Lattice Boltzmann Simulation of Droplets Impacting on

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A Lattice Boltzmann Simulation of Droplet Impacting on Superhydrophobic Surfaces with Randomly Distributed Rough Structures Wuzhi Yuan, and Lizhi Zhang Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b04041 • Publication Date (Web): 30 Dec 2016 Downloaded from http://pubs.acs.org on January 2, 2017

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A Lattice Boltzmann Simulation of Droplet Impacting on Superhydrophobic Surfaces with Randomly Distributed Rough Structures

Wu-Zhi Yuan1, Li-Zhi Zhang1,2∗ 1. 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, China 2. State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou 510640, China

ABSTRACT Superhydrophobic surfaces have attracted much attention in environmental control due to their excellent water-repellent property. A successful design of superhydrophobic surfaces requires a correct understanding of the influences of surface roughness on water repellent behaviors. Here a new approach, a mesoscale lattice Boltzmann simulation approach, is proposed and used to model the dynamic behavior of droplet impacting on surfaces with randomly distributed rough micro-structures. A Fast Fourier Transformation method is used to generate non-Gaussian randomly distributed rough surfaces with the skewness and kurtosis obtained from real surfaces. Then the droplets impacting on the rough surfaces are modeled. It is found that the shape of droplet spreading is obviously affected by the distributions of surface asperity. Decreasing the skewness and keeping the kurtosis around three is an effective method to enhance the ability of droplet rebound. The new approach gives more detailed insights into the design of superhydrophobic surfaces.

∗

Author for correspondence. Tel/fax: 86-20-87114264; Email: [email protected] 1

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INTRODUCTION Droplet impingement on solid surfaces is a behavior encountered in many natural processes as well as many industrial applications, such as rain drops falling on the ground, spray cooling of hot surfaces and ink-jet printing.1 Understanding of the dynamics of droplets impacting will be of great importance to the design of functional surfaces with well-bouncing ability. Several parameters affect the droplet behaviors including roughness, wettability, impacting velocity, physical properties of the droplet and droplet size. Numerous studies including experiments, theoretical investigations and numerical simulations were performed on the interactions between droplets and surfaces.2- 5 As for experiment, it is still difficult to prepare solid surfaces with well controlled structures and chemical compositions. What’s more, due to the optical opacity of the surface and the very short timescale in droplet impacting, it’s difficult to clearly observe the details of solid-liquid contacting. Based on energy conservation, Mao et.al5 investigated the maximum spreading factors and the critical conditions for droplet bouncing on smooth surfaces. These studies are very interesting. Relatively, physics of droplet impacting on smooth and regularly textured surfaces have been well understood. However the impacting behavior of droplets on surfaces with randomly distributed rough structures, which are typical for real superhydrophobic surfaces, remains unclear. The main difficulty comes from the non-ideality of the randomly distributed rough surfaces which are usually heterogeneous both in structures and in chemical compositions. 6-8 Experimental investigations on the random surfaces are quite limited due to the small time scale. 9, 10 According to their investigations, whether the droplets are pinned, break up, or bounce off depends on the droplet velocity as well as the surface topography, but the reason is not clearly disclosed. As an alternative, numerical simulation can overcome these difficulties and provide more detailed information under various operating conditions with heterogeneous surface structures. There have been some studies in this area, 2


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mainly on well-organized surfaces. Relevant numerical studies can be classified into following three categories: 1. Molecular Dynamics Simulation (MDS). Several molecular dynamics simulations were carried out to investigate the spreading of droplet on flat surfaces11-13 and porous surfaces.14 Although this approach discloses very detailed mechanisms of the interactions between the wetting with structure and the chemistry of surfaces, the limitation is that the computational load is rather heavy even for a small fraction of solid surface with simple structure. Therefore, MD simulations are unable to study real heterogeneous surfaces. 2. Macroscopic CFD modeling. Numerous studies show the possibility of using CFD to simulate droplet impacting, with different models to track the interface, such as volume of fluid (VOF),15, 16 SOLA-VOF,17 CLSVOF18 and Level Set.19 However, it is a challenging task to model the heterogeneous micro boundary conditions with this technique. 3. Lattice Boltzmann method (LBM). Unlike traditional CFD, it can precisely capture the surface based on the density of lattice nodes. What’s more, LBM is just such a mesoscale modeling approach to model the micro-nano structures in boundaries. As a result, it would provide a promising alternative. In recent years, LBM has attracted considerable attention in simulating the droplet impingement on various solid surfaces like the smooth, curved and textured surfaces.20-23 However investigations of droplet impinging on surfaces of randomly distributed structures have seldom been performed. The effects of rough surface parameters, such as the skewness and the kurtosis on droplet bouncing have not been exactly investigated. The spreading dynamics are commonly formulated using two dimensionless parameters: the Weber number (We=ρD0v2/σ) describing the ratio between the kinetic and capillary energy, and the Reynolds number (Re=ρD0v/µ) describing the ratio between the kinetic and viscous energy. D0 is the initial diameter of droplet, v is impacting velocity, ρ is liquid density, σ is surface tension and µ is dynamic viscosity. Another important 3


