Numerical simulation of coalescence-induced jumping of multi

Numerical simulation of coalescence-induced jumping of. 1 multi-dropletson superhydrophobic surfaces: initial. 2 droplet arrangements effect. 3. Kai W...
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Numerical Simulation of Coalescence-Induced Jumping of Multidroplets on Superhydrophobic Surfaces: Initial Droplet Arrangement Effect Kai Wang, Qianqing Liang, Rui Jiang, Yi Zheng, Zhong Lan, and Xuehu Ma* State Key Laboratory of Fine Chemicals, Liaoning Key Laboratory of Clean Utilization of Chemical Resources, Institute of Chemical Engineering, Dalian University of Technology, Dalian 116024, China S Supporting Information *

ABSTRACT: The coalescence-induced droplet jumping on superhydrophobic surfaces (SHSs) has attracted considerable attention over the past several years. Most of the studies on droplet jumping mainly focus on two-droplet coalescence events whereas the coalescence of three or more droplets is actually more frequent and still remains poorly understood. In this work, a 3D lattice Boltzmann simulation is carried out to investigate the effect of initial droplet arrangements on the coalescence-induced jumping of three equally sized droplets. Depending on the initial position of droplets on the surface, the droplet coalescence behaviors can be generally classified into two types: one is that all droplets coalesce together instantaneously (concentrated configuration), and the other is that the initial coalesced droplet sweeps up the third droplet in its moving path (spaced configuration). The critical Ohnesorge number, Oh, for the transition of inertial-capillary-dominated coalescence to inertially limited-viscous coalescence is found to be 0.10 for droplet coalescence on SHSs with a contact angle of 160°. The jumping droplet velocity for concentrated multidroplet coalescence at Oh ⩽ 0.10 still follows the inertial-capillary scaling with an increased prefactor, which indicates a viable jumping droplet velocity enhancement scheme. However, the droplet jumping velocity is drastically reduced for the spaced configuration compared to that for the aforementioned concentrated configuration. Because Oh exceeds 0.10, the effects of initial droplet arrangements on multidroplet jumping become weaker as viscosity plays a key role in the merging process. This work will provide effective guidelines for the design of functional SHSs with enhanced droplet jumping for a wide range of industrial applications. coalescence.20,27,28,30 However, the well-established understanding of droplet jumping is mostly based on two equally sized droplet coalescence cases. The coalescence of three or more droplets is actually more frequent because the nucleation process is spatially random and the basic physical process lacks more indepth understanding.8,19,31 Unlike the two-droplet coalescence case, initial droplet arrangements should be considered for multidroplet coalescence. The effects of initial droplet arrangements on the droplet jumping behavior are not clear. Although several studies show that multidroplet coalescence has the potential to enhance the droplet jumping velocity, the corresponding conversion efficiency of excess surface energy into translational kinetic energy is not investigated.32−35 The Lattice Boltzmann method (LBM) is a mesoscopic simulation method that achieves great success in revealing droplet dynamics.36,37 In this article, a 3D multiphase isothermal lattice Boltzmann simulation based on the pseudopotential LB model is carried out to investigate the effect of initial droplet arrangements on the coalescence-induced jumping of three

1. INTRODUCTION A tiny liquid bridge will be formed when two liquid droplets approach each other. The liquid bridge expands quickly because of its large curvature and pulls two droplets together. Eventually, the two droplets will merge into a larger droplet with a smaller surface area, which is called droplet coalescence. When condensed droplets coalesce on superhydrophobic surfaces during condensation, the self-propelled droplet jumping occurs as a result of the low adhesion of superhydrophobic surfaces.1,2 Coalescence-induced droplet jumping is observed not only on natural materials, i.e., lotus leaves,3,4 cicada wings,5 and gecko skins,6 but also on fabricated superhydrophobic surfaces.7−12 The jumping droplets show great potential for performance enhancement in many fields such as condensation heat transfer,10,13,14 thermal diodes,15 energy harvesting,16 and so forth. Systematic research on droplet jumping has been conducted through experimental observations,17−21 theoretical analysis,3,22−24 and numerical simulations.14,25−30 Droplet jumping is triggered by the released surface energy during the merging process that surpasses other dissipations. It is demonstrated that the jumping velocity of the coalesced droplet follows the capillary-inertial scaling law for low Ohnesorge number droplet © XXXX American Chemical Society

