Crossover scaling of apparent first-order wetting in two dimensional systems with short-ranged forces
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Crossover scaling of apparent first-order wetting in two dimensional systems withshort-ranged forces
Andrew O. Parry
Department of Mathematics, Imperial College London, London SW7 2B7, UK
Alexandr Malijevsk´y
Department of Physical Chemistry, University of Chemical Technology Prague, Praha 6, 166 28, Czech Republic;Institute of Chemical Process Fundamentals of the Czech Academy of Sciences, v. v. i., 165 02 Prague 6, Czech Republic
Recent analyses of wetting in the semi-infinite two dimensional Ising model, extended to includeboth a surface coupling enhancement and a surface field, have shown that the wetting transitionmay be effectively first-order and that surprisingly the surface susceptibility develops a divergencedescribed by an anomalous exponent with value γ eff11 = . We reproduce these results using aninterfacial Hamiltonian model making connection with previous studies of two dimensional wettingand show that they follow from the simple crossover scaling of the singular contribution to the surfacefree-energy which describes the change from apparent first-order to continuous (critical) wetting dueto interfacial tunnelling. The crossover scaling functions are calculated explicitly within both thestrong-fluctuation and intermediate-fluctuation regimes and determine uniquely and more generallythe value of γ eff11 which is non-universal for the latter regime. The location and the rounding of a lineof pseudo pre-wetting transitions occurring above the wetting temperature and off bulk coexistence,together with the crossover scaling of the parallel correlation length, is also discussed in detail. I. INTRODUCTION
Abraham’s exact solution of the semi-infinite planarIsing model showed a wetting transition which was con-tinuous and strictly second-order i.e. the surface spe-cific heat exponent takes the value α s = 0 [1]. Sub-sequent studies based on interfacial Hamiltonian mod-els, and also random walk arguments gave strong sup-port that this is the general result for 2D wetting in sys-tems with short ranged forces and describes a universalityclass, referred to as the strong-fluctuation (SFL) regime[2–4]. In particular renormalization group analyses ofinterfacial models show that for systems with strictlyshort-ranged forces the flow is described by only two non-trivial fixed points describing a bound phase (character-ising the SFL regime) and an unbound phase respectively[5, 6]. While first-order wetting transitions are possible in2D they require the presence of sufficiently long-rangedintermolecular forces [7–9]. However, very recently ex-act and numerical studies of the wetting transition inthe Ising model, but now including an additional short-ranged field representing the enhancement of the surfacecoupling constant, have shown that the wetting transi-tion is effectively first-order when the coupling constantis large [10].This enhancement of the surface coupling,which acts in addition to a surface field, is similar tothe well known mechanism which drives wetting tran-sitions first-order in mean-field treatments of Ising andlattice-gas models [11]. What is most surprising here isthat it was observed that on approaching the wettingtemperature the surface susceptibility and specific heatappear to diverge and are characterised by an anomalousexponent equal to 3 / II. SCALING AND FLUCTUATION REGIMESFOR 2D CRITICAL WETTING
Background:
