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Alberto
Giacomello
*^{ab},
Lothar
Schimmele
^{b},
Siegfried
Dietrich
^{bc} and
Mykola
Tasinkevych
*^{bc}
^{a}Sapienza Università di Roma, Dipartimento di Ingegneria Meccanica e Aerospaziale, 00184 Rome, Italy. E-mail: alberto.giacomello@uniroma1.it; Tel: +39 06 44585200
^{b}Max-Planck-Institut für Intelligente Systeme, 70569 Stuttgart, Germany. E-mail: miko@mf.mpg.de; Tel: +49 711 689-1949
^{c}IV. Institut für Theoretische Physik, Universität Stuttgart, 70569 Stuttgart, Germany

Received
27th July 2016
, Accepted 5th October 2016

First published on 7th October 2016

A liquid droplet placed on a geometrically textured surface may take on a “suspended” state, in which the liquid wets only the top of the surface structure, while the remaining geometrical features are occupied by vapor. This superhydrophobic Cassie–Baxter state is characterized by its composite interface which is intrinsically fragile and, if subjected to certain external perturbations, may collapse into the fully wet, so-called Wenzel state. Restoring the superhydrophobic Cassie–Baxter state requires a supply of free energy to the system in order to again nucleate the vapor. Here, we use microscopic classical density functional theory in order to study the Cassie–Baxter to Wenzel and the reverse transition in widely spaced, parallel arrays of rectangular nanogrooves patterned on a hydrophobic flat surface. We demonstrate that if the width of the grooves falls below a threshold value of ca. 7 nm, which depends on the surface chemistry, the Wenzel state becomes thermodynamically unstable even at very large positive pressures, thus realizing a “perpetual” superhydrophobic Cassie–Baxter state by passive means. Building upon this finding, we demonstrate that hierarchical structures can achieve perpetual superhydrophobicity even for micron-sized geometrical textures.

On rough hydrophobic surfaces the Cassie–Baxter state can be either stable or metastable depending on the thermodynamic conditions (e.g., pressure and temperature).^{6} The transition from the (meta)stable Cassie–Baxter state to the Wenzel state requires to overcome free energy barriers much larger than the thermal energy k_{B}T, even for surface asperities of a few nanometers in size.^{7,8} These metastabilities generate strong hysteresis in the wetting and dewetting processes on such surfaces.^{9} On the other hand, the large free energy barriers also imply that once the system has occupied the Wenzel state, superhydrophobicity cannot be restored without supplying free energy to the system.

In order to restore superhydrophobicity from the Wenzel state the most effective strategy to date seems to be the application of an electric field to the system.^{10–12} Other methods include a magnetically driven Wenzel to Cassie–Baxter transition^{13} and heating of the surface until boiling restores the vapor pockets.^{14} All these active methods require special preparations of the surface and/or of the liquid as well as a free energy source; in addition, they cannot be easily applied to large surfaces.

Passive methods for restoring superhydrophobicity are in principle more economical and general than active ones. One strategy for trying to passively “stabilize” the Cassie–Baxter state is to shrink the size of surface decorations down to the nano-scale.^{15} For instance, by using carefully designed hydrophobic nano-textures of ca. 10 nm size it was possible to support pressures up to several tens of atmospheres before mechanical destabilization of the Cassie–Baxter state takes place at a certain critical pressure P_{c}.^{9} Verho et al. realized hierarchical structures via a regular texture (on the scale of ca. 10 μm) which is decorated by superhydrophobic silicon filaments (on the scale of 100 nm, grown on the micro-topography).^{16} With this two-level topography the transitions between nano- and micron-sized Cassie–Baxter states offer the opportunity that they can be reversibly switched by using either local suction (recovering the micron-sized Cassie–Baxter state from the nano-sized one) or a jet of water at low pressures (creating the nano-sized Cassie–Baxter state from the micron-sized one). Such passive approaches provide several advantages, but they do not eliminate the potential occurrence of the transition to the Wenzel state, which can always be realized by thermally activated events or pressure changes.

