Contact Angle Hysteresis on Randomly Rough Surfaces - American

18 Mar 2013 - A free energy-based computational simulation is used to study the effect of roughness ... The wetting of solid surfaces by liquids is ub...
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Contact Angle Hysteresis on Randomly Rough Surfaces: A Computational Study Robert David and A. Wilhelm Neumann* Department of Mechanical & Industrial Engineering, Room RM502A, University of Toronto, 5 King’s College Rd., Toronto, ON, Canada M5S 3G8 S Supporting Information *

ABSTRACT: Wetting is important in many applications, and the solid surfaces being wet invariably feature some amount of surface roughness. A free energy-based computational simulation is used to study the effect of roughness on wetting and especially contact angle hysteresis. On randomly rough, self-affine surfaces, it is found that hysteresis depends primarily on the value of the Wenzel roughness parameter r, increasing in proportion with r − 1. Micrometer-level roughness causes hysteresis of a few degrees. analyzed by Wenzel,3 who showed that the equilibrium contact angle θW on a rough surface differs from the Young contact angle on a flat surface made of the same material, as follows:

1. INTRODUCTION The wetting of solid surfaces by liquids is ubiquitous. Wetting is an important process in many industries and research areas, such as coatings, spray cooling, oil recovery, adhesives, and microfluidics. The extent to which a liquid wets a solid is quantified by the contact angle θ, a material property of the liquid−solid combination (as well as, in some cases, the surrounding vapor). The classical Young’s equation1 gives a unique value for the equilibrium contact angle θY in terms of the interfacial energies (or tensions) between the three adjoining phases: γsv = γsl + γlv cos θY

cos θ W = r cos θY

where r ≥ 1 is the ratio of true to projected surface area of the rough solid. Equation 2 means that roughness has no effect on the equilibrium contact angle if θY = 90°, while otherwise it pushes the equilibrium angle away from 90°. The second effect of surface roughness on wetting is not captured by Wenzel’s equation. The varying local slope of a rough surface changes the local equilibrium contact angle, as measured relative to the average plane of the surface. This local variation creates energy barriers that impede the spreading or retraction of a liquid front (not unlike speed bumps). Because of the energy barriers, the macroscopic contact angle depends on the history of the contact line, with different contact angles for advancing and receding liquid fronts.4 The energy barriers do not change the global minimum in free energy, which is still the Wenzel angle θW; rather, they make it more difficult for the system to reach this equilibrium. In fact, it is the advancing and receding angles that are generally observable and that have the greatest impact in practice. For example, the difference between the advancing and receding

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where γsv is the solid−vapor interfacial energy, γsl is the solid− liquid interfacial energy, and γlv is the liquid−vapor interfacial energy. While Young’s equation is always valid locally,2 its use for an entire drop requires an ideal solid surface that is perfectly flat, rigid, homogeneous, and nonreactive. However, practical surfaces are not ideal and usually to an extent that produces significant departures from the simple wetting behavior predicted by Young’s equation. In this contribution, the influence of surface roughness on wetting is investigated. It is known that roughness can affect contact angles in two ways. First, roughness increases the surface area of the solid. As a liquid spreads on a rough surface, the amount of solid−vapor interface that is replaced by solid−liquid interface is larger than it would be on a flat surface. This effect of roughness was © 2013 American Chemical Society

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Received: January 23, 2013 Revised: March 18, 2013 Published: March 18, 2013 4551