The fluctuation theory of wetting tran-sitions, particularly those occurring in 2D systems, wassuccessfully developed several decades ago; see for ex-ample the excellent and comprehensive review articles[2–4]. Wetting transitions refer to the change from par-tial wetting (finite contact angle) to complete wetting(zero contact angle) which occurs at a wetting tem-perature T w . Viewed in the grand canonical ensemblethe wetting transition, occurring at a wall-gas interfacesay, is associated with the change from microscopic tomacroscopic adsorption of liquid as T → T − w at bulkcoexistence. The transition is therefore equivalent tothe unbinding of the liquid-gas interface, whose ther-mal fluctuations are resisted by the surface tension σ .The transition may be first-order or continuous (oftentermed critical wetting) as identified from the vanishingof the singular contribution to the wall-gas surface ten-sion σ sing ≡ σ (cos θ − ∝ − ( T w − T ) − α s . Thus instandard Ehrenfest classification the value α s = 1 cor-responds to first-order wetting and is usually associatedwith the abrupt divergence of the equilibrium adsorp-tion (proportional to the wetting film thickness h ℓ i ) as T → T − w . In 3D the transition is also associated with apre-wetting line of thin-thick transitions extending above T w and off coexistence which terminates at a pre-wettingcritical point. For critical wetting the exponent α s < h ℓ i ∝ ( T w − T ) − β s , and parallel cor-relation length, ξ k ∝ ( T w − T ) − ν k , which diverge con-tinuously on approaching the transition. In the nearvicinity of the transition, the free-energy shows scaling σ sing = t − α s W ( ht − ∆ s ) where t ∝ ( T w − T ) and h (mea-suring the bulk ordering field or deviation from liquid-gascoexistence) are the two relevant scaling fields for criti-cal wetting. Here W ( x ) is a scaling function, ∆ s is thesurface gap exponent and we have suppressed metric fac-tors for the moment. As is well known the scaling of thefree-energy is a powerful constraint on the critical singu-larities. For example it follows that the exponents satisfystandard relations such as the Rushbrooke-like equality2 − α s = 2 ν k − β s . With the additional assumption ofhyperscaling, which in 2D implies 2 − α s = ν k the gapexponent follows as ∆ s = 3 ν k / α s = 0, β s = 1 and ν k = 2 (and hence ∆ s = 3) inkeeping with Abraham’s exact Ising model results. Morerecently, studies of fluid adsorption in other geometries,in particular wedge filling, have revealed a number of un-expected geometry invariant properties of wetting [12]whose microscopic origins have been illuminated by verypowerful field theoretic formulations of phase separation[13]. Finally we note that scaling theories pertinent tofirst-order wetting transitions have also been developedand been used in particular to analyse the critical singu-larities associated with the line tension [14, 15]. We shallreturn to this later.These remarks are completely supported by analysesof wetting based on interfacial Hamiltonians which havebeen used extensively and very successfully to determinethe specific values of the critical exponents and their moregeneral dependence on the range of the intermolecularforces present [16]. In 2D the energy cost of an inter-facial configuration can be described by the mesoscopiccontinuum model H [ ℓ ] = Z dx Σ2 (cid:18) dldx (cid:19) + V ( ℓ ) ! (1)where ℓ ( x ) is a collective co-ordinate representing the lo-cal height of the liquid-gas interface above the wall. HereΣ is the stiffness coefficient, equivalent to the tension σ for isotropic fluid interfaces, while V ( ℓ ) is the bindingpotential which models the direct interaction of the in-terface with the wall arising from intermolecular forces.The binding potential V ( ℓ ) can be thought of as describ-ing the underlying bare or mean-field wetting transition which would occur if the stiffness were infinite and inter-facial fluctuation effects are suppressed. To account forfluctuations it is necessary to evaluate the partition func-tion for the model (1). In 2D the scaling properties ofthe interfacial roughness are insensitive to the choice ofmicroscopic cut-off which is reflected by a universal (notdepending on microscopic details) relation between theroughness and the parallel correlation length. With an“infinite momentum” cut-off the evaluation of the parti-tion function Z