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Stability Analysis of Printed Liquid Elbows Kuan-Ming Huang, Tsung-Yen Tsou, Chia-Wei Chang, and Ying-Chih Liao Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b03906 • Publication Date (Web): 14 Dec 2016 Downloaded from http://pubs.acs.org on December 20, 2016

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Impact of Viscous Droplets on Superamphiphobic Surfaces Binyu Zhao,† Xiang Wang,‡ Kai Zhang,§ Longquan Chen,*,† Xu Deng,*,ǁ †

State Key Laboratory of Traction Power, Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China ‡ Department of Mechanical Engineering, Dongguan University of Technology, Dongguan 523808, China § Wood Technology and Wood Chemistry, Georg-August-Universität Göttingen, Büsgenweg 4, Göttingen D-37077, Germany ǁ Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China

Abstract The impact of a liquid droplet on a solid surface is one of the most common phenomena in nature and frequently encountered in numerous technological processes. Despite the significant progress on understanding droplet impact phenomena in the past one century, the impact dynamics, especially those coupling effects between liquid property and surface wettability on the impact process, is still less understood. In this work, we experimentally investigated the impact of viscous droplets on superamphiphobic surfaces, with the viscosity of liquids ranging from 0.89 mPa s to

150 mPa s. We show that the increase of liquid viscosity will slow down the impact

process and cause bouncing droplets to rebound lower and fewer times. The critical impact velocity, above which droplets can rebound from the superamphiphobic surface, is found to linearly increase with the liquid viscosity. We also show that the maximum spreading factor increases with Weber number or Reynolds number but decreases with liquid viscosity. Scaling analyses based on energy conservation were carried out to explain these findings and they were found to be in good agreement with our experimental results.

Keywords: droplet impact, viscosity, viscous dissipation, critical velocity, maximum spreading factor

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1. Introduction The impact of liquid droplets on solid surfaces is of prominent importance as it is involved in numerous technological processes1, such as ink-jet printing2, spray painting and spin coating3, pesticides deposition4, and also recently in bloodstain pattern analysis5. Driven by potential applications in these fields aforementioned, droplet impact has been extensively investigated for more than one century since the pioneering work of Worthington in 1876

1, 6, 7

. Various impact phenomena, including

deposition, complete rebound, partial rebound and splashing, have been identified1, 8, 9. In particular, the rebound of droplets from solid surfaces has attracted considerable attention of surface engineers due to its significant role in self-cleaning10,

11

,

anti-icing12-14, anti-fogging11, 15 and water harvesting16, 17. It has been demonstrated that surface superhydrophobicity18,

19

, Leidenfrost

effect20, 21, substrate sublimation22, and ambient pressure reduction23 can facilitate droplet rebound. Among them, surface superhydrophobicity is the one that has been widely studied. Superhydrophobic surfaces are low energy surfaces decorated with micro-, nano- or micro-/nanostructures, which can entrap a thin layer of air beneath the droplet and cause it to rebound after impact24-28. The impact process of a droplet on a superhydrophobic surface can be described as follows. Upon contact with the surface, the inertial force forces the droplet to expand up to a maximum extent. In this spreading process, most of the kinetic energy is converted into the interfacial energy, which triggers the subsequent droplet retraction. It is noted that both the spreading and retraction processes are associated with viscous energy dissipation within the droplet29-34. Therefore, a droplet can rebound only when its kinetic energy is high enough to compensate the energy dissipated during impact9, 35-37. For low viscosity liquid, e.g. water, a critical impact velocity for droplet rebound on superhydrophobic surfaces can be obtained by balancing the kinetic energy with the stored interfacial energy

35

. In industrial applications, however, highly viscous liquids, such as silicon

oils and paints, are commonly encountered. To the best of our knowledge, the impact of highly viscous droplets on superhydrophobic or superamphiphobic surfaces has received little attention33.


