Critical behaviour of the contact angle within nonwetting gaps

Recent density functional theory and simulation studies of fluid adsorption near planar walls in systems where the wall–fluid and fluid–fluid interactions have different ranges, have shown that critical point wetting may not occur and instead nonwetting gaps appear in the surface phase diagram, separating lines of wetting and drying transitions, that extend up to the critical temperature Tc . Here we clarify the features of the surface phase diagrams that are common, regardless of the range and balance of the forces, showing, in particular, that the lines of temperature driven wetting and drying transitions, as well as lines of constant contact angle π>θ>0 , always converge to an ordinary surface phase transition at Tc . When nonwetting gaps appear the contact angle either vanishes or tends to π as t≡(Tc−T)/Tc→0 . More specifically, when the wall–fluid interaction is long-ranged (dispersion-like) and the fluid–fluid short-ranged we estimate π−θ∝t0.16 , compared with θ∝t0.77 when the wall–fluid interaction is short-ranged and the fluid–fluid dispersion-like, allowing for the effects of bulk critical fluctuations. The universal convergence of the lines of constant contact angle implies that critical point filling always occurs for fluids adsorbed in wedges.

The modern theory of wetting transitions began when Cahn [1] and Ebner and Saam [2] showed that a wall-gas interface may change from a state of partial wetting (contact angle θ > 0) to complete wetting (θ = 0) as the temperature T is increased.For * Authors to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.systems where the contact angle θ > π/2 a similar transition from partial to complete drying (θ = π) may occur on increasing T. For reviews see [3][4][5][6][7].Cahn also proposed that wetting/drying transitions must occur before the bulk critical temperature T c .While this argument is flawed, his prediction of 'critical-point wetting' is indeed borne out for systems where the wall-fluid and fluid-fluid interactions are short-ranged (SR) [8][9][10][11][12][13] (hereafter SR/SR systems) where the global surface phase diagrams were first elucidated by Nakanishi and Fisher (NF) using Landau theory and scaling arguments [10].Recent density functional theory (DFT) studies have shown that critical point wetting also occurs when the wall-fluid and fluid-fluid interactions are long-ranged (LR) [14].Indeed the surface phase diagrams for SR/SR and LR/LR systems are similar with each showing lines of wetting and drying transitions that converge to an ordinary surface phase transition at T c .
When there is a mismatch between the wall-fluid and fluidfluid forces however, wetting and drying transitions need not happen [15].Recently, Evans, Stewart and Wilding (ESW) [11] have shown that when the wall-fluid interaction is LR and the fluid-fluid SR (hereafter LR/SR) a nonwetting gap separates the lines of wetting and drying transitions, which do not converge at T c .A nonwetting gap also occurs when the wall-fluid interaction is SR and the fluid-fluid LR (hereafter SR/LR), where the surface phase diagram shows only a line of drying transitions.In the present paper we extend ESW and also Parry and Malijevský (PM) [14] and ask two specific questions.Firstly, what happens to θ as T approaches T c within a nonwetting gap?Equivalently, what replaces critical point wetting when it is absent?Secondly, what features of the four surface phase diagrams are common for critical point wetting and nonwetting as might be expected in the critical region when the bulk correlation length is large?
Consider the interface between a planar solid, situated in the z = 0 plane, and a fluid at temperature T and chemical potential µ.The solid is considered inert, exerting an external field V(z) on the fluid particles, while the fluid shows bulk coexistence between gas and liquid phases, with number densities ρ g and ρ l , along a saturation curve µ sat (T) up to T c .On approaching T c , at saturation, we identify bulk critical exponents for ∆ρ = ρ l − ρ g ∝ t β and the bulk gas correlation length ξ g ∝ t −ν where t = (T c − T)/T c .We shall also need the exponents for the (liquid) compressibility χ l = ∂ρ l /∂µ ∝ t −γ , the surface tension γ lg ∝ t μ, and finally for the anomalous decay of correlations G(r) ≈ 1/r d−2+η at the critical point.Below T c , and at bulk coexistence, the contact angle is identified from Young's equation γ wg = γ wl + γ lg cos θ involving the wall-gas and wall-liquid tensions.The surface phase diagram, in the (ϵ, T) plane (with ϵ, the strength of the attractive part of V(z)), will show lines of wetting transitions (where θ vanishes) and drying transitions (where θ approaches π) which may be continuous (referred to as critical wetting) or first-order.To answer the questions above, we turn to DFT and minimise a grand potential functional [16] which determines the equilibrium density profile ρ(z), surface tensions and hence θ.The Helmholtz potential F[ρ] models the fluid-fluid interactions, which we split into a repulsive, hardsphere contribution [17], and an attractive part which treats the fluid-fluid potential ϕ(z) (integrated along the plane of the wall) in a reliable mean-field (MF) fashion as described in [11,14].Our MF studies are complemented by generalised Landau theory and scaling arguments which, we believe, allow correctly for the influence of bulk critical fluctuations [18].
