideal exp ession as a
consequence o he in e ac ion be ween associa ed pai s. The
local maximum showed by he dis ibu ion unc ion indica es
ha he mos bound pai s a e sol a ed by less bound pai s, o
o m s able neu al e ame s 共and highe o de clus e s, bu
hei inclusion does no seem o a ec he conclusions o his
sec ion兲. This ac is in ag eemen wi h he ea u es obse ed
in he pai co ela ion unc ions in he p eceding sec ion. We
mus s ess ha he s uc u e ha p( ) p esen s is only due o
he Coulombic in e ac ions.
I is ins uc i e o compa e ou esul s in he coexis ence
egion o he highe empe a u e ones. We ha e compu ed
p( )a T*⫽1 and he same ange o densi ies 共see Fig. 9兲.
The quali a i e beha io o p( ) a high empe a u es is
simila o ha p edic ed by S illinge and Lo e 关39兴. Fi s ,
i shows a maximum localized in ⬃
␳
⫺1/3. On he o he
hand, p( ) akes signi ican alues in a wide ange han he
co esponding unc ions a lowe empe a u es, decaying o
FIG. 8. P obabili y dis ibu ion p( *) o ha ing a pai an in e -
nal sepa a ion dis ance *be ween ions a T*⫽0.0443 and educed
densi ies
␳
*⫽0.0064(7) 共solid line兲, co esponding o he gas
phase, and
␳
*⫽0.149(3) 共dashed line兲, co esponding o he liquid
phase. The do -dashed line co esponds o he ideal dime luid
app oach 共see ex 兲. The inse shows he cumulan dis ibu ion
㜷( *)⬅
兰
1
*p( )d . The meaning o he symbols is he same as
be o e.
FIG. 9. P obabili y dis ibu ion p( *)a T*⫽1. A *⫽1,
om bo om o op he lines co espond o educed densi ies
␳
*
⫽0.00113(1),0.00618(1),0.01453(2),0.0456(3), and 0.1566(6).
ROMERO-ENRIQUE, RULL, AND PANAGIOTOPOULOS PHYSICAL REVIEW E 66, 041204 共2002兲
041204-6
la ge alues o as ⫺n, wi h n⬇3. The la e p edic ion
di e s om he alue n⫽4 gi en in Re . 关39兴and equi es
u he s udy o elucida e i .
The p obabili y dis ibu ion pRc( ) o a pai ha ing an
in e nal sepa a ion dis ance be ween ions when he cu o
dis ance o de ine a pai is equal o Rccan be ob ained om
p( )as
pRc共 兲⫽p共 兲
㜷共Rc兲⫽p共 兲
冕
␴
⫾
Rcp共 兲d
共17兲
whe e 㜷( )⬅
兰
␴
⫾
p( )d is he cumulan dis ibu ion co e-
sponding o p( ). The densi y o associa ed pai s is equal o
␳
㜷(Rc)/2, and he o al densi y o ee ions is
␳
关1
⫺㜷(Rc)兴, whe e
␳
is he o al densi y o ions in he physical
pic u e. I can be seen om Fig. 8 ha o Rcⲏ3
␴
⫾, mo e
han 95% o ions a e associa ed in o pai s in bo h apo and
liquid phases. So, i he associa ed pai luid phase sepa a e
in he same (T,
␳
) egion as he ionic luid, i is expec ed ha
he ee ions do play a me e pe u ba i e ole in he phase
coexis ence. Ac ually, he ee ionic subsys em is a nonaddi-
i e bina y cha ged ha d-sphe e mix u e, whe e Rcplays he
ole o he unlike-ion collision diame e . Howe e , he in e -
ac ions be ween like ions a e pu ely epulsi e, and we do no
expec di e ences be ween he beha io o his mix u e and
he RPM as soon as
␴
⫹and
␴
⫺a e less han Rc. Fu he -
mo e, his sys em is in a pola en i onmen gi en by he
associa ed pai subsys em, and consequen ly he e ec i e di-
elec ic cons an De o he backg ound will be inc eased.
