Droplet Sliding on an Inclined Substrate with a Topographical Defect

Jun 26, 2017 - In comparison, the case with a residual droplet on the bump shows a sudden drop in Δx (Figure 12a) due to the coalescence, followed by...
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Droplet Sliding on an Inclined Substrate with a Topographical Defect Joonsik Park and Satish Kumar* Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, United States ABSTRACT: Pinning and depinning of droplets on heterogeneous substrates are widely seen in nature and need to be carefully controlled in industrial processes such as substrate cleaning and spray coating. In this work, a two-dimensional droplet sliding on an inclined substrate with a topographical defect is studied with a thinfilm evolution equation. Using results from time-dependent finite-difference calculations, we focus our discussion on the dynamic interactions between the sliding droplet and the topographical defect. For a Gaussian defect shape, we find that droplet pinning is primarily determined by the advancing contact line pinning at the defect surface where the topography slope is minimum. We demonstrate that with certain combinations of defect heights and widths, residual droplets can form on the defect as a result of geometric constraints involving the receding droplet meniscus and the defect shape. We show that the delay in sliding caused by the defect is mainly due to the pinning and depinning of the receding contact line, and less affected by the dynamic behavior of the advancing contact line. This topography-induced delay in sliding of an individual droplet may have important implications for controlling the collective sliding behavior of multiple droplets.

1. INTRODUCTION Careful observations of droplet motion in nature (e.g., superhydrophobicity of lotus leaves,1 dew collection of Namib Desert grass,2 and beetles3,4) often bring inspiration to new technologies. In industrial applications, the wetting dynamics between a droplet and a surface need to be controlled to keep surfaces clean from stains (e.g., building windows, solar panels), enhance heat-exchange rates (e.g., air conditioners), and remove water efficiently (e.g., dehumidifiers, diesel engine filters).5−8 In the many studies devoted to gravity-driven droplet motion, substrates are often idealized as smooth homogeneous surfaces such that droplets spread on flat substrates without pinning or slide at any nonzero inclination angle of the substrates.9−12 Most real substrates, however, exhibit chemical and topographical heterogeneities that induce stick−slip motion from contact-line pinning.13−16 Understanding how the transient pinning process influences droplet dynamics is crucial to designing realistic substrates for industrial applications. In previous work, we have studied the influence of pinning at a topographical defect on droplet spreading on horizontal substrates.17 The aim of the present work is to study the influence of such pinning on the gravity-driven motion of droplets on inclined substrates. Experimental observations of a droplet sliding on a inclined substrate are characterized by droplet retention until a critical inclination angle is reached and a constant sliding speed after the droplet reaches a steady shape.18−20 As the inclination angle increases beyond the critical value, not only does the constant sliding speed increase, but the droplet undergoes a wetting transition from an ellipsoidal to a teardrop shape. At higher speeds, the tail of the teardrop breaks into smaller droplets in a process called “pearling”.18 © 2017 American Chemical Society

The onset of droplet sliding has been commonly described by a force-balance approach, where the gravitational driving force is matched by the retention force arising from contactangle hysteresis. Macdougall and Ockrent first introduced a formula to compute the critical sliding angle αc from contactangle hysteresis,21 where the generalized formula is kBo sin αc = cos θadv − cos θrec

(1)

Here, k is a geometric factor, θadv is the advancing contact angle, θrec is the receding contact angle, and Bo = ρga2/σ is a Bond number that measures the importance of gravity relative to capillarity, where ρ is the liquid density, g is the gravitational acceleration, and a is the characteristic length of the droplet. Equation 1 has been extensively studied and modified to describe the motion of droplets having an equilibrium contact angle θe > 45°.22−25 Several studies have clarified that θadv and θrec are not intrinsic properties between a liquid and substrate but rather quantities that can dynamically change within a range of possible contact angles on a given substrate.20,25−27 When the equilibrium contact angle of a droplet is relatively small, the droplet can be treated as a thin film under the lubrication approximation, in which the moving contact line is described with various approaches including precursor-film models,17,28−30 diffuse-interface models,10,11 and slip models.31 In the first two approaches, the equilibrium contact angle is imposed through the balance between capillarity and disjoining pressure.10,28,32,33 Unlike the force-balance approach with the assumption of contact-angle hysteresis, droplet sliding on inclined, chemically homogeneous substrates shows no critical Received: May 22, 2017 Revised: June 26, 2017 Published: June 26, 2017 7352

DOI: 10.1021/acs.langmuir.7b01716 Langmuir 2017, 33, 7352−7363

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ε = h*/l* ≪ 1, allowing us to apply the lubrication approximation. We use a precursor film with a thickness b and a twoterm disjoining pressure to model the contact-line behavior. Following prior works,17,35 the characteristic speed, time, pressure, and disjoining-pressure scales are defined, respectively, as u* = ε3σ/3μ, t* = l*/u*, p* = h*σ/l*2, and (* = p*, where σ and μ are the surface tension and viscosity of the liquid, respectively. With these scales, we define the following dimensionless variables (denoted with primes):

