a Xi :cond-ma /0510623 1 [cond-ma .s a -mech] 24 Oc 2005
T ic i ical wedge illing ansi ions wi h
sho - anged o ces
J. M. Rome o-En ique†and A. O. Pa y‡
†Depa 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
‡Depa men o Ma hema ics, Impe ial College 180 Queen’s Ga e, London SW7
2BZ, Uni ed Kingdom
Abs ac . We show ha he 3D wedge illing ansi ion in he p esence o sho -
anged in e ac ions can be i s -o de o second o de depending on he s eng h
o he line ension associa ed wi h o he wedge bo om. This ac implies he
exis ence o a ic i ical poin cha ac e ized by a sho -dis ance expansion which
di e s om he usual con inuous illing ansi ion. Ou analysis is based on an
e ec i e one-dimensional model o he 3D wedge illing which a ises om he
iden i ica ion o he b ea he modes as he only ele an in e acial luc ua ions.
F om such analysis we ind a co espondence be ween con inuous 3D illing a
bulk coexis ence and 2D we ing ansi ions wi h andom-bond diso de .
PACS numbe s: 68.08.Bc, 05.70.Np, 68.35.C , 68.35.Rh
T ic i ical wedge illing ansi ions wi h sho - anged o ces 2
αx
yl0
ξ
lW(y)
WW
ξy
Figu e 1. Schema ic illus a ion o a ypical in e acial con igu a ion and ele an
leng hscales o a luid adso p ion in a 3D wedge.
Fluid adso p ion in mic opa e ned and sculp ed geome ies has become he
subjec o in ense s udy o e he las decade. Highly imp essi e echnological ad ances
which allow he ailo ing o mic o-pa e ned and s uc u ed solid su aces on he
nanome e o mic ome e scale [1] a e a landma k in he de elopmen o he eme ging
mic o luidic indus y [2] which aims a mini ua izing chemical syn hesis plan s o
biological analysis equipmen in much he same way he silicon chip b ough abou
he elec onics e olu ion. Howe e , he heo e ical unde s anding o his phenomenon
is a om being comple e. Recen s udies o illing ansi ions o luids in 3D wedges
show ha in e acial luc ua ions a e g ea ly enhanced compa ed wi h we ing a la
subs a es [3, 4]. The con ol o such enhanced in e acial luc ua ions is c ucial o
he e ec i eness o he mic o luidic de ices. Fo una ely, he e a e simple heo e ical
app oaches which ake in o accoun hese e ec s. Fo example e ec i e Hamil onian
p edic ions o he c i ical exponen s a con inuous (c i ical) wedge illing wi h sho -
anged o ces ha e been con i med in la ge scale Ising model simula ion s udies [5].
Simila expe imen al e i ica ion o he p edic ed geome y-domina ed adso p ion
iso he ms a comple e wedge illing [6] aise hopes ha he illing ansi ion i sel and
ela ed luc ua ion e ec s will be obse able in he labo a o y. We u he de elop
he heo y o wedge illing in his pape , ocussing on he eme gence o a new ype o
con inuous illing: ic i ical illing.
Fi s we b ie ly e iew he luc ua ion heo y o 3D wedge illing. Conside he
in e ace be ween a bulk apou a empe a u e Tand sa u a ion p essu e wi h a
3D wedge cha ac e ised by a il angle α. Mac oscopic a gumen s dic a e ha he
wedge is pa ially illed by liquid i he con ac angle θ > α and comple ely illed i
θ < α [7]. The illing ansi ion e e s o he change om mic oscopic o mac oscopic
liquid adso p ion as T→T , a which θ(T ) = α, and may be i s -o de o con inuous
(c i ical illing). Bo h o hese ansi ions can be iewed as he unbinding o he liquid-
apou in e ace om he wedge bo om. Cha ac e is ic leng h scales a e he mean
in e acial heigh abo e he wedge bo om lW, he oughness ξ⊥and he longi udinal
co ela ion leng h ξy, measu ing luc ua ions along he wedge (see Figu e 1). The
ele an scaling ields a c i ical illing a e θ−αand he bulk o de ing ield h(which
is p opo ional o he p essu e di e ence wi h he sa u a ion alue). A coexis ence
(h= 0) we de ine c i ical exponen s by lW∼(θ−α)−βWand ξy∼(θ−α)−νy. The
oughness can be ela ed o ξyby he scaling ela ionship ξ⊥∼ξζW
y, whe e ζWis he
wedge wande ing exponen . Fo sho - anged o ces, ζW= 1/3.
