Critical adsorption at chemically structured substrates

0.49 MB·English

by  Monika Sprenger

Critical adsorption at chemically structured substrates
Monika Sprenger, Frank Schlesener, and S. Dietrich
Max-Planck-Institut fu¨r Metallforschung, Heisenbergstr. 3, D-70569 Stuttgart, Germany, and
Institut fu¨r Theoretische und Angewandte Physik,
Universit¨at Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany
(Dated: February 2, 2008)
5 We consider binary liquid mixtures near their critical consolute points and exposed to geometri-
0 callyflatbutchemicallystructuredsubstrates. Thechemicalcontrastbetweenthevarioussubstrate
0 structures amounts to opposite local preferences for the two species of the binary liquid mixtures.
2 Order parameters profiles are calculated for a chemical step, for a single chemical stripe, and for a
periodic stripe pattern. The order parameter distributions exhibit frustration across the chemical
n
steps which heals upon approaching the bulk. The corresponding spatial variation of the order
a
parameteranditsdependenceontemperaturearegovernedbyuniversalscalingfunctionswhichwe
J
calculatewithinmeanfieldtheory. Thesescalingfunctionsalsodeterminetheuniversalbehaviorof
5
theexcess adsorption relative to suitably chosen reference systems.
2

PACSnumbers: 64.60.Fr,68.35.Rh,61.20.-p,68.35.Bs
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I. INTRODUCTION (a) y (b) y (c) y
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(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)+(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) − (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)+(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)− xz
o have applications in micro-reactors, for the “lab on a
c chip”, and in chemical sensors [1, 2]. They can oper-
[ ate with small amounts of reactants which is important FIG.1: Thedifferentchemicalsubstratestructuresstudiedin
1 when investigating expensive substances and substances this article: (a) chemical step (cs), (b) single stripe (ss), (c)
periodic stripes (ps). All systems are translationally invari-
v which are available only in small amounts like biological
1 material or when dealing with toxic or explosive mate- ant in the y-direction. The shading indicates different local
9 preferences for, e.g., the two components of a binary liquid
rials. At these small scales the interaction of the fluids
5 mixture exposed to the geometrically flat surface of macro-
with the substrate becomes important and there is the
1 scopicly large lateral extension B.
challenge to control the distribution and the flow of the
0
5 fluids on these structures.
0 In the following we investigate three different types
t/ of chemically structured substrates, as shown in Fig.1. transition. This is either the liquid-vapor critical point
a First we analyze fluid structures at a chemical step (see of a one-component liquid or the critical demixing tran-
m
Fig. 1(a)) which is important for understanding the lo- sitionofabinaryliquidmixture. Inthesecasestheensu-
- cal properties of fluids at the border of stripes. Next ing critical phenomena are to a large extent universal in
d
weconsiderasinglechemicalstripe(seeFig.1(b))asthe character,i.e.,theyrendermoleculardetailsirrelevantin
n
o simplest chemical surface pattern, and finally we study favor of universal scaling functions by involving spatial
c a periodic stripe pattern (see Fig. 1(c)) as the paradig- variationsonthescaleofthedivergingcorrelationlength.
: matic case for the investigation of adsorption at hetero- Near the critical point the surface patterning acts like a
v
i geneous surfaces. laterally varying surface field of alternating sign. This
X
The chemical contrast on the substrates acts on the generates an order parameter profile characterizing crit-
r adjacent liquid [3, 4]. In order to impress the chemi- ical adsorption of opposite sign such that the system is
a
cal pattern of the substrate on a one-component liquid, frustrated across the chemical steps. Upon approaching
the chemical structure has to be chosen as a pattern of the bulk of the fluid this frustration is healed and the
lyophobic and lyophilic regions. For binary liquid mix- healing is expected to be governed by universal scaling
tures one chooses a pattern of two different substrate functions. In addition, the substrate patterning results
types, such that one component is preferred by one sub- in a change of the excess amount of adsorbed fluid with
strate type and the second component by the other sub- respect to the case of critical adsorption on a homoge-
strate type. This lateral structuring of the substrate neoussubstrate. Theexcessadsorptionisexpectedtobe
causes a rich fluid-substrate interface structure which governedby universal scaling functions, too.
typically depends on the molecular details of the local It is the purpose of this contribution to describe this
forcefields. However,inthisstudywefocusonthepartic- scaling in terms of general renormalization group argu-
ular case of the fluid being close to a second-orderphase ments and to calculate the corresponding universal scal-
2
ing functions to lowest order, i.e., within mean field the- 2. Experimental methods
ory. If, as it is actually the case, the size of the lateral
structures is comparable with the range of the correla-
Different techniques to produce chemical structures in
tion length, which can reach up to 100nm close to the
therangeofµmandnmhavebeenestablished[7]. Inor-
criticalpoint,onecanexpectarichinterplaybetweenthe
dertoobtaintopologicallyflatbutchemicallystructured
externallyimprintedpatternsandthecriticalphenomena
substrates one can take advantage of the self-ordering
characterized by the correlationlength.
mechanism of self-assembled monolayers (SAM) in us-
ingblock-copolymersormixturesofpolymerswhichform
structures while demixing [3, 8, 9]. The morphologyand
domainsize ofthese structuresdepend ondifferent char-
acteristics of the materials and of the formationprocess.
