Interfacial structure at a two-dimensional wedge filling transition: Exact results and a renormalization group study

In e acial s uc u e a a wo-dimensional wedge illing ansi ion: Exac esul s and a eno maliza ion g oup s udy J. M. Rome o-En ique,1,2 A. O. Pa y,1and M. J. G eenall1 1Depa men o Ma hema ics, Impe ial College, 180 Queen’s Ga e, London SW7 2BZ, Uni ed Kingdom 2Depa amen o de Física A ómica, Molecula y Nuclea , A ea de Física Teó ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, 41080 Se illa, Spain (Recei ed 11 No embe 2003; published 2 June 2004) In e acial s uc u e and co ela ion unc ions nea a wo-dimensional wedge illing ansi ion a e s udied using e ec i e in e acial Hamil onian models. An exac solu ion o sho ange binding po en ials and esul s o K a ze binding po en ials show ha su icien ly close o he illing ansi ion a new leng h scale eme ges and con ols he decay o he in e acial p o ile ela i e o he subs a e and he co ela ions be ween in e acial posi ions abo e di e en posi ions. This new leng h scale is much la ge han he in insic in e acial co ela- ion leng h, and i is ela ed geome ically o he a e age alue o he in e acial posi ion abo e he wedge midpoin . The in e acial beha io is consis en wi h a b ea he mode luc ua ion pic u e, which is shown o eme ge om an exac decima ion unc ional eno maliza ion g oup scheme ha keeps he geome y in a ian . DOI: 10.1103/PhysRe E.69.061604 PACS numbe (s): 68.08.Bc, 05.70.Np, 68.35.Md, 68.35.Rh I. INTRODUCTION Fluid adso p ion in wedge and cone-shaped nonplana ge- ome ies has a ac ed much a en ion in he las ew yea s [1–5]. Geome y plays an impo an ole in he su ace phase diag am, and new phase ansi ions as he illing ansi ion a ise. The modynamic conside a ions [6–8]p edic ha he gas-liquid in e ace unbinds om he wedge be o e he we - ing empe a u e Twco esponding o he subs a es. So, he wedge is comple ely illed by liquid o empe a u es highe han he illing empe a u e T ⬍Tw, whe e T is gi en by he condi ion ␪ 共T 兲= ␣ 共1兲 and ␪ 共T兲is he empe a u e-dependen con ac angle o a liquid d op on he plana subs a e and ␣ is he il angle (see Fig. 1). Capilla y wa e models show ha he illing ansi- ion can be c i ical e en hough he we ing ansi ion co e- sponding o he subs a e is i s o de , and ha in e acial luc ua ions a e enhanced wi h espec o he we ing case [3,4]. Fo he wo-dimensional (2D)wedge illing ansi ion in shallow wedges cha ac e ized by a small angle ␣ wi h espec o he xaxis (see below), he e exis s a ema kable co a iance ela ionship be ween he wedge midpoin p ob- abili y dis ibu ion unc ion Pw 1共l0兲in he illing luc ua ion egime and he plana 1-poin p obabili y dis ibu ion unc- ion P ␲ 1共l0兲cha ac e is ic o a s ong- luc ua ion egime c i i- cal we ing ansi ion: Pw 1共l0; ␪ , ␣ 兲=P ␲ 1共l0; ␪ − ␣ 兲.共2兲 This exp ession es ablishes a connec ion be ween wo appa - en ly un ela ed phenomena, he deep o igin o which is s ill elusi e. The co a iance ela ionship has been obse ed also in acu e wedges [9], Ising model exac calcula ions [10], and compu e simula ions [11]. Al hough he co a iance ela ion- ship is es ic ed o he in e acial beha io abo e he wedge midpoin , some o he quan i ies, such as he local suscep i- bili y, which is ela ed o he 2-poin co ela ion unc ion, also showed a modi ied co a iance ela ionship [5]. Conse- quen ly, i is in e es ing o see i he co a iance ex ends o highe -o de p obabili y dis ibu ion unc ions. In his pape we s udy he s uc u e o he in e acial p o- ile and co ela ions o 2D wedge illing phenomena. Exac esul s o he capilla y wa e e ec i e Hamil onian heo y in he illing luc ua ion egime a e ob ained as an ex ension o he analysis p esen ed in Re . [12]. The exac esul s show he appea ance o a new leng h scale ␰ Fac oss he wedge close o he c i ical illing ansi ion. This scale con ols he decay o he in e acial p o ile, local oughness, and co ela- ions, and is ela ed geome ically o he wedge midpoin a e age in e ace posi ion. Fo he local p ope ies, we ound a e y in e es ing ela ionship be ween he wedge 1-poin p obabili y dis ibu ion unc ion and he co esponding unc- ions in he plana geome y, which can enligh en he o igin o he wedge co a iance. FIG. 1. Schema ic illus a ion o a ypical in e acial con igu a- ion in he wedge geome y. The ele an co ela ion leng h scales ␰ xand ␰ ⬜共x兲a e also highligh ed. O he no a ion is de ined in he ex . PHYSICAL REVIEW E 69, 061604 (2004) 1539-3755/2004/69(6)/061604(14)/$22.50 ©2004 The Ame ican Physical Socie y69 061604-1 Rega ding he wo-poin co ela ion unc