Submillimeter-Sized Bubble Entrapment and a High-Speed Jet

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Submillimeter-sized bubble entrapment and highspeed jet during droplet impact on solid surfaces Longquan Chen, Long Li, Zhigang Li, and Kai Zhang Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b01506 • Publication Date (Web): 29 Jun 2017 Downloaded from http://pubs.acs.org on July 4, 2017

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Submillimeter-sized bubble entrapment and high-speed jet during droplet impact on solid surfaces Longquan Chen,*, † Long Li,‡, § Zhigang Li,‡ and Kai Zhangǁ †

State Key Laboratory of Traction Power, Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, and School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, P. R. China ‡

Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong §

Qian Xuesen Laboratory of Space Technology, China Academy of Space Technology, 100094, Beijing, China

ǁ

Wood Technology and Wood Chemistry, Georg-August-Universität Göttingen, Büsgenweg 4, Göttingen D-37077, Germany

Abstract When a droplet impacts on a solid surface, the entrapment of a submillimeter-sized bubble and the emission of a high speed jet can be observed at low impact velocities. In this work, we show that the bubble entrapment only occurs on sufficiently hydrophobic surfaces within a narrow range of impact velocities. The bubble is entrapped on hydrophobic surfaces, whereas it is trapped into the top of the droplet on superhydrophobic surfaces. The collapse of the air cavity formed during droplet impact, which is dominated by inertia and influenced by surface wettability, is the cause for the bubble entrapment. The velocity of liquid jets emitted after cavity collapse for drop impact with and without bubble entrapment scales with their sizes according to different power laws, which is explained by simple scaling analyses.

June 2017 1

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1. Introduction Motivated by numerous applications, significantly for spray cooling,1 internal combustion engines,2 painting and coating,3 ink-jet printing4 and recently also bloodstain pattern analysis,5 the impact of liquid droplets onto solid surfaces has attracted researchers in various disciplines for more than one century.6 It was only in recent years, however, that this multiphase flow can be investigated at different timescales in detail with the availability of high speed imaging techniques.7-9 Depending on impact conditions, several impact phenomena, including deposition, complete rebound, partial rebound, splashing and bubble entrapment, have been reported in the literature.7-9 Among them, the bubble entrapment during droplet impact is of great interest due to its key role for controlling droplet deposition in the applications aforementioned.7-8, 10 Generally, two types of bubble entrapment, which occur at different timescales, have been identified during droplet impact on solid surfaces. The first type takes place in the early spreading stage (the impact timescale is below 1 ms) and its formation mechanism can be simply described as follows.7-8, 11-12 When an impinging droplet approaches a solid surface, the compression of the air between the droplet and surface causes a pressure buildup in the air, which deforms the lower cap of the droplet in turn. As a result, a small bubble with a size ranging from a few tens to hundreds of micrometers is trapped beneath the impinging droplet once the droplet contacts the surface.13-14 The other type of bubble entrapment happens at impact timescale from ~ 1 ms to ~ 10 ms.15-16 It is promoted by the development of a capillary wave that propagates along the impinging droplet. The oscillation of the capillary wave leads to the formation of a cylindrical air cavity at droplet center,17-18 which is sealed during the subsequent droplet recoiling, forming a submillimeter-sized air bubble and emitting a singular jet simultaneously.15-16,

19

This type of bubble entrapment is

usually observed at low impact velocities.16, 19-24 The entrapment of microbubbles upon impact is a ubiquitous phenomenon in the course of droplet impact, and it has been extensively investigated via experimental,12-14, 18, 25-26 theoretical and numerical approaches.11, 27-30 In particular, 2