To begin, we recall results for SR/SR and LR/LR systems to highlight the connection between wetting and surface criticality.Figure 1 shows three similar schematic phase diagrams: (a) the NF phase diagrams for Ising-like systems (with h 1 the surface field and for positive surface enhancement, −g > 0) [10], (b) DFT results for SR/SR systems, where ϵ SR is the strength of the SR potential V(z) [11], and (c) DFT results for LR/LR systems with dispersion forces decaying as V(z) = −ϵ 3 /(z + σ) 3  and ´∞ z dxϕ(x) ≈ −ϕ 3 /z 3 [14].In these diagrams the lines of wetting/drying transitions converge to an ordinary surface phase transition C ∞ ord at T c .For Ising-like systems (figure 1(a)) and fluid SR/SR systems (figure 1(b)) the lines of critical wetting and critical drying converge symmetrically, while for LR/LR systems there is asymmetry reflecting the difference between the orders of the transitions.Either side of C ∞ ord are lines of critical desorption, C ∞ − , and critical adsorption, C ∞ + , along which the profile decays to the bulk critical density as ρ(z) − ρ c ≈ ±z −β/ν .At MF level β/ν = 1 while more correctly β/ν ≈ 0.52.The adsorption, Γ wl = ´∞ 0 dz[ρ(z) − ρ l ], defined for the wall-liquid interface, diverges as Γ wl ∝ t β−ν on approaching C ∞ + (and similarly Γ wg ∝ −t β−ν on approaching C ∞ − ).At C ∞ ord the profile exhibits a weaker decay.Indeed, it is flat within NF Landau theory, while for LR/LR systems it is characterised by an exponent x ord ⩾ β/ν.PM determined that for dispersion forces x ord = 2 at MF level and estimate x ord ≈2.14 when bulk fluctuations are allowed for.The ordinary surface phase transition is characterised by a surface gap exponent ∆ 1 with ∆ 1 = 1/2 at MF level close to the true value ∆ 1 ≈0.47 [19].NF showed that in the critical region the contact angle scales as θ = Θ(h 1 t −∆1 ) implying that the lines of critical wetting (and drying) approach T c as t w ∝ h 1/∆1 1 .This also applies for LR/LR forces, consistent with the universality class of the ordinary surface phase transition, although at MF level, the location of the line of first-order wetting, which extends up to T c in figure 1(c), has a logarithmic correction [14].We mention that while the location of critical wetting/drying phase boundaries is determined by ∆ 1 , the critical singularities associated with these transitions are distinct [4][5][6][7][20][21][22][23].Finally, we note that within NF scaling theory the lines of constant contact angle also converge to C ∞ ord following paths We now turn to surface phase diagrams which display nonwetting gaps beginning with LR/SR systems.ESW studied a wall-fluid potential decaying as V(z) ≈ −ϵ 3 /z 3 so that when ϵ 3 = 0 the potential corresponds to a pure hard-wall.DFT and simulation studies indicate that the surface phase diagram shows a line of first-order wetting transitions, which meets the critical line at T c , tangentially, at a finite value of ϵ c 3 -similar to earlier studies of wetting in Ising systems with a LR surface field [24].Additionally, ESW show that a line of continuous drying transitions occurs at ϵ 3 = 0, over the whole temperature range up to T c at which θ = π.PM clarified that the line of wetting transitions terminates at an ordinary surface phase transition, C ∞ ord , while the line of continuous drying transitions at ϵ 3 = 0, which is not driven by temperature, simply ends at a point on the line of critical desorption C ∞ − .It is useful to generalise to potentials that decay as V(z) ≈ −ϵ p /z p before specialising to p = 3.We also keep the hardwall condition for z < 0 so the wall is completely dry when ϵ p = 0 for all T < T c .When p = 0, the field ϵ 0 is equivalent to a shift in the chemical potential, or bulk field h for Ising systems, associated, in the critical region, with a bulk gap exponent ∆.Therefore we can anticipate that near T c , and for weakly attractive walls, the surface tensions and contact angle depend on a scaling variable ϵ p t −∆ϵ where ∆ ϵ = ∆ − pν is a LR gap exponent.The surface phase diagrams then fall into three regimes: For p < 1 wetting transitions are not possible