APS/123-QED

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Wetting phase transitions and critical phenomena in condensed matter∗

J.O. IndekeuInstitute for Theoretical Physics, Katholieke Universiteit Leuven, BE-3001 Leuven, Belgium

(Dated: August 30, 2009)

Equilibrium wetting phase transitions and critical phenomena are discussed from a phenomenologicalpoint of view. The ubiquitous character of the wetting phase transition is illustrated through itsoccurrence in a variety of condensed matter systems, ranging from classical fluids to superconductorsand Bose-Einstein condensates. The intriguing behaviour of the three-phase contact line and its linetension, at wetting, is an example of a fundamental problem in this field on which much progresshas been made.

I. SCOPE

In these lectures three examples of systems that display equilibrium wetting phase transitions are discussed. Thefirst of these involves mixtures of classical fluids. In this context, which follows the lines of experimental investigations,the basic concepts of the phenomenology of wetting are introduced. The second example deals with type-I supercon-ductors, solid state systems which already in the 1960’s were the subject of debates around wetting-like problems, andwhich are now fully recognized to possess a rich wetting phase diagram, a small part of which has been observed. Thethird illustration treats mixtures of dilute ultracold gases that undergo Bose-Einstein condensation (BEC) and forwhich theoretical work convincingly predicts unusual wetting behaviour, which can be manipulated by tuning basicatomic properties. So far no experimental demonstration has been given for wetting in this system. The lecturesclose with recalling the fundamental problem of the behaviour of the three-phase contact line and its line tension τ ,at or very near to a wetting phase transition (for an artist’s impression, see Fig.1). The decisive role of the order ofthe wetting transition (first-order or critical) and of the range of the forces (with exponential or algebraic decay) isclarified and the effect of fluctuations (in thermal equilibrium) on the singular behaviour of τ is outlined.

FIG. 1: Cartoon of a three-phase contact line (dashed line) where solid, liquid and vapour phases meet, and some line tensionproperties – see Section V. Drawing: Frank-Ivo Van Damme; caligraphy: Joke van den Brandt; design by the author (1994).

A new review on wetting and spreading [1] can serve to complement what is being presented here. Although thereis some overlap with the section on classical fluids, most of our topics (type-I superconductors, BEC and line tensionat wetting) are not covered in the review. It is hoped that these lectures can provide a new incentive for recognizingand studying equilibrium wetting phenomena in other systems than hitherto envisaged and that they can stimulatefurther experimental efforts to demonstrate wetting phase transitions and critical phenomena in condensed matter.

∗ Version Nr 2 of lecture notes for the Summer School “Fundamental Problems in Statistical Physics XII” to be held in Leuven, Belgium,September 2009.

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II. CLASSICAL FLUIDS

A. Contact angle and film thickness

Binary liquid mixtures of linear alkanes and methanol constitute an interesting soft condensed matter systemfor demonstrating a variety of wetting phase transitions [2, 3]. In Fig.2 droplets of methanol inside linear alkaneCn ≡ CnH2n+2 are shown, suspended from the liquid-vapour interface. The contact angle θ which the liquid-liquidinterface makes with the liquid-vapour interface is larger (smaller) than 90 for C6 (C11), while it is close to 90 forC8. This contact angle is the first of two key variables in the wetting problem. Its value in thermal equilibrium isdetermined by a force balance, known as Young’s law, for the interfacial tensions acting on the circular three-phasecontact line along which methanol, alkane and vapour meet [4]. This balance of forces per unit length, or free energiesper unit area, reads

γV,A = γV,M + γM,A cos θ, (1)

where V , A and M denote vapour, alkane and methanol, respectively.

FIG. 2: Methanol droplets immersed in n-alkane and adsorbed at the liquid-vapour interface, as first shown in [2]. The vapourphase (black) is at the top.

All the configurations shown in Fig.2 are partial or incomplete wetting states. For such states the three phasesconcerned can be in pairwise mutual contact along interfaces that intersect at a line where all three phases meet.However, simultaneous direct contact between every two out of the three phases is not possible when a condition arisesthat we term complete wetting. One of the phases then fully intrudes, or spreads, between the other two, forming amacroscopic wetting layer. If the drop of methanol in Fig.1 would completely wet the V-A interface, its (equilibrium)contact angle would be identically zero, and we would obtain the special case of Young’s law called Antonov’s rule [4]

γV,A = γV,M + γM,A, complete wetting by M (2)

Alternatively, complete wetting of the V-M interface by A (θ = π) is also conceivable, and could be termed “completedrying” if A were a gas and V a solid substrate. The transition, from partial to complete wetting, is a thermodynamicphase transition of surface type [5, 6], which can be brought about by varying, for example, the temperature T .

Fig.3 shows experimental data for the T -dependence of 1− cos θ for C11 and C9. One observes the vanishing of thecontact angle at some particular temperature Tw below the consolute point Tc of the binary liquid mixture [2, 7] andfor T closer to Tc, θ remains zero (always assuming thermal equilibrium). The wetting transition at Tw is characterizedby specific singularities in surface thermodynamic functions. For example, the surface excess free energy singularityat wetting is contained in the behaviour of the spreading coefficient S, defined as the difference of surface free energiesγV,A of partial and complete wetting states, (1) and (2),

S = γM,A(cos θ − 1) (3)

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FIG. 3: Variation of 1− cos θ with temperature, after [1, 2]. On the left are data for C11 and on the right for C9.

Note that, in equilibrium, S ≤ 0, since a state with S > 0 relaxes to S = 0. Since γA,M is perfectly smooth near Tw,the singularity of S is directly apparent from the behavior of 1 − cos θ shown in Fig.3. By analogy with bulk phasetransitions and critical phenomena, one writes

−S ∝ |T − Tw|2−αs , (4)

where αs is the surface specific heat exponent. For C11 Fig.3 (left) suggests that S vanishes in a linear fashion, sothat αs = 1. This kind of wetting transition is of first order, because the first derivative of S w.r.t. T is discontinuous.In contrast, for C9 (Fig.3; right) S appears to approach zero tangentially (with vanishing derivative), or αs < 1. Thisis typical of a critical wetting transition. For α = 0 this criticality is of second order, while for α = −1 it is of thirdorder, etc. Also non-integer values of αs are possible.

FIG. 4: Methanol film thickness versus temperature, after [1, 2].

The second key variable is the thickness ` of the adsorbed methanol-rich thin film, with a composition close to thatof the wetting phase (M), which decorates the V-A interface away from the droplets. This film is too thin to beperceptible in Fig.2, but its thickness can be inferred from ellipsometry measurements [7]. Fig.4 shows `(T ) for C9

and C11. The data for C11 clearly suggest a jump in ` upon traversing Tw, while those for C9 show a continuousincrease. The film thickness, which is roughly proportional to the adsorption, acts as an order parameter for thewetting transition. Its singular behaviour is characterized by the critical exponent βs in the manner

` ∝ |T − Tw|βs , (5)

where, in most conventions used, the minus sign is absorbed in the definition of βs, so βs ≤ 0. The value βs = 0is frequently found because a logarithmic divergence of ` at critical (or complete) wetting is common, but also for afirst-order transition βs = 0(jump). The two critical exponents encountered so far are independent, and combine withthe surface susceptibility exponent into the surface critical exponent equality

αs + 2βs + γs = 2, (6)

which can be proven by means of standard scaling hypotheses for surface thermodynamic functions [8], in full analogywith its counterpart α + 2β + γ = 2 for bulk critical points.

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B. Prewetting and the approach to complete wetting

So far we discussed states in which the three phases (V, M and A) coexist. We now extend our discussion tostates in which only two phases are simultaneously stable in bulk, say, V and A. Phase M is unstable in bulk andconsequently, adsorption of M at the V-A interface is necessarily limited to a thin film. Nonequilibrium droplets maynevertheless be present and disappear by diffusion (after a few weeks). A macroscopic wetting layer of M is onlypossible when M is stable in bulk. Consider now the thin film of M at the V-A interface and denote its thickness by`. We can ask in which manner ` increases when we change the thermodynamic conditions, by varying a chemicalpotential or temperature, in such a way that we approach three-phase coexistence. There are two possibilities. If thesurface state at three-phase coexistence is partial wetting, the increase of ` will be modest and ` will reach a finitevalue, similar to the value of ` shown in Fig.4 for C11 at the bottom of the jump. On the other hand, if the surfacestate at three-phase coexistence is complete wetting, ` increases and, if no other mechanism limits its extent, divergesto a value comparable to the system size (which is by default macroscopic). This divergence of ` upon approach tocomplete wetting is also characterized by a (negative) critical exponent,

` ∝ |∆µ|βcos , (7)

where ∆µ is a short notation for the difference between the chemical potential (of one of the species) and its valueat bulk coexistence, and the superscript co of βs refers to complete wetting at coexistence. When complete wettingis approached starting from states off of bulk coexistence, not only ` diverges, but also the correlation length(s)describing interface fluctuations like thermal capillary waves. The insight that complete wetting is a surface criticalphenomenon with diverging (correlation) lengths is due to Lipowsky [9] and has led to the terminology, which we willalso adopt, that critical and complete wetting are examples of continuous wetting transitions, to be contrasted withfirst-order ones.

As regards first-order transitions, states off of bulk coexistence can undergo a so-called prewetting transition in whichthe adsorption (of M) jumps between two finite amounts. The locus of such first-order transitions from a thin to athick film in the phase diagram of temperature and chemical potential is a line which starts at the first-order wettingtransition at bulk coexistence and ends at a prewetting critical point. This critical point is akin to an ordinary criticalpoint, but in dimension d− 1 (d is the bulk dimensionality). Experimentally, this first-order prewetting scenario withd− 1 = 2 Ising prewetting criticality, has been observed in binary liquid mixtures [10].

The critical reader will have noticed that neither for the first-order, nor for the critical wetting transition depictedin Fig.4, does the film thickness attain macroscopic proportions but rather stays limited to a modest 100A. Thecommonly accepted explanation for this is that, strictly speaking, the film of M adsorbed at the V-A interfaceundergoes gravitational thinning. This effect is equivalent to taking phase M slightly away from bulk coexistencewith V and A, by an amount ∆µ ∝ ∆ρgL, where ∆ρ is the mass density difference between phases A and M, g thegravitational acceleration, and L the elevation or height of the V-A interface above the A-M interface. In fact, strictlyspeaking, bulk phase coexistence between A and M is only achieved right at the A-M interface and not at any otherheight, such as that of the V-A interface. In conclusion, the first-order transition for C11 apparent in Fig.4 is in realitya prewetting transition very close to the theoretical first-order wetting transition and similar reservations apply to thecritical wetting transition for C9 [11].

C. Mean-field theories

In some exceptional cases wetting phase transitions can be studied in microscopic statistical mechanical modelsthat can be solved exactly [12]. For most systems approximate methods are available, whose accuracy can varysignificantly from case to case. For fluids, and in particular liquids, the usual starting point is a mean-field density-functional theory, which in itself is not quantitatively reliable for two main reasons. Most liquids involve van der Waalsintermolecular forces, which must be included explicitly in the model in order for realistic forces between surfaces andinterfaces to emerge from the calculations. If one just assumes short-range forces like, e.g., in a lattice-gas model ofIsing type, the resulting forces between interfaces decay exponentially with their separation. In reality, these forcesdecay algebraically and the precise decay exponent ultimately affects the critical exponents of a continuous wettingtransition. Moreover, the presence of these long-range forces can affect the order and even the very occurrence of thewetting transition.

