Research Article

A unified description of hydrophilic and superhydrophobic surfaces in terms of the wetting and drying transitions of liquids

Robert Evans, Maria C. Stewart, and Nigel B. Wilding
PNAS November 26, 2019 116 (48) 23901-23908; first published October 14, 2019; https://doi.org/10.1073/pnas.1913587116

  1. Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved September 24, 2019 (received for review August 8, 2019)

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Significance

Whether a drop of liquid such as water is repelled from a solid substrate yielding a large angle of intersection θ between the drop’s surface and the substrate or is strongly attracted, small θ, is key to the function of a host of physical and biological systems. We elucidate the physics of hydrophobic and hydrophilic substrates in terms of wetting and drying surface phase diagrams. These display a surprising variety of forms that depend upon the nature of the substrate–liquid and liquid–liquid attractive forces. Liquids near weakly attractive substrates exhibit critical drying, a phenomenon occurring as θ→180○ that is accompanied by divergent density fluctuations. Our findings provide a conceptual framework for tailoring the properties of superhydrophobic and hydrophilic materials.

Abstract

Clarifying the factors that control the contact angle of a liquid on a solid substrate is a long-standing scientific problem pertinent across physics, chemistry, and materials science. Progress has been hampered by the lack of a comprehensive and unified understanding of the physics of wetting and drying phase transitions. Using various theoretical and simulational techniques applied to realistic fluid models, we elucidate how the character of these transitions depends sensitively on both the range of fluid–fluid and substrate–fluid interactions and the temperature. Our calculations uncover previously unrecognized classes of surface phase diagram which differ from that established for simple lattice models and often assumed to be universal. The differences relate both to the topology of the phase diagram and to the nature of the transitions, with a remarkable feature being a difference between drying and wetting transitions which persists even in the approach to the bulk critical point. Most experimental and simulational studies of liquids at a substrate belong to one of these previously unrecognized classes. We predict that while there appears to be nothing particularly special about water with regard to its wetting and drying behavior, superhydrophobic behavior should be more readily observable in experiments conducted at high temperatures than at room temperature.

The ability to control the behavior of a liquid in contact with a solid substrate is crucial for the functional properties of a host of physical and biological systems (1). For instance, plant leaves need to remain dry during rain to allow gas exchange through their pores whereas liquids such as paints, inks, and lubricants are required to spread out to coat surfaces. The key quantity characterizing the range of different possible behaviors is the contact angle θ that a liquid drop makes with a solid substrate. A hydrophobic (or more generally, solvophobic) substrate yields a large contact angle and when 90○<θ<180○, one refers to the system as partially dry (Fig. 1). Substrates for which θ is close to the limit of complete drying, θ→180○, are termed superhydrophobic and are of interest for many important potential applications involving liquid-repellant materials (2). Occupying the opposite extreme are hydrophilic (or solvophilic) surfaces for which θ is small. The regime 0○<θ<90○ is referred to as partially wet (Fig. 1), with complete wetting occurring when θ→0○.

Fig. 1.
Fig. 1.

Schematic representation of a liquid drop on a solid substrate (or “wall”). The contact angle can take a range of values 0○≤θ≤180○, depending on the combinations of interfacial tensions that feature in Young’s equation.

Young’s equation γlv⁡cos(θ)=γwv−γwl provides the macroscopic (thermodynamic) basis for the contact angle: θ is determined by 3 interfacial tensions (surface excess free energies) between the substrate (or wall, w), liquid (l), and vapor (v). Intuitively, the physical factors that control cos(θ) seem clear. The primary role is played by the strength of the substrate–fluid interaction: Strengthening the attraction decreases θ and promotes wetting, while weakening the attraction increases θ and thus promotes drying.

The phenomenology associated with wetting and drying is most profitably characterized in terms of the physics of surface phase transitions. The current understanding of these transitions derives largely from extensive simulation studies on simple lattice-gas models of fluids which possess the special “particle–hole” symmetry of the Ising model, e.g., refs. 3⇓–5. It is commonly, albeit tacitly, assumed that the resultant picture of phase behavior [having its origins in a seminal paper by Nakanishi and Fisher (6)] is generic and therefore should apply to real fluids. At first sight this is not unreasonable given the close similarity between the bulk phase behavior of fluids and Ising magnets, as well as the universality linking their bulk critical behavior (7). However, as has recently become apparent, the lack of particle–hole symmetry in realistic (i.e., off-lattice) fluid models engenders important qualitative differences in surface phase behavior compared to their lattice-based counterparts. Most pertinent is the distinction in the essential character of wetting and drying transitions. In lattice models with particle–hole symmetry, wetting and drying are formally equivalent. By contrast, simulation studies of the wetting transition, where cos(θ)→1, in realistic fluid models such as Lennard-Jonesium or SPC/E water (8, 9) find this to be a strongly first-order surface phase transition, while simulations of the same models reveal drying (cos(θ)→−1) to be a critical (continuous) transition with accompanying divergent length scales (10⇓–12). Ref. 12 provides a brief review of simulations of drying. Accordingly, the physics of drying in realistic fluids (and, by extension, the phenomenon of superhydrophobicity) are far richer than those of wetting, a fact illustrated by the simulation snapshot and Movie S1 described in SI Appendix which display the large-length-scale fractal-like configurations of “bubbles” of incipient vapor phase that develop for a Lennard-Jones liquid close to complete drying.