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the thickness evolution of the air layer during the entire bubble entrapment process has been resolved with a time resolution of 200 ns recently. 31 The underlying physics is well understood and a universal mechanism has been established.29 The entrapment of submillimeter-sized bubbles during impact, however, has received little attention so far. From the fundamental point of view, surface wettability is the key factor determining droplet impact dynamics in long timescales,7-8 and thus it should affect the bubble entrapment during impact. Indeed, the entrapment of bubbles on solid surfaces has been observed on hydrophobic surfaces,16, 19, 23 while floating bubbles trapped into the top of the impinging droplet was reported on superhydrophobic surfaces.20-22, 24, 32 However, how surface wettability influences the mechanism and occurrence of the bubble entrapment, as well as the singular jet, still remains unresolved. The goal of this work is to investigate the entrapment of submillimeter-sized bubbles and the emission of singular jets during droplet impact on solid surfaces with diverse wettabilities. Our experimental observations show that the bubble entrapment only occurs on sufficiently hydrophobic surfaces within a narrow range of impact velocities, and the entrapment process is determined by surface wettability. We also demonstrate that the jet velocity scales with jet radius according to power laws, and the corresponding exponents depend on the bubble entrapment behaviors.

2. Experimental Section We prepared ten solid surfaces with water contact angles of 40° − 161° using a variety of coating materials as shown in Table 1. Smooth substrates of glass, polyurethane and silicon were first cleaned with isopropyl alcohol and then ethanol in an ultrasonic bath for 5 min each. After drying with pure nitrogen, we obtained hydrophilic surfaces (surface i-iii). By silanizing glass substrates using a standard procedure,33 two hydrophobic surfaces (surface iv & surface v) were prepared. The other hydrophobic surface (surface vi) was fabricated by spin-coating a 300 μm-thickness polydimethylsiloxane (PDMS) mixture (monomer to cross-linker 3


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ratio of 10:1, Sylgard 184, Dow Corning, Wiesbaden, Germany) on the glass substrates and curing at 75 °C for 12 h. To further increase surface hydrophobicity, PDMS surfaces of square microcavities (surface vi-ix) were fabricated via soft lithography.34 The width ( ) and height ( ) of the cavities are 8 m and 2 m,

respectively. The spacing () between two neighboring cavities is varied from 2 m

to 10 m to adjust surface wettability (Table 1). The superhydrophobic surface

(surface x) in Table 1 was prepared by coating glass substrates with a layer of candle soot, which owns a fractal-like network.35

Table 1 The equilibrium ( ), advancing ( ) and receding ( ) contact angles of 4-L water droplets on diverse surfaces. Water contact angles (°)

Surfaces i. Glass

40±4

51±3

13±2

ii. Polyurethane

73±1

75±1

49±3

iii. Silicon

79±1

94±2

66±1

iv. Hexamethyldisilazane

91±1

97±2

79±1

v. 1H, 1H, 2H, 2H-perfluorodecyltriethoxysilane 103±2 111±1

95±1

vi. Polydimethylsiloxane

114±2 122±3

87±3

vii. PDMS-W8H2S10

122±2 132±2

69±2

viii. PDMS-W8H2S6

127±3 135±2

76±3

ix. PDMS-W8H2S2

140±1 143±1

93±3

x. Candle soot

161±2 163±1 159±1

Droplet impact experiments were carried out with pure water. The equilibrium ( ), advancing ( ) and receding ( ) contact angles of 4-L water droplets on these solid surfaces were measured using the sessile droplet technique with a commercial goniometer (Krüss DSA 30, Germany) and are summarized in Table 1. Water droplets with radii = 1.0 ± 0.02 mm were released at the tip of a

212 m-diameter needle and impacted on the solid surfaces placed underneath. Since 4


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the droplet size is smaller than the capillary length (~2.7 mm for water), the effects of the gravity on drop impact can be ignored, and the impact dynamics is mainly controlled by inertial forces, viscous forces and capillary forces. Therefore, the Webber number =

! /#

and the Reynolds number =

employed to describe these effects, where

/

are

is the impact velocity; , # and

are the density, surface tension and dynamic viscosity of the liquid, respectively. By varying the impact height, the impact velocity was adjusted from 0.25 to 0.8 m/s,

which corresponds to a Webber number range of 0. 9 − 8.8 and Reynolds number

range of 200 − 899. In order to visualize the cavity and bubble dynamics in the

impinging droplet, the incident light was illuminated on a printing paper (100 × 100 mm! ), which served as a light background for the impact experiment. The droplet impact process was recoded using a high-speed camera (Fastcam SAZ, Photron, Japan) at 60,000-100,000 fps. The camera was equipped with a macro lens (100F28D, Macro, Tokina) and teleconverter (2X, TC-201, Nikon), and tilted down for ~5° to observe the cavity dynamics near the solid-liquid interface. A spatial

resolution of 10 m pixel-1 was eventually achieved. Image processing and data analysis were accomplished using a custom-programmed MATLAB (MathWorks, Inc.) algorithm.