is then particularly straightforward sinceit is equivalent to a path integral and we can immediatelywrite [17, 18] Z ( ℓ, ℓ ′ ; L ) = X n ψ ∗ n ( ℓ ) ψ n ( ℓ ′ )e βE n L (2)where β = 1 /k B T , L is the lateral extent of the sys-tems while ℓ, ℓ ′ are the end point interfacial heights. Here ψ n and E n are the eigenfunctions and eigenvalues of thecontinuum transfer matrix which takes the form of theShr¨odinger-like equation [19] − β Σ ψ ′′ n ( ℓ ) + V ( ℓ ) ψ n ( ℓ ) = E n ψ n ( ℓ ) . (3)In the thermodynamic limit ( L → ∞ ) of an infinitelylong wall the ground state identifies the singular contri-bution to the wall-gas surface tension σ sing = E andthe probability distribution for the interface position fol-lows as P ( ℓ ) = | ψ ( ℓ ) | . Similarly the parallel correlationlength describing the decay of the height-height corre-lation function along the wall is determined within thetransfer-matrix formulation as ξ k = k B T / ( E − E ).The analysis of 2D wetting transitions using thistransfer-matrix approach has already been done in agreat deal of detail by Kroll and Lipowsky [19]. Supposethe bare wetting transition is continuous as described bythe binding potential V ( ℓ ) = aℓ − p + bℓ − q + hℓ where q > p and the coefficient a is considered negative at lowtemperatures. Provided that b > a = 0(and h = 0) represents the mean-field critical wettingphase boundary [20]. Solution of the Shr¨odinger equa-tion shows that the critical wetting transition falls intoseveral fluctuation regimes with the SFL regime, repre-sentative of short-ranged wetting holding for p >
2. For p < a = 0 although criti-cal exponents are non-classical if q >
2. However in theSFL regime, the wetting temperature is lowered below itsmean-field value since the interface is able to tunnel awayfrom the potential well in V ( ℓ ) even though a <
0. Cal-culation shows that the singular part to the free-energyexhibits the anticipated scaling behaviour [2–4] σ sing = t W ( h | t | − ) (4)identifying the universal values of the critical exponents α s = 0 and ∆ s = 3 as quoted above. Implicit here is thatthe scaling function W ( x ) is different below and abovethe wetting temperature and we have replaced t with | t | in the argument for convenience. The scaling of the free-energy determines that the film thickness h ℓ i ∝ ∂σ sing ∂h andcorrelation length ξ k ∝ ∂ σ sing ∂h must diverge as h ℓ i ∝ t − and ξ k ∝ t − as T → T − w at bulk coexistence. Thesealso follow from direct calculation. Indeed, the interfacialmodel (1) goes further and recovers precisely the scalingproperties of energy density and magnetization correla-tion functions known from the exact solution of the Isingmodel [21, 22]. Above the wetting temperature the scal-ing of σ sing also identifies, the correct singular behaviour σ sing ∝ h − α cos where α cos = 4 / complete wetting transition occurring as h → p = 2), referred toas the intermediate-fluctuation (IFL) regime, is marginaland the critical behaviour subdivides into three furthercategories [9].In a related article Zia, Lipowsky and Kroll [7] alsodiscussed what happens if the binding potential V ( ℓ )has a form pertaining to a mean-field first-order wet-ting transition. Suppose that, at bulk coexistence, thepotential has a long-ranged repulsive tail V ( ℓ ) = aℓ − p (with a >
0) which competes with a short-ranged attrac-tion close to the wall. They showed that if p > V ( ℓ ) when T issufficiently close to T w . For p < α s = 1) and the adsorption diverges discontin-uously at the wetting temperature. The latter followsfrom (3) since at T w there is a zero energy bound statewavefunction which determines that the probability dis-tribution decays (ignoring unimportant constant factors)as P ( ℓ ) ∝ exp ( − ℓ − p ). Explicit results for p = 1 confirmthis for a restricted solid-on-solid model [8]. The case p = 2 is marginal but displays first-order wetting with α s = 1 for a > / β Σ corresponding to sub-regime C ofthe IFL regime [9]. In this case a zero energy bound statewavefunction also exists at T w and determines that theprobability distribution decays as P ( ℓ ) ∝ ℓ − √ β Σ a .This algebraic decay means, rather unusually, that notall moments of the distribution exist at T w [9]. Thus,for example, for 1 /β Σ > a > / β Σ the adsorptiondiverges continuously as T → T w even though the transi-tion is strictly first-order. For 3 / β Σ > a > − /β Σ thewetting transition is continuous with non-universal expo-nents (sub-regime B ) to which we shall return shortly.Note that the parallel correlation length for all 2D first-order wetting transitions also diverges continuously witha universal power-law ξ k ∼ t − independent of p . Thisis equivalent to the statement of hyperscaling, whichalso holds in the SFL regime, since near T w the nextwavefunction above the groundstate lies at the bottomof the scattering spectrum ( E = 0) and hence σ sing = − k B T /ξ k . This scenario is subtly different to first-order wetting in 3D where ξ k , as defined through the decayof the height-height correlation function, remains finiteas T → T − w . However a continuously diverging parallelcorrelation length, very similar to that occurring in 2D,can still be identified for 3D first-order wetting by con-sidering the three phase region near a liquid droplet oralternatively by approaching the wetting temperature T w from above along the prewetting line [14, 15]. III. APPARENT FIRST-ORDER BEHAVIOURIN THE SFL AND IFL REGIMES
One issue that has not been addressed concerns the sizeof the asymptotic critical region in either the SFL regimeor sub-regime B of the IFL regime when the interface hasto tunnel through the potential barrier in V ( ℓ ). Let usconsider the SFL regime first. For systems with short-ranged forces and in zero bulk field, h = 0, this can bemodelled by the very simple potential V ( ℓ ) = − U Θ( R − ℓ ) + cδ ( ℓ − R ) (5)together with the usual hard-wall repulsion for ℓ < x ) is the Heaviside step function. With c ≫ U >
0) and a large but alsoshort-ranged repulsion similar to that arising in the Isingmodel studies where the surface enhancement term com-petes with a surface field. We emphasise that preciselythe same crossover scaling described below emerges if weuse a square-shoulder repulsion in place of the delta func-tion. This choice of local binding potential is the simplestone that incorporates a short ranged attraction and a re-pulsive potential barrier. It therefore has the same quali-tative features as binding potentials describing first-orderwetting constructed from more microscopic continuummodels [20]. Here the coefficient c is regarded simplyas an adjustable parameter in order to tune the size ofthe critical region but, more generally, will increase ex-ponentially with the size and width of the potential bar-rier. Without loss of generality we work in units where R = 1 and also set 2 β Σ = 1 for simplicity. Ratherthan vary the temperature we equivalently decrease thedepth of the attractive short-ranged contribution untilthe interface unbinds from the wall. Elementary solu-tion of the Shr¨odinger equation for the potential (4) de-termines that the ground state wavefunction behaves as ψ ( ℓ ) ∝ sin( √ U + E ℓ ) for ℓ < R and ψ ( ℓ ) ∝ e − √ | E | ℓ for ℓ > R . The delta function contribution to the po-tential necessitates that ψ ′ ( R − ) − ψ ′ ( R + ) = cψ ( R ) andcontinuity of the wavefunction immediately gives − p − E − p U + E cot p U + E = c . (6)Therefore the wetting transition occurs when U = U w where −√ U w cot √ U w = c . For large c ≫ U w ≈ c π / (1 + c ) . Writing U ≡ U w + t , it follows that if t and c − are small then theequation for the ground state energy simplifies to p − E ≈ c π ( E + t ) (7)and solution of this quadratic equation determines thatthe singular part to the free-energy (recall that σ sing = E ) behaves as σ sing = − t Gi (cid:18) − r tt Gi (cid:19) (8)Here we have introduced a thermal Ginzburg scaling field t Gi = π /c which measures the size of the asymp-totic critical regime [25]. For t/t Gi ≪ σ sing ≈ − t / t Gi consistent with uni-versal critical behaviour characterising the SFL regime( α s = 0). However for t/t Gi ≫
