Numerical and Analytical Study of The Impinging and Bouncing

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Numerical and Analytical Study of The Impinging and Bouncing Phenomena of Droplets on Superhydrophobic Surfaces with Microtextured Structures Yunyun Quan† and Li-Zhi Zhang*,†,‡ †

Key Laboratory of Enhanced Heat Transfer and Energy Conservation of Education Ministry, School of Chemistry and Chemical Engineering, South China University of Technology, Guangzhou 510640, People’s Republic of China ‡ State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou 510640, People’s Republic of China S Supporting Information *

ABSTRACT: The dynamics of droplets impinging on different microtextured superhydrophobic surfaces are modeled with CFD combined with VOF (Volume of Fluid) technique. The method is validated by experimental data and an analytical model (AM) that is used to predict the penetrating depth and the maximum spreading diameter of an impinging droplet. The effects of geometrical shapes and operating conditions on the spreading and bouncing behaviors of impinging droplets are investigated. Six surfaces with different shapes of pillars are considered, namely, triangular prism, square pillar, pentagonal prism, cylindrical pillar, and crisscross pillar surfaces. The bouncing ability of an impinging droplet on textured surfaces can be illustrated from three aspects, namely, the contact time, the ranges of velocities for rebound and the penetrating depth of liquid in the maximum spreading stage. The surface with crisscross pillars exhibits the best ability to rebound, which can be attributed to its large capillary pressure (PC) and its special structures that can capture air in the gaps during the impinging process.

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cos θ W = r cos θY

INTRODUCTION The phenomenon of droplets impacting on solid surfaces has been studied for more than a century. The impinging process is related to many practical applications1 such as spray coating, inkjet painting, rapid cooling, and delayed frosting. The understanding of the dynamics of droplets impacting will be of great importance to design surfaces that could improve the painting effect of inkjet, minimize droplet erosion, etc. In the frosting process, the reduced contact time and the bouncing behavior of impinging droplets can minimize ice formation if the droplets could bounce off before ice nucleation occurs.2 A larger number of experiments and theoretical investigations on droplets and surface interaction have been reported previously.1,3−6 However, these investigations are based on smooth surfaces. The concept of superhydrophobicity originates from the “lotus effect” in nature. Barthlott7 shows for the first time that the interdependence between surface roughness and water repellency is the keystone in the self-cleaning mechanism of many biological surfaces. When droplets contact with superhydrophobic surfaces, two wetting states can be observed on microstructured hydrophobic surfaces. (i) Wenzel (W) state:8 liquid impales the structures and follows the topography of surfaces: © XXXX American Chemical Society

(1)

where r is the roughness of textured surfaces, θY is the young contact angle on a smooth surface. (ii) Cassie-Baxter (CB) state:9 liquid rests on the top of structures: cos θC = − 1 + f (cos θY + 1)

(2)

where f refers to the solid−liquid contact fraction. A layer of air cushion captured inside the structures enables the droplet to keep an almost spherical shape. The dynamic behaviors of droplets impinging on textured superhydrophobic surfaces are different from those on smooth surfaces. Liquid will penetrate into microcracks under external forces, exhibiting complex wetting states on textured surfaces. According to the theory of Deng,10 there are three wetting states when a droplet impacts on textured surfaces, i.e., the total nonwetting state, the partial wetting state, and the total wetting state. The bouncing behavior of the impinging droplet can occur in the former two states, i.e., the total nonwetting state and/or in the partially impaled state, as long as the remaining kinetic energy is sufficient to overcome the dissipative energy in the receding Received: July 17, 2014 Revised: September 7, 2014

A

dx.doi.org/10.1021/la502836p | Langmuir XXXX, XXX, XXX−XXX

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stage.11,12 A qualitative description of the three impinging states cannot reflect the interactions between the kinetic energy and the dissipative energy. Thus, the wetting theory cannot account for the bouncing behavior of impinging droplets clearly. In addition, a direct theoretical investigation of the bouncing behavior of droplets is difficult because of some uncertain factors, such as the complicated viscous dissipation in impinging and recoiling stages, the unknown penetrating depth of impinging droplets, and the complicated geometric configurations of the recoiling droplets. Numerical simulation can overcome these difficulties and provide more detailed information under varied initial and operating conditions. There have been several studies in this direction; however, the effects of surface microstructures on the bouncing behaviors have not been classified.1,13−15 VOF methods are able to model the dynamic behaviors of liquids impacting on surfaces on a small scale.16−18 In this paper, a CFD (Computational Fluid Dynamics) coupled with VOF (Volume of Fluid) approach is used to study the spreading and bouncing behaviors of droplets impinging on different microstructured superhydrophobic surfaces. Further, an analytical model is proposed to predict the penetrating depth and the maximum spreading diameter of droplets. This part of the work can be a supplement to the previous work that is unable to predict the dynamics of droplets in a partially impaled state.19−21 The CFD simulated results are validated by experimental data and the analytical model. The effects of geometrical shapes and the initial and operating conditions on the bouncing behaviors of impinging droplet are then investigated. Six surface structures are considered. The bouncing ability, i.e., the antiwetting ability of an impinging droplet on textured surfaces, can be quantitatively illustrated from three aspects: the contact time, the ranges of velocities for rebound, and the penetrating depth of liquid in the maximum spreading stage. The contact time refers to the total time for a droplet to undergo impinging, spreading, and complete rebound on surfaces.

