Su ace and capilla y ansi ions in an associa ing bina y mix u e model
J. M. Rome o-En ique,1,2,*L. F. Rull,2and U. Ma ini Be olo Ma coni3
1Depa men o Ma hema ics, Impe ial College o Science, Technology and Medicine, 180 Queen’s Ga e,
London SW7 2BZ, Uni ed Kingdom
2Depa amen o de Fı
´sica A o
´mica, Molecula y Nuclea , A ea de Fı
´sica Teo
´ ica, Uni e sidad de Se illa,
Apa ado de Co eos 1065, 41080 Se illa, Spain
3Dipa imen o di Fisica and Is i u o Nazionale di Fisica della Ma e ia, ia Madonna delle Ca ce i,
Uni e si a
`di Came ino, 62032 Came ino, I aly
共Recei ed 3 Decembe 2002; published 11 Ap il 2003兲
We in es iga e he phase diag am o a wo-componen associa ing luid mix u e in he p esence o selec-
i ely adso bing subs a es. The mix u e is cha ac e ized by a bulk phase diag am ha displays peculia
ea u es such as closed loops o immiscibili y. The p esence o he subs a es may in e e e wi h he physical
mechanism in ol ed in he appea ance o hese phase diag ams, leading o an enhanced endency o phase
sepa a e below he lowe c i ical solu ion poin . Th ee di e en cases a e conside ed: a plana solid su ace in
con ac wi h a bulk luid, while he o he wo ep esen wo models o po ous sys ems, namely, a sli and an
a ay on in ini ely long pa allel cylinde s. We con i m ha su ace ansi ions, as well as capilla y ansi ions
o a la ge su ace a ea o olume a io, a e s abilized in he one-phase egion. Applicabili y o ou esul s o
expe imen s epo ed in he li e a u e is discussed.
DOI: 10.1103/PhysRe E.67.041502 PACS numbe 共s兲: 64.75.⫹g, 05.70.Np, 68.08.Bc, 68.35.Rh
I. INTRODUCTION
In he las ew yea s he e has been a conside able
p og ess in he unde s anding o he physics o luids in e-
s ic ed geome ies 关1兴. The p ope ies o liquids o gases
con ined in na ow po es appea di e en om hose in he
bulk. A common end o hese sys ems is he ac ha con-
densa ion occu s a p essu es di e en om he sa u a ion
bulk p essu e and he c i ical empe a u e o phase sepa a-
ion is gene ally lowe han he co esponding bulk empe a-
u e. Acco ding o Nakanishi and Fishe 关2–5兴, an incom-
p essible luid mix u e 共wi h iso opic in e ac ions兲unde he
e ec o con inemen displays a educed endency owa d
phase sepa a ion a a ixed empe a u e, because he e ec i e
a ac i e o ces esul weake wi hin a po e. The con ine-
men also de e mines an e ec i e dimensional educ ion hus
enhancing luc ua ion e ec s. In addi ion, he p esence o
adso bing walls may induce a su ace ansi ion in a subc i i-
cal luid, e en when his is in a single-phase egion in he
bulk. Such a beha io , which is now well unde s ood by
means o a mean- ield app oach 关6–8兴is due essen ially o
he educ ion o he con ibu ion o he ee ene gy om he
cohesi e luid- luid o ces caused by he p esence o he con-
ining walls. The e exis s, howe e , a class o nonsimple
luid mix u es o which he si ua ion is di e en : hese a e
he s ongly associa ing mix u es. S ong di ec ional a ac-
i e in e ac ions such as hyd ogen bonding o cha ge ans e
complexing be ween molecules a ec d ama ically he p op-
e ies o he luid o solid. An in e es ing phenomenon ha
occu s in such a mix u e is he een an miscibili y: he mix-
u e is comple ely miscible a low empe a u es, up o a em-
pe a u e whe e i sepa a es in o wo liquid phases. As he
empe a u e is u he inc eased, he comple e miscibili y e-
appea s. The nico ine-wa e mix u e is a pa adigm o such a
beha io 关9兴. The empe a u e-composi ion phase diag am
shows closed loops o immiscibili y, cha ac e ized by he
exis ence o a lowe and an uppe c i ical solu ion poin
共LCSP and UCSP, espec i ely兲. Hi sch elde e al. 关10兴p o-
posed a mechanism o explain he een an miscibili y. The
essen ial ing edien is he exis ence o highly di ec ional
s ong a ac ion 共as he hyd ogen bond兲be ween he unlike
pa icles. The dispe si e 共iso opic兲 o ces be ween hese pa -
icles a e assumed o be so weak as o a o seg ega ion o
he species. A low empe a u es, he ene gy a o s comple e
mixing. Howe e , as he empe a u e inc eases, he ene gy
dec emen due o he o ma ion o di ec ional bonds com-
pe es wi h he dec ease o o ien a ional en opy. As a conse-
quence, he sys em phase sepa a es when he empe a u e
aises. This mechanism has been con i med by heo e ical
s udies on la ice 关11–13兴and con inuous models 关14兴,as
well as in ecen compu e simula ion s udies 关15,16兴.
How he p esence o subs a es o he con inemen in a
po e a ec he beha io o hese mix u es 共i in a manne
simila o ha o simple luids, o i on he con a y hese
mix u es a e peculia 兲is an issue o g ea p ac ical and he-
o e ical in e es . The p esence o selec i ely adso bing sub-
s a es can in e e e wi h he o ma ion o bulk hyd ogen
bonding, leading o a s abiliza ion o he wo-phase egion
wi h espec o he comple ely miscible phase. Compu e
simula ion s udies on con ined la ice models 关17兴show he
appea ance o su ace ansi ions in a ange o empe a u es
below he LCSP. The same mechanism has been in oked o
explain some expe imen s 关18,19兴 ha seemed o con adic
he Nakanishi-Fishe pic u e.
In he p esen pape we shall s udy a la ice model o
associa ing bina y mix u es, in oduced ew yea s ago by Lin
and Taylo 关20,21兴and econside ed ecen ly by he p esen
au ho s 关22–25兴. In spi e o i s idealized na u e, he model
has he me i o allowing o an explici analysis o i s non-
*Co esponding au ho . Email add ess: jose.en ique@
impe ial.ac.uk
PHYSICAL REVIEW E 67, 041502 共2003兲
1063-651X/2003/67共4兲/041502共11兲/$20.00 ©2003 The Ame ican Physical Socie y67 041502-1
i ial phase diag am. As an ins ance, he s udy o i s bulk
p ope ies p edic s he exis ence o closed loops o immisci-
bili y unde cons an p essu e condi ions. This analysis will
be ex ended o he beha io o he mix u e in p esence o
subs a es and unde con inemen .
The s uc u e o he pape is o ganized as ollows. In Sec.
