LINE TENSION AT WETTING

Transcription

1 International Journal of Modern Physics B, Vol. 8, No. 3 (1994) World Scientific Publishing Company LINE TENSION AT WETTING J. O. INDEKEU Laboratorium voor Vaste Stof-Fysika en Magnetisme, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium Received 10 September 1993 A review is presented of recent theoretical advances on a fundamental problem in statistical mechanics that concerns the three-phase contact line C and its tension r near a wetting phase transition. In addition to answering the intriguing question whether or not C and r vanish at wetting, recent work has also revealed that r displays universal singular behavior, reflecting critical phenomena associated with the wetting transition. Three factors are crucial for determining the fate of C and r at wetting: the order of the wetting transition, the range of the intermolecular forces, and the upper critical dimension d u, above which mean-field theory holds and below which fluctuations dominate. For most systems studied experimentally, d u < 3, so that the mean-field predictions should be correct in d = 3. In the thermal fluctuation regime, for d < d u, hyperscaling predicts the value 2(d 2)/(d 1) for the critical exponent of r(0), in the limit that the contact angle 6 approaches Introduction The subjects of this review are some remarkable properties associated with a threephase contact line, as, for example, the solid-liquid-vapor contact line shown in Fig. 1. We distinguish three bulk phases (solid, liquid and vapor), three interfaces (solid-liquid, solid-vapor, and liquid-vapor), and a contact line, along which all these phases and interfaces meet. The phases are characterized by bulk free energy densities /, the interfaces by surface free energies, or interfacial tensions, 7 (respectively, 7SL> 7SV> and 7LV)> and the excess free energy (per unit length) associated with C is denoted by the line tension r. There is no restriction on the sign of r, in contrast with the positivity requirement that is known for fluid-fluid interfacial tensions, such as 7LV- Alternatively to the example in Fig. 1, one may consider a liquid-liquid-liquid contact line, and although our discussion is tailored to the geometry of Fig. 1, we certainly expect that the main results apply much more generally, on the basis of the universality of critical phenomena. An excellent introduction to the statistical mechanics of C and r is found in Refs. 1 and 2. Experimental measurement of r has proven very difficult, but the fact that r can take both signs, and that its typical magnitude is of the order of the product of PACS Nos.: Gd, Fr, Dp.

2 310 J. O. Indekeu VAPOR LIQUID SOLID Fig. 1. Solid-liquid-vapor contact line, and dihedral contact angle 0. a typical interfacial tension and a microscopic (correlation) length, was verified to some extent. 3-5 In order to discuss what happens to the line tension at wetting, a short introduction to wetting phenomena is useful. Consider the bulk pressure-temperature phase diagram of a simple substance (say, CO2) in Fig. 2. We concentrate on the liquid-vapor two-phase coexistence line, which runs from the triple point t to the critical point c. Now suppose we bring a second substance into play, which acts like an inert "spectator" phase, and serves as a substrate, or wall, which preferentially adsorbs the liquid phase. We can then examine the wetting behavior of the substrate-adsorb ate system, as we approach liquid-vapor coexistence from the lowpressure side. Two scenarios are possible. The first is achieved following arrow 1 in Fig. 2. When the pressure is increased towards the saturated vapor pressure p sat, a solid liquid vapor Fig. 2. Bulk pressure-temperature phase diagram of an ordinary substance, featuring triple point t, critical point c, and wetting transition point w.

3 Line Tension at Wetting 311 PARTIAL WETTING Fig. 3. Liquid drop resting on a substrate, in the partial wetting regime. COMPLETE WETTING Fig. 4. Wetting layer in the complete wetting regime. thin liquid film condenses on the wall, and when crossing liquid-vapor coexistence, bulk liquid condenses in the form of droplets making a contact angle 6 with the substrate (see Fig. 3). This is called partial wetting. The second scenario corresponds to arrow 2. As p p~ at the liquid condenses in the form of a uniform wetting layer of eventually macroscopic thickness (see Fig. 4). This is called complete wetting, and we have 0 = 0. In some systems both scenarios can occur, and a surface phase transition from partial to complete wetting takes place at T = T w : the wetting transition (located at w in Fig. 2). For reviews see, for example, Refs. 2 and 6. In the case of partial wetting (Fig. 3) Young's law expresses the mechanical equilibrium of the forces (per unit length) that the interfaces exert on the contact line. (Recall that surface free energies are energies per unit area and thus forces per unit length.) Balancing the horizontal components of these forces leads to This equality is consistent with some basic thermodynamic inequalities that apply to wetting phenomena. Firstly, our assumption that the substrate prefers the liquid phase, implies

4 312 J. O. Indekey We could equally well have assumed a substrate that prefers the vapor phase. The inequality would then be reversed, and instead of "wetting" we would then speak of "drying" phenomena. Note that (1.2) implies 0 < 90 in (1.1). Next, the general inequality applies to any three phases in contact and in thermodynamic equilibrium. We apply it with i = 5, j = L, and k = V. An attempt at violating (1.3), by assuming the > sign, fails because 7sv can always be lowered to equal the sum 7SL + 7LV } simply by wetting the solid-vapor interface by an intruding liquid layer. In non-equilibrium systems, however, (1.3) may be violated. 2 We now see that partial wetting (0 > 0) corresponds to the < sign in (1.3), and complete wetting (0 = 0) to the = sign. In other words, at partial wetting, although the wall prefers the liquid, direct contact with the vapor is still tolerated (because a SV interface is still more favorable in free energy than the combination of the other two interfaces). At complete wetting, direct contact with the vapor is excluded and a liquid layer separates S from V. It is useful to introduce the so-called spreading coefficient which measures the surface free energy difference between the competing states. (Note that S < 0 in equilibrium.) It now follows from (1.1) that so that for partial wetting S < 0, and for complete wetting 5 = 0. The wetting transition can now be characterized by the way in which S tends to zero, or, equivalent^, the way in which cos# tends to 1. A typical case is sketched in Fig. 5. The singularity in S at T w can be written as or, equivalently where a s is the surface specific heat exponent. If S vanishes with a discontinuous first derivative (as it appears in Fig. 5), the wetting transition is of first order and a s = 1. On the other hand, if a s < 1 the transition is smoother and said to be continuous, or critical. It is conspicuous that critical phenomena are associated with continuous wetting transitions. It is less well known, however, that first-order wetting also involves a critical phenomenon. We will return to this important point in detail.

5 Line Tension at Wetting 313 Fig. 5. Sketch of cos 6 versus temperature in the vicinity of a first-order wetting transition. Experimental studies of first-order wetting transitions have been abundant, following the pioneering work of Moldover and Cahn. 7 Around 1980, these authors studied a wetting transition in a ternary liquid mixture against a spectator phase consisting of air, and recorded a plot of cos 9 versus concentration (rather than temperature). The plot is qualitatively similar to Fig. 5, in that it features an apparent jump in slope. However, it was not until 1992 that the first-order character of a similar wetting transition was demonstrated in a more convincing way, not by examining a cos# plot, but by observing hysteresis in the measured wetting layer thicknesses near T w, 8 and by measuring the elusive pre wetting extension of the wetting transition. 9 The first-order wetting transition was thus seen to consist unambiguously of a surface two-phase coexistence between a microscopically thin film and a macroscopically thick layer. Thus, the distinction between first-order and continuous wetting is clearly seen in a plot of the condensed film thickness /, achieved in the limit p» p~^t) versus a suitable thermodynamic density or field. As Fig. 6 illustrates, at first-order wetting the equilibrium thickness / jumps from a small value (a few A) to a large (about 10 3 A for gravity-thinned layers), and in principle infinite, value. In contrast, at continuous wetting / diverges in a continuous manner (Fig. 6), described by where /? s is the surface order parameter exponent. We remark that together with a surface susceptibility exponent, 7*$, the surface exponents satisfy a s -f 2/? s + 7s = 2, analogously to a relation that is well known for bulk critical exponents. We will see that a similar relation holds at the level of the line excess quantities. In our opinion, a proper understanding of the question of what becomes of the contact line C at wetting, derives from a correct appreciation of the following problem. Suppose that the wetting transition is of first order (as in most experiments). Then, as we approach T w the contact line C in Fig. 1 gradually transforms into a transition zone, or in homogeneity, connecting a thin film to a thick layer. The

