Induced Detachment of Coalescing Droplets on ... - ACS Publications

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Induced Detachment of Coalescing Droplets on Superhydrophobic Surfaces Mehran M. Farhangi, Percival J. Graham, N. Roy Choudhury, and Ali Dolatabadi* Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC, Canada ABSTRACT: Coalescence of a falling droplet with a stationary sessile droplet on a superhydrophobic surface is investigated by a combined experimental and numerical study. In the experiments, the droplet diameter, the impact velocity, and the distance between the impacting droplets were controlled. The evolution of surface shape during the coalescence of two droplets on the superhydrophobic surface is captured using high speed imaging and compared with numerical results. A two-phase volume of fluid (VOF) method is used to determine the dynamics of droplet coalescence, shape evaluation, and contact line movement. The spread length of two coalesced droplets along their original center is also predicted by the model and compared well with the experimental results. The effect of different parameters such as impact velocity, center to center distance, and droplet size on contact time and restitution coefficient are studied and compared to the experimental results. Finally, the wetting and the self-cleaning properties of superhydrophobic surfaces have been investigated. It has been found that impinging water drops with very small amount of kinetic impact energy were able to thoroughly clean these surfaces.

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INTRODUCTION Water accumulation can lead to serious issues in many natural and engineering systems, such as inhibited data transfer for telecommunications systems, corrosion of structures exposed to the elements, absorption of water into porous building material, decreased aerodynamic performance due to ice accumulation on aircrafts, and damage to buildings and similar structures caused by increased loading from buildup of snow and ice. A popular solution is the application of coatings or surface treatments to drastically decrease the wettability of the surfaces to render them superhydrophobic. Research has progressed in this field rapidly over recent years, in two major domains, developing superhydrophobic surfaces and characterizing the behavior of droplets on these surfaces.1−3 To date, the work done related to understanding the mechanisms of reduced water accumulation has focused on the impingement of a single droplet,1,3,4 the condensation of droplets in a humid environment, and the adhesive strength between the droplet and the surface.5 Mitigating ice accumulation requires both the study of droplet behavior for isothermal systems and the study of the freezing mechanism. The effect of surface geometry, wettability, and environmental conditions is seen in the works of Varanasi et al.,6Meuler et al.,7 and Mishchenko et al.8 Furthermore, these works help to distinguish between ice-phobic and hydrophobic surfaces. The current investigation presents a potential mechanism for reduced droplet accumulation on superhydrophobic surfaces: induced detachment of a sessile droplet due to an impinging droplet. A similar mechanism has been observed by Boreynko et al.9 where condensing droplets coalesce on a superhydrophobic surface, then as a result of the coalescence proceed to detach from the surface. © 2011 American Chemical Society

The main characteristic of a superhydrophobic surface is its low surface energy, which presents itself as a high static contact angle typically larger than 150°, and the hysteresis is less than 5°. In these surfaces, the droplet rests on numerous nanostructures, as a result of which air is present under the droplet. To take advantage of these phenomena, numerous artificial superhydrophobic surfaces have been manufactured using rough polymer textures,10−13 polymer nanocomposites,14 metaloxide nanostructures,15,16 and carbon nanotubes.17,18 Inspiration for fabrication of superhydrophobic surfaces comes from nature in the form of the lotus leaf and the legs of water spiders, to name a few. As an example, a lotus leaf’s surface consists of dual scale micro/nanostructures and a sheet of hydrophobic resin. Cassie, Baxter, and Wenzel studied the principles of superhydrophobic surfaces.19,20 Cassie and Wenzel states refer to whether the droplet is embedded in the roughness of the surface or resting on top of the various peaks on the surface. Whether a droplet exists in a Cassie or Wenzel state on a rough surface was studied in the works of Xia et al.21 Furthermore, the effect of impingement velocity and surface geometry was studied in the works of Smyth et al.22 They found that, at low ratios or spacing to feature size, the sample is likely to remain in Cassie state during the impingement process. The self-cleaning aspect of superhydrophobic surfaces has been examined extensively.12The spreading of a single droplet on a solid surface was studied to understand a fundamental mechanism of the complex problem by Chandra and Avedisian.23 It has been found Received: October 6, 2011 Revised: December 10, 2011 Published: December 15, 2011 1290

