Interfacial structure at a two-dimensional wedge filling transition: Exact results and a renormalization group study

Interfacial structure at a two-dimensional wedge filling transition: Exact results and a renormalization group study J. M. Romero-Enrique,1,2 A. O. Parry,1and M. J. Greenall1 1Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2BZ, United Kingdom 2Departamento de Física Atómica, Molecular y Nuclear, Area de Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain (Received 11 November 2003; published 2 June 2004) Interfacial structure and correlation functions near a two-dimensional wedge filling transition are studied using effective interfacial Hamiltonian models. An exact solution for short range binding potentials and results for Kratzer binding potentials show that sufficiently close to the filling transition a new length scale emerges and controls the decay of the interfacial profile relative to the substrate and the correlations between interfacial positions above different positions. This new length scale is much larger than the intrinsic interfacial correlation length, and it is related geometrically to the average value of the interfacial position above the wedge midpoint. The interfacial behavior is consistent with a breather mode fluctuation picture, which is shown to emerge from an exact decimation functional renormalization group scheme that keeps the geometry invariant. DOI: 10.1103/PhysRevE.69.061604 PACS number(s): 68.08.Bc, 05.70.Np, 68.35.Md, 68.35.Rh I. INTRODUCTION Fluid adsorption in wedge and cone-shaped nonplanar geometries has attracted much attention in the last few years [1–5]. Geometry plays an important role in the surface phase diagram, and new phase transitions as the filling transition arise. Thermodynamic considerations [6–8]predict that the gas-liquid interface unbinds from the wedge before the wetting temperature Twcorresponding to the substrates. So, the wedge is completely filled by liquid for temperatures higher than the filling temperature Tf⬍Tw, where Tfis given by the condition ␪ 共Tf兲= ␣ 共1兲 and ␪ 共T兲is the temperature-dependent contact angle of a liquid drop on the planar substrate and ␣ is the tilt angle (see Fig. 1). Capillary wave models show that the filling transition can be critical even though the wetting transition corresponding to the substrate is first order, and that interfacial fluctuations are enhanced with respect to the wetting case [3,4]. For the two-dimensional (2D)wedge filling transition in shallow wedges characterized by a small angle ␣ with respect to the xaxis (see below), there exists a remarkable covariance relationship between the wedge midpoint probability distribution function Pw 1共l0兲in the filling fluctuation regime and the planar 1-point probability distribution function P ␲ 1共l0兲characteristic of a strong-fluctuation regime critical wetting transition: Pw 1共l0; ␪ , ␣ 兲=P ␲ 1共l0; ␪ − ␣ 兲.共2兲 This expression establishes a connection between two apparently unrelated phenomena, the deep origin of which is still elusive. The covariance relationship has been observed also in acute wedges [9], Ising model exact calculations [10], and computer simulations [11]. Although the covariance relationship is restricted to the interfacial behavior above the wedge midpoint, some other quantities, such as the local susceptibility, which is related to the 2-point correlation function, also showed a modified covariance relationship [5]. Consequently, it is interesting to see if the covariance extends to higher-order probability distribution functions. In this paper we study the structure of the interfacial profile and correlations for 2D wedge filling phenomena. Exact results for the capillary wave effective Hamiltonian theory in the filling fluctuation regime are obtained as an extension of the analysis presented in Ref. [12]. The exact results show the appearance of a new length scale ␰ Facross the wedge close to the critical filling transition. This scale controls the decay of the interfacial profile, local roughness, and correlations, and is related geometrically to the wedge midpoint average interface position. For the local properties, we found a very interesting relationship between the wedge 1-point probability distribution function and the corresponding functions in the planar geometry, which can enlighten the origin of the wedge covariance. FIG. 1. Schematic illustration of a typical interfacial configuration in the wedge geometry. The relevant correlation length scales ␰ xand ␰ ⬜共x兲are also highlighted. Other notation is defined in the text. PHYSICAL REVIEW E 69, 061604 (2004) 1539-3755/2004/69(6)/061604(14)/$22.50 ©2004 The American Physical Society69 061604-1 Regarding the two-point correlation functions, we