We show that the 3D wedge filling transition in the presence of short-ranged
interactions can be first-order or second order depending on the strength of
the line tension associated with to the wedge bottom. This fact implies the
existence of a tricritical point characterized by a short-distance expansion
which differs from the usual continuous filling transition. Our analysis is
based on an effective one-dimensional model for the 3D wedge filling which
arises from the identification of the breather modes as the only relevant
interfacial fluctuations. From such analysis we find a correspondence between
continuous 3D filling at bulk coexistence and 2D wetting transitions with
random-bond disorder.Comment: 7 pages, 3 figures, 6th Liquid Matter Conference Proceedings (to be
  published in J. Phys.: Condens. Matter

Parry, A. O.

Romero-Enrique, J. M.

English

arXiv

We show that the 3D wedge filling transition in the presence of short-ranged interactions can be first order or second order depending on the strength of the line tension associated with the wedge bottom. This fact implies the existence of a tricritical point characterized by a short-distance expansion which differs from the usual continuous filling transition. Our analysis is based on an effective one-dimensional model for the 3D wedge filling, which arises from the identification of the breather modes as the only relevant interfacial fluctuations. From such analysis we find a correspondence between continuous 3D filling at bulk coexistence and 2D wetting transitions with random-bond disorder.Comisión Europea, MEIF-CT-2003-50104

Romero Enrique, José Manuel

Parry, Andrew O.

