A New Theoretical Model for Coalescence-induced Droplet Jumping

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A New Theoretical Model for Coalescenceinduced Droplet Jumping on Hydrophobic Fibers Bingbing li, Feng Xin, Wei Tan, and Guorui Zhu Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00716 • Publication Date (Web): 24 May 2018 Downloaded from http://pubs.acs.org on May 24, 2018

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Industrial & Engineering Chemistry Research

A New Theoretical Model for Coalescence-induced Droplet Jumping on Hydrophobic Fibers Bingbing Li 1, 2, Feng Xin 1, Wei Tan 1 *, and Guorui Zhu1*

(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)

KEYWORDS: theoretical model, coalescence-induced jumping, hydrophobic fiber, asymmetric coalescence ABSTRACT

First, a new theoretical model that accounts for the excess surface energy, viscous dissipation, surface adhesion, the droplet wetting state, the droplet and fiber radii, and the fiber surface property is developed to predict the coalescence-induced droplet jumping velocity on hydrophobic fibers. Then, the effects of the droplet and fiber radii on the energy terms, the jumping velocity and the jumping critical contact angle of symmetric and/or asymmetric coalescence are investigated. The results indicate that for the symmetric coalescence, the jumping velocity falls after increases first along with the raise of droplets radii, despite the larger excess surface energy. However, for the asymmetric coalescence, the highest jumping velocity is achieved when the

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droplets have the same size. With the increase of the radii difference between the two coalesced droplets, the highest jumping velocity reduces gradually, even if the greater excess surface energy released upon coalescence.

1. Introduction

Hydrophobic surfaces are characterized by a high water contact angle and low adhesion. When a droplet coalesces on a superhydrophobic surfaces, coalescence-induced droplet jumping occurs owing to the low adhesion of these surfaces1. Due to these properties and their wide range of potential applications, such as self-cleaning2, self-repelling3, anti-icing4, and condensation heat-transfer enhancement5 and so on, hydrophobic surfaces have attracted great attention in recent years. Hydrophobic fibers, which is one of the most widely used hydrophobic surface materials, are well known to have many advantages besides hydrophobic surface properties, i.e., small diameter, large surface area and high porosity, and have been utilized for a variety of applications, such as emulsion and aerosol purification6,7, energy harvesting5,8, environmental protection9, 10, and superhydrophobic nanofibrous membrane fabrication11. Hence, studies of the coalescence-induced droplet jumping on hydrophobic fiber are very important.

The coalescence-induced droplet jumping on superhydrophobic surface was first reported by Broreyko and Chen1. Based on the experimental observation, they proposed a scaling model for the jumping velocity of the coalesced droplet which assumed that all the excess surface energy during the merged process was converted 2


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into kinetic energy. After that, many efforts have been done based on experiments12-16 and numerical simulation17-23 to study the influencing factors for coalescence-induced droplet jumping and improve the jumping motion of coalesced droplet on superhydrophobic surfaces. Moreover, some theoretical analysis focused on the physical mechanism of spontaneous droplet jumping from a superhydrophobic surface also have been conducted. Wang et al.24 analyzed the self-propelled jumping of coalesced droplets over a superhydrophobic rough surface. They derived a theoretical model for the coalescence-induced jumping velocity of symmetric coalescence based on the energy balance of surface energy, kinetic energy, and the viscous dissipation energy. Lv et al.8 improved the theoretical model with consideration of the energy dissipation due to the change of droplet morphology. Cha et al.25 studied the coalescence-induced removal of water nanodroplets from superhydrophobic carbon nanotube surfaces. They introduced an efficiency term η to correct the incomplete conversion of excess surface energy to the kinetic energy of the jumping droplet and viscous effects. Chen et al. 26 observed the coalescence-induced jumping of condensate droplets from a hierarchical superhydrophobic surface, and considered the influences of the superhydrophobic surface morphology and the line tension-induced dissipation energy in their model27.