The aim of the present study is to introduce a completely passive method, which involves the thermodynamic destabilization of the Wenzel state over a wide range of pressures above the bulk liquid–vapor coexistence pressure. In this way, a virtually perpetual superhydrophobic Cassie–Baxter state is realized which is the only one allowed thermodynamically within that pressure range. As shown below, this can be achieved by properly choosing the surface chemistry and by scaling the size of the surface roughness, here modeled as an array of rectangular grooves, down to the nanometer range. The required size of the roughness actually depends on the contact angle of the surface; in principle the approach could work even with groove widths w of the order of 7 nm. For such surfaces, the Wenzel state can be eliminated for pressures as high as 5 atm.

The idea of eliminating the Wenzel state is based on the well-known fact that liquids have a spinodal, i.e., thermodynamic conditions for which the liquid state is unstable and disappears via cavitation. For pure liquids far from the critical point, the spinodal is observed at strongly negative pressures P (i.e., tensile stresses applied to liquid). For example, for bulk water at ambient temperature this is estimated to be around −150 MPa.^{17,18} However, by strongly confining the liquid it is possible to shift this spinodal to positive pressures P. This is known, e.g., for macroscopically extended hard^{19} and lyophobic slit pores^{20} (see also ref. 21–23). In ref. 20 the authors estimate that for confined water the liquid spinodal reaches, upon varying the pore width w, the bulk liquid–vapor coexistence line P_{0}(T) at w ≈ 5 nm. Recent studies have proposed a special texture composed of a 2D square lattice of rectangular nano-pillars with added nano-particles at the centers of the lattice cells.^{24} Since in the fully wet Wenzel state the liquid penetrates into the surface geometric features, the following question arises: is it possible to destabilize thermodynamically the liquid confined within the surface features at positive values of the pressure by decreasing the spatial extent of the features? This would provide a means to achieve a perpetual superhydrophobic Cassie–Baxter state. The present systematic study provides a positive answer to the above question.

DFT is based on the minimization of the grand potential functional

(1) |

The geometry of the system is illustrated schematically in Fig. 1(a). Before introducing surface structures, the substrate occupies the lower half-space and is associated with Young's contact angle θ_{Y}. For each particular system, θ_{Y} is computed via independent calculations of the solid–vapor (γ_{sv}), solid–liquid (γ_{sl}), and liquid–vapor (γ_{lv}) interfacial tensions following Young's law cosθ_{Y} ≡ (γ_{sv} − γ_{sl})/γ_{lv}. In a second step, a rectangular groove of height h and width w is excavated from the solid material. Calculations of the wetting and dewetting processes in such grooves are performed using a DFT code^{28} applying periodic boundary conditions in the x and y directions. In these calculations the same mesh width of the computational grid has been used as in those carried out in order to determine θ_{Y}. The temperature is kept fixed at T = 0.71T_{c}, where T_{c} is the critical temperature of the bulk fluid.

Fig. 1(b) shows that at the critical pressure P_{c} = ΔP_{c} + P_{0} the intrusion branch jumps from the suspended Cassie–Baxter state (small Φ) to the completely filled Wenzel one (large Φ): P_{c} is the maximum pressure at which superhydrophobicity can survive, before it becomes mechanically unstable. It is well known that narrow grooves exhibit larger values of P_{c} (see the discussion below). What is more surprising in Fig. 1 is that for w = 6σ the extrusion branch jumps from the Wenzel state to the Cassie–Baxter state: this is the sought liquid spinodal pressure P_{sp} > 0 in confinement (for w = 21σ (red) ΔP_{sp} is negative). For P ≤ P_{sp}, the Wenzel state is unstable and superhydrophobicity becomes perpetual.

In order to understand how P_{sp} depends on the characteristics of the surface, we performed DFT calculations similar to those leading to Fig. 1 for grooves with various widths and contact angles. The results reported in Fig. 2 show that by increasing the lyophobicity of the surface (i.e., upon increasing Young's contact angle θ_{Y}) the Wenzel state can still be eliminated at ΔP_{sp} > 0 even from increasingly wide grooves, up to w = 21σ (see red squares in Fig. 2). In summary, confinement and lyophobicity cooperate towards increasing P_{sp} and can be engineered in order to obtain perpetual superhydrophobicity. By adopting a typical parameter value σ = 0.34 nm corresponding to the Lennard-Jones potential for Argon^{29} and setting T = 300 K, one can estimate that, for θ_{Y} = 134°, in the case w = 6σ the Wenzel state remains eliminated up to ΔP = ΔP_{sp}(θ_{Y}) ≃ 9.3 MPa and in the case w = 21σ up to ΔP ≃ 0.5 MPa.