6 314 J. O. Indektu FIRST-ORDER CONTINUOUS Fig. 6. Equilibrium wetting layer thickness / versus temperature, along bulk liquid-vapor coexistence, near a first-order or, respectively, a continuous wetting transition. AT FIRST-ORDER WETTING Fig. 7. In this figure the following question is raised: "What is the structure of the transition zone between a thin film and a macroscopically thick wetting layer, at a first-order wetting transition?" latter is bounded by the SL and LV interfaces, which become parallel to one another as 8 0. Not only do they become parallel, but they also become infinitely separated. At T w the problem therefore looks as sketched in Fig. 7, and the basic questions we would like to answer are now conspicuous. Assuming it exists, how can we determine the profile l(x) of the transition zone between the thin film at x oo, with / = /i, and the wetting layer at x oo, with / oo? And, next, what is the excess free energy associated with this inhomogeneity, that is, what is the line tension r at wetting? For continuous wetting transitions the problem is qualitatively different, since there are no coexisting states, but only a single uniform wetting layer, in the limit 0 0. We may therefore expect the answer to depend strongly on the character of the wetting transition. Perhaps the reason why the behavior of r near T w has attracted a lot of attention recently, is the frustrating history of conflicting results that were obtained between 1980 and The question what happens to the line tension when the

7 Line Tension at Wetting 315 contact angle approaches zero was probably first addressed around 1980 by Starov, Churaev, and Derjaguin The proposed result was that r is negative near T w and vanishes at T w, as the first power of 0. Although the Derjaguin-type theory that was employed has proven to be extremely useful (see further), in the course of the actual calculations unnecessary and crude approximations for the disjoining pressure isotherm were invoked. The aforementioned result for r is therefore incorrect in general (but fortuitously gives the correct answer in a special case that we will discuss). Apparently independently of the earlier work by the Russian School, the next move was made in 1986 by Joanny and de Gennes. 13,14 These authors employed a de Gennes-type theory equivalent to that of Churaev et a/., but made a more refined analysis and arrived at very different results. For example, for the important class of systems with (non-retarded) van der Waals forces, they predicted that r is positive near T w and diverges in the limit T T w, in the manner r(0) oc ln(l/#). We note that van der Waals forces and other forces with algebraic decay at large distances are termed long-range forces in the field of wetting phenomena. Local interactions (e.g., nearest-neighbor) or exponentially decaying ones are termed short-range forces. Again independently of earlier works, Widom and Clarke argued in 1990 that, at least at the level of mean-field theory and in systems with short-range forces, C and r may be expected to vanish at wetting. 15 In the simple phenomenological description they employed, r is negative near T w, and vanishes at T w, as the first power of 9. Note that this is identical to the result proposed by Churaev et ail For the first time the issue of the order of the wetting transition was addressed, and Widom and Clarke expected that the vanishing of the r is fairly general, applicable also to first-order wetting. In particular they suggested that the analogue of r along the prewetting line, referred to as the boundary tension (which we denote by f), may also be expected to vanish when T w is approached. In subsequent work, Widom and Widom 16 employed a more detailed mean-field or Landau-type free-energy functional and confirmed, albeit within an approximate calculation, the earlier expectation. That is, r{9) < 0 and r oc 0, near T w. The incipient concensus about the vanishing of r at wetting in mean-field theories for short-ranged forces was suddenly shattered in 1992 by two works. Varea and Robledo, 17 and, almost simultaneously, Szleifer and Widom 18 discovered, on the basis of improved and fairly accurate mean-field theories, that, instead of vanishing, r increases without apparent bound and perhaps diverges at wetting, towards +oo. This conclusion was reached, independently, in the lattice mean-field theory of the nearest-neighbor 3d Ising model, 17 and in a further refinement of a Landau-type density functional theory. 18 Both works dealt with a first-order wetting transition. Furthermore, Varea and Robledo also found evidence for a divergence of the boundary tension f along prewetting. It so happened that this whole subject began to appear very chaotic, as totally different answers appeared to result from supposedly innocent changes in the models or approximation schemes.

8 316 J. O. Indekeu But, also in 1992 it was shown on the basis of an interface displacement model, equivalent to Derjaguin's and de Gennes' theory, that r displays interesting singular behavior at wetting. 19 The results are detailed and quantitative, and give the dependence of the regular and singular part of r on (i) the order of the wetting transition, and (ii) the range of the forces. One of the predictions is that in the mean-field theory for short-range forces, and assuming a first-order wetting transition, r neither vanishes nor diverges, but reaches a finite positive limit at wetting! The mean-field results from this interface displacement model, and how they can serve to reinterpret and reconcile the foregoing conflicts, is the subject of the next section. In closing this introduction, and in order to set the stage for a proper appreciation of the singular behavior of r, we draw attention to a hierarchy of singular behavior, from the bulk down to the line. The bulk free energy density displays the following well-known singularity near the bulk critical point c (see Fig. 2), where a is the bulk specific heat exponent (a «0.1 for ordinary fluids). Next, the interfacial tension is characterized, for T T c ~, by the following singularity, where a t - is the interfacial specific heat exponent. Usually 2 c* t - is denoted by /i (/i «1.26 for ordinary fluids). Note that at the bulk critical point the interface disappears, and in such a way that its tension vanishes. To proceed further in this hierarchy, we turn to the surface excess free energy near wetting, which shows the following singular behavior for T T~ y as discussed already in (1.6). At this stage it is plausible that the line tension r can be characterized by an analogous form, where T(Ty,) is a regular background term (which may well be non-zero!) and a\ the line specific heat exponent. Note that, using (1.7), we can also write Since there was no need to assume the wetting transition to be continuous, we expect this formalism to apply equally well to first-order wetting. We now see that we can ask already two important questions concerning r: (i) what is the value of r(tyf) y and, perhaps more importantly, (ii) what is the value of a\l

9 Line Tension at Wetting Mean-Field Theory In this section we describe the interface displacement model and its results at the mean-field level. In the next section the effect of fluctuations will be incorporated in this model by invoking standard scaling arguments. Although the model is in many respects an oversimplification of the physical problem under study, it should be capable of producing correct answers in as far as universal features (critical exponents, amplitude ratios,... ) are concerned. Indeed, these quantities are insensitive to many microscopic details that are included in more realistic models. At the mean-field level we assume that the line inhomogeneity can be described by a single-valued interface displacement profile /(#), where a? is a coordinate parallel to the substrate. Translational invariance is assumed in the remaining d 2 directions parallel to the d 1 dimensional substrate. Of course, we will mostly be interested in d = 3. Figure 8 shows a typical l(x) for partial wetting. Note that the geometry is that of a liquid wedge, rather than a drop. In the interface displacement model, 19 which is equivalent to de Gennes' approach, 2 ' 13 and to Derjaguin's theory, 11,12 the line tension is first written as a functional, where integration is done from the thin film state (at x = oo) to the bulk liquid (at x = oo). The integrand is the surface excess free energy due to the inhomogeneity, and l x = dl/dx as usual. We will henceforth neglect gravitational contributions to the energy (and thus set the capillary length to infinity), but gravity can be included in a simple way in this model. 2 Fig. 8. Typical interface displacement profile l(x) in the partial wetting regime, for a system disposed to undergo a first-order wetting transition. For a more quantitative plot, see Fig. 4 in Ref. 20. In order to construct a physically plausible surface excess free energy G we distinguish two essential contributions, (i) the excess free energy of a uniform film of thickness /, and (ii) the free energy cost of non-uniformity. The first of these is given by the interface potential V(/), related to Derjaguin's disjoining pressure II(/)