found a confirmation in the scaling limit of the breather mode picture [3,4], which states that the interface is effectively infinitely stiff in the filled region and is driven by fluctuations of the wedge midpoint interfacial position, i.e., critical effects at 2D wedge filling arise simply from local translations in the height of the flat, filled interfacial region. Finally, we explain the critical behavior of the filling transition in the functional renormalization group approach. As the geometry is fundamental in the understanding of the critical filling transition, we choose a scheme that leaves the wedge geometry invariant. We show that the breather mode picture emerges as a straightforward consequence. The predictions for the critical behavior are in complete agreement with exact solutions. Our paper is organized as follows. In Sec. II we describe the continuous transfer-matrix formalism and the definition of the wedge n-point interfacial probability distribution functions. We apply this formalism to the case of contact binding potentials in Sec. III and, in particular, calculate analytically the 1-point probability distribution function and the 2-point correlation functions. Some results for Kratzer binding potentials will be presented in Sec. IV. In Sec. V we analyze the breather mode picture and derive a relation between two important scaling functions. Section VI is devoted to the development of a renormalization group theory of 2D critical filling transition, which requires a generalization of previous approaches for critical wetting. A brief conclusion is presented in Sec. VII. II. THE FORMALISM Consider a two-dimensional wedge formed by the intersection of two equal planar substrates at angles ± ␣ with respect to the horizontal (see Fig. 1). We suppose that the wedge is in contact with a bulk vapor phase at saturation conditions, i.e., in equilibrium with the liquid phase, and the substrates preferentially adsorb the liquid phase. Our starting point is the effective interfacial Hamiltonian for shallow wedges: ␤ H关l兴= 冕 −X/2 X/2 dx 再 ⌺ 2 冉 dy dx 冊 2+W共y共x兲− ␣ 兩x兩兲 冎 ,共3兲 where y共x兲is the interfacial local height measured with respect to the horizontal, Xis the interfacial horizontal length, kBT⌺is the interfacial stiffness, kBTW共l兲is the local binding potential, and ␤ ⬅1/kBT. We impose periodic boundary conditions at the ends, i.e., y共−X/2兲=y共X/2兲. While the model assumes that the wedge angle is shallow 共tan ␣ ⬇ ␣ 兲, this does not influence the universal properties occurring in the asymptotic critical limit ␪ → ␣ at fixed ␣ . Studies of filling in acute wedges based on more refined interfacial [9]and microscopic, Ising models [10]yield identical results for universal quantities. Defining the local relative height between the vapor-liquid interface and the substrate l共x兲=y共x兲− ␣ 兩x兩, Eq. (3)can be rewritten as [2] ␤ H关l兴=X⌺ ␣ 2 2+ 冕 −X/2 X/2 dx 再 ⌺ 2 冉 dl dx 冊 2+⌺ ␣ 冉 dl dx 冊 关2⍜共x兲−1兴 +W„l共x兲… 冎 ,共4兲 where ⍜共x兲is the Heaviside step function. Integrating by parts to eliminate the term proportional to 共dl/dx兲, the effective Hamiltonian can be expressed as ␤ H关l兴=X⌺ ␣ 2 2+2⌺ ␣ l共X/2兲−2⌺ ␣ l共0兲 + 冕 −X/2 X/2 dx 再 ⌺ 2 冉 dl dx 冊 2+W„l共x兲… 冎 .共5兲 The first two terms in the equation are irrelevant constants for the interfacial properties in the wedge, the third one is the origin of the boost factor that decreases the pinning effect of the binding potential [2], and the fourth one corresponds to the effective Hamiltonian of an equivalent planar interface problem. As the probability distribution of an interfacial configuration is proportional to exp共− ␤ H兲we can relate the wedge and planar probability distributions in a straightforward way. In particular, the n-point wedge correlation functions can be related to 共n+1兲-point correlation functions in the planar case by adding the wedge midpoint position. However, the presence of the boost factor will alter significantly the behavior of the wedge correlation functions with respect to their planar counterparts. Our approach is based on a standard application of transfer-matrix methods [13]. The partition function Z ␲ 共l1,l2,x1,x2兲of the interface with fixed end points 共x1,l1兲 and 共x2,l2兲with x2⬎x1in the presence of a planar substrate is defined as the following path integral: Z ␲ 共l1,l2,x1,x2兲⬅Z ␲ 共l1,l2;x2−x1兲 = 冕 Dlexp 冉 − 冕 x1 x2dx 冋 ⌺ 2 冉 dl dx 冊 2+W共l兲 册 冊 . 