1, that is outsidethe critical regime, the surface free-energy vanishes as σ sing ≈ − t in accord with the expectations of a first-orderphase transition. The expression (8) has a form consis-tent with phenomenological theories of crossover scaling σ sing = − tA cr ( t/t Gi ) with the scaling function behavingas A cr ( x ) → x → ∞ and A cr ( x ) ∼ x/ x → σ sing w.r.t t determines that the surface specific heat orequivalently the surface susceptibility behaves as χ ∝ t Gi (1 + tt Gi ) / (9)which outside the critical regime, tt Gi ≫
1, shows thesame apparent power-law χ ∝ t − γ eff11 with γ eff11 = 3 / aℓ − , for ℓ > R , to the potential V ( ℓ ) shown in(5). Recall that for a > / / > a > − / β Σ = 1) it is continuous. This sub-regime B is characterised by strongly non-universal critical ex-ponents with, for example, 2 − α s = 2 / √ a fromwhich all other exponents follows using hyperscaling etc[9]. Setting a = 0 recovers the results for the SFLregime described above. For completion we note thatfor a < − / a = − / A of the IFL regime [9]). These wettingtransitions, which display essential singularities, are nolonger induced by variation of the short-ranged field U and crossover scaling cannot be considered. Within sub-regime B the presence of the delta function repulsion at ℓ = R does not affect the asymptotic critical singulari-ties but once again significantly reduces the size of theasymptotic regime. In this case the wetting transitionoccurs when −√ U w cot √ U w = c − (1 − √ a ) and writing U = U w + t , it is straightforward to show that forsmall t and small c − the ground-state energy E satis-fies an equation similar to (7) but with the LHS replacedwith − E raised to the power ( √ a ) /
2. In this waywe can see that the crossover from first-order behaviour σ sing ≈ − t occurring for t/t Gi ≫ σ sing = − t Gi ( t/t Gi ) − α s , with α s <
1, is de-scribed by the implicit equation (up to an unimportantmultiplicative constant) (cid:18) − σ sing t Gi (cid:19) − αs = σ sing + tt Gi (10)which recovers trivially (8) when we set α s = 0. Thisnow shows the role played by the exponent α s in deter-mining the crossover from apparent first-order to crit-ical wetting in two dimensions. In particular for fixed t , and in the limit t Gi →
0, this has the expansion σ sing = − t + O ( t − αs ) where the coefficient of the singularcorrection term depends on t Gi . With α s = 0 this is thesame expansion of the free-energy, σ sing = − t + O ( √ t )found in the Ising model calculations in the strong sur-face coupling limit; see in particular equations (15) and(17) of [10]. As noted by these authors it is the presenceof the non-analytic correction to the pure first-order sin-gularity, σ sing = − t , which determines the apparent di-vergence of the surface susceptibility and specific heat.It follows that, more generally, the value of the expo-nent γ eff11 characterising the apparent divergence of χ satisfies the exponent relation(2 − γ eff11 )(2 − α s ) = 1 (11)Thus in sub-regime B of the IFL regime, for t/t Gi ≫ χ ∝ t − γ eff11 with a non-universal exponent γ eff11 = 2 − r
14 + 2 β a Σ (12)and we have reinstated the dependence on the stiffnesscoefficient Σ for completion. This recovers the Isingmodel result on setting a = 0 corresponding to strictlyshort-ranged interactions. Note that as a is increasedtowards the boundary with sub-regime C the value of γ eff11 approaches unity. This means that exactly at the B / C regime border the apparent divergence of χ , oc-curring for t/t Gi ≫
1, is near indistinguishable from theasymptotic divergence χ ∝ /t ( lnt ) occurring as t → / α s = 0) [2, 4, 27].Returning to the case of short-ranged forces perti-nent to the SFL regime we note that the expression (8)also determines the apparent and asymptotic divergencesof the parallel correlation length. First note that thefirst excited state is bound to the wall ( E <