Received: March 16, 2017 Revised: April 27, 2017 Published: May 31, 2017 A

DOI: 10.1021/acs.langmuir.7b00901 Langmuir XXXX, XXX, XXX−XXX

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Langmuir droplets. The fitting expressions of droplet jumping velocities and the corresponding energy conversion efficiencies are given. The critical Ohnesorge number Oh for the transition of inertialcapillary-dominated coalescence to inertially limited-viscous coalescence is found to be 0.10. The extra velocities generated in the horizontal plane due to initial droplet arrangements are responsible for the discrepancy in the jumping velocities. When Oh exceeds 0.10, the effects of initial droplet arrangements on the jumping velocity become weaker as viscosity plays a key role in the merging process. Both the droplet jumping velocity and the corresponding energy conversion efficiency are enhanced for the concentrated three-droplet coalescence. However, the droplet jumping velocity is decreased for the spaced three-droplet coalescence compared to that for a concentrated configuration, which is not beneficial to droplet jumping.

The evolution equation of the LB model with the MRT collision operator is written as38 fi (x + eiδt , t + δt ) − fi (x , t ) = −Λij(f j (x , t ) − f jeq (x , t )) + Δfi (x , t )

(1)

where f i(x, t) is the density distribution function, x is the spatial position, and ei (i = 0, 1, . . .18) is the discrete velocity along the ith direction. f eq j (x, t) is the equilibrium distribution function that can be written as39 ⎡ e ·u (e ·u)2 u2 ⎤ ⎥ f ieq = wiρ⎢1 + i 2 + i 4 − 2cs cs 2cs 2 ⎦ ⎣

where u is the macroscopic velocity, cs is the lattice speed of sound, and wi represents the weights that are given as follows: w0 = 1/3, w1−6 = 1/18, and w7−18 = 1/36. For the D3Q19 lattice model, ei can be given as

2. MODEL DESCRIPTION In this work, the three-dimensional 19-velocity (D3Q19) lattice Boltzmann model with an MRT collision operator is considered.

⎡ 0 1 −1 0 0 0 0 1 −1 1 −1 1 −1 1 −1 0 0 0 0 ⎤ ⎢ ⎥ ei = ⎢ 0 0 0 1 −1 0 0 1 1 −1 −1 0 0 0 0 1 −1 1 −1⎥ ⎣ 0 0 0 0 0 1 −1 0 0 0 0 1 1 −1 −1 1 1 −1 −1⎦

Collision matrix Λij in eq 2 is given by Λ = M−1SM in which M is the orthogonal transformation matrix and S is a diagonal matrix given by (for the D3Q19 lattice)38

ρ= (4)

∑ eifi

(9)

i

ρ U = ρ u + 0.5δt ·F

(10)

The force acting on the fluid F consists of the fluid−fluid force Fint, the fluid−solid interaction force Fads, and the body force Fg. For single-component multiphase flows, the interaction force Fint mimicking molecular interactions is given by41 Fint = −βGψ (x) ∑ wiψ (x + ei)ei − i

G ∑ wiψ (x + ei)ei

1−β 2

2

i

(11)

where β is the weight coefficient and G is the interaction strength. ψ(x) is the interaction potential, which is a function of the local density and is determined by the equation of state, (5)

ψ (x) = 2(p − ρcs 2)/cs 2G .42 The wettability of the solid wall can be conveniently implemented by introducing the interaction force between the solid and the fluid, given by43

where jx = ρux,jy = ρuy and jz = ρuz are the momentum fluxes. Thus, the evolution equation of the density distribution function can be rewritten as (6)

Fads = −Gadsψ (x) ∑ wsi (x + eiδt )ei

The collision step is implemented in moment space, and the distribution functions after collision is given by f* = M−1m*. Then, the streaming process is implemented as

i

(12)

where Gads is the fluid−solid interaction strength for adjusting the contact angles. s(x) is the indicator function; s(x) = 1 when x is in a solid, and s(x)= 0 when x is in a fluid. The contact angle hysteresis is not able to be implemented in this model, so only the static contact angle is assumed in this work. Body force Fg is calculated by26

(7)

where force term Δf i(x, t) is incorporated through the exact difference method (EDM) as40 Δfi (x , t ) = f ieq (ρ(x , t ), u + Δu) − f ieq (ρ(x , t ), u)