The second reason why mean-field DFT for wetting in liquids is often not sufficient is that thermal fluctuationscan have important effects on wetting behaviour. Functional renormalization group studies have shed light on thisproblem, which is especially subtle in systems with short-range forces in d = 3 [13]. For systems with van der Waalsforces, thermal fluctuations are less important, at least in d = 3, since the upper critical dimension, above which mean-field critical behaviour is observed, is less than 3. If we move away from liquids and, for example, turn to classical

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superconductors as we will do in the next Section, the status of the mean-field DFT can change from approximate tovery precise. Incidentally, the Bardeen-Cooper-Schrieffer (BCS) theory is a mean-field theory.

The first mean-field DFT we discuss is a model, proposed by Widom et al., for our three-phase fluid system [14, 15].The model is basic, has very few parameters, and is probably the best tool for appreciating the remarkable feature of(all) wetting theories that seemingly small changes in the model parameters can lead to qualitatively different wettingtransitions. The model features two spatially varying densities ρ1(r) and ρ2(r). These could be the number densitiesof alkane and methanol molecules. The surface free energy, associated with, for example, the interface between phasesA and M, is the following functional of these densities,

γ[ρ1, ρ2] =∫ ∞

−∞dx

12(∇ρ1(r))2 +

12(∇ρ2(r))2 + F (ρ1(r), ρ2(r); b)

, (8)

where b is a parameter with which one can vary, e.g., the location of the bulk densities of the three phases in the(ρ1, ρ2)-plane. In order to find the equilibrium surface free energies associated with the various interfaces one wishesto study, as well as the structure and tension of the three-phase contact line, one minimizes functionals akin to (8)w.r.t. ρ1(r) and ρ2(r). Especially in the case of three phases in mutual contact with finite dihedral angles betweenany two interfaces (partial wetting), for which the line tension functional is a double integral, over x and y, andthe densities are inhomogeneous in these two spatial directions, this problem can be very demanding numerically forstates close to a wetting transition.

Recently, two interesting aspects of this model have become clear [14, 15]: i) depending on the choice of the functionF (ρ1, ρ2; b), which is only required to possess three local (non-colinear) minima in the (ρ1, ρ2)-plane representing thethree bulk phases, first-order or second-order wetting transitions can occur; and ii) precise numerical investigationsof the model have allowed to verify quantitatively a number of predictions concerning the singular behaviour of ther-modynamic functions at or very near the wetting transition, predictions based on simpler and more phenomenologicaldensity functional theories.

The second mean-field DFT we draw attention to can be viewed as a special case of the first. One of the phasesis considered to be an inert “spectator” phase or “wall”. The vapour in our V-A-M system could be this phase, or asolid substrate could assume this role. In such situations, the adsorbate consists of two phases, occupies a half-spacex > 0, and is described by a single density m(r), which is often modeled by the magnetization variable of the Isingmodel, so that m ∈ [−1, 1]. The properties of the spectator phase are incorporated in a surface contact energy. Theparadigm of this DFT is the Nakanishi-Fisher model [16], which is based on the Cahn-Landau DFT proposed by Cahnin his seminal study of wetting phase transitions near bulk critical points [5] and on the alternative DFT proposed byEbner and Saam for studying prewetting [17]. The surface free-energy functional reads

γ[m] = −h1m(0)− g

2m(0)2 +

∫ ∞

0

dx

12(∇m(r))2 + F (m(r))

, (9)

where the surface contact energy features h1, the surface field that can induce wetting, and g, the surface enhancementwhich affects the order of the wetting transition. The function F (m) is usually taken to be the quartic double wellpotential which gives the bulk free energy density for an order parameter constrained to take the value m. For adetailed discussion of this DFT the reader is invited to consult earlier lecture notes by the author [18]. This DFTpredicts first-order and critical wetting, with a tricritical wetting transition as cross-over point [16]. Interestingly, thewetting transitions observed for mixtures of alkanes and methanol [19] fit quite well in the scenario predicted by thisDFT, as Fig.5 illustrates. The reason for this good agreement is that van der Waals forces, which are neglected inthe DFT, are less influential when the wetting transition is close to the bulk critical point. It is then mainly the largebulk correlation length which is the important length scale, and within the window of experimentally accessible filmthicknesses, critical wetting behavior for systems with short-range forces is observed for C9. In particular, αs is foundto be close to zero and the divergence of ` is found to be logarithmic [2].

Thirdly, a more microscopic DFT has been developed in which van der Waals forces between molecules are includedfrom the start [20–22]. From such DFT interface Hamiltonians for wetting have been derived [23] and along this routeeffects of thermal capillary wave fluctuations can be studied, starting from the intrinsic mean-field density profile. Weclose this section with the remark that van der Waals forces are responsible for the critical wetting transition (withαs = −1) observed for alkanes on water [3, 24].

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FIG. 5: Global wetting phase diagram for mixtures of alkanes and methanol against their vapour, in effective chain length neff

and reduced temperature, after [1, 19].

III. TYPE-I SUPERCONDUCTORS

A. Interfaces and “wetting”

The seminal paper of Landau and Ginzburg of 1950 [25], which lies at the basis of (most) mean-field theories forcritical phenomena, focused on superconductivity and in particular on the properties of the interface between normaland superconducting phases in type-I materials. In this class of materials, which includes the pure metals Al, In, Sn,Ta and Pb, the bulk transition to superconductivity in an external magnetic field H is of first order. The bulk phaseboundary ends at a critical temperature Tc in zero field. This first-order phase boundary is denoted by Hc(T ), whereHc is, misleadingly, termed “critical” field. The state in which normal (N) and superconducting (SC) phases coexistas macroscopic domains, separated by an interface, is called the “intermediate state”.

Already in the 1960’s the possibility was discussed “that either the SC or the N phase wets the specimen surface”[26]. The specimen surface is simply the outer surface of the sample or “wall” (W) against vacuum or some otherinsulator. However, this possibility was considered unlikely since Pippard had argued [27] that the difference in surfacefree energies |γW,SC − γW,N | is almost certainly small compared to the interfacial tension γSC,N . Notwithstandingthis apparent impossibility of “wetting”, it was recognized in 1963, in a famous paper by Saint James and de Gennes[28], that superconductivity may start at the specimen surface, and that a surface superconductivity transition canprecede the transition in bulk, when T is lowered in a non-zero external field applied parallel to the surface. Thesurface transition is denoted by Hc3(T ).

Thus, for type-I materials which display surface superconductivity, a thin superconducting surface sheath forms infields Hc(T ) < H < Hc3(T ). The thickness ` of this sheath is of the order of the bulk superconducting coherence lengthξ. From a wetting perspective the natural question to ask is whether ` remains of order ξ or becomes macroscopicallylarge instead, when bulk two-phase coexistence is approached by lowering H or lowering T . Calculations withinGinzburg-Landau (GL) theory by Speth in 1986 [29] confirmed Pippard’s anticipation: ` remains finite and the SCphase does not wet the surface. (Also the N phase does not wet the surface.)

In the mid 1990’s, however, it was discovered that the GL theory predicts the possibility of complete wetting bythe SC phase for surfaces which are different from the standard choices of surfaces against vacuum or insulators(which feature the Neumann boundary condition (bc) for the superconducting wave function ψ), or against metalsor ferromagnets (Dirichlet bc, ψ(x = 0) = 0). For yet a different type of boundary condition at the surface, knownas Robin bc, corresponding to enhancement of superconductivity, wetting phase transitions were predicted [30, 31].Before discussing the theoretical and experimental implications, we comment briefly on the relevant length scales ofour problem. A broad and thorough introduction to the main properties of superconductors can be found in widelyused textbooks [32, 33].

The main bulk lengths are the coherence length ξ and the magnetic penetration depth λ, both of which diverge atTc in zero field in the manner |Tc−T |−1/2. Their ratio (when evaluated near Tc) is a material constant termed the GLparameter κ = λ/ξ. Type-I superconductors lie in the interval 0 < κ < 1/

√2, while the technologically more useful

type-II materials have larger κ. Type-I superconductors are characterized by a positive interfacial tension γSC,N > 0,

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FIG. 6: Left: H-T phase diagram for surface superconductivity with Neumann bc. Hc(T ) marks the bulk transition from thenormal (N) to the Meissner state (SC) and Hc3(T ) marks the surface transition. Right: superconducting surface sheath ofthickness `.

leading to stable phase separation between normal and Meissner phases, whereas type-II materials have γSC,N < 0,implying that N and SC phases do not segregate but mix down to the quantum limit of the magnetic flux line orvortex (“mixed state”). A superconducting surface sheath can arise whenever κ > 0.41. Strongly type-I materialshave large ξ (e.g., for Al, ξ is of the order of a micron and κ ≈ 0.01).

The main surface length for our purposes is the so-called surface extrapolation length b. For the standard Neumannbc we have b = ∞, and for the Dirichlet bc, b = 0+. The more general Robin bc is [32]

dψ

dx

∣∣∣∣0

=ψ(0)

b, (10)

and given that the sample occupies x ≥ 0, surface enhancement of superconductivity corresponds to b < 0, whilesurface suppression of superconductivity occurs for b > 0. The latter is achieved for a surface in contact with anormal metal. The question now arises under which physical circumstances the rather unusual condition b < 0 canbe obtained.

Surface enhancement of superconductivity has been studied, theoretically and experimentally, already since 1969 byFink and Joiner [34], who achieved it by cold working of the specimen surface. This method was recently investigatedmore systematically by Kozhevnikov et al. [35]. Enhancement of superconductivity at twinning planes (or more gen-

FIG. 7: Top: Wall-Normal surface possibly decorated with a superconducting surface sheath of thickness `. Middle: Wall-Superconducting surface. Bottom: Superconducting-Normal interface.

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erally, grain boundaries) was observed by Khlyustikov in the early 1980’s and developed theoretically by Buzdin. Fora review, see [36]. Several other works have dealt with b < 0 and therefore surface enhancement of superconductivityis not an academic curiosity but an experimental reality, and it is not limited to classical superconductors. Let usgive an example in the arena of so-called unconventional superconductors. Enhancement of superconductivity wasinvoked to explain the strongly increased transition temperature observed at the interface between the p−wave tripletsuperconductor Sr2RuO4 and Ru inclusions embedded in the bulk material [37, 38]. Observations of and mechanismsfor surface enhancement of superconductivity continue to be provided in current-day research [39].

To set the stage for the wetting problem, we recall the main thermodynamic (in-)equalities concerned. Firstly, weassume - and calculations and experiments confirm - that almost always the SC phase is preferred by the wall, so that

γW,SC < γW,N (11)

This signifies that there is preferential adsorption of the SC phase. As long as

γW,N < γW,SC + γSC,N (12)

at most a microscopically thin superconducting sheath is stable at the wall and partial wetting is observed. (Notethat a film of zero thickness is also included here as a possibility.) We henceforth refer to this state as one with alocalized or bound SC-N interface. On the other hand, complete wetting by the SC phase takes place if

γW,N = γW,SC + γSC,N , (13)

implying a macroscopically thick SC wetting layer. We henceforth refer to this state as one with a delocalized orunbound SC-N interface. The wetting transition is termed interface delocalization transition, which is a more generalterm without (misleading) allusion to liquids and commonly used in wetting studies in lattice spin models.