Anotherpossibilitytoproducestructuredsubstratesisto
1. Critical phenomena changethe functionality ofthe SAMs partiallyby irradi-
atinga homogeneousSAM whichis coveredwitha mask
that carries the desired structure [10]. A third method
In order to describe critical phenomena [5, 6] one dis- is the so-called micro-contact printing [11] where sam-
tinguishes properties to classify them. Starting with un- ples, e.g., produced with lithography,are used as a mold
confined systems one introduces the so-called bulk uni- for an elastomer stamp. The structure of the original is
versalityclasseswhicharecharacterizedbycriticalexpo- copied by stamping a “thiol ink” on gold covered sub-
nents and amplitudes which describe the dependence of strates which chemisorbs there and forms a SAM. The
variousquantities,e.g.,the orderparameterandthe cor- emptyspacesbetweenthepatternsofthestampedstruc-
relation length, on the reduced temperature t = T−Tc ture can be filled with a second thiol with a different
Tc
upon approaching the critical temperature Tc. These functional end-group such that a topologically flat but
critical exponents are universal, i.e., the same for all chemically structured substrate is built up. As a last
members of a universality class. The amplitudes usually method we mention the exposure of oxidated titanium
are non-universal, whereas the number of independent surfaces to UV-light [12, 13, 14].
non-universal amplitudes is limited; any non-universal Inordertoinvestigatethestructuralpropertiesoffluid
amplitude can be expressed in terms of these indepen- systemsnearsurfacesvariousexperimentalmethodshave
dent non-universal amplitudes and universal amplitude beendeveloped: Ellipsometryandespecially phasemod-
ratios. One-componentfluidsneartheirliquid-vaporcrit- ulated ellipsometry have been established in Refs. [15]
ical point and binary liquids near their demixing critical and [16] more than 20 years ago and are still powerful
point belong to the Ising universality class like uniax- tools [17, 18]. An incident light beam is reflected by the
ial ferromagnets. These systems have two independent surfaceofinterestandthe ratioofthe complexreflection
non-universal amplitudes. In this sense our subsequent amplitudes for polarizations parallel and perpendicular
analysis holds for all systems encompassed by the Ising to the plane ofincidence (“coefficientofellipsometry”or
universality class. “ellipticity”) is measured. The order parameter is mod-
eledandrelatedtotheellipticityviathespatiallyvarying
Theorderparameterφindicatesthedegreeoforderin
dielectric constant and can then be compared with the
the system and has to vanish above the critical temper-
measured value. In neutron or X-ray reflectometry one
ature in the absence of an ordering field while below the
measures the reflectivity as a function of the momentum
criticaltemperatureittakesafinitevalue. Itisdescribed
transfernormalto the surfaceofthe sample. Thisreflec-
by the universal critical exponent β and a non-universal
tivity spectrum is related to the refractive index profile
amplitude a: φ(t) = atβ. For a one-component fluid φ
| | whichitselfis associatedwiththe profileofthe orderpa-
is the difference between the density and its value at the
rameter [18, 19, 20, 21, 22]. A more direct measurement
critical point. In the case of a binary liquid mixture it
of the adsorption of a component to a substrate can be
is chosen as the difference between the concentration of
carriedoutwithadifferentialrefractometer[23,24]. Here
one fluid component andits concentrationat the critical
alaserbeampassesa measurementcellconsistingoftwo
demixing point.
compartments filled with the fluid of interest and a ref-
The correlation length is defined by the exponential erence liquid, respectively. The intensity of the deflected
decayofthe bulk two-pointcorrelationfunction G(r) for beam is proportional to their difference in refractive in-
large distances (r ) at temperatures off criticality dex. By comparing this measured intensity with the one
(T =T ) and it is d→en∞oted as ξ+ above and as ξ− below of a beam deflected by a measurement cell filled with a
c
the6criticaltemperature,respectively. Itdivergesaccord- liquid with known interfacial properties and the refer-
ing to the power law ξ± = ξ± t−ν with the universal ence liquid one is able to infer the properties of the fluid
criticalexponentν andthenon0-u|n|iversalamplitudesξ±. of interest.
0
Their ratio ξ+/ξ− is, however, universal. If the critical With these methods critical adsorption on homoge-
0 0
medium is broughtnear a substrate, surface universality neous substrates has been experimentally investigated
classes come into play which we shall discuss in Sec.II. (seeRefs. [25]and[26]andreferencestherein). Onchem-
3
ically structured substrates mostly wetting experiments which is larger at the surface than its value in the bulk
have been performed [27, 28, 29]. The present paper canbe obtained in the absence of surface fields but for a
extends theoreticalwork on criticaladsorption on chem- negative surface enhancement (h =0,c<0). Since this
1
icallyhomogeneous[30,31, 32,33]andtopologically[34] is rather uncommon for magnetic systems, the normal
structured surfaces and work on wetting phenomena at surface universality class is also referred to as the so-
chemically structured substrates [35, 36, 37, 38, 39] to called extraordinary surface universality class. The fact
the case of critical adsorption on geometrically flat, but that the normal and the extraordinary cases are equiva-
chemically structured surfaces. lentandidenticalattheirfixedpoints,(h ,c=0)
1
| |→∞
The remainder of this paper is organized as follows: for fluid systems and (h = 0,c ) for magnetic
1
→ −∞
In Sec.II we introduce our model and in order to set systems, and thus identical with respect to their asymp-
the stagewerecallpreviousresultsoncriticaladsorption totic behavior, was predicted by Bray and Moore [40]
at homogeneous substrates. In Sec.III we present our and later proven by Burkhardt and Diehl [41]. In the
resultsforthecriticaladsorptionatchemicallystructured followingweshallfocusonthenormalandextraordinary
substrates. We summarize our findings in Sec.IV. surface universality classes, respectively.
