Beyond-Cassie Mode of Wetting and Local Contact Angles of Droplets

Article pubs.acs.org/Langmuir

Beyond-Cassie Mode of Wetting and Local Contact Angles of Droplets on Checkboard-Patterned Surfaces Jan Carmeliet,†,‡ Li Chen,§,∥ Qinjun Kang,§ and Dominique Derome*,‡ †

Chair of Building Physics, ETH Zürich (Swiss Federal Institute of Technology in Zürich), Zürich, Switzerland Laboratory of Multiscale Studies in Building Physics, Empa (Swiss Federal Laboratories for Materials Science and Technology), Dübendorf, Switzerland § Earth and Environment Sciences Division (EES-16), Los Alamos National Laboratory, Los Alamos, New Mexico 87544, United States ∥ Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy and Power Engineering, Xi’an Jiaotong University, Shaanxi, China ‡

S Supporting Information *

ABSTRACT: Droplet wetting and distortion on flat surfaces with heterogeneous wettability are studied using the 3D Shan− Chen pseudopotential multiphase lattice Boltzmann model (LBM). The contact angles are compared with the Cassie mode, which predicts an apparent contact angle for flat surfaces with different wetting properties, where the droplet size is large compared to the size of the heterogeneity. In this study, the surface studied consists in a regular checkboard pattern with two different Young’s contact angles (hydrophilic and hydrophobic) and equal surface fraction. The droplet size and patch size of the checkboard are varied beyond the limit where Cassie’s equation is valid. A critical ratio of patch size to droplet radius is found below which the apparent contact angle follows the Cassie mode. Above the critical value, the droplet shape changes from a spherical cap to a more distorted form, and no single contact angle can be determined. The local contact angles are found to vary along the contact line between minimum and maximum values. The droplet is found to wet preferentially the hydrophilic region, and the wetted area fraction of the hydrophilic region increases quasi-linearly with the ratio between patch and droplet sizes. We propose a new equation beyond the critical ratio, defining an equivalent contact angle, where the wetted area fractions are used as weighting coefficients for the maximum and minimum local contact angles. This equivalent contact angle is found to equal Cassie’s contact angle. where θY is Young’s contact angle and γ the interfacial tensions: solid−gas γSG, solid−liquid γSL, and liquid−gas γLG. Young’s equation is only valid for ideal solids, meaning smooth, chemically homogeneous, rigid, and nonreactive solids. However, surfaces in nature are often chemically heterogeneous showing wettability dependence on location. On smooth but chemically heterogeneous surfaces, Cassie’s equation is widely used.3 When a smooth surface is composed of two materials showing Young’s contact angles θY,1 and θY,2 and with surface

1. INTRODUCTION Wetting is a ubiquitous phenomenon of interest in many applications and processes.1 A simple wetting system consists in a liquid droplet (L) on a solid surface (S) immersed in its gas phase (G). Such wetting systems are commonly characterized by a contact angle θ at the three-phase contact line. In equilibrium state between liquid, gas, and solid phases, the contact angle is given by Young’s equation:2 cos θY =

γSG − γSL γLG

Received: April 29, 2017 Published: May 31, 2017

(1) © 2017 American Chemical Society

6192

DOI: 10.1021/acs.langmuir.7b01471 Langmuir 2017, 33, 6192−6200

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Langmuir fractions f1 and f 2, Cassie’s equation gives the apparent contact angle θC as cos θC = f1 cos θY,1 + f2 cos θY,2