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parameter, the spread factor D*=D/D0, is defined as the ratio of the total diameter D of the spreading droplet and the initial droplet diameter D0. The rest of the present paper is organized as follows. In section 2, rough surface are fabricated and characterized. Then, the randomly distributed rough surfaces are reconstructed by the FFT (Fast Fourier Transform) method.24-26 A brief introduction of the lattice Boltzmann method is given, which is based on the Shan-Chen (SC) model27 and the improvement in equation of state (EOS) by Yuan and Schaefer.28 In section 3, the simulation method are verified by the experiment and the simulated results are discussed. Finally, a brief conclusion will be made in section 4. EXPERIMENTAL AND NUMERICAL MODEL Fabrication and Characterization of the Rough Surfaces The substrate material used in this study is aluminum sheet. Due to its good properties at high temperature and high mechanical strength, it is widely used in daily life and industry. Sandblasting is applied to surfaces prior to the application of a paint coating to increase roughness and durability. In this study, the operations are performed using the sandblasting machine. Two kinds of grits, marked with A and B, are used to prepare the rough surfaces at pressures 0.2MPa, 0.3MPa and 0.4MPa, respectively. The sandblasting process is performed at right angle to the aluminum sheet at a distance of about 100 mm from the nozzle for the duration of about 40s. The sizes of grits A and B are 12µm and 18µm, respectively. Six surfaces are fabricated by sandblasting at different conditions. After the sandblasting treatment, the rough surfaces are modified with a low-surface-energy material to improve its hydrophobicity. Then the fabricated surfaces are observed by the atomic force microscope (AFM) to obtain their basic surface topography and roughness parameters. They are listed in Table 1, from A-2 to B-4. Examples of the

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two-dimensional and three-dimensional surface profile of the treated surface A-3 by experimental measurement and reconstruction, which will be discussed below, are shown in Figure 1.

Figure 1. The surface profile of the treated surface A-3. (a) Two-dimensional cross-sectional surface at red line by experiment. (b) Two-dimensional cross-sectional surface at y=10µm and the amplified profile between 5-10µm by reconstruction. (c) Three-dimensional surface topography by experiment. (d) Three-dimensional surface topography by reconstruction. After the measurement of surfaces by atomic force microscope, following parameters like the skewness Sk, the kurtosis K and the root mean square Rq are obtained. The above parameters are usually sufficient to identify a surface. Skewness measures the amplitude asymmetry, which is negative for surfaces with deeper valleys and lower peaks and positive for surfaces with higher peaks and shallower valleys. The larger the absolute value of skewness is, the larger the asymmetry of the roughness is. Kurtosis is a measurement of the peakedness of a surface roughness distribution. It is smaller than 3 if the statistical distribution is more compressed than the Gaussian distribution. It is 0 when the surface is flat. A higher kurtosis indicates an

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increasing tendency to form deep, sharp valleys or high peaks. In order to understand intuitively the effects of skewness and kurtosis on surfaces, the Sk and K are graphically illustrated in Figure 2.

Figure 2. Schematic diagram of skewness and kurtosis. The parameters of the rough surfaces are summarized in Table 1. At least three different sections are measured on each surface to obtain the structural parameters, which are then averaged. As the sizes of grit decrease or the pressure of sandblasting increase, the absolute values of skewness and kurtosis tend to increase. The Rq is not related to the types of grit, which only depends on the pressure of sandblasting. With a pressure increasing, the Rq also increases. Table 1. Parameters for the Rough Surfaces. Pressure(*0.1MPa)

Symbol

K

Sk

Rq(µm)

2

A-2

4.825

-0.54

0.675

3

A-3

8.115

-0.885

0.795

4

A-4

7.18

-0.635

0.91

Sandblasting with

2

B-2

3.47

-0.10

0.665

abrasive B

3

B-3

4

-0.13

0.885

4

B-4

5.81

-0.19

0.915

Sandblasting variant Sandblasting with abrasive A

Reconstruction of the Randomly Distributed Rough Surfaces There are two ways to obtain randomly distributed rough surfaces. One is obtained by experimental measurement and the other is based on the computer reconstruction. Because the measurement of surface roughness is a time-consuming process and it can only measure some limited sections at a time. The 6