B. Ginzburg-Landau theory for interface delocalization

Classical superconductors are condensed matter systems that are exceptional in that the mean-field approximation(BCS theory) describes the experimental data very well, and the GL mean-field theory, which can be derived fromBCS theory, is ubiquitously used as a reliable theoretical tool [32, 33]. Within GL theory the surface free energyfunctional for a semi-infinite sample is given by

γ[ψ,A] =∫ ∞

0

dx

1

2m

∣∣∣∣(~i∇− 2eA

)ψ

∣∣∣∣2

+∣∣∣∣∇×A− µ0H

2µ0

∣∣∣∣2

+ α | ψ |2 +β

2| ψ |4

+~2

2m

| ψ(0) |2b

, (14)

in a parallel magnetic field H = Hez. The complex order parameter ψ(r), or wave function, is coupled to theelectromagnetic vector potential A(r), which determines the magnetic induction through B = ∇×A. The coefficientsα and β are standardly derived from BCS theory

α =~2

2mξ(0)2T − Tc

Tc, and β =

8π~2e2

m2c2κ2, (15)

where ξ(0) = ξ(T = 0) and m is the electron mass. Note that derivations of b from microscopic material constantsare not standardly available, and therefore we treat b as a phenomenological parameter whose value can be inferred,e.g., by calculating or measuring ξ(Tcs), where Tcs is a surface critical temperature to be discussed below.

In the following we recall briefly how the GL equations, which are the Euler-Lagrange equations of the functional(14), are derived. Details and a full discussion of the calculations leading to the interface delocalization phase diagramcan be found in [31]. While the Maxwell equations require continuity of B at the surface x = 0, implying B(0) = µ0H,the bc for ψ compatible with (14) is the gauge-invariant condition

(∇− 2ieA/~)xψ|0 =ψ(0)

b, (16)

where x denotes the component normal to the wall. The GL equations now follow by minimizing (14) w.r.t. ψ andA,

12m

(~i∇− 2eA

)2

ψ + αψ + βψ | ψ |2 = 0, GL I (17)

∇× (∇×A) = µ0J, GL II, (18)

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with J the current density. Note that GL II is a Maxwell equation. Also note that the normal current Jx is zero andthat shielding currents, if any, are parallel to the wall. The problem can now be considerably simplified by assumingtranslational invariance in y and z directions (appropriate for studying Meissner states in type-I materials) and byfixing the gauge so that only Ay(x) is nonzero. Further, one can take the wave function to be a real and positivefunction of x. After suitable scaling of the variables, in particular ψ → φ, Ay → a and Hz → h , the GL equationsand bc’s reduce to

φ = ±φ + a2φ + κ2φ3, GL I (19)a = aφ, GL II, (20)

where +(−) refers to T > (<)Tc, and

φ(0) =ξ

bφ(0) (21)

a(0) = h (22)

We also need to specify the bulk conditions at x →∞. These depend on whether the normal phase N is imposed inbulk, in which case φ(∞) = 0 and a(∞) = h, or the SC phase, in which case φ(∞) = 1/κ and a(∞) = 0.

Our problem can be handled elegantly using the phase portrait method, which exploits the dynamical analogybetween our GL equations and Newton’s equations of motion of a particle with coordinates φ and a moving in amechanical potential V = ∓φ2/2 − a2φ2/2 − κ2φ4/4 as a function of time x. Energy conservation implies φ2/2 +a2/2 + V = E, where the constant E depends on the imposed bulk state. We can now compute trajectories in the(a, φ)-plane corresponding to W-N, W-SC and SC-N interfaces, and evaluate their free energies using (14). We can thencheck Young’s equation and determine the interface delocalization properties. The phase portrait technique allows toobtain curves of initial conditions φ(0)(φ(0)) and to identify first-order phase transitions of interface delocalizationor pre-delocalization (“prewetting”) character from an equal-areas rule in the (φ(0), φ(0))-plane. This is a powerfulapproach because it allows a fairly direct inspection of the structure of the solutions to the GL equations and theirrelative stability. Also for the determination of critical interface delocalization transitions the phase portraits arevery useful, especially when they are complemented with an analytical study of the trajectories near the SC fixedpoint in the contour map of V (a, φ). A combination of these and other analytical and numerical methods (includingperturbation expansions in φ for monitoring nucleation of superconductivity and rapidly converging expansions in κfor obtaining global analytic solutions in the type-I regime) has allowed to obtain accurate interface delocalizationphase diagrams of which we recall the most representative ones here.

The global interface delocalization phase diagram of type-I superconductors with a surface characterized by a surfaceextrapolation length b is presented in Fig.8. This diagram pertains to the condition of bulk two-phase coexistencebetween SC and N phases, that is, it assumes H = Hc(T ). The variable ξ/b is temperature-dependent through theT -dependence of ξ, while b is supposed to be approximately constant. The sign of b is crucial. For b > 0 surfacesuperconductivity is suppressed and no interface delocalization is possible. For b < 0, at fixed κ (fixed bulk material),an interface delocalization transition occurs when T is increased towards Tc (when |ξ/b| is increased). For κ < 0.374the transition is of first order (segment FD), while in the remainder of the type-I regime it is critical (segment CD).For critical delocalization the wall-interface distance ` diverges in the manner, for fixed κ and along bulk coexistence,H = Hc(T ),

`(T ) ∝ ln(

1TD − T

), (23)

where TD denotes the interface delocalization temperature. This behaviour (implying βs = 0) is typical for systemswith short-range forces, implicit in the GL theory. The line CD ends on the line FD at a critical endpoint (CEP).Incidentally, a similar CEP topology has been observed experimentally in the wetting phase diagram of linear alkaneson water with added glucose [40]. This is an interesting variation on the more standard wetting tricritical-pointtopology found in the 3d Ising model [16] and experimentally for alkane-methanol mixtures against their vapour [19],which we discussed in the classical fluids section (Fig.5). Besides the interface delocalization transitions the phasediagram also features continuous and first-order (equilibrium) nucleation transitions of surface superconductivity,joined at a tricritical point. In these transitions the thickness ` of the superconducting surface sheath remains finiteand the interface localized. In this global phase diagram we understand clearly why the standard bc (Neumann;b = ∞) does not lead to “wetting”. The only phenomenon that emerges at ξ/b = 0 is the nucleation of surfacesuperconductivity at κ = 0.41, leading to the property Hc3(T ) > Hc(T ) for materials with κ > 0.41 with a surfaceagainst vacuum or an insulator.

Although the global phase diagram is the best tool for obtaining an overview of the various possible scenario’s ofinterface delocalization, more system-specific phase diagrams are directly relevant to experiments. The most useful

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ML

FD

I =

I = 0

∞

∞

I > 0CD

TCP

FNCN

RA

CEP

κC

ξ/b

l =

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.10

0.1

0.2

0.3

0.4

0.5

0.6

κ

FIG. 8: Global interface delocalization phase diagram for type-I superconductors, composed from the phase diagrams reportedin [30, 31, 54]. The variables ξ/b and κ contain temperature and material constants, respectively (κc =

√2/2). The line

FD, which starts for κ = 0 at ξ/b = −0.6022, is a first-order interface delocalization line, while CD is a critical interfacedelocalization line (with non-universal critical exponents). The complete wetting regime is indicated by ` = ∞. FD meets CDat a critical endpoint (CEP) and FD continues into a short first-order nucleation line (FN), which, at a tricritical point (TCP),changes into critical nucleation (CN). RA is the line of reversal of preferential adsorption (to the right of this line the N phaseis preferred by the wall instead of the SC phase.

representation is the H − T phase diagram, as in Fig.6 (left), which was valid for b = ∞ and κ > 0.41. We now askhow this phase diagram is modified for surfaces with b < 0. For κ < 0.374 the phase diagram of Fig.9 replaces thatof Fig.6 for b < 0. Instead of the surface transition line Hc3(T ) we find a line of “prewetting” transitions, emanatingfrom the first-order interface delocalization point D, and terminating at a surface critical point at H = 0 and T = Tcs.The equilibrium thickness of the superconducting surface sheath jumps from ` = 0 to a finite value (of order ξ), whenH or T are lowered across the transition line. Note that a short segment of the prewetting line is of second order.The cross-over occurs at a surface tricritical point. Note also that in contrast to the case b = ∞ (Fig.6) the surfacetransition in zero field no longer coincides with the bulk critical temperature Tc, but takes place at a distinct highertemperature Tcs, which in GL theory satisfies

ξ(Tcs)/b = −1, (24)

from which b can be determined once Tcs is known. The function ξ(T ) is the zero-field coherence length, in this casefor temperatures close to and above Tc. Once the prewetting line has been crossed, the sheath thickness ` divergesupon approach of the bulk transition at Hc(T ) in the manner, for fixed T > TD,

`(H) ∝ ln(

1H −Hc

), (25)

characteristic for the approach to complete wetting for systems with short-range forces. Needless to say, all thefundamental thermodynamic properties of the prewetting line near D, found for fluids and lattice systems, carry overto type-I superconductors [41]. At least two independent experiments have confirmed the scenario of Fig.9, as we willdiscuss in the next subsection. For κ > 0.374 qualitatively different physics is found, since the delocalization transitionis now critical. The phase diagram which now replaces Fig.6 is shown in Fig.10. The interface delocalization point Dat bulk two-phase coexistence is now entirely disconnected from the surface transition which is now a second-ordernucleation line ending at Tcs in zero field. We may call this line Hc3(T ) as long as we remember that it is differentfrom the Hc3(T ) found for b = ∞, which ends at Tc and H = 0.

In closing this subsection on interface delocalization phase diagrams predicted by GL theory we draw attention tothe fact that the topology of the phase diagram depends on the precise form of the imposed bc’s. In particular, ifwe assume, instead of the semi-infinite system with a surface equipped with b < 0, a grain boundary (e.g., twinningplane) with identical enhancement of superconductivity on both sides of the defect, and in addition a continuous orderparameter ψ at the defect plane, which corresponds to full transparancy for electrons [42], a qualitatively differentglobal phase diagram is obtained. For low κ the difference with Fig.9 is minor, but at larger κ there is no criticalwetting (!) and the line FD continues smoothly up till the point (ξ/b = 0, κ = 1/

√2) [43]. Similar calculations, for

the transparent twinning plane with enhancement, were essentially already done by Buzdin in the 1980’s, but at thattime the significance of the phase diagram in terms of (pre-)wetting and interface delocalization was not recognized[36].

Another qualitative modification of the phase diagram results when the surface of the superconductor is in contactwith another material, located at z < 0, that is already superconducting in bulk. Then, by the proximity effect [32, 33],

11

FIG. 9: Interface delocalization phase diagram for a type-I superconductor with κ = 0.2. D = (TD, HD) denotes the first-orderinterface delocalizaton transition. The prewetting line is of first order between D and TCP, and critical between TCP andH = 0.

FIG. 10: Interface delocalization phase diagram for a type-I superconductor with κ = 0.5. D = (TD, HD) denotes the criticalinterface delocalizaton transition and CN is the critical nucleation line of surface superconductivity.

a superconducting surface sheath is a priori present at the surface while the bulk, at x → ∞, is in the normal state(phase N). The investigation of the question whether the thickness of the sheath diverges when Hc(T ) is approachedcan then lead to an interface delocalization phase diagram which resembles very closely the standard prewetting phasediagram with a first-order prewetting line terminating at a simple prewetting critical point [44, 45]. This proximity-induced variant of interface delocalization in superconductors was rediscovered, fully independently, in 2005 [46]. Thisenhances the reality of wetting transitions in superconductors, since, in the words of Sherlock Holmes: “When youfollow two separate chains of thought, Watson, you will find some point of intersection which should approximate thetruth.” [47]

12

FIG. 11: Experimental first-order interface delocalization phase diagram for surface enhanced Sn. The point W correspondsto D in Fig.9. Enhancement was achieved through polishing or ion irradiation of the surface. The solid line is the bulkcoexistence curve Hc(T ); the dashed line is the surface transition Hs(T ). The line in the inset is a theoretical curve fitted tothe ∆H(T ) ≡ Hs(T )−Hc(T ) data for the ion-implanted sample.