Different computational methods have been used to simulate droplet configurations. For example, Karapetsas et al. use a finite-element implementation of Young−Laplace and Derjaguin pressures to study a tridimensional droplet sliding on a vertical flat surface with checkerboard or striped patterns. The patterns are seen to affect the contact line configuration during sliding, and only advancing and receding contact angles are reported. The multiphase lattice Boltzmann method (LBM) has previously been used to study droplet on heterogeneous surfaces in 2D15 and in 3D.16,17 In the latter, droplets are deposited on patterned flat surfaces, and the advancing and receding global contact angle hysteresis is analyzed. The variation of contact angles along the perimeter of the droplet is however not documented, which is the focus of the present study. In this paper, we study the droplet behavior on checkboardpatterned heterogeneous surfaces with two different contact angles where the patch size of a checkerboard cell is in similar order of magnitude as the droplet radius. The droplet equilibrium shape and position are determined using 3D multiphase lattice Boltzmann method (LBM). The aim of this study is to determine the local contact angles of droplets deposited on such surfaces and to identify the evolution of the local contact angles beyond Cassie mode. As a general note, we mention that for droplet size sufficiently larger than the size of the molecules, line tension effects may be disregarded and only surface tension effects can be considered. Gravity is neglected since the droplet diameter is sufficiently smaller than the capillary length. We do not consider contact angle hysteresis, nor advancing nor receding contact angles, as we focus on equilibrium contact angles. The paper is organized as follows. In section 2, we briefly describe the lattice Boltzmann method and the computational domain used in this study. In section 3, the simulation results of local contact angle variation along the contact line of droplets deposited on checkboard-patterned heterogeneous surfaces are presented, and the evolution of the local contact angles as a function of the ratio of the typical size of the heterogeneity versus the droplet radius is captured. In section 4, conclusions are drawn.

(2)

where f1 + f 2 = 1. Cassie’s equation can be derived from the thermodynamic equilibrium condition, where the Gibbs free energy of a spherical cap droplet on a solid surface attains its global minimum.4 The validity of Cassie’s equation has been investigated and questioned for many years, and still controversies exist on the conditions when the equation can be used.5−8 Different modified equations have been proposed.9 Marmur and Bittoun demonstrate that Cassie’s equation holds for droplets sufficiently large compared with the wavelength of the chemical heterogeneity. They suggest that the droplet should be about 3 orders of magnitude larger than the heterogeneity size for Cassie’s equation to hold. This statement will be studied in detail in this paper. The use of Cassie’s equation is further complicated when taking into account contact angle hysteresis.4 A general approach to find the apparent contact angle is to consider the change in free energy at the three-phase contact line by considering an infinitesimal displacement of the contact line over a typical portion of the material containing both types of surface.10 In equilibrium state, the droplet shape and apparent contact angle are then found at minimal free energy. On the basis of such an approach, McHale proposed that Cassie’s equation can be used for more complex surface geometries when the surface fractions and apparent contact angle are reinterpreted and defined locally. It is argued that in this case the properties of the surface along the contact line of the droplet, and not of the full contact area, determine the observed contact angle, as observed by experimentally5,11 and derived theoretically.12 However, McHale also notes that the determination of a contact line on a complex surface does not simplify to a problem of finding a set of independent threephase contact points along this line as solutions along distinct two-dimensional profile sections. The different contact points are found to behave as connected points, where the motion of one triple point influences the motion of the others. This interplay is governed by a competition between liquid−gas surface tensions at the droplet edge and local solid−liquid interaction forces exerted by the chemically different patches.11,13 Moreover, when the droplet and heterogeneity sizes are of similar order of magnitude, the droplet may become confined to or repelled from particular regions of the surface. Local droplet deformations determine then the local contact angles, which may vary from one location to the other attaining values between the two Young’s contact angles present on that surface.7,8 Two main questions remain open. First, what is the critical ratio between heterogeneity size and droplet size for Cassie mode to be valid? Second, how can the resulting local contact angles be understood in deformed droplet configurations beyond the Cassie mode? It has to be remarked that due to the complex geometry of the deformed droplet beyond the Cassie mode, Gibbs free energy cannot be expressed anymore as a simple function of the droplet geometry, and a straightforward derivation of the equilibrium droplet geometry using a Gibbs energy minimum becomes more difficult. This means that for such complex cases droplet wetting is not amenable to a general analytical treatment anymore, and numerical methods have to be used to determine the equilibrium droplet geometry and the local contact angles along the contact line.