sliding angle as a solution at any nonzero inclination angle is always found in a comoving frame.11,23,34 In thin-film models, contact-angle hysteresis is realized as a consequence of the surface-energy heterogeneity.12,28,31 Several studies have focused on stability of a droplet on a chemically heterogeneous defect to understand the mechanism of pinning and the resulting apparent contact-angle hysteresis.10−12,31,34 However, surprisingly less attention has been given to pinning due to a topographical defect. An experimental study of contact-line pinning on a topographical defect showed that a significant retention force arises while the contact line pins, distorts, and slides across the defect.16 Using a two-dimensional lubrication model with a slip assumption, Savva and Kalliadasis studied the sliding behavior of droplets on inclined substrates with periodic topographical defects and demonstrated how stick−slip motion, contact-angle hysteresis, and a critical sliding angle can be induced.31 The study, however, did not consider a droplet interacting with a single topographical defect, which can reveal what geometric features of the defect play key roles in determining the critical sliding angle. The present work focuses on revealing the interactions between a sliding droplet and a topographical defect on an inclined substrate using a thin-film evolution equation with a precursor-film assumption. The defect shapes are idealized as Gaussian functions with variable width and height to systematically understand how contact-line pinning stems from the defect geometry. We also examine cases where the receding meniscus of the sliding droplet interacts with the defect to form residual droplets and discuss how temporary pinning at the defect delays droplet sliding. The rest of the paper is organized as follows. The governing equations and our model are explained in section 2, and simulations of droplet sliding on an inclined flat substrate without a defect are presented in section 3. The interaction between a droplet and a defect is discussed in section 4, the formation of residual droplets is examined in section 5, and the delay in droplet sliding from a defect is analyzed in section 6. Finally, conclusions are presented in section 7.

x = l*x′, z = h*z′, u = u*u′, w = εu*w′⎫ ⎬ t = (l* /u*)t ′, p = p*p′, Π = (*Π′ ⎭ ⎪ ⎪

(2)

where Π is the disjoining pressure. At leading order in ε, the momentum and mass conservation equations are 1 0 = −p′x ′ + u′z ′ z ′ + Gp (3) 3

0 = −p′z ′ − Gn

(4)

u′x ′ + w′z ′ = 0

(5)

where Gp = Bo sin(α)/ε and Gn = Bo cos(α) are parallel and normal gravitational parameters, respectively, with ρ being the density of the liquid and g the gravitational acceleration. The Bond number Bo = ρgl*2/σ measures the strength of gravitational forces relative to surface-tension forces. The kinematic condition at the liquid−air interface h′(x′,t′) is h′t ′ = w′ − u′h′x ′

(6)

The normal and tangential components of the interfacial stress balance are, at leading order,

−p′ = h′x ′ x ′ + Π′

(7)

u′z ′ = 0

(8)

At the substrate, the no-slip and no-penetration conditions are

u′ = w′ = 0

2. MODEL 2.1. Governing Equations. In a two-dimensional Cartesian coordinate system, we consider a Newtonian droplet moving over a flat substrate inclined at a tilt (inclination) angle α from the horizontal (Figure 1). Downstream from the droplet is a topographical defect. The height of the droplet at time t is denoted by h(x, t) . The droplet flows with velocity (u, w), which denote the horizontal (x) and vertical (z) components of the velocity field, respectively. The droplet with its characteristic height h* and half width l* is long and thin such that

(9)

The shape of a topographical defect, η(x) = h*η′, is described 2 2 by a Gaussian function η(x) = hde−(x−xd) /(2wd), where hd, xd, and wd denote the height, location, and width of the defect, respectively. The use of a Gaussian function conveniently enables investigation of the influence of topographical variables (width, height, and slope of the defect) on droplet dynamics while ensuring a smooth transition from the flat parts of the substrate to the defect. We use the following form of a two-term disjoining pressure:28,29 ⎡⎛ b′ ⎞m ⎛ b′ ⎞n⎤ Π(h) = (⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝ h′ ⎠ ⎦ ⎣⎝ h′ ⎠

(10)

where b′ = b/h* is the dimensionless precursor-film thickness. Following prior works,17,28−30 we choose the powers m = 2 and n = 3 to provide an appropriate physical description of contactline behavior at a moderate computational cost. Here, the Hamaker constant ( is related to the equilibrium contact angle θe,28,29 2 1 ⎛ θ ⎞ (n − 1)(m − 1) ( = − ⎜ e⎟ 2⎝ ε ⎠ b′(n − m)

Figure 1. Schematic of a droplet sliding on an inclined substrate with a topographical defect (gray). 7353

(11) DOI: 10.1021/acs.langmuir.7b01716 Langmuir 2017, 33, 7352−7363

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Figure 2. (a) Terminal sliding speed expressed in terms of Ca (magnitude shown in color) as a function of equilibrium contact angle θe and sliding force (Bo sin α) for droplets (V = 1) on inclined flat substrates. The dashed line at S = 1 indicates transitions from nearly pinned droplets to sliding droplets. The dashed-dotted line at S = π/2 marks transitions from droplets to elongated films. (b) Ca vs Bo sin α at θe = 20°; the S = 1 and S = π/2 transitions are shown with dashed lines. (c) Droplet profiles at four values of Bo sin α (solid circles in (b)) are shown from left to right, respectively.