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the liquid bridge was shown, and the bridge radius follows a scaling law. The internal motion in droplets has been studied by Kaneda et al.33 Sessile droplets have been identified to exhibit the existence of symmetric circulation flows during evaporation, which either increase or decrease at the axis of symmetry with respect to the movement of the line of contact. Quantitative measurement techniques such as particle image velocimetry (PIV) on either coalescing or sessile droplets experience many limitations such as the optical distortion effect which are the result of differences in the refractive indices between the substrate material, droplet fluid, and the air which surrounds the fluid. Using typical dual-field and tomographic PIV systems, experimentation within index-matched liquids have been performed. The experiments performed have provided a precise understanding into the internal flow in colliding and coalescing drops.26,27Along with these studies, some quantitative experimental works have been conducted on the internal flow in droplets during collision and coalescence in air. In addition to the flow study of inside the droplets, PIV has been used to study the flow of the medium for droplet collisions in liquids. The medium in which a droplet resides is an important parameter in the coalescence process. Both OrtizDuenas et al.46 and Jungyong Kim et al.47 studied the effect of the medium during droplet coalescence by studying the coalescence process of a water−glycerin mixture in a silicon−oil mixture by using tomographic and dual field PIV, respectively. Ortiz-Duenas et al.46 studied the effect of the coalescence between a droplet and a bulk of liquid, while Jungyong Kim et al.47 studied the coalescence of two droplets. Based on the flow filed measurements inside droplets, both works discuss the presence of a film separating the droplet and present the specific instant and location along the interface where the film is depleted and the droplets begin to merge. The present investigation examines the impact, coalescence, and induced detachment of a falling droplet with a stationary sessile droplet lying on superhydrophobic substrate. We have studied the physical mechanism governing the phenomenon of coalescence of a falling droplet with a sessile droplet both experimentally and numerically. A two-phase volume of fluid (VOF) method is used to determine the dynamics of droplet coalescence, shape evaluation, and contact line movement. An unstructured numerical grid is used along with an adaptive local grid refinement technique, which improves the accuracy of numerical results along the liquid−gas interface and decreases the computational cost. Parameters studied to quantify the degree of induced detachment include the restitution coefficient and the contact time. These parameters were studied for different droplet sizes, impact velocities, and center to center distances.

experimentally that, when a droplet impacts on a solid surface, it may spread, rebound, partially rebound, and splash due to different wetting properties of the surface and various impact velocities.24 On a superhydrophobic surface, the droplet can fully rebound with remarkable elasticity.25−27 The restitution coefficient and the contact time are two parameters studied experimentally in the works of Richard et al.,25 Reyssat et al.,28 and Chen et al.29 Richard et al.25 found that the contact time of an impinging droplet does not depend on impact velocity. Reyssat et al.28 examined the deformation and energy losses during the bouncing of the impinging droplet which results in a lower restitution coefficient. Chen et al.29,30 showed that the microstructure of the surface influences the behavior of the restitution coefficient and the contact time with respect to velocity. Substantial research already exists regarding the coalescence of a sessile droplet with an impinging droplet since it is important to a number of industrial processes such as solid-inkjet printing, rapid prototyping, microfabrication, and electronic packaging.24,31−35 These applications require accurate placement of polymer solution or metal droplets on substrates to build images, liquid lines, or electrically conductive lines. Neighboring droplets must overlap and coalesce during the impact process to avoid breaks in the pattern being fabricated. The evolution of surface-shape during the coalescence of two mercury drops on a glass surface, driven only by surface tension, was examined by Menchaca-Rocha et al.36 The final shape of molten wax droplets after they had been deposited along a straight line was investigated experimentally by Li et al.37 They observed that, when two overlapping droplets are deposited on a surface, surface tension might pull the second toward the first, a process recognized as “drawback”. The spacing of subsequent droplets is changed by this motion and can cause breaks in the line. Thoroddsen et al.38 investigated the coalescence of a pendant droplet with a sessile droplet deposited on a surface. Their experimental setup focuses on the initial coalescence motions, by growing a pendent and a sessile drop on vertically aligned metal tubes, until they come into contact. Theoretical investigation of droplet coalescence has been mostly focused on two-dimensional configurations. The coalescence shapes for two-dimensional inviscid wedges, based on the selfsimilar solutions for capillary-driven flows, have been calculated by Keller et al.,39 and Oguzand Prosperetti40 studied the coalescence of rain drops. They observed that a finite velocity of approach will increase the probability of bubble entrapment during coalescence. Ristenpart et al.41 studied the coalescence dynamics of two droplets on a highly wettable substrate and measured the width of a growing meniscus bridge between the two droplets. Andrieu et al.42 studied experimentally and theoretically the description of the coalescence kinetics of two water drops on a plane solid surface. Their results showed that the drops grow by condensation and eventually touch each other and coalesce. Duchemin et al.43 theoretically investigated the surface tension driven coalescence of two inviscid drops. The impact of multiple drops onto a dry substrate and their interactions were studied experimentally and theoretically by Roisman et al.44 They showed that their experimental results and analytical model are valid for inertia-dominated impact, when the Reynolds and Weber numbers are large and when droplets are weakly interacting. Eggers et al.45 studied the coalescence of two drops surrounded by a viscous fluid when the Reynolds number Re ≪ 1. They defined Reynolds number of the surface tension driven flow using the radius of the bridge along the initial contact line as the characteristic length. The early time evolution of the shape of