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roughness data sets are quite finite. The computer reconstruction is relatively simple and rough surfaces with arbitrary size and roughness can be generated. As a result, based on the limited measured surface data, a statistical method is applied to reconstruct the randomly distributed rough surfaces. Based on previous studies, 24-26

a fast and convenient tool in generating rough surfaces, the Fast Fourier Transformation (FFT), is used to

generate non-Gaussian randomly distributed rough surfaces with the skewness and kurtosis obtained from real surfaces. The detail of the reconstruction will be discussed in the following. The surface, which is represented by arrays of N by M surface heights z (i, j), can be described in terms of central moments of the height distribution and an exponential auto-correlation function (AFC).29-31 It is generally sufficient to reconstruct randomly distributed rough surfaces by the first four moments, including mean, standard deviation (Rq), skewness (Sk) and kurtosis (K). Mean is generally forced to zero and the other three will be part of the basis set. The first four moments and an exponential auto-correlation function (AFC, Rz) are listed as follows −

z=

1 MN

Rq2 =

Sk =

K=

M −1 N −1

∑∑ z j = 0 i =0

1 MN

(1)

ij

M −1 N −1

∑∑ z j =0 i =0

1 MNRq3

M −1 N −1

1 MNRq4

M −1 N −1

2

∑∑ z j =0 i = 0

∑∑ z j = 0 i =0

R z ( k , l ) = R q2 exp[ − 2 . 3 (

(2)

ij

3 ij

4 ij

k 2 l 2 ) +( ) ] βx βy

7


(3)

(4)

(5)

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where k=1, 2,…, N, l=1, 2,…, M. βx and βy are the auto-correlation lengths of anisotropic surface in the x and y direction.31 If βx and βy are equal, the surface is isotropic. Eqs 1 to 4 give the information of roughness distribution vertically in a surface, while eq 5 governs the distribution in x-y plane. Non-Gaussian surfaces with various Sk and K values for given Rq and β are generated on the computer using a two-dimensional digital filter technique suggested by Hu and Tonder.32 An output sequence z(i, j) with given amplitude distribution can be obtained from an input sequence η(i, j) of N by M by a filter function h(k,l), which is a linear transformation system defined as N M i + k   j+l  z (i , j ) = ∑ ∑ h( k , l )η(i + k −  − 1 N , j + l −  − 1 M )  N   N  k =1 l =1

(6)

where i=1, 2, . . ., N, j=1, 2, . . ., M and h is a filter function. The symbol   refers to the operation to obtain the nearest integer that is greater than itself. To obtain the filter function h, which reflects the relations between the input random η(i, j) and real roughness z(i, j), following steps are required. Firstly, we take the Fourier transform of eq 6:

Z (I , J ) = H (I , J ) A(I , J )

(7)

where I=1, 2, . . ., N, J=1, 2, . . ., M. Z and A are the Fourier transformation of the z and η, respectively. H is the transfer function of the system given by

H (I , J ) =

N

M

∑∑

h ( k , l )e −

jk ω

x

e

− jl ω

y

(8)

k =1 l =1

where ωx=2πI/N and ωy=2πJ/M are discrete spatial frequencies. Secondly, the power spectral density (PDS) of the surface is deduced by the Fourier transformation of auto-correlation function eq 5 Sz(I , J ) =

N

M

∑∑

R z ( k , l ) e − jk ω x e

− jl ω

k =1 l =1

8


y

(9)

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Since input sequence η(i, j) consists of random numbers, its PSD function Sη is a constant of 1.0. The relation between Sη and Sz for a linear system is given as 33

S z ( I , J ) =| H ( I , J ) |2 S η ( I , J )

(10)

Using eqs 9 and 10, the H can be obtained. At last, based on the relation between H and h in eq 8, the filter function h(k, l) are calculated as follows

h( k , l ) =

1 MN

N

M

∑∑ H ( I , J )e

jkω x

e

jlω y

(11)

I =1 J =1

Now h(k, l) from eq 11 is used in eq 6 to obtain z(i, j). So far, a technique has been described to generate Gaussian random surfaces. However, the height distributions of surfaces are commonly non-Gaussian in reality. To generate non-Gaussian randomly distributed surfaces, the Gaussian input sequence η must be transformed to an non-Gaussian input sequence η’ with appropriate Skη’ and Kη’ by using the Johnson translator system.34, 35 The detail for transformation can be found in Supporting Information. Then steps of the digital filter technique outlined earlier with eqs 6 to 11 are used to transform the non-Gaussian input sequence

η’(i, j) to z(i, j). To summarize, the rough surfaces are generated using the following steps: (1) Measure the rough parameters from the real rough surface: Rq, Sk, K. (2) Generate a Gaussian input sequence η(i, i) using a random number generator. (3) Transform the Gaussian input sequence η(i, j) to the non-Gaussian input sequence η’(i, j) by using the Johnson translator system. (4) Calculate the filter function h(k, l) using eqs 5 to11. (5) Obtain the randomly distributed rough surface z(i, j) by the filtering operation using eq 6.