3. Results and Discussions Figure 1 summarizes the impact conditions for the occurrence of bubble entrapment. We found that a submillimeter-sized air bubble can only be entrapped during droplet impact on sufficiently hydrophobic surfaces with an impact velocity between a lower and upper threshold. On hydrophobic surfaces with 103° ≲ ≲ 140°, the air bubbles are entrapped on the solid surfaces, and the snapshots of the entrapment process for such a case are shown in Fig. 2a. When the droplet impacts on the solid surface, the inertial force causes the droplet to spread out immediately. It is noted that a microbubble is always entrapped beneath the impinging droplet upon impact,

7-8

although it is barely visible in Fig. 2a due to its small size and the thick

droplet. The contact with the surface generates a capillary wave at the bottom of the 5


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droplet, which is indicated by the red arrow in Fig. 2a. The capillary wave can only be observed for low-viscosity droplets when its wave length, which scales as #/

!

is smaller than the droplet radius, i.e. > 1, regardless of surface properties.17 In

the experiments, we found the generation of capillary waves at ≳ 1.6 on all the

solid surfaces. It propagates along the drop surface and deforms the droplet into a pyramidal structure with several steps (see 1.2 ms in Fig. 2a). As the drop spreads, these staircase steps subsequently merge into each other, and the droplet takes a pancake structure with a spire on top of its center (see 2.8 ms in Fig. 2a) at the maximum spreading. The downward oscillation of the spire leads to the formation of a cylindrical air cavity at the droplet center, which is very close to the entrapped microbubble near the maximum drop spreading. When the droplet starts to recoil, the air cavity is squeezed in the direction normal to the surface and the liquid film separating the cavity and microbubble ruptures, i.e. the air cavity merges with the microbubble. As a result, a capillary wave is generated at the bottom of the cavity and propagates along the cavity surface (denoted by the blue arrow in Fig. 2a). This process is similar to the formation of a dry-patch at the center of a splashing droplet reported by Thoroddsen et al. 36 In the further drop recoiling, the top of the air cavity retracts faster than the bottom since the contact line of the cavity is pined on the surface (see 4.1-4.4 ms in Fig. 2a). This asymmetric retraction results in the formation of a neck at the cavity center, as denoted by the white arrows in Fig. 2a. The neck is further stretched and eventually ruptured. This pinch-off process has some similarity with the detachment of an air bubble from a nozzle.37-38 As a result, an air bubble with a contact radius of ~0.15 mm is entrapped on the surface (denoted by red circle in Fig. 2a), and simultaneously a thin liquid jet is emitted.

6


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8 No Bubble

6 Bubble

We

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4 Bubble 2 40

80

120

160

o

θeq ( ) Figure 1. The bubble entrapment behavior as a function of Weber number and surface

wettability. On the superhydrophobic surface with = 161° , we also observed the entrapment of an air bubble inside the impinging droplet (Fig. 1). Similar to the observations on hydrophobic surfaces, an air cavity is formed at the drop center due to the oscillation of the capillary wave, and it reaches deeply into the droplet (3.5 ms in Fig. 2b & 2c). As shown in Fig. 2b & 2c, the cavity top recoils faster than the bottom, necking the cavity top. When the cavity top reconnects, an air bubble is entrapped. At low impact velocities (3.4 ≲ ≲ 4.6), a high-speed, thin jet is launched after the closeness of the air cavity near its top (indicated by white arrows in Fig. 2b), and the cavity is further constrained into a spherical bubble (the radius is ~0.30 mm) under the surface tension. This process was also observed on other superhydrophobic and superhydrophobic-like soft surfaces.15-16,