∂ρ + ∇ × (ρυ ⃗) = 0 ∂t

(5)

∂(ρυ ⃗) + ∇ × (ρυυ⃗ ⃗) = −∇P + ∇ × [μ∇υ ⃗ + (∇υ ⃗)T ] ∂t + ρg ⃗ + Fs⃗ (6)

where P is pressure and ρ and μ are the volume averaged density and viscosity, respectively. These averaged values are calculated using the volume fraction of each phase, ρ=

∑ ρq αq

(7)

μ=

∑ μq αq

(8)

In the Navier−Stokes equation, Fs is a source term related to the surface tension according to the continuum surface force model of Brackbill,24 F=γ

κρ∇α 0.5(ρl + ρg )

(9)

where γ is the surface tension and κ is the mean curvature of the interface. The curvature is defined in terms of the divergence of the unit normal n⃗:

⎛ n⃗ ⎞ κ = ∇·⎜ ⎟ ⎝ |n ⃗ | ⎠

(10)

where n is defined as the gradient of αq, n ⃗ = ∇αq

(11)

Computational Domain. Figure 1 shows six surfaces with different types of microstructures. For each surface, the pillars are arrayed with the same intervals. The simulations of droplets impinging and recoiling processes are implemented in 3D computational domains shown in Figure 2(a). A droplet resting on the top of the pillars with a certain impacting velocity is

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NUMERICAL METHOD AND ANALYTICAL MODEL Numerical Model. CFD commercial software Fluent 6.3.2622 in conjunction with VOF surface tracking technique is adopted to investigate the impinging behavior of droplets on microstructured surfaces. The pressure-based segregated solver is chosen for the transient laminar flow of droplet impinging process. The Pressure-Implicit with Splitting of Operators (PISO) scheme is used for the pressure−velocity coupling.22 The liquid−air interface is tracked by the explicit VOF formulation. In the VOF model,23 the tracking of the interface between two phases is accomplished by solving a continuity equation for one of the two phases in each computational cell: ∂(αqρq ) ∂t

+ ∇·(αqρq υq⃗ ) = 0

(3)

where t is time, ρ is density and αq is a volume fraction function of the qth fluid. The sum of αq in a cell is unit: m

∑ αq = 1 i=1

Figure 1. Superhydrophobic surfaces with different micropillars: (a) Surface-triangle surface, triangular prisms; (b) Surface-square surface, square pillars; (c) Surface-pentagon surface, pentagonal prisms; (d) Surface-cylinder surface, cylindrical pillars; (e) Surface-cross surface, crisscross pillars; and (f) Surface-sphere surface, spherical pillars.

(4)

where m is total number of phases. The continuity and Navier− Stokes equations are solved as follows: B


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height separating the liquid interface and the basal surface of the substrate diminishes, whereas for the depinning mechanism,11,12 the contact line will spontaneously slide downward along the pillars to reach the floor of the valleys. The surfaces tested in this work are designed such that the depinning transition can occur by sufficiently increasing the height of pillars. The liquid−gas interface will depin from the structures and slide down with increasing outside pressures. The schematic diagram of a partially penetrated droplet in the maximum spreading stage is listed in Figure 3. The droplet is Figure 2. (a) A 3D computational domain for a droplet impinging on a surface with cylindrical pillars. (b) Geometrical parameters of a textured surfaces.