A key feature of surface phase diagrams is their extreme sensitivity to the range of the relevant interactions. Such dependence contrasts starkly with the situation for bulk fluids where irrespective of whether interatomic potentials are truncated beyond some cutoff radius (as in simulation studies) or retain true long-ranged power-law decay (characteristic of dispersion/van der Waals forces), principal features such as the phase diagram topology and critical point behavior (including critical exponents) are universal. Aspects of the importance of interaction range for surface phase behavior have been recognized previously, notably in the context of how the nature of wetting in lattice models is influenced by the range of wall–particle forces (13, 14). Several other studies, e.g., refs. 15 and 16, have considered long-range forces. However, to date there has been no wider elucidation for realistic fluid models of how the choice of interaction ranges for both wall–fluid (wf) and fluid–fluid (ff) forces determines the overall form of surface phase diagrams. Here we provide the requisite theoretical framework. We investigate a simple model system that captures all of the features of real fluids and which allows us to address fundamental questions concerning how fluids wet or dry at substrates across the whole range of bulk liquid–vapor coexistence, i.e., from near the triple point to the bulk critical point at temperature Tc. The model (Eqs. 1 and 2) enables us to treat short-ranged (SR) interactions, for which the wf or ff potential is truncated, and long-ranged (LR) interactions, for which the full power-law tail is retained, thereby incorporating the correct nonretarded dispersion/van der Waals forces.

The imperative for establishing such a framework is clear: In computer simulation studies of liquids, dispersion/power-law interactions are typically truncated on grounds of computational tractability, prompting the question of how this limitation affects the resultant surface phase behavior and what other scenarios can emerge. The same question is of relevance in experiments. Increasingly, experimentalists have the ability to control substrate–liquid interaction (1), e.g., by tailoring the choice of substrate material (17), the surface structure (2, 18), and substrate flexibility (19) or by functionalizing the substrate surface with special coatings (20). Soft matter systems provide particularly rich possibilities for controlling the form of interactions, e.g., by tuning the refractive index difference between colloidal particles and a solvent to modify or eliminate the dispersion tail (21) or by exploiting the depletion mechanism to induce intrinsically short-ranged colloidal interactions (22). For electrolytic liquids, the substrate–liquid interactions are tunable by means of an applied potential difference (23).

Our approach harnesses sophisticated classical density functional theory (DFT) methods and phenomenological binding potential calculations, supported by state-of-the-art Monte Carlo (MC) simulation. We focus on the phase behavior in the plane of wf attractive strength, measured by the dimensionless parameter ϵw, and temperature. Depending on whether or not the ff and wf potentials are SR or LR, we find 4 distinct classes of phase diagram which differ greatly in character and even in topology. These are displayed in Fig. 2. Fig. 2C, SR ff and SR wf, corresponds closely to the class identified as pertinent to fluids by Nakanishi and Fisher (6) and studied in detail in simulations of Ising models (3⇓–5) subject to a SR surface magnetic field. Such studies determine lines of critical drying and critical wetting merging at Tc and vanishing surface field. However, Fig. 2C differs greatly from the previously unrecognized phase behavior shown for the other 3 classes in Fig. 2. In Fig. 2A relevant for simulation and in Fig. 2B, the one most relevant to experiment, we find critical drying and first-order wetting lines that do not merge at Tc; there is a “gap.” Fig. 2D has no true wetting transition. In the sections below we explain the genesis of these surface phase diagrams. Our findings challenge some of the conventional “wisdom” regarding wetting and drying and should have broad relevance to future theoretical, experimental, and simulational studies of superhydrophobic and hydrophilic surfaces.

Fig. 2.
Fig. 2.