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The maximum spreading diameter ( ) is an important parameter for the

control of droplet deposition4, 38. Normalizing it by the initial droplet diameter ( ) yields a dimensionless parameter - the maximum spreading factor - =

/ . A number of theoretical models have been proposed to explain its relationship to the Weber number ( = /, where , , and are the

density, radius, velocity and surface tension of the droplet, respectively) or Reynolds number ( = / with being the viscosity) in the literature7,

39-41

. If

capillary force dominates the dynamics, a balance between the kinetic energy and interfacial energy results in ∝ /

predicts ∝ /

41, 42

39

, while momentum conservation

. On the other hand, if viscous force dominates the

dynamics, a scaling of ∝ /

39, 43, 44

or ∝ /

45

can be obtained

by balancing the kinetic energy and viscous dissipation energy. Recently, Laan et al.29 rescaled the spreading factor using ∝ ⁄ g ( #/ ), based on which

Lee et al.46 further developed a scaling of ( − )⁄ ∝ / g (), where

g is a function of #/ , g is a function of and is the equilibrium spreading factor at zero impact velocity. With these two models, they collapsed experimental data obtained from the impact of low viscosity droplets ( ≤ 51 mPa s)

on different partial wetting surfaces (the equilibrium contact angle '() is in the range of 23° − 110°). However, if these scaling analyses can be applied to predict

of the impact of viscous droplets on superamphiphobic surfaces is still an open question. In this work, we thus carried out an experimental investigation on the impact of

various viscous droplets on superamphiphobic surfaces. The viscosity of the liquids ranges from 0.89 mPa s to 150 mPa s and the impact velocity is varied from 0.05 m/s to 2.73 m/s, which correspond to of 0.03 − 131 and of 1.31 − 2275. We found that the impact process is slowed down by viscous dissipation and the critical

velocity for droplet rebound linearly increases with the liquid viscosity. The maximum spreading factor of various viscous droplets on superamphiphobic surfaces cannot be described by the scaling model of Laan et al.29 and Lee et al.46. Preliminary models based on energy conservation were proposed to explain these findings.


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2. Experimental Section 2.1 Surface and Liquids Superamphiphobic surfaces were fabricated on the silicon substrate by soot-templated structure comprising a fractal-like network with hydrophobized silica shell. Detailed fabrication information is described elsewhere47. Scanning electron microscopy (SEM, Quanta 200, FEI) images reveal that these nanoparticles form a fractal-like network (Fig. 1a), which is a desirable structure for superamphiphobicity 48

. Droplet impact experiments were carried out with pure water and various

glycerol-water mixtures. These liquids have slightly different surface tensions (63.0 – 71.8 mN/m) but much different viscosities (0.89 – 150 mPa s), which are obtained from the literatures49-51 and shown in Table 1. The equilibrium ('() ), advancing (' ) and receding ('. ) contact angles of these liquids on the superamphiphobic surface

were measured by the sessile drop technique using a commercial goniometer (DSA4,

Krüss GmbH, Germany). A 5 µL droplet was carefully deposited on the surface. The static (equilibrium) contact angle was obtained by fitting the drop profile using Laplace-Young method. By inflating or deflating the drop, we determined the advancing and receding contact angles when the contact line started to move. Each measurement was repeated at least at three different places on the surface. The error

bar is defined by the standard deviation. As shown in Table 1, '() is larger than 150° for all liquids (Fig. 1b) and the contact angle hysteresis (' − '. ) is always below 10°.

Fig. 1 The superamphiphobic surface. (a) SEM images of the superamphiphobic surface coated on silicon substrate. The brace indicates protrusions and the spacing B in between. (b) Contact angles of 5 µL droplets of pure water (Top) and 60 wt% glycerol-water mixture (Bottom) on the


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superamphiphobic surface shown in (a).

