Tricritical wedge filling transitions with short-ranged forces

arXiv:cond-mat/0510623v1 [cond-mat.stat-mech] 24 Oct 2005 Tricritical wedge filling transitions with short-ranged forces J. M. Romero-Enrique†and A. O. Parry‡ †Departamento de F´ısica At´omica, Molecular y Nuclear, Area de F´ısica Te´orica, Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain ‡Department of Mathematics, Imperial College 180 Queen’s Gate, London SW7 2BZ, United Kingdom Abstract. We show that the 3D wedge filling transition in the presence of shortranged 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. PACS numbers: 68.08.Bc, 05.70.Np, 68.35.Ct, 68.35.Rh Tricritical wedge filling transitions with short-ranged forces 2 αx yl0 ξ lW(y) WW ξy Figure 1. Schematic illustration of a typical interfacial configuration and relevant lengthscales for a fluid adsorption in a 3D wedge. Fluid adsorption in micropatterned and sculpted geometries has become the subject of intense study over the last decade. Highly impressive technological advances which allow the tailoring of micro-patterned and structured solid surfaces on the nanometer to micrometer scale [1] are a landmark in the development of the emerging microfluidic industry [2] which aims at minituarizing chemical synthesis plants or biological analysis equipment in much the same way the silicon chip brought about the electronics revolution. However, the theoretical understanding of this phenomenon is far from being complete. Recent studies of filling transitions for fluids in 3D wedges show that interfacial fluctuations are greatly enhanced compared with wetting at flat substrates [3, 4]. The control of such enhanced interfacial fluctuations is crucial for the effectiveness of the microfluidic devices. Fortunately, there are simple theoretical approaches which take into account these effects. For example effective Hamiltonian predictions for the critical exponents at continuous (critical) wedge filling with shortranged forces have been confirmed in large scale Ising model simulation studies [5]. Similar experimental verification of the predicted geometry-dominated adsorption isotherms at complete wedge filling [6] raise hopes that the filling transition itself and related fluctuation effects will be observable in the laboratory. We further develop the theory of wedge filling in this paper, focussing on the emergence of a new type of continuous filling: tricritical filling. First we briefly review the fluctuation theory of 3D wedge filling. Consider the interface between a bulk vapour at temperature Tand saturation pressure with a 3D wedge characterised by a tilt angle α. Macroscopic arguments dictate that the wedge is partially filled by liquid if the contact angle θ > α and completely filled if θ < α [7]. The filling transition refers to the change from microscopic to macroscopic liquid 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 liquidvapour interface from the wedge bottom. Characteristic length scales are the mean interfacial height above the wedge bottom lW, the roughness ξ⊥and the longitudinal correlation length ξy, measuring fluctuations along the wedge (see Figure 1). The relevant scaling fields at critical filling are θ−αand the bulk ordering field h(which is proportional to the pressure difference with the saturation value). At coexistence (h= 0) we define critical exponents by lW∼(θ−α)−βWand ξy∼(θ−α)−νy. The roughness can be related to ξyby the scaling relationship ξ⊥∼ξζW y, where ζWis the wedge wandering exponent. For short-ranged forces, ζW= 1/3. For shallow wedges, i.e. α≪1, the free energy of an interfacial configuration can be modelled by an effective Hamiltonian based on the capillary wave model for wetting of planar substrates [8]. However, an analysis of this model [3] shows that the liquidvapour interface across the wedge is aproximately flat and soft-mode fluctuations arise Tricritical wedge filling transitions with short-ranged forces 3 from local translations in the height of the filled region along the wedge. These breather modes are the only relevant fluctuations in the continuous filling phenomena, and can be taken into account by the following effective Hamiltonian [3] HW[l0] = Zdy (Σl0 αdl0 dy 2 +VW(l0))(1) where l0(y) is the local height of the interface at position yalong the wedge bottom and Σ is the