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EXPERIMENTAL METHOD

The profile of the coalescing droplets was identified by using a setup similar to shadowgraphy: backlighting and edge detection software from MATLAB. Experiments were performed at 20 °C and a relative humidity of 25%. As seen in the works of Yin et al.48 and Mockenhaupt et al.,49 humidity and temperature can have a significant impact when environmental conditions deviate significantly from standard room temperature conditions. However, these works show that in the range 30−60% relative humidity and temperatures ranging 10−30 °C the wettability of superhydrophobic surfaces does not show a substantial change. The shadowgraphy images were processed using image processing tools included in MATLAB to track the boundaries of the two droplets and quantitatively study the coalescences process. 1291


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In addition, qualitative images related to surfaces features were obtained using an angled view and front lighting. Experimental Apparatus. The experimental setup includes a positioning system, imaging equipment, and substrate and droplet generator which are mounted on an optical bench and breadboard. The sessile droplet is aligned using a screw drive micropositioning stage. The setup is shown in Figure 1.

static contact angle of 156° and advancing and receding contact angle of 162/148°. It can be observed qualitatively from Figure 2 that the ratio between spacing to feature size is much less than 1; therefore, the droplets should behave in a Cassie state. Advancing and receding contact angles were measured by a tilting plate method, where a plate is inclined until the droplet begins to slide; at the moment preceding sliding, the advancing and receding angles was measured. This process is observed using a high speed camera to capture the correct moment of sliding. Pinning was not observed, since the required inclination of the substrate was on the order of a few degrees. Image Processing. Performing quantitative analysis of the droplet coalescence was done using the image processing toolbox in MATLAB. The air−water interface was identified by comparing each recorded image to a prerecorded background image so that the background can be removed, then converting the resulting image to a binary image. An example of the detected interface is shown in Figure 3. Obtaining the binary image required two parameters to be adjusted for each run, a tolerance between the background and the images of the run and a threshold to convert the grayscale image to binary. One parameter investigated experimentally is the restitution coefficient, which is defined as the ratio of the detachment velocity to impinging droplet velocity. This parameter was studied by Chen et al.29 for the case of a single droplet impinging on a lotus leaf and on an artificially prepared surface; no mention of how the detaching velocity is obtained is present in their works. It is easier to determine to restitution coefficient for the case of a single droplet than for a merging droplet, because there is less deformation of the droplet while it is detaching. Four different methods of determining the detaching velocity were compared. Each of the methods involved tracking a certain point on the droplet, then using linear regression analysis of twenty to thirty frames to determine the detaching velocities of the points of interests. The points studied for each method are as follows: (a) studying the geometric center of the 2D side view; (b) studying the average of the top and bottom velocities of the droplet; (c) inferring a 3D geometry based on the 2D profile and tracking the centroid; and (d) assuming a velocity profile based on the top and bottom and taking a weighted average based on the inferred 3D geometry used when tracking the centroid. The fourth option was found to give the most consistent results. This method is ideal for head-on impacts due to the axisymmetric configuration. A result was deemed invalid if the standard deviation of the inferred volume was more than 15% of the mean inferred volume for a head-on impacts and more than 25% for offset impacts.

Figure 1. Schematic of the experimental setup. Droplets are generated by detachment from the tip of a syringe of three different diameters to allow for three different droplet sizes. The droplets are then accelerated by gravity from heights between 30 mm and 50 mm. Flow is driven by a pressurized water tank at a pressure below 10 psi and controlled by a solenoid valve which is synchronized to the high speed camera. The number of runs and sizes and speeds of the droplets used to study the effect of spacing are shown in Table 1.

Table 1. Droplet Properties D0 (mm) case

mean

standard deviation

1 2 3 4 5

3.57

0.15

2.59

0.06

U0 (mm/s) mean

standard deviation

number of runs

670 935 744 847 955

44 20 25 23 23

24 27 38 40 28

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Imaging equipment used includes a Photoron SA1.1 (Photron, California USA) high speed camera, an LED light, and a high magnification zoom lens (Navitar, New York USA). A magnification up to 10.5× can be achieved with this lens, which is composed of a microscopic zoom, a tube adaptor, and lens attachment. The camera was run at its full resolution of 1024 × 1024 pixels and frame rates between 3000 and 5400 fps. A shutter speed of 500 000 1/s was used to ensure clear images for the shadowgraphy (replace by backlight image). The shutter speed was decreased when a large field of view is required, since light was diffused to a greater degree. In order to provide enough light, and capture the coalescence from an angled view, the camera was run at 1000−2000 fps with a shutter speed of 1/4000 s. Preparation of the Surface. The superhydrophobic surface was fabricated with a commercially available superhydrophobic WX2100 spray coating (made by Cytonix, Maryland, USA), on a polished aluminum sample. This coating is composed of microparticles embedded in a hydrophobic liquid; the induced small scale roughness causes the surface to become superhydrophobic. Surface image of rough superhydrophobic sample is shown in Figure 2. SEM micrographs were made with a JSM840 JEOL scanning electron microscope equipped with an X-ray spectrometer selection of energy (EDS) for performing chemical analysis. The particle sizes vary in the range of 1−20 μm. Based on the SEM measurements, the average roughness of the coated sample is 1.72 μm. This results in a superhydrophobic surface with

