Pancake Bouncing - American Chemical Society

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Pancake Bouncing: Simulations and Theory and Experimental Verification Lisa Moevius,† Yahua Liu,‡ Zuankai Wang,‡,§ and Julia M. Yeomans*,† †

The Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, United Kingdom Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong § Shenzhen Research Institute of City University of Hong Kong, Shenzhen, China ‡

ABSTRACT: Drops impacting superhydrophobic surfaces normally spread, retract, and leave the surface in an approximately spherical shape, with little loss of energy. Recently, however, it was shown that drops can leave the substrate before retracting while still in an extended pancake-like form. We use mesoscale simulations and theoretical arguments, compared to experimental data, to show that such “pancake bouncing” occurs when impacting fluid that enters the surface is slowed and then expelled by capillary forces. For the drop to bounce as a pancake, two criteria must be satisfied: the fluid must return to the surface at the appropriate time, and it must do so with sufficient kinetic energy to lift the drop. We argue that this will occur for superhydrophobic surfaces with topological features having dimensions of ∼200 μm, larger than those normally considered. The contact time of pancake bouncing events is reduced by up to 5-fold compared to that of conventional bouncing, suggesting relevance to drop shedding and anti-icing applications.

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tension energies. The Weber number is given by We = ρv02r0/γ, where ρ is the density and γ is the surface tension of the drop, r0 is its radius, and v0 is the impact velocity. We have recently shown19 that a liquid drop can also directly rebound from a superhydrophobic surface at, or close to, its maximum extension before undergoing lateral retraction. Figure 1 compares conventional bouncing on a superhydrophobic surface (Figure 1a) to a case where the drop leaves the surface at its maximum lateral extension (Figure 1b). We shall call this pancake bouncing. As shown in Figure 1c, the surface used in this experiment was patterned with square posts of side length b = 100 μm in a square lattice with center-to-center spacing of w = 300 μm and height h = 1.4 mm. The surface was coated with a thin, rough hydrophobic layer of flowerlike protrusions, with the scale of roughness being ∼3μm. The post coating leads to very strong hydrophobicity: a flat surface covered with the flower structure alone shows an intrinsic contact angle of 160°. With posts in place, the apparent contact angle of a suspended

INTRODUCTION The study of liquid drops impacting solid surfaces has been of interest for more than a century because of its scientific and practical importance to, for example, crop spraying, inkjet printing, spray cooling, and soil erosion. As a drop lands, it spreads across the substrate because of its inertia and then retracts under the influence of surface tension.1−5 If the drop still has sufficient energy, then the horizontal motion is rectified into vertical motion by internal flows and the drop can bounce off the surface.6−8 However, complete bouncing is uncommon for drops on flat surfaces because of frictional losses. Recently, it has been possible to fabricate surfaces that are structured on micro- and nanoscales.9−12 Surfaces that are hydrophobic and patterned by a lattice of posts exhibit superhydrophobic behavior characterized by large contact angles and easy runoff.9,13−15 Several investigations have shown that droplet bouncing can be greatly enhanced on superhydrophobic surfaces because of the reduced friction.6,7 Typically, a drop spreads, retracts, and then bounces in an approximately spherical shape.6,8,16−18 The dimensionless variable that governs the bouncing dynamics is the Weber number, which measures the ratio of inertial and surface © 2014 American Chemical Society

Received: August 25, 2014 Revised: September 30, 2014 Published: October 6, 2014 13021

dx.doi.org/10.1021/la5033916 | Langmuir 2014, 30, 13021−13032

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Figure 1. Drop impact experiments on square posts.(a) Drop impact experiments for b = 100 μm, wx,y = 300 μm, h = 1.4 mm, and r0 = 1.45 mm at We = 4.7. The drop retracts before bouncing. (b) Pancake bouncing is observed for the same drop radius and surface at We = 8.7. (c) SEM micrograph of the substrate used in the experiments showing square posts covered with smaller-scale, flower-like roughness.

drop was over 165°, with a contact angle hysteresis of 3.3°.19 The impacting drop in Figure 1a,b has a radius of r0 = 1.45 mm. For this substrate, pancake bouncing occurred for 6.3 < We < 9.5. Similar pancake bouncing has also been seen on surfaces with posts with a diameter that decreases from the base to tip and on a more disordered surface comprising layers of hexagonal rings topped with posts. (See previous work.19) We shall concentrate on straight post surfaces here as the most direct comparison to the simulations and theory. In previous research,19 we argued that pancake bouncing occurs because, as the drop lands on the superhydrophobic posts, inertial forces push liquid into the structure, transferring kinetic energy to surface energy. The penetrated fluid slows down, stops, and then is accelerated upward because the post surfaces are strongly hydrophobic, thus rectifying momentum to propel the drop upward. If the substrate empties at, or close to, the time at which the fluid above the surface completes its outward motion and if sufficient energy is supplied by the penetrated fluid, the drop can jump off the surface directly as a pancake. Here we present simulation results to support this interpretation and a theoretical description allowing us to predict the substrate parameters needed for the two criteria of matching time scales and sufficient energy for bouncing to be satisfied. In particular, we explain why post dimensions of ∼200 μm, which are large compared to the majority of superhydrophobic substrates,20−23 are needed for pancake bouncing. Section 2 of this article describes our numerical model, a diffuse interface (phase field) representation of a coexisting liquid and gas, and the geometry used in the simulations. In section 3, we present numerical results that track the fluid motion below the surface of the posts and hence explain the conditions necessary for bouncing as a pancake. We demonstrate the match between simulations and experiments

and use the numerics to motivate a theoretical description of the bouncing, which is presented in section 4. This in turn allows us to predict the drop and surface parameters necessary to observe pancake bouncing. In section 5, we summarize and discusses our results.

