Nematic wetting and filling of crenellated surfaces

PHYSICAL REVIEW E 86, 011703 (2012) Nema ic we ing and illing o c enella ed su aces N. M. Sil es e,1,2,*Z. Eskanda i,2P. Pa ´ ıcio,2,3J. M. Rome o-En ique,4and M. M. Telo da Gama1,2 1Depa amen o de F´ ısica da Faculdade de Ciˆ encias 2Cen o de F´ ısica Te´ o ica e Compu acional, Uni e sidade de Lisboa, A enida P o esso Gama Pin o 2, P-1649-003 Lisboa, Po ugal 3Ins i u o Supe io de Engenha ia de Lisboa, Rua Conselhei o Em´ ıdio Na a o 1, P-1959-007 Lisboa, Po ugal 4Depa amen o de F´ ısica A ´ omica, Molecula y Nuclea , A ea de F´ ısica Te´ o ica Uni e sidad de Se illa, Apa ado de Co eos 1065, 41080 Se illa, Spain (Recei ed 30 Ap il 2012; published 9 July 2012) We in es iga e nema ic we ing and filling ansi ions o c enella ed su aces ( ec angula g a ings) by nume ical minimiza ion o he Landau–de Gennes ee ene gy as a unc ion o he ancho ing s eng h, o a wide ange o he su ace geome ical pa ame e s: dep h, wid h, and sepa a ion o he c enels. We ha e ound a ich phase beha io ha depends in de ail on he combina ion o he su ace pa ame e s. By compa ison o simple fluids, which unde go a con inuous filling o unbending ansi ion, whe e he su ace changes om a d y o a filled s a e, ollowed by a we ing o unbinding ansi ion, whe e he hickness o he adso bed fluid becomes mac oscopic and he in e ace unbinds om he su ace, nema ics a c enella ed su aces e eal an in iguingly ich beha io : in shallow c enels only we ing is obse ed, while in deep c enels, only filling ansi ions occu ; o in e media e su ace geome ical pa ame e s, a new class o filled s a es is ound, cha ac e ized by ben iso opic-nema ic in e aces, which pe sis o su aces s uc u ed on la ge scales, compa ed o he nema ic co ela ion leng h. The global phase diag am displays wo we and ou filled s a es, all sepa a ed by fi s -o de ansi ions. Fo c enels in he in e media e egime e-en an filling ansi ions d i en by he ancho ing s eng h a e obse ed. DOI: 10.1103/PhysRe E.86.011703 PACS numbe (s): 61.30.Dk, 61.30.Hn, 61.30.J I. INTRODUCTION As design and echnology ocus on he manipula ion o ma e ials a inc easingly smalle scales, he ole o su aces and in e aces becomes inc easingly impo an . A he mi- c ome e scale he mic oscopic (nanome e ) and mac oscopic (millime e ) scales become en angled and new phenomena eme ge. Thus, he unde s anding o su aces is c ucial o ou abili y o u he minia u ize componen s and de ices, wi h ele an applica ions o mic ofluidics and we ing a s uc u ed su aces. A he mac oscopic le el, we ing is desc ibed by a o ce balance a gumen , which ela es he equilib ium con ac angle θo he sa u a ed liquid d ople on he plana su ace o he solid- apo σs , solid-liquid σsl, and liquid- apo σsu ace ensions, h ough Young’s equa ion: cos θ=σs −σsl σ.(1) When he con ac angle is ze o, he liquid sp eads on he su ace and is said o we he plana su ace. In his con ex , awe ing ansi ion occu s when he con ac angle changes om a fini e alue o ze o, as he empe a u e o he su ace p ope ies a y [1–4]. A he mic oscopic le el, he we ing ansi ion is cha ac e ized by an o de pa ame e ela ed o he (in e se) hickness o he liquid film in con ac wi h he su ace. In ac , a he nanome e scale, he sa u a ed liquid d ople is accompanied by a film o adso bed liquid, he hickness o which di e ges a he we ing ansi ion. In e ms o he posi ion o he liquid- apo in e ace he we ing ansi ion may be desc ibed as he unbinding o he fluid in e ace wi h espec o he su ace [5]. *[email p o ec ed] Mo e ecen ly, a sys ema ic in es iga ion o simple fluids a s uc u ed su aces was ca ied ou , unco e ing new phase ansi ions be ween d y and we s a es [6]. On s uc u ed su aces, a he mac oscopic le el, he e ec i e su ace a ea inc eases due o he su ace s uc u e. I he a io o eal su ace a ea S o appa en (i.e., p ojec ed) a ea Ais defined as he oughness pa ame e =S/A, hen he in e acial ee ene gies o he solid- apo and solid-liquid in e aces inc ease by he same amoun , leading o cos θ= cos θπ,(2) whe e θπis he con ac angle a he plana su ace. This is known as he Wenzel equa ion and esul s om simple he mo- dynamic a gumen s [7]. Equa ion (2) implies ha he su ace s uc u e amplifies he su ace we ing p ope ies: hyd ophilic su aces become mo e hyd ophilic, and hyd ophobic su aces become mo e hyd ophobic. Beyond he modynamics, a he mic oscopic le el, su ace geome y has mo e d as ic e ec s on we ing phenomena: he na u e o he ansi ion may change and new ansi ions, e.g., filling, may occu , p eceding he we ing ansi ions [6]. In line wi h ea lie wo k [8] we dis inguish h ee in e acial s a es: d y, illed, and we . We e e o illing as he p ocess h ough which a d y s a e becomes filled, i.e., a mesoscopic liquid laye g ows on he g oo es o he su ace. In his way, his mechanism can be unde s ood as he unbending o he in e ace be ween he coexis ing fluids, which emains a ached o he su ace [5]. On he o he hand, we will e e o we ing as he p ocess h ough which a d y o filled s a e becomes we . In his a icle we epo he exis ence o se e al nema ic we and filled su ace s a es, sepa a ed by fi s -o de phase ansi ions, a c enella ed ( ec angula g a ing) su aces. The nema ic phase is cha ac e ized by long- ange o ien a ional 011703-1 1539-3755/2012/86(1)/011703(12) ©2012 Ame ican Physical Socie y N. M. SILVESTRE e al. PHYSICAL REVIEW E 86, 011703 (2012) o de and posi ional diso de . The di ec ion o molecula align- men , he nema ic di ec o n, is a bi a