ρu =

The whole fluid velocity is given by

⎛ ⎞T 19 2 11 2 (jx + jy 2 + jz 2 ), 3ρ − (jx + jy 2 + jz 2 ), ⎟ ⎜ ρ , − 11ρ + ρ 2ρ ⎜ ⎟ ⎜ ⎟ 2 2 2 1 2 2 2 − − − − − , , , , , , (2 ), j j j j j j j j j ⎜x ⎟ y z 3x y 3y z 3z ρ x ⎜ ⎟ =⎜ ⎟ 1 1 2 1 2 2 2 2 2 2 ⎜− ⎟ (2jx − jy − jz ), (jy − jz ), − (jy − jz ), ρ 2ρ ⎜ 2ρ ⎟ ⎜ ⎟ 1 1 1 ⎜⎜ j j , j j , j j , 0, 0, 0 ⎟⎟ x y yz xz ρ ρ ⎝ρ ⎠

fi (x + eiδt , t + δt ) = f i* (x , t ) + Δfi (x , t )

∑ fi , i

In the present work, we set Se = 0.3, Sε = 0.7, Sq = 1.1, Sυ = 1/τ, Sπ = 1.1, and St = 1.8, where τ is the relaxation time. Through transformation matrix M,38 the density distribution function f i and its equilibrium distribution f eq i can be projected onto moment space via m = Mf and meq = Mfeq, respectively. Equilibrium distribution functions meq in the moment space are given by

m* = m − S(m − m eq)

(3)

with Δu = F·δt/ρ being the velocity change due to force term F during time step δt. The density and velocity of the fluid can be obtained by

S = diag(0, se , sε , 0, sq , 0, sq , 0, sq , sυ , sπ , sυ , sπ , sυ , sυ , sυ , st , st , st )

m eq

(2)

Fg (x) = (ρ(x) − ρv )g

(8) B

(13) DOI: 10.1021/acs.langmuir.7b00901 Langmuir XXXX, XXX, XXX−XXX

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Figure 1. Top and side views of the simulated jumping process of two water droplets with Oh = 0.0258 (R0 = 20.7 μm).

where g is the acceleration of gravity and ρv is the density of the vapor phase. In the simulations, the Peng−Robinson (P−R) equation of state is used29 and is given by p=

aρ 2 α(T ) ρRT − 1 − bρ 1 + 2bρ − b2ρ2

the capillary length, so the gravitational effect is neglected.27,28,44 The properties of the droplet are taken to be the same as those of water at 20 °C, such that ρ = 998.23 kg/m3, σ = 72.75 mN/m, and μ = 1.0087 mPa·s.22 3.1. Numerical Validation. The coalescence of two identical, initially static droplets on a superhydrophobic surface is simulated. The simulated results are compared with the available numerical data to validate our developed numerical model. The detailed jumping process is analyzed from different perspectives, such as top x−y and side x−z views. Figure 1 shows the spontaneous jumping process induced by the coalescence of two identical water droplets with Oh = 0.0258 (R0 = 20.7 μm). Time is represented as a dimensionless variable with an asterisk,

(14)

where α(T ) = [1 + (0.37464 + 1.54226ω − 0.26992ω2)(1 −

T /Tc )]2

and ω is the acentric factor, which equals 0.344 (the acentric factor of water). The critical properties can be obtained by a=

0.45724R2Tc 2 0.0778RTc ,b = pc pc

(15)

t* =

where Tc and pc are the critical temperature and critical pressure, respectively.

t = τc

t ρl R 0 3 σlv

3. RESULTS AND DISCUSSION In this section, the coalescence-induced droplet jumping phenomena on superhydrophobic surfaces are investigated via the aforementioned 3D multiphase LBM simulation. Following Liu and Cheng,29 the saturation temperature of water is chosen to be T0 = 0.85Tcr, which corresponds to a liquid/vapor density ratio of 19.5. The parameters in the PR equation of state are set to a = 2/49, b = 2/21, and R = 1. The simulation domain is Nx × Ny × Nz = 200 × 200 × 250. The surface is set to be flat and homogeneous with a contact angle of 160° in order to eliminate the morphological effects on droplet jumping. The halfway bounce-back boundary condition is applied on the solid surfaces. All of the variables are represented by the lattice unit in the LBM simulations. To relate them to real physical properties, the constant Ohnesorge number, Oh, in the lattice unit is equal to that in the real unit, i.e., [Oh]lu= [Oh]real. The Ohnesorge number, Oh = μ l / ρl σlvR 0 (μl, ρl, σlv, and R0 are the water dynamic viscosity, density, surface tension, and initial droplet radius, respectively), denotes relative effects of viscous and capillary-inertial effects that dominate the merging process. The radii of droplets considered in this work are much smaller than

