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Article pubs.acs.org/JPCC

Connection of Intrinsic Wettability and Surface Topography with the Apparent Wetting Behavior and Adhesion Properties Michail E. Kavousanakis, Nikolaos T. Chamakos, and Athanasios G. Papathanasiou* School of Chemical Engineering, National Technical University of Athens, Athens 15780, Greece S Supporting Information *

ABSTRACT: The need for connecting the intrinsic material wettability with surface geometry, adhesion to liquids, and the apparent wettability is of primary importance when aiming to design advanced functional materials. Here, by solving the Young−Laplace equation, augmented with a Derjaguin pressure, we tackle the necessity for implementing the Young angle boundary condition at the contact line, and thus we are able to compute multiple and reconfigurable threephase contact lines in equilibrium. Using the finite element method and special parameter continuation techniques, we highlight the highly nonlinear dependence of the apparent contact angle on the Young angle, which quantifies the material wettability. By computing equilibrium shapes of entire droplets, we find multiple Cassie and Wenzel type states in certain wettability regimes. We, for the first time, find a material wettability regime where Cassie, Wenzel, and partially impregnated states are (meta)stable. The energy barriers for transitions between these states are computed, and their dependence on certain surface geometric features is shown. The “rose petal effect” as well as the “lotus effect” are illuminated through free and adhesion energy computations, and certain geometries are suggested that favor one state or the other.

1. INTRODUCTION When designing surfaces with desirable and even switchable wettability, the surface patterning is of great importance. Proper patterning design can render a slightly hydrophobic material as super water- or oleo-repellant.1−4 However, two important aspects have to be seriously accounted for: (a) the dependence of the apparent wettability of a patterned surface on the material wettability, quantified as the Young’s contact angle, is highly nonlinear, and (b) once the surface is fabricated it cannot be changed (with very few exceptional cases5,6). The nonlinearity is related to multiplicity of (meta)stable equilibrium states with extreme and even with heterogeneous features. Nonsticking Cassie (or Cassie-Baxter, CB) states, where air is trapped underneath a droplet footprint, can coexist (in the sense that the particular geometry can accommodate both states) with sticking Wenzel states where the liquid has impaled the surface patterning. The case is even more complicated when the surface protrusions are partially filled, and the resulting apparent wetting behavior lies between those of the Cassie and the Wenzel. The complete picture of the states’ space can be notably complex with an extended range of the possible apparent contact angle, θa, as well as of the adhesion properties. This picture cannot be derived by simplified Cassie or Wenzel equations, unless one considers limiting or asymptotic cases. To compute the different attainable wetting states on rough solid surfaces, which potentially can be observed in nature, several computational approaches have been proposed. The © 2015 American Chemical Society

simplest and most efficient computational approach for the determination of equilibrium wetting states is the solution of the Young−Laplace (YL) equation,7 with limited applicability for the computation of droplets wetting geometrically heterogeneous surfaces, and in particular for the computation of CB states. Computations of such composite liquid/air/solid states are feasible in a unit cell of the surface pattern;8 however, they disregard the significant pinning effects at the droplet endings. To compute the entire droplet shape, one can apply fine scale computational techniques, such as Molecular Dynamics9−11 and mesoscopic Lattice Boltzmann models.12−15 In this case, the computational power requirements increase dramatically when the ratio of the droplet size over roughness is large (millimeter size droplets). In a recent publication,16 we presented a new parametrization of the YL equation, which is augmented with a Derjaguin pressure (disjoining pressure)17−20 term modeling the solid/liquid interactions. Young’s angle in this framework emerges as the result of the combined action of the Derjaguin pressure (active at the vicinity of solid/liquid interface) and the surface tension (liquid/vapor interface), enabling the computation of entire droplet shapes with an unknown cardinality of three phase contact lines (TPLs). By exploiting the computational merits of this continuum-level approach, we can track Received: January 23, 2015 Revised: May 4, 2015 Published: June 5, 2015 15056

DOI: 10.1021/acs.jpcc.5b00718 J. Phys. Chem. C 2015, 119, 15056−15066

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The Journal of Physical Chemistry C

where γLV is the liquid/vapor (LV) interfacial tension, and Rs is a characteristic length (e.g., radius of a circular disc of area equal to the droplet cross-sectional area); C is the dimensionless local mean curvature, K is a reference pressure, constant along the droplet surface, and pLS is the Derjaguin pressure, which is practically active only at the liquid/solid interface. In our computations, we study droplets with translational invariance along the direction perpendicular to the xy-plane (see Figure 1). The droplet surface is defined in cylindrical

efficiently all wetting states that can be observed on a rough surface,21−23 which correspond to local minima of the surface free energy. Furthermore, we also compute nonobservable (experimentally intractable) unstable wetting states, which correspond to local maxima of the free energy landscape, and are utilized for the determination of wetting and dewetting transition energy barriers; in particular, the (minimum) required energy to induce a transition between two (meta)stable states is determined by subtracting their free energy from the unstable state (saddle), which separates them. The performance of parametric analysis with respect to the material wettability of a structured solid surface reveals interesting findings. In particular, certain material wettability regimes allow the coexistence of suspended CB, collapsed Wenzel states, and mixed states, all featuring the same apparent wettability, when wetting the same number of protrusions. By comparing their relative surface energy, we can characterize a wetting state as stable or metastable; this information can be particularly useful for the understanding of the so-called “rose petal effect”, where droplets wet rough surfaces with high apparent contact angles, accompanied by strong adhesion to the substrate.24−26 Our computations predict that equilibrium CB and Wenzel states with the same high θa can be observed on corrugated solid surfaces; however, their relative stability changes with the shape of the protrusions. This suggests that certain geometries promote CB over Wenzel states, as observed in the lotus effect,27 whereas different shapes favor Wenzel states, even in cases where θa ≫ 90° (rose petal effect). In addition, by efficiently computing all attainable equilibrium wetting states on a structured surface, we can identify the thermodynamically stable state for a particular material wettability and different droplet sizes. The thermodynamically stable θa can be approximated with limited accuracy by the Wenzel and CB equations for small droplet sizes,28 whereas for larger droplet sizes the stable θa practically coincides with the Wenzel and CB models.29−31 It should be stressed at this point that θa cannot be used as a sole criterion for the characterization of a wetting state. For a complete characterization, it is also required to determine the adhesion properties of the wetting state: does it correspond to a suspended CB or a collapsed Wenzel state? Such questions cannot be answered by the CB and Wenzel models. This Article is organized as follows: In the following section, we describe the mathematical formulation of the augmented YL equation for the computation of droplets with multiple TPLs on structured solid surfaces. Following this, we present the solution space of droplets wetting solid surfaces of gradually increasing roughness factor, starting from simple single- and three-striped surfaces to multistriped surfaces. Different protrusion shapes are examined, and we demonstrate that different geometries promote hydrophobic states with different adhesion properties. In the final section, we summarize the main findings of this work and report future research directions.