10 318 J. O. Indekeu by The typical shape of V(l) for a system at partial wetting is shown in Fig. 9. Since only differences in V are relevant, we define V as the surface excess free energy of the film, 7, relative to the equilibrium value 7sv- Consequently, for the equilibrium solid-vapor interface, corresponding to a thin film of thickness /1 (minimum of V), we have V(h) = 0. For the (local) minimum at / = 00, corresponding to a metastable wetting layer, we have V^oo) = 7SL + TLV TSV) which equals 5, as is seen in (1.4). Now suppose that this V(l) changes smoothly, for S 0, to a potential with two equal minima (at I = l\ and / = 00) with a barrier in between. This potential then describes a first-order wetting transition. Fig. 9. Typical interface potential V(/) in the partial wetting regime, for a system disposed to undergo a first-order wetting transition. For sketches of V(l) in other regimes, see Fig. 2 in Ref. 19. As we will discuss, the model calculations show that the details of V(l) at small / are largely irrelevant. What matters is the asymptotic part for large /. For intermolecular potentials <f> that decay, for large distances r, as <j>(r) oc r~( d + a \ simple dimensional analysis shows Indeed, integration of <f> over d dimensions gives the so-called substrate-adsorb ate potential, or the energy of a unit volume of adsorbate interacting with a half-space of substrate, and one more integration (perpendicular to the substrate) gives an energy per unit area. The prefactor A in (2.3) is a non-trivial combination of properties (usually polarizabilities per unit volume) of the different phases, and is often referred to as effective Hamaker constant. Note that for non-retarded van der

11 Line Tension at Wetting 319 Waals forces in d 3, we have a 3, and for retarded forces a = 4. On the other hand, for short-range forces, We note that / is a dimensionless variable, which measures length in units of a microscopic reference length (e.g., an atomic diameter or a bulk correlation length). The second contribution is the free energy cost of non-uniformity. Since distortions of the interface lead to a local increase in area, this cost can to a first approximation be estimated as the product of the interfacial tension and the area increment per unit area, which is simply given by the line element ds = dx ds/dx. Combining the two contributions gives where a piecewise constant function c(x) must be included to ensure that the integrand properly vanishes for \x\ oo and to guarantee that the line tension is independent of the location of the so-called dividing line, 19 which is the analogue of the dividing surface for interfaces. 1 The final step is then to find the profile l(x) that minimizes the line tension functional. Associated with this equilibrium profile is the equilibrium line tension r, defined as subject to the boundary conditions and where x^ is the (arbitrary) location of the dividing line. In practice, one may expand the square root in (2.5) and work in the so-called gradient-squared approximation. By including the gradient to all orders it was verified explicitly that this approximation leaves the interesting singular behavior of r unmodified. 20 The basic result is

12 320 J. O. Indekeu for partial wetting, including the transition point S = 0. A very similar expression was probably first derived already in 1972 by de Feijter and Vrij, 21 who studied C and r near a Plateau border, which has a wedge-type geometry that is more symmetric than that of Fig. 1 (see Fig. 3 in Ref. 21). These authors also noted the crucial fact that r is sensitive to the tail of V(l)\ We now proceed to discuss the main results of the interface displacement model, and will compare with alternate theories in due course. For more detailed results, we refer to Refs. 19 and Mean-field results for first-order wetting Along partial wetting When the contact angle 6 tends to zero, the contact line C develops into a macroscopic transition zone connecting a thin to an infinitely thick film. An example of a limiting profile at first-order wetting is shown in Fig. 10. This profile has evolved smoothly from the partial wetting profile of Fig. 8. The equilibrium displacement profile l(x) that describes C at wetting decays exponentially fast in the direction of the thin film at x oo, whereas in the direction of the wetting layer at x» oo it has the form 2 ' 19 long-range forces short-range forces Fig. 10. Typical interface displacement profile l(x) at a first-order wetting transition. For a more quantitative plot, see Fig. 2 in Ref. 20. Since the contact line by no means disappears at first-order wetting, it is not surprising that the line tension does not vanish either. For long-range forces with a >3, and for *->0", with t = (T-T W )/T W1

13 Line Tension at Wetting 321 which is a realization of (1.12) indeed. The fact that r(0) is positive and finite follows immediately from inspecting (2.9) at S = 0, using (2.3). The assumption a > 3 then guarantees convergence of the integral. The second interesting point is that the amplitude r_ comes out negative, and that the exponent 2 a\ is sufficiently small, so that r increases towards r(0) with diverging slope. (This remains true when r is plotted as a function of 9). In conclusion, a typical plot of r for a > 3 takes the form of Fig. 11. FIRST-ORDER WETTING Fig. 11. Typical behavior of the line tension r(t) versus temperature, approaching a first-order wetting transition from the partial wetting regime T < T w. At T w a finite positive value is reached from below, with diverging slope. In some respects short-range forces are similar to the a - oo limit (and we therefore discuss them now), but a number of properties are not reproduced in this limit (e.g., for continuous wetting 22 ; see further). The result for short-range forces is qualitatively similar to (2.11), and resembles Fig. 11. Explicitly, for t 0~,which features a leading exponent 2 a\ = 1/2, but with a logarithmic correction factor. Since, as a function of 0, r smg oc 01n#, the logarithm causes the slope of r(9) to diverge weakly at 9 0. We stress once more that these results are insensitive to the precise form of V(l) at small /. 19 > 20 These quantitative predictions for short-range forces can be compared with more qualitative results of alternative mean-field calculations, that start either from the microscopic Ising model Hamiltonian, 17 or from a Landau-type density functional theory with two spatially varying densities. 18 Due to computational limitations, however, the approach to the wetting transition along partial wetting states could not be completed, and the smallest value for the contact angle that could be reached is 9 «13 in both works. Nevertheless, qualitatively there is an overall consistency in that T{9) becomes positive and increases with decreasing 9', and also its slope increases in magnitude. Although in both plots, in Refs. 17 and 18, it may seem that

14 322 J. O. Indekeu r diverges at wetting, the data are certainly also consistent with an approach to a finite limiting value, with infinite slope, as predicted by the interface displacement model. We will come back to this issue when we discuss more recent works on the boundary tension f along prewetting. We now return to long-range forces and discuss the important case a = 3 (nonretarded van der Waals forces in d = 3). In this case, for t 0~, the line tension diverges towards + oo in the manner a result probably first uncovered by Joanny and de Gennes, 13 and confirmed independently in calculations specific to first-order wetting. 19 For a < 3 the divergence of T becomes more pronounced, and takes the form and for a < 2, r(9) equals positive infinity already at non-zero 6. These divergences are easily seen to be due to the slow decay of V(l) at large /, using (2.3) and (2.9). Strictly speaking the line tension does not exist in these cases, but the divergence can be curtailed by introducing a macroscopic system size cutoff L. For a system in d 3 with linear size L, we would then arrive at the following decomposition of the total free energy F into bulk, surface and line contributions, (where the dots stand for additional contributions from corners). If a is sufficiently small, T(L) diverges in the thermodynamic limit L oo, but as long as a > 1, which in view of (2.3) is necessary in order that the interaction between interfaces decrease with increasing separation, we have T(L)/L 0. Consequently, the line contribution remains small compared with the interfacial one and there is no confusion. In fact, divergences in the thermodynamic limit are quite common when working with long-range forces. It is well-known that the bulk energy density, akin to /, diverges as soon as a < 0. For example, for potentials <j>(r) oc r~ 3 in d = 3 the energy density diverges as InL. Next, interfacial contributions, akin to 7, diverge for a < 1. This is well exemplified by recent work by Flament and Gallet, who studied linear interfaces in d = 2 and obtained divergent tensions for r~ n potentials, with n < 3 (that is, a < l). 23 The divergence of r for a < 2 is just one further step in this hierarchy. The precise form and importance of finite-size effects on r has been studied in detail recently Along prewetting A first-order wetting transition is expected to have a prewetting extension into the bulk vapor phase, consisting of the surface phase coexistence of a thin and a