The 4 different classes of surface phase transitions obtained from DFT plotted in the plane of wall–fluid attraction strength ϵw versus reduced temperature T. The wetting transition cos(θ)=1 refers to vapor as the bulk phase at coexistence (co), i.e., the chemical potential μ→μco−, while the drying transition cos(θ)=−1 refers to the bulk liquid at coexistence μ=μco+. Taking a horizontal path, at fixed T, one passes from a dry state at small wf attraction through a phase transition to partially dry and partially wet regimes to a phase transition, at larger attractive strength, to the wet state. Tc denotes the bulk critical temperature. (A) SR fluid–fluid and LR wall–fluid. (B) LR fluid–fluid and LR wall–fluid. (C) SR fluid–fluid and SR wall–fluid. (D) LR fluid–fluid and SR wall–fluid. The symbols are the results of DFT calculations. In cases C and D the wf potential 2 is truncated at zc=2.0σ. In A and B we note that complete drying is a critical surface phase transition and wetting is a first-order surface phase transition; there is a gap at Tc between critical drying and first-order wetting. In C we find lines of critical wetting and critical drying transitions merging at Tc, in the same fashion as in the Ising model. In D drying is first order and wetting occurs formally only for an infinitely attractive wall–fluid potential. These results confirm the topologies of phase diagrams predicted by the binding potential analysis.

Choice of Model Potentials

The simplest model system that incorporates all of the key physical ingredients that we wish to investigate is a Lennard-Jones (LJ) 12-6 fluid with particle diameter σ adsorbed at a substrate/wall described by a planar 9-3 wf potential. As is well known, the latter is generated by integrating LJ wall particle–fluid particle pair interactions, with diameter σw over a uniform half-space so that the resulting wf potential depends only on z, the coordinate normal to the wall. Specifically, the ff potential used in our present DFT and MC simulation studies isϕff(r)=4ϵσr12−σr6, r≤rc,0, r>rc,[1]and we consider different values of the cutoff rc. Setting rc=∞ defines our LR ff potential. The SR case usually corresponds to truncating (and leaving unshifted) ϕff(r) at rc=2.5σ. We note that changing rc affects the bulk phase diagram, altering the liquid–vapor coexistence and the location of the critical point.

The planar LR wf potential isWLR(z)=∞, z≤0ϵwϵ215σz9−σz3, z>0,[2]where for simplicity we have taken σw=σ. ϵw is the dimensionless parameter measuring the strength of wf attraction. We also consider SR wf potentials where Eq. 2 is truncated to zero at some finite cutoff zc (but not shifted).

Binding Potential Analysis for Different Choices of Interactions

The standard phenomenological treatment of wetting and drying transitions, e.g., ref. 24, considers contributions to ωex(l), the excess grand potential per unit surface area, as a function of l, the thickness of the wetting/drying layer. l serves as an order parameter. Deriving ωex(l), often termed the effective interface potential, from a microscopic description of the fluid usually begins with a simple DFT treatment for a grand potential functional accompanied by a (sharp-kink) parameterization of the one-body density profile ρ. This is modeled as a layer of constant density (the coexisting liquid density for wetting) adsorbed at the substrate with the liquid–gas interface treated as a Heaviside step function located at a distance l from the surface. Inputting more realistic smoothed density profiles contributes additional terms to ωex(l) (25, 26). Sometimes empirical contributions are invoked. Often drying is considered equivalent to wetting and focus is placed on the latter. We emphasize the differences.

For the case of drying we set the chemical potential μ=μco+, so that the bulk liquid far from the substrate/wall is infinitesimally close to coexistence, and introduce the binding potential ωB(l) which measures the free energy associated with a layer of the metastable phase (in this case vapor v):ωex(l)=γwv+γlv+ωB(l) :drying.[3]If the equilibrium thickness leq→∞, ωB(leq)→0 and then a macroscopically thick layer of vapor intrudes between the weakly attractive wall and the liquid, the wl interface becomes a composite of the wv and lv interfaces, and the wl surface tension is γwl=ωex(∞)=γwv+γlv. It follows from Young’s equation that in this limit cos(θ)=−1. Wetting is equivalent with Eq. 3 replaced byωex(l)=γwl+γlv+ωB(l) :wetting.[4]Now the bulk is vapor and μ=μco−. For a sufficiently attractive wall one expects the equilibrium thickness leq of the adsorbed liquid layer to diverge and ωB(leq)→0 so that the wv interface is a composite of the wl and lv interfaces with γwv=ωex(∞)=γwl+γlv. In this limit cos(θ)=+1. The nature of the transitions to complete drying and complete wetting depends sensitively on the shape of the binding potential. This depends in turn on the form of the ff and wf potentials. We consider 4 different combinations of SR (finite range or exponentially decaying) and LR (retaining the full power-law tail) potentials.