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MODEL AND SIMULATIONS Free Energy and Equations of Motion. We consider a diffuse interface model of a binary fluid in contact with a solid substrate. The equilibrium properties of the fluid are described by a Landau free-energy functional that is minimized in thermodynamic equilibrium: Ψ=

⎛

⎞

∫V ⎝ψb + κ2 (∇ϕ)2 ⎠ dV + ∫S ψS dS ⎜

⎟

(1)

where ⎛ 1 1 ⎞ ψb = cs 2ρ ln ρ + A⎜ − ϕ2 + ϕ4⎟ ⎝ 2 4 ⎠

(2)

represents the bulk free energy of the fluid. The first term of eq 2 helps to enforce the compressibility, and the total density ρ is constrained close to unity everywhere in the system. cs = Δx/ (31/2Δt) is the lattice speed of sound where Δx and Δt denote the discretization in time and space. In the simulations, we use Δx = Δt = 1. The second term represents a double-well potential that allows phase separation into two coexisting bulk phases with concentrations ϕ = ±1 that we identify as liquid and gas. The term in κ in eq 1 gives rise to a surface tension associated with an interface between the two phases by penalizing nonuniformities in ϕ. The interface width is ξ = (8κ/ A)1/2, and the surface tension is γ = (8κA/9)1/2. The third term is a surface integral that models the wetting energy of fluid in contact with the surface, where ψS denotes the surface energy 13022


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Figure 2. Simulation geometry, showing (a) an unperturbed cylindrical drop of radius r0 above the substrate and (b) the drop interacting with the surface.

density. Following Cahn, we let ψS take the form of ψS = −hwetϕS, with ϕS being the value of the concentration field at the surface. Minimization of the free energy at the solid surface leads to the equilibrium boundary condition κ∂⊥ϕ|S = −hwet. The wetting potential parameter hwet is related to the equilibrium contact angle of the flat substrate θY by hwet =

μ = − Aϕ + Aϕ 3 − κ

(3)

where α = cos (sin θY) and sign(...) returns the sign of its argument. The hydrodynamic equations of the two-phase fluid are 2

Φshear

∂t(ρvα) + ∂β(ρvαvβ) = −∂βPαβ + ∂β[ρν(∂βvα + ∂αvβ) + ρλδαβ ∂γvγ ]

(5)

∂tϕ + ∂α(ϕvα) = M ∇2 μ

(6)

Φdiff = M(∇μ)2

(7)

where pb is the bulk pressure ⎛ 1 c2 3 ⎞ ρ + A ⎜ − ϕ2 + ϕ 4 ⎟ ⎝ 2 3 4 ⎠

(10)

(11)

Simulation Geometry and Parameters. Performing a large number of 3D simulations of spreading drops is demanding of computer time. However, 2D geometry is not sufficient because it does not allow for fluid to move horizontally once it enters the substrate. A useful compromise is to model a cylindrical drop on a square lattice of posts as shown in Figure 2. Using periodic boundary conditions, we can then limit the simulation box size along the drop axis (y direction) to the period of the post lattice. The substrate was set up as a rectangular lattice of square posts of side length b, lattice constants wx and wy in the x and y directions, respectively, and height h. The size of the simulation box was Lx = 500, Ly = wy, Lz = 600 lattice points in the x, y, and z directions, respectively, and periodic boundary conditions were applied in each direction. The height of the posts h was chosen to be sufficiently large that the impacting liquid could penetrate the posts without reaching the bottom, and the substrate was set up without a base to allow gas to be displaced more easily by the incoming liquid. Simulations were run with

the Navier−Stokes equation, the continuity equation, and the advection−diffusion equation, respectively. v is the local velocity vector, and λ, ν, and M are the bulk viscosity, the shear viscosity, and the mobility associated with diffusion. The pressure tensor P and the chemical potential μ follow from the free energy:

pb =

2 ∂vj ⎞ νρ ⎛ ∂vi ⎜ ⎟ = + 2 ⎜⎝ ∂xj ∂xi ⎟⎠

and dissipation from diffusion (4)

∂tρ + ∂α(ρvα) = 0

⎛ ⎞ κ Pαβ = ⎜pb − κϕ∂γγϕ − (∂γϕ)2 ⎟δ δαβ + κ(∂αϕ)(∂βϕ) ⎝ ⎠ 2

(9)

The equations of motion (eqs 4−6) were solved using a multiple relaxation time lattice Boltzmann algorithm on a regular D3Q19 grid. No-slip boundary conditions were applied on the solid surface. Details of the lattice Boltzmann algorithm are given in refs 24 and 25. It is possible to calculate the approximate dissipation of energy in this system. Following Qian et al.,26 the dissipation rate (per unit volume) is the viscous dissipation from shear flow

⎛ α ⎞⎛ ⎛ α ⎞⎞ ⎛π ⎞ 2κA sign⎜ − θY ⎟ cos⎜ ⎟⎜1 − cos⎜ ⎟⎟ ⎝2 ⎠ ⎝ 3 ⎠⎝ ⎝ 3 ⎠⎠ −1

d2ϕ dx 2

(8)

and 13023


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Table 1. Simulation Parameters, in Lattice Units, and Corresponding Symbols Used in Figures 5 and 6a

a Approximate values are given for the impact velocities v0 for each data set. However, measured values for each run are used for all results presented later in the paper. The parameters that were not varied were kept at the default values of r0 = 100, γ = 3.23 × 10−2, θY = 150°, wx,y = 24, and b = 8.