modeled. The domain size is decided according to the maximum spreading diameter of the impinging droplet. The geometrical parameters of textured surfaces are listed in Figure 2(b). dp is the width and H is the height of a pillar. P is the centroid distance between two pillars. dx is the space between two pillars not including the pillar width. The impinging behaviors of droplets occur in an opening system. Thus, the side and top surfaces of the 3D computational domain are considered to be pressure outlets. The bottom of the valley and the microsurfaces comprising of the pillars themselves are considered to be smooth walls. They are set to wall boundary conditions. Wall adhesion is realized by the values of surface tension and contact angle. In the section of model validation with experimental data, the initial conditions of impinging droplets are varied according to the test conditions in literature,12 while in other sections, the initial diameter and surface tension of the impinging droplet are fixed to normal conditions of 44.8 μm and 0.073 N/m, respectively. A mesh independence test is checked for all surfaces. For example, a domain size of 0.065 × 0.065 × 0.072 mm3 is used for the 3D simulation of a surface-cylinder surface with a pillar pitch (P) of 9 um. The initial velocity (V) and diameter (D0) of the impinging droplet before contacting with the surface are 1.89 m/s and 44.8 μm, respectively. The domain is discretized by about 294 562, 562 409, 1 771 219 grids, respectively. The differences in the morphological evolution of the simulated impinging droplet with 562 409 and 1 771 219 grids are minimal. Hence the domain composed of 562 409 grids is applied to simulate the impinging process. Analytical Model. Due to the lack of quantitative experimental data, such as the penetrating depth of droplets,12 experimental validation of the CFD method is limited. Thus, an analytical model (AM) is established to help to validate the model. The penetrating depths and the maximum spreading diameters of impinging droplets can be predicted by this model. Many researchers have established theoretical models to estimate the maximum spreading sizes by considering the texturing effects and viscosity dissipations.19,21 However, these models can only predict the maximum spreading diameter when the wetting states of impinging droplets are in Cassie state and/or Wenzel state. The dynamics in a partially penetrated state is unknown. The penetrating depths of droplets still remain unclear. The model proposed in this paper can overcome these drawbacks and can be a supplement to the previous work. There are two possible mechanisms for liquid penetrating process under pressure: the sagging impalement and the depinning impalement. For the sagging mechanism,8 the curvature of the liquid−gas interface increases and the minimal

Figure 3. A schematic diagram of a partially penetrated droplet in the maximum spreading state. The droplet is divided into three parts. abgha: liquid left above the pillars (part 1), rectangular bcfgb: liquid penetrated inside the pillars (part 2), cdefc: truncated spherical wetting area inside the pillars (part 3).

divided into three parts. The first part is the liquid above the pillars with a height of H1. The second is the area inside the pillars with a liquid height of (Hx − H2). The third is the truncated spherical wetting area inside the pillars with a liquid height of H2. Considering the volume conservation of the impinging droplet, a formula can be obtained as follows: 1 3 1 2 1 πD0 = πDmH1 + πDC2(Hx − H2) 6 4 4 1 2 + πH2 (3R L−H2) (12) 3 where D0 is the initial diameter, Dm is the maximum spreading diameter, and DC is the contact diameter in the maximum spreading stage:25 DC = (2x + 1)d p + 2xdx

(13)

where 2x+1=1, 2, 3··· is the number of pillars beneath the droplet. In this work, x = 2 is used according to the initial conditions. In the depinning transition, the sliding of liquid-gas interface occurs when the θP equals to θe.25 θP is the angle that the curved interface underneath the drop makes with the sides of the posts in Figure 3. θe is the intrinsic contact angle of the solid surface. RL is the radius of the curvature of part three: RL =

1 D 2 C

(14)

cos α

The angle α is marked in Figure 3. The value of H2 can be expressed by the following:

H2 = RL − RL sin α C

(15)


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PWH = KCsρV

The total energy of droplet just before the impinging (the initial stage) is endowed by the kinetic energy (EK,0) and surface energy (ES,0), which are given by the following: π E K,0 = ρD03V 2 (16) 12

ES,0 = πD02γ

where K is a constant that depends on impact conditions, which can be found empirically for waterdrop-to-solid collisions. At very high impact velocities, K is probably close to unity. In this paper, K is empirically determined to be 0.003.12 Cs is sound speed, ρ is the liquid density and V is the initial velocity of droplets. In the spreading stage, the dynamic pressure is dominant,

(17)

After spreading, the kinetic energy is consumed and transformed into the surface energy and dissipation energy with increasing in spreading diameter. In the maximum spreading stage, the surface energy (ES,1) is as follows: ES,1 = E LG + ESL − SG

PD = 1/2ρV 2

(18)

W1 =

Re =

(20)

k=

F≈

(22)

2πRH2 P2

(23)

2πRLH2 π

πDCH2 + 4 DC2

(24)

where r and rr are the fractions of the liquid−solid contact area inside pillars. The lower surface inside the pillars is regarded as a segment of sphere rather than cylinder. Thus, rr has to be multiplied by a proportion coefficient k. Hx is the penetrating depth of the impinging droplet which can be obtained by the Hagen−Poiseuille’s law:26,27 Hx =

(PWH

2 dx D0 + PD − PC) · 32μ V

W2 ≈

μVxhD 2

P ln(P /b)

(31)

π 2μVDC4 H 6[ln(P /b)]D0 P 2

(32)

(25)

W2 =

π 2μVDC4 H 6[ln(P /R c) − 1.31]D0 P 2

(33)

Rc is the radius of the cylindrical pillar. The kinetic energy in the maximum spreading stage is considered as zero. The energy conservation between the initial stage and the maximum spreading stage can be established as follows:

−γ cos θa·L P 2 − A top

(30)

For the surfaces with cylindrical pillars, the energy dissipation is given by the following:21

dx is the mean pitch between two pillars. PC is the capillary pressure of the textured surface to impede the impalement transition.11,12 The definition of PC based on the depinning mechanism is given by the following:12

PC =

ρD0V μ

where Vx is the velocity, and D is the spreading distance of the liquid front, b is the distance that the velocity gradients exist for a stokes flow. According to the literature,30 the value of b for a cylindrical pillar is the order of half the pillar diameter. Thus, b is not an exact value. In this paper, the shape of pentagonal pillar is the most close to cylinder, b is the order of half the distance from one vertex to it’s opposite edge. For other textured surfaces, b is the order of half of the equivalent diameters. Thus, b varies with germetric shapes of pillars. The energy dissipation by liquid fraction is expressed as follows:

2πR(Hx − H2)

rr =

(29)

The liquid friction against the pillars in the penetrating process is scaled as follows:30

where Atop is the top area of a pillar and Acell is the cell area surrounded by four pillars. ESL−SG is the surface energy of solid−liquid interface inside pillars: π ESL − SG = DC2(fs + r + k·rr )(γSL − γSG) (21) 4 P2

π μVDm2fs Re 8

where Re is the Reynolds number:

where fs is the fraction of liquid−solid contact area. For these surfaces with well-ordered pillars, the fraction can be given by the equation,

r=

(28)

The viscous dissipative energy of the impinging droplet spreading against the pillars’ top can be described as follows:29

ELG is the surface energy of liquid−gas interface including the upper surface, the lateral surface and the lower surface (Figure 3): π E LG = Dm2γ + πDmγH1 + 2πRLH2(1 − fs )γ (19) 4

fs = A top /Acell

(27)

(26)

where L is perimeter, P is the pitch of pillar, γ is surface tension and θa is the advancing contact angle. PWH and PD are the two wetting pressures in different stages. In the initial contact stage, the water hammer pressure (PWH) experienced by the contact region is given by28

E K,0 + ES,0 = E K,1 + ES,1 + W1 + W2

(34)

Finally, the equation accounting for the maximum spreading coefficient of the surface-cylinder surface can be expressed as D


Langmuir 2π CaD0 (ln(P / R c) − 1.31)P 2

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(Hx − H2 + H2·k) ·C 4

+ 3C 2( −cos θe· (r + fs + rr ·k)) + 3 β 2(1 + 0.5fs Re Ca) + +

12CH2(1 − fs ) D0cos α

1⎛ 12C 2 (Hx − H2) − H22/D03 · ⎜8 − D0 β⎝

⎛ 24DC ⎞⎞ − 16H2⎟⎟ − We − 12 = 0 ⎝ cos α ⎠⎠

⎜

μV γ

Ca =

We =

ρV 2D0 γ

(35)

(36)

(37)

C=

DC D0

(38)

β=

Dm D0

(39)

Ca is the capillary number and β is the maximum spreading coefficient. We is used to describe the relationship between the kinetic energy and the surface energy of impinging droplets. For other textured surfaces, the equations can be established in the same way. When fs = 1, Hx = 0 and Dm = DC, eq 35 turns into the spreading model of droplet impinging on smooth surfaces, which is consistent with the Ukiew and Kwok model:4

Figure 4. Comparison of the spreading and recoiling processes of droplets on surfaces with cylindrical pillars under different initial conditions. (a, c, e, g, i, color) CFD simulational results, (b, d, f, h, j, black-white) experimental images by a high-speed camera.12 The geometrical parameters and the operating conditions of the simulations are listed in SI Table S1.

3β 3(1‐cos θe + 0.5 Re Ca) − β(We + 12) + 8 = 0

■

(40)

It can be seen from the images (Figure 4) that the CFD simulation can capture the dynamics of the impinging process effectively. The simulated final wetting states of impinging droplets under different initial and operating conditions are consistent with the experiment results. Small deviations in the contact time exist between experimental and simulated results (Figure 4g,h). This may be attributed to several factors. First, in the simulations, the micro surfaces to form the pillars are considered to be absolutely smooth, which is not the real situation in experiments. Some defects may exist during the preparation of samples, including in the formation of micropillars and during the process of chemical modification. Second, the value of intrinsic contact angle of 110° is referred from other literatures.32,33 The contact angle of smooth surfaces with a thin coating of (tridecafluoro-1,1,2,2-tetrahydrooctyl) trichlorosilane is unknown in the literature.12 The Verification of CFD Method with the Analytical Model. Figure 5 compares the present CFD method with the analytical model in predicting the penetrating depths in the maximum spreading stage. Five surfaces, including surfacetriangle, surface-square, surface-pentagon, surface-cylinder, surface-cross surfaces, are used in this section. The geometrical parameters of various surfaces are listed in SI Table S2. The values of fs are constant for these surfaces. The penetrating depths obtained by the two methods are consistent to some extent (Figure 5a). The trends are in agreement. It can be