Figure 1. Cylindrical droplet on a multistriped solid surface. The liquid free surface is defined by means of polar coordinates (r, φ), which are parametrized by the arc-length, s.

coordinates (r, φ), and when parametrized with respect to the angular coordinate, φ, eq 1 becomes one-dimensional. For droplets wetting surfaces of increased topographic complexity, the function r(φ) may no longer remain single valued; in Chamakos et al.16 we applied an alternative arc-length parametrization of the liquid surface and the augmented YL (eq 1), which, while preserving the problem one-dimensional, enables the computation of the entire surface of droplets, including the liquid/solid and liquid/vapor interfaces, in a unified framework (see Supporting Information for a detailed description). Because we are interested in tracking the entire solution space, that is, the dependence of the solutions on various physical parameters, such as the Young’s contact angle, we apply a pseudo arc-length parametric continuation method.33 This allows the computation of both stable and unstable equilibrium droplet solutions. The Young’s CA, directly reflecting the material wettability, is incorporated in our formulation through the Derjaguin pressure term, which models the liquid/solid interactions and is discussed in the next section. 2.1. Derjaguin Pressure Isotherm. The Derjaguin pressure is the sum of a dispersion (van der Waals) component, as well as an electrostatic component due to the overlapping of electrical double layers of neighboring charged surfaces.32 The resulting pressure isotherm has a Lennard-Jones form with a strong repulsion exerted to the liquid phase by the solid at close proximity and attraction at intermediate distances from the solid boundary. To model these interactions, we adopt the following formula:

2. THEORETICAL METHODS The computation of the equilibrium droplet shape is performed by solving an augmented YL equation, which states the force balance between surface tension and the Derjaguin pressure, reading:32 R s LS p +C=K γLV

(1) 15057


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Figure 2. Solution space for simple structured solid surfaces. (a) Single-striped solid surface featuring a single S-shaped bifurcation diagram. (b) Wave-like, solid surface with a double S-shaped bifurcation diagram. Stable and unstable solutions are depicted with solid and dashed lines, respectively; critical turning points are depicted with open circles. (Parameter values: Adroplet = π (see Supporting Information for details), and p1 = 0.6, p2 = 10, p3 = 5, p4 = 3, and p5 = 1.8.)

⎛ δ⎞ R s LS l 2 + δl0 − δ 2 p (δ) = w LS 0 exp ⎜− ⎟ γLV δ 2l0 ⎝ l0 ⎠

where LLV and LLS are the length of the liquid/vapor and liquid/solid interface, respectively, and Ltot = LSV + LLV is the total solid length. If we denote with (A) and (B) two (meta)stable states separated by an intermediate unstable state, (un), then the energy barrier for (A) → (B) and (B) → (A) transitions is computed from (see also Figure S2 in the Supporting Information):

(2)

where wLS is a parameter controlling the wettability of the solid by the liquid; by increasing wLS the wettability is enhanced; δ is the signed distance of the droplet surface from the solid surface. This quantity can be computed from the solution of the Eikonal equation,16,34 practically for any kind of solid surface geometry, structured or unstructured. Finally, the parameter l0 regulates the minimum distance between the liquid and the solid phase, δmin. When δ = δmin, then pLS = 0, and from eq 2 the minimum distance is δmin = l0(1 +

5 )/2

EA → B = Fun − FA ⇒

− cos θY(L LS,un − L LS,A ) E B → A = Fun − FB ⇒

(3)

⎛ 3− 5 5 + exp⎜ − ⎝ 2 2

1⎞ ⎟−1 ⎠

(7)

In the following section, we solve the augmented YL equation to compute the entire solution space of equilibrium droplets wetting structured solid surfaces (from single-striped to multistriped surfaces). The computation of the entire solution space enables the determination of the energy barriers required to induce transitions between distinct wetting states, as well as the characterization of the stability of the different wetting states.

(4)

Equation 4 (whose rigorous derivation is described in the Supporting Information) is also used for the determination of transition energy barriers separating coexisting equilibrium states, that is, wetting steady states that are computed for the same Young’s angle, θY. 2.2. Computation of Wetting Transition Energy Barriers. The computed solution space shows highly nonlinear features when the solid surface is not smooth and flat. The nonlinearity concerns solutions, that is, equilibrium states multiplicity for a certain range of parameter values. Coexisting (meta)stable equilibrium states correspond to distinct minima of the free energy landscape implying the existence of at least one intermediate unstable state, which corresponds to a saddle of the free energy landscape. The difference of the free energy between the saddle and the basins of attraction (stable states) determines the energy barrier one needs to surpass to induce transitions between the (meta)stable states. The free energy per unit depth, X (for cylindrical droplets), is given from γ F = L LV − cos θYL LS + SV L tot γLVX γLV

(6)

EB→ A = (L LV,un − L LV,B) γLVX

− cos θY(L LS,un − L LS,B)

In the computations presented below, the dimensionless minimum distance is δmin = 5 × 10−3, much lower than the (dimensionless) roughness dimensions (order of ∼10−1 dimensionless units). The wetting parameter, wLS, is related to the Young’s angle, θY, through: cos θY = w LS

EA → B = (L LV,un − L LV,A ) γLVX

3. RESULTS AND DISCUSSION 3.1. Effect of Solid Surface Structure on the Multiplicity of Wetting States. The structure of the solution space, that is, the connectivity of the solution branches and the corresponding computed droplet shapes, is strongly affected by the solid surface geometry. To show this effect, we first present results for a single-striped surface, then for a surface decorated with smaller but still with relatively big stripes, and, finally, for a “realistic” multistriped surface structure. The single-striped solid surface is the simplest solid surface geometry, in which both stable and unstable solutions can be computed. In this case, the stripe shape is given by the following formula: y=−

1 + erf((x − p1 )p2 ) p3

(8)

where erf denotes the error function, and the parameters p1, p2, and p3 regulate the width, the lateral walls curvature, and the maximum height, respectively, of the stripe. The continuation