ions, we ound a con i ma ion in he scaling limi o he b ea he mode pic u e [3,4], which s a es ha he in e ace is e ec i ely in ini ely s i in he illed egion and is d i en by luc ua ions o he wedge midpoin in e acial posi ion, i.e., c i ical e ec s a 2D wedge illing a ise simply om local ansla ions in he heigh o he la , illed in e acial egion. Finally, we explain he c i ical beha io o he illing an- si ion in he unc ional eno maliza ion g oup app oach. As he geome y is undamen al in he unde s anding o he c i i- cal illing ansi ion, we choose a scheme ha lea es he wedge geome y in a ian . We show ha he b ea he mode pic u e eme ges as a s aigh o wa d consequence. The p e- dic ions o he c i ical beha io a e in comple e ag eemen wi h exac solu ions. Ou pape is o ganized as ollows. In Sec. II we desc ibe he con inuous ans e -ma ix o malism and he de ini ion o he wedge n-poin in e acial p obabili y dis ibu ion unc- ions. We apply his o malism o he case o con ac binding po en ials in Sec. III and, in pa icula , calcula e analy ically he 1-poin p obabili y dis ibu ion unc ion and he 2-poin co ela ion unc ions. Some esul s o K a ze binding po- en ials will be p esen ed in Sec. IV. In Sec. V we analyze he b ea he mode pic u e and de i e a ela ion be ween wo im- po an scaling unc ions. Sec ion VI is de o ed o he de el- opmen o a eno maliza ion g oup heo y o 2D c i ical ill- ing ansi ion, which equi es a gene aliza ion o p e ious app oaches o c i ical we ing. A b ie conclusion is p e- sen ed in Sec. VII. II. THE FORMALISM Conside a wo-dimensional wedge o med by he in e - sec ion o wo equal plana subs a es a angles ± ␣ wi h e- spec o he ho izon al (see Fig. 1). We suppose ha he wedge is in con ac wi h a bulk apo phase a sa u a ion condi ions, i.e., in equilib ium wi h he liquid phase, and he subs a es p e e en ially adso b he liquid phase. Ou s a ing poin is he e ec i e in e acial Hamil onian o shallow wedges: ␤ H关l兴= 冕 −X/2 X/2 dx 再 ⌺ 2 冉 dy dx 冊 2+W共y共x兲− ␣ 兩x兩兲 冎 ,共3兲 whe e y共x兲is he in e acial local heigh measu ed wi h e- spec o he ho izon al, Xis he in e acial ho izon al leng h, kBT⌺is he in e acial s i ness, kBTW共l兲is he local binding po en ial, and ␤ ⬅1/kBT. We impose pe iodic bounda y con- di ions a he ends, i.e., y共−X/2兲=y共X/2兲. While he model assumes ha he wedge angle is shallow 共 an ␣ ⬇ ␣ 兲, his does no in luence he uni e sal p ope ies occu ing in he asymp o ic c i ical limi ␪ → ␣ a ixed ␣ . S udies o illing in acu e wedges based on mo e e ined in e acial [9]and mi- c oscopic, Ising models [10]yield iden ical esul s o uni- e sal quan i ies. De ining he local ela i e heigh be ween he apo -liquid in e ace and he subs a e l共x兲=y共x兲− ␣ 兩x兩, Eq. (3)can be ew i en as [2] ␤ H关l兴=X⌺ ␣ 2 2+ 冕 −X/2 X/2 dx 再 ⌺ 2 冉 dl dx 冊 2+⌺ ␣ 冉 dl dx 冊 关2⍜共x兲−1兴 +W„l共x兲… 冎 ,共4兲 whe e ⍜共x兲is he Hea iside s ep unc ion. In eg a ing by pa s o elimina e he e m p opo ional o 共dl/dx兲, he e ec- i e Hamil onian can be exp essed as ␤ H关l兴=X⌺ ␣ 2 2+2⌺ ␣ l共X/2兲−2⌺ ␣ l共0兲 + 冕 −X/2 X/2 dx 再 ⌺ 2 冉 dl dx 冊 2+W„l共x兲… 冎 .共5兲 The i s wo e ms in he equa ion a e i ele an cons an s o he in e acial p ope ies in he wedge, he hi d one is he o igin o he boos ac o ha dec eases he pinning e ec o he binding po en ial [2], and he ou h one co esponds o he e ec i e Hamil onian o an equi alen plana in e ace p oblem. As he p obabili y dis ibu ion o an in e acial con- igu a ion is p opo ional o exp共− ␤ H兲we can ela e he wedge and plana p obabili y dis ibu ions in a s aigh o - wa d way. In pa icula , he n-poin wedge co ela ion unc- ions can be ela ed o 共n+1兲-poin co ela ion unc ions in he plana case by adding he wedge midpoin posi ion. Howe e , he p esence o he boos ac o will al e signi i- can ly he beha io o he wedge co ela ion unc ions wi h espec o hei plana coun e pa s. Ou app oach is based on a s anda d applica ion o ans e -ma ix me hods [13]. The pa i ion unc ion Z ␲ 共l1,l2,x1,x2兲o he in e ace wi h ixed end poin s 共x1,l1兲 and 共x2,l2兲wi h x2⬎x1in he p esence o a plana subs a e is de ined as he ollowing pa h in eg al: Z ␲ 共l1,l2,x1,x2兲⬅Z ␲ 共l1,l2;x2−x1兲 = 冕 Dlexp 冉 − 冕 x1 x2dx 冋 ⌺ 2 冉 dl dx 冊 2+W共l兲 册 冊 . 