20-22, 24, 32

At higher impact velocities

( 4.6 ≲ ≲ 6.1 ), the rather fast recoiling of the cavity top compresses the cylindrical cavity into a bottle shape and a high-speed jet is emitted from the cavity bottom before it seals (denoted by red arrows at 4.1 ms in Fig. 2c). The further growth of the water jet inflates the cavity bottle (4.2 ms in Fig. 2c) and it drives the cavity to rise up (4.3 ms in Fig. 2c). Eventually, a slightly smaller bubble (the radius is ~0.15 mm) is entrapped. To the best of our knowledge, it is the first time to visualize 7


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this type of jet formation during droplet impact on solid surfaces.

Figure 2. Snapshots of bubble entrapment and jet emission on a structured PDMS surface

(PDMS-W8H2S6,

= 127° )

superhydrophobic surface ( = 161°) at

at

= 0.50m/s

= 0.58 m/s (b) and at

(a),

on

a

= 0.67 m/s

(c). (d) Jet emission during droplet impact on the superhydrophobic surface at

= 0.71 m/s. All the scale bars represent 1.0 mm. The entrapment processes observed above suggest that two compulsory conditions

are required to achieve bubble entrapment: formation of a deep air cavity close to the surface and fast recoiling of the droplet to seal the cavity at its center or top. Therefore, it is reasonably expect that the occurrence of bubble entrapment is highly dependent on the impact velocity and surface wettability (Fig. 1), which control the impact dynamics.7-8 If the impact velocity is lower than a lower threshold or the surface hydrophobicity does not reach a critical value, the downwards oscillation of the capillary wave cannot reach deep into the droplet, and the formed cavity collapses as the droplet recoils. On the other hand, if the impact velocity is higher than an upper threshold, the fast recoiling of the droplet can cause air cavity collapse before it seals (Fig. 2d), which also leaves no bubbles in the droplet. It is noted that liquid jets can 8


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also be observed after air cavity collapse in these two cases mentioned above. Neck radius Rmin

θeq=122o, V0=0.50 m/s θeq=122o, V0=0.60 m/s

Rmin, Rc (µm)

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θeq=127o, V0=0.50 m/s θeq=127o, V0=0.58 m/s

100 Cavity radius Rc

θeq=161o, V0=0.51 m/s

(Rmin, Rc)=16.5t0.5 10

θeq=161o, V0=0.58 m/s

100

tc-t (µs)

Figure 3. Log-log plot of the minimum neck radius -./ and the cavity radius 0

as a function of time (20 − 2) on three different surfaces at two impact velocities.

The collapse of the air cavity during droplet recoiling, which induces the axial implosion flow and initiates the bubble entrapment and jet formation, can be described by the standard Rayleigh-Plesset equation39-40 4(5) − 40 1 # + 29 = 67 + 9 ! : ln = > + 9 ! − 5 2 where 40 and 4 (5) are the pressures in the cavity and at a distance 5 of the order

of the drop radius ; is the radius of the cavity. The above equation is derived

under the assumption that the external fluid flow is purely radial and there is no vorticity, i.e. the fluid flow can be considered as a potential flow. It has a power-law solution in time, which depends on the type of force dominating the dynamics. If the inertial force resists the collapse, the logarithmic term can be neglected and the cavity radius scales as (# /)?/@ (20 − 2)?/! ,15, 38, 41 where 20 is the collapse time. On the other hand, if the viscous force is the main source resisting collapse, the last term in the equation diverges and the cavity collapses according to (#/2)(20 − 2).38, 41 In our experiments, the cylindrical air cavity was found to collapse either through pinch-off (Fig. 2a) or symmetrically axial implosion (Fig. 2b-c), depending on the 9


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impact velocity and surface wettability. Figure 3 plots the variation of the minimum radius of the neck region of the cavity -./ for hydrophobic surfaces with

