Asymmetric coalescence-induced droplet jumping on hydrophobic fibers

Accepted Manuscript Asymmetric Coalescence-Induced Droplet Jumping on Hydrophobic Fibers Bingbing Li, Feng Xin, Guorui Zhu, Wei Tan PII: DOI: Reference:

S0009-2509(19)30209-X https://doi.org/10.1016/j.ces.2019.02.041 CES 14825

To appear in:

Chemical Engineering Science

Received Date: Revised Date: Accepted Date:

29 October 2018 21 January 2019 16 February 2019

Please cite this article as: B. Li, F. Xin, G. Zhu, W. Tan, Asymmetric Coalescence-Induced Droplet Jumping on Hydrophobic Fibers, Chemical Engineering Science (2019), doi: https://doi.org/10.1016/j.ces.2019.02.041

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Asymmetric Coalescence-Induced Droplet Jumping on Hydrophobic Fibers Bingbing Li 1, 2, Feng Xin 1, Guorui Zhu1, * and Wei Tan 1, *

(1 School of Chemical Engineering and Technology, Tianjin University, Tianjin 300350, China

2 Department of Chemical Engineering, Renai College of Tianjin University, Tianjin 301636，China)

ABSTRACT Coalescence of two unequal-sized droplets on hydrophobic fibers is more common in nature. To elucidate the mechanism of merged droplets jumping on hydrophobic fibers, a new theoretical model for two unequal-sized droplets coalescence was first developed, which considered the effects of contact angle hysteresis and droplet-fiber contact area. Then, the influences of Ohnesorge number (Oh), droplet radius ratio (n), droplet-fiber radius ratio (k) and contact angle (CA) on the dimensionless jumping velocity (U/vci) were studied theoretically. The value of U/vci increased with the increase of n, k and CA but decreased as Oh increased. The dimensionless jumping velocity for two unequal-sized droplets coalescence on hydrophobic fibers still followed the capillary-inertial scaling law with U/vci≈0.45 1

when n=0.5. In addition, the effects of k and CA on the critical jumping radius ratio were the critical jumping radius ratio being reduced as k and CA increased, and the minimal critical jumping radius ratio was approximately 0.1. These findings will help to characterize coalescence-induced droplet jumping on hydrophobic fibers.

KEYWORDS: asymmetric coalescence, theoretical model, hydrophobic fiber, capillary-inertial scaling law, critical jumping radius ratio

1. INTRODUCTION When two or more droplets coalescence on a superhydrophobic surface (SHS), the excess surface energy will be partially converted into kinetic energy, which will result in an out-of-plane jumping motion of the merged droplet. This phenomenon is called coalescence-induced droplet jumping, which was first reported by Boreyko et al.1, and has recently received significant attention due to its potential biological and industrial applications, such as self-cleaning2, self-repelling3, anti-icing 4, electrostatic energy harvesting5, and condensation heat-transfer 6. Over the past several years, a considerable amount of work has been performed to improve the jumping of droplets coalescence on SHSs, and many fabrication methods were used to produce SHSs 7-10. For example, Sarkar et al.7 used an advanced electrospinning method to fabricate a dual-layer SHS by comprising polyvinylidene fluoride (PVDF) and fluorinated silane molecules (FSM) onto the glass. Miljkovic et al.8 applied a simple chemical etching and chemical vapor deposition to prepare

2

superhydrophobic copper surfaces to enhance condensation heat transfer, and they also used chemical etching and plasma method to fabricate SHS for electrostatic energy harvesting5. Tian et al.9 presented a chemical crystal growth method to prepare an SHS of closely packed ZnO nanoneedles. To explore the involved mechanisms and principles of merged droplets jumping on SHSs, many studies examining the coalescence-induced droplet jumping on SHSs have been conducted based on experiments10-13. For example, Mulroe et al.11 studied the effects of the surface topography on the dynamic behavior of merged droplets jumping on six different SHSs, and they found that the critical jumping diameter was highly dependent upon the topography of nanopillars. However, since the merged processes of the droplets on SHSs are very rapid and complex, they cannot be easily explored by experimental studies. To better understand the detailed dynamics of the event, many numerical methods have been successfully applied to droplet jumping studies and simulations14-19, such as the finite-element-based computational method (FEM) 14, 15, lattice Boltzmann method (LBM) 16, 17, molecular dynamics simulation (MD)18, and phase-field method19. Recently, some theoretical models for the jumping velocity of two equal-sized droplets coalescence on the SHSs have been proposed10, 20-24. Wang et al.20 presented a theoretical model for the symmetric coalescence-induced droplet jumping on a rough SHS based on the energy balance, the variation trend of the predicted jumping velocity was similar to that of Boreyko et al.’s experiments. Then, Peng et al.21 proposed a new theoretical model based on the multiphase LBM, and their predicted

3

jumping velocities of merged droplets showed better agreement with Boreyko et al.’s experiments than Wang’s model. After that, Lv et al. 10 established a new model based on the previous models presented by Wang et al. 20 and Peng et al. 21. They considered the impact of the changes in droplet morphology during the merging process; hence, the predicted jumping velocity was very consistent with the Boreyko et al.’s experiments. In addition, given that the excess surface energy is only partially converted into kinetic energy during the merging process of merged droplets, an efficiency term η was introduced to correct this incomplete conversion by Cha et al.22. The influence of the SHS morphology on the energy terms and jumping velocity were also considered by other researchers23, 24.

Regarding the coalescence-induced jumping of two unequal-sized droplets coalescence on SHSs, researchers also have performed studies on them in recent years25-32. He et al.25 studied two unequal-sized droplets coalescence on superhydrophobic porous aluminum surfaces with different adhesion energy between the droplets and surface. The results showed that the radius ratio of the two merged droplets jumping on the SHSs was raised with increasing adhesion energy. Liu et al.26 investigated two droplets coalescence on a superhydrophobic two-tier structure surface by theoretical analysis, and found that an unsuitable size ratio could not lead to jumping, which was consistent with the results of Yanagisawa et al.27. This finding is observed because the conversion efficiency of the released surface energy to translational kinetic energy is reduced as the size difference of two merged droplets increased28, 29. The critical size ratios for merged droplets jumping on SHSs was 4

studied by Wang et al.30, they found that the merged droplets jumping on flat SHSs would not occur when the radius ratio of merged droplets was less than a certain threshold value31. This law was also fit for the coalescence-induced jumping of nanodroplets on SHSs32.

