Joseph O Indekeu

Kenichiro Koga

B Widom

Wetting transitions of continuously varying or infinite order from a mean-field density-functional theory *

Keywords:

Wetting transitions of continuously varying or infinite

order from a mean-field density-functional theory

The sensitivity of wetting phenomena and in particular (equilibrium) wetting phase transitions to details of the physico-chemical properties of substrate and adsorbate is remarkable and has been the subject of many investigations [1]. Recently it has been shown that the order of the wetting transition can be varied over an impressive range, from first to infinite order, as a result of small modifications in the parameters of a "minimal" mean-field density functional model for a three-phase equilibrium with two spatially varying densities [2]. While some aspects of these findings are reminiscent of earlier discoveries, such as the possibility of wetting transitions with continuously varying order and nonuniversal critical exponents in systems with a two-component order parameter [3], the appearance of infinite-order transitions in a standard model for wetting is unexpected. The present paper aims at a deeper understanding of these findings, especially through a more detailed analysis of the analytical calculations and approximations that quantitatively reproduce, or predict, and explain the main computational (numerical) results.

Consider three phases, 𝛼, 𝛽 and 𝛾, and their mutual interfaces, that meet along a line of common contact (Fig. 1a). Alternatively, one of the three phases, say 𝛽, may intrude between the other two, and only two interfaces result, which do not intersect (Fig. 1b). The first possibility is termed partial wetting (or incomplete wetting), or simply nonwet, and the second complete wetting, or simply wet. In the latter case, the phases are ordered along the 𝑧-direction, which is perpendicular to the interfaces. The transition between the two possibilities is termed the wetting phase transition and it arises when one of the dihedral contact angles in the configuration of Fig. 1a, for example the angle between the 𝛼𝛽 and 𝛽𝛾 interfaces, becomes zero. The following triangle inequality between the interfacial tensions [START_REF] Rowlinson | Molecular theory of capillarity[END_REF] is satisfied in all considered configurations,

𝜎 𝛼𝛾 ≤ 𝜎 𝛼𝛽 + 𝜎 𝛽𝛾 , (1)
the equality being reserved for complete wetting and the strict inequality for partial wetting. As usual, the spreading coefficient is defined through

𝑆 = 𝜎 𝛼𝛾 -(𝜎 𝛼𝛽 + 𝜎 𝛽𝛾 ) (2)
In thermodynamic equilibrium, 𝑆 ≤ 0, in view of (1). Density-functional theories of interfacial structure and tension and of surface phase transitions (wetting transitions) have been brilliantly applied and reviewed by Bob Evans [START_REF] Evans | [END_REF][START_REF] Roth | [END_REF][7]. The mean-field density-functional model we study here is characterized by the following functional of two spatially varying densities 𝜌 1 (r) and 𝜌 2 (r), giving the excess free energy per unit area of an interface, oriented perpendicular to 𝑧,

σ[𝜌 1 , 𝜌 2 ] = ∫ ∞ -∞ 𝑑𝑧 〈{ 1 2 (∇𝜌 1 (r)) 2 + 1 2
(∇𝜌 2 (r)) 2 + 𝐹 (𝜌 1 (r), 𝜌 2 (r); 𝑎, 𝑏)

}〉 {𝑥,𝑦} , (3)
where the outer brackets denote that the integrand is averaged over the directions 𝑥 and 𝑦 parallel to the interface and the free-energy density (per unit volume) 𝐹 is the following 6-th order polynomial function

𝐹 (𝜌 1 , 𝜌 2 ; 𝑎, 𝑏) = ( (𝜌 1 + 1) 2 + 𝜌 2 2 ) ( (𝜌 1 /𝑎) 2 + (𝜌 2 -𝑏) 2 ) ( (𝜌 1 -1) 2 + 𝜌 2 2 ) (4)
The model features (only) two parameters, 𝑎 and 𝑏. The latter, 𝑏, is a control parameter that allows wetting to be induced. Although, dimensionally, it stands on the same footing as the density 𝜌 2 , it is physically more akin to a field variable through its (possible) dependence on temperature, chemical potentials, external magnetic or electric fields, or some combination of these. The former, 𝑎, is in general an asymmetry parameter, 𝑎 = 1 being its symmetric value. This asymmetry parameter can, in some systems, be related to spatial anisotropy. We will discuss concrete examples of asymmetry and anisotropy further on.

Clearly, for arbitrary 𝑎 and 𝑏, 𝐹 reaches its minimum value, equal to zero, when the densities reach their values appropriate to each of the bulk phases 𝛼, 𝛽 and 𝛾, at 𝑧 = ±∞. These densities are

𝛼 phase : 𝜌 1 = -1, 𝜌 2 = 0 𝛽 phase : 𝜌 1 = 0, 𝜌 2 = 𝑏 (5) 𝛾 phase : 𝜌 1 = 1, 𝜌 2 = 0
We treat our problem in mean-field theory and neglect the dependence of the densities on the directions 𝑥 and 𝑦 parallel to the interface. Then, the equilibrium density profiles 𝜌 1 (𝑧), 𝜌 2 (𝑧) that minimize the functional (3) can be viewed, in a mechanical analogy [START_REF] Rowlinson | Molecular theory of capillarity[END_REF], as least-action trajectories of a particle traveling in the two-dimensional (𝜌 1 , 𝜌 2 )space at time 𝑡 ≡ 𝑧, on a triple-hill potential 𝑉 ≡ -𝐹 . The "boundary conditions" or "initial" and "final" conditions correspond to the two bulk phases that are separated by the interface that is being considered. For example, the interfacial tension 𝜎 𝛼𝛽 is obtained by minimizing σ[𝜌 1 , 𝜌 2 ] subject to the boundary conditions that the bulk 𝛼 phase is reached at 𝑧 = -∞ and the bulk 𝛽 phase at 𝑧 = ∞.

We consider the wetting or nonwetting of the 𝛼𝛾 interface by the 𝛽 phase. Fig. 2 shows typical equilibrium density profiles, corresponding to the 𝛼𝛽 interface (Fig. 2, a and b), the nonwet 𝛼𝛾 interface (Fig. 2, c andd) and the wet 𝛼𝛾 interface (Fig. 2, e andf).

It is instructive to study the interfacial trajectories in the (𝜌 1 , 𝜌 2 )-plane. Fig. 3 shows the trajectories for the nonwet and the wet 𝛼𝛾 interface. Using this picture, it is possible to distinguish straightforwardly between a first-order and In (c) and (d) 𝜌1(𝑧) and 𝜌2(𝑧) are shown in the 𝛼𝛾 interface when it is not wet by the 𝛽 phase. In (d) 𝜌2 reaches a maximum value 𝑏0 that is less than its value 𝑏 in the bulk 𝛽 phase. In (e) and (f) 𝜌1(𝑧) and 𝜌2(𝑧) are shown in the 𝛼𝛾 interface when it is wet by the 𝛽 phase. The dotted lines, bridging the gap between the solid lines, in (e) and (f), indicate the macroscopic (infinite) thickness of the wetting layer of 𝛽. a continuous (higher-order) wetting transition. At a first-order wetting transition, the interfacial tensions 𝜎 𝛼𝛾 for the nonwet (dashed) and for the wet (solid) trajectory are equal, while the trajectories do not coincide, i.e., 𝑏 0 < 𝑏. Thus, the equilibrium structure of the 𝛼𝛾 interface is discontinuous at first-order wetting, in the sense that the equilibrium density profiles change discontinuously from the "non-wet" to the "wet" profiles. In contrast, at a continuous wetting transition at bulk three-phase coexistence, for which we adopt the familiar term "critical wetting", the interfacial tensions for the wet and non-wet trajectories become equal only when the two trajectories coincide, asymptotically, in the limit 𝑏 0 → 𝑏.

This paper is organized as follows. In Section II we present an exploratory survey of various possible wetting transitions, classified according to their (increasing) order, and provide some concrete examples sampled from a diversity of systems. In Section III we turn to a detailed discussion of analytical approximations useful for extracting the asymptotic properties, close to wetting, of our density-functional model. In Section IV a somewhat simpler density-functional model is introduced, for which wetting transitions of infinite order are possible. The paper closes with conclusions and an outlook on a possible experimental system that might display infinite-order wetting.

II. RECONNAISSANCE

A. First-order wetting

The paradigm of a first-order wetting transition is that which was studied by Moldover and Cahn [8] and by Bonn and co-workers [9] in binary liquid mixtures "adsorbed" at their common vapor (𝛾 phase). When a small amount of water is added to a mixture of methanol (𝛽 phase) and cyclohexane (𝛼 phase), the liquid-vapor interface undergoes a qualitative change. At very low water concentration 𝑥 ≪ 1 a gravity-thinned [10] wetting layer of methanol is present at the cyclohexane-vapor interface (wet state), while a genuine three-phase contact is apparent at higher water content (nonwet state). In this nonwet state the 𝛼𝛽 interface makes a finite contact angle 𝜃 with the (nearly horizontal) 𝛼𝛾 interface. At the transition between the two regimes, at water concentration 𝑥 = 𝑥 𝑤 , the spreading coefficient vanishes, with a discontinuity in its slope, as a function of 𝑥. We can characterize the singularity in 𝑆 at wetting by writing

𝑆 = 𝜎 𝛼𝛽 (cos 𝜃 -1) ∝ |𝑥 -𝑥 𝑤 | 2-𝛼𝑠 ; with 𝛼 𝑠 = 1 (6)
The critical exponent 𝛼 𝑠 is usually referred to as the "surface specific heat exponent" associated with the wetting transition. In principle, it can take any value less than or equal to 1. Clearly, for first-order wetting, 𝛼 𝑠 = 1, since 𝑆 vanishes proportionally to the first power of the distance to the transition, measured in a suitable field variable (in this case water concentration).

In the density-functional model defined through (3) the wetting transition is found to be of first order for 𝑎 > 1. Its locus is given by a curve 𝑏 𝑤 (𝑎) or 𝑎 𝑤 (𝑏), depending on which parameter, 𝑏 or 𝑎, is used to control the distance to the transition. An example of interfacial trajectories at first-order wetting, computed in our model, is shown in Fig. 3. It can be proven that a sufficient condition for the wetting transition to be of first order is 𝑎 > 2 [2]. In this regard we note that, in particular, for one of the mean-field density functional models studied earlier [11,12] one can calculate the corresponding value of 𝑎 and verify that 𝑎 > 2 at the wetting transition. The transition was found to be of first order, indeed.

B. Second-order wetting

Probably the best known second-order wetting transition is the critical wetting transition of the two-dimensional Ising model discovered by Abraham [13]. For that phenomenon one finds, e.g., upon approach of the critical wetting temperature 𝑇 𝑤 ,

𝑆 = 𝜎 𝛼𝛽 (cos 𝜃 -1) ∝ |𝑇 -𝑇 𝑤 | 2-𝛼𝑠 ; with 𝛼 𝑠 = 0, (7)
implying a "parabolic" vanishing of the spreading coefficient at wetting (and a discontinuity in the surface specific heat, which is defined here through the second derivative of the spreading coefficient with respect to temperature). For any continuous wetting transition, in particular this second-order one, there is another critical exponent, 𝛽 𝑠 , which is independent of 𝛼 𝑠 and which describes the divergence of the equilibrium wetting layer thickness ℓ upon approach of the complete wetting state. The exponent value 𝛽 𝑠 = -1 is not only found for critical wetting in the 2d Ising model, for which thermal fluctuations are important, but also for critical wetting in 3d systems with long-range forces (with algebraic decay), for which thermal fluctuations are unimportant (mean-field theory applies) [14]. In contrast, if we study second-order wetting (𝛼 𝑠 = 0) in systems with short-range forces, then, typically the wetting layer thickness diverges more weakly (logarithmically),

ℓ ∝ log(|𝑇 -𝑇 𝑤 | -1 ); so that 𝛽 𝑠 = 0(log) (9)
However, it was quite a surprise when, in 1999, in binary liquid mixtures of nonane and methanol, mean-field and short-range critical wetting signatures were observed experimentally, i.e., 𝛼 𝑠 = 0 and 𝛽 𝑠 = 0(log) [15]. Apparently, the critical wetting transitions in those systems occur so close to the bulk critical point that effects of long-range (van der Waals) forces are negligible. Furthermore, thermal fluctuation effects on short-range critical wetting were not observed. This is consistent with the current insight that these effects are expected to show up only extremely close to the transition point [18].