(16)

where τc is the characteristic merging time. The coalescence-induced droplet jumping process can be divided into four stages based on the time-lapse images: (I) expansion of the liquid bridge between two droplets; (II) impact of the liquid bridge on the surface; (III) retraction of the droplet base area; and (IV) deceleration of the coalesced droplet in air. The two droplets are initially placed close to each other at t* = 0, and a tiny liquid bridge will be formed and expands quickly as a result of the relatively large curvature near the liquid bridge at stage I. The expanding liquid bridge hits the substrate at 0.91. Because of the counteraction of the solid substrate, the droplet spreads laterally after the impact of the liquid bridge until the droplet base area reaches a maximum at 1.67. At the beginning of stage III, the droplet starts to reduce the solid−liquid contact area in the x direction while the droplet shape in the y direction is elongated under the action of surface tension. Meanwhile, a portion of the liquid mass is forced to move upward by the substrate as the droplet moves upward in the z direction (t* = 2.28). Once the merged droplet jumps away from the surface at 2.71, the bottom contact area reduces to zero. The droplet keeps oscillating between oblate (t* = 4.55) and prolate (t* = 6.37) C

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Figure 2. (a) Comparison of current numerical results with available data. (b) Variation of energy conversion efficiency with the Ohnesorge number. The red arrow indicates the critical Ohnesorge number for droplet jumping.

⎡ ΔEs = 2πσlv ⎢2(1 − cos θ) + (1 − φ − φ cos θY )sin 2 θ ⎢⎣

features while maintaining the upward motion due to the inertial effect in stage IV. The mass-averaged velocity of a droplet is defined as follows v=

⎛ 2 − 3 cos θ + cos3θ ⎞2/3⎤ 2 −2×⎜ ⎟ ⎥R 0 2 ⎝ ⎠ ⎥⎦

∫Ω ρl vz dΩ ∫Ω ρl dΩ

(17)

where θY is the Young contact angle and φ is the solid−liquid fraction. The energy conversion efficiency η is defined as the ratio of the translational kinetic energy Ek to the released surface energy for two-droplet coalescence,

where Ω represents the entire droplet computational domain. In this work, we shall refer to v as simply the droplet velocity. The dimensionless droplet velocity is given by v∗ =

v = U

v σlv ρl R 0

(19)

η=

(18)

Ek 2 ∗2 = (v ) ΔEs 3A 0

(20)

where A 0 = 2(1 − cos θ) + (1 − φ − φ cos θY )sin 2 θ . The

where U is the inertial-capillary velocity. The droplet jumping velocity is extracted as the point of departure when the bottom of the coalesced droplet leaves the solid substrate. Figure 2a shows the comparison of current simulated droplet jumping velocities with published numerical results.14,29,30 The Ohnesorge number is varied from Oh = 0.011 to 0.35 corresponding to the variation of liquid and vapor viscosity. The other parameters are fixed at μl/μg = 19.5, ρl/ρg = 19.5, and contact angle θ = 160°, where μ is the dynamic viscosity, ρ is the density, and subscripts l and g denote the liquid and vapor, respectively. The corresponding initial droplet radius varies from R0 = 113 to 0.11 μm at 20 °C. From Figure 2a, it can be found that our simulated jumping velocities agree well with the available numerical results. It should be noted that there is a discrepancy between the jumping velocities in the current work and that in ref 29. Liu et al.29 numerically simulated the coalescence-induced droplet jumping on textured superhydrophobic surfaces, whereas the solid substrate considered here is homogeneous. It has been demonstrated that the spacing between microstructures weakens the droplet jumping velocity in which the lower contour of the merging droplet falls into gaps between microstructures.20,29,45 For two identical droplets coalescing on a superhydrophobic surface with a contact angle θ, the released surface energy ΔEs before and after droplets coalesce can be given by

−2×

(

2 − 3 cos θ + cos3 θ 2

2/3

)

surface is set to be flat and homogeneous here with φ = 1.0 and θY = θ = 160°. The energy conversion efficiencies corresponding to the jumping velocities in Figure 2a are shown in Figure 2b. As shown in Figure 2b, the coalescence-induced droplet jumping process is inherently inefficient because the energy conversion efficiency is rather low with less than 6% of the released surface energy converting to translational kinetic energy. An increasing proportion of the released surface energy was dissipated by viscous effects with increasing Oh. Both the droplet jumping velocities in Figure 2a and the corresponding energy conversion efficiencies in Figure 2b decrease steeply as Oh increases to 0.10. In other words, the inertial-capillary effects dominate the coalescence hydrodynamics, with viscosity playing a limited role for Oh ⩽ 0.10, which is called inertial-capillarydominated coalescence.27 For 0.10 < Oh < 1, the coalescence is governed by a balance among viscous, inertial, and surface tension, which is in the inertially limited viscous regime.46 For droplet coalescence on superhydrophobic surfaces with a contact angle of 160°, a two-order polynomial fit captures the numerical data well for inertial-capillary-dominated coalescence, η = 1.9531 × Oh2 − 0.5765 × Oh + 0.05403 for Oh ⩽ 0.10 D