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Based on the above proposed method, the rough parameters in Table 1 are used to generate the non-Gaussian randomly distributed rough surfaces. Considering that the lattice Boltzmann method is a mesoscopic method, a 20µm*20µm area is reconstructed with A-3 in Figure 1(d). The kurtosis K is 8.115, the root mean square Rq is 0.795µm, and the skewness Sk is -0.885. Lattice Boltzmann Method Based on the simple and famous Bhatnagar-Gross-Krook approximation, the general form of the lattice Boltzmann equation can be written as

1 f i ( x + e i ∆t , t + ∆t ) − f i ( x, t ) = − [ f i ( x, t ) − f i eq ( x, t )] τ

(12)

where fi(x,t) is the density distribution function at the lattice site x and time t, feq is the equilibrium distribution function, and τ is the dimensionless relaxation time. The equilibrium distribution function f i eq can be calculated from

(

 e ⋅ u eq e ⋅ u eq f i (x, t ) = ρwi 1 + i 2 + i 4 cs 2cs  eq

)

2

−

(u eq ) 2   2cs2 

(13)

where wi is the weight factors. In order to reduce the load of calculation, a two-dimensional modeling of the cross section of above reconstructed three-dimensional rough surface is chosen to investigate the dynamic behavior of droplet. As a result, multiphase Shan-Chen (SC) type LB method36 based on the D2Q9 lattice37 is implemented. The discrete velocities are given by

0 1 0 − 1 0 1 − 1 − 1 1  [e0 , e1 , e 2 , e3 , e4 , e5 , e6 , e7 , e8 ] =   0 0 1 0 − 1 1 1 − 1 − 1 and weight factors wi are w0=4/9, w1-4=1/9, and w5-8=1/36.

10


(14)

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The corresponding macroscopic fluid density ρ and velocity u can be obtained from the first and second moments of the particle distribution functions

ρ = ∑ fi

(15)

i

ρu = ∑ ei f i

(16)

i

The viscosity in the LB model is given by

υ = (τ − 0.5)cs2 ∆t

(17)

where cs = c / 3 is the lattice sound speed, and c = ∆x / ∆t is the lattice speed with ∆x and ∆t is the lattice spacing and time step, respectively. The equilibrium value of the velocity in eq 13 equals

u eq = u +

τF ρ

(18)

The force acting on the fluid consists of fluid-fluid force, fluid-solid force and body force. The fluid-fluid force Fl 8

Fl (x, t ) = −Gψ ( x, t )∑ wiψ (x + e i , t )ei

(19)

i =1

and the fluid-solid force Fs. 8

Fs ( x, t ) = −Gadsψ ( x, t )∑ wi s ( x + ei , t )ei

(20)

i =1

and the body force Fb

Fb ( x, t ) = ρ(x, t ) g

(21)

where g is the gravitational acceleration, G and Gads are parameters that control the strength of inter-particles and the strength of interaction between the fluid and the solid wall. s is an indicator function equals 1 or 0 for

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solid nodes and fluid nodes, respectively. ψ(x,t), the effective mass, is a function of the local density and it depends on the equation of state. It can be expressed as

ψ (x, t ) = 2( p − ρcs2 ) /(Gcs2 )

(22)

Based on Yuan and Schaefer’s investigation,28 the Peng-Robinson (P-R) equation of state is adopted, which provides the maximum density ratio and maintains small spurious currents around the interface. The P-R equation of state can be expressed as

p=

ρRT aρ2 κ(T ) − 1 − bρ 1 + 2bρ − b2ρ2

(23)

where

κ (T ) = [1 + (0.37464 + 1.5422ω − 0.26992ω2 )(1 − T / Tc )]2

(24)

2 2 ω is the acentric factor, the attraction parameter a=0.45724R Tc /Pc and the repulsion parameter

b=0.0778TRc/Pc. Tc and Pc are the critical temperature and pressure, respectively. In the present study, we set a=2/49, b=2/21 and R=1. 28 Thus ψ can be obtained by

ψ=

6 ρRT aρ 2 κ (T ) ρ ( − − ) G 1 − bρ 1 + 2bρ − b 2 ρ 2 3

(25)

RESULTS AND DISCUSSION The two-dimensional computational domain used in the simulation is shown in Figure 3. The simulation domain is nx× ny = 1001×1001 (LB units), which are proved to be appropriate after grid dependency tests with three sets of meshes of 800*800, 1000*1000 and 1200*1200. A liquid droplet with initial radius D0=200 (lattice units) is placed at the center of the domain and it moves towards the randomly distributed rough surface with an initial velocity v. In the following simulation, the parameters in lattice units are fixed at: density of liquid ρl=7.205, density of gas ρv=0.197, which are set as the theoretical densities predicted by Maxwell equal-area construction at the corresponding temperature T=0.8Tc, the surface tension σ=0.183 and 12


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gravitational acceleration g=5.0×10-13. Periodic boundary conditions are implemented at the left and right boundaries. According to this concept, the unknown distributions at left boundary are set to those at right boundary. No-slip boundary condition, which uses a bounce-back scheme in this study, is employed at the top boundary of the domain and the fluid/solid interfaces.