RESULTS AND DISCUSSION The Verification of AM with Experimental Data. The analytical model is established to predict the maximum spreading diameters of impinging droplets in partially penetrated state. Thus, the experimental values β31 of surface T5.8 are used to compare the analytical computed results (Supporting Information (SI) Figure S1). It can be observed that the AM can effectively predict the spreading diameters of impinging droplets of partially penetrated state. The Verification of CFD Method with Experimental Data. To verify the simulation method, the experimental data of literature12 are used. Figure 4 shows a series of images of droplets undergoing shape deformation upon impinging by experimental observation and CFD simulation. The geometrical parameters and the initial and operating conditions (SI Table S1) are the same as the literature.12 Wall adhesion is a significant factor that influences the droplet impinging and recoiling processes, which can be set by adjusting the values of contact angle and surface tension. The value of intrinsic contact angle (θe) on the smooth surface modified with (tridecafluoro1,1,2,2-tetrahydrooctyl) trichlorosilane is unknown (there is only an advancing contact angle of 114° in the literature12). In the simulation, the contact angle is set to 110° by referring to other literatures.32,33 Grid independence has been checked in the simulations. E


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Figure 6. Comparison between the maximum spreading coefficients obtained by CFD simulations and the analytical models (AM) for droplets impinging on different textured surfaces. ⊕ and red line: the surface-cross surface; ▲ and purple line: the surface-triangle surface; ■ and blue line: the surface-square surface; ⬟ and green line: the surface-pentagon surface;● and black line: the surface-cylinder surface.

simulations under small velocities (Figure 5). The energy dissipation by liquid friction in AM is larger. The surface energy converted from the impact kinetic energy and surface energy of the droplet is less. Thus, the maximum spreading coefficients β predicted by AM is smaller than that by CFD simulations at lower We. Similarly, as the increase of velocities, the Hx predicted by AM is gradually smaller than that by CFD method, resulting in smaller energy dissipations. Thus, the β predicted by AM is larger than that by CFD at higher We. In addition, the dynamics of droplets impinging on textured surfaces are complex because of a lack of comprehensive knowledge about viscous dissipation.21 Accurate energy conservational equations are difficult. Some hypotheses are made to simplify the modeling process, which may also have led to the deviations. Effects of Capillary Pressure (PC). According to the Cassie and Wenzel wetting theory, contact angle is independent of the shapes of microstructures as long as the solid−liquid contact fraction or the surface roughness is constant. However, the contact angle alone is not enough to describe the dynamic process of droplets impinging on microstructured superhydrophobic surfaces. The initial and operating conditions including the geometrical shapes will significantly influence the final states of droplets. The superhydrophobic surfaces with different pillar shapes have been shown in Figure 1. The definition of PC for the surface-sphere surface34 is different from that for other textured surfaces.12 The surface-sphere surface is not compared with other five textured surfaces here. The geometrical parameters of the five surfaces are listed in SI Table S2. The common features of these surfaces are that they have the same pitches, the same heights of pillars, and the same liquid−solid contact fractions. When droplets impact on textured surfaces, the dynamic behaviors of droplets are different from those on smooth surfaces. Liquid will penetrate into the inner spaces among pillars under large impinging velocities. The bouncing behavior occurs within a range of impinging velocities. Figure 7a shows the ranges of impinging velocities under which the droplets can rebound on different superhydrophobic surfaces. There are two critical impinging velocities for the droplets to rebound, i.e., the

Figure 5. Comparison of the penetration depth (Hx) obtained by the CFD simulation and the analytical model. (a) Hx (CFD) vs Hx (AM) and (b) Hx (CFD and AM) vs impinging velocities V. ⊕ and red line: the surface-cross surface; ▲ and black line: the surface-triangle surface; ■ and blue line: the surface-square surface; ⬟ and purple line: the surface-pentagon surface; ● and green line: the surface-cylinder surface.

observed that the penetrating depths of liquid increase with impinging velocities (Figure 5b). In addition, the penetrating depths on different textured surfaces with the same impacting velocities are different. The Hx of the surface-cylinder surface is the largest and that of the surface-cross surface is the smallest. The maximum spreading coefficients (β, eq 39) obtained by CFD simulation and by AM method are also compared. The value of β vs We is plotted in Figure 6. It can be observed that the trends of β for various textured surfaces obtained by the two methods are consistent. The spreading diameter of the surface-cross surface is the largest and that of the surfacecylinder surface is the least, which may be attributed to the different penetrating depths of these surfaces. CFD simulation results show that the increase in β is not obvious with an increase in impinging velocities. This is because that more liquid penetrates into the inner spaces among pillars under larger velocities. The deviations between the CFD simulation and the analytical model can be explained as follows. The droplet in the maximum spreading state is divided into three parts in the analytical model. The energy dissipation due to liquid friction against pillars mainly takes place in the second part (rectangular region). By observing the results of CFD simulations, it can be found that when impinging velocities are small, the liquid−solid interfaces beneath the droplet slightly deform. The rectangular wetting regions (part 2) in CFD simulations are not obvious. However, in the AM, the depth of part 2 (Hx − H2) is related to the predicted penetrating depth (Hx) which is larger than the value obtained by CFD F