2. SIMULATION METHOD Lattice Boltzmann Method. In this study, the 3D Shan− Chen pseudopotential multiphase lattice Boltzmann method is used to simulate the equilibrium configuration of a droplet deposited on a surface with heterogeneous wettability. LBM considers flow as a collective behavior of pseudoparticles residing on a mesoscopic level and solves the Boltzmann equation using a small number of velocities adapted to a regular grid in space.18 In single-component multiphase LBM,19−21 the interaction between liquid particles is described by a cohesive force, which depends on a parameter G reflecting the interaction strength. For G < 0, the cohesive force of the liquid phase is stronger than the force of the gas phase leading to phase separation and surface tension phenomena. The liquid−gas interaction value G in this study is put to −1 for automatic phase separation, which is one of the important advantages of the pseudopotential LBM. The cohesive force is characterized by a nonideal equation of state (EOS), which describes the relation between the density of the gas and liquid phases for a given pressure and temperature. The Carnahan−Starling (C−S) EOS is selected because it reduces the spurious currents at the phase interface 6193


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Langmuir and allows the application of a wider range of temperature T/Tc (Tc is the critical temperature) and density ρ/ρc (where ρc is the critical density) ratios. The adhesive force reflects the interaction between the liquid and solid phases, and its strength is governed by the solid−liquid interaction parameter w. It is noted that LBM does not require the contact angle be explicitly prescribed because, by adjusting values for w, different contact angles can be obtained. When combining the LBM with a proper forcing scheme for the interaction forces, accurate predictions of surface tension independent of the relaxation time and density ratio are obtained. The exact-difference method ENREF-12 22 is used as the forcing scheme guaranteeing numerical accuracy and stability. For numerical stability, the relaxation time τ is chosen to be 1 in this study. A full description of this model is found in our previous papers.23,24 Geometry and Domain Size. The surface with heterogeneous wettability has a regular and periodic pattern of a checkboard with patch size a (Figure 1). For the base case, the

patches are alternating hydrophilic and hydrophobic with Young’s contact angles of 23.3° and 120.5°, respectively (solid−liquid interaction parameter w of −0.1 and 0.05). In a parametric study, other pairs of Young’s contact angles are considered, namely 23.3° and 84.0° (w: −0.1 and −0.01) and 65.8° and 102.0° (w: −0.04 and 0.02). The droplet radius R refers to the initial radius of a hemispheric droplet before deposition. Running different initial configurations, i.e., a hemisphere, sphere, or pancake geometry, results in the same contact angle variations along the perimeter of the equilibrated droplet. The hemisphere and spherical geometry refer to an advancing contact angle configuration, while the pancake geometry simulates a receding contact angle. So, the hemispherical configuration is used for all further simulations. Two domain sizes are used: 300 × 300 × 300 lattices3 for the droplet radii of 50 or 100 lattices and a domain of 400 × 400 × 400 lattices3 for the droplet radius of 150 lattices. With the resolution Δx = 1 μm, the physical sizes of the two domains are thus 300 × 300 × 300 μm3 or 400 × 400 × 400 μm3. The droplet radius is of 50, 100, or 150 μm, which is sufficiently smaller than the capillary length (2 mm) allowing gravity to be neglected. Six different patch sizes are considered ranging from 3 to 150 lattices. The ratio of patch size a to droplet radius R is represented by α = a/R. Bounce-back boundary conditions are imposed at the top and bottom sides, while the other sides are treated as periodic boundary conditions. The densities of liquid and gas are 0.28 and 0.0299 lattice units, corresponding to a density ratio ρ/ρc = 9.4 at T/Tc = 0.85. To reach the equilibrium state, all cases are run for 50 000 time steps, where the time step equals 1 in lattice units. All numerical simulations are run in parallel computing based on Message Passing Interface (MPI) on the high performance computing cluster of Los Alamos National Laboratory (LANL). The cluster aggregate performance is 352 TFlop/s with 102.4 TB of memory for 38 400 cores. Each simulation is run on 1000 (10 × 10 × 10) or 4000 (20 × 20 × 10) processor cores for the two different domain sizes and requires 16 h to run the 50 000 time steps.

3. RESULTS AND DISCUSSION Dependence on Location of Droplet Deposition. When the droplet size R is sufficiently large compared to the patch size a (small α = a/R), the surface fractions f1 and f 2 defined in eq 2 for a surface with a regular pattern would not change with the precise location of droplet deposition.

Figure 1. Schematic of the computational domain showing a droplet on a checkboard-patterned heterogeneous surface with patch size a. The patches are alternating hydrophilic (magenta) and hydrophobic (green).