C. Experimental verification, theoretical refinements and open problems

It is an interesting fact that the first experimental observation of prewetting was not made in fluids but in supercon-ductors, although it was not recognized as being prewetting (see, e.g., Fig.11 in [36]), and also the wetting transitionat bulk coexistence was missed. In fluids, the first successful observation of prewetting, after several fierce attemptsin the early 1980’s [48], was reported in 1993 [10]. It should be added, however, that first-order wetting transitionshad already been observed in fluids [6, 49] but it turned out to be very difficult to observe also the prewetting line.In superconductors the converse appears to be true, since experimental detection of surface superconductivity aboveHc(T ) is straightforward by very sensitive SQUID magnetometer measurements of the (small) diamagnetic momentarising from field expulsion in the surface sheath [35] or defect plane sheath [36].

The experimental study of interface delocalization in type-I superconductors has been facilitated by preliminarydevelopments of techniques for obtaining controllable and reproducible surface enhancement. A lot of progress hasbeen made since the pioneering efforts of Fink and Joiner [34] who cold worked InBi samples and obtained both anincrease in the critical field and in the transition temperature. Mechanical polishing and subsequent annealing wasapplied more systematically in recent years for demonstrating reproducible surface transition temperatures exceedingbulk Tc in Sn samples [35], similar to those used by Khlyustikov who attributed the enhancement of superconductivityto twinning planes. That interpretation was, however, challenged once it became clear that annealing can suppressthe effect almost entirely, while twinning planes are robust and cannot be annealed. Moreover, even more systematiccontrol of surface enhancement was achieved by ion bombardment of the surface, paving the way for a more thoroughobservation of the first-order interface delocalization transition and prewetting in surface enhanced Sn samples [50].Fig.11 shows the results of recent measurements.

Measurement of the total diamagnetic moment of the sample is an integral approach, capable of revealing the totalextent of the superconducting surface sheath without, however, giving direct information about the spatial magneticinduction profile B(x). Spin polarized neutron reflectometry [51] and low-energy muon spin rotation techniques[52] allow mapping a spatially resolved diamagnetic response which can reveal the sheath thickness ` as well asgive information on deviations from the exponential decay of the penetrating magnetic field. For strongly type-Isuperconductors Pippard predicted important deviations from the predictions of local London electrodynamics, sincethe size of the Cooper pairs, or Pippard coherence length ξ0, is large compared to λ. Consequently, the current densityJ depends on A in a nonlocal manner, and the current at r samples the influence of A over a volume of linear sizeξ0 about r. Nonlocal electrodynamics effects are thus expected to affect in a fundamental way the magnetic fieldpenetration B(x) in the strongly type-I superconductors Al, In and Sn. It is therefore imperative that the theoreticalstudy of the interface delocalization problem for ultra-low κ materials be extended beyond GL theory, to includenonlocal electrodynamics effects. To our knowledge this problem is still open.

The analogy between interface delocalization in superconductors and in fluids goes very far. For partial wettingstates, superconducting “droplets” can form at the sample surface, taking the shape of one-dimensional sausages

13

elongated parallel to the direction of the externally applied magnetic field. Their interface displacement profile `(y)shows a typical S-shape, which can be calculated exactly within GL theory in the κ → 0 limit [45, 53]. Experimentalmethods capable of revealing the spatial magnetic induction profile may verify this. Also in the limit κ → 0 theinterface potential for the first-order interface delocalization transition can be obtained analytically [53] and variousproperties can be studied using series expansions in κ, for small κ [54]. One of the successes of the perturbationapproach lies in the elegant fundamental problem of the calculation of the SC-N interfacial tension posed by Landauand Ginzburg in their seminal 1950 paper [25], and for which they outlined the basic procedure. Decisive progress onthe κ-expansion of γSC,N was obtained by Mishonov [55] and a finishing touch was delivered a few years later [56]. Thispiece of analytic progress was reviewed a decade ago by Ginzburg at age 80, in a paper entitled “Superconductivityand superfluidity (what was done and what was not)” [57]. The result for the reduced interfacial tension is

ΓSC,N = 4πγSC,N/(ξH2c ) = 2

√2/3− 1.02817

√κ− 0.13307κ

√κ +O(κ2

√κ) (26)

The expansion was shown to be so accurate in the entire type-I regime (0 ≤ κ < 1/√

2) that derivation of higher-orderterms is unnecessary.

In the late 1990’s it was realized that the critical wetting transition in type-I superconductors must be governedby non-universal critical exponents, which depend on the ratio of the two relevant bulk lengths λ and ξ [58]. Thisis to be expected for mean-field theories with more than one diverging length at criticality [59]. Consequently, thespreading coefficient singularity at critical wetting is characterized by

2− αs = (1−√

2κ)−1, (27)

while βs = 0 according to (23). This provides another incentive for further experimental exploration of the globalphase diagram (Fig.8).

For finite systems, in particular mesoscopic samples, surface enhancement of superconductivity leads to interestingeffects, including capillary condensation phenomena [60] and an overall increase of Tc [61–63]. The size-dependentcritical temperature (in zero field) Tc(L) can be significantly larger than the bulk Tc and the surface transitiontemperature for the semi-infinite system Tcs. Also, depending on the geometry and surface curvature of the sample(sphere, cylinder, slab), the manner in which Tc(L) scales and decreases towards Tcs for L → ∞ can take differentforms, algebraic or exponential. However, in spite of extensive theoretical work, few experiments have been devotedto this route for increasing Tc. Also vortex configurations in thin films and disks change qualitatively when surfaceenhancement is allowed for [63].

Although thermal fluctuation effects are fully negligible for most properties of classical superconductors, the questionwhether capillary-wave fluctuations affect the physics of critical interface delocalization in type-I superconductors ispertinent and has been investigated [64]. The external magnetic field is oriented along one of the two directionsparallel to the SC-N interface. For interface fluctuations which cause the magnetic field lines to bend, with wavevector q parallel to H, the interface is stiffer than for perpendicular fluctuations, which leave the field lines straight.It turns out that for the former the upper critical dimension above which the GL theory is valid is less than 3, sodu < 3, while for the latter du = 3. In either case, the interface width remains finite when the interface unbinds fromthe wall and the predictions of the GL theory for the critical exponents αs and βs remain correct in d = 3.

A final point is concerned with a question, raised by de Gennes (private communication), addressing the possibleimportance of dispersion forces, with algebraic decay at long distances and not included in GL theory. Such forcesare expected to arise because the frequency- and wave-vector-dependent dielectric functions ε(q, ω) are not identicalin SC and N phases of the same metal. Even neglecting spatial dispersion and taking into account only the frequencydependence already leads to contributions to the free energy per unit area of the interaction between interfaces atseparation ` that decay as `−5 [65]. Note that for fluids dispersion forces lead to contributions that decay much moreslowly, as `−2. Blossey predicted that such algebraic contributions add on to the exponentially decaying ones thatGL theory predicts, and may lead to modifications of the critical wetting scenario (for κ ≈ 0.5) while the effect isnegligible for κ ¿ 1. Since the calculations performed so far are incomplete, this difficult and interesting problemremains to a large extent unexplored.

IV. BOSE-EINSTEIN CONDENSATES

A. Physical motivation

We are used to think that interactions between particles are responsible for phase transitions in large systems.Solid-fluid phase transitions are possible for systems with short-range repulsive forces between particles, the paradigm

14

being hard spheres. Further, if we add attractive tails to the forces, a liquid-gas phase transition becomes possible inaddition to solid-liquid and solid-gas ones. It thus comes as quite a surprise that an ideal gas of particles, for whichthe interactions are zero, can nevertheless display a phase transition. The purely quantum statistical phase transitionthat comes to mind here is the Bose-Einstein condensation (BEC) transition for a dilute gas of N bosons of mass min a volume V , which takes place when the temperature is low enough for the thermal de Broglie wavelength λT ofthe atoms to span the interparticle distance,

λT ≡√

2π ~√mkT

> 1.38 (V/N)1/3 (28)

Another way to formulate this criterion is to say that the transition occurs when the confinement energy exceeds thethermal energy. In reality, particles do interact, but for very dilute gases the interactions can successfully be treatedas perturbations and the BEC is modified, even qualitatively, but persists. Also the fact that in the experimentsparticles are confined in a (harmonic) magneto-optical trap potential does not spoil the phase transition [66].

BEC is often viewed as a phase transition in momentum space (the condensate particles being in the k = 0 state,for an infinite system, and in the quantum ground state of the trap, for a confined system) rather than in realspace. Therefore, phase separation in real space, with an interface between the condensate and the normal fraction,is difficult to envisage (unless an external potential is invoked [67]). This leads us to consider a binary mixture ofBE condensates, as the adsorbate for the wetting problem, since phase separation of mixtures of BE condensates ofalkali atoms was experimentally established [68]. The two species in this mixture are remarkably alike, being twodifferent hyperfine states of the same isotope of the same atom, 87Rb. A surface phase transition in which one of thecondensates is excluded from contact with the outer region of a cylindrical trap was predicted in [69, 70], and this wassoon followed by the prediction of a genuine wetting phase transition (at zero contact angle) in standard semi-infinitesystem geometry [71]. The wetting phase transition in BEC mixtures has to our knowledge not yet been observedexperimentally. We discuss its theoretical derivation here, limiting ourselves to T = 0. For an extension to finite T ,see [72].

B. Mean-field theory

We employ the Bogoliubov mean-field theory for BEC, which is a very good approximation in the dilute limit

ρa3 ¿ 1, (29)

where ρ = N/V is the particle number density and a the so-called s-wave scattering length, which is a characteristiclength that can be significantly larger than the atomic diameter, entering in the amplitude of the effective δ-functionFermi pseudo-potential associated with the interatomic forces. More precisely, the effective potential is

Vps(rij) =4πaij~2

mijδ(ri − rj), (30)

with 2m−1ij = (m−1

i + m−1j ). Note that there are two different species and therefore in principle two masses and

three scattering lengths. There is no simple relation between a12 and the like-particle scattering lengths a11 and a22.We recall that the usefulness of the pseudo-potential stems from the irrelevance of the precise shape of the true pairpotentials in ultracold dilute gases, when the thermal wavelength λT exceeds the interparticle distance. Under thesecircumstances the true potential V is smeared out to Vps according to

Vps(rij) ≈∫

λ3T

dr′i

∫

λ3T

dr′j |ψi(r′i − ri)|2V (r′ij) |ψj(r′j − rj)|2 (31)

At ultralow T the (kinetic) energy of the particles is so small that the entire particle structure relevant for thescattering can be subsumed in a single length scale, the s-wave scattering length a. It takes typical values of, e.g., 49A for 23Na and 100 A for 87Rb. Positive (negative) a signifies repulsive (attractive) forces. Note that for ideal hardspheres, a equals the particle diameter. It is worth emphasizing that very few atomic constants are needed in ourdescription, just the atomic mass and the scattering length.