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The Journal of Physical Chemistry C parameter, here, is wLS; however, results are presented using the mapping between wLS and θY (see eq 4). In the S-shaped bifurcation diagram shown in Figure 2a, one can observe a range of θY values with three coexisting equilibrium states, two of them being stable (branches (I) and (III)) and one unstable (dashed branch (II)). Such type of a solution space practically denotes that the transitions due to modification of the Young’s angle, θY, between suspended states (the ones in branch (I)) and the collapsed ones (in branch (III)) are hysteretic. The number of the hysteretic loops increases with the number of the surface stripes. We next examine the case of a three-striped (wave-like) patterned surface. Here, the solid surface height is given from ⎛ sin(p x) ⎞2 4 ⎟⎟ y = −⎜⎜ ⎝ p5 ⎠

(9) Figure 3. Bifurcation diagram of equilibrium wetting states on a multistriped solid surface. The stripe shape is shown in the inset with height, h = 0.1, the upper base, d1 = 0.08, the lower base, d2 = 0.14, and the corner radius of curvature, R = 0.01. The wavelength of the periodic stripe array is l = 0.2.

where p4 is a parameter controlling the wavelength and p5 regulates the maximum height of the stripes. The resulting solution space resembles a double S-shaped curve presented in Figure 2b. Each of the hysteretic loops corresponds to a transition where the number of the covered stripes, by the liquid, changes. By further roughening the solid surface, that is, by making the stripes smaller, it is expected that the number of attainable wetting steady states and the corresponding hysteretic loops will increase accordingly, as presented in Chamakos et al.16 for sinusoidal shaped structured surfaces; therein, the number of solution branches in a multiple S-shaped bifurcation curve was directly related to the number of wetted stripes. In that particular geometry, that is, with a moderate number of macroscopically smooth stripes, Cassie-like states (the droplet is suspended on the stripes) were computed for large Young’s angles, whereas Wenzel-like states were computed for low Young’s angles. The distance between the corrugations considered in our computations is significantly larger (approximately 10 times larger) as compared to the action range of the Derjaguin pressure (for the particular choice of δmin), enabling the formation of menisci in cases of Cassie like states. The hysteretic transitions corresponded to increasing or decreasing the number of wetted stripes either in a suspended (Cassie-like) or in a collapsed (Wenzel-like) configuration. By making the stripes sharper and even smaller, another interesting aspect arises. In particular, besides the parameter ranges where solely Cassie or Wenzel states are computed, there is an intermediate regime of Young’s angle values where metastable Cassie, Wenzel, or impregnated states are found. The impregnated states are those where underneath the droplet the solid structure is partially filled with liquid.35 In Figure 3, the solution space for a multistriped surface is presented. The cross section of each stripe has a trapezoidal shape (see inset in Figure 3). For this geometry, the intermediate regime of Young’s angles where Cassie, Wenzel, and impregnated states are metastable is bounded by two critical Young’s CA values, θ1 = 105.7° and θ2 = 137.9°, and shown with two vertical dashed guidelines. Outside this regime, only Wenzel (for low Young’s contact angles, i.e., on the left of the left guideline) or only Cassie states (for large Young’s CA, i.e., on the right of the right guideline) are found. Practically, for θY < θ1, wetting transitions can be observed by changing the number of the wetted stripes retaining the Wenzel state. Accordingly, for θY > θ2, transitions can happen in the Cassie state, only by changing the number of the covered stripes. The

limiting angle, θ2, sets also another important limit regarding reversibility of wetting transitions. As Figure 3 shows, to realize spontaneous dewetting transitions the Young’s angle needs to be larger than θ2. If not, then it is always required to supply a finite energy amount that corresponds to an energy barrier separating intermediate states. In the intermediate region, transitions between Cassie, Wenzel, and impregnated states can be observed either by changing the number of the wetted stripes or/and by changing the number of the filled grooves of the stripes. It should be reported at this point that the bounds of the intermediate regime can be correlated with the groove geometric features. For a simple trapezoidal shape, the minimum Young’s angle that renders the groove unable to accommodate Cassie (suspended) states is the one in which the liquid interface is horizontal. Smaller angle results in an unphysical convex liquid/vapor interface shape (see the dashed line in Figure 4a). For the case under study, the liquid/vapor interface is horizontal (see Figure 4a) when θY = θsolid = 104°, that is, an angle very close to the computed θ1 = 105.7°. Similarly, the maximum Young’s angle that renders the groove unable to accommodate Wenzel (collapsed) states is the θmax = 180 − θIso, where θIso is the acute angle of the isosceles triangle shown in Figure 4b. In this limiting case, the liquid can just touch the bottom of the trapezoidal groove. A larger angle makes the droplet recede and detach from the bottom. The limiting material wettability is then computed from θmax = 180° − θIso = 90° + θ1/2. For the trapezoid shaped stripe, θmax is expected to be equal to 142°, which is close enough to the computed value of θ2 = 137.9°. The above analysis provides an estimation of the limiting angles θ1 and θ2, based on simple geometric arguments. It is, however, obvious that this simplified analysis is of limited applicability when the stripes are of more complicated shape. In what follows, we present representative solutions computed for specific Young’s angles values in the three distinct regimes. 3.1.1. Wenzel States Regime. Indicatively, we present in Figure 5a a few of the (meta)stable equilibrium wetting states computed, for θY = 100° < θ1. This “picture” suggests that for the specific Young’ angle and the geometry, one can observe a 15059


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Figure 4. Limiting cases for accommodating (a) only Wenzel and (b) only Cassie states, respectively.