y in bulk nema ics bu he p esence o su aces selec s a pa icula di ec ion, known as su ace ancho ing. Typical si ua ions include homeo opic (o ien a ion pe pendicula o he su ace), andom plana ( andom o ien a ion pa allel o he su ace), and degene a e plana (o ien a ion along an a bi a y di ec ion on he su ace). Nema ic we ing o plana su aces was p edic ed in he amewo k o he Landau–de Gennes (LdG) heo y [9–12] and has been obse ed expe imen ally using a a ie y o echniques [13–17]. Nema ic we ing o s uc u ed su aces has a ac ed much less a en ion [18,19] al hough i is o conside able undamen- al in e es , wi h ele an applica ions in bis able liquid-c ys al displays and mic ofluidic de ices. The su ace s uc u e causes o ien a ional us a ion ha con ibu es an elas ic e m o he ee ene gy, which may lead o he nuclea ion o opological de ec s ha will, in u n, a ec he nema ic o de a he su ace. The nema ic elas ici y and he nuclea ion o de ec s, wi h complex s uc u e and dynamics, is a new ing edien , no p esen in he we ing o simple liquids a s uc u ed su aces no in nema ic we ing a s uc u eless ones. Like simple fluids, nema ic we ing phenomena is es- sen ially an in e play o h ee possible s a es: d y, filled, and we . In nema ics, he filled and we s a es may be cha ac e ized by a numbe o dis inc ex u es o symme ies (“uni o m,” “symme ically,” o “asymme ically” dis o ed, wi h o wi hou de ec s, e c.) inc easing he numbe o s a es he in e play o which dic a es he global su ace phase diag am. In mos cases hese a e ue su ace s a es sepa a ed by (fi s -o de ) phase ansi ions. In he we s a e, he sys em has a single leng h scale l, se by he su ace geome y. The elas ic ene gy o smoo h dis o ions is (KS/l)˜ Fd, whe e Kis he F ank elas ic cons an o he nema ic, and ˜ Fdis a educed elas ic con ibu ion o he in e acial ee ene gy, dependen only on he su ace geome y, bu no on i s scale l. The we ing ansi ion is hen gi en by he gene alized Wenzel equa ion [20] 1=cos θ= cos θπ−K˜ Fd lσni ,(3) whe e σni is he nema ic-iso opic (NI) su ace ension. In he limi o la ge l, he elas ic con ibu ion is i ele an , and he Wenzel equa ion is eco e ed. Fo simple fluids, he we ing ansi ion a s uc u ed su aces occu s a a lowe empe a u e o a a lowe su ace field han he we ing ansi ion a fla su aces; o nema ics, he we ing h esholds may be dep essed (as in simple fluids) i he geome ical s uc u e domina es o inc eased i he elas ic ene gy domina es. When he elas ic ene gy domina es, de ia ions om he Wenzel equa ion may be significan . In he s ong ancho ing egime, he o ien a ional us a ion, due o he geome ical su ace pa e n, may lead o he nuclea ion o opological de ec s close o he su ace, wi h a cu o leng h (de ec size) se by he bulk co ela ion leng h ξ. The p esence o bulk and/o su ace de ec s will con ibu e a singula ln(l/ξ) e m o he escaled elas ic ee ene gy ˜ Fd[21]. Beyond he modynamics, a he mic oscopic le el, he su ace geome y has an e en mo e p o ound e ec on nema ic we ing. As o simple fluids, i may lead o new ansi ions such as filling. Howe e , he ansi ion om a filled o a we s a e may be supp essed, as he pinned NI in e ace in he filled s a e may p e en he nuclea ion o opological de ec s in he bulk nema ic, which inc ease he elas ic ene gy by an amoun which is no compensa ed by he dec ease in he su ace ancho ing ene gy. In his pape we con inue a sys ema ic s udy o he in e play be ween su ace geome y and he elas ic ene gy o he dis o ions caused by he spa ially a ying su ace di ec o field. P e ious s udies based on he LdG heo y we e ca ied ou o pe iodic iangula [20,21] and sinusoidal [22] su aces, and gene al ules begin o eme ge. We ecall hese ules and p esen sys ema ic esul s, based on he LdG heo y, o nema ic we ing a c enella ed su aces. P elimina y esul s we e published in [8]. C enella ed su aces ha e been in es iga ed, nume ically and expe imen ally, in he con ex o zeni hal bis able swi ch- ing o liquid c ys al (LC) de ices [23,24]. These s udies ocused on he elas ic configu a ions, deep in he nema ic phase. We ing ansi ions we e no discussed as hese occu a coexis ence wi h he iso opic phase. We ing o simple fluids on ec angula g a ings was also s udied in he con ex o capilla y condensa ion [25]. On a capped capilla y, filling o he su ace u ns condensa ion in o a con inuous ansi ion, which is subsequen ly ollowed by he we ing ansi ion. In wha ollows, we show ha c enella ed su aces exhibi a ich nema ic we ing beha io due o he p esence o a la ge numbe o me as able o de ed s a es in he we ing ange o pa ame e s. In pa icula , we find mul iple we and filled s a es, wi h di e en nema ic ex u es. Some o he filled s a es a e cha ac e ized by a cu ed o ben NI in e ace e en o su aces wi h la ge leng h scales, when compa ed o he nema ic co ela ion leng h, while o he s exhibi he usual unben in e ace. Thus on hese su aces filling and unbending do no occu simul aneously. In he in e media e ange o su ace pa ame e s we find mul iple filling and we ing ansi ions and e-en an in e acial beha io , which is no obse ed, in gene al, a pe iodic iangula and sinusoidal su aces. The emainde o his pape is o ganized as ollows. In Sec. II we summa ize he esul s o nema ic we ing o iangula and sinusoidal su aces, se ing he s age o he esul s on c enella ed su aces epo ed he e. In Sec. III we desc ibe b iefly he LdG heo y and he nume ical me hod used o minimize i . In Sec. IV we p esen and discuss he esul s o he global phase