Controlling the order of wedge filling transitions: the role of line tension

arXiv:cond-mat/0702658v2 [cond-mat.soft] 31 May 2007 Controlling the order of wedge filling transitions: the role of line tension J. M. Romero-Enrique 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 A. O. Parry Department of Mathematics, Imperial College 180 Queen’s Gate, London SW7 2BZ, United Kingdom Abstract. We study filling phenomena in 3D wedge geometries paying particular attention to the role played by a line tension associated with the wedge bottom. Our study is based on transfer matrix analysis of an effective one dimensional model of 3D filling which accounts for the breather-mode excitations of the interfacial height. The transition may be first-order or continuous (critical) depending on the strength of the line tension associated with the wedge bottom. Exact results are reported for the interfacial properties near filling with both short-ranged (contact) forces and also van der Waals interactions. For sufficiently short-ranged forces we show the lines of critical and first-order filling meet at a tricritical point. This contrasts with the case of dispersion forces for which the lines meet at a critical end-point. Our transfer matrix analysis is compared with generalized random-walk arguments based on a necklace model and is shown to be a thermodynamically consistent description of fluctuation effects at filling. Connections with the predictions of conformal invariance for droplet shapes in wedges is also made. PACS numbers: 68.08.Bc, 05.70.Np, 68.35.Rh, 05.40.-a Controlling the order of wedge filling transitions: the role of line tension 2 1. Introduction Fluid adsorption on micropatterned and sculpted solid substrates exhibit novel phase transitions compared to wetting behaviour at planar, homogeneous walls [1, 2, 3]. The simple 3D wedge geometry has been extensively studied in the past decade theoretically [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], experimentally [15, 3, 16, 17] and by computer simulation [18, 19, 20]. Thermodynamic arguments [21, 22, 23] show that the wedge is completely filled with liquid provided the contact angle θis less than the tilt angle α. These studies show that the conditions for continuous wedge filling transition are less restrictive than for critical wetting at planar walls [5, 6]. Close to critical filling, the substrate geometry enhances interfacial fluctuations, which become highly anisotropic. We refer to these as breather modes excitations [5, 6]. However, most of these studies neglect the presence of a line tension associated with the wedge bottom. Previous studies by the authors [13, 14] for short-ranged binding potentials show that the line tension may play an important role in filling phenomena and may drive the transition first-order if it exceeds a threshold value. We extend our analysis to arbitrary binding potentials, in particular to van der Waals dispersive interactions. Again, this shows that we can induce first-order filling by tailoring (micro-patterning) the substrate close to the wedge bottom. This may provide a practical means of reducing the fluctuation effects which would otherwise dominate any continuous filling transition. This finding may have technological implications for microfluidic devices. However, the borderline between first-order and critical filling depends on the specific range of the interactions. If the binding potential between the interface and the flat wall decays faster than 1/z4, where zis the local interfacial height above the substrate, both regimes are separated by a tricritical point, as in the case of contact binding potentials [13, 14]. On the other hand, for longer-ranged binding potentials, a critical end point separates the first-order and critical filling transitions. We note that these two situations correspond exactly to the fluctuation-dominated and mean-field regimes for critical filling [5, 6]. Our Paper is arranged as follows: In Section II we review briefly the phenomenology of wedge filling and introduce the breather mode interfacial model used in our study. The definition of the path integral used in our transfer matrix analysis is discussed in some detail. While other formalisms have been forwarded they suffer from a number of problems. As we shall show our definition is consistent with thermodynamic requirements (exact sum-rules), generalized random walk arguments and also the predictions of conformal invariance. Section III is devoted to the analysis of wedge filling for contact binding potentials. Some of these results have been previously reported, without derivation, in a brief communication [13, 14]. Section IV extends the transfer matrix analysis to the important practical case of filling with long-ranged van der Waals forces. We conclude with a brief discussion and summary. 