Figure 3. Cross sections of a bouncing drop obtained from lattice Boltzmann simulations. (a) Evolution of the drop shape during an impact at We = 9.5, θY = 150°. The lines indicate the contour where ϕ = 0, the middle of the diffuse interface. Layer y = 4 cutting through the center of the posts is shown. Time runs from light to dark blue in steps of 3000 (in lattice units). (b−d) Sequential time snapshots of the drop cross section showing (b) pancake bouncing (θ = 170°, We = 1.4), (c) the penetrated fluid returning to the surface too late to produce pancake bouncing (θ = 150°, We = 6.6), and (d) the penetrated fluid returning to the surface when the drop is still in a pancake configuration but does not have enough energy to cause detachment (θ = 140°, We = 1.4). The times corresponding to each frame are given in simulation units (measured relative to the first frame). The blue drop area corresponds to the lattice points where ϕ > 0, that is, the liquid component.

simulation. The drop’s initial position had to be sufficiently high above the surface to allow the liquid−gas interface to relax into its equilibrium configuration before interacting with the substrate. This is because the two phases are initiated with ϕ = ±1, with a sharp boundary, that takes a certain time to equilibrate to the natural tanh configuration. We confirmed this by checking that the process of interface equilibration took only about 1000 simulation time steps whereas the time to reach the surface for a nonzero initial velocity was between 3000 and 9000 simulation time steps. Gravity was not included in our simulations.

different surface parameters, liquid parameters, and impact velocities, as listed in Table 1. Only one parameter was varied at a time, and the default values were r0 = 100, γ = 3.23 × 10−2, θY = 150°, wx,y = 24, b = 8, liquid viscosity, ηl = 1/6, and gas viscosity ηg = 1/30 in lattice units. Table 1 also lists the approximate impact velocity for each parameter set. (The exact value for each run was measured and used for all results presented later in the article.) To simulate drop impact, the cylindrically shaped drop has to hit the substrate with a certain impact velocity. This was done by starting the drop above the surface, but with a velocity vinit given to both the liquid and the gas phase (to avoid the liquid being slowed by the surrounding gas) at the beginning of the 13024


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Figure 4. Evolution of (I) drop width d(t), (II) penetration depth l(t), and (III) penetration depth with squared time at two different impact velocities v0 ≈ 0.02 (copper-shaded graphs) and v0 ≈ 0.03 (blue-green-shaded graphs) for (a) varying intrinsic contact angle θY and (b) varying surface tension γ. The square symbols indicate the moment of detaching from the surface, and dashed lines denote the evolution after lift-off. All axis labels and parameter values are given in simulation units.

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SIMULATION RESULTS

bouncing. Fluid penetrates the posts and then is pushed back to the surface, imparting momentum that pushes the drop off the substrate while it is still extended horizontally. Figure 3c,d shows examples of parameters that prevent pancake bouncing. In Figure 3c, where θY = 150° and a high We = 6.6 is used, there is a much greater surface penetration and a larger upward impulse as the fluid leaves the substrate. However, the time the liquid takes to enter and leave the surface is considerably longer than in Figure 3b, and by the time the momentum has been rectified into upward motion, the drop has retracted to an almost spherical shape. Another reason that pancake bouncing fails is illustrated in Figure 3d. Here the Weber number of the impact is We = 1.4, similar to that in Figure 3b, but a lower contact angle of θY = 140° is used. The

Typical simulation results are summarized by Figure 3. In Figure 3a, we show a drop impact at We = 9.5. We plot the contour showing the position of the drop interface in the cross section cutting through the centers of the posts as a function of time. This figure clearly highlights how the liquid penetrates the posts and then is driven out again by the highly hydrophobic nature of the post surfaces. Figure 3b−d summarizes the different bouncing pathways that we shall distinguish below. Here we plot separately sequential images of the interface profile from the moment of impact to immediately after the drop has left the surface. In Figure 3b, the contact angle is θ = 170° at We = 1.4. Impact with these parameters shows pancake 13025


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Table 2. Parameters Used in Experiments and Corresponding Symbol Used in Figures 5 and 6a

a

In each case, the impact velocity was varied. The post edge width was b = 100 μm, and the intrinsic contact angle was θY = 160°.

for the high contact angles considered. Panels II and III show that, by contrast, the speed of recoiling from below the surface depends on θY. Consider the blue-green curves. The moment at which the liquid reaches the top of the posts t↑, which appears as a kink in the l(t) graph (panels II) and corresponds to the moment of jumping, depends on θY. Therefore, the drop detaches at different stages of the lateral motion. For high contact angles corresponding to the green curves, jumping happens just after the drop has reached its maximum extension, and the drop jumps in a pancake shape. For the blue curves, however, jumping occurs after significant retraction. Figure 4b indicates that both the horizontal and the vertical spreading depend on surface tension. Again, the width of the drop upon jumping is governed by the relative time scales for horizontal and vertical motion. To quantify this criterion, we define

degree of penetration is similar to that in Figure 3b, and the penetrated liquid returns to the surface while the drop is still flattened out. However, the fluid leaving the superhydrophobic structure has insufficient energy to lift the drop, and it fails to leave the surface in the elongated shape. It detaches only after the drop has retracted to a spherical shape, with the bouncing driven by the inward, horizontal flow being rectified to upward motion. These sequences strongly suggest that pancake bouncing is a consequence of the penetration of the liquid into the superhydrophobic substrate. When the drop hits the surface, a portion of the liquid is driven between the posts because of the liquid’s downward momentum. The penetrated fluid is decelerated and comes to rest because of the capillary force exerted by the hydrophobic posts. Thus, part of the drop’s kinetic energy is stored as surface energy because of the interaction of the liquid with the hydrophobic solid substrate. The adsorbed liquid is then accelerated upward and pushed back out of the surface, and the surface energy is converted back into kinetic energy during the process of capillary emptying. This provides sufficient kinetic energy to lift the drop off the surface once it has reached the top of the posts. We now argue that for bouncing to occur with the drop in a pancake shape two criteria must be satisfied. The first concerns time scales: the substrate must empty as the spreading drop is close to its maximum lateral extension. The second is that sufficient energy must be imparted by the penetrating fluid returning to the surface to lift the drop. To quantify the difference between a drop bouncing as a pancake and after retraction, it will be helpful to define the ratio

