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
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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).
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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
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