(21) DOI: 10.1021/acs.langmuir.7b00901 Langmuir XXXX, XXX, XXX−XXX

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jumping is mostly based on the coalescence of two equally sized droplets. However, the coalescence of three or more droplets is actually more frequent. Taking the coalescence of three equally sized droplets as an example, droplet jumping behaviors triggered by multidroplet coalescence are studied in this section. The research methods proposed here can also be applied to other situations (the coalescence of four or more droplets). For three-droplet coalescence, the initial droplet arrangements should be considered, which is different from the two-droplet coalescence case. Depending on the initial position of droplets on the surface, the droplet coalescence behaviors can be generally classified into two types: concentrated configurations and spaced configurations, as presented in Figure 4.

Cha et al. assume an exponentially decaying efficiency due to the increasing role of viscous effects at large Oh,47 η = ηmax [e−bOh − (1 − e−bOh)e−bOhc]

(22)

where ηmax is the maximum efficiency as Oh approaches 0, b is the fit parameter, and Ohc is the critical Ohnesorge number beyond which droplet jumping does not occur. A double-exponential fit to the numerical data for inertially limited viscous coalescence gives η = 0.05403[e−13.8096 × Oh − (1 − e−13.8096 × Oh)e−13.8096 × 0.35] for 0.10 < Oh ⩽ 0.35

(23)

For inertial-capillary-dominated coalescence, if the released surface energy is totally converted to translational kinetic energy for the case of two identical inviscid droplets coalescing on a surperhydrophobic surface with negligible adhesion, then the merged droplet can jump with a velocity (v0) that follows inertialcapillary scaling, v0 = C0 σlv where C0 is the prefactor. ρl R 0

The droplet jumping velocity can be calculated by rearranging eq 20, v0 =

3A 0η × 2

σlv ρl R 0

(24)

Figure 4. Schematic of initial droplet distribution configurations.

The droplet jumping velocities in the inertial-capillary regime (Oh ⩽ 0.10) are obtained after substituting eq 21 into eq 24 as shown in Figure 3.

In concentrated configurations, all droplets get close to each other and coalesce together instantaneously. In spaced configurations, one droplet is a small distance away from the other two droplets and the initial coalesced droplet sweeps up the third droplet in its moving path. 3.2.1. Concentrated Configurations. 3.2.1.1. Transient Variations of Vertical Velocity during the Jumping Process. Droplet distribution angle α is defined as the included angle between the two lines that connect the droplet mass centers (Figure 4a). Five distribution angles are tested, namely, 60, 90, 120, 150, and 180°. Figure 5 shows the spontaneous jumping process induced by the coalescence of three identical water droplets with Oh = 0.0258 (R0 = 20.7 μm) in which droplet distribution angle α is 60°. The visuals of the coalescence dynamics for other values of α are included in the Supporting Information. When droplets approach each other, coalescence occurs rapidly with the growth of liquid bridges between every two droplets at t* = 0.15. The remaining part of the droplets stays stationary despite the growth of the liquid bridges. The liquid mass is driven by capillary pressure toward the center until the expansion of the liquid bridge is hindered by the solid substrate at 0.76. At t* = 1.67, the initial three droplets merge into a larger one and start to retract in the horizontal plane (x−y plane). Meanwhile, a portion of the liquid mass is forced to move upward by the substrate as the droplet moves upward in the z direction (t* = 2.43). The bottom contact area reduces to zero when the merged droplet jumps away from the surface. The droplet keeps oscillating while maintaining its upward motion due to the inertial effect. The transient variation of droplet velocity corresponding to the jumping process in Figure 5 is shown in Figure 6a. It is observed that the v*(t*) curve of a three-droplet coalescence event with α = 60° is qualitatively quite similar to that of a two-droplet coalescence case. Thus, the coalescenceinduced droplet jumping process in Figure 6a can also be divided

Figure 3. Droplet jumping velocity as a function of initial droplet radius.

