Surface and capillary transitions in an associating binary mixture model

Surface and capillary transitions in an associating binary mixture model J. M. Romero-Enrique,1,2,*L. F. Rull,2and U. Marini Bettolo Marconi3 1Department of Mathematics, Imperial College of Science, Technology and Medicine, 180 Queen’s Gate, London SW7 2BZ, United Kingdom 2Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear, Area de Fı ´sica Teo ´rica, Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain 3Dipartimento di Fisica and Istituto Nazionale di Fisica della Materia, via Madonna delle Carceri, Universita `di Camerino, 62032 Camerino, Italy 共Received 3 December 2002; published 11 April 2003兲 We investigate the phase diagram of a two-component associating fluid mixture in the presence of selectively adsorbing substrates. The mixture is characterized by a bulk phase diagram that displays peculiar features such as closed loops of immiscibility. The presence of the substrates may interfere with the physical mechanism involved in the appearance of these phase diagrams, leading to an enhanced tendency to phase separate below the lower critical solution point. Three different cases are considered: a planar solid surface in contact with a bulk fluid, while the other two represent two models of porous systems, namely, a slit and an array on infinitely long parallel cylinders. We confirm that surface transitions, as well as capillary transitions for a large surface area to volume ratio, are stabilized in the one-phase region. Applicability of our results to experiments reported in the literature is discussed. DOI: 10.1103/PhysRevE.67.041502 PACS number共s兲: 64.75.⫹g, 05.70.Np, 68.08.Bc, 68.35.Rh I. INTRODUCTION In the last few years there has been a considerable progress in the understanding of the physics of fluids in restricted geometries 关1兴. The properties of liquids or gases confined in narrow pores appear different from those in the bulk. A common trend of these systems is the fact that condensation occurs at pressures different from the saturation bulk pressure and the critical temperature for phase separation is generally lower than the corresponding bulk temperature. According to Nakanishi and Fisher 关2–5兴, an incompressible fluid mixture 共with isotropic interactions兲under the effect of confinement displays a reduced tendency toward phase separation at a fixed temperature, because the effective attractive forces result weaker within a pore. The confinement also determines an effective dimensional reduction thus enhancing fluctuation effects. In addition, the presence of adsorbing walls may induce a surface transition in a subcritical fluid, even when this is in a single-phase region in the bulk. Such a behavior, which is now well understood by means of a mean-field approach 关6–8兴is due essentially to the reduction of the contribution to the free energy from the cohesive fluid-fluid forces caused by the presence of the confining walls. There exists, however, a class of nonsimple fluid mixtures for which the situation is different: these are the strongly associating mixtures. Strong directional attractive interactions such as hydrogen bonding or charge transfer complexing between molecules affect dramatically the properties of the fluid or solid. An interesting phenomenon that occurs in such a mixture is the reentrant miscibility: the mixture is completely miscible at low temperatures, up to a temperature where it separates into two liquid phases. As the temperature is further increased, the complete miscibility reappears. The nicotine-water mixture is a paradigm for such a behavior 关9兴. The temperature-composition phase diagram shows closed loops of immiscibility, characterized by the existence of a lower and an upper critical solution point 共LCSP and UCSP, respectively兲. Hirschfelder et al. 关10兴proposed a mechanism to explain the reentrant miscibility. The essential ingredient is the existence of highly directional strong attraction 共as the hydrogen bond兲between the unlike particles. The dispersive 共isotropic兲forces between these particles are assumed to be so weak as to favor segregation of the species. At low temperatures, the energy favors complete mixing. However, as the temperature increases, the energy decrement due to the formation of directional bonds competes with the decrease of orientational entropy. As a consequence, the system phase separates when the temperature raises. This mechanism has been confirmed by theoretical studies on lattice 关11–13兴and continuous models 关14兴,as well as in recent computer simulation studies 关15,16兴. How the presence of substrates or the confinement in a pore affect the behavior of these mixtures 共if in a manner similar to that of simple fluids, or if on the contrary these mixtures are peculiar兲is an issue of great practical and theoretical interest. The presence of selectively adsorbing substrates can interfere with the formation of bulk hydrogen bonding, leading to a stabilization of the two-phase region with respect to the completely miscible phase. Computer simulation studies on confined lattice models 关17兴show the