Figure 3. Schematic diagram of the droplet impacting on the randomly distributed rough surface. The red, green and blue colors represent rough surface, liquid droplet and saturating vapor, respectively. The diameter of droplet is 200 lattice units. The amplified grid between two grooves is placed in the left corner. In order to compare LBM simulation results with experimental data, the lattice time tLB should be converted to physical time tp. The time scale is calculated by comparing the inertia-capillary time t=(ρR03/σ)1/2, with t0=[t]real/[t]LB. Thus, given tLB (the number of time steps), we obtain the corresponding physical time tp=tLB*t0. Verification of LB Method with Experimental Data Distilled water droplets of around 12µl are dropped onto horizontal rough surfaces from a height of 8mm. The schematic for the experimental setup is shown in Figure S1. A video of droplet impinging on the rough

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surfaces is recorded at 3000 fps using a camera (PCO.dimax). At least five impingements are measured for each sample. The ambient conditions are set to 30 °C and 55% relative humidity. To verify the LB simulation method, rough surfaces are reconstructed based on measured rough parameters in Table 1. Then the droplets impacting on these surfaces are investigated by experimental observation and LB simulation. Two sample surfaces are plotted here. Figure 4 shows the processes of droplets impacting on sample A-3 and B-3. Snapshots of the different stages of droplets after impinging are clearly shown. They include contacting, maximum spreading, departing and complete rebound stages. It can be seen from the images that the LB simulation can capture the dynamics of impinging process effectively. Due to the existence of rough structures, the shapes of droplets are asymmetrical both in experiment and simulation. (a)

(b)

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Figure 4. Comparision of the dynamic behavior of droplet impacting on the randomly distributed rough surfaces with different structures A-3 (a) and B-3 (b). The black-white images are obtained by a high-speed camera and the color images are simulated results by LBM. The intrinsic contact angle θY=126° and droplet impacting velocity v=0.08. As seen, LBM simulation demonstrates excellent agreement with the existing experimental data. The simulated shapes of droplet on the rough surfaces are consistent with experiment results. The small deviation in the contact time (Figure 4(a)) between experimental and simulated results may be attributed to the structural differences at the re-bounce location. The maximum spreading factor D*max=Dmax/D0 is also similar (see Figure 4, the second column). In the present study, Laplace law, one of the most well-established benchmark, is also simulated to validate the two-dimensional multiphase Shan-Chen (SC) type lattice Boltzmann model in Supporting Information. Then, we will investigate rough surfaces of random structures with the modeling in order to gain a more comprehensive understanding of bouncing oﬀ ability from these surfaces. The Dynamic Behavior of Droplet Impinging on Rough Surfaces With the LBM model, droplets impacting under various artificially designed operating conditions are modeled to investigate the effects of surface structure on droplets rebound abilities. The snapshots at the different stages of the droplet impinging on a surface with Sk=-0.75 and K=3.0 are presented in Figure 5. 15


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Owing to the small shear force, the shape of the droplet is spherical before impinging (see Figure 5,I). As the droplet impingement proceeds, the droplet spreads radially due to the inertial force and extends to the maximum diameter gradually (see Figure 5,II-III). At the start time, the total energy of the droplet is the sum of kinetic energy and surface energy. During spreading, part of the kinetic energy is converted into the droplet’s surface energy, which will be converted to kinetic energy at retracting. And other fractions of kinetic energy will be consumed by the means of irreversible dissipation, which will be transformed to heat energy finally. As the kinetic energy decreases, the surface tension and viscous force will be able to balance the inertial force. So the movement of three-phase contact line becomes slower and it stops at last. At this moment, the droplet achieves the maximum spreading diameter (see Figure 5, III). Due to the contact angle hysteresis, the three-phase contact line will stand still for a moment, during which the contact angle changes from the advancing contact angle to the receding contact angle. Then, the lamella begins to retract due to gas-liquid surface tension (see Figure 5,III-V). During retraction, the surface energy converts to kinetic energy and viscous dissipation. If the converted surface energy exceeds the viscous dissipation, the extra energy will result in droplet bouncing from the randomly distributed rough surface (see Figure 5,VI).