A homogeneous substrate confining a binary liquid
mixture inevitably has a preference for one of the two
II. CRITICAL ADSORPTION AT components. Thispreferencebecomespronouncedatthe
HOMOGENEOUS SUBSTRATES criticalpointandleadstoanenrichmentofthepreferred
component at the substrate. At the critical temperature
the localorder parameterprofile decays algebraicallyto-
Boundaries, which come into play when investigating
wards its bulk value. This phenomenon has been called
confined systems, induce deviations from the bulk be-
critical adsorption [32].
havior. Near a critical point the Ising bulk universality
In the case of planar homogeneous substrates besides
classsplitsintothreepossiblesurfaceuniversalityclasses
the correlation length ξ± the distance z from the sub-
(denotedasnormal,special,andordinarysurfaceuniver-
strate is the other relevant length scale. If this length
sality class, respectively) characterized by surface criti-
is scaled with the bulk correlation length ξ±, the or-
cal exponents and amplitudes [30, 31]. The boundary
der parameter profile φ(z,t) takes the following scal-
conditions applied to the system determine the surface
ing form at the fixed points (h ,c = 0) and
universalityclassthe systembelongsto. Thebehaviorof | 1| → ∞
(h =0,c ), respectively:
the bulk is not affected by the boundaries. 1 →−∞
It has turned out that in the sense of renormalization z
grouptheoryitissufficienttodescribethepresenceofthe φ(z,t)=atβP± w= for t≷0, (1)
| | ξ±
substrate by a surface field h and the so-called surface (cid:18) (cid:19)
1
enhancementc[31]. (Thesenamesoriginatefromthecor- with the scaling variable w = z/ξ± which describes the
responding description of surface magnetic phenomena distance from the substrate in units of the correlation
[30, 31], see also,c.f., Eq.(7).) The surface enhancement length ξ±.
is related to the couplings between the ordering degrees The scaling functions P±(w) are universal after fix-
of freedom at the surface. ing the non-universalamplitude a and the non-universal
The so-called ordinary surface universality class is amplitude ξ+ of the correlation length ξ. (Recall that
0
characterized by a vanishing surface field and a positive the ratio ξ+/ξ− is universaland therefore the amplitude
0 0
surface enhancement (h1 = 0,c > 0) which suppresses ξ0− is fixed along with the amplitude ξ0+.) The ampli-
the order parameter at the surface below its bulk value. tude a is chosen in such a way that the scaling function
For magnetic systems this effect of missing bonds is the P−(w)belowthecriticaltemperaturetendsto1forlarge
generic case. distances from the substrate, i.e., the amplitude a cor-
Thespecialsurfaceuniversalityclass,describingamul- responds to the amplitude of the bulk order parameter
ticritical point, requires in addition to a vanishing sur- φ(z ,t < 0)= atβ. With this choice one finds the
→∞ | |
facefieldasurfaceenhancementwhichwithinmeanfield following behavior of the scaling functions:
theory vanishes (h = 0,c = 0) and causes a flat order
1
parameter profile in the vicinity of the substrate; fluctu- P±(w 0) w−βν , (2)
→ ∼
ations induce a divergenceofthe orderparameterprofile P+(w ) e−w, (3)
at the surface. →∞ ∼
P−(w ) 1 e−w. (4)
The normal surface universality class is characterized →∞ − ∼
by a non-vanishing surface field and the absence of the Away from the renormalization group fixed point the
surface enhancement (|h1| > 0,c = 0) which leads to an surface field h1 and the surface enhancement c have fi-
order parameter value at the surface larger than in the nitevalues. Theyappearasadditionalparametersinthe
bulk,evenabovethe criticaltemperaturewherethebulk scalingfunctionswithpowersofthereducedtemperature
value of the order parameter is 0. For fluid systems the as prefactor due to scaling with the correlation length:
normal surface universality class is the generic case. In
contrast,inthecontextofmagnetismanorderparameter φ(z,t) = a|t|βP±(|t|ν(ξ0±)−1z,|t|−∆1(ξ0±)d2 h1,|t|−Φξ0±(c5)),
4
where∆ andΦ aresurfacecriticalexponents[31]andd the correctcritical exponents are used, which are known
1
denotes the spatial dimension of the system. with high accuracy [42] (see Eq.(12)).
Theexplicitcalculationoftheorderparameterprofiles Taking the functional derivative of the Hamiltonian
φ(z,t) starts from the following fixed point Hamiltonian with respect to the order parameter (see Eqs. (6), (7),
[φ]= [φ]+ [φ] which separates into the bulk part and(10)) yieldsadifferentialequationforthe meanfield
b s
H H H
[φ] inthe volumeV andthe surface part [φ]onthe profile m(z) of the order parameter [30, 31],
b s
H H
surface S [30, 31]:
∂2 u
m+τm+ m3 =0, (13)
[φ] = dd−1r dz 1( φ)2+ 1τφ2+ uφ4 ,(6) − ∂z2 3!
Hb k 2 ∇ 2 4!