1. Order parameter

The next point of particular attention is concerned with the order parameter of BEC. Ideally, a good order parameterwould be the fraction of particles, N0/V , in the condensate, being the k = 0 or zero-momentum state in a (semi-)

15

infinite system. A condensate quantum amplitude ψ0 can be defined so that

N0

V= |ψ0|2 =< O| a†0√

V

a0√V|O >, (32)

where |O > denotes the ground state of the interacting, i.e., a 6= 0, system. In this ground state, not all particleshave zero momentum. There is a depletion consisting of particles with k 6= 0. For instance, some pairs of particleswith opposite momenta are generated through the interparticle interactions. These particles are said to be “out ofthe condensate”. To lowest order in a the condensate depletion is given by the following singular contribution

N −N0

N=

83

√Na3

πV(33)

For a large system (N,V →∞) and for a finite condensate fraction, the following approximation is justified,

N0

V≈ | < O| a0√

V|O > |2 (34)

Now, in view of the property that the total momentum of the ground state is zero, the right hand side is preciselythe modulus squared of the ground state expectation value of the Boson field operator

ψ(r) =1√V

∑

k

eik.rak, (35)

which is related as follows to the total number of particles per unit volume,

N

V=< O|ψ†(r)ψ(r)|O > (36)

We thus conclude

N0

V≈ | < O|ψ(r)|O > |2 (37)

This means that the condensate amplitude or wave function is well approximated by the ground state expectationvalue of the Boson field operator itself. In case we deal with a spatially inhomogeneous system we can thus define theBogoliubov condensate order parameter as

ψ(r) ≡< O|ψ(r)|O >, (38)

which will be our best approximation to the true ψ0(r).The basic grand potential Ω = H −µN for our semi-infinite system, with a planar substrate occupying z ≤ 0, then

becomes the functional

Ω[ψ1, ψ2] =∫

z≥0

dr∑

i=1,2

[ψi(r)∗

(− ~2

2mi∇2 − µi

)ψi(r) +

Gii

2|ψi(r)|4

]+ G12|ψ1(r)|2|ψ2(r)|2 (39)

In this functional we recognize the quantum kinetic energy associated with the condensate wave functions for con-densates 1 and 2, with respective chemical potentials µ1 and µ2, and the potential energy for interactions betweenparticles belonging to like and unlike species. The positive coupling constants are proportional to the scatteringlengths through

Gij =4π~2

mijaij (40)

2. Wall boundary condition

The simple Dirichlet boundary condition (bc) standardly considered in the textbooks in the context of BEC in ahalf-space,

ψ1(x, y, 0) = ψ2(x, y, 0) = 0, (41)

16

Blue

detuned

laser

light

Phase 1

Phase 2

PrismPrism

Phase 1

Phase 2

FIG. 12: Left: partial wetting state of BEC mixtures adsorbed at an optical wall. Right: complete wetting.

is physically already a very reasonable approximation for our adsorbate against an optical wall. Indeed, the closestexperimentalists have come to a “hard wall” bc for a BEC is to use an evanescent wave emanating from a planar prism.A repulsive potential results when the frequency of the (totally internally reflected) laser light is blue detuned relativeto the main absorption frequency of the atoms in the gas. This leads to a potential which turns on exponentiallywith a characteristic length of roughly 80nm (λ/2π). The gas is then trapped for z ≤ 0 by this optical wall, and forz > 0 it is confined by a conventional magnetic trap which corresponds to a 3d harmonic potential about the origin.The characteristic length of this harmonic potential is typically at least 5000nm, which is two orders of magnitudelarger, and this is why the optical wall acts more or less like an impenetrable wall at z = 0 for which the Dirichlet bc(vanishing of the order parameter) is a reasonable first approximation [73]. Nevertheless, it is worthwhile to extendour present point of view and to consider also softer walls; this was done and the results are very interesting in thatthey provide a qualitatively richer wetting phase diagram than that for the hard wall [72].

The setting for observing the transition is thus schematically shown in Fig.12, which is inspired by the optical surfacetrap employed by Grimm et al. [74]. The adsorbate consists of two BEC phases in a phase-separated binary BECmixture. The substrate is an optical wall or “atom mirror”, formed by an evanescent electromagnetic wave emanatingfrom a prism in which laser light is totally internally reflected. On the left the partial wetting state is depicted, whilecomplete wetting is illustrated on the right. Note that gravity plays little or no role here in configurational aspectsinvolving the relative positions of the two phases, since the particles of all species concerned have (almost exactly)equal mass. The capillary length is consequently very large. However, gravity generally influences the confinement ofthe gas mixture as a whole [74].

C. Wetting phase transition

The variational principle applied to the surface grand potential functional (39) evidently leads to two coupledEuler-Lagrange equations, which were studied by Gross and, independently, by Pitaevskii in 1961, and bear theirnames. Bulk phase separation in the binary mixture is found to occur only if the repulsion between unlike atoms issufficiently stronger than that between like atoms. Specifically, two-phase coexistence is possible for

K ≡ G12√G11G22

> 1 (42)

Further, the specific pressures of the pure species condensates 1 and 2 are required to be equal at two-phase coexistence,i.e.,

P1 = P2, with Pi =µ2

i

2Gii(43)

We recall that, by definition, the actual pressure P is the grand potential per unit volume of the entire system.Interestingly, the two phases that coexist are pure 1 and pure 2, whereas more generally impure phases rich in 1 andrich in 2 might have been expected.

At K = 1 a bulk triple point is found, where a mixed phase (stable for K < 1) coexists with the pure phases 1and 2. This bulk triple point is anomalous in at least one respect. The interfacial tension between pure 1 and pure2 goes to zero when K is lowered towards 1. This behavior is normally expected at a bulk critical point, but at ourtriple point at K = 1 phases 1 and 2 remain fully distinct in bulk, at z = −∞ and z = ∞, respectively. However, theinterface width diverges and concomitantly the order parameter gradients vanish for all z. Therefore, in practice, asregards wetting phenomena, it is convenient to imagine the point (K = 1, P1 = P2) to play the role of an ordinarybulk critical point.

17

The bulk condensate number density for species i is given by

ρ0i ≡ |ψi(∞)|2 =µi

Gii(44)

Besides the relevant length scale set by the scattering length a there is also the healing length ξ, defined for eachspecies as

ξi =~√

2miρ0iGii, (45)

which is typically of the order of 100nm. The healing length, whence its name, is the typical length scale over whichthe order parameter profile that solves the Gross-Pitaevskii (GP) equation(s), recovers from its value zero at the wallto a value close to its asymptotic value ρ0 in bulk. The fact that this length is of the same order of magnitude asthe turn-on length of the optical wall, which is 80nm as we discussed previously, suggests that the hard wall bc maynot be sufficient for describing the experimental system in a fully self-consistent way. We come back to this when wecomment on the results for soft walls.

Although the healing length is apparently a property of an inhomogeneous order parameter profile near a surface,its value is entirely determined by bulk parameters and not by any surface characteristics. Indeed, ξ is identical tothe cross-over length that enters the bulk dispersion relation for excitations of wave vector k in the Bose gas, forρ0a

3 ¿ 1,

ω(k) ≈ ~k2m

√k2 + 16πρ0a (46)

For long wavelengths (low energy), i.e., k ¿ ξ−1, the excitations are phonon-like with energy linear in k, and forshort wavelengths, k À ξ−1, the free-particle energy spectrum, which is quadratic in k, takes over. Note that theslope of the phonon spectrum determines the superfluid velocity cs ∝ ~/(mξ). Note that cs is finite by virtue of theinteractions, since cs ∝

√a. In other words, the ideal (non-interacting) BEC is not a superfluid because its critical

velocity, below which frictionless flow is possible, is zero. Superfluidity in ultracold dilute Bose gases is a consequenceof the (weak) interactions.

Although the complex condensate order parameter ψi(r) depends on all three spatial coordinates x, y, and z, wecan consider its phase to be constant (e.g., zero) if we limit ourselves to static configurations without fluid flow, whichis sufficient to study equilibrium wetting. Also, we can consider its modulus to depend only on the distance z fromthe wall, if we ignore transverse fluctuations in the usual mean-field spirit. All calculations were thus performed with“one-dimensional” scalar order parameters f1(z) and f2(z). The coupled non-linear GP equations then take the form

ξ21

d2f1

dz2= −f1 + f3

1 + Kf1f22

ξ22

d2f2

dz2= −f2 + f3

2 + Kf2f21 (47)

Already the standard textbooks provide, starting from the GP equation for a single condensate (or for K = 0), thesimple and intuitively clear expressions for the wall-adsorbate surface excess energies, or wall energies (at T = 0) [75],

γwi =4√

23

Pξi, (48)

which simply express that the wall energy is a positive quantity equal to the characteristic bulk energy per unitvolume, which is the pressure P , multiplied by a length scale which is of the order of the healing length. We concludethat the healing lengths of the two condensates are the relevant quantities for deciding which condensate is preferredby the (optical) wall. A smaller healing length corresponds to a lower surface energy. This leads us to introduce asurface field proportional to the difference of the two healing lengths, or, in dimensionless form, ξ1/ξ2 − 1. The ratioξ1/ξ2 thus plays the role of the parameter which may induce wetting.

1. First-order wetting

Two qualitatively different regimes deserve our attention. The first is the strong segregation limit, K → ∞, inwhich the interface between condensates 1 and 2 is quite sharp, with very little mutual penetration of order parameter

18

0

f f2 11

zξ ξ ξΛΛ22 1 12

FIG. 13: Order parameter profiles of two BE condensates in the strong segregation limit. An optical wall is assumed on theleft, at z = 0.

0

1

z

f f2 1

ξ Λ2 1 2

Λ

FIG. 14: Order parameter profiles of two BE condensates in the weak segregation limit. An optical wall is assumed on the left,at z = 0.

profiles. Fig.13 shows calculated order parameter profiles for this regime, with at z = 0 a contact between the walland condensate 2, and further on, an interface between condensates 2 and 1. The GP equations imply that, near theinterface, condensate 2 penetrates into 1 over a depth

Λ2 =ξ2√

K − 1, (49)

which is tiny compared to the healing length, in the large-K limit. Consequently, for large K the GP equations implythe following equality of wall and interfacial energies relevant to the wetting problem,

γ12 ≈ γw1 + γw2, (50)

where γ12 is the interfacial tension between 1 and 2. Since complete wetting requires the interfacial tension to becomeequal to the difference of the two wall energies, we can already conclude that complete wetting is practically impossiblein the strong segregation limit. The 1-2 interface is just too costly in energy.

More luck is on our side in the so-called weak segregation limit K ↓ 1, which is incidentally the regime encounteredin experiments [69]. As (49) and Fig.14 show, the penetration depth is now much larger than the healing length.The condensates are strongly interpenetrating and from the GP theory one can derive that the interfacial tension isgreatly reduced relative to its value in the strong segregation limit, roughly in the manner

γ12,weak ≈ γ12,strong

√K − 1, (51)

which can now easily become equal to the difference |γw1 − γw2|. The wetting phase boundary can be calculatedexactly, and reads

K − 1 =29

(ξ1

ξ2− ξ2

ξ1

)2

, (52)

which opens up parabolically at the bulk triple point at K = 1 and extends into the half-space K > 1. Fig.15 displaysthe wetting phase diagram for BEC mixtures adsorbed at a hard wall for T = 0, within GP theory, in the space ofsurface field and relative interaction strength. Note that, if indeed we take the liberty to perceive the triple point atK = 1 as a critical point, this phase diagram is compatible with the “critical-point wetting” scenario advocated byCahn [5]. In this scenario one of the two phases must wet the interface between the wall and the other phase, uponapproach of the bulk critical point where γ12 vanishes.