In the context of our mean-field density-functional model, in which long-range forces are fully ignored, a (shortrange) second-order wetting transition was found for 𝑎 = 1 (symmetric model), when 𝑏 is lowered towards 𝑏 𝑤 = 0.681... [16]. For that transition, 2 -𝛼 𝑠 = 2, whence the qualification "second-order wetting", and we verified numerically that 𝛽 𝑠 = 0(log), as expected.

C. Third-order wetting

It may seem odd that third-order wetting transitions are the subject of a separate subsection in our Reconnaissance, but it is fully justified, not only because transitions of higher than second order are found in our density-functional model for 𝑎 < 1, as we will discuss shortly, but also because, remarkably, third-order wetting transitions were predicted to occur in density functional models for systems with long-range (van der Waals) forces, and they have been observed experimentally in liquid mixtures (e.g., pentane on water) [17]. For that system, with phases 𝛼 (water), 𝛽 (pentane liquid) and 𝛾 (moist pentane vapor),

𝑆 = 𝜎 𝛽𝛾 (cos 𝜃 -1) ∝ |𝑇 -𝑇 𝑤 | 2-𝛼𝑠 ; with 𝛼 𝑠 = -1, (10)
so that 2 -𝛼 𝑠 = 3, which is characteristic for critical wetting in systems with non-retarded van der Waals forces, and

ℓ ∝ |𝑇 -𝑇 𝑤 | 𝛽𝑠 ; with 𝛽 𝑠 = -1 (11)
The latter result is quite generally a consequence of the algebraic decay of the intermolecular forces. Within our mean-field density-functional model, a third-order wetting transition does not arise in an isolated manner but rather as a non-special member of a family of critical wetting transitions of continuously varying order, as will be discussed next. For these transitions the exponent 𝛼 𝑠 varies as a function of the asymmetry parameter 𝑎 and the divergence of the wetting layer thickness is found to be logarithmic (𝛽 𝑠 = 0(log)).

D. Non-universal wetting, with continuously varying order

Since the pioneering work of Hauge it is well known that inhomogeneous systems in which two different length scales occur, can display non-universal critical wetting phenomena [3]. These two different lengths can pertain to decay lengths associated with two distinct order parameters or order parameter components. This is the case we will be interested in here. Alternatively, even in a single order parameter theory non-universal wetting can arise when the decay length of the "wall potential" varies relative to the bulk correlation length [19].

Before turning to the manifestation of non-universal wetting in our density-functional model, we discuss briefly two concrete physical systems for which non-universal critical wetting exponents have been identified. The first of these is a two-component magnetic vector model with cubic anisotropy [20] and the second is a type-I superconductor with surface enhancement of superconductivity [21].

Ferromagnet with cubic anisotropy

The order parameters in this anisotropic vector model of a semi-infinite ferromagnet are the components 𝑀 𝑥 and 𝑀 𝑦 of a magnetization function M(𝑧). They play roles similar to those of our densities 𝜌 1 and 𝜌 2 . Particularly relevant for us is to examine how the anisotropy is defined through parameters entering the free-energy density 𝐹 (M), which in zero external magnetic field is taken to be of the form

𝐹 (M; 𝜖) = 𝑡 2 M.M + 1 4 (M.M) 2 + 𝜖𝑀 2 𝑥 𝑀 2 𝑦 , (12)
with 𝑡 ≡ (𝑇 -𝑇 𝑐 )/𝑇 𝑐 (< 0) the reduced temperature and 𝑇 𝑐 the Curie temperature.

In the absence of crystal lattice anisotropy the isotropic form, with 𝜖 = 0, applies. In the presence of cubic anisotropy the magnetization preferentially aligns along a crystal axis, 𝑥 or 𝑦, for 𝜖 > 0, and along a body diagonal for 𝜖 < 0. We now examine how the anisotropy parameter 𝜖 can be related to our asymmetry variable 𝑎 introduced in (4). From now on we focus on the regime 𝜖 > 0. In this regime, four bulk phases, with magnetizations M = (± √ -𝑡, 0) and M = (0, ± √ -𝑡), coexist for 𝑡 < 0 and zero bulk field (see Fig. 1 in [20]). We reconsider the example discussed in [20] and assume that, without loss of generality, the "A" phase with bulk order parameter M 𝐴 = ( √ -𝑡, 0) is the wetting phase. To define the relevant length scales it suffices to expand the free energy density (12) to quadratic order about its minimum at M 𝐴 . In terms of the "potential energy" 𝑉 (M) ≡ 𝐹 (M 𝐴 ) -𝐹 (M) this gives

𝑉 (M; 𝜖) ≈ - 1 2
( ( 𝜉 0 𝜉 𝑥 (𝜖) ) 2 (𝑀 𝑥 - √ -𝑡) 2 + ( 𝜉 0 𝜉 𝑦 (𝜖) ) 2 𝑀 2 𝑦 ) , ( 13
)
where 𝜉 0 is a constant reference length and 𝜉 𝑖 a length related to the (inverse) curvature of the potential along the 𝑖-axis in the order parameter plane, which sets the decay length for the 𝑖-component of the order parameter towards its value in bulk. Consistently with ( 4) we now define the asymmetry parameter 𝑎 through

1 𝑎 ≡ 𝜉 𝑥 𝜉 𝑦 = √ 𝜖 2 ≡ √ 𝜆 -1 2 , ( 14
)
where we defined 𝜆 ≡ 1 + 𝜖 to allow easy comparison with the anisotropy variable, 𝜆 -1, used in [20]. Clearly, the "symmetric" case for the wetting problem corresponds to 𝑎 = 1, or 𝜆 = 3 (𝜖 = 2). Note that this is different from the value 𝜆 = 1 (𝜖 = 0), for which the bulk free energy density is isotropic in M.

For 𝑎 > 1 first-order wetting was predicted, while for 𝑎 < 1 critical wetting results, with the standard (universal) second-order wetting exponent 𝛼 𝑠 = 0 for 0 < 𝑎 ≤ 1/2 and a non-universal exponent [20]. Note that the order of the transition varies continuously between 2 (the universal value valid for 𝑎 ≤ 1/2) and arbitrarily high order (for 𝑎 → 1). However, for the symmetric case 𝑎 = 1 a standard second-order transition (𝛼 𝑠 = 0) was established [22]. In all critical wetting regimes (0 < 𝑎 ≤ 1), a logarithmic divergence of the wetting layer thickness was reported (𝛽 𝑠 = 0(𝑙𝑜𝑔)).

𝛼 𝑠 = (1 -2𝑎)/(1 -𝑎) or 2 -𝛼 𝑠 = 1/(1 -𝑎) for 1/2 < 𝑎 < 1

Type-I superconductors with surface enhancement

The two order parameters in this case are the complex superconducting wave function 𝜓 and the magnetic vector potential A. In a suitable semi-infinite geometry the Meissner state and the normal state of a type-I superconductor can be described by the scaled real (and positive) superconducting order parameter 𝜙(𝑧) and a scaled vector potential component 𝒜(𝑧), where 𝑧 is the distance from the surface into the material. The wetting transition in this material (in which the Meissner phase intrudes between the surface and the normal phase) has been studied within Ginzburg-Landau theory [21] and verified experimentally for Sn [23]. A necessary condition for this transition to occur is the presence of surface enhancement of superconductivity, which can be induced by polishing or ion bombardment of the surface [23]. The "asymmetry" in this system is fundamental and arises from the qualitative difference between the superconducting coherence length 𝜉 and the magnetic penetration depth 𝜆. The ratio 𝜅 = 𝜆/𝜉 is a material constant (near the bulk critical temperature 𝑇 𝑐 ). In order to see how the asymmetry parameter 𝑎 can be defined, we examine the expansion of the bulk free energy density about the order parameter values in the bulk superconducting phase, 𝜙 = 1/𝜅 and 𝒜 = 0, where we follow the scaling of the variables as employed in the second paper in [21]. The potential energy is given by

𝑉 (𝜙, 𝒜; 𝜅) = 1 2 𝜙 2 - 1 2 𝒜 2 𝜙 2 - 1 4 𝜅 2 𝜙 4 - 1 4𝜅 2 (15)
Expanding, to second order, about the bulk superconducting phase point, leads to

𝑉 (𝜙, 𝒜; 𝜅) ≈ - ( 𝜙 - 1 𝜅 ) 2 - 𝒜 2 2𝜅 2 (16)
𝑎 = √ 2𝜅 (17)
Guided by the seminal insight of Hauge [3] concerning the possible dependence of critical wetting singularities on the ratio of two competing lengths, the critical wetting transition in type-I superconductors [21] was scrutinized along similar lines of reasoning [24]. Mathematically, a branch with universal second-order wetting exponents and a branch with non-universal exponents that vary continuously with 𝜅 are possible. Specifically, for 𝑎 < 1 critical wetting is possible with the standard (universal) second-order wetting exponent 𝛼 𝑠 = 0 for 0 < 𝑎 ≤ 1/2 and with a non-universal exponent 𝛼 𝑠 = (1 -2𝑎)/(1 -𝑎), or 2 -𝛼 𝑠 = 1/(1 -𝑎) for 1/2 < 𝑎 < 1, corresponding to critical wetting transitions of arbitrarily high order. However, accurate model computations have shown that a first-order wetting transition takes place for 0 < 𝜅 < 0.374 [21], which corresponds to 0 < 𝑎 < 0.529. Consequently, only the non-universal critical wetting transition is physically relevant in this system. Further, a logarithmic divergence of the wetting layer thickness at critical wetting was established (𝛽 𝑠 = 0(𝑙𝑜𝑔)) [21].

The symmetric point in this model, 𝑎 = 1, or 𝜅 = 1/ √ 2, corresponds to the bulk multicritical transition between type-I and type-II superconductivity. Consequently, for 𝑎 ≥ 1 a discussion of wetting transitions is meaningless since a stable interface between Meissner and normal phases does not exist. Notwithstanding the presence of this physical "frontier" at 𝑎 = 1, the wetting layer thickness displays an interesting divergence when, coming from a partial wetting, the limit 𝑎 → 1 is considered [24][25][START_REF] Dietrich | Phase transitions and critical phenomena[END_REF][START_REF] Clarysse | [END_REF]. In particular, ℓ ∝ (1 -𝑎) -1 , in this limit, but a macroscopic wetting layer is thermodynamically unstable for 𝑎 = 1. Therefore, in this physical system this divergence is not associated with a wetting transition.

After this overview of the non-universal continuous wetting transition in these two concrete systems, we return to our density-functional model described by (3). In this model a non-universal critical wetting transition is found for 𝑎 < 1, upon lowering 𝑏 towards 𝑏 𝑤 = 0 [2]. For this transition,

𝛼 𝑠 = (1 -2𝑎)/(1 -𝑎), or 2 -𝛼 𝑠 = 1/(1 -𝑎).
Note that the order of the transition varies continuously and increases to an arbitrarily high value for 𝑎 → 1. It is furthermore interesting to note that, in contrast with expectations based on previous models and scaling arguments [3,20,21,24], also for 𝑎 < 1/2, and at least down to values of 𝑎 ≈ 0.2, non-universal behavior is found instead of the universal second-order transition. Further, for 𝑎 < 1, our computations confirm a logarithmic divergence of the wetting layer thickness (𝛽 𝑠 = 0(log)) for 𝑏 → 𝑏 𝑤 = 0, in line with the predictions of the analytical approximations outlined in Section III.