A. SR ff and LR wf.

This choice is pertinent to the majority of simulation studies of simple atomic liquids. It corresponds to model fluids with truncated LJ ff potentials as in Eq. 1 adsorbed at a wall exhibiting −z−3 wf attraction as in Eq. 2. This class includes models of ionic liquids and electrolytes where Coulomb interactions are screened so that effective ff interactions decay exponentially. It should also include models of water that truncate oxygen–oxygen dispersion interactions and tackle Coulomb interactions using Ewald methods. The binding potential isωB(l)=a⁡exp(−l/ξb)+bl−2+H.O.T.,[5]where the higher-order terms include higher inverse powers of l and more rapidly decaying exponentials (12). The exponential contributions arise from SR ff interactions and, in the case of drying, ξb is the true correlation length of the bulk vapor v, the phase that is intruding at or wetting the wl interface. The coefficient a>0 depends on the strength of the ff attraction. The second term in Eq. 5 reflects the leading −ϵwz−3 wf attraction arising from dispersion interactions. This term and the higher-order power-law contributions are proportional to ϵw. For the wf potential 2 a calculation for drying using standard methods, e.g., ref. 24, yields b=−b0ϵϵw with b0≡(ρl−ρv)σ3/2>0 (12), where ρl and ρv are the coexisting liquid and vapor number densities at temperature T. Minimizing ωex(l) w.r.t. l yields the following equation for the equilibrium thickness leq of the vapor layer:−leq/ξb=lnϵw−3⁡ln(leq/ξb)+consts,ϵw→0.[6]leq is finite for all T<Tc, provided ϵw>0. However, in the limit ϵw→0, where the wf potential reduces to that of a hard wall, the equilibrium thickness diverges continuously, and one has critical drying. In ref. 12 we determined the critical exponents characterizing the singular behavior of surface thermodynamic quantities and the divergence of the correlation length ξ∥ measuring the extent of density fluctuations parallel to the wall. Here we recall how the contact angle θ approaches 180○:1+cos(θ)∼ϵw(−lnϵw)−2,ϵw→0.[7]This prediction remains valid, within the binding potential picture, provided b0>0; i.e., critical drying should occur for T right up to the bulk critical temperature Tc.

Turning to wetting, Eq. 4 applies with Eq. 5 for the binding potential but b is replaced by −b so the leading decay of the binding potential is positive. If the wf potential is sufficiently attractive (large ϵw) the denser (liquid) phase must eventually wet the wv interface. However, it is clear from the sign of b that there must be a maximum of ωex(l) at some intermediate l and that any wetting transition must be first order; there can be no continuous (critical) wetting transition at any temperature.

B. LR ff and LR wf.

This scenario pertains to real systems where LR dispersion interactions are present between ff and wf particles; i.e., we retain the full −r−6 tail in the ff pair potential. The binding potential in Eq. 5 is replaced byωB(l)=b(T)l−2+cl−3+H.O.T.,[8]where, for drying, the coefficients obtained from a sharp-kink input density profile, a Hamaker-like approximation, are b(T)=b0ϵ(2πρv(T)σ3/3−ϵw) and c=2b0σϵϵw>0 (27). Beyond the sharp-kink approximation, c and the coefficients of the higher-order power-law terms are likely to depend on the detailed form assumed for the density profile but b(T), the coefficient of the leading term, is expected to be unchanged (25, 26). The important ingredient is the presence in b(T) of the ff contribution, proportional to the vapor density ρv(T). At low T, ρv(T) is very small and b(T)<0. Minimization of the excess grand potential yields a layer thickness leq=−3c/(2b(T)) that is finite. Increasing T, one reaches the situation where b(T)→0− and then leq diverges continuously, corresponding to critical drying. The drying temperature TD for a fixed ϵw is given by the simple formulaϵw=2πρv(TD)σ3/3.[9]Alternatively, one can fix T and decrease ϵw to induce the transition. Such a scenario was presented in a previous DFT study of drying at a single low temperature (27) but its repercussions were not appreciated fully. Here we emphasize that Eq. 9 implies critical drying will persist up to bulk Tc. This equation defines a line of critical drying transitions in the (ϵw,T) plane terminating at the point (ϵwbc,Tc), where ϵwbc=2πρcσ3/3, and ρc is the bulk critical density. The critical exponents pertaining to any drying point are easily calculated. For the contact angle we find1+cos(θ)=−ωB(leq)γlv=−4b(T)3/(27γlvc2)∼|t|3,t→0−,[10]where t≡(T−TD)/TD. The case of wetting by liquid is very different. The binding potential takes the same form (Eq. 8), but now b(T)=−b0ϵ(2πρlσ3/3−ϵw) and c=−2b0σϵϵw<0. As in case A, first-order wetting can occur provided the wf attraction is sufficiently strong but critical wetting cannot occur since this requires c>0.