It is easy to verify that the wetting transition predicted here is of first order. Indeed, for ξ1 > ξ2, for instance, thesurface energy curves γw1 and γw2 + γ12 meet at a corner (and their metastable extensions intersect) as a function

19

FIG. 15: Global wetting phase diagram for mixtures of BE condensates at a hard optical wall, at T = 0.

of K at fixed ξ1/ξ2. Thus, the first derivative of the equilibrium surface grand potential is discontinuous at wetting.Two configurations that are in equilibrium at the wetting transition are, on the one hand, a wall in contact withcondensate 1 and, on the other hand, a wall in contact with condensate 2, which, at z = ∞, goes over into condensate1 via a 2-1 interface.

However, the grand potential is infinitely degenerate at wetting. There exists a one-parameter family of config-urations, corresponding to a continuously varying amount of phase 2 adsorbed between the wall and phase 1. Inother words, the grand potential at wetting is invariant under inflating the adsorption of the wetting phase. An exactsymmetry in the GP equations that could prove this invariance has hitherto not been found, but interface potentialcalculations confirm the degeneracy of Ω [72, 76]. It turns out that the interface potential is fully flat at wetting, asa function of the wetting layer thickness `, while infinitesimally away from wetting the minimum of this constrainedenergy lies at ` = 0 and ` = ∞, for partial and complete wetting, respectively.

2. Critical prewetting

More anomalies lie waiting for us, as soon as we consider also prewetting transitions off of bulk two-phase coexistence.By varying the chemical potentials of the two species, or varying the interaction strengths between like atoms, onecan change the specific pressure ratio P1/P2, which allows one to go away from bulk coexistence into the one-phaseregion. Upon approach of bulk coexistence the GP theory reveals a prewetting transition of second order, in whichan infinitesimal amount of the wetting phase is “nucleated” at the wall. Fig.16 shows a line of continuous prewettingtransitions in the space of relative interaction strength K and bulk field

√P1/P2. The latter equals 1 at bulk

coexistence. Along the dashed path, approaching bulk coexistence, a wetting layer develops after the prewetting lineis traversed, and the wetting layer thickness diverges continuously as bulk coexistence (P1 = P2) is reached. This isconform to what is standardly called complete wetting, which is a continuous transition.

The global phase diagram, however, does not follow the rather general expectation that a first-order wettingtransition is accompanied by a first-order prewetting line which meets the bulk two-phase coexistence line tangentiallyat the wetting point. Instead, we uncover a second-order prewetting line, which meets bulk coexistence at a finiteangle. Our extraordinary scenario, as unexpected as it may be, does not violate any (surface) thermodynamical laws[71].

D. Outlook to experiments and conclusions

We now ask how the global wetting phase diagram can be explored experimentally. A first observation is that thepertinent variables ξ1/ξ2 and K depend in a simple way on very few atomic constants. We can write

ξ1

ξ2=

(m2a22

m1a11

)1/4

, (53)

20

FIG. 16: Prewetting phase diagram for mixtures of BE condensates at a hard optical wall, at T = 0.

and

K =m1 + m2

2√

m1m2

(a12√a11a22

)(54)

Since the masses are fixed atomic constants, we ask whether it is possible to manipulate the scattering lengths.Interestingly, it is possible to change all three scattering lengths largely independently by a factor of 10 to 100 byemploying the technique of Feshbach resonances [77]. A Feshbach resonance arises when the energy of a bound stateof two particles almost coincides with the energy of a colliding pair of the same but freely scattering particles. Themagnetic moments being different in the two states, an external magnetic field allows to tune the energy of one staterelative to the other, and to create a resonance. We conclude that BEC mixtures provide unique opportunities forstudying wetting phase transitions, in that fundamental atomic parameters that can be tuned by the experimentalist,directly control the basic variables appropriate for exploring the phase diagram.

We come back to the issue of the reliability of the hard wall approximation for the optical atom mirror potential.Ideally, the turn-on length of the potential should be much smaller than all the length scales that enter in theinhomogenous GP solutions. This is not really the case. Therefore, a study was carried out which extends ourcalculations to soft (exponentially rising) walls [72]. The most important findings in this study were i) the confirmationof the robustness of the wetting phase transition scenario and ii) the prediction of critical wetting transitions in additionto first-order ones, in the global phase diagram which now also features a length scale associated with the characterof the wall.

We conclude that the possibility of wetting phase transitions in mixtures of Bose-Einstein condensates has beenpredicted based on Gross-Pitaevskii theory. At T = 0 the wetting transition is of first-order for hard walls, andcan be critical when the wall is soft. Extraordinary features, or anomalies, include the infinite degeneracy of thegrand potential at wetting, the critical character of the prewetting transition, and the non-tangential approach of theprewetting line to the bulk two-phase coexistence line at the wetting point. Important open problems, which havebeen addressed to some extent in hitherto unpublished work [72], are the role of thermal fluctuations at T > 0, andthe role of quantum fluctuations beyond GP theory, even at T = 0.

V. LINE TENSION AT WETTING

The problem of the line tension at wetting is not only a fundamental one in statistical mechanics, it is also a veryintriguing one. The reason for its fundamental nature is that the behaviour of the three-phase contact line at wettingdisplays the full physics of phase transitions and critical phenomena, including universal and non-universal aspects.The reason for its exceptional intrigue is that the effects of the order of the wetting transition on the one hand,and of the range of the forces on the other hand, are strongly amplified at the level of the (one-dimensional) contactline thermodynamics with respect to their significance at the level of (two-dimensional) surface thermodynamics ofwetting. This amplification is so strong that, for roughly a decade, the answer to the question “How big is the line

21

tension at wetting?”, seemed to be randomly wandering between zero and infinity, even between publications by thesame research group.

Our understanding of the three-phase contact line and its tension has considerably deepened over the last threedecades, and as a result more subtle issues and more sophisticated settings have entered in the focus of the lateststudies. For a recent review of state-of-the-art line tension research, with emphasis on conceptual aspects, see [78].For a review of experimental and theoretical determinations of the magnitude and sign of the line tension, we referthe reader to [79].

Higher numerical precision has recently permitted experts to settle once and for all some dangling uncertaintiespertaining to the line tension at wetting [14]. In these lectures, we limit ourselves to discussing the main questionsand providing an intuitive understanding of the main answers concerning this specific problem. More details of thecrucial developments, which took place in the early 1990’s, can be found in an earlier review [80] . For a thoroughunderstanding, also nonequilibrium phenomena associated with first-order wetting transitions must be taken intoconsideration [81].

A. The intriguing fate of the contact line and its tension at wetting

The main question here is concerned with the fate of the three-phase contact line when the dihedral contact angleθ approaches zero. What happens to the contact line, and to its excess free energy per unit length, called line tensionτ , when the wetting transition is approached? The question seems to be ill-defined, because in a complete wettingstate (θ = 0) the wetting phase intrudes between the other two phases and consequently the contact line seems tohave disappeared. It is therefore not surprising that several theoretical studies performed between 1980 en 1991 cameto the plausible conclusion the line tension vanishes at wetting. This can be expected for the excess free energy of astructure that ceases to exist. However, several other works between 1986 and 1992, based on similar models, cameto the unexpected conclusion that the line tension appears to increase without bound as θ → 0. Clearly, the behaviorof the line tension at wetting is sensitive, very sensitive, to details in the models.

In 1992 it was shown [82] that a simple interface displacement model suffices to elucidate that the order of thewetting transition is one such detail that is all-important. If the transition is of first order, the contact line does notdisappear at wetting, but develops into an extended transition zone connecting a thin adsorbed film to a macroscopicwetting layer. Consequently, τ does not vanish. If the transition is critical, on the other hand, the contact linegradually disappears, and so does its tension, when the adsorbed film thickness diverges in a continuous mannerfor θ → 0. Another detail that was shown to be all-important as well, is the range of the forces. Intermolecularvan der Waals forces lead to very different behavior of τ than short-range forces (e.g., of finite range or exponentialdecay). Some examples may illustrate these relevant aspects. For first-order wetting and non-retarded van der Waalsforces, τ diverges at wetting, in the manner τ ∝ log(1/θ), while for short-range forces τ reaches a finite positive valueτ(0) at wetting. For critical wetting and non-retarded van der Waals forces, τ vanishes at wetting, in the mannerτ(θ) ∝ −θ1/3, while for short-range forces τ(θ) ∝ −θ.

The quantitative predictions of the simple interface displacement model (IDM) allowed to interpret the variousseemingly contradictory results from other models, some of which are more microscopic and therefore more difficultto analyze numerically. In later years it was shown that i) the local approximation inherent in the functional of theIDM provides results that compare very well to those of a full nonlocal theory, as long as the interface displacement isnot varying too rapidly in space [83], and ii) the phenomenological collective coordinate approximation also inherentin the IDM leads to results that are in very good agreement with the most accurate computations to date based ona more microscopic density functional theory with two spatially varying order parameters [14].

In the remainder of this subsection we aim at understanding in simple terms why the order of the transition and therange of the forces play such a crucial role in τ . To do this, let us start from the simple mean-field theory embodiedin the basic IDM. The line tension τ [`] is given as the following functional of the interface displacement `(x), with xrunning parallel to the substrate plane and perpendicular to the contact line,

τ [`] =∫ ∞

−∞dx

γ

√1 +

(d`

dx

)2

− 1

+ V (`(x)) + c(x)

(55)

This functional features the free energy cost, per unit length of the contact line, of interface deformations whichincrease the interfacial area relative to that of a flat plane. This cost is proportional to the (liquid-vapor) interfacialtension γ and the increment ds − dx, with ds2 = d`2 + dx2. The second contribution concerns the free energy costof imposing an interface displacement ` different from the equilibrium value. For a uniform configuration, with `independent of x, this cost is given by the interface potential V (`), per unit area. This interface potential thus gives

22

FIG. 17: Interface displacement profile for partial wetting and anticipating a first-order wetting transition, after [82].

FIG. 18: Interface displacement profile for partial wetting and anticipating a critical wetting transition, after [82].

the constrained surface free energy of an adsorbed film of arbitrary imposed thickness `. Naturally, the minimum ofV (`) is reached for the equilibrium thickness of the adsorbed liquid film, which we denote by `1. For partial wetting,`1 is finite, while for complete wetting the minimum occurs at ` = ∞. A first-order wetting transition is described byan interface potential with two equal minima, one at ` = `1 and one at ` = ∞, separated by a free energy barrier. Acritical wetting transition on the other hand corresponds to an interface potential without barrier and with a singleminimum, which continuously moves out to infinity, `1 → ∞, in the limit θ → 0. We adopt the local approximationand extend this free energy cost to non-uniform displacements simply by replacing ` by `(x) in the argument of V .Finally we add a piece-wise constant function c(x) which ensures that the integrand approaches zero smoothly in thetwo limits, x → −∞, where the displacement reaches the value of the thin equilibrium adsorbed film, and x → ∞,where the displacement assumes the slope dl/dx ∼ tan θ appropriate for a liquid wedge with contact angle θ. Thepresence of c(x) reveals that this model features an intrinsic dividing line, where the asymptotic planar configurationsintersect. For more details, the reader is invited to look up [82].