ZV (cid:16) (cid:17) with boundary conditions
[φ] = dd−1r 1cφ2 h φ . (7)
Hs ZS k 2 − 1 ∂m =cm(z =0) h (14)
(cid:0) (cid:1) 1
∂z −
Hereτ isproportionaltothereducedtemperaturet,u> (cid:12)z=0
(cid:12)
0stabilizestheHamiltonian [φ]fortemperaturesbelow (cid:12)
the critical point (T < T ) aHnd ( φ)2 penalizes spatial and (cid:12)
c
variations; rk is a vector parallel∇to the substrate. The ∂m
=0. (15)
order parameter φ is fluctuating around a mean value
∂z
φ . Each configuration φ contributes to the partition (cid:12)z→∞
(cid:12)
fhunictionZ withthe statisticalBoltzmannweighte−H[φ]: (cid:12)
(cid:12)
1. Infinite surface fields
Z = Dφ e−(Hb[φ]+Hs[φ]) , (8)
Z (cid:16) (cid:17) For systems that belong to the extraordinary surface
1
φ = Dφ φe−(Hb[φ]+Hs[φ]) . (9) universalityclass (h1 =0,c<0) the boundary condition
h i Z
atthesurface(14)simplifiesandthedifferentialequation
Z (cid:16) (cid:17)
(13) has an analytical solution [31]. Together with the
In the present work we shall provide general scaling
scalingbehaviorofthe orderparameter(1) andthe non-
properties with the quantitative results for the scaling
universal amplitude a which within the present model
functions determined within mean field approximation,
and within mean field (MF) approximation equals a =
i.e.,onlythe orderparameterprofilem(z)withthemax-
imum statistical weight will be consideredand all others 6 12 ξ± −1 this leads to scaling functions P± (w) of
u 0 MF
will be neglected: the following form:
(cid:0) (cid:1) (cid:0) (cid:1)
δ [φ] √2
H =0. (10) P+ (w) = ; coth(w )= c˜ , (16)
δφ MF sinh(w+w ) 0 | |
(cid:12)φ=m 0
(cid:12) w+w 1
This mean field approxim(cid:12)(cid:12)ation is valid above the upper PM−F(w) = coth 2 0 ; sinh(w0)= c˜ ,(17)
critical dimension dc = 4. this approximation the afore- (cid:18) (cid:19) | |
mentioned critical exponents take the following values: where c˜ = ξ0±|t|−21c is the scaled and dimensionless
β(d 4)= 1 and ν(d 4)= 1, (11) surface enhancement. The parameter w0 vanishes at
≥ 2 ≥ 2 the extraordinary fixed point (h˜ = 0,c˜ ) –
1
→ −∞
whereasthecriticalexponentsatphysicaldimensiond= with the scaled and dimensionless surface field h˜1 =
3 are [42]: u 12 ξ±2 t−1h – so that Eqs. (16) and (17) reduce
6 0 | | 1
to the scaling functions
β(d=3)=0.3265 and ν(d=3)=0.6305. (12) (cid:0) (cid:1)
√2
The mean field approximation is important because it P∞+(w)= sinh(w) (18)
is the zeroth-order approximation in a systematic Feyn-
mangraphexpansiononwhichthe (ǫ=d 4)-expansion and
−
andhencethe renormalizationgroupapproacharebased
[6, 31, 43]. The higher orders in the Feynman graph P−(w)=coth w , (19)
∞ 2
expansion require integrations over the mean field order
parameter profile and the two-point correlation function which in the following are referr(cid:0)ed(cid:1)to as “half-space pro-
(compare Eq.(3.209) in Ref. [31]) which can be car- files” above and below the critical temperature, respec-
ried out reasonably if they are available in an analytical tively. As already mentioned at the beginning of Sec.II
form. However,themeanfieldapproximationisexpected the extraordinaryfixedpoint is equivalent to the normal
to yield the qualitatively correct behavior of the scaling fixedpoint(h˜ ,c˜=0)andthusEqs.(18)and(19)
1
| |→∞
functionsifforthevariablesformingthescalingvariables represent the latter fixed point as well.
5
2. Finite surface fields
25
Since in experimental systems the surface fields are fi- (a) P1+
nite,wealsodiscussthecaseofahomogeneoussubstrate
+
with a surface field 0 < h1 < ∞. This provides the ) 20 Ph1
starting point for discussing the case of a chemical step w
(
with finite - albeit strong - surface fields which we shall + h1 15
P
consider in Subsec. IIIA2. Off criticality (i.e., t = 0)
6 ); T >T
and for a finite surface field h the scaled surface field w
h˜ = u 12 ξ±2 t−1h isalsofin1ite. Atthesubstrate,for + (1 10
1 6 0 | | 1 P
finite surface fields the order parameter profiles P±(w)
(cid:0) (cid:1) h1 5
above/below the critical temperature have finite values
P+(w =0)=P+ and P−(w =0)=P− , respectively.
h1 sub h1 sub
These values are determined by the following equations 0
whichareobtainedbyperformingthefirstintegralofthe 0 0.5 1 1.5 2 2.5 3
differential equations for PM+F and PM−F, corresponding w
to Eq.(13) [30]:
P+ 4+2(1 c˜2) P+ 2+4c˜˜h P+ 2h˜2 =0 (20) 25
sub − sub 1 sub− 1 (b) P1(cid:0)
and 20 Ph(cid:0)1
)
P− 4+2(1+2c˜2) P− 2 8c˜h˜ P− +4h˜2 =0.(21) (w
sub sub − 1 sub 1 (cid:0) h1 15
P
For the normal surface universality class (i.e., c˜ = 0) ); T <T
w
Eqs.(20) and (21) simplify and we find for the order pa- (cid:0) (1 10
rameter Ps+ub and Ps−ub at the substrate above and below P
the critical temperature, respectively:
5
P+ = 1+2h˜2 1 , h˜ ≶0 (22)
sub ±r 1− 1 0
q 0 0.5 1 1.5 2 2.5 3
and
w
P− = 1+4h˜2+1 , ˜h ≶0. (23) FIG. 2: (a) The half-space profile P+(w = z/ξ+) for an in-
sub ±r 1 1 finite surface field h˜ and P+(w = z∞/ξ+) for a finite scaled
q 1 h1
Using these equations together with the boundary con- surfacefieldh˜1 =100abovethecriticaltemperature. (b)The
dition (14) and the differential equation (13) the half- same below Tc with w=z/ξ−.
spaceprofileforfinitesurfacefieldsP±(w)=P±(w,h˜ <
h1 1
,c˜= 0) can be calculated numerically (see Appendix
∞
A). In Fig.2 the half-space profiles for infinite and fi- (x = z = 0) of the two halves (see Fig.1(a)). We in-
nite surface fields are shown for temperatures above as
troduce the scaled coordinates v = x/ξ and w = z/ξ
well as below the critical temperature. In the following describing the distance x from the contact line and z
section we shall consider inhomogeneous substrates, i.e., from the substrate, respectively, in units of the correla-
substrates with a laterally varying surface field h1 for tion length ξ. The system is translationally invariant in
which further length scales come into play, which also the direction perpendicular to the x-z – plane.
scale with the correlationlength ξ±.