The jumping velocity is expected to follow inertial-capillary scaling with v0 = 0.18

6σlv (1 ρl R 0

− n−1/3) = 0.20

σlv ρl R 0

(n = 2)

with prefactor C0 = 0.20, which is consistent with recent experimental and numerical results.1,14,20,21,26,27,30,44 It should be noted that the Ohnesorge number for the transition of inertial-capillary-dominated coalescence to inertially limited viscous coalescence is 0.10. For water at 20 °C, this corresponds to an initial droplet radius of R0 = 1.30 μm, which is much smaller than the previously observed jumping droplet sizes (∼10−100 μm) under standard laboratory conditions. This indicates that a smaller range of jumping droplet sizes is possible. 3.2. Droplet Jumping Triggered by the Coalescence of Three Equally Sized Droplets. The understanding of droplet E

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Figure 5. Top and side views of the simulated jumping process of three water droplets with α = 60° (Oh = 0.0258, R0 = 20.7 μm).

Figure 6. (a) Time evolution of the droplet velocity during the jumping process of Figure 5. (b) Effect of droplet distribution angle on the instantaneous droplet velocity with Oh = 0.0258. (c) Time evolution of horizontal velocity in the x direction during the jumping process with Oh = 0.0258. (d) Time evolution of horizontal velocity in the y direction during the jumping process with Oh = 0.0258.

coalescence event. There exists a strong deceleration of the

into four stages: expansion of the liquid bridge between droplets, impact of the liquid bridge on the surface, retraction of the droplet base area, and deceleration of the coalesced droplet in air. The vertical velocity increases faster than that of a two-droplet

coalesced droplet after the velocity reaches a maximum. Later, the velocity transitions to a milder deceleration. The droplet F

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Figure 7. (a) Jumping velocity as a function of Oh for concentrated three-droplet coalescence. (b) Energy conversion efficiency as a function of the Ohnesorge number for concentrated three-droplet coalescence. The red arrow indicates the critical Ohnesorge number for droplet jumping.

jumping velocity is also improved and is increased from v* = 0.223 to 0.388. The temporal evolution of the droplet velocity for various distribution angles with Oh = 0.0258 (R0 = 20.7 μm) is plotted in Figure 6b. The curves in Figure 6b take on a similar character that indicates that each curve can be divided into the four stages illustrated in Figure 6a. The final jumping droplet velocities for different distribution angles show some discrepancies, the * > v180° * > v150° * > v90° * ≈ v120° * . The jumping relation of which is v60° velocity decreases with increasing distribution angle first, and then it increases with the distribution angle. We find no clear or explicit relationships between the distribution angles and droplet jumping velocities. To explore the reason for the discrepancies in vertical jumping velocities, the temporal evolution of the mass-averaged velocities in the x and y directions for various distribution angles with Oh = 0.0258 (R0 = 20.7 μm) is also plotted in Figure 6c,d, respectively. The droplets stay almost stationary during the jumping process in the horizontal plane with α = 60 and 180°. However, for the other distribution angles, extra velocities are generated in both the x and y directions. It is interesting that the curves oscillate quite like the one shown in Figure 6a. The kinetic energy of the coalesced droplet can be decomposed into translational (vertical jumping direction) and oscillatory components.25,27 The translational kinetic energy is only a portion of the total kinetic energy that is responsible for droplet jumping. The oscillatory component does not contribute to the jumping motion and is mainly dissipated by the internal liquid viscosity. With different distribution angles, the oscillatory kinetic energy is different, which finally leads to various jumping velocities. A more systematic study considering the energetics of droplet jumping is required to confirm our point of view. 3.2.1.2. Droplet Jumping Velocity for Concentrated ThreeDroplet Jumping. The effects of viscosity on the droplet jumping velocity for three-droplet coalescence with concentrated configurations are illustrated in Figure 7a. It is observed that the effect of the original arrangement of three droplets on the droplet jumping velocity becomes weaker when Oh increases. At Oh ⩽ 0.10, the initial droplet arrangements show a certain effect on droplet jumping as the jumping velocities are more dispersed. There are few differences in the jumping velocities for different distribution angles when Oh exceeds 0.10 because the viscous effects dominate the

coalescence hydrodynamics. As a result, there is insufficient energy available for upward-moving kinetic energy. We notice that three-droplet coalescence results in drastically increased jumping velocities compared to that triggered by twodroplet coalescence. The critical Ohnesorge number for droplet jumping is increased from 0.35 to 0.47 in Figure 7a. The corresponding critical droplet radius for jumping decreases from R0 = 0.11 to 0.06 μm for water at 20 °C. This indicates that droplet jumping not only happen more easily but also could be realized on a smaller scale as a result of the concentrated multidroplet jumping mode, which will contribute greatly to generating large dry areas on superhydrophobic surfaces. For n identical droplets coalescing on a superhydrophobic surface with contact angle θ, the released surface energy ΔEs before and after droplets coalesce can be given by ⎧ ⎪ ΔEs = πσlv ⎨n × [2(1 − cos θ ) + (1 − φ − φ cos θY ) ⎪ ⎩ ⎡ n × (2 − 3 cos θ + cos3 θ ) ⎤2/3⎫ ⎪ sin θ ] − 4 × ⎢ ⎥ ⎬R 0 2 4 ⎦ ⎪ ⎣ ⎭ 2

