Surface and capillary transitions in an associating binary mixture model

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 041502-3 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