For laterally inhomogeneous systems one has to re-
consider whether the surface Hamiltonian (Eq.(7))
s
H
should contain terms like φ∂ φ and φ∂ φ. However, the
k ⊥
III. CRITICAL ADSORPTION AT
term φ∂ φ would favor order parameter profiles which
INHOMOGENEOUS SUBSTRATES k
are non-symmetric with respect to (x =0,y) even with-
out surface fields. Therefore such a term is ruled out.
A. Chemical step Thetermφ∂ φleadsonlytoaredefinitionofthesurface
⊥
enhancement [31] and therefore can be neglected for ho-
First we consider an infinite substrate which is di- mogeneous as well as for the inhomogeneous substrates.
vided into two halves with opposing surface fields h so ThusthesurfaceHamiltonian forinhomogeneoussur-
1 s
H
that there is a chemical step at the straight contact line facefieldshasthesameformastheoneforhomogeneous
6
P
+s(v;w) 1
(a)
10 w) 0.5
(
+ 1
0 P
=
)
w 0 T >T
-10 v; w=0:1
0.6 + (s w=1:1

0.4 P w=2:2
6 4 2 0.2 w -0.5 w=4:4
0 w=9:9
-2
v -4 -6 0 w=1
-1
-10 -5 0 5 10
FIG. 3: Scaling function P+(v = x/ξ+,w = z/ξ+) for the
cs
v
order parameter profile of a system above the critical tem-
perature and confined by a substrate with a chemical step
located at x=z =0. Dueto symmetry P+(v=0,w)=0.
cs 1
(b)
)
surface fields (Eq.(7)). w 0.5
(
(cid:0) 1
P
=
)
1. Infinite surface fields w 0 T <T
;
v
(
(cid:0) s w=0:1

First we analyze the case of a homogeneous infinite P w=1:1
-0.5
surface field on both halves of the substrate but with w=2:2
opposite sign and a vanishing surface enhancement, i.e., w=4:4
w=1
weconsiderastep-likelateralvariationofthesurfacefield
h : h = for x≷0. The actual smooth variation of -1
1 1
±∞ -10 -5 0 5 10
h onamicroscopicscaleturnseffectivelyintoastep-like
1
variation if considered on the scale ξ±. v
a. Order parameter profiles
The orderparameterprofilefor asystemwitha chem- FIG. 4: Cuts through the order parameter profiles Pc+s(v,w),
ical step (cs) exhibits the following scaling property as shown in Fig.3, and Pc−s(v,w) for w = z/ξ = const with
thenormalization Pc±s(v,w) (a)aboveand(b)belowthecritical
P∞±(w)
x z temperature. Uponincreasingthedistancefromthesubstrate
φ(x,z,t)=atβP± v = ,w= for t≷0(24)
| | cs ξ± ξ± theslopeofthecurvesatv=0decreases. Thesolidlinesrep-
(cid:18) (cid:19) resent the limiting curves for w→∞: (a) Above the critical
which generalizes Eq.(1). This scaling function shows temperature the limiting slope at v = 0 of the normalized
the following limiting behavior: For large distances from curves is given by Ps∞++((ww→∞)) = 2A√+22 ≃ 0.200 with the slope
the chemical step (|v| → ∞) the profile approaches the s+(w) = ∂Pc+s(v,w) o→f∞the unnormalized scaling function
corresponding order parameter profile of a system with ∂v
v=0
a homogeneous substrate, whose asymptotic behavior is and its amplitude A(cid:12)(cid:12) +2 ≃0.566 (see, c.f., Eq.(28) and Fig.5).
given by Eqs.(2)-(4), Pc±s(|v| → ∞,w) = P∞±(w). For (b) Below Tc the li(cid:12)miting curvecorresponds to tanh(v2).
large distances from the substrate (w ) and above
→ ∞
the criticaltemperature the orderparameterprofile van-
ishes for all values of v, i.e., P+(v,w ) = 0 while
cs → ∞ merically determined (see Appendix A) order parame-
below the critical temperature the order parameter pro-
ter profile for T > T . In Fig.4 we show cuts through
file tends to the profile P−(v) of a free liquid-vapor in- c
lv the order parameter profile parallel to the substrate as
terface P−(v,w ) = P−(v) [44]. the mean field
cs → ∞ lv it changes with increasing distance from the substrate.