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Figure 5. Typical snapshots for droplet impinging on a randomly distributed rough surface. (a) Sk=-0.75 and K=3.0. The intrinsic contact angle θY=126° and droplet impacting velocity v=0.1. As retraction goes on, a convex liquid column is observed under the action of surface tension and adhesive force. Due to the effect of rough structures, the shape of convex liquid column is obviously asymmetrical at the moment of leaving the rough surface (see Figure 5,V). The adhesive force prohibits the droplet from leaving the wall while the free liquid surface on the top of droplet continues to move upward. These contradicting actions contribute to the droplet deformation and the formation of liquid column. Finally the droplet bounces back. After bouncing back, the droplet continues to move with inertial force and surface tension. The surface tension will induce the droplet to a shape with minimum surface energy, thus the droplet almost becomes spherical again (see Figure 5,VI). Before investigating the effects of asperity distributions, the sensitivity of the simulation results to the location of contact between the droplet and the rough surface is verified by droplet impacting various surfaces with the same rough parameters. Five surfaces with skewness -1.0 and kurtosis 3.0 are reconstructed and droplets impact those surfaces at the velocity of 0.1. During the impacting, several important parameters, such as the maximum spreading time (from the beginning of the contact with the surface to the maximum spreading diameter tmax), the maximum spreading factor D*max and the contact time, are recorded in the Table 2. As shown in the Table 2, the relative standard deviation (RSD) of those parameters is less than 2%. Therefore, we can conclude that the simulated results are not sensitive to the impacting locations. The surfaces of different locations can be regarded as statistically the same. In the subsequent simulations, droplets impact one location for each surface.

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Table 2. Simulated Results of the Rough Surfaces with the Same Skewness and Kurtosis. Surface 1

Surface 2

Surface 3

Surface 4

Surface 5

RSD

tmax (ms)

6.23

6.23

6.23

6.23

6.23

0.0%

D*max

3.31

3.35

3.21

3.23

3.26

1.69%

tcontact (ms)

26.42

27.17

27.42

26.67

26.93

1.51%

Effects of Skewness on the Dynamic Behavior of Droplet Impinging. According to the initial and operating conditions, different scenarios can be observed after droplet impinging the randomly distributed rough surfaces. In order to investigate the effects of asperity distributions on the ability of droplet bouncing, a series of surfaces with various skewness and kurtosis are investigated, while other conditions are fixed, such as the root mean square (Rq), auto-correlation length, the diameter of droplet and its velocity. The bouncing abilities of impinging droplets on randomly distributed rough surfaces can be characterized by two parameters. The first one is the contact time (from the beginning of the contact with the surface to the time of rebound off the surface), which is an important factor for the design of ice-free surfaces.38 The other parameter is the range of velocities for droplet bouncing.39 To disclose the effect of skewness on the droplet rebound, eleven rough surfaces with the same kurtosis but different skewness are investigated. Figure 6 shows the effects of skewness on the contact time. The impacting droplet spreads to the maximum spreading diameter and then retracts and bounces off the rough surface. Therefore, the contact time consists of two components: spreading time and retracting time. According to the simulations in Table S1, spreading times have a little change with the skewness increases at the same impacting velocity 0.1, which is in a good agreement with other experimental and theoretical studies.10, 40 The main differences in contact time are attributed to the retracting time. With the skewness increasing (in negative part), there are more sharp edges on the top of the rough surfaces, which greatly limit 18


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the retraction of the droplets. The contact time increases correspondingly when the skewness varies from -1.25 to -0.25. As the skewness increases further, obvious structure change can be found from insets in Figure 6 on the both sides of skewness 0.0. When the skewness is 0.25, more gas is trapped in a groove between the droplet and the solid surface at the maximum spreading diameter. Therefore, the retracting resistance is much smaller than that of surface with the skewness -0.25. The retracting time decreases quickly when the skewness varies from -0.25 to 0.25. As the skewness continues to increase (in positive part), the sharp edges increase and the retracting time increases to the maximum value at the skewness 0.75. After that, the maximum spreading diameter decreases significantly in Table S1 and the retracting time decreases correspondingly.

Figure 6. Effects of skewness on the contact time. Kurtosis is kept at of 3.0 and the intrinsic contact angle is 126°. The insets show the maximum spreading state of droplets on the rough surfaces skewness -0.25 (left) and 0.25 (right), respectively.