NUMERICAL METHOD The governing equations are described by the mixture continuity and momentum equations

∇⃗ · V⃗ = 0

(1)

∂ρV⃗ ⃗ ⃗ ) = −∇⃗p * + ∇⃗ ·(μ·∇V⃗ ) + (∇⃗V⃗ ) ·∇⃗μ + ∇⃗ ·(ρVV ∂t − g ·x∇⃗ρ + γk∇⃗f

(2)

where V⃗ is the velocity vector, t is time, x is the coordinate vector, p* = p − ρg·x is the modified pressure, γ is the surface tension, k is the curvature of the interface, and f is the volume fraction which is used to compute the mixture density and viscosity

ρ = f ρl + (1 − f )ρg μ = f μl + (1 − f )μg

(3)

VOF method uses a scalar field, f, whose value is unity in the liquid phase and zero in the gas. When a cell is partially filled 1292


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Figure 2. SEM micrographs of the tested superhydrophobic surface. The inset picture shows the micro particles in detail.

facilitates a sharp interface between phases. This is achieved by introducing an extra artificial compressive term into the volume fraction equation

∂f + (V⃗ ·∇⃗)f + ∇⃗ · [Vr⃗ f (1 − f )] = 0 ∂t

(6)

→ ⎯ where Vr , is a velocity field normal to the interface which is used to compress the interface ⎯→ ⎯ Vr

|n · V ⃗ | |S|2

(7)

where kc is an adjustable coefficient used to adjust the amount of compression. S is the surface area vector. We used kc = 1.5 which was also shown by Rusche50 to provide sharp interface between phases. The interface unit normal n is computed by taking the gradient of a smoothed (via elliptic relaxation) volume fraction f * at the cell faces

Figure 3. Depiction of raw image (a) and traced boundaries (b) at three distinct stages, before merging, during spreading, and prior to detachment.

with liquid, f will have a value between zero and one.

⎧1 in liquid ⎪ f = ⎨> 0, < 1 at the liquid − gas interface ⎪ in gas ⎩0

= kcn max

n=

∇f *f |∇f *f | + δ

(8)

It is noted that, due to the function of f(1 − f), the artificial compressive term is only active near the interface. The continuum surface force (CSF) method51 is used to model surface tension as a body force (Fb) that acts only on interfacial cells. The overall solution strategy is based on the pressure implicit with splitting of operators (PISO) algorithm;52 however, it departs from its typical implementation since the implicit solution of the momentum equations is not utilized. After the time step is adjusted to meet the specified maximum courant number, the volume fraction equation is solved, then the resulting field is smoothed and the unit normal vector and interface curvature are calculated based on that. Finally, the PISO algorithm is used to calculate the pressure and velocity.

(4)

The discontinuity in f is propagating through the computational domain according to the following

∂f + (V⃗ · ∇⃗)f = 0 (5) ∂t The governing equations are discretized using finite volume scheme for general unstructured polyhedral cells. Following the method outlined by Rusche,50 the volume fraction equation is formulated with a bounded compression scheme, which 1293


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Figure 4. Computational domain for (a) unrefined and (b) adaptive local refined mesh.

is replaced by either the advancing contact angle, θA, or the receding contact angle θR depending on the direction of the velocity vector at the contact line.

The nonlinear system of the flow equations is solved numerically on a three-dimensional unstructured mesh using a recently developed adaptive local mesh refinement technique,53 which enhances the accuracy at the interface region and achieves low computational cost compared to the case of a uniform mesh with the same resolution. A base mesh is used and the cells at the region of the interface are subdivided according to the levels of local mesh refinement, while the interface lies always in the densest mesh region since the mesh is reconstructed at every time steps. The cell size used in this study is set based on a mesh refinement study in which the grid size is progressively decreased until no significant changes are observed in the simulation results. The mesh resolution is characterized by the number of cells per the droplet diameter. From the mesh refinement study, the optimum mesh size was found to be 25 cells per drop diameter and 2 levels of local grid refinement at the interface region. This resulted in a resolution of 50 μm at the interface region. Therefore, this mesh size was used for all simulations throughout this article. Figure 4 shows the comparison of an unrefined mesh with adaptive local refined mesh. Dynamic Contact Angle. The dynamic contact angle is assumed to be a function of the contact line velocity. There are several empirical formulas expressing the dependence of the dynamic contact angle on the contact line velocity.54−56 Most of them are in the form of the Hoffman−Tanner− Voinov law, or in a more general form derived by Cox.55 The predicted values of the dynamic contact angle are very similar for all of these models. In this study, the correlation of Kistler57 is used to calculate the dynamic contact angle for each time step.