idUS. Depósito de Investigación Universidad de Sevilla

arXiv:cond-mat/0510623v1  [cond-mat.stat-mech]  24 Oct 2005Tricritical wedge filling transitions withshort-ranged forcesJ. M. Romero-Enrique† and A. O. Parry‡† Departamento de F́ısica Atómica, Molecular y Nuclear, Area de F́ısica Teórica,Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain‡ Department of Mathematics, Imperial College 180 Queen’s Gate, London SW72BZ, United KingdomAbstract. We show that the 3D wedge filling transition in the presence of short-ranged interactions can be first-order or second order depending on the strengthof the line tension associated with to the wedge bottom. This fact implies theexistence of a tricritical point characterized by a short-distance expansion whichdiffers from the usual continuous filling transition. Our analysis is based on aneffective one-dimensional model for the 3D wedge filling which arises from theidentification of the breather modes as the only relevant interfacial fluctuations.From such analysis we find a correspondence between continuous 3D filling atbulk coexistence and 2D wetting transitions with random-bond disorder.PACS numbers: 68.08.Bc, 05.70.Np, 68.35.Ct, 68.35.Rhbrought to you by COREView metadata, citation and similar papers at core.ac.ukprovided by idUS. Depósito de Investigación Universidad de SevillaTricritical wedge filling transitions with short-ranged forces 2αxyl0ξlW (y)ξ yFigure 1. Schematic illustration of a typical interfacial configuration and relevantlengthscales for a fluid adsorption in a 3D wedge.Fluid adsorption in micropatterned and sculpted geometries has become thesubject of intense study over the last decade. Highly impressive technological advanceswhich allow the tailoring of micro-patterned and structured solid surfaces on thenanometer to micrometer scale [1] are a landmark in the development of the emergingmicrofluidic industry [2] which aims at minituarizing chemical synthesis plants orbiological analysis equipment in much the same way the silicon chip brought aboutthe electronics revolution. However, the theoretical understanding of this phenomenonis far from being complete. Recent studies of filling transitions for fluids in 3D wedgesshow that interfacial fluctuations are greatly enhanced compared with wetting at flatsubstrates [3, 4]. The control of such enhanced interfacial fluctuations is crucial forthe effectiveness of the microfluidic devices. Fortunately, there are simple theoreticalapproaches which take into account these effects. For example effective Hamiltonianpredictions for the critical exponents at continuous (critical) wedge filling with short-ranged forces have been confirmed in large scale Ising model simulation studies [5].Similar experimental verification of the predicted geometry-dominated adsorptionisotherms at complete wedge filling [6] raise hopes that the filling transition itself andrelated fluctuation effects will be observable in the laboratory. We further developthe theory of wedge filling in this paper, focussing on the emergence of a new type ofcontinuous filling: tricritical filling.First we briefly review the fluctuation theory of 3D wedge filling. Consider theinterface between a bulk vapour at temperature T and saturation pressure with a3D wedge characterised by a tilt angle α. Macroscopic arguments dictate that thewedge is partially filled by liquid if the contact angle θ > α and completely filled ifθ < α [7]. The filling transition refers to the change from microscopic to macroscopicliquid adsorption as T → Tf , at which θ(Tf ) = α, and may be first-order or continuous(critical filling). Both of these transitions can be viewed as the unbinding of the liquid-vapour interface from the wedge bottom. Characteristic length scales are the meaninterfacial height above the wedge bottom lW , the roughness ξ⊥ and the longitudinalcorrelation length ξy, measuring fluctuations along the wedge (see Figure 1). Therelevant scaling fields at critical filling are θ − α and the bulk ordering field h (whichis proportional to the pressure difference with the saturation value). At coexistence(h = 0) we define critical exponents by lW ∼ (θ − α)−βW and ξy ∼ (θ − α)−νy . Theroughness can be related to ξy by the scaling relationship ξ⊥ ∼ ξζWy , where ζW is thewedge wandering exponent. For short-ranged forces, ζW = 1/3.For shallow wedges, i.e. α≪ 1, the free energy of an interfacial configuration canbe modelled by an effective Hamiltonian based on the capillary wave model for wettingof planar substrates [8]. However, an analysis of this model [3] shows that the liquid-vapour interface across the wedge is aproximately flat and soft-mode fluctuations ariseTricritical wedge filling transitions with short-ranged forces 3from local translations in the height of the filled region along the wedge. These breathermodes are the only relevant fluctuations in the continuous filling phenomena, and canbe taken into account by the following effective Hamiltonian [3]HW [l0] =∫dy{Σl0α(dl0dy)2+ VW (l0)}(1)where l0(y) is the local height of the interface at position y along the wedge bottomand Σ is the liquid-vapor surface tension. Note that the effective bending termresisting fluctuations along the wedge is proportional to the local interfacial height.The effective binding potential VW (l0) for l0 ≫ lπ, where lπ is the mean wetting filmthickness for a