Fo shallow wedges, i.e. α≪1, he ee ene gy o an in e acial con igu a ion can
be modelled by an e ec i e Hamil onian based on he capilla y wa e model o we ing
o plana subs a es [8]. Howe e , an analysis o his model [3] shows ha he liquid-
apou in e ace ac oss he wedge is ap oxima ely la and so -mode luc ua ions a ise
T ic i ical wedge illing ansi ions wi h sho - anged o ces 3
om local ansla ions in he heigh o he illed egion along he wedge. These b ea he
modes a e he only ele an luc ua ions in he con inuous illing phenomena, and can
be aken in o accoun by he ollowing e ec i e Hamil onian [3]
HW[l0] = Zdy (Σl0
αdl0
dy 2
+VW(l0))(1)
whe e l0(y) is he local heigh o he in e ace a posi ion yalong he wedge bo om
and Σ is he liquid- apo su ace ension. No e ha he e ec i e bending e m
esis ing luc ua ions along he wedge is p opo ional o he local in e acial heigh .
The e ec i e binding po en ial VW(l0) o l0≫lπ, whe e lπis he mean we ing ilm
hickness o a plana subs a e, is gi en (up o i ele an addi i e cons an s) by [3]:
VW(l0)≈h(l0−lπ)2
α+Σ(θ2−α2)l0
α+Z(l0−lπ)/α
−(l0−lπ)/α
dxW(l0−α|x|)(2)
whe e W(l) is he binding po en ial be ween he gas-liquid in e ace and a plana
subs a e. No e ha he mean ield esul o lWis eco e ed by minimizing VW(l0)
o l0> lπ, which is an impo an check on he sel -consis ency o he me hod [3]. In
addi ion o a ha d wall epulsion o l0<0, he po en ial VW(l0) con ains a sho -
anged a ac ion which may be modi ied by mic opa e ning a s ipe along he wedge
bo om, so as o weaken he local wall- luid subs a e and he e o e s eng hen he
in e acial binding. This obse a ion will be c ucial o he exis ence o ic i ical
illing, since wi h his deco a ion i may be possible o bind he in e ace o he wedge
bo om a he illing bounda y θ=αand h= 0. Fo la e con enience, he ea e we
will se h= 0 in ou discussion o con inuous illing.
The quasi-one-dimensional na u e o he e ec i e Hamil onian Equa ion (1)
allows us o use he ans e -ma ix o malism. In he con inuum limi he pa i ion
unc ion is de ined as a pa h in eg al [9] (se ing kBT= 1 o con enience)
Z[lb, la, Y ] = ZDl0exp(−HW[l0]) (3)
whe e Yis he wedge leng h and laand lba e he endpoin heigh s. The posi ion-
dependen s i ness in oduces some ambigui y in he de ini ion o he pa h in eg al.
This p oblem was al eady poin ed ou in Re . [10] and is ela ed o he well-
known o de ing p oblem in he quan iza ion o classical Hamil onians wi h posi ion-
dependen masses. Simila issues also a ise in solid s a e physics [11]. Bo owing om
he me hods used o o e come hese di icul ies we use he ollowing de ini ion
Z[lb, la, Y ] = lim
N→∞Zdl1...dlN−1
N
Y
j=1
K(lj, lj−1, Y/N) (4)
whe e l0≡laand lN≡lb, and K(l, l′, y) is de ined as:
K(l, l′, y) = sΣ√ll′
απy exp −Σ√ll′
αy (l−l′)2−yVW(l)!(5)
In he con inuum limi he pa i ion unc ion becomes
Z(lb, la, Y ) = X
n
ψn(lb)ψ∗
n(la)e−EnY(6)
T ic i ical wedge illing ansi ions wi h sho - anged o ces 4
h
θ−α
b)a) 1/u
00
1/w
(i)
(ii)
(iii)
BoundUnbound
(i)
(ii)
(iii)
Figu e 2. Phase diag ams o (a) illing and (b) we ing ansi ions. The
hick and dashed lines in bo h diag ams co espond o con inuous and i s -o de
bounda ies be ween bound and unbound in e acial s a es, espec i ely. The
a ows show ep esen a i e pa hs along which con inuous unbinding occu : (i)
and (ii) o ic i ical illing (c i ical we ing) and (iii) o c i ical illing (comple e
we ing), espec i ely. The illed ci cles ep esen he ic i ical illing and c i ical
we ing poin s, espec i ely. See ex o explana ion.
whe e he comple e o hono mal se sa is y
−α
4Σ
∂
∂l 1
l
∂
∂l+VW(l)−3α
16Σl3!ψ=Eψ (7)
In he he modynamic limi Y→ ∞ we ob ain he p obabili y dis ibu ion unc ion
(PDF) o he midpoin in e acial heigh PW(l0) = |ψ0(l0)|2, he wedge excess ee
ene gy W=E0and he longi udinal co ela ion leng h ξy= 1/(E1−E0). A his
poin we mus ema k ha any de ini ion o he pa h in eg al which is in a ian upon
exchanging laand lbleads o an Sch ¨odinge equa ion simila o Equa ion (7) bu wi h
a di e en coe icien o he ex a 1/l3 e m in he e ec i e binding po en ial [10].