15 Line Tension at Wetting 323 thick (but finite) film (see Ref. 6 for a review). The prewetting line terminates at a prewetting critical point, which belongs to the universality class of d 1- dimensional bulk critical phenomena. At its other end, the prewetting line meets the bulk two-phase coexistence line tangentially, at T w. A typical first-order wetting phase diagram is shown in Fig. 12, where h is a bulk field that takes the bulk away from liquid-vapor coexistence. Fig. 12. Typical phase diagram for a first-order wetting transition. Bulk liquid-vapor coexistence is at h 0. Partial wetting (T < T w and h = 0) corresponds to an interface triple line (3), prewetting (T > T w and h = h pyv (T) > 0) corresponds to coexistence of two interfaces (2), and complete wetting (T > T w and h = 0) corresponds to interface criticality (C). In region (l) only one (solid-vapor) interface is stable. The prewetting critical point is denoted by T pwc. The linear interface or boundary (assuming d 3) between the thin and the thick film is a particular case of a contact line C and is characterized by a boundary tension f, which, in contrast to r, is subject to a positivity requirement. Indeed, f > 0, because the boundary is equivalent to a fluid-fluid interface within ad-1- dimensional bulk. 15 Also, f vanishes at the prewetting critical point, in the manner described by the critical exponent \i = 2 a,- for dimension d 1. The interface displacement model predicts the following main results for f near T w and near h 0. For long-range forces with a > 3, and for h * 0 +, where f(0) = T(T W ) > 0, the same finite value encountered when approaching T w along partial wetting, and f + < 0. For short-range forces the results agree with taking a * oo in (2.16), and the exponent of h, which we denote by 2 &i, equals 1/2. Unlike for partial wetting, see (2.12), there is now no logarithmic correction factor. In sum, for forces that are of sufficiently short range, a typical plot of f versus h (or t) looks similar to Fig. 11 near the wetting transition. A finite positive value is reached with diverging slope.

16 324 J. O. Indekeu In a pioneering calculation of the boundary tension along pre wetting, in lattice mean-field theory for the 3d Ising model, Varea and Robledo saw what seemed like a weak divergence of f at wetting, but it could also be consistent with a finite limit. 17 Their subsequent refined analysis appears to confirm the prediction of the interface displacement model quantitatively, including the exponent 1/2. 25 Very recently, applying the density-functional theory of Szleifer and Widom to prewetting, Perkovic et al. computed r close to the wetting transition, and found that it increases with increasing slope. 26 They noted that, together with the earlier results on r, 18 the combined plot of f and r, as a function of, is strikingly similar to a "lambda point". Fitting the data to an assumed divergence, they concluded that the precision was not good enough to distinguish a cusp (finite limit) from a divergence. We observe, however, that the data of Perkovic et al. are well fitted by the detailed form of the cusp predicted by the interface displacement model. This is (including the latest results for amplitude ratios 27 ), and, after making use of the equation h h pv/ (t) converting f(h) to T( ), with now t > 0, for the pre wetting line, and thus where the non-universal constants r(0)(> 0), T_(< 0), and co(> 0) can be treated as fitting parameters. The universal amplitude ratio Q\ = T+/T_ is known and approximately equals (see further). Note that the logarithmic correction factors are different in (2.17) and (2.18). This induces an extra asymmetry in the cusp! Near t = 0, the resulting combined plot for r along pre wetting and partial wetting is remarkably similar to the "lambda"-shaped peak seen in Fig. 7 of Ref. 26, already with the simple choices r(0) = 1, r_ = 1, and c 0 = 1. Figure 13 illustrates this. We now return to long-range forces and address the case a = 3, for which the interface displacement model predicts a (weak) divergence, which can be compared with (2.13). On the basis of this prediction Schick and Taborek deduced that the lifetime of thick wetting films when taken out of equilibrium will be extraordinarily long, in systems with the usual van der Waals forces. 28 They observed that the divergence of f implies the vanishing of the probability to nucleate a drop of the thermodynamically stable thin film phase when the system is prepared in the wetting layer phase, near prewetting. They suggested that measurement of the increase of relaxation times of the thick film would elucidate the form of the divergence of the boundary tension.

17 Line Tension at Wetting 325 prewetting partial wetting Fig. 13. Calculated plot of a typical "lambda"-shaped line tension cusp predicted by the interface displacement model for short-range forces and first-order wetting (see text for the relevant functions). Data for partial wetting (T < T w ) refer to the line tension r, and those for prewetting (T > T w ) to the boundary tension f. In this model r(0) is finite (and equals 1 in this calculation, in suitable units). Finally, for forces of longer range, with 1 < a < 3 the boundary tension diverges as a power law, for h + 0 +, Out of equilibrium: The nucleation route Up to now we have encountered the line tension in two physically distinct situations: along partial wetting, as r, and along prewetting, as f. An interesting third situation arises when a thin film, which is stable at T < T w, is suddenly subjected to a temperature rise into the complete wetting regime at T > T w, where T w locates a first-order wetting transition. If T is not too far from T w, the thin film is metastable and is characterized by a positive spreading coefficient S. The equilibration of the system, consisting of the relaxation towards a thick wetting layer, can then occur through a nucleation process. Dome-shaped liquid droplets greater than a critical droplet size grow, whereas smaller ones shrink. The critical droplet is thus unstable and corresponds to a maximum of the free energy. Joanny and de Gennes pointed out that this nucleation process is characterized by a free energy barrier that reflects a compromise between S > 0, promoting thick film formation, and a line tension cost, TCD, associated with critical droplets. 14 A generalized Laplace equation,

18 326 J. O. Indekeu determines the radius R of the contact line along which the critical droplet is in contact with the substrate and the vapor. Note that this is analogous to the standard Laplace equation for spherical critical droplets in bulk nucleation, Ap = JLV/R, and one may interpret S as a "surface pressure" in (2.21). For sufficiently small 5, R will be very large, and the free energy barrier AF embodied in the critical droplet of nucleated wetting, then consists of distinct surface and line contributions, Bausch and Blossey calculated the shape of the critical droplet explicitly, using an interface Hamiltonian equivalent to the interface displacement model, and discussed its dynamical stability properties from a Langevin model. 29 They found that the critical droplet has a single unstable growth mode but its shape is stable with respect to all other fluctuations. Subsequently, these authors determined that the cylindrically symmetric critical droplets are characterized by two critical length scales: a parallel length iz, the contact line radius, and a perpendicular length / max > the droplet height. 30 They calculated how these quantities, and also the barrier AF, depend on S near the wetting transition. The implications for TCD a r e in harmony with the predictions of the interface displacement model for r. To leading order in 5, and recalling that S oc T T w in the metastable continuation of the thin film above T w, const ant (> 0), for which was already conjectured in Ref. 14, and We believe that the concerted behavior of these non-equilibrium results and the previous ones for equilibrium states indicates that the mean-field theory is internally consistent and at the same time widely applicable. In a further application, Bausch and Blossey studied the crossover behavior of critical droplets from the complete wetting to the partial wetting and prewetting regimes. 31,32 The droplet shape changes, respectively, from (generalized) ellipsoidal cap to spherical cap and cylindrical pancake. Mean-field scaling forms, which feature the crossover exponent A = <T/(<T 1) that governs the prewetting line through h pyf (t) oc tf A, were found to describe R, / max > ar *d AF in the various non-equilibrium regimes in a unifying way. These results are, again, in accord with the scaling theory for equilibrium, described in the next section. Very recently, the precise macroscopic shape of the critical droplet of nucleation has been obtained analytically for the special case that the dimension d coincides with the upper critical dimension d u for for for

19 Line Tension at Wetting 327 first-order wetting (explained in the next section), by Burschka et a/. 33 The result a generalized ellipsoid, which reduces to an ellipse for a 3, for which d u = 2, as we will discuss. Blossey and Bausch also uncovered a dynamical critical exponent z, associated with the late-stage growth of supercritical droplets. 31 They found, with z equal to 2, and where t denotes time and not temperature. Note that the spatial anisotropy, reflected by the exponent 2/(cr+ 1), is also conspicuous in the interface displacement profile at the wetting transition, given in (2.10). We will come back to this when we discuss the wandering or roughness exponent C in the next section Mean-field results for continuous wetting The simplest description of a continuous wetting transition involves an interface potential V(I) that displays a single minimum at / = h and approaches its asymptotic value -S from below. 19 The absence of a free energy barrier in V(l) then leads to a contact line profile l(x) without inflection point, as illustrated in Fig. 14 for partial wetting. As 0 tends to zero the profile flattens while the interface gradually unbinds from the substrate, and the resulting picture is that of a uniform thick wetting layer, as in Fig. 4. This is in sharp contrast with the picture of the two coexisting layer thicknesses near first-order wetting. Therefore, in mean-field theory, the contact Fig. 14. Typical interface displacement profile l(x) in the partial wetting regime, for a system disposed to undergo a continuous wetting transition. For a more quantitative plot, see Fig. 5 in Ref. 20.