Although many researchers have extensively studied on various aspects of the coalescence-induced droplet jumping on SHSs in recent years, the studies were mostly concentrated on the jumping of merged droplets on flat SHSs, and few studies were devoted to coalescence-induced droplet jumping on hydrophobic fibers. Zhang et al.33 reported the jumping motions of two equalized droplets coalescence on hydrophobic fibers, and observed that the hydrophobic fiber with a water contact angle (CA) larger than 90° enabled to occur the self-propelled process. Their investigation for the asymmetric coalescence-induced droplet jumping on the hydrophobic fibers was limited in measuring the jumping velocities of merged droplets on the Teflon-coated copper fiber. Subsequently, Li et al.

34

derived a

comprehensive theoretical model for predicting the jumping velocity of two droplets coalescence on hydrophobic fibers from the energy balance, and predicted jumping velocity of two equal-size droplets coalescence on the hydrophobic fibers to fit well with Zhang et al.'s experiments33.

In fact, the surface morphology of many flat SHSs is a fibrous structure, which has become a focus of current research. Hydrophobic fibers also have many practical applications, such as droplet filtration/separation35, 36, environmental protection37, 38,

5

health and safety. Therefore, studies on the jumping motions of merged droplets on hydrophobic fibers are of great practical significance. Moreover, the coalescence of two droplets that commonly occur in nature involve two unequal-sized droplets. Hence, it is necessary to study the coalescence-induced jumping of two unequal-sized droplets on hydrophobic fibers systematically.

Based on this background, we developed a physical model for the coalescence-induced jumping of two unequal-sized droplets, i.e., asymmetric coalescence-induced droplet jumping on hydrophobic fibers based on our previous work. Then, we validated our physical model by comparing the predicted jumping velocity with the experimental data available in the literature. Next, to investigate the capillary-inertial scaling law for coalescence-induced droplet jumping on hydrophobic fibers, we studied the effects of the Ohnesorge number (Oh), droplet radius ratio (n=RS/RL), droplet-fiber radius ratio (k=RL/rf) and the contact angle (CA) on the dimensionless jumping velocity. Finally, we examined the critical jumping radius ratio for coalescence-induced droplet jumping on hydrophobic fibers.

2. THEORETICAL MODEL For two droplets coalescence on an SHS, if they want to detach from the SHS, the excess surface energy must overcome the adhesion energy and other dissipation energies. However, the excess surface energy can be affected by many factors, such as surface roughness39, surface morphology40, and surface shape41. Moreover, according to our previous reports34, we found that the effects of the contact areas between the

6

droplets and fibers on the excess surface energy cannot be ignored, especially for small droplets. The excess surface energy of asymmetric coalescence is governed by the smaller droplet radius. Hence, in this section, we will develop a new theoretical model on the basis of our previous model34, which considers the influences of the contact areas between the droplets and fibers on the excess surface energy. In addition, unlike the superhydrophobic surface, the contact angle of water on hydrophobic fibers is always less than 150° due to wettability. Therefore, the influence of contact angle hysteresis on the coalescence-induced droplet jumping on hydrophobic fiber cannot be ignored. This effect is also implemented in our model.

Figure 1. Schematic of coalescence-induced jumping of two unequal-sized droplets on hydrophobic fibers

On the basis of our previous model34, we write the dynamic equation for two droplets coalescence on a hydrophobic fiber as

 2 3 lU 2  RL3  Rs3     f   lv (1  cos Y )( Alf ,L  Alf ,S )  l lv   RL3/2  RS3/2  (1) 3 2 l in which the left term is the whole kinetic energy (Ek) of the droplet immediately after jumping, U is the droplet jumping velocity, ρl is the liquid density, R is the droplet 7

radius, and the subscripts S and L denote small and large, respectively. The first term on the right is the total excess surface energy (ΔП), the second and third terms are the total interfacial adhesion energy (Eadh) and viscous dissipation energy (Evis), respectively. f is the roughness of the fiber surface. γlv is the liquid/gas surface tension. θY is the equilibrium contact angle of liquid on the fiber. μl is the liquid dynamic viscosity. Alf is the droplet-fiber contact area, which can be estimated by

Alf ,i  4rf  ci / bi   

bi

0

bi  rf sin(r ,i  0 )





bi2  x 2 / rf2  x 2 dx ,

and

where

rf

is

the

fiber

radius,

ci  R i sin 0 are the radii of the ellipse in the radial

direction and the axial direction of the fiber, respectively. The ellipse is the projection of the droplet-fiber contact area on the axial cross-section of the fiber. θ0 is the liquid apparent

contact

angle

along

the

axial

direction

of

the

fiber.

r ,i  arc tan sin(0 ) / (cos(0 )  Ri / rf )  is the droplet apparent contact angle along the radial direction of the fiber.

According to equation (1), if we want to accurately predict the jumping behavior of merged droplets on hydrophobic fibers, the calculations of the interface adhesion energy (Eadh) and total excess surface energy (ΔП) are critical.