planar substrate, is given (up to irrelevant additive constants) by [3]:VW (l0) ≈h(l0 − lπ)2α+Σ(θ2 − α2)l0α+∫ (l0−lπ)/α−(l0−lπ)/αdxW (l0 −α|x|)(2)where W (l) is the binding potential between the gas-liquid interface and a planarsubstrate. Note that the mean field result for lW is recovered by minimizing VW (l0)for l0 > lπ, which is an important check on the self-consistency of the method [3]. Inaddition to a hard wall repulsion for l0 < 0, the potential VW (l0) contains a short-ranged attraction which may be modified by micropatterning a stripe along the wedgebottom, so as to weaken the local wall-fluid substrate and therefore strengthen theinterfacial binding. This observation will be crucial for the existence of tricriticalfilling, since with this decoration it may be possible to bind the interface to the wedgebottom at the filling boundary θ = α and h = 0. For latter convenience, hereafter wewill set h = 0 in our discussion of continuous filling.The quasi-one-dimensional nature of the effective Hamiltonian Equation (1)allows us to use the transfer-matrix formalism. In the continuum limit the partitionfunction is defined as a path integral [9] (setting kBT = 1 for convenience)Z[lb, la, Y ] =∫Dl0 exp(−HW [l0]) (3)where Y is the wedge length and la and lb are the endpoint heights. The position-dependent stiffness introduces some ambiguity in the definition of the path integral.This problem was already pointed out in Ref. [10] and is related to the well-known ordering problem in the quantization of classical Hamiltonians with position-dependent masses. Similar issues also arise in solid state physics [11]. Borrowing fromthe methods used to overcome these difficulties we use the following definitionZ[lb, la, Y ] = limN→∞∫dl1 . . . dlN−1N∏j=1K(lj , lj−1, Y/N) (4)where l0 ≡ la and lN ≡ lb, and K(l, l′, y) is defined as:K(l, l′, y) =√Σ√ll′απyexp(−Σ√ll′αy(l − l′)2 − yVW (l))(5)In the continuum limit the partition function becomesZ(lb, la, Y ) =∑nψn(lb)ψ∗n(la)e−EnY (6)Tricritical wedge filling transitions with short-ranged forces 4hθ−αb)a) 1/u0 01/w(i)(ii)(iii)BoundUnbound(i)(ii)(iii)Figure 2. Phase diagrams for (a) filling and (b) wetting transitions. Thethick and dashed lines in both diagrams correspond to continuous and first-orderboundaries between bound and unbound interfacial states, respectively. Thearrows show representative paths along which continuous unbinding occur: (i)and (ii) for tricritical filling (critical wetting) and (iii) for critical filling (completewetting), respectively. The filled circles represent the tricritical filling and criticalwetting points, respectively. See text for explanation.where the complete orthonormal set satisfy(− α4Σ∂∂l[1l∂∂l]+ VW (l)−3α16Σl3)ψ = Eψ (7)In the thermodynamic limit Y → ∞ we obtain the probability distribution function(PDF) for the midpoint interfacial height PW (l0) = |ψ0(l0)|2, the wedge excess freeenergy fW = E0 and the longitudinal correlation length ξy = 1/(E1 − E0). At thispoint we must remark that any definition of the path integral which is invariant uponexchanging la and lb leads to an Schrödinger equation similar to Equation (7) but witha different coefficient for the extra 1/l3 term in the effective binding potential [10].The change of variables λ =√8Σ/αl3/2/3 and ψ(l) = (2Σl/α)1/4φ(λ(l)) [12]transforms the Equation (7) to:− 12d2φ(λ)dλ2+(VW [l(λ)]−572λ2)φ(λ) = Eφ(λ) (8)with l(λ) =(3λ/√8Σ/α)2/3. In general, there will be an interfacial bound state atbulk coexistence for θ = α if the strength of the small l0 attraction between the gas-liquid interface and the substrates, which we will denote as u, is greater than somevalue uc. Consequently, the filling transition is first-order if u > uc and critical ifu < uc. Tricritical filling is observed when u − uc emerges as a new relevant field (inthe renormalization-group sense). IfW (l) ∼ −a/lp+b/lq, different scenarios may ariseas the range of the binding potential is varied. In particular, for p > 4 the long-rangebehaviour of VW is dominated by the 1/l3 term for θ = α, so the filling phenomena arefluctuation-dominated. This finding is consistent with the existence of two differentfluctuation regimes for critical filling: mean-field if p < 4 and fluctuation-dominatedregime if p > 4 [3]. In the fluctuation-dominated regime, examination of Equation(8) shows that there is an analogy between 3D continuous filling and 2D continuouswetting (see Figure 2), where the role at wetting of the bulk ordering field h and thepotential strength w are played by θ−α and u for filling phenomena, respectively. Inparticular, 3D tricritical (critical) filling is analogous to 2D critical (complete) wetting,respectively. Different critical exponents which characterize the divergence of lengthscales can be defined. In addition to the critical filling critical exponents βW and νyTricritical wedge filling transitions with short-ranged forces 5defined along route (ii) in Figure 2(a) (see above), we can define new critical exponentsfor tricritical filling at θ = α (route (i) in Figure 2(a)) as:lw ∼ (u− uc)−β∗W , ξy ∼ (u − uc)−ν∗y (9)and ξ⊥ ∼ ξζ∗Wy , where ζ∗W is the tricritical wandering exponent which in general may bedifferent from ζW (in contrast with the wetting case). More generally, in the vicinity ofthe tricritical point we anticipate scaling e.g. ξy ∼ |u − uc|−ν∗yΛ[(θ − α)|u− uc|−∆∗]with the gap exponent ∆∗. Thus