Article pubs.acs.org/Langmuir

Depinning of Drops on Inclined Smooth and Topographic Surfaces: Experimental and Lattice Boltzmann Model Study Stefan Bommer,† Hagen Scholl,†,‡ Ralf Seemann,*,†,‡ Krishan Kanhaiya,§ Vivek Sheraton M,§ and Nishith Verma*,§ †

Experimental Physics, Saarland University, Saarbrücken, Germany Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany § Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur, India ‡

S Supporting Information *

ABSTRACT: In this study, the dynamics of initially stationary liquid drops on smooth and topographic inclined silicon surfaces was investigated experimentally and by lattice Boltzmann simulations. The transient contact angles and the critical angle of inclination were measured systematically for different liquids, drop sizes, and surfaces having different wettability and surface roughness. In general, the critical angle of inclination is larger for hydrophilic than for hydrophobic surfaces, irrespective of the liquids, and increases with increasing contact angle hysteresis and decreasing drop sizes. A two-phase liquid−vapor lattice Boltzmann model based on the Shan and Chen approach was developed for two dimensions which incorporates the wetting and topographic characteristics of the surface. The simulation results matched the experimentally found features quantitatively and allowed one to explore the roll-off behavior even in cases that can hardly be accessed experimentally.

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where θeq is the equilibrium contact angle, and σSV, σSL, and σLV are the surface tensions of the solid−vapor, solid−liquid, and liquid−vapor interfaces, respectively. However, if the droplet is placed on a real surface with microscopic physical and chemical heterogeneities,5−8 the droplet exhibits a contact angle hysteresis, and the contact angle of the sessile droplet assumes an angle between the advancing and the receding contact angle, θadv and θrec. When such a real surface is inclined, the drop loses its initially symmetric shape and the contact angle at the front increases, whereas the contact angle at the rear side decreases (see Figure 1). This contact angle hysteresis prevents the drop from sliding as, long as neither the limiting advancing nor receding contact angles are reached. A force balance on such a drop along the direction of the inclined plane can be expressed as

INTRODUCTION The stability and dynamics of a liquid drop on a surface have interested scientists and engineers for several decades because of its biological relevance and industrial applications. In many natural and technical examples it is of importance that sessile droplets are easily removed from surfaces like lotus leaves, insect wings, windshields, air wings, solar panels, and greenhouses.1,2 Whereas the equilibrium condition of droplets on ideal surfaces is presently well understood, there are still open questions about the roll-off behavior of droplets on real surfaces in the presence of external forces. If a drop is placed on a horizontal surface, it relaxes and eventually assumes an equilibrium shape, i.e., a symmetric spherical cap for small drops if gravitational effects can be neglected or a flattened circular pancake for drop sizes exceeding the capillary length. If the surface is perfectly smooth and chemically homogeneous, the equilibrium contact angle for the drop is defined by Young’s equation:3,4 cos θeq = (σSV − σSL)/σLV © 2014 American Chemical Society

mg sin α = σLVRk(cos θfront − cos θrear)

(2)

Received: April 25, 2014 Revised: August 21, 2014 Published: August 25, 2014

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Extending along these lines, we study the dynamic contact angles for different types of liquids and for surfaces having different wettability (neutrally wetting and hydrophilic) and patterned surface. The observed behavior of the drops is reproduced via simulations based on the lattice Boltzmann methods (LBMs), which allow the prediction of the shape and critical angle over the range of initial contact angles between θadv and θrec, based even on parameters that could not be probed experimentally. To simulate such a two-phase liquid−vapor flow, the LBM has been increasingly applied.11−15 Using the mesoscopic scale, LBM-based models are easier to implement than the conventional computational fluid dynamics based models for a wide variety of fluid flow problems. The salient feature of the models based on LBM is that they can simulate the interaction between different phases via the collision operator of the Boltzmann transport equation and the phase boundaries of complex geometries by using simple boundary conditions. Various approaches have been used in the LBM simulations for incorporating the regular and randomly distributed heterogeneities on the solid surface. Iwahara et al.11 have modeled the solid surface by assigning different densities to the solid nodes. A model developed by Davies and co-workers12 introduces a solid af finity parameter to study the wetting of the surface. The LBM-based models to simulate dynamics of a sliding drop are limited. Varnik et al.13 have simulated the drop behavior on a neutrally wetting rough surface similar to lotus leaves, using LBM, by constructing the periodic square micropillars on the solid surface. Moradi et al.14,15 used LBM to simulate the motion of a drop under the influence of an external body force and correlated the velocity of the center of the mass of the drop to its density, radius, and the external force. These studies are limited to the understanding of the motion of drops and do not elaborate the effect of the contact angle dynamics on the depinning behavior of drops before sliding on a tilted plane. In general, the salient advantages of the LBM-based models are the simple ways of implementing boundary conditions for complex geometries and the simulation of multiphase flow and phase separations. A comparative study of the LBM-based models and Surface Evolver is also available.16 As mentioned in this study, the major advantages of the LBM-based models over the Surface Evolver are its ability to model the hydrodynamics of a drop and establish changes in the drop topologies, such as a change in drop’s shape or length. The current studies described in the literature address the dynamics of contact angle and the depinning behavior, specific to certain types of liquids and the surface. Thus, a general theory to explain the phenomena of the drop roll-off on a surface due to external forces is missing. The motivation to establish a common theory to describe the depinning behavior of liquid drops regardless of the types of liquids and surface forms the core idea of the present study. Although the drop roll-off has been studied using experimental techniques, there are often limitations, such as practically unattainable initial contact angles and the generation of an ideal surface, which prohibit us from gaining complete insight into the contact angle dynamics. The present study comprising experimental techniques and the LBM-based models has attempted to fill in the present gaps in the understanding of drop roll-off on surfaces due to external forces, in particular, the depinning of a drop (onset of the sliding or motion of the drop). The simulation or the discussion of contact line motion after depinning is beyond the scope of this work.