20 328 J. O. Indekeu line C smoothly disappears at continuous wetting, and we have reason to suspect that T vanishes. This is indeed confirmed by the calculations, at least if the forces are of short or moderately long range (a > 2). We first discuss critical wetting and will turn to multicritical wetting at the end. For more details we refer to Refs. 19 and 22. For critical wetting and long-range forces, the surface specific heat exponent takes the value a s = 2 a. As regards the line tension, for a > 2, and t 0~, with r_ < 0. So, r vanishes but with diverging slope. This remains true when r is expressed as a function of 0, using (1.13), which gives, for 0» 0, In sum, for long-range forces, a sketch of r near critical wetting looks like Fig. 15. CONTINUOUS WETTING Fig. 15. Typical behavior of the line tension r(t) versus temperature, approaching a continuous wetting transition from the partial wetting regime T < T w. At T w the value 0 is reached from below, with diverging slope (except for short-range forces, for which the slope remains finite). The case of short-range forces does not simply correspond to letting a + oo. For critical wetting, a s = 0, implying a simple second-order transition. The leading behavior of the line tension is given by the linear forms and

21 Line Tension at Wetting 329 with negative amplitudes. Note that this linear behavior constitutes an exception to the general trend that the slope of r diverges at wetting. Also note that (2.32) coincides with the early conjecture of Churaev et a/ and Widom et a/. 15,16 We now see that it gives the correct answer in the special case of continuous wetting in systems with short-range forces, and we now also understand qualitatively why it takes a continuous transition for the line tension to vanish. Let us now return to long-range forces and consider a < 2. As already announced earlier, the line tension ceases to be well defined in the thermodynamic limit if the forces are too long ranged. In this case r(0) (with 6 > 0) diverges, towards oo, as the system size L is taken towards oo. The precise finite-size dependence of r in the vicinity of a continuous wetting transition is the subject of further study, along the lines of Ref. 24. We finally address briefly how r behaves near a multicritical wetting transition. The interface displacement model predictions for tricritical and higher-order multicritical wetting transitions are qualitatively similar to those for critical wetting, but there are some noteworthy quantitative differences. 22 For long-range forces a s = (n <x)/(n 1), where n = 2 for critical wetting, n = 3 for tricritical wetting, etc. The vanishing of r, from negative values towards zero, is described by the multicritical exponent On the other hand, for short-range forces a s = (n 2)/(n 1), and Consequently, using (1.13) we find that T(0) OC 9 for short-range forces and continuous wetting (critical and multicritical). In closing the mean-field theory section, we note that in all cases considered so far the line tension is predicted to increase as the wetting transition is approached. We therefore put forward as a general conjecture, r is maximal at wetting. (2.35) Now that in our opinion the mean-field theory for the line tension at wetting is fairly complete, it is time to ask: what about fluctuations? 3. Fluctuations, Scaling and Universality In this section we ask what the effect of fluctuations (of thermal or other nature) on the singular behavior of r at wetting 34 is. Specifically, referring back to (1.12), we will be most interested in the effect of fluctuations on the value of a\. It is important to emphasize that this problem is of principal interest from the point of view of

22 330 J. O. Indekeu critical phenomena and universality. It is only recently that this point of view has been adopted in studying contact line phenomena. Previously, the focus of interest was on non-universal aspects. Among these earlier developments we cite especially the work of Clarke, which discusses the effect of the line tension, and in particular its limiting value at wetting, the regular background T(T W ), on the fluctuations of the contact line. 35 This topic, which is central to understanding the wandering of C on random surfaces, 2 merits a review in its own right, and is outside our present scope. Adopting, as we do, the point of view of critical phenomena, two questions are immanent. First, what do first-order wetting transitions have to do with critical phenomena, and, secondly, what is the upper critical dimension d u above which the mean-field results for a\ should be correct? 3.1. Fluctuations at first-order wetting The observation that first-order wetting can, and perhaps should, be regarded as a critical phenomenon, came in several respects as a surprise. 27,34 At the phenomenological level, two steps were involved in this insight: (i) as Lipowsky demonstrated, complete wetting can be regarded as a critical phenomenon with a diverging correlation length, 36 and (ii) a first-order wetting transition, consisting of the coexistence of a thin film and a macroscopically thick wetting layer, can thus be regarded as the interface analogue of a critical endpoint (where a non-critical and a critical phase coexist). At a more quantitative level, the singular behavior of r at first-order wetting reveals that a critical phenomenon akin to complete wetting is taking place, through the recently derived exponent equality 34 This relation is a special case of the more general result ot\ = Qf s + i/, valid also for continuous wetting transitions (see further). Here, v\\ is the exponent of the divergence of the parallel interface correlation length y, as a function of temperature. Normally, this divergence is expressed as a function of the field A, which measures the deviation from complete wetting (at bulk coexistence), and the correlation length and its exponent are then denoted by and J>, respectively. When the usual relation is applied to the thick film along the prewetting line, the /i-dependence can be converted to a ^-dependence, using the equation for the prewetting line, h = h pvv (t) oc t A, where A is the crossover exponent. We thus obtain i/y = Ai>, and the same can be done for the surface free energy, implying 2 a s = 1 = A(2 d s ).

23 Line Tension at Wetting 331 The key observation leading to (3.1) is that both paths t 0~ (along partial wetting), and t 0 + (along prewetting), involve an incipient wetting layer as a constituent part of the structure of the contact line C. Therefore, the characteristic parallel length scale over which the contact line inhomogeneity is spread out, along the x direction, is singular and given by. This leads to the simple scaling argument, or dimensional analysis argument, that the singular part of r is the product of the singular part of the surface free energy at wetting and the singular parallel length scale, where 7 s ing S on the partial wetting side. From this the exponent equality (3.1) follows immediately, using a s = 1 for first-order wetting, and, equivalently, we can write In addition to r s i ng there is, in general, a non-vanishing regular part r(0), due to the fact that at first-order wetting the contact line also features a second nonsingular length scale, on the side of the thin film, which remains finite as \t\ 0. Since this length scale is microscopic, r(0) is highly non-universal. In sum, we obtain for first-order wetting. The scaling argument we have presented suggests that the exponent 2 c*i is the same for t > 0 and t < 0. Although this is certainly plausible, some caution is in order because we have already met one case, in (2.17) and (2.18), where the corrections to scaling (in the form of logarithmic factors) have different exponents above and below T w. A more detailed scaling argument is given in compact form in Ref. 34 and we would like to present a more pedagogical version here. Excellent expositions of analogous scaling arguments at the level of the surface free energy at wetting are found in Refs. 37 and 38. What follows is a straightforward extension of these well-established ideas to the line excess free energy. We begin with recalling the anisotropy of the length scales parallel and perpendicular to the substrate. This is reflected, already at the mean-field level, in the shape of the profile at first-order wetting, as is evident from (2.10). As regards interface fluctuations, the correlation length parallel to the interface,, and that perpendicular to it, j_, scale differently. For example, when the interface unbinds at complete wetting, these lengths diverge in the manner