E. Infinite-order wetting

Infinite-order wetting transitions have hitherto only been encountered under rather special circumstances. For example, a subregime of the intermediate (thermal) fluctuation regime for wetting transitions in two dimensions consists of infinite-order transitions [28]. It was therefore quite a surprise that they show up in a mean-field density functional model with a multi-component order parameter [2]. Previous studies of similar models [3,20] missed them, presumably because they do not occur when the obvious control parameter of wetting is varied, but instead, they occur upon varying the asymmetry parameter 𝑎, related to a ratio of length scales associated with the decay of the order parameter components. The obvious point of view is to study wetting at arbitrary but fixed value of 𝑎. Ironically, in the case of type-I superconductors an infinite-order transition was the first to be uncovered while studying wetting [25], but it is not a wetting transition [21,24,[START_REF] Clarysse | [END_REF].

In our density-functional model a segment of infinite-order critical wetting transitions is found for 𝑎 = 1 (symmetric case) and for 0 < 𝑏 < 𝑏 𝑤 (1) = 0.681... [2]. This 𝑏 𝑤 (1) is defined as the value of 𝑏 at the second-order wetting transition found for 𝑎 = 1. The singularity in the spreading coefficient near wetting is predicted, based on analytic arguments, to be of the form

-𝑆 ∝ (𝑏/𝑏 𝑤 (1)) 2𝑎/(1-𝑎) (18)
This can be cast in a form which more conspicuously brings out the exponential singularity arising in the limit 𝑎 ↑ 1,

-𝑆 ∝ 𝑒 -𝐶/(1-𝑎) ; with 𝐶 > 0 ( 19
)
This has been confirmed numerically. Interestingly as well, we verified numerically and by analytical arguments that the wetting layer thickness diverges in the algebraic manner, for 𝑎 ↑ 1,

ℓ ∝ (1 -𝑎) -1 , ( 20
)
implying 𝛽 𝑠 = -1. This represents a much stronger divergence than the logarithmic law found in the other regimes discussed in this paper. Figure 4 presents the global wetting phase diagram of our model in the (𝑎, 𝑏)-plane. The segments of first-order (thin line; numerically computed), infinite-order (thick line at 𝑎 = 1) and continuously-varying-order wetting transitions (line with dots at 𝑏 = 0) are shown as solid curves, together with the isolated second-order transition point at 𝑎 = 1. In states above and to the left of the wetting phase boundary the 𝛼𝛾 interface is not wet by 𝛽, while in the complementary sector it is wet by 𝛽. The dashed curve in the figure for 𝑎 > 1 is the numerically determined limit of metastability of the non-wet states in the region where the equilibrium configuration is that of complete wetting. At points (𝑎, 𝑏) in the region below the dashed line we no longer find a solution to the Euler-Lagrange equations for a non-wetting trajectory such as in Figs. 2c-d or the dashed curve in Fig. 3.

III. ANALYTICAL APPROXIMATIONS TO THE MODEL

Here we study our model from the point of view of applying a constraint on the order parameter profiles, and subsequently minimizing an effective free energy function with respect to the constraint. Due to the approximation of restricting the order parameter profiles to a judiciously chosen subset of all suitable functions, our approach is not exact for our model but may still capture the correct singular behavior of the spreading coefficient 𝑆 and the layer thickness ℓ near a continuous wetting transition (𝑎 ≤ 1). Therefore, the predictions of this analytical approximation must be checked against the results of "numerically exact" model computations.

A. Interface potential approach

In this approach we follow closely the lines of reasoning of, and experience gathered in, previous works on twocomponent order parameters [3,20,24]. Two main assumptions enter: i) an accurate description of the trajectories in the (𝜌 1 ,𝜌 2 )-plane in the vicinity of the 𝛽-phase point is essential for capturing the correct thermodynamic singularities at critical wetting; and ii) a non-wet state can be parameterized by a single "collective coordinate", ℓ, which "measures" the thickness of the wetting layer and serves as an effective scalar order parameter for wetting.

Assumption i) prompts us to expand (4) to second order about (0, 𝑏), which provides the approximation 𝐹 (𝜌 1 , 𝜌 2 ; 𝑎, 𝑏) ≈ 𝐹 (2) (𝜌 1 , 𝜌 2 ; 𝑎, 𝑏) ≡ 𝐴 2 ≡ 𝜌 2 (0), where the superscript refers to the harmonic approximation. Symmetry considerations lead us to fix the position 𝑧 = 0 to the "middle" of the layer of 𝛽 and to restrict our attention to finding solutions for 𝑧 ∈ [0, ∞) with the boundary conditions, or "initial conditions" at 𝑧 = 0, 𝜌 1 (0) = 0 (24) 𝑑𝜌 2 (𝑍)/𝑑𝑍| 0 = 0 These imply 𝑏 1 = -𝑎 1 and 𝑏 2 = 𝑎 2 . This still leaves two parameters undetermined. We cannot (yet) impose the "final conditions" for 𝑧 → ∞ because our approximation 𝐹 (2) to 𝐹 is not useful in this limit. However, following [3] we can eliminate one more parameter by requiring that the first integral of the Euler-Lagrange equation, 𝐸, be a "constant of the motion" along the entire trajectory. This constant can be determined, e.g., in the bulk 𝛾 phase, where the 𝑧-derivatives of the densities as well as the function 𝐹 attain the value zero,

( (𝜌 1 /𝑎) 2 + (𝜌 2 -𝑏) 2 ) , ( 21
𝐸 = 1 2 ρ2 1 + 1 2 ρ2 2 -𝐹 = 0 (25)
Applying this at 𝑧 = 0, where our approximation 𝐹 (2) to 𝐹 is accurate (provided 𝑏

(2)

0 is sufficiently close to 𝑏) implies (𝑎 1 /𝑎) 2 = 𝑎 2 2 . Thus we have 𝑎 2 = -𝑎 1 /𝑎 since 𝑏 (2)
0 -𝑏 = 2𝑎 2 < 0 and 𝑎 1 > 0 (because ρ1 (0) > 0). Further, note that (𝜌 2 -𝑏 (2) 0 )/𝜌 1 ∼ -𝑍/2 for small 𝑧, so that the trajectory starts parallel to the 𝜌 1 -axis, from the point (0, 𝑏

(2) 0 ), in the (𝜌 1 , 𝜌 2 )-plane.
Provided 𝑎 1 is chosen sufficiently small, the approximate (non-wet) trajectory that we are now constructing, intersects, at a position denoted by 𝑧 = 𝑧 × , the exact wet trajectory associated with the interface between the 𝛽 and 𝛾 phases, in the (𝜌 1 , 𝜌 2 )-plane [29]. We now define a basically new model by imposing that, for 𝑧 > 𝑧 × , from the point of intersection up till the 𝛾 phase point at (1, 0) our approximate trajectory be identical to the exact wet trajectory in the (𝜌 1 , 𝜌 2 )-plane. Fig. 5 illustrates our strategy. It is important to stress that, in the original model, the non-wet trajectory does not intersect the exact wet trajectory before both meet at the 𝛾 phase point (see Figs. 3 or 5). The intersection we consider is thus entirely an artifact of the different starting point (at 𝜌 2 (0) somewhere between 𝑏 and the exact 𝑏 0 ) of the non-wet trajectory. The intersection is a property of the new model, not present in the original model.

The interfacial trajectory, denoted by 𝜌 0 , takes the following form in the vicinity of the 𝛽 phase,

𝜌 0,1 (𝑍 ′ ) = 𝑎 0,1 𝑒 𝑍 ′ /𝑎 ; with 𝑎 0,1 > 0 𝜌 0,2 (𝑍 ′ ) = 𝑏 + 𝑎 0,2 𝑒 𝑍 ′ ; with 𝑎 0,2 < 0, (26)
with the convention that 𝑧 ′ = -∞ in the 𝛽 phase and 𝑧 ′ = 0 corresponds to the "middle" of the 𝛽𝛾 interface.

Assuming that 𝑧 × remains finite when the wetting transition is approached -an assumption which will be checked on its self-consistency further on -we define the scaled wetting layer thickness 𝐿 ≡ √ 𝐴 ℓ as the scaled distance from the point 𝑧 = 𝑧 × to the point 𝑧 ′ = 0. This then defines how the scaled distances 𝑍 ′ and 𝑍 are related to one another,

𝑍 ′ = 𝑍 -(𝑍 × + 𝐿), ( 27
)
so that, in particular, 𝑍 ′ × = -𝐿. Further, we will soon establish how 𝐿 depends on 𝑍 × . Thus, at 𝑧 = 𝑧 × we match our trajectory with the wet solution and for 𝑧 > 𝑧 × we follow the latter up till the 𝛾 phase. The matching conditions at 𝑧 = 𝑧 × read, The solid line, from phase point 𝛼 to 𝛾 through 𝛽, is a composite of the 𝛼𝛽 and 𝛽𝛾 interfaces and is therefore also the exact trajectory for the 𝛼𝛾 interface when it is wet by 𝛽. Note that, because 𝑎 < 1, this trajectory forms a cusp near 𝛽, in contrast with typical wet trajectories for 𝑎 > 1, which are flat near 𝛽 (Fig. 3). The (lower) dashed curve is the exact trajectory for the 𝛼𝛾 interface when it is not wet by 𝛽. The (upper) thin dashed curve is one of many possible non-wet trajectories in the harmonic approximation to the potential, which can be used only in the neighborhood of the 𝛽 phase point. It crosses the solid line in two points (black dots). The approximate non-wet trajectory, on which our analytic arguments are built, is -by definition -composed of the thin dashed curve, in between the crossings, and the solid line, on either side of the intersections.

𝜌 1 (𝑍 × ) = 2𝑎 1 sinh(𝑍 × /𝑎) = 𝜌 0,1 (𝑍 ′ × ) = 𝑎 0,1 𝑒 -𝐿/𝑎 𝜌 2 (𝑍 × ) = 𝑏 -2(𝑎 1 /𝑎) cosh 𝑍 × = 𝜌 0,2 (𝑍 ′ × ) = 𝑏 + 𝑎 0,2 𝑒 -𝐿 (28)
In between the crossings, the approximate non-wet trajectory is based on 𝜌1(𝑍) = 2𝑎1 sinh 2𝑍 and 𝜌2(𝑍) = 𝑏 -4𝑎1 cosh 𝑍, with, for illustrative purposes in this Figure ,𝑏 = From suitable combinations of these two conditions it follows how 𝐿 depends on 𝑍 × and vice versa, or, how 𝑎 1 depends on 𝑍 × and vice versa. For example, to illustrate the former we can write

𝑒 -𝐿( 1 𝑎 -1) = 𝑎 |𝑎 0,2 | 𝑎 0,1 sinh(𝑍 × /𝑎) cosh 𝑍 × , ( 29
)
which shows that, for fixed 𝑎 and 𝑏, besides the (fixed) parameters associated with the wet trajectory, only the "crossing" distance 𝑍 × controls the wetting layer thickness 𝐿 associated with our one-parameter family of approximations to the non-wet trajectory. It is therefore useful to think of 𝐿 as the (only) free parameter left in our approach. It is important to keep in mind that ( 29) is an approximate relationship, valid only to leading order, for large 𝐿. Only in that limit it is sufficient to work with just the leading terms, for 𝑍 ′ → -∞, given in [START_REF] Dietrich | Phase transitions and critical phenomena[END_REF].