appearance of surface transitions in a range of temperatures below the LCSP. The same mechanism has been invoked to explain some experiments 关18,19兴that seemed to contradict the Nakanishi-Fisher picture. In the present paper we shall study a lattice model of associating binary mixtures, introduced few years ago by Lin and Taylor 关20,21兴and reconsidered recently by the present authors 关22–25兴. In spite of its idealized nature, the model has the merit of allowing for an explicit analysis of its non- *Corresponding author. Email address: jose.enrique@ imperial.ac.uk PHYSICAL REVIEW E 67, 041502 共2003兲 1063-651X/2003/67共4兲/041502共11兲/$20.00 ©2003 The American Physical Society67 041502-1 trivial phase diagram. As an instance, the study of its bulk properties predicts the existence of closed loops of immiscibility under constant pressure conditions. This analysis will be extended to the behavior of the mixture in presence of substrates and under confinement. The structure of the paper is organized as follows. In Sec. II we briefly present the model and write explicitly the equations for the order parameter profiles. Section III is devoted to analyze in detail the wetting behavior of the mixture in the presence of adsorbing substrates. In Sec. IV we study the capillary behavior of the same system when confined between two parallel plates 共a slit兲and in Sec. V we consider the same system confined in a network of very narrow cylinders. Capillary transitions 共and surface transitions for the slab geometry兲will be analyzed in terms of the physical parameters of the system. Finally, in Sec. VI we present our conclusions. II. THE MODEL Some years ago, Lin and Taylor proposed a lattice model to describe a binary mixture, in which the two unlike species can form highly directional bonds. In more detail, the LinTaylor 共LT兲model 关20,21兴is an A-Bbinary mixture lattice model, defined on a generic D-dimensional lattice of coordination number ␯ . The Aparticles are allowed to singly occupy any of the lattice cells. Every cell contains ␯ equal subcells, one for each face of the mother cell. Again, each subcell can host at most one Bparticle, provided the mother cell is not occupied by an Aparticle. In addition to the repulsive interactions 共which determine the previous occupation rules兲, one includes short-ranged interactions 共see Fig. 1兲. The interaction strength between a pair of nearestneighbor Aparticles is ⑀ AA , ⑀ AB between unlike species and ⑀ BB between B-Bparticles. All these interactions are nonvanishing only if the particles share a face. The bulk properties, as well as the global phase diagram, have been exhaustively analyzed in previous work 关22–25兴. One of the main conclusions of such studies is that the appearance of closed loops of immiscibility occurs when the interaction between A-Aand A-Bpairs are attractive ( ⑀ AA ⬍0, ⑀ AB⬍0), and ⑀ ABⱗ关 ⑀ AA⫹min( ⑀ BB,0)兴/2. The latter condition is in complete agreement with the mechanism proposed by Hirschfelder, Stevenson, and Eyring 关10兴for the reentrant miscibility. In this work, we assume the bulk fluidfluid parameters to have values: ⑀ AA⫽⫺1共it defines the energy scale兲, ⑀ AB⫽⫺0.65, and ⑀ BB⫽0, so that we are in the regime where closed loops of immiscibility appear. The presence of substrates introduces new interactions 共see Fig. 1兲. We shall assume that the substrates do not affect the fluid-fluid interactions, i.e., there is no surface enhancement. The interaction strength between an Aparticle and the substrate is VA, if the particle is in a cell in contact with the substrate, and zero otherwise. The interaction between the B particles and the substrate is orientationally dependent, i.e., its strength VBis nonvanishing only if the face of the B particle touches the substrate. The form of the substrate-fluid interactions are consistent with the bulk interactions. Hereafter, the substrate-fluid interactions will be assumed as attractive, i.e., VA⭐0 and VB⭐0. For absolute values of VBlarge enough, the substrate-Binteractions compete with the bulk A-Binteractions, so that one expect this type of substrate to enhance the demixing tendency for temperatures lower than the corresponding LCSP. To study the LT model in presence of substrates, we consider the grand-canonical ensemble, where the system is in thermal and chemical equilibrium with a reservoir at fixed temperature Tand chemical potentials ␮ Aand ␮ B. The activities zAand zBare defined via zA⬅exp( ␤␮ A) and zB ⬅exp( ␤␮ B), where ␤ ⫽1/kBT. After tracing over all configurations of the Bparticles, the system results to be isomorphic to a monocomponent lattice gas, with renormalized interactions and chemical potential. The details on this procedure can be found in Refs. 