approximation this profile is given by
In order to provide a clearer comparison the cross sec-
v tions are normalized to 1 at the lateral boundaries via
Pl−v(v)=tanh 2 . (25) dividing by the exponentially decaying half-space profile
(cid:16) (cid:17) P∞±(w). From the fact that the cross sections do not fall
Figure3 provides a three-dimensional plot of the nu- ontoonecurvefollowsthatthescalingfunctionP±(v,w)
cs
7
doesnotseparateintoa v-dependentandaw-dependent
part. It shows that the slope s±(w) = ∂Pc±s(v,w) of 3
∂v 10
v=0
thescalingfunctionatthestepdecreaseswithincr(cid:12)easing t<0
(cid:12)
distance from the substrate which visualizes the h(cid:12)ealing t>0
of the frustration upon approaching the bulk. Note that
theslopesinFig.4mustbemultipliedbyP∞±(w)inorder (w) 101
to obtain s±(w). From Eq.(24) it follows: (cid:0)
s
); (cid:0)2
∂φ(x,z,t) tβ ∂P±(v,w) w (cid:24)w
= a| | cs + (
∂x ξ ∂v s 10(cid:0)1
(cid:12)x=0 (cid:12)v=0
(cid:12) a (cid:12)
(cid:12) = tβ+νs±(w) .(cid:12) (26)
(cid:12) ξ± | | (cid:12)
0 (a)
Since there is a non-vanishing order parameter profile 10(cid:0)3
even at the critical point (φ(x,z,t = 0) = 0) the overall 0.1 1 10
6
temperature dependence of the slope ∂φ in Eq.(26) has w
∂x
to vanish. Therefore one finds for w= z tν 0:
ξ± →
0
s±(w 0)=A±w−βν−1. (27) 103 t<0
→ 1
t>0
For T >Tc the slope s+(w) vanishes exponentially upon slv 100
approaching the bulk: (cid:0)
)
w
s+(w →∞)=A+2e−w . (28) (cid:0) s( 10(cid:0)3
Below T the slope s−(w ) approaches the slope );
c w (cid:0)6
s = ∂Pl−v(v) of the sc→alin∞g function for the liquid- + ( 10
lv ∂v s
v=0
vapor profile ((cid:12)Eq.(25)) at most e−w or slower: (cid:0)9
(cid:12) ∼ 10 (b)
(cid:12)
s−(w ) s =A−e−Cw, 0<C 1. (29)
→∞ − lv 2 ≤
0 5 10 15 20 25
As the half-space profiles P+(v ,w) = P+(w)
(Eq.(3)) and P−(v ,cws) =→ P±−∞(w) (Eq.±(4)∞) de- w
cs → ±∞ ± ∞
cayexponentially e−w forlargedistancesfromthesub-
strate, the slope s∼±(w) cannot decay faster because this fFuInGc.tio5n:sSPloc±ps(evs =s±(xw/)ξ±a,twth=e czh/eξm±i)caflorsttehpeoofrdthere pscaaralimng-
would require that the slope s±(w) is smaller than the eter profile of a system with a chemical step at x = 0.
slope ∂Pc+s for a certain v = 0. On the other hand (a) For small distances from the substrate the slopes di-
∂v v0 0 6 verge as A±1w−2 with amplitudes A+1 = 2.367±0.006 and
a decay of(cid:12)(cid:12)the slope s+(w) slower than e−w would lead A−1 =3.366±0.005 (seeEq.(27)). (b)Forlargedistancesthe
to anunph(cid:12)ysicalincreasingslopeofthe normalizedcross slopes decay exponentially, above Tc s+(w→∞)=A+2e−w
sections at v = 0 with increasing distance from the sub- withA+2 =0.566±0.001(seeEq. (28));belowTc s−(w→∞)
strate. However,forthe slope s−(w) a decaytowardsslv approaches its limiting value slv = 21 slower, i.e., as A−2e−Cw
slower than e−w cannot be ruled out. withA−2 =2.338±0.001andC =0.858±0.001(seeEq.(29)).
∼
Within mean field theory Eqs.(27) and (25) yield
s±(w 0) w−2 (see also Eq.(11)) and s−(w )=
s =→1, res∼pectively, which is in agreement with→th∞e nu-
lv 2 and
mericalresultsshowninFig.5. Theseresultsfortheslope
s±(w) of the scaled order parameter profile Pc±s(v,w) ∂φ ∂φlv tβ+νe−Cξz− , T <Tc, z ξ+,
transform into the following findings for the slope of the ∂x − ∂x ∼ | | ≫
(cid:12)x=0
unscaled order parameter profile φ(x,z): (cid:12)(cid:12) ξ −βν−1e−Cξz− . (32)
(cid:12) ∼ | |
∂φ z−βν−1, T =Tc, (30) As anexample we investigate more closely the healing
∂x ∼ effect above the critical temperature. To this end we
(cid:12)x=0
∂φ(cid:12)(cid:12)(cid:12) tβ+νe−ξz+ , T >Tc, z ξ+, rtheseciralselotpheeantorvm=al0izebdeccormosesss1e,ctsieoenFsiogf.6F.igI.t4shsouwchs tthhaatt
∂x ∼ ≫
(cid:12)x=0 these rescaled normalized cross sections for different
(cid:12)(cid:12)(cid:12) ξ+ −βν−1 e−ξz+ , (31) distances from the substrate differ basically only in the
∼
(cid:0) (cid:1)
8
)
w 1
(
+ 1
P
=
)
w 0.5
);
)
w
(
+ 1
P 0 1:4 1:6 1:8 2
=
)
w
( w=8:6
+ 0:95
(s -0.5
(cid:1) w=0:1
v
(
+ s 0:85

P
-1
-3 -2 -1 0 1 2 3
v
FIG. 7: The shaded area indicates the contribution to the
FIG. 6: Rescaled normalized cross sections with slope 1 at excess adsorption Γ˜±ex,cs as defined in Eq.(37) for w = w∗ =
v=0 for w=0.1,1.5,2.9,4.3,8.6. 6.0 for a system above Tc and with a chemical step. Γ˜±ex,cs is
obtained bysumming these shaded areas over w.
region where the curvature of these curves is largest.
From the fact that these cross sections do not fall onto extension of the system in the y -...-y -directions.)