II we b ie ly p esen he model and w i e explici ly he equa-
ions o he o de pa ame e p o iles. Sec ion III is de o ed
o analyze in de ail he we ing beha io o he mix u e in he
p esence o adso bing subs a es. In Sec. IV we s udy he
capilla y beha io o he same sys em when con ined be-
ween wo pa allel pla es 共a sli 兲and in Sec. V we conside
he same sys em con ined in a ne wo k o e y na ow cyl-
inde s. Capilla y ansi ions 共and su ace ansi ions o he
slab geome y兲will be analyzed in e ms o he physical
pa ame e s o he sys em. Finally, in Sec. VI we p esen ou
conclusions.
II. THE MODEL
Some yea s ago, Lin and Taylo p oposed a la ice model
o desc ibe a bina y mix u e, in which he wo unlike species
can o m highly di ec ional bonds. In mo e de ail, he Lin-
Taylo 共LT兲model 关20,21兴is an A-Bbina y mix u e la ice
model, de ined on a gene ic D-dimensional la ice o coo di-
na ion numbe
␯
. The Apa icles a e allowed o singly oc-
cupy any o he la ice cells. E e y cell con ains
␯
equal
subcells, one o each ace o he mo he cell. Again, each
subcell can hos a mos one Bpa icle, p o ided he mo he
cell is no occupied by an Apa icle. In addi ion o he e-
pulsi e in e ac ions 共which de e mine he p e ious occupa-
ion ules兲, one includes sho - anged in e ac ions 共see Fig.
1兲. The in e ac ion s eng h be ween a pai o nea es -
neighbo Apa icles is
⑀
AA ,
⑀
AB be ween unlike species and
⑀
BB be ween B-Bpa icles. All hese in e ac ions a e non a-
nishing only i he pa icles sha e a ace.
The bulk p ope ies, as well as he global phase diag am,
ha e been exhaus i ely analyzed in p e ious wo k 关22–25兴.
One o he main conclusions o such s udies is ha he ap-
pea ance o closed loops o immiscibili y occu s when he
in e ac ion be ween A-Aand A-Bpai s a e a ac i e (
⑀
AA
⬍0,
⑀
AB⬍0), and
⑀
ABⱗ关
⑀
AA⫹min(
⑀
BB,0)兴/2. The la e con-
di ion is in comple e ag eemen wi h he mechanism p o-
posed by Hi sch elde , S e enson, and Ey ing 关10兴 o he
een an miscibili y. In his wo k, we assume he bulk luid-
luid pa ame e s o ha e alues:
⑀
AA⫽⫺1共i de ines he en-
e gy scale兲,
⑀
AB⫽⫺0.65, and
⑀
BB⫽0, so ha we a e in he
egime whe e closed loops o immiscibili y appea .
The p esence o subs a es in oduces new in e ac ions
共see Fig. 1兲. We shall assume ha he subs a es do no a ec
he luid- luid in e ac ions, i.e., he e is no su ace enhance-
men . The in e ac ion s eng h be ween an Apa icle and he
subs a e is VA, i he pa icle is in a cell in con ac wi h he
subs a e, and ze o o he wise. The in e ac ion be ween he B
pa icles and he subs a e is o ien a ionally dependen , i.e.,
i s s eng h VBis non anishing only i he ace o he B
pa icle ouches he subs a e. The o m o he subs a e- luid
in e ac ions a e consis en wi h he bulk in e ac ions. He ea -
e , he subs a e- luid in e ac ions will be assumed as a ac-
i e, i.e., VA⭐0 and VB⭐0. Fo absolu e alues o VBla ge
enough, he subs a e-Bin e ac ions compe e wi h he bulk
A-Bin e ac ions, so ha one expec his ype o subs a e o
enhance he demixing endency o empe a u es lowe han
he co esponding LCSP.
To s udy he LT model in p esence o subs a es, we con-
side he g and-canonical ensemble, whe e he sys em is in
he mal and chemical equilib ium wi h a ese oi a ixed
empe a u e Tand chemical po en ials
␮
Aand
␮
B. The ac-
i i ies zAand zBa e de ined ia zA⬅exp(
␤␮
A) and zB
⬅exp(
␤␮
B), whe e
␤
⫽1/kBT. A e acing o e all con igu-
a ions o he Bpa icles, he sys em esul s o be isomo phic
o a monocomponen la ice gas, wi h eno malized in e ac-
ions and chemical po en ial. The de ails on his p ocedu e
can be ound in Re s. 关21,23,25兴. The g and-canonical pa i-
ion unc ion ⌶o he LT model can be w i en in e ms o
an equi alen Ising model as
⌶⫽共1⫹2zB⫹zB
2e⫺
␤⑀
BB兲
␯
N/2 exp
冋
N
冉
H⫺
␯
K
2
冊
册
⫻
冉
1⫹zBe⫺
␤
VB
冑
1⫹2zB⫹zB
2e⫺
␤⑀
BB
冊
␯
⬜Nb
⫻exp
冋
Nb
冉
⌬H1⫹
␯
⬜K
2
冊
册
ZD,
␯
Ising共K,H,⌬H1兲,共1兲
whe e Nis he o al numbe o la ice cells, Nbis he numbe
o cells adjacen o he subs a e,
␯
⬜is he numbe o nea es
neighbo s in he di ec ion pe pendicula o he subs a e;
␯
储
is de ined as he numbe o nea es -neighbo cells in di ec-
ions pa allel o he subs a e. Ob iously,
␯
⫽
␯
储
⫹2
␯
⬜. Fi-
nally, ZD,
␯
Ising(K,H,⌬H1) is he canonical pa i ion unc ion o
he Ising model on he same la ice. Le us name K he e -
ec i e coupling cons an , H he e ec i e bulk magne ic
ield, and ⌬H1 he e ec i e su ace magne ic ield ac ing
only on he bounda y si es,
K⫽⫺
␤⑀
AA
4⫹1
2ln
冉
冑
1⫹2zB⫹zB
2e⫺
␤⑀
BB
1⫹zBe⫺
␤⑀
AB
冊
,共2兲
FIG. 1. Schema ic ep esen a ion o he A-BLT model in he
wo-dimensional squa e la ice 共 he unde lying la ice is no shown
o cla i y兲. Rele an luid- luid and luid-subs a e in e ac ions a e
also highligh ed. 共See ex o explana ion.兲
ROMERO-ENRIQUE, RULL, AND BETTOLO MARCONI PHYSICAL REVIEW E 67, 041502 共2003兲
041502-2
H⫽1
2lnzA⫺
␯
␤⑀
AA
4⫺
␯
4ln共1⫹2zB⫹zB
2e⫺
␤⑀
BB兲,共3兲
⌬H1⫽
␯
⬜
冋
1
2ln
冉
冑
1⫹2zB⫹zB
2e⫺
␤⑀
BB
1⫹zBe⫺
␤
VB
冊
⫹
␤
4
冉
⑀
AA⫺2VA
␯
⬜
冊
册
.