2. The model Our starting point is the interfacial Hamiltonian pertinent to filling in shallow wedges (small tilt angle α) [4]: H[l] = Z Z dxdy Σ 2(∇z)2+W(z−α|x|, x)(1) Controlling the order of wedge filling transitions: the role of line tension 3 αx yl ξx ξξy Figure 1. Schematic illustration of a typical interfacial configuration in the 3D wedge geometry. The diverging lengthscales at the filling transition are highlighted. where z(x, y) is the local height of the liquid-vapour interface relative to the horizontal, Σ is the liquid-vapour surface tension and W(z, x) is the binding potential between the liquid-vapour interface and the substrate (see figure 1). Here after we assume the temperature defines the energy scale and set kBT= 1. To first approximation we may suppose that W(z, x) is independent of the position across the wedge section, i.e. W(z, x)≈Wπ(z), where Wπ(z) is the binding potential due to a single flat substrate. Corrections may arise when the liquid adsorption is small enough however - a point we shall return to later. A mean-field analysis shows that locally the interface across the wedge is flat and that fluctuation effects are dominated by pseudo-one-dimensional local translations in the height of the filled region along the wedge (the breather modes) [5, 6]. Fluctuation effects at filling can be studied by using an effective pseudo-one-dimensional wedge Hamiltonian which accounts only for the breather-mode excitations [5, 6]: HW[l] = Zdy (Λ(l) 2dl dy 2 +VW(l))(2) where l(y) = z(0, y) is the local height of the interface above the wedge bottom. The effective bending term Λ(l) resisting fluctuations along the wedge can be expressed as [5, 6]: Λ(l)≈2Σl α+2τ α2≈2Σl α(3) where τis the line tension associated with the contact lines between the filled region and the substrate far from the wedge bottom. Note that for large l, we may neglect the τcontribution since Λ(l) is proportional to the local interfacial height. Similarly, for large lthe effective binding potential VW(l) takes the form [5, 6]: VW(l)≈h(l2−l2 π) α+Σ(θ2−α2)(l−lπ) α +2τ+τ′+Z(l−lπ)/α −(l−lπ)/α dx Wπ(l−α|x|) (4) The first two terms corresponds to the bulk and surface thermodynamic contributions required to form the filled liquid region [7]. Here hdenotes the bulk ordering field measuring deviations from bulk two-phase coexistence, θis the contact angle of the liquid drop at the planar wall-vapour interface and lπis the equilibrium liquid layer thickness for a single planar wall. The line tension τis defined as above, and τ′is the line tension associated with the wedge bottom. Note that the line tension contributions Controlling the order of wedge filling transitions: the role of line tension 4 are essentially independent of lfor l≫lπ, so they become irrelevant in that limit. Finally, the last term corresponds to the binding potential contribution to VW. Upon minimisation of VW(l), we recover the mean-field expression for the mid-point height (at bulk coexistence) [5, 6]: Σα2 2=Wπ(l) + Σθ2 2≡∆Wπ(l) (5) However, we stress that the form (4) for VW(l) is only valid for l≫lπ. For l.lπ, both Λ(l) and VW(l) will behave in a different manner. Furthermore, we may control the adsorption properties for small lby micropatterning a stripe along the wedge bottom, so as to weaken the local wall-fluid intermolecular potential. The interfacial binding potential is consequently strengthened, and under some conditions it may bind the liquid-vapour interface to the wedge bottom even at the filling transition boundary θ=α. Thus by introducing a line tension associated with the wedge bottom one may induce first-order filling in the modified wedge provided the modification is strong enough. As we shall see the lines of first-order and continuous filling transitions are separated by either a tricritical point or a critical end point depending on the range of the intermolecular forces. The quasi-one-dimensional character of the Hamiltonian means it is amenable to a transfer-matrix analysis. The partition function corresponding to this Hamiltonian can be expressed as the following path integral [24]: Z(lb, la, Y ) = ZDlexp(−HW[l]) (6) However, the presence of a position-dependent stiffness coefficient makes the definition of the partition function ambiguous. This problem was pointed out, but not satisfactorily resolved, in [8] and is intimately related to issues associated with the canonical quantization of classical systems with a position-dependent mass [25, 26]. In this paper we use the following definition of the partition function: Z(lb, la, Y ) = lim N→∞Zdl1...dlN−1 N Y j=1 K(lj, lj−1, Y/N) (7) where l0≡laand lN≡lb, and K(l, l′, y) is defined as: K(l, l′, y) = (Λ(l)Λ(l′))1/4 √2πy e−√Λ(l)Λ(l′) 2y(l−l′)2−yVW(l)(8) The partition function Z(lb, la, Y ) satisfies the differential equation HWZ(lb, la, Y ) = −∂Z(lb, la, Y ) ∂Y (9) with initial condition Z(lb, la, Y )→δ(lb−la) as Y→0. The operator HWis defined as [26]: HW≡ −1 2 ∂ ∂lb1 Λ(lb) ∂ ∂lb+VW(lb) + ˜ VW(lb) (10) where ˜ VW(l) is given by ˜ VW(l) = −1 2Λ(l)"3 4Λ′(l) Λ(l)2 −Λ′′(l) 2Λ(l)#(11) Controlling the order of wedge filling transitions: the role of line tension 5 and prime denotes differentiation with respect to argument. The solution of (9) can be expressed via the spectral expansion [24]: Z(lb, la, Y ) = X α ψα(lb)ψ∗ α(la)e−EαY(12) where {ψα(l)}is a complete orthonormal set of eigenfunctions of the Hamiltonian operator HW, with associated eigenvalues Eα. Analogous to discussion of 2D wetting [24] we can now obtain the interfacial properties from the knowledge of the propagator Z(lb, la, Y ). In particular, the probability distribution function (PDF), PW(l), can be obtained as: PW(l, Y ) = lim L→∞ Z(l, l−L/2, Y +L/2)Z(lL/2, l, L/2−Y) Z(lL/2, l−L/2, L)(13) while the joint probability P(2) W(l1, l2, Y1, Y2) of finding the interface at midpoint heights l1and l2at positions Y1and Y2(> Y1), respectively, is: P(2) W(l1, l2, Y1, Y2) = Z(l2, l1, Y2−Y1) ×lim L→∞ Z(l1, l−L/2, Y1+L/2)Z(lL/2, l2, L/2−Y2) Z(lL/2, l−L/2, L)(14) Further simplifications arises if we assume the existence of a bounded ground eigenstate ψ0. Then, substitution of the spectral expansion (12) into (13) reads (for an infinitely long wedge) PW(l) = |ψ0(l)|2(15) Similarly the excess wedge free-energy per unit length of an infinitely long wedge is identified with the ground eigenvalue E0. The two-point correlation function h(l1, l2, Y ) can be obtained in a similar manner: h(l1, l2, Y )≡P(2) W(l1, l2,0, Y )−PW(l1)PW(l2) =X α6=0 ψ∗ α(l1)ψ0(l1)ψα(l2)ψ∗ 0(l2) exp [−(Eα−E0)Y] (16) For large separations this vanishes exponentially allowing us to determine the correlation length ξy= (E1−E0)−1, where E1is eigenvalue corresponding to the first excited eigenstate. This result still holds even if E1corresponds to the lower limit of the continuous part of the spectrum of HW, although now the leading order of h(l1, l2, Y ) is not purely exponential but is modulated by a power of Y. An interesting connection with 2D wetting problems can now be seen. Introducing the change of variables [27] η(l) = ZdlpΛ(l) ; φ(η) = Λ(l)−1/4ψ(l(η)) (17) the eigenvalue problem HWψα(l) = Eαψα(l) transforms to the following Schr¨odingerlike equation: −1 2 d2φα(η) dη2+VW[l(η)] + ˜ V∗ W(η)φα(η) = Eαφα(η) (18) where ˜ V∗ Wis defined as: ˜ V∗ W(η) = −3 32Λ2(l(η)) dΛ(l(η)) dη 2 +1 8Λ(l(η)) d2Λ(l(η)) dη2(19) Controlling the order of wedge filling transitions: the role of line tension 6 (18) shows that the filling problem can be mapped onto a 2D wetting problem in the η variable, under an effective binding potential. In our case Λ(l) = 2Σl/α, so we obtain ˜ VW(l) = −3α/16Σl3. The change of variables (17) leads to η=p8Σ/9αl3/2and thus ˜ V∗ W(η) = −5/72η2. Note that the use of the variable η∝l3/2as the appropriate collective coordinate has been previously recognized in the literature [8, 7]. However, our definition of the path integral (7) and (8) leads to a novel term in the wedge binding potential, which will be essential in our study and ensures thermodynamic consistency. The filling potential VW(l) must fulfill some requirements. In order that the interface does not penetrate the substrate, VW(l) has a hard-wall repulsion for l < 0. Consequently we need to impose an appropriate boundary condition on the eigenfunctions of HWat l= 0. The analytical expression of the boundary condition is obtained by a regularization procedure: we assume that the filling potential VW and the position-dependent stiffness Λ are constant for l < ξ0, where ξ0is some microscopic scale. Furthermore, we impose that Λ(l) must be continuous at l=ξ0, so Λ(l < ξ0) = 2Σξ0/α. On the other hand, the filling potential can take an arbitrary value −U. The latter square-well potential models the modification of the filling potential VW(l) for small ldue to the line tension associated with the wedge bottom. The eigenstates must fulfill the usual matching conditions that φαand ∂φα/∂l are continuous at l=ξ0. Finally, we consider the appropriate scaling limit as ξ0→0. The qualitative form of the filling potential depends on the order of the mean-field phase transition [5, 6]. For critical filling with long-ranged intermolecular forces, the binding potential at bulk coexistence behaves as VW(l) = Σ(θ2−α2)l α+2A (p−1)αl1−p+... (20) where Ais a Hamaker constant while the exponent pdepends on the range of the forces. Specifically, for non-retarded van der Waals forces p= 2. For systems with short-ranged forces this is replaced by an exponential decay ∼exp(−κl) where κis an inverse bulk correlation length. The presence of the fluctuation-induced filling potential ˜ VW(l)∝l−3in (9) gives rise to two distinct scenarios. For p > 4 and large l, the direct contribution in (9) ∝l1−pis negligible compared to the fluctuationinduced potential ˜ VWarising from the position dependent stiffness. In this case we anticipate universal, fluctuation dominated behaviour. On the other hand, for p < 4 and large lwe can neglect ˜ VW. Since the l1−pcontribution to the binding potential is now repulsive, we expect a qualitatively different phase diagram. It is worthwhile noting that both situations exactly correspond to the existence of a mean-field and a filling fluctuation-dominated regime for the filling transition predicted from heuristic arguments [5, 6]. Fig. 2 shows the schematic filling phase diagrams we expect for short-ranged (Fig. 2a) and long-ranged forces (Fig. 