关21,23,25兴. The grand-canonical partition function ⌶of the LT model can be written in terms of an equivalent Ising model as ⌶⫽共1⫹2zB⫹zB 2e⫺ ␤⑀ BB兲 ␯ N/2 exp 冋 N 冉 H⫺ ␯ K 2 冊 册 ⫻ 冉 1⫹zBe⫺ ␤ VB 冑 1⫹2zB⫹zB 2e⫺ ␤⑀ BB 冊 ␯ ⬜Nb ⫻exp 冋 Nb 冉 ⌬H1⫹ ␯ ⬜K 2 冊 册 ZD, ␯ Ising共K,H,⌬H1兲,共1兲 where Nis the total number of lattice cells, Nbis the number of cells adjacent to the substrate, ␯ ⬜is the number of nearest neighbors in the direction perpendicular to the substrate; ␯ 储 is defined as the number of nearest-neighbor cells in directions parallel to the substrate. Obviously, ␯ ⫽ ␯ 储 ⫹2 ␯ ⬜. Finally, ZD, ␯ Ising(K,H,⌬H1) is the canonical partition function of the Ising model on the same lattice. Let us name Kthe effective coupling constant, Hthe effective bulk magnetic field, and ⌬H1the effective surface magnetic field acting only on the boundary sites, K⫽⫺ ␤⑀ AA 4⫹1 2ln 冉 冑 1⫹2zB⫹zB 2e⫺ ␤⑀ BB 1⫹zBe⫺ ␤⑀ AB 冊 ,共2兲 FIG. 1. Schematic representation of the A-BLT model in the two-dimensional square lattice 共the underlying lattice is not shown for clarity兲. Relevant fluid-fluid and fluid-substrate interactions are also highlighted. 共See text for explanation.兲 ROMERO-ENRIQUE, RULL, AND BETTOLO MARCONI PHYSICAL REVIEW E 67, 041502 共2003兲 041502-2 H⫽1 2lnzA⫺ ␯ ␤⑀ AA 4⫺ ␯ 4ln共1⫹2zB⫹zB 2e⫺ ␤⑀ BB兲,共3兲 ⌬H1⫽ ␯ ⬜ 冋 1 2ln 冉 冑 1⫹2zB⫹zB 2e⫺ ␤⑀ BB 1⫹zBe⫺ ␤ VB 冊 ⫹ ␤ 4 冉 ⑀ AA⫺2VA ␯ ⬜ 冊 册 . 共4兲 All equilibrium properties can be obtained from the knowledge of the partition function Eq. 共1兲by using standard thermodynamical relationships. In particular, the bulk pressure is obtained as p⫽lim V→⬁ kBTln⌶ V⫽kBT v0 冋 ␯ 2ln共1⫹2zB⫹zB 2e⫺ ␤⑀ BB兲⫺ ␯ K 2 ⫹H⫹lim N→⬁ 1 NlnZD, ␯ Ising 册 ,共5兲 where v0is the volume of lattice cell, and N⬅V/v0. The mole fraction XA(i) profile is obtained as XA共i兲⬅nA共i兲 nA共i兲⫹兺 s⫽1 ␯ nB共i,s兲 ,共6兲 where nA(i), the A-particle density in the cell i, is defined in terms of the local magnetization mias nA共i兲⫽1⫹mi 2v0共7兲 and nB(i,s), the Bparticle density in the subcell sof the cell i,is nB共i,s兲⫽1⫺mi 2v0 zBe⫺ ␤ VB 1⫹zBe⫺ ␤ VB,共8兲 if the subcell is by the substrate 共see Fig. 1兲. Otherwise, nB(i,s) is defined as nB共i,s兲⫽1 4v0 冋 zBe⫺ ␤⑀ AB 1⫹zBe⫺ ␤⑀ AB 共1⫺mi⫹mj⫺ 具 sisj 典 兲 ⫹zB⫹zB 2e⫺ ␤⑀ BB 1⫹2zB⫹zB 2e⫺ ␤⑀ BB 共1⫺mi⫺mj⫹ 具 sisj 典 兲 册 , 共9兲 where 具 sisj 典 is the two-spin correlation function between the magnetization on the cell iand its nearest-neighbor jthat it is associated to the subcell s. It is easy to see that in the homogeneous case these equations reduce to the expressions given in Refs. 关23,25兴. The equivalence between the LT model and the Ising model, allows one to obtain the surface behavior of the mixture from the knowledge of the behavior of the latter. However, such a mapping, is not at all trivial and may lead in some cases to unexpected results. III. WETTING TRANSITIONS IN THE LT MODEL Let us consider an A-Bbinary mixture in the presence of a planar, infinite, and structureless substrate. Such a substrate confines the fluid to the half-space z⭓0. When the fluid 共e.g., the A-particle rich phase, ␣ ) is at bulk coexistence, the substrate can promote the formation of drops of the opposite coexisting phase 共the B-particle rich phase ␤ ) on it, characterized by a contact angle ␪ . When ␪ ⫽0, the drops spread over the substrate, i.e., the phase ␤ wets completely the ␣ -substrate interface. A finite value of ␪ , instead, corresponds to a partial wetting situation. The crossover between a partial to a complete wetting regime is called the wetting transition 关26兴. The surface tensions involved in the problem ( ␴ ␣ s, ␴ ␤ sand ␴ ␣␤ , corresponding to the ␣ -substrate, ␤ -substrate, and ␣ - ␤ interfaces, respectively兲are related to the contact angle via the Young equation 关27兴 ␴ ␣ s⫽ ␴ ␤ s⫹ ␴ ␣␤ cos ␪ .共10兲 Thermodynamically the surface tensions correspond to the excess grand-canonical-free energy per interfacial unit area 共respect to the bulk兲关27兴. Since both ␴ ␣␤ and ␴ ␤ shave an analytical behavior at this thermodynamic state, the wetting transition is a 共surface兲thermodynamic phase transition, as it can be seen from Eq. 共10兲and the dependence of ␪ on the temperature. As usual, appropiate order parameters corresponding to the wetting transition can be defined as first derivatives of the surface tension respect to thermodynamic fields. In simple fluids, the excess number of particles per interfacial unit area or adsorption ⌫关that corresponds to ⫺( ⳵␴ ␣ s/ ⳵ ␮ )T, ␮ being the chemical potential兴is the natural order parameter choice. In mixtures, other choices are available, e.g., the volume integrated excess mole fraction per interfacial unit area. Microscopically, the wetting transition manifests itself as the growth of a macroscopically thick ␤ layer intruding in the ␣ -substrate interface composition profile. Three different types of wetting transition may arise. If the layer width jumps suddenly from a microscopically finite value, the wetting transition is of first order, and an off-coexistence surface transition 共prewetting兲is associated to it. If the layer width diverges continuously, the wetting transition is named critical. Finally, the wetting transition can occur as the accumulation point of an infinite sequence of first-order transitions in the layering wetting case 共these transitions extend to the off-coexistence region兲. The conditions for such a transition depend subtly on the fluid-fluid and substrate-fluid interactions, and interfacial fluctuations can play an important role 关28兴. In the LT model two-phase coexistence occurs when H ⫽0 and K⬎Kc, where Kcis