共4兲
All equilib ium p ope ies can be ob ained om he knowl-
edge o he pa i ion unc ion Eq. 共1兲by using s anda d he -
modynamical ela ionships. In pa icula , he bulk p essu e is
ob ained as
p⫽lim
V→⬁
kBTln⌶
V⫽kBT
0
冋
␯
2ln共1⫹2zB⫹zB
2e⫺
␤⑀
BB兲⫺
␯
K
2
⫹H⫹lim
N→⬁
1
NlnZD,
␯
Ising
册
,共5兲
whe e 0is he olume o la ice cell, and N⬅V/ 0. The
mole ac ion XA(i) p o ile is ob ained as
XA共i兲⬅nA共i兲
nA共i兲⫹兺
s⫽1
␯
nB共i,s兲
,共6兲
whe e nA(i), he A-pa icle densi y in he cell i, is de ined in
e ms o he local magne iza ion mias
nA共i兲⫽1⫹mi
2 0共7兲
and nB(i,s), he Bpa icle densi y in he subcell so he cell
i,is
nB共i,s兲⫽1⫺mi
2 0
zBe⫺
␤
VB
1⫹zBe⫺
␤
VB,共8兲
i he subcell is by he subs a e 共see Fig. 1兲. O he wise,
nB(i,s) is de ined as
nB共i,s兲⫽1
4 0
冋
zBe⫺
␤⑀
AB
1⫹zBe⫺
␤⑀
AB 共1⫺mi⫹mj⫺
具
sisj
典
兲
⫹zB⫹zB
2e⫺
␤⑀
BB
1⫹2zB⫹zB
2e⫺
␤⑀
BB 共1⫺mi⫺mj⫹
具
sisj
典
兲
册
,
共9兲
whe e
具
sisj
典
is he wo-spin co ela ion unc ion be ween he
magne iza ion on he cell iand i s nea es -neighbo j ha i is
associa ed o he subcell s. I is easy o see ha in he homo-
geneous case hese equa ions educe o he exp essions gi en
in Re s. 关23,25兴.
The equi alence be ween he LT model and he Ising
model, allows one o ob ain he su ace beha io o he mix-
u e om he knowledge o he beha io o he la e . How-
e e , such a mapping, is no a all i ial and may lead in
some cases o unexpec ed esul s.
III. WETTING TRANSITIONS IN THE LT MODEL
Le us conside an A-Bbina y mix u e in he p esence o
a plana , in ini e, and s uc u eless subs a e. Such a subs a e
con ines he luid o he hal -space z⭓0. When he luid
共e.g., he A-pa icle ich phase,
␣
) is a bulk coexis ence, he
subs a e can p omo e he o ma ion o d ops o he opposi e
coexis ing phase 共 he B-pa icle ich phase
␤
) on i , cha ac-
e ized by a con ac angle
␪
. When
␪
⫽0, he d ops sp ead
o e he subs a e, i.e., he phase
␤
we s comple ely he
␣
-subs a e in e ace. A ini e alue o
␪
, ins ead, co e-
sponds o a pa ial we ing si ua ion. The c osso e be ween
a pa ial o a comple e we ing egime is called he we ing
ansi ion 关26兴. The su ace ensions in ol ed in he p oblem
(
␴
␣
s,
␴
␤
sand
␴
␣␤
, co esponding o he
␣
-subs a e,
␤
-subs a e, and
␣
-
␤
in e aces, espec i ely兲a e ela ed o
he con ac angle ia he Young equa ion 关27兴
␴
␣
s⫽
␴
␤
s⫹
␴
␣␤
cos
␪
.共10兲
The modynamically he su ace ensions co espond o he
excess g and-canonical- ee ene gy pe in e acial uni a ea
共 espec o he bulk兲关27兴. Since bo h
␴
␣␤
and
␴
␤
sha e an
analy ical beha io a his he modynamic s a e, he we ing
ansi ion is a 共su ace兲 he modynamic phase ansi ion, as i
can be seen om Eq. 共10兲and he dependence o
␪
on he
empe a u e. As usual, app opia e o de pa ame e s co e-
sponding o he we ing ansi ion can be de ined as i s
de i a i es o he su ace ension espec o he modynamic
ields. In simple luids, he excess numbe o pa icles pe
in e acial uni a ea o adso p ion ⌫关 ha co esponds o
⫺(
⳵␴
␣
s/
⳵
␮
)T,
␮
being he chemical po en ial兴is he na u-
al o de pa ame e choice. In mix u es, o he choices a e
a ailable, e.g., he olume in eg a ed excess mole ac ion
pe in e acial uni a ea. Mic oscopically, he we ing ansi-
ion mani es s i sel as he g ow h o a mac oscopically hick
␤
laye in uding in he
␣
-subs a e in e ace composi ion
p o ile. Th ee di e en ypes o we ing ansi ion may a ise.
I he laye wid h jumps suddenly om a mic oscopically
ini e alue, he we ing ansi ion is o i s o de , and an
o -coexis ence su ace ansi ion 共p ewe ing兲is associa ed
o i . I he laye wid h di e ges con inuously, he we ing
ansi ion is named c i ical. Finally, he we ing ansi ion can
occu as he accumula ion poin o an in ini e sequence o
i s -o de ansi ions in he laye ing we ing case 共 hese an-
si ions ex end o he o -coexis ence egion兲. The condi ions
o such a ansi ion depend sub ly on he luid- luid and
subs a e- luid in e ac ions, and in e acial luc ua ions can
play an impo an ole 关28兴.
In he LT model wo-phase coexis ence occu s when H
⫽0 and K⬎Kc, whe e Kcis he la ice dependen Ising
c i ical coupling. Mo eo e , he we ing beha io is con-
olled by he a io ⌬H1/
␯
⬜K, ha is he a io be ween he
su ace ene gy and he loss o bulk ene gy due o he p es-
ence o he su ace. I such a pa ame e is posi i e he sub-
s a e a o s he
␣
phase o in ude be ween he subs a e and
SURFACE AND CAPILLARY TRANSITIONS IN AN . . . PHYSICAL REVIEW E 67, 041502 共2003兲
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he
␤
phase. On he o he hand, i ⌬H1/
␯
⬜K⬍0, he phase
␤
in udes be ween he subs a e and he phase
␣
. We shall
ocus on he coexis ence side H⫽0⫹(H⫽0⫺) o ⌬H1
⬍0(⌬H1⬎0), espec i ely 共o he wise, only pa ial we ing
can occu 兲. I he absolu e alue o he pa ame e is g ea e
han one (
兩
⌬H1/
␯
⬜K
兩
⬎1), hen all coexis ence su ace co -
esponds o a comple e we ing si ua ion. I such a condi ion
is no ul illed, one will obse e a pa ial we ing si ua ion up
o a alue Kw共dependen on
兩
⌬H1/
␯
⬜K
兩
) and beyond i a
comple e we ing si ua ion up o K⫽Kc.