diag am o nema ics a c enella ed su aces. Finally, in Sec. Vwe summa ize he heo e ical esul s. We highligh he new esul s a small and in e media e oughness and con as hem wi h hose ound a iangula and sinusoidal su aces and conclude wi h a b ie discussion o u u e heo e ical and expe imen al wo k. II. RESULTS FOR NEMATIC WETTING OF TRIANGULAR AND SINUSOIDAL SURFACES The LdG ee ene gy exhibi s a fi s -o de we ing ansi ion a s uc u eless su aces [12,20]. A ough su aces (on he mesoscale) he phase beha io is, in gene al, iche . The su - aces in es iga ed so a a e ansla ionally in a ian (along z) 011703-2 NEMATIC WETTING AND FILLING OF CRENELLATED ... PHYSICAL REVIEW E 86, 011703 (2012) FIG. 1. Schema ic ep esen a ion o a su ace cusp. The hick con inuous line depic s he su ace close o a cusp, cha ac e ized by an opening angle φ. S ong ancho ing condi ions a e en o ced a he su ace, wi h θ1and θ2 hei asymp o ic alues as he cusp is app oached om ei he side. The in eg a ion con ou Cis shown as a hin cu ed line, wi h ηi s ou wa d no mal. See ex o de ails. wi h he s uc u e defined on he xy plane. These su aces exhibi we ing beha io , which di e s om ha epo ed o simple fluids. In his sec ion we will summa ize he esul s. De ails may be ound elsewhe e [8,20–22]. Fo iangula su aces he e is only a we ing ansi ion [20], in ag eemen wi h he phenomenology epo ed o simple fluids [26]. While o simple fluids he we ing ansi ion occu s a a lowe empe a u e han he ansi ion a plana su aces, o nema ics he we ing field may exceed he plana su ace ansi ion field. This is a esul o he nuclea ion o opological de ec s a he su ace cusps [21,27], which lead o bis abili y o he we ing s a es o opening angles close o α=π/2 (inside he g oo e) and an ex emely slow app oach o he we ing diag am p edic ed by he gene alized Wenzel equa ion, Eq. (3) [8,20]. The o igin o hese de ia ions is he d as ic inc ease o he we ing ancho ing ene gy, e en in he limi o long wa eleng h, due o he geome ical su ace singula i ies [21]. The analysis o iangula su aces can be ex ended s aigh o wa dly o gene al cusped su aces as ollows. The leading con ibu ion o he elas ic ee ene gy om he opological de ec s may be ob ained om he F ank- Oseen ee ene gy wi h a single elas ic cons an K. A ound each su ace de ec , he nema ic di ec o field θcan be w i en as θ=θ0( ,φ)+qφ, whe e θ0is a nonsingula backg ound and ( ,φ) a e he adial and azimu hal pola coo dina es, wi h o igin a he de ec co e. Finally, qis he opological cha ge associa ed o he disclina ion line, which depends on he nema ic ex u e and he opening angle φ o he cusp (see Fig. 1). I we deno e by θ1and θ2≡θ1+θ he asymp o ic alues o he nema ic di ec o field on he su ace, on ei he side o he cusp, as shown in Fig. 1, he opological cha ge can be w i en as q=θ/φ. Using he di e gence heo em, we may ob ain he elas ic ee ene gy pe uni leng h in he zdi ec ion, el, as a con ou in eg al unning pa allel o he su ace [21]: el =K 2C ds θ(s)η·∇θ(s),(4) whe e ηis he ou wa ds no mal o he in eg a ion con ou . Le us ocus on he con ibu ion o el close o he cusp. The di ec o field close o one disclina ion sa isfies ∇θ≈quφ/ . In o de o a oid he singula i y a he cusp, he in eg a ion con ou is ounded o by an a c o ci cle wi h adius o he o de o he bulk co ela ion leng h ξcen e ed a he cusp, as shown in Fig. 1. The con ibu ion o his a c o el anishes when ξis much smalle han he o he ele an leng h scales as in his limi ηis o hogonal o ∇θ. Close o he cusp, η≈±uφ, as shown in Fig. 1, and θis well app oxima ed by he asymp o ic alues θ1and θ2on ei he side. Subs i u ing hese es ima es in o Eq. (4), he leading con ibu ion o el close o he cusp is gi en by K 2C ds θ(s)η·∇θ(s)≈K 2qθln l ξ=K 2q2φln l ξ(5) whe e lis a leng h which cha ac e izes he su ace pe iodici y. Fo la ge l/ξ, he leading con ibu ion o he elas ic ee ene gy pe pe iodic cell and uni leng h along he cusp axis is gi en by he supe posi ion o he con ibu ions om he di e en de ec s. As o iangula su aces, he nex - o-leading con ibu ion o he elas ic ee ene gy is expec ed o depend on geome y, ancho ing ene gy, and nema ic ex u e, bu no on he su ace pe iodici y [21]. On sinusoidal su aces he si ua ion is di e en as he de ec s, i p esen , a e necessa ily bulk opological de ec s. I is ound, in gene al, ha filling and we ing p e-emp each o he , i.e., he e is a single ansi ion—filling o we ing— om d y o a phase wi hou opological de ec s. Fo a gi en oughness, he ansi ion is ei he om d y- o-filled o om d y- o-we and he sequence d y-filled-we ound in simple fluids is ha dly e e obse ed. Whe he he s able phase is he we o he filled s a e depends on a combina ion o he su ace oughness and ancho ing condi ions a he in e aces. The compe ing we and filled phases a e me as able o e a ela i ely wide ange o pa ame e s bu he nuclea ion o bulk opological de ec s— equi ed o ealize one o he o he —is no compensa ed by a dec ease in he ancho ing ene gy in he we ing ange o pa ame e s. As a consequence, o a gi en su ace, he sequence o filling o we ing ansi ions does no occu , excep in a e y na ow ange o pa ame e s [22]. III. LANDAU–DE GENNES THEORY The nema ic liquid c ys al is modelled by he Landau– de Gennes (LdG) ee ene gy, which is adequa e o desc ibe inhomogenei ies, including egions o educed nema ic o de , on leng h scales o he o de o he bulk co ela ion leng h, such as nema ic-iso opic in e aces and opological de ec s. The o de pa ame e is ep esen ed by a aceless, symme ic