2.2 Droplet Impact Experiments & Data Analysis The superamphiphobic surface was placed on a horizontal stage, above which

5 − 8 μL droplets ( ≈ 2 mm) were generated by pumping liquids through a steel

needle using a syringe pump. Needles with an outer diameter of ~0.24 mm were applied throughout the experiments unless otherwise specified. The impact velocity was varied from 0.05 m/s to 2.73 m/s by changing the releasing height. The corresponding Weber number is 0.03 − 131 and the Reynolds number is 1.31 −

2275. We recorded the impact process with a high-speed camera (Photron, Fastcam

Mini UX100) at 12,800 fps. Each impact experiment was repeated at least three times. The

recorded

videos

of

droplet

impact

were

processed

using

a

custom-programmed MATLAB (Math_Works Inc., USA) algorithm. To quantify the impact process, we evaluated the variation of contact angle ('), contact diameter ( ) and centroid height (2) of the impinging droplets (Fig. 2 & Fig. 3).

Table 1. The viscosity (3), density (4) and surface tension (5) of various glycerol−water mixtures and the corresponding advancing (67 ), receding (68) and equilibrium (69: ) contact angles of these liquids on the superamphiphobic surfaces at 25°C. 3 (mPa s)49

4 (kg/m3)50

5 (mN/m)51

67 (°)

68 (°)

69: (°)

30

2.2

1071

66.5

163±1

153±3

158±4

60

8.8

1151

64.6

162±1

152±2

157±4

80

46

1205

63.3

163±1

152±2

157±5

85

82

1219

63.1

160±1

153±2

156±5

87

150

1227

63.0

159±1

152±1

156±4

Fraction (wt%) 0

0.89

997

71.8

163±1

3. Results and Discussion 3.1 Effects of liquid viscosity on the impact process


159±1

161±1

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When a liquid droplet impacts a non-wetting surface, it generally spreads on the surface and then retracts back. Eventually, the droplet can stick to or rebound from the surface depending on the impact velocity, surface wettability and liquid viscosity1, 9, 33, 36, 52

. We conducted a systematic study of droplet impact of pure water and various

glycerol-water mixtures on the superamphiphobic surface. We found that droplets could only rebound from the surface above a critical impact velocity, below which droplet deposition was observed. However, further increase of the impact velocity led to the occurrences of partial rebound, sticking and splashing of droplets. Fig. 2 shows the snapshots of droplets of water (left), 60 wt% (middle) and 85 wt% glycerol-water mixtures (right) impacting on the superamphiphobic surface at = 0.24 8.8 mPa s . Fig. 5a & b compares the impact process of three liquids at

= 0.89 m/s. Similar to bouncing droplets, a droplet with higher viscosity spreads

and retracts slower than droplets with low viscosities, and a smaller contact diameter is attained. It is also seen that the dynamic contact angle of all droplets is smaller than

150° during the whole impact process (Fig. 5b), indicating the penetration of liquids

into surface structure. Similar to other studies9, 33, droplets splashing was observed at very high velocity in our experiment. It happens at ≳ 2.2 m/s for water and 2.5

m/s for 30 wt% glycerol-water mixture ( = 2.2 mPa s), while no splashing was observed for other viscous liquids in the range of impact velocity we studied. 2

b 160

0.89 mPa s 8.8 mPa s 82 mPa s

120

θ (°)

a Dc/D0 (mm)

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1

80 0.89 mPa s 8.8 mPa s 82 mPa s

40

0

0

2

4

6

8

10

t (ms)

12

14

16

0

0

2

4

6

8

10

t (ms)

12

14

16

Fig. 5 Time evolution of HI /HC (a) and 6 (b) for the impact of droplets with different


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viscosities on the superamphiphobic surfaces at BC = C. ab F/G.

3.2 The critical velocity for droplet rebound

A droplet can rebound from a solid surface when its kinetic energy is high enough to compensate energy dissipated during impact. Therefore, there should exist a critical

impact velocity ? , above which droplet rebound is possible to occur. In this study,

this critical velocity is much smaller than the threshold velocity for liquid penetration into surface structures (0.40-0.69 m/s, depending on liquid properties) and splashing

( ≥2.2 m/s, depending liquid viscosity) during impact. Fig. 6 summarizes ? for

various viscous droplets on the superamphiphobic surface. It is seen that ? linearly increases with .