Le us conside i s he case VB⫽0, when ⌬H1depends
only on T, ega dless he alue o zB. Mo eo e , he sign o
⌬H1is uniquely de e mined by he di e ence be ween VA
and
␯
⬜
⑀
AA/2 o e he whole empe a u e ange. One can en-
isage a ious cases 共see Fig. 2兲:i VA⫽0, he coexis ence
co esponds o a si ua ion o comple e we ing o he in e -
ace subs a e-
␣
phase by phase
␤
, deno ed by he symbol
s
兩
␤
兩
␣
.I
␯
⬜
⑀
AA/2⬍VA⬍0, he e exis s a egion o comple e
we ing s
兩
␤
兩
␣
and a lowe p essu e egion o pa ial we ing.
As VAdec eases, he egion o comple e we ing s
兩
␤
兩
␣
sh inks and e en ually disappea s o VA⫽
␯
⬜
⑀
AA/2, because
⌬H1⬅0. Fo VA⬍
␯
⬜
⑀
AA/2, he sign o ⌬H1changes and a
egion o comple e we ing o he in e ace subs a e-phase
␤
by phase
␣
appea s (s
兩
␣
兩
␤
). As VAinc eases, he egion o
pa ial we ing educes and disappea s o VA⭓
␯
⬜
⑀
AA .
When VB⫽0, i is possible ha in co espondence wi h
di e en s a e poin s, he subs a e a o s ei he he
␤
o he
␣
phase. To s udy such an e ec i is necessa y o loca e he
coexis ence s a es, whe e ⌬H1⫽0. Since VBand
⑀
AB a e
nega i e, o a empe a u e Tall he coexis ence s a es wi h
p essu e pla ge han pnw(T)⬅p(T,H⫽0,⌬H1⫽0) co e-
spond o ⌬H1⬎0, whe eas s a es wi h lowe p essu es co -
espond o ⌬H1⬍0. I is easy o p o e om Eq. 共4兲 ha a
necessa y condi ion o he exis ence o his line is ha VB
⫹
⑀
AA/2⬍(VA/
␯
⬜)⭐
⑀
AA/2. Howe e , ⌬H1changes sign o
a empe a u e Tonly i he alue o he p essu e which co -
esponds o ⌬H1⫽0 is less han he c i ical p essu e a such
empe a u e.
In o de o illus a e such a phenomenology, we shall
s udy he 2D squa e la ice case (
␯
⫽4,
␯
⬜⫽1). In his case,
only c i ical we ing can occu 共 luc ua ion e ec s p eclude
any o he possibili y兲and he alue o Kw, a which he
ansi ion o c i ical we ing occu s is gi en by solu ion o
he equa ion 关29兴
exp共2Kw兲关cosh共2Kw兲⫺cosh共2⌬H1兲兴⫽sinh共2Kw兲.
共11兲
Le us i s conside he case
␯
⬜
⑀
AA/2⭐VA⬍0: ⌬H1⬍0 o
a bi a y alues o T,zB, and VB. Mo eo e , ⌬H1becomes
mo e nega i e as VBdec eases, a ixed Tand zB共o p essu e
p). The ypical diag ams o comple e we ing a e displayed
in Fig. 3. The egion be ween he we ing ansi ion line and
he c i ical line co esponds o a si ua ion o we ing s
兩
␤
兩
␣
,
and he emaining coexis ence egion co esponds o a si u-
a ion o pa ial we ing. The comple e we ing zone inc eases
as VBbecomes mo e nega i e.
When (VA/
␯
⬜)⬍
⑀
AA/2, he subs a e also a o s he ad-
so p ion o pa icles A. This de e mines a compe i ion be-
ween Aand Bspecies. In gene al, when (VA/
␯
⬜)⫺
⑀
AA/2
⬍0.1
⑀
AA
␤
-phase p e e en ial adso p ion can only occu i
VB⫺(VA/
␯
⬜)⬍
⑀
AB⫺
⑀
AA . Figu es 4 and 5 display he we -
ing diag ams o
⑀
AA⬍(VA/
␯
⬜)⬍
⑀
AA/2 and (VA/
␯
⬜)
⭐
⑀
AA , espec i ely. In he i s case, he comple e we ing
ansi ion line o igina es a T⫽0 and e mina es a he em-
pe a u e o comple e we ing o he A-pu e liquid in he
in e ace subs a e-A apo . Fo alues o VB⯝0, ⌬H1⬎0
o all condi ions o coexis ence, and he con igu a ions a e
s
兩
␣
兩
␤
. Fo VB⬍(VA/
␯
⬜)⫹
⑀
AB⫺
⑀
AA , he e appea s a es
cha ac e ized by ⌬H1⬍0. As a consequence, wi hin he co-
FIG. 2. Quali a i e we ing phase diag ams o he LT model o
VB⫽0. Thick solid lines co espond o c i ical lines, do ed lines o
he A-pu e gas-liquid coexis ence lines, and hin solid lines o he
we ing ansi ion lines. The symbol s
兩
␣
兩
␤
(s
兩
␤
兩
␣
) means he bulk
condi ions whe e he e exis we ing o he
␤
-subs a e
(
␣
-subs a e兲in e ace by he
␣
(
␤
) phase, espec i ely 共see ex 兲.
The bulk condi ions unde which he e is pa ial we ing a e also
shown.
FIG. 3. We ing phase diag am o he squa e la ice LT model.
The bulk pa ame e s a e ixed o
⑀
AB⫽0.65
⑀
AA⬍0,
⑀
BB⫽0, and
VA⫽0.25
⑀
AA . The hick solid line co esponds o he bulk c i ical
line, and he hick dashed line o he liquid- apo ansi ion line o
he A-pu e luid. The hin solid lines co espond o he we ing
ansi ion lines o di e en alues o VB. He ea e , he educed
uni s a e de ined ia T*⬅kBT/
兩
⑀
AA
兩
and p*⬅p 0/
兩
⑀
AA
兩
, whe e 0
is he olume o a la ice cell. 共See explana ion in ex .兲
ROMERO-ENRIQUE, RULL, AND BETTOLO MARCONI PHYSICAL REVIEW E 67, 041502 共2003兲
041502-4
exis ence egion a he le o he line pnw(T), he subs a e
adso bs p e e en ially pa icles B, and he we ing is o ype
s
兩
␤
兩
␣
. On he o he hand, a he igh o he cu e pnw(T),
he we ing is o ype s
兩
␣
兩
␤
. The line o we ing ansi ion
ouches angen ially he c i ical cu e a he poin whe e he
cu e in u n pnw(T) c osses he c i ical line. Such a poin
mo es mono onically owa d highe empe a u es as VBde-
c eases, d i ing wi h i he we ing ansi ion line and he
pnw(T) cu e. As also happened o he
␯
⬜
⑀
AA/2⬍VA⬍0
case, he we ing ansi ion line om a s
兩
␤
兩
␣
comple e si u-
a ion o pa ial we ing mo es o lowe p essu es, con e ging
owa d he A-pu e liquid- apo coexis ence line o VB→
⫺⬁共howe e , along his line he e is a comple e we ing by
liquid o he apo -subs a e in e ace o e all he empe a-
u e ange兲.