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angles, called the contact angle hysteresis, determines the force needed to actuate a stationary liquid drop on a solid surface.5 This force is important in applications such as electrowetting.6 Shuttleworth and Bailey4 derived the advancing and receding contact angles for a model two-dimensional drop/surface system in which the surface roughness was symmetrical about the drop. The advancing angle is θY + α, and the receding angle is θY − α, where α is the maximum local angle of inclination of the surface. Also in two dimensions, for surfaces with random roughness, similar results have been found more recently using statistical methods.7 Predicting the advancing and receding angles in three dimensions is more complicated because the contact line can now contort around surface features. Cox8 used analytical methods to calculate the advancing and receding angles on several classes of rough surfaces, including surfaces with doubly periodic roughness. He represented the surface topography by a two-dimensional Fourier series and found the contact angle hysteresis to first order in the height of the roughness. The influence of more general and realistic patterns of roughness on the wetting properties of a solid surface is still unknown. Considering the inevitability of roughness on any solid surface, and the wide range of applications for wetting, the relevance of further research is clear. In this paper, we introduce a computational method to analyze wetting on rough surfaces in three dimensions. The simulated systems are in the Wilhelmy plate geometry, i.e., a vertical plate dipped into a pool of liquid. Our method can handle weak roughness in a wider variety of patterns than are accessible by analysis, including random patterns. It is based on an earlier-described wetting simulation for flat, chemically heterogeneous surfaces.9 In the next section, the theoretical background for the method is presented. In section 3, details of the computation are given. In section 4, we show computed results for surfaces with periodic roughness and verify them against analytical results. In section 5, results are presented for random, self-affine surfaces, and the main features of these results are analyzed in section 6. In section 7, our results for random surfaces are compared with experimental data from the literature, and conclusions are drawn in section 8.

Figure 1. Schematic of the rough plate with the liquid meniscus rising to meet it (side view). The x-axis extends into the page. Gray represents liquid.

where ρ is the liquid density, g is the gravitational acceleration, and V is the volume of liquid. For the purpose of evaluating the integral in eq 3, the volume of liquid can be broken into two parts. The first part is the volume if the surface were flat (i.e., if S(x,y) = 0). The integral over this part can be evaluated analytically from the Laplace equation of capillarity:11

∫V ρgy dV

2] (4)

where L is the width of the surface in the x-direction. The second part of the liquid volume is the liquid that fills crevices or is displaced by bumps on the rough surface. This extra volume, which can be positive or negative, can be included as follows: L

E1b =

η(x)

S(x , η(x))

∫0 ∫0 ∫S(x ,y)

ρgy dz dy dx

(5)

A second contribution to the system’s free energy arises from the replacement of solid−vapor interface with solid−liquid interface as liquid wets the plate: L

E2 =

∫0 ∫0 L

=−

η(x)

∫0 ∫0

γsl − γsv

cos α1 cos α2 η(x) γ cos θ Y lv

dy dx

cos α1 cos α2

dy dx

(6)

where α1 and α2 are the local angles of inclination of the surface about the x- and y-axes, relative to the plane z = 0 (Figure 1). The cosines in eq 6 account for the extra surface area of the rough plate relative to a flat plate (i.e., they account for the effect described by Wenzel). The second equality follows from Young’s equation, applied locally. The final contribution to the free energy of the liquid meniscus is the energy of the liquid−vapor interface. The liquid surface energy can be split into three parts: one relating to a flat plate and the other two accounting for the actual plate’s roughness. On a flat plate, the contact line is horizontal and straight, and the liquid surface energy is11

2. FREE ENERGY We consider one side of a rough, solid plate that is dipped vertically into a pool of liquid. Starting from some initial position, the three-phase contact line will move until it rests in a local minimum of the thermodynamic free energy. We consider surfaces with gentle roughness (r < 1.01) on which wetting occurs in the Wenzel state, i.e., with no formation of air pockets between the liquid and solid. Let x be the horizontal coordinate along the plate, y the vertical coordinate, and z the horizontal coordinate normal to the plate (Figure 1). The rough surface of the plate is described by the function S(x,y) and the rise height of the meniscus by η(x). On average, because of gravity, the contact line follows the x-direction. The free energy of this system can be expressed as the sum of several terms.10 The gravitational potential energy is

E1 =

2γlv 1 Lγlv [(2 − sin θY ) 1 + sin θY − 3 ρg

E1a =

E3a = Lγlv

2γlv ρg

( 2 −

1 + sin θY )

(7)

On a rough plate, the contact line may be displaced, on average, in the z-direction. This requires a correction10 E3b = −Lγlv⟨S(x , η(x))⟩

(8)

where ⟨ ⟩ denotes an average along the contact line.