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Figure 7. Effects of skewness on the range of velocities for droplet bouncing. Kurtosis is kept at 3.0 and the intrinsic contact angle is 126°. Droplets bouncing is a necessity for self-cleaning. The ability to re-bounce droplets is a criteria to select good surfaces. In Figure 7, we can find that the skewness has different effects on the range of velocities for droplet bouncing. For negative skewness, the range of velocities for bouncing becomes larger as skewness decreases. The droplet bouncing can be witnessed at the velocity of 0.04 when the skewness is -1.25 and/or -1.0. By contrast, when the skewness is positive, the surface becomes non-bouncing under velocity of 0.04. So the rough surfaces with smaller skewness have a larger range of velocity for droplet bouncing. They are beneficial. This may be attributed to the energy consumption in the process of interaction between the droplet and the rough surface.40-42 The energy consumption are mainly composed of two parts, which are described as follows. The first part is the viscous dissipation caused by viscous force at the solid-liquid interface. The viscous dissipation occurs both on the top of the surface and between the grooves. After the droplet impinges the

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rough surface, the droplet propagates along the rough surface and part of the liquid permeates the grooves along the geometrical shape (see Figure 5,II ). This part of liquid is pushed back to the surface while it is still extended horizontally. There are no liquid in the grooves at the maximum spreading state (see Figure 5 III). With the skewness increases, the spaces of grooves become larger and more liquid would permeates into the grooves in the process of spreading, which leads to more viscous dissipation during droplet spreading and retracting. The second part is structure-induced dissipation when the droplet overcomes the pinning at the sharp edges on the randomly distributed rough surfaces.22, 43 The pinning effect occurs at the sharp edges, which can pin the triple line at a position far from the equilibrium state. When the contact line is far away from the sharp edge, the droplet moves along the rough surface with advancing contact angle. As the contact line reaches the sharp edge, the triple line will be pinned at the edge until it proceeds along the neighboring inclined edges. Based on Nosonovsky and Bhushan,43 the change of the surface slope at the sharp edge is the reason for pinning. With the skewness increasing, sharp edges on the top of the randomly distributed rough surface are observed more frequently, which also results in more energy dissipation after droplet impingement. Effects of Kurtosis on the Dynamic Behavior of Droplet Impinging The bouncing ability of the randomly distributed rough surface may also be affected by the rough surface’s kurtosis, which reflects the peakedness of the distribution. In order to eliminate the influences of skewness, surfaces which have the same skewness but different kurtosis are investigated. As shown in Figure 8, the droplets display different bouncing abilities on rough surfaces with various kurtosis. For the same skewness, surfaces can be classified into three groups: non-bouncing, bouncing and non-bouncing, when the kurtosis varies from 2.0 to 5.0. Our simulations do indicate that the kurtosis is of great importance to the ability of droplet bouncing when the skewness of surfaces are kept constant. The surfaces with a kurtosis 21


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around 3.0 show the best bouncing ability. When the kurtosis is too small, the height distributions of rough surfaces are compressed and the rough surfaces are relatively smooth. So the roughness-induced hydrophobicity is weak. When the kurtosis is too large, there are more sharp edges on the rough surface and the pinning effect will lead to more energy consumption, which has been proved to bring about main energy consumption in the process of spreading and retracting. 9, 41

Figure 8. Effects of kurtosis on the dynamic behaviors of droplets impacting. The velocity of droplet is 0.1 and the intrinsic contact angle is 121°. Effects of Intrinsic Contact Angle (CA) on the Dynamic Behavior of Droplet Impinging. The inherent hydrophobicity of the randomly distributed rough surface, which is represented by the intrinsic contact angle, has significant effects on the contact times. The CAs are calculated by changing Gads. As observed in Fig. S2 in the supporting information, though the relation is non-linear in nature, within the narrow range of contact angles considered, the CA changes with Gads almost linearly. Similar trends have been observed in Zhang et al.

44

The relations between Gads (intrinsic contact angle) and contacting time are

presented in Figure 9(a). It can be seen that with intrinsic contact angle increasing, the contact time decreases

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on all surfaces, regardless of other surface parameters. When the contact angle exceeds a certain value, the contact time is almost unchanged.

Figure 9. (a) Variations of the contact time with contact angles under the same impacting velocity of 0.1 and kurtosis of 3.0. The skewness is -0.75, -1.0 and -1.25, respectively. (b) States of droplets after impinging with various velocities and intrinsic angles. The skewness and kurtosis are -0.75 and 3.0, respectively.