the lattice dependent Ising critical coupling. Moreover, the wetting behavior is controlled by the ratio ⌬H1/ ␯ ⬜K, that is the ratio between the surface energy and the loss of bulk energy due to the presence of the surface. If such a parameter is positive the substrate favors the ␣ phase to intrude between the substrate and SURFACE AND CAPILLARY TRANSITIONS IN AN . . . PHYSICAL REVIEW E 67, 041502 共2003兲 041502-3 the ␤ phase. On the other hand, if ⌬H1/ ␯ ⬜K⬍0, the phase ␤ intrudes between the substrate and the phase ␣ . We shall focus on the coexistence side H⫽0⫹(H⫽0⫺) for ⌬H1 ⬍0(⌬H1⬎0), respectively 共otherwise, only partial wetting can occur兲. If the absolute value of the parameter is greater than one ( 兩 ⌬H1/ ␯ ⬜K 兩 ⬎1), then all coexistence surface corresponds to a complete wetting situation. If such a condition is not fulfilled, one will observe a partial wetting situation up to a value Kw共dependent on 兩 ⌬H1/ ␯ ⬜K 兩 ) and beyond it a complete wetting situation up to K⫽Kc. Let us consider first the case VB⫽0, when ⌬H1depends only on T, regardless the value of zB. Moreover, the sign of ⌬H1is uniquely determined by the difference between VA and ␯ ⬜ ⑀ AA/2 over the whole temperature range. One can envisage various cases 共see Fig. 2兲:ifVA⫽0, the coexistence corresponds to a situation of complete wetting of the interface substrate- ␣ phase by phase ␤ , denoted by the symbol s 兩 ␤ 兩 ␣ .If ␯ ⬜ ⑀ AA/2⬍VA⬍0, there exists a region of complete wetting s 兩 ␤ 兩 ␣ and a lower pressure region of partial wetting. As VAdecreases, the region of complete wetting s 兩 ␤ 兩 ␣ shrinks and eventually disappears for VA⫽ ␯ ⬜ ⑀ AA/2, because ⌬H1⬅0. For VA⬍ ␯ ⬜ ⑀ AA/2, the sign of ⌬H1changes and a region of complete wetting of the interface substrate-phase ␤ by phase ␣ appears (s 兩 ␣ 兩 ␤ ). As VAincreases, the region of partial wetting reduces and disappears for VA⭓ ␯ ⬜ ⑀ AA . When VB⫽0, it is possible that in correspondence with different state points, the substrate favors either the ␤ or the ␣ phase. To study such an effect it is necessary to locate the coexistence states, where ⌬H1⫽0. Since VBand ⑀ AB are negative, for a temperature Tall the coexistence states with pressure plarger than pnw(T)⬅p(T,H⫽0,⌬H1⫽0) correspond to ⌬H1⬎0, whereas states with lower pressures correspond to ⌬H1⬍0. It is easy to prove from Eq. 共4兲that a necessary condition for the existence of this line is that VB ⫹ ⑀ AA/2⬍(VA/ ␯ ⬜)⭐ ⑀ AA/2. However, ⌬H1changes sign for a temperature Tonly if the value of the pressure which corresponds to ⌬H1⫽0 is less than the critical pressure at such temperature. In order to illustrate such a phenomenology, we shall study the 2D square lattice case ( ␯ ⫽4, ␯ ⬜⫽1). In this case, only critical wetting can occur 共fluctuation effects preclude any other possibility兲and the value of Kw, at which the transition of critical wetting occurs is given by solution of the equation 关29兴 exp共2Kw兲关cosh共2Kw兲⫺cosh共2⌬H1兲兴⫽sinh共2Kw兲. 共11兲 Let us first consider the case ␯ ⬜ ⑀ AA/2⭐VA⬍0: ⌬H1⬍0 for arbitrary values of T,zB, and VB. Moreover, ⌬H1becomes more negative as VBdecreases, at fixed Tand zB共or pressure p). The typical diagrams of complete wetting are displayed in Fig. 3. The region between the wetting transition line and the critical line corresponds to a situation of wetting s 兩 ␤ 兩 ␣ , and the remaining coexistence region corresponds to a situation of partial wetting. The complete wetting zone increases as VBbecomes more negative. When (VA/ ␯ ⬜)⬍ ⑀ AA/2, the substrate also favors the adsorption of particles A. This determines a competition between Aand Bspecies. In general, when (VA/ ␯ ⬜)⫺ ⑀ AA/2 ⬍0.1 ⑀ AA ␤ -phase preferential adsorption can only occur if VB⫺(VA/ ␯ ⬜)⬍ ⑀ AB⫺ ⑀ AA . Figures 4 and 5 display the wetting diagrams for ⑀ AA⬍(VA/ ␯ ⬜)⬍ ⑀ AA/2 and (VA/ ␯ ⬜) ⭐ ⑀ AA , respectively. In the first case, the complete wetting transition line originates at T⫽0 and terminates at the temperature of complete wetting for the A-pure liquid in the interface substrate-Avapor. For values of VB⯝0, ⌬H1⬎0 for all conditions of coexistence, and the configurations are s 兩 ␣ 兩 ␤ . For VB⬍(VA/ ␯ ⬜)⫹ ⑀ AB⫺ ⑀ AA , there appear states characterized by ⌬H1⬍0. As a consequence, within the coFIG. 2. Qualitative wetting phase diagrams for the LT model for VB⫽0. Thick solid lines correspond to critical lines, dotted lines to the A-pure gas-liquid coexistence lines, and thin solid lines to the wetting transition lines. The symbol s 兩 ␣ 兩 ␤ (s 兩 ␤ 兩 ␣ ) means the bulk conditions where there exist wetting of the ␤ -substrate ( ␣ -substrate兲interface by the ␣ ( ␤ ) phase, respectively 共see text兲. The bulk conditions under which there is partial wetting are also shown. FIG. 3. Wetting phase diagram for the square lattice LT model. The bulk parameters are fixed to ⑀ AB⫽0.65 ⑀ AA⬍0, ⑀ BB⫽0, and VA⫽0.25 ⑀ AA . The thick solid line corresponds to the bulk critical line, and the thick dashed line to the liquid-vapor transition line of the A-pure fluid. The thin solid lines correspond to the wetting transition lines for different values of VB. Hereafter, the reduced units are defined via T*⬅kBT/ 兩 ⑀ AA 兩 and p*⬅pv0/ 兩 ⑀ AA 兩 , where v0 is the volume of a lattice cell. 