Q=

k=

t↑ tmax

where t↑ is the time the liquid takes to penetrate and then empty the substrate and tmax is the time required for maximum horizontal spreading, with all times measured relative to the moment of impact. For k = 1, there is no retraction before liftoff; therefore, pancake quality Q = 1 corresponds to a “perfect” pancake. With increasing k > 1, the drop will have time to retract partially and Q will decrease from unity. An example of a rebound where the penetrated fluid returns to the surface too late to lift the drop is shown in Figure 3c. Energy Criterion. Even if the time-scale criterion is satisfied, the drop may not bounce as a pancake. This is because the fluid ejected from beneath the surface may not have enough energy to lift the drop. In this case, the drop will continue to retract across the surface until the surface tension energy released is sufficient to lift the drop as for conventional bouncing. An example of this behavior is shown in Figure 3d, which has a low We = 1.4. We shall refer to this condition as the energy criterion. The energy criterion can be observed in Figure 4a,b in the copper-shaded plots of panels I and II. Consider Figure 4b. The copper-shaded graphs are at a lower velocity (or an equivalently lower We) than are the blue-green ones. The liquid returns to the surface very quickly because of the strong capillary force. However, lift-off occurs at a later time, denoted by the square symbol. By then the drop has retracted considerably, suppressing pancake bouncing. In Figure 4a, the same behavior is observed for the lower contact angles. Here the Weber number of impact does not differ from the other curves, but

d jump dmax

where dmax is the maximum lateral extension of the drop and djump is its lateral extension on jumping. Q = 1 corresponds to bouncing at maximum extension. Time-Scale Criterion. We compare the lateral and vertical fluid motion in Figure 4 where we plot the time evolution of the horizontal spread d(t) (panels I) of the drop and its penetration depth l(t) (panels II). Figure 4a,b show results for varying contact angle and surface tension, respectively, for each of two different impact velocities. Squares denote where the drop detaches from the surface, and dashed lines indicate the evolution after lift-off. Panel I of Figure 4a shows that the drop spreads further across the surface with increasing contact angle but that the time for spreading is almost independent of the contact angle 13026


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Figure 5. (a) Simulation results and (b) experimental results describing pancake bouncing. (I) Pancake quality Q = djump/dmax as a function of timescale ratio k = t↑/tmax. (II) Pancake quality when the substrate empties Q↑ = d↑/dmax as a function of time-scale ratio k. Pancake bouncing is defined to occur between Q = 1 and Q = Q* (blue dashed line) or, equivalently, k = 1 (gray dashed line) and k = k* (gray dotted line). (III) Relative waiting time on top of the structure, tjump/t↑, as a function of the Weber number We. Pancake bouncing is suppressed as a result of the lack of energy to the left of the red solid line. The parameters for the different simulation and experimental data sets indicated in the legend are listed in Tables 1 and 2, respectively.

To demonstrate the time-scale criterion more immediately, we define an intermediate pancake quality

there is increased dissipation within the superhydrophobic structure and thus there is insufficient energy to push the drop from the surface. Comparison to Experiment. We next compare simulation and experimental results and discuss further the time scale and energy criteria for pancake bouncing. In particular, we shall show that, to a good approximation, these can be treated independently. The simulation data used in the plots is listed in Table 1, and the experimental data is listed in Table 2. In panel I of Figure 5, we plot the pancake quality Q = djump/ dmax as a function of k = t↑/tmax. The plots in the left-hand panels show simulation results, and those in the right-hand panels, experimental data. The shape of the curves are broadly similar but not easy to interpret. This is because both the time scale and energy criteria are relevant. Their effects can be disentangled by plotting slightly different quantities.

Q↑ =

d↑ dmax

where d↑ is the lateral extension of the drop at time t↑ that the substrate empties. Q↑ measures whether the liquid returns to the surface while the drop is still in a pancake shape, irrespective of whether it can jump. It is plotted as a function of the time-scale ratio k in panel II of Figure 5. For both simulations and experiments, Q↑ = 1 corresponds to k = 1. Deviations from k = 1 cause a decrease in the pancake quality. For panels I and II in Figure 5, the dashed gray vertical line marks the k = 1 boundary. The dashed blue horizontal line shows the cutoff Q* above which we choose to define the drop as a pancake; for Q < Q*, the rebound is driven primarily by 13027


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horizontal fluid retraction and is not considered to be pancake bouncing. In simulations, we choose Q* = 0.9, and for the experimental data, Q* = 0.8. This is so that the cutoff in k (gray, dotted line) is at the same value of k* = 1.7 for both. The difference in the slope of the graphs arises because we are comparing cylindrical and spherical drops. The physical 3D drops retract faster because a decrease in d corresponds to a larger relative decrease in surface area than for the cylindrical drops used in the simulations, or equivalently, in 3D there are two curvatures that contribute to the Laplace pressure instead of one in 2D. Figure 5III displays the energy criterion. If there is sufficient energy for lift-off as a pancake, then the drop will leave the top of the posts when they empty. Therefore, we plot tjump/t↑, where tjump is the time to lift-off, as a function of the Weber number. For sufficiently large We, this time-scale ratio is close to unity. However, when the Weber number (or, equivalently, the kinetic energy of impact) falls below a threshold value Wec there is an abrupt increase in the time the drop spends on the surface after emptying and pancake bouncing disappears. Wec is indicated by the solid red line in Figure 5. Combining the time-scale and energy criteria from panels II and III, respectively, gives the data in the top panel of Figure 5. For drops that obey the energy criterion, Q = Q↑. For those that do not, the drops retract between t↑ and tjump and Q < Q↑. These data points are distinguished with green circles. In the simulations, we could not achieve values of k < 1. In the experiments this was possible. Such drops never showed pancake bouncing, which we interpret as being due to the fluid returning to the surface being entrained by the outward flow. These points, seen in Figure 5 in panels Ib and IIb to the left of the dashed gray line, are displayed in gray in panel IIIb. To summarize, matching time scales and sufficient energy are both necessary conditions for the occurrence of pancake bouncing. The two criteria are, to a good approximation, independent. Therefore, we now present a theoretical framework for pancake bouncing considering each criterion in turn.