Although numerous studies on the self-propelled jumping of coalesced droplets have been conducted and several theoretical models for predicting the jumping velocities of coalesced droplets have been proposed1, 8, 24-27, most of them were focused on the jumping of coalesced droplets on superhydrophobic flat substrates, and 3



few models could be used for the coalescence-induced jumping of droplets on hydrophobic fibers. Only Zhang et al.28 reported the jumping phenomenon of coalesced droplets on Teflon-coated copper fiber and presented a semiempirical model for predicting the jumping velocity of these coalesced droplets. However, the semiempirical model was proposed based on only the experiment results of water droplets jumping on Teflon-coated copper fiber; thus, some ambiguities may remain in other situations.

Here, we develop a physical model for predicting the coalescence-induced jumping velocity of two droplets on a hydrophobic fiber based on the conservation of energy. Unlike the situation of droplets jumping on superhydrophobic flat substrates, the jumping motion of a coalesced droplet on a hydrophobic fiber is not only affected by the nature of the fiber, but also by the droplet radius, the radius and surface properties of fiber, all of which were taken into account in our model. In the following sections, a comprehensive theoretical model based on the conservation of energy is proposed first and then validated by comparison with the experimental data available in the literature. Next, the effects of the fiber and droplet radii on the energy terms and the jumping critical contact angle of coalesced droplets are analyzed. Finally, some conclusions are drawn, which might be useful for understanding the mechanism of self-propelled droplet jumping on textured surfaces.

2. Theoretical Model

In this section, we propose a new theoretical model for predicting the jumping 4


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velocity of coalesced droplet on hydrophobic fiber based on the energy conservation law. And the effects of the coalescing droplet sizes, droplet wetting states, the fiber radius and fiber surface properties are all taken into account in our model.

When two sessile droplets coalesce on a hydrophobic fiber, the coalesced droplets will display jumping behavior only if the excess surface energy ∆Π exceeds the interfacial adhesion energy Eadh and the viscous dissipation energy Evis during the merging process. Hence, the kinetic energy of the coalesced droplet is given by

Ek = ∆Π − Eadh − Evis

(1)

First, the excess surface energy Π of one droplet on a substrate with a certain contact angle θ0 is given by8 Π i = Esurf ,i − ∆pi ⋅ Vi , where Esurf,i is the droplet surface energy and ∆pi ⋅Vi is the energy dissipated due to the change in droplet morphology with the capillary pressure ∆p = 2γ lv / Ri , in which Ri is the radius of the droplet with volume Vi. The surface energy of one droplet is given by Esurf ,i = γ lv Alv ,i + (γ sl − γ sv ) Asl ,i , where γlv, γsl, and γsv are the liquid/vapor, solid/liquid and solid/vapor interfacial tensions, respectively, and Alv,i and Asl,i are the liquid/vapor and solid/liquid contact areas of the droplet, respectively. For a sessile droplet on a hydrophobic fiber, considering that the scale of the droplets is much smaller than the capillary length

lc = γ lv / ρ g (~2.7 mm for water droplets at 20°C) 21, the effect of gravity can be ignored. In addition, the droplets that can self-remove from a hydrophobic fiber are always three times larger than the fiber28, the contact area between the droplet and fiber is very small relative to the liquid-vapor surface area. 5


Thus, the equilibrium


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shape of a droplet on a hydrophobic fiber can be seen as a spherical cap and the effect of the droplet-fiber contact area on the surface energy can be ignored. The surface energy of the droplet can be estimated by Esurf ,i = γ lv Alv ,i = 3γ lvVi / Ri . Accordingly, the total excess surface energy of two droplets before and after coalescence is given by 2   4 ∆Π = πγ lv ( R12 + R22 ) − ( R13 + R23 ) 3  3  

(2)

where R1 and R2 are the radii of droplets with volumes V1 and V2, respectively.

Second, the adhesion energy of one droplet on a hydrophobic fiber can be estimated according to the Young-Dupré Equation

Eadh = γ lv (1 + cos θ 0 )rf Alf ,i

(3)

where Alf,i is the droplet-fiber contact area of a droplet with volume Vi, rf is the roughness of the fiber surface.