共6兲 The partition function, Eq. (6), is the solution of the following Schrödinger equation: 冋 ⳵ ⳵ x+W共l2兲−1 2⌺ ⳵ 2 ⳵ l2 2 册 Z ␲ 共l1,l2;x兲=0, 共7兲 with the initial condition Z ␲ 共l1,l2;0兲= ␦ 共l2−l1兲,共8兲 where ␦ 共x兲is the Dirac delta function. Formally, the partition function can be expressed as Z ␲ 共l1,l2;x兲=兺 i ␺ i ⴱ共l1兲 ␺ i共l2兲exp共−Eix兲,共9兲 where ␺ 共l兲and Eiare the eigenfunctions and eigenvalues of the time-independent Schrödinger equation: ROMERO-ENRIQUE, PARRY, AND GREENALL PHYSICAL REVIEW E 69, 061604 (2004) 061604-2 −1 2⌺ ␺ n ⬙共l兲+W共l兲 ␺ n共l兲=En ␺ n共l兲,共10兲 with appropriate boundary conditions. In the thermodynamic limit Z ␲ ⬃exp共− ␤ fX兲as X→⬁, where ␤ f=⌺共cos ␪ −1兲is the excess free energy per interfacial length. Consequently, Eq. (9)implies that E0= ␤ f, so that in the low contact angle limit, E0⬇−⌺ ␪ 2/2. The n-point distribution functions can be obtained in terms of Z ␲ 共l1,l2;x兲as P ␲ 共1;...;n兲= lim X→⬁ 兿 i=0 n Z ␲ 共li,li+1;xi+1 −xi兲 Z ␲ 共l−X/2,lX/2;X兲 = ␺ 0共l1兲 ␺ 0 ⴱ共ln兲兿 i=1 n−1 Z ␲ 共li,li+1;xi+1 −xi兲eE0共xi+1−xi兲, 共11兲 where i⬅共li;xi兲,xn+1=−x0⬅X/2, and l0=ln+1=lX/2. For n =1, P ␲ 共i兲⬅兩 ␺ 0共li兲兩2. From Eqs. (11)and (9)it is clear that if the distance between two subsets 兵x1,...,xm其and 兵xm+1,...,xn其is much greater than the planar correlation length ␰ 储 ⬅1/共E1−E0兲(with E1the first excited state eigenvalue), the distribution function factorizes and the two subsets become uncorrelated. The n-point wedge distribution functions Pw共1;...;n兲 can be expressed, in general, in terms of 共n+1兲-point planar distribution functions. So, for a set 兵x−m⬍¯⬍x−1⬍0⬍x1⬍¯xn其, they can be expressed as Pw共−m;...;n兲= 冕 0 ⬁ dl0e2⌺ ␣ l0 具0兩e2⌺ ␣ l0兩0典P ␲ 共−m;...; −1;0;1; ... ;n兲 =Pw共−1;1兲P ␲ 共−m;...;n兲 P ␲ 共−1;1兲,共12兲 where 具n兩f共l兲兩m典⬅兰0 ⬁dl ␺ n共l兲f共l兲 ␺ m ⴱ共l兲.If0艋x1⬍¯⬍xn, the expression of Pw共1;...;n兲is slightly simpler: Pw共1;...;n兲= 冕 0 ⬁ dl0e2⌺ ␣ l0 具0兩e2⌺ ␣ l0兩0典P ␲ 共0;1; ... ;n兲 =Pw共1兲P ␲ 共1;...;n兲 P ␲ 共1兲.共13兲 A similar expression is found if x1⬍¯⬍xn艋0. Finally, if x=0 is included in the xset, the wedge n-point distribution function reduces to Pw共−m;...;n兲=e2⌺ ␣ l0 具0兩e2⌺ ␣ l0兩0典P ␲ 共−m;...;n兲.共14兲 Although this approach is general for arbitrary binding potentials, we will restrict ourselves to some special cases. The first case will be contact potentials, in which W共l兲=0 for l⬎0, W共l兲=+⬁for l⬍0 and at the wall the eigenfunctions fulfill the boundary condition [13] 冏 ⳵ ⳵ lln ␺ 共l兲冏l=0 =− ␶ ,共15兲 where ␶ is proportional to the deviation from the critical wetting temperature. For ␶ ⬎0 the contact angle is related to ␶ via ␶ =⌺ ␪ [13]. These potentials can be understood as the limiting case of a square-well binding potential when the well width tends to zero. Its importance is threefold. First, this case corresponds to the filling fluctuation regime, which previous studies show to be the relevant one for potentials which decay faster than 1/l. Second, there is an analytical expression for Z ␲ 共l1,l2;x兲[13]given by Z ␲ 共l1,l2;x兲=冑⌺ 2 ␲ x共e−⌺共l2−l1兲2/2x+e−⌺共l1+l2兲2/2x兲 + ␶ e ␶ 2x/2⌺− ␶ 共l1+l2兲erfc 冉 冑⌺ 2x共l1+l2兲− ␶ 冑x 2⌺ 冊 . 共16兲 Finally, this case can be compared to more microscopic results, such as the exact solutions of the interfacial properties of the corner filling of an Ising model. Another interesting case is the Kratzer binding potential [14] W共l兲=− ␾␪ l+w l2,共17兲 where ␾ =共1+冑1+8⌺w兲/2 and we assume Dirichlet boundary conditions at the origin. Previous studies indicate that this class of binding potentials corresponds to the marginal case between the mean-field and fluctuation-dominated regimes for the critical filling transition. The Laplace transform of Z ␲ 共l1,l2;x兲,Z ˜ ␲ 共l1,l2,E兲is given by [14] Z ˜ ␲ 共l1,l2,E兲 = 冕 0 ⬁ dx eExZ ␲ 共l1,l2;x兲 =冑E0 E⌫ 冋 ␾ 冉 1−冑E0 E 冊 册 ␪ ⌫关2 ␾ 兴W ␾ 冑E0/E, ␾ −1/2共冑−8⌺El⬎兲 ⫻M ␾ 冑E0/E, ␾ −1/2共冑−8⌺El⬍兲,共18兲 where E0=−⌺ ␪ 2/2, l⬎=max共l1,l2兲,l⬍=min共l1,l2兲,⌫共x兲is the Gamma function, and finally M ␬ ,m共z兲and W ␬ ,m共z兲are Whittaker functions, related to confluent hypergeometric functions. III. EXACT RESULTS FOR CONTACT BINDING POTENTIALS In this section we will obtain and analyze some relevant wedge distribution functions for contact binding potentials. In particular, we will revisit the 1-point distribution function (considered previously by our group [12]) and the 2-point height-height correlation function between the midpoint and INTERFACIAL STRUCTURE AT A TWO-DIMENSIONAL…PHYSICAL REVIEW E 69, 061604 (2004) 061604-3 any other interfacial positions. Related quantities as the average interfacial profile 具l共x兲典w, the local roughness ␰ ⬜共x兲, and the correlation length across the wedge ␰ x(see Fig. 1) will be also obtained. Some results are already known for the 1-point distribution functions. The probability distribution function for the midpoint x=0 interfacial height is given by [2] Pw 1共l0; ␪ , ␣ 兲⬅Pw共l0,0兲=2⌺共 ␪ − ␣ 兲e−2⌺共 ␪ − ␣ 兲l0,共19兲 which verifies the remarkable covariance relationship, Eq. (2). For arbitrary x艌0 the 1-point distribution function has the expression [12] Pw共l,x兲=⌺ ␪ e−2⌺ ␪ lerfc 冉 −冑⌺x 2 ␪ +冑⌺ 2xl 冊 +⌺共 ␪ − ␣ 兲e2⌺共 ␣ − ␪ 兲le2⌺ ␣ x共 ␣ − ␪ 兲 ⫻erfc 冉 冑⌺x 2共 ␪ −2 ␣ 兲−冑⌺ 2xl 冊 −⌺ ␣ e−2⌺ ␣ le2⌺ ␣ x共 ␣ − ␪ 兲 ⫻erfc 冉 冑⌺x 2共 ␪ −2 ␣ 兲+冑⌺ 2xl 冊 .