103° ≲ ≲ 140° (defined in Fig. 2a) and the air cavity radius 0 for the

superhydrophobic surface with = 161° (defined in Fig. 2b) as a function of time

in the log-log scale. Here the collapse time 20 refers to the time when the cavity neck ruptures, e.g. ~4.5 ms in Fig. 2a. It is noted that the cavity size measured from the

recorded image is slightly smaller than the actual value due to light refraction by the curved surface of the droplet. However, the complex imaging system and droplet shape during impact cause difficulties to correct this optical effect. Nevertheless, by measuring the cavity size in air and within droplet close to the cavity top, we estimated an experimental error of ~6% or less for the experimental data. As shown in Fig. 3, both 0 and -./ follow a power law with an exponent of ~0.5, which is

in good agreement with the inertial collapse mechanism. Moreover, the linear fit in Fig. 3 provides a prefactor of 0.015 − 0.017 m ∙ s C?/! , which is also very close to

the theoretical value of ~0.016 m ∙ sC?/! ( # = 0.073 N/m , = 1 mm and = 997 kg/mG ). We also point out that the early cavity collapse of impinging

droplets on the superhydrophobic surface with 4.6 ≲ ≲ 6.1 is still dominated by inertia. However, the later formation of the bottle-shaped cavity and emission of a high speed jet from the cavity bottom (Fig. 2c) lead to more complex dynamics. We further investigate the liquid jet formed after air cavity collapse by measuring its radius H (defined in Fig. 2a,) and velocity

H

as it emerges from the impinging

droplets. Figure 4 shows the results obtained on three surfaces with different bubble entrapment behaviors, i.e. no bubble entrapment on hydrophobic surface with = 91°, bubble entrapped on surface on structured PDMS surface with = 127°, and floating bubble entrapped in droplet on superhydrophobic surface with = 161°. Although the experimental data is scattered, one can still identify three

distinct regimes on these surfaces, which are denoted by solid and dotted arrows in Fig. 4 a & b. At low impact velocities, the jet radius decreases while the jetting velocity increases with increasing impact velocity. H is typically a few hundreds of 10


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micrometers and

is of the order of ~1 m/s, which is about 2-5 times of the

H

impact velocity. At intermediate impact velocities, the jet radius decreases to a small value, which is ~0.25 mm for surface with = 91° and 15 − 40 m for

surfaces with ≳ 103°. For surfaces with ≳ 103°, this regime corresponds to

the emission of thin liquid jets, resulting from bubble entrapment (Fig. 1 & Fig. 2), and the jet velocity can be up to ~18 times of the impact velocity (see Fig. 4b). At

high impact velocities, H increases while

H

decreases with the increase of the

impact velocity as air bubbles cannot be entrapped any more. A similar jet formation dynamics was also reported by Bartolo et al. on superhydrophobic surface. 15 0.4

(a)

Rj (mm)

0.3

Regime III

0.2

Regime II

0.1 0.0 12

Regime I

(b)

9

Vj (m/s)

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Regime III 6

Regime I Regime II

3 0 0.3

0.4

0.5

0.6

0.7

0.8

V0 (m/s)

Figure 4. Plot of the jet radius (a) and jet velocity (b) as a function of the impact velocity obtained on three surfaces with = 91° (□), 127° (○,●) and 161° (△, ▲), respectively. Open symbols denotes impact without bubble entrapment; closed symbols denotes impact with bubble entrapment. The solid arrows indicates the transition from regime I to regime II, and the dotted arrows indicates the transition from regime II to regime III.

The different dependence of the jet characteristics (i.e. jet radius and jet velocity) 11


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on impact velocity for droplet impact without and with bubble entrapment suggests that their jet dynamics are different. Figure 5 plots the jet velocity as a function of the jet radius for surfaces with ≥ 91°. In general, the jet velocity decreases with increasing jet radius and all the experimental data collapse onto two master curves, which can be fitted by a power law

H

∝ HCK with a correlation coefficient of ~0.7,

where the exponent L depends on the bubble entrapment behavior. Other linear and non-linear functions were also applied to fit the experimental results in Fig. 5. However, a smaller correlation coefficient was obtained, indicating that the best fit for these data is the power law function. For drop impact without bubble entrapment, thick jets with H ≳ 0.1 mm are generated by the axial squeezing flow (Fig. 2d) and we find L = 0.94 ± 0.2. The relation between

H

and H can be explained by a

scaling analysis based on mass and energy conservation.15 At the onset of jet emission, the mass loss due to cavity collapse scales as 2M0 09 and the jet mass is 2M0! H , where is the thickness of the air cavity and it is comparable to the drop radius (see Fig. 1), i.e. ~ . Balancing these two terms one obtains

H

∝ 0C! . Moreover, the

kinetic energy flux passing through the cavity surface, which can be estimated as M90G 0 , should be equal to the kinetic energy of thin liquid jet M with the mass conservation equation, it yields

H

G ! H H .