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By integrating eq 9 to find the liquid surface area, it can be shown12,13 that the additional free energy due to surface undulations is in general

Furthermore, on a rough plate the contact line is not straight but meanders in both the y- and z-directions. The meandering of the contact line creates undulations in the liquid surface that extend a short distance away from the plate.12 These undulations increase the liquid surface area beyond the area accounted for in eqs 7 and 8. The free energy due to this additional surface area must therefore be included as an additional term. For convenience, in the following analysis we introduce new coordinate axes (Figure 2), although the final result, which is an

E3c =

(12)

3. COMPUTATION The computation was performed in MATLAB. The plate surface was discretized as a grid of points (xi, yi), with grid spacings Δx = xi+1 − xi and Δy = yi+1 − yi (note that we now revert to the coordinate system of Figure 1). The surface of the plate was represented in the program as a matrix, with each matrix element giving S(xi, yi). Between the specified grid points, the surface was modeled as a triangular mesh. For example, one triangular facet in the mesh connected the surface points at (xi, yi, S(xi, yi)), (xi, yi+1, S(xi, yi+1)), and (xi+1, yi+1, S(xi+1, yi+1)) and another the surface points at (xi, yi, S(xi, yi)), (xi+1, yi, S(xi+1, yi)), and (xi+1, yi+1, S(xi+1, yi+1)) (Figure 3). Note that Figure 3 is drawn with Δx = Δy for clarity, while in the program Δy was smaller than Δx in order to allow the contact line to be gently sloped, as required for eq 11.

even function of x only, will also be valid in the original axes (Figure 1). The liquid surface must everywhere obey the Laplace equation of capillarity. The liquid surface near the plate can then be modeled as12,13 1 2π

∞

∫−∞ |q|αq2 dq

For our case of a rough surface, αq is given by eq 11. Thus, for a given contact line, the free energy contribution E3c can be calculated from eqs 11 and 12 using the known functions ξ1(x) and ξ2(x) that describe the contour of the contact line (Figure 2). Note that only ξ1(x) need be given, since ξ2(x) is then enforced by the condition of contact with the plate. The total free energy for a given contact line on the rough plate is then the sum of E1a (eq 4), E1b (eq 5), E2 (eq 6), E3a (eq 7), E3b (eq 8), and E3c (eq 12).

Figure 2. Close-up schematic of the region near the contact line. The x-axis extends out of the page, and the origin is where the asymptotic liquid−vapor interface would intersect the average plane of the plate. Gray represents liquid.

h(x , y) ≈

1 γ 4π lv

∞

∫−∞ αqeiqxe−|q|y dq

(9)

where h is measured in the z-direction, q is spatial frequency, and the αq are Fourier coefficients (Figure 2). The liquid surface in eq 9 must intersect the rough plate along the contact line: h[x , −ξ1(x) cos θY + ξ2(x) sin θY ] ≈ ξ1(x) sin θY + ξ2(x) cos θY

(10) Figure 3. Schematic of a small part of a modeled plate viewed along the z-axis, showing the triangular mesh. The shading is intended to convey that the center point is raised out of the page. For clarity, not all points are labeled.

if the asymptotic contact angle is not far from θY (i.e., if hysteresis is small). In eq 10, ξ1(x) is the deviation of the contact line (from its average position) in the plane of the plate, and ξ2(x) is its deviation normal to the plane of the plate (Figure 2). On a flat but chemically heterogeneous plate, ξ2(x) would be zero and eq 10 would reduce to the theory of Robbins and Joanny.13 On a flat and chemically homogeneous plate, both ξ1(x) and ξ2(x) would be zero, and the contact line would be horizontal. Substituting eq 10 into eq 9 leads to αq ≈ ξ1̃ sin θY + ξ2̃ cos θY