共6兲 The pa i ion unc ion, Eq. (6), is he solu ion o he ol- lowing Sch ödinge equa ion: 冋 ⳵ ⳵ x+W共l2兲−1 2⌺ ⳵ 2 ⳵ l2 2 册 Z ␲ 共l1,l2;x兲=0, 共7兲 wi h he ini ial condi ion Z ␲ 共l1,l2;0兲= ␦ 共l2−l1兲,共8兲 whe e ␦ 共x兲is he Di ac del a unc ion. Fo mally, he pa i ion unc ion can be exp essed as Z ␲ 共l1,l2;x兲=兺 i ␺ i ⴱ共l1兲 ␺ i共l2兲exp共−Eix兲,共9兲 whe e ␺ 共l兲and Eia e he eigen unc ions and eigen alues o he ime-independen Sch ödinge equa ion: ROMERO-ENRIQUE, PARRY, AND GREENALL PHYSICAL REVIEW E 69, 061604 (2004) 061604-2 −1 2⌺ ␺ n ⬙共l兲+W共l兲 ␺ n共l兲=En ␺ n共l兲,共10兲 wi h app op ia e bounda y condi ions. In he he modynamic limi Z ␲ ⬃exp共− ␤ X兲as X→⬁, whe e ␤ =⌺共cos ␪ −1兲is he excess ee ene gy pe in e acial leng h. Consequen ly, Eq. (9)implies ha E0= ␤ , so ha in he low con ac angle limi , E0⬇−⌺ ␪ 2/2. The n-poin dis ibu ion unc ions can be ob ained in e ms o Z ␲ 共l1,l2;x兲as P ␲ 共1;...;n兲= lim X→⬁ 兿 i=0 n Z ␲ 共li,li+1;xi+1 −xi兲 Z ␲ 共l−X/2,lX/2;X兲 = ␺ 0共l1兲 ␺ 0 ⴱ共ln兲兿 i=1 n−1 Z ␲ 共li,li+1;xi+1 −xi兲eE0共xi+1−xi兲, 共11兲 whe e i⬅共li;xi兲,xn+1=−x0⬅X/2, and l0=ln+1=lX/2. Fo n =1, P ␲ 共i兲⬅兩 ␺ 0共li兲兩2. F om Eqs. (11)and (9)i is clea ha i he dis ance be ween wo subse s 兵x1,...,xm其and 兵xm+1,...,xn其is much g ea e han he plana co ela ion leng h ␰ 储 ⬅1/共E1−E0兲(wi h E1 he i s exci ed s a e eigen- alue), he dis ibu ion unc ion ac o izes and he wo sub- se s become unco ela ed. The n-poin wedge dis ibu ion unc ions Pw共1;...;n兲 can be exp essed, in gene al, in e ms o 共n+1兲-poin plana dis ibu ion unc ions. So, o a se 兵x−m⬍¯⬍x−1⬍0⬍x1⬍¯xn其, hey can be exp essed as Pw共−m;...;n兲= 冕 0 ⬁ dl0e2⌺ ␣ l0 具0兩e2⌺ ␣ l0兩0典P ␲ 共−m;...; −1;0;1; ... ;n兲 =Pw共−1;1兲P ␲ 共−m;...;n兲 P ␲ 共−1;1兲,共12兲 whe e 具n兩 共l兲兩m典⬅兰0 ⬁dl ␺ n共l兲 共l兲 ␺ m ⴱ共l兲.I 0艋x1⬍¯⬍xn, he exp ession o Pw共1;...;n兲is sligh ly simple : Pw共1;...;n兲= 冕 0 ⬁ dl0e2⌺ ␣ l0 具0兩e2⌺ ␣ l0兩0典P ␲ 共0;1; ... ;n兲 =Pw共1兲P ␲ 共1;...;n兲 P ␲ 共1兲.共13兲 A simila exp ession is ound i x1⬍¯⬍xn艋0. Finally, i x=0 is included in he xse , he wedge n-poin dis ibu ion unc ion educes o Pw共−m;...;n兲=e2⌺ ␣ l0 具0兩e2⌺ ␣ l0兩0典P ␲ 共−m;...;n兲.共14兲 Al hough his app oach is gene al o a bi a y binding po en ials, we will es ic ou sel es o some special cases. The i s case will be con ac po en ials, in which W共l兲=0 o l⬎0, W共l兲=+⬁ o l⬍0 and a he wall he eigen unc ions ul ill he bounda y condi ion [13] 冏 ⳵ ⳵ lln ␺ 共l兲冏l=0 =− ␶ ,共15兲 whe e ␶ is p opo ional o he de ia ion om he c i ical we ing empe a u e. Fo ␶ ⬎0 he con ac angle is ela ed o ␶ ia ␶ =⌺ ␪ [13]. These po en ials can be unde s ood as he limi ing case o a squa e-well binding po en ial when he well wid h ends o ze o. I s impo ance is h ee old. Fi s , his case co esponds o he illing luc ua ion egime, which p e ious s udies show o be he ele an one o po en ials which decay as e han 1/l. Second, he e is an analy ical exp ession o Z ␲ 共l1,l2;x兲[13]gi en by Z ␲ 共l1,l2;x兲=冑⌺ 2 ␲ x共e−⌺共l2−l1兲2/2x+e−⌺共l1+l2兲2/2x兲 + ␶ e ␶ 2x/2⌺− ␶ 共l1+l2兲e c 冉 冑⌺ 2x共l1+l2兲− ␶ 冑x 2⌺ 冊 . 共16兲 Finally, his case can be compa ed o mo e mic oscopic e- sul s, such as he exac solu ions o he in e acial p ope ies o he co ne illing o an Ising model. Ano he in e es ing case is he K a ze binding po en ial [14] W共l兲=− ␾␪ l+w l2,共17兲 whe e ␾ =共1+冑1+8⌺w兲/2 and we assume Di ichle bound- a y condi ions a he o igin. P e ious s udies indica e ha his class o binding po en ials co esponds o he ma ginal case be ween he mean- ield and luc ua ion-domina ed e- gimes o he c i ical illing ansi ion. The Laplace ans o m o Z ␲ 共l1,l2;x兲,Z ˜ ␲ 共l1,l2,E兲is gi en by [14] Z ˜ ␲ 共l1,l2,E兲 = 冕 0 ⬁ dx eExZ ␲ 共l1,l2;x兲 =冑E0 E⌫ 冋 ␾ 冉 1−冑E0 E 冊 册 ␪ ⌫关2 ␾ 兴W ␾ 冑E0/E, ␾ −1/2共冑−8⌺El⬎兲 ⫻M ␾ 冑E0/E, ␾ −1/2共冑−8⌺El⬍兲,共18兲 whe e E0=−⌺ ␪ 2/2, l⬎=max共l1,l2兲,l⬍=min共l1,l2兲,⌫共x兲is he Gamma unc ion, and inally M ␬ ,m共z兲and W ␬ ,m共z兲a e Whi ake unc ions, ela ed o con luen hype geome ic unc ions. III. EXACT RESULTS FOR CONTACT BINDING POTENTIALS In his sec ion we will ob ain and analyze some ele an wedge dis ibu ion unc ions o con ac binding po en ials. In pa icula , we will e isi he 1-poin dis ibu ion unc ion (conside ed p e iously by ou g oup [12]) and he 2-poin heigh -heigh co ela ion unc ion be ween he midpoin and INTERFACIAL STRUCTURE AT A TWO-DIMENSIONAL…PHYSICAL REVIEW E 69, 061604 (2004) 061604-3 any o he in e acial posi ions. Rela ed quan i ies as he a - e age in e acial p o ile 具l共x兲典w, he local oughness ␰ ⬜共x兲, and he co ela ion leng h ac oss he wedge ␰ x(see Fig. 1) will be also ob ained. Some esul s a e al eady known o he 1-poin dis ibu- ion unc ions. The p obabili y dis ibu ion unc ion o he midpoin x=0 in e acial heigh is gi en by [2] Pw 1共l0; ␪ , ␣ 兲⬅Pw共l0,0兲=2⌺共 ␪ − ␣ 兲e−2⌺共 ␪ − ␣ 兲l0,共19兲 which e i ies he ema kable co a iance ela ionship, Eq. (2). Fo a bi a y x艌0 he 1-poin dis ibu ion unc ion has he exp ession [12] Pw共l,x兲=⌺ ␪ e−2⌺ ␪ le c 冉 −冑⌺x 2 ␪ +冑⌺ 2xl 冊 +⌺共 ␪ − ␣ 兲e2⌺共 ␣ − ␪ 兲le2⌺ ␣ x共 ␣ − ␪ 兲 ⫻e c 冉 冑⌺x 2共 ␪ −2 ␣ 兲−冑⌺ 2xl 冊 −⌺ ␣ e−2⌺ ␣ le2⌺ ␣ x共 ␣ − ␪ 兲 ⫻e c 冉 冑⌺x 2共 ␪ −2 ␣ 兲+冑⌺ 2xl 冊 .