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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
Schematic representation of a liquid drop on a solid substrate (or “wall”). The contact angle can take a range of values
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
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
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
The 4 different classes of surface phase transitions obtained from DFT plotted in the plane of wall–fluid attraction strength
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
The planar LR wf potential is
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
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
Turning to wetting, Eq. 4 applies with Eq. 5 for the binding potential but b is replaced by
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
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
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
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
Results from DFT
DFT is a microscopic theory based on constructing a grand potential functional of the average one-body particle (fluid) density
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
(A) DFT results for
In contrast, the curves 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
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
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
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
The hockey-stick shape of the
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
Results from Simulation
Turning now to our MC simulations, these focus on the properties of the probability distribution of the fluctuating density
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
The form of
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
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
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
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
- ↵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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