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For the surface-cross surface, the velocity range for rebound is wider and the contact time is obviously shorter than for other textured surfaces (Figure 7a). In addition, the penetrating depth on surface-cross surface is the smallest among the five surfaces under the same velocities (Figure 5b). Thus, the bouncing ability, or the antiwetting property of surface-cross is the best compared to other textured-surfaces. For surface-triangle, surface-square, surface-pentagon, and surface-cylinder surfaces, the effects of surface texture on velocity ranges and contact time are not significant. The bouncing ability of surface-triangle is slightly better than other four surfaces. It can be explained by the different capillary pressures of textured surfaces. When fs is constant, the capillary pressure (PC) of surface-cross is significantly larger than others (SI Table S2). While for surface-square, surface-pentagon, and surface-cylinder surfaces, the differences of PC are not obvious, leading to comparable antiwetting abilities. One conclusion obtained from SI Table S2 is that to increase the antiwetting ability of a superhydrophobic surface, micropillars with top surfaces of larger perimeters are preferred when fs is constant. In Figure 7b, the ratio of PWH/PC is plotted vs. Weber number (We). It can be observed that as We increases, the transition from nonbouncing (NB) to bouncing (B) occurs (PWH/PC < 1), resulting in a minimum bouncing velocity. More and more liquid penetrates into the cavities before the droplet rebounds in the situation of PWH/PC > 1. This case is called partially penetrated bouncing (PPB). Further increasing We, the transition from PPB to nonbouncing (2NB, the second nonbouncing) occurs when the penetrated liquid is too much to be rebounded back, resulting in a maximum bouncing velocity. The ranges of velocities for bouncing are widened as PC increases. The transition from 2NB to Wenzel state will occur when PD > PC, which is not discussed in this paper. Effects of Geometrical Shapes. The bouncing abilities for textured surfaces may be also due to the different geometrical shapes of pillars. To eliminate the influence of PC, SI Table S3 lists the geometrical parameters of five surfaces with the same values of PC. It can be seen that when PC is constant, the Vmax of the surface-cross surface is still the largest, which means that the excellent bouncing ability of the surface-cross surface is not only due to its large PC, but also due to its geometrical shape. The remained air in the gaps among crisscross pillars can decrease the adhesion between the liquid and the solid. Thus, the contact time for surface-cross is smaller than other surfaces. For surface-triangle, surface-square, surface-pentagon, and

Figure 7. (a) Contact time vs impinging velocities, (b) the ratio of wetting pressure (PWH) to antiwetting pressure (PC) vs the Weber number (We) for different microtextured superhydrophobic surfaces. ▲, Triangular prism; ■, square pillar; ⬟, pentagonal prism; ●, cylindrical pillar; and ⊕, crisscross pillar. NB, nonbouncing; B, bouncing; PPB, partially penetrated bouncing; and 2NB, 2nd nonbouncing.

minimum velocity and the maximum velocity, on various kinds of structures. For every surface, the contact time is not constant as initial velocity varies. The trends of contact time with velocities are parabolic for various surfaces. The minimum contact time appears in the middle of the range of velocities for rebound. Under smaller velocities, the contact time is longer. The reason is that the kinetic energy of the droplets is small under a small velocity, leading to a slower recoiling behavior. Under larger velocities, the penetrating depths of the droplets increase. Thus, the viscous dissipation is greater in the spreading and receding stages, resulting in a longer contact time.

Table 1. Geometrical Parameters of Surfaces with Different Pillar Size and the Same Value of PC