Figure 5. (a) Wenzel and (b) Cassie equilibrium droplet profiles that can be admitted on a solid surface of material wettability θY = 100° and θY = 140°, respectively, decorated with periodically arrayed trapezoid stripes (for stripe size, see Figure 3 and the relative discussion).

wide range of apparent CA values, from θa = 151.5° to 79.2°. All of these droplet profiles correspond to Wenzel type states; that is, the liquid completely penetrates the grooves; θa multiplicity is attributed to the different number of stripes covered by the liquid (from 5 to 17 stripes, as shown in Figure 5a). As the droplet spreads to wet an increased number of stripes, θa reduces accordingly. The issue concerning their relative stability will be illuminated in the discussion about the states energetics (see section 3.2). 3.1.2. Cassie States Region. As reported above, for θY > θ2, only Cassie-type droplets can be accommodated on the studied surface. However, in reality, Young’s angles larger than 120° (the wettability of Teflon) have not been found yet. Thus, we could assume that θY > θ2 (see Figure 3) could be achieved indirectly by secondary scale roughness (probably at the nanoscale). In Figure 5b, we present coexisting stable droplet profiles wetting a solid surface with material wettability θY = 140° > θ2. Here, the variation of θa values is not as significant as in the Wenzel droplets, ranging from 164° to 133.2°. Again, the energetics of the Cassie states will be discussed later. 3.1.3. Regime of Cassie, Wenzel, and Mixed States Coexistence. When the material wettability lies in the intermediate regime (see Figure 3), that is, when θ1 < θY < θ2, metastable Cassie, Wenzel, and mixed/impregnated states (partially collapsed, partially suspended) are computed with almost identical apparent CA for identical θY, when the number of the covered stripes is preserved. This is better seen when part of the solution space, shown in Figure 3, is plotted separately and in particular when the droplet perimeter length, 2·Smax (see also Figure 1), instead of the apparent CA (Figure 6c), is plotted versus the θY.

Figure 6. (a) Blow-up of the bifurcation diagram showing the dependence of apparent wettability on the material wettability on a multistriped solid surface. (b) Stable droplet equilibrium shapes for various material wettabilities. (c) Dependence of the droplet perimeter length on the material wettability in the range delimited by points A and B. (d) Coexisting CB (state (5)), mixed (states (6),(7)), and Wenzel (state (8)) solutions for θY = 114°, all featuring the same θa = 152°.

We focus (for the sake of clarity) on the most hydrophobic states (upper branches of the bifurcation diagram in Figure 3). In Figure 6a and c, stable solution branches are depicted with solid lines, and unstable branches are depicted with dashed lines. The upper stable branch in Figure 6a corresponds to Cassie states wetting one stripe (see shape (1) in Figure 6b), and the intermediate stable branch corresponds to Cassie states wetting three stripes (shape (2) in Figure 6b). The lower stable branch concerns either Cassie states wetting five stripes for sufficiently large material wettability, that is, θY > 137.9° (solution (3)), or fully collapsed Wenzel states wetting (also) five stripes when θY < 105.7° (solution (4)). Several solution branches intervene between points A and B, which are visually indistinguishable, due to the fact that for the same material wettability the coexisting equilibrium states feature approximately the same θa. In Figure 6d, we present four coexisting states for material wettability θY = 114° (marked with an “△” in Figure 6a), all featuring an apparent contact angle of θa = 152°. The different intervening solution branches between points A and B (in Figure 6a) can be visualized by plotting the droplet 15060


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The Journal of Physical Chemistry C perimeter as a function of the material wettability (see Figure 6c). The lowest stable branch in Figure 6c corresponds to suspended Cassie states (state (5)). The intermediate stable branch corresponds to mixed states, where the liquid penetrates into the two internal (state (6)) or two external (state (7)) grooves of the striped surface. The upper stable branch corresponds to collapsed Wenzel states (state (8)). The dashed lines in Figure 6c correspond to unstable steadystate solutions with at least one groove beneath the droplet, which is only partially penetrated by the liquid; the liquid wets the lateral walls of the groove and leaves its ground partially dry. In particular, the lowest unstable branch concerns states where the liquid partially penetrates into the two internal or the two external grooves and remains suspended on the two remaining grooves. The intermediate unstable branch concerns states with a pair of grooves partially penetrated by the liquid, and a pair of grooves fully wetted. The upper unstable branch concerns states with the liquid partially penetrating all grooves beneath the droplet. It is noteworthy to report that the number of coexisting stable and unstable mixed states increases substantially, as the droplet wets an increased number of stripes, because there is a large number of possible combinations of filled and not filled grooves. If we consider the case of a droplet wetting N-grooves at each of its symmetric parts, and that in each groove there exist three possible states (stable suspended, stable collapsed, and unstable partially penetrating), then the total number of stable and unstable coexisting states is 3N (2N of them being stable). 3.2. Energetics of Wetting States and Wetting Transitions Energy Barrier Computations. As reported above, the number of wetting states that can be observed on a patterned surface for a particular material wettability can be significantly large. By computing the surface free energy of the coexisting equilibrium states, we can answer the question “which state is energetically most favored”, and thus, most probably observable in an experiment. In particular, we can determine the energetically favorable state, by comparing the normalized free energy of the (meta)stable equilibrium droplet profiles: F̃ ≡

γ F − SV L tot = L LV − cos θYL LS γLVX γLV

Figure 7. Normalized free energy (F̃ ) of coexisting Wenzel states on a periodic array of trapezoid stripes solid surface with material wettability, θY = 100°.

as compared to the size of the protrusions. In the Supporting Information, we present cases of larger droplets featuring stable θa, which are very close to the theoretically (by Wenzel’s or CB’s equations) predicted values. Similarly, when performing free energy computations for the Cassie droplets presented in Figure 5b, we find that the energetically most favorable state features an apparent CA of θa = 153.1°. By applying the CB equation: cos θCB = rf f cos θY + f − 1

(11)

with rf the ratio of the real surface in contact with the liquid over its projection to the horizontal plane, and f the fraction of the projected area of the solid surface wetted by the liquid, then θCB = 150.1°, which is very close to the computed θa corresponding to the minimum energy. It should be reported here that the energetically favored state is not the only one state that can be observed on the studied structured surface; when this state is appropriately perturbed with sufficient amount of energy, a transition toward states wetting fewer or larger number of stripes can be realized. Such states are energetically less favorable; still they require proper actuation to induce transitions toward the most stable state. The observable wetting state strongly depends on the way the droplet is deposited on the surface (e.g., droplet impact velocity), as well as from characteristics that affect the wetting dynamics (e.g., liquid viscosity). To induce transitions between the distinct metastable states, we need to surpass certain energy barriers, and their magnitude is determined by the local energy maxima, that is, the energy of the unstable states. In particular, for transitions between CB (Figure 8a) and Wenzel (Figure 8c) states, the intermediate unstable state that sets the barrier is the one where all grooves beneath the droplet have been partially penetrated by the liquid (Figure 8b). By using eqs 6 and 7, we then can determine the corresponding energy barrier for transitions between CB and Wenzel states with (approximately) the same θa. In Figure 9a, we plot the dependence of the wetting (CB to Wenzel) and dewetting (Wenzel to CB) transition energy barriers on the material wettability (θY) within the range set by points A and B as shown in Figure 6a (i.e., for θY ∈ [105.7°, 137.9°]). Therein, it is shown that as we increase θY, the wetting energy barriers increase, whereas dewetting energy barriers decrease. It can be also observed that higher energy is required to induce wetting transitions as compared to