共20兲 Fo x⬍0, we ha e he symme y Pw共l,x兲=Pw共l,−x兲, so he e- a e we will conside only he case x艌0. The momen s 具ln共x兲典wcan be ob ained a e some algeb a. The a e age in e acial posi ion p o ile eads 具l共x兲典w=1 2⌺ ␪ +冑x 2 ␲ ⌺e−共⌺ ␪ 2/2兲x+ 冋 ␪ ␪ − ␣ − ␪ ␣ 册 ⫻e2⌺ ␣ x共 ␣ − ␪ 兲 4⌺ ␪ e c 冉 冑⌺x 2共 ␪ −2 ␣ 兲 冊 + 冋 1 4⌺ ␪ 冉 ␪ ␪ − ␣ + ␪ ␣ −2 冊 − ␪ x 2 册 e c 冉 冑⌺ ␪ 2 2x 冊 . 共21兲 The wedge excess adso p ion ⌫wmeasu ed wi h espec o he plana case can be ob ained as ⌫w=2共 ␳ l− ␳ g兲 冕 0 ⬁ 冉 具l共x兲典w−1 2⌺ ␪ 冊 dx = ␳ l− ␳ g 2⌺2 冋 1 ␪ 共 ␪ − ␣ 兲2−1 ␪ 3 册 ,共22兲 whe e ␳ gand ␳ la e he coexis ence densi ies o he apo and liquid phases, espec i ely. Close o he illing ansi ion 共 ␪ → ␣ 兲,⌫w⬃2共 ␳ l− ␳ g兲具l共0兲典w 2/ ␣ . The oughness p o ile ␰ ⬜共x兲(see Fig. 1)is de ined as 冑具l2共x兲典w−具l共x兲典w 2, whe e 具l2共x兲典wis gi en by 具l2共x兲典w=1 2⌺2 ␪ 2− 冉 −1 ⌺共 ␪ − ␣ 兲−1 ⌺ ␣ +1 ⌺ ␪ + ␪ x 冊 ⫻冑x 2 ␲ ⌺e−共⌺ ␪ 2/2兲x+ 冋 ␪ 2 共 ␪ − ␣ 兲2− ␪ 2 ␣ 2 册 ⫻e2⌺ ␣ x共 ␣ − ␪ 兲 4⌺2 ␪ 2e c 冉 冑⌺x 2共 ␪ −2 ␣ 兲 冊 − 冋 1 4⌺2 ␪ 2 冉 − ␪ 2 共 ␪ − ␣ 兲2− ␪ 2 ␣ 2+2 冊 − ␪ x 2⌺ 冉 ␣ ␪ − ␣ − ␪ − ␣ ␣ 冊 +x2 ␪ 2 2 册 e c 冉 冑⌺ ␪ 2 2x 冊 . 共23兲 Fo gene al n, he ollowing exp ession can be ob ained by induc ion: 具ln共x兲典w=具ln典 ␲ 冋 1+1 2 冉 ␪ n 共 ␪ − ␣ 兲n− ␪ n ␣ n 冊 e2⌺ ␣ x共 ␣ − ␪ 兲 ⫻e c 冉 冑⌺x 2共 ␪ −2 ␣ 兲 冊 册 +Pn共x兲冑x 2 ␲ ⌺e−共⌺ ␪ 2/2兲x +Qn共x兲e c 冉 冑⌺ ␪ 2 2x 冊 ,共24兲 whe e 具ln典 ␲ =n!/共2⌺ ␪ 兲nand Pn共x兲and Qn共x兲a e polynomi- als in xo o de n−1 and n, espec i ely. These exp essions a e only alid i ␪ ⬎ ␣ ( o smalle al- ues o ␪ he in e ace is unbound om he wedge). Fo x →0, Eq. (20) educes o Eq. (19). On he o he hand, o 兩x兩→⬁,Pw共l,x兲decay o P ␲ 共l兲⬅2⌺ ␪ exp共−2⌺ ␪ l兲. Howe e , he scale o e which his decay occu s depends on he alue o ␣ .I ␪ 艌2 ␣ , his scale is he plana co ela ion leng h ␰ 储 ⬅2/⌺ ␪ 2. Howe e , i ␣ ⬍ ␪ ⬍2 ␣ , he decay leng h is ␰ F ⬅1/2⌺ ␣ 共 ␪ − ␣ 兲(ou no a ion di e s sligh ly om he one used in Re . [12]). No e ha ␰ Fis always la ge han ␰ 储 , and di e ges on app oaching he illing ansi ion. On he o he hand, ␰ Fis ela ed geome ically wi h he wedge midpoin a e age in e acial heigh ia ␰ F=具l共0兲典w/ ␣ ⬇具l共0兲典w/ an ␣ o small ␣ . I is amusing o no e ha Eq. (20) e i ies he ollowing di e en ial ela ion: Pw共l,x兲+ ␰ F 冉 ⳵ Pw共l,x兲 ⳵ x 冊 =L ␲ 共l,x兲 ⬅P ␲ 共l兲+1 ␪ ⳵ ⳵ x 冕 0 ⬁ dl0l0P ␲ 共l0,0;l,x兲,共25兲 whe e L ␲ 共l,x兲is o con ac binding po en ials, ROMERO-ENRIQUE, PARRY, AND GREENALL PHYSICAL REVIEW E 69, 061604 (2004) 061604-4 L ␲ 共l,x兲=⌺ ␪ e−2⌺ ␪ le c 冉 −冑⌺x 2 ␪ +冑⌺ 2xl 冊 +2 冑⌺ 2 ␲ xe−共冑共⌺x/2兲 ␪ +冑共⌺/2x兲l兲2.共26兲 No e ha he igh hand side (RHS)o Eq. (25)depends only on he plana p ope ies and, consequen ly, is independen o ␣ . I can be shown ha Eq. (25)is ob ained o any binding po en ial i he le hand side (LHS)is expanded in powe s o ␣ and unca ed a he lowes -o de e m, which is indepen- den o ␣ . Consequen ly, his di e en ial ield equa ion im- plies an in ini e hie a chy o in eg o-di e en ial ela ionships o he 2-poin plana co ela ion unc ion. Al e na i ely, Eq. (25)p o ides an elegan ou e o he calcula ion o any mo- men o he in e acial heigh . Mul iplying Eq. (25)by a bi- a y powe o land in eg a ing o e all he possible alues o l, he ollowing di e en ial equa ions a e ob ained: 具ln共x兲典w+ ␰ Fd具ln共x兲典w dx = 冕 0 ⬁ dl lnL ␲ 共l,x兲 ⬅具ln典 ␲ +1 ␪ d dx具l共0兲ln共x兲典 ␲ ,共27兲 whe e 具¯典wand 具¯典 ␲ mean he a e age wi h he wedge and he plana dis ibu ion unc ion, espec i ely. The RHS o Eq. (27)depends only on he plana dis ibu ion unc ions, and consequen ly decays o 具ln典 ␲ o dis ances la ge han ␰ 储 . Close o he illing ansi ion, ␰ FⰇ ␰ 储 , and we can app oxi- ma e Eq. (27) o xⲏ ␰ Fby 具ln共x兲典w+ ␰ Fd具ln共x兲典w dx ⬇具ln典 ␲ ,共28兲 which has as a solu ion 具ln共x兲典w⬇具ln典 ␲ +关具ln共0兲典w −具ln典 ␲ 兴exp 共−x/ ␰ F兲. Taking in o accoun ha 具ln共0兲典wⰇ具ln典 ␲ close o he illing ansi ion, he app oxima e solu ion can be simpli ied e en u he o 具ln共x兲典w⬇具ln共0兲典wexp 共−x/ ␰ F兲 [which is equi alen o se 具ln典 ␲ =0 in Eq. (28)]. These ind- ings a e ob iously in ag eemen wi h Eq. (24)and he asymp o ic beha io o Pw共l,x兲 o la ge xand ␪ ⬍2 ␣ [12]. I is in e es ing o no e ha he momen s ob ained om he ac ual 1-poin dis ibu ion unc ion a e he only solu ions o Eq. (27) ha (a)decay exponen ially wi hin a leng h scale ␰ 储 o 0⬍ ␣ / ␪ Ⰶ1 and x→⬁;(b)a e analy ical as a unc ion o ␣ o 0艋 ␣ ⬍ ␪ , in pa icula , a he diso de poin . The exis ence o he ela ionship, Eq. (25), om which co a i- ance o he momen s o he in e acial posi ion p o ile a x =0 can be in e ed p o ided he (a)and (b) egula i y condi- ions a e ul illed, leads us o specula e on he exis ence o a hidden symme y o he Hamil onian ha explains wedge co a iance. Howe e , he na u