enso Qwi h componen s: Qij =S 2(3ninj−δij )+B 2(lilj−mimj),(6) whe e nia e he Ca esian componen s o he di ec o field n, Sis he nema ic o de pa ame e , which measu es he o ien a- ional o de along he nema ic di ec o , and Bis he biaxiali y pa ame e , which measu es he o de ing o he molecules in di ec ions pe pendicula o n, cha ac e ized by he eigen ec- o s land m. We conside only in-plane dis o ions. Al hough, o some sys ems, wis (ou -o -plane) dis o ions may ha e an impo an ole [28,29], o he LC and ancho ing condi ions epo ed in his pape hese dis o ions a e no ene ge ically 011703-3 N. M. SILVESTRE e al. PHYSICAL REVIEW E 86, 011703 (2012) a o able. The e o e, we es ic his s udy o ansla ionally in a ian sys ems in he zdi ec ion. Then, n=(cos θ,sin θ,0), and he enso o de pa ame e has only h ee independen componen s, namely Qxx,Qyy, and Qxy. The LdG ee ene gy may be w i en as F= ( b+ el)dx+∂ sds, (7) whe e bis he bulk ee-ene gy densi y, el is he elas ic ee-ene gy densi y, and sis he su ace ee ene gy, defined as [30] b=aT Q2−bT Q3+c[T Q2]2,(8) el =L1 2∂kQij ∂kQij +L2 2∂jQij ∂kQik,(9) s=−2 3wT [Q·Qs],(10) whe e adepends linea ly on he empe a u e, band ca e posi i e cons an s, and L1and L2a e posi i e pa ame e s ela ed o he elas ic cons an s. I we escale all he a iables as in [31], ˜ Q=6cQ/b, he posi ions ˜ = /ξ, whe e he co ela ion leng h ξis ξ2=8c(3L1+2L2)/b2, and ˜ FLdG = 242c3FLdG/ξ3b4, we ob ain ˜ F=˜ (˜ b+˜ el)d˜ x+∂˜ ˜ sd˜ s, wi h escaled ee-ene gy densi ies: ˜ b=2 3τT ˜ Q2−8 3T ˜ Q3+4 9[T ˜ Q2]2,(11) ˜ el =1 3+2k(˜ ∂l˜ Qij ˜ ∂l˜ Qij +k˜ ∂j˜ Qij ˜ ∂l˜ Qil),(12) ˜ s=−2 3˜wT [ ˜ Q·˜ Qs].(13) He e τ=24ac/b2is a dimensionless empe a u e, k=L2/L1 is a dimensionless elas ic pa ame e ( o s abili y easons, he elas ic pa ame e is es ic ed o k>−3/2), and ˜w= 16wc/b2ξis he dimensionless ancho ing s eng h. He ea e , we will conside hese escaled a iables and we will d op he ilde no a ion. We se he empe a u e a NI coexis ence, τ=1, whe e he bulk ee-ene gy densi y has wo minima, co esponding o b=0 o escaled scala o de pa ame e s Si=0 (iso opic phase) and Sn=1 (nema ic phase). I is impo an o no e ha he o de pa ame e Sin he coexis ing nema ic phase is escaled, and hus i s alue in eal uni s is b/6c, which mus be smalle han 1 ( ypically ≈0.4). I he elas ic pa ame e kis posi i e (nega i e), he nema ic aligns pa allel (pe pendicula ) o he nema ic-iso opic in e ace. Finally, Qsdefines he p e e ed enso o de pa ame e a he su ace. We conside homeo opic alignmen by se ing Qs=(3ν⊗ν−1)/2, wi h ν he no mal o he su ace, in line wi h p e ious wo k [9,10,12]. IV. CRENELLATED SURFACES We conside c enella ed su aces (see Fig. 2) ha a o pe pendicula (homeo opic) alignmen o he nema ic. The su ace is ansla ionally in a ian in he zdi ec ion and has a pe iodic ec angula c oss sec ion, cha ac e ized by h ee leng hs: he sepa a ion be ween c enels ( op pla eau) l1, he wid h l2, and he dep h ho he c enels. The pe iodic s uc u e FIG. 2. (Colo online) C enella ed su aces. l1is he sepa a ion be ween c enels, and l2and ha e he wid h and dep h o he c enels. The s uc u e is pe iodic wi h wa eleng h λ=l1+l2. has wa eleng h λ=l1+l2. In wha ollows, leng hs will be measu ed in uni s o he co ela ion leng h ξo he nema ic. The use o a mesoscopic heo y, such as he Landau–de Gennes heo y, equi es ha he su ace cha ac e is ic leng hs a e a leas one o de o magni ude la ge han he bulk co ela ion leng h. We se k=2, which implies ha he nema ic alignmen is pa allel o he NI in e ace (i he e is one), away om he su ace bounda ies. The oughness o he su ace depends on he a ios l1/l2and h/l2, ≡ (l1/l2,h/l2), and is gi en by =1+2h/(l1+l2). A fixed a io o he sepa a ion and wid h o he c enels ( a io o he op and bo om pla eaux), l1/l2, he oughness inc eases wi h he a io o he dep h and wid h o he c enels, h/l2. Fo a gi en su ace geome y, we calcula e he ee ene gy o he in e acial s a es: d y (D), filled (F), and we (W), using he ancho ing s eng h as he con ol pa ame e . A fixed ancho ing s eng h, nema ic filled and we s a es exhibi , in gene al, dis inc di ec o configu a ions o ex u es (see Fig. 3), which a e in es iga ed using di e en ini ial condi ions. In addi ion o he d y s a e [Fig. 3(D)], we ha e ound a uni- o m [Fig. 3(Wu)] and wo dis o ed we s a es dis inguished by he symme y o he nema ic di ec o . The uni o m we s a e is he plana su ace limi , oughness =1, whe e he nema ic ex u e is uni o m and pe pendicula o he su ace. A low oughness, he nema ic di ec o bends symme ically a ound he bo om co ne s o he c enel [Fig. 3(Ws d)], in o de o ollow he homeo opic ancho ing a o ed by he su ace e e ywhe e. This dis o ion is mo e e iden a high ancho ing s eng hs whe e he conflic ing o ien a ions nea he co ne s lead o he nuclea ion o opological de ec s ha adop a symme ic configu a ion. When he oughness inc eases u he he nema ic akes a dis o ed asymme ic ex u e [Fig. 3 (Wa d)] which is bis able, as he di ec o bends a ound one o he bo om co ne s and splays a ound he o he . We ha e also ound dis inc filled s a es. A in e media e oughness, he filled s a e has a NI in e ace ha is cu ed, o ben , inside he c enels [Fig. 3(Fb)], while a ough su aces he NI in e ace is fla , o unben , a he op o he c enels [Fig. 3(Fu)]. In he unben filled s a e, he nema ic di ec o a om he bo om o he c enels is pe pendicula o he la e al su aces, in line wi h he o ien a ion a o ed by he fla NI in e ace. Nea he bo om, howe e , he nema ic is dis o ed as a esul