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

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

## 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

## 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

Young’s equation

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 *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 **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 *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. 2*C*, 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 *C* differs greatly from the previously unrecognized phase behavior shown for the other 3 classes in Fig. 2. In Fig. 2*A* relevant for simulation and in Fig. 2*B*, the one most relevant to experiment, we find critical drying and first-order wetting lines that do not merge at *D* 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.

## 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 *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 is*ff* potential. The SR case usually corresponds to truncating (and leaving unshifted)

The planar LR *wf* potential is*wf* attraction. We also consider SR *wf* potentials where Eq. **2** is truncated to zero at some finite cutoff

## Binding Potential Analysis for Different Choices of Interactions

The standard phenomenological treatment of wetting and drying transitions, e.g., ref. 24, considers contributions to

For the case of drying we set the chemical potential **3** replaced by*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 *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 is*ff* interactions and, in the case of drying, *wl* interface. The coefficient *ff* attraction. The second term in Eq. **5** reflects the leading *wf* attraction arising from dispersion interactions. This term and the higher-order power-law contributions are proportional to *wf* potential **2** a calculation for drying using standard methods, e.g., ref. 24, yields *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

Turning to wetting, Eq. **4** applies with Eq. **5** for the binding potential but b is replaced by *wf* potential is sufficiently attractive (large

### 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 *ff* pair potential. The binding potential in Eq. **5** is replaced by*ff* contribution, proportional to the vapor density **9** implies critical drying will persist up to bulk **8**), but now *wf* attraction is sufficiently strong but critical wetting cannot occur since this requires

### 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 *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 form**5** with *ff* interactions, is *wf* attraction is sufficiently weak. Wetting is a very different scenario. The coefficient of the leading term is now *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

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 **9** for critical drying in case B remains valid up to

## Results from DFT

DFT is a microscopic theory based on constructing a grand potential functional of the average one-body particle (fluid) density *SI Appendix*. Minimizing the grand potential functional with respect to ρ with suitable boundary conditions permits the direct determination of the 3 interfacial tensions

Results are shown in Fig. 3*A* for case A: SR *ff* and LR *wf*, the situation encountered in most simulations of fluids. One observes that the curves of *B* shows a plot of the numerical binding potential

In contrast, the curves of *A* and shows that the value of

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 *SI Appendix*). Thus

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. 2*C* there is a tricritical point near *ff* and *wf* potentials are SR with identical decay length. It yields lines of critical wetting and critical drying transitions merging at *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 *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. 2*A*. 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

The hockey-stick shape of the *A* is important. The curves exhibit a well-defined “crossing point” for *ff* and LR *wf*) the crossing point (not shown) is close to

The increasingly vertical shape of the hockey-stick curves as T increases in Fig. 3*A* points to the onset of a near jump from partial drying to first-order wetting, as *SI Appendix*, Fig. S4, where we plot the so-called “neutral” line for which

## Results from Simulation

Turning now to our MC simulations, these focus on the properties of the probability distribution of the fluctuating density *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 *wf* interactions and at *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

## 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

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

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 *wf* interaction is always present. The fact that critical drying occurs for increasing

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 *wf* well depths and temperatures close to *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, **9** holds close to the bulk critical point then critical drying requires

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 *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

## Footnotes

- ↵
^{1}To 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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