When VA⭐
␯
⬜
⑀
AA 共see Fig. 5兲, o alues VBnea ze o,
he coexis ence co esponds o comple e we ing s
兩
␣
兩
␤
.
Only when VB⬍(VA/
␯
⬜)⫹
⑀
AB⫺
⑀
AA s a es o pa ial we -
ing eappea , nea he line pnw(T). The beha io o he la -
e as well as he we ing ansi ion line is simila o ha
obse ed in he p e ious case, wi h he di e ence ha he
igh b anch o he we ing ansi ion line 共 ha co esponds
o a ansi ion om pa ial o s
兩
␣
兩
␤
comple e we ing兲begins
a ze o empe a u e.
One expec s ha he beha io o he we ing ansi ion o
he h ee-dimensional case will be quali a i ely simila o he
wo-dimensional case. Howe e , he comple e we ing phase
diag am o he h ee-dimensional Ising model is no o ally
unde s ood ye . Theo e ical 关30兴and simula ion s udies 关31兴
show, in he case o s ic ly sho - anged o ces and in he
absence o an enhanced su ace spin-spin coupling, ha he
we ing ansi ion is con inuous only i Kwis smalle han
KR, whe e KRis he alue o he coupling cons an co e-
sponding o he oughening ansi ion (KR⬇1.85Kc o he
simple cubic la ice兲. Howe e , i such condi ion is no ul-
illed, he we ing ansi ion will occu h ough a sequence o
i s -o de laye ing ansi ions. Theo e ical s udies ha e
claimed ha he we ing ansi ion can be weakly i s o de
ins ead o second o de 关32兴. In he p esen wo k we will
conside he Ising model in he amewo k o he mean- ield
B agg-Williams app oxima ion,
lim
N⬜→⬁
1
N⬜
lnZIsing共K,H,⌬H1兲⫽⌬H1m1⫹兺
j⫽1
⬁
冋
␯
储
2Kmj
2⫹
␯
⬜Kmjmj⫹1⫹Hmj⫺
冉
1⫹mj
2
冊
ln
冉
1⫹mj
2
冊
⫺
冉
1⫺mj
2
冊
ln
冉
1⫺mj
2
冊
册
,共12兲
whe e N⬜is he numbe o cells in a laye , an
兵
mj
其
,j
⫽1,2,...⬁is he equilib ium magne iza ion p o ile ob-
ained by sol ing he Eule -Lag ange equa ions
mj⫽ anh共
␯
储
Kmj⫹
␯
⬜K共mj⫺1⫹mj⫹1兲⫹H兲,共13兲
whe e m0⬅⌬H1/
␯
⬜K. This se o equa ions mus be sol ed
in conjunc ion wi h he asymp o ic condi ion mj→mb o
j→⬁, whe e mbis he spon aneous magne iza ion pe si e in
he bulk case wi h coupling cons an Kand magne ic ield H.
In gene al, di e en solu ions o Eq. 共13兲a e ound. The
equilib ium p o ile is he solu ion ha maximizes unc ional
共12兲. This app oach neglec s capilla y luc ua ions, ha i is
equi alen o assume KR⬅Kcand he we ing ansi ion oc-
cu s as a sequence o laye ing ansi ions 关33,34兴. Once he
equilib ium magne iza ion p o ile is compu ed, he su ace
ension
␴
is ob ained ia
FIG. 4. The same as in Fig. 3 bu o VA⫽0.75
⑀
AA . The do ed
lines co espond o he coexis ence poin s a which ⌬H1⫽0. 共See
ex o explana ion.兲
FIG. 5. The same as in Fig. 4 bu o VA⫽
⑀
AA .共See ex o
explana ion.兲
SURFACE AND CAPILLARY TRANSITIONS IN AN . . . PHYSICAL REVIEW E 67, 041502 共2003兲
041502-5

␴
⫽共⍀⫹pV兲/A,共14兲
whe e ⍀⫽⫺kBTln⌶, wi h ⌶de ined by Eqs. 共1兲and 共12兲,
and he p essu e pby Eq. 共5兲. The adso p ion ⌫, ha i is he
ele an o de pa ame e in he we ing ansi ion, is de ined
as
⌫⫽兺
j⫽1
⬁
共XA共j兲⫺XA
b兲,共15兲
whe e XA(j) is he Amola ac ion p o ile, Eq. 共6兲, by using
he app oxima ion
具
sisj
典
⬇mimj. Finally, XA
bis i s bulk
alue.
We ha e s udied he simple cubic la ice, o which
␯
⫽6 and
␯
⬜⫽1. The magne iza ion p o iles ha e been ob-
ained by an i e a i e me hod o Eq. 共13兲 o j
⫽1,...,Nmax , imposing he condi ion mj⫽mb o j
⬎Nmax . In o de o sample he solu ion space, we use sha p
kink p o iles 关mj⫽⫹1(⫺1) o ⌬H1⬎0(⌬H1⬍0), e-
spec i ely, i j⭐j0and mj⫽mb o j⬎j0] as ini ial magne-
iza ion p o iles. S a ing he i e a ion wi h di e en ini ial
p o iles, in gene al, he algo i hm con e ges owa d di e en
local minima o he ee ene gy. As al eady men ioned, he
global ee ene gy minimum gi es he ue equilib ium mag-
ne iza ion p o ile. Finally, we ha e conside ed di e en al-
ues o Nmax , since he equilib ium p o ile in a comple e
we ing si ua ion has a s ong size dependence. On he con-
a y, no app eciable ini e wid h e ec s occu in a pa ial
we ing si ua ion o Nmax la ge enough. In o de o loca e
he we ing ansi ion, we s udied he equilib ium p o iles a
bulk coexis ence (H⫽0) and cons an empe a u e, a ying
he p essu e.
The we ing phase diag am o VA⫽
⑀
AA and VB
⫽1.25
⑀
AA is plo ed in Fig. 6. As one can see, he quali a i e
beha io o he we ing ansi ion is simila o ha obse ed
in he wo-dimensional case. Howe e , he we ing ansi ion
is associa ed o se ies o i s -o de laye ing ansi ions be-
ween su ace phases wi h he same su ace ension, bu di -
e en adso p ions. The beha io o he laye ing ansi ions a
low empe a u es shows su p ising ea u es. Figu e 7 shows
he i s ou laye ing ansi ions o T*⬅kBT/
兩
⑀
AA
兩
⫽0.25.