2b). Previous work for the contact potential case [13, 14] showed a similarity between 3D wedge filling (for h= 0) and 2D wetting, with θ−αplaying the role of the ordering field and the effective line tension the role of the wetting binding potential. The borderline between the first-order and the secondorder transition lines corresponds to a tricritical point (analogous to the 2D critical wetting case). For long-ranged forces the analogy between 3D filling and 2D wetting still holds (see above). However, as the effective binding potential is repulsive for large lat θ=α, the similarity must be established with 2D first-order wetting. As for contact binding potentials, we expect the interface to be bound to the wedge bottom Controlling the order of wedge filling transitions: the role of line tension 7 a) b) θ−α Bound Unbound θ−α Bound Unbound U✖ cep U tc UU Figure 2. Schematic phase diagrams for 3D wedge filling pertinent to: (a) shortrange forces, (b) long-range forces. Uis the effective line tension strength, θis the contact angle for the liquid on a planar substrate, and αthe wedge angle. The filled circle locates the borderline between the first-order transition line (dashed line) and the second-order transition line (thick continuous line) between the bound and unbound states. These correspond to a tricritical point and a critical end-point for short-ranged and long-ranged forces, respectively. In the latter case, the first-order transition line continues to a first-order pseudo-transition line in the partial filling region (dotted line), which terminates at a pseudo-critical point (cross). See text for explanation. at θ=αfor a large well depth U(corresponding to the line-tension contribution). As Udecreases, the interface will unbind along the θ=αpath. However, the interface must tunnel through a free-energy barrier to become unbound. Consequently, the borderline between the first-order and second-order wedge filling is a critical end-point, where the spectator phase is the bound state at θ=α. Actually, the connection with wetting phenomena also leads naturally to this picture, since first-order wetting was previously recognized as an interfacial critical-end point scenario [28]. In principle, we may expect the first-order transition line to continue in the partially filled region as the coexistence between two bound states with different adsorptions. This thin-thick transition is analogue to the prewetting line, and it should terminate at a critical point. However, the quasi-one-dimensional character of the wedge geometry rules out this transition, as it is destroyed by breather-mode fluctuations. Nevertheless traces of this smeared transition may be found in the bimodal form of the interfacial height PDF (see later). We will consider two cases which correspond to different regimes of the filling transition. The contact interaction will be studied as the paradigm of the filling fluctuation-dominated regime. On the other hand, the van der Waals (p= 2) case is analysed as a prototypical case of the mean-field regime. 3. Results for contact interactions We consider first the case of short-ranged potentials. Some of our results have been reported in a brief communication but without detailed explanation [13, 14]. Here we provide full details of our transfer matrix solution and present new results for the form of the propagator. At lengthscales much larger than that of the bulk correlation length ξbwe may write VW(l) = Σ(θ2−α2)l/α for l > 0 and allow for line tension arising Controlling the order of wedge filling transitions: the role of line tension 8 00.5 11.5 22.5 3 l/ξ u 0 0.2 0.4 0.6 0.8 PW(l/ξu) 0 0.1 0.2 0.3 ε -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 ε∗ ∼ ∼ Figure 3. Scaled probability distribution function PW(l/ξu) as a function of the scaled wedge midpoint interfacial height l/ξufor θ=α. Inset: Plot of ˜ǫ1/3Ai′(˜ǫ1/3)/Ai(˜ǫ1/3) (continuous line) and √u−˜ǫcot √u−˜ǫ−1/2 for u= 0.5< uc(dotted line), u=uc= 1.358 (dashed line) and u= 2.0> uc(dotdashed line). For the latter case, the value of the reduced ground eigenvalue ˜ǫ= ˜ǫ∗is highlighted. from the wedge bottom via a suitable boundary condition at the origin. Analysis of (9) shows that the short-distance behaviour of the eigenfunctions will be dominated by the l−3contribution to the effective filling potential. In fact, ψα(l)∼l1/2or l3/2 as l→0, so the PDF function PW(l, θ)∼lor l3as l→0. We anticipate that the latter behaviour corresponds to the critical filling transition as predicted by scaling arguments [7], but the former will be a completely new situation which, as we shall see, is related to the possibility of tricriticality. Turning to the critical filling transition first we note that the short-distance behaviour of the PDF, PW(l, θ)∼l3, which emerges from our analysis, is indeed the required, thermodynamically consistent, result. This short-distance expansion ensures that the local density of matter near the wedge bottom contains a scaling contribution that vanishes ∝T−Tf, where Tfis the filling temperature. This is the required singularity which emerges from an analysis of sum-rules connecting the local density near the wedge bottom to (derivatives of) the excess wedge free-energy [29]. This leads us to conclude that our definition of the path integral in (8) is the correct one for the 3D wedge filling problem. As we shall see it also ensures that our model is conformally invariant. 