For our present purposes we can consider d`/dx small enough to linearize the functional in the squared gradient.Minimization of the line tension functional is then equivalent to finding least action trajectories for a particle withposition ` at time x, moving in a potential −V . The Euler-Lagrange equation is then akin to Newton’s law,

γd2`

dx2=

dV

d`, (56)

which is to be solved respecting the boundary conditions ` → `1 for x → −∞, and ` → tan θ x ≈ θ x, for x → ∞.A simple convention is to choose V (`1) = 0, so that V (∞) = −S, S ≤ 0 being the spreading coefficient, which givesthe excess free energy of a macroscopic wetting layer relative to that of the equilibrium film. One readily checks that,with −S = γ(1− cos θ) ≈ γθ2/2, the “constant of the motion”

γ

2

(d`

dx

)2

− V (`) = 0 (57)

is satisfied by the entire profile `(x). In the mechanical analogy the particle moves with zero total energy and its

23

velocity is proportional to√

V (`(x)). We illustrate the resulting interface displacement profiles for two cases. Thefirst is a partial wetting state in the vicinity of a first-order wetting transition (Fig.17), for which `(x) displays thecharacteristic S-shape due to the presence of a barrier in V (`). The second is a partial wetting state in the vicinity ofa critical wetting transition (Fig.18), for which `(x) has a monotonic slope.

Extension of this calculation beyond the gradient-squared approximation is straightforward and allows to discusswedges with contact angles up till 90 [84]. The equilibrium line tension is the value τ [`] attains in the optimal profile,and is easily found to be

τ(S) =√

2γ

∫ ∞

`1

d`√

V (`)−√−S

, (58)

from which the value τ(0) is immediately seen to be proportional to the integral over the square root of the barrierin V (`), for first-order wetting, or equal to zero, for critical wetting, provided

√V (`) is integrable for large `. Let

us illustrate this result for the concrete case of a d-dimensional bulk system with intermolecular pair potentials φ(r)which for large r decay as

φ(r) ∝ r−(d+σ), (59)

which lead to an interface potential decaying as [85]

V (`) ∝ `−(σ−1) (60)

The reason why the exponent of ` is raised by d+1 w.r.t. that of r is that the free energy per unit area of the interfaceat a distance ` from the substrate is obtained by integrating contributions φ(r12)ρ(r1)ρ(r2) first over a half-space, sayz2 ≤ z, and then integrating the resulting energy density, or wall potential, from, say ` to ∞. Here, the ρ’s are particlenumber densities. Note that σ = 3 (4) for non-retarded (retarded) van der Waals forces in d = 3. We conclude thatwhether the line tension diverges or not depends in a crucial way on the tail of V (`). This hardly surprising pointwas already emphasized by de Feyter and Vrij in 1972 [86] who studied borders of thin soap films.

The IDM teaches us first of all that, already for partial wetting, the line tension diverges if σ ≤ 2. This is notunexpected since bulk energy densities diverge for σ ≤ 0 and interfacial tensions diverge for σ ≤ 1. The line tension isjust the next thermodynamic quantity in a hierarchy of ever more sensitive excess energies. It is well known that, inorder for the thermodynamic limit to exist, the forces must be of sufficiently short range. Further, we now reach theinteresting conclusion that, for a first-order wetting transition (i.e., when a barrier in V (`) persists), the line tensionreaches a finite and positive limit provided σ > 3 and diverges for σ ≤ 3. The divergence of τ(0) for the very physicalcase of non-retarded van der Waals forces may surprise us, but it was already anticipated by Joanny and de Gennesin 1986 [87] although the crucial additional condition that the wetting transition be of first order was not addressedin their paper. In contrast, for critical wetting, τ(0) = 0 for van der Waals forces. More precisely, τ ∝ −θ(σ−2)/σ forθ → 0 and for σ > 2 [82].

Perhaps the most surprising prediction from this simple model is the fact that for short-range forces and first-orderwetting a finite limit τ(0) > 0 results, because previous works had not speculated on this possibility of an intermediateanswer between either zero or positive infinity. Interestingly, the microscopic density functional theory (DFT) withtwo order parameters turned out to display, upon ever more thorough scrutiny, singular behavior of the line tensionat wetting in excellent agreement with the asymptotic analytic forms predicted by the IDM [14]. This is the case forthe approach to the first-order wetting transition along three-phase coexistence as well as along the prewetting line.The characteristic cusp-like behavior of τ is accurately displayed in Fig.19.

Experimental measurements of the line tension near first-order wetting are scarce. In 1999 Wang, Betelu andLaw [88] studied liquid drops on a solid substrate, varying the temperature towards first-order wetting transitiontemperatures. Their results show that τ is negative far from the transition, then becomes positive and continues toincrease as θ is reduced, in qualitative agreement with the IDM and the microscopic DFT.

B. Transition zone between a thin film and a macroscopic wetting layer

While the vanishing of the contact line and the line tension near critical wetting can be understood fairly straightfor-wardly from the fact that the spatial configuration, in particular `(x), becomes more and more uniform and featureless,the problem of the transition zone between the thin film and the macroscopic wetting layer at first-order wetting ismuch more subtle and its solution much less predictable. In this subsection we devote special attention to this in-teresting variational problem, and disentangle its structure to an extent which goes beyond what was presented inprevious works. Our purpose in this subsection is to gain new physical insights rather than to bring new results.

24

τb

τb

τ

τ

b

-0.6

0

0.6

0.4 1 1.6

0.2

0.3

0.4

0.5

0.4 0.5

FIG. 19: Microscopic DFT model predictions for line tension τ and prewetting boundary tension τb versus a temperature-likevariable b, approaching a first-order wetting transition, originally presented in [14].

At a first-order wetting transition two distinct surface phases coexist. In order to have them present simultaneously,a one-dimensional transition zone must be established that connects them. This problem seems analogous to thatof a two-dimensional interface connecting two bulk phases. However, the special challenge in the case of the surfacestates at first-order wetting is that a thin film is to be connected to an infinitely thick layer. From a geometrical pointof view, there are several ways to do this. It can be done dramatically by invoking a vertical liquid wall, just like inthe movie “The ten commandments” when Moses prepares the passage through the Red Sea for his people (Fig.20).We can estimate what this solution costs in terms of line tension. For this it suffices to examine the functional (55)and to realize that a liquid wall of macroscopic height H and width ∆x = ξ (where ξ may well be thought of asa microscopic length of the order of the bulk correlation length) costs roughly γH in energy per unit length. Thiscontribution comes entirely from the square root of the gradient squared term, since d`/dx ≈ H/ξ is very large, andsince the potential line energy cost V (`)ξ is bounded by the finite amount Vmaxξ, with Vmax the value of V (`) at thetop of the interface potential barrier. We conclude that this liquid wall costs an unbounded amount of line tension.Note that the gradient squared approximation to (55) cannot be used to evaluate this profile.

A radically alternative solution to Moses’ problem would be to replace the vertical liquid wall by a linear liquidwedge with very small slope, and to let the slope, tan θ ≈ θ, tend to zero, while keeping the height H, to which `(x)rises, macroscopic. The line tension cost of such a piece-wise linear solution, with ` = `1 for x < 0 and ` = θx, forx > 0, can be evaluated using the linearization of (55) in the squared gradient. Most surprisingly, the result is againunbounded. It is now not the gradient contribution which is the most costly ingredient, since the kinetic term iscancelled exactly by the piece-wise constant c(x), which is zero for x < 0, but for x > 0 corresponds to the kineticenergy of the linear wedge `(x) = θx. Instead, the interface potential cost is now the important one. Indeed, as θ → 0,larger and larger intervals ∆x are “spent” at values of ` in a finite range ∆` in the vicinity of the potential barrier.Thus the potential line energy cost amounts to Vmax∆`/θ. This diverges for θ → 0, even for fixed H. We concludethat also this gradual solution leads to a diverging line tension at first-order wetting. It is all the more remarkable thatthe optimal solution, which solves the Euler-Lagrange equation (56), avoids both kinds of divergences and rendersτ finite, by traversing “rapidly” (i.e., on a finite interval ∆x) the region of film thicknesses where V (`) is not small,and at the same time keeping the profile gradient sufficiently small everywhere. This optimal profile takes the form`(x) ∝ log x for short-range forces, and `(x) ∝ x2/(σ+1) for algebraically decaying forces. Fig.21 shows a sketch of thethree interface displacement profiles we have discussed, the vertical, linear and optimal ones.

In closing this subsection we note that the problem of calculating the transition zone between two surface phasesalso arises at a first-order prewetting transition, in which a thin and a thicker adsorbed film coexist. But in this casethe problem is less challenging because the transition zone is now of finite height and finite width, and corresponds to aone-dimensional boundary that is fully analogous, in all thermodynamic respects to an interface in a two-dimensionalsystem. The line tension, conventionally called boundary tension, τb, is now necessarily positive in order for the phaseseparation between the thin and the thick film to be thermodynamically stable. Lattice mean-field calculations inthe Ising model provided an early demonstration of how the prewetting boundary tension τb behaves [89]. The IDMpredicts, and more microscopic models confirm (see Fig.19), that the transition zone at prewetting and its boundarytension smoothly converge to the transition zone and the line tension at first-order wetting, respectively, when onemoves along the prewetting line towards the wetting point [14, 82].

25

FIG. 20: Moses and the transition zone (fictitious).

FIG. 21: Qualitative comparison of the three interface displacement profiles approaching a first-order wetting transition: (top)vertical liquid wall trial profile, (middle) optimal profile, (bottom) piece-wise linear trial profile.

C. Fundamental singular behavior

In the previous subsections our attention has been entirely devoted to the intriguing fate of the line tension atwetting. We have seen under which circumstances the line tension vanishes or diverges at wetting, and when afinite positive limit is predicted. We have especially appreciated that the result depends on the order of the wettingtransition and on the range of the forces. As a general conclusion we may state that the line tension is maximal atwetting, since in all cases the line tension increases towards its value τ(0), regardless of whether this limiting value is

26

zero, finite or infinite. This maximum is to be interpreted as a local maximum, since still higher values may possiblybe reached away from the wetting transition. For example, the line tension of the contact line where the interfacebetween two Bose-Einstein condensates meets a hard optical wall is found in the Gross-Pitaevskii mean-field theory,with short-range forces, to be maximal at θ = 90[72]. This physical system, in the limit T = 0, displays anotherinteresting peculiarity in that τ(0) = 0 even though the wetting transition is of first-order!

In this subsection our focus will shift to a more fundamental problem, which is concerned with the singular behaviorof the line tension at wetting. This problem is different from the previous one, in that we are now dealing with the(possible) singularity, and not with the regular background contribution to the line tension. For example, we are nownot so much interested in whether τ(0) is zero or finite, but rather in how this zero or finite value is approached inthe limit θ → 0. Also, in case τ(θ → 0) diverges, we are now interested in the precise form of the singularity. Inorder to get a proper intuitive picture of this problem and a feeling for its fundamental nature, we first broaden thecontext and recall briefly the singular behavior of bulk and surface free energies at bulk critical points and at wettingtransitions.

A hierarchy of singular behavior is apparent, starting from the properties of the bulk free energy density near abulk critical point c. The singular part of that thermodynamic quantity is standardly written as

fsing ∝ |Tc − T |2−α, (61)

where α is the critical exponent of the bulk specific heat. Next, we inspect the (liquid-gas) interfacial tension near c.This excess free energy vanishes in the following singular manner

γ ∝ |Tc − T |2−αi , (62)

where αi is the critical exponent of the interfacial specific heat. Standardly, 2 − αi is denoted by µ. At the bulkcritical point c, the interface between the bulk phases disappears (in spite of its diverging width) and so does itstension [4]. The third member of our hierarchy is the surface excess free energy near wetting transitions w. For thespreading coefficient, we recall

S = γSG − (γSL + γ) ∝ |Tw − T |2−αs , (63)

where αs is the surface specific heat exponent at wetting. Note that also first-order wetting transitions are includedin this description by setting αs = 1.