24 332 J. O. Indekeu The anisotropy is characterized by the wandering or roughness exponent, so that For thermal capillary wave fluctuations, a liquid-vapor interface is rough in 1 < d < 3 (C = (3 - d)/2 > 0) and smooth in d > 3 (C = 0), d = 3 being a borderline case with marginal roughness ( = 0(log)). There are many equivalent ways to derive the upper critical dimension d u, which is the borderline dimension between the mean-field and the fluctuation regimes. In the context of the contact line at first-order wetting it is appropriate to determine d u as that value of d below which the broadening of the interface by fluctuations overwhelms the increase of l(x) with x implied in the mean-field profile (2.10). Conversely, above d Ui the fluctuations are merely superimposed (as ripples) on the mean-field contact line shape. This line of reasoning leads to which gives the correct answers long-range forces short-range forces long-range forces, short-range forces, for first-order wetting. We note that this d u is the same as that found by Lipowsky for complete wetting. 36 This is not surprising, since it has been argued that firstorder wetting is analogous to a critical endpoint and complete wetting is analogous to the critical line attached to this endpoint. 27 The physically most important consequence of this is that d u < 3 for long-range forces, so that mean-field theory is valid in d = 3! The effect of fluctuations on the singular behavior of r is found heuristically as follows. The physical picture that is assumed is that of a wandering interface at a finite average distance from the substrate. Let us assume that the fluctuations are important (d < d u ). After a height excursion j_, achieved over a distance, the interface collides with the substrate. As a result the fluctuations lose their coherence, which justifies the term correlation lengths. We now obtain the singular part of the surface excess free energy as The first term is the gradient-squared contribution or bending energy, and expresses that between two collisions l(x) changes typically by an amount j_ over an x- distance M, the range of coherence of the capillary waves. The second term is

25 Line Tension at Wetting 333 the entropic repulsion and reflects that the wandering interface is excluded from the half-space occupied by the substrate. The area involved in one collision is the correlation area ^n -1 and the entropy loss is of order &B- The sum of the first two terms gives the so-called fluctuation-induced repulsion of the interface. To a first approximation this fluctuation contribution is then simply added to the mean-field contribution, which is the last term. For example, for partial wetting this term is the mean-field result for the spreading coefficient S. The game to play is now simply to keep track of which term dominates in (3.11), that is, which term constitutes the leading singularity as criticality (complete wetting, first-order wetting, or, also, continuous wetting) is approached. This leads to the general result 37,38 In order to obtain 2 a\, all we need to do if we consistently proceed with these scaling arguments, is to multiply the surface excess free energy in (3.11), which was obtained as energy per correlation area, by the correlation length in the x direction,. This gives and also implies the exponent equality ; This relation was first checked in the mean-field theory for first-order as well as continuous wetting, and it was found to hold throughout. It is interesting to note that the mean-field value of I/JJ is non-trivial (for first-order wetting I/JJ = (a + l)/[2(<7-1)], and for critical wetting i/ = (cr + 2)/2). The scaling argument now suggests that (3.14) should also hold in all fluctuation regimes. This includes thermal wandering, stronger ("superthermal") fluctuations as in random media (quenched random fields or bonds), and weaker ("subthermal") fluctuations as in quasiperiodic systems. Superthermal (with > (3 d)/2) and subthermal (with < (3 d)/2) fluctuations were considered in some detail in Ref. 34. In order to avoid proliferation of this pedagogical presentation we will restrict our attention to thermal fluctuations from now on. When thermal fluctuations dominate, hyperscaling holds, and the basic hyperscaling relation for wetting is which implies i/ = l/(rf 1) for first-order wetting. In fact, since C = (3 d)/2 and I/_L = C^ll i the bending energy and the entropic repulsion have the same critical exponent: the first two on the right of (3.12) are equal. The hyperscaling relation

26 334 J. O. Indekeu for the line follows from (3.13), which implies 2 a\ = (d 2)/(d 1) for first-order wetting. The exponents for first-order wetting and for long-range forces are summarized in Table 1 (which has been derived using Table III of Ref. 34). The results for short-range forces can, in this case, be obtained simply by letting <r oo wherever (7 occurs in this Table (MF regime). The results in the fluctuation regime are universal with respect to the range of the forces. We recall that these temperaturelike exponents can be converted to field-like exponents, using the crossover exponent A. We define a new exponent p\ s as the ratio (2 <*i)/(2 <*s)«this is convenient for describing the singular part of r as a function of 0 in the partial wetting regime. In view of (1.13), We will see shortly that p\ s merits special attention. Table 1. Exponents for first-order wetting. The mean-field regime (MF) applies for d > d u = 3 4/(a + 1), and the thermal fluctuation regime for d < d u. The results for short-range forces agree with taking the limit a» oo in this Table. In particular, for short-range forces d u = 3. We have already alluded to the fact that a first-order wetting transition is the analogue for interfaces of what a critical endpoint is for a bulk system. This powerful analogy was recently demonstrated and served as a guide to explore novel universal features associated with the behavior of r. 27 We give a concise account of the main results. The first-order wetting phase diagram can be interpreted as being composed of a triple line, where three interfaces coexist (the partial wetting line), its extension where two interfaces coexist (the prewetting line), and a line of interface criticality (along complete wetting). Criticality here signifies a diverging correlation length. In this scheme, T = T w and h = 0 is an interface critical endpoint, where a thin ("non-critical") film coexists with a ("critical") wetting layer. This makes a lot of sense, especially if we recall that the profile l(x) decays exponentially into the thin film, but algebraically slowly into the thick wetting layer, just like the bulk order

27 Line Tension at Wetting 335 parameter does at a bulk critical endpoint. As soon as this critical-endpoint scheme was accepted, some of the latest results on singularities and universal quantities at bulk critical endpoints provided inspiration for gaining further insight in the line tension at wetting. The most straightforward next step was the calculation, in mean-field theory, of the universal line amplitude ratio T+/T-. For a review of universal criticalpoint amplitude relations, see Ref. 44. The outcome is a function T+/T- = Q\((r) } obtained analytically in closed form. For the physically important case of van der Waals forces in d 3, Qi(3) = 1. For short-range forces simple scaling must be corrected to include logarithmic factors, as is clearly seen in (2.17) and (2.18), but the ratio r + /r_ is universal and takes the value Q\ « (The number is known exactly in the form of an integral). It was furthermore noted that one may define a new line order parameter m\ and an associated line susceptibility xi- The associated critical exponents, combined with the line specific heat exponent, satisfy the analogue of the familiar exponent equality a + 2/3 + y = 2, (We have already mentioned that a similar law holds for the surface exponents.) Attention was also devoted to universal combinations of r and the interface correlation lengths and j_> by an extension of two-scale factor universality or hyperscaling. 44 Closely related to this development was the finding that finite-size amplitudes of r at wetting display a high degree of universality. These results are best understood in the context of an explicit finite-size scaling analysis, 24 and we close this discussion of fluctuations at first-order wetting with a brief summary of that study. We assume the wedge geometry of Fig. 1 and let the system size L\\ in the indirection, to the right of, be finite. We then define the perpendicular system size simply as L± = /(Z/ ), where l(x) is the interface displacement. We are interested in large L\\ and large II (close to wetting), but the ratio / i may vary. We distinguish the thermodynamic limit, where this ratio is large, and the critical limit, where it is small. Plausible finite-size scaling arguments, that correctly embody the spatial anisotropy, then imply Here, /f (:E ) is a universal scaling function. We emphasize that y pertains to the infinite system and diverges at T w. The finite-size dependence becomes conspicuous when T smgl is rewritten as A similar scaling law describes how L± depends on L\\.