Figure 2. Shapes of the contact area of a droplet on a surface when deposited at different locations: (a) centered on the middle of a hydrophilic patch, (b) centered on the crossing of hydrophilic and hydrophobic patches, and (c) centered on the middle of a hydrophobic patch. The droplet has a radius R of 50 lattices and the patch size a is of 10 lattices, or α = a/R = 0.2. 6194


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Figure 3. Schematic representation of the procedure to determine the local contact angle of the distorted droplet on the checkboard-patterned heterogeneous surface. (a) Droplet on checkboard. (b) The contact line is extracted, and circles are fitted to the contact line. (c) Based on the center of these circles, vertical planes normal to the contact line are determined. (d) Finally, the local contact angle is determined on the cross section of the droplet.

Figure 4. Droplet shapes and contact areas of droplets on checkboard-patterned heterogeneous surfaces. The droplet has a radius R of 50 lattices; different patch sizes a result in different ratios α = a/R. The part of a hydrophilic patch covered with gas phase is colored in light blue, covered with liquid phase in red, while the part of a hydrophobic patch covered with gas phase is colored in dark blue and covered with liquid phase in yellow.

Three different locations of droplet deposition are evaluated. The droplet is successively centered first on the middle point of a hydrophilic patch, second at the crossing point of hydrophilic and hydrophobic patches, and third on the middle point of the hydrophobic patch. Figure 2 presents the contact area of the droplet with the surface when deposited at the different locations. We observe clearly that the equilibrium shape of the contact area depends on the location of droplet deposition. The

Following Cassie’s equation, the apparent contact angle would not depend on the location of droplet deposition. Since the aim of this paper is to analyze the droplet behavior beyond Cassie’s assumptions, we consider the droplet radius R and patch size a in similar order of magnitude where the exact location of droplet deposition may play a role in the final droplet shape. Therefore, we study the equilibrium shape of a droplet with patch size to droplet radius ratio α = a/R equal to 10/50 = 0.2. 6195


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Figure 5. Cosine of contact angle θloc plotted versus droplet perimeter angle in the anticlockwise direction for different ratios of patch size to droplet radius, (a) α = 0.1, (b) α = 0.2, (c) α = 0.5, and (d) α = 1, for a droplet radius of 50 lattices. Also represented are the cosines of the Young’s contact angle of the hydrophilic patch of 23.3° (blue dashed line) and of the hydrophobic patch of 120.5° (red dashed line), the Cassie contact angle of 78.2° (black dashed line), and the cosine of the maximum and minimum local contact angles, cmin and cmax, respectively in black long-dashed lines.

ADSA in ImageJ.25 We note that the location along the contact line is given below by the angle in the anticlockwise direction along the droplet perimeter. When the droplet shows a spherical shape, an apparent contact angle is determined as the average contact angle for several cross sections going through the center of the circular droplet contact area. As will be shown later, there exists a critical ratio of patch size versus droplet radius αcrit below which an apparent contact angle exists and above which only local contact angles can be determined. Droplet Shapes and Local Contact Angles. Figure 4 shows the droplet shapes at equilibrium and the contact areas of the droplet with the surface for six patch sizes a (3, 5, 10, 25, 50, and 150 lattices) for the base case of pair of contact angles of 23.3° and 120.5°. The droplet radius R equals 50 lattices, resulting in values of α = a/R ranging from 0.06 to 3. In Figure 4, the part of a hydrophilic patch covered with gas phase is shown in light blue and the part covered with liquid in red. The part of a hydrophobic patch covered with gas phase is shown in dark blue and the part covered with liquid in yellow. At small ratio α = 0.06 (Figure 4a, case a), the contact area shows a circular shape, and an apparent contact angle can be determined (θA = 78°), which is very close to the Cassie’s contact angle predicted by eq 2 (θY,1 = 23.3°, θY,2 = 120.5°, f1 = f 2 = 0.5 results in θC = 78.2°). With increasing ratio α = a/R, the droplet first shows deviations from a spherical shape in vicinity of the contact line (Figure 4b, case b) and loses then its axisymmetric shape and circular contact area (Figure 4c, case c). The droplet distortion increases further with increasing ratio α, and the droplet shows a tendency to wet preferentially the hydrophilic patches (red compared to yellow area, Figure 4d,e,