By integrating eqs 3−5 with respect to z, and using the boundary conditions (eqs 6−10), the thin-film evolution equation is (primes dropped) ht =

We use the equilibrium droplet shape on a horizontal substrate as the initial condition for the droplet on an inclined substrate. In all cases, we set V = 1, x0 = L/3, l0 = 1.4, b = 0.001, and L = 35 with the discretization size between 100 and 500 points per unit length. The choice of these parameters allows us to explore a rich variety of behavior at a reasonable computational cost. When a droplet slides on an perfectly wetting (θe = 0) inclined substrate without a defect, the droplet spreading fol-

⎞ d ⎛⎜ 3 d h ( −Gpx + Gn(h + η) − Π − (h + η)xx )⎟ ⎝ ⎠ dx dx (12)

2.2. Solution Method. Equation 12 is solved numerically on the domain 0 ≤ x ≤ L subject to the boundary conditions h(0, t ) = b , h(L , t ) = b ⎫ ⎪ ⎬ hx(0, t ) = 0, hx(L , t ) = 0 ⎪ ⎭

lows the self-similar solution38 h(x , t ) = (3Gp)−1/2

1/2

( x −tx ) min

,

where h, x, and t are dimensionless and xmin is the leftmost side of the droplet. In the simulations, the spreading of numerical droplet profiles is well approximated by the self-similar solution, except close to the right contact-line region (results not shown). Since the self-similar solution is obtained when gravitational forces dominate over surface-tension forces (Bo ≫ 1), it is expected to hold only away from the contact-line region.35,38 From the simulations on an inclined partially wetting substrate (θe ≠ 0) without a defect, at some large t, a sliding droplet reaches a steady shape with a constant sliding speed. The condition for a steady shape is determined when the change in the maximum height during a time interval becomes smaller than the numerical accuracy. If the right-most side of the droplet reaches close to the downslope end of the domain at x = L − 2l0 before the steady shape is achieved, the droplet is translated close to the upper end of the domain such that the left-most side of the droplet is placed at x = 2l0. The terminal base width of the droplet and the numerical convergence of the final droplet shape are discussed in Appendix A. The values of the apparent advancing contact angle θadv and receding contact angle θrec are extracted from the minimum and maximum slopes of the interface with respect to the flat substrate, respectively.28,29 (These are the angles between the lines tangent to the slopes and a line parallel to the x-axis.) It must be emphasized that even though the equilibrium contact angle θe is the same on the advancing and receding sides of the droplet, the values of θadv and θrec will, in general, be different from θe and each other. The positions of the apparent advancing and receding contact lines are computed by extrapolations of the corresponding tangents to the substrate.17 The apparent contact angles are then rescaled by multiplying by ε to obtain the actual values.17

(13)

Numerical solutions of eq 12 with the above conditions (eq 13) are obtained with a fully implicit finite-difference scheme with all spatial derivatives approximated by second-order centered differences. Time-integration is performed with the DDASPK iterative solver.36 Initially, the droplet shape on a horizontal flat substrate (α = 0) is solved for using the initial condition given by a fourthorder polynomial between x = x0 − l0 > 0 and x = x0 + l0 < L that satisfies the conditions given by eq 13. This corresponds to a droplet having a center at x = x0 and an initial width of 2l0. The system volume is V + 2l0b, where V is the dimensionless volume of the droplet and 2l0b is the dimensionless volume of the precursor film below the droplet. The base width of the droplet d is found by subtracting the left-most side of the droplet (defined as min(x) where h(x) > b) from the rightmost side (max(x), where h(x) > b). These simulations are used to confirm the spreading dynamics of a two-dimensional droplet37 (results not shown for brevity), where the width of the spreading droplet on a perfectly wetting substrate (θe = 0) follows a power law d ∼ t1/7. On partially wetting substrates (θe ≠ 0), the power law holds until at some large t the equilibrium d is reached with a change less than the numerical accuracy (10−6) of the simulation. In this state, the value of the apparent equilibrium contact angle is obtained from the maximum slope of the interface with respect to the substrate28,29 and then rescaled by multiplying by ε. Since it is nearly equal to θe to within a degree, we use θe to imply both numerically imposed (via eq 11) and apparent equilibrium (obtained from the maximum interface slope) contact angles in the remainder of the paper. 7354

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Figure 3. (a) Advancing contact angles θadv and (b) receding contact angles θrec (magnitudes shown in color) as a function of θe and Bo sin α with the threshold lines at S = 1 and S = π/2. (c) At θe = 20°, θadv (circles) and θrec (squares) are shown with expected angles from the Cox-Voinov law (dotted lines) from eq 15. The vertical dashed lines mark the S = 1 and S = π/2 transitions.

Figure 2a shows two lines that clearly mark the sliding transitions. We have chosen to draw these lines at the values S = 1 and S = π/2. The former value corresponds to a balance between the gravitational and retention forces, whereas the latter value was chosen for convenience and does not have any special physical significance. When S < 1, the droplet remains nearly pinned since the retention force dominates over the sliding force. We note that the terminal sliding speed within S < 1 is nonzero when the tilt angle α > 0 and is not clearly visible from the color of the plot. When 1 < S < π/2, the terminal sliding speed increases monotonically as Bo sin α increases at a given θe. When S > π/2, the terminal sliding speed remains nearly constant as Bo sin α increases since the droplet base width d increases significantly (shown in Figure 13a in Appendix A) to compensate for the increase in Bo sin α in eq 14. The change in the terminal sliding speed (Ca) is more clearly displayed in Figure 2b, which shows the horizontal crosssection of Figure 2a at θe = 20°. When 1 < S < π/2, the terminal sliding speed increases linearly as predicted from a theoretical model with a two-dimensional approximation and experimentally confirmed with three-dimensional droplets by Kim et al.19 When S > π/2, the terminal sliding speed plateaus to a nearly constant value. Figure 2c displays the four droplet profiles corresponding to the solid symbols in Figure 2b, from left to right. When the sliding force Bo sin α is relatively low at 0.1, the droplet profile has a shape resembling a circular segment. As the sliding force increases, the profiles become more elongated with the advancing side bulging and the receding side tapering. For a large enough sliding force (Bo sin α ∼ 0.38), the droplet profile starts to resemble a liquid film. For these relatively large values of S, the droplet becomes significantly elongated (Figure 13a in the Appendix) such that the dynamics of the advancing and receding contact lines are decoupled12 and the retention force is not well estimated from contact-angle hysteresis of the droplet (eq 14). The profiles in Figure 2c undergo similar shape transitions as the droplet profiles on an inclined flat substrate obtained from a two-dimensional lubrication model with a slip assumption by Savva and Kalliadasis.31 We now turn our attention to the changes in the advancing contact angle θadv and receding contact angle θrec shown