The contact line was constrained to lie on points belonging to the mesh of the solid surface. In the program, the contact line was represented as a vector, with each entry giving its ycoordinate at that x location (Figure 1). For any given contact line, a function within the program calculated the free energy according to discretized forms of the equations given in the previous section. To save computing time, certain quantities that depended only on the surface itself were precalculated when the surface was generated: the two inner integrals in the term E1b (eq 5) for a point on a contact line at any (x, y) location on the surface and the contribution of each triangular facet of the solid surface to the free energy should it become covered with liquid (cf. eq 6)in other words, the area of each triangular facet times −γlv cos θY. The contact line was initialized as a line of constant y near the bottom of the plate. Arbitrary contiguous sections of the

(11)

where ξ1̃ and ξ2̃ are the Fourier transforms of ξ1 and ξ2, respectively. Equation 11 is valid if |ξ1q| ≪ 1 and |ξ2q| ≪ 1, i.e., if all the deviations of the contact line away from its average position involve gentle slopes. This will be the case as long as the surface is not too rough, that is, as long as the heights of any features measured normal to the surface are much smaller than their lateral dimensions, a condition that is often met in practice. 4553

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contact line were then allowed to move by ±Δy on a trial basis, and the movement producing the greatest decrease in the calculated free energy was chosen. This process was iterated until the contact line reached a position from which no decrease in the free energy was possible, signifying equilibrium. Runs in which the contact line attempted to move below the bottom of the plate were discarded. To simulate an advancing contact angle, the plate was shifted downward by Δy after each contact line equilibration, and a new equilibrium was found. To simulate a receding contact angle, the plate was shifted upward by Δy after each contact line equilibration, and a new equilibrium was found. Further details of the modeled contact line motion are available elsewhere.9 The output of the program was the equilibrium position of the contact line on the rough plate, i.e., the equilibrium rise height of the liquid meniscus, as a function of the plate’s displacement. The apparent contact angle was determined from the rise height η as follows: ⎛ ρg ⟨η⟩2 ⎞ ⎟⎟ θ = sin−1⎜⎜1 − 2γlv ⎠ ⎝

Figure 4. Output for a horizontally grooved surface.

A similar surface, on which the horizontal ridges had sinusoidal rather than V-shaped cross sections, was also tested. The equation of the surface was S(x,y) = A sin (2πy/a). The analytical solutions for the advancing and receding contact angles on this surface are θa = θY + 2πA/a and θr = θY − 2πA/a, respectively.8 For a surface with θY = 70°, A = 2 μm, and a = 200 μm, the computed behavior of the contact angle was qualitatively similar to that for the sawtooth surface of Figure 4, with advancing and receding contact angles of θa = 73.6° and θr = 66.4°, respectively, matching the theoretical values. Next, a sinusoidal surface with the grooves oriented vertically was examined. The equation of the surface was S(x,y) = A sin(2πx/a). Since this surface did not vary in the y-direction, the forces experienced by the contact line did not change as it advanced or receded over the surface, and therefore no contact angle hysteresis was expected (θa = θr). On the other hand, the variation of the surface in the x-direction along the contact line meant that different parts of the contact line experienced different forces, so that, unlike the cases considered so far, the equilibrium contact line was not straight. The computed equilibrium shape of the contact line, for a surface with θY = 30° and A/a = 0.0278, is shown in Figure 5. Liquid climbed higher in the troughs of the surface. The computed shape closely followed the superimposed, analytical solution by Cox.8 The computed contact angle hysteresis was zero, with both the advancing and receding contact angles equal to the equilibrium (Wenzel) contact angle of θW = 29.24°. Finally, a surface that was periodic in both the x- and ydirections was studied. The equation of the surface was S(x,y) = A sin(2πx/a) sin(2πy/a). This surface combined the features of the previous two: contact angle hysteresis was nonzero, and the contact line was wavy. The analytical values for the advancing and receding contact angles on this doubly periodic surface are θa = θY + √2πA/a and θr = θY − √2πA/a, respectively.8 For a surface with θY = 30° and A/a = 0.025, the computed advancing and receding contact angles were θa = 36.0° and θr = 23.5°, respectively, close to the theoretical values of θa = 36.4° and θr = 23.6°. The computed amplitude of the wavy contact line was ∼14% less than the amplitude obtained by Cox.8 The small discrepancies may have been due to the approximations in the theory presented above or the neglect of terms higher than the first order in the analytical solution.8