contact area is most symmetric when the droplet is deposited in the middle of the hydrophilic patch (Figure 2a), while it is most asymmetric for the droplet deposited at the crossing of the hydrophilic and hydrophobic patches (Figure 2b). Figure 2c shows that deposition at the center of a hydrophobic patch also yields a contact area that departs from a circle. The patterns of contact area in Figure 2 are given for one specific ratio of patch size versus droplet radius α. The contact area patterns at different droplet deposit locations will also change depending on the ratio α. However, such a detailed study is beyond the scope of this paper, and for the remainder of this paper, we limit our study to the case where the droplet deposition location is at the crossing of hydrophilic and hydrophobic patches (case b). Contact Angle Determination. Figure 3 shows an example of the final shape of a droplet deposited on a checkboard-patterned heterogeneous surface with a ratio α = 0.2. We observe that the droplet is nonspherical and is locally distorted at the contact line (Figure 3). As a result, it is impossible to define a single apparent contact angle. Local contact angles along the contact line can be determined, but care should be taken as the contact angle determination highly depends on the profile view direction used for the cross section. Therefore, a three-step procedure is followed as illustrated in Figure 3. First, the distorted contact line is determined and locally approximated by a set of circles (Figure 3b). Using the center of each circle, a vertical plane normal to the contact line can be determined (Figure 3c). This plane is further used to determine the cross section of the droplet (Figure 3d). Finally, the local contact angle is determined using the method LB6196


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Langmuir cases d and e). For a ratio α = 3 (Figure 4f, case f), the droplet breaks into two parts and wets only the hydrophilic patches. The contact angle in the middle of the hydrophilic patch is 32°, showing the contact angle tends toward the Young’s contact angle of the hydrophilic patch (θY,1 = 23.3°). We remark that for completeness the images of droplet shapes and contact areas for the two other droplet radii of 100 and 150 lattices for the base case of contact angle pair are presented in the Supporting Information, sections A1 and A2. Figure 5 shows the variation of the cosine of the local contact angle cos(θloc) at regular intervals along the contact line for cases b−e of Figure 4. The yellow dots mark the local contact angles on the hydrophilic patches, while the red dots on the hydrophobic patches. We remark that for completeness supplementary graphs are given for the two other droplet radii of 100 and 150 lattices in the Supporting Information, sections A3 and A4. The cosines of the Young’s contact angles for the hydrophilic (θY,1 = 23.3°) and hydrophobic (θY,2 = 120.5°) surfaces are also represented as well as the cosine of the apparent contact angle predicted by Cassie’s equation (θC = 78.2°). In Figure 5a−c (cases b−d of Figure 4), the cosine of the local contact angle varies between maximum and minimum values, referred to as cmax and cmin, respectively. The maximum cmax is defined as cmax = cos(θloc,min), where θloc,min is the minimal local contact angle reached on the hydrophilic patches. The minimum cmin is defined as cmin = cos(θloc,max), where θloc,max is the maximal local contact angle reached on the hydrophobic patches. The maximum local contact angle θloc,max is higher than the Cassie’s apparent contact angle θC but lower than Young’s contact angle θY,2, or cos(θY,2) < cmin < cos(θC). A similar observation is made for the minimum contact angle, or cos(θC) < cmax < cos(θY,1). In Figure 5d (case e of Figure 4, a/R = 1), the cosine of the local contact angle follows a particular behavior, which is attributed to the specific shape of the droplet and contact line (as seen in Figure 4e, case e). The droplet shape is a result of the confinement of the droplet in between the borders of the hydrophilic patches attaining a quite square contact area on the hydrophilic patches. The local contact angle in the middle of this confined area is found to be very high (73° or a cosine value of 0.3) compared to the Young’s contact angle for hydrophilic patches (θY,1 = 23.3°). This high local contact angle is due to the confinement, and as a result, the cmin and cmax values are not symmetrically spread around Cassie’s contact angle. Figure 6 shows that the cosines of the minimum and maximum local contact angles, cmax and cmin, versus the ratio of the patch size a to the droplet radius R, α = a/R, for all three sizes of droplet radii. The data for all cases studied almost collapse onto a single curve. Some spread is observed and is attributed to the uncertainty in determining the local contact angles and their maximum (minimum) values. It is clear that cmin decreases with increasing α, while cmax increases with increasing α. This means that the range of possible local contact angles increases with α. This behavior can be explained by the fact that for higher α, the contact line covers relatively a longer length on the patches and shows thus a higher degree of freedom to adjust its position. As a result, the droplet deforms and the local contact angle tends toward the Young’s contact angle of the patch. At low α, the cosine of the maximum (minimum) local contact angle converges to the cosine of Cassie’s contact angle θC. The critical ratio αcrit, below which