(25)

The energy conversion efficiency for multidroplet coalescence events is

η=

Ek 2 ∗2 (v ) = 3A n ΔEs

(26)

where A n = 2(1 − cos θ) + (1 − φ − φ cos θY )sin 2 θ −

1/3

( 4n )

× (2 − 3 cos θ + cos3 θ )2/3

The energy conversion efficiencies corresponding to the droplet jumping velocities in Figure 7a are shown in Figure 7b. Obviously, the inertial-capillary energy conversion is more efficient for the concentrated three-droplet coalescence mode as a result of the higher surface energy associated with multiple microdroplets. The critical droplet size for jumping will be decreased for multidroplet coalescence compared to that for twodroplet coalescence as more released surface energy is converted to upward-moving translational kinetic energy. G

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Langmuir A two-order polynomial fit captures the averaged energy conversion efficiencies well for inertial-capillary-dominated coalescence in Figure 7b,

coalescence in which the third droplet is located on the line perpendicular to the line of the droplet centers with a offset distance L. Figure 9 shows the spontaneous jumping process induced by the spaced three-droplet coalescence with Oh = 0.0258 (R0 = 20.7 μm); offset distance L is 8 lu (lattice unit). The droplet coalescence first begins with two droplets that get close to each other while the third one remains stationary at t* = 0.30. When the capillary waves propagate in the x direction, the length in the y direction also changes in order to maintain a constant mass. At t* = 1.06, the elongated droplet in the y direction touches the third droplet, which results in the second coalescence. The oscillation keeps running for many periods as the droplet length in both the x and y directions fluctuates until the jumping of the droplet at 5.30. The launched droplet continues to oscillate while maintaining its upward motion. The transient variation of droplet velocity corresponding to the jumping process in Figure 9 is shown in Figure 10a. It is observed that the v*(t*) curve of spaced three-droplet coalescence is also qualitatively similar to that of two-droplet coalescence. The maximum instantaneous droplet velocity for the spaced configuration is suppressed. The droplet departs at t* = 3.12 for concentrated three-droplet coalescence with α = 60°, whereas it is postponed to 5.30 for spaced three-droplet coalescence with L = 8 lu. The time it takes to jump from the solid substrate increases with offset distance L. The temporal evolution of horizontal (x and y) component velocities for different offset distances with Oh = 0.0258 is plotted in Figure 10b. The droplets stay almost stationary during the jumping process in the horizontal plane for concentrated threedroplet coalescence with α = 60°. However, extra velocities are generated in the y direction for spaced three-droplet coalescence. The amplitude in the y direction fluctuates more strongly for larger offset distances. The generated horizontal velocities due to the original droplet arrangement result in reduced jumping speeds, as a portion of the released surface energy is converted to oscillatory kinetic energy. 3.2.2.2. Droplet Jumping Velocity for Spaced Three-Droplet Jumping. The droplet jumping velocities for spaced threedroplet coalescence at various Ohnesorge numbers are shown in Figure 10c. The corresponding energy conversion efficiencies are plotted in Figure 10d. As we discuss above, the initial droplet arrangements influence the jumping velocity in the inertial-capillary regime (Oh ⩽ 0.10). It is observed that spaced three-droplet coalescence results in drastically reduced jumping velocities, which shows a negative effect on the jumping velocity for spaced configurations. The jumping velocities and corresponding energy conversion efficiencies for spaced three-droplet coalescence with L = 8 lu are even lower than that of the two-droplet coalescence. As Oh exceeds 0.10, the initial droplet arrangements have a weaker effect on the multidroplet jumping as viscosity plays a key role in the merging process. The current work mainly focuses on the coalescence-induced jumping of three equally sized droplets. In fact, the coalescence of unequally sized droplets is actually studied more frequently but remains poorly understood. The effect of contact angle hysteresis on droplet jumping needs to be further probed because the contact angle hysteresis plays a more important role than the static contact angle. Besides coalescence configurations (concentrated and spaced configurations) considered in this work, the multihop coalescence configuration also needs further research.