The surface free energy cost of the approximate non-wet trajectory on the interval 𝑧 ∈ [0, 𝑧 × ] is now found using (3), ( 21), ( 23) and ( 28), and reads σ (2) [0,𝑧×] =

∫ 𝑧× 0 𝑑𝑧{ 1 2 ρ2 1 + 1 2 ρ2 2 + 𝐹 (2) (𝜌 1 , 𝜌 2 )} = √ 𝐴 2 ( 𝑎 2 0,2 tanh 𝑍 × 𝑒 -2𝐿 + 𝑎 2 0,1 𝑎 coth 𝑍 × 𝑎 𝑒 -2𝐿/𝑎 ) , ( 30
)
while on the interval 𝑧 ∈ [𝑧 × , ∞] the surface free energy cost of the approximate non-wet trajectory can be denoted by 𝜎 ×𝛾 ≡ σ[𝑧×,∞] and corresponds to a large fraction of the equilibrium interfacial tension 𝜎 𝛽𝛾 . We need not compute this fraction, because, as can be seen by inspecting (2), it contributes equally to the surface free energies of both the approximate non-wet state and the exact wet state so that it gets canceled out in the spreading coefficient 𝑆.

Therefore, what remains to be calculated is the harmonic approximation [START_REF]non-wet and wet trajectories near 𝛽 intersect[END_REF] to the remaining fraction, 𝜎 𝛽× , of the equilibrium interfacial tension 𝜎 𝛽𝛾 ≡ 𝜎 𝛽× + 𝜎 ×𝛾 . This is achieved through

𝜎 (2) 𝛽× = ∫ -ℓ -∞ 𝑑𝑧 ′ { 1 2 ρ2 0,1 + 1 2 ρ2 0,2 + 𝐹 (2) (𝜌 0,1 , 𝜌 0,2 )} = √ 𝐴 2 ( 𝑎 2 0,2 𝑒 -2𝐿 + 𝑎 2 0,1 𝑎 𝑒 -2𝐿/𝑎 ) (31)
In sum, we obtain the following harmonic approximation to the spreading coefficient 𝑆 associated with our oneparameter family of non-wet states for given 𝑎 and 𝑏, 𝑆 (2) This expression can be interpreted as a "constrained" surface free energy excess of the non-wet state relative to the wet state, the constraint being a fixed, or imposed, value of the wetting layer thickness ℓ. This constrained surface free energy difference is the so-called interface potential. Once more, we emphasize that this harmonic approximation is only valid to leading order(s) in exp(-2𝐿) and exp(-2𝐿/𝑎). Corrections terms, not calculated in our approach, should be expected. Presumably these corrections are of order exp(-4𝐿) and exp(-4𝐿/𝑎). Whether such possible corrections are relevant must be assessed case by case.

(𝐿) = 2(σ (2) [0,𝑧×] -𝜎 (2) 𝛽× ) = √ 𝐴 ( 𝑎 2 0,2 (tanh 𝑍 × (𝐿) -1)𝑒 -2𝐿 + 𝑎 2 0,1 𝑎 (coth 𝑍 × (𝐿) 𝑎 -1)𝑒 -2𝐿/𝑎 ) (32)
We may now attempt to obtain the harmonic approximation to the equilibrium spreading coefficient by minimizing this interface potential 𝑉 (ℓ) ≡ 𝑆 (2) (𝐿) with respect to ℓ. Note that in view of (32) we have 𝑉 (∞) = 0. The equilibrium wetting layer thickness l follows from

𝑑𝑉 (ℓ) 𝑑ℓ l = 0 ( 33
)
To proceed analytically is in general a nontrivial task since, as we have made explicit in (32), the crossing distance 𝑍 × depends on 𝐿. Note that, for 𝑎 > 1, the second term in ( 32) is the leading one and its prefactor is positive. Therefore, for 𝑎 > 1 this approximation predicts the existence of a barrier in the interface potential and a (local) minimum at 𝐿 = ∞. This implies a first-order wetting transition, consistently with our numerical computations for 𝑎 > 1. On the other hand, for 𝑎 < 1, the first term leads in (32) and it is negative, while the second term is positive. In this case the interface potential allows us to describe critical wetting in the limit |𝑎 0,2 | → 0, as we will now show in more detail.

B. Non-universal wetting, with continuously varying order: The case 𝑎=1/2

It is instructive to explore first the special case 𝑎 = 1/2, for which (29) greatly simplifies to

𝑒 -𝐿 = |𝑎 0,2 | 𝑎 0,1 sinh 𝑍 × , (34)
to leading order in exp(-𝐿) << 1. Using this, we can eliminate 𝑍 × in favor of 𝐿, 𝑎 0,1 and 𝑎 0,2 in the interface potential. Subsequently, we calculate the leading terms, for large ℓ, in the expansion of 𝑉 (ℓ) in powers of exp(-𝐿). This leads to the result

𝑉 (ℓ)/ √ 𝐴 = -𝑎 2 0,2 𝑒 -2𝐿 -2𝑎 0,1 𝑎 0,2 𝑒 -3𝐿 + 𝒪(𝑒 -4𝐿 ), (35)
and we recall 𝑎 0,2 < 0. We do not know the amplitude of the correction term, of order exp(-4𝐿), because it may contain a contribution from an "anharmonic" correction to (32). Nevertheless, using only the two leading terms it is possible to derive an approximate equilibrium wetting layer thickness using (33). We find

𝑒 -L = |𝑎 0,2 | 3𝑎 0,1 (36)
Note that, within this approximation, 𝑧 × ( l) is finite since sinh 𝑍 × ( L) = 1/3 for 𝑎 = 1/2, independently of 𝑏. The precise value of this constant (1/3) is unimportant, as it does not affect the form of the critical singularities at wetting. Important is that our approximation is self-consistent because for finite 𝑧 × the intersection point of the trajectories lies close to the 𝛽 phase point where our harmonic approximation is justified. Making the observation that |𝑎 0,2 | = 𝒪(𝑏), in view of the second of Eqs. [START_REF] Dietrich | Phase transitions and critical phenomena[END_REF], we obtain a logarithmic divergence of the wetting layer thickness

l ∝ ln(1/𝑏) (37)
implying a critical wetting transition in the limit 𝑏 → 0. Besides the exponent 𝛽 𝑠 = 0(log) we can also obtain the critical exponent of the surface free energy singularity, after inserting the equilibrium expression (36) into (either of) the two terms in (35). We obtain

-𝑆 ∝ 𝑏 4 (38)
In order to obtain 𝛼 𝑠 explicitly we now generalize our calculations to 𝑎 ∕ = 1/2. For 𝑎 ∕ = 1/2 we assume that 𝑧 × ( l) is also a finite constant, as our inspection of the case 𝑎 = 1/2 suggested. The outcome of the calculations, in as far as the critical exponents at wetting are concerned, are quite robust to changing the value of this constant and to the approximations made to derive it. In the following we will accordingly simplify the algebra considerably by assuming that 𝑧 × and even 𝑧 × /𝑎 are smaller than unity in such a way that the hyperbolic functions can be safely approximated by their expansions to first order in these variables. We will check the self-consistency of these Ansätze after we get the results.

We approximate (29) by

𝑎 0,1 |𝑎 0,2 | 𝑒 -𝐿( 1 𝑎 -1) = 𝑎 sinh(𝑍 × /𝑎) cosh 𝑍 × ≈ tanh 𝑍 × ≈ 𝑍 × (39)
This soon leads to the following asymptotic form for the interface potential

𝑉 (ℓ)/ √ 𝐴 ≈ -𝑎 2 0,2 𝑒 -2𝐿 -2𝑎 0,1 𝑎 0,2 𝑒 -𝐿(1+1/𝑎) -(𝑎 2 0,1 /𝑎) 𝑒 -2𝐿/𝑎 + 𝒪(𝑒 -4𝐿 ), (40)
where we have no calculational evidence for a possible correction of order exp(-4𝐿) but prudently allow for it on general grounds (cf. our remarks following (35)). Note that this possible correction can, depending on the value of 𝑎, move forward in our expansion and even become the next-to-leading term. In any case, for 1/3 < 𝑎 < 1 the first two terms in (40) suffice to obtain a clear prediction for the equilibrium wetting layer thickness l and the spreading coefficient singularity. The generalization of (36) reads

𝑒 -L( 1 𝑎 -1) ≈ |𝑎 0,2 | (1 + 1/𝑎)𝑎 0,1 ∝ 𝑏 (41)
We can now verify that 𝑧 × ( l) and 𝑧 × ( l)/𝑎 are indeed bounded below 1. Note that (41) and (39) imply

𝑎 0,1 |𝑎 0,2 | 𝑒 -L( 1 𝑎 -1) ≈ 𝑎 𝑎 + 1 = 𝑍 × ( L), (42)
in our (linear) approximation for small 𝑍 × . On the other hand, if at this stage we revert back to the hyperbolic functions, we obtain values of 𝑍 × that satisfy

sinh(𝑍 × ( L)/𝑎) cosh 𝑍 × ( L) = 1 𝑎 + 1 , ( 43
)
which is solved rather well by our approximation (39), i.e., 𝑍 × ( L)/𝑎 ≈ 1/(𝑎 + 1) < 1. In particular, for 𝑎 → 0, 𝑍 × ( L/𝑎) ≈ 0.8814 solves (43). Next, we discuss the surface free energy singularity at wetting. Eliminating 𝐿 in any of the terms in (40) with use of (41) we find that (38) generalizes to

-𝑆 ∝ 𝑏 2 1-𝑎 (44)
If we now inspect the results and notational conventions featured in our survey (see Section II.D, final paragraph) we arrive at the conclusion that

2 -𝛼 𝑠 = 1 1 -𝑎 (45)
is consistent with the interpretation that in our model not 𝑏 but 𝑏 2 is the natural variable to measure the distance to the wetting transition when this transition takes place for 𝑏 → 0. Remarkably, for 𝑎 < 1/2 the wetting transition in our model does not "lock in" to a standard second-order transition with 𝛼 𝑠 = 0, as in the other models discussed in Section II.D. Instead, the order of the transition decreases continuously from 2 to 3/2, when 𝑎 is lowered from 1/2 towards 1/3. This is due to the particular form of the exponents of the leading and subleading terms in our interface potential (40), which is somewhat different from that in the other models. High-precision numerical computations for our model corroborate the conclusions (values of 𝛼 𝑠 and 𝛽 𝑠 ) reached on the basis of the analytic interface potential approach and the accompanying approximations we used. What happens, then, for 𝑎 < 1/3? The analytical arguments we presented leave open three possibilities. First, if the amplitude, which we denote by 𝐵, of the correction term of order exp(-4𝐿) to (40) is positive, the order of the transition takes a universal value for 𝑎 ≤ 1/3. Secondly, if 𝐵 = 0 the transition remains non-universal also below 𝑎 = 1/3, until a higher-order correction becomes relevant. Thirdly, if 𝐵 < 0 it is less obvious what to expect, but a (non-universal) critical wetting transition in the limit 𝑏 → 0 is still possible provided 𝐵 → 0. Our numerical computations are consistent with the second (or third) scenario. The non-universal critical wetting transition appears to persist, at least down to 𝑎 = 1/5, at which point we obtain -𝑆 ∝ 𝑏 5/2 , in agreement with (44). Fig. 6 and Fig. 7 illustrate the results of our numerical computations for selected values of 𝑎, for the original model defined through (3) and ( 4). Fig. 6 shows the behavior of the spreading coefficient when the critical wetting transition at 𝑏 𝑤 = 0 is approached for 𝑎 = 0.2, 𝑎 = 0.5 and 𝑎 = 0.9. The results are in quantitative agreement with the analytic prediction (44) that the order of the transition is, respectively, 2 -𝛼 𝑠 = 1.25, 2 and 10. Fig. 7 shows the behavior of the wetting layer thickness approaching the critical wetting transition at 𝑏 𝑤 = 0, for 𝑎 = 0.5, and approaching the second-order critical wetting transition at 𝑏 𝑤 = 𝑏 𝑤 (1) = 0.618... for 𝑎 = 1. In both cases the divergence of ℓ is logarithmic, in accord with the analytical arguments which we presented for 𝑎 < 1 and with general expectations for 𝑎 = 1 (see our discussion in Section II.B, in particular result (9)). Incidentally, these computations for the wetting layer thickness, for 𝑎 = 1, complement the earlier paper which uncovered the second-order wetting transition point in this model and in which the behavior of the spreading coefficient was discussed [16]).