liquid-vapor surface tension. Note that the effective bending term resisting fluctuations along the wedge is proportional to the local interfacial height. The effective binding potential VW(l0) for l0≫lπ, where lπis the mean wetting film thickness for a planar substrate, is given (up to irrelevant additive constants) 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) where W(l) is the binding potential between the gas-liquid interface and a planar substrate. Note that the mean field result for lWis recovered by minimizing VW(l0) for l0> lπ, which is an important check on the self-consistency of the method [3]. In addition to a hard wall repulsion for l0<0, the potential VW(l0) contains a shortranged attraction which may be modified by micropatterning a stripe along the wedge bottom, so as to weaken the local wall-fluid substrate and therefore strengthen the interfacial binding. This observation will be crucial for the existence of tricritical filling, since with this decoration it may be possible to bind the interface to the wedge bottom at the filling boundary θ=αand h= 0. For latter convenience, hereafter we will 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 partition function is defined as a path integral [9] (setting kBT= 1 for convenience) Z[lb, la, Y ] = ZDl0exp(−HW[l0]) (3) where Yis the wedge length and laand lbare the endpoint heights. The positiondependent 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 wellknown ordering problem in the quantization of classical Hamiltonians with positiondependent masses. Similar issues also arise in solid state physics [11]. Borrowing from the methods used to overcome these difficulties we use the following definition Z[lb, la, Y ] = lim N→∞Zdl1...dlN−1 N Y j=1 K(lj, lj−1, Y/N) (4) where l0≡laand lN≡lb, and K(l, l′, y) is defined as: K(l, l′, y) = sΣ√ll′ απy exp −Σ√ll′ αy (l−l′)2−yVW(l)!(5) In the continuum limit the partition function becomes Z(lb, la, Y ) = X n ψn(lb)ψ∗ n(la)e−EnY(6) Tricritical wedge filling transitions with short-ranged forces 4 h θ−α b)a) 1/u 00 1/w (i) (ii) (iii) BoundUnbound (i) (ii) (iii) Figure 2. Phase diagrams for (a) filling and (b) wetting transitions. The thick and dashed lines in both diagrams correspond to continuous and first-order boundaries between bound and unbound interfacial states, respectively. The arrows show representative paths along which continuous unbinding occur: (i) and (ii) for tricritical filling (critical wetting) and (iii) for critical filling (complete wetting), respectively. The filled circles represent the tricritical filling and critical wetting points, respectively. See text for explanation. where the complete orthonormal set satisfy −α 4Σ ∂ ∂l 1 l ∂ ∂l+VW(l)−3α 16Σl3!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 free energy fW=E0and the longitudinal correlation length ξy= 1/(E1−E0). At this point we must remark that any definition of the path integral which is invariant upon exchanging laand lbleads to an Schr¨odinger equation similar to Equation (7) but with a different coefficient for the extra 1/l3term in the effective binding potential [10]. The change of variables λ=p8Σ/αl3/2/3 and ψ(l) = (2Σl/α)1/4φ(λ(l)) [12] transforms the Equation (7) to: −1 2 d2φ(λ) dλ2+VW[l(λ)] −5 72λ2φ(λ) = Eφ(λ) (8) with l(λ) = 3λ/p8Σ/α2/3. In general, there will be an interfacial bound state at bulk coexistence for θ=αif the strength of the small l0attraction between the gasliquid interface and the substrates, which we will denote as u, is greater than some value uc. Consequently, the filling transition is first-order if u > ucand critical if u < uc. Tricritical filling is observed when u−ucemerges as a new relevant field (in the renormalization-group sense). If W(l)∼ −a/lp+b/lq, different scenarios may arise as the range of the binding potential is varied. In particular, for p > 4 the long-range behaviour of VWis dominated by the 1/l3term for θ=α, so the filling phenomena are fluctuation-dominated. This finding is consistent with the existence of two different fluctuation regimes for critical filling: mean-field if p < 4 and fluctuation-dominated regime 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 continuous wetting (see Figure 2), where the role at wetting of the bulk ordering field hand the potential strength ware played by θ−αand ufor filling phenomena, respectively. In