o he homeo opic ancho ing a o ed by he su aces. This s a e is bis able as he nema ic ex u es may be ben owa ds he le o he igh bo om co ne (and splayed a ound he o he ). Unde s ong ancho ing condi ions a nema ic d ople o film nuclea es on he uppe ho izon al su aces, co esponding o 011703-4 NEMATIC WETTING AND FILLING OF CRENELLATED ... PHYSICAL REVIEW E 86, 011703 (2012) FIG. 3. (Colo online) In e acial s a es o c enella ed su aces o nema ics wi h posi i e elas ic aniso opy. (D) d y s a e, (Wu) uni o m we s a e,(Ws d) symme ic we s a e, (Wa d) asymme ic we s a e, (Fb) ben filled s a e, (F+ b) ben filled s a e wi h d ople s, (Fu) unben filled s a e, (F+ u) unben filled wi h d ople s. new filled s a es [Fig. 3(F+ b) and (F+ u)], wi h ben and unben NI in e aces. The unben s a e is also bis able. A. C enel wid h = c enel sepa a ion We s a by conside ing a su ace whe e he wid h o he c enels is equal o hei sepa a ion, l1=l2=l. Figu e 4illus- a es he global phase diag am o l=10ξ. The lines ep esen fi s -o de ansi ions be ween in e acial s a es (see Fig. 3). A low and high oughness, co esponding o shallow and deep c enels, he phase diag am esembles ha o sinusoidal su aces [22]. Fo low oughness, when he c enels a e shallow, 11.2 1.4 1.6 1.8 2 2.2 2.4 0 0.1 0.2 0.3 0.4 w W d s W d a Fb Fb + Fu F u + D FIG. 4. Phase diag am o a c enella ed su ace wi h he same c enel wid h and sepa a ion, l1=l2=10ξ. The lines co espond o fi s -o de phase ansi ions ha sepa a e d y (D), we (W), and filled (F) s a es. Two dis inc we s a es and ou dis inc filled s a es a e obse ed. A he we ing ansi ion, he nea ly uni o m we s a e (Wu) a low oughness changes con inuously o a symme ic dis o ed we s a e (Ws d) as he ancho ing inc eases. The e is a bis able asymme ic nema ic we s a e (Wa d) a in e media e oughness. The we s a es a e sepa a ed by a weak fi s -o de ansi ion indica ed by he g ey line. The e a e also filled s a es, cha ac e ized by a ben (Fb)o unben NI in e ace (Fu). The unben filled s a es a e bis able. A s ong ancho ing he op su aces p omo e he nuclea ion o nema ic d ople s o films o ben (F+ b) and unben (F+ u) filled s a es. See Fig. 3 o a ske ch o he co esponding s a es. he nema ic unde goes a we ing ansi ion o a symme ic we s a e Ws d. A he ansi ion he dis o ions a e small, and become mo e p onounced as he ancho ing s eng h inc eases. These dis o ions anish in he limi ing case o a plana su ace, whe e he nema ic akes a uni o m homeo opic alignmen . Fo in e media e alues o he oughness he nema ic de o ma ions a he ansi ion become mo e p onounced and inc easing he ancho ing s eng h leads o a configu a ional ansi ion om Ws d o he asymme ic we s a e Wa d. This ansi ion is weakly fi s o de , and we could no esol e he c ossing o he wo b anches wi h he cu en nume ical echnique. We ha e, howe e , es ima ed he ansi ion, depic ed in g ey in he phase diag ams, by inspec ion o he nema ic ex u es. In he limi o highly ough su aces, when he c enels a e deep, a filling ansi ion is obse ed o a s a e whe e he nema ic fills he c enels, Fu. The NI in e ace is fla and is pinned a he c enels op co ne s. This s a e is cha ac e ized by a uni o m pa allel di ec o nea he NI in e ace which bends asymme ically nea he bo om su ace. This configu a ion is bis able as he nema ic di ec o can bend in wo equi alen ways. A in e media e oughness, howe e , when he wid h o he c enels lis simila o hei hei dep h h, he su ace exhibi s a no el filling ansi ion, whe e he NI in e ace emains ben , Fb, wi h a symme ic nema ic ex u e inside he c enels. These ben nema ic filled s a es we e no obse ed a iangula o sinusoidal su aces. In bo h filled s a es, Fbo Fu, he op su aces p omo e he nuclea ion o nema ic d ople s (o films) as he ancho ing 011703-5 N. M. SILVESTRE e al. PHYSICAL REVIEW E 86, 011703 (2012) s eng h inc eases, h ough he mechanism ha d i es we ing on plana su aces. On c enella ed su aces, howe e , he NI in e ace emains pinned a he op co ne s o p e en he nuclea ion o opological de ec s, he ene gy o which is no compensa ed by depinning he NI in e ace. As a esul , he we ing ansi ion is p e-emp ed. Recen La ice Bol zmann calcula ions indica e ha i is possible o swi ch be ween ben and unben filled s a es h ough he coupling o weak ex e nal elec ic fields [32]. 1. We ing ansi ion, D ↔Wu S uc u ed su aces wi h low oughness and weak ancho ing induce smoo h de o ma ions o he nema ic di ec o field. As he ancho ing inc eases a we ing ansi ion akes place, in line wi h he gene alized Wenzel equa ion (3). This equa ion has been es ed o sinusoidal and iangula pe iodic su aces and i was ound o ag ee wi h he LdG heo y when he loga i hmic con ibu ions om he opological de ec s nuclea ed a he iangula su ace singula i ies a e aken in o accoun [21]. A low oughness (hl) and weak ancho ing (w1), c enella ed su aces a e we ed by a nea ly uni o m (weakly dis o ed) we s a e as he gain in elas ic ene gy ha esul s om he (nea ly) uni o m ex u e compensa es he cos o he nema ic misalignmen a he la e al su aces. As he ancho ing s eng h inc eases, he (nea ly) uni o m we s a e changes smoo hly in o a symme ic dis o ed we s a e Ws d. The ee ene gy pe uni leng h and uni cell, =F/(Lzn), whe e Lzis he leng h o he sys em pe pendicula o he su ace, and nis numbe o pe iodic cells, in he d y s a e [Fig. 3 (D)] is p opo ional o he iso opic fluid-su ace in e acial ension σis and is gi en by (D)=σis (l1+l2+2h)=σis (l1+l2),(14) whe e is he oughness pa ame e . The ee ene gy o he (nea ly) uni o m we s a e is a weigh ed sum o he NI σni and he nema ic-su ace σns in e acial ensions. Nema ic in e acial ee ene gies depend on he nema ic o ien a ion, and hus we w i e (Wu)=(σ ni +σ⊥ ns)(l1+l2)+2σ nsh, =(σ ni +σ⊥ ns +σ ns( −1))(l1+l2).