In this paper, we focus on the situation of a liquid droplet on the top of a hydrophobic cylindrical fiber and assume that the droplet is axisymmetric around the Z-axis, which passes through the center of the droplet.

6


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(A)

Z

(B)

Z

R

Water Droplet O

R

c

i

Water Droplet

θ0

i

G

θr E

D A

bi

ci

C

C

F

H

Y

B

θ0 X

r

Fiber

r

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


bi

I

D

Fiber

X

Figure 1. (A) The front view of a liquid droplet adhering on the top of a fiber. The shaded part on the axial cross-section of the fiber is the projection of the droplet-fiber contact area. (B) The side view of a liquid droplet adhering on the top of a fiber.

For a sessile droplet on a hydrophobic fiber, it is well known that the droplet-fiber contact area Alf,i is difficult to calculate. Specifically, the apparent contact angle of a liquid droplet on the cylindrical fiber surface changes along the contact line in a complicated manner: it depends not only on the nature of the fiber but also on the droplet radius (Ri) and the droplet-to-fiber radius ratio (Ri/r). To calculate the droplet-fiber contact area, we assumed that the projection of the curved surface on the axial cross-section of cylindrical fiber is an ellipse determined by the droplet apparent contact angles along the X-axis and Y-axis and by the droplet and fiber radii, as shown in Figure 1(A). Hence, the droplet-fiber contact area can be estimated by the surface integral of the ellipse.

At this point, we assumed that the standard equation of the ellipse on the axial

7



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cross-section of a cylindrical fiber is x2 y 2 + =1 bi2 ci2

(4)

where bi and ci are the radii of the ellipse along the X-axis and Y-axis, respectively.

Equation (4) can also be written as

y=

ci bi

bi2 − x 2

(5)

For the apparent contact angle of the droplet on the fiber (θr,i) along the X-axis relative to the horizontal, we found that it is greater than the apparent contact angle on a flat surface (θ0) due to the interaction with the curvature of fiber, as shown in Figure 1. The effect of sharp edges on the apparent contact angle was first considered by Gibbs29 while analyzing the equilibrium of the three-phase (solid/liquid/air) contact line at a solid edge. And the droplet apparent contact angle on the fiber (θr,i) along the X-axis can be calculated from a given droplet radius (Ri), fiber radius (r) and the apparent contact angle of the liquid on a flat surface (θ0) with the same wettability30, 31

θr ,i ≈ Arc tan [sin(θ0 ) / (cos(θ0 ) − Ri / r )]

(6)

As shown in Figure 1(B), the line segment EH equals the ellipse radius bi, and the line segment EI equals the fiber radius r. In the right-angled triangles GIE and GEH, the angle GIE equals GEH, and the value is equal to θr,i −θ0 . Therefore, the value of bi can be given as

8


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bi = r sin(θr,i −θ0 )

(7)

Besides, the droplet apparent contact angle (θ0) on the fiber along the Y-axis is equal to that on a flat surface. Hence, the radius of the ellipse along the Y-axis (ci) can be calculated as

ci = R i sin θ0

(8)

Here, the equation of the cylindrical fiber is assumed as

x2 + y2 = r 2

(9)

where r is the radius of the cylindrical fiber.

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

z x' =

r 2 − x2

z 'y = 0

(10)

(11)

The unit surface area of the droplet-fiber contact area, dS, is given by

dS = 1 + ( z x' )2 + ( z 'y )2 dxdy

(12)

By substituting Equations (10) and (11) into Equation (12), we obtain

dS =

r r − x2 2

dxdy

9


(13)


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Hence, the droplet-fiber contact area of one droplet, Alf,i, can be estimated by

Alf ,i = ∫∫ dS = 4∫∫ D

D1

r r 2 − x2

bi

ci 2 2 bi − x bi 0

r

rc dy = 4 i 2 2 bi r −x

dxdy = 4∫ dx∫ 0

bi

bi2 − x2

0

r 2 − x2

∫

dx

(14)