particular, 3D tricritical (critical) filling is analogous to 2D critical (complete) wetting, respectively. Different critical exponents which characterize the divergence of length scales can be defined. In addition to the critical filling critical exponents βWand νy Tricritical wedge filling transitions with short-ranged forces 5 defined along route (ii) in Figure 2(a) (see above), we can define new critical exponents for 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∗ W y, where ζ∗ Wis the tricritical wandering exponent which in general may be different from ζW(in contrast with the wetting case). More generally, in the vicinity of the 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 the fluctuation-dominated regime. In addition to the hard-wall condition, VW(l0) can be 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∼l0 or PW∼l3 0. We anticipate that the former corresponds to tricritical behaviour and the latter to critical filling. It is remarkable that thermodynamic consistency at critical filling is ensured as the local density at the wedge bottom is non-singular, i.e. ρw(0)−ρl∼T−Tf, where ρlis the bulk liquid density [14]. This property is only obtained 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 condition on the short-distance expansion of the interfacial height PDF. We report now our explicit results (details will be presented elsewhere). Along route (i) we find that there is only one bound solution to Equation (7) for u > uc≈ 1.358 with E0∝(u−uc)3and associated PDF PW(l0) = 6√3π ξu l0 ξuAi l0 ξu2 (10) where Ai(x) is the Airy function and in the scaling limit ξu∼ |u−uc|−1. Thus lW∼ξ⊥∝(u−uc)−1and ξy∝(u−uc)−3identifying β∗ W= 1, ν∗ y= 3 and obtaining ζ∗ W=ζW= 1/3. As predicted, the short-distance behaviour of the PDF is linear with l0. On the other hand, the scaling of the PDF for θ > α is given by (see also Figure 3): PW(l0)∝l0exp 2ǫl0 ξθ−2l2 0 ξ2 θH2 ν√2l0 ξθ−ǫ √2(11) where ξθ= Σ−1/2[(θ/α)2−1]−1/4,ǫ= ΣE0ξ3 θ/α,ν=ǫ2/4−1/2 and Hν(x) is the Hermite function [13]. The value of ǫis obtained as the smallest solution of the following equation: ±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) 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 that scaling is obeyed in the vicinity of the tricritical point as the wedge excess free energy fW∼ξ−3 θF±(ξθ/ξu). For u > ucand ξθ/ξu≫1, we have checked numerically that the PDF (11) converges to the expression given by Equation (10) for the corresponding value of ξugiven by Equation (12). Thus, the interface remains bound to the substrate when θ→α, in agreement to the first-order character of the filling transition. The thermodynamic path (ii) to the tricritical point corresponds to ξu→ ∞, which corresponds to ǫ≈1.086. Thus along this route lW∼ξ⊥∝(θ−α)−1/4similar Tricritical wedge filling transitions with short-ranged forces 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) Figure 3. Plot of the scaled PDF for ǫ=−1.5 (thick dashed line), and along routes (ii) and (iii) in Figure 2(a), i.e. for ǫ≈1.086 (thick continuous line) and ǫ≈1.639 (thick dot-dashed line), respectively. For comparison, the PDF from Equation (10) with ξu≈1.968ξθ(which corresponds to ǫ=−1.5, see inset) is also plotted (thin dashed line). Finally, the scaled PDF obtained in Ref. [10] is also shown (thin dot-dashed line). Inset: Plot of ǫas a function of ξθ/ξufor u < uc(continuous line) and u > uc(dashed line). to critical filling. From analysis of the spectrum it is also possible to show that ξy∝(θ−α)−3/4, so the tricritical gap exponent ∆∗= 4. Finally, for thermodynamic paths (iii) far from the tricritical point, i.e. ξu→0, we have found that the scaling of the PDF is of the form shown in Equation (11) with ǫ≈1.639. Note that PW∼l3 0as l0→0, in agreement with our previous statement. This condition is not fullfilled by the solution presented in Ref. [10], although globally it does not differ too much from our exact solution (see Figure 3). We finish by mentioning a remarkable connection for short-ranged forces between 3D wedge filling and 2D wetting with random-bond disorder [15]. The critical exponents corresponding to tricritical and critical wedge filling can be obtained from generalized random-walk methods [16] in terms of the wedge wandering exponent ζW. 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