because of the unphysically large density of the gas in the simulations, decreases the upward acceleration of the drop. The liquid moving down between the posts is decelerated by a capillary force Fc = −4bγ cos θYn

(12)

where the perimeter of each square post is 4b and n is the number of columns between posts that are wetted. The deceleration during filling and the acceleration during emptying are then a↓ =

−γn 4b cos θY −γn 4b cos θY , a↑ = f↓ m f↑ m

(13)

where f↑m and f↓m are effective masses and m is the mass of the drop. The effective masses need to be introduced to account for large dissipation and gas inertia in the simulations as we will discuss in section 5. In the experiments, the substrate filling and emptying take approximately equal times and f↓ ≈ f↑ ≈ 1. The total time for filling and emptying the substrate is therefore t↑ =

v0m(f↓ +

f↓ f↑ )

−n 4b cos θY

(14)

which, using n ≈ 2r0/wx for the 2D simulations or n ≈ πr02/ (wxwy) for the 3D experiments, leads to sim: t↑ ≈ −

π v0ρr0wxwy f 8 γb cos θY t

(15a)

exp: t↑ ≈ −

1 v0ρr0wxwy f 3 γb cos θY t

(15b)

where f t = f↓ + (f↓ f↓)1/2. The derivation makes several assumptions. First, we have assumed that the flow of the fluid beneath the surface can be decoupled from that of the rest of the drop, and we have ignored any difference between the advancing and receding contact angles. We have not accounted for gravity, which is, by construction, zero in the simulations but will have a small effect in the experiments. For a typical experiment, gravitational acceleration is 1 order of magnitude smaller than a↓, a↑ and is therefore neglected in the estimate of t↑. We have also not considered the Laplace pressure that will tend to oppose the emptying of the substrate but can be estimated to be an order of magnitude smaller than the capillary pressure. Moreover, we have assumed that the posts are sufficiently tall that the liquid can penetrate between them without hitting the substrate base. To obtain the ratio of time scales k, we also need an expression for the spreading time tmax. The time scale over which a drop spreads and retracts on a surface is determined by the balance of inertia and surface tension and hence is governed by a time scale7,16,27

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THEORY OF PANCAKE BOUNCING Time Scale Matching. In this section, we relate the time scale end energy criteria to surface and drop parameters to identify the conditions that lead to pancake bouncing. First, we estimate the time for fluid to penetrate and then empty the substrate, t↑, and the time the drop takes to spread across the substrate, tmax. We consider both experiments and simulations pointing out the different approximations needed to describe each. In Figure 4II, we plotted the variation of the penetration depth of the liquid below the surface as a function of time. In Figure 4III, the same data is plotted as a function of ±{t − t(lmax)}2, where lmax is the maximum penetration depth. These curves show a quadratic dependence of penetration depth on time corresponding to a constant acceleration, suggesting that we may use a simple kinematic argument to describe the vertical motion. Note, however, that the magnitude of the acceleration (slope of the curves in Figure 4III) differs for the capillary filling and the capillary emptying. This is due to a combination of effects. There is deceleration of the droplet and gas due to dissipation caused by the posts, which is stronger for the large velocities across the system as the drop enters the post substrate. Additionally, the fluid being pushed out of the substrate has to accelerate the gas phase around it, which,

tmax = ξτ

ρ 3 r0 γ

(16)

where ξτ is a number of order unity that will depend on dimensionality. When we combine eqs 15 and 16 and identify ρr0v02/γ as the Weber number We, k = t↑/tmax becomes 13028


Langmuir ⎛ f ⎞ πwxwy sim: k ≈ ⎜⎜ t ⎟⎟ We ⎝ ξτ ,2d ⎠ −8b cos θYr0

⎛ f ⎞ wxwy exp: k ≈ ⎜⎜ t ⎟⎟ We ⎝ ξτ ,3d ⎠ −3b cos θYr0

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data show that the prefactors in these expressions are close to unity. Therefore, eq 19 is well approximated by (17a)

sim: We >

(20a)

(17b)

exp: We >

In section 3.3, we identified 1 < k < k* with k* = 1.7 as the range of time-scale ratios leading to pancake bouncing (see also Figure 5). Equation 17 allows us to rewrite these inequalities in terms of the range of Weber numbers that will give pancake bouncing, Wemin < We < Wemax, where

(18a)

⎛ ξ 3b cosθ r ⎞2 τ ,3d Y 0⎟ < We exp: Wemin : = ⎜⎜ wxwy ⎟⎠ ⎝ ft ⎛ ξ 3b cosθ r ⎞2 τ ,2d Y 0⎟ < 1.72⎜⎜ = : Wemax wxwy ⎟⎠ ⎝ ft

(18b)

Our theoretical model involves only linear scaling expressions; therefore, when the measured experimental or simulation data is plotted against the theoretical prediction, the data points should lie approximately on a straight line of the form y = cx. Fitting the data points to this gives a value for the prefactor c. The scaling factors f t and ξτ in eq 18 were determined by fitting the simulation and experimental data for t↑ and tmax to the theoretical predictions for t↑, eq 15, and tmax, eq 16, respectively. We found f t = 3.99 and ξτ = 1.36 for simulations and f t = 1.34 and ξτ = 0.82 for experiments. We shall compare these formulas to the simulations and experimental results and discuss their implications after obtaining an expression for the energy criterion. Energy Criterion. We now estimate the minimum Weber number needed for pancake bouncing by relating the energies at impact and for jumping at k = 1. At impact, the energy is the sum of the initial kinetic energy E0 and the initial surface tension energy of the spherical impinging drop. We assume that a fraction f of the initial kinetic energy is dissipated during the spreading process as a result of velocity gradients and surface friction. For a drop that jumps at its maximum extension, we assume that the final kinetic energy Ef is due solely to vertical motion. There is also extra interface energy because of the drop’s elongated shape. For pancake bouncing, we require Ef > 0 so that the drop can leave the surface and therefore sim: (1 − f )E0 + {2πr0}γ > (π 2dmax + π 2hmax )γ