e o such a symme y (i any) is comple ely unknown. In he mean- ield app oxima ion, he a e age in e acial posi ion p o ile o binding po en ials cha ac e ized by a c i ical exponen ␣ s=0 ul ills he ollowing gene alized co- a iance ela ionship [15]: l共x兲=l ␲ 冉 ␪ −冏dl共x兲 dx 冏 冊 ,共29兲 whe e l共x兲 ep esen s he (a e aged)in e acial posi ion a x, and l ␲ 共 ␪ 兲is he plana (a e aged)in e acial posi ion o a gi en con ac angle ␪ . Making he subs i u ion l共x兲 →具l共x兲典w, i is clea om Eq. (21) ha his ex ended co a i- ance is no e i ied o x⫽0(e en asymp o ically when x →0o 兩x兩→⬁). Howe e , i is ema kable ha he e exis s an analogous o Eq. (29), gi en by Eq. (27) o n=1. To inish ou discussion abou he 1-poin dis ibu ion unc ions, we compa e ou esul s wi h compu e simula ions o he 2D Ising model [11]. Close o he illing ansi ion poin , we expec ha he app oxima e solu ion o Eq. (28) o n=1 will be gene alized o a bi a y ␣ o 具l共x兲典w⬇具l典 ␲ cos ␣ + 冉 具l共0兲典w−具l典 ␲ cos ␣ 冊 e−x/ ␰ F,共30兲 whe e now ␰ Fis de ined as 具l共0兲典w/ an ␣ . We ha e es ed his app oxima ion wi h he simula ion esul s epo ed in Re . [11](see Fig. 2). The symbols co espond o he simula ion da a ob ained o a squa e 64⫻64 Ising la ice wi h ze o bulk magne ic ield and bounda y magne ic ields +h o he bounda y ows ending a he lowe le co ne , and −h o he emaining bounda y ows. In his geome y, ␣ = ␲ /4. The empe a u e is se o T=Tc/2, whe e Tcis he bulk c i ical empe a u e. Fo his empe a u e and ␣ he c i ical illing ansi ion occu s a hc/J=0.606. Figu e 2 shows he com- pu e simula ion esul s o h/J=0.595,0.597, and 0.599. We ha e no di ec es ima ion o 具l典 ␲ . Howe e , we ha e ob ained 具l典 ␲ by i ing he simula ion da a wi h 兩x兩艋16 la ice spac- ings (in o de o minimize he e ec o he uppe le and lowe igh he e ogeneous wedges) o Eq. (30). The bes i - ing alues a e, in la ice spacing uni s, 具l典 ␲ =0.314,0.335, FIG. 2. Compa ison be ween 具l共x兲典wob ained in Re . [11]by Ising model compu e simula ions o bounda y magne ic ields h/J=0.595 (ci cles),h/J=0.597 (squa es), and h/J=0.599 (dia- monds); and he app oxima ion gi en by Eq. (30)(con inuous lines). The Ising model pa ame e s a e he ollowing: ␣ = ␲ /4, he empe a u e T=Tc/2, and he bulk magne ic ield Hbulk=0. The bounda y magne ic ield a he c i ical illing is hc/J=0.606. The leng hs 兩x兩and 具l共x兲典wa e measu ed in la ice spacing uni s. See ex o explana ion. INTERFACIAL STRUCTURE AT A TWO-DIMENSIONAL…PHYSICAL REVIEW E 69, 061604 (2004) 061604-5 and 0.436 o h/J=0.595,0.597, and 0.599, espec i ely. As i can be seen, he i ing o he simula ion da a is qui e good, despi e he c ude app oxima ions in ol ed in Eq. (30). Now we wan o cha ac e ize he 2-poin co ela ions, in pa icula , he co ela ions be ween he in e acial posi ion abo e he wedge midpoin and he co esponding o an a bi- a y x, which a e gi en by he ollowing unc ion: 具关l共x兲−具l共x兲典w兴关l共0兲−具l共0兲典w兴典w ⬅具l共x兲l共0兲典w−具l共x兲典w具l共0兲典w =1 2⌺ 冉 ⳵ 具l共x兲典w ⳵ ␣ 冊 .共31兲 Subs i u ing Eq. (21)in o Eq. (31), we ob ain 具l共x兲l共0兲典w−具l共x兲典w具l共0兲典w =冑x 2 ␲ ⌺ 共2 ␣ − ␪ 兲e−共⌺ ␪ 2/2兲x 2⌺ ␣ 共 ␪ − ␣ 兲+1 8⌺2 ␪ 2 冉 ␪ 2 共 ␪ − ␣ 兲2− ␪ 2 ␣ 2 冊 ⫻e c 冉 冑⌺ ␪ 2 2x 冊 +e2⌺ ␣ x共 ␣ − ␪ 兲 8⌺2 ␪ 冋 ␪ 共 ␪ − ␣ 兲2+ ␪ ␣ 2 +2⌺共 ␪ −2 ␣ 兲2 ␪ x ␣ 共 ␪ − ␣ 兲 册 e c 冉 冑⌺x 2共 ␪ −2 ␣ 兲 冊 .共32兲 This unc ion decays exponen ially o ze o o la ge x. How- e e , he cha ac e is ic co ela ion leng h ␰ x(see Fig. 1)de- pends on ␣ :i is ␰ 储 o ␪ ⬎2 ␣ and ␰ Fi ␣ ⬍ ␪ ⬍2 ␣ . Conse- quen ly, he diso de poin no only in oduces a new leng h scale o he a e age in e acial p o ile, bu also o he in- e acial luc ua ions. IV. RESULTS FOR THE KRATZER BINDING POTENTIALS The K a ze binding po en ial [see Eq. (17)] is he bo de - line be ween he illing mean- ield and illing luc ua ion e- gimes. While no o di ec physical signi icance, i is ins uc- i e o conside his case in o de o unde s and he in luence o a ma ginal ope a o on he c i ical p ope ies. Fo such po en ials he wedge midpoin p obabili y dis ibu ion unc- ion also obeys wedge co a iance, Eq. (2): Pw 1共l0; ␪ , ␣ 兲=关2⌺共 ␪ − ␣ 兲兴2 ␾ +1l0 2 ␾ ⌫关2 ␾ +1兴exp关−2⌺共 ␪ − ␣ 兲l0兴 =P ␲ 1共l0; ␪ − ␣ 兲.