(15) The we ing ansi ion occu s when (D)= (Wu), esul ing in he balance equa ion (σ⊥ ns −σis) +(σ ns −σ⊥ ns)( −1) +σ ni =0.(16) The second e m depends on he aniso opy o he su ace po en ial, which a o s a pa icula nema ic o ien a ion. In he weak ancho ing egime he fi s wo e ms can be cal- cula ed (see Appendix A) yielding σ⊥ ns −σis =−w+O(w3) and σ ns −σ⊥ ns =3w/2+O(w2). Consequen ly, he we ing ansi ion occu s when w =2σ ni 3− .(17) This implies ha (nea ly) uni o m we s a es [Fig. 3(Wu)] do no occu a e y ough su aces, ⩾3. Wi hin he LdG heo y he NI in e acial ension σ ni is [20] σ ni =1 66+k 3+2k,(18) which may be used in Eq. (17). 2. Uni o m o dis o ed we ing ansi ion Wu↔Wd In gene al, s uc u ed su aces wi h homeo opic ancho ing p omo e smoo hly dis o ed we s a es, whe e he nema ic o i- en a ion ollows he su ace s uc u e. A c enella ed su aces, howe e , he p esence o geome ical singula i ies (co ne s) implies ha nema ic configu a ions which ollow he su ace ancho ing will ha e a numbe o opological de ec s. Fo shallow c enels, Wu, he uni o m we s a e analyzed in he p e ious Sec. IV A1, as an app oxima ion o he nea ly uni o m we s a e, is ene ge ically a o able. As he dep h o he c enels inc eases, he con ibu ion om he su ace alignmen a he la e al su aces inc eases and induces significan dis o ions. As in pe iodic iangula su aces [20,21], he ee ene gy o a dis o ed we configu a ion may be analyzed using he a gumen s ou lined in Sec. II. In each cell, he co ne s o he c enella ed su ace gene a e opological de ec s wi h cha ge ±1/3 a he op (wi h opening angle φ =3π/2), and ±1 a he bo om (wi h opening angle π/2) o he c enels. Using Eq. (4), he ee ene gy pe uni leng h and cell o he dis o ed we s a e may be w i en: (Wd)=σ⊥ ns (l1+l2+2h)+σ ni(l1+l2) +2πK 3ln h ξ+Bw,l1 h,l2 h,(19) whe e K(k)=(9/2)(2 +k)/(3 +2k) is an e ec i e elas ic cons an [21]. The second- o-las e m accoun s o he leading con ibu ion o he elas ic ee ene gy associa ed wi h he opological de ec s. Bco esponds o he nex - o-leading o de con ibu ion o he elas ic ee ene gy which, as discussed in Sec. II, is expec ed o depend on he ancho ing s eng h wand he a ios o he ele an leng h scales o he su ace s uc u e, bu no on he su ace scale. In addi ion, his con ibu ion will be di e en o symme ic we Ws dand asymme ic we Wa ds a es, by con as o he leading con ibu ion which is independen o he ex u e. In o de o check his scaling, we sub ac ed he su ace ension and loga i hmic con ibu ions o he ee ene gy ob ained om he LdG heo y, o di e en cell sizes keeping he aspec a ios fixed. As expec ed, we obse e a collapse o mode a e alues o he ancho ing s eng h w. The collapse occu s a smalle alues o was he dep h o he c enels hinc eases (see Fig. 5). The ansi ion be ween he wo we s a es Wuand Wdis gi en by he ee-ene gy balance (Wu)= (Wd), and yields 2h(σ ns −σ⊥ ns)≈3wh =2πK 3ln h ξ+Bw,l1 h,l2 h.(20) In deep c enels, la ge h/l2, we may neglec B. Then, he dis o ed we s a e is s able o su ficien ly s ong ancho ing, w, in deep c enels, h. The balance equa ion also implies ha 011703-6 NEMATIC WETTING AND FILLING OF CRENELLATED ... PHYSICAL REVIEW E 86, 011703 (2012) 0.1 0.2 0.3 0.4 0.5 -10 -5 0 = 1.2, = 20 ξ = 1.2, = 30 ξ = 1.2, = 40 ξ = 1.2, = 50 ξ = 1.5, = 20 ξ = 1.5, = 30 ξ = 1.5, = 40 ξ B(w,l 1 /h,l 2 /h) l l l l l l l w FIG. 5. (Colo online) Scaling o he ee ene gy, F−σ⊥ nw(l1+ l2+2h)−(2π/3)Kln(h/ξ) o dis o ed we s a es as a unc ion o he ancho ing s eng h wwi h he sys em size, o oughness =1.2 and =1.5. K(k) is an e ec i e elas ic cons an . The cu es o equal oughness collapse a mode a e o high alues o he ancho ing s eng h. The collapse occu s a lowe alues o he ancho ing as he dep h o he c enels hinc eases. by inc easing he scale o he sys em (by inc easing bo h l and h), he le -hand side inc eases linea ly, while he igh - hand side inc eases loga i hmically. As a esul , dis o ed we s a es a e a o ed wi h espec o (nea ly) uni o m we s a es in la ge sys ems. We no e, howe e , ha he analysis does no disc imina e be ween symme ic Ws dand asymme ic Wa dwe s a es. A symme ic dis o ion implies a configu a ion wi h opological de ec s wi h he same cha ge on he le and igh co ne s: −1/3 a he op, and +1 a he bo om co ne s, as depic ed in Fig. 3(Ws d). By con as , he asymme ic dis o ed we s a e is cha ac e ized by +1/3 and −1/3 cha ges a he op and −1 and +1 a he bo om co ne s, as depic ed in Fig. 3 (Wa d). The ee ene gy o he dis o ed s a es has he same loga i hmic con ibu ion, as his does no depend on he sign o he opological cha ges, bu he nex - o-leading con ibu ion is di e en . Ou de ailed calcula ions sugges ha , in gene al, he ee ene gy is lowe o he Ws ds a e, which is hus he s able dis o ed s a e. As men ioned p e iously, we ha e no obse ed a ansi ion be ween he nea ly uni o m we s a e and he dis o ed symme ic we s a e. Howe e , he ac ha in a small egion o he phase diag am (Fig. 4) heWa ds a e is s able indica es ha nex - o-leading con ibu ion Bis indeed di e en o each we s a e and s ongly dependen on he geome ic pa ame e s and he ancho ing s eng h. 3. Filling ansi ion o he unben s a e D ↔Fu We conside now he limi o high oughness (Fig. 4), which co esponds o su aces wi h deep c enels. In his egime, we ha e ound a filling ansi ion, om he d y (D) o a filled s a e cha ac e ized by a fla o unben NI in e ace (Fu). In deep c enels, he nema ic is dis o ed only a he bo om. Elsewhe e, he nema ic configu a ion is uni o m, pe pendicula o he la e al su aces and pa allel o he NI in e ace. Neglec ing he elas ic con ibu ions om he dis o ions, we can w i e he ee ene gy pe uni leng h and cell as (Fu)=σisl1+(σ ni +σ ns)l2+2σ⊥ nsh, =(σis −σ⊥ ns)l1+(σ ns −σ⊥ ns)l2+σ⊥ ns (l1+l2)+σ nil2. (21) The filling ansi ion occu s when (D)= (Fu), which leads o he balance equa ion (σ⊥ ns −σis)l1 l2 ( −1) + +σ ns −σ⊥ ns +σ ni =0,(22) he solu ion o which is w =σ ni l1 l2( −1) + −3 2 .