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After impinging the rough surface, the final states of droplets are determined from our extensive simulations. They are plotted in the CA-We plot in Figure 9(b). The figure indicates that larger intrinsic contact angles (CA) are beneficial for droplet bouncing. The minimum velocity for droplet bouncing decreases from 0.08 to 0.02 when the CA changes from 121° to 131°. As a result, increasing the intrinsic contact angle is an effective way to widen the range of velocity for droplet bouncing. In summary, increasing the intrinsic contact angle has two benefits: on one hand, the contact time could be reduced; on the other hand, the range of velocity for droplet bouncing is extended. They are both important for increasing the hydrophobicity of rough surfaces. Comparisons to well-textured Surfaces The above investigation shows that roughness has a strong effect on droplet bouncing. Roughness are mostly random, however sometimes they can be well-textured by artificial fabrication. Previously, rough surfaces with man-made well-textured structures were studied. For a textured surface consisting of a regular array of pillars, the top of the surface is relatively flat and the spacing of pillars is equal. The droplet would spread freely along the surface and partial liquid would impale into the structures. 39, 45, 46 When the roughness is random, there are many sharp edges on the top of the surface and the spacings of rough structures are random. During spreading, droplet tends to fall into the larger grooves and the sharp edges also bring bad effects on the droplet spreading. In order to compare the effects of the two types of rough surfaces on droplet bouncing, two surfaces, one well textured and the other random, are modeled. The rough parameters, Sk, K and Rq, for the random surface are selected as the same as the ones for the well-textured surface, which has been employed for the investigation of droplets impinging previously in our research group.39 By changing the auto-length, the spacing of the grooves on the rough surface can be adjusted. In the simulation, the contact angle is set to the 24


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same value as in the literature, 110°. Figure 10 reports the states of droplet after impinging the two types of rough surfaces for a range of Weber numbers. When the We varies from 1.0 to 3.5, partially penetrated bouncing are observed on the well-textured surface. However, no bouncing occurs on the random surface under the same We. Therefore, to obtain robust superhydrophobicity, proper measures should be taken to get random surfaces with relatively equal spacing and less sharp edges. Random surfaces are inferior to well-textured surfaces, but they are convenient in large-scale manufacture.

Figure 10. States of droplets after impinging on the random surface and textured surface under different Weber numbers. The insets show surfaces with different types of roughness. T-s and R-s refer to random surface and textured surface, respectively. Even though the two types of rough surfaces show obviously different bouncing abilities, the rough parameters have a similar effect on the trend of bouncing. Surfaces with a lateral pillar size A and a pillar spacing B are studied.

45

When the B/A is 10, the water droplet is eventually pinned to the surface due to the

positive skewness. When the B/A is 1/3, the water droplet eventually pinched off the surface due to the

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negative skewness. The textured surface with smaller skewness also possesses a better performance in droplet bouncing, which supports our conclusion obtained in random surface.

CONCLUSIONS Superhydrophobic surfaces are accepted as the surfaces of self-cleaning. Due to their complexities in surface structure and the microscopic scale in droplets, it is difficult to model the dynamic behavior of droplet impinging on real surfaces with randomly distributed rough structures. Previous studies mainly focused on the smooth and textured surfaces, while in this study, a mesoscale LBM simulation method are used to investigate droplets impacting on randomly structured surfaces. The proposed LBM simulations disclose the details in the processes of droplet spreading, retracting and bouncing on the randomly distributed rough surfaces. The effects of roughness parameters, surface wettability and impinging velocities on the droplet bouncing abilities are studied. The rebound mechanisms are revealed. Some conclusions can be obtained as follows: (1) The surfaces with smaller skewness and a kurtosis around 3.0 show the best bouncing ability, which can be attributed to the smaller energy consumption in the process of droplet spreading and retracting. When the skewness is less, the viscous dissipation and structure-induced dissipation, pining effect, becomes smaller. Appropriate kurtosis is also needed. When the kurtosis is too small, the surface is relatively smooth and roughness-induced hydrophobicity is week. When the kurtosis is too large, the frequent sharp edges give rise to more dissipation by pinning effect. (2) Surfaces with larger CAs are beneficial for widening the ranges of velocities for droplet bouncing and shortening the contact time. The bouncing ability of rough surfaces can be enhanced by decreasing the number of sharp edges and decreasing the spacing between grooves.

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(3) Apart from the skewness and kurtosis, the root mean square and the auto-correlation length of rough surface also have significant impacts on the bouncing ability, which will be discussed in the future work.

ASSOCIATED CONTENT Supporting Information Experimental setup for the process of droplet impacting on the rough surface. The relation between the equilibrium contact angle and Gads. Simulated results of the rough surfaces with various skewness and the same kurtosis. The input sequence transformation by the Johnson translator system. Validation of two-dimensional multiphase Shan-Chen (SC) type lattice Boltzmann model. This information is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] Notes The authors declare no competing financial interest

Acknowledgements This Project was supported by the National Science Fund for Distinguished Young Scholars of China, No. 51425601. It was also supported by Natural Science Foundation of China, No. 51376064.

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Lattice Boltzmann Simulation of Droplets Impacting on

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