Fig. 3 Summary of the main findings for lyophobic (θ_{Y} = 121°) confined systems (rectangular grooves and slit pores). (a) Bulk liquid–vapor coexistence (orange) as well as capillary liquid and capillary vapor coexistence for slits at ΔP_{cc} (red crosses, interpolated by the dashed red line); Kelvin–Laplace law (eqn (2), solid black line); corrected Kelvin–Laplace law with l = 0.3σ (eqn (3), dash-dotted black line). Concerning rectangular grooves, the interpolated open blue and solid blue squares represent ΔP_{c} and ΔP_{sp}, respectively; a second solid blue square occurs at ΔP_{sp} < 0 and is not shown. Enlarged views of the liquid layering (b) near the top corners in the Cassie–Baxter state and (c) at the bottom corners in the Wenzel state for rectangular grooves with w = 6σ at the same pressure ΔP ≳ ΔP_{sp}. |

The shift of the spinodal is caused by the strong layering in the liquid induced by the confinement (Fig. 3(c)): the layers form “interference-like” patterns within the groove, which become “destructive” for the liquid at the bottom corners. This triggers a density depletion at the bottom corners, which is further enhanced by the increased lyophobicity of the walls near the corners due to the “missing” fluid–fluid interactions at the corners as compared to a planar wall. For narrow grooves these depletion zones at the bottom corners become connected (see Fig. 3(c)) favoring the growth of a bubble and thus the formation of the Cassie–Baxter state. This mechanism has some resemblance to dewetting via the growth of unstable surface waves (see, e.g., Herminghaus et al.^{31}). Evans and Stewart recently discussed how the local compressibility is enhanced at a lyophobic wall, thereby enhancing the density fluctuations;^{32} the present results suggest that the enhancement is even stronger at corners.

Another important parameter to characterize superhydrophobicity is the critical pressure P_{c} for which the Cassie–Baxter state ceases to be mechanically stable (Fig. 1(b)). At the macroscopic level of description P_{c} is determined by balancing the hydrostatic pressure, as given by the Laplace law, and the z component of the capillary forces acting upon the contact line due to the liquid–vapor surface tension γ_{lv}. In the case of rectangular grooves this force balance renders the Kelvin–Laplace law:

(2) |

It is known that the Kelvin–Laplace law can be improved on the mesoscale by accounting for the formation of wetting films at the walls of the slit pore.^{33,34} This gives rise to the correction

(3) |

As shown in Fig. 3(a), the Kelvin–Laplace law, eqn (2), accurately predicts ΔP_{c} for grooves with w ≥ 15σ. In the same range, eqn (2) also captures ΔP_{cc} (dashed red line). For grooves and slit pores with w ≤ 15σ both ΔP_{c} and ΔP_{cc}, as obtained by DFT, are larger than what is predicted by eqn (2). More specifically, ΔP_{cc} falls in between ΔP_{c} and the Kelvin–Laplace law. By using l = 0.3σ, which is within a plausible range, as a fit parameter in eqn (3), we obtain a good agreement between ΔP_{cc} and ΔP_{c,meso} for w > 4σ (black dash-dotted line). In contrast to the macroscopic predictions, we find that the critical pressure ΔP_{c} for intrusion is different from the capillary condensation pressure ΔP_{cc}. Actually ΔP_{c} is significantly higher than ΔP_{cc} for small groove widths w. Before we discuss the physics behind this observation we also have to exclude potential numerical errors which could influence this result. First, ΔP_{cc} is calculated by equating the free energies of a slit either filled with capillary liquid or with capillary vapor. Although within this treatment there is no explicit force balance at a liquid–vapor interface, the coexistence pressure, defined via the pressure in the reservoir, must be the same as one would obtain from a search for an indifferent liquid–vapor interface position in the slit. Numerics should not spoil this identity if otherwise the same conditions are chosen. A more serious source of error could be in the determination of ΔP_{c} because the iterative computation of the fluid number densities close to the critical intrusion pressure converges very slowly. In order to ensure that metastable configurations have indeed been found and that, in the vicinity of the critical pressure, the appropriate intrusion pressure has not been missed, we carried out computations starting from distinct initial positions of the liquid–vapor interface and we increased the number of calculated pressures (see Fig. 1(b)). The error in ΔP_{c} is estimated to be considerably smaller than the difference between ΔP_{c} and ΔP_{cc}.