The interface adhesion energy depends on the equilibrium contact angle (θY) and the droplet-fiber contact area (Alf,i). If the equilibrium contact angle and droplet-fiber contact area are available, the adhesion energy can be calculated. However, the equilibrium contact angle is always not easy to determine experimentally due to the contact angle hysteresis. Recently, a theoretical model to calculate the equilibrium

8

contact angle (θY) was published. This angle can be calculated from the advancing (θA) and receding (θR) contact angles as follow42  rA cos  A  rR cos  R   rA  rR  

Y  arcos 

(2)

where   sin 3  A rA   , 3  2  3cos  A  cos  A    sin 3  R rR    3  2  3cos  R  cos  R 

Hence, the interface adhesion energies (Eadh) between the droplets and fiber can be calculated. The excess surface energy of one droplet on a hydrophobic fiber can be estimated by i

  lv Alv ,i  2 lvVi / Ri

10

, where Alv,i and Ri are the droplet-gas contact area and

the radius of the droplet with volume Vi, respectively. From this equation, if we want to calculate the excess surface energy of merged droplet jumping on hydrophobic fiber, we must know the droplet-gas contact area (Alv,i) first. According to previous studies20, 22- 24, for a sessile droplet on hydrophobic fiber, the actual droplet-gas contact area (Alv,i) is always smaller than the surface area of the spherical droplet which has the same volume due to the influence of the droplet-fiber contact area. Hence, the influence of droplet-fiber contact area on the droplet-surface contact area (Alv,i) must be included when calculating the excess surface energy for the coalescence-induced droplet jumping on the hydrophobic fiber.

9

Given that the size of water droplet on hydrophobic fibers is always smaller than the water capillary length (~2.7 mm), the water droplet can be considered as a spherical droplet15. Hence, the droplet-gas contact area (Alv, i) of a spherical droplet with volume Vi on a hydrophobic fiber can be estimated by Alv,i  3Vi / Ri  Si

(3)

where Si is the surface area of the droplet segment that is cut off by the fiber. Based on the calculation of the droplet-fiber contact area in our previous paper34, the surface area of the droplets segment, Si, can also be obtained by the surface integral. In this instance, the standard equation of the spherical droplet can be written as x 2  y 2  z 2  Ri2

(4)

The corresponding partial derivatives of z=f (x, y) with respect to x and y are obtained as

z x' 

z 'y 

x R  x2  y 2 2 i

y R  x2  y 2 2 i

(5)

(6)

The unit surface area of the droplet segment that is cut off by the fiber is dS, which is given by

dS  1  (z'x )2  (z'y )2 dxdy By substituting equations (5) and (6) into equation (7), we obtain 10

(7)

dS 

Ri R  x2  y 2 2 i

(8)

dxdy

Hence, the surface area of the droplet segment, Si, can be estimated by the surface integral

bi

ci

Si   ds  4 dx  bi 0

bi2  x 2

0

D

Ri R x y 2 i

2

2

bi

ci bi2  x 2

0

bi R  x

dy  4 Ri  arc sin

2 i

2

dx

(9)

As previously mentioned, the excess surface energy of one droplet on a hydrophobic fiber is given by i   lv Alv,i  2 lvVi / Ri 10, and the droplet-gas contact area of a spherical droplet on a hydrophobic fiber is given by Alv,i  3Vi / Ri  Si . Hence, the excess surface energy of one droplet on a hydrophobic fiber can be written as: i   lv  3Vi / Ri  Si   2 lvVi / Ri   lvVi / Ri   lv Si

(10)

The excess surface energy of a merged droplet immediately after jumping on hydrophobic fiber is given by

0   lv 3V0 / R0  2 lvV0 / R0   lvV0 / R0

(11)

where R0 is the radius of the merged droplet and can be estimated by R0  3 RL3  RS3 .

Therefore, the change in the total excess surface energy of two droplets before and after coalescence is 2   4    lv  RL2  RS2    RL3  RS3  3    lv  S L  S S  3  

(12)

Finally, the jumping velocity of the merged droplet immediately after coalescence

11

is written as follows:

1/2

2 4   2 2 3 3 3  R  R  R  R  1/2  lv  L S   L S     lv  S L  S S     3 3    U    3 3   lv 3  2l ( RL  RS )   3/2 3/2   f   lv (1  cos Y )( Alf ,L  Alf ,S )  2 l    RL  RS   l  

(13)

Thus, for the coalescence-induced jumping of two unequal-sized droplets on hydrophobic fibers, the jumping velocity can be calculated if we know the radii of larger and smaller droplets, the fiber radius and the liquid contact angle. This information is of great importance to many industrial applications.

3. RESULTS AND DISCUSSION In this section, we initially confirmed our physical model by comparing the numerical results with available experimental evidence and then discussed the factors that influenced the jumping behaviors of two unequal-sized droplets coalescence on hydrophobic fibers systematically. Since the experiments of Zhang et al.33 were performed on a hydrophobic Teflon-coated conical copper fiber at 7°C, we made a simplified assumption that the surface morphologies of fibers were smooth with a surface roughness of exactly f=1 to facilitate comparisons with Zhang's data. The properties of water at 7°C (ρ=999.8 kg/m3, γlv=75.22 mN/m, and μl=1.647 mPa.s) 43 were used for the following calculation.

In this instance, the Ohnesorge number Oh is defined as the initial larger droplet radius with Oh  l / l  lv R L , which reflects the relative contribution of viscous

12

versus capillary-inertial effects. The dimensionless droplet velocity is given by

U *  U / vci , where vci   lv / l Rave

is the capillary characteristic velocity,

Rave  ( RL  RS ) / 2 is the average radius of initial droplets. The kinetic conversion efficiency is defined as the ratio of the jumping kinetic energy (Ek) to the excess surface energy (ΔП),   Ek /  .