D. Infinite-order wetting in the limit 𝑎 ↑ 1

The observation that the wetting phase boundary in the global wetting phase diagram (Fig. 4) features a vertical segment at 𝑎 = 1 has prompted us to study what happens as this segment is approached from the left. Numerical computation reveals that the wetting transition approached in this manner is of very high, possibly infinite, order. Concomitantly, the wetting layer thickness diverges as the first inverse power of the distance to the phase boundary, a much more dramatic increase than the ubiquitous logarithmic growth found under almost all other circumstances in our and other models with short-range forces.

Analytical support for the occurrence of a genuine infinite-order wetting transition is gathered by examining the interface potential (40) under a different angle. Instead of varying 𝑎 0,2 , or, equivalently, 𝑏, towards zero, we keep this parameter constant and vary 𝑎 towards 1. Recalling (40) and the result for the equilibrium wetting layer thickness (41), which we rewrite as follows FIG. 7: Equilibrium wetting layer thickness ℓ versus wetting parameter 𝑏, in semi-log representation, for selected values of the asymmetry parameter 𝑎. For 𝑎 = 0.5 the critical wetting transition takes place in the limit 𝑏 → 0, while for 𝑎 = 1 it takes place at 𝑏 = 𝑏𝑤(1) ≈ 0.681. In both cases the divergence of ℓ is logarithmic.

l √ 𝐴 ≡ L ≈ 𝑎 1 -𝑎 ln ( (1 + 1/𝑎)𝑎 0,1 |𝑎 0,2 | ) , (46)
the algebraic divergence of l for 𝑎 ↑ 1 is conspicuous. Likewise, embedding this result in (40) leads to

-𝑆 ∝ (𝑏/𝑏 𝑤 (1)) 2𝑎/(1-𝑎) ∝ 𝑒 -𝐶/(1-𝑎) , (47)
with 𝐶 > 0, which is possible provided 𝑏 < 𝑏 𝑤 (1) = 0.681..., where 𝑏 𝑤 (1) corresponds to the upper end of the infinite-order transition line in the (𝑎, 𝑏)-plane at 𝑎 = 1. The infinite-order character of the transition (exponential singularity) is thus clearly supported by analytical arguments.

In closing this Section we draw attention to the fact that the variational approach which underlies our analytical results was presented in a different way in our Letter [2]. There, no derivation was presented of the interface potential. In fact, the behavior of the wetting layer thickness was not discussed. The constraint was formulated in terms of the order parameter 𝜌 1 rather than in terms of the distance 𝑧. That approach is equivalent to the one we have employed in this paper.

IV. A SIMPLE(R) MODEL WITH AN INFINITE-ORDER WETTING TRANSITION

Instead of the 6th-order free energy density function (4) we now consider the 4th-order form

𝐹 (𝜌 1 , 𝜌 2 ; 𝑐, 𝑏) = 𝑐𝜌 2 1 𝜌 2 2 + (𝜌 2 -𝑏 + 𝑏𝜌 2 1 ) 2 (48)
The bulk phases correspond, as before, to the points (-1, 0), (1, 0) and (0, 𝑏) in the (𝜌 1 ,𝜌 2 )-plane. When this free energy density is expanded about the wetting phase, at (𝜌 1 , 𝜌 2 ) = (0, 𝑏), we obtain the "potential energy"

𝑉 (𝜌 1 , 𝜌 2 ; 𝑐, 𝑏) ≡ -𝐹 (𝜌 1 , 𝜌 2 ; 𝑐, 𝑏) ≈ -𝑐 𝑏 2 𝜌 2 1 -(𝜌 2 -𝑏) 2 (49)
so that we can define the asymmetry parameter 𝑎 through

1 𝑎 = 𝑏 √ 𝑐 (50)
Chronologically, before we began to explore the properties of the two-parameter model defined through (4), we investigated numerically the model (48) for a fixed value of 𝑐, e.g., 𝑐 = 16. We did this while searching for models FIG. 9: Equilibrium wetting layer thickness ℓ versus wetting parameter 𝑏, in log-log representation, for the model defined in (48), with 𝑐 = 16 and assuming 𝑏𝑤 equals the analytically predicted value 1/4. Although the data appear to settle along a straight line, the slope still tends to increase slightly as the wetting transition is approached. With our present accuracy we conclude 𝛽𝑠 = -0.85 ± 0.2, with a downward trend. This is not inconsistent with the predicted value 𝛽𝑠 = -1.

with a critical wetting transition and with an 𝐹 -polynomial of order lower than six, knowing that for a 6th-order 𝐹 a critical wetting transition was already established [16]. Surprisingly, numerical analysis of this quartic-polynomial model with one parameter, 𝑏, suggested a wetting transition of "very high and possibly infinite" order at 𝑏 𝑤 ≈ 1/4. Assuming that the character of the singularity of the surface free energy at wetting is governed by the properties of the model in the vicinity of the 𝛽 phase point in the (𝜌 1 , 𝜌 2 )-plane, it suffices to insert 𝑐 = 16 in (50) in order to understand that this model "lies on" a hyperbola in the (𝑎, 𝑏) wetting phase diagram (Fig. 4), which crosses the wetting phase boundary in the segment of infinite-order transitions located at 𝑎 = 1, in the point (1, 𝑏 𝑤 ) with 𝑏 𝑤 = 1/4. More generally, this argument predicts an infinite-order transition for the two-parameter model (48) provided 𝑐 is chosen large enough so that 𝑏 𝑤 = 1/ √ 𝑐 < 𝑏 𝑤 (1), where 𝑏 𝑤 (1) is a model-dependent threshold.

The prediction of an infinite-order transition is in accord with the numerical computations for this model, illustrated in Fig. 8 and Fig. 9. Fig. 8 shows the behavior of the spreading coefficient when the critical wetting transition at 𝑏 𝑤 = 1/4 is approached for 𝑐 = 16. The results are in quantitative agreement with the analytic prediction of an exponential singularity (47), implying that the order of the transition is infinity. Fig. 9 shows the behavior of the wetting layer thickness approaching the infinite-order wetting transition. The divergence of ℓ appears to be algebraic, in agreement with the analytical prediction (46), which can be written in the form in view of (50). However, the slope of the line formed by the data in Fig. 9 has apparently not yet converged. With our present accuracy we cannot yet conclusively confirm the value -1 predicted by (51).

l ∝ 𝑎 1 -𝑎 ∝ 𝑏 𝑤 𝑏 -𝑏 𝑤 , (51)

V. CONCLUSIONS AND OUTLOOK

We close this paper with a concise recapitulation and some speculations. The phase diagram (Fig. 4) teaches us that fairly basic choices of density-functional models for three-phase equilibria possess an astonishingly rich variety of wetting phase transitions. The infinite-order transitions, to which we draw special attention, and which were almost not recognized until very recently, appear to be fairly ubiquitous! Indeed, while they seem to emerge for the first time, within mean-field theory, in the two-parameter model studied in our Letter [2], we conjecture that a segment of infinite-order wetting also occurs in the global wetting phase diagram of two previously studied multi-component order parameter models [3,20]. From a theoretical point of view, a segment of infinite-order transitions (Fig. 4) appears to be a new alternative to known scenarios for connecting first-order to critical wetting, such as the familiar tricritical point [START_REF] Nakanishi | [END_REF] and the less familiar critical endpoint scenario [21,32].

Moreover, we suggest that the ferromagnet with cubic anisotropy might well be the most obvious experimental system in which to look for the infinite-order transition. Since in general the (cubic) magnetic anisotropy, and consequently the parameter 𝑎, is temperature dependent [33], varying the temperature implies following an oblique path in the global temperature-anisotropy phase diagram (Fig. 4). Here we assume that 𝑏 is a temperature-like or at least temperature-dependent field, which is explicitly realized in (13), where 𝑏 ∝ √ -𝑡. Therefore, the infinite-order wetting transition ought to be accessible in this system, no less than the wetting transitions of finite order and the experiment can, in principle, be performed on a single sample by varying 𝑇 . Further, note that the path traced by varying the control parameter in the simpler model studied in Section IV, corresponds to an oblique path in the (𝑎, 𝑏)-plane.

How do thermal fluctuations, not included in our mean-field treatment, affect our results, knowing that the upper critical dimension for wetting and short-range forces is 𝑑 = 3 [14,[START_REF] Dietrich | Phase transitions and critical phenomena[END_REF]? We follow the reasoning advocated in [3] and acknowledge that, when the wetting transition takes place sufficiently far from bulk (multi-)criticality, the results are not significantly altered. Otherwise, exponent renormalization is in order, and the results presented in [19] can provide a guideline, to a large extent also for our present model. In any case, this is an open problem outside the scope of our paper.

In this paper we have gone beyond what was presented in our Letter [2] in several respects. We have, inter alia, derived new results for the behavior of the wetting layer thickness ℓ upon approach of an infinite-order wetting transition, in particular regarding the algebraic divergence (𝛽 𝑠 = -1), and we have outlined how analytic approximations can be obtained and used to anticipate, or explain, the findings of accurate numerical analysis of the full density-functional model. It is peculiar to the infinite-order transition that analytic arguments are indispensable to complement "exact" numerical computations, because it is subtle to distinguish a transition of very high but still finite order from one of infinite order.

  Wetting transitions of continuously varying or infinite order from a mean-field density-functional theoryI. INTRODUCTION

FIG. 1 :

 1 FIG. 1: Two possible configurations for three-phase equilibrium. (a) Partial wetting: the three phases 𝛼, 𝛽 and 𝛾 are in mutual contact. The three interfaces, with interfacial tensions 𝜎 𝛼𝛽 , 𝜎 𝛽𝛾 and 𝜎𝛼𝛾 meet at a common contact line. (b) Complete wetting. Phase 𝛽, called the wetting phase, intrudes between phases 𝛼 and 𝛾.

Fig. 1

 1 Fig. 1 Koga et al.

FIG. 2 :

 2 FIG. 2: Density profiles as a function of the spatial coordinate 𝑧. In (a) and (b) 𝜌1(𝑧) and 𝜌2(𝑧) are shown in the 𝛼𝛽 interface.In (c) and (d) 𝜌1(𝑧) and 𝜌2(𝑧) are shown in the 𝛼𝛾 interface when it is not wet by the 𝛽 phase. In (d) 𝜌2 reaches a maximum value 𝑏0 that is less than its value 𝑏 in the bulk 𝛽 phase. In (e) and (f) 𝜌1(𝑧) and 𝜌2(𝑧) are shown in the 𝛼𝛾 interface when it is wet by the 𝛽 phase. The dotted lines, bridging the gap between the solid lines, in (e) and (f), indicate the macroscopic (infinite) thickness of the wetting layer of 𝛽.