By substituting Equations (7) and (8) into Equation (14), we obtain

Alf ,i

2 2 2 Ri sin θ0 r sin(θ r ,i −θ0 ) r sin (θ r ,i − θ 0 ) − x dx =4 sin(θ r ,i − θ0 ) ∫0 r 2 − x2

(i=1, 2)

(15)

Hence, the adhesion energy of two droplets can be given as follows

Eadh = γ lv (1 + cosθ0 )rf ( Alf ,1 + Alf ,2 )

(16)

where Alf,1 and Alf,2 are the droplet-fiber contact areas of droplets with volumes V1 and V2, respectively.

Third, the viscous dissipation energy of one droplet caused by the internal fluid τi

flow during the merging process is estimated as Evis,i = ∫

0

∫ φ dVdt ≈φτ V Vi

i

i i

8, 32

, where

φ = 2τ xxτ x, x ≈ µl (u / Ri ) 2 / 2 is the dissipation function and τ i ≈ ρ Ri3 / γ lv is the characteristic capillary time. The average velocity of each droplet during the merging process can be obtained as u ≈ τ ⋅ ∆p ⋅ Asec / ρ lVi ≈（3 / 2） γ lv / ρ l Ri , where Asec is the droplet cross-section. Hence, the viscous dissipation energy of two droplets is given as

E vis ≈

3 γ πµ l lv ⋅ R13 / 2 + R23 / 2 2 ρl

(

)

where µl and ρl are the viscosity and density of liquid, respectively.

10


(17)

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Fourth, the kinetic energy of the coalesced droplet is obtained by Ek =

1 1 mU 2 = ρ lU 2 (V1 + V2 ) 2 2

(18)

where U is the jumping velocity of the coalesced droplet just after coalescence.

Finally, the jumping velocity of the coalesced droplet just after coalescence is written as follows 1/ 2

 2(∆Π − Evis − Eadh )  U =  ρ (V1 + V2 )  

1/ 2

 3(∆Π − Evis − Eadh )  =  3 3  2πρ(R1 + R2 ) 

(19)

In this case, due to the jumping droplets are larger than the microstructures on the surface of the fiber, the contribution of line tension can be ignored33.

As mentioned above, for a small droplet on a substrate, the effect of gravity can be ignored, moreover, the energies are scalar values (meaning that they have magnitude but no direction). Thus, the influence of the position of droplet on fiber surface on the coalescence-induced jumping velocity can be ignored28. Our model has no special requirements for the position of coalesced droplets on fiber.

3. Results and discussion

In this section, we first confirm the theoretical model by comparing the theoretically predicted jumping velocities with the available experimental data. Then, the effects of the fiber and liquid droplet radii on the energy terms and the jumping critical contact angle of the coalesced droplet are discussed. In this paper, we focused on fibers with a smooth surface, that is, with a fiber roughness rf =1. The properties of 11



water at 7°C (ρ=999.8 kg/m3, γlv=75.22 mN/m, and µl=1.647 mPa.s) 34 were used for the following calculation.

3.1 Numerical validation

To validate our developed theoretical model, the coalescence-induced jumping velocities given by Equation (19) were compared with the experimental results of the r=25-35 µm data series from Zhang et al.28, whose experiments were conducted on a Teflon-coated conical copper fiber at 7°C. The parameters used in the calculations were the same as the conditions of these experiments, that is, the water apparent contact angle θ0=120°, fiber radius r=30 µm, fiber roughness rf =1.0, and the properties of water at 7°C.