because of the energy cost of having a macroscopic liquid layer.Such phase diagrams only show a line of drying at ϵ p = 0.For p > 1, both wetting and drying transitions can occur and the surface phase diagram divides into two further classes depending on the sign of ∆ ϵ , i.e. if the field ϵ p is relevant (∆ ϵ > 0) or irrelevant (∆ ϵ < 0) or equivalently if the power p is lower or greater than a marginal value p * = ∆/ν.Within MF theory ∆ = 3/2 and ν = 1/2 giving p * = 3 so that dispersion forces are marginal while more correctly p * ≈2.51.When 1 < p < p * , the wall-fluid interaction dominates and there is no nonwetting gap.In this case a line of first-order wetting transitions, occurring along the curve t w ∝ ε1/∆ϵ p , with εp = ϵ p /(p − 1), meets the line of continuous drying, at ϵ p = 0, at T c .We note A crucial feature of the phase diagram is that, apart from the line of drying transitions, at ϵ p = 0, where θ = π, all other lines of constant contact angle, π > θ ⩾ 0 converge to C ∞ ord on approaching T c as shown in figures 2(b) and (c).The only impact of the marginality of dispersion forces at MF level is that this convergence is only apparent very close to T c -which we explain below.An important consequence of this convergence is that the limiting value of θ as T → T c within the nonwetting gap is π.The necessity of this can be understood without detailed calculation.Suppose that we truncate the potential V(z) so that it vanishes for z > R. In this case the surface phase diagram resorts to being of SR/SR type (figure 1(b)), and the line of critical drying transitions occurs for ϵ d p (T) > 0 which converges to C ∞ ord as T → T c .As we increase R the line of drying transitions deforms as ϵ d p ∝ R p e −R/ξg recovering, smoothly, in the limit R → ∞ the ESW phase diagram for LR/SR systems [14].However, for all finite R the whole region, ϵ p ⩽ ϵ d p (T), is completely dry with θ = π.This region shrinks as R increases but the limiting value of the contact angle is still π along ϵ p = 0 + for T < T c and for T = T − c for ϵ p < ϵ c p .These features must be preserved in the limit R → ∞ when we recover the phase diagram for LR/SR systems with the nonwetting gap.We can expect that, in general, as T → T c the contact angle approaches π as a power-law.Moreover, since no other phase boundaries are encountered, this should be same along the whole nonwetting gap and we introduce a nonwetting exponent, π − θ ∝ t ψLS , to characterise this.To determine ψ LS we follow ESW and construct a binding potential W LS (ℓ), for fixed T < T c and ϵ p > 0, corresponding to the surface free-energy of a drying layer of gas which is constrained to be of thickness ℓ.Provided that ℓ ≫ ξ g this is given by incorporating a LR attraction, due to the wall-fluid potential, and a SR repulsion due to the hard-wall.The coefficient a is positive with MF studies showing that a ∝ t μ−∆1 which vanishes, on approaching T c , more slowly than the surface tension γ lg ∝ t μ [25].Minimizing W LS (ℓ) determines the equilibrium drying film thickness as ℓ eq /ξ g ≈ − ln ϵ p .Then Young's equation where we have ignored logarithmic terms, which are only of importance at the marginal value p = p * .We stress that the combination of thermodynamic quantities εp (which must be conjugate to ∆ρ), γ lg and ξ g in this expression is the only one that is consistent with the requirement that the contact angle is dimensionless and vanishes as √ εp as found by ESW and verified in their simulation studies.We can therefore be very confident that this general expression must be valid in all descriptions of LR/SR systems at MF and beyond.Substituting for the anticipated bulk critical behaviours, ξ g ∝ t −ν , ∆ρ ∝ t β and γ lg ∝ t μ then gives on using standard bulk critical exponent relations and identifying that ψ LS = (pν − ∆)/2.Notice that this is precisely in accord with the scaling theory developed earlier which implies that the dependence of the contact angle on the surface field strength must appear through the scaling variable ϵ p t −∆ϵ .At MF level ∆ ϵ = (3 − p)/2 so that for dispersion forces ψ LS = 0 in which case