1 d−2
one curve it followsthat the scaling function P±(v,w) is Equation(33) leads to the following temperature depen-
not simply given by the knowledge of one cross section, dence of the excess adsorption:
the slope s(w), and the half space profile P+(w) but
∞
requires the full numerical analysis. Nonetheless Fig.6 Γ =Γ˜± aξ±2 H tβ−2ν, (35)
demonstrates that to a large extent the rescaling used ex ex 0 | |
there reduces the full scaling function Pc+s(v,w) to a withtheuniversalamplitudeΓ˜± andthreenon-universal
single function of v only. amplitudes a, ξ±, andH. Withexin meanfieldapproxima-
0
b. Excess adsorption otiforneftehriesnycieelsdysstΓeemxs=asΓ˜g±exivaenξ0±be2loHw|Pt|−±21.(vF,owr)olueradcshoeivceens
Itisachallengetodetermineexperimentallythefullor- ref
within mean field theory to a cancellation of the diver-
der parameter profile φ(x,z). Therefore in the following
weanalyzetheadsorptionΓ= dVφ(r)atthesubstrate genceofthecorrespondingintegralsoverthescalingfunc-
tion P±(v,w) caused by small distances from the sub-
which as an integralquantity is more easily accessible to
R strate. This waythe numericalmeanfielddata for d=4
experiments. To this end for any system with an or-
allow one to make meaningful approximate contact with
derparameterprofileφ(x,z)andacorrespondingscaling
potential experimental data for d=3.
function P±(v,w) we introduce a suitable reference sys-
Specifically, for the chemical step we introduce a ref-
tem with an order parameter profile φref(x,z) and the erence system P± (v,w) which can be interpreted as a
corresponding scaling function Pref(v,w). This allows cs,0
system with a chemical step for which no healing at the
us to introduce an excess adsorptionΓ with respect to
ex
chemical step occurs:
this reference system
P±(w) , v <0
Γex = Hx dxdz(φ(x,z) φref(x,z)) P± (v,w)= − ∞ (36)
= atβ ξ±2HΓ˜± − (33) cs,0 (+P∞±(w) , v >0.
| | ex
Since this choice of the reference systems leads to a van-
with its universal part Γ˜± defined as (see Eqs.(1) and
ex ishing excess adsorptionfor any antisymmetric (with re-
(24)) spect to v = 0) scaling function P±(v,w) upon integra-
cs
tion over the whole half space w > 0, we restrict the
Γ˜±ex = x dvdw P±(v,w)−Pr±ef(v,w) . (34) integration to a quarter of the space (see Fig.7):
(cid:16) (cid:17)
H denotes the extension of the system perpendicular to ∞ ∞
Γ˜± = dv dw P±(v,w) P± (v,w) . (37)
the x-z – plane. (In the three-dimensional case H corre- ex,cs cs − cs,0
spondstotheone-dimensionalextensionofthesystemin Z0 Z0 (cid:0) (cid:1)
y-direction. Within mean field theory, which is valid for For a binary liquid mixture confined by a substrate with
dimensionsd 4,H correspondstothed 2-dimensional a chemical step Γ± can be interpreted as the amount
≥ − ex,cs
9
ofparticlesofonetype removedacrossthe chemicalstep B. Single stripe
from the substrate which prefers them.
Below Tc the scaling function Pl−v(v)=Pc−s(v,w =∞) The second system we focus on consists of a laterally
(Eq.(25)) of the liquid-vapor profile gives rise to a non- extended substrate with a negative surface field h
1
vanishing universal excess adsorption Γ˜b with respect to in which a stripe of width S with a positive surfa→ce
the chosen reference system P− : −fie∞ld h is embedded (see Fig.1(b)). The surface
cs,0 1
→ ∞
field h is assumed to vary step-like:
1
∞
Γ˜b == Z0∞ddvv(cid:0)PPl−−v((vv))−1Pc−s,0(v,w =∞)(cid:1) (38) h1(x)=(cid:26)+−∞∞ ,, xx∈∈/ [[00,,SS]]. (43)
lv − We introduce the coordinates v and w scaled in units of
Z0
= 2ln2(cid:0). (cid:1) the correlation length, where v denotes the scaled dis-
−
tance from the left border of the stripe at x = 0 and w
Thus Γ˜− can be written as the scaled distance from the substrate. The influence of
ex,cs
the stripe width S and of the reduced temperature t are
Γ˜− =L˜Γ˜ +Γ˜− (39) captured by the scaled stripe width S˜ = S/ξ±. The or-
ex,cs b ex,f
der parameter profile for the system with a single stripe
where L˜ =L/ξ− denotes the extension of the sys- (ss) exhibits the scaling property
0 →∞
tem perpendicular to the substrate and the system size
φ(x,z,t,S)
independent contribution Γ˜− characterizes the influ-
ex,f x z S
ence of the chemical step. From our numerical analysis =atβP± v = ,w = ,S˜= (44)
we find the following universal excess adsorption ampli- | | ss ξ± ξ± ξ±
(cid:18) (cid:19)
tudes:
with the limiting behavior
Γ˜+ = 1.457 0.001, (40) P±(v,w,S˜=0) = P±(w), (45)
ex,cs − ± ss − ∞
Γ˜−ex,f = 1.299±0.001. (41) Ps±s(v,w,S˜→∞) = Pc±s(v,w). (46)
Figures8(a) and8(b) showcrosssections(parallelto the
substrate) through the order parameter scaling function
2. Non-antisymmetric finite surface fields
P+(v,w,S˜)forsystemswithdifferentscaledstripewidth
ss
S˜attwodifferentscaleddistanceswfromthesubstratein
Next we study an infinite substrate consisting of two
comparisonwithacrosssectionthroughtheorderparam-
halves with surface fields of opposite sign but different
eter profile P+(v,w) at a single chemical step at v = 0
aisbhsionlgutseufirfnaictee veanlhuaenschepmaenndt.hnA,srebspeefocrteivewlye,aconndsiadverana- (seeSubsec.IIcIsA). WithincreasingscaledwidthS˜
→∞
step-like variation of the surface field h : ofthestripetheleftpartofthecrosssectionofthestripe
1
systemmergeswiththecrosssectionofthesystemwitha
h , x<0 single step. Further away from the substrate the mutual
n
h1(x)= (42) influence of the two step structures onto each other is
h , x>0.
(cid:26) p morepronounced(see Fig.8(b)) andstrongerforsmaller
stripes.