Figure 1. Schematic of a drop on an inclined surface. The droplet is shown at the onset of motion at the critical tilt angle αc, where θfront = θadv and θrear = θrec.

where m is the mass of droplet, g is the acceleration due to gravity, α is the angle of inclination of the plane surface with the horizontal direction, and θfront and θrear are the contact angles at the front and rear part of drop, respectively. R may be considered as a length scale similar to the drop radius in the plane of the solid substrate, i.e., the radius of the drop base touching the solid substrate, and k is a dimensionless factor that accounts for the contour length and “absorbs” the evolving nonspherical shape of the drop.9 The right side of eq 2 can be considered as the retention force of the droplet, which reaches a maximum at the critical angle of inclination (αc), when the contact angles at the front and rear end of a droplet assume the limiting advancing and receding contact angles. Fmax = σLVRk(cos θrec − cos θadv)

(3)

Beyond the critical angle of inclination, the gravity exceeds the maximum retention force and the drop is depinned from its position and starts to slide downward. The various experimental observations9,10 suggest that partial depinning might occur from either the front or the rear side before the drop slides as a whole. Whether the depinning of a droplet starts from the rear or the front side is attributed to different initial contact angles. Chou et al.10 observed that if the initial contact angle was made equal to the limiting contact angle θadv, the front edge moved at a constant contact angle θfront = θadv, while θrear continuously decreased to θrec. Vice versa, the rear edge moved at a constant contact angle of θrear = θrec, and θfront continuously increased to θadv if the initial contact angle was set to θrec. Santos et al.9 reported in a theoretical study that the front and rear angles continuously varied to θadv and θrec, respectively, if the initial contact angle was equal to (θadv + θrec)/2. However, these studies do not provide sufficient information regarding the evolution of contact angle dynamics having intermediate initial contact angles, i.e., between θadv and θrec, and are limited to a specific type of smooth surfaces exhibiting mostly hydrophobic behavior. In this context, it may be also mentioned that these studies9,10 used the software package Surface Evolver for simulating the drop dynamics. Surface Evolver minimizes the surface area/energy of a triangulated tree interface and thus generically computes static liquid morphologies. To compute dynamic aspects with Surface Evolver is not straightforward and might be done incrementally by computing quasistatic morphologies. 11087

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Figure 2. LBM d2q9 square lattice. 5×) and CCD cameras (PCO 1600, 1600 × 1200 pixel; PCO pixelfly, 1340 × 1040 pixel) during continuous inclination of 0.114°/s. One of the objectives was mounted in the axis of rotation to monitor the side view of the droplets, and the other objective was mounted perpendicular to the axis of rotation to monitor the top view of the droplets. Depending on the expected unpinning behavior, images were captured at rates of 5−30 fps. The contact angles at the front and the rear side of the droplets were analyzed from the side view images using a self-written segmentation routine in Matlab. The analysis routine detected the baseline of the surface and the contour of the droplets and fits separate ellipsoids to both sides of the droplet. The advancing and receding angles were taken at the points where ellipsoids and baseline intercept. As criteria for the length of the elliptic arc we chose 85% of the total drop height. This condition was chosen to guarantee a good and stable fit for all measured droplets. The determined intercepts, however, do not vary with the exact details of the fitting procedure and could be determined with subpixel precision. The obtained contact angles were written in a data file as a function of the angle of inclination together with the position of the points of intercept. The top view images were mainly used to determine the droplet width shortly before depinning and to guarantee a mirror symmetric droplet shape with respect to a plane parallel to the tilt direction. 2D LBM Development. LBM has grown into a stand-alone subject of research in the past decade. The basic concepts of LBM for the simulation of fluid flows have been comprehensively explained in the literature.19,20 In this simulation, we have selected the d2q9 (two dimensions and nine speeds) square lattice schematically described in Figure 2. The lattice allows three types of particles: (1) particles moving along the horizontal and vertical lines, (2) particles moving in a diagonal direction, and (3) particles at rest in the center of the lattice. The equilibrium distribution function of the particle ( f i0(u⃗)) expressed in terms of the dynamic collision invariants, particle number density, momentum, kinetic energy, and momentum flux tensor is as follows

The content of our work is organized as follows. We first describe the experimental setup and procedure and the LBMbased model developed in this study. The experimental drop dynamics is studied and analyzed for three different surfaces hydrophilic, neutrally wetting, and topographically patterned and different liquids. Finally, the results of LBM simulation and experiments are compared and discussed, and the simulation study of the depinning behavior is extended to initial conditions that could not be realized experimentally.