0.3

0.2

Vc (m/s)

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0.1

0.0

0

40

80

120

160

µ (mPa s)

Fig. 6 Plot of the critical impact velocity BI for droplet rebound as a function of the liquid

viscosity 3. The dashed line is the best linear fitting.

For

the

impact

of

low

viscosity

droplets

on

superhydrophobic

or

superamphiphobic surfaces, viscous dissipation inside the droplet is always negligible as the Reynolds number is much larger than 1

58

. The kinetic energy of impinging

droplet scales as [ and the surface energy can be stored during impact is estimated as |cos' − cos '. |

9, 35, 36

. The initial kinetic energy should be higher

than the stored surface energy for the droplet to rebound. Thus, one obtains a critical velocity for droplet rebound ? = e|cos' − cos '. |/ . For water, the above

scaling analysis predicts ? = 0.029 m/s on our superamphiphobic surface, which is consistent with the value (~0.031 m/s) that we measured in the experiment (see Fig.


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6). However, for liquids with higher viscosities, the initial kinetic energy is mainly dissipated by viscosity, so the above model will underestimate the critical impact velocity. Herein, we derive a critical impact velocity for viscous droplets based on energy

conservation. We assume that an impinging droplet with ? takes off from the

superamphiphobic surface with a much lower velocity, i.e. the kinetic energy of

rebounding droplet is negligible. Moreover, at low impact velocity a droplet slightly deforms during impact and always takes a spherical shape after impact 9. Therefore, we can also assume that droplet’s surface energy is the same before and after impact. Furthermore, although the flow field inside impinging droplet during spreading and retraction is different, the characteristic properties of the flow, e.g. the thickness of the boundary layer, the duration, should be in the same order of magnitude. As a result, the kinetic energy of an impinging droplet is dissipated by liquid viscosity. The viscous dissipation energy fg can be estimated as59

fg = h o hm i jk jA ≈ ikl? n

in which i is the viscous dissipation function i = p

qgr

qs

+

qgs

u

qgr

qr qs

≈ v Ox L

w

(1)

(2)

where k is the characteristic volume for viscous dissipation, l? is the contact time,

and δ is a characteristic length in the vertical direction.

Pasandideh-Fard et al.45 demonstrated that viscous dissipation mainly occurs in

the boundary layer with a thickness of y = 2 ⁄√2 and thus the characteristic volume can be estimated as k = z y. They scaled the time needed for an

impinging droplet to reach the maximum spreading as A ≈ 8 ⁄3 , and thus one could estimate the contact time as l? ≈ 2A . However, we found this scaling

strongly overestimates the contact time, e.g. it predicts a value of ~44 ms at an impact velocity of 0.24 m/s while the actual l? is 9.5 – 11.5 ms as shown in Fig. 3. In contrast, we found that the contact time can be well estimated by l? ≈ ⁄. Thus,

the viscous dissipation energy can be further expressed as


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fg = √2 { |

(3)

To promote a droplet rebound from the superamphiphobic surface, the kinetic energy,

f} = z [ /12, should be higher than the viscous dissipation energy, i.e. f} > fg ,

which leads to

> ? =

~

? ≈

RO

(4)

where the maximum spreading factor = / . At low impact velocity, is close to 1 (see Fig. 7), and thus we obtain ~

RO

(5)

This linear relationship between and matches our experimental results in

Fig. 6. A linear regression of the data in Fig. 6 provides a slope of 1.24 m/(Pa s2), which is also close to the value in Eq. (5).

3.3 The maximum spreading factor

The maximum spreading factor, , which is defined as the ratio of the

maximum spreading diameter ( ) to the initial droplet diameter before impact ( ), is the key parameter controlling droplet dynamics in practical applications4, 5, 60

and much effort has been devoted to understand the maximum spreading during droplet impact29, 32, 38-40, 42, 43, 45, 46. A number of theoretical models, which depend on the type of dissipation energy resisting impact, have been proposed to describe the relationship between and or Re

38, 39, 43, 45

. However, these models are

only applicable for specific liquids or surfaces. Recently, Laan et al.29 and Lee et al.46

developed a universal rescaling of droplet impact respectively, with which they collapsed of different viscous droplets ( ≤ 51 mPa s) on hydrophilic and

hydrophobic surfaces (23° ≤ '() ≤ 110°) into one master curve. Here, we will check

if these scaling analyses are also applicable for impact of various viscous droplets (

is up to 150 mPa s) on the superamphiphobic surfaces ('() = 156°~161°).