Figure 6. Cosine of the maximum (minimum) local contact angles, respectively cmax and cmin, versus ratio α = a/R. The cosines of the contact angles of the hydrophilic patch of 23.3° (blue dashed line), of the hydrophobic patch of 120.5° (red dashed line), and of Cassie’s contact angle of 78.2° (black dashed line) are also shown.

the apparent contact angle equals Cassie’s contact angle and above which only local contact angles can be determined, is found to range around 0.06. A further analysis covers the determination of the wetted contact areas of the hydrophobic and hydrophilic surfaces. The ratio of the wetted area on the hydrophilic surface A1,w versus the total wetted surface Atot,w is defined as the wetted area fraction f1,w = A1,w/Atot,w. Analogously, we define the wetted area fraction f 2,w = A2,w/Atot,w for the hydrophobic wetted surface. Remark that f1,w + f 2,w = 1. Figure 7a plots the wetted area fraction f1,w versus the ratio of patch size to droplet radius α = a/R. Again we observe that all results for the different droplet radii almost collapse onto a single curve. The results show that when α increases, the droplet will wet more the hydrophilic patches, resulting in larger f1,w values. This behavior can again be explained by the fact that at high α the droplet and its contact line show higher degree of freedom resulting in an increasing wetted area of the hydrophilic patches. A quasi-linear relation can be observed for wetted area fraction versus α. Figure 7b gives a zoom of the data at small α in a semilog plot. For low α, the wetted area fractions attain values of 0.5, which is the fraction used in Cassie’s equation for a checkboard. Let us now define a criterion for the data to agree with Cassie’s equation, i.e., when the wetted area fraction falls into a 1% range of the Cassie fraction, or we assume Cassie mode to be attained in the range [0.5, 0.505]. We find again that the critical ratio αcrit ranges around 0.06, which is consistent with the findings of Figure 6. Since the critical ratio αcrit may depend on the pair of Young’s contact angles of the patches of the checkerboard, the wetted area fractions f1,w are determined for three different contact angle pairs: [23.3°, 120.5°], [23.3°, 84.0°], and [65.8°, 102.0°] respectively for solid−liquid interactions values w of [−0.1, 0.05], [−0.1, −0.01], and [−0.04, 0.02], as shown in the Supporting Information, sections A5 and A6. Figure 8a shows that the data follow quasi-linear functions for α > αcrit, where the slope depends on the values of the pair of Young’s contact angle values. A detailed study of the dependence of the slope of the curves on the values of the pair of contact angles is beyond the scope of this paper. Figure 8b gives a zoom of the data at low α in a semilog plot. Values of the critical ratio αcrit are found to vary between 0.06 and 0.1, but more simulations are needed to determine the dependence of αcrit on the pair of the 6197


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Figure 7. Fraction of wetted area on hydrophilic patches f1,w versus ratio between patch size a and droplet radius R for three droplet radii R = 50, R = 100, and R = 150: (a) linear plot; (b) zoom-in semilog plot. Black solid lines indicate the 1% range for f1,w around the Cassie fraction of 0.5, or [0.5− 0.505].