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EXPERIMENTAL AND NUMERICAL DETAILS

Experimental Details. We used ethylene glycol with a purity of 98.3%, glycerol with a purity of 99.5%, and ultrapure water as liquids having different polarity and surface tension. As smooth substrates, we used polished silicon wafers (SI-MAT) with a ⟨100⟩ surface orientation and a natural oxide layer as smooth substrates. These silicon samples were cut in rectangular pieces of about 1 cm2 and precleaned in ethanol, acetone, and toluene using an ultrasonic bath. The remaining organic contaminations were removed by piranha etch ([peroxymonosulfuric acid H2SO5, i.e., 50% H2SO4 and 50% H2O2 (30%)] followed by a thorough rinse in hot ultrapure water. A selfassembly monolayer of octadecyltrichlorosilane (OTS, Sigma-Aldrich) was grafted onto the cleaned silicon wafers using standard protocols.17 The OTS monolayer coating results in contact angles of 114° ± 9° for water, 91° ± 6° for glycerin, and 81° ± 8° for ethylene glycol. In the following text, we will refer to this type of sample as neutrally wetting. To achieve a surface with lower contact angles, the OTS-coated silicon samples were exposed for about 12 s to oxygen plasma (Femto, Diener Electronics), which partially removes the OTS coating. The contact angles achieved by this procedure lie between 81° ± 8° for ethylene glycol, 47° ± 14° for water, and 32° ± 16° for glycerin. We will refer to this type of surface as hydrophilic. The error of the contact angles indicates the difference between advancing and receding contact angle. The individual values for the other experimental situations are shown in Table 1S of the Supporting Information. The surface roughness of both types of silicon wafers is similar to the surface roughness of the polished and cleaned silicon sample and was determined to be smaller than 0.1 nm RMS roughness measured on an area of 1 μm2 using atomic force microscopy. A third type of surface having a well-defined topographic structure was fabricated by wet chemical etching (Zentrum für Mikrotechnologien, TU Chemnitz) and consisted of parallel triangular grooves with a wedge angle of 54.7°, a groove depth of 3 μm, and a spacing of 4.2 μm. The samples were cleaned in ethanol, acetone, and toluene using an ultrasonic bath. This cleaning procedure results in a contact angle averaged over the surface roughness18 of 117° ± 45° for water, 104° ± 56° for glycerin, and 102° ± 53° for ethylene glycol. The large contact angle hysteresis for these substrates results from the additional pinning at the acute edges of the topographies. In the following, we will refer to this topographic sample as the structured surface. To investigate the unpinning behavior of droplets from the abovedescribed samples, we used a motorized and computer-controlled goniometer (Multiscope, Optrel). The sample was placed in the rotation center of the goniometer, and droplets of 50, 20, and 10 μL volumes were deposited using a Pipetman precision microliter pipet. The droplet was monitored by two zoom objectives (magnification 1−

⎡ ⎤ 3 9 f i0 (u ⃗) = fi (0)⎢1 + 3u ⃗ · eȋ − u ⃗ · u ⃗ + (eȋ · u ⃗)2 ⎥ ⎣ ⎦ 2 2

(4)

where u⃗ is the velocity of the particle at the nodes, ȇi is the unit vector along the ith direction, as shown in Figure 2, f i(0) = 4ρ/9 for i = 0, f i(0) = ρ/9 for i = 1−4, and f i(0) = ρ/36 for i = 5−8, ρ(x⃗,t) = ∑i f i(x⃗,t), f i is the discrete distribution function at the nodes, x⃗ is the position vector of the lattice node, and t is the lattice time. Once the equilibrium distribution function is obtained, the lattice Boltzmann transport equation based on the Bhatnagar−Gross−Krook (BGK) approximation for the collision rules is adopted:21

1 fi (x ⃗ + eȋ , t + 1) − fi (x ⃗ , t ) = − (fi − f i0 ) + Si(x ⃗ , t ) τ

(5)

where τ is the time scale associated with collisional relaxation to the local equilibrium whose distribution function f 0i is obtained from eq 5. The tilting angle “α” and the gravity as the external body force (g sin α) acting on the drop due to the incline is incorporated in the lattice Boltzmann eq 5 as the source term Si(x⃗,t) = Δt(wiρ(g⃗·ȇi/cs2)), where wi is the weighting factor for d2q9 lattice 11088