共See explanation in text.兲 ROMERO-ENRIQUE, RULL, AND BETTOLO MARCONI PHYSICAL REVIEW E 67, 041502 共2003兲 041502-4 existence region at the left of the line pnw(T), the substrate adsorbs preferentially particles B, and the wetting is of type s 兩 ␤ 兩 ␣ . On the other hand, at the right of the curve pnw(T), the wetting is of type s 兩 ␣ 兩 ␤ . The line of wetting transition touches tangentially the critical curve at the point where the curve in turn pnw(T) crosses the critical line. Such a point moves monotonically toward higher temperatures as VBdecreases, driving with it the wetting transition line and the pnw(T) curve. As also happened for the ␯ ⬜ ⑀ AA/2⬍VA⬍0 case, the wetting transition line from a s 兩 ␤ 兩 ␣ complete situation to partial wetting moves to lower pressures, converging toward the A-pure liquid-vapor coexistence line for VB→ ⫺⬁共however, along this line there is a complete wetting by liquid of the vapor-substrate interface over all the temperature range兲. When VA⭐ ␯ ⬜ ⑀ AA 共see Fig. 5兲, for values VBnear zero, the coexistence corresponds to complete wetting s 兩 ␣ 兩 ␤ . Only when VB⬍(VA/ ␯ ⬜)⫹ ⑀ AB⫺ ⑀ AA states of partial wetting reappear, near the line pnw(T). The behavior of the latter as well as the wetting transition line is similar to that observed in the previous case, with the difference that the right branch of the wetting transition line 共that corresponds to a transition from partial to s 兩 ␣ 兩 ␤ complete wetting兲begins at zero temperature. One expects that the behavior of the wetting transition for the three-dimensional case will be qualitatively similar to the two-dimensional case. However, the complete wetting phase diagram for the three-dimensional Ising model is not totally understood yet. Theoretical 关30兴and simulation studies 关31兴 show, in the case of strictly short-ranged forces and in the absence of an enhanced surface spin-spin coupling, that the wetting transition is continuous only if Kwis smaller than KR, where KRis the value of the coupling constant corresponding to the roughening transition (KR⬇1.85Kcfor the simple cubic lattice兲. However, if such condition is not fulfilled, the wetting transition will occur through a sequence of first-order layering transitions. Theoretical studies have claimed that the wetting transition can be weakly first order instead of second order 关32兴. In the present work we will consider the Ising model in the framework of the mean-field Bragg-Williams approximation, lim N⬜→⬁ 1 N⬜ lnZIsing共K,H,⌬H1兲⫽⌬H1m1⫹兺 j⫽1 ⬁ 冋 ␯ 储 2Kmj 2⫹ ␯ ⬜Kmjmj⫹1⫹Hmj⫺ 冉 1⫹mj 2 冊 ln 冉 1⫹mj 2 冊 ⫺ 冉 1⫺mj 2 冊 ln 冉 1⫺mj 2 冊 册 ,共12兲 where N⬜is the number of cells in a layer, an 兵 mj 其 ,j ⫽1,2,...⬁is the equilibrium magnetization profile obtained by solving the Euler-Lagrange equations mj⫽tanh共 ␯ 储 Kmj⫹ ␯ ⬜K共mj⫺1⫹mj⫹1兲⫹H兲,共13兲 where m0⬅⌬H1/ ␯ ⬜K. This set of equations must be solved in conjunction with the asymptotic condition mj→mbfor j→⬁, where mbis the spontaneous magnetization per site in the bulk case with coupling constant Kand magnetic field H. In general, different solutions of Eq. 共13兲are found. The equilibrium profile is the solution that maximizes functional 共12兲. This approach neglects capillary fluctuations, that it is equivalent to assume KR⬅Kcand the wetting transition occurs as a sequence of layering transitions 关33,34兴. Once the equilibrium magnetization profile is computed, the surface tension ␴ is obtained via FIG. 4. The same as in Fig. 3 but for VA⫽0.75 ⑀ AA . The dotted lines correspond to the coexistence points at which ⌬H1⫽0. 共See text for explanation.兲 FIG. 5. The same as in Fig. 4 but for VA⫽ ⑀ AA .共See text for explanation.兲 SURFACE AND CAPILLARY TRANSITIONS IN AN . . . PHYSICAL REVIEW E 67, 041502 共2003兲 041502-5 ␴ ⫽共⍀⫹pV兲/A,共14兲 where ⍀⫽⫺kBTln⌶, with ⌶defined by Eqs. 共1兲and 共12兲, and the pressure pby Eq. 共5兲. The adsorption ⌫, that it is the relevant order parameter in the wetting transition, is defined as ⌫⫽兺 j⫽1 ⬁ 共XA共j兲⫺XA b兲,共15兲 where XA(j) is the Amolar fraction profile, Eq. 共6兲, by using the approximation 具 sisj 典 ⬇mimj. Finally, XA bis its bulk value. We have studied the simple cubic lattice, for which ␯ ⫽6 and ␯ ⬜⫽1. The magnetization profiles have been obtained by an iterative method of Eq. 共13兲for j ⫽1,...,Nmax , imposing the condition mj⫽mbfor j ⬎Nmax . In order to sample the solution space, we use sharp kink profiles 关mj⫽⫹1(⫺1) for ⌬H1⬎0(⌬H1⬍0), respectively, if j⭐j0and mj⫽mbfor j⬎j0] as initial magnetization profiles. Starting the iteration with different initial profiles, in general, the algorithm converges toward different local minima of the free energy. As already mentioned, the global free energy minimum gives the true equilibrium magnetization profile. Finally, we have considered different values of Nmax , since the equilibrium profile in a complete wetting situation has a strong size dependence. On the contrary, no appreciable finite width effects occur in a partial wetting situation for Nmax large enough. In order to locate the wetting transition, we studied the equilibrium profiles at bulk coexistence (H⫽0) and constant