1 FIG. 3 :

 13 FIG.3: Computed trajectories in the (𝜌1, 𝜌2)-plane, obtained by eliminating 𝑧 from the density profiles 𝜌1(𝑧) and 𝜌2(𝑧). The model parameters are 𝑎 = 1.6 and 𝑏 = 0.973. They correspond to a first-order wetting transition. The solid line is a composite of the 𝛼𝛽 and 𝛽𝛾 interfaces and is therefore also the trajectory for the 𝛼𝛾 interface when it is wet by 𝛽. The dashed curve is the trajectory for the 𝛼𝛾 interface when it is not wet by 𝛽. The non-wet and wet trajectories are clearly distinct, but the corresponding interfacial structures have the same surface free energy.

  [START_REF] Rowlinson | Molecular theory of capillarity[END_REF] allows us to define the asymmetry parameter as

FIG. 4 :

 4 FIG.4: Computed global wetting phase diagram in the (𝑎, 𝑏)-plane. The phase boundary between wet and non-wet states consists of a segment of first-order wetting (𝑎 > 1; solid line), a second-order wetting point (𝑎 = 1, 𝑏 = 𝑏𝑤(1) ≈ 0.681), a segment of infinite-order wetting (𝑎 = 1; vertical line), and a segment of wetting transitions of continuously varying order (𝑎 < 1, 𝑏 = 𝑏𝑤 = 0; line decorated with dots). The thin dashed line 𝑎 > 1 is the spinodal of the non-wet states in the wet regime.

FIG. 5 :

 5 FIG.5: Computed trajectories in the (𝜌1, 𝜌2)-plane for 𝑎 = 1/2, obtained by eliminating 𝑧 from the density profiles 𝜌1(𝑧) and 𝜌2(𝑧). The solid line, from phase point 𝛼 to 𝛾 through 𝛽, is a composite of the 𝛼𝛽 and 𝛽𝛾 interfaces and is therefore also the exact trajectory for the 𝛼𝛾 interface when it is wet by 𝛽. Note that, because 𝑎 < 1, this trajectory forms a cusp near 𝛽, in contrast with typical wet trajectories for 𝑎 > 1, which are flat near 𝛽 (Fig.3). The (lower) dashed curve is the exact trajectory for the 𝛼𝛾 interface when it is not wet by 𝛽. The (upper) thin dashed curve is one of many possible non-wet trajectories in the harmonic approximation to the potential, which can be used only in the neighborhood of the 𝛽 phase point. It crosses the solid line in two points (black dots). The approximate non-wet trajectory, on which our analytic arguments are built, is -by definition -composed of the thin dashed curve, in between the crossings, and the solid line, on either side of the intersections. In between the crossings, the approximate non-wet trajectory is based on 𝜌1(𝑍) = 2𝑎1 sinh 2𝑍 and 𝜌2(𝑍) = 𝑏 -4𝑎1 cosh 𝑍, with, for illustrative purposes in this Figure, 𝑏 =0.3 and 𝑎1 = 𝑏 2 /5 = 0.018; cf. [29].

  FIG.5: Computed trajectories in the (𝜌1, 𝜌2)-plane for 𝑎 = 1/2, obtained by eliminating 𝑧 from the density profiles 𝜌1(𝑧) and 𝜌2(𝑧). The solid line, from phase point 𝛼 to 𝛾 through 𝛽, is a composite of the 𝛼𝛽 and 𝛽𝛾 interfaces and is therefore also the exact trajectory for the 𝛼𝛾 interface when it is wet by 𝛽. Note that, because 𝑎 < 1, this trajectory forms a cusp near 𝛽, in contrast with typical wet trajectories for 𝑎 > 1, which are flat near 𝛽 (Fig.3). The (lower) dashed curve is the exact trajectory for the 𝛼𝛾 interface when it is not wet by 𝛽. The (upper) thin dashed curve is one of many possible non-wet trajectories in the harmonic approximation to the potential, which can be used only in the neighborhood of the 𝛽 phase point. It crosses the solid line in two points (black dots). The approximate non-wet trajectory, on which our analytic arguments are built, is -by definition -composed of the thin dashed curve, in between the crossings, and the solid line, on either side of the intersections. In between the crossings, the approximate non-wet trajectory is based on 𝜌1(𝑍) = 2𝑎1 sinh 2𝑍 and 𝜌2(𝑍) = 𝑏 -4𝑎1 cosh 𝑍, with, for illustrative purposes in this Figure, 𝑏 =0.3 and 𝑎1 = 𝑏 2 /5 = 0.018; cf. [29].

  -universal wetting, with continuously varying order: The more general case 𝑎 < 1

2 FIG. 6 :

 26 FIG.6: Equilibrium spreading coefficient 𝑆 versus wetting parameter 𝑏, in log-log representation, for selected values of the asymmetry parameter 𝑎. The slopes, which correspond to twice the order of the wetting transition, according to (44) and (45), are 2.5 ± 0.1 for 𝑎 = 0.2, 4 ± 0.2 for 𝑎 = 0.5 and 20 ± 1 for 𝑎 = 0.9. The wetting transition takes place in the limit 𝑏 → 0.

FIG. 8 :

 8 FIG.8: Equilibrium spreading coefficient 𝑆 versus wetting parameter 𝑏, in semi-log representation, for the model defined in (48), with 𝑐 = 16 and assuming 𝑏𝑤 equals the analytically predicted value 1/4. Clearly, the slope converges rapidly to a finite value, supporting the conjectured exponential singularity associated with a wetting transition of infinite order at 𝑏𝑤 = 1/4.

  𝑏 2 ) 2 a (finite) prefactor. Solving the Euler-Lagrange equations∂𝐹 (2) /∂𝜌 1 = 𝑑 2 𝜌 1 /𝑑𝑧 2 , ∂𝐹 (2) /∂𝜌 2 = 𝑑 2 𝜌 2 /𝑑𝑧 2 , (22)within this "harmonic" approximation, leads to the (general) solutions, expressed in the scaled distance 𝑍 ≡ √ 𝐴 𝑧,𝜌 1 (𝑍) = 𝑎 1 𝑒 𝑍/𝑎 + 𝑏 1 𝑒 -𝑍/𝑎 (23) 𝜌 2 (𝑍) = 𝑏 + 𝑎 2 𝑒 𝑍 + 𝑏 2 𝑒 -𝑍 ,

	with 𝐴 = 2(1 + and we define 𝑏
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	)

VI. ACKNOWLEDGEMENT

We wish to express our deep appreciation and admiration of Bob Evans' pioneering works in statistical and soft condensed matter physics which have been a continuous source of inspiration for us and countless others. We also gratefully acknowledge Bob's generous critical opinion and plentiful spiritual remarks on many of our own works. It is with pleasure and excitement that we dedicate this contribution to him on the occasion of his 65th birthday, since wetting phenomena have enjoyed his special attention since the late 70's when they became irrevocably entangled with phase transitions and critical phenomena.


Rogier Aletta Adriana Verhoeff
email: a.a.verhoeff@uu.nl

Paul Brand

Henk Lekkerkerker

R P Brand

H N W Lekkerkerker Van '

Tuning the birefringence of the nematic phase in suspensions of colloidal gibbsite platelets

Keywords: liquid crystals, colloids, birefringence, interfaces, magnetic fields liquid crystals, colloids, birefringence, interfaces, magnetic fields

Tuning the birefringence of the nematic phase in suspensions of colloidal gibbsite platelets

Introduction

Bob Evans has given seminal and lasting theoretical contributions to the statistical physics of liquids, interfacial phenomena and fluids in confining geometries [START_REF] Evans | The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluidsAdv[END_REF][START_REF] Tarazona | A simple density functional theory for inhomogeneous liquids. Wetting by gas at a solid-liquid interfaceMol[END_REF][START_REF] Tarazona | Phase equilibria of fluid interfaces and confined fluids. Non-local versus local density functionalsMol[END_REF]. His interest ranges from simple (atomic) fluids, to complex liquids such as colloidal suspensions. He also has a keen interest in experiments, which frequently serve as a source of inspiration for his theoretical work. As a result, his work is equally well appreciated in the theoretical physics community [START_REF] Evans | Pair correlation function decay in models of simple fluids that contain dispersion interactionsJ[END_REF] as in the circles of physical chemists [START_REF] Schmidt | Density functional for a model colloid-polymer mixturePhys[END_REF] and chemical engineers [START_REF] Evans | Fluids adsorbed in narrow pores: phase equilibria and structureJ[END_REF].

Less well known is that Bob in his free time (and at his own expense!) is experimenting with a whole series of liquids which in the terminology introduced by Daan Frenkel can be qualified as "nasty" liquids. In the course of these experiments he has recently become interested in the wetting properties of macroscopic plates at glass walls [START_REF] Evans | Personal homepage[END_REF]. We felt it therefore fitting to contribute to this volume that is dedicated to him at the occasion of his 65 th birthday a paper which deals with a case where wetting of colloidal plates leads to an explanation of birefringence in liquid crystal phases and droplets.

Starting with the work of Einstein [START_REF] Einstein | On the motion, required by the molecular-kinetic theory of heat, of particles suspended in a fluid at restAnn[END_REF] and Perrin [START_REF] Perrin | Mouvement brownien et réalité moléculaireAnn[END_REF][START_REF] Perrin | Les atomes[END_REF] on the application of statistical mechanics to mesoscopic particles, the analogy between atomic and molecular fluids and colloidal suspensions has been appreciated. A formal basis for this analogy was provided by Onsager [START_REF] Onsager | Theories of Concentrated ElectrolytesChem[END_REF] and McMillan and Mayer [START_REF] Mcmillan | The statistical thermodynamics of multicomponent systemsJ[END_REF]. Apparently, the former was the first to coin the phrase "potential of the average force" which for colloidal suspensions replaces the energy in the Maxwell-Boltzmann distribution.

Why study colloidal systems if they simply mimic the behaviour of atoms? One answer is that the interactions between colloidal particles can be tuned from steeply repulsive to deeply attractive and their shapes vary from spherical to highly anisometric rods and plates. This gives rise to unique and fascinating properties ranging from novel phases to remarkable long-lived metastable and non-equilibrium states. These states have been used for detailed studies of fundamental processes of broad relevance such as crystallisation and glass formation [START_REF] Pusey | Phase behaviour of concentrated suspensions of nearly hard spheres[END_REF]14].

Suspensions of highly anisometric particles are known to form a nematic (N) liquid crystalline phase at remarkably low concentration. The formation of the nematic phase was addressed theoretically by Onsager in the 1940s [START_REF] Onsager | The effect of shape on the interaction of colloidal particlesAnn[END_REF], who showed that the I-N phase separation arises from the competition between the orientational entropy and the packing entropy.

Gibbsite (γ-Al(OH) 3 ) particles represent a unique plate-like system, which displays a nematic phase, both in suspensions of sterically stabilized gibbsite particles in organic solvents [START_REF] Van Der Kooij | Formation of nematic liquid crystals in suspensions of hard colloidal plateletsJ[END_REF] as well as in suspensions of charged stabilized gibbsite particles in water [START_REF] Van Der Beek | Nematic ordering vs. gelation in suspensions of charged plateletsEurophys[END_REF]. In the former case the interactions are nearly hard, while in the latter case the interactions are soft repulsive and can be tuned by the ionic strength.

Due to the ordering of the particles, the nematic phase displays birefringence. This birefringence is due to form or shape birefringence and in case the particles themselves are optically anisotropic also to intrinsic birefringence [18,[START_REF] Bragg | The Form Birefringence of MacromoleculesActa Cryst[END_REF]. In case of gibbsite these contributions have an opposite sign, and moreover, their relative contribution can be tuned by varying the optical contrast between between the particles and the suspending medium.