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4 , 9

variation in the nonspherical shape of the drop and that in the length of the contact line with increasing angles of inclination

i=0

⃗ (xpin,s , t ) = k′F (⃗ xpin,s , t ) Fmod

i = 1, 2, 3, 4

where F⃗mod is the modified interaction force at the three-phase contact points (front and rear edge of the contact line on the surface) and xpin,s is the location of the front or rear edge of the contact line on the surface. The parameter k′ of the LBM-based model is analogous to the product of the length scale R and the shape factor k of the physical force balance (eq 2). In the model simulations the value of k′ was varied at each time step of calculations, as the shape transformation and the length scale of the drop varied during tilting. The value of k′ was linearly increased with increasing angles of inclination until the critical angle of inclination in the simulation equaled the experimentally measured inclination angle (αc). Also, variation in the front and rear contact angles obtained from the simulation matched with that of the experimental values during tilting of the plane. Even if the substrate in the simulations is perfectly smooth and homogeneous, the shape of the drop changes with increasing angles of inclinations. This is reflected in the increasing values of the factor k′. In this instance, the model predicted advancing and receding contact angles, and the contact angle hysteresis are compared to the corresponding experimental data. Thus, the model parameter k′ can be construed as a link between the 2D description using the LBM-based model and the 3D behavior of a real drop. The quantitative variation in k′ is discussed later. In the present simulations, the following values were used for different parameters: G = 6.0, Δx = Δy = Δt = 1, 2D computational grids = 1000 × 1000. The size of the drop used in the simulation was proportional to that used in the experimental study. For instance, if 20 grid-radius was used to simulate the 20 μL drop of radius 1682 μm, 16 grid-radius was used to simulate the 10 μL drop of radius 1336 μm, thus maintaining the ratio of radii, 16.82/13.36 ≈ 20/16, in the simulation. Similarly, the appropriate number of computational grids was used to simulate the ridge size used in the experiments for topographic surface and is discussed later. The contact angles were measured from the drop-images generated from the simulations results, using Image-J software. Boundary Conditions. The bounce back scheme proposed by Zou and He26 has been implemented at the solid boundary of the inclined plane in contact with the liquid drop for applying the “no-slip” boundary condition at the continuum scale. Briefly, the incoming particles to the solid surface are deflected at the solid boundary nodes to the parent nodes from which the particles originated. Therefore, the corresponding particle distribution functions for those links are known from streaming. For instance, consider the lattice node shown in Figure 2 to be present on an underlying solid surface. The particle distribution functions except f 2, f5, and f6 are known from streaming. These distribution functions and density are determined by applying mass and x−y momentum conservation equations. Further, the nonequilibrium part of the distribution function for the vertical link to the solid surface ( f 2 and f4 in the present case) is assumed to be conserved. At the curvilinear (slant) boundaries of the patterned surface, the scheme originally proposed by Filippova and Hanel27 was applied for the lattice boundary condition. Briefly, by introducing the concept of “rigid nodes” within the slant solid surface and “fluid nodes” in the liquid/vapor field, the method defines the distribution function for the fluid nodes located adjacent to the solid boundary lying between the nodes of the uniform rectangular lattice.

i = 5, 6, 7, 8

and cs is the velocity of sound in the lattice. As the plane is gradually tilted, the tilting angle (α) varies. Equation 5 is accordingly updated with the modified force at every time step of the model calculation. The liquid−vapor phase separation or transition is simulated using the Shan−Chen (SC) model.22,23 Mechanistically, the phase separation is caused by interparticle forces within the nonideal fluid, which does not occur in an ideal fluid. The SC model calculates the long-range interparticle forces (F⃗) for four nearest-neighbor sites in a two-dimensional (2D) space, according to the following equation 4

F (⃗ x ⃗ , t ) = − ψ (x ⃗ , t ) ∑ Gψ (x ⃗ + eȋ , t )eȋ i=0

(6)

where G is the numerical parameter of the model. This parameter serves the same role in the lattice Boltzmann equation as temperature does in the van der Waals theory of phase transitions and ψ(ρ(x⃗,t)) is the local density distribution function. Yaun and Schaefer24 showed that the interaction force for nearest-neighbor particles can be extended to include other neighboring particles, i.e., to the particles at all eoght sites of the d2q9 lattice, by properly assigning the weighting coefficients for the nearest and next-nearest neighbor sites F (⃗ x ⃗ , t ) = − c0ψ (x ⃗ , t )G∇ψ (x ⃗ , t )

(7)

where c0 is an arbitrary constant and was assigned the value of 6 for the d2q9 lattice. The Shan−Chen model incorporates the interaction force by shifting the velocity in the equilibrium distribution function, f 0i (u⃗), by the following ueq ⃗ = u⃗ +

τF ⃗ ρ

where ue⃗ q is the modified equilibrium velocity. The equation of state for a nonideal fluid is c P = cs 2ρ + 0 G[ψ (ρ)]2 2

(8)

(9)

The SC model uses the function ψ(ρ) = 1 − exp(−ρ) to yield a nonmonotonic pressure−density relationship. According to the model proposed by Borcia et al.,25 the wetting of the surface by the drop may be controlled by using the density field at the solid wall. The solid density is assumed to be larger than the densities of the liquid or the vapor. Thus, by assuming different density values at the computational nodes for the solid, the interaction force between the solid and liquid particles is modified according to the following equation for the local density distribution function at the solid−liquid interface:

ψ (xs , t ) = ρ0 (1.0 − exp(− ρs /ρ0 ))

(11)

(10)