The absence of antisymmetry for these systems is re-
For systems with a single chemical stripe on the
flected by the “zero-line” v (w), where the order param-
0
eter vanishes: P(v ,w) = 0. For ratios h˜ /h˜ = 1 of substrate a reference system with the scaling function
0 − p n 6 P+ (v,w,S˜) can be introduced similar to the one for a
the scaled surface fields (see Subsec. II 2) the zero-line ss,0
singlechemicalstepwhichcanbeinterpretedasasystem
is shifted towardsthe regionof the surface field with the
with a single chemical stripe on the substrate where no
smaller absolute value. Furthermore the zero-line is not
healing occurs:
straight,buttendstoincreasingvalues v forincreasing
0
| |
distance w from the substrate. However, the deviation +P+(w), v [0,S˜]
of the zero-line from the line (v = 0,w) decreases for P+ (v,w,S˜)= ∞ | |∈ (47)
−h˜p/h˜n → 1. For constant ratios −h˜p/h˜n the deviation ss,0 ( −P∞+(w), |v|∈/ [0,S˜].
of the zero-line from the line (v = 0,w) decreases with
This reference system allows us to define the excess ad-
increasing absolute values of the scaled surface fields h˜p sorption Γ˜+ for the striped systems:
andh˜ . Theseresultsdemonstratehowtheorderparam- ex,ss
n
eter structure for the fixed point fields h˜ , h˜ ∞ ∞
| p| | n| → ∞ Γ˜+ (S˜)= dv dw P+(v,w,S˜)
emerges smoothly from the general case of finite fields. ex,ss ss
In this sense in the following we focus on the case of Z−∞ Z0 (cid:16)
infinite surface fields, i.e., strong adsorption. −Ps+s,0(v,w,S˜) . (48)
(cid:17)
10
1
(a) S~= 8:1 0
) S~=12:0
(w 0.5 S~=16:0
+ 1 S~= 1 -0.2
P
= s
~w;S) 0 +~(cid:0)ex;s -0.4
;
v
(
+ s
s -0.6
P -0.5
w=2:7 -0.8
-1
-10 0 10 20 30 0 5 10 15
v S~
FIG. 9: Universal excess adsorption Γ˜+ (Eq.(48)) for a
1 ex,ss
single stripe with respect to a system with no healing as a
(b) S~= 8:1 function of the stripe width S˜. For S˜ = 0 the system cor-
) S~=12:0 responds to a system with a homogeneous substrate whose
(w 0.5 S~=16:0 correspondingexcessadsorptionis0. InthelimitS˜→∞the
+ P1 S~= 1 stripesystem corresponds totwoindependentchemical steps
= whoseexcessadsorptionalsovanishesduetoantisymmetryof
)
~S theprofilesaroundv=0andv=S˜→∞. Thenon-vanishing
w; 0 excess adsorption for intermediate stripe widths S˜ indicates
;
v theeffectofthestripewithasurfacefieldoppositetotheone
(
+ Pss -0.5 So˜f≃th0e.3e.mbedding substrate. Γ˜+ex,ss attains its minimum at
w=13:5
-1
-10 0 10 20 30 of Γ˜+ is discussed in Ref. [33] and in Ref. [25] where it
∞
v is denoted as Γ˜+ =g /(ν β) (see Eq. (2.9) and Fig.5
∞ + −
in Ref. [25]; in d = 3 one has Γ˜+ = 2.27). Equation
∞
FIG. 8: Comparison between the normalized cross sections (49) describes how in an operational sense the univer-
oftheorderparameterscalingfunctionnearasubstratewith sal function Γ˜+ (S˜) (see Fig.9) can be obtained from
negative surface field in which a stripe with positive surface ex,ss
the measurements of the excess adsorption (relative to
fieldof scaled width S˜isembeddedandthenormalized cross
the bulk order parameter) at a striped surface and from
sections of theorder parameter profileneara single chemical
step (which emerges as limiting case for S˜ → ∞), (a) at a those of the excess adsorption at the corresponding ho-
mogeneous surface.
scaled normal distance w = 2.7, (b) at w = 13.5. Here we
consider the case T > Tc. The chemical steps are located at Figure9 shows the dependence of the excess adsorp-
x = 0 and x = S corresponding to v = 0 and v = S˜ (see tion Γ˜+ on the scaled stripe width S˜. For S˜ the
ex,ss →∞
vertical lines). structuresassociatedwiththetwochemicalstepsforming
the stripe decouple so that Γ˜ (S˜ ) = 0. Upon
ex,ss
→ ∞
construction Γ˜ (S˜ = 0) = 0. The decrease of the
ex,ss
Equation (48) can be rewritten as excess adsorption at S˜ 1 arises due to the fact that
≈
for sufficiently small stripes the spatial region where the
B˜/2 ∞
order parameter profile is positive does no longer resem-
Γ˜+ (S˜)= lim dv dwP+(v,w,S˜)
ex,ss B˜→∞(Z−B˜/2 Z0 ss ble a rectangular but a tongue-like shape (see Fig.10).
Since for the reference system the region with a positive
+(B˜ 2S˜)Γ˜+ , (49) order-parameter resembles still a rectangle this leads to
− ∞
a negative excess adsorption.
o
where B = B˜ξ S is the overall lateral extension of FordifferentstripewidthS˜thetongue-likeregionsde-
≫
the substrate surface in x-direction (see Fig.1(b)) and fined by the zero-lines v (w) where the order parameter
0
Γ˜+ = ∞dwP+(w) is a universal number characteriz- vanishes, P(v ,w) = 0, are shown in Fig.10. The width
∞ 0 ∞ 0
ing the amplitude of the excess adsorption at a homoge- ofthetongueatthesubstrateisgivenbythestripewidth
nous suRbstrate: ∞dzφ(z,t) = Γ˜+aξ+tβ−ν. The value S˜duetotheinfinitelystrongsurfacefields. Withincreas-
0 ∞ 0
R