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RESULTS AND DISCUSSION As a first step, the numerical methods were validated by comparing the spread diameter and free surface with the results for different impact velocities, droplet sizes and separations. Experiments were carried out by varying the separation between the sessile and the impacting droplet from the axisymmetric drop on drop to sufficiently large offsets to allow the impinging droplet to spread along the surface prior to merging. The governing parameters for the collision of isothermal droplets include the droplet diameter D0, the impact velocity U0, the contact angle of droplet with the substrate θe, the contact angle hysteresis, the droplet density ρ, the surface tension γ, the droplet kinetic viscosity μ, the gravitational acceleration g, and offset distance L.

θd = [ca + fH−1 (θe)] (9) where f H−1 is the inverse function of the “Hoffman’s” empirical function which is given in the following form. ⎧ ⎡ ⎡ ⎤0.706⎤⎫ ⎪ ⎪ x ⎥⎬ fH = arccos⎨1 − 2 tanh⎢5.16⎢ ⎥ 0.99 ⎪ ⎪ ⎣ 1 + 1.31x ⎦ ⎢⎣ ⎥⎦⎭ ⎩

(10)

The Capillary number is defined as

ca =

μVcl γ

Figure 5. Definition of the main dimensions characterizing the droplet coalescence. (11)

To quantify the overlap between droplets (see Figure 5), an overlap ratio is defined as

where μ is the dynamic viscosity, Vcl the contact line velocity, and γ the surface tension. Most surfaces are not ideally smooth and are subject to the effect of contact angle hysteresis. In order to consider this effect in the numerical model, the equilibrium contact angle θe in eq 9

λ= 1294

L D0

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Figure 6. Time evolution of head-on impact and coalescence of two water droplets on a superhydrophobic surface (We = 22, Re = 2020). The experimental images (left side) are compared with simulated images (right side).

where D0 is the droplet diameter. An offset distance of zero results in a head-on collision. To quantify deviations from ideal behavior, a nondimensional spread length similar to the works done by Li et al.58 is defined as

ψ=

DY D0 + L

Figure 6 the shows time evolution images of a head-on impact and coalescence of the impinging droplet and the sessile droplet. The diameter of each droplet is 2.6 mm, and the impact velocity is 0.774 m/s, resulting in Re = 2027 and We = 19.8. It can be clearly observed from this figure that, after the second droplet impacts and merges with the sessile droplet, the fluid spread out horizontally forming a flattened toroidal disk shape at around 9 ms. After the maximum spreading is achieved, surface tension force pulls the two edges back. The motion is radially inward at around 13 ms after the impact. A jet rises in the center (Worthington jet),59 caused by the kinetic energy developed during the recoiling process, which can lead to a lift off of the whole merged drop. Experiments along with numerical simulations were carried out varying the separation between the sessile and the impacting droplets from the axisymmetric drop-on-drop case up to the point of merging while spreading. The first case presented when studying the effect of offset is a side view of a near head-on collision seen in Figure 7. The diameter of each droplet is 2.6 mm, with Re = 2027 and We = 19.62. As shown in Figure7, after the impact of the second droplet on the sessile droplet the merged drop spreads out, increasing the surface area of the drop, thus increasing its surface energy. Once the restoring force of surface tension becomes dominant over the inertial force, at approximately 18 ms, the merged droplet recoils. The droplet height rises until it detaches from the surface. Figure 8 serves to quantify the spreading and bouncing behavior of a merged droplet by plotting the dimensionless spread length, ψ, versus dimensionless time. Both numerical and experimental results show that, after the impinging droplet

(13)

The spreading times are nondimensionalized as

U τ=t 0 D0

(14)

In addition, the investigation of the coalescence phenomenon requires examination of the most significant dimensionless numbers, such as the Weber number (We), and the Reynolds number (Re), defined as

We =

ρ lU0 2D0 γ

Re =

ρ lU0D0 μl

(15)

Superhydrophobic surfaces represent a class of special surfaces which have very low surface energy and are strongly water-repellant. In order to study the coalescence behavior of two droplets on superhydrophobic surfaces, experimental tests along with simulations are conducted. Coalescence dynamics can be observed from time-resolved images. The air−water interface was tracked for each image allowing the time evolution of droplet position and velocity. Three quantitative parameters are presented: the spread diameter, contact time, and restitution coefficient. 1295