C. SR ff and SR wf.

This case, like case A above, is encountered in simulations of fluids and is the one that corresponds to the (many) Ising/lattice gas studies in which a field h1 is applied in the first (surface) layer only. Such models have been treated within mean-field (MF) Landau theory (6) and in great detail by MC simulations (3, 4). The binding potential for such models consists of 2 exponential terms,ωB(l)=a1⁡exp(−l/ξb)+a2⁡exp(−2l/ξb)+H.O.T.,[11]with a2>0. ξb refers to the correlation length of the bulk phase that intrudes/wets. Critical drying can occur if the sign of a1 changes on varying T (or ϵw). First-order drying can also occur. For strongly attractive wf potentials it is well known that both critical and first-order wetting transitions can occur with a tricritical point separating the two (6). Critical wetting for SR interactions has attracted much attention because of predictions of novel, nonuniversal critical exponents (SI Appendix).

D. LR ff and SR wf.

Although less relevant to physical situations, this case is important in understanding the overall genesis of drying and wetting phase diagrams. For drying the binding potential takes the formωB(l)=awf⁡exp(−l/ξb)+bffl−2+H.O.T.,[12]similar to Eq. 5 with awf∝ϵw. However, the physical consequences are quite different. The coefficient of the leading power-law term, now associated with ff interactions, is bff=b0ϵ2πρvσ3/3. Since this is positive, it follows that there can be no critical drying but first-order drying can occur provided the wf attraction is sufficiently weak. Wetting is a very different scenario. The coefficient of the leading term is now −b0ϵ2πρlσ3/3 and the long-ranged negative tail of the ff contribution to the binding potential always limits the thickness of the wetting layer: Complete wetting cannot occur for finite wall–fluid attraction. Of course, in the limit ϵw→∞, we expect leq to diverge but this is slow: leq∼ξb⁡lnϵw, and a straightforward calculation shows1−cos(θ)∼(lnϵw)−2, ϵw→∞;[13]i.e., the approach to complete wetting is very slow.

Two remarks are in order: 1) It is important to note that the analysis presented in all 4 cases is strictly MF; we simply minimize the binding potential. If we treat the binding potential as an effective Hamiltonian, we must consider fluctuations of the order parameter l. The effects of fluctuations are described in SI Appendix. For case A fluctuations have little effect. For case B fluctuations play no role, and we expect the location of the transition and the critical exponents to be predicted correctly within MF. In case D there is no criticality. Case C, the Ising-like case, is where fluctuations play a significant role. 2) The binding potential analysis can be viewed as a low-temperature approximation. Very close to the bulk critical temperature Tc the bulk correlation length ξb is long and one should take into account the broadening of the liquid–vapor interface as set by the (diverging) ξb. This leads to the regime of critical adsorption. In contrast, our microscopic DFT calculations, described below, incorporate fully the interplay between wetting/drying and critical adsorption, albeit within MF. We emphasize these considerations come into play only when ξb approaches the thickness leq of a drying or wetting layer. And we speculate that the criterion 9 for critical drying in case B remains valid up to Tc.

Results from DFT

DFT is a microscopic theory based on constructing a grand potential functional of the average one-body particle (fluid) density ρ≡ρ(r). The particular approximate functional we employ is described, and justified, in SI Appendix. Minimizing the grand potential functional with respect to ρ with suitable boundary conditions permits the direct determination of the 3 interfacial tensions γlv,γwv,γwl at bulk vapor–liquid coexistence and hence, via Young’s equation, cos(θ). We have calculated cos(θ) as a function of ϵw for a selection of temperatures in the range 0.75Tc≲T≤Tc and for the 4 combinations of LR and SR interactions considered in cases A to D above. The results complement, confirm, and extend the insight gained from the binding potential analysis.

Results are shown in Fig. 3A for case A: SR ff and LR wf, the situation encountered in most simulations of fluids. One observes that the curves of cos(θ)+1 versus ϵw approach drying (cos(θ)=−1) tangentially, indicating critical drying. A previous careful analysis of the location of the drying point (12) at the low temperature T=0.775Tc shows this to occur at ϵw=0, i.e., in the limit of a hard wall. We find within DFT, and as predicted by the binding potential analysis, that this is true for all temperatures T≤Tc. Fig. 3B shows a plot of the numerical binding potential ωB(l) obtained from our DFT calculations via the procedure of ref. 28 for T=0.999Tc and ϵw=10−3. Even at this very small value of ϵw and the near critical temperature, there is a clear minimum in ωB(l) at a large but finite leq, demonstrating that the wall is not yet completely dry.