3.1. Wedge filling along the θ=αpath At the filling phase boundary (θ=α), the problem can be mapped onto the intermediate fluctuation regime of 2D wetting [30] via (18). We apply the regularization method described above, and consider the scaling limit ξ0→0, U→ ∞ and 4ΣUξ3 0/α →u. Under these conditions, the ground eigenvalue E0satisfies √u−˜ǫcot √u−˜ǫ−1 2= ˜ǫ1/3Ai′˜ǫ1/3 Ai ˜ǫ1/3(21) Controlling the order of wedge filling transitions: the role of line tension 9 where ˜ǫ=−4ΣE0ξ3 0/α and Ai(x) is the Airy function. Graphical solution of (21) (see inset of figure 3) shows that there is a bound state with E0<0 for u > uc, where uc≈1.358. Otherwise E0= 0 and there is no bound state. The existence of a bounded ground state at θ=αimplies that the filling transition is first-order for u > uc, and critical for u < uc. The explicit form of the PDF for u > ucin the thermodynamic limit is PW(l, θ =α) = 6√3π ξu l ξuAi l ξu2 (22) where ξu=ξ0/˜ǫ1/3∝ξ0/(u−uc) as u→uc. Note that the lengthscale ξucan be arbitrary large as u→uc. Figure 3 plots the PDF in terms of the scaling variable l/ξu. For small l,PW(l, α)∼lγu, with a short-distance exponent (SDE) γu= 1. Asymptotically PW(l, α)∼√lexp −4(l/ξu)3/2/3as l→ ∞. The mean interfacial mid-point height lW≡ hliand roughness ξ⊥≡phl2i−hli2 satisfy lW∼ξ⊥∼ξushowing that, in the scaling limit, there is only one lengthscale controlling the fluctuations of the interfacial height. Finally, the correlation length along the wedge axis ξyclose to the filling transition can be obtained as ξy=−1/E0≡ 4Σξ3 u/α ∝(u−uc)−3since we can identify E1≡0. These observations indicate the emergence of a new relevant field (in the renormalization group sense) tu∝(u−uc), in addition to tθ∝θ−αand the bulk ordering field h. Thus the conditions θ=α,u=ucand h= 0 correspond to a tricritical point which separates the lines of first-order and critical filling transitions. The excess wedge free energy density E0vanishes as E0∼t2−αu W u∼t3 uas utends to ucfrom above. Critical exponents for the divergence of the characteristic lengthscales can be defined as the tricritical point is approached along the θ=αpath: lw∼t−βu W u, ξ⊥∼t−νu ⊥ u, ξy∼t−νu y u(23) Our results show that βu W=νu ⊥= 1 and νu y= 3. Finally, the effective wedge wandering exponent ζW=νu ⊥/νu y= 1/3, which coincides with its value for the critical filling transition [5, 6]. For u≤ucand θ=α, the interface is unbounded in the thermodynamic limit. However, we can study the finite-size behaviour of the droplet shape when the interface is pinned very close to the wedge bottom at positions y=±L/2. Making use of results presented in the Appendix, in particular (A.11) and (A.12), we find that the PDF at the tricritical (u=uc) and critical (u≪uc) wedge filling transition are given by Pu=uc,L W(l, θ =α) = λ2 Ll 31/3Γ(2/3) exp −(λLl)3 9(24) Pu≪uc,L W(l, θ =α) = λ4 Ll3 35/3Γ(4/3) exp −(λLl)3 9(25) where λL=16Σ αL 1/3"1−2Y L2#−1/3 (26) The typical droplet shape may be characterized by the most-probable position lmp(y) which follows from the relation ∂PL W(lmp)/∂l = 0, or by the average shape lav(y) via the definition lav =R∞ 0dllPL W(l) [24]. In all cases we find that the typical droplet shape follows λLl=c, where cis a number which depends on the definition of the Controlling the order of wedge filling transitions: the role of line tension 16 bound state is the same as for the contact binding potential. Now we expand (52) for uaround ucand ˜ǫaround zero. In the scaling limit, we obtain that: ˜ǫ∼ξ2 0 (l∗)2(u−uc) (53) Finally, the PDF in this regime can obtained from substitution of the asymptotic relationship (51) into (49). After some algebra, the PDF reads: PW(l) = 9l (l∗)2exp −3l l∗(54) (see also figure 6). Since l∗remains finite at uc, the interface remains bound as u approaches ucfrom above, and suddenly unbinds for u < uc. This identifies the point h= 0, θ=αand u=ucas a critical end point in the surface phase diagram. However, although uis not a relevant field in the renormalization-group sense, the longitudinal correlation length ξy=|E0|−1diverges as (u−uc)−1. A similar behaviour occurs within the subregime C of the intermediate fluctuation regime for 2D wetting transitions [30]. 