The change o a iables λ=p8Σ/αl3/2/3 and ψ(l) = (2Σl/α)1/4φ(λ(l)) [12]
ans o ms he Equa ion (7) o:
−1
2
d2φ(λ)
dλ2+VW[l(λ)] −5
72λ2φ(λ) = Eφ(λ) (8)
wi h l(λ) = 3λ/p8Σ/α2/3. In gene al, he e will be an in e acial bound s a e a
bulk coexis ence o θ=αi he s eng h o he small l0a ac ion be ween he gas-
liquid in e ace and he subs a es, which we will deno e as u, is g ea e han some
alue uc. Consequen ly, he illing ansi ion is i s -o de i u > ucand c i ical i
u < uc. T ic i ical illing is obse ed when u−uceme ges as a new ele an ield (in
he eno maliza ion-g oup sense). I W(l)∼ −a/lp+b/lq, di e en scena ios may a ise
as he ange o he binding po en ial is a ied. In pa icula , o p > 4 he long- ange
beha iou o VWis domina ed by he 1/l3 e m o θ=α, so he illing phenomena a e
luc ua ion-domina ed. This inding is consis en wi h he exis ence o wo di e en
luc ua ion egimes o c i ical illing: mean- ield i p < 4 and luc ua ion-domina ed
egime i p > 4 [3]. In he luc ua ion-domina ed egime, examina ion o Equa ion
(8) shows ha he e is an analogy be ween 3D con inuous illing and 2D con inuous
we ing (see Figu e 2), whe e he ole a we ing o he bulk o de ing ield hand he
po en ial s eng h wa e played by θ−αand u o illing phenomena, espec i ely. In
pa icula , 3D ic i ical (c i ical) illing is analogous o 2D c i ical (comple e) we ing,
espec i ely. Di e en c i ical exponen s which cha ac e ize he di e gence o leng h
scales can be de ined. In addi ion o he c i ical illing c i ical exponen s βWand νy
T ic i ical wedge illing ansi ions wi h sho - anged o ces 5
de ined along ou e (ii) in Figu e 2(a) (see abo e), we can de ine new c i ical exponen s
o ic i ical illing a θ=α( ou e (i) in Figu e 2(a)) as:
lw∼(u−uc)−β∗
W, ξy∼(u−uc)−ν∗
y(9)
and ξ⊥∼ξζ∗
W
y, whe e ζ∗
Wis he ic i ical wande ing exponen which in gene al may be
di e en om ζW(in con as wi h he we ing case). Mo e gene ally, in he icini y o
he ic i ical poin we an icipa e scaling e.g. ξy∼ |u−uc|−ν∗
yΛ(θ−α)|u−uc|−∆∗
wi h he gap exponen ∆∗. Thus along ou e (ii) ξy∼(θ−α)−ν∗
y/∆∗.
We ocus now on he case o sho - anged o ces as he p o o ype o he
luc ua ion-domina ed egime. In addi ion o he ha d-wall condi ion, VW(l0) can
be modelled as a con ac -like a ac ion wi h s eng h u. An analysis o Equa ion (7)
o l0→0 shows ha he sho -dis ance expansion o he PDF is ei he PW∼l0
o PW∼l3
0. We an icipa e ha he o me co esponds o ic i ical beha iou
and he la e o c i ical illing. I is ema kable ha he modynamic consis ency
a c i ical illing is ensu ed as he local densi y a he wedge bo om is non-singula ,
i.e. ρw(0)−ρl∼T−T , whe e ρlis he bulk liquid densi y [14]. This p ope y is only
ob ained i he pa i ion unc ion is de ined by Equa ion (4) and Equa ion (5). Thus,
he ambigui y in i s de ini ion can be emo ed by imposing his egula i y condi ion
on he sho -dis ance expansion o he in e acial heigh PDF.
We epo now ou explici esul s (de ails will be p esen ed elsewhe e). Along
ou e (i) we ind ha he e is only one bound solu ion o Equa ion (7) o u > uc≈
1.358 wi h E0∝(u−uc)3and associa ed PDF
PW(l0) = 6√3π
ξu
l0
ξuAi l0
ξu2
(10)
whe e Ai(x) is he Ai y unc ion and in he scaling limi ξu∼ |u−uc|−1. Thus
lW∼ξ⊥∝(u−uc)−1and ξy∝(u−uc)−3iden i ying β∗
W= 1, ν∗
y= 3 and ob aining
ζ∗
W=ζW= 1/3. As p edic ed, he sho -dis ance beha iou o he PDF is linea wi h
l0.