It is now natural to suppose that by analogy a similar critical exponent α` is to be associated with a line specificheat, so that the singular part of the line tension at wetting takes the form [90]

τsing ∝ |Tw − T |2−α` ∝ θ2(2−α`)/(2−αs) (64)

We stress that the behavior of τsing, which is determined by the yet unknown exponent α`, is a problem that is apriori distinct from the question what is the limiting value of τ at wetting, since in general we have

τ(θ) = τ(0) + τsing(θ), for θ → 0 (65)

The determination of α`, to which we now turn, puts the line tension study at the same footing as the already knownfundamental problems of the critical behavior of thermodynamic functions, with all its repercussions in terms ofuniversality and universality classes, especially those related to surfaces, interfaces and wetting [8? ].

We can now cast the main results obtained with the IDM in the form (65). This will bring out clearly the singularbehavior of the line tension at the level of the mean-field theory. Thus, the critical exponents we obtain are mean-fieldlike. How these are modified when fluctuations are taken into account (mainly thermal capillary wave-like fluctuationsof the liquid-gas interface) is discussed in the next subsection.

For first-order wetting and forces characterized by dimension d and decay exponent d + σ as in (59) the IDMpredicts, for θ → 0,

τ(θ) ∼ τ(0) + τ−θ(σ−3)/(σ−1), (66)

with τ(0) positive and finite and the amplitude τ− negative. For forces of strictly finite range or with exponentialdecay, the result is

τ(θ) ∼ τ(0) + τ−θ ln θ (67)

Note that for these cases, dτ/dθ → −∞, although this divergence of the slope is extremely weak for short-range forces.The exponent α` is easy to read off since the power of θ is 2(2− α`), given that αs = 1 for first-order wetting.

27

FIG. 22: Wandering interface with correlation lengths ξ⊥ and ξ‖.

For first-order wetting and non-retarded van der Waals forces (d = 3 and σ = 3) the IDM predicts that τ divergestowards positive infinity at wetting in the manner

τ(θ) ∼ τ− ln(1/θ) (68)

When this divergence is scrutinized using an anisotropic finite-size scaling approach [91], it turns out that

τ(0) ∝ ln(L⊥/a), (69)

where L⊥ is the linear system size in the direction perpendicular to the substrate plane and a some microscopic length.We will not give details of the finite-size effects here, but just emphasize that whenever thermodynamic quantitiesdiverge in the thermodynamic limit (i.e., infinite system size), this can be elucidated by examining their dependenceon longitudinal and transverse system lengths in a finite system in which all infrared divergencies are cut off. Forσ < 3 the divergencies in the thermodynamic limit become stronger and finite-size scaling is then all the more useful.

The mean-field results for the line tension singularity at critical wetting predict by the IDM are the following. Forσ > 2 and θ → 0,

τ(θ) ∼ τ−θ(σ−2)/σ, (70)

while for short-range forces,

τ(θ) ∼ τ−θ (71)

Note that τ(0) = 0 and τ− < 0 for these cases. For σ ≤ 2 the line tension diverges in the thermodynamic limit (alsoaway from the wetting transition).

We conclude that the mean-field values for the new “critical” exponent α` are known. For long-range forces theydepend on the (auxiliary) force decay exponent σ.

D. Fluctuation effects and a critical exponent equality

A study of fluctuation effects on the singular behavior of the line tension at wetting naturally begins with consideringthe effects of thermal wandering of the interface displacement `(r‖) about the mean-field profile `MF (x), as a functionof the d−1 coordinates parallel to the substrate plane, among which x is the one perpendicular to the contact line. Theinterface fluctuations, which can be decomposed into capillary waves, are correlated over parallel and perpendicularlengths ξ‖ and ξ⊥, respectively (Fig.22). These correlation lengths both diverge at continuous wetting transitions andat first-order wetting transitions approached along the prewetting line (thick film side), according to

ξ‖ ∝ |Tw − T |−ν‖ , and ξ⊥ ∝ |Tw − T |−ν⊥ (72)

provided d ≤ du, where du = 3 − 4/(σ + n) is the upper critical dimension. For first-order and complete wettingtransitions, n = 1, and for (second-order) critical wetting, n = 2. For short-range forces du = 3, and very interestingfluctuation effects on wetting behavior arise in the marginal dimension d = 3 [92]. For d ≤ du, the roughness exponentζ = (3− d)/2 relates the interface correlation lengths in the manner

ξ⊥ ∝ ξζ‖ , (73)

28

FIG. 23: Fluctuating interface and contact line.

while for d > du, ζ locks in to its mean-field value ζMF = 2/(σ + n) and is no longer a roughness exponent butrather an anisotropy exponent which describes, e.g., the shape of the transition zone between the thin film and themacroscopic wetting layer at first-order wetting (Subsection B) [91]. Likewise, the exponents ν‖ and ν⊥ lock in totheir mean-field values, ν‖ = (σ + 1)/(2(σ − 1)), for complete wetting, and ν‖ = (σ + 2)/2, for second-order criticalwetting, while ν⊥ = ζMF ν‖.

Note that the exponent ν‖ is a fundamental surface critical exponent associated with the wetting transition itself.It has a priori nothing to do with the contact line or its properties. Also note that we employ the term “critical”,even when the wetting transition is of first order. Indeed, first-order wetting is actually an endpoint of a line ofcontinuous wetting transitions. Since complete wetting is a critical phenomenon with a diverging correlation length[9], the macroscopic wetting layer at first-order wetting shares these critical properties. In fact, first-order wetting isan interface critical endpoint [93].

When d < du thermal fluctuations govern the critical exponents at wetting. A hyperscaling relation links αs to ν‖and the surface dimensionality d− 1. It reads

2− αs = (d− 1)ν‖ (74)

This is in fact the surface analogue of hyperscaling for the bulk exponents at a critical point c,

2− α = dν, (75)

where ξ ∝ |Tc − T |−ν describes the divergence of the (isotropic) bulk correlation length at c. It is now natural topostulate, by analogy, that hyperscaling holds at the level of the contact line, which is embedded in the fluctuatinginterface (see Fig.23), in the form [94]

2− α` = (d− 2)ν‖ (76)

This hypothesis implies that α` is not an independent new exponent. This becomes even more visible when d iseliminated from (74) and (76), which leads to the result [94]

α` = αs + ν‖ (77)

Since d is no longer present in this exponent equality, its validity may well be much more general than that ofhyperscaling. Indeed, we can verify that (77) is exact in d ≥ du because the mean-field exponents satisfy it. Thusthe exponent equality, which was based on a hyperscaling hypothesis valid for d < du, appears to apply for all d. Wenow give a reformulation of the original derivation, using heuristic scaling arguments, and infer that (77) is likely tohold under all circumstances, including non-thermal fluctuations and including first-order wetting [94].

We start by assuming that the singular part of τ can be simply decomposed as

τsing = τMF − τ(θ = 0) + τfluc (78)

This supposed additivity of mean-field and fluctuation contributions is a handy tool for discussing the cross-overbetween mean-field and fluctuation regimes, but in reality non-additivity is to be expected. However, the equality(77) survives when a more general derivation is given, as was done in 2002 by Parry et al. in the frame-work of wedgecovariance relations [95]. A very recent review of Parry’s work on wedge wetting is included in [1].

Inspired by the structure of the mean-field functional (55) and by standard scaling assumptions for fluctuationeffects [85, 96] we write

τsing ≈ L‖

[γ

2<

(d`

dx

)2

> +kBT

Ld−1‖

+ V (`eq)− V (∞)

](79)

29

The first term in the square brackets is the so called elastic energy cost (per unit area) of an interface displacement.The brackets < . > signify that an average value is to be estimated corresponding to correlated interface fluctuations.Across a parallel distance ξ‖ the interface wanders a transverse amount ξ⊥. Therefore, we find that the elastic energyper unit area scales as γξ2

⊥/ξ2‖ . The next term corresponds to the so called entropy loss per unit area of interface

“collisions” with the boundaries which confine it. This entropic repulsion arises when the interface hits a hard wall,imposed by, e.g., the constraint ` > 0, but more often a soft wall potential V (`) will have a similar effect of confiningthe interface displacement to a “channel” of width ξ⊥. The interface bounces back and its wandering becomesdecorrelated whenever it reaches the sides of this channel. The typical parallel distance between such collisions is ξ‖.In fact, ξ‖ is standardly defined through the curvature of the wall potential at its minimum,

γ

ξ2‖≡ d2V (`)

d`2

∣∣∣∣`eq

. (80)

We thus obtain a “scaled” version of the second term in (79) by replacing L‖ by ξ‖. The last term V (`eq) − V (∞)represents the mean-field spreading coefficient. It vanishes at wetting in the manner

V (`eq)− V (∞) = SMF ∝ |Tw − T |2−αMFs (81)

In sum, we obtain the scaling form of the singular part of the line tension at wetting

τsing =γ

2ξ2⊥ξ‖

+kBT

ξd−2‖

+ ξ‖SMF = ξ‖S, (82)

where S is the actual spreading coefficient including fluctuation effects, so S ∝ |Tw − T |2−αs .The leading singularity of τ will clearly be determined by the smallest of the critical exponents corresponding to

the three terms in this expression, so

2− α` = minν‖ − 2ν⊥, (d− 2)ν‖, 2− αMF

s − ν‖

(83)

One can now discuss various cases [94]. For example, for thermal (or weaker than thermal) fluctuations and for d < du,the entropic repulsion dominates and we are in the hyperscaling regime with 2− α` = (d− 2)ν‖. On the other hand,for random-field disorder, ζ = (5− d)/3. The fluctuation effects are now stronger than thermal ones, and the elasticenergy dominates [85]. But whatever regime we are in, we invariably have

2− α` = 2− αs − ν‖ (84)

by virtue of the second equality in (82). Therefore, (77) should be expected to be a very robust exponent equality.Note that our considerations include also first-order wetting simply by setting αs = αMF

s = 1.Consequently, we conclude that the line tension near first-order wetting is characterized by the form

τ(T ) = τ(Tw) + τ±|Tw − T |1−ν‖ + ... (85)

and, near critical wetting, by

τ(T ) = τ±|Tw − T |2−αs−ν‖ + ... (86)

These predictions can be tested against some exact results. In 1993 Abraham et al. [97] showed that in the 2-dimensional Ising model, where the contact line reduces to a contact point, the point tension at critical wettingdisplays the following divergence (!),

τ(T ) ∝ ln(1

Tw − T) ∝ ln

1θ

(87)

While this result is in accord with the equality (77), since αs = 0 and ν‖ = 2 and hence 2 − α` = 0, the exponentequality is obviously insufficiently informative for predicting whether τ(0) is zero, finite or infinite! Other veryinteresting analytic results are to be found in Parry’s work [95].

30

VI. ACKNOWLEDGEMENTS

Special thanks goes to the graduate students and colleagues at Okayama University, Cornell University and Mas-sachusetts Institute of Technology, in front of whom various parts of these lectures matured in recent years, and tomy respective hosts Ken Koga, Ben Widom and Nihat Berker for their generous hospitality. Bert Van Schaeybroeckis kindly acknowledged for help in composing figures and Ken Koga for providing Fig.19.

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