The bulk liquid crystal ordering is modified at interfaces (such as the solidfluid and fluid-fluid interface) over length scales that are considerably larger than the particle size, as was shown theoretically [START_REF] Harnau | Fluids of platelike particles near a hard wallPhys[END_REF][START_REF] Bier | Free isotropic-nematic interfaces in fluids of charged platelike colloidsJ[END_REF], experimentally [START_REF] Van Der Beek | Isotropic-nematic interface and wetting in suspensions of colloidal plateletsPhys[END_REF] and in simulations [START_REF] Pineiro | Surface ordering and capillary phenomena of confined hard cut-sphere particlesSoft[END_REF]. Furthermore, the ordering of liquid crystals can be modified by the aligning action of an externally applied magnetic field [START_REF] Van Der Beek | Influence of a magnetic field on the nematic phase of hard colloidal plateletsPhys[END_REF]. In the case of gibbsite with its tunable birefringence these aligning effects of interfaces and external fields give rise to fascinating phenomena that can be conveniently studied with polarised light microscopy. These effects will be addressed in this paper, which is organized as follows.

In the next section we describe the preparation of the colloidal gibbsite platelets that are subject of investigation. In section 3 we will discuss the theoretical background of intrinsic and form birefringence and show that in the case of gibbsite platelets we have the rather unique situation that depending on the suspending medium, either the one or the other dominates. Experimental details on the polarised light microscopy experiments are given in section 4. Finally, in section 5 we describe results obtained with gibbsite nematic phase aligned at interfaces and in a magnetic field, as well as confined to droplets, and discuss what these experiments teach us about the optical properties of gibbsite liquid crystals.

Preparation of colloidal gibbsite

Colloidal gibbsite (γ-Al(OH) 3 ) platelets were synthesized by hydrothermal treatment of aluminum alkoxides in acidic environment, a procedure developed in our laboratory [START_REF] Wierenga | Aqueous dispersions of colloidal gibbsite platelets: synthesis, characterisation and intrinsic viscosity measurementsColloids Surf[END_REF]. To that end, an acidic aqueous solution containing hydrochloric acid (HCl 0.09 M, 37%, Merck), aluminium sec-butoxide (0.08 M, 95%, Fluka Chemika) and aluminium iso-propoxide (0.08 M, 98+%, Acros Organics) was pre- pared and mechanically stirred for 10 days. Subsequently, the mixture was heated in a glass reaction vessel in a water bath at 85 • C for 72 h. Then, the colloidal dispersion was centrifuged at 1000g (overnight, 15-20 h) in order to remove the smallest particles and decrease the polydispersity. The resulting dispersion was dialyzed against demineralized water in tubes of regenerated cellulose (Visking, MWCO 12 000-14 000) until the conductivity dropped to 20 µS cm -1 . The obtained particles were characterised with transmission electron microscopy (Tecnai 10, FEI Company), see Figure 1(a). The hexagonal platelets typically have a diameter of about 200 nm and a thickness of 8 nm.

Subsequently, the obtained particles were treated either to enhance the stability in aqueous dispersions, or to enable dispersion in apolar solvents.

For enhanced stability in aqueous dispersions, 0.5 w/w % aluminum chlorohydrate (Locron P, Hoechst AG, Germany) was added to the dispersion. The enhanced stability is associated with the presence of aluminum polycations (Al 13 Keggin ions, [Al 13 O 4 (OH) 24 (H 2 O) 12 ] 7+ ) that form at pH ≈ 4 [26] and adsorb on the surface of the gibbsite particles. This gives rise to a steep repulsive particle interaction which prevents the platelets from aggregating [START_REF] Van Bruggen | Anomalous stability of aqueous boehmite dispersions induced by hydrolyzed aluminium poly-cationsColloids Surf[END_REF]. After three days, the dispersion was centrifuged to remove excess aluminum chlorohydrate and the particles were subsequently redispersed in a 10 -2 M NaCl solution.

To enable dispersion of the gibbsite platelets in apolar solvents the particles were sterically stabilized with a amino-modified poly-iso-butylene (SAP 230 TP, Infineum, UK), by following two different procedures.

The first procedure involves a gradual substitution of the solvent upon addition of the stabilizing polymer, as described by [START_REF] Van Der Kooij | Formation of nematic liquid crystals in suspensions of hard colloidal plateletsJ[END_REF]. The particles obtained through this method were dispersed in 2-bromotoluene (Acros, 99%).

Following an alternative method, the gibbsite platelets were freeze-dried in the presence of poly-iso-butylene in 1-propanol (Acros, 99%) [START_REF] Heux | Nonflocculating and Chiral-Nematic Self-ordering of Cellulose Microcrystals Suspensions in Nonpolar[END_REF][START_REF] Mourad | Formation of Nematic Liquid Crystals of Sterically Stabilized Layered Double Hydroxide PlateletsJ[END_REF]. The thus stabilized particles were dispersed in toluene (J.T. Baker, 99.5%).

Finally, the dispersions were concentrated with centrifugation and subsequent redispersion to reach the desired particle concentration. At sufficiently high concentration suspensions of gibbsite particles form nematic liquid crystal phases in which the platelets possess orientational ordering. As a result, the nematic phase is birefringent, and as is shown in Figure 1(b) light is transmitted when it is placed between crossed polarisers. The isotropic phase is dark, but can be made birefringent by gently moving the sample so that the platelets align in the resulting flow.

Birefringent materials have two or more refractive indices. Here, we will focus on uniaxial materials, with two refractive indices. If linearly polarised light enters such a material, it is refracted into two components with orthogonal vibration directions, the ordinary ray (o-ray) with the polarisation perpendicular and the extraordinary ray (e-ray) parallel to the axis of anisotropy. The birefringence is defined as ∆n ≡ n e -n o , with the n e and n o the refractive indices for the e-ray and the o-ray. Because of the different refractive indices for these rays, they travel with a different velocity through the material, giving rise to a phase difference δ:

δ = 2π λ (n e -n o )t (1)
with λ the wavelength of the incident light and t the path length through the material [START_REF] Born | Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light[END_REF]. The value | (n e -n o ) | t is referred to as retardation or optical path difference (OPD) [START_REF] Robinson | Qualitative Polarized-Light Microscopy[END_REF]. When the refracted light arrives at the second polariser (also, analyser) that is at a 90 • angle with respect to the first one, only the component with the vibration direction parallel to the polariser will pass through. The resulting intensity is given by:

I = E 2 sin 2 2θsin 2 δ 2 (2)
with E the amplitude of the electro-magnetic vector and θ the angle between the polarisation of the incident light and the direction in the plane perpendicular to the direction of propagation in which the refractive index corresonds to n o [START_REF] Born | Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light[END_REF]. We will find a minimum intensity behind the analyser if θ = 0, π/2, π, ... and relative maxima for θ = π/4, 3π/4, 5π/4, ... with values of:

I max = E 2 sin 2 δ 2 . (3)
Due to the wavelength dependence of δ, the use of white light leads to beautiful interference (or retardation) colours, as part of the wavelengths will interfere constructively and others destructively. These colours occur in a specific sequence, known as Newton's colour scale and are often presented as a function of retardation in the form of a Michel-Levy chart. Figure 2 shows the colour sequence visualised with a Babinet compensator, a tunable retardation plate that is composed of two opposed quartz wedges [START_REF] Hartshorne | The microscopy of liquid crystals Microscope series[END_REF].

One can distinguish two types of birefringence: intrinsic and form birefringence. Intrinsic birefringence is caused by the anisotropy of the crystal lattice and therefore a fixed material parameter.

Gibbsite (γ-Al(OH) 3 ) has a monoclinic crystal lattice with the following refractive indices: natural gibbsite α 1.56-1.58, β 1.56-1.58, and γ 1.58-1.60. For synthetically prepared gibbsite α = β = 1.577 and γ = 1.595 has been reported [START_REF] Bowles | Rock-Forming Minerals Volume 5A: Non-Silicates: Oxides, Hydroxides and Sulphides[END_REF]. The optic axial plane (where the refractive index does not depend on the polarization direction of the incident light) is perpendicular to the (010) plane, and the extinction angle makes a angle of -21 • with the z-axis [START_REF] Bowles | Rock-Forming Minerals Volume 5A: Non-Silicates: Oxides, Hydroxides and Sulphides[END_REF].

For colloidal gibbsite, which consists of hexagonal platelets, this implies that the largest refractive index, or slow axis, has an angle of -21 • with respect to the particle director (see Figure 1). However, in the nematic phase, although the platelets have an net average orientation of the particle director, in the particle plane all orientations are allowed. The resulting intrinsic birefringence of a platelet is therefore effectively parallel to the director, with ∆n i ≈ (γ -α) cos[21 • ] = 0.016. Following [START_REF] Neveu-Prin | Magnetic birefringence of cobalt ferrite ferrofluidsColloids Surf[END_REF][START_REF] Lemaire | Physical properties of aqueous suspensions of goethite nanorodsEur[END_REF] the birefringence of a suspension of platelets due to the intrinsic particle birefringence is given by ∆n intr = ∆n i φS 2 [START_REF] Evans | Pair correlation function decay in models of simple fluids that contain dispersion interactionsJ[END_REF] with φ the particle volume fraction and S 2 the nematic order parameter.

On the other hand, form (or shape) birefringence arises when anisotropic molecules or particles are dispersed in an ordered arrangement in a liquid with a different refractive index. It depends on the particle shape and the solvent refractive index and can therefore be manipulated.

Wiener [18] showed that an assembly of thin parallel plates in a medium with a different refractive index behaves like a negative uniaxial crystal. Bragg and Pippard [START_REF] Bragg | The Form Birefringence of MacromoleculesActa Cryst[END_REF] extended his work to ellipsoids and obtained for the form birefringence of an assembly of particles with refractive index n p in a solvent with n s :

∆n f orm = n a -n b = φ(n p -n s ) 9k (3 + k) 2 (L b -L a ) (5)
with φ the particle volume fraction, k = (1 -φ)(n 2 p /n polarizing factors that depend on the particle symmetry. To obtain a reasonable estimate for the form birefringence of gibbsite where the particles have intrinsic birefringence as well, we can use this equation with an average of the three refractive indices, n p = 1.58. Although this is not exact, according to Twersky [START_REF] Twersky | Form and intrinsic birefringenceJ[END_REF] it should introduce little error because the refractive indices are close. In their derivation Bragg and Pippard considered an assembly with perfectly aligned particles (orientational order parameter S 2 = 1), which is obviously not true for the gibbsite nematic phase. To account for this, we multiply by S 2 [START_REF] Neveu-Prin | Magnetic birefringence of cobalt ferrite ferrofluidsColloids Surf[END_REF][START_REF] Lemaire | Physical properties of aqueous suspensions of goethite nanorodsEur[END_REF], analogous to Equation 4. Previous experiments on the alignment of the gibbsite nematic phase in a magnetic field established typically S 2 ≈ 0.8 [START_REF] Van Der Beek | Magnetic-field-induced orientational order in the isotropic phase of hard colloidal plateletsPhys[END_REF].

In Figure 3(b) we plot the intrinsic and form birefringence as a function of solvent refractive index n s , for a volume fraction φ of 0.26. The depolarizing factors can be calculated by considering the particles as ellipsoids [START_REF] Osborn | Demagnetizing Factors of the General EllipsoidPhys[END_REF], resulting in L a = 0.03 and L b = 0.94 for a bare platelet with an aspect ratio of 25. ∆n tot is obtained by simply taking the sum of the two contributions. We find that the birefringence of a gibbsite nematic phase is negative for most solvents, except if the refractive index is close to that of gibbsite itself.

Polarised light microscopy

Birefringence of the gibbsite nematic phase was studied with polarised light microscopy, using a Nikon LV100Pol microscope, equipped with a 2× Nikon CFI Plan UW objective and a 10× Nikon CFI Plan Fluor ELWD objective. The microscope was in a tilted position with the focal plane along gravity in order to allow investigation of the samples in vertical position. Images were captured with a MicroPublisher 5 megapixel ccd camera (MP5, QImaging).