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Equation 13 describes the contact angle that would be measured in a capillary rise experiment. Unless otherwise noted, the following parameter values were used in the computations: ρ = 1000 kg/m3, γlv = 72.5 mJ/m2, Δx = 10 μm, and Δy = 1 μm. Surfaces were normally N = 101 points across in the x-direction, for a total width of L = (N − 1)Δx = 1 mm. Other input parameters such as the intrinsic contact angle of the surface θY will be specified below. The runtime for most of the simulations (corresponding to individual points in Figures 8−11) was ∼1 day on a 2.5 GHz desktop computer.

4. PERIODIC ROUGHNESS Analytical results are available for equilibrium contact angles and contact angle hysteresis on a number of surfaces with periodic roughness. Some of these results were used to check the accuracy of the simulation. For flat surfaces with various intrinsic contact angles θY, the liquid meniscus reached equilibrium at the correct contact angle of θY, with zero hysteresis. A surface with a sawtooth pattern of roughness was simulated. The grooves of the sawtooth were oriented horizontally, parallel to the contact line. On such a surface, the contact line should remain straight, with advancing contact angle of θY + α and receding contact angle of θY − α, where α is the angle of inclination of the grooves.4,10 The computed results are shown in Figure 4, for a surface with θY = 40° and α = 5°. The abscissa shows the simulated displacement of the plate relative to the pool of liquid; it was pushed 800 μm into the liquid and then withdrawn the same distance, in steps of 10 μm. Each point represents the equilibrated contact angle after a step. The advancing and receding contact angles agreed (within the discretization error of ∼0.02°) with their predicted values of 45° and 35°, respectively. During the first, advancing stage, the sudden changes in contact angle occurred when the contact line jumped over the downward-facing sides of the ridges, attempting to reach the local apparent contact angle of 35°. During the second, receding stage, the jumps occurred in the opposite direction over the upward-facing sides of the ridges, on which the local apparent contact angle was 45°.

5. RANDOM ROUGHNESS On a self-affine rough surface, the height of the roughness is proportional to a power of its spatial wavelength. Random, self4554

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Figure 5. Equilibrium contact line on a sinusoidally patterned surface: gray, liquid; heavy line, analytical solution.8

Figure 6. A self-affine surface with H = 0.8, λroll‑off = 200 μm, and Rq = 1 μm.

determined as the ending section of values over which a best-fit line had a slope smaller than a suitable, fixed value. To estimate the reproducibility of the results, four runs were performed with the same input parameters (other than the random number seed). The results are listed in Table 1. The reproducibility was of order ±0.1°.

affine surfaces were generated using the recipe of Persson et al.14 Surfaces had a power spectrum ⎧ 2(H + 1) ⎪λ λ < λroll‐off C(λ) ∼ ⎨ ⎪ ⎩C(λroll‐off ) λ ≥ λroll‐off

(14)

where λ is wavelength and H is known as the Hurst exponent. If H = 1, the surface is self-similarit looks the same under any magnification. The leveling of C(λ) at the roll-off wavelength λroll‑off models the fact that many real surfaces become relatively smoother at long wavelengths. The magnitude of roughness was set by adjusting its root-mean-square Rq. A typical surface with random, self-affine roughness is shown in Figure 6. Figure 7 shows the output for a self-affine surface with θY = 40°, H = 0.6, λroll‑off = 200 μm, and Rq = 1 μm. The computed

Table 1. Results for Four Surfaces with θY = 40°, H = 0.8, λroll‑off = 200 μm, Rq = 1 μm, and N = 101 run

θa (deg)

θr (deg)

hysteresis (deg)