Fig. 3.
Fig. 3.

(A) DFT results for cos(θ) for case A. SR ff and LR wf demonstrating critical drying and first-order wetting occur for all reduced temperatures kBT/ϵ. The reduced critical temperature is Tc=1.31944. Note that all of the curves cross at a point, as discussed in the text. Also note that we display values of cos(θ) that slightly exceed unity. These correspond to metastable states reflecting the first-order character of the wetting transition. (B) DFT results for the binding potential ωB(l) for SR ff and LR wf at T=0.999Tc. The extremely shallow minimum at a large but finite leq demonstrates that the wall is still not completely dry at ϵw=10−3.

In contrast, the curves of cos(θ)+1 versus ϵw cross the wetting point (cos(θ)=1) with a nonzero gradient indicating first-order wetting and we return to the resulting “hockey-stick” shape below. The full phase diagram of drying and wetting transitions is displayed in Fig. 2A and shows that the value of ϵw at which wetting occurs decreases with increasing T. However, it remains nonzero as T→Tc; i.e., the line of first-order wetting transitions does not merge with the drying line as T→Tc. This leaves a substantial gap between drying and wetting points at Tc which has not been identified previously and which should be related to the physics of critical adsorption of fluids, e.g., refs. 29 and 30. That a distinction between drying and wetting survives even at Tc is certainly counterintuitive given that the bulk phases become identical there. The effect is related to the fact that cos(θ(T))=(γwv(T)−γwl(T))/γlv(T) and that both the numerator and the denominator vanish in a singular fashion as T→Tc.

DFT calculations were also performed for cases B to D and the corresponding phase diagrams are shown in Fig. 2 B–D. Case B pertains to most experimental situations for which both ff and wf interactions are LR; it exhibits features in common with case A except that critical drying occurs at nonzero ϵw(T)=2πρv(T)σ3/3 as predicted by the binding potential calculations and verified numerically via DFT (SI Appendix). Thus ϵw for critical drying (supersolvophobicity) increases with T in this case. The gap between drying and wetting at T=Tc seen in case A occurs here too.

For case C, in which both ff and wf interactions are SR, the phase diagram as calculated by DFT exhibits critical drying, while wetting can be either first order or critical depending on the temperature. As shown in Fig. 2C there is a tricritical point near T=1.27. Above this temperature wetting is critical. A major distinction to cases A and B is that there is no gap between drying and wetting at Tc. This type of phase diagram, where wetting and drying are critical on approaching Tc, was known previously from Ising model studies and from an insightful early DFT treatment by Sullivan (31) and was considered universal. Importantly, this differs dramatically from cases A and B which pertain, respectively, to most simulations and experiments on real fluids. The Sullivan model (31) treats an attractive Yukawa fluid subject to an exponentially decaying wall potential, of strength ϵwS: the ff and wf potentials are SR with identical decay length. It yields lines of critical wetting and critical drying transitions merging at Tc; there is no first-order transition. Remarkably the Sullivan criterion for critical drying ϵwS(T)=ρv(T)α/2, where α is the integrated strength of the ff attraction, is identical in form to our result 9 for case B.

For the final case D with LR ff and SR wf interactions, we again observe very different behavior. This is the only case in which the drying transition is first order and it occurs at nonzero ϵw. Wetting is essentially absent, occurring formally only for infinite attractive wall strength, although thick (but finite) liquid layers are expected to occur for strongly attractive wf potentials as described in the previous section.

Several observations are germane to these findings. Critical drying is found in cases A, B, and C. At first sight, case A, i.e., SR ff and LR wf, might be considered equivalent to the lattice gas model treated in ref. 14. Indeed, the argument that any wetting transition must be first order is also confirmed by the results presented in ref. 14. Moreover, the shape of the calculated wetting line is close to what we display in Fig. 2A. However, in the lattice treatment of ref. 14 there is no line of critical drying transitions, in sharp contrast to our present treatment pertinent to a “real” fluid where the imposition of the hard-wall limit as ϵw→0 guarantees the occurrence of critical drying, with its accompanying signatures.

The hockey-stick shape of the cos(θ) vs. ϵw plots for various T shown in Fig. 3A is important. The curves exhibit a well-defined “crossing point” for ϵw≈0.75 where θ≃130○. For ϵw>0.75, cos(θ) increases with T, but decreases with T for smaller ϵw. While these results pertain to case A, we find similar-shaped plots for other cases. In case B (LR ff and LR wf) the crossing point (not shown) is close to ϵw=1.23 where θ≃110○. Earlier studies, notably for realistic models of water (32), also found similar plots, which implies there is nothing special about the overall surface phase behavior of water. These observations are pertinent for the design of wetting engines (33) which rely upon knowledge of the T dependence of cos(θ). One interesting thermodynamic cycle requires the propensity for wetting to decrease with increasing T. From our results for case B, this might occur for contact angles ≳110○.