After the steady droplet shape is reached, the dimensionless terminal contact-line speed u′ is obtained from linear regression of the advancing contact-line positions versus t. The corresponding capillary number Ca = μu/σ is found from u* = ε3σ/3μ and u = u*u′ as 3u′/ε3. For the simulations with a defect, the Gaussian-shaped defect is placed at x = 15. When the tilt angle α is incrementally increased from α = 0, the droplet slides toward the defect and remains pinned until when α becomes greater than a critical sliding angle αc. Droplet pinning is determined by checking whether both changes in the maximum height and the right side of the droplet become smaller than the numerical accuracy.

3. DROPLET SLIDING ON AN INCLINED FLAT SURFACE To facilitate understanding of droplet behavior when a topographical defect is present, we first discuss the case where the defect is absent. We use characteristic values from silicon oil (μ = 9.15 cP, σ = 20.5 mN/m, ρ = 924 kg/m3, θe = 45°) to simulate droplet sliding behavior. We set the Bond number to Bo = 0.9 and the corresponding droplet width to l* = 1.43 mm, while the tilt angle α varies from 0 to 40°.18 Figure 2a displays the terminal sliding speed in terms of Ca, where the magnitude is shown in color. When the sliding force from gravity is relatively high (Bo sin α ∼ 0.3), as the equilibrium contact angle θe increases, the terminal sliding speed (Ca) also increases monotonically. However, when the sliding force is relatively low (Bo sin α ∼ 0.1), as θe increases beyond θe = 19°, the terminal sliding speed decreases such that Ca becomes nearly zero. To better understand the qualitative change in the terminal sliding speed, we introduce the sliding factor S, which compares the sliding force from gravity (∼ρgl*2 sin α) with the retention force from capillarity (∼σ(cos θrec − cos θadv)/h̅). Here, h̅ = d−1∫ xminxmin+dh(x)dx is the dimensionless mean droplet height, where h, d, and x are all dimensionless. By assuming V ≈ h̅d = 1, S is defined as S=

Bo sin α d(cos θrec − cos θadv)

(14)

which follows from the expressions given above for the gravitational and retention forces. 7355

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Figure 4. When the advancing contact line is pinned, (a) the critical sliding angle αc as a function of defect height hd at defect width wd = 0.1, (b) αc as a function of wd at hd = −0.05 (squares) and hd = 0.05 (circles). (c) Pinning locations on the normalized defect (bold lines) from the cases shown as filled symbols in (a) (solid lines) and (b) (dashed lines). Circles indicate the locations where the slope to the defect surface is the most negative.

Figure 5. Schematics of a defect (bold line) and a pinned advancing contact line (solid line) with key parameters such as the advancing contact angle θadv, mesoscopic angle θm, and maximum defect angle γmax for a (a) dent (hd < 0) and (b) bump (hd > 0). Insets show a magnification of the pinned region.

in Figures 3a and b, respectively. Interestingly, when S < 1 or S > π/2, θadv remains nearly constant with respect to changes in Bo sin α for a given θe. In contrast, θrec rapidly decreases when S < π/2 and decreases more gradually when S > π/2. In general, the change in θrec is much greater than the change in θadv when the sliding force is increased. This observation can be explained geometrically from Figure 2c. As the sliding force is increased, the advancing side bulges in such a way that there are only small changes in the maximum profile height and θadv. In comparison, the receding side shows a more rapid decrease in θrec until the point at which a thin film forms with a nearly constant θrec. The observations of both θadv and θrec qualitatively agree with the theoretical result from Krasovitski and Marmur,25 which shows that, on a hydrophilic surface, the change in θrec is much greater than that of θadv when the tilt angle α is increased. Figure 3c displays the horizontal cross-section of Figure 3a and b at θe = 20°. The changes in θadv and θrec are qualitatively consistent with predictions from the Cox-Voinov model,26,27 which relates the dynamic contact angle to the contact line speed as 3 θadv,rec = θe3 ± 9Ca ln λ

where the dimensionless precursor-film thickness b is used in place of the dimensionless slip parameter λ.39 The Cox-Voinov model can be derived from an asymptotic analysis that assumes Ca is small.27 Experimental studies also report that contactangle measurements of a droplet sliding on an inclined flat substrate are closely described by the Cox-Voinov model, with potentially different values of λ for advancing and receding contact angles.40,41

4. DROPLET PINNING ON A DEFECT 4.1. Pinning at the Advancing Contact Line. We now examine how droplet pinning can occur from a topographical defect by evaluating the critical sliding angle αc for droplet motion for various combinations of defect heights and widths. In the simulations, a droplet having V = 1, θe = 19°, and Bo = 0.9 is placed before a defect. For a given defect shape, the tilt angle α is incrementally varied by 0.2° until the droplet begins to slide over the defect. We define the critical sliding angle αc as the mean angle between the maximum angle at which the droplet still remains pinned and the minimum angle at which the droplet begins to slide. In Figure 4a, when the defect width wd is fixed, an increase in the magnitude of the defect height (or depth) hd results in a higher critical sliding angle αc for both a dent (hd < 0) and