3.1 Numerical Validation To validate the accuracy and applicability of the present physical model, the numerical results of two unequal-sized droplets coalescence on hydrophobic fibers were compared with the experimental data from Zhang et al.33 and the numerical data calculated by our previous model, which did not consider the effect of droplet-fiber contact area on the excess surface energy34. The advancing and receding contact angles used here were θA=120° and θR=108°, respectively, which were the same as that used in Zhang's experiments. We only chose the experimental results of rf=25-35 μm data series for comparison here. 0.24

0.6

(A)

0.16

0.4

n0.51

n = 0.51



0.12

0.3

U

U (m/s)

(B)

0.5

0.20

0.2

0.08 Experimental Data (Zhang et al.) Numerical resultes (Li et al. ) Current Prediction

0.04 0.00

6

8

10

12

14

16

Experimental Data (Zhang et al.) Numerical resultes (Li et al. ) Current Prediction

0.1 0.0

18

k = RL /rf

8

10

12

k = RL /rf

13

14

16

0.5

0.8

(C)

0.4

(D)

0.6



0.2

n =1.0

n =1.0

0.1 0.0

0.4

U

U (m/s)

0.3

0.2

Experimental Data (Zhang et al.) Numerical resultes ( Li et al.) Current Prediction

0

5

10

15

20

k = RL/rf

0.0

Experimental Data (Zhang et al.) Numerical resultes (Li et al.) Current Prediction

0

5

10 k = RL /rf

15

20

Figure 2. Comparisons of current numerical results with available data. (A) and (B) are the jumping velocity and dimensionless jumping velocity of n=0.51, respectively. (C) and (D) are the jumping velocity and dimensionless jumping velocity of n=1.0, respectively. (■: Experimental data (Zhang et al.33); ▼: Numerical results predicted by our previous model34; —: Predicted by equation (13)). Adapted in part with permission from ref 33. Copyright 2015 American Physical Society. Adapted in part with permission from ref 34. Copyright 2018 American Chemical Society.

Figures 2(A) and 2(B) compare the jumping velocities and dimensionless jumping velocities of current numerical results with Zhang's experiments data and the data calculated by our previous model. The radius ratio of the smaller droplet to larger droplet (n=RS/RL) was fixed at n=0.51. The two graphs depicted the same result in two different manners. As noted in Figure 2(A), as the radius ratio of the larger droplet to fiber increases (k=RL/rf), the jumping velocity predicted by our current model (—: red line) quickly increases at first and then slows down. The result exhibits good agreement with the experimental data (■: black square) because the kinetic energy conversion efficiency increases quickly at first then slows down as the value of 14

k=RL/rf increases (the reasons will be discussed in section 3.2.3). Hence, the increment of jumping velocity is reduced as k=RL/rf increases. Furthermore, for the asymmetric coalescence with n≈0.51, the jumping behavior that can occur is approximately k=8 for the minimum value of k, and this value is larger than that of symmetric coalescence (approximately k=3 as shown in Figure 2(D)). The main reason for this finding is that the excess surface energy of merged droplet for the asymmetric coalescence is lower than that of symmetric coalescence, even if the size of larger droplets is equal to that of symmetric coalescence. Hence, in this case, the minimum value of k that reliably indicated that the merged droplets can jump on hydrophobic fibers is approximately k=8. However, as noted in Figure 2(A) for the numerical results predicted by our previous model (▼: blue down-triangle), when the value of k decreases from 16 to 8, the jumping velocity consistently increases and exhibits an approximately linear increase, which is clearly contrary to the variations of experimental data. This finding is attributed to the fact that the influence of the droplet-fiber contact area on the excess surface energy of merged droplet is ignored in our previous model. Nevertheless, for a droplet on a hydrophobic fiber, the droplet surface energy is always reduced due to the influence of the droplet-fiber contact area. This effect can be negligible for a larger droplet. However, for a small droplet, this effect is amplified. In addition, for asymmetric coalescence, the excess surface energy of a merged droplet is governed by the smaller droplet. Thus, the effect of droplet-fiber contact area on the excess surface energy of the merged droplet cannot be ignored. Thus,

15

numerical results predicted by our current model show better agreement with Zhang et al.'s experiments than our previous model. The dimensionless jumping velocity of numerical results and experimental data are also compared in Figure 2(B). As noted in the graph, the numerical results predicted by our present model are obviously better than that of our previous model, which are also consistent with the experimental data. To verify the applicability of our present model, the situation of n=1, i.e., symmetric coalescence was also examined in this study, and the results are presented in Figure 2(C) and 2(D). As illustrated in the two graphs, although the numerical results predicted by our present model are somewhat lower than Zhang et al.’s experimental results and the numerical results predicted by our previous model, the variation trends are consistent. The main reason for this deviation is probably that when the surface area of a droplet segment that was cut off by fiber (Si) was calculated, we made some simplified assumptions, which may result in the discrepancy between the predicted surface area Si and the actual values. Although there are some deviations between the predicted results of the current model and the experimental data, the predictions are completely acceptable. In particular, the predictions around n=0.50 are very consistent with the experiments. Therefore, our present model can predict the jumping velocities of two droplets coalescence on hydrophobic fibers and can also qualitatively analyze the jumping behaviors of asymmetric coalescence cases on hydrophobic fibers. This information is very useful for many practical applications, such as providing a useful guideline for the design of droplet fibrous filters, health and safety equipment, and fog harvesting

16

equipment. 3.2 Capillary-Inertial Scaling Law For two equal-size droplets coalescence on a flat superhydrophobic surface (SHS), when Oh ≤ 0.10, the coalescence-induced dimensionless jumping velocity follows a capillary-inertial scaling law U *  U / vci  C0 , where C0≈0.2 is a constant19. However, for two unequal-sized droplets coalescence on hydrophobic fibers, is the capillary-inertial

scaling

law

still

suitable?