Then he ee ion subsys em is expec ed o ha e he same
beha io as he RPM a he e ec i e educed empe a u e T
˜
T
˜
⫽De kBTRc
q2⫽De
D
Rc
␴
⫾
T*,共18兲
whe e T*is he educed empe a u e. Fo Rcⲏ3
␴
⫾, he
c i ical empe a u e educes a leas o one- hi d o he RPM
educed c i ical empe a u e. Consequen ly, any possible ee
ion-d i en apo -liquid ansi ion should occu a much
lowe empe a u es, and hus he ee ion subsys em should
play no ole in he apo -liquid ansi ion. In o de o check
his hypo hesis, he phase diag am o he associa ed pai luid
has o be ob ained. This issue will be add essed in he ol-
lowing sec ion.
IV. PHASE BEHAVIOR OF THE ASSOCIATED PAIRS
FLUID
The esul s ob ained in he p eceding sec ion sugges ha
he chemical pic u e in oduced abo e is a e y con enien
desc ip ion o he ionic luid s uc u e. Fu he mo e, we can
analyze he ole played by he associa ed pai s in he phase
equilib ium by elimina ing he ee ions. We ha e pe o med
g and-canonical Mon e Ca lo simula ions o he associa ed
dime sys em o ␭⫽3 and di e en alues o Rc. No ee
ions a e allowed (N ⬅0) and he po en ial ene gy o one
con igu a ion is gi en by Eq. 共10兲. The g and-canonical ee
ene gies co esponding o hese sys ems will consequen ly
p o ide an uppe bound o he ionic luid g and-canonical
ee ene gy a he same empe a u e and pai chemical po en-
ial. As o he ionic sys em, a ine-disc e iza ion app oach
wi h a e inemen pa ame e
␨
⫽10, and Ewald summa ion
echnique wi h conduc ing bounda y condi ions is used o
ake in o accoun he long ange cha ac e o he Coulombic
in e a ions. The basic s eps a e ei he inse ion o dele ion o
associa ed pai s 共chosen andomly wi h he same p obabil-
i y兲, biased wi h a Bol zmann dis ibu ion ha depends on
he sepa a ion be ween he ions ha compose he dime . An
associa ed pai is inse ed in he ollowing way: he nega i e
ion is placed a a andom place in he box, and i s coun e ion
is placed a a ela i e posi ion ⫾(
␴
⫾⭐
兩
⫾
兩
⬍Rc) ollowing
a p obabili y dis ibu ion p opo ional o exp关
␤
0q2
␾
0( ⫾)兴,
whe e ⫺
␾
0( ) is he Ewald po en ial ene gy be ween wo
unlike ions o uni cha ge. On he o he hand, a pai is de-
le ed wi h a p obabili y p opo ional o exp关
␤
0q2
␾
0( ⫾)兴, ⫾
being he ela i e posi ion o he posi i e ion o he pai wi h
espec o he nega i e one. The alue o
␤
0has been ad-
jus ed o imp o e he sampling du ing he simula ion. In his
wo k we ha e se
␤
0⫽8.
The accep ance p obabili ies o a pai inse ion Wij
iand a
pai dele ion, Wij
da e he ollowing:
Wij
i⫽min
冉
1,共L*兲3exp共
␤␮
⫺
␤
⌬U兲
⫻
冋
兺
␴
⫾⭐
兩
⫾
兩
⬍Rc
exp关
␤
0q2
␾
0共 ⫾兲兴
␨
3
册
⫻
冋
兺
k⫽1
Np⫹1
exp关
␤
0q2
␾
0共 ⫾
k兲兴
册
⫺1
冊
,共19兲
Wij
d⫽min
冉
1,exp共⫺
␤␮
⫺
␤
⌬U兲
共L*兲3
⫻
冋
兺
␴
⫾⭐
兩
⫾
兩
⬍Rc
exp关
␤
0q2
␾
0共 ⫾兲兴
␨
3
册
⫺1
⫻
冋
兺
k⫽1
Np
exp关
␤
0q2
␾
0共 ⫾
k兲兴
册
冊
,共20兲
whe e iand ja e he ini ial and inal con igu a ions, espec-
i ely, Npis he numbe o pai s a he con igu a ion i,
␤
⫽1/(kBT),
␮
is he con igu a ional chemical po en ial, and
⌬Uis he chemical po en ial ene gy a ia ion in he mo e-
men , including he s e ic hind ance e ms gi en by Eq. 共12兲.