η = 1.4786 × Oh2 − 0.4641 × Oh + 0.07031 for Oh ⩽ (27)

0.10

The maximum efficiency is 0.07031 as Oh approaches 0, which is larger than that for two-droplet coalescence (0.05403). For inertially limited viscous coalescence, a double-exponential fit to the numerical data gives η = 0.07031[e−5.5067 × Oh − (1 − e−5.5067 × Oh)e−5.5067 × 0.47] for 0.10 < Oh ⩽ 0.47

(28)

3.2.1.3. Inertial-Capillary-Velocity Scaling for Concentrated Three-Droplet Jumping. The droplet jumping velocities for ndroplet coalescence can be calculated by rearranging eq 26, v0 =

3A nη × 2

σlv ρl R 0

(29)

After substituting eq 27 into eq 29, the droplet jumping velocities for concentrated three-droplet coalescence with Oh ⩽ 0.10 are obtained as plotted in Figure 8.

Figure 8. Droplet jumping velocity as a function of the initial droplet radius for concentrated three-droplet coalescence.

At low Ohnesorge number (Oh ⩽ 0.10), the jumping velocity for concentrated three-droplet coalescence still follows the inertial-capillary scaling, v0 = C0 σlv Prefactors C0 for α = 60, ρl R 0

90, 120, 150, and 180° are 0.37, 0.28, 0.27, 0.31, and 0.33, respectively. The value of α decreases with increasing distribution angle first, and then it increases with the distribution angle. The averaged prefactor for concentrated three-droplet coalescence is 0.31 ± 0.04. Prefactor C0 is increased from 0.20 to 0.31, which indicates the enhanced inertial-to-capillary energy conversion compared to than for two-droplet jumping. The results demonstrate the concentrated multidroplet coalescence as an alluring avenue to enhancing the jumping velocity breaking the two-droplet coalescence velocity limit. 3.2.2. Spaced Configurations. 3.2.2.1. Transient Variations of Vertical Velocity during the Jumping Process. Figure 4b shows a simplified schematic diagram of spaced three-droplet H

DOI: 10.1021/acs.langmuir.7b00901 Langmuir XXXX, XXX, XXX−XXX

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Langmuir

Figure 9. Top and side views of the simulated jumping process of three spaced water droplets with L = 8 lu.

Figure 10. (a) Effect of offset distance on the instantaneous droplet velocity with Oh = 0.0258. (b) Time evolution of horizontal velocity during the jumping process with Oh = 0.0258. (c) Jumping velocity as a function of Oh for spaced three-droplet coalescence. (d) Energy conversion efficiencies corresponding to the jumping velocities shown in plot c.

4. CONCLUSIONS

critical Ohnesorge number Oh for the transition of inertialcapillary-dominated coalescence to inertially limited viscous coalescence is found to be 0.10. (2) At Oh ⩽ 0.10, the droplet jumping velocities for concentrated three-droplet coalescence are more dispersed, which indicates that the initial droplet arrangements show a certain effect on droplet jumping. The averaged droplet jumping velocity for concentrated three-droplet coalescence still follows

In this work, the effect of initial droplet arrangements on droplet jumping triggered by three-droplet coalescence is investigated numerically using a three-dimensional MRT pseudopotential lattice Boltzmann model. The following conclusions can be drawn from this article. (1) For droplet coalescence on superhydrophobic surfaces with a contact angle of 160°, the numerical results show that the I

DOI: 10.1021/acs.langmuir.7b00901 Langmuir XXXX, XXX, XXX−XXX

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Langmuir the inertial-capillary scaling (v0 = 0.31

σlv ρl R 0

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). Prefactor C0 is

increased from 0.20 to 0.31 compared to that for two-droplet jumping, which indicates a viable jumping droplet velocity enhancement scheme. (3) At 0.10 < Oh < 1, the droplet jumping velocities for threedroplet coalescence no longer depend strongly on initial droplet arrangements because viscous effects play a key role in the coalescence hydrodynamics. (4) It is found that spaced multidroplet coalescence results in drastically reduced jumping velocities compared to that for concentrated multidroplet coalescence as a result of the weakened conversion of released surface energy to translational kinetic energy.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.7b00901. Top and side views of the simulated jumping process of three water droplets with different values of α (Oh = 0.0258, R0 = 20.7 μm) (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Kai Wang: 0000-0001-6450-381X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful for the financial support of the National Natural Science Foundation of China (nos. 51236002 and 51476018).

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REFERENCES

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K

DOI: 10.1021/acs.langmuir.7b00901 Langmuir XXXX, XXX, XXX−XXX