temperature, varying the pressure. The wetting phase diagram for VA⫽ ⑀ AA and VB ⫽1.25 ⑀ AA is plotted in Fig. 6. As one can see, the qualitative behavior of the wetting transition is similar to that observed in the two-dimensional case. However, the wetting transition is associated to series of first-order layering transitions between surface phases with the same surface tension, but different adsorptions. The behavior of the layering transitions at low temperatures shows surprising features. Figure 7 shows the first four layering transitions for T*⬅kBT/ 兩 ⑀ AA 兩 ⫽0.25. It is observed that the first layering transition does occur for pressures higher than the bulk critical pressure. Consequently, for a fixed pressure between the first layering transition critical point and the bulk critical point corresponding to this temperature, the layering transition will extend to lower temperatures than the critical temperature corresponding to the LCSP. This behavior is found only for s 兩 ␤ 兩 ␣ complete wetting in the low-temperature zone. The pressure is an increasing function on Hfor fixed temperature Tand zB共so both Kand ⌬H1are fixed兲. Consequently, as K⬎Kcfor the layering transition critical point, the corresponding pressure can only be bigger than the bulk critical pressure if the transition occurs for H⬎0. This implies that ⌬H1must be negative, and thus the wetting is of type s 兩 ␤ 兩 ␣ . The physical interpretation of this phenomenon is that the surface adsorbs preferentially Bparticles. Such a mechanism competes with the A-Bbonding 共at least close to the substrate兲. Therefore, the tendency to phase separate in a fluid layer close to the substrate is enhanced 共because the A-Bbonding favors mixing兲for temperatures lower than the 共lower兲critical temperature. This fact was also observed in computer simulations of a different model of associating binary mixture in Ref. 关17兴. However, we must stress that this phenomenon is not related to an enhanced coupling 共i.e., the interactions between particles in the first layer close to the substrate are increased respect the bulk values兲, that it is known to stabilize surface transitions to temperatures higher to the bulk critical one in the Ising model 关2兴. On the other hand, the competition between substrate preferential adsorption and the A-Bbonding should be also relevant for prewetting transitions. IV. CONFINEMENT BETWEEN SYMMETRIC WALLS We turn to consider the behavior of the Lin-Taylor model when confined between two identical parallel substrates. FIG. 6. Wetting phase diagram for the simple cubic 共sc兲LT model in the mean-field approximation. The substrate-fluid interactions are VA⫽ ⑀ AA and VB⫽1.25 ⑀ AA . The critical line 共thick solid兲, A-pure vapor-liquid transition line 共dashed兲, and pnw(T) line 共dotted兲are plotted. The wetting transition points 共diamonds兲are also represented. The line that joins them is only to guide the eye. FIG. 7. Layering transitions for T*⫽0.25 for the sc LT model in the mean-field approach. The interaction couplings are the same as in Fig. 6. The dashed lines correspond to the bulk coexistence, and the solid lines correspond to the bulk conditions corresponding to the layering transitions. In the inset, the coexistence adsorptions for the layering transition 共only the first four transitions are plotted兲. The arrow shows the bulk critical pressure. ROMERO-ENRIQUE, RULL, AND BETTOLO MARCONI PHYSICAL REVIEW E 67, 041502 共2003兲 041502-6 Confinement in the two-dimensional case corresponds to a quasi-one-dimensional situation. Thus fluctuations preclude the existence of capillary transitions if the interactions are short ranged. Consequently, only the 3D case will be studied. We shall consider a simple cubic lattice within the meanfield approximation. The substrate-fluid interactions are set to VA⫽ ⑀ AA and VB⫽1.25 ⑀ AA , that stabilize the first 共surface兲layering transition at temperatures lower than the lower critical temperature. We also assume that the separation between plates is NDlattice units. The free energy of the confined mixture is given by Eq. 共1兲, with the mean-field partition function ZIsing(K,H,⌬H1) given by lim N⬜→⬁ 1 N⬜ lnZIsing共K,H,⌬H1兲⫽⌬H1共m1⫹mND兲⫹兺 j⫽1 ND 冋 ␯ 储 2Kmj 2⫹ ␯ ⬜Kmjmj⫹1⫹Hmj⫺ 冉 1⫹mj 2 冊 ln 冉 1⫹mj 2 冊 ⫺ 冉 1⫺mj 2 冊 ln 冉 1⫺mj 2 冊 册 .共16兲 The equation for the equilibrium profile is given by Eq. 共13兲with the boundary condition Nmax⫽ND/2 for NDeven, or (ND⫹1)/2 for NDodd. The values of mjfor j⭓Nmax are obtained from the symmetry of the problem: i.e., mj ⫽mND⫺j⫹1. In addition to the layering transitions 共related to the phenomenology observed on the semi-infinite case兲, the capillary transitions results from the shift of the bulk coexistence under confinement. Near the center of the pore both phases have different magnetizations, in contrast with the layering and prewetting transitions where the profiles differ in a localized surface region and they have the same values far enough from that region. In the Ising model, the shift is always toward higher values of K共i.e., lower effective temperature兲and Hnegative (Hpositive兲when ⌬H1positive (⌬H1negative兲, respectively. The