To determine the sign of the birefringence a full wave (λ = 530 nm) retardation plate was used, which is a birefringent filter with a known birefringence and orientation. It is placed in the light path between the polariser and analyser at an angle of 45 • for maximum effect, and it adds a fixed retardation to all the wave fronts in the optical path. Depending on the respective orientation of the slow axes of the liquid crystal and the retardation plate, the retardation is added (parallel orientation) or subtracted (perpendicular orientation).

To align the nematic phase in a magnetic field a home-built polarization microscope setup was used, assembled from the head of a Zeiss Axiolab microscope, equipped with crossed polarisers, and a 2× Edmund Optics objective. The magnetic field was generated with a Bruker BE25v Electromagnet, equipped with large flat pole shoes producing a very homogeneous magnetic field in the sample volume. Field strengths ranged from B = 0 -1.8 T and were monitored with a LakeShore 421 Gaussmeter. The samples could be rotated along their vertical axis with adjustable velocity (usually ∼ 0.25 Hz).

Samples were prepared in flat optical capillaries (VitroCom, NJ, USA), with internal dimensions of 0.1 × 2 × 40 mm 3 for the experiments at the isotropic-nematic interface and in magnetic field and 0.2 × 4 × 40 mm 3 for study of tactoids, and flame-sealed to avoid evaporation of the solvent. The samples were homogenized and left to phase separate, which evolved via the formation, sedimentation and subsequent coalescence of tactoids.

Results

Anchoring at a planar wall and the isotropic-nematic interface

The isotropic-nematic phase separated sample shown in Figure 1(b), was imaged just after phase separation. The nematic phase displays bright colourful birefringence, indicating the presence of many domains with different orientations. Over a period of days to weeks (depending, among other things, on the sample thickness) these domains coalesce and align in such a way that most platelets have the preferred anchoring with the interface. For entropic reasons, colloidal platelets exhibit homeotropic anchoring, which means that they tend to align with the director perpendicular to the isotropic-nematic interface as well as to hard walls [START_REF] Van Der Beek | Isotropic-nematic interface and wetting in suspensions of colloidal plateletsPhys[END_REF][START_REF] Otten | Capillary rise of an isotropic-nematic fluid interface: surface tension and anchoring versus[END_REF], as illustrated in Figure 4(a).

We will now use this preferential alignment to study the birefringence of the gibbsite nematic phase in toluene with refractive index n s = 1.4969 and compare that to the gibbsite nematic phase in water with n s = 1.333.

Figure 4(b) and (c) show two gibbsite systems, both one week after phase separation. The sample in Figure 4(b) is a suspension in toluene with an overall volume fraction φ = 0.22, while the sample in Figure 4(c) is a dispersion of gibbsite in an aqueous salt solution (10 -2 M NaCl) at φ = 0.25.

The appearance of the nematic phase is clearly different from the freshly phase separated sample in Figure 1(b). The bulk nematic phase now appears with the same colour as the isotropic phase, the typical magenta colour of the retardation plate independent of the orientation of the polarisers, which implies that it does not exhibit birefringence. Thus the platelets are oriented with their director pointing into the focal plane, which indeed corresponds to homeotropic alignment with the capillary walls. On the other hand, along the isotropic-nematic interface and along the capillary wall at the side we observe birefringent bands with different colours. This indicates that the nematic director points in the vertical and horizontal direction, corresponding to the expected alignment shown in Figure 4(a).

In the suspension with sterically stabilized gibbsite in toluene (Figure 4(b)) the nematic phase near the isotropic-nematic interface appears orange, which corresponds to a subtraction of the retardation with respect to the retardation plate (see Figure 2). On the other hand, a blue retardation colour can be observed near the capillary wall, corresponding to an addition of the optical path difference. From these observations we can deduce that the slow axis is parallel to the platelet director and hence that intrinsic birefringence is dominant.

If we now compare this to the aqueous suspension shown in Figure 4(c), we observe that the retardation colours are reversed: blue at the isotropic-nematic interface and orange near the capillary wall. This implies that the slow axis is now perpendicular to the platelet director and thus that in water, with a much larger difference between the refractive index of particle and solvent, form birefringence wins over the intrinsic birefringence.

Nematic phase aligned in a magnetic field

One could argue that the difference in particle stabilisation between the two systems could influence the anchoring preference. For instance, in the case of thermotropic liquid crystals it is well-known that chemical interaction rather than molecular shape determines the preferred anchoring [START_REF] De Gennes | The Physics of Liquid Crystals[END_REF]. A convenient way to exclude the influence of this effect, is to align the nematic phase of both samples in a magnetic field.

Colloidal gibbsite platelets have an anisotropic magnetic susceptibility, which implies that an external magnetic field gives rise to an induced magnetic moment in the particles. Previous studies on sterically stabilized gibbsite suspensions [START_REF] Van Der Beek | Isotropic-nematic interface and wetting in suspensions of colloidal plateletsPhys[END_REF][START_REF] Van Der Beek | Influence of a magnetic field on the nematic phase of hard colloidal plateletsPhys[END_REF] established that the magnetic anisotropy of these platelets is negative (∆χ = -10 -22 J/T 2 ), and therefore the platelets tend to align with their director perpendicular to an externally applied magnetic field. This implies that the director orientation is restricted to the plane perpendicular to the magnetic field direction. If instead of a static field a rotating field is used, the platelets will align in a single direction, with the director perpendicular to the rotation plane. Rotation of the sample in a static field (like sketched in Figure 5(a)) has the same effect, and it has been shown that one can obtain a single domain nematic phase with a well-defined orientation [START_REF] Van Der Beek | Influence of a magnetic field on the nematic phase of hard colloidal plateletsPhys[END_REF].

Figure 5(b) shows the two isotropic-nematic phase separated suspensions after rotation at 0.25 Hz for one hour in a magnetic field with a field strength of 1.8 T. There is a distinct difference in the retardation colours of the samples, in the isotropic phase and even more clearly in the nematic phase. With the retardation plate in the horizontal direction the retardation colour of the sample in toluene corresponds to first order orange, implying a subtraction of the optical path difference with roughly 140 nm (see Figure 2). On the other hand, the aqueous sample displays a second order blue tint, corresponding to an addition of the optical path difference with about 210 nm. With a capillary inner thickness of 100 µm this corresponds to a ∆n for toluene of 1.4 • 10 -3 and ∆n = -2.1 • 10 -3 for aqueous gibbsite. By combining Equation 4 and 5 in section 3, we find ∆n = 2 • 10 -3 for gibbsite dispersed in toluene, in line with the experimentally obtained value. For aqueous gibbsite we obtain ∆n = -8 • 10 -3 , which is somewhat lower than the experimental value, but still in reasonable agreement taking into account the approximations with regard to the order parameter and the volume fraction of the nematic phase.

The fact that the isotropic phases of the two samples also display a colour difference, both deviating from the typical magenta colour of the retardation plate, indicates that the particles in the isotropic phase also align in the magnetic field, in that way forming a paranematic phase. The paranematic phase in toluene displays an orange-red tint, corresponding to a subtraction of the retardation, whereas the paranematic phase in the aqueous sample is purple-blue, corresponding to an addition of the retardation.

From these results we find again that for gibbsite dispersed in toluene the birefringence is positive, corresponding to the intrinsic birefringence of the platelets. On the other hand, for aqueous gibbsite, with a much larger difference between the refractive index of particle and solvent, the birefringence is negative, implying that form birefringence is dominant over the intrinsic particle birefringence. This is in line with what we obtained from the anchoring at the wall and the isotropic-nematic interface, implying that particle stabilisation does not affect the anchoring preference and is indeed a result of the particle shape. However, the absolute magnitude of the anchoring strength does depend on particle stabilisation and solvent, as we have shown in previous publications [START_REF] Verhoeff | Magnetic field effects on tactoids of plate-like colloidsJ[END_REF][START_REF] Verhoeff | Tactoids of plate-like particles: size[END_REF].

Tactoids with radial director field

An alternative method to study the birefringence of the nematic phase is to investigate the nematic phase confined to micrometer-sized droplets. The properties of such droplets, or tactoids, are determined by a competition between surface anchoring and bulk elastic free energy [START_REF] Verhoeff | Magnetic field effects on tactoids of plate-like colloidsJ[END_REF]. The balance of these two depends on the droplet size, which in the case of plate-like particles leads to oblate shaped droplets with a uniform director field for tactoids smaller than the ratio of the splay elastic constant and the anchoring strength K 1 /w. However, tactoids larger than K 1 /w are spherical with a radial director field and a point defect in the center, a so-called hedgehog, as depicted schematically in Figure 6(a). Here, we focus on the latter type of tactoid.

Figures 6(b) and (c) display two radial tactoids, one in a suspension in bromotoluene and the other in an aqueous gibbsite suspension. The refractive index of bromotoluene is very close to that of gibbsite (n s = 1.556 and n s ≈ 1.58 respectively), so that the gibbsite platelets are almost index matched. We observe the typical birefringence pattern of a radial tactoid, consisting of four parts with the opposite quarters the same retardation colour. The retardation colours of the tactoids in the two solvents are reversed: the quarters that are blue in the one are yellow/orange in the other and vice versa. This confirms again the inversion of the birefringence observed above, with in bromotoluene the slow axis along and in water perpendicular to the nematic director.

Conclusions

We have demonstrated that it is possible to tune the birefringence of the nematic phase in suspensions of gibbsite platelets by varying the solvent refractive index. In solvents with a refractive index close to that of the gibbsite platelets, such as toluene and bromotoluene, the positive intrinsic birefringence of the particles determines the birefringence of the nematic phase, while in solvents with a considerably lower refractive index, such as water, form birefringence wins over the intrinsic birefringence, resulting in a net negative birefringence of the nematic phase. In conjunction with the anchoring properties of interfaces and the aligning influence of an external magnetic field this leads to fascinating birefringence patterns.

3 Figure 1 .

 31 Figure 1. (a) Transmission electron micrograph of colloidal gibbsite platelets. (b) Isotropic-nematic phase separated gibbsite suspension between crossed-polarisers. The birefringence visible in the isotropic phase is flow birefringence caused by gentle movement. Scale: the cuvette is 1 cm wide.

Figure 2 .

 2 Figure 2. Newton's colour scale, here visualised with a Babinet compensator, represents the retardation colour sequence as a function of retardation.

Figure 3 .

 3 Figure 3. (a) Crystallographic and optical axes of gibbsite. Reprinted from [33] with permission of The Geological Society London. (b) Intrinsic, form and total birefringence of a nematic phase of colloidal gibbsite as a function of solvent refractive index for φ = 0.26, S 2 =0.8, La = 0.03 and L b = 0.94.

Figure 4 .

 4 Figure 4. Alignment of the nematic phase at the isotropic-nematic interface and the capillary walls; schematically depicted in (a) and observed with polarised light microscopy in a system of gibbsite dispersed in toluene (b) and water (c).

Figure 5 .

 5 Figure 5. (a) Rotation of the sample around its vertical axis in a magnetic field aligns the platelets with the director in the vertical direction. (b) Isotropic-nematic phase separated suspension of sterically stabilized gibbsite (left) and charge stabilized gibbsite (right) after one hour rotation in a magnetic field with a strength of 1.8 T. Solid arrows indicate the polarisers, dashed arrow the orientation of the retardation plate. Scale: the capillaries are 2 mm wide.

  Taylor & Francis and I.T. Consultant

Figure 6 .

 6 Figure 6. Tactoid with radial director field and hedgehog defect; schematically depicted in (a) and observed with polarised light microscopy in a gibbsite suspension in bromotoluene (b) and in water (c).

Acknowledgement

A.A.V. would like to thank the Royal Netherlands Academy of Arts and Sciences (KNAW) for financial support.