(23) By con as o he we ing ansi ion D↔Wu, he filling ansi ion o he unben s a e depends no only on he su ace oughness bu also on he a io be ween he c enel sepa a ion and wid h, l1/l2. Equa ion (23) implies ha , a fixed l1/l2, he filling ansi ion o he unben s a e occu s only a su aces wi h oughness abo e a ce ain h eshold, >(l1/l2+3/2)/(l1/l2+1). 4. Filling ansi ion o he ben s a e D ↔Fb The filling ansi ion p edic ed by Eq. (23) does no accoun o he nema ic dis o ions ha a e, o cou se, induced by he geome ic s uc u e o he su ace. The elas ic ene gy a ising om hese dis o ions is no easy o ake in o accoun analy ically, as discussed p e iously. He e we gi e a c ude desc ip ion o he ben filled s a e. In shallow c enels he ben filled s a e is app oxima ed by wo ci cula domains o nema ic ben a ound he bo om co ne s, wi h a bounda y ha is ci cula o adius h, connec ed by a fla laye o uni o m homeo opic nema ic, o heigh h <h(see Fig. 6). As he c enel dep h inc eases he wo ci cula domains inc ease, and he uni o m fla laye ha connec s hem dec eases. We app oxima e he ee ene gy pe uni leng h and cell o his s a e by (Fb)≈σ⊥ ns(2h+l2)+σisl1+σ ni(πh−2h ) +σ⊥ ni l2−2h2−h2 + e(h),(24) whe e he heigh o he in e ace is, de e mined by mini- miza ion, h =σ ni/σ⊥2 ni +σ2 ni h. e(h) is he elas ic ee ene gy pe uni leng h and cell associa ed wi h he di ec o de o ma ions. Fo simplici y, we assume ha hese a ise om he ci cula domains cen e ed a he bo om co ne s, which co espond o +1 opological su ace de ec s. Wi hin he hh FIG. 6. Ben filled s a e app oxima ed by wo ci cula domains, o adius h, o nema ic ben a ound he bo om co ne s connec ed by a fla laye o uni o m homeo opic nema ic, o heigh h . 011703-7 N. M. SILVESTRE e al. PHYSICAL REVIEW E 86, 011703 (2012) F ank-Oseen elas ic heo y e(h)isgi enby e(h)=πK 2lnh ξ+Bw, h l1 ,h l2.(25) The filling ansi ion occu s when he d y and he ben filled s a es coexis , (D)= (Fb), which occu s a w = σ niπh l2−2h l2+σ⊥ ni 1−2√h2−h2 l2+ e(h)/l2 l1 l2( −1) + .(26) This ansi ion may occu a su aces wi h oughness > 1/(1+l1/l2), which is lowe han he oughness h eshold o he ansi ion o he unben s a e ob ained in Sec. IV A3. The analysis also shows ha he ben filled s a e is s abilized by wide c enels, i.e., la ge l2. 5. T ansi ion be ween illed s a es wi h and wi hou nema ic d ople s Fu(b)↔F+ u(b) When he s able configu a ion is a filled s a e, inc easing he su ace ancho ing wp omo es he nuclea ion o nema ic d ople s on he op su aces, sugges ing he g ow h o a we ing laye . In o de o p e en he nuclea ion o opological de ec s, howe e , he NI in e ace s ays pinned a he op co ne s, and does no de ach om he su ace. Figu e 7 depic s he ancho ing s eng h w , a he ansi ion—which is discon inuous—as a unc ion o he c enel sepa a ion l1. I dec eases apidly a small c enel sepa a ions, and decays slowly owa ds a sa u a ion alue a la ge c enel sepa a ions. In he ange o c enel sepa a ion l1, which was in es iga ed, he nema ic di ec o in he d ople s is uni o m and homeo opic e e ywhe e, including a he NI bounda y (which is plana a he ee NI in e ace). The pinning a he co ne s does no allow he NI in e ace o de ach and he wo in e aces in e ac s ongly, wi h he su ace ancho ing p e ailing a he bound NI 20 40 60 80 100 l1 0.24 0.26 0.28 0.3 0.32 w nume ical σ niR(l1)/l1 FIG. 7. (Colo online) Filling Fu(b) o filling F+ u(b) ansi ion. Ancho ing s eng h w as a unc ion o he c enel sepa a ion l1.The ull-black line co esponds o he ancho ing s eng h w ob ained nume ically. The dashed- ed line is he analy ical es ima e w = σ⊥ ni R(l1)ξ/l1,whe eR(l1)=l1/ξ +4+O(1/l1). The inse depic s he nema ic d ople on he op su ace. The nema ic di ec o is s ongly ancho ed o he su ace and is homeo opic e e ywhe e. in e ace. The shape o hese uni o mly aligned d ople s, z(x) can be calcula ed analy ically using an e ec i e Hamil onian (see Appendix B). A la ge c enel sepa a ions l1, he shape o he d ople z(x) is gi en by z(x)≈√2ξlnl1 πw σ⊥ ni ξ2sin πx l1.(27) This indica es ha he d ople heigh inc eases loga i hmically wi h l1up o he midpoin o he op su ace, whe e i a ains a maximum gi en by z(l1/2) ≃√2ξlnl1+O(1). As l1inc eases he d ople s sp ead and become fla e and fla e esembling a film, he hickness o which di e ges loga i hmically wi h l1. The ee ene gy pe uni leng h and uni cell o he d ople s, in he limi o la ge l1, is gi en by Eq. (B11). A he ansi ion his ee ene gy is equal o he ee ene gy o he s a e wi hou a d ople , σisl1, yielding he ancho ing ansi ion s eng h w , w =σ⊥ ni +4σ⊥ ni ξ l1+O1 l2 1.