To

investigate

whether

the

capillary-inertial scaling law still holds for the coalescence-induced jumping of two unequal-sized droplets on hydrophobic fibers, the dimensionless jumping velocity (U*) is studied here. However, for the cases of two unequal-sized droplets coalescence on hydrophobic fibers, they are more complex than that on SHSs. Numerous factors can affect the coalescence-induced jumping behavior, such as the radius ratio of the smaller droplet to larger droplet (n=RS/RL), the radius ratio of the larger droplet to the fiber (k=RL/rf), and the contact angle (CA). Thus, the influences of Oh, n, k and CA on the dimensionless jumping velocity (U*) were studied here. 3.2.1 Effect of Ohnesorge Number (Oh) To understand the influence of Ohnesorge numbers (Oh) on the jumping behaviors of two unequal-sized droplets on hydrophobic fibers, the variations in the dimensionless jumping velocity (U*) and the kinetic energy conversion efficiency (ɳ) with Oh on four different hydrophobic fibers were studied here. In this work, the radius ratio of the smaller droplet to the larger droplet was fixed at n=0.5 first. Then, Oh was increased by reducing the initial droplets’ radii. Ohnesorge numbers were

17

varied from Oh=0.003 to the maximums that indicated that the merged droplets could not detach from the fibers. The advancing and receding contact angles of water droplets on fibers used here were the same as that in Zhang's experiments, in which θA=120° and θR=108°, respectively. 0.5

1.0

(A)

(B) n=0.5

0.4

0.8

0.3

0.6 η

U*

n=0.5

rf=30μm

0.2

0.4

rf=50μm

rf=50μm

rf=70μm

0.1

0.2

rf=90μm 0.0 0.002

0.004

rf=30μm rf=70μm rf=90μm

0.006

0.008

0.010

0.012

Oh number

0.0 0.002

0.004

0.006

0.008

0.010

0.012

Oh number

Figure 3. (A) Variations in dimensionless jumping velocity with Ohnesorge Number. (B) Variations in kinetic energy conversion efficiency with Ohnesorge Number.

As noted in Figure 3(A), the variations of U* with Oh on four fibers are significantly different. As Oh increases, the dimensionless jumping velocity on larger fibers decreases faster than that on smaller fibers, but the values of U* at Oh=0.003 are approximately equal. For example, when Oh increases from 0.003 to 0.067, the value of U* on the small fiber of rf=30μm (■: black square) decreases from approximately 0.450 to 0.365, which is a reduction of approximately 0.085. For the large fiber of rf=90μm (▼: blue down-triangle), the value of U* sharply decreases from approximately 0.416 to 0, which is approximately 4.9-fold more than that of small fiber. The main reason for these changes is that the coalescence-induced droplet jumping on hydrophobic fibers is in the capillary-inertial regime. Under small droplet 18

radius, the viscous effect is dominant, the released excess surface energy is quickly dissipated through viscous dissipation, and little kinetic energy is obtained. As the droplet radius increased (that is, with the decrease in Oh), the viscous effect is gradually reduced, and more kinetic energy is obtained. Hence, the dimensionless jumping velocity of the merged droplets increases as the Ohnesorge number decreases. When the value of Oh is sufficiently small (that is, the droplets are sufficiently large), the effect of adhesion energy can be ignored, and the kinetic energy conversion efficiency (ɳ) will approach the maximum, which is approximately 1 (as shown in Figure 3(B)). This situation is similar to the situations of droplets on non-wetting surfaces. Hence, the final values of U* at Oh=0.003 were approximately equal. As noted in Figure 3(A), as the fiber radius increases, the maximum Oh that the merged droplets can jump from fibers decreases. For example, the maximum of Oh for the small fiber rf=30 μm is approximately Oh=0.0113, whereas that for the large fiber rf=90 μm is only approximately Oh=0.0067. According to our model, the adhesion energy is governed by the droplet-fiber contact area. The droplet-fiber contact area increases as the fiber radius increases. Hence, the maximal Ohnesorge number that indicated that the merged droplet cannot detach from the fiber must decrease with increasing the fiber radius. In other words, if the merged droplets want to detach from the fiber, the minimum radius of the initial droplets also must increase with increasing fiber radius. Moreover, as noted in Figure 3(A), as Oh increases, the dimensionless jumping velocities U* increase and all finally approach a constant C0. At Oh=0.003, the

19

dimensionless jumping velocities U* for fiber rf=30 μm, 50 μm, 70 μm and 90 μm are approximately 0.46, 0.44, 0.43 and 0.42, respectively. These values were approximately equal and all approach a constant value of C0≈0.45 as Oh decreases. Hence, at a low Ohnesorge number (Oh ≤ 0.10), the jumping velocity for two unequal-sized droplets coalescence on hydrophobic fiber still follows the capillary-inertial scaling law. The constant C0 for n=0.5, θ0=120° on hydrophobic fibers is approximately 0.45. This value was larger than that of two equal-size droplets coalescence on flat SHSs (C0≈0.219). The reasons are still unclear, and further investigations are needed in our future studies.

3.2.2 Effect of Droplet Radius Ratio (n=RS/RL) For two droplets coalescence on hydrophobic fibers, the droplet radius ratio can significantly affect the excess surface energy and kinetic energy conversion efficiency 34

. Hence, to deeply understand the capillary-inertial scaling law of two unequal-sized

droplets jumping on hydrophobic fibers, the study of the variation in dimensionless jumping velocity with droplet radius ratio is necessary. Here, the fiber radius rf=30, and the water contact angles θA=120° and θR=108°. To investigate the influence of n on the dimensionless jumping velocity, the value of Oh was calculated by Oh  l / l  lv R L and fixed at a certain value, i.e., the corresponding initial larger

droplet radius was fixed, and the value of n decreased only by reducing the initial smaller droplet radius. The droplet radius ratios here varied from n=1.0 to the minimums that indicated the jumping motions of merged droplets on fibers could not

20

occur. We investigated four series of coalescence-induced droplet jumping on fibers in this study: Oh=0.00849, 0.00601, 0.00425 and 0.003. 1.0

0.6

(A)

(B) 0.8

0.5

0.6

rf=30μm

0.3 0.2 0.1 0.0

0.2

0.4

0.6

rf=30μm

η

U*

0.4

0.4

Oh=0.00849 Oh=0.00601 Oh=0.00425 Oh=0.00300 0.8 1.0

Oh=0.00849 Oh=0.00601 Oh=0.00425 Oh=0.00300

0.2 0.0

0.2

0.4

0.6

0.8

1.0

n=RS /RL

n=RS /RL

Figure 4. (A) Variations in dimensionless jumping velocity with droplet radius ratio. (B) Variations in kinetic energy conversion efficiency with droplet radius ratio.