4.2. Wedge filling for θ > α We now extend our results to θ > α proceeding in the same way we did for contact interactions. Again we can analytically obtain the ground state ψ0(l) of HWas: ψ0(l)∝√lexp l ξθǫ0−l ξθHǫ2 0 4−1 2−9ξ2 θ 16(l∗)2√2l ξθ−ǫ0 √2(55) where ǫ0,ξθand Hs(x) are defined as in (36) for the contact binding potential. Note that the dependence on the dispersion forces only appears at the order sof the Hermite function Hs. The regularization procedure leads to the following equation for the reduced eigenvalues ǫ(for uclose to uc): ξθ ξ0√ucot√u−1 2≈ ±Γ−1 33−2/3 Γ1 3 ξθ ξu−3ξθ 2l∗ =ǫ+ǫ2 √2−√2−9ξ2 θ 4√2(l∗)2Hǫ2 4−3 2−9ξ2 θ 16(l∗)2−ǫ √2 Hǫ2 4−1 2−9ξ2 θ 16(l∗)2−ǫ √2(56) and ǫ0is the minimum of the solutions. We can see that ǫ0will depend now not only on the ratio ξθ/ξuand the sign of u−uc, but also on ξθ/l∗. Nevertheless as for the case with a contact binding potential, we are mainly interested in the limit θ→α, so that the lengthscale ξθis very large. For u > uc, saddle-point asymptotic techniques analogous to those applied in the previous Section [35] show that the ground state of HWconverges to the expression (49) as θ→α. Consequently we recover the results obtained earlier for the special case θ=α. This indicates that, for these values of u, the wedge filling transition must be first-order. For u < ucthe non-existence of ground state for θ=αindicates that the filling transition is critical. We anticipate that mean-field theory will describe faithfully singularities at the critical wedge filling [5, 6]. Thus we expect that the interfacial Controlling the order of wedge filling transitions: the role of line tension 17 height PDF is centered around lMF Wwith Gaussian fluctuations on the scale of ξMF ⊥ representing the breather mode excitations. The mean-field value of the excess free energy per unit length is given by VW(lMF W). Consequently, the mean-field value of the reduced ground eigenvalue ǫMF 0is: ǫMF 0=3ξθ 2l∗(57) The shift of ǫ0with respect to ǫMF 0due to breather-mode fluctuations ∆ǫ0can be estimated in the following way. We expand VWaround its minimum up to quadratic order, so the shift may be estimated via: ∆E0∼1 2V′′ W(lMF W)(ξMF ⊥)2(58) implying that ∆ǫ0∼1 ǫMF 0∼l∗ ξθ (59) which vanishes as θ→α. We can now proceed with a more formal derivation. Equation (55) can be written as: ψ0(l)∝√lexp(−x2/2)Hs(x) (60) where we have defined: x=√2∆l ξθ−∆ǫ0 √2(61) s=ǫMF 0∆ǫ0−1 2+(∆ǫ0)2 4(62) with ∆l≡l−lMF Wand ∆ǫ0=ǫ0−ǫMF 0. Note that (60) is the wavefunction of the harmonic oscillator in the xcoordinate (in units of p~/mω), modulated by the factor √l. As in the latter case, ψ0will increase exponentially as x→ −∞, i.e. l→0 and large lMF W, unless sis a non-negative integer. This result is independent of the explicit value of ξθ/ξu. Consequently, the shift ∆ǫfor the lowest eigenvalues is given by: ∆ǫ=s(ǫMF 0)2+ 4 n+1 2−ǫMF 0≈2n+ 1 ǫMF 0 (63) with na non-negative integer. The ground eigenstate will correspond to the case n= 0. The corresponding PDF becomes a Gaussian: PW(l)≈s2 πξ2 θ exp −2(l−hli)2 ξ2 θ(64) with hli=lMF W+ 2l∗/3≈lMF W, and roughness ξ⊥=ξθ/2. Finally, the longitudinal correlation length ξy= (E1−E0)−1∼ξ3 θǫMF 0/2∼(θ−α)−1. Consequently, the explicit transfer matrix solution is in complete agreement with the predicted mean-field values for the critical exponents when p= 2 [5, 6]: 2−αθ W= 1/2, βθ W= 1/2, νθ y= 1 and νθ ⊥= 1/4. Finally, we search for the existence of the thin-thick transition line. Our calculations show that there is no sharp phase transition for θ > α. However, we observe that the interfacial PDF becomes bimodal for u < ucand some range of values of θ > α. We can identify a first-order pseudo-transition line when the areas Controlling the order of wedge filling transitions: the role of line tension 18 0 0.01 0.02 0.03 (l*/ξθ)4 0 0.2 0.4 0.6 0.8 l*/ξu 0 1 2 3 4 5 l/ξθ 0 0.5 1 1.5 2 PW(l/ξθ) Figure 7. Interfacial height PDF at the first-order pseudo-transition for ξθ/l∗= 4.216 (continuous line), 2.981 (dot-dashed line), 2.749 (dashed line) and 2.494 (dotted line). Inset: location of the first-order pseudo-transition line. Note that l∗/ξu∝uc−u(here u < uc) and (l∗/ξθ)4∝θ−α. The diamonds represent our calculated pseudo-coexistence values. The continuous line serves only as a guide for the eye. under each maximum of the PDF are equal (see Fig. 7). This line is the continuation to the partial filling region of the first-order filling transition line for u < uc, and touches tangentially the filling transition borderline at u=uc. As udecreases, the two maxima become closer, and eventually merge (the pseudo-critical point). 5. Discussion and conclusions In this paper we have reported analytical results for 3D wedge filling transitions in the presence of short-ranged and long-ranged (van der Waals) interactions based on exact solution of the continuum transfer equations for a pseudo one-dimensional interfacial Hamiltonian. First-order and continuous (critical) filling are possible for both types of force depending on the strength of the line tension associated with the decoration of the wedge bottom. Our analytical solution for the interfacial height PDFs at critical filling recovers the known values of the critical exponents for critical filling for short and long-ranged forces. In addition we have elucidated the nature of the cross-over from first to critical filling which occurs at a specific value of the wedge bottom line tension. This is qualitatively different for systems with short and long ranged forces whose surface phase diagrams are shown to have tricritical and critical end-points respectively. To finish