I is obse ed ha he i s laye ing ansi ion does occu o
p essu es highe han he bulk c i ical p essu e. Conse-
quen ly, o a ixed p essu e be ween he i s laye ing an-
si ion c i ical poin and he bulk c i ical poin co esponding
o his empe a u e, he laye ing ansi ion will ex end o
lowe empe a u es han he c i ical empe a u e co espond-
ing o he LCSP. This beha io is ound only o s
兩
␤
兩
␣
com-
ple e we ing in he low- empe a u e zone. The p essu e is an
inc easing unc ion on H o ixed empe a u e Tand zB共so
bo h Kand ⌬H1a e ixed兲. Consequen ly, as K⬎Kc o he
laye ing ansi ion c i ical poin , he co esponding p essu e
can only be bigge han he bulk c i ical p essu e i he an-
si ion occu s o H⬎0. This implies ha ⌬H1mus be nega-
i e, and hus he we ing is o ype s
兩
␤
兩
␣
. The physical
in e p e a ion o his phenomenon is ha he su ace adso bs
p e e en ially Bpa icles. Such a mechanism compe es wi h
he A-Bbonding 共a leas close o he subs a e兲. The e o e,
he endency o phase sepa a e in a luid laye close o he
subs a e is enhanced 共because he A-Bbonding a o s mix-
ing兲 o empe a u es lowe han he 共lowe 兲c i ical empe a-
u e. This ac was also obse ed in compu e simula ions o
a di e en model o associa ing bina y mix u e in Re . 关17兴.
Howe e , we mus s ess ha his phenomenon is no ela ed
o an enhanced coupling 共i.e., he in e ac ions be ween pa -
icles in he i s laye close o he subs a e a e inc eased
espec he bulk alues兲, ha i is known o s abilize su ace
ansi ions o empe a u es highe o he bulk c i ical one in
he Ising model 关2兴. On he o he hand, he compe i ion be-
ween subs a e p e e en ial adso p ion and he A-Bbonding
should be also ele an o p ewe ing ansi ions.
IV. CONFINEMENT BETWEEN SYMMETRIC WALLS
We u n o conside he beha io o he Lin-Taylo model
when con ined be ween wo iden ical pa allel subs a es.
FIG. 6. We ing phase diag am o he simple cubic 共sc兲LT
model in he mean- ield app oxima ion. The subs a e- luid in e ac-
ions a e VA⫽
⑀
AA and VB⫽1.25
⑀
AA . The c i ical line 共 hick solid兲,
A-pu e apo -liquid ansi ion line 共dashed兲, and pnw(T) line 共do -
ed兲a e plo ed. The we ing ansi ion poin s 共diamonds兲a e also
ep esen ed. The line ha joins hem is only o guide he eye.
FIG. 7. Laye ing ansi ions o T*⫽0.25 o he sc LT model in
he mean- ield app oach. The in e ac ion couplings a e he same as
in Fig. 6. The dashed lines co espond o he bulk coexis ence, and
he solid lines co espond o he bulk condi ions co esponding o
he laye ing ansi ions. In he inse , he coexis ence adso p ions o
he laye ing ansi ion 共only he i s ou ansi ions a e plo ed兲.
The a ow shows he bulk c i ical p essu e.
ROMERO-ENRIQUE, RULL, AND BETTOLO MARCONI PHYSICAL REVIEW E 67, 041502 共2003兲
041502-6
Con inemen in he wo-dimensional case co esponds o a
quasi-one-dimensional si ua ion. Thus luc ua ions p eclude
he exis ence o capilla y ansi ions i he in e ac ions a e
sho anged. Consequen ly, only he 3D case will be s udied.
We shall conside a simple cubic la ice wi hin he mean-
ield app oxima ion. The subs a e- luid in e ac ions a e se
o VA⫽
⑀
AA and VB⫽1.25
⑀
AA , ha s abilize he i s 共su -
ace兲laye ing ansi ion a empe a u es lowe han he lowe
c i ical empe a u e. We also assume ha he sepa a ion be-
ween pla es is NDla ice uni s. The ee ene gy o he con-
ined mix u e is gi en by Eq. 共1兲, wi h he mean- ield pa i-
ion unc ion ZIsing(K,H,⌬H1) gi en by
lim
N⬜→⬁
1
N⬜
lnZIsing共K,H,⌬H1兲⫽⌬H1共m1⫹mND兲⫹兺
j⫽1
ND
冋
␯
储
2Kmj
2⫹
␯
⬜Kmjmj⫹1⫹Hmj⫺
冉
1⫹mj
2
冊
ln
冉
1⫹mj
2
冊
⫺
冉
1⫺mj
2
冊
ln
冉
1⫺mj
2
冊
册
.共16兲
The equa ion o he equilib ium p o ile is gi en by Eq.
共13兲wi h he bounda y condi ion Nmax⫽ND/2 o NDe en,
o (ND⫹1)/2 o NDodd. The alues o mj o j⭓Nmax a e
ob ained om he symme y o he p oblem: i.e., mj
⫽mND⫺j⫹1. In addi ion o he laye ing ansi ions 共 ela ed o
he phenomenology obse ed on he semi-in ini e case兲, he
capilla y ansi ions esul s om he shi o he bulk coex-
is ence unde con inemen . Nea he cen e o he po e bo h
phases ha e di e en magne iza ions, in con as wi h he
laye ing and p ewe ing ansi ions whe e he p o iles di e
in a localized su ace egion and hey ha e he same alues
a enough om ha egion. In he Ising model, he shi is
always owa d highe alues o K共i.e., lowe e ec i e em-
pe a u e兲and Hnega i e (Hposi i e兲when ⌬H1posi i e
(⌬H1nega i e兲, espec i ely.
The cases ND⫽1 and ND⫽2 can be s udied analy ically
since mj⫽m o all j. The bulk condi ions, o which he
capilla y ansi ion akes place a e shown in Fig. 8. In spi e
o he ac ha he empe a u e ange o which he capilla y
ansi ion occu s is educed, a low empe a u es he p essu e
ange is la gely inc eased. Hence, a a gi en p essu e, he
LCSP shi s o lowe empe a u es and can e en disappea .
This means ha o ce ain alues o he p essu e ange he e
a e no immiscibili y islands in he con ined sys em, bu he
usual bell shaped coexis ence line e mina ing in a UCSP.
This is a di ec consequence o he di ec ional cha ac e o
he subs a e-pa icle Bin e ac ion. The kink ha is obse ed
in he capilla y c i ical line co esponds o he c ossing o he
pnw(T) line.