共20兲 For x⬍0, we have the symmetry Pw共l,x兲=Pw共l,−x兲, so hereafter we will consider only the case x艌0. The moments 具ln共x兲典wcan be obtained after some algebra. The average interfacial position profile reads 具l共x兲典w=1 2⌺ ␪ +冑x 2 ␲ ⌺e−共⌺ ␪ 2/2兲x+ 冋 ␪ ␪ − ␣ − ␪ ␣ 册 ⫻e2⌺ ␣ x共 ␣ − ␪ 兲 4⌺ ␪ erfc 冉 冑⌺x 2共 ␪ −2 ␣ 兲 冊 + 冋 1 4⌺ ␪ 冉 ␪ ␪ − ␣ + ␪ ␣ −2 冊 − ␪ x 2 册 erfc 冉 冑⌺ ␪ 2 2x 冊 . 共21兲 The wedge excess adsorption ⌫wmeasured with respect to the planar case can be obtained as ⌫w=2共 ␳ l− ␳ g兲 冕 0 ⬁ 冉 具l共x兲典w−1 2⌺ ␪ 冊 dx = ␳ l− ␳ g 2⌺2 冋 1 ␪ 共 ␪ − ␣ 兲2−1 ␪ 3 册 ,共22兲 where ␳ gand ␳ lare the coexistence densities of the vapor and liquid phases, respectively. Close to the filling transition 共 ␪ → ␣ 兲,⌫w⬃2共 ␳ l− ␳ g兲具l共0兲典w 2/ ␣ . The roughness profile ␰ ⬜共x兲(see Fig. 1)is defined as 冑具l2共x兲典w−具l共x兲典w 2, where 具l2共x兲典wis given by 具l2共x兲典w=1 2⌺2 ␪ 2− 冉 −1 ⌺共 ␪ − ␣ 兲−1 ⌺ ␣ +1 ⌺ ␪ + ␪ x 冊 ⫻冑x 2 ␲ ⌺e−共⌺ ␪ 2/2兲x+ 冋 ␪ 2 共 ␪ − ␣ 兲2− ␪ 2 ␣ 2 册 ⫻e2⌺ ␣ x共 ␣ − ␪ 兲 4⌺2 ␪ 2erfc 冉 冑⌺x 2共 ␪ −2 ␣ 兲 冊 − 冋 1 4⌺2 ␪ 2 冉 − ␪ 2 共 ␪ − ␣ 兲2− ␪ 2 ␣ 2+2 冊 − ␪ x 2⌺ 冉 ␣ ␪ − ␣ − ␪ − ␣ ␣ 冊 +x2 ␪ 2 2 册 erfc 冉 冑⌺ ␪ 2 2x 冊 . 共23兲 For general n, the following expression can be obtained by induction: 具ln共x兲典w=具ln典 ␲ 冋 1+1 2 冉 ␪ n 共 ␪ − ␣ 兲n− ␪ n ␣ n 冊 e2⌺ ␣ x共 ␣ − ␪ 兲 ⫻erfc 冉 冑⌺x 2共 ␪ −2 ␣ 兲 冊 册 +Pn共x兲冑x 2 ␲ ⌺e−共⌺ ␪ 2/2兲x +Qn共x兲erfc 冉 冑⌺ ␪ 2 2x 冊 ,共24兲 where 具ln典 ␲ =n!/共2⌺ ␪ 兲nand Pn共x兲and Qn共x兲are polynomials in xof order n−1 and n, respectively. These expressions are only valid if ␪ ⬎ ␣ (for smaller values of ␪ the interface is unbound from the wedge). For x →0, Eq. (20)reduces to Eq. (19). On the other hand, for 兩x兩→⬁,Pw共l,x兲decay to P ␲ 共l兲⬅2⌺ ␪ exp共−2⌺ ␪ l兲. However, the scale over which this decay occurs depends on the value of ␣ .If ␪ 艌2 ␣ , this scale is the planar correlation length ␰ 储 ⬅2/⌺ ␪ 2. However, if ␣ ⬍ ␪ ⬍2 ␣ , the decay length is ␰ F ⬅1/2⌺ ␣ 共 ␪ − ␣ 兲(our notation differs slightly from the one used in Ref. [12]). Note that ␰ Fis always larger than ␰ 储 , and diverges on approaching the filling transition. On the other hand, ␰ Fis related geometrically with the wedge midpoint average interfacial height via ␰ F=具l共0兲典w/ ␣ ⬇具l共0兲典w/tan ␣ for small ␣ . It is amusing to note that Eq. (20)verifies the following differential relation: Pw共l,x兲+ ␰ F 冉 ⳵ Pw共l,x兲 ⳵ x 冊 =L ␲ 共l,x兲 ⬅P ␲ 共l兲+1 ␪ ⳵ ⳵ x 冕 0 ⬁ dl0l0P ␲ 共l0,0;l,x兲,共25兲 where L ␲ 共l,x兲is for contact binding potentials, ROMERO-ENRIQUE, PARRY, AND GREENALL PHYSICAL REVIEW E 69, 061604 (2004) 061604-4 L ␲ 共l,x兲=⌺ ␪ e−2⌺ ␪ lerfc 冉 −冑⌺x 2 ␪ +冑⌺ 2xl 冊 +2 冑⌺ 2 ␲ xe−共冑共⌺x/2兲 ␪ +冑共⌺/2x兲l兲2.共26兲 Note that the right hand side (RHS)of Eq. (25)depends only on the planar properties and, consequently, is independent of ␣ . It can be shown that Eq. (25)is obtained for any binding potential if the left hand side (LHS)is expanded in powers of ␣ and truncated at the lowest-order term, which is independent of ␣ . Consequently, this differential field equation implies an infinite hierarchy of integro-differential relationships for the 2-point planar correlation function. Alternatively, Eq. (25)provides an elegant route to the calculation of any moment of the interfacial height. Multiplying Eq. (25)by arbitrary power of land integrating over all the possible values of l, the following differential equations are obtained: 具ln共x兲典w+ ␰ Fd具ln共x兲典w dx = 冕 0 ⬁ dl lnL ␲ 共l,x兲 ⬅具ln典 ␲ +1 ␪ d dx具l共0兲ln共x兲典 ␲ ,共27兲 where 具¯典wand 具¯典 ␲ mean the average with the wedge and the planar distribution function, respectively. The RHS of Eq. (27)depends only on the planar distribution functions, and consequently decays to 具ln典 ␲ for distances larger than ␰ 储 . Close to the filling transition, ␰ FⰇ ␰ 储 , and we can approximate Eq. (27)for xⲏ ␰ Fby 具ln共x兲典w+ ␰ Fd具ln共x兲典w dx ⬇具ln典 ␲ ,共28兲 which has as a solution 具ln共x兲典w⬇具ln典 ␲ +关具ln共0兲典w −具ln典 ␲ 兴exp 共−x/ ␰ F兲. Taking into account that 具ln共0兲典wⰇ具ln典 ␲ close to the filling transition, the approximate solution can be simplified even further to 具ln共x兲典w⬇具ln共0兲典wexp 共−x/ ␰ F兲 [which is equivalent to set 具ln典 ␲ =0 in Eq. (28)]. These findings are obviously in agreement with Eq. (24)and the asymptotic behavior of Pw共l,x兲for large xand ␪ ⬍2 ␣ [12]. It is interesting to note that the moments obtained from the actual 1-point distribution function are the only solutions of Eq. (27)that (a)decay exponentially within a length scale ␰ 储 for 0⬍ ␣ / ␪ Ⰶ1 and x→⬁;(b)are analytical as a function of ␣ for 0艋 ␣ ⬍ ␪ , in particular, at the disorder point. The existence of the relationship, Eq. (25), from which covariance for the moments of the interfacial position profile at x =0 can be inferred provided the (a)and (b)regularity conditions are fulfilled, leads us to speculate on the existence of a hidden symmetry of the Hamiltonian that explains wedge covariance. However, the