the logarithmic corrections must be included leading to π − θ ∝ 1/| ln t|.This very weak singularity is qualitatively consistent with our DFT findings since the lines of constant contact angle, θ > π/2, only show a significant temperature dependence, where they converge to C ∞ ord , extremely close to the bulk critical temperature (see figure 2(b)).This contrasts with the behaviour seen for p = 4 where the convergence of the lines of constant contact angle is much more apparent.We have checked, using a simpler Landau theory (with a LR external field), that for p = 3 the contact angle indeed vanishes as an inverse logarithm.However we do not dwell on these MF considerations since they are quantitatively invalid close to T c and the correct values of the bulk critical exponents must be used i.e. beyond MF dispersion forces are non marginal.In this case the prediction, that for dispersion forces, ψ LS = (3ν − ∆)/2, simplifies to, leading to the estimate that ψ LS ≈ 0.16, when we allow correctly for the non-classical values of the bulk critical exponents.Writing h 1 ≡ (ϵ c p − ϵ p )/ϵ c p , we note that near C ∞ ord the lines of constant contact angle close to π follow paths in accord with both NF scaling theory and the presence of a nonwetting gap when θ = π.Finally, we turn to the remaining SR/LR case, where the wall-fluid potential is SR, for example V(z) = −ϵ SR exp(−z/σ) and the fluid-fluid interaction is dispersionlike decaying as ´∞ z dz ′ ϕ(z ′ ) ≈ −ϕ 3 /z 3 .ESW showed that the phase diagram contains only a line of first-order drying transitions at ϵ d SR (T) > 0, which meets T c tangentially at ϵ c SR .This behaviour is fully consistent with the universal aspects of fluid adsorption that we expect at T c and we identify the terminus of the line of drying as an ordinary surface phase transition, C ∞ ord , which separates lines of critical adsorption and desorption.No wetting transitions occur for the wall-gas interface, although ESW point out that the contact angle θ must vanish as ϵ SR → ∞, leading to a nonwetting gap of infinite extent.A similar truncation argument, but now applied to the fluid-fluid potential ϕ(z) shows that the limiting value of the contact angle must be zero as T → T c for ϵ SR > ϵ c SR and that all lines of constant contact angle π ⩾ θ > 0 converge to C ∞ ord .For SR/LR systems therefore we may also define a nonwetting exponent θ ∝ t ψSL characterising the vanishing of the contact angle on approaching T c in the (infinite) nonwetting gap.To determine ψ SL we construct the binding potential for a wetting layer of liquid of thickness ℓ, at the wall-gas interface, which is given by This is the exact analytic expression for the binding potential within the DFT model used by ESW.The first term is the energy cost of integrating the fluid-fluid potential over the wetting layer [21].The second term arises from two contributions, which also have a clear physical interpretation-one comes from the overlap of the tail of the fluid-fluid potential with the local adsorption near the wall [21] and the other from the overlap of the decay of the density profile from the liquid-gas interface, which contains a term 1/(ℓ − z) 3 , with the wall-fluid potential [26,27].The latter dominates in the critical region since χ l ∝ t −γ diverges faster than Γ wl ∝ t β−ν on approaching C ∞ + .The form of the binding potential is not affected by interfacial fluctuations.Minimizing the binding potential determines the equilibrium film thickness as ℓ eq ∝ ϵ SR χ l .The contact angle then follows from γ lg θ 2 /2 = −W SL (ℓ eq ), giving valid for ϵ SR ≫ ϵ c SR .This vanishes for fixed T < T c as ϵ SR → ∞ and also for T → T c , in the nonwetting gap.At MF level, ψ SL = 1/2, while allowing for the correct bulk critical exponents, we find specific to dispersion forces.Near C ∞ ord the lines of constant contact angle close to zero follow (10) where now h 1 = (ϵ SR − ϵ c SR )/ϵ c SR , which is again consistent with NF scaling theory and the nonwetting gap when θ = 0.The estimates ψ LS ≈ 0.16 and ψ SL ≈ 0.77 are the central predictions of our paper and answer our first question.