共33兲 I is possible o ex end he ans e analysis and ob ain exac esul s o o he quan i ies o in e es . Conside , o example, he 1-poin p obabili y dis ibu ion unc ion Pw共l,x兲. The Laplace ans o m P ˜ w共l;E兲can be exp essed as P ˜ w共l;E兲= 冕 0 ⬁ dl0e⌺共2 ␣ − ␪ 兲l0共2⌺ ␪ 兲2 ␾ +1共l0l兲 ␾ ⌫关2 ␾ +1兴 ⫻exp共−⌺ ␪ l兲Z ˜ ␲ 共l0,l,E−⌺ ␪ 2/2兲,共34兲 whe e Z ˜ ␲ 共l0,l,E兲is gi en by Eq. (18). This educes o P ˜ w共l;E兲=l ␾ 共2⌺ ␪ 兲2 ␾ +1e−⌺ ␪ l ␪ ⌫关2 ␾ +1兴⌫关2 ␾ 兴 ␬ ⌫关 ␾ 共1− ␬ 兲兴 ⫻ 再 冕 0 ⬁ l0 ␾ e⌺共2 ␣ − ␪ 兲l0W ␬␾ , ␾ −1/2 冉 2⌺ ␪ l0 ␬ 冊 ⫻M ␬␾ , ␾ −1/2 冉 2⌺ ␪ l ␬ 冊 − 冕 0 ll0 ␾ e⌺共2 ␣ − ␪ 兲l0 ⫻ 冋 W ␬␾ , ␾ −1/2 冉 2⌺ ␪ l0 ␬ 冊 M ␬␾ , ␾ −1/2 冉 2⌺ ␪ l ␬ 冊 −W ␬␾ , ␾ −1/2 冉 2⌺ ␪ l ␬ 冊 M ␬␾ , ␾ −1/2 冉 2⌺ ␪ l0 ␬ 冊 册 冎 , 共35兲 whe e ␬ ⬅1/冑1−2E/⌺ ␪ 2. The poles o P ˜ w共l;E兲in he E eal posi i e semiaxis a e he cha ac e is ic in e se leng h scales ac oss he wedge o Pw共l,x兲. Since he second in eg al is o e a ini e in e al and he in eg and does no di e ges in ha ange, no new leng h scale eme ges om i . Fo he i s in eg al, we ake in o accoun ha [16] 冕 0 ⬁ x ␯ −1exp共−px兲W ␬ , ␮ 共ax兲dx =⌫关 ␮ + ␯ + 1/2兴⌫关 ␯ − ␮ + 1/2兴a ␮ +1/2 ⌫关 ␯ − ␬ +1兴共p+a/2兲 ␮ + ␯ +1/2 ⫻2F1 冢 ␮ + ␯ +1 2, ␮ − ␬ +1 2; ␯ − ␬ +1; p−a 2 p+a 2 冣 , 共36兲 whe e 2F1共a,b,c;x兲is a hype geome ic unc ion. I ␪ ⬎2 ␣ , he in eg al does no in oduce any new cha ac e is ic leng h. Howe e , o ␣ ⬍ ␪ ⬍2 ␣ a new singula i y eme ges o ⌺共 ␪ −2 ␣ 兲+⌺ ␪ / ␬ =0, i.e., E=2⌺ ␣ 共 ␪ − ␣ 兲=1/ ␰ F. Rema kably, ␰ F has he same exp ession as o con ac binding po en ials, and is p opo ional (bu no equal) o 具l共0兲典w/ ␣ . F om his i ollows ha he non he modynamic singula - i y occu ing a ␪ =2 ␣ men ioned in he p eceding sec ion is no speci ic o con ac po en ials. A simple geome ical a gu- men gi en in Re . [12]explains why. The mos ele an in e acial luc ua ions a e hose whe e he in e ace lea es he subs a e wi h a con ac angle ␪ ( ela i e o he il ed wall)a an a bi a y subs a e poin . I ␪ ⬎2 ␣ , he o he side o he wedge does no play any ole and we can an icipa e ha he only leng h scale ha con ols he 1-poin dis ibu- ion decay is ␰ 储 . Howe e , i ␪ ⬍2 ␣ , he in e ace will e en- ually each he o he subs a e, and consequen ly we can expec he geome y o play an impo an ole leading o he eme gence o a new leng h scale. Fo mally, his non he mo- dynamic singula i y occu s when he ollowing in eg als ha a ise om he spec al expansion o Z ␲ 共l1,l2;x兲, ROMERO-ENRIQUE, PARRY, AND GREENALL PHYSICAL REVIEW E 69, 061604 (2004) 061604-6 冕 0 ⬁ ␺ 0共l兲exp共2⌺ ␣ l兲 ␺ p ⴱ共l兲,共37兲 become ill de ined. The e, ␺ p共l兲a e he sca e ing eigens a es wi h eigen alues E=p2/2⌺and ␺ 0共l兲is he g ound eigen- s a e. A s aigh o wa d WKB asymp o ic analysis o he eigen unc ions shows ha , o p⫽0, he in eg als gi en by Eq. (37)become ill de ined o ␪ ⬍2 ␣ o qui e a bi a y choices o binding po en ial. As ␪ / ␣ dec eases, ␰ Fexceeds he in insic in e acial leng h scales 1/共Ei−E0兲, and becomes he ue co ela ion leng h ac oss he wedge ␰ xa an ano he diso de poin when ␰ F= ␰ 储 ( ecall ha ␰ x= ␰ 储 o ␪ / ␣ la ge han he alue a he diso de poin ). Fo he case o con ac binding po en ials bo h non he modynamic singula i ies occu a he same alue ␪ =2 ␣ . Howe e , in gene al, he non he modynamic singu- la i ies a e dis inc p o ided he e a e a leas wo bounded eigens a es o Eq. (10). Fo he pu e Coulomb case 共 ␾ =1兲 he second diso de poin occu s a ␪ =4 ␣ /3. Close o he new singula i y ␰ F −1 we ound ha Pw共l;E兲⬃ 1 共 ␰ F −1 −E兲1+2 ␾␣ /共2 ␣ − ␪ 兲E ␰ F→1−,共38兲 so Pw共l,x兲beha es asymp o ically o la ge alues o xas x2 ␾␣ /共2 ␣ − ␪ 兲exp共−x/ ␰ F兲, p o ided ha ␰ 储 ⬍ ␰ F. A ield equa ion analogous o Eq. (25)can be ound o K a ze po en ials. T ans e -ma ix calcula ions o a bi a y binding po en ials lead o he ela ion 具0兩e2⌺ ␣ l兩0典 再 ␪ − ␣ ␪ 冋 Pw共l,x兲+ ␰ F 冉 ⳵ Pw共l,x兲 ⳵ x 冊 册 − 冕 0 ⬁ dl0 ␺ ¯ 0 ⬘共l0兲 ⌺ ␪␺ ¯ 0共l0兲Pw共l0,0;l,x兲 冎 =L ␲ 共l,x兲− 冕 0 ⬁ dl0 ␺ ¯ 0 ⬘共l0兲 ⌺ ␪␺ ¯ 0共l0兲P ␲ 共l0,0;l,x兲,共39兲 whe e ␺ ¯ 0共l0兲⬅ ␺ 0共l0兲exp共⌺ ␪ l0兲and ␺ ¯ 0 ⬘共l0兲is i s de i a i e wi h espec o l0[ ecall ha ␺ 0共l0兲is he g ound s a e eigen- unc ion]. Fo K a ze po en ials, ␺ ¯ 0共l0兲⬀l0 ␾ ,soEq.(39)can be exp essed as 冉 ␪ ␪ − ␣ 冊 ␾ ⳵ Pw共l,x兲 ⳵ ␣ + ⳵ ⳵ ␣ 冋 ␰ F 冉 ␪ ␪ − ␣ 冊 ␾ ⳵ Pw共l,x兲 ⳵ x 册 =0. 共40兲 As o he con ac binding po en ial case, some in e es ing quan i ies can be e alua ed om his exp ession. Fo ex- ample, he wedge adso p ion is ound o be ⌫w=共2 ␾ +1兲共 ␾ +1兲⌫CP,共41兲 whe e ⌫CP is he adso p ion co esponding o he con ac binding po en ial, Eq. (22). V. THE BREATHER MODE PICTURE In o de o unde s and he o igin o he new co ela ion leng h ␰ Fwe iden i ied in p e ious sec ions, we ecall he de ini ion o he 2-poin dis ibu ion unc ion o x2⬎x1艌0, Eq. (13). This exp ession can be w i en in he ollowing way: Pw c共l2,x2兩l1,x1兲=P ␲ c共l2,x2兩l1,x1兲⬅P ␲ c共l2,x2−x1兩l1,0兲, 共42兲 whe e Pw c共l2,x2兩l1,x1兲and P ␲ c共l2,x2兩l1,x1兲a e, espec i ely, he wedge and he plana condi ional p obabili y o he in e - ace being a a ela i e heigh l2 om he subs a e a x2, p o ided ha he in e ace is pinned a a ela i e heigh l1a x1, de ined as Pi c共l2,x2兩l1,x1兲=Pi共l1,x1;l2,x2兲 Pi共l1,x1兲,共43兲 whe e he subsc ip iindica es i his p obabili y is consid- e ed in he wedge 共i=w兲o in he plana 共i= ␲ 兲geome y. In iew o he iden i y be ween he wedge and plana condi ional p obabili y dis ibu ion unc ions we i s con- side he case o a plana subs a e. The condi ional p obabil- i y can be ob ained as P ␲ c共l2,x兩l1,0兲= ␺ 0 ⴱ共l2兲 ␺ 0 ⴱ共l1兲e−共⌺ ␪ 2/2兲xZ ␲ 共l1,l2;x兲.