1 1/2 −1/4 − 6} {3(We) + 3 2 (We) 1−f

(20b)

Thus, for pancake bouncing the initial kinetic energy of the drop, or equivalently the Weber number, must be above a certain threshold, Wec, which solves eq 20 as an equality. In 3D (eq 20b), for no dissipation, f = 0 and Wec = 2.32. In the experiments, the threshold can be identified from Figure 5bIII, from the position of the sharp increase in the scaled residence time tjump/t↑ as Wec ≈ 3. This implies f ≈ 0.08, a low value for the dissipation as expected on superhydrophobic surfaces. From the simulation data, we find f = 0.23, which agrees with direct measurements of the dissipation28 according to eqs 10 and 11, and Wec = 1.4. Parameters for Pancake Bouncing. We now want to relate the time-scale and energy criteria, eqs 18 and 20, to the drop and substrate properties necessary for pancake bouncing. Both equations give a condition for the Weber number. The energy criterion, eq 20, states that We > Wec is necessary, where Wec is a constant for all data sets. Equation 18 provides upper and lower bounds on We that depend on the substrate parameters and the drop radius r0. These conditions are plotted in Figure 6 by choosing the Weber number, scaled by the

⎛ ξ 8b cosθ r ⎞2 τ ,2d Y 0⎟ < We sim: Wemin : = ⎜⎜ πwxwy ⎟⎠ ⎝ ft ⎛ ξ 8b cosθ r ⎞2 τ ,2d Y 0⎟ < 1.72⎜⎜ = : Wemax πwxwy ⎟⎠ ⎝ ft

⎫ 1 ⎧8 4 2 ⎨ (We)1/2 + (We)−1/2 − 2⎬ ⎭ 1 − f ⎩π π

Figure 6. Design diagram for pancake bouncing. Full symbols denote a successful pancake bounce (Q > Q*), and empty symbols denote a conventional rebound. The parameters for the different data sets indicated in the legend are listed in Table 1 for the simulations and in Table 2 for the experiments. For We/Wec > 1 (red line), the energy criterion holds. The time-scale criterion is met in the region between the gray dashed and the gray dotted lines 1 < k < k*(green shaded area). k on the y-axis label contains the structural information and is given by eq 17.

(19a)

⎛ πd 2 ⎞ exp: (1 − f )E0 + {4πr02}γ > ⎜ max + πdmaxhmax ⎟γ ⎝ 2 ⎠ (19b)

critical Weber number Wec, as the x axis and We/(k2Wec) as the y axis where k, which contains the structural information, is given by eq 17. The solid red line in Figure 6 represents the energy criterion We = Wec. The dashed gray line represents We = Wemin (eq 18), corresponding to an optimal pancake, k* = 1, and the dotted

where we have assumed that the drop jumps as a disc of diameter dmax and height hmax (a rectangle of width dmax and height hmax in 2D). Following the arguments of Clanet et al.16 we assume that hmax ∼ ((2γr0)/(ρv02))1/2 and, by volume conservation, dmax ∼ We1/4. Similar arguments lead to dmax ∼ We1/2 in 2D. Fits to the 13029


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gray line indicates the upper boundary We = Wemax (eq 18), corresponding to k* = 1.7, the crossover to conventional rebound. This defines a region, shaded in green, within which pancake bouncing is predicted to occur. The symbols on the design diagram are simulation and experimental data. Filled symbols denote a successful pancake bounce, and empty symbols correspond to a conventional rebound. Most successful pancake bounces lie convincingly in the expected region, giving us confidence that the theory can be used to explain the link between the surface parameters and the bouncing. Agreement is not perfect, but this is not surprising given the approximations and assumptions inherent in the model. Note also that we have used fitting parameters ( f t, f, ξτ), but these are necessary on physical grounds (or because of the limitations of the simulations) and take physically reasonable values. It is apparent from Figure 6 that once the impact energy is high enough, that is, when We > Wec, pancake bouncing depends strongly on the surface parameters and r0. As the diagram shows, a higher y coordinate on the diagram results in a wider We region that allows pancakes. According to eq 18, decreasing the post spacing wx,y increases y, as does increasing the post width b, the intrinsic contact angle θY, or the drop size r0. In turn, if wx,y becomes too large, or b, r0, or θY become too small, then the pancake effect will vanish because matching time scales will be possible only for Weber numbers We < Wec that are too small to give sufficient energy for jumping while in a pancake shape. The dimensionless parameter D = wxwy/(−br0 cos θY) helps define the region of parameters for pancake bouncing. In our experiments, its value ranged from 0.45 to 1.5 for pancake events, and in simulations, it was between 0.58 and 0.90. By comparison, bouncing experiments on superhydropobic surfaces reported in the literature were typically performed with D between 0.02 and 0.11.21,29−32 Thus, the substrate features were simply too small (microposts rather than submillimeter posts) when using millimeter drops because they were designed to suppress liquid penetration in the spirit of conventional superhydrophobicity. This had the effect that the time scale for any imbibition was too small for time-scale matching. In our design diagram, data points for D between 0.02 and 0.11 would correspond to very high y values (as 1/k2 ∼ 1/D2, see eq 17), showing that the pancake region could be accessed only for very high Weber numbers, for which impacting drops undergo splashing. Those few experimental studies that featured a D comparable to ours (0.33 to 1.222,23,29) or larger (D > 1.5,29,32,33 including other lattice Boltzmann studies34,35) used post heights of ≪r0, much smaller than for our pancake experiments. Here the base substrate suppresses imbibition, returning the fluid to the surface too early or causing an unwanted Cassie to Wenzel transition. Also, if D is too high, then the data points would correspond to y values in Figure 6 that are too low, where the energy criterion prevents pancake bouncing via We c , irrespective of post height. Moreover, the design diagram explains why the jumping pancakes are more impressive for certain surfaces, in the sense that they have a wider lateral extension at detachment. Not only does a higher y coordinate allow a wider We region that shows pancake jumping, the region also extends to higher Weber numbers. Because We affects the maximum drop spreading according to dmax ∼ We1/4,16 this will result in the drops being wider when they jump.