cases ND⫽1 and ND⫽2 can be studied analytically since mj⫽mfor all j. The bulk conditions, for which the capillary transition takes place are shown in Fig. 8. In spite of the fact that the temperature range for which the capillary transition occurs is reduced, at low temperatures the pressure range is largely increased. Hence, at a given pressure, the LCSP shifts to lower temperatures and can even disappear. This means that for certain values of the pressure range there are no immiscibility islands in the confined system, but the usual bell shaped coexistence line terminating in a UCSP. This is a direct consequence of the directional character of the substrate-particle Binteraction. The kink that is observed in the capillary critical line corresponds to the crossing of the pnw(T) line. For larger values of ND, the behavior of the capillary transitions are studied numerically. As ND→⬁the capillary critical line must move toward the bulk critical curve. We shall consider the isotherm T*⫽0.25, as a representative case of the low-temperature regime. The excess free energy ␻ reads ␻ ⫽共⍀⫹pV兲/A共17兲 and converges to 2 ␴ when ND→⬁, with ␴ defined by Eq. 共14兲. The relevant order parameter for the capillary transitions is the average molar fraction X ¯ A, defined as X ¯ A⬅1 ND兺 j⫽1 ND XA共j兲共18兲 that converges to the bulk coexistence molar fractions as ND→⬁. We have obtained the constant temperature phase diagrams for T*⫽0.25 共see Fig. 9兲. Various transitions are observed for ND⭓3. One of these transitions converges toward the bulk phase transition for large separations and is identified with the true capillary transition. The remaining transitions are characterized by having similar average molar fractions and correspond to the layering transitions observed in the presence of a single substrate. The coverages 关as defined by Eq. 共15兲兴 corresponding to the coexisting phases in these transitions converge very fast to the semi-infinite values, as expected from their surface nature. However, the number of these transitions depends strongly on ND, since the capillary FIG. 8. Capillary transitions of the mean-field sc LT model (VA⫽ ⑀ AA ,VB⫽1.25 ⑀ AA) for ND⫽1共thin solid line兲and ND⫽2 共dot-dashed line兲. In both cases, the upper line is the capillary critical line, and the lower line is the A-pure capillary gas-liquid transition. The bulk conditions under which there is ␣ - ␤ capillary coexistence are the regions enclosed by their respective lines, and the T⫽0 axis. By comparison, the bulk critical line 共thick solid line兲 and the A-pure gas-liquid transition line 共thick dashed line兲. SURFACE AND CAPILLARY TRANSITIONS IN AN . . . PHYSICAL REVIEW E 67, 041502 共2003兲 041502-7 transition, which depends sensibly on ND, competes with them 关35,36兴. For instance, in the case of 3⭐ND⬍10 only the first layering transition appears, while the second appears for ND⭓10. The values of the capillary critical pressure for small ND are larger than the corresponding bulk critical values. By increasing ND, the capillary critical pressure decreases up to a minimum, corresponding to a pressure less than the bulk critical pressure. From such value it converges monotonically to the bulk critical value 共see Fig. 10兲. This fact means that along an isobaric the LCSP shifts to lower temperatures only for very narrow pores. Consequently, the stabilizing mechanism of the coexistence region below the bulk LCSP seems to be only effective for small values of ND. The deviations of the capillary critical pressure with respect to the bulk value in the range 5⬍ND⭐15 follow a power law ⌬p ⬃ND ⫺x, with x⫽1.04⫾0.09. This behavior is consistent with the findings for the confined lattice gas for intermediate values of NDreported in Ref. 关35兴. The scaling behavior ⌬p ⬃ND ⫺1/ ␯ (ND ⫺2in the mean-field approach兲predicted by the Nakanishi-Fisher theory should be obeyed for larger values of ND. However, our results do not extend to that region due to the numerical error in estimating the capillary critical pressure close to the bulk critical point. Nevertheless, we do not expect a change in the tendency for large values of ND and we expect our results to be qualitatively correct for all values of ND. It is interesting to note that the slab approximation 关35兴, which corresponds to set mj⫽mfor all j, predicts that for sufficiently small T*the capillary critical pressure is larger than the bulk value for all values of ND. The failure of this approach is not surprising, since it gives poor results for the confined lattice gas close to the capillary critical point as it overestimates both Kand Hat the capillary critical point 关35兴. This fact alters the balance between the different mechanisms involved in the shift of the critical point, leading to the wrong result mentioned above. V. THE CYLINDRICAL PORE NETWORK Finally, we consider the confinement of the LT mixture in a network of parallel infinitely long, cylindrical pores. This model can be understood as a crude approximation for adsorption in zeolites. It represents another instance in which the substrate can prevent the formation of hydrogen bonds: the geometrical hindrance due to the confinement in very narrow nanopores. We shall assume that the pore diameter is so small that the A-Bbonds can form only along the direction zparallel to the cylinder axis 共see Fig. 11兲. The cylinder diameter is taken to be less than twice the A-Bcollision diameter. In addition, the pores have diameters less than two A-particle diameters and they form a two-dimensional netFIG. 