The dependence of the dimensionless jumping velocity and kinetic energy conversion efficiency on droplet radius ratio n are plotted in Figure 4. As shown in Figure 4(A), as n increases, all the dimensionless jumping velocities continuously increase and reach the maximums at n=1.0. The finding is attributed to the fact that as n increases, the kinetic energy conversion efficiency also increases and reaches the maximum at n=1.0 (as shown in Figure 4(B)). In addition, as noted in Figure 4(A), as Oh decreases, the discrepancy of the dimensionless jumping velocity between different Oh at the same n also decreases. For example, under the droplet radius ratio n=1.0, when the value of Oh is reduced from Oh=0.00849 to Oh=0.00601, the value of U* increases from approximately 0.523 to 0.581, which represents an increase of 0.058. When Oh decreases from Oh=0.00601 to Oh=0.00425, the increase in U* is only approximately 0.029, which is less than half of that when Oh reduced is from 21

Oh=0.00849 to Oh=0.00601. The main reason for these variations is that as the droplet radius increases, the adhesion energy effect is reduced. The droplet-fiber contact area is governed by the fiber when the radius of the droplet is larger than that of fiber. If the fiber radius is fixed, as the droplet radius increases, the increase in the droplet-fiber contact area will be slower and slower, and the adhesion energy effect continually decreases. Therefore, the discrepancy of the dimensionless jumping velocity between different Oh values is reduced as the value of Oh decreases.

Figure 5. (A) Dimensionless jumping velocity for various droplet radius ratios with different Ohnesorge numbers. (B) Kinetic energy conversion efficiency for various droplet radius ratios with different Ohnesorge numbers.

To better understand the influences of the droplet radius ratio and Ohnesorge number, the droplet dimensionless jumping velocities and the corresponding kinetic energy conversion efficiencies for various droplet radius ratios with different Ohnesorge numbers are plotted in Figure 5. As noted in Figure 5(A), as Oh increases, the minimal droplet radius ratio n must

22

increase. Otherwise, the excess surface energy cannot overcome the adhesion energy, and the coalescence-induced jumping behavior of merged droplet cannot occur. For example, when the Ohnesorge number Oh=0.004 (the corresponding initial larger droplet radius is approximately 2250 μm), the minimum value of n is only approximately 0.16. When the Ohnesorge number increases to Oh=0.0152 (the corresponding initial larger droplet radius is 156 μm), the minimum value of n increases sharply up to 0.992. In other words, when the Ohnesorge number increases to Oh=0.0152, if the merged droplet wants to detach from hydrophobic fiber, the smaller droplet radius must be greater than 99.2% of larger droplet radius. Otherwise, the jumping behavior of merged droplet on hydrophobic fibers will not occur. As the Ohnesorge number increases, the influence of adhesion energy on the jumping

motions

of

merged

droplets

increases,

which

will

cause

the

coalescence-induced jumping to become more difficult. In this case, if the merged droplet wants to detach from the fiber, more kinetic energy is needed, which can be achieved by increasing the droplet radius. This notion can be confirmed by the variations of kinetic energy conversion efficiency with droplet radius ratio n as shown in Figure 5(B), where the kinetic energy conversion efficiency ɳ can be raised by decreasing the value of Oh or increasing the value of n. Therefore, increasing the droplet radius ratio or decreasing the Ohnesorge number is both conducive to the jumping of two unequal-sized droplets coalescence on hydrophobic fibers. 3.2.3 Effect of Droplet-Fiber Radius Ratio (k=RL/rf) In contrast to the coalescence-induced jumping of merged droplets on SHSs, the

23

jumping of merged droplet on hydrophobic fibers is also affected by the fiber radius. Hence, the effect of the droplet-fiber radius ratio on the coalescence-induced jumping behaviors of merged droplets was studied. In this instance, we studied the coalescence-induced jumping behaviors of merged droplets on three fibers, which were rf=30 μm, 90 μm and 200 μm. The droplet radius ratio of the smaller droplet to the larger droplet was fixed at n=0.5, and the water contact angles were θA=120° and θR=108°, respectively. Figure 6 shows the variations of the dimensionless jumping velocities and the corresponding kinetic energy conversion efficiency as a function of k=RL/rf. Here, the fiber radius was fixed at a certain value at first, and then the radii of the smaller and larger droplets were increased simultaneously with a fixed ratio of n=0.5. 1.0

0.5

(B)

(A)

0.4

0.8

0.35

0.8

0.30

0.6 

0.20 0.15



U*

U*

0.6

0.25

0.3

0.10

0.2

0.00

rf=90μm

8

12

16

n=0.5 0

20

rf=30μm

0.2

0.4

rf=200μm

0.0

20

k=RL/rf

0.1 0.0

rf=30μm

0.05

rf=90μm 60

n=0.5 0.0

80

rf=30μm

rf=90μm

rf=200μm 8

10

12

14

16

18

20

k=RL/rf

0.2

rf=200μm 40

0.4

0

20

rf=30μm

rf=90μm

rf=200μm 40

60

80

k=RL/rf

k=RL/rf

Figure 6. (A) Variations in dimensionless jumping velocity with droplet-fiber radius ratio. (B) Variations in kinetic energy conversion efficiency with droplet-fiber radius ratio.