we argue that these results, which are proto-typical of the fluctuation dominated and mean-field regimes respectively, are qualitatively valid for any type of binding potential. As already mentioned, the 3D wedge filling phenomenon can be mapped onto a 2D wetting problem with a new collective coordinate η∝l3/2. If we suppose that, at the filling phase boundary, l3VW(l)→0 for large l, we can make use of a renormalization-group arguments to determine the allowed behaviour. As h and tθ∝θ−αare always relevant operators, we will restrict ourselves to the filling Controlling the order of wedge filling transitions: the role of line tension 19 transition boundary θ=α,h= 0. The effective 2D wetting binding potential decays faster than 1/η2, so it becomes irrelevant in the renormalization-group sense [36, 37]. The renormalization-group flows are dominated by the unstable fixed point and the stable fixed point for the 2D wetting binding potential −5/72η2, which correspond to the tricritical and critical filling transition, respectively. Thus, for large scales the only effect of such binding potentials that decay faster than 1/l3is to renormalize the line tension associated with the wedge bottom. For long-ranged binding potentials i.e. those for which, at θ=α,l3VW(l)→ ∞ for large l, we may resort to a mean-field analysis [5, 6]. We expect that close to the filling transition the PDF is asymptotically a Gaussian characterized by the meanfield interfacial height lMF Wand roughness ξMF ⊥. However, if the next-to-leading order to the wedge binding potential is determined by the short-distance repulsive part of Wπ(z), we find a similar scenario to the one depicted in figure 5. In particular, it is possible to bind the interface to the wedge bottom. The interfacial roughness will be controlled by the (microscopic) lengthscale l∗∼lπcorresponding to the maximum of the total effective binding potential. The existence of such a bound state, as well as the threshold to the critical filling transition will depend on the specific details of the interfacial binding potential. The predicted filling phenomenology presented in this paper can hopefully be checked experimentally or by computer simulations of more microscopic models. Of course our predictions for contact (strictly short-ranged) forces requires the elimination of van der Waals forces which are ubiquitous for simple fluids. Here our results are most easily tested using large scale Ising model simulations. In this case it should be straightforward to induce first-order filling by weakening the local spin-substrate interaction near the wedge bottom. In contrast our predictions for first-order filling with van der Waals forces may well be amenable to experimental verification sometime in the near future. Taking an even broader perspective it may be that the chemical decoration of a wedge bottom will provide a practical means of eliminating large scale interfacial fluctuations. This may be of relevance to the construction of microfluidic devices whose efficiency will depend crucially on the control of fluctuation effects. Acknowledgments J.M.R.-E. acknowledges partial financial support from Secretar´ıa de Estado de Educaci´on y Universidades (Spain), co-financed by the European Social Fund, and from the European Commission under Contract MEIF-CT-2003-501042. A “Ram´on y Cajal” Fellowship from the Spanish Ministerio de Educaci´on y Ciencia is also gratefully acknowledged. Appendix A. Evaluation of Z(lb, la;Y)at θ=αfor short-ranged forces. The evaluation of the interfacial properties at the filling boundary θ=αis based on the knowledge of the partition function (6). In this Appendix we will evaluate explicitly the partition function Z(lb, la, Y ) at the tricritical and critical points for short-ranged forces, and we will discuss some properties of the partition function for arbitrary u. Our starting point is the solution of (18), for Λ(l) = 2Σl/α and VW(l) = 0, so the effective potential is V∗ W(η) = −5/72η2. This Hamiltonian has (1,1) deficiency indexes Controlling the order of wedge filling transitions: the role of line tension 20 on the (0,∞) interval, so the solution is not unambiguosly defined, but instead depends on a parameter cwhich defines the short-distance behaviour of the eigenfunctions, i.e. φα(η)∼cη5/6+22/3Γ(1/3) Γ(−1/3) η1/6η→0 (A.1) We define the partition function ˜ Z(ηb, ηa, Y ) via the spectral expansion: ˜ Z(ηb, ηa, Y ) = X α φα(ηb)φ∗ α(ηa)e−EαY(A.2) where the summation must be understood as an integral for the continuous part of the spectrum. From (17), this partition function is related to Z(lb, la, Y ) via: Z(lb, la, Y ) = r2Σ α(lbla)1/4˜ Z(η(lb), η(la), Y ) (A.3) The continuous part of the spectrum of any self-adjoint extension of our interfacial Hamiltonian corresponds to E > 0. The corresponding scattering states can be expressed as [38]: φE(η) = √ηhcJ1/3(√2Eη)−(2E)1/3J−1/3(√2Eη)i pc2−c(2E)1/3+ (2E)2/3(A.4) where Js(x) is the s-order Bessel function of first kind. In addition, for c > 0 there is a bounded eigenstate: φ0(η) = r33/2c3η πK1/3(c3/2η) (A.5) with