0) for t > t NT but lies at the bottom of the scattering spec-trum ( E = 0) for t < t NT . Here t NT is the location of anon-thermodynamic singularity at which ξ k has a discon-tinuity in its derivative w.r.t t similar to that reported in[28]. For large c ≫ t NT ≈ π far fromthe wetting transition and crossover scaling region. Thismeans that for t < t NT the same hyperscaling or ratherhyperuniversal relation ξ k = k B T / | σ sing | applies equallyinside ( t/t Gi ≪
1) and outside ( t/t Gi ≫
1) the asymp-totic critical regime. Thus implies that the correlationlength shows crossover between two different power-laws; ξ k ∝ t − valid for t/t Gi ≫
1, characteristic of 2D first-order wetting, to ξ k ∝ t Gi t − for t/t Gi ≪ IV. ROUNDED PRE-WETTING TRANSITIONSFOR
T > T w Further insight into the crossover scaling behaviourin the SFL regime can be seen off bulk-coexistence byadding a term hℓ or h ( ℓ − R ) to (5). In this case forsmall t , c − and h the ground state energy is determinedfrom solution of − h Ai ′ ( − E h ) Ai ( − E h ) ≈ c π ( E + t ) (13)which, for t > h = 0 + . Here Ai ( x )is the Airy function which determines the decay of thewavefunction for ℓ > R [23, 24]. It follows that the sin-gular part of the free-energy scales as σ sing = tW cr (cid:18) h | t | ; t/t Gi (cid:19) (14)which is the more general result involving a crossoverscaling function of two variables and applies bothabove and below the wetting temperature. In theasymptotic critical regime t/t Gi ≪
1, the scalingfunction W cr ( x ; y ) → yW ( xy − ) so that σ sing = − t t Gi W ( ht Gi / | t | ). This is precisely the same scalingshown in (4) but now including a dependence on t Gi ,which recall determines the size of the asymptotic criti-cal regime, appearing via metric factors. It follows thaton approaching the wetting transition, T → T − w at bulkcoexistence, the adsorption ultimately diverges as h ℓ i ∝√ t Gi t − while for the parallel correlation length we re-cover the expression ξ k ∝ t Gi t − quoted above. These arethe standard critical singularities for the SFL regime butnow reveal the dependence of the critical amplitudes on t Gi . In particular the amplitude for the divergence of theadsorption vanishes as t Gi → t Gi in σ sing , h ℓ i and ξ k are allconsistent with the relation σ sing ∝ −A σ h ℓ i /ξ k where,within the SFL regime, A = 8 is a universal critical am-plitude independent of t Gi . This is reminiscent of the“bending energy” contribution to the free-energy in theheuristic scaling theory wetting transitions [16] and leadsdirectly to the Rushbrooke equality 2 − α s = 2 ν k − β s discussed earlier.The crossover scaling of σ sing shown in (14) depends onthe scaling variable h | t | − which is different to that ap-pearing in (4) characteristic of the SFL regime. Howeverthis power-law dependence is in complete agreement withthe predictions of the phenomenological scaling theoryof first-order wetting developed by Indekeu and Robledo[14, 15]. Indeed setting α s = 1 determines ν k = 1 (fromhyperscaling) and hence ∆ s = 3 / s = 3 ν k / σ sing and ξ k for t/t Gi ≫
1. Note also that above thewetting temperature, and for | t | /t Gi ≫
1, we may ap-proximate σ sing ≈ tW cr ( h | t | − ; −∞ ). The value 3 / h → σ sing ∝ h the amplitude of which must not dependon t . Thus the crossover scaling form (14) provides aconsistent link between previous scaling theories of con-tinuous and first-order wetting.More explicitly, above the wetting transition and for | t | /t Gi ≫
1, that is away from the immediate vicinity of T w , the approximate solution of (13) can be determinedfrom simple expansion of the Airy function around itsfirst zero. In this way it follows that the singular part tothe free-energy behaves as σ sing ≈ (cid:18) λh + | t | − q ( λh − | t | ) + 8 ht Gi (cid:19) (15)where here λ ≈ .