0.5

0.7

(A)

(B) 0.6

0.4

R1=R2

0.5

0.3

0.4

U∗

Jumping velocity U (m/s)

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

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0.2

R1=R2

0.3 0.2

0.1 0.0

Ref.[28], r =25-35µm Prediction, r =30µm 0

100

200

300

400

Ref.[28] r =25-35µm Prediction r =30µm

0.1 0.0

500

0

2

4

Droplet radius Ri (µm）

6

8 10 Ri /r

12

14

16

18

Figure 2. (A) The variation in jumping velocity with droplet radius. (B) The variation in nondimensional jumping velocity (U*) with the droplet-to-fiber radius ratio (Ri/r), where U * =

ρU 2 Ri /γ lv

28

. (The parameters used in the calculation are θ0=120°,

r=30µm, rf =1.0, and the properties of water at 7°C). Adapted in part with permission from [Zhang, K. G.; Liu, F. J.; Williams, A. J.; Qu, X. P.; Feng, J. J.; Chen, C. H. 12


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Self-Propelled Droplet Removal from Hydrophobic Fiber-Based Coalescers. Phys. Rev. Lett. 2015, 115, 074502]. Copyright [2015] [Physical Review Letters]

As seen in Figure 2 (A), the variations in the jumping velocities predicted by the model and the experimental results coincided well, especially for large droplets. For example, when the droplet radius ranged from 100 µm to 140 µm, the average error of the jumping velocity was reduced to 26%, whereas when the droplet radius was between 150 µm and 500 µm, the average error of the jumping velocity was less than 10%. As the droplet size increased, the jumping velocity increased initially, reached a maximum, and then started to decrease. This behavior is observed because the coalescence-induced jumping process of liquid droplets is in the capillary-inertial regime. When the droplet radius was small, the viscous effect was dominant, the adhesion energy and viscous dissipation energy values increased as the droplet radius decreased, and less kinetic energy was obtained. Hence, the jumping velocity of the coalesced droplet was reduced, and the coalescence-induced jumping behavior would not even happen if the coalesced droplet radius was less than a critical value. However, when the droplet radius was large, the jumping process was controlled by the gravitational effect, and although the kinetic energy of the coalesced droplet increased with increasing droplet radius, the jumping velocity of the merged droplet was still reduced due to the increased droplet mass. The variations in the energy terms with the droplet radius are discussed in the next section.

The values predicted by the model and the experimental data for the 13



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nondimensional jumping velocity were also compared. The nondimensional jumping velocity, which was obtained by normalizing the jumping velocity with the capillary-inertial velocity vci = γ lv / ρR , can be given by U * =

ρU 2 Ri /γ lv . As seen

in Figure 2 (B), as the droplet-to-fiber radius ratio (Ri/r) increased, the nondimensional jumping velocity predicted by our model varied similarly to the experimental results. Additionally, with a small droplet-to-fiber radius ratio (Ri/r), the coincidence of the predicted and experimental nondimensional jumping velocity was obviously superior to that of the predicted and experimental jumping velocity (as shown in Figure 2 (A)). This is because the capillary-inertial velocities increased quickly with the decline of droplet radius, and the differences of the nondimensional jumping velocities between the predicted and experimental data were reduced by normalizing the jumping velocity with the capillary-inertial velocity.

Although the jumping velocity and nondimensional jumping velocity predicted by our model deviated from the experimental data for smaller droplets, the predicted and experimental values had the same variation trends. In addition, Zhang et al.28 pointed out that the critical radius of droplets that can self-remove from hydrophobic fibers was approximately three times larger than that of the fiber. Therefore, our theoretical predictions were consistent with the experimental data for cases of coalescence-induced droplet jumping on superhydrophobic fibers.

The main reasons for the discrepancy between the predicted and experimental results when the droplet radius was small are as follows. First, the water apparent

14


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contact angle on a hydrophobic fiber varied with the droplet-to-fiber radius ratio and the position along the contact line because they depended on the fiber diameter and droplet size, especially for the small droplet-to-fiber radius ratio, while in our model, the contact angle was a constant value by model input, and we only considered only the variations in the contact angle along the axial and radial directions of the fiber. Second, the shape of the droplet-fiber contact area was not the same in different situations, as the shape depended on both the apparent contact angle of the liquid droplet and the droplet-to-fiber radius ratio. Third, when the droplet radius was small, the influences of droplet deformation on the liquid-vapor area and the excess surface energy of the droplet could not be ignored. Finally, the surface structure of the fiber also had a significant effect on the adhesion energy for small droplets.