The variable ρs/ρ0 assumes a value between 0 and 1 for a superhydrophobic surface and a completely wetting surface, respectively. ρ0 is the reference density and is generally the maximum density value used in the system. In the present simulations it was assumed to be 1.0. With this approach, the different degrees of wetting of the solid surface by different liquids can be incorporated in the lattice models. Briefly, the numerical methodology included initializing a degree of wetting (ρs) corresponding to the experimentally observed initial contact angles in between the advancing and receding angles. The value of interaction force F⃗ at the front and rear edge of the contact line on the surface, calculated from eq 7, was modified at every time step of the calculation using a factor k′ to take into consideration

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RESULTS AND DISCUSSIONS Experimental Results. The unpinning behavior of droplets on various substrates was observed experimentally following the procedure which is explained in the methods section. A drop of liquid was initially placed on a horizontal substrate obtaining a static contact angle that is between the limiting advancing and the receding contact angles. When the plane is gradually tilted, the contact angles at front and rear sides of the drop change. 11089

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advancing contact angle upon further tilting of the sample. During this partial depinning the droplet slightly elongates in the tilt direction, whereas the extension of the droplet parallel and perpendicular to the tilt direction never differs more than 10% (cf. Figure 3a). When finally the contact angle at the rear side of the drop reached the receding angle at the critical tilt angle αc, also the rear side depins and the drop slides down the inclined plane. While sliding downward the droplet footprint becomes more circular again. A somewhat different behavior is observed for the droplet on the topographically structured surface (cf. Figure 3c). In contrast to the smooth substrates, the three-phase contact line is pinned at the peaks of the grooves and cannot move freely on the topographic substrates. Note that for flat substrates the apparent contact angle corresponds to the material contact angle, whereas the apparent contact angle on topographic substrates is varied by the increased effective surface area and corresponds to the so-called Wenzel angle18 when assuming a random surface roughness. A moving three-phase contact line jumps by one or several periodicities of the topographies, keeping the receding, respectively advancing Wenzel angle about constant. This pinning of the three-phase contact line also causes an increased contact angle hysteresis of about 75° for the topographic surface, compared to the contact angle hysteresis of 16° and 37° for the same liquid on the hydrophilic and neutrally wetting surfaces, respectively (cf. Table 1).

Figure 3 shows the experimental data for the variation of contact angles at the front and rear edges of the contact line

Table 1. Summary and Comparison of Experimental and Simulation Results for Ethylene Glycol on the Hydrophilic, Neutrally Wetting, and Topographic Surfaces experiment

Figure 3. Experimental data for contact angles θfront and θrear vs the angle of inclination α. The data are plotted from the initial value until depinning of the droplet at the critical tilt angle αc for (a) 50 μL glycerol drop on a hydrophilic substrate, (b) 10 μL ethylene glycol drop on a neutrally wetting substrate, and (c) 50 μL ethylene glycol drop on a topographical structure. The arrows in the graph indicate the depinning of the front or rear contact line and the optical micrographs at the right display the side and top view of the corresponding experiments just before complete depinning of a droplet.

vol (μL)

θini (deg)

10 20 50

27.7 28.9 27.2

10 20 50

83.4 82.5 83.5

50

89.6

αc (deg)

θadv (deg)

simulation θrec (deg)

θini (deg)

αc (deg)

θadv (deg)

ethylene glycol on hydrophilic surface 26.2 37.0 20.6 27.3 26.9 34.3 18.9 37.7 16.7 27.3 18.0 38.8 14.1 36.1 16.2 27.2 14.4 43.9 ethylene glycol on neutrally wetting surface 15.3 86.0 76.9 83.4 14.2 87.4 12.8 86.1 76.3 82.5 11.1 86.7 7.8 87.8 73.3 83.5 6.7 86.6 ethylene glycol on topographic surface 47.8 150.4 52.9 89.2 46.1 145.8

θrec (deg) 19.2 19.7 22.9 77.0 76.2 70.4 55.3

Moreover, in our experimental study, the initial contact angle of about 90° was always closer to the receding contact angle, and the front contact angle θfront increases by ∼55° and the rear contact angle θrear decreases by just ∼32° upon tilting the sample. Accordingly, the receding contact angle is reached before the advancing contact angle and the depinning and the sliding of the droplets start from the rear side. However, the difference in tilt angle between the partial and the complete depinning is very small and we only observe a short plateau for the rear contact angle upon tilting the sample. Another qualitative difference is that droplets on grooved substrates with a groove orientation perpendicular to the tilt direction elongate perpendicular to the tilt direction and not parallel to the tilt direction, as was found for droplets on smooth substrates. Depinning experiments were conducted for smooth hydrophilic and neutrally wetting surfaces, as explained in the methods section using ethylene glycol, glycerol, and water and droplet volumes of 10, 20, and 50 μL each. On topographically

with increasing angles of inclination, for three types of surfaces, namely, hydrophobic, neutrally wetting, and rough (topographic), and different liquids. In all our experiments on smooth substrates (Figure 3a,b), the initial contact angle of the deposited droplets was closer to the advancing contact angle for both the hydrophilic and the neutrally wetting substrates. Accordingly, the front contact angle reached the advancing contact angle before the rear side of the droplet reached the receding contact angle upon tilting the sample. During the substrate tilting and the adjustments of the contact angles, the droplet footprint remains circular until one of the three phase contact lines starts to move. When the front side of the drop reached the advancing contact angle, the three-phase contact line at the front depins and adjusts itself to maintain the 11090