associated eigenvalue E0=−c3/2, where Ks(x) is the s-order modified Bessel function of second kind. Transforming back to the original variable l= (9α/8Σ)1/3η2/3, the associated eigenfunction ψ0(l) is: ψ0(l)∝sl ξu Ai l ξu(A.6) where Ai(x) is the Airy function, and ξuis defined: ξu=α 2Σ1/31 c=−4ΣE0 α−1/3 (A.7) On the other hand, the longitudinal correlation length along the wedge axis reads ξy= 2/c3. Comparison with the results obtained by the regularization procedures in the text allows us to identify c∝(u−uc). For general c, we cannot perform the spectral integral. However, for c= 0 (u=uc) and c=−∞ (u≪uc) we have closed form expressions for ˜ Z(ηb, ηa, Y ) [39, 40]: ˜ Zc=0 =√ηaηbZ∞ 0 dEe−EY J−1 3(√2Eηa)J−1 3(√2Eηb) =√ηaηb Yexp −η2 b+η2 a 2YI−1 3ηaηb Y(A.8) ˜ Zc=−∞ =√ηaηbZ∞ 0 dEe−EY J1 3(√2Eηa)J1 3(√2Eηb) =√ηaηb Yexp −η2 b+η2 a 2YI1 3ηaηb Y(A.9) Controlling the order of wedge filling transitions: the role of line tension 21 with the scaling property: ˜ Zc=0,−∞ =1 √Y ˜ Ucηa √Y,ηb √Y(A.10) Substituting into (A.3) we obtain Zc=0 =l∗ al∗ b 3ξuY∗exp −(l∗ a)3+ (l∗ b)3 9Y∗I−1 32(l∗ al∗ b)3/2 9Y∗(A.11) Zc=−∞ =l∗ al∗ b 3ξuY∗exp −(l∗ a)3+ (l∗ b)3 9Y∗I1 32(l∗ al∗ b)3/2 9Y∗(A.12) where l∗≡l/ξuand Y∗≡Y/ξy. Now the lengthscale ξuis arbitrary but ξy= 4Σξ3 u/α. In both cases we have scaling such that Zc=Y−1/3Ucla Y1/3,lb Y1/3(A.13) for c= 0 and −∞. For Y→ ∞,Uchas the following asymptotic behaviour: Uc=0 ∝rla Y1/3rlb Y1/3(A.14) Uc=−∞ ∝la Y1/33/2lb Y1/33/2 (A.15) To obtain results for arbitrary c, we will make use of the Krein formula [41], which relates the Green functions of different self-adjoint extensions of a closed symmetric operator. In our case, the Green function Z(lb, la;E) is basically the Laplace transform of Z(lb, la;Y) with respect to Y: Z(lb, la;E) = Z∞ 0 dY exp(EY )Z(lb, la;Y) (A.16) For c= 0 and c=−∞ we have the closed expressions for the Green function [39]: Zc=0 =2ξyl∗ al∗ b 3ξu K1 32 3q−E∗(l∗ >)3I−1 32 3q−E∗(l∗ <)3(A.17) Zc=−∞ =2ξyl∗ al∗ b 3ξu K1 32 3q−E∗(l∗ >)3I1 32 3q−E∗(l∗ <)3(A.18) where l∗ >and l∗ <are the largest and smallest between l∗ aand l∗ b, respectively, and E∗≡Eξy. To continue, we define the lengthscales ξuand ξyfor each cas: ξu=α 2Σ1 31 |c|(A.19) ξy=2 |c|3(A.20) Note that ξuand ξyreduce to the relevant correlation lengthscales for c > 0. Application of the Krein formula leads to the following expression for the Green function corresponding to an arbitrary c: Zc(lb, la, E) = Z−∞(lb, la, E) +gl∗ al∗ bK1 32 3p−E∗(l∗ a)3K1 32 3q−E∗(l∗ b)3(A.21) Controlling the order of wedge filling transitions: the role of line tension 22 where g≡g(E, Σ, α) is obtained by imposing that the short-distance behaviour of Z is consistent with the boundary condition (A.1). After some algebra, we obtain the following expression: Zc=Z−∞ 1∓(−E∗)1/3∓(−E∗)1/3Z0 1∓(−E∗)1/3(A.22) where the negative sign corresponds to c > 0, and the positive sign to c < 0. It is worthwhile to note that the dependence on the boundary condition, i.e. c, has been absorbed into the lengthscales ξuand ξy. We can formally invert (A.22): Zc(lb, la, Y ) = 1 2πi Zγ+i∞ γ−i∞ dE exp(−EY )Zc(lb, la, E) (A.23) with γ < min(0,−c3/2). Note that there is a branch point of Zcat E= 0. In addition, there is a single pole for E=−c3/2 for c > 0. The scaling properties of Zc(A.22) lead to the following scaling: Zc(lb, la, Y ) = 1 ξu Z±lb ξu ,la ξu ,Y ξy(A.24) (A.23) provides a means of obtaining the asymptotic behaviour of Zcas Y→ ∞. For c > 0, the dominant contribution comes from the pole of Zcat E=−c3/2, so: Zc(lb, la, Y )∼6√3π ξupl∗ al∗ bAi(l∗ a)Ai(l∗ b) exp(Y∗) (A.25) in complete agreement with our previous analysis. Thus ξuand ξyare true correlation lengths. For c < 0, the branch point of Zcat E= 0 controls the large-Ybehaviour of Zc. We expand Zcaround E= 0: Zc∼ξyl∗ al∗ b ξu" sl∗ < l∗ >−32/3Γ1 3 Γ−1 3pl∗ al∗ b! + (−E)1/3 Γ−1 3pl∗ al∗ b 32/3Γ1 3−sl∗ < l∗ >−sl∗ > l∗ < +32/3Γ1 3 Γ−1 3pl∗ al∗ b! +Oh(−E)2/3i#(A.26) Thus Zchas for large Ythe following asymptotic behaviour: Zc∼ − l∗ al∗ b ξuΓ−1 3(Y∗)4/3"Γ−1 3pl∗ al∗ b 32/3Γ1 3−sl∗ < l∗ >−sl∗ > l∗ < +32/3Γ1 3 Γ−1 3pl∗ al∗ b#+... (A.27) Consequently, we find that for l∗ a, l∗ b≫1 but pl∗ al∗ b/(Y∗)1/3small, the asymptotic behaviour of Zcis the same as for the c=−∞ case, i.e. critical filling. For the c < 0 case ξuand ξyare not correlation lengths, but the (microscopic) lengthscales for l and Y, respectively, so the critical filling behaviour is observed over larger scales than these. In order to obtain the asymptotics of the correlation function h(lb, la, Y ) for c > 0, we also need the next-to-leading order dependence of Ythat emerges from Zc(lb, la, Y ). Controlling the order of wedge filling transitions: the role of line tension 23 Our approach provides a systematic means of obtaining such correction which can be obtained in a similar manner as before. If we denote by Zl cthe leading contribution to Zcgiven by (A.25), the first correction is given by: Zc−Zl c∼ − l∗ al∗ b ξuΓ−1 3(Y∗)4/3"Γ−1 3pl∗ al∗ b 32/3Γ1 3+sl∗ < l∗ > +sl∗ > l∗ < +32/3Γ1 3 Γ−1 3pl∗ al∗ b#+... 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