(15) 7356

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Figure 6. When the advancing contact line is critically pinned, (a) critical sliding angle αc for a dent (squares) and a bump (circles), (b) receding contact angle θrec, (c) advancing contact angle θadv (solid lines) and mesoscopic angle θm (dashed lines), and (d) sliding factor S as a function of the maximum defect angle γmax.

bump (hd > 0). Figure 4b shows the change in αc when the defect height is fixed and the defect width is varied. For a bump, as the width increases, the critical sliding angle decreases since the bump becomes flatter. In the case of a dent, when the width becomes too small (wd = 0.02), capillary action causes the advancing contact line to fill the dent quickly at a small critical sliding angle. The effect of capillary action disappears at slightly higher width (wd = 0.04) where the maximum critical sliding angle is achieved. As the width increases above wd = 0.04, the critical sliding angle decreases. One intriguing observation is that when the defect geometries are normalized by their widths and heights in Figure 4c, the pinned locations of the advancing contact line at the critical sliding angles collapse to nearly the same points for a dent and bump, respectively. These points (shown as circles in Figure 4c) are where the slope to the defect surface ηx is the most negative (minimum). The good collapse of the pinning locations implies that the critical sliding angle is strongly related to the minimum slope of the defect rather than its height or width. To analyze how pinning at the critical sliding angle occurs at the defect surface where the slope is the most negative (min(εηx)), we introduce the maximum defect angle γmax = |atan(min(εηx ))|

Since both the receding contact angle and mesoscopic angle are very close to the equilibrium contact angle (θe = 19°), we conclude that the change in the advancing contact angle stems from the defect topography while θm ≈ θe. As the advancing contact line moves toward the location of the maximum defect angle γmax, the slope of the defect surface (illustrated in Figure 5) gradually becomes steeper (in the case of a bump, the increase in magnitude starts from x = xd), causing the advancing contact angle to rise (which follows from the observation above that at the point of pinning, θadv = θm + γmax). Because the increase in θadv results in a stronger droplet retention force from eq 14, the Gaussian-shaped topographical defect creates a type of a potential well from which the advancing contact line can only escape above a critical sliding angle. A similar mechanism was discussed by Thiele and Knobloch in their study of droplets near a chemically heterogeneous defect.34 As a result, the critical pinning occurs at the location of the minimum slope (i.e., maximum defect angle γmax) where θadv and subsequently the retention force are maximized. For a dent, when the tilt angle becomes greater than the critical sliding angle, the advancing contact line slides from the negative-slope side (x < xd) of the dent to the positive-slope side (x > xd). We note that additional pinning does not occur on the positive-slope side since geometrically θadv on that side is smaller than θadv on the negative-slope side, as illustrated in Appendix B. Similarly for a bump, pinning does not occur on the positive-slope side (x < xd) before the advancing contact line reaches the negative-slope side (x > xd). Figure 6d displays the change in the sliding factor S from eq 14 for the critically pinned droplets. For both a dent and a bump, S ≪ 1, implying that the estimated retention force dominates over the sliding force from gravity at the critical sliding angle. This result suggests that although eq 14 can qualitatively describe the sliding transitions in Figure 2a, it overestimates the retention force and thus cannot be used to accurately predict the critical sliding angle (which would be expected to correspond to S ∼ 1). 4.2. Pinning at the Receding Contact Line. We also evaluate pinning at the receding contact line. To rule out the effect of pinning at the advancing contact line, we initially place a droplet on top of a defect, so that only the receding contact line interacts with the defect as the droplet slides.

(16)

which is |atan(ε(hd/wd)e−1/2)| for a Gaussian defect and is found at both x = xd − wd and x = xd + wd. Figure 5 shows γmax at x = xd − wd for a dent and x = xd + wd for a bump. We also define the mesoscopic angle θm as the angle between the intersection of the tangent to the liquid−air interface and the tangent to the substrate surface.17,42 When the advancing contact line is pinned at the location of the maximum defect angle γmax, it is easy to see from the insets of Figure 5 that θadv = θm + γmax. Figure 6a shows the monotonic increase in the critical sliding angle as the maximum defect angle increases for the cases shown as solid symbols in Figure 4a. Figure 6b shows that the receding contact angle of the critically pinned droplet remains nearly constant when γmax is varied. In comparison, Figure 6c shows that the advancing contact angle increases when γmax increases, whereas the mesoscopic angle stays relatively constant. 7357

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when the receding contact line with θrec > γmax approaches a bump from left to right, it glides over the bump without leaving a residual droplet. In contrast, when θrec < γmax as shown in Figure 8b, eventually the liquid−air interface interacts with the defect via disjoining pressure and “pinches off”, leaving a residual droplet behind. For a dent, the geometric argument is slightly more complicated. In Figure 8c and d, a straight line with a slope corresponding to θrec is shown with a dashed line. Assuming that the mesoscopic angle θm is close to the value of θrec on a flat part of the substrate, the slope of this straight line m2 can be computed as