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Figure 7. Coalescence of two water droplets on a superhydrophobic surface with a small offset of λ = 0.15 (We = 19.62, Re = 2027). The experimental images (left side) compared to calculated images (right side).

which can be attributed to viscous dissipation; thus, less mechanical energy in the system was observed. This decreased detached time ultimately translates to the droplet failing to detach from the surface. In addition to the decrease in detached time, the maximum spreading also decreases with successive bounces. These two trends can be simultaneously explained through the analogy of damped harmonic motion where potential energy is transferred from surface energy to gravitational potential energy. As time progresses, energy is dissipated causing the droplet to have less surface energy expressed as the wetting length, and less gravitational energy expressed as a shorter detached time. After multiple bounces, the discrepancy between experimental and numerical results begins to increase, likely due to accumulated error from inhomogeneity in the surface. To understand the mechanism of droplet merging and bouncing, The surface and kinetic energy are calculated at various stages. Patankar2 described the stable surface free energy, Gs, of a sessile droplet in contact with a textured surface as

Gs

Figure 8. Temporal evolution of dimensionless spread length for droplet coalescence on a superhydrophobic surface (We = 19.62, Re = 2027, λ = 0.15).

3

9π V

2/3

γ

= (1 − cos θe)2/3 (2 + cos θe)1/3

(16)

Before impact, the kinetic energy KE1 and surface energy SE1 of a spherical droplet are given by

impacts and merges with the sessile droplet, the spread length increases rapidly until it reaches its maximum value, which is about four times the equilibrium spread length of the sessile droplet. After this point, surface tension forces cause retraction of the contact line until the droplet detaches from the surface. Once the kinetic energy of the detached droplet is converted into potential energy the merged droplet starts to fall and spreads out on surface. This process continues until the droplet reaches the equilibrium position. The effect of viscous dissipation on the maximum spreading and bouncing of the droplets can be deduced from Figure 8. A decrease in detached time for successive bounces can be seen,

KE1 =

⎛ 1 2⎞⎛ π 3⎞ ⎜ ρU ⎟⎜ D ⎟ ⎝ 2 0 ⎠⎝ 6 0 ⎠

SEI = πD02 γ

(17) (18)

Therefore, the total surface energy of the sessile and impacting droplets can be calculated by

SE1 = SEI + Gs

(19)

After impact, when the merged droplet is at its maximum extension diameter Dmax, the kinetic energy is zero, and the surface 1296


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energy SE2 is60

SE 2 =

π 2 Dmax γ(1 − cos θA ) 4

The evolution of the free surface for a moderate spacing of 0.65 is shown in Figure 10. The diameter of each droplet is 2.55 mm, and the impact velocity is 0.757 m/s, giving Re = 2027 and We = 19.62. In the figure, the shape evolution of the two droplets during the coalescence phenomena from simulation is shown together with the images of the experimentally observed drop shapes. As seen from Figure 10, the impinging droplet lands partially on the sessile droplet and partially on a dry surface at 2.6 ms. After merging with the sessile droplet, the inertia of the second droplet causes the merged drop to deform and spread. After reaching the maximum spread diameter (at 6 ms), surface forces acting on the drop reduce the surface area of the merged drop. Developed inertia during the recoiling process lifts the merged drop from the surface at about 26 ms. Figure 8 shows that the merged droplet rotates in air between 26 ms and 35.6 ms after bouncing. Figure 11 shows the comparison of simulated results with experiments for the evolution of dimensionless spread length versus dimensionless time. It can be observed from the figure that the hanging time for the merged drop after bounce off from surface for the first impact is much longer than that for the second and third ones, which indicates a large amount of dissipation during the detached time and subsequent impact. The deviation of the simulation results from experiments magnifies after the second impact. Simulated results of the coalescence of two droplets at a large offset are compared with the experimental images in Figure 12. The diameter of each droplet is 2.6 mm, and the impact velocity is 0.786 m/s, giving Re = 1977 and We = 21.8 at an overlap ratio of 1.1. As shown in Figure 12, droplet impinges on the dry section of the surface. Then, the inertia causes the droplet to spread out and during the spreading the impinging droplet merges with the sessile droplet. Afterward, the kinetic energy of the merged droplet is converted into surface energy,

(20)

The calculated variation of the surface energy and kinetic energy of the merged droplet at the moment of droplet impact and maximum spreading on superhydrophobic surface is shown together in Figure 9. This figure shows that kinetic and surface

Figure 9. Variation of surface and kinetic energy of the merged droplet during bouncing on superhydrophobic surface (We = 19.62, Re = 2027, λ = 0.15).

energy were interchanging between each other and loss of energy occurs through viscous dissipation.