As i can be seen, he accep ance p obabili ies educe o he
usual ones as
␤
0→0.
E ec i e c i ical poin s o di e en alues o L*we e
es ima ed by using mixed- ield ini e-size scaling me hods
关34兴and assuming Ising-like c i icali y. Al hough ecen e-
sul s 关40兴indica e ha he p essu e should also en e he ield
mixing, ou app oach should be sa is ac o y o disce n he
dependence o he c i ical pa ame e s on Rc. Mo eo e , a
DIPOLAR ORIGIN OF THE GAS-LIQUID . . . PHYSICAL REVIEW E 66, 041204 共2002兲
041204-7
sys ema ic s udy o he RPM case has shown ha he as-
sumed uni e sali y class is e y likely o be he igh one
关16兴. We use his og am eweigh ing echniques 关33兴 o com-
bine he his og ams om di e en uns 共 ypically h ee兲and
es ima e he c i ical pa ame e s and hei s anda d e o s.
Ve y long uns a e needed in o de o o e come he c i ical
slowing down and he low accep ance a ios due o he low
empe a u es in ol ed. De ining a s ep as a y o a pai
inse ion 共o dele ion兲, we ha e pe o med simula ions o
abou (2⫻107)–(4⫻107) equilib a ion s eps and (2
⫻108)–(4⫻108) sampling s eps o L*⫽12; (5⫻107)–(2
⫻108) equilib a ion s eps and (6⫻108)–(1.2⫻109) sam-
pling s eps o L*⫽15; and 2⫻108equilib a ion s eps and
(1.2⫻109)–(1.8⫻109) sampling s eps o L*⫽18, in o de
o ge good s a is ics o he his og ams. The 共e ec i e兲c i i-
cal pa ame e s a e ob ained by minimizing he de ia ion o
he app op ia ely scaled mixed- ield M⬀
␳
⫺su ma ginal
p obabili y dis ibu ion (uis he po en ial ene gy densi y and
sis he mixing pa ame e 兲wi h espec o he co esponding
c i ical h ee-dimensional Ising uni e sal unc ion 关34,41兴.
As also occu s in he ionic and e he ed dime sys ems, he
ma ching imp o es as L*inc eases 共see Fig. 10兲. Ou es i-
ma ions o he c i ical pa ame e s o di e en alues o L*
and Rc
*⬅Rc/
␴
⫾a e lis ed in Table I and plo ed in Figs. 11
and 12. The smalles alue o Rc
*is he same as he maxi-
mum allowed sepa a ion in he e he ed dime sys em, and as
expec ed he esul s ob ained o he associa ed pai s luid a e
p ac ically indis inguishable om he e he ed dime sys em.
As Rc
*inc eases, we obse e a sha p inc ease in he c i ical
empe a u e bu he c i ical densi y emains almos una -
ec ed. Howe e , o Rc
*⬇3–4 he c i ical empe a u e
eaches a maximum and hen dec eases, con e ging owa ds
he ee ion c i ical empe a u e. On he o he hand, he c i i-
cal densi y also dec eases owa ds he ee ion c i ical den-
si y, al hough he s a is ical unce ain ies a e bigge o his
pa ame e and he exac pa h o con e gence is less de ined.
We mus also poin ou ha he s a is ical unce ain ies a e
big enough no o make possible an es ima ion o co ec ion-
o-scaling e ec s.