In general, our theoretical model was very useful for predicting of the jumping velocity of coalesced droplets on hydrophobic fibers, and might provide useful guideline for the design of jumping-droplet systems to maximize the efficiency through coalescence-induced droplet jumping on superhydrophobic fibers, which is very important to more industrially relevant internal flow situations such as fibrous and membrane filters.

3.2 The influences of the droplet and fiber radii on the energy terms of a coalesced droplet

As mentioned above, we know that the droplet and fiber radii have a significant influence on the jumping motion of coalesced droplet on hydrophobic fibers; thus, the 15



effects of the fiber and droplet radii on the energy terms and jumping velocity were analyzed in detail.

3.2.1 Effect of the fiber radius

To understand the effect of the fiber radius on the jumping motion of a coalesced droplet, the variations in all energy terms of the coalesced droplet with the fiber radius were predicted. The radii of two droplets R1= R2=200 µm, and the water apparent contact angle θ0=120°. As shown in Figure 3 (A), when the fiber radius was increased from 1 µm to 200 µm, the excess surface energy and viscous dissipation energy remained constant at 5.01×10-9 J and 6.39×10-11 J, respectively, while the adhesion energy increased from 3.9×10-11 J to 4.07×10-9 J and the kinetic energy was reduced from 4.01×10-9 J to 8.77×10-10 J.

U

0.40

50

0.35

40

0.30

30

0.25

20

0.20 R1=R2=200µm

10 0

0

50

100 150 Fiber radius r (µm)

12

0.45

0.15 200

10 8 6 4 2 0

0.10

(B)

Total contact area × 10 -8 (m 2)

Evis

Total contact area × 10 -8 (m 2 )

Eadh

Ju m p in g velocity U (m /s)

∆Π Ek

60 (A)

E nergy × 1 0 − 10 (J)

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

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1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

2

4

6

8

10

Fiber radius r (µm)

0

50

100

150

200

250

Fiber radius r (µm)

Figure 3. (A) The changes in all energy terms and jumping velocity with the fiber radius. (B) The variation in the total droplet-fiber contact area with the fiber radius.

16


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The effect of the droplet-fiber contact area on the excess surface energy was ignored in our model. The viscous dissipation energy was the lost in deforming the drop against viscosity by the internal flow during the merging process, which was also independent to the droplet-fiber contact area. Hence, the fiber radius had no effects on excess surface energy and viscous dissipation energy. In contrast, according to Equation (16), the adhesion energy was dominated by the droplet-fiber contact area. As the fiber radius increased, the total droplet-fiber contact area increased quickly (as shown in Figure 3(B)), which would result in a fast increase in the adhesion energy. As a result, the kinetic energy and the jumping velocity of the coalesced droplet decreased as the fiber radius increased. Therefore, reducing the fiber radius could improve the coalescence-induced jumping of the coalesced droplets.

3.2.2 Effect of the droplet radius

To better understand the jumping process of coalesced droplets, we discuss two different situations of the coalescence-induced jumping of two droplets, namely, symmetric coalescence and asymmetric coalescence. And the influences of the radii of two droplets on each energy term and the energy conversion efficiency are provided in this section. The energy conversion efficiency ηi was defined as the ratio of the excess surface energy ∆Π converted into the kinetic energy Ek, adhesion energy Eadh or viscous dissipation energy Evis, i.e., ηi= Ei/∆Π.

The fiber radius and water apparent contact angle values used in the following calculation were the same as those used in section 3.1, i.e., r=30 µm and θ0=120°, 17



respectively.

3.2.2.1 Symmetric coalescence

The effects of droplet radius on each energy term and the energy conversion efficiency of coalesced droplet under the symmetric coalescence are shown in Figure 4. In this study, the radius of the fiber r=30 µm, the water apparent contact angle θ0=120°, and the radii of the two droplets ranged from 30 µm to 500 µm.