Figure 10. Coalescence of two water droplets on a superhydrophobic surface at a moderate offset of λ = 0.65 (We = 19.62, Re = 2027). The experimental images (left side) compared to simulated images (right side). 1297


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Figure 11. Temporal evolution of dimensionless spread length for droplet coalescence on a superhydrophobic surface at a moderate offset of λ = 0.65 (We = 19.62, Re = 2027).

Figure 13. Temporal evolution of dimensionless spread length for droplet coalescence on a superhydrophobic surface (We = 21.8, Re = 1977, λ = 1.10).

which will be used for the retraction and rebound of the merged drop. Figure12 shows that the merged droplet rotates in the air after bouncing-off (between 24.2 ms and 44.4 ms). As spacing increases, the merging process imposes more viscous dissipation. Figure 13 displays the comparison of the simulated results with the experiments for the evolution of

dimensionless spread length versus time. The maximum spread diameter decreased abruptly with successive bounces as compared to the results of Figure 11, which has moderate overlap ratio. It has been observed that by increasing the overlap ratio the number of detachments of the droplet from surface decreased. This would imply more viscous dissipation

Figure 12. Coalescence of two water droplets on a superhydrophobic surface at a large offset of λ = 1.10 (We = 21.8, Re = 1977). The experimental images (left side) compared to the simulated images (right side). 1298


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Figure 14. Velocity fields calculated from simulation of the coalescence of two droplets for We = 19.62, Re = 2027, at overlap ratios of (a) 0.15, (b) 0.65, (c) 1.1.

while the droplet is detached from the surface. The surface energy released upon coalescence of two drops can be as high as 20% of the original energy Figure 14 illustrates the effect of offset on the internal velocity of the droplets in order to understand the rotational behavior of the detached droplet. The largest vector size in each case corresponds to the velocities of 0.6 ms−1, 0.5 ms−1, and 0.5 ms−1, respectively. For the near head-on offset, a cavity at the center of the droplets was formed and the merged droplet took a toroidal shape at 7.2 ms. The droplet begins to detach at about 12 ms. During detachment, the largest velocities are upward and at the center of the droplet. At a moderate overlap ratio of 0.65, the droplet begins to detach at around 18 ms; in contrast to the near head-on case which has its largest velocity at its center and in an upward direction, the moderate overlap

ratio has two dominant velocities, one mostly upward and one parallel to the surface and toward the impinging droplet, which causes the rotation of the merged droplet after detachment. At a large overlap ratio of 1.1, more rotation is observed, which is consistent with the greater horizontal velocity of the base of the droplet during detachment. It is important to note that the delay in change of velocity of the sessile droplet increases with increasing offset. This increase in delay is seen when comparing the large offset at 7.4 ms to the moderate offset at 6 ms; furthermore, no delay is observed for the near head-on scenario. The horizontal velocity component of the base of the droplet increases with increasing offset because the sessile droplet only begins to flow in the later stages of detachment. The contact time and restitution coefficient are two crucial parameters for droplet−surface interaction. The contact time is 1299


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the time that the merged droplet is in contact with the surface before detaching, and the restitution coefficient, ε, is the ratio of impact velocity to the rebound velocity of the merged droplet. In this section, the effects of impact velocity, droplet sizes, and droplet separation on these two parameters are investigated both numerically and experimentally. Our investigation may help measure the efficiency of water repellent (superhydrophobic) surfaces and improve the cooling of hot solids, for which drop rebounds are a severe limitation. The effect of impact velocity and droplet diameter on contact time for head-on collisions is shown in Figure 15. As seen in

Figure 16. Contact time versus droplet diameter.

Figure 15. Contact time versus impact velocity for different droplet sizes.

this figure, the contact time is found to be independent of the impact velocity in the range studied; however, it largely depends on droplet diameter. This is similar to a harmonic spring, where the natural frequency of the droplet depends on its size and surface tension and not any initial conditions imposed on it. Figure 16 shows the effect of droplet diameter on the contact time of head-on coalescence of two droplets on the superhydrophobic surface. As can be observed from this figure, upon taking a power regression to the data, it was determined that the least-squares fit had an exponent of 1.678, which compares well to the work of Richard et al.4 who developed a model by balancing inertia (of order ρR/tc2) with capillarity (γ/R2), which yields tc ∼ (ρR3/γ)1/2. Uncertainty in the measurement of droplet size is based on the standard deviation of droplets, while the temporal uncertainty is based on the recording rate of the camera. The resulting power law relating the diameter of the drop to its contact time is given by tc ∞ D01.5. The dimensionless contact time of the merged droplets as a function of overlap ratio is displayed for different Weber number in Figure 17. As can be seen from this figure, by increasing the separation between the centers of the droplets, the dimensionless contact time remains almost constant, while increasing the Weber number increases the contact time. The restitution coefficient, ε, is the ratio of the impact velocity U0 to the rebound velocity of the droplet. Considering that the final droplet volume is twice the impacting droplet volume, the ideal value of the restitution coefficient is 0.5. The restitution coefficient for the studied superhydrophobic surface for head-on impact is plotted as a function of impact velocity for different