Figure 7 shows that compared to the critical sliding angle αc from pinning at the advancing contact line, αc from pinning at

m2 =

tan θrec + m1 1 − m1 tan θrec

(17)

where m1 = εηx(x)|x=xrec is the slope of the defect surface at the receding contact line xrec. As the receding contact line moves along the dent, if the straight line defined as p(x) = m2(x − xrec) + η(xrec) intersects with the dent profile only once at x = xrec, as is the case in Figure 8c, the liquid−air interface leaves no residual droplet. But if the straight line intersects with the dent profile more than once at some xrec, as is the case in Figure 8d, the liquid−air interface breaks off and leaves a residual droplet inside the dent. Based on the above geometric arguments, we determine a threshold defect height at which a residual droplet starts to form. Figure 9a shows the threshold hd for a bump, computed from eq 16 by setting γmax = θrec. The simulation results from Figure 7 for a bump are also shown in Figure 9a; in some cases, a residual droplet forms, whereas in others it does not. As can be seen in Figure 9a, the threshold defect height is correctly predicted by the geometric argument. For a dent, the threshold defect height is numerically solved for by finding hd at which f(x) has more than one root at some xrec ∈ (xd − wd, xd) for a given θrec, where f(x) = p(x) + η(x) has the form

Figure 7. Critical sliding angle αc from pinning at the receding contact line (solid line) as a function of defect heights hd at a defect width wd = 0.1. For comparison, αc from pinning at the advancing contact line (dashed line) from Figure 4a is shown.

the receding contact line is always lower. Therefore, in general, the critical sliding angle of a droplet sliding toward a defect is determined by pinning at the advancing contact line. At higher defect heights (hd ≥ 0.05), αc increases rapidly. This phenomenon is related to the receding side of the droplet adhering to the defect and becoming elongated, which will be discussed in section 5.2.

5. FORMATION OF RESIDUAL DROPLETS 5.1. Residual Droplet behind a Defect. When the tilt angle is greater than the critical sliding angle, a sliding droplet leaves a residual droplet behind a defect for certain combinations of defect heights and widths. Typical cases of residual droplet formation are shown in Figure 8b and d for a bump and a dent, respectively. We explain the formation of the residual droplet geometrically by comparing the receding contact angle with the defect topography. In the case of a bump, the geometric argument of how a residual droplet forms is made by comparing θrec with the maximum defect angle γmax (eq 16) at x = xd − wd. In Figure 8a,

2

f (x) = m2(x − xrec) − hd(e−(x − xd)

/(2wd2)

2

− e−(xrec − xd)

/(2wd2)

) (18)

with m2 computed from eq 17. The geometric argument can be simplified by fixing the location of the straight line at xrec = xd − wd, so that for a given θrec, only hd is numerically varied to

Figure 8. When the receding contact line (solid lines) of a droplet (θe = 19° and Bo = 0.9) moves from left to right over a bump (bold line) with a width wd = 0.1, a residual droplet does not form at (a) a defect height hd = 0.03 and a tilt angle α = 1.2° but forms at (b) hd = 0.05, α = 1.8°. Similarly for a dent, a residual droplet does not form at (c) hd = −0.03, α = 1.6°, but forms at (d) hd = −0.05, α = 3.2°. In panels (c) and (d), the dashed line has a slope corresponding to θrec. 7358

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Figure 9. Threshold hd at which a residual droplet forms as a function of θrec when (a) hd > 0 (dashed line) and (b) hd < 0 (solid line) from eqs 18 and 19 (dashed-dotted line). From Figure 7, cases with and without a residual droplet are shown with circles and crosses, respectively. (c) Threshold maximum defect angle γmax as a function of θrec.

Figure 10. Profiles of receding side of a droplet (θe = 19° and Bo = 0.9) moving from left to right over a bump with wd = 0.1 and hd = 0.06 at (a) α = 6° and (b) α = 5.8°. Dashed lines have slopes corresponding to θrec, and the circles mark the location of the maximum defect angle γmax.

In contrast, Figure 10b shows what happens when α is only 0.2° less than in the case shown in Figure 10a. The receding contact line moves more slowly so that the four profiles are now displayed at a longer time interval Δt = 120. Here, the bottleneck is broader and does not move as much compared to the case in Figure 10a. As time progresses, the influence of disjoining pressure at the bottleneck becomes more pronounced and the liquid−air interface there pinches off, leaving a new residual droplet in front of the bump. We note that since the receding contact line moves faster at the higher tilt angle, the influence of viscous bending is more pronounced, as can be seen by comparing the slope of the line corresponding to θrec to the interface slope at the location of the maximum defect angle γmax (t = 30 in Figure 10a and t = 120 in Figure 10b). As a consequence, the bottleneck is narrower and tends to move faster when the tilt angle is higher. Interestingly, the size of the additional residual droplet is similar to the size of the residual droplet formed behind the bump. The whole pinch-off process delays the sliding of the droplet considerably compared to the case in Figure 10a. These results indicate that the formation of residual droplets depends not only on the difference between the receding contact angle and the maximum defect angle, but also on the dynamic evolution of the liquid−air interface.

determine the threshold hd at which g(x) has more than one root, where g(x) is 2

g (x) = m2(x − xd + wd) − hd(e−(x − xd)

/(2wd2)

− e−1/2) (19)