With this we turn to the physical mechanism which can lead to the observed difference between ΔP_{c} and ΔP_{cc}. It has to be linked to the fact that a groove, in contrast to an infinite slit, has an open upper end and a capped bottom. Therefore, the wetting properties of a segment of a groove side wall depend on its depth in the groove, i.e., on its distance, say, from the upper corner. In a coarse picture, the wetting properties of a wall are determined by a balance of the loss of fluid–fluid interactions versus the gain in fluid–solid interactions due to the presence of the wall. For the lyophobic walls, as discussed here, one expects that the wall segments close to the upper corner are less lyophobic than an infinitely extended wall because less of the fluid–fluid interaction is replaced by the weaker fluid–solid interaction. Deeper in the groove the side wall segments become more lyophobic and, for a very deep groove, the wetting properties approach those of an infinitely extended slit. The precise characteristics of these effective properties of the wall depend on the details of the system. These generic considerations explain our observation of a partial intrusion. Liquid intrusion starting from the open upper end of the groove may progress to some depth below the open upper end where the side walls are less lyophobic, but it is stopped deeper in the groove as a result of the side walls becoming increasingly lyophobic (Fig. 3(b)). Whether the critical intrusion pressure ΔP_{c} is larger than ΔP_{cc} or not might depend on details of the shapes of the fluid–solid as well as of the fluid–fluid interaction. In our model the attractive part of both interactions is of the van der Waals type with, however, the fluid–fluid interaction cut off at 2.5σ. Therefore, at some distance below the upper corner of the groove the side wall of the groove becomes more lyophobic than an infinitely extended wall of the same material, because the missing fluid–solid interactions are still appreciable whereas the fluid–fluid interactions are cut off. This would lead to an enhancement of ΔP_{c} over ΔP_{cc} with the latter being based on the properties of infinitely extended walls. Of course intrusion into grooves is influenced additionally by specific confinement effects which become very pronounced for very small groove widths. For instance, the liquid–vapor interface at the open upper end of the groove has a structure (see Fig. 3(b)) which deviates significantly not only from that of a free liquid–vapor interface, but also from that between capillary liquid and capillary vapor deep inside the groove, which is a transient unstable configuration.

In order to summarize our remarks, deviations between ΔP_{cc} and the Kelvin–Laplace law arise because for very narrow slits, due to various confinement effects, the force balance at a liquid–vapor interface in an infinitely extended slit cannot be reliably characterized in terms of size independent surface and interfacial tensions. Deviations between ΔP_{c} and ΔP_{cc} can occur because even quite deep into a groove the force balance at a liquid–vapor interface might still deviate from that in an infinitely extended slit; ΔP_{c} is determined by the highest pressure required to push the liquid through the open upper end and further into the groove. In the present case, the two upper corners effectively enhance the lyophobicity of the groove, resulting in ΔP_{c} > ΔP_{cc}. In the general case, whether ΔP_{c} is effectively enhanced over ΔP_{cc} and to which extent will presumably depend on the detailed shapes of the fluid–solid and fluid–fluid interactions.