nature of such a symmetry (if any) is completely unknown. In the mean-field approximation, the average interfacial position profile for binding potentials characterized by a critical exponent ␣ s=0 fulfills the following generalized covariance relationship [15]: l共x兲=l ␲ 冉 ␪ −冏dl共x兲 dx 冏 冊 ,共29兲 where l共x兲represents the (averaged)interfacial position at x, and l ␲ 共 ␪ 兲is the planar (averaged)interfacial position for a given contact angle ␪ . Making the substitution l共x兲 →具l共x兲典w, it is clear from Eq. (21)that this extended covariance is not verified for x⫽0(even asymptotically when x →0or兩x兩→⬁). However, it is remarkable that there exists an analogous to Eq. (29), given by Eq. (27)for n=1. To finish our discussion about the 1-point distribution functions, we compare our results with computer simulations of the 2D Ising model [11]. Close to the filling transition point, we expect that the approximate solution to Eq. (28)for n=1 will be generalized for arbitrary ␣ to 具l共x兲典w⬇具l典 ␲ cos ␣ + 冉 具l共0兲典w−具l典 ␲ cos ␣ 冊 e−x/ ␰ F,共30兲 where now ␰ Fis defined as 具l共0兲典w/tan ␣ . We have tested this approximation with the simulation results reported in Ref. [11](see Fig. 2). The symbols correspond to the simulation data obtained for a square 64⫻64 Ising lattice with zero bulk magnetic field and boundary magnetic fields +hfor the boundary rows ending at the lower left corner, and −hfor the remaining boundary rows. In this geometry, ␣ = ␲ /4. The temperature is set to T=Tc/2, where Tcis the bulk critical temperature. For this temperature and ␣ the critical filling transition occurs at hc/J=0.606. Figure 2 shows the computer simulation results for h/J=0.595,0.597, and 0.599. We have no direct estimation of 具l典 ␲ . However, we have obtained 具l典 ␲ by fitting the simulation data with 兩x兩艋16 lattice spacings (in order to minimize the effect of the upper left and lower right heterogeneous wedges)to Eq. (30). The best fitting values are, in lattice spacing units, 具l典 ␲ =0.314,0.335, FIG. 2. Comparison between 具l共x兲典wobtained in Ref. [11]by Ising model computer simulations for boundary magnetic fields h/J=0.595 (circles),h/J=0.597 (squares), and h/J=0.599 (diamonds); and the approximation given by Eq. (30)(continuous lines). The Ising model parameters are the following: ␣ = ␲ /4, the temperature T=Tc/2, and the bulk magnetic field Hbulk=0. The boundary magnetic field at the critical filling is hc/J=0.606. The lengths 兩x兩and 具l共x兲典ware measured in lattice spacing units. See text for explanation. INTERFACIAL STRUCTURE AT A TWO-DIMENSIONAL…PHYSICAL REVIEW E 69, 061604 (2004) 061604-5 and 0.436 for h/J=0.595,0.597, and 0.599, respectively. As it can be seen, the fitting to the simulation data is quite good, despite the crude approximations involved in Eq. (30). Now we want to characterize the 2-point correlations, in particular, the correlations between the interfacial position above the wedge midpoint and the corresponding to an arbitrary x, which are given by the following function: 具关l共x兲−具l共x兲典w兴关l共0兲−具l共0兲典w兴典w ⬅具l共x兲l共0兲典w−具l共x兲典w具l共0兲典w =1 2⌺ 冉 ⳵ 具l共x兲典w ⳵ ␣ 冊 .共31兲 Substituting Eq. (21)into Eq. (31), we obtain 具l共x兲l共0兲典w−具l共x兲典w具l共0兲典w =冑x 2 ␲ ⌺ 共2 ␣ − ␪ 兲e−共⌺ ␪ 2/2兲x 2⌺ ␣ 共 ␪ − ␣ 兲+1 8⌺2 ␪ 2 冉 ␪ 2 共 ␪ − ␣ 兲2− ␪ 2 ␣ 2 冊 ⫻erfc 冉 冑⌺ ␪ 2 2x 冊 +e2⌺ ␣ x共 ␣ − ␪ 兲 8⌺2 ␪ 冋 ␪ 共 ␪ − ␣ 兲2+ ␪ ␣ 2 +2⌺共 ␪ −2 ␣ 兲2 ␪ x ␣ 共 ␪ − ␣ 兲 册 erfc 冉 冑⌺x 2共 ␪ −2 ␣ 兲 冊 .共32兲 This function decays exponentially to zero for large x. However, the characteristic correlation length ␰ x(see Fig. 1)depends on ␣ :itis ␰ 储 for ␪ ⬎2 ␣ and ␰ Fif ␣ ⬍ ␪ ⬍2 ␣ . Consequently, the disorder point not only introduces a new length scale for the average interfacial profile, but also for the interfacial fluctuations. IV. RESULTS FOR THE KRATZER BINDING POTENTIALS The Kratzer binding potential [see Eq. (17)] is the borderline between the filling mean-field and filling fluctuation regimes. While not of direct physical significance, it is instructive to consider this case in order to understand the influence of a marginal operator on the critical properties. For such potentials the wedge midpoint probability distribution function also obeys wedge covariance, Eq. (2): Pw 1共l0; ␪ , ␣ 兲=关2⌺共 ␪ − ␣ 兲兴2 ␾ +1l0 2 ␾ ⌫关2 ␾ +1兴exp关−2⌺共 ␪ − ␣ 兲l0兴 =P ␲ 1共l0; ␪ − ␣ 兲.