1 2 3 4

40.48 40.66 40.57 40.59

39.03 39.17 39.34 39.07

1.45 1.48 1.23 1.52

Figure 8 shows the departures of computed advancing and receding contact angles from the global equilibrium angle θW. In each run, one of the surface parameters H, λroll‑off, or Rq was varied from the “default” parameters of Table 1, always with θY = 40° and N = 101. As hysteresis increased, the advancing and receding angles separated in an approximately symmetric manner from θW. The situation of θa − θW ≈ θW − θr would result from a free energy that is approximately symmetric about the global minimum θW. For rougher surfaces (the right-hand end of Figure 8), however, the receding angles began to depart slightly more than the advancing angles from θW. (A similar pattern was seen if the cosines were plotted instead of the angles, due to the small differences involved.) The asymmetry at high hysteresis could be attributed to the memory effectan advancing contact line can only access the roughness immediately ahead of it, whereas sections of a receding contact line can cling to roughness for some distance after the main part of the contact line has moved on. Thus, roughness (or high-energy chemical defects9) affects the receding angle more strongly than the advancing angle, drawing it farther from the global equilibrium θW. If θY were greater than 90° (or for low-energy chemical defects), roughness would decrease wettability instead of increasing it

Figure 7. Output for a random, self-affine surface.

advancing and receding contact angles are indicated by solid lines. The advancing angle θa was calculated as the average contact angle during the period in which the plate was virtually lowered into the liquid (from 0 to 0.6 mm in Figure 7). The receding angle θr was calculated as the average contact angle during the period in which the plate was virtually raised out of the liquid, and the contact angle was stable. This period was 4555

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affected the spatial frequency distribution of the roughness (cf. eq 14). The surfaces for which r was increased via manipulation of H or λroll‑off contained relatively more high-frequency roughness. Thus, in addition to the primary dependence of hysteresis on the value of r, there appeared to be a secondary dependence on the amount of high-frequency roughness. Figure 9 also shows the results from two additional checks on the computation. As described above (cf. Figure 3), surfaces were normally discretized into points spaced horizontally by Δx = 10 μm and vertically by Δy = 1 μm. When the point spacings were altered (with other parameters at their default, Table 1 values), the contact angle hysteresis remained close to the main trend line. Figure 10 shows the calculated values of hysteresis when the surface width N was varied, with all other parameters fixed.

Figure 8. Differences between the advancing and receding angles and the Wenzel angles on random surfaces. The straight line denotes equality. Stars from Table 1; numerical results in Table S1.

(cf. eq 2), and the situation should be reversed, with the advancing angle more affected due to the memory of the advancing contact line for nonwetting surface features. Figure 9 shows the computed contact angle hystereses θa − θr for the runs from Figure 8, plotted against the values of the Figure 10. Contact angle hysteresis as a function of surface width, with other parameters at values from Table 1. Stars from Table 1.

Despite the fact that r was essentially constant (1.002 19 < r < 1.002 27) between all runs, a weak dependence of hysteresis on surface width was clear. (The scatter suggested that the results from the third line of Table 1 were a statistical outlier.) The dependence of hysteresis on N was consistent with the secondary trend of hysteresis with high-frequency roughness described above. For wider surfaces (higher N), lower frequencies of roughness were admitted, reducing the relative amount of high-frequency roughness on the surface (at fixed Rq), therefore reducing hysteresis. Finally, a series of runs were performed with different intrinsic contact angles θY and all the other parameters fixed at their Table 1 values. The results are plotted in Figure 11. The contact angle hysteresis was relatively insensitive to θY, consistent with the concentric groove model of Shuttleworth and Bailey,4 in which hysteresis is twice the maximum surface inclination, independent of θY (see above). Figure 11 still shows some dependence of hysteresis on θY. This dependence can be explained from a different point of view that, unlike the concentric groove model, takes into account roughness parallel to the contact line. A rough surface can be regarded as a heterogeneous surface on which the local equilibrium contact angle is given by inserting the local value of r into Wenzel’s equation. For a surface with a given pattern of r(x,y), the variation of the local effective contact angle is then greater the farther is θY from 90° (cf. eq 2). Therefore, for the same statistical roughness, as θY departs from 90° the contact angle hysteresis should increase, as suggested in Figure 11.