Figure 9b shows only a minor difference between the threshold hd for a dent from the two arguments. Also plotted in Figure 9b are simulation results from Figure 7, and it is seen that the geometric arguments again correctly predict the threshold defect height. Figure 9c shows the value of γmax at the threshold conditions for both a bump and dent. Given the same value of θrec, residual-droplet formation requires a lower value of γmax for a dent. 5.2. Residual Droplet in Front of a Defect. Recall that for a bump, a residual droplet can form behind the bump when θrec < γmax. In this section, we discuss how an additional droplet can be formed in front of the bump. Figure 10a shows four profiles of the receding side of a droplet at a tilt angle α = 6° at a time interval Δt = 30. We note that the liquid−air interface has a bottleneck-like shape on the negativeslope side of the dent. This bottleneck follows the receding side of the droplet as time progresses, and the liquid−air interface moves over the bump without leaving a residual droplet in front. 7359

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Figure 11. Profiles of a droplet (θe = 19° and Bo = 0.9) sliding from left to right at α = 6° over (a) a dent with γmax = 10° (hd = −0.03, wd = 0.1) and (b) a bump with γmax = 10° (hd = 0.03, wd = 0.1). The insets show the residual droplet at the defect which is deposited by a previous sliding droplet.

Figure 12. (a) Differences in advancing contact-line positions Δx of a droplet (θe = 20°, Bo = 0.9) sliding on an inclined substrate at α = 6° without a defect (dotted line at Δx = 0) and with a dent (γmax = 10°, solid line) or a bump (γmax = 10°, dashed-dotted line). The Δx values for a dent with a residual droplet (dashed line) and for a bump with a residual droplet (dashed-double-dotted line) are also shown. (b) The final Δx as a function of the maximum defect angle γmax without and with a pre-existing residual droplet.

6. DELAYED SLIDING FROM A DEFECT We describe the transient depinning process of a droplet sliding over a defect at a tilt angle above the critical sliding angle and quantify the delay compared to when the droplet slides on a flat substrate. The comparison helps characterize how the sliding behavior of a droplet is modified due to the presence of a defect. The delay is expressed as a spatial difference Δx between advancing contact-line positions of a droplet sliding on a substrate without and with a defect, respectively, where a positive Δx implies a lag. We also consider whether there is a noticeable difference when a droplet slides over a defect without or with a residual droplet. This is relevant to determining whether droplet sliding is altered after a previous sliding droplet leaves a residual droplet. Figure 11 shows the interaction of a droplet sliding over defects with residual droplets. In Figure 11a, a droplet sliding at a constant speed coalesces with the residual droplet inside a dent between t = 620 and t = 640 and experiences a sudden jump in the advancing contact-line position. When the dent is within the droplet base at t = 720, the droplet slides at a constant speed again until the receding contact line eventually gets temporarily pinned at the dent and causes the droplet to

elongate at t = 2400. Finally, the receding side of the droplet pinches off and releases a residual droplet at t = 2560. The volume of the residual droplet stays approximately constant for the rest of the simulation, and its value will depend on the dent geometry, disjoining pressure, and substrate inclination angle. The interaction of a sliding droplet with a bump with a residual droplet (Figure 11b) shows similar behavior except for temporary pinning at the advancing contact line at t = 900, which causes a contraction of the droplet base width. When a droplet slides over a dent without a residual droplet, Δx increases as the advancing contact line gets temporarily pinned at the location of the maximum defect angle (Figure 12a). During this time, the droplet base width contracts while both θadv and θrec increase, forming a more spherical-caplike droplet shape. Once released from pinning, the advancing contact line temporarily accelerates as θadv relaxes to its original value on the flat part of the inclined substrate. As a result, Δx after the release decreases nearly to zero at t = 2400 until the time the receding contact line gets temporarily pinned. Subsequently, as the droplet becomes elongated (similar to Figure 11a at t = 2400), Δx increases and eventually plateaus to a constant value when the detached droplet relaxes to a steady shape on the flat part of the substrate. 7360

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Langmuir The delay from a bump shows a similar change in Δx as the advancing contact line undergoes pinning and depinning, followed by the receding contact-line pinning and depinning, with differences being the exact pinned location and the magnitude of the change in Δx (Figure 12a). Since the delay induced from the advancing contact-line pinning can be quickly recovered after depinning as the deformed droplet shape (illustrated in Figure 11b at t = 900) relaxes to a steady shape on the flat part of the substrate, the final Δx is mainly determined by the receding contact-line pinning and depinning processes. Surprisingly, the presence of a pre-existing residual droplet on the defect does not make a significant difference in the final Δx for both the dent and bump. When there is a residual droplet inside the dent, the incoming droplet coalesces with the residual droplet, and its advancing contact line quickly jumps inside the dent, as shown in the initially negative Δx in Figure 12a (also in Figure 11a at t = 640). As the advancing contact line moves beyond the dent, the lead in Δx gained from the coalescence is about 0.3 times the dent width wd = 0.1, much smaller than the lag induced from the receding contactline pinning and depinning. In comparison, the case with a residual droplet on the bump shows a sudden drop in Δx (Figure 12a) due to the coalescence, followed by a quick rise in Δx due to the temporary pinning of the advancing contact line (shown in Figure 11b). Similarly, the final Δx for the bump with a residual droplet does not differ significantly from the final Δx without one. Figure 12b summarizes the final sliding delay Δx induced by interactions between a droplet and a Gaussian-shaped defect with various maximum defect angles γmax at t = 8000. Again, the cases with a pre-existing residual droplet do not exhibit a significant difference from the cases without one. Figure 12b shows that in general, given the same γmax, a dent induces a higher delay than a bump. The bump with γmax = 16.9° has a higher Δx than the dent with the same γmax, because this is when θrec < γmax such that the receding contact line gets temporarily pinned on the bump (section 5.2). The knowledge of the sliding delay Δx caused by interactions between a sliding droplet and a defect suggests several design principles for modulating droplet sliding behavior, such as promoting droplet coalescence or controlling spacing between multiple droplets. For example, when two droplets with spacing s < Δx approach a defect, the front droplet would remain pinned until the droplet behind approaches and coalesces. Similarly, when a series of droplets with s > Δx approaches a series of defects, the spacing will be reduced to s − Δx. In a more complex scenario, when uniformly sized droplets with s < Δx randomly adhere to a substrate with a series of identical defects (e.g., spraying droplets on a bumpy surface), the droplets eventually would coalesce to form larger droplets and slide more rapidly. In nature, Namib Desert Darkling beetles have been known to use such a design to collect water by condensing fog and dew droplets on their bumpy body surface.3,4 By inducing coalescence through a bumpy surface, droplet sliding can be enhanced by increasing the relative effect of gravity over contact-line pinning.