共33兲 It is possible to extend the transfer analysis and obtain exact results for other quantities of interest. Consider, for example, the 1-point probability distribution function Pw共l,x兲. The Laplace transform P ˜ w共l;E兲can be expressed as P ˜ w共l;E兲= 冕 0 ⬁ dl0e⌺共2 ␣ − ␪ 兲l0共2⌺ ␪ 兲2 ␾ +1共l0l兲 ␾ ⌫关2 ␾ +1兴 ⫻exp共−⌺ ␪ l兲Z ˜ ␲ 共l0,l,E−⌺ ␪ 2/2兲,共34兲 where Z ˜ ␲ 共l0,l,E兲is given by Eq. (18). This reduces to P ˜ w共l;E兲=l ␾ 共2⌺ ␪ 兲2 ␾ +1e−⌺ ␪ l ␪ ⌫关2 ␾ +1兴⌫关2 ␾ 兴 ␬ ⌫关 ␾ 共1− ␬ 兲兴 ⫻ 再 冕 0 ⬁ l0 ␾ e⌺共2 ␣ − ␪ 兲l0W ␬␾ , ␾ −1/2 冉 2⌺ ␪ l0 ␬ 冊 ⫻M ␬␾ , ␾ −1/2 冉 2⌺ ␪ l ␬ 冊 − 冕 0 ll0 ␾ e⌺共2 ␣ − ␪ 兲l0 ⫻ 冋 W ␬␾ , ␾ −1/2 冉 2⌺ ␪ l0 ␬ 冊 M ␬␾ , ␾ −1/2 冉 2⌺ ␪ l ␬ 冊 −W ␬␾ , ␾ −1/2 冉 2⌺ ␪ l ␬ 冊 M ␬␾ , ␾ −1/2 冉 2⌺ ␪ l0 ␬ 冊 册 冎 , 共35兲 where ␬ ⬅1/冑1−2E/⌺ ␪ 2. The poles of P ˜ w共l;E兲in the Ereal positive semiaxis are the characteristic inverse length scales across the wedge of Pw共l,x兲. Since the second integral is over a finite interval and the integrand does not diverges in that range, no new length scale emerges from it. For the first integral, we take into account that [16] 冕 0 ⬁ x ␯ −1exp共−px兲W ␬ , ␮ 共ax兲dx =⌫关 ␮ + ␯ + 1/2兴⌫关 ␯ − ␮ + 1/2兴a ␮ +1/2 ⌫关 ␯ − ␬ +1兴共p+a/2兲 ␮ + ␯ +1/2 ⫻2F1 冢 ␮ + ␯ +1 2, ␮ − ␬ +1 2; ␯ − ␬ +1; p−a 2 p+a 2 冣 , 共36兲 where 2F1共a,b,c;x兲is a hypergeometric function. If ␪ ⬎2 ␣ , the integral does not introduce any new characteristic length. However, for ␣ ⬍ ␪ ⬍2 ␣ a new singularity emerges for ⌺共 ␪ −2 ␣ 兲+⌺ ␪ / ␬ =0, i.e., E=2⌺ ␣ 共 ␪ − ␣ 兲=1/ ␰ F. Remarkably, ␰ F has the same expression as for contact binding potentials, and is proportional (but not equal)to 具l共0兲典w/ ␣ . From this it follows that the nonthermodynamic singularity occurring at ␪ =2 ␣ mentioned in the preceding section is not specific to contact potentials. A simple geometrical argument given in Ref. [12]explains why. The most relevant interfacial fluctuations are those where the interface leaves the substrate with a contact angle ␪ (relative to the tilted wall)at an arbitrary substrate point. If ␪ ⬎2 ␣ , the other side of the wedge does not play any role and we can anticipate that the only length scale that controls the 1-point distribution decay is ␰ 储 . However, if ␪ ⬍2 ␣ , the interface will eventually reach the other substrate, and consequently we can expect the geometry to play an important role leading to the emergence of a new length scale. Formally, this nonthermodynamic singularity occurs when the following integrals that arise from the spectral expansion of Z ␲ 共l1,l2;x兲, ROMERO-ENRIQUE, PARRY, AND GREENALL PHYSICAL REVIEW E 69, 061604 (2004) 061604-6 冕 0 ⬁ ␺ 0共l兲exp共2⌺ ␣ l兲 ␺ p ⴱ共l兲,共37兲 become ill defined. There, ␺ p共l兲are the scattering eigenstates with eigenvalues E=p2/2⌺and ␺ 0共l兲is the ground eigenstate. A straightforward WKB asymptotic analysis for the eigenfunctions shows that, for p⫽0, the integrals given by Eq. (37)become ill defined for ␪ ⬍2 ␣ for quite arbitrary choices of binding potential. As ␪ / ␣ decreases, ␰ Fexceeds the intrinsic interfacial length scales 1/共Ei−E0兲, and becomes the true correlation length across the wedge ␰ xat an another disorder point when ␰ F= ␰ 储 (recall that ␰ x= ␰ 储 for ␪ / ␣ larger than the value at the disorder point). For the case of contact binding potentials both nonthermodynamic singularities occur at the same value ␪ =2 ␣ . However, in general, the nonthermodynamic singularities are distinct provided there are at least two bounded eigenstates of Eq. (10). For the pure Coulomb case 共 ␾ =1兲 the second disorder point occurs at ␪ =4 ␣ /3. Close to the new singularity ␰ F −1 we found that Pw共l;E兲⬃ 1 共 ␰ F −1 −E兲1+2 ␾␣ /共2 ␣ − ␪ 兲E ␰ F→1−,共38兲 so Pw共l,x兲behaves asymptotically for large values of xas x2 ␾␣ /共2 ␣ − ␪ 兲exp共−x/ ␰ F兲, provided that ␰ 储 ⬍ ␰ F. A field equation analogous to Eq. (25)can be found for Kratzer potentials. Transfer-matrix calculations for arbitrary binding potentials lead to the relation 具0兩e2⌺ ␣ l兩0典 再 ␪ − ␣ ␪ 冋 Pw共l,x兲+ ␰ F 冉 ⳵ Pw共l,x兲 ⳵ x 冊 册 − 冕 0 ⬁ dl0 ␺ ¯ 0 ⬘共l0兲 ⌺ ␪␺ ¯ 0共l0兲Pw共l0,0;l,x兲 冎 =L ␲ 共l,x兲− 冕 0 ⬁ dl0 ␺ ¯ 0 ⬘共l0兲 ⌺ ␪␺ ¯ 0共l0兲P ␲ 共l0,0;l,x兲,共39兲 where ␺ ¯ 0共l0兲⬅ ␺ 0共l0兲exp共⌺ ␪ l0兲and ␺ ¯ 0 ⬘共l0兲is its derivative with respect to l0[recall that ␺ 0共l0兲is the ground state eigenfunction]. For Kratzer potentials, ␺ ¯ 0共l0兲⬀l0 ␾ ,soEq.(39)can be expressed as 冉 ␪ ␪ − ␣ 冊 ␾ ⳵ Pw共l,x兲 ⳵ ␣ + ⳵ ⳵ ␣ 冋 ␰ F 冉 ␪ ␪ − ␣ 冊 ␾ ⳵ Pw共l,x兲 ⳵ x 册 =0. 共40兲 As for the contact binding potential case, some interesting quantities can be evaluated from this expression. For example, the wedge adsorption is found to be ⌫w=共2 ␾ +1兲共 ␾ +1兲⌫CP,共41兲 where ⌫CP is the adsorption corresponding to the contact binding potential, Eq. (22). V. THE BREATHER MODE PICTURE In order to understand the origin of the new correlation length ␰ Fwe identified in previous sections, we recall the definition of the 2-point distribution function for x2⬎x1艌0, Eq. (13). This expression can be written in the following way: Pw c共l2,x2兩l1,x1兲=P ␲ c共l2,x2兩l1,x1兲⬅P ␲ c共l2,x2−x1兩l1,0兲, 共42兲 where Pw c共l2,x2兩l1,x1兲and P ␲ c共l2,x2兩l1,x1兲are, respectively, the wedge and the planar conditional probability of the interface being at a relative height l2from the substrate at x2, provided that the interface is pinned at a relative height l1at x1, defined as Pi c共l2,x2兩l1,x1兲=Pi共l1,x1;l2,x2兲 Pi共l1,x1兲,共43兲 where the subscript iindicates if this probability is considered in the wedge 共i=w兲or in the planar 共i= ␲ 兲geometry. In view of the identity between the wedge and planar conditional probability distribution functions we first consider the case of a planar substrate. The conditional probability can be obtained as P ␲ c共l2,x兩l1,0兲= ␺ 0 ⴱ共l2兲 ␺ 0 ⴱ共l1兲e−共⌺ ␪ 2/2兲xZ ␲ 共l1,l2;x兲.