Figure 9. Contact angle hysteresis on random surfaces as a function of roughness. Stars from Table 1.

Wenzel roughness parameter for each surface. Hysteresis was well-fit by a straight line passing through zero at r = 1 (a flat surface). The point with “negative” hysteresis was an artifact of the way the receding angle was calculated (see above); the actual hysteresis for this surface was virtually zero. The principal trend in Figure 9 is that, for the most part, the surface parameters H, λroll‑off, and Rq affected the contact angle hysteresis only insofar as they affected the Wenzel roughness parameter r. It is also noticeable that hysteresis was generally lower for runs in which Rq was varied than for runs in which H or λroll‑off were varied. Unlike Rq, the latter two parameters 4556

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r−1∝

∫ q2S 2̃ dq

(18)

Comparing eqs 16 and 18, it is seen that, like the defect energy in eq 15, the corrugation energy scales approximately with r − 1, but also increases faster when the surface roughness is concentrated at higher spatial frequencies (larger q). Contact angle hysteresis is proportional to the total energy dissipated by the contact line (Edefect + Ecorrugation) in a complete cycle of advancing and receding motion.12 Thus, the approximated scaling behavior is consistent with the primary and secondary trends for the hysteresis observed in Figure 9.

7. DISCUSSION Experimental measurements of contact angle hysteresis on random surfaces of known r are relatively scarce. Oliver, Huh, and Mason15 measured advancing and receding contact angles for ethylene glycol and silicone oil on smooth and bead-blasted Teflon substrates. The value of r on the roughened surfaces was about 1.001. For silicone oil, the effect of roughness on hysteresis was within the measurement resolution of ∼2°, in agreement with our computations (Figure 9). For ethylene glycol, roughness had a similarly small effect on advancing contact angles, but it reduced receding angles by ∼25°. This may have been due to surface contamination, as surfaces were not rigorously cleaned.15 Kamusewitz and Possart16 measured advancing and receding contact angles on polypropylene surfaces manually roughened with sandpaper. The value of r for each surface was measured by scanning force microscopy. The symmetric behavior of θa and θr about θW seen in Figure 8 was consistent with their observations. For r < 1.3, the hysteresis measured by Kamusewitz and Possart appeared to be linear in r, as found here (Figure 9). However, for higher r (well outside the range considered here), hysteresis increased much more quickly. Meiron et al.17 made similar measurements on beeswaxcoated abrasive papers. They found somewhat asymmetric behavior of θa and θr about θW and an increase of hysteresis with r that was slower than linear. The values of r examined were again much higher than those considered here, and the roughness on their surfaces appeared to be dominated by features of a certain size, unlike our modeled self-affine surfaces, which would look similar (up to a vertical scaling factor) under any magnification. The order of magnitude of the relationship between r and hysteresis, with an increase in r of 0.01 responsible for a few degrees of hysteresis, was consistent between our results and the experimental measurements.15−17 In terms of rms roughness, our computations showed a few degrees of hysteresis arising from micrometer-level roughness, suggesting that nanometer-level roughness would generally produce insignificant hysteresis. This implication is consistent with experiments on very flat polymeric surfaces with nanometer-level roughness.18,19 For such surfaces, it has been concluded that observed hysteresis arises almost entirely from liquid sorption and not from surface roughness.19 All in all, the situation on experimental surfaces appears to be complicated. Larger values of r may introduce higher order effects, and the frequency spectrum of the roughness may be strongly influential for surfaces that are not self-affine. It is nevertheless hoped that the computational results presented here can serve as a first step toward the understanding of wetting on practical rough surfaces.

Figure 11. Contact angle hysteresis as a function of Young contact angle. Stars from Table 1.