Fig. 7a & b plot of liquids with various viscosities as a function of

and , respectively. increases with and but decreases with the ACS Paragon Plus Environment

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increase of liquid viscosity, which is similar to the results obtained on hydrophobic and hydrophilic surfaces39,

. We also found that we cannot collapse of

43

droplets with different viscosity on superamphiphobic surfaces by either the scaling model of Laan et al.29 or Lee et al.46 (see Fig. S1 & Fig. S2 in the Supporting Information). This is plausible as a simple scaling analysis is hard to capture all energy involved in droplet impact. To understand the impact dynamics of viscous liquids on superamphiphobic surfaces, we balance all energy terms involved before impact and at the maximum spreading. Prior to impact, the kinetic energy is f} = z [ /12, the surface

energy of the droplet is f = z and the interfacial energy of the surface in air is f ≈ z /4, where is the solid fraction of the nanostructure and is

the solid/air interfacial tension. At the maximum spreading, we treated the droplet as a

circular cylindrical disc with a thickness of ℎ. Thus, the surface energy of the droplet scales as fP ≈ (2 − )z /4 + z ℎ and the interfacial energy of the

surface in liquid is fP ≈ z /4, where is the solid/liquid interfacial

tension. With energy conservation equation f} + f + f = fP + fP + fg , we obtain [

(

(

√(

[ + 1 − cos 'U ) − ( + 6) + 4 = 0

(6)

with cos 'U = (cos' + 1) − 1, where 'U is the contact angle of a droplet stays on

the structured surface in the Cassie state53, and ' is the Young’s contact angle of

the liquid on the smooth surface. On our superamphiphobic surfaces, static droplets always stay in the Cassie state and thus 'U ≈ '() . Solving the above equation, we

find

= cos [ #

in which = arccos 18√N [

(()

√N

#

(()

.


(7)

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a

6 5

βmax

3

1/2

2

1 -2 10

b

We

0.89 mPa s 2.2 mPa s 8.8 mPa s 46 mPa s 82 mPa s 150 mPa s

4

10

6 5

-1

10

0

10

We

3

1

10

2

1/4

Re


4

βmax

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2

1 10

0

10

1

10

Re

2

10

3

Fig. 7 The maximum spreading factors as a function of We (a) and Re (b). The dashed lines are determined from Eq. (7). The solid lines in (a) and (b) are depicted for scaling behaviors of

= ⁄ and = ⁄ , respectively.

As shown in Fig. 7a & b, Eq. (7) agrees our experimental data both in trend and order of magnitude. The agreement is the best for highly viscous droplets at high impact velocity, while for low viscosity droplets or low impact velocity, Eq. (7)

overestimates . This phenomenon may be due to two reasons. First, for low viscosity droplets, a capillary wave can be generated after contact with the solid surface at 1 ≲ ≲ 10 9. This capillary wave propagates along the droplet and

leads to the formation of an air cavity at the droplet center when the droplet reaches the maximum spreading. In our model, additional surface energy and viscous dissipation due to this strong deformation were not considered, which would result in

a larger . Second, at the maximum spreading an internal circular flow exists near the contact line 39. This internal kinetic energy was also not considered in our model


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and thus would cause an overestimation of .

Eq. (7) can be further simplified to the scaling in the literature. If ≪

√2(1 − cos '() ), the viscous dissipation within the droplet can be neglected during impact. At high impact velocity with ≫ 6, we obtain = £ cos ∝ / [ #

(

(8)

On the other hand, if ≫ 6 and ≫ √2(1 − cos '() ) , the viscous

dissipation dominates the dynamics and Eq. (7) reduces to = e2√2 ∝ / [

(9)

For comparison, the above scaling is also plotted in Fig. 7 (the solid lines). We found that the maximum spreading factor of all liquids we used cannot be described

by ∝ /. In contrast, the scaling of ∝ / can capture the trend

of for highly viscous liquids. These findings highlight the importance to consider viscous dissipation in the course of droplet impact dynamics.