The increasingly vertical shape of the hockey-stick curves as T increases in Fig. 3A points to the onset of a near jump from partial drying to first-order wetting, as ϵw increases, in the limit where T approaches Tc. This is elucidated in SI Appendix, Fig. S4, where we plot the so-called “neutral” line for which cos(θ)=0 (34) that separates the regimes of partial drying and partial wetting, alongside the line for cos(θ)=1, complete wetting. Note that both lines meet at the same value, ϵw=0.75, at Tc, implying the disappearance of the partial wetting regime in this limit.

Results from Simulation

Turning now to our MC simulations, these focus on the properties of the probability distribution of the fluctuating density P(ρ) at μ=μco(T) within a slit geometry (2 identical planar walls) for the LJ system as described in ref. 12. Detailed studies were performed for the computationally tractable cases A and C in which the ff interactions are truncated. The results serve to corroborate the picture emerging from theory. Specifically, drying is found to be critical in all cases and to occur at ϵw=0 for LR wf interactions and at ϵw>0 for SR wf interactions. Wetting is first order in case A and either first order or critical in case C, depending on the temperature.

The simulations also confirm for case A the presence of a gap separating wetting and drying at bulk criticality, i.e., that the wetting line intersects the critical isotherm at nonzero ϵw. For temperatures close to Tc, a large bulk correlation length ξb pertains and this engenders finite-size effects which complicate accurate estimation of the quantity γlv that is required to estimate cos(θ) from Young’s equation. However, the attractive wall strength ϵw that corresponds to a neutral wall, cos(θ)=0, does not require knowledge of γlv and is simply determined (12) by the equality of the peak heights in P(ρ). Thus the neutral wall strength can be obtained accurately right up to Tc and because this value provides a lower bound on ϵw for complete wetting, we have been able to confirm by simulation that the first-order wetting line indeed has the form shown in the DFT results of Fig. 2; i.e., it meets the line T=Tc at a nonzero value of ϵw and does not bend to meet the drying line at ϵw=0. Fig. 4 shows the form of P(ρ) at the neutral wall strength.

Fig. 4.
Fig. 4.

The form of P(ρ) corresponding to a neutral wall having cos(θ)=0, measured in a GCE MC simulation of the LJ fluid in a slit of dimensions V=L2D. The temperature is T=0.985Tc and the system size is L=15σ,D=40σ (SI Appendix). The LR wf potential has the form 2 with ϵw=0.957. The SR ff interactions are truncated at rc=2.5σ. Note the equal peak heights: The lower-density peak corresponds to vapor and the higher one to liquid.

Discussion and Outlook

We have identified the types of wetting and drying behavior that can occur across the full temperature range of bulk liquid–vapor coexistence for a realistic fluid model. The presence of LR interactions leads to 3 previously unrecognized classes of surface phase diagram which differ dramatically from the SR ff and SR wf class characterizing the Ising model with finite-ranged surface fields, hitherto assumed to be universal. In the latter, the lines of (critical) drying and wetting transitions merge as T approaches Tc and in this limit drying and wetting are equivalent. However, the presence of LR interactions leads to wetting and drying lines that can have different character and which do not merge at Tc. Rather there is a gap between wetting and drying at Tc where critical adsorption occurs. Most experiments and simulations of liquids at an interface, including studies of water at hydrophilic or superhydrophobic substrates, belong to one of these previously unrecognized surface phase diagrams. Accordingly our results are widely applicable and should provide a firmer foundation for future developments.

Our study also relates directly to the confinement of fluids by hydrophobic entities. Many studies, e.g., refs. 35⇓⇓–38, emphasize the usefulness of macroscopic (capillarity) approximations, i.e., generalizations of the well-known Kelvin equation, in understanding phenomena such as capillary evaporation, the formation of vapor bridges, and solvent-mediated forces arising under nanoscale confinement. Key to such approaches is the product γlv⁡cos(θ). Given our results it would be instructive to investigate the temperature dependence of this quantity for a variety of substrates and adsorbates. Note that the characteristic length scale for evaporation in water at room temperature and pressure is much longer than in most common organic liquids but this is simply due to the large surface tension γlv rather than any special feature of water (37, 39).