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and Jacobi,29 but differs significantly from the values k = 1, k = 4/π, or k = π as reported in refs 30−32, although they use similar assumptions about the geometry of the tilted drop. However, despite providing a description of the roll-off angle, the analytical formula cannot describe the details of the depinning behavior or a possible dependence of the critical inclination angle from the initial conditions. In the following we will describe the depinning behavior for different starting configurations using numerical simulation as explained in the methods section. First, the numerical results are quantitatively compared to the experimental data and the thus verified numerical scheme is used to discuss the situations which could hardly be realized experimentally. Simulation Results. The model developed in this study describes the dynamics of the drop both before and after the surface is tilted. Figure 5 shows the transformations undergone

structured substrates we restricted our experiments to study the depinning behavior of 50 μL droplets due to the large retention force. The results in terms of initial contact angle, critical tilt angle, and advancing and receding contact angle are summarized on the left side of Table 1 for the example of ethylene glycol droplets. From these experimental results, we observe two general trends: (1) A larger contact angle hysteresis is observed for the hydrophilic surface than that for the neutrally wetting surface, which is the largest for topographic surfaces. (2) The critical angle of inclination αc decreases with increasing drop size as the relative importance of the gravitational force is increased. In Figure 4 the maximum retention force Fmax = σLVRk(cos θrec − cos θadv) (eq 3) normalized with the gravitational force Fg

Figure 4. Retention force Fmax = σLVRk(cos θfront − cos θrear) normalized by the kFg plotted for all experimental data as a function of the critical inclination angle. Equation 2 is fitted to the data (red solid line), whereas the dimensionless shape factor k is the only fit parameter; the error of k is shown by the green lines.

Figure 5. Time series showing the spreading of a liquid drop on a neutrally wetting horizontal surface under the influence of gravity as obtained by LBM simulations. The solid surface density ρs was set to 0.38 to initialize the static contact angle equal to the experimentally observed angle of about 88° for a 10 μL glycerol droplet deposited on an OTS-coated silicon surface.

= mg and the dimensionless shape factor k is plotted as the function of the critical inclination angle αc. The graph includes all our experimental data for water, glycerol, and ethylene glycol drops having different volumes of 10, 20, and 50 μL on hydrophilic, neutral, and structured substrates bridging the range of 16° and 162° between the smallest receding contact angle and the largest advancing contact angle. The radius R of the droplet, which was used to calculate the retention force, was determined as half the drop width perpendicular to the tilt direction just before roll off; the definition of R is shown in the top view image of Figure 3a. It turned out that regardless of the exact details of the depinning behavior, i.e., if the depinning starts from front or rear side, the critical tilt angle for the sliding of droplets can be described with eq 3 within the experimental uncertainties for all liquids having different droplet size on all tested smooth substrates. Surprisingly, this applies also to liquid drops on the topographic substrate when considering their Wenzel angles with respect to the horizontal substrate. However, due to the local pinning at the acute edges, the experimental error, i.e. the contact hysteresis, is larger than the error on the smooth surfaces. By fitting eq 2 with respect to eq 3, i.e., a line through the origin, to all data including the topographic substrates the dimensionless shape factor can be determined as k = 1.85 ± 0.11. Within experimental error this value is in reasonable agreement with k = π/2 ≈ 1.571 and k = 1.548, which were predicted by Brown et al.28 and El Sherbini

by a drop after it is placed on a horizontal surface and allowed to reach its equilibrium shape, before the plane is tilted. When the drop is initially placed on a horizontal surface and allowed to settle under the influence of gravity, it may undergo several elongation and contraction cycles, before assuming a static contact angle (see Figure 5). The number of elongation and contraction cycles increases with increasing contact angle of the liquid on the substrate. We elucidate the transience of attaining the initial contact angle through the model simulation results. The final (equilibrium) shape and contact angle of the drop (t = 1000) are used as the input to the model’s initial condition at the onset of tilting. In Figure 6, three examples of the simulation results of transient contact angles upon substrate tilting are presented together with the corresponding experimental data for comparison. The degree of wetting (ρs) was initialized in the model, corresponding to the experimentally observed initial contact angles as described in the methods section on and k′ was varied from 1.000 to 1.015 with increasing tilting angles until the critical angle of inclination in the simulation equaled the experimentally measured inclination angle αc. Within the experimental accuracy, we thus obtain a perfect agreement between the experimental, analytical, and simulated results for the critical angle of inclination. Moreover, we obtain perfect 11091