Several theoretical models for the maximum spreading factor could also be found in the literature

30, 38, 45, 59

. However, we found that none of these models could

provide a good estimation for on the superamphiphobic surface (see Fig. S3 S5 in the Supporting Information).

Pasandideh-Fard et al.45 suggested that the viscous dissipation in a spreading droplet mainly occurs within a boundary layer. They neglected part of the surface energy of the droplet at the maximum spreading and calculated the viscous dissipation with a much longer time scale as we discussed before. As shown in Fig. S3 in the

Supporting Information, their model provides a good estimation of for low viscosity liquids at high , but it underestimates for highly viscous liquids.

The maximum deviation between the theoretical prediction and our experimental

result is up to 42% (see liquid with = 150 mPa s in Fig. S3). Li et al.30 assumed that the viscous dissipation occurs within the entire nanodroplet and estimated the

spreading time as the time to change the droplet height from to h. However, such

assumption is not appropriate for the impact of millimeter-sized droplet in our


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experiment, and thus their model is not applicable for our study. Indeed, a maximum deviation of 55% is found between their model and our experimental data (see Fig. S4 in the Supporting Information). Ukiwe et al.38 modified the theoretical model of Pasandideh-Fard et al.45 by considering the complete surface energy of the impinging

droplet. This model overestimates for low viscosity liquids with a maximum

deviation of 28% at high , but it underestimates for highly viscous liquids with a maximum deviation of 41% (see Fig. S5 in the Supporting Information).

In contrast, we refined the model of Pasandideh-Fard et al.45 by using an appropriate characteristic time for droplet spreading and a complete estimation of the surface energy at the maximum spreading. As a result, a good agreement between the model and experimental result for , especially for high viscosity droplets, is

reached. The maximum deviation between the theoretical value and the experimental result is 25% for water, and it decreases to 11% for liquid with viscosity of 150 mPa s, as shown in Fig. 7.

4. Conclusion In conclusion, we comparatively investigated the impact dynamics of various viscous droplets on superamphiphobic surfaces. We found that droplets with high viscosity spread and retract slower, take off the surface later and eventually rebound lower and less times than droplets with low viscosity. This phenomenon demonstrates that the viscous dissipation within droplet can strongly affect the impact process. Experiments also showed that the critical impact velocity for droplet rebound linearly increases with the liquid viscosity, which can be explained by a scaling argument. Furthermore, we found that the maximum spreading factor of various viscous droplets on superamphiphobic surfaces cannot be described by the scaling analyses recently reported in the literatures29, 46. Based on energy conservation, we developed a simple model to explain our experimental data. Since the impact of liquid droplets on solid surfaces is commonly encountered in industry, our results can shed light on the use of superamphiphobic surfaces in practical applications.


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Author information

Corresponding Author *E-mail: [email protected] (L.Q.C.); [email protected] (X.D.) Notes The authors declare no competing financial interest.

Acknowledgements

This research was supported by the National Young 1000 Talents Plan, the Young 1000 Talents Plan of Sichuan province, Sichuan Province Science Foundation for Youths (Grant No. 2016JQ0050) and the University Initiative Grant No. A0920502051607-1 of SWJTU. X.D acknowledges the National Natural Science Foundation of China (Grant No.21603026).

Associated Content

Supporting information is available free of charge

via the Internet at

http://pubs.acs.org/.

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TOC

Vc (m/s)

0.3 0.2

Vc≈(9/4ρD0)/µ

0.1 0.0 0

βmax

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

40

6 5 4 3

100

101

120

160

Re1/4


2

1

80

µ (mPa s)

102

Re

103

TOC. The critical velocity for droplet rebound linearly increases with droplet viscosity, and the maximum spreading factor increases with Reynolds number.


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