共44兲 Fo con ac binding po en ials, Eq. (44)can be w i en ex- plici ly as P ␲ c共l2,x兩l1,0兲=冑⌺ 2 ␲ xe−⌺共l2−l1+ ␪ x兲2/2x +e−2⌺ ␪ l2 冋 冑⌺ 2 ␲ xe−⌺共l1+l2− ␪ x兲2/2x +⌺ ␪ e c 冉 冑⌺ 2x共l1+l2− ␪ x兲 冊 册 .共45兲 I l1is e y la ge compa ed wi h 具l典 ␲ ⬅1/2⌺ ␪ , we can iden- i y wo di e en beha io s o P ␲ c共l2,x兩l1,0兲as a unc ion o l2(see Fig. 3).I x⬍l1/ ␪ , he condi ional p obabili y is ba- sically he ee in e ace condi ional p obabili y ha luc u- a es a ound an a e age alue 具l2共x兲典=l1− ␪ x, wi h a s anda d de ia ion o he o de o 冑x/⌺. Fo x⬎l1/ ␪ , he condi ional p obabili y becomes he 1-poin plana dis ibu ion unc ion P ␲ 共l2兲=2⌺ ␪ exp共−2⌺ ␪ l2兲, comple ely unco ela ed o he alue o l1. The ansi ion be ween he wo egimes occu in an xin e al a ound x =l1/ ␪ which has a wid h o he o de o 冑2l1/⌺ ␪ 3⬅冑x ␰ 储 . These esul s a e con i med by he exac e alua ion o he i s momen s o he condi ional p obabil- i y: 具l2 n典c共l1,x兲= 冕 0 ⬁ dl2l2 nP ␲ c共l2,x兩l10兲.共46兲 The a e age condi ional in e acial p o ile, which co e- sponds o n=1, is gi en by INTERFACIAL STRUCTURE AT A TWO-DIMENSIONAL…PHYSICAL REVIEW E 69, 061604 (2004) 061604-7 具l2典c共l1,x兲=共l1− ␪ x兲+冑⌺ 2 ␲ xe−⌺共l1− ␪ x兲2/2x + 冋 1 4⌺ ␪ −l1−x ␪ 2 册 e c 冉 冑⌺ 2x共l1− ␪ x兲 冊 −e2⌺ ␪ l1 4⌺ ␪ e c 冉 冑⌺ 2x共l1+ ␪ x兲 冊 共47兲 and he condi ional oughness ␰ ⬜ c共l1,x兲is de ined as 冑具l2 2典c−共具l2典c兲2, whe e 具l2 2典c共l1,x兲can be w i en as 具l2 2典c共l1,x兲= 冋 共l1− ␪ x兲2+x ⌺ 册 − 冉 1 ⌺ ␪ −l1+ ␪ x 冊 ⫻冑⌺ 2 ␲ xe−⌺共l1− ␪ x兲2/2x+ 冋 x 2⌺−1 4⌺2 ␪ 2 +共l1−x ␪ 兲2 2 册 e c 冉 冑⌺ 2x共l1− ␪ x兲 冊 + 冋 x ␪ +l1−1 2⌺ ␪ 册 e2⌺ ␪ l1 2⌺ ␪ e c 冉 冑⌺ 2x共l1+ ␪ x兲 冊 . 共48兲 We ob ain wo main conclusions om hese esul s when l1Ⰷ具l典 ␲ . Fi s , he in e acial posi ions a e highly co ela ed o he cen al one o 兩x兩⬍l1/ ␪ . Second, he in insic in e a- cial luc ua ions a e small in his x ange compa ed o he condi ional a e age alue. Ac ually, i we se l1as he leng h scale, he escaled condi ional p obabili y dis ibu ion unc- ion P ˜ ␲ c共l2/l1,x/l1兩1,0兲⬅l1P ␲ c共l2,x兩l1,0兲beha es as P ˜ ␲ c共l2/l1,x/l1兩1,0兲→ ␦ 冉 l2−l1+ ␪ x l1 冊 ⍜共l1− ␪ x兲 + ␦ 冉 l2 l1 冊 ⍜共 ␪ x−l1兲,共49兲 when ⌺ ␪ l1→⬁. We expec his esul o be alid o any po en ial and also o andom bond diso de , since in all cases he wande ing exponen o he ee in e ace ␨ ⬍1. This can be checked o he ma ginal 1/lpo en ial. The Laplace ans o m o he condi ional p obabili y dis ibu ion is L关P ␲ c共l2,x兩l1,0兲兴 ⬅ 冕 0 ⬁ dx eExP ␲ c共l2,x兩l1,0兲 = ␺ 0 ⴱ共l2兲 ␺ 0 ⴱ共l1兲Z ˜ ␲ 共l1,l2,E−⌺ ␪ 2/2兲.共50兲 Fo ⌺→⬁a ixed E, ␪ ,l1, and l2, and aking in o accoun Eq. (18)and ha he g ound s a e eigen unc ion ␺ 0共l兲 ⬀l ␾ exp共−⌺ ␪ l兲, we ob ain he ollowing beha io o he Laplace ans o m o he condi ional p obabili y dis ibu ion unc ion: L关P ␲ c共l2,x兩l1,0兲兴 →1 ␪ ⍜共l1−l2兲eE共l1−l2兲/ ␪ − ␦ 共l2兲 EeEl1/ ␪ . 共51兲 The Laplace ans o m can be in e ed, leading o Eq. (49). To p oceed, we e u n o ou discussion abou he wedge geome y. Due o he p esence o he boos ac o exp共2⌺ ␣ l兲 in he midpoin p obabili y dis ibu ion unc ion, he mid- poin in e acial heigh is almos always u he om he sub- s a e han he mean we ing laye hickness 具l典 ␲ o any binding po en ial. I we assume ha he condi ional p obabil- i y dis ibu ion unc ion is gi en by Eq. (49), which co e- sponds o neglec ing he in insic in e acial luc ua ions a ound he condi ional in e acial p o ile, we can cap u e he main ea u es o bo h he a e age in e acial p o ile and he co ela ions along he wedge o con ac binding po en ials. Ac ually, his pic u e is comple ely equi alen o he 2D wedge b ea he mode model [3,4]. The a e age in e acial p o ile can be w i en as 具l共x兲典w= 冕 0 ⬁ dl1Pw共l1,0兲 冋 冕 0 ⬁ dl2l2P ␲ c共l2,x兩l1,0兲 册 ⬇ 冕 ␪ x ⬁ dl1Pw共l1,0兲共l1− ␪ x兲 = 冕 0 ⬁ sPw共s+ ␪ x,0兲ds.共52兲 The beha io o 具l共x兲典w o la ge xis domina ed by he la ge lasymp o ics o Pw共l,0兲. The la e can be ob ained by ak- ing in o accoun Eq. (14) o m=n=0 and making use o he WKB app oxima ion o he 1-poin plana dis ibu ion unc- ion: FIG. 3. Illus a ion o a ypical in e acial con igu a ion pinned a l1Ⰷ具l典 ␲ o x=0 ( hin con inuous line). We ha e se ⌺=1 (i de ines he leng h scale), ␪ =0.2, and l1=500. The hick con inuous line co esponds o he condi ional a e age p o ile 具l2典c共l1,x兲, and he do ed lines co espond o max(0,具l2典c共l1,x兲±3 ␰ ⬜ c共l1,x兲), whe e ␰ ⬜ c共l1,x兲is he condi ional oughness. Any in e acial con igu a ion has a p obabili y o a leas 95% o being be ween he do ed lines. Inse : an enla gemen o he a ea a ound x =l1/ ␪ . O he cha ac e - is ic leng h scales a e ep esen ed. See ex o explana ion. ROMERO-ENRIQUE, PARRY, AND GREENALL PHYSICAL REVIEW E 69, 061604 (2004) 061604-8 P ␲ 共l兲⬃ 1 冑1+2W共l兲 ⌺ ␪ 2 exp 冉 −2⌺ ␪ 冕 ld 冑1+2W共 兲 ⌺ ␪ 2 冊 ⬃e−2⌺ ␪ lexp 冉 −2 冕 ld W共 兲 ␪ 冊 ,l→⬁.