Figure 8. Fraction of wetted area on hydrophilic patches f1,w versus ratio between patch size a and droplet radius R for three contact angle pairs (a) linear plot; (b) zoom-in semilog plot. Black solid lines indicate the 1% range for f1,w around the Cassie fraction of 0.5, or [0.5−0.505].

the ratios with respect to the entire surface, or in our case f1 = f 2 = 0.5. We propose for the deformed droplet to use the actual wetted area fractions f1,w and f 2,w of the hydrophilic and hydrophobic surfaces and the cosine of the minimum and maximum local contact angles. We can then define an equivalent contact angle θE as

Young’s contact angles of the patches of the checkerboard more accurately. The use of wetted area fraction could be questioned in discussion on local versus full area dependence of the triplephase contour line mentioned above.5,11,12 We remark that in this work the underside of the droplet is in contact with a checkboard surface, presenting the same general characteristics as the surface in the vicinity of the contact line. In this way, the heterogeneity is uniformly distributed under the droplet, and as such, a region far from the triple contact line cannot be uniquely determined in the scope of the present study. We observed that the form of a droplet deposited on a checkerboard results from two competing phenomena: on one side, the droplet wants to accommodate its bulk shape to the average wettability of the full contact area, and on the other side, the droplet wants to adjust its local contact angle to the wettability of the local patch. This competition depends on the ratio a/R: at low ratio a/R the contact angle adjusts to the full contact area, while at increasing a/R ratio the local wettability of the patch becomes more important. For the range of patch sizes used in this paper, the full contact area is more at play in cases as documented in Figures 4a and 4b and also in parts c and d, while in cases e and f the local wettability determines the local contact angle. Interpretation of Results beyond Cassie Mode. In this part, we study whether a relation exists between the cosine of the maximum (minimum) local contact angles, cmax and cmin, and the cosine of Cassie’s apparent contact angle cos(θC). In the original Cassie approach, surface fractions f i are defined as

cos θE = f1,w cos θ loc,min + f2,w cos θ loc,max

(3)

Figure 9 shows the predicted cosine of the equivalent contact angle and compares it with the cosine of Cassie’s apparent contact angle. A good agreement is observed between the equivalent contact angle obtained from eq 3 and Cassie’s contact angle obtained from eq 2. We conclude that although the validity of Cassie’s equation breaks down, the actual behavior of the droplet follows a similar weighted relationship when the actual fractions of wetted area and local minimum and maximum contact angles are used. The fractions f1,w and f 2,w of the wetted area of hydrophobic and hydrophilic patches are not known beforehand but can be determined from the quasi-linear relationship as observed in Figure 7a,b. Finally, Figure 10 plots the fractions of wetted area on hydrophilic or hydrophobic patches versus the cosine of minimum and maximum local contact angles. This figure is obtained by combining the data from Figures 6 and 7a. The results for different droplet radii coincide into a single curve, which is slightly asymmetric with respect to the hydrophobic and hydrophilic regions. In the hydrophobic region, the cosine of the local contact angle increases gradually attaining Cassie’s contact angle at a fraction of wetted area of 0.5. In the 6198


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4. CONCLUSION In this study, the wetting of a droplet on a checkboardpatterned heterogeneously wetting surface with regular hydrophilic and hydrophobic patches is studied with the 3D pseudopotential lattice Boltzmann model (LBM). Different patch sizes and droplet radii are considered, and local contact angles and fractions of wetted area are analyzed beyond the Cassie mode. The ratio between patch size and droplet radius is found to be a key parameter controlling the degree of droplet deformation and the resulting local contact angle of a droplet deposited on the checkboard-patterned surface. Below a critical ratio, the apparent contact angle follows the Cassie’s equation. The critical ratio was found to be in the range of 0.1 and shown to have similar values for different pairs of Young’s contact angles of the different surfaces. Above the critical contact ratio, the droplet becomes more and more distorted showing a tendency to preferentially wet the more hydrophilic regions. This behavior is attributed to the higher degree of freedom of the contact line for increasing ratio, resulting in larger differences between the minimum and maximum local contact angles. Above the critical ratio, the fraction of wetted area of the hydrophilic surface increases quasi-linearly with the ratio between patch size and droplet radius. At lower ratio, the fraction of wetted area tends to the Cassie fraction. Although the validity of the Cassie’s equation breaks down above the critical ratio, a new equation for an equivalent contact angle could be derived using a weighted relationship of the actual fractions of wetted area on the hydrophilic and hydrophobic surfaces and their minimum and maximum local contact angles. It was shown that this equivalent contact angle above the critical ratio equals the Cassie’s apparent contact angle. This study was limited to a regular checkboard-patterned surface, and only three pairs of Young’s contact angles for the patches were considered. Before being able to further generalize our findings regarding critical ratio value and equivalent contact angle beyond Cassie’s limit, other surface patterns and more contact angle pairs have to be considered.