28 336 J. O. Indekeu The scaling functions could be calculated in both the thermodynamic and the critical limit, in the mean-field regime. The effect of finite size on r is, of course, especially important in systems with forces that are sufficiently long-ranged for r to diverge at wetting. For example, for non-retarded van der Waals forces we recall that r diverges approaching the wetting transition, according to (2.13). Now, connected to this is the divergence with system size, after taking the critical limit. At t = 0, Another interesting example is a = 2, relevant to elastic forces between interfaces bounding nematic liquid crystals. 45 Here, since a < 2, r diverges also away from t = 0. This is seen in the thermodynamic limit, where which diverges logarithmically with system size Lj_, at fixed (small) t. Conversely, in the critical limit the divergence is stronger. At t = 0, since = 2/{a + 1) = 2/3 is the mean-field value for the anisotropy exponent, which relates L± to Ljj. The slope dr/do also shows interesting universal behavior near wetting. Its divergence (towards oo), generally observed on approach to a first-order wetting transition, is also curtailed by finite-size effects, in the manner independently of the range of the forces. This remarkably simple result is consistent with finite-size scaling and was therefore argued to be valid also in the fluctuation regime Fluctuations at continuous wetting Continuous wetting is a critical (or multicritical) phenomenon for which a meanfield regime, a weak fluctuation regime, and a strong fluctuation regime have been identified. 6,37 ' 38 This classification extends to the level of the line tension, through the recently derived exponent equality 34

29 Line Tension at Wetting 337 which implies that universality classes for the surface singularities at continuous wetting are in one-to-one correspondence with those for the line. Here, i/y is the exponent of the divergence of the parallel interface correlation length y, as t * 0~, approaching T w from partial wetting. One may also consider approaching the transition along the field direction, with T fixed at T w, and h» 0 +, and invoke a crossover exponent A (see, e.g. Ref. 37), but this will not be pursued here. Scaling arguments that lead to (3.25) run parallel to those invoked at firstorder wetting. In particular, (3.3) is again the key hypothesis. The important difference with respect to first-order wetting, is the absence of a regular background contribution r(0). Indeed, since continuous wetting involves a single and divergent length scale, the line tension and its singular part (3.3) should coincide near T w. Note that this does not necessarily mean that r should vanish at continuous wetting! Whether it vanishes, remains finite, or diverges, depends on the critical exponents that conspire in (3.3) to produce 2 ct\, and on possible logarithmic corrections, should a\ 2 (and on the behavior of the amplitude r_). But it almost certainly means that, if r s i ng vanishes at T w, then r vanishes also. Clearly, this was not true at first-order wetting, as, for example, (2.17) and (2.18) testify. In sum, we obtain for continuous wetting. A more detailed scaling argument 34 also runs parallel to that given for first-order wetting, and is largely inspired by Refs. 37 and 38. The anisotropy of the interfacial correlations is as described by (3.7), featuring the thermal wandering exponent (d) (3 c?)/2 in 1 < d < 3. The upper critical dimension is modified, however, and depends on the order n of the multicriticality of the wetting transition, 6,22,37 ' 38 for long-range forces, where n = 2 for critical wetting, n = 3 for tricritical wetting, etc. Note that d u for first-order wetting (and for complete wetting) is incidentally recovered by setting n = 1 in this expression. Furthermore, d u = 3, for short-range forces. We are again fortunate that for most systems d u < 3, so that mean-field results apply correctly in d = 3. The singular part of r can now be obtained using (3.11), (3.12), and (3.13), which apply also to continuous wetting. The result is (3.14), which we expect to hold in all fluctuation regimes (and which is also satisfied by the mean-field results). In the following we will restrict attention to thermal fluctuations. For applications to other types of fluctuations, we refer to Ref. 34. For thermal fluctuations the familiar hyperscaling relation for the surface exponents, (3.15), holds also for continuous

30 338 J. O. Indekeu wetting, and (3.13) then implies hyperscaling for the line, The exponents for continuous wetting and long-range forces are summarized in Table 2 (which has been derived using Table I of Ref. 34). Note that the fluctuation regime, in which the previous hyperscaling relations apply, is split in two subregimes. In the so-called weak fluctuation subregime, the fluctuation-induced repulsion of the interface can be inserted, as an additional term, precisely between the leading and next-to-leading terms in the mean-field interface potential. As a consequence, it is possible to evaluate i/y directly from the effective V^/). 34 ' 37,38 In the strong fluctuation regime, on the other hand, the fluctuation-induced repulsion is more important than the leading mean-field contribution, which results in the inadequacy of calculations based on minimization of an effective V(l). Tedious functional renormalization group or exact calculations are then needed to calculate 37,38,46 Table 2. Exponents for nth order multicritical wetting and long-range forces (for critical wetting, n = 2). The mean-field regime applies for d > d u = 3 4/(a + n). The thermal fluctuation regime is divided into a weak-fluctuation regime (WF), valid for d u > d > d* = 3 4/(<T + 1), and a strong-fluctuation regime, which holds for d < d*. In the SF regime i> j is not known explicitly, except in d = 2, where i/ = 2. The results for short-range forces are this time quite different from the a» oo limit! (For details, see Ref. 22). They are given in Table 3 (based on Table II in Ref. 34). We remark that the fluctuation regime features the same exponents as the strong fluctuation subregime for long-range forces. In these regimes the exponents are universal with respect to the precise form of the (sufficiently) shortranged interactions. (Note that a > (d + l)/(3 d) is required for the SF regime to apply.) An important recent advance has been the exact calculation by Abraham et al. of T in the vicinity of the critical wetting transition in the Ising model in d = Since the line is effectively zero-dimensional in this case, r in d = 2 was provisionally referred to as "point tension" 5 34 > 47 but "point excess free energy" may be preferrable, since it is more difficult to associate a physical tension to a point than to a line or a surface.

31 Line Tension at Wetting 339 Table 3. Exponents for nth order multicritical wetting and short-range forces. The mean-field regime (MF) applies for d > d u = 3, and the thermal fluctuation regime for d < 3. Note that some of these results do not agree with taking the limit <r oo in Table 2. The result is very intriguing, which implies a logarithmic divergence, and 2 a\ = 0. This teaches us a number of things. (1) The plausible expectation that r vanishes at continuous wetting, suggested intuitively by the disappearance of the inhomogeneity C and confirmed quantitatively in the mean-field theory, may be invalid in the fluctuation regime. The present divergence of r at critical wetting appears to be purely fluctuation-induced. It is to our knowledge the first example of this sort. (2) The exponent value 2 a\ = 0 is consistent with the exponent equality a\ = CKs+^H i predicted by scaling arguments. Indeed, for critical wetting in the d = 2 Ising model, a s = 0 and v\\ = 2. This is probably the first non-trivial verification of the exponent equality in the fluctuation regime. Note that the exponent is very different from the mean-field result 2 a\ = 1. (3) The simple and general scaling hypothesis (3.3), which is a sufficient condition for the exponent equality, is probably only approximately correct! Indeed, the logarithmic divergence in (3.30) is not anticipated on the basis of (3.3), since neither S nor y were reported to possess a logarithmic correction factor. 46,47 The logarithm in (3.30) thus appears to be an important correction to scaling. The exponent equality a\ = a s 4- i/h merits further comment in the context of the hierarchy of singularities described in (1.9) (1.12). The closest analogue that we can think of is the well-known Widom exponent equality 48 relating the bulk exponents to that of the interfacial tension, near bulk T c. Also, the simple scaling assumption (3.3) is analogous to Widom's scaling argument 49 where / is the bulk free energy density and the bulk correlation length. Near T c, oc T c -T -".

32 340 J. O. Indekeu These analogies are beautiful, but can be misleading! For example, (3.32) is sometimes used to argue why 7LV should be expected to vanish at the bulk critical point. The argument goes as follows. "A priori we do not know if 7LV vanishes at T c, and since /sing vanishes but diverges, what actually happens depends on the combination of critical exponents, 2 a v. Since this equals (d \)v (hyperscaling 49 ), which is positive, we conclude that 7LV vanishes at T c ". However, this line of reasoning presumes that a possible regular background term in 7LV is zero. While this is correct for the interfacial tension at T c, it would be incorrect for the line tension at first-order wetting, as we have shown in detail A super universal exponent for T(9) In closing this section, we draw special attention to a feature that, to our knowledge, was not previously noted, and that concerns the exponent p\ s. This exponent describes the dependence of r sing on the contact angle 0, expressed in (3.17). Throughout Tables 1, 2 and 3, in the fluctuation regimes, it is seen to take the surprisingly simple form, so that (The mean-field result is obtained by setting d d u, as usual.) The universality of this result extends much further than what is normally expected. Indeed, (3.34) is independent of the order of the wetting transition! It applies equally well to first-order as to continuous (critical or multicritical) transitions. It furthermore applies to all fluctuation regimes for which hyperscaling holds. This includes not only thermal wandering, but also weaker than thermal fluctuations as found in quasiperiodic systems (see Ref. 34 for hyperscaling predictions for r s j ng in these systems). We conclude that the critical exponent of the line tension at wetting is superuniversal, provided the line tension is expressed as a function of the contact angle. This is of utmost physical interest, because the contact angle is a directly measurable variable, experimentally. Furthermore, for the Ising model the new prediction can be made that, in d 3, for first-order, critical, and tricritical wetting. In order to appreciate better the universality implied in this result, it is useful to mention that at critical wetting in the d = 3 Ising model, the exponents 2 a s and therefore also 2 a\ are highly non-universal, because the correlation length exponent j/j depends sensitively on a