Here we propose the concept of perpetual superhydrophobicity, i.e., complete thermodynamic elimination of the Wenzel state, also for micron-sized geometrical textures. The basic idea is to utilize hierarchical surface structures, which exploit the perpetual superhydrophobicity of the nano-scale textures described above, in combination with the so-called wedge drying phenomenon^{38–40} at larger scales. Micron-sized wedges immersed in a liquid at bulk liquid–vapor coexistence, i.e., at ΔP = 0, exhibit spontaneous drying when the condition

(4) |

(5) |

Within our model we computed θ_{eff}via DFT for a nano-groove with w = 6σ, θ_{Y} = 121°, and a solid fraction ϕ_{s} = 1/2 yielding θ_{eff} = 141°. This result is in fair agreement with the macroscopic estimate θ_{CB} ≃ 139° obtained by using the well-known macroscopic Cassie–Baxter equation^{42} cosθ_{CB} = ϕ_{s}(cosθ_{Y} + 1) − 1 for describing the contact angle θ_{CB} of sessile droplets in the so-called “fakir regime”.^{43} In deriving this Cassie–Baxter equation, in the first step, effective solid–fluid interfacial tensions are calculated as area weighted averages of the solid–fluid and vapor–fluid interfacial tensions, which characterize the patches forming the actual surface. In the second step, these effective interfacial tensions are used in Young's law leading to the above expression for cosθ_{CB}. In order to support the proposed wedge-drying mechanism for restoring the Cassie–Baxter state on the micro-scale, the nano-sized surface structure must be in the Cassie–Baxter state which is guaranteed if it is perpetual. Otherwise, once the liquid has intruded the nano-texture, a nucleus of vapor forming at the edge of a wedge cannot grow, even if it is favored thermodynamically for θ_{Y} > 135°, as the result of strong pinning of the liquid–vapor interface near the intruded liquid pockets. Therefore, in this case the proposed wedge-drying mechanism becomes ineffective. In contrast to that, in the nano-sized Cassie–Baxter state this pinning is much weaker and we expect this mechanism to be effective.

Here we describe the computational domains used for obtaining the intrusion and extrusion curves presented above. The densities have been discretized on a grid and in most calculations a mesh size of 0.05σ has been used. For comparison some computations have been repeated with a smaller mesh size of 0.025σ. From a semi-infinite wall occupying the lower half space, a parallelepipedic groove is excavated (Fig. 5). The height of the groove is fixed to the value h = 20σ while various widths are considered: w = 6σ, w = 11σ, w = 16σ, w = 21σ, and w = 41σ. Since for technical reasons we have used a 3D DFT, despite the translational invariance of the system in the y direction, our computational box has a finite extent in this direction for which we have chosen periodic boundary conditions. In order to speed up the calculations, actually only one half of the domain is considered, applying symmetric boundary conditions to the left side of the domain (see Fig. 5). To the right side also reflecting boundary conditions are applied which are equivalent to having a mirror symmetry plane along the y–z plane, i.e., we effectively treat a periodic array of grooves aligned along the y axis. The box dimensions in the x direction have been chosen such that the ridge separating the grooves has a width of 20σ. A constant number density is imposed at the top boundary of the computational box, which is equivalent to prescribing the pressure in the system far away from the wall. At the bottom boundary there is the wall.

The external substrate potential V(r) is the sum of a repulsive contribution V_{rep}(r) and an attractive one V_{att}(r): V(r) = V_{rep}(r) + V_{att}(r). We account for V_{rep}(r) in terms of a hard-sphere repulsion, chosen such that the distance of closest approach is σ/2, i.e., the radius of the fluid particles. V_{rep}(r) is set to infinity inside the zone of closest approach. V_{att}(r) is taken as the linear superposition of the attractive part Φ_{w} of a Lennard-Jones potential between the fluid particles and the particles forming the wall with a number density ρ_{w}:

(6) |

The pressure is calculated from the following bulk equation of state (which corresponds to the density functional which is used here):

(7) |

(8) |

The density (or, equivalently, the pressure) is always chosen to lie either on the liquid side or at liquid–vapor coexistence of the bulk phase diagram. The computational domain has a total extent of Δx × Δy × Δz = (w/2 + 10σ) × 5σ × 35.5σ. This domain is discretized with 20 points per σ for a total of ca. 40 millions of points for the domain in Fig. 5.

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## Footnote |

† This generally requires a Young's contact angle θ_{Y} > 90°. However, special reentrant^{2} and doubly reentrant^{4} textures can achieve a “superomniphobic” behavior, i.e., they also repel liquids with θ_{Y} < 90°. |

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