Figure 17. Dimensionless contact time with varied overlaps and impact velocity.

droplet sizes. As shown in Figure 18, the restitution coefficient decreases for both increasing size and impact velocity due to an increase in energy dissipation. An increase in energy dissipation is due to either an increase in the deformation of the merged droplets, such as droplet elongation, or oscillation of the detached droplet. It was difficult to obtain accurate experimental results for the coefficient of restitution, since inferring a threedimensional geometry based on the two-dimensional profile has some inaccuracies. However, a good qualitative agreement is observed between predictions from the numerical model and measurements from the experiments. Figure 19 illustrates the effect of overlap ratio on the restitution coefficient for the studied superhydrophobic surface for different Weber numbers. As seen from the figure, increasing the overlap ratio decreases the restitution coefficient. Moreover, the figure shows that increasing the Weber number decreases the restitution coefficient. This can be attributed to two effects, an increase in dissipation within the droplet and an increase in the rotational momentum at the expense of translation 1300


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distances between the center of sessile and impacting droplet, the merged droplet rotates in the air after bouncing off from the surface. The effect of droplet size and impact velocity on contact time was also investigated and compared well with numerical simulations. It has been found that the contact time (tc) is independent of the impact velocity in a wide range of velocities; however, it largely depends on droplet diameter. It was determined that contact time is well-described by the power law tc ∼ D01.68. Finally, the effect of droplet size, impact velocity, and overlap ratio on restitution coefficient was investigated. It was found that increasing the impact velocity, droplet size, and overlap ratio reduces restitution coefficient due to either the increased deformation of the merged droplets or increased rationality. Based on the evolution of the free surface and the spread diameter of the merging droplets, the use of dynamic contact angle and bounded compression in the numerical works was validated. Since the numerical simulations proved accurate, they were used to obtain internal flow velocity. The partnership between experimental and numerical works allowed for understanding of the mechanism of induced detachment of droplets on the studied superhydrophobic surface.

Figure 18. Restitution coefficient ε for head-on coalescence as a function of impact velocity U0 for different droplet sizes.

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AUTHOR INFORMATION

Corresponding Author

*Author E-mail Address [email protected]. D0 U0 k g θe θd θR θA p t tc f We Re Ca ψ kc L Dy Ds Vr Vcl V Gs SEI KE g

Figure 19. Restitution coefficient ε, as a function of overlap ratio λ for different Weber numbers.

momentum. At larger spacing, a larger moment is imposed on the droplet system due to a greater moment arm caused by the spacing.

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CONCLUSION The coalescence of two droplets on a solid superhydrophobic surface was experimentally and numerically investigated. The evolution of surface shape during the coalescence of two droplets on a superhydrophobic surface is observed using high speed imaging and compared with the numerical results. Experiments along with numerical simulations were carried out varying the separation between the sessile and the impacting droplet from the axisymmetric drop-on-drop case up to the point where the impinging droplet fully impacts a dry surface and coalesces while spreading. It has been observed that, after the second droplet impacts and merges with the first droplet, the inertia of the second droplet causes the merged drop to deform and spread. Developed inertia during the recoiling process lifts the merged drop from the surface. It was found that for larger

NOMENCLATURE Drop diameter, [m] Impact velocity Curvature of interface gravitational acceleration [m/s2] Equilibrium contact angle [deg] Dynamic contact angle [deg] Receding contact angle [deg] Advancing Contact angle [deg] pressure [N/m2] time [s] Contact time [ms] Fractional amount of liquid (dimensionless) Weber number (dimensionless) Reynolds number (dimensionless) Capillary number (dimensionless) Spread length (dimensionless) Adjustable coefficient Center to center distance [m] Spread length [m] Sessile spread diameter [m] velocity [m/s] Contact line velocity [m/s] Droplet volume [m3] Surface energy of sessile droplet [N-m] Surface energy of impacting droplet [N-m] Kinetic energy [N-m] Gravitational acceleration [m/s2]

Greek Letters

ρ ρl ρg Δρ μ μl μg 1301

density [kg/m3] liquid density [kg/m3] gas density [kg/m3] density difference [kg/m3] dynamic viscosity [kg/(m s)] liquid dynamic viscosity [kg/(m s)] gas dynamic viscosity [kg/(m s)] dx.doi.org/10.1021/la203926q | Langmuir 2012, 28, 1290−1303

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surface tension [kg/s2] pressure difference [N/m2] Restitution coefficient Overlap ratio Dimensionless time Dimensionless contact time

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Subscript and Superscript

l g cl

Liquid Gas Contact line

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