G


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velocity of 2.4 m/s is listed in SI Table S4. The antiwetting ability of spherical pillar surfaces degenerates with larger φ. The penetrating depth increases and the range of velocities for rebound narrows under the same operating conditions. The liquid easily penetrates into the inner spaces of pillars, leading to more energy dissipation in the impinging and receding stages. The air captured in the gaps between two stacked spheres decreases and disappears when φ increase, leading to a longer contact time. Thus, to obtain robust superhydrophobicity for surface-sphere surface, the value of φ should be decreased. Effects of Intrinsic Contact Angle (CA). When droplets impact on surface-cross surfaces, larger contact angles will significantly enhance the bouncing behavior of droplets. The variations of contact time and penetrating depths with impacting velocities are listed in Figure 9a. The maximum

surface-cylinder surfaces, the effects of geometrical shapes on bouncing abilities are not obvious. Effects of Geometrical Dimensions. In this section, surfaces of surface-triangle, surface-square, surface-cylinder, and surface-sphere have been considered. A separate analysis for surface-sphere is provided in the next paragraph because of the different definitions of PC. For surface-triangle, surface-square, and surface-cylinder surfaces, three sizes of pillars are designed (Table 1). For each kind of structured surface, the value of PC is constant by adjusting geometrical parameters, such as the perimeters and areas of top surfaces and the pitches of pillars. The contact time vs initial velocities for surface-triangle, surface-square, and surface-cylinder surfaces is plotted in Figure 8. For the surfaces with the same shape of pillars, the smaller

Figure 8. Contact time vs initial impinging velocity on surfaces with different pillar sizes. The value of PC is constant for each kind of textured surface. (a) the surface-triangle surface; (b) the surface-square surface; and (c) the surface-cylinder surface. P = 8 μm: a surface with a pitch of 8 μm, P = 9 μm: a surface with a pitch of 9 μm, P = 10 μm: a surface with a pitch of 10 μm.

the pillar sizes are, the wider the range of velocities is, and the shorter the contact time will be. Thus, it will be an effective way to obtain a superhydrophobic surface with robust bouncing property by reducing the geometrical sizes. For the surface with spherical pillars, the calculation of PC is different from other textured surfaces. The schematic of a surface-sphere surface consisting of pillars of stacked spheres is listed in SI Figure S2. The radius of one sphere is 1.91 μm and the pitch is 9 μm. The pitch (P) is much larger than the radius (R) with the value of (P − 2R)/R of 2.71. Thus, surface-sphere can be regarded as a surface with large pillar spacings. The PC is defined as follows:26

Figure 9. Effects of intrinsic contact angle (CA) on the dynamic behaviors of droplets. (a) Variations of contact time and penetrating depth with impacting velocities surface-cross surfaces. (b) Variations of penetrating depth with CA on surface-cross, surface-triangle, and surface-cylinder surfaces under the same velocity of 1.68 m/s.

velocities for rebound increase from 3.2 to 3.8 m/s when CA changes from 110° to 115°. In addition, the contact time and penetrating depths of droplets decrease markedly with a larger CA. The significant decrease in penetrating depths with larger CAs can also be observed on other textured surfaces in Figure 9b. For the surface-cross surface, the depth declines more rapidly than for other two surfaces. It also shows that under the same optimized conditions, the bouncing and antiwetting abilities of the surface with crisscross pillars are still superior to other surfaces.

θ 2πRγ sin 2 a (41) 2 P2 where θa is the advancing contact angle. Three different sizes of structures are designed by changing the values of φ, where φ is the position of the connecting part of two spheres. Other parameters, including the heights and pitches of pillars and the radii of spheres are constant (SI Table S4). According to eq 41, the capillary pressures of the three surfaces are constant. The effects of φ on the bouncing behavior of impinging droplets are significant. An example of the surface impacting under the same PC =