As noted in Figure 6(A), the value of k=RL/rf has a great influence on the dimensionless jumping velocity U*. As k increases, the value of U* increases sharply 24

at first, then slows down and approaches the maximum. For example, for the fiber of rf=30 μm, when the value of k=RL/rf increases from k=8 to 20, the dimensionless jumping velocity U* increases sharply from U* =0 to 0.33. When the value of k increases from k=20 to 80, the dimensionless jumping velocity U* only increases from approximately U*=0.33 to 0.44, which is only approximately one-third of that when k ranged from k=8 to 20. The main reason for these changes of U* with k is that kinetic energy conversion efficiency increases quickly at first then slows down (as shown in Figure 6(B)). As mentioned in section 3.2.2, for a sessile droplet on a hydrophobic fiber, the droplet-fiber contact area is governed by the radii of both the droplet and the fiber. If the fiber radius is fixed, the adhesion energy increases very slowly when the value of k=RL/rf is exclusively increased. However, the excess surface energy increases quickly, and more kinetic energy is obtained. Therefore, with the increase of k=RL/rf, the adhesion energy effect decreases sharply at first, but the rate of decrease in the adhesion energy effect will become slower and slower. The kinetic energy conversion efficiency shows an opposite trend as that of adhesion energy effect, which increases quickly at first then slows down34. Moreover, as shown in Figure 6(A), if the value of k=RL/rf is fixed at a certain value, the dimensionless jumping velocity U* increases as the fiber radius rf increases. However, the increase is not significant, and this increase is reduced as the value of k increases. For example, at k=12, the value of U* on fibers rf=30 μm and rf=200 μm are approximately 0.20 and 0.23, respectively. The difference is approximately 15%. When the value of k increases to k=20, the value of U* on fibers rf=30 μm and rf=200

25

μm are approximately 0.33 and 0.34, respectively, and the difference is only approximately 3.0%. According to the definition of dimensionless jumping velocity, the dimensionless jumping velocity is given by U *  U / vci , where the capillary-inertial velocity vci is obtained by the assumption that all the released surface energy (ΔEs) is converted to kinetic energy. Hence, based on equation (1) and the kinetic energy equation, the dimensionless jumping velocity can also be written as the energy ratio of

U *  U / vci     Evis  Eadh  / Es . Moreover, for larger droplets coalescence on a small hydrophobic fiber, the viscous dissipation energy (Evis) can be negligible relative to the excess surface energy (ΔП)

34

. Therefore, the dimensionless jumping

velocity can be simplified to U *  U / vci  (   Eadh ) / Es . As mentioned in section 3.2.1, the coalescence-induced droplet jumping on hydrophobic fibers is in the capillary-inertial regime. Although the adhesion energy increases as the fiber radius increases, the influence of adhesion energy on the jumping process decreases as the droplet-fiber radius ratio increases34. Under larger k values, the influence of adhesion energy (Eadh) on the jumping velocity can be ignored relative to the excess surface energy (ΔП). In addition, although increasing the fiber radius can reduce the excess surface energy, its effect is limited and will continue to decrease as the value of k increases. Therefore, the influence of the fiber radius on the dimensionless jumping velocity U* is very small, and this influence is reduced as the value of k is increased and can even be ignored when the value of k is sufficiently large. In this case, the dimensionless jumping velocities U* of merged droplet jumping on hydrophobic

26

fibers had the same maximum of approximately 0.44, which was approximately C0≈0.45 compared with that obtained in section 3.2.1.

3.2.4 Effect of Contact Angle (CA) The influences of liquid contact angle (CA) on the jumping behaviors of two unequal-sized droplets coalescence on hydrophobic fibers were also studied here, and the results are presented in Figure 7. To facilitate research, we made a simplified assumption that there was no contact angle hysteresis in the solid/liquid/air system. In other words, the advancing contact angle was equal to the receding angle, and both were equal to that of the equilibrium contact angle,  A   R  Y . 1.0

0.50 (A)

(B) 0.9

0.45

0.8 rf=30μm

η

U*

0.40 0.35 Oh=0.00849 Oh=0.00601 Oh=0.00425

n=0.5 0.30 0.25

120

140

160

rf =30μm 0.7 Oh=0.00849 Oh=0.00601 Oh=0.00425

n=0.5 0.6 0.5

180

120

140

160

180

Contact angle (°)

Contact angle (°)

Figure 7. (A) Variations in dimensionless jumping velocity with contact angle. (B) Variations in kinetic energy conversion efficiency with contact angle.

As shown in Figure 7(A), the CA can significantly affect the dimensionless jumping velocity U*, and a larger contact angle is conducive to the jumping behavior of merged droplet coalescence on fibers. In addition, the influence of CA on the dimensionless jumping velocity U* of smaller droplets is more pronounced than that 27

of larger droplets. However, as CA increases, the differences between them are continuously reduced due to the diminishing impact of adhesion energy. For example, in Figure 7(A), when the CA ranges from 120° to 180°, the value of U* of the smallest droplet (Oh=0.00849) increases from approximately 0.3 to 0.45, which is an increase of approximately 0.15. Regarding the largest droplet (Oh=0.00425), the value of U * increases from approximately 0.43 to 0.46, which is only 20% of the increase noted for the smallest droplet. Moreover, as noted in Figure 7(A), when the CA=180°, the values of U* for the three cases are approximately equal to U*=0.45. This value was also equal to C0=0.45, which was obtained in section 3.2.1. This finding is attributed to the fact that when the CA=180°, the adhesion energy between the droplets and fibers can be ignored. This situation is similar to the situation of merged droplet jumping on hydrophobic fiber under a larger Ohnesorge number, where the influence of adhesion energy is also insignificant. Thus, the dimensionless jumping velocity of merged droplets coalescence on non-wetting fibers was approximately equal to the maximum dimensionless jumping velocity obtained in section 3.2.1, and both were equal to C0≈0.45.

3.3 Critical Jumping Radius Ratio (CJR) The critical jumping radius ratio (CJR) is defined as follows: for a given fiber, the critical jumping radius ratio, which is the minimum radius ratio of the smaller droplet to larger droplet ( CJR  RS,min / RL ), is that the merged droplet can just jump

28

spontaneously from the fiber. In this work, to determine the critical jumping radius ratio, the contact angle and fiber radius were fixed at first. Subsequently, we fixed the larger droplet at a certain value and then decreased the smaller droplet radius to the critical value that the merged droplet could not be self-propelled to detach from the fiber. In this case, the minimal droplet radius ratio was the critical jumping radius ratio. In this work, the initial larger droplets radii varied from RL=146 μm to 2500 μm, and these values were less than the water capillary length (~2.7 mm). 1.2

CJR

0.8

(B)

Non-jumping (Zhang et al.) Jumping Regime (Zhang et al.)