(10)

In Figure 7, we present the normalized free energy of the surface profiles shown in Figure 5a and their corresponding normalized contact radius, R̃ = Rc/Ro; Ro is the radius of the droplet when θY = 180° (circular droplet), and Rc is the contact radius, that is, the droplet radius at the asperity apex. As it can be seen, there are two states with almost the same energy corresponding to the minimum, and the associated apparent CAs are found as 100° and 113°, respectively. Interestingly enough, the two most stable wetting states feature an apparent wettability that is closest to that predicted by Wenzel’s equation: cos θW = r cos θY, with r as the surface roughness factor, which is defined as the ratio of the real surface in contact with the liquid over its projection on the horizontal plane. In particular, for the studied multistriped surface, r = 1.74. By applying Wenzel’s equation, the expected θa is θW = 107.6°, which is almost the mean value of the apparent CAs of the two most stable states. The difference between the computed θa and the Wenzel one (predicted by the Wenzel equation) can be attributed to the relatively small droplet size 15061


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observation also holds for the computed CB states; the energetically favorable CB state with contact radius, R̃ = 0.83 (wetting nine stripes), has θa = 127°, which is the closest approximated by the CB equation: θCB ≈ 132°. In the next section, we investigate the effect of stripe shape on the range of each of the computed regimes, accommodating Wenzel, Cassie, or mixed states. Their energetics are also presented. 3.3. Effect of Stripe Shape. In this section, we study cases of solid surfaces, which are decorated with periodically arranged stripes of different shapes, and in particular trapezoid shape, orthogonal, and stripes with inverse trapezoid shape (see Figure 10a).

Figure 8. Blow-up of liquid/solid interface of coexisting (a) stable suspended CB, (b) intermediate unstable, and (c) stable collapsed Wenzel states featuring the same θa. Material wettability θY = 114°.

Figure 9. (a) Dependence of normalized energy barrier on the material wettability for transitions between CB and Wenzel droplets wetting five trapezoid stripes. (b) Dependence of the normalized free energy of CB and Wenzel states on the normalized contact radius. Material wettability: θY = 114°.

dewetting transitions for θY > 116°. This suggests that CB states, within the range set by points A and B in Figure 6a, are energetically favored (as compared to Wenzel states) for sufficiently hydrophobic materials (here for θY > 116°). To provide estimates of the energy barriers computed, physical units for the wetting and dewetting transitions are obtained as follows. If we consider a droplet with volume 1 μL, then the characteristic length is Rs = 0.62 mm. By considering a water/air interface, the surface tension is γLV = 0.072 N/m. The one (1) dimensionless unit of energy per unit length then corresponds to

Figure 10. (a) Cross section of (i) trapezoid shaped, (ii) orthogonal, and (iii) inverse trapezoid shaped stripes. (b) Dependence of θa on the material wettability for equilibrium droplets wetting multistriped solid surfaces. The blue, red, and green lines correspond to orthogonal, trapezoid, and inverse trapezoid shaped stripes, respectively. (c) Material wettability regimes accommodating CB, Wenzel, and mixed wetting states for (i) trapezoid, (ii) orthogonal, and (iii) inverse trapezoid shaped stripes.

N ·1m = 0.045 mJ m In the intermediate regime, among the coexisting CB, Wenzel, and mixed states, one can determine which of them is energetically most favorable by checking how their energies depend on the number of stripes covered, for a fixed θY, that is, for a given material. In Figure 9b, we plot the dependence of the normalized free energy of CB and Wenzel states on the number of the wetted stripes and for material wettability, θY = 114°. In all cases, the Wenzel states are more stable as compared to the CB ones. Furthermore, the most stable Wenzel state (among all coexisting Wenzel states) is the one with a normalized contact radius, R̃ = 0.83 (wetting nine stripes for this particular surface), with θa = 127°. The predicted θa from Wenzel’s equation is θw = 132.9°, which is close enough to the computed thermodynamically stable θa for material wettability, θY = 114°. In fact, Wenzel’s equation provides the best approximation of the θa of the most stable36 among all Wenzel states for material wettability, θY = 114°, as shown in Figure 9b. The same R s·γLV ·X = 0.62 mm· 0.072

In the computations presented below, the dimensions of the stripes (see Figure 10a) are: h = 0.1, d1 = 0.08, d2 = 0.14. d3 = 0.08, d4 = 0.08, d5 = 0.05, and R = 0.01. The wavelength in all cases is l = 0.2, which results in a roughness factor of r = 1.74, 1.91, and 2.27 for the trapezoid, orthogonal, and inverse trapezoid stripes, respectively. In Chamakos et al.,16 we performed computations of wetting steady states on sinusoidal multistriped surfaces with approximately the same height and wavelength, yet with lower roughness factor, r = 1.52. Figure 10b shows the dependence of θa on the material wettability for the most hydrophobic states. In particular, we choose to present the upper branches of the computed bifurcation diagrams, rather than the whole picture of them, which is messy and arduous to read. A comparative study of the three superimposed solution curves shows that the left turning points for inverse trapezoidal shaped stripes are shifted to the left, that is, to lower Young’ s angles. That means that to induce a transition that increases the covered stripes (wetted by the 15062