The adsorption of colloid–polymer mixtures can provide examples of wetting and drying for micrometer-sized particles. A simple glass wall favors wetting by the “liquid” phase rich in colloid because of the depletion mechanism (22). However, one might tailor substrates so that the interface between the substrate and the liquid phase is wet by the gas phase, dilute in colloid; this corresponds to drying (40). We note that in colloid–polymer mixtures the solvent is refractive-index matched to the colloidal particles so that the relevant interactions are short range, mimicking case C.

Returning to the possibility of observing the surface criticality associated with complete drying, we emphasize this requires a very weakly attractive substrate: small ϵw. In experiments one works with a given substrate material and liquid adsorbate and hence a fixed ϵw which is always nonzero; an attractive wf interaction is always present. The fact that critical drying occurs for increasing ϵw as T increases, in the experimentally relevant case B, leads to the possibility that one might attain values of cos(θ) close to −1 if one explores higher temperatures. By increasing T one can approach the drying line from the partially dry side, potentially allowing observation of the strong density fluctuations associated with the approach to critical drying (10⇓–12). For experiments on water at superhydrophobic surfaces this would entail use of a pressure vessel to maintain coexistence conditions at high T. In practice, it may be easier to access this regime by considering fluids for which the critical temperature is close to room temperature, such as CO2, NH3, or Rn.

It is well known that some of the most weakly adsorbing systems at the atomic scale are the inert (noble) gases on alkali metal substrates. Ne at a Cs substrate is considered a particularly weakly adsorbing combination; see Chizmeshya et al. (41). A classical DFT investigation (42) using a functional equivalent to SI Appendix, Eq. S2 and the wf potential of ref. 41 found no drying transition for Cs-Ne. Ref. 42 also considered wf potentials which had the 9-3 form of Eq. 2 and made these “ultraweak” by changing the coefficient of the repulsive z−9 term, thereby reducing the well depth. Even for very small wf well depths and temperatures close to Tc they found no drying transition. At first sight this contradicts our analysis of case B. However, ref. 42 did not vary the coefficient C3 of the −z−3 attraction; they kept this fixed to the Chizmeshya value. Had they reduced C3, which is proportional to our ϵw, sufficiently they should have observed critical drying. From our analysis it is clear ϵw, the strength of the dispersion-force wf attraction, rather than the well depth, determines the leading-order contribution to the binding potential and therefore the resulting surface phase behavior. This is relevant in the context of an important microbalance measurement of the adsorption of liquid Ne on Cs (43) that provided firm evidence for a significant density depletion, interpreted as a vapor-like layer close to the Cs substrate. What does our current theory say for this system? For Ne, Tc=44.4 K and the reduced critical density is ρcσ3=0.305. Assuming Eq. 9 holds close to the bulk critical point then critical drying requires ϵw<0.63. It is likely that the value of C3 used by ref. 42 corresponds to ϵw>0.63, explaining why no drying transition was found in their DFT calculations. Clearly it would be worthwhile to reexamine the coefficient C3 for Cs-Ne. Should this turn out to be smaller, critical drying could occur near Tc which might account for the experimental observation (43).

This example illustrates the importance of understanding the underlying phenomenology of surface phase transitions. To emphasize further, the MC studies mentioned in ref. 42 employed a SR ff (truncated LJ) and a LR wf (10-4) potential which corresponds to case A. It is not surprising that no drying transition was observed—this occurs only for ϵw=0. Wetting and drying transitions are characterized by the divergence of the layer thickness l, i.e., by the asymptotic decay of terms in the binding potential, reflecting the range of competing interactions. This raises major challenges for simulations where accessing the experimentally relevant regimes is demanding, requiring special techniques to simulate inhomogeneous fluids with LR (power-law) ff interactions and locate the onset and character of the transitions.

Although we have focused on planar substrates, the simulation and DFT techniques that we employ can be applied to substrates structured at the nanoscale. These allow us to address how critical drying depends on ϵw and the form of critical interfaces and density fluctuations for model structures pertinent to (super)hydrophobic substrates.

Footnotes

  • ↵1To whom correspondence may be addressed. Email: Bob.Evans{at}bristol.ac.uk or nigel.wilding{at}bristol.ac.uk.

  • Author contributions: R.E. and N.B.W. designed research; R.E., M.C.S., and N.B.W. performed research; and R.E. and N.B.W. wrote the paper.

  • The authors declare no competing interest.

  • This article is a PNAS Direct Submission.

  • See Commentary on page 23874.

  • This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1913587116/-/DCSupplemental.

Published under the PNAS license.

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A unified description of hydrophilic and superhydrophobic surfaces in terms of the wetting and drying transitions of liquids
Robert Evans, Maria C. Stewart, Nigel B. Wilding
Proceedings of the National Academy of Sciences Nov 2019, 116 (48) 23901-23908; DOI: 10.1073/pnas.1913587116
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