Figure 6 also reveals why the onset of pancake bouncing depends strongly on the substrate. The minimum Weber number for pancake bouncing is determined by Wec (red solid line) for small y but by the k = 1 line (dashed) for larger y. The energy criterion may be met, but for k < 1, there is still no pancake bouncing because the fluid returns to the surface while there is still outward flow from the spreading.

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DISCUSSION Drops impacting superhydrophobic surfaces normally spread, retract, and leave the surface in an approximately spherical shape with little loss of energy because of the low friction between the drop and the surface. In this article, we have discussed a new impact route, pancake bouncing, where the drops leave the surface close to maximum extension without retracting. Pancake bouncing requires that the impacting fluid can partially penetrate the solid structure. The resulting capillary energy is then rectified into upward momentum sufficient to lift the drop as the liquid is pushed out of the solid. Thus, to observe the effect, relatively large topographical features are necessary (posts heights on the order of millimeters and edge dimensions of ∼200 μm), together with a high intrinsic contact angle. These requirements may be the reason that pancake bouncing has not been reported before: usually liquid penetration is not desired; therefore, typical superhydrophobic surfaces are engineered with much smaller post dimensions, commonly between a few micrometers and 100 μm. We argue that two criteria must be satisfied for pancake bouncing to be observed. First, the time scales for lateral spreading and capillary emptying must match. Second, the impact energy must be sufficiently large that the upward momentum can overcome surface adhesion and any dissipation due to the mixing of the upward and lateral flows. Estimates of surface and drop parameters that lead to pancake bouncing are summarized in the design diagram in Figure 6. A number of simplifications were necessary to develop a model for the pancake effect. First, in all our scaling estimates we treated the vertical penetration motion of the drop as separate from the horizontal spreading motion, when in reality they are coupled, and the depth of liquid penetration affects the time for horizontal spreading. Second, the width of penetrating liquid was approximated by the size of the unperturbed drop. Moreover, we neglected the effects of gravity and Laplace pressure over the interfacial forces from wetting the posts because they are an order of magnitude smaller. Our theoretical development was motivated by simulations of drop impact using a lattice Boltzmann solution of a diffuse interface model of two-phase hydrodynamics (model H). These reproduced pancake bouncing very well, but there is an important limitation. Because of stability issues, the density and viscosity of the gas phase must be taken to be unphysically large, and this must be accounted for in the design of the simulations and the interpretation of the data. In particular, the post substrate had to be set up without a base such that the gas component could flow between the posts when displaced by the incoming liquid instead of acting like a cushion and preventing drop impact, and both the liquid and the gas components were given the same initial velocity to avoid the incoming drop being slowed by the surrounding gas. Moreover, during the imbibition viscous dissipation was much more important than in the experiments (Reynolds number Re ≈ 5 in the simulations and Re ≈ 100 in the experiments). This could 13030


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(6) Richard, D.; Quéré, D. Bouncing water drops. Europhys. Lett. 2000, 50, 769−775. (7) Richard, D.; Clanet, C.; Quéré, D. Contact time of a bouncing drop. Nature 2002, 417, 811−811. (8) Biance, A.-L.; Chevy, F.; Clanet, C.; Lagubeau, G.; Quéré, D. On the elasticity of an inertial liquid shock. J. Fluid Mech. 2006, 554, 47− 66. (9) Quéré, D. Surface chemistry: Fakir droplets. Nat. Mater. 2002, 1, 14−15. (10) Shirtcliffe, N. J.; Aqil, S.; Evans, C.; McHale, G.; Newton, M. I.; Perry, C. C.; Roach, P. The use of high aspect ratio photoresist (SU-8) for super-hydrophobic pattern prototyping. J. Micromech. Microeng. 2004, 14, 1384−1389. (11) He, B.; Patankar, N. A.; Lee, J. Multiple equilibrium droplet shapes and design criterion for rough hydrophobic surfaces. Langmuir 2003, 19, 4999−5003. (12) Quéré, D. Non-sticking drops. Rep. Prog. Phys. 2005, 68, 2495− 2532. (13) Lafuma, A.; Quéré, D. Superhydrophobic states. Nat. Mater. 2003, 2, 457−460. (14) Koch, K.; Barthlott, W. Superhydrophobic and superhydrophilic plant surfaces: an inspiration for biomimetic materials. Philos. Trans. R. Soc., A 2009, 367, 1487−1509. (15) Reyssat, M.; Richard, D.; Clanet, C.; Quéré, D. Dynamical superhydrophobicity. Faraday Discuss. 2010, 146, 19−33. (16) Clanet, C.; Béguin, C.; Richard, D.; Quéré, D. Maximal deformation of an impacting drop. J. Fluid Mech. 2004, 517, 199−208. (17) Okumura, K.; Chevy, F.; Richard, D.; Quéré, D.; Clanet, C. Water spring: A model for bouncing drops. Europhys. Lett. 2003, 62, 237−243. (18) Wang, Z.; Lopez, C.; Hirsa, A.; Koratkar, N. Impact dynamics and rebound of water droplets on superhydrophobic carbon nanotube arrays. Appl. Phys. Lett. 2007, 91, 023105. (19) Liu, Y.; Moevius, L.; Xu, X.; Qian, T.; Yeomans, J. M.; Wang, Z. Pancake bouncing on superhydrophobic surfaces. Nat. Phys. 2014, 10, 515−519. (20) Moulinet, S.; Bartolo, D. Life and death of a fakir droplet: Impalement transitions on superhydrophobic surfaces. Eur. Phys. J. E: Soft Matter Biol. Phys. 2007, 24, 251−260. (21) Reyssat, M.; Pépin, A.; Marty, F.; Chen, Y.; Quéré, D. Bouncing transitions on microtextured materials. Europhys. Lett. 2006, 74, 306− 312. (22) Hee Kwon, D.; Joon Lee, S. Impact and wetting behaviors of impinging microdroplets on superhydrophobic textured surfaces. Appl. Phys. Lett. 2012, 100, 171601. (23) Bartolo, D.; Bouamrirene, F.; Verneuil, É.; Buguin, A.; Silberzan, P.; Moulinet, S. Bouncing or sticky droplets: Impalement transitions on superhydrophobic micropatterned surfaces. Europhys. Lett. 2006, 74, 299−305. (24) Briant, A. J.; Yeomans, J. M. Lattice Boltzmann simulations of contact line motion. II. Binary fluids. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2004, 69, 031603. (25) Pooley, C.; Furtado, K. Eliminating spurious velocities in the free-energy lattice Boltzmann method. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2008, 77, 046702. (26) Qian, T.; Wang, X.-P.; Sheng, P. A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 2006, 564, 333− 360. (27) Vincent, F.; Le Goff, A.; Lagubeau, G.; Quéré, D. Bouncing Bubbles. J. Adhes. 2007, 83, 897−906. (28) Moevius, L. Droplet Dynamics on Superhydrophobic Surfaces. Ph.D. Thesis, St. Hugh’s College Oxford, The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 2014. (29) Jung, Y. C.; Bhushan, B. Dynamic Effects of Bouncing Water Droplets on Superhydrophobic Surfaces. Langmuir 2008, 24, 6262− 6269. (30) McCarthy, M.; Gerasopoulos, K.; Enright, R.; Culver, J. N.; Ghodssi, R.; Wang, E. N. Biotemplated hierarchical surfaces and the