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Figure 7 shows the experimental and simulation results for the transition in shape and position of a 50 μL water drop on the structured topographic surface. In the present simulation, the exact resolution (exact number of grooves within the drop on the topographic surface) could not be maintained the same as that in the experiment, because of the computational constraints in using a significantly large number of grids or small grid size. For example, the 50 μL droplet (∼4 mm diameter) placed on a topographic surface having 3 μm depth × 4 μm spacing grooves would contain approximately 1000 ridges on the surface. To exactly resolve so many ridges within a droplet on the surface, one would require a significantly large number of 2D computational grids for the simulation. It may be mentioned that so many grids are, however, not required for studying contact angles at two edges of the drop during depinning, in which case only one of the edges moves and the other edge remains pinned. The exact number of ridges on the surface confined within the contact line would surely affect sliding of the drop after depinning, which is not the case here, as the simulation is stopped when the drop begins to slide or both ends are depinned at the critical angle of inclination. In the simulation, however, the ratio of the maximum width or largest expansion of the initially settled drop to the length of the contact line on the surface was maintained the same as that in the experiment, and 3 × 4 lattice nodes were used for the simulation, in proportion to (3 × 4) μm2 dimensions of the grooves used in the experiment. Therefore, the total number of grooves which were in contact with the drop used in the simulation would displace the same volume of liquid as that by the total number of grooves used in the experiment having the proportionate size. The drop is initially assumed to rest on the horizontal surface following the same procedure as for the smooth substrates (Figure 7a). The numbers in Figure 7b (bottom tile) indicate the transient shapes and positions of the drop with increasing inclination angles: The rear and front edge of the drops remain pinned during positions 1−3, whereas θ front and θ rear continuously vary, as observed on smooth substrates. For further increase of the tilt angle, θrear equals θrec and the rear contact line depins. However, as the contact line cannot move continuously, as observed for a smooth substrate, it jumps by one or several grooves to position 4 once the contact angle falls slightly below the receding contact angle. At this position, θrear is slightly larger than or equal to θrec. Upon further tilting the substrate, the rear contact line jumps to position 5 following the same procedure, while θrear remained approximately constant at θrec. Throughout the transition through positions 1−5, the contact line at the front of the drop stayed pinned and θfront continuously increases. The droplet finally slides down the substrate when the advancing contact angle is reached at the critical tilt angle. The same situation is observed experimentally. However, due to a lack of time resolution, the individual jumps of the three-phase contact line could not be observed with the same precision in the experimental results. Figure 7c shows the position of the drop at the critical angle of inclination, with θadv = 149.4° and θrec = 74.6°. It is remarkable that despite the microscopic differences in the depinning behavior on topographic surfaces, the macroscopic depinning behavior of a liquid drop on a topographic substrate is similar to the depinning on a smooth substrate provided the size of the droplet is much larger than the typical periodicity and amplitude of the topography.

Figure 6. Experimental data and simulation results for contact angles θfront and θrear vs the angle of inclination α. The data are plotted from the initial value until depinning of the droplet at the critical tilt angle αc. Symbols and solid lines represent experimental data and simulation results, respectively. (a) 50 μL glycerol drop on a hydrophilic substrate, (b) 10 μL ethylene glycol drop on a neutrally wetting substrate, and (c) 50 μL ethylene glycol drop on a topographically structured substrate.

agreement between experimental and simulation results considering the development of the contact angles at the front and the rear side of the droplet upon tilting the sample, both for the smooth and the topographically structured substrates. The trend of the variation of front/rear angles as a function of the tilt angle in our simulations is also qualitatively similar to that of the variation observed in the studies of Santos et al.9 and Chou et al.10 Specifically, the partial depinning behavior of the edge, which has the contact angle closer to the θadv or θrec, was observed commonly in all studies. Having considered the depinning from smooth substrates, we will now discuss the simulation results for the depinning of a liquid drop on a substrate structured with triangular grooves. 11092

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Figure 7. Experiments (top) and simulation (bottom) results for the dynamics of 50 μL water drop on a topographic surface: (a) static drop, (b) drop at the inclined plane, and (c) drop at the critical angle of inclination.

between the advancing and the receding contact angle, the contact angle at the front and rear side reached their limiting advancing and receding contact angle at the same angle of inclination, and the depinning started simultaneously at both edges, confirming the results from ref 9. For smooth substrates, the critical angle of inclination turned out to be independent of the starting configuration; the slight differences in the advancing and receding angles for different initial contact angles were in the range of ±2.5° (±7%) and arose because of the numerical errors in calculating the contact angles of the drop. In the case of a drop resting on the topographic substrate, it is evident that the variation of receding and advancing Wenzel angles follows a similar trend as that of the contact angles on the smooth surface (see Figure 8c). As the angle of inclination is increased, the front or rear edge of the drop is depinned, depending upon the nearness of initial contact angle to the advancing or receding angle, respectively. Irrespective of the details of the (partial) depinning behavior, the critical angle of inclination and the maximum retention force (cf. eq 3) were obviously not affected by the initial conditions, which confirmed the collapse of all experimental data points onto a line through the origin, as shown in Figure 4. From the model simulation results shown in Figure 8, it may be inferred that irrespective of the drop size, the type of liquid, and the type or topography of the surface, the contact angle dynamics can be described solely by the initial contact or Wenzel angles that the drop makes with the surface. The difference between the depining dynamics of the drop for the smooth and topographic surface is the variation of contact angles during tilting. For smooth surfaces, a nonlinear variation in contact angles, with the asymptotic approach to the advancing or receding contact angles, is observed in cases where the initial contact angles are closer to the advancing or receding contact angles, whereas for the topographic surfaces the variation is approximately linear with increasing angles of inclination. We have earlier mentioned that the value of k′ is linearly varied at every time step of calculation in such a way that the critical angle of inclination of the plane obtained by the