共44兲 For contact binding potentials, Eq. (44)can be written explicitly as P ␲ c共l2,x兩l1,0兲=冑⌺ 2 ␲ xe−⌺共l2−l1+ ␪ x兲2/2x +e−2⌺ ␪ l2 冋 冑⌺ 2 ␲ xe−⌺共l1+l2− ␪ x兲2/2x +⌺ ␪ erfc 冉 冑⌺ 2x共l1+l2− ␪ x兲 冊 册 .共45兲 If l1is very large compared with 具l典 ␲ ⬅1/2⌺ ␪ , we can identify two different behaviors of P ␲ c共l2,x兩l1,0兲as a function of l2(see Fig. 3).Ifx⬍l1/ ␪ , the conditional probability is basically the free interface conditional probability that fluctuates around an average value 具l2共x兲典=l1− ␪ x, with a standard deviation of the order of 冑x/⌺. For x⬎l1/ ␪ , the conditional probability becomes the 1-point planar distribution function P ␲ 共l2兲=2⌺ ␪ exp共−2⌺ ␪ l2兲, completely uncorrelated to the value of l1. The transition between the two regimes occur in an xinterval around xt=l1/ ␪ which has a width of the order of 冑2l1/⌺ ␪ 3⬅冑xt ␰ 储 . These results are confirmed by the exact evaluation of the first moments of the conditional probability: 具l2 n典c共l1,x兲= 冕 0 ⬁ dl2l2 nP ␲ c共l2,x兩l10兲.共46兲 The average conditional interfacial profile, which corresponds to n=1, is given by INTERFACIAL STRUCTURE AT A TWO-DIMENSIONAL…PHYSICAL REVIEW E 69, 061604 (2004) 061604-7 具l2典c共l1,x兲=共l1− ␪ x兲+冑⌺ 2 ␲ xe−⌺共l1− ␪ x兲2/2x + 冋 1 4⌺ ␪ −l1−x ␪ 2 册 erfc 冉 冑⌺ 2x共l1− ␪ x兲 冊 −e2⌺ ␪ l1 4⌺ ␪ erfc 冉 冑⌺ 2x共l1+ ␪ x兲 冊 共47兲 and the conditional roughness ␰ ⬜ c共l1,x兲is defined as 冑具l2 2典c−共具l2典c兲2, where 具l2 2典c共l1,x兲can be written as 具l2 2典c共l1,x兲= 冋 共l1− ␪ x兲2+x ⌺ 册 − 冉 1 ⌺ ␪ −l1+ ␪ x 冊 ⫻冑⌺ 2 ␲ xe−⌺共l1− ␪ x兲2/2x+ 冋 x 2⌺−1 4⌺2 ␪ 2 +共l1−x ␪ 兲2 2 册 erfc 冉 冑⌺ 2x共l1− ␪ x兲 冊 + 冋 x ␪ +l1−1 2⌺ ␪ 册 e2⌺ ␪ l1 2⌺ ␪ erfc 冉 冑⌺ 2x共l1+ ␪ x兲 冊 . 共48兲 We obtain two main conclusions from these results when l1Ⰷ具l典 ␲ . First, the interfacial positions are highly correlated to the central one for 兩x兩⬍l1/ ␪ . Second, the intrinsic interfacial fluctuations are small in this xrange compared to the conditional average value. Actually, if we set l1as the length scale, the rescaled conditional probability distribution function P ˜ ␲ c共l2/l1,x/l1兩1,0兲⬅l1P ␲ c共l2,x兩l1,0兲behaves as P ˜ ␲ c共l2/l1,x/l1兩1,0兲→ ␦ 冉 l2−l1+ ␪ x l1 冊 ⍜共l1− ␪ x兲 + ␦ 冉 l2 l1 冊 ⍜共 ␪ x−l1兲,共49兲 when ⌺ ␪ l1→⬁. We expect this result to be valid for any potential and also for random bond disorder, since in all cases the wandering exponent for the free interface ␨ ⬍1. This can be checked for the marginal 1/lpotential. The Laplace transform of the conditional probability distribution is L关P ␲ c共l2,x兩l1,0兲兴 ⬅ 冕 0 ⬁ dx eExP ␲ c共l2,x兩l1,0兲 = ␺ 0 ⴱ共l2兲 ␺ 0 ⴱ共l1兲Z ˜ ␲ 共l1,l2,E−⌺ ␪ 2/2兲.共50兲 For ⌺→⬁at fixed E, ␪ ,l1, and l2, and taking into account Eq. (18)and that the ground state eigenfunction ␺ 0共l兲 ⬀l ␾ exp共−⌺ ␪ l兲, we obtain the following behavior for the Laplace transform of the conditional probability distribution function: L关P ␲ c共l2,x兩l1,0兲兴 →1 ␪ ⍜共l1−l2兲eE共l1−l2兲/ ␪ − ␦ 共l2兲 EeEl1/ ␪ . 共51兲 The Laplace transform can be inverted, leading to Eq. (49). To proceed, we return to our discussion about the wedge geometry. Due to the presence of the boost factor exp共2⌺ ␣ l兲 in the midpoint probability distribution function, the midpoint interfacial height is almost always further from the substrate than the mean wetting layer thickness 具l典 ␲ for any binding potential. If we assume that the conditional probability distribution function is given by Eq. (49), which corresponds to neglecting the intrinsic interfacial fluctuations around the conditional interfacial profile, we can capture the main features of both the average interfacial profile and the correlations along the wedge for contact binding potentials. Actually, this picture is completely equivalent to the 2D wedge breather mode model [3,4]. The average interfacial profile can be written as 具l共x兲典w= 冕 0 ⬁ dl1Pw共l1,0兲 冋 冕 0 ⬁ dl2l2P ␲ c共l2,x兩l1,0兲 册 ⬇ 冕 ␪ x ⬁ dl1Pw共l1,0兲共l1− ␪ x兲 = 冕 0 ⬁ sPw共s+ ␪ x,0兲ds.共52兲 The behavior of 具l共x兲典wfor large xis dominated by the large lasymptotics of Pw共l,0兲. The latter can be obtained by taking into account Eq. (14)for m=n=0 and making use of the WKB approximation for the 1-point planar distribution function: FIG. 3. Illustration of a typical interfacial configuration pinned at l1Ⰷ具l典 ␲ for x=0 (thin continuous line). We have set ⌺=1 (it defines the length scale), ␪ =0.2, and l1=500. The thick continuous line corresponds to the conditional average profile 具l2典c共l1,x兲, and the dotted lines correspond to max(0,具l2典c共l1,x兲±3 ␰ ⬜ c共l1,x兲), where ␰ ⬜ c共l1,x兲is the conditional roughness. Any interfacial configuration has a probability of at least 95% of being between the dotted lines. Inset: an enlargement of the area around xt=l1/ ␪ . Other characteristic length scales are represented. See text for explanation. ROMERO-ENRIQUE, PARRY, AND GREENALL PHYSICAL REVIEW E 69, 061604 (2004) 061604-8 P ␲ 共l兲⬃ 1 冑1+2W共l兲 ⌺ ␪ 2 exp 冉 −2⌺ ␪ 冕 ldt冑1+2W共t兲 ⌺ ␪ 2 冊 ⬃e−2⌺ ␪ lexp 冉 −2 冕 ldtW共t兲 ␪ 冊 ,l→⬁.