be modeled in the simulations by introducing effective masses in eq 13. Note that in the simulations m↑ is significantly larger than m↓. This again reflects the additional difficulty of accelerating the displaced gas. For vertical square posts, pancake bouncing occurs over a restricted range of Weber numbers and geometries. However, when tapered posts with widths that increase linearly with depth are used, the time scales for spreading and penetration both become independent of velocity.19 Therefore, time-scale matching and hence pancake bouncing can occur over a wide range, e.g., 10 < We < 60 for 200 μm post spacing. For larger We, the bouncing is particularly impressive as the maximum drop extension is large. Moreover, pancake bouncing has been observed on surfaces comprising porous, hexagonal layers topped by posts,19 suggesting that it may be a rather widespread phenomenon. There are still many questions to be answered in understanding the mechanism and prevalence of pancake bouncing. In the simulations, we assume very long posts, and the drop does not hit the base substrate. In the experiments, impacting the surface does not appear to seriously affect the bouncing if the posts are sufficiently long, but this is not yet quantified. The experiments suggest that the drop does not spread laterally between the posts, which we attribute to pinning on the vertical edges of the posts, and this effect has not yet been fully explained or explored. The drop sizes considered are close to the capillary length, so it is of interest to ask how large the drops can be before gravity destroys the bouncing. Experimental and numerical studies of different surface structures and how they affect the flow and pancake quality will help to understand the pancake bouncing in more detail. Moreover, considering the practical applications of superhydrophobicity, contact times with the surface are reduced by a factor of ∼5 compared to bouncing after retraction, which may be of relevance to the design of fast-water-shedding surfaces and surfaces resistant to the buildup of ice from impacting raindrops.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS J.M.Y. acknowledges support from ERC Advanced Grant MiCE. Z.W. is grateful for support from the University Research Council (grant 112134) and the National Natural Science Foundation of China (no. 51276152).

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REFERENCES

(1) Rioboo, R.; Marengo, M.; Tropea, C. Time evolution of liquid drop impact onto solid, dry surfaces. Exp. Fluids 2002, 33, 112−124. (2) Hartley, G.; Brunskill, R. In Surface Phenomena in Chemistry and Biology; Danielli, J., Pankhurst, K., Riddiford, A., Eds.; Pergamon: New York, 1958; pp 214−226. (3) Rein, M. Phenomena of liquid drop impact on solid and liquid surfaces. Fluid Dyn. Res. 1993, 12, 61−93. (4) Yarin, A. Drop impact dynamics: splashing, spreading, receding, bouncing. Annu. Rev. Fluid Mech. 2006, 38, 159−192. (5) Roisman, I. V.; Rioboo, R.; Tropea, C. Normal impact of a liquid drop on a dry surface: model for spreading and receding. Proc. R. Soc. London, Ser. A 2002, 458, 1411−1430. 13031


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role of dual length scales on the repellency of impacting droplets. Appl. Phys. Lett. 2012, 100, 263701. (31) Tran, T.; Staat, H. J. J.; Susarrey-Arce, A.; Foertsch, T. C.; van Houselt, A.; Gardeniers, H. J. G. E.; Prosperetti, A.; Lohse, D.; Sun, C. Droplet impact on superheated micro-structured surfaces. Soft Matter 2013, 9, 3272−3282. (32) Deng, T.; Varanasi, K. K.; Hsu, M.; Bhate, N.; Keimel, C.; Stein, J.; Blohm, M. Nonwetting of impinging droplets on textured surfaces. Appl. Phys. Lett. 2009, 94, 133109. (33) He, M.; Zhou, X.; Zeng, X.; Cui, D.; Zhang, Q.; Chen, J.; Li, H.; Wang, J.; Cao, Z.; Song, Y. Hierarchically structured porous aluminum surfaces for high-efficient removal of condensed water. Soft Matter 2012, 8, 6680−6683. (34) Hyväluoma, J.; Timonen, J. Impalement transitions in droplets impacting microstructured superhydrophobic surfaces. Europhys. Lett. 2008, 83, 64002. (35) Hyväluoma, J.; Timonen, J. Impact states and energy dissipation in bouncing and non-bouncing droplets. J. Stat. Mech.: Theory Exp. 2009, 2009, P06010.

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