In figure 3 we present a schematic illustration of the four different surface phase diagrams from which we can read off the answers to our second question.At T c all the surface phase diagrams have an ordinary surface phase transition separating lines of critical adsorption and desorption, and the limiting value of the contact angle at T c is always θ = 0 and θ = π either side of C ∞ ord .Another common feature is that all the lines of constant contact angle, π > θ > 0, converge to C ∞ ord regardless of the presence of critical point wetting (SR/SR and LR/LR) or nonwetting gaps (LR/SR and SR/LR).The appearance of nonwetting gaps for the latter two diagrams can be seen as a limit in which the lines of drying and wetting are skewed to ϵ 3 = 0 and ϵ SR = ∞, respectively and hence become temperature independent.Only in this case do those lines not converge to C ∞ ord .
To finish we point out a consequence for related phase transitions when a fluid is adsorbed near a wedge, formed by two planar walls that meet with opening angle π − 2α.Thermodynamic arguments dictate that a wedge-gas interface is completely filled with liquid when θ ⩽ α, while a wedgeliquid interface is completely filled with gas when π − θ ⩽ α [28][29][30][31][32][33].The surface phase diagrams for this can be read directly from figure 3 since the wedge-wetting and wedgedrying phase boundaries correspond to the loci of lines of constant contact angle θ(T ww ) = α and π − θ(T wd ) = α.Since these always converge to C ∞ ord it follows that critical point wedge wetting/wedge drying always occurs, universally for all systems-there are no 'nonfilling gaps'.The inevitability of critical point wedge filling, predicted here, is the most immediate practical consequence of the universal convergence of the lines of constant contact angle, and may be studied in experiments similar to those described in [34,35].Finally, for LR/SR (and similarly SR/LR) systems the location of the wedge drying transition, in shallow edges, is determined by the nonwetting exponent as (T c − T wd )/T c ∝ α 1/ψLS .This may be an alternative of measuring this exponent in DFT and simulation studies.

Figure 1 .
Figure 1.Schematic illustration of surface phase diagrams showing critical point wetting: (a) NF (Landau theory) for positive surface enhancement (−g > 0) [10], (b) DFT results for SR wall-fluid and fluid-fluid forces [11], and (c) DFT results for LR wall-fluid and fluid-fluid forces [14].Lines of first-order wetting/drying are shown in red and critical wetting/drying in blue.These lines always converge to an ordinary surface phase transition, C ∞ ord , at Tc, which separates lines of critical desorption, C ∞ − , and critical adsorption C ∞ + (shown as thick black).

Figure 2 .
Figure 2. Numerical mean-field DFT results for the surface phase diagrams for LR/SR systems with (a) p = 2 (b) p = 3 and (c) p = 4, showing lines of continuous drying, at ϵp = 0, and first-order wetting.A nonwetting gap appears in cases (b) and (c) which correspond to the regime p ⩾ p * = 3.The loci of lines of constant angle, π > θ ⩾ 0, which must converge to C ∞ ord , at ϵ c p , are also shown for these two cases.For the marginal case p = 3 these show only a weak dependence on T for θ > π/2.
that the shape of the wetting curve changes qualitatively as p → 1 (the amplitude diverges) and as p → p * (since 1/∆ ϵ diverges) signalling the change of regimes.Numerical MF DFT results for p = 2 are shown in figure 2(a) illustrating that the line of wetting transitions and the line of constant contact angle θ = 3π/4 converge towards ϵ c 2 = 0, which can be established analytically.For p ⩾ p * on the other hand, the surface phase diagram is the same as ESW and shows a nonwetting gap separating a line of first-order wetting, terminating at C ∞ ord from the line of continuous drying transitions at ϵ p = 0illustrated in figures 2(b) and (c) for p = 3 and p = 4.As established previously by PM, precisely at C ∞ ord the density profile decays more weakly than for critical adsorption/desorption as ρ(z) − ρ c ≈ −ϵ c p z −x ord (p) .Generalised Landau theory determines that x ord (p) = (p − 2)/(1 − ην/β), which implies that for dispersion forces x ord (3)≈1.07,very close to the MF value x ord (3) = 1.

Figure 3 .
Figure 3. Schematic illustration of the four surface phase diagrams for SR/SR, LR/LR, LR/SR and SR/LR systems, showing how the lines of temperature driven wetting and drying transition (red for first-order and blue for critical wetting) and the lines of constant angle (dashed), all converge to C ∞ ord .Nonwetting gaps appear when there is a mismatch of the range of the wall-fluid and fluid-fluid forces in which case either the line of drying transitions (at ϵ 3 = 0) or wetting transitions (at ϵ SR = ∞) is temperature independent.In the nonwetting gap the contact angle tends to π (LR/SR) and 0 (SR/LR) as T → Tc.