On he o he hand, he scaling o he PDF o θ > α is gi en by (see also Figu e 3):
PW(l0)∝l0exp 2ǫl0
ξθ−2l2
0
ξ2
θH2
ν√2l0
ξθ−ǫ
√2(11)
whe e ξθ= Σ−1/2[(θ/α)2−1]−1/4,ǫ= ΣE0ξ3
θ/α,ν=ǫ2/4−1/2 and Hν(x) is he
He mi e unc ion [13]. The alue o ǫis ob ained as he smalles solu ion o he
ollowing equa ion:
±Γ−1
33−2/3
Γ1
3
ξθ
ξu
=ǫ+ǫ2
√2−√2Hǫ2
4−3
2−ǫ
√2
Hǫ2
4−1
2−ǫ
√2(12)
whe e he posi i e (nega i e) sign co esponds o u > uc(u < uc), espec i ely.
The inse o Figu e 3 plo s he solu ion o his equa ion. As an icipa ed, no e ha
scaling is obeyed in he icini y o he ic i ical poin as he wedge excess ee ene gy
W∼ξ−3
θF±(ξθ/ξu). Fo u > ucand ξθ/ξu≫1, we ha e checked nume ically ha he
PDF (11) con e ges o he exp ession gi en by Equa ion (10) o he co esponding
alue o ξugi en by Equa ion (12). Thus, he in e ace emains bound o he subs a e
when θ→α, in ag eemen o he i s -o de cha ac e o he illing ansi ion. The
he modynamic pa h (ii) o he ic i ical poin co esponds o ξu→ ∞, which
co esponds o ǫ≈1.086. Thus along his ou e lW∼ξ⊥∝(θ−α)−1/4simila

T ic i ical wedge illing ansi ions wi h sho - anged o ces 6
0 1 2 3 4
ξθ/ξu
-3
-2
-1
0
1
2
ε
00.5 11.5 22.5 3
l0/ξθ
0
0.4
0.8
1.2
1.6
2
PW(l0/ξθ)
Figu e 3. Plo o he scaled PDF o ǫ=−1.5 ( hick dashed line), and along
ou es (ii) and (iii) in Figu e 2(a), i.e. o ǫ≈1.086 ( hick con inuous line) and
ǫ≈1.639 ( hick do -dashed line), espec i ely. Fo compa ison, he PDF om
Equa ion (10) wi h ξu≈1.968ξθ(which co esponds o ǫ=−1.5, see inse ) is
also plo ed ( hin dashed line). Finally, he scaled PDF ob ained in Re . [10]
is also shown ( hin do -dashed line). Inse : Plo o ǫas a unc ion o ξθ/ξu o
u < uc(con inuous line) and u > uc(dashed line).
o c i ical illing. F om analysis o he spec um i is also possible o show ha
ξy∝(θ−α)−3/4, so he ic i ical gap exponen ∆∗= 4. Finally, o he modynamic
pa hs (iii) a om he ic i ical poin , i.e. ξu→0, we ha e ound ha he scaling o
he PDF is o he o m shown in Equa ion (11) wi h ǫ≈1.639. No e ha PW∼l3
0as
l0→0, in ag eemen wi h ou p e ious s a emen . This condi ion is no ull illed by
he solu ion p esen ed in Re . [10], al hough globally i does no di e oo much om
ou exac solu ion (see Figu e 3).
We inish by men ioning a ema kable connec ion o sho - anged o ces be ween
3D wedge illing and 2D we ing wi h andom-bond diso de [15]. The c i ical
exponen s co esponding o ic i ical and c i ical wedge illing can be ob ained om
gene alized andom-walk me hods [16] in e ms o he wedge wande ing exponen ζW.
In pa icula , hey a e ound o ha e he same dependence o he c i ical exponen s o
c i ical and comple e we ing, espec i ely, bu in e ms o an e ec i e 2D wande ing
exponen equal o 2ζW. Fo sho - anged o ces (ζW= 1/3), his implies ha he
se o c i ical exponen s is he same as o 2D andom-bond diso de [17]. These
p edic ions may ce ainly be es ed in Ising model simula ion s udies and would be a
s ingen es o he heo y o 3D wedge illing.
Acknowledgmen s
J.M.R.-E. acknowledges inancial suppo om he Eu opean Commission unde
Con ac MEIF-CT-2003-501042.
T ic i ical wedge illing ansi ions wi h sho - anged o ces 7
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