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The Journal of Physical Chemistry C liquid), lower Young’s angles are required. Thus, inverse trapezoidal stripes favor limited spreading, that is, higher apparent hydrophobicity. No significant changes are observed for receding (dewetting) transitions because the right turning points’ positions are not considerably affected by the stripe geometry modification. Other geometry features, such as the wavelength and/or the height of stripes,37 might need to be modified to affect dewetting transitions. As in the case of trapezoid stripes, where CB, Wenzel, and mixed states coexist within a certain range of θY values (see section 3.1), orthogonal and inverse trapezoid stripes also share this feature. In Figure 10c, we mark with black dashed lines the wettability regions of coexisting CB, Wenzel, and mixed states for the different cases of stripe shapes. In particular, trapezoid stripes (Figure 10c-i) can admit coexisting CB, Wenzel, and mixed states within the interval θY ∈ [105.7°, 137.9°]; for orthogonal stripes (Figure 10c-ii), this range is shifted toward lower θY values, that is, [92.4°, 135.8°]. When inverse trapezoid stripes decorate the solid surface (Figure 10c-iii), the range spans over even more hydrophilic θY values, that is, [72.4°, 126.7°]. The corresponding limits when we consider the analysis presented in section 3.1 for orthogonal stripes are θ1 = 90° and θ2 = 90° + θ1/2 = 135°, which agree well with the computed interval [92.4°, 135.8°]; for inverse trapezoids, the respective values are θ1 = 69.4° and θ2 = 90° + θ1/2 = 124.7°, which again approximate well the limits of the computed interval: [72.4°, 126.7°]. In practice, this suggests that to induce transitions through material wettability modification, a collapse transition would require higher modification for the inverse trapezoid case; the material wettability in this case needs to drop below 72.4° to induce a CB to Wenzel transition, whereas in the trapezoid shape wetting transitions can be observed at θY < 105.7°. In a reverse experiment, a lifting transition would require lowering of wettability of as much as 137.9° for the trapezoid, 135.8° for the orthogonal, and 126.7° for the inverse trapezoid stripe. This suggests that dewetting transitions through wettability modification are infeasible for the studied geometries, considering that a perfectly smooth hydrophobic material cannot exhibit θY > 120°. Such superhydrophobic behavior could be eventually featured by nanoroughened materials.38−40 3.4. CB ↔ Wenzel Transitions Energy Barrier Computations. In this section, we examine the effect of stripe shape on the energy barrier required to induce CB to Wenzel state (and reverse) transitions preserving the number of covered stripes, thus preserving (approximately) the apparent wettability. In particular, we perform the computations for θY = 114° (corresponding to a Teflon coated surface). In Figure 11, we show the dependence of the normalized energy barrier on the normalized contact radius for CB to Wenzel transitions. For all studied cases of striped shapes, transitions between CB and Wenzel states feature an almost linear dependence on the contact radius. Among the three cases under study, the inverse trapezoid shaped stripes require the highest energy to induce a CB to Wenzel transition, followed by orthogonal stripes; trapezoid shaped stripes require the least energy. The realization of a reverse Wenzel to CB transition requires approximately the same energy for inverse trapezoid and orthogonal stripes. Trapezoid shaped stripes, as in the wetting transition case, require the lowest energy. This suggests that reversibility between CB to Wenzel states is favored when using

Figure 11. Dependence of (a) CB to Wenzel and (b) Wenzel to CB energy barrier magnitude with the number of wetted stripes for different shapes of stripes (orthogonal, trapezoid, and inverse trapezoid shape). Material wettability θY = 114°.

trapezoid stripes because it is this shape, among all studied cases, that requires the least amount of energy to induce both CB to Wenzel and Wenzel to CB transitions. Energy computations on different stripe shapes can also be utilized for the better understanding of complex wetting phenomena on rough surfaces. A characteristic example is the so-called “rose petal effect”, which has recently attracted the interest of many research groups,24,26,41,42 due to the unusual behavior of droplets featuring large θa accompanied by high adhesion, when deposited on rose petals. In particular, we demonstrate how specific protrusion shapes favor collapsed over suspended states, even in cases where the wetting state is highly hydrophobic. 3.5. Rose Petal versus Lotus Effect. Low apparent wettabilities cannot be considered as a sole criterion for superhydrophobicity.24−26 It needs to be accompanied by the criterion of low adhesion energy of the droplet on the wetted area, which is related to contact angle hysteresis.43 The existence of superhydrophobic states exhibiting high sticking properties was first reported in Feng et al.24 This phenomenon is commonly known as the “rose petal effect” featuring high θa and high CA hysteresis, as opposed to the most popular “lotus effect” with droplets of high θa and low CA hysteresis.25 In a hierarchical structure of rough surfaces, high θa values emerge when water does not penetrate the smaller (nanometric) scale roughness. High or low solid/water adhesion is the result of water penetration into the larger (micro-) scale roughness. In this work, we do not investigate the effect of secondary scale roughness; however, free energy computations of equilibrium states on different shapes of striped geometries can be useful for the better understanding of the rose petal effect. In Figure 12a and b, we plot the (normalized) free energy of coexisting CB and Wenzel states admitted on multistriped surfaces with trapezoid and inverse trapezoid stripes, respectively. The selected material wettability is θY = 110°. The most stable equilibrium state (i.e., the one with the lowest energy) in both cases wets nine stripes featuring an θa ≈ 127° (hydrophobic); however, the energetically favored state that can be admitted on the trapezoid shaped stripes is the Wenzel state (high stickiness), whereas the lower adhesion energy CB state is the most stable one on inverse trapezoid stripes. In Figure 12c,d, we plot the two most stable equilibrium states that can be admitted on the two cases of multistriped solid surfaces. Despite the fact that they both feature approximately the same θa (θa ≈ 127°), their adhesion energies are different, as presented in the computations below. 15063


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compared to the adhesion of the inverse trapezoidal ones. It should be noted here that one could design surface adhesion properties by stripe shape design optimization aiming to minimum or maximum adhesion energy.

4. CONCLUSIONS We apply an efficient continuum-level augmented YL formulation,16 which enables the computation of entire droplet profiles with multiple and reconfigurable TPLs on patterned solid surfaces. With the aid of the finite element method and a pseudo arc-length parameter continuation technique,33 we track a remarkably wealthy solution space of stable and unstable solution branches, which correspond to distinct valleys and ridges of the free energy landscape. Our approach enables (i) the determination of wetting and dewetting transition energy barriers, and (ii) the characterization of the various wetting states’ relative stability by comparing their surface free energy. The reversibility of the transitions is discussed, and the effect of the surface geometry is highlighted. Special attention is given to the dependence of the existence and multiplicity of solutions on the material wettability modification and on certain geometric features of the solid surface. We present, for the first time, distinct wettability regimes where only Wenzel, only Cassie, and, interestingly enough, both Cassie, Wenzel, and mixed states (Cassie impregnated, Wenzel impregnated) are (meta)stable. Not only apparent wettabilities can be predicted but also surface adhesion properties can be estimated, thus predicting whether a surface geometry favors droplet sticky or slippery behavior. We deliberately show how a wetting behavior reported as “rose petal effect”24 can be directly related to the solid surface geometry. While small scale roughness has been shown to ensure superhydrophobicity,24,26 the larger scale roughness can be appropriately designed to achieve desirable levels of adhesive or nonadhesive properties. We successfully correlate our predictions with those of Cassie and of Wenzel equations especially for small-scale geometric protrusions. We find that Cassie and Wenzel equations successfully predict the apparent wettability of the most stable (the one with the lowest free energy) suspended and collapsed states, respectively. However, because our approach goes beyond idealized simplifications concerning unit cell computations (computations for single surface protrusion), we are able to fully map the solution space. This full mapping is of particular importance for the design of functional surfaces, where the material wettability can be actively altered, for example, by electric or thermal fields (using the electrowetting or the thermocapillary effect). Finally, we report that, although we present results for cylindrically shaped droplets, the same methodology can be extended for the computation of droplets resting on threedimensional structured surfaces. In this case, proper parametrization of the two-dimensional free droplet surface is required, using either unstructured or elliptical structured meshes for its discretization. The study of wetting transition kinetics is also a subject of ongoing research. In particular, standard Navier−Stokes equations are used to model the flow in the interior of the liquid droplet; the boundary condition applied at the droplet free surface expresses the local force balance between the surface tension, the viscous stresses, and the liquid/solid interactions by incorporating the Derjaguin pressure term. This way, the stress singularity at any contact line is alleviated, enabling the modeling of droplet dynamics on structured surfaces.