0.50

∆Π

Eadh

EK

U

0.45

300

0.40 0.35

200

0.30 100

0

R1=R2

0

100

0.25

200 300 400 Droplet radius Ri (µm)

100

Evis

500

0.20

Energy conversion efficiency η i (% )

(A)

Jum ping velocity U (m /s)

400

Energy ×10 -10 (J)

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

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(B) 80 60

ηadh 40

ηvis

R1=R2

ηk

20 0

0

100

200 300 400 Droplet radius Ri (µ µm)

500

Figure 4. (A) The variations in the energy terms and jumping velocity with the droplet radius. (B) The variations in the energy conversion efficiencies with the droplet radius.

As seen in Figure 4, with the increase of the droplet radius, all the energy terms were increased, especially the excess surface energy and kinetic energy, which displayed approximately exponential growth. When the droplet radius increased from 30 µm to 500 µm, the viscous dissipation energy and adhesion energy raised only around 4.2 ×10-10 J and 28×10-10 J, respectively, whereas the excess surface energy 18


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and kinetic energy increased approximately 325×10-10 J and 292×10-10 J, respectively. This is because the slow increase in the fiber-droplet contact area and the rapid increase in the surface area of coalesced droplets as the droplet radius increased, as shown in Figure 5. When the radii of the two droplets increased from 30 µm to 500 µm, the total fiber-droplet contact area, Alf, , increased only approximately by 25 times from 0.0025×10-6 m2 to 0.073×10-6 m2, whereas the surface area, ∆S, increased by 277 times from 0.0047×10-6 m2 to 1.3×10-6 m2. Hence, as the raise of the droplet radius, the excess energy increased quickly, and a greater amount of excess surface energy was converted into kinetic energy, i.e., the kinetic energy conversion efficiency ηk increased sharply and the adhesion energy conversion efficiency ηadh decreased as increasing the droplet radius, as shown in Figure 4 (B).

0.08

(B) 0.10

1.2

0.06

0.08

0.04 0.03 0.02

R1=R2

0.01

2

)

0.9 0.06 0.6 0.04 R1=R2

0.3

R1=100µm

R1=100 µm 0

100

200

300 Ri (µm)

400

0.0

500

0

100

200 300 400 Droplet radius R (µm)

500

∆ S×10 -6 ( m

∆ S×10 -6 (m 2 )

0.05

0.00

0.12

1.5

(A)

0.07 Contact area A lf,i× 10 -6 (m 2 )

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


0.02 0.00

i

Figure 5. (A) The changes in the total fiber-droplet contact areas of two droplets with the droplet radius. (B) The increase in the surface area during the merging process of the droplet against the droplet radius, where ∆S = R12 + R22 − ( R13 + R13 )2/3 .

19



The jumping velocity of the coalesced droplet is also plotted in Figure 4(A). From this figure, we noticed that the jumping velocity did not increase continually with the kinetic energy and that the maximum jumping velocity was reached at approximately Ri =70 µm. This result is observed because the kinetic energy depends on both the droplet mass and the jumping velocity. Hence, controlling the size of the droplet so that it is in a moderate range promotes the detachment of the coalesced droplet from the hydrophobic surface.

3.2.2.2 Asymmetric coalescence

Until now, the understanding of droplet jumping has mostly been based on the coalescence of two equally sized droplets. However, the coalescence of two droplets with different sizes is actually more frequent. Hence, we studied the influences of droplet radius on the energy terms of two droplets with different sizes, as shown in Figure 6. In this study, we assumed that the radius of one liquid droplet (R1) was 100 µm and that the radius of the other liquid droplet (R2) was changed; and other initial conditions were the same as those used for the simulation of symmetric coalescence.

∆Π Ek

Eadh U

0.4

25 20

0.3

R1=100µm

15

0.2 10

0.1

5 0

0

100

200 300 400 Droplet radius R2 (µm)

100

Evis 0.5

500

0.0

Energy conversion efficiency η i (% )

(A) 30 R1>R2 R1R2 R1

A New Theoretical Model for Coalescence-induced Droplet Jumping

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