Fo la ge alues o ND, he beha io o he capilla y
ansi ions a e s udied nume ically. As ND→⬁ he capilla y
c i ical line mus mo e owa d he bulk c i ical cu e. We
shall conside he iso he m T*⫽0.25, as a ep esen a i e
case o he low- empe a u e egime. The excess ee ene gy
␻
eads
␻
⫽共⍀⫹pV兲/A共17兲
and con e ges o 2
␴
when ND→⬁, wi h
␴
de ined by Eq.
共14兲. The ele an o de pa ame e o he capilla y ansi-
ions is he a e age mola ac ion X
¯
A, de ined as
X
¯
A⬅1
ND兺
j⫽1
ND
XA共j兲共18兲
ha con e ges o he bulk coexis ence mola ac ions as
ND→⬁.
We ha e ob ained he cons an empe a u e phase dia-
g ams o T*⫽0.25 共see Fig. 9兲. Va ious ansi ions a e ob-
se ed o ND⭓3. One o hese ansi ions con e ges owa d
he bulk phase ansi ion o la ge sepa a ions and is iden i-
ied wi h he ue capilla y ansi ion. The emaining ansi-
ions a e cha ac e ized by ha ing simila a e age mola ac-
ions and co espond o he laye ing ansi ions obse ed in
he p esence o a single subs a e. The co e ages 关as de ined
by Eq. 共15兲兴 co esponding o he coexis ing phases in hese
ansi ions con e ge e y as o he semi-in ini e alues, as
expec ed om hei su ace na u e. Howe e , he numbe o
hese ansi ions depends s ongly on ND, since he capilla y
FIG. 8. Capilla y ansi ions o he mean- ield sc LT model
(VA⫽
⑀
AA ,VB⫽1.25
⑀
AA) o ND⫽1共 hin solid line兲and ND⫽2
共do -dashed line兲. In bo h cases, he uppe line is he capilla y c i i-
cal line, and he lowe line is he A-pu e capilla y gas-liquid an-
si ion. The bulk condi ions unde which he e is
␣
-
␤
capilla y co-
exis ence a e he egions enclosed by hei espec i e lines, and he
T⫽0 axis. By compa ison, he bulk c i ical line 共 hick solid line兲
and he A-pu e gas-liquid ansi ion line 共 hick dashed line兲.
SURFACE AND CAPILLARY TRANSITIONS IN AN . . . PHYSICAL REVIEW E 67, 041502 共2003兲
041502-7
ansi ion, which depends sensibly on ND, compe es wi h
hem 关35,36兴. Fo ins ance, in he case o 3⭐ND⬍10 only
he i s laye ing ansi ion appea s, while he second appea s
o ND⭓10.
The alues o he capilla y c i ical p essu e o small ND
a e la ge han he co esponding bulk c i ical alues. By
inc easing ND, he capilla y c i ical p essu e dec eases up o
a minimum, co esponding o a p essu e less han he bulk
c i ical p essu e. F om such alue i con e ges mono oni-
cally o he bulk c i ical alue 共see Fig. 10兲. This ac means
ha along an isoba ic he LCSP shi s o lowe empe a u es
only o e y na ow po es. Consequen ly, he s abilizing
mechanism o he coexis ence egion below he bulk LCSP
seems o be only e ec i e o small alues o ND. The de-
ia ions o he capilla y c i ical p essu e wi h espec o he
bulk alue in he ange 5⬍ND⭐15 ollow a powe law ⌬p
⬃ND
⫺x, wi h x⫽1.04⫾0.09. This beha io is consis en wi h
he indings o he con ined la ice gas o in e media e al-
ues o ND epo ed in Re . 关35兴. The scaling beha io ⌬p
⬃ND
⫺1/
␯
(ND
⫺2in he mean- ield app oach兲p edic ed by he
Nakanishi-Fishe heo y should be obeyed o la ge alues
o ND. Howe e , ou esul s do no ex end o ha egion due
o he nume ical e o in es ima ing he capilla y c i ical
p essu e close o he bulk c i ical poin . Ne e heless, we do
no expec a change in he endency o la ge alues o ND
and we expec ou esul s o be quali a i ely co ec o all
alues o ND. I is in e es ing o no e ha he slab app oxi-
ma ion 关35兴, which co esponds o se mj⫽m o all j, p e-
dic s ha o su icien ly small T* he capilla y c i ical p es-
su e is la ge han he bulk alue o all alues o ND. The
ailu e o his app oach is no su p ising, since i gi es poo
esul s o he con ined la ice gas close o he capilla y c i i-
cal poin as i o e es ima es bo h Kand Ha he capilla y
c i ical poin 关35兴. This ac al e s he balance be ween he
di e en mechanisms in ol ed in he shi o he c i ical
poin , leading o he w ong esul men ioned abo e.
V. THE CYLINDRICAL PORE NETWORK
Finally, we conside he con inemen o he LT mix u e in
a ne wo k o pa allel in ini ely long, cylind ical po es. This
model can be unde s ood as a c ude app oxima ion o ad-
so p ion in zeoli es. I ep esen s ano he ins ance in which
he subs a e can p e en he o ma ion o hyd ogen bonds:
he geome ical hind ance due o he con inemen in e y
na ow nanopo es. We shall assume ha he po e diame e is
so small ha he A-Bbonds can o m only along he di ec-
ion zpa allel o he cylinde axis 共see Fig. 11兲. The cylinde
diame e is aken o be less han wice he A-Bcollision
diame e . In addi ion, he po es ha e diame e s less han wo
A-pa icle diame e s and hey o m a wo-dimensional ne -
FIG. 9. X
¯
A⫺pphase diag ams o T*⫽0.25 and di e en al-
ues o ND:ND⫽1共dashed-double do ed line兲,ND⫽2共dashed
line兲,ND⫽5共do ed line兲, and ND⫽15 共 hick solid line兲, unde he
same condi ions as in Fig. 8. The bulk ansi ion 共 hin solid line兲is
also plo ed o compa ison. The inse co esponds o an enla ge-
men o he zone a ound he bulk c i ical poin .
FIG. 10. Capilla y educed c i ical p essu es p* e sus ND o
T*⫽0.25 共same condi ions as in Fig. 8兲. The do ed line co e-
sponds o he bulk c i ical p essu e. Inse , enla gemen o he mini-
mum zone.
FIG. 11. Schema ic ep esen a ion o he LT model adso bed in
a ne wo k o na ow in ini ely long cylinde s. The Apa icles a e
ep esen ed as sphe ical pa icles, and he Bpa icles as sphe ical
pa icles wi h an in e ac ion side 共shadow ci cle兲. In his case, he
coo dina ion numbe is
␯
⬜⫽6.