Fo a heu is ic explana ion o such a beha io , we ha e
o conside he ene ge ically a o ed con igu a ions. I is im-
po an o conside he in e ac ion be ween wo di e en as-
socia ed pai s, which has i s minimum ene gy con igu a ion
in a squa e ing con o ma ion o low alues o ␭, wi h a
close-ene gy seconda y minimum con o ma ion o he linea
(⫹,⫺) chain 关17兴.I Rcis oo small, by de o ming he
squa e i is possible o ind ene ge ically ele an con igu a-
ions 共i.e., he po en ial ene gy di e ence pe ion o such
con igu a ions wi h espec o he minimum one is o he
same o de as he he mal ene gy兲in which one o he asso-
cia ed pai s has an in e nal ionic sepa a ion less han Rc, bu
he o he one does no ul il such a condi ion. Consequen ly,
a highe alue o Rcalso inc eases he numbe o ene ge i-
cally ele an con igu a ions be ween wo associa ed pai s.
Then he e ec i e in e ac ion be ween associa ed pai s is en-
hanced, and hus an inc ease o he c i ical empe a u e
should be expec ed un il all he ele an con igu a ions a e
allowed 共 ha occu s o Rc⬃3
␴
⫾). On he o he hand, he
TABLE I. Dependence on Rc
*o he es ima ed c i ical pa am-
e e s. 共The 1
␴
s a is ical unce ain ies e e o he las decimal
places.兲
Rc
*L*Tc
*⫻102⫺
␮
c
*
␳
c
*⫻102
1.02 12 4.24共2兲1.2512共4兲7.66共12兲
15 4.25共1兲1.2504共2兲7.56共9兲
18 4.26共1兲1.25077共8兲7.34共10兲
1.5 12 4.44共1兲1.30489共10兲7.41共12兲
15 4.46共1兲1.30565共5兲7.52共10兲
18 4.47共1兲1.30580共8兲7.33共17兲
2.0 12 4.53共2兲1.3112共3兲7.45共14兲
15 4.53共1兲1.3109共2兲7.53共18兲
18 4.53共2兲1.3107共3兲7.54共17兲
3.0 12 4.65共2兲1.3187共4兲7.23共10兲
15 4.60共1兲1.31792共13兲6.93共12兲
18 4.62共3兲1.3182共4兲6.69共10兲
4.0 12 4.56共4兲1.3202共5兲7.05共22兲
15 4.59共2兲1.3207共5兲6.45共14兲
18 4.62共3兲1.3213共5兲6.4共2兲
5.0 12 4.58共1兲1.32210共14兲6.17共21兲
15 4.57共1兲1.32230共7兲6.46共24兲
18 4.56共4兲1.3220共7兲6.17共10兲
6.0 12 4.55共1兲1.3222共3兲6.41共2兲
15 4.51共2兲1.3214共3兲6.33共10兲
18 4.59共3兲1.3222共6兲6.1共2兲
7.0 15 4.51共2兲1.3216共5兲6.33共10兲
18 4.50共2兲1.3211共3兲6.04共17兲
8.0 18 4.53共4兲1.3224共6兲6.1共4兲
9.0 18 4.49共1兲1.3215共3兲5.6共3兲
F ee ion casea12 4.49共1兲1.3200共1兲6.37共7兲
15 4.48共1兲1.3199共1兲6.04共11兲
18 4.49共2兲1.3199共4兲6.05共11兲
aI is equi alen o conside Rc
*⬅
冑
3L*/2.
FIG. 10. Rescaled ma ginal p obabili y dis ibu ions pM
c(x) o
he associa ed pai s luid cha ac e ized by ␭⫽3 and Rc
*⫽3. The
solid line is he uni e sal unc ion co esponding o he h ee-
dimensional Ising uni e sali y class, and he symbols co espond o
he bes -ma ching simula ion esul s: he squa es co espond o L*
⫽12, he diamonds o L*⫽15, and he iangles o L*⫽18.
ROMERO-ENRIQUE, RULL, AND PANAGIOTOPOULOS PHYSICAL REVIEW E 66, 041204 共2002兲
041204-8
a e age size o he associa ed pai s and hei agg ega es is
inc eased, leading o a dec ease o he c i ical densi y 共in
o de o keep he educed densi y in e ms o he e ec i e
pa icle size cons an 兲. These a gumen s a e no longe alid
o Rc
*ⲏ3. Ou esul s show ha he inclusion o associa ed
pai s wi h la ge in e nal ionic sepa a ions is no c ucial o
he gas-liquid phase ansi ion. Also hey ba he phase an-
si ion, as can be in e ed om he c i ical pa ame e s de-
c ease.