so does the critical sliding angle. The advancing contact angle is largest at this point, suggesting that the droplet retention force is maximized. Second, for some combinations of defect heights and widths, residual droplets can form above the critical sliding angle. Some aspects of residual-droplet formation can be rationalized by geometric arguments, whereas others depend on the dynamic evolution of the liquid−air interface. Third, the presence of a defect introduces a delay in droplet sliding due to pinning and depinning of the receding contact line. The delay is less affected by the dynamic behavior of the advancing contact line, a conclusion that could not readily be inferred without the calculations performed here. Although computational limitations prevented a more extensive exploration of the problem parameter space, we expect that qualitatively similar behavior would be observed in other parameter regimes. Each of these conclusions has important implications for the study of droplet sliding on inclined surfaces. First, the pinning of the droplet at the maximum defect angle suggests that in addition to using a mean roughness to characterize a surface (e.g., as done in several papers43−46), it would be useful to also determine the mean of the local topography slope minima (or mean of the local maximum defect angles). Second, the formation of residual droplets indicates that for certain topographical shapes, it may be difficult to fully clean surfaces. Third, the delay in sliding caused by the presence of a defect could exploited to control the collective sliding behavior of multiple droplets, as discussed in section 6. The modeling approach used here can readily be extended to consider three-dimensional droplets, other defect shapes and disjoining-pressure functions, and the presence of multiple droplets and multiple defects (including defects with different size and spatial distributions). Examination of these issues along with comparisons to results from complementary experiments would advance the design of surfaces for industrial applications such as self-cleaning surfaces and filtration where dropletsurface interactions play a key role.

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APPENDIX A In Figure 13a, the final droplet base width at the corresponding final sliding speed (shown as Ca) in Figure 2a is displayed. Significant droplet elongation is observed in the region S > π/2, where the gravitational driving force dominates over capillarity (which promotes a more spherical-cap-like droplet shape). Interestingly, the temporal evolution of the droplet base width d closely resembles an exponential relation, d = 1 − e −t / τ dmax

(20)

where τ is the transient time scale and dmax is the maximum droplet base width when the steady droplet shape is achieved. Figure 13b shows the change in the transient time scale. In the region S < π/2, the transient time scale is shorter compared to the region S > π/2, where the elongation from gravity takes a longer time, especially at small θe. The fitting based on eq 20 to extract τ is illustrated in Figure 13c. At the same Bo sin α, both cases at θe = 15° and 20° show excellent agreement with the fitted functions. In all cases, the simulations to obtain dmax and Ca were run for at least t = 6τ.

7. CONCLUSIONS Three important conclusions arise from our study of droplet sliding on an inclined substrate with a topographical defect. First, at the critical sliding angle, droplets pin at the point where the slope of the defect surface is most negative. This point can be represented as a maximum defect angle, and as it increases

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APPENDIX B The geometric argument behind why pinning of an advancing contact line does not occur at the location where the slope to 7361

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Figure 13. (a) Droplet base width, d, and (b) transient time scale, τ (magnitudes shown in color), as a function of θe and Bo sin α. (c) The change of d/dmax versus t at Bo sin α = 0.34 and θe = 15° or 20° are shown with exponential fits using eq 20, from which values of τ are determined.

Figure 14. Schematics of a defect (bold line) and an advancing contact line (solid line) (a) pinned on a dent at x = xd − wd, (b) pinned on a bump at x = xd + wd, (c) sliding on a dent at x = xd + wd, and (d) sliding on a bump at x = xd + wd with the advancing contact angle θadv, mesoscopic angle θm, and maximum defect angle γmax.

the National Science Foundation under Grant No. CBET1449337.

the defect surface is positive is illustrated in Figure 14. As discussed in section 4.1, when pinning of the advancing contact line occurs at the location where the slope is the most negative, the advancing contact angle is computed as θadv = θm + γmax (Figure 14a,b). When the advancing contact line is at the location where the slope is the most positive, Figure 14c,d shows that θadv = θm − γmax. Since in section 4.1 we found that the mesoscopic angle is close to the equilibrium contact angle for a pinned droplet, it follows that θadv at the location where the defect slope is negative is always higher than θadv at the location where the defect slope is positive. Consequently, pinning of the advancing contact line on a defect at a given tilt angle always occurs at the location where the slope to the defect surface is negative.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Satish Kumar: 0000-0003-0829-6355 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge financial support from Cummins Filtration for this work. This material is based on work partially supported by 7362

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