33 Line Tension at Wetting 341 microscopic and temperature-dependent capillary-wave parameter (see Ref. 6 for a review). In other words, the universality is completely lost when r is expressed as a function of temperature! It should be possible, and it would be very interesting, to test (3.35) in Monte Carlo simulations. The explanation of this superuniversality is surprisingly simple, if one accepts basic hyperscaling ideas. The crucial point is that 0 is in all cases simply and directly related to the surface excess free energy (or spreading coefficient), through S oc 0 2, near wetting. We thus obtain implying that p\ s is a dimensional crossover exponent between the surface, of dimensionality d 1, and the line, of dimensionality d 2. Simple dimensional analysis then implies (3.33). In fact, a fully analogous relation holds between the bulk and the interface, at bulk criticality, which follows from the hyperscaling relations 2 a = dv and 2 Oi\ \x (d V)v. 4. Conclusions and Outlook In this review we have attempted to bring the reader up to date on both the history and the latest results concerning the behavior of the line tension at wetting transitions, in a comprehensive and (hopefully) comprehensible way. We have emphasized that the line tension at wetting consists of two parts. There may be a non-zero regular background term T(T W ), which is always expected if the wetting transition is of first order. This background term is, of course, important for answering the question whether or not r vanishes at wetting. However, if one is interested in the universal aspects associated with critical phenomena, the non-universal background term plays only a minor role. Quantities of primary interest then are the critical exponent 2 c*i, and the amplitude ratio T+/T- } both being universal numbers describing the singular behavior of r at wetting. We have discussed that the behavior of r depends crucially on the order of the wetting transition. At first-order wetting, r does not vanish, and the contact line converges to a macroscopic inhomogeneity. This inhomogeneity and its line tension have various features in common with the non-critical interface and its surface tension at a bulk critical endpoint. In fact, in the same way that complete wetting can be regarded as a critical phenomenon for the interface, first-order wetting can be identified with an interface critical endpoint. On the other hand, at continuous wetting transitions the contact line disappears and its tension vanishes, at least in mean-field theory. Thermal fluctuations may modify this picture drastically. There is evidence for a fluctuation-induced divergence of r at critical wetting in the d = 2 Ising model.

34 342 J. O. Indekeu We have also described how r depends on the range of the intermolecular forces. If the forces are sufficiently long-ranged, the thermodynamic limit for the line tension does not exist. However, in those cases it is interesting to study the finite-size dependence of r, and one may hope that experimental measurement of r is easier in systems where r is anomalously large and diverges, as the logarithm, or as a small positive power, of the system size. Finally, we have examined the effect of thermal fluctuations on the critical exponent 2 orj, for first-order as well as continuous wetting. For systems with algebraically decaying forces, as in most physical cases, the mean-field results should remain valid, because the upper critical dimension d u is less than 3. For systems with short-range forces (Ising models, etc.) d u = 3, and fluctuation effects are more important. In d 2, and for short-range forces, the effect of fluctuations can be extremely important, as is well illustrated by the exact result in the Ising model. In general, in mean-field and fluctuation regimes alike, we believe that the exponent equality ot\ = a s + v\\ relates the line to the surface exponents at wetting. Also in general, we propose that the superuniversal exponent 2p\ s = 2(rf 2)/(rf 1) describes how the singular part of r depends on the contact angle 0. (In this result, d must be fixed at d u if the physical dimension d exceeds d u.) We now turn to mentioning a few open problems for further research. We limit ourselves to the most important issues, and do not elaborate on suggestions for further specialistic advances (such as the calculation of finite-size effects at continuous wetting, etc.). 1) While the interface displacement model has proven to be a very practical tool for studying the behavior of r at wetting, the model is still too phenomenological to be satisfactory from a more fundamental point of view (see, for example Ref. 50, and references therein). In other words, on the one hand the strength of the model is probably that it produces correct results at least at the mean-field level and that it can be easily adapted to include the effect of weak fluctuations. On the other hand, its weakness is that a description based on the profile l(x) is a caricature of a more realistic density-functional theory of Landau type. More work is needed to demonstrate that the model can be derived from a more fundamental description. Progress along this line has recently been made by Blokhuis. 51 He has found that the prediction that the line tension reaches a finite positive limit at first-order wetting, is supported by the more detailed Landau theory for wetting, from which he has been able to derive an explicit realization of the interface displacement model. This provides new evidence in support of the prediction that in the work of Perkovic et a/. 26 the "lambda"-shaped plot of r hides a cusp rather than a divergence. 2) In this review we have focussed attention on the fate of C and r at a wetting transition, where C either remains on the scene as an extended transition zone, or disappears. Alternate physical phenomena, like the possible dissociation of a contact line into two distinct contact lines, C\ and 2, also occur and have been the subject of early theoretical anticipation (by J. W. Gibbs), recent experimental observations

35 Line Tension at Wetting 343 (by J. W. Cahn), and lattice mean-field calculations in Ising models. 52 It would be interesting to study this contact-line wetting transition, where an interface intrudes between two contact lines, from the point of view of critical phenomena and to examine the mean-field and fluctuation regimes. There are indications that a similar contact-line wetting transition is involved in the static description of contact angle hysteresis using the interface displacement model, and, in particular, in connection with the determination of the receding contact angle. This topic also provides a natural link to the dynamics of wetting. 2 We refer to Ref. 20 for a critical discussion. 3) The possible interplay between wetting and bulk criticality has been left undiscussed, as well as the effect of gravity on contact line phenomena. We have set the bulk correlation length equal to a microscopic constant, and the capillary length to infinity. Naturally, new regimes and crossover phenomena should be expected as soon as these lengths interfere with the wetting layer thickness or the interface correlation lengths that we discussed. This is an interesting research area, which has remained largely unexplored. Progress along this line has recently been made in a study of the three-phase contact line in an Ising model at two-phase coexistence against a neutral wall (with zero surface field). Then, under sufficient surface coupling enhancement, so that the extraordinary transition is achieved, the contact line persists at and even above bulk criticality, in the form of a boundary between ordered surface phases. The critical behavior of r under these circumstances is interesting, and results in new exponent equalities. 53 4) When three phases meet in a line of common contact, and if this contact line has significant curvature (as is the case for a tiny droplet on a substrate), then the line tension r affects the contact angle 0. Preliminary phenomenological studies of this interesting phenomenon were made independently by Clarke, Nijmeijer, and Widom, and led to the unusual result that, under the constraint of fixed volume for the droplet, the wetting transition in the finite system is accompanied by a jump of 0, from a finite value to zero. 54 An accompanying jump in r/r, where R is the radius of the contact circle, then keeps the free energy continuous, as it should be./furthermore, a variety of interesting power laws, expressing the sensitivity of 9 to r, have been uncovered. We believe it would be worthwhile to investigate this problem in more detail (taking into account the order of the infinite-system wetting transition, the range of the forces, and fluctuation effects), using the interface displacement model supplemented with an appropriate fixed volume constraint. 5) On the experimental side, an investigation of the line tension at wetting is almost within the range of current expertise. Direct measurement of r has proven to be possible but fairly difficult (for three-phase fluid contact lines, see Refs. 3 and 4, for solid-liquid-vapor contact lines, see Ref. 5, and for the boundary line between domains within a two-dimensional Langmuir monolayer, see Ref. 55). Recently, special attention has been devoted to the possibility of measuring r indirectly, through determinations of the relaxation time of metastable thin films in the complete wetting regime T > T w. 28 ' 56 This strategy looks promising, especially for systems with forces of sufficiently long range for r to diverge at wetting. Through the relation