A small mismatch observed in the geometrical similarities of the model and experiment is attributed to the limitation of the simulations, i.e., using the coarser grid size. Furthermore, the simulation results shown in Figure 7b are magnified in comparison to the experimental results (top panel). The agreement between the model simulations and experimental observations may be considered to be reasonable within the limitations of the experimental measurements and numerical analysis. Model Parametric Study for the Depinning Behavior of Drops. Following the strategy above, the model developed in this study was calibrated by the experimental data. Using the calibrated numerical scheme we explored the depinning behavior for the cases that can hardly be obtained experimentally or calculated analytically, including the movement of the three-phase contact lines at the front and rear edges of the drops. Figure 8 shows the model simulation results for the variation of contact angles at the front and rear edges of the contact line with increasing angles of inclination on the smooth (hydrophilic and neutrally wetting) and patterned surface for different initial contact angles of an ethylene glycol drop. The range of initial contact angles spanned between the limiting advancing and receding contact angles. For the smooth surface (Figure 8a,b), the front and rear edge of the drops remained pinned until the angle of inclination increased to a value where either the contact angle at the front side or the rear side reached the limiting advancing or receding contact angle. Upon further increasing the tilt angle of the substrate, the depinned contact line moved, with the advancing or the receding contact angle remaining constant. The critical angle of inclination αc is reached when both the contact angle at the front side of the droplet and the contact angle at the rear side reached the respective advancing and receding contact angles. Whenever the initial contact angle of a liquid drop was closer to its advancing contact angle, the contact angle at the front side reached the advancing contact angle first and the depinning of the drop started from the front side of the droplet upon tilting the substrate. Conversely, the depinning started from the rear side when the initial contact angle was closer to the receding contact angle. Provided the initial contact angle was the mean 11093

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and by lattice Boltzmann modeling, including systematic parameter studies, which can hardly be realized experimentally. The critical angle of inclination and the respective maximum retention force was found to agree with the analytically calculated retention force when determining the drop size from half of the drop width perpendicular to the tilt direction. A common shape factor independent of drop size, contact angle, contact angle hysteresis, start configuration, and surface topography was determined to k ≈1.85, in agreement with analytical predictions.26,27 During tilting, partial depinning and displacement of the three-phase contact line occurred before the onset of drop sliding at either the front or rear side of the drop, depending if the initial contact angle was closer to the advancing or the receding contact angle. Simultaneous depinning of front and rear contact line occurs only if the initial contact angle is the mean between the advancing and receding contact angle. The same macroscopic depinning behavior was found for both smooth and topographically structured substrates, provided the structure size is small compared to the drop size. This finding was surprising, as the three-phase contact line on topographically structured substrates cannot move continuously as for smooth substrates but is arrested at the acute edges of the topographies and thus moves by stepwise jumping across the grooves by one or several periodicities upon tilting the sample. During the stepwise contact line motion the receding or advancing Wenzel angle at the “depinned” contact line is about constant. The difference in the depinning behavior between smooth and topographic substrates is the contact angle hysteresis, which is much larger for topographic substrates, and the partial depinning of front or rear contact line before the onset of drop sliding is much less pronounced for topographic substrates. Both experimentally and numerically observed features can be attributed to the strong pinning at the acute edges of the topographic features.

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ASSOCIATED CONTENT

S Supporting Information *

Figure 8. Model parametric study of the depinning behavior of ethylene glycol droplets having various initial contact angles upon tilting the surface: (a) 20 μL drop on a hydrophilic substrate, (b) 20 μL drop on a neutrally wetting substrate, and (c) 50 μL drop on the structured topographic substrate.

Table S1 and Figures S1−S3. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

simulation matches with that of the experimental values. Also, variation in the front and rear contact angles obtained from the simulation matches with that of the experimental values during tilting of the plane. The Figure 1S Supporting Information shows variation in k′ with increasing inclination angles, corresponding to the simulation cases shown in Figure 8a,b. The Figures 2S and 3S (Supporting Information) depict the effect of k′ over the contact angle dynamics, including critical angles of inclination. The predicted contact angles with and without variation in k′ vary in approximately similar fashion. However, the corresponding angles of inclination are significantly different, with the drop depinning at a smaller critical angle if assuming a constant k′ rather than a linear variation in k′ with increasing tilting angles, as applied in the simulation.

*R.S. e-mail: [email protected]. *N.V. e-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS N.V. acknowledges the Alexander Humboldt Foundation for support in conducting part of the study at the Saarland University, Saarbrücken, Germany, from May to July 2011. The experimental work was supported by the GRK 1276.

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REFERENCES

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CONCLUSIONS The depinning behavior of liquid drops having different sizes and surface tensions sliding down inclined smooth surfaces of different wettability and topography was studied experimentally 11094

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