共53兲 The i s hing we can see is ha , o la ge x, he decay o 具l共x兲典win his app oxima ion is con olled by an exponen ial e m exp关−2⌺ ␪ 共 ␪ − ␣ 兲x兴. So, a new leng h scale ␰ F ⴱis de ined as 1/2⌺ ␪ 共 ␪ − ␣ 兲. Close o he illing ansi ion, ␰ F ⴱ= ␰ F −1/2⌺ ␣␪ ⬃ ␰ F+O共1兲. Depending on he la ge lbeha io o he (a ac i e)bind- ing po en ials, di e en si ua ions can a ise [5]. The illing mean- ield egime is cha ac e ized by binding po en ials ha decay o ze o as 1/lpwhe e p⬍1/ ␨ −1, implying ␨ ⬍1 o he mal diso de ( he wande ing exponen ␨ =1/2). A saddle poin calcula ion shows ha close o he illing ansi ion 具l共0兲典w⬃1/⌺共 ␪ − ␣ 兲p.As ␪ → ␣ , he ele an leng h scale in he xdi ec ion, 具l共0兲典w/ ␪ Ⰷ ␰ F ⴱ, so he la e leng h scale is i ele an (in ac , in insic in e acial luc ua ions ha we neglec ed can be mo e impo an ). Fo p=1, bo h leng h scales become o he same o de , and consequen ly 具l共x兲典w⬃具l共0兲典w 共x/ ␰ F ⴱ兲exp共−x/ ␰ F ⴱ兲, whe e 共x兲di e ges a mos algeb aically, and depends on he de- ailed s uc u e o he binding po en ial h ough he sho dis ance ldependence o Pw共l,0兲. Fo a pu e 1/lpo en ial, 共x兲=共1+2x/3+x2/6兲. This exp ession e i ies he di e en- ial equa ion o 具l共x兲典w ha a ises om Eq. (40)in he scal- ing limi . The illing luc ua ion egime co esponds o po en ials wi h p⬎1, and is cha ac e ized by uni e sal c i ical expo- nen s and scaling unc ions. Indeed in he c i ical egime he scaling beha io is he same as ha ound o con ac binding po en ials. Fo x→⬁, we ind ha asymp o ically 具l共x兲典w ⬃具l共0兲典wexp共−x/ ␰ F ⴱ兲. This solu ion ag ees wi h he asymp o - ics o 具l共x兲典w o con ac binding po en ials when ␪ → ␣ , al- hough wi h a decay leng h sligh ly smalle . Howe e , he beha io is asymp o ically co ec i we assume ha ␰ F ⴱ⬅ ␰ F. Fo he co ela ion unc ions, we ha e 具l共x兲l共0兲典w−具l共x兲典w具l共0兲典w= 冕 0 ⬁ dl1l1Pw共l1,0兲⌬共l1,x兲, 共54兲 whe e ⌬共l1,x兲is de ined as ⌬共l1,x兲= 冕 0 ⬁ dl2l2关P ␲ c共l2,x兩l1,0兲−Pw共l2,x兲兴.共55兲 In he b ea he mode app oxima ion, ⌬共l1,x兲can be ob ained as ⌬共l1,x兲⬇共l1− ␪ x兲⍜共l1− ␪ x兲−具l共x兲典w.共56兲 We ind di e en beha io s depending on he alue o p.In he illing mean- ield egime, ⌬共l1,x兲is negligible in his scale. Fo he illing luc ua ion egime, he co ela ion unc- ion decays as 具l共x兲l共0兲典w−具l共x兲典w具l共0兲典w⬃具l共0兲典w 2 冉 1+ x ␰ F ⴱ 冊 e−x/ ␰ F ⴱ, 共57兲 and again is in ag eemen wi h he beha io o he exac co ela ion unc ion o con ac binding po en ials, Eq. (32), when x→⬁and ␪ → ␣ (assuming again ha ␰ F ⴱ⬅ ␰ F). Finally, o he ma ginal case p=1 he beha io o he co ela ion unc ion is p edic ed o be o x→⬁as 具l共0兲典w 2g共x/ ␰ F ⴱ兲exp共−x/ ␰ F ⴱ兲, whe e g共x兲is a unc ion ha di e ges a mos algeb aically. Ano he quan i y o in e es is he midpoin local suscep- ibili y ␹ w共l兲de ined as ␹ w共l兲=冏 ⳵␳ 共l兲 ⳵ h冏h=0 =2共 ␳ l− ␳ 兲 冕 l ⬁ dsPw共s,0兲⌬共s兲, 共58兲 whe e ⌬共l兲⬅兰0 ⬁dx⌬共l,x兲. In he b ea he mode app oxima- ion and in he illing luc ua ion egime, ⌬共s兲has he ollow- ing exp ession: ⌬共l兲=1 ␪ 冉 l2 2−具l共0兲典w 2 冊 ,共59兲 which is exac o con ac binding po en ials. This exp es- sion, oge he wi h he midpoin wedge co a iance, Eq. (2), leads o he co a iance ela ionship be ween he local suscep- ibili ies [5]: ␹ w共l; ␪ , ␣ 兲= ␪ − ␣ ␪ ␹ ␲ 共l, ␪ − ␣ 兲,共60兲 whe e ␹ ␲ 共l, ␪ 兲is he local suscep ibili y co esponding o he plana geome y o a con ac angle ␪ . Finally, we no e ha he b ea he mode pic u e has di ec consequences o he scaling o he in e acial p o ile in he illing luc ua ion egime. To see his, ecall ha he wedge midpoin p obabili y dis ibu ion unc ion scales as [5] Pw共l兲=1 具l共0兲典w ⌳ 冉 l 具l共0兲典w 冊 ,共61兲 whe e ⌳共s兲is a uni e sal unc ion and, due o co a iance, is he same as he scaling unc ion o he co esponding plana 1-poin p obabili y dis ibu ion unc ion. Complemen ing he scaling o he p obabili y dis ibu ion unc ion is he posi ion dependence o he in e acial p o ile, which we an icipa e sa is ies 具l共x兲典w=具l共0兲典w ␾ 冉 ␪ x 具l共0兲典w 冊 ,共62兲 whe e ␾ 共s兲is ano he uni e sal unc ion. In he b ea he mode pic u e, he in e ace is in ini ely s i in he illed e- gion implying ha he scaling unc ions ⌳共s兲and ␾ 共s兲a e ela ed ia INTERFACIAL STRUCTURE AT A TWO-DIMENSIONAL…PHYSICAL REVIEW E 69, 061604 (2004) 061604-9