9. X ¯ A⫺pphase diagrams for T*⫽0.25 and different values of ND:ND⫽1共dashed-double dotted line兲,ND⫽2共dashed line兲,ND⫽5共dotted line兲, and ND⫽15 共thick solid line兲, under the same conditions as in Fig. 8. The bulk transition 共thin solid line兲is also plotted for comparison. The inset corresponds to an enlargement of the zone around the bulk critical point. FIG. 10. Capillary reduced critical pressures p*versus NDfor T*⫽0.25 共same conditions as in Fig. 8兲. The dotted line corresponds to the bulk critical pressure. Inset, enlargement of the minimum zone. FIG. 11. Schematic representation of the LT model adsorbed in a network of narrow infinitely long cylinders. The Aparticles are represented as spherical particles, and the Bparticles as spherical particles with an interaction side 共shadow circle兲. In this case, the coordination number is ␯ ⬜⫽6. ROMERO-ENRIQUE, RULL, AND BETTOLO MARCONI PHYSICAL REVIEW E 67, 041502 共2003兲 041502-8 work with coordination number ␯ ⬜. We allow attenuated interactions between particles in nearest-neighbor pores, in order to allow a capillary transition. A similar zeolite model has been employed to study the capillary transition of methane adsorbed in ALPO4-5 关37兴. The in-pore fluid-fluid interactions have the same form as in bulk, with coupling constants ⑀ AA , ⑀ AB , and ⑀ BB . However, the interactions between particles in nearest-neighbor pores are attenuated to ⑀ ˜ AA , ⑀ ˜ AB , and ⑀ ˜ BB ( 兩 ⑀ ˜ ij 兩 ⬍ 兩 ⑀ ij 兩 ), although the A-Band B-Binteractions directional character remains the same as before. This system behaves as a bulk lattice model with anisotropic interactions. The grandcanonical-free energy per unit volume reads ⍀ V⫽⫺ kBT v0 冋 ␯ ⬜ 2ln共1⫹2z ˜ B⫹z ˜ B 2e⫺ ␤⑀ ˜ BB兲 ⫹ln共1⫹2zB⫹zB 2e⫺ ␤⑀ BB兲⫹H⫺ ␯ ⬜K⬜ 2⫺K 储 ⫹1 NlnZIsing共K⬜,K 储 ,H兲 册 ,共19兲 where z ˜ B⬅zBe⫺ ␤ VBis the effective activity of the Bparticles oriented in the plane perpendicular to the cylinder axis, and ZIsing is canonical partition function of an anisotropic Ising model characterized by couplings K 储 in the parallel directions to the cylinder axis and K⬜in the perpendicular directions, and an effective magnetic field H.K 储 has the same expression as the bulk coupling constant, Eq. 共2兲, while K⬜ and Hare defined as K⬜⫽⫺ ␤⑀ ˜ AA 4⫹1 2ln 冉 冑 1⫹2z ˜ B⫹z ˜ B 2e⫺ ␤⑀ ˜ BB 1⫹z ˜ Be⫺ ␤⑀ ˜ AB 冊 ,共20兲 H ˜ ⫽H⫹ ␯ ⬜ 冋 ␤ 4 冉 ⑀ AA⫺ ⑀ ˜ AA⫺2VA ␯ ⬜ 冊 ⫹1 4ln 冉 1⫹2zB⫹zB 2e⫺ ␤⑀ BB 1⫹2z ˜ B⫹z ˜ B 2e⫺ ␤⑀ ˜ BB 冊 册 ,共21兲 with Hgiven by Eq. 共3兲. One can see that confinement shifts the effective magnetic field Hwith respect to its bulk value with a contribution analogous to that found for planar substrates 关compare Eqs. 共4兲and 共21兲兴. In addition, the confinement determines an anisotropy in the couplings. The A-particle molar fraction XAreads XA⫽nA nA⫹nB 1⫹nB 2,共22兲 where nA⫽(1⫹m)/2v0is the A-particle density, and nB 1 (nB 2) is the density of Bparticles oriented in a perpendicular 共parallel兲direction to the cylinder axis, respectively, nB 1⫽ ␯ ⬜ 4v0 冉 z ˜ B⫹z ˜ B 2e⫺ ␤⑀ ˜ BB 1⫹2z ˜ B⫹z ˜ B 2e⫺ ␤⑀ ˜ BB ⫹z ˜ Be⫺ ␤⑀ ˜ AB 1⫹z ˜ Be⫺ ␤⑀ ˜ AB 冊 ⫺ ␯ ⬜ 4v0 冉 z ˜ B⫹z ˜ B 2e⫺ ␤⑀ ˜ BB 1⫹2z ˜ B⫹z ˜ B 2e⫺ ␤⑀ ˜ BB ⫺z ˜ Be⫺ ␤⑀ ˜ AB 1⫹z ˜ Be⫺ ␤⑀ ˜ AB 冊 冉 ⳵ lnZIsing /N ⳵ K⬜ 冊 ⫺ ␯ ⬜ 2v0 z ˜ B⫹z ˜ B 2e⫺ ␤⑀ ˜ BB 1⫹2z ˜ B⫹z ˜ B 2e⫺ ␤⑀ ˜ BB m,共23兲 nB 2⫽1 2v0 冉 zB⫹zB 2e⫺ ␤⑀ BB 1⫹2zB⫹zB 2e⫺ ␤⑀ BB ⫹zBe⫺ ␤⑀ AB 1⫹zBe⫺ ␤⑀ AB 冊 ⫺1 2v0 冉 zB⫹zB 2e⫺ ␤⑀ BB 1⫹2zB⫹zB 2e⫺ ␤⑀ BB ⫺zBe⫺ ␤⑀ AB 1⫹zBe⫺ ␤⑀ AB 冊 冉 ⳵ lnZIsing /N ⳵ K 储 冊 ⫺1 v0 zB⫹zB 2e⫺ ␤⑀ BB 1⫹2zB⫹zB 2e⫺ ␤⑀ BB m.共24兲 Phases characterized by different molar fractions XAmay coexist when the spontaneous 共i.e., H ˜ ⫽0) magnetization m ⫽⫾ 兩 m 兩 ⫽0. We consider the case ⑀ AA⫽⫺1, ⑀ AB⫽⫺0.6, and ⑀ BB⫽0, that corresponds to a case that presents closed loops of immiscibility in the bulk. On the other hand, as the hydrogen bond interactions are very short ranged, we will take ⑀ ˜ AB⫽ ⑀ ˜ BB⫽0 as physically sensible values. Moreover, ⑀ ˜ AA is set to be ␣⑀ AA , where ␣ is a positive number less than one. In order to discriminate between different mechanisms, we chose the substrate-fluid interactions in such a way that H ˜ ⫽H, i.e., VA⫽ ␯ ⬜(1⫺ ␣ ) ⑀ AA/2 and VB⫽0. Hence, the preferential substrate adsorption effect 共very similar to that studied in the slab geometry兲is suppressed and we can focus on the coupling anisotropy. We have studied the 2D square lattice case, for which analytical expressions for ZD, ␯ are known in the anisotropic case 关38兴. However, we expect that the features obtained for this case will be qualitatively correct also for the 3D case. From the exact solution, the values of (K⬜,K 储 ) that correspond to the phase coexistence satisfy the following relationship: SURFACE AND CAPILLARY TRANSITIONS IN AN . . . PHYSICAL REVIEW E 67, 041502 共2003兲 041502-9