Figure 12. Dependence of the normalized free energy of CB and Wenzel states on the normalized contact radius for (a) trapezoid and (b) inverse trapezoid shaped stripes. Stable equilibrium states on periodic arrays of (c) trapezoid and (d) inverse trapezoid stripes. Both states feature the same apparent wettability and different adhesion properties. (e) The disconnected liquid cap with θa is used for the computation of the Young−Dupré work of adhesion for the detachment of stable wetting states (c) and (d).

To quantify the adhesion strength, we compute the Young− Dupré work of adhesion43,44 for each state shown in Figure 12c,d. For this computation, we determine the free energy of the equilibrium droplet wetting the structured surface (Figure 12c,d), and the free energy of the liquid cap (Figure 12e) featuring the same θa. The work of adhesion can be then computed from43,45 Wad = Fdis − Feq

(12)

where Fdis is the surface free energy of the disconnected droplet (liquid cap), and Feq is the free energy of the stable equilibrium state. Taking into account eq 10, the work of adhesion (for cylindrical droplets) can be computed from W ̃ ≡ ad = L LV,dis − L LV,eq + cos θYL LS,eq Wad γLVX (13) where LLV,dis, LLV,eq are the liquid/vapor interface lengths of the disconnected and equilibrium droplet, and LLS,eq is the length of liquid/solid interface of the equilibrium droplet. Assuming that the disconnected droplet has the same θa, with the stable equilibrium droplet, then we can compute (when Adroplet = π): L LV,dis = 2

2π (θa + sin θa) 2θa − sin(2θa)

(14)

The (dimensionless) normalized work of adhesion for the stable Wenzel state (Figure 12c) is W̃ ad,Wenzel = 0.79, and the work of adhesion for the stable CB state (Figure 12d) wetting inverse trapezoid stripes is ∼20% lower: W̃ ad,CB = 0.65. Thus, trapezoidal stripes promote higher adhesion to droplets as 15064


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(14) Dupuis, A.; Yeomans, J. M. Lattice Boltzmann Modelling of Droplets on Chemically Heterogeneous Surfaces. Future Gen. Comput. Syst. 2004, 20, 993−1001. (15) Colosqui, C. E.; Kavousanakis, M. E.; Papathanasiou, A. G.; Kevrekidis, I. G. Mesoscopic Model for Microscale Hydrodynamics and Interfacial Phenomena: Slip, Films, and Contact-Angle Hysteresis. Phys. Rev. E 2013, 87, 013302. (16) Chamakos, N. T.; Kavousanakis, M. E.; Papathanasiou, A. G. Enabling Efficient Energy Barrier Computations of Wetting Transitions on Geometrically Patterned Surfaces. Soft Matter 2013, 9, 9624−9632. (17) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Surface Forces; Consultants Bureau: New York, 1987. (18) Berim, G. O.; Ruckenstein, E. Microcontact and Macrocontact Angles and the Drop Stability on a Bare Surface. J. Phys. Chem. B 2004, 108, 19339−19347. (19) Snoeijer, J. H.; Andreotti, B. A Microscopic View on Contact Angle Selection. Phys. Fluids 2008, 20, 057101. (20) Ruckenstein, E.; Berim, G. O. Microscopic Description of a Drop on a Solid Surface. Adv. Colloid Interface Sci. 2010, 157, 1−33. (21) He, B.; Patankar, N. A.; Lee, J. Multiple Equilibrium Droplet Shapes and Design Criterion for Rough Hydrophobic Surfaces. Langmuir 2003, 19, 4999−5003. (22) Bormashenko, E.; Pogreb, R.; Stein, T.; Whyman, G.; Erlich, M.; Musin, A.; Machavariani, V.; Aurbach, D. Characterization of Rough Surfaces with Vibrated Drops. Phys. Chem. Chem. Phys. 2008, 10, 4056−4061. (23) Marmur, A. Soft Contact: Measurement and Interpretation of Contact Angles. Soft Matter 2006, 2, 12−17. (24) Feng, L.; Zhang, Y.; Xi, J.; Zhu, Y.; Wang, N.; Xia, F.; Jiang, L. Petal Effect: a Superhydrophobic State with High Adhesive Force. Langmuir 2008, 24, 4114−4119. (25) Bhushan, B.; Nosonovsky, M. The Rose Petal Effect and the Modes of Superhydrophobicity. Philos. Trans. R. Soc., A 2010, 368, 4713−4728. (26) Bormashenko, E.; Stein, T.; Pogreb, R.; Aurbach, D. “Petal Effect” on Surfaces Based on Lycopodium: High-Stick Surfaces Demonstrating High Apparent Contact Angles. J. Phys. Chem. C 2009, 113, 5568−5572. (27) Barthlott, W.; Neinhuis, C. Purity of the Sacred Lotus, or Escape from Contamination in Biological Surfaces. Planta 1997, 202, 1−8. (28) Gao, L.; McCarthy, T. J. How Wenzel and Cassie Were Wrong. Langmuir 2007, 23, 3762−3765. (29) McHale, G. Cassie and Wenzel: Were They Really so Wrong? Langmuir 2007, 23, 8200−8205. (30) Nosonovsky, M. On the Range of Applicability of the Wenzel and Cassie Equations. Langmuir 2007, 23, 9919−9920. (31) Brandon, S.; Haimovich, N.; Yeger, E.; Marmur, A. Partial Wetting of Chemically Patterned Surfaces: The Effect of Drop Size. J. Colloid Interface Sci. 2003, 263, 237−243. (32) Starov, V. M. Surface Forces Action in a Vicinity of Three Phase Contact line and other Current Problems in Kinetics of Wetting and Spreading. Adv. Colloid Interface Sci. 2010, 161, 139−152. (33) Keller, H. Lectures on Numerical Methods in Bifurcation Problems. Appl. Math. 1987, 217, 50. (34) Evans, L. C. Partial Differential Equations; American Mathematical Society: Washington, DC, 1998. (35) Søgaard, E.; Andersen, N. K.; Smistrup, K.; Larsen, S. T.; Sun, L.; Taboryski, R. Study of Transitions between Wetting States on Micro-Cavity Arrays by Optical Transmission Microscopy. Langmuir 2014, 30, 12960−12968. (36) Marmur, A.; Bittoun, E. When Wenzel and Cassie are Right: Reconciling Local and Global Considerations. Langmuir 2009, 25, 1277−1281. (37) Kavousanakis, M. E.; Colosqui, C. E.; Papathanasiou, A. G. Engineering the Geometry of Stripe-Patterned Surfaces toward Efficient Wettability Switching. Colloids Surf., A 2013, 436, 309−317.