0.8

1.0

0.6

0.4

4

8

12 k=RL/rf

16

rf=30μm

rf=50μm

rf=70μm

rf=90μm

20

CJR

0.6

0.2 20

RL=1000μm

0.3

RL=2000μm

0.1

Non-jumping

0

RL=500μm

0.2

Non-jumping (Zhang et al.) Jumping (Zhang et al.)

0.0

RL=300μm Jumping Regime

0.4

Non-jumping 0.2

rf=30μm

0.5

Jumping Regime

0.4

Jumping-Regime

CJR

0.6

1.0

(A)

Non-jumping

40

60

0.0

80

k = RL/rf

120

130

140

150

160

170

180

Contact Angle (°)

Figure 8. (A) Variations in critical jumping radius ratio (CJR) with the droplet-fiber radius ratio. (B) Variations in critical jumping radius ratio (CJR) with contact angle.

First, the changing trends of CJR as a function of the value of k=RL/rf on four different fibers were studied, and the results are presented in Figure 8(A). The advancing and receding contact angles of water used in the calculation were θA=120° and θR=108°, respectively. It is clear from this graph that the fiber radius has no effect on the CJR, but the CJR is significantly affected by the value of k=RL/rf. As such, when the value of k=RL/rf increases from k=5 to 80, CJR reduces from approximately 29

1 to 0.1. In addition, when the value of k=RL/rf ranges from k=5 to 20, CJR decreases sharply from about CJR=1 to 0.3 as the value of k is reduced. When the value of k is larger than 20, the variation of CJR is very small. As the valuof k increases from k=20 to 80, CJR only decreases by approximately 0.2 and reaches the minimum CJR≈0.1 at k=80. As k=RL/rf increases, the kinetic energy conversion efficiency increases sharply at first and then slows down. Hence, CJR decreases sharply at first and then slows down as the value of k=RL/rf increases, which shows the opposite trend as that of kinetic energy conversion efficiency. Moreover, in Figure 8(A), the predicted jumping regime is compared with the available experimental data of Zhang et al.'s33, and the results show that our predicted results are in good agreement with the experimental data. For example, for the asymmetric coalescence, when the droplet radius ratio n≈0.5, the predicted minimal droplet-fiber radius ratio is approximately k≈0.92, which is consistent with the experimental value (k≈0.91). For the symmetric coalescence, the predicted minimal droplet-fiber radius ratio is approximately k≈4.6, which is also close to the value of k=4.0 that is reported by Zhang et al.33. Figure 8(B) shows the variation trends of CJR as a function of contact angle. It is apparent that as CA increases, the values of CJR decreases quickly, especially for small droplets. This finding is attributed to the fact that the influences of the adhesion energy on the jumping motions of small droplets are larger than that of larger droplets. However, as the contact angle increases, the adhesion energy is reduced, and it can be ignored when the CA=180°. Hence, the values of CJR are reduced with increasing CA

30

and reaches the minimum at CA=180°. As noted in this figure, the minimal CJR for droplets jumping on hydrophobic fibers is equal to CJR≈0.1. This value was consistent with that obtained in Figure 8(A). Hence, for capillary-inertial dominated coalescence, the minimal critical jumping radius ratio for merged droplets jumping on hydrophobic fibers is CJR≈0.1, which is much smaller than that on SHS. The CJR on SHS is approximately 0.5630. This finding may be attributed to the fact that the contact areas between the droplets and fibers are much smaller than that of a droplet on a flat superhydrophobic surface. As mentioned above, the CJR of two droplets coalescence on hydrophobic fibers is governed by the value of k=RL/rf and the water contact angle. Increasing the value of k or the liquid contact angle (CA) can both improve the coalescence-induced droplet jumping on hydrophobic fibers. Moreover, for capillary-inertial dominated coalescence, the minimal critical jumping radius ratio for droplets jumping on hydrophobic fibers is much smaller than that on SHS. Thus, the jumping of merged droplets on hydrophobic fibers is much easier than that on SHSs. Hence, a fabricated superhydrophobic fibrous structure surface is an effective method to improve the coalescence-induced jumping of merged droplets on SHSs. 4. CONCLUSIONS A new theoretical model for coalescence-induced jumping of two droplets on hydrophobic fibers was developed first, and the effects of contact angle hysteresis and droplet-fiber contact area were considered. Then, the influences of the Ohnesorge number (Oh), the droplet radius ratio (n=RS/RL), droplet-fiber radius ratio (k= RL/rf)

31

and the contact angle (CA) on the dimensionless jumping velocity were all studied theoretically. Theoretical calculations revealed that the value of the dimensionless jumping velocity increased as n, k, and CA increased but was reduced as Oh increased. The dimensionless jumping velocity for two unequal-sized droplets coalescence on hydrophobic fibers still followed the capillary-inertial scaling law (U/vci=C0) with C0≈0.45 when n=0.5. However, the constant C0≈0.45 was larger than that of two equal-size droplets coalescence on flat superhydrophobic surfaces. In addition, the effects of droplet-fiber radius ratio (k= RL/rf) and the contact angle (CA) on the critical jumping radius ratio were also investigated. The results showed that the critical jumping radius ratio was reduced as the value of k and CA decreased, but the fiber radius had no effect on the critical jumping radius ratio. For capillary-inertial dominated coalescence, the minimal critical jumping radius ratio for merged droplets jumping on hydrophobic fiber was approximately 0.1, which was smaller than that on flat superhydrophobic surface. These findings are conducive to deepening our understanding of the mechanism of coalescence-induced droplet jumping on hydrophobic fibers, which is helpful for many practical applications. ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (NO. 21606164). REFERENCES (1) Boreyko, J. B., Chen, C. H., 2009. Self-propelled dropwise condensate on superhydrophobic surfaces. Phys. Rev. Lett. 103, 184501.

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Highlight: 1.

A model for asymmetric coalescence of droplets jumping on fiber was established.

2.

Capillary-inertial scaling law of merged droplet jumping on fiber was researched.

3.

Jumping critical radius ratio for droplet on fiber was determined by the model.

39