Together

∝ HC? , which is consistent with the

experimental observations in Fig. 5.

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10

Vj (m/s)

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θeq No bubble Bubble

1

91o 103o 114o 122o 127o 140o 160o

0.01

0.1

Rj (mm)

Figure 5. Log-log plot of the jet velocity as a function of the jet radius. The dashed line and solid line are the power fits with

H

= N? HC.O@ and

H

= N! HC.PQ ,

respectively.

In contrast, for droplet impact with bubble entrapment, singular jets with H ≲ 0.1 mm were observed and we obtain

H

∝ HC.PQ±.? (Fig. 5). Here we carry

out a dimensional analysis to understand this finding. It is known that thin liquid jets are always generated by focusing capillary waves at an apex,42 which has been observed in a number of capillary phenomena such as oscillating droplets, bubble collapse,

44

and also droplet impact.

15-16, 24

43

cavity

With respect to thin jets generated

during droplet impact, they are formed through the convergence of capillary waves that associate with bubble entrapment (see Fig. 2a-c). Since the gravity is negligible, the dynamics is thus only dependent on the physical parameters of the liquid jet, i.e. the viscosity, surface tension, density and jet radius, and the phenomenon is governed by the Ohnesorge number Rℎ =

T

UVWXY

. Out of these physical parameters one can

define two characteristic velocities, the capillary-inertial velocity the capillary-viscous velocity

Z

.

= U#/H and

= #/. In the experiments, the thin jet velocity is in

the same order of magnitude with

.,

while

Z

is 1-2 orders of magnitude larger than

13


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H

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and is independent on jet radius. This indicates that the capillary and inertial forces

predominate the dynamics. Thus, dimensional analysis gives velocity in the form [Y [\

= ](Rℎ), where ]() is an unknown function. In the high-speed jetting process, it

is reasonable to ignore the viscous effects and one obtains

H

=N

.

∝ HC.P with N

as a coefficient. This scaling law matches with the experimental results in Fig. 5.

4. Conclusion In summary, we experimentally investigated the effects of surface wettability on bubble entrapment during droplet impact on solid surfaces. We show that the bubble entrapment is the consequence of the air cavity collapse due to capillary wave propagation, and it is influenced by droplet recoiling. As a result, the entrapment of air bubbles was only observed on sufficiently hydrophobic surfaces within a narrow range of impact velocities. On hydrophobic surfaces (103° ≲ ≲ 140°), the air cavity can be very close to the entrapped microbubble underneath and merges with it during droplet recoiling. Eventually, a bubble is entrapped on the surface as its contact line is pinned on the surface. On the superhydrophobic surface ( = 161°), the cavity bottom is driven away from the surface by surface tension or the singular jet, which causes a floating bubble to be entrapped in the impinging droplet. It was also found that the collapse of air cavity is dominated by inertial force and the axial squeezing flow is responsible for the formation of thick liquid jets for droplet impact without bubble entrapment. In contrast, the thin jets launched after bubble entrapment are due to the convergence of capillary waves along the liquid-air interface.

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

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6. 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 Fundamental Research Funds for the Central Universities.

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TOC 8 No Bubble

10

6

Vj (m/s)

Bubble

We

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

4

θeq No bubble Bubble o

Bubble

1

2 40

80

120

91 o 103 o 114 o 122 o 127 o 140 o 161

0.01

160

θeq (o)

0.1

Rj (mm)

TOC. The impact conditions for the occurrence of bubble entrapment and the relationship between the jet velocity and jet radius.

19


Submillimeter-Sized Bubble Entrapment and a High-Speed Jet

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