338 is the negative of the first zeroof the Airy function. If we could set t Gi = 0, whichcorresponds of course to an artificial infinite potentialbarrier, then σ sing =Min( | t | , λh ). This determines aline of first-order phase transition extending away frombulk coexistence located at | t | = λh . For small t Gi these transitions are rounded on a scale set by h t Gi .Taking the derivative of σ sing w.r.t h determines that h ℓ i ≈ | t | < λh while h ℓ i ≈ h − for t > λh . Thesharp increase in the film thickness therefore correspondssimply to a line of pseudo pre-wetting transitions. Thisline meets the bulk coexistence axis tangentially and thepower-law dependence on h is in precise accord with thestandard thermodynamic prediction for its location basedon the Clapeyron equation [20]. Sitting at a given pointalong this line the parallel correlation length scales as ξ k = t − ˜Λ( t/t Gi ) which follows from (14) and also directcalculation of the spectral gap E − E . For | t | /t Gi ≫ ξ k = | t | − ( | t | /t Gi ) which is very largeif t Gi is small. This lengthscale determines the round-ing of the pre-wetting phase transition equivalent to thecharacteristic size of the domains of the thick and thinprewetting states which are in pseudo phase coexistence.Moving along the pre-wetting line away from the wettingtemperature the lengthscale ξ k , and hence the size of thedomains simply decreases, indicating that the thin-thicktransition is eventually smoothed away by fluctuationsi.e. no pre-wetting critical point is encountered. Onthe hand moving towards the wetting transition, whileremaining along the pseudo pre-wetting line, the paral-lel correlation length eventually crossovers to ξ k ∝ / | t | .This is not indicative of any pseudo thin-thick phase co-existence but rather the usual thermal wandering of theunbinding interface when T w is approaching along thethermodynamic path h ∝ | t | . The above remarks areall consistent with the general theory of the rounding offirst-order phase transitions in pseudo one dimensionalsystems [29] V. CONCLUSIONS
In this paper we have shown that recent Isings modelstudies which show apparent first-order wetting transi-tions are consistent with analysis of an interfacial Hamil-tonian model which also allows us to consider propertiesof the transition in the presence of marginal long-rangedforces and occurring off bulk coexistence. Our studyhas revealed that the singular contribution to the sur-face free-energy shows a simple crossover scaling due tothe tunnelling of the interface through a potential barrierwhich generalises the standard scaling theory of criticalwetting linking it consistently with scaling predictionsfor first-order wetting. The form of the scaling func-tion is explicitly calculated above and below the wetting transition and illustrates the rounding of pseudo first-order phase transition in this low dimensional system.The crossover scaling occurring below T w , which is deter-mined both within the SFL and IFL regimes, allows us totrace the value 3 / γ eff11 high-lighted in the Ising model studies directly to the strictsecond-order nature of the critical wetting transition i.e.that α s = 0. It would be interesting to test the pre-dicted non-universality of γ eff11 in the IFL by adding along-ranged external field to the Ising model i.e. decay-ing as the inverse cube from the distance to the wall.Even for systems with short-ranged forces our predic-tions for the location of a pseudo pre-wetting line abovethe wetting temperature can also be tested in numericalstudies of the Ising model with a strong surface couplingenhancement similar to that described in [10]. Finally wemention that similar apparent first-order behaviour andcrossover scaling should also occur in 2D for the interfa-cial delocalization transition near defect lines in the bulkif these too are now modified to include enhanced cou-plings [4, 30]. Scenarios involving apparent first-orderinterfacial unbinding or delocalization in three dimen-sions are more challenging. However similar behaviourmay occur at wedge filling transitions where fluctuationeffects are enhanced compared to wetting and interfacialtunnelling through a potential barrier can occur [31, 32]. Acknowledgments
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