Figure 9. Cosine of an equivalent contact angle defined in eq 3 versus ratio between patch size and droplet radius α. The cosines of the contact angles of the hydrophilic patch of 23.3° (blue dashed line), of the hydrophobic patch of 120.5° (red dashed line), and of Cassie’s contact angle of 78.2° (black dashed line) are also shown.

Figure 10. Fractions of the wetted area on hydrophilic and hydrophobic patches versus the cosines of the minimum and maximum local contact angles. The contact angle (black dashed line) and fraction of wetted area of 0.5 (black dashed dotted line) according to Cassie’s equation are also shown as well as the cosines of the contact angles of the hydrophilic patch of 23.3° (blue dashed line) and of the hydrophobic patch of 120.5° (red dashed line). The dotted red line is added to guide the eye.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.7b01471. Droplet shapes and wetted areas of droplets for droplet radii of 100 and 150 lattices; cosine of the local contact angle θloc for droplet radii of 100 and 150 lattices; droplet shapes and wetted areas of droplets for droplet radii of 50 and 100 lattices, for 23.3°/84.0° and 65.8°/102.0° contact angle pairs (PDF)

hydrophilic region, the cosine of the local contact angle is almost constant at high cosines of contact angle and decreases rapidly toward the fraction of wetted area of 0.5 at Cassie’s contact angle. This asymmetric curve may be explained by the fact that wetting of hydrophilic patches is favorable from an energetic point, and as a result, the local contact angle on the hydrophilic patches remains quite constant for different wetted area fractions. The wetting of hydrophobic patches is much more difficult since it is only driven by the amount of area wetted on the hydrophilic patches next to them and therefore increases more gradually. In all of the above, it is clearly shown that the ratio between patch size and droplet radius, α = a/R, is a key parameter that controls the amount of droplet deformation and the resulting local contact angle of a droplet deposited on checkboardpatterned heterogeneously wetting surface. It is also shown that the Cassie’s equation is a limit only attained when the ratio a/R is lower than a critical value αcrit.

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AUTHOR INFORMATION

Corresponding Author

*(D.D.) E-mail [email protected]. Funding

This work has been supported by Swiss National Science Foundation project no. 200021-143651. Notes

The authors declare no competing financial interest. 6199


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Methods and applications. Int. J. Heat Mass Transfer 2014, 76, 210− 236. (22) Kupershtokh, A.; Medvedev, D.; Karpov, D. On equations of state in a lattice Boltzmann method. Computers & Mathematics with Applications 2009, 58 (5), 965−974. (23) Son, S.; Chen, L.; Derome, D.; Carmeliet, J. Numerical study of gravity-driven droplet displacement on a surface using the pseudopotential multiphase lattice Boltzmann model with high density ratio. Comput. Fluids 2015, 117, 42−53. (24) Son, S.; Chen, L.; Kang, Q.; Derome, D.; Carmeliet, J. Contact Angle Effects on Pore and Corner Arc Menisci in Polygonal Capillary Tubes Studied with the Pseudopotential Multiphase Lattice Boltzmann Model. Computation 2016, 4 (1), 12. (25) Stalder, A. F.; Melchior, T.; Müller, M.; Sage, D.; Blu, T.; Unser, M. Low-bond axisymmetric drop shape analysis for surface tension and contact angle measurements of sessile drops. Colloids Surf., A 2010, 364 (1), 72−81. (26) Son, S. Lattice Boltzmann modeling of two-phase flow in macroporous media with application to porous asphalt. ETHZ PhD Thesis, 2016.

ACKNOWLEDGMENTS We acknowledge the important contribution of the PhD work of Soyoun Son26 to this paper. Her PhD was made possible through Swiss National Science Foundation project no. 200021-143651. L. Chen and Q. Kang acknowledge the support from LANL’s LDRD Program. We acknowledge the fruitful discussion with K. Kulasinski on local angle determination and thank B. Coasne and R. Guyer for helpful discussions. Finally, we thank Dr. Ali Mazloomi Moqaddam for providing complementary simulations in the review process.

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Beyond-Cassie Mode of Wetting and Local Contact Angles of Droplets

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