along route (ii) ξy ∼ (θ − α)−ν∗y/∆∗.We focus now on the case of short-ranged forces as the prototype of thefluctuation-dominated regime. In addition to the hard-wall condition, VW (l0) canbe modelled as a contact-like attraction with strength u. An analysis of Equation (7)for l0 → 0 shows that the short-distance expansion of the PDF is either PW ∼ l0or PW ∼ l30. We anticipate that the former corresponds to tricritical behaviourand the latter to critical filling. It is remarkable that thermodynamic consistencyat critical filling is ensured as the local density at the wedge bottom is non-singular,i.e. ρw(0)−ρl ∼ T −Tf , where ρl is the bulk liquid density [14]. This property is onlyobtained if the partition function is defined by Equation (4) and Equation (5). Thus,the ambiguity in its definition can be removed by imposing this regularity conditionon the short-distance expansion of the interfacial height PDF.We report now our explicit results (details will be presented elsewhere). Alongroute (i) we find that there is only one bound solution to Equation (7) for u > uc ≈1.358 with E0 ∝ (u − uc)3 and associated PDFPW (l0) =6√3πξul0ξu[Ai(l0ξu)]2(10)where Ai(x) is the Airy function and in the scaling limit ξu ∼ |u − uc|−1. ThuslW ∼ ξ⊥ ∝ (u− uc)−1 and ξy ∝ (u− uc)−3 identifying β∗W = 1, ν∗y = 3 and obtainingζ∗W = ζW = 1/3. As predicted, the short-distance behaviour of the PDF is linear withl0.On the other hand, the scaling of the PDF for θ > α is given by (see also Figure 3):PW (l0) ∝ l0 exp[2ǫl0ξθ− 2l20ξ2θ]H2ν(√2l0ξθ− ǫ√2)(11)where ξθ = Σ−1/2[(θ/α)2 − 1]−1/4, ǫ = ΣE0ξ3θ/α, ν = ǫ2/4 − 1/2 and Hν(x) is theHermite function [13]. The value of ǫ is obtained as the smallest solution of thefollowing equation:± Γ[− 13]3−2/3Γ[13]ξθξu= ǫ+(ǫ2√2−√2) H ǫ24− 32(− ǫ√2)H ǫ24− 12(− ǫ√2) (12)where the positive (negative) sign corresponds to u > uc (u < uc), respectively.The inset of Figure 3 plots the solution of this equation. As anticipated, note thatscaling is obeyed in the vicinity of the tricritical point as the wedge excess free energyfW ∼ ξ−3θ F±(ξθ/ξu). For u > uc and ξθ/ξu ≫ 1, we have checked numerically that thePDF (11) converges to the expression given by Equation (10) for the correspondingvalue of ξu given by Equation (12). Thus, the interface remains bound to the substratewhen θ → α, in agreement to the first-order character of the filling transition. Thethermodynamic path (ii) to the tricritical point corresponds to ξu → ∞, whichcorresponds to ǫ ≈ 1.086. Thus along this route lW ∼ ξ⊥ ∝ (θ − α)−1/4 similarTricritical wedge filling transitions with short-ranged forces 60 1 2 3 4ξθ/ξu-3-2-1012ε0 0.5 1 1.5 2 2.5 3l0/ξθ00.40.81.21.62P W(l0/ξ θ)Figure 3. Plot of the scaled PDF for ǫ = −1.5 (thick dashed line), and alongroutes (ii) and (iii) in Figure 2(a), i.e. for ǫ ≈ 1.086 (thick continuous line) andǫ ≈ 1.639 (thick dot-dashed line), respectively. For comparison, the PDF fromEquation (10) with ξu ≈ 1.968ξθ (which corresponds to ǫ = −1.5, see inset) isalso plotted (thin dashed line). Finally, the scaled PDF obtained in Ref. [10]is also shown (thin dot-dashed line). Inset: Plot of ǫ as a function of ξθ/ξu foru < uc (continuous line) and u > uc (dashed line).to critical filling. From analysis of the spectrum it is also possible to show thatξy ∝ (θ − α)−3/4, so the tricritical gap exponent ∆∗ = 4. Finally, for thermodynamicpaths (iii) far from the tricritical point, i.e. ξu → 0, we have found that the scaling ofthe PDF is of the form shown in Equation (11) with ǫ ≈ 1.639. Note that PW ∼ l30 asl0 → 0, in agreement with our previous statement. This condition is not fullfilled bythe solution presented in Ref. [10], although globally it does not differ too much fromour exact solution (see Figure 3).We finish by mentioning a remarkable connection for short-ranged forces between3D wedge filling and 2D wetting with random-bond disorder [15]. The criticalexponents corresponding to tricritical and critical wedge filling can be obtained fromgeneralized random-walk methods [16] in terms of the wedge wandering exponent ζW .In particular, they are found to have the same dependence of the critical exponents forcritical and complete wetting, respectively, but in terms of an effective 2D wanderingexponent equal to 2ζW . For short-ranged forces (ζW = 1/3), this implies that theset of critical exponents is the same as for 2D random-bond disorder [17]. Thesepredictions may certainly be tested in Ising model simulation studies and would be astringent test of the theory of 3D wedge filling.AcknowledgmentsJ.M.R.-E. acknowledges financial support from the European Commission underContract MEIF-CT-2003-501042.Tricritical wedge filling transitions with short-ranged forces 7References[1] Herminghaus S, Gau H and Monch W 1999 Adv. Mater. 11 1393Service R F 1998 Science 282 399[2] Terray A, Oakey J and Marr D W M 2002 Science 296 1841Whitesides G M and Stroock A D 2001 Physics Today 54 42[3] Parry A O, Rascón C and Wood A J 2000 Phys. Rev. Lett. 85 345Parry A O, Wood A J and Rascón C 2001 J. Phys.: Condens. Matter 13 4591[4] Greenall M J, Parry A O and Romero-Enrique J M 2004 J. Phys.: Condens. Matter 16 2515[5] Milchev A, Müller M, Binder K and Landau DP 2003 Phys. Rev. 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Tricritical wedge filling transitions with short-ranged forces

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