(28) This es ima e o w is compa ed o he nume ical esul s o he LdG ee ene gy in Fig. 7. The ag eemen is excellen o he whole ange o c enel sepa a ions. 6. Inc easing he size o he sys em Inc easing he size o he sys em o l=20ξ esul s in a phase diag am simila o ha depic ed in Fig. 4. Figu e 8 illus a es he di e ences be ween he wo. A low oughness, he we ing ansi ion D↔Wuis una ec ed by he scale o he sys em, as expec ed. Fo ough su aces he filling ansi ion D↔Fucon e ges o he same asymp o ic alue, which is also scale independen . The ansi ion om d y o a ben filled s a e D↔Fbis s abilized and is obse ed a lowe oughness and ancho ing s eng h. Likewise, he ansi ions be ween filled s a es wi h and wi hou nema ic d ople s on he op su aces, (Fu(b)↔F+ u(b)), occu a lowe alues o he 11.2 1.4 1.6 1.8 2 2.2 2.4 0 0.1 0.2 0.3 0.4 =20 ξ =10 ξ l l w FIG. 8. (Colo online) Phase diag ams o su aces wi h l1=l2= 10ξ( ull-black line) and 20ξ(dashed- ed line). Re-en an filling is enhanced. The g ey lines ep esen he weak fi s -o de ansi ion be ween he wo we s a es (see Fig. 4). 011703-8 NEMATIC WETTING AND FILLING OF CRENELLATED ... PHYSICAL REVIEW E 86, 011703 (2012) 11.2 1.4 1.6 1.8 2 2.2 2.4 0 0.1 0.2 0.3 0.4 w l=30 ξ l=25 ξ FIG. 9. (Colo online) Phase diag ams o sys ems wi h l1=l2= 25ξ,30ξ. A low and high oughness he phase diag am is simila o he p e ious ones. A in e media e oughness, he e is a clea enhancemen o he e-en an egions, as he we s a e is s able be ween he filled s a es Fband F+ b. The g ey lines ep esen he weak fi s -o de ansi ion be ween he wo we s a es (see Fig. 4). Blue ( e ical) and g een (ho izon al) a ows indica e e-en an pa hs by a ying he ancho ing and he oughness, espec i ely. ancho ing s eng h, in line wi h he discussion o he p e ious sec ion. As a esul , he egion whe e e-en an filling is obse ed inc eases. As he size o he sys em inc eases u he , he phase diag am changes significan ly. Figu e 9depic s he phase diag ams o sys ems wi h l/ξ =25,30. The low and high oughness egimes exhibi he same beha io , as hey a e independen o he sys em size. The mos s iking changes occu a in e media e oughness. The e is a clea enhancemen o he filled s a es a in e media e oughness. The e-en an egion is also mo e p ominen as he we s a e is clea ly sepa a ed om he filled s a es, wi h and wi hou nema ic d ople s, Fband F+ b. In his egime, he e is a ansi ion om he filled o he we s a e, Fb↔Wd, a low ancho ing s eng h, wi hou he nuclea ion o opological de ec s. As he ancho ing inc eases, he nema ic dis o ions become s onge leading o he nuclea ion o de ec s. As a esul he sys em e-en e s he filled s a e—wi h nema ic d ople s— h ough he pinning o he NI in e ace a he op co ne s o he c enels, Fig. 3(F+ b). Fo la ge sys em sizes he e is a second na ow egion, a highe oughness, whe e he e-en an sequence d y →filled →we →filled is obse ed. The filled s a es, howe e , a e unben filled s a es Fu. In a na ow ange o he ancho ing s eng h, inc easing he oughness esul s in geome ically induced e-en an sequences whe e he sys em goes h ough we →(ben ) filled →we →(unben ) filled s a es, howe e , a e unben filled s a es Fu. In a na ow ange o he ancho ing s eng h, inc easing he oughness esul s in geome ically induced e-en an sequences whe e he sys em goes h ough we →(ben ) filled →we →(unben ) filled s a es, a sequence ha is no obse ed o simple fluids and appea s o be a ea u e o nema ic liquid c ys als a c enella ed su aces. 11.2 1.4 1.6 1.8 2 2.2 2.4 0 0.1 0.2 0.3 0.4 w l1=10 ξ ,l2=20 ξ l1=20 ξ ,l2=40 ξ FIG. 10. (Colo online) Phase diag ams o sys ems wi h l1= 10ξ,l2=20ξand l1=20ξ,l2=40ξ. The s abili y o he ben filled, Fband F+ b, is enhanced. The g ey lines ep esen he weak fi s -o de ansi ion be ween he wo we s a es (see Fig. 4). B. C enel wid h = c enel sepa a ion Dis inc phase diag ams a e ob ained when he sepa a ion be ween c enels is di e en om hei wid h. In his sec ion we add ess c enella ed su aces whe e l1= l2. We conside wo cases: (1) wide o closely spaced c enels l2=2l1, and (2) na ow o widely sepa a ed c enels l1=2l2. 1. Wide o closely spaced c enels: l2=2l1 When he wid h o he c enel is wice he c enel sepa a ion, he s abili y o he ben filled s a e inc eases, i.e., ben filled s a es occu in a wide ange o pa ame e s. The phase diag ams o l1=10ξ,l2=20ξand l1=20ξ,l2=40ξa e illus a ed in Fig. 10. The enhanced s abili y o he symme ic filled s a es esul s om he ac ha in wide c enels bending o he NI is less se e e and he co esponding elas ic ene gy is lowe . As he dep h o he c enels inc eases he NI in e ace becomes fla o unben . We no e ha he ansi ions o unben filled s a e occu a a highe oughness >4/3. As he sys em size inc eases he s abili y o he ben filled s a es also inc eases, as hey occu a lowe oughness and ancho ing s eng h. O he wise, he phase diag am esembles ha o small sys ems wi h l1=l2. 2. Na ow o widely sepa a ed c enels: l1=2l2 Finally, we conside na ow o widely sepa a ed c enels, on su aces whe e he dis ance be ween c enels is wice hei wid h. Figu e 11 compa es he phase diag ams o sys ems wi h l1=20ξ,l2=10ξand l1=40ξ,l2=20ξ. In his case, he s abili y o he ben filled s a es is educed, i.e., ben filled s a es occu in a smalle ange o pa ame e s. By con as , he ansi ion o unben filled s a es wi h fla NI in e aces occu s a lowe oughness >7/6. We ing is also enhanced, and as a consequence, he e-en an egion becomes mo e p ominen . The phase diag am esembles ha o la ge sys ems wi h l1=l2(same wid h and sepa a ion) in he sense ha he we egion in udes in o he filled egions. In his case, he ben filled s a es—wi hou nema ic d ople s—a e educed o small pocke s o islands o s abili y ha seem o anish o he 011703-9