共53兲 The first thing we can see is that, for large x, the decay of 具l共x兲典win this approximation is controlled by an exponential term exp关−2⌺ ␪ 共 ␪ − ␣ 兲x兴. So, a new length scale ␰ F ⴱis defined as 1/2⌺ ␪ 共 ␪ − ␣ 兲. Close to the filling transition, ␰ F ⴱ= ␰ F −1/2⌺ ␣␪ ⬃ ␰ F+O共1兲. Depending on the large lbehavior of the (attractive)binding potentials, different situations can arise [5]. The filling mean-field regime is characterized by binding potentials that decay to zero as 1/lpwhere p⬍1/ ␨ −1, implying ␨ ⬍1 for thermal disorder (the wandering exponent ␨ =1/2). A saddle point calculation shows that close to the filling transition 具l共0兲典w⬃1/⌺共 ␪ − ␣ 兲p.As ␪ → ␣ , the relevant length scale in the xdirection, 具l共0兲典w/ ␪ Ⰷ ␰ F ⴱ, so the latter length scale is irrelevant (in fact, intrinsic interfacial fluctuations that we neglected can be more important). For p=1, both length scales become of the same order, and consequently 具l共x兲典w⬃具l共0兲典wf共x/ ␰ F ⴱ兲exp共−x/ ␰ F ⴱ兲, where f共x兲diverges at most algebraically, and depends on the detailed structure of the binding potential through the short distance ldependence of Pw共l,0兲. For a pure 1/lpotential, f共x兲=共1+2x/3+x2/6兲. This expression verifies the differential equation for 具l共x兲典wthat arises from Eq. (40)in the scaling limit. The filling fluctuation regime corresponds to potentials with p⬎1, and is characterized by universal critical exponents and scaling functions. Indeed in the critical regime the scaling behavior is the same as that found for contact binding potentials. For x→⬁, we find that asymptotically 具l共x兲典w ⬃具l共0兲典wexp共−x/ ␰ F ⴱ兲. This solution agrees with the asymptotics of 具l共x兲典wfor contact binding potentials when ␪ → ␣ , although with a decay length slightly smaller. However, the behavior is asymptotically correct if we assume that ␰ F ⴱ⬅ ␰ F. For the correlation functions, we have 具l共x兲l共0兲典w−具l共x兲典w具l共0兲典w= 冕 0 ⬁ dl1l1Pw共l1,0兲⌬共l1,x兲, 共54兲 where ⌬共l1,x兲is defined as ⌬共l1,x兲= 冕 0 ⬁ dl2l2关P ␲ c共l2,x兩l1,0兲−Pw共l2,x兲兴.共55兲 In the breather mode approximation, ⌬共l1,x兲can be obtained as ⌬共l1,x兲⬇共l1− ␪ x兲⍜共l1− ␪ x兲−具l共x兲典w.共56兲 We find different behaviors depending on the value of p.In the filling mean-field regime, ⌬共l1,x兲is negligible in this scale. For the filling fluctuation regime, the correlation function decays as 具l共x兲l共0兲典w−具l共x兲典w具l共0兲典w⬃具l共0兲典w 2 冉 1+ x ␰ F ⴱ 冊 e−x/ ␰ F ⴱ, 共57兲 and again is in agreement with the behavior of the exact correlation function for contact binding potentials, Eq. (32), when x→⬁and ␪ → ␣ (assuming again that ␰ F ⴱ⬅ ␰ F). Finally, for the marginal case p=1 the behavior of the correlation function is predicted to be for x→⬁as 具l共0兲典w 2g共x/ ␰ F ⴱ兲exp共−x/ ␰ F ⴱ兲, where g共x兲is a function that diverges at most algebraically. Another quantity of interest is the midpoint local susceptibility ␹ w共l兲defined as ␹ w共l兲=冏 ⳵␳ 共l兲 ⳵ h冏h=0 =2共 ␳ l− ␳ v兲 冕 l ⬁ dsPw共s,0兲⌬共s兲, 共58兲 where ⌬共l兲⬅兰0 ⬁dx⌬共l,x兲. In the breather mode approximation and in the filling fluctuation regime, ⌬共s兲has the following expression: ⌬共l兲=1 ␪ 冉 l2 2−具l共0兲典w 2 冊 ,共59兲 which is exact for contact binding potentials. This expression, together with the midpoint wedge covariance, Eq. (2), leads to the covariance relationship between the local susceptibilities [5]: ␹ w共l; ␪ , ␣ 兲= ␪ − ␣ ␪ ␹ ␲ 共l, ␪ − ␣ 兲,共60兲 where ␹ ␲ 共l, ␪ 兲is the local susceptibility corresponding to the planar geometry for a contact angle ␪ . Finally, we note that the breather mode picture has direct consequences for the scaling of the interfacial profile in the filling fluctuation regime. To see this, recall that the wedge midpoint probability distribution function scales as [5] Pw共l兲=1 具l共0兲典w ⌳ 冉 l 具l共0兲典w 冊 ,共61兲 where ⌳共s兲is a universal function and, due to covariance, is the same as the scaling function for the corresponding planar 1-point probability distribution function. Complementing the scaling of the probability distribution function is the position dependence of the interfacial profile, which we anticipate satisfies 具l共x兲典w=具l共0兲典w ␾ 冉 ␪ x 具l共0兲典w 冊 ,共62兲 where ␾ 共s兲is another universal function. In the breather mode picture, the interface is infinitely stiff in the filled region implying that the scaling functions ⌳共s兲and ␾ 共s兲are related via INTERFACIAL STRUCTURE AT A TWO-DIMENSIONAL…PHYSICAL REVIEW E 69, 061604 (2004) 061604-9