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Surfaces Based on Repulsion of Impacting Water Droplets. ACS Nano 2010, 4, 7699−7707. (3) 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. A 2002, 458, 1411−1430. (4) Ukiwe, C.; Kwok, D. Y. On the Maximum Spreading Diameter of Impacting Droplets on Well-Prepared Solid Surfaces. Langmuir 2005, 21, 666−673. (5) Mao, T.; Kuhn, D. C. S.; Tran, H. Spread and Rebound of Liquid Droplets upon Impact on Flat Surfaces. AIChE J. 1997, 43, 2169− 2179. (6) Gunjal, P. R.; Ranade, V. V.; Chaudhari, R. V. Dynamics of Drop Impact on Solid Surface: Experiments and VOF simulations. AIChE J. 2005, 51, 59−78. (7) Barthlott, W.; Neinhuis, C. Purity of the Sacred Lotus, or Escape From Contamination in Biological Surfaces. Planta 1997, 202, 1−8. (8) Wenzel, R. N. Resistance of Solid Surfaces to Wetting by Water. Ind. Eng. Chem. 1936, 28, 988−994. (9) Cassie, A. B. D.; Baxter, S. Wettability of Porous Surfaces. Trans. Faraday Soc. 1944, 40, 546−551. (10) 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. (11) 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. (12) Kwon, D. H.; Lee, S. J. Impact and Wetting Behaviors of Impinging Microdroplets on Superhydrophobic Textured Surfaces. Appl. Phys. Lett. 2012, 100, 171601. (13) Zhang, J. F.; Kwok, D. Y. Contact Line and Contact Angle Dynamics in Superhydrophobic Channels. Langmuir 2006, 22, 4998− 5004. (14) Hyväluoma, J.; Timonen, J. Impalement Transitions in Droplets Impacting Microstructured Superhydrophobic Surfaces. Europhys. Lett. 2008, 83, 64002. (15) Gac, J. M.; Gradoń, L. Modeling of Axial Motion of Small Droplets Deposited on Smooth and Rough Fiber Surfaces. Colloids Surf., A 2012, 414, 259−266. (16) Mead-Hunter, R.; King, A. J.; Mullins, B. J. Plateau Rayleigh Instability Simulation. Langmuir 2012, 28, 6731−6735. (17) Roberts, N. A.; Fowlkes, J. D.; Mahady, K.; Afkhami, S.; Kondic, L.; Rack, P. D. Directed Assembly of One- and Two-Dimensional Nanoparticle Arrays from Pulsed Laser Induced Dewetting of Square Waveforms. ACS Appl. Mater. Interfaces 2013, 5, 4450−4456. (18) Gumulyaa, M.; Utikar, R. P.; Pareek, V.; Tade, M. O.; Mitra, S.; Evans, G. M. Modelling of the Interaction Between a Falling nHeptane Droplet and Hot Solid Surface. Chem. Eng. Sci. 2014, 116, 23−37. (19) Lee, J. B.; Lee, S. H. Dynamic Wetting and Spreading Characteristics of a Liquid Droplet Impinging on Hydrophobic Textured Surfaces. Langmuir 2011, 27, 6565−6573. (20) Li, X. Y.; Ma, X. H.; Lan, Z. Dynamie Behavior of Water Droplet Impacting onTextured Hydrophobic/Superhydrophobic. Langmuir 2010, 26, 4831−4838. (21) Li, X. Y.; Mao, L. J.; Ma, X. Dynamic Behavior of Water Droplet Impact on Microtextured Surfaces: The Effect of Geometrical Parameters on Anisotropic Wetting and the Maximum Spreading Diameter. Langmuir 2013, 29, 1129−1138. (22) I. Fluent 6.3 User’s Guide; Fluent, 2006. (23) Hirt, C. W.; Nichols, B. D. Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries. J. Comput. Phys. 1981, 39, 201−225. (24) Brackbill, J. U.; Kothe, D. B.; Zemach, C. A Continuum Method for Modeling Surface Tension. J. Comput. Phys. 1992, 100, 335−354. (25) Kusumaatmaja, H.; Blow, M. L.; Dupuis, A.; Yeomans, J. M. The Collapse Transition on Superhydrophobic Surfaces. Europhys. Lett. 2008, 81, 36003. (26) Pittoni, P. G.; Lin, Y. C.; Lin, S. Y. The Impalement of Water Drops Impinging onto Hydrophobic/Superhydrophobic Graphite

CONCLUSIONS In the present study, both the CFD method and an analytical approach are used to investigate the impinging, spreading, recoiling, and rebound of droplets on different microtextured superhydrophobic surfaces. Experimental data are used to validate the model. The AM can be a supplement to the previous work that was unable to predict the dynamics of partially penetrated wetting state. Effects of geometrical parameters and initial and operating conditions on the bouncing behavior of impinging droplets have been investigated. The bouncing abilities of impinging droplets on textured surfaces can be illustrated from three aspects, namely, the contact time, the ranges of impacting velocities for rebound and the penetrating depth of liquid in the maximum spreading stage. Six surfaces with different pillar shapes are considered, namely, triangular prism, square pillar, pentagonal prism, cylindrical pillar, crisscross pillar, and spherical pillar surfaces. Some conclusions can be obtained as follows: (1) The surfaces with crisscross pillars show the best bouncing ability, which can be attributed to their larger capillary pressures (influenced by perimeters of pillar tops when fs are constant) and their special structures that can capture air in the gaps in the impinging process. (2) When droplets impinge on surfaces with the same geometrical shapes and capillary pressures, the smaller the sizes of the pillars are, the wider the ranges of inlet velocities for rebound are, and the shorter the contact time becomes. (3) For spherical surfaces, the bouncing ability can be enhanced by decreasing the value of φ. The air captured in the gaps among stacked pillars decreases and even disappears with an increase in φ. (4) The effect of intrinsic contact angle on the bouncing behavior is significant. Surfaces with larger CA can widen the ranges of velocities, shorten the contact time, and decrease the penetrating depth of the liquid.

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ASSOCIATED CONTENT

S Supporting Information *

Description of the geometrical parameters of various surfaces. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel/fax: 86-20-87114264; e-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This project is supported by Natural Science Foundation of China, No. 51376064. The authors gratefully acknowledge Prof Sang Joon Lee and his postgraduate student, for the friendly supply of experimental images by a high-speed camera.

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REFERENCES

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Numerical and Analytical Study of The Impinging and Bouncing

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