As o he ionic and e he ed dime models, his og am
eweigh ing echniques 关33,41兴allow us o ob ain he coex-
is ence cu e up o Tⱗ0.98Tc. As he ini e-size e ec s a e
no impo an a om he c i ical poin , we ha e conside ed
L*⫽15. Taking ad an age o he wide ange o densi ies
co e ed by nea -c i ical his og ams, we combine hem wi h
liquid subc i ical s a e his og ams in o de o ex end he den-
si y ange. The ex a simula ions in ol e sho e uns 共 ypi-
cally 2⫻108s eps a e equilib a ion兲. The gas-liquid coex-
is ence cu es o di e en alues o Rc
*a e plo ed in Fig.
13, showing he same endency as he obse ed one in he
c i ical pa ame e s. I is in e es ing o no e ha he apo
b anches coincide 共excep close o he c i ical poin 兲 o Rc
⬎3
␴
⫾. We conclude om his obse a ion ha is in he
liquid b anch whe e he ionic luid di e s mos ly om he
associa ed pai sys em.
V. DISCUSSION AND CONCLUSIONS
In summa y, an exac ‘‘chemical’’ ep esen a ion o he
ionic sys em as a mix u e o (⫹,⫺) associa ed ion pai s and
ee ions has been in oduced. This ep esen a ion, closely
ela ed o he S illinge -Lo e pai ing p ocedu e o he
RPM, has he ad an age o no only p o iding an exac
ma ching o he physical and chemical ep esen a ion he -
modynamics, om a mic oscopic poin o iew; i also
a oids an en opy ca as ophe ha occu s o he e he ed
dime model s udied a la ge alues o he e he leng h. The
addi ion o he physical Hamil onian o some new pai wise
ha d-co e in e ac ions be ween he chemical componen s
p o ides a sui able Hamil onian o he chemical ep esen a-
ion. In he low empe a u e and low densi y egime, in
which he ionic apo -liquid ansi ion occu s, such a ep e-
sen a ion p o ides a ai h ul cha ac e iza ion o he mic o-
scopic s uc u e o he ionic luid, and i can be he basis o
imp o ed heo e ical app oaches.
The analysis o he phase beha io o he sys em only
composed o associa ed pai s, indica es s ongly ha he
ionic luid apo -liquid ansi ion is d i en by hem. Fo Rc
ⱗ3
␴
⫾, he c i ical empe a u e inc eases wi h Rc, as mo e
ene ge ically a o ed con igu a ions a e allowed. Fo Rc
ⲏ3
␴
⫾, he c i ical pa ame e s con e ge smoo hly om
abo e owa ds he ionic luid c i ical pa ame e s as Rcin-
c eases. This ac indica es no only ha he ee ions do no
d i e he phase ansi ion, bu also hey ha e he opposi e
e ec in he ansi ion. The alue o Rc⬇3
␴
⫾co esponding
o he maximum on he c i ical empe a u e, can be ega ded
as he op imal size o he associa ed pai .
Some ema ks on he limi a ions o he p esen wo k a e
FIG. 11. Dependence o he c i ical empe a u e Tc
*on he e-
duced cu o dis ance Rc
*, o L*⫽12 共squa es兲,L*⫽15 共dia-
monds兲, and L*⫽18 共 iangles兲.
FIG. 12. Dependence o he c i ical densi y
␳
c
*on he educed
cu o dis ance Rc
*, o L*⫽12 共squa es兲,L*⫽15 共diamonds兲, and
L*⫽18 共 iangles兲.
FIG. 13. Gas-liquid coexis ence cu es o Rc
*⫽1.02 共ci cles兲,
Rc
*⫽3共squa es兲,Rc
*⫽5共diamonds兲, and he ee ion sys em 共 i-
angles兲.
DIPOLAR ORIGIN OF THE GAS-LIQUID . . . PHYSICAL REVIEW E 66, 041204 共2002兲
041204-9