ASSOCIATED CONTENT

S Supporting Information *

Arc-length parametrization of the augmented Young−Laplace equation. Correlation of wetting parameter, wLS, with the Young’s angle, θY. Determination of energy barriers for wetting and dewetting transitions. Derivation of the CB equation for cylindrical droplets. Effect of droplet size on the stable ACA. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b00718.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +30 210 772 3234. Fax: +30 210 772 3298. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We kindly acknowledge funding from the European Research Council under the Europeans Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. [240710].

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REFERENCES

(1) Quéré, D. Surface Chemistry: Fakir Droplets. Nat. Mater. 2002, 1, 14−15. (2) Lafuma, A.; Quere, D. Superhydrophobic States. Nat. Mater. 2003, 2, 457−460. (3) Paven, M.; Papadopoulos, P.; Schöttler, S.; Deng, X.; Mailänder, V.; Vollmer, D.; Butt, H.-J. Super Liquid-Repellent Gas Membranes for Carbon Dioxide Capture and Heart−Lung Machines. Nat. Commun. 2013, 4, 2512. (4) Liu, T.; Kim, C. Repellent Surfaces. Turning a Surface Superrepellent Even to Completely Wetting Liquids. Science (New York, NY) 2014, 346, 1096−1100. (5) Xia, F.; Zhu, Y.; Feng, L.; Jiang, L. Smart Responsive Surfaces Switching Reversibly between Super-Hydrophobicity and SuperHydrophilicity. Soft Matter 2009, 5, 275−281. (6) Zhang, J.; Han, Y. Active and Responsive Polymer Surfaces. Chem. Soc. Rev. 2010, 39, 676−693. (7) Pujado, P.; Huh, C.; Scriven, L. On the Attribution of an Equation of Capillarity to Young and Laplace. J. Colloid Interface Sci. 1972, 38, 662−663. (8) Lobaton, E.; Salamon, T. Computation of Constant Mean Curvature Surfaces: Application to the Gas−Liquid Interface of a Pressurized Fluid on a Superhydrophobic Surface. J. Colloid Interface Sci. 2007, 314, 184−198. (9) Koishi, T.; Yasuoka, K.; Fujikawa, S.; Ebisuzaki, T.; Zeng, X. C. Coexistence and Transition between Cassie and Wenzel State on Pillared Hydrophobic Surface. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 8435−8440. (10) Savoy, E. S.; Escobedo, F. A. Simulation Study of Free-Energy Barriers in the Wetting Transition of an Oily Fluid on a Rough Surface with Reentrant Geometry. Langmuir 2012, 28, 16080−16090. (11) Yuan, Q.; Zhao, Y.-P. Multiscale Dynamic Wetting of a Droplet on a Lyophilic Pillar-Arrayed Surface. J. Fluid Mech. 2013, 716, 171− 188. (12) Kavousanakis, M. E.; Colosqui, C. E.; Kevrekidis, I. G.; Papathanasiou, A. G. Mechanisms of Wetting Transitions on Patterned Surfaces: Continuum and Mesoscopic Analysis. Soft Matter 2012, 8, 7928. (13) Vrancken, R. J.; Blow, M. L.; Kusumaatmaja, H.; Hermans, K.; Prenen, A. M.; Bastiaansen, C. W.; Broer, D. J.; Yeomans, J. M. Anisotropic Wetting and De-Wetting of Drops on Substrates Patterned with Polygonal Posts. Soft Matter 2013, 9, 674−683. 15065


Article

The Journal of Physical Chemistry C (38) Quere, D.; Reyssat, M. Non-Adhesive Lotus and other Hydrophobic Materials. Philos. Trans. R. Soc., A 2008, 366, 1539− 1556. (39) Sun, T.; Feng, L.; Gao, X.; Jiang, L. Bioinspired Surfaces with Special Wettability. Acc. Chem. Res. 2005, 38, 644−652. (40) Lau, K. K.; Bico, J.; Teo, K. B.; Chhowalla, M.; Amaratunga, G. A.; Milne, W. I.; McKinley, G. H.; Gleason, K. K. Superhydrophobic Carbon Nanotube Forests. Nano Lett. 2003, 3, 1701−1705. (41) Bhushan, B.; Her, E. K. Fabrication of Superhydrophobic Surfaces with High and Low Adhesion Inspired from Rose Petal. Langmuir 2010, 26, 8207−8217. (42) Bormashenko, E.; Starov, V. Impact of Surface Forces on Wetting of Hierarchical Surfaces and Contact Angle Hysteresis. Colloid Polym. Sci. 2013, 291, 343−346. (43) Bormashenko, E.; Bormashenko, Y. Wetting of Composite Surfaces: When and Why Is the Area Far from The Triple Line Important? J. Phys. Chem. C 2013, 117, 19552−19557. (44) Dupré, A.; Dupré, P. Théorie Mécanique de la Chaleur; GauthierVillars: 1869. (45) Schrader, M. E. Young-Dupre Revisited. Langmuir 1995, 11, 3585−3589.

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