6. ANALYSIS In this section, we develop some simple scaling arguments for the results on randomly rough surfaces. As just discussed, roughness can act like a chemical defect in changing the local equilibrium contact angle according to Wenzel’s equation (eq 2). For isolated defects, the scaling of the pinning energy12 would then be Edefect ∝ cos θW − cos θY = (r − 1) cos θY

(15)

On our random surfaces, the roughness does not consist of isolated bumps but is continuous. In this case, the “defects” would actually correspond to regions in which the local value of θW (calculated from a local r) differs greatly from the average. The scaling of eq 15 is still plausible, since the variation of r over a random surface would likely increase together with its global, average value. The defect mode of pinning is not operative when θY = 90° (eq 15). Since hysteresis is still present when θY = 90° (Figure 11), a second mode of pinning must be active as well. The second way for roughness to pin the contact line is by increasing its length. A corrugated contact line produces additional liquid−vapor surface area, increasing the free energy of the system. The rise height η of a liquid meniscus on a vertical surface is given by eq 13. It can be shown via Taylor expansion that if the surface is tilted from the vertical by an angle α ≪ 1, the change in rise height is proportional to α. Thus, it is reasonable to expect that the corrugation Δη of a contact line on a surface with weak, random roughness scales with an average value of surface inclination |∂S/∂y|. The free energy due to corrugation then scales as12 Ecorrugation ∝

∫ qΔη 2̃ dq ∝ ∫ q3S 2̃ dq

(16)

where a tilde denotes a Fourier transform. On the other hand, the Wenzel roughness parameter for an isotropic surface can be written as r=

≈

1 A 1 A

∫A

1+

2 ⎛ ∂S ⎞2 ⎛ ∂S ⎞ ⎜ ⎟ + ⎜ ⎟ dA ⎝ ∂x ⎠ ⎝ ∂y ⎠

⎡ ⎛ ⎞2 ⎤ ⎢1 + ⎜ ∂S ⎟ ⎥ dA A⎢ ⎝ ∂y ⎠ ⎥⎦ ⎣

∫

(17)

Using Parseval’s relation 4557

dx.doi.org/10.1021/la400294t | Langmuir 2013, 29, 4551−4558

Langmuir

Article

(15) Oliver, J. F.; Huh, C.; Mason, S. G. An experimental study of some effects of solid surface roughness on wetting. Colloids Surf. 1980, 1, 79−104. (16) Kamusewitz, H.; Possart, W. Wetting and scanning force microscopy on rough polymer surfaces: Wenzel’s roughness factor and the thermodynamic contact angle. Appl. Phys. A: Mater. Sci. Process. 2003, 76, 899−902. (17) Meiron, T. S.; Marmur, A.; Saguy, I. S. Contact angle measurement on rough surfaces. J. Colloid Interface Sci. 2004, 274, 637−644. (18) Extrand, C. W.; Kumagai, Y. An experimental study of contact angle hysteresis. J. Colloid Interface Sci. 1997, 191, 378−383. (19) Lam, C. N. C.; Wu, R.; Li, D.; Hair, M. L.; Neumann, A. W. Study of the advancing and receding contact angles: liquid sorption as a cause of contact angle hysteresis. Adv. Colloid Interface Sci. 2002, 96, 169−191.

8. CONCLUSIONS Our computations allowed three general trends to be identified in the wetting of self-affine surfaces with gentle, random roughness: 1. Contact angle hysteresis is primarily determined by the value of the Wenzel parameter r and is proportional to r − 1. 2. Hysteresis increases weakly with the high-frequency content of the roughness. 3. Hysteresis increases weakly as the intrinsic contact angle departs from 90°. All of these trends can be qualitatively understood in terms of the defect and corrugation modes of pinning.

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ASSOCIATED CONTENT

S Supporting Information *

Table S1, containing the parameter values and numerical results from Figure 8. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was funded by the Natural Sciences and Engineering Research Council of Canada, Grant 8278.

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REFERENCES

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dx.doi.org/10.1021/la400294t | Langmuir 2013, 29, 4551−4558