ROMERO-ENRIQUE, RULL, AND BETTOLO MARCONI PHYSICAL REVIEW E 67, 041502 共2003兲
041502-8
wo k wi h coo dina ion numbe
␯
⬜. We allow a enua ed
in e ac ions be ween pa icles in nea es -neighbo po es, in
o de o allow a capilla y ansi ion. A simila zeoli e model
has been employed o s udy he capilla y ansi ion o me h-
ane adso bed in ALPO4-5 关37兴.
The in-po e luid- luid in e ac ions ha e he same o m as
in bulk, wi h coupling cons an s
⑀
AA ,
⑀
AB , and
⑀
BB . How-
e e , he in e ac ions be ween pa icles in nea es -neighbo
po es a e a enua ed o
⑀
˜
AA ,
⑀
˜
AB , and
⑀
˜
BB (
兩
⑀
˜
ij
兩
⬍
兩
⑀
ij
兩
), al-
hough he A-Band B-Bin e ac ions di ec ional cha ac e
emains he same as be o e. This sys em beha es as a bulk
la ice model wi h aniso opic in e ac ions. The g and-
canonical- ee ene gy pe uni olume eads
⍀
V⫽⫺ kBT
0
冋
␯
⬜
2ln共1⫹2z
˜
B⫹z
˜
B
2e⫺
␤⑀
˜
BB兲
⫹ln共1⫹2zB⫹zB
2e⫺
␤⑀
BB兲⫹H⫺
␯
⬜K⬜
2⫺K
储
⫹1
NlnZIsing共K⬜,K
储
,H兲
册
,共19兲
whe e z
˜
B⬅zBe⫺
␤
VBis he e ec i e ac i i y o he Bpa icles
o ien ed in he plane pe pendicula o he cylinde axis, and
ZIsing is canonical pa i ion unc ion o an aniso opic Ising
model cha ac e ized by couplings K
储
in he pa allel di ec-
ions o he cylinde axis and K⬜in he pe pendicula di ec-
ions, and an e ec i e magne ic ield H.K
储
has he same
exp ession as he bulk coupling cons an , Eq. 共2兲, while K⬜
and Ha e de ined as
K⬜⫽⫺
␤⑀
˜
AA
4⫹1
2ln
冉
冑
1⫹2z
˜
B⫹z
˜
B
2e⫺
␤⑀
˜
BB
1⫹z
˜
Be⫺
␤⑀
˜
AB
冊
,共20兲
H
˜
⫽H⫹
␯
⬜
冋
␤
4
冉
⑀
AA⫺
⑀
˜
AA⫺2VA
␯
⬜
冊
⫹1
4ln
冉
1⫹2zB⫹zB
2e⫺
␤⑀
BB
1⫹2z
˜
B⫹z
˜
B
2e⫺
␤⑀
˜
BB
冊
册
,共21兲
wi h Hgi en by Eq. 共3兲. One can see ha con inemen shi s
he e ec i e magne ic ield Hwi h espec o i s bulk alue
wi h a con ibu ion analogous o ha ound o plana sub-
s a es 关compa e Eqs. 共4兲and 共21兲兴. In addi ion, he con ine-
men de e mines an aniso opy in he couplings. The
A-pa icle mola ac ion XA eads
XA⫽nA
nA⫹nB
1⫹nB
2,共22兲
whe e nA⫽(1⫹m)/2 0is he A-pa icle densi y, and nB
1
(nB
2) is he densi y o Bpa icles o ien ed in a pe pendicula
共pa allel兲di ec ion o he cylinde axis, espec i ely,
nB
1⫽
␯
⬜
4 0
冉
z
˜
B⫹z
˜
B
2e⫺
␤⑀
˜
BB
1⫹2z
˜
B⫹z
˜
B
2e⫺
␤⑀
˜
BB
⫹z
˜
Be⫺
␤⑀
˜
AB
1⫹z
˜
Be⫺
␤⑀
˜
AB
冊
⫺
␯
⬜
4 0
冉
z
˜
B⫹z
˜
B
2e⫺
␤⑀
˜
BB
1⫹2z
˜
B⫹z
˜
B
2e⫺
␤⑀
˜
BB
⫺z
˜
Be⫺
␤⑀
˜
AB
1⫹z
˜
Be⫺
␤⑀
˜
AB
冊
冉
⳵
lnZIsing /N
⳵
K⬜
冊
⫺
␯
⬜
2 0
z
˜
B⫹z
˜
B
2e⫺
␤⑀
˜
BB
1⫹2z
˜
B⫹z
˜
B
2e⫺
␤⑀
˜
BB m,共23兲
nB
2⫽1
2 0
冉
zB⫹zB
2e⫺
␤⑀
BB
1⫹2zB⫹zB
2e⫺
␤⑀
BB
⫹zBe⫺
␤⑀
AB
1⫹zBe⫺
␤⑀
AB
冊
⫺1
2 0
冉
zB⫹zB
2e⫺
␤⑀
BB
1⫹2zB⫹zB
2e⫺
␤⑀
BB
⫺zBe⫺
␤⑀
AB
1⫹zBe⫺
␤⑀
AB
冊
冉
⳵
lnZIsing /N
⳵
K
储
冊
⫺1
0
zB⫹zB
2e⫺
␤⑀
BB
1⫹2zB⫹zB
2e⫺
␤⑀
BB m.共24兲
Phases cha ac e ized by di e en mola ac ions XAmay
coexis when he spon aneous 共i.e., H
˜
⫽0) magne iza ion m
⫽⫾
兩
m
兩
⫽0. We conside he case
⑀
AA⫽⫺1,
⑀
AB⫽⫺0.6,
and
⑀
BB⫽0, ha co esponds o a case ha p esen s closed
loops o immiscibili y in he bulk. On he o he hand, as he
hyd ogen bond in e ac ions a e e y sho anged, we will
ake
⑀
˜
AB⫽
⑀
˜
BB⫽0 as physically sensible alues. Mo eo e ,
⑀
˜
AA is se o be
␣⑀
AA , whe e
␣
is a posi i e numbe less han
one.
In o de o disc imina e be ween di e en mechanisms,
we chose he subs a e- luid in e ac ions in such a way ha
H
˜
⫽H, i.e., VA⫽
␯
⬜(1⫺
␣
)
⑀
AA/2 and VB⫽0. Hence, he
p e e en ial subs a e adso p ion e ec 共 e y simila o ha
s udied in he slab geome y兲is supp essed and we can ocus
on he coupling aniso opy.
We ha e s udied he 2D squa e la ice case, o which
analy ical exp essions o ZD,
␯
a e known in he aniso opic
case 关38兴. Howe e , we expec ha he ea u es ob ained o
his case will be quali a i ely co ec also o he 3D case.
F om he exac solu ion, he alues o (K⬜,K
储
) ha co e-
spond o he phase coexis ence sa is y he ollowing ela ion-
ship:
SURFACE AND CAPILLARY TRANSITIONS IN AN . . . PHYSICAL REVIEW E 67, 041502 共2003兲
041502-9