Sliding behaviour of water-ethanol mixture droplets on inclined low-surface-energy solid

International Journal of Heat and Mass Transfer 120 (2018) 1315–1324

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Sliding behaviour of water-ethanol mixture droplets on inclined lowsurface-energy solid Yukihiro Yonemoto a,⇑, Shosuke Suzuki b, Sae Uenomachi b, Tomoaki Kunugi c a

Priority Organization for Innovation and Excellence, Kumamoto University, 2-39-1, Kurokami, Chuo-ku, Kumamoto-shi, Kumamoto 860-8555, Japan Department of Mechanical System Engineering, Kumamoto University, 2-39-1, Kurokami, Chuo-ku, Kumamoto-shi, Kumamoto 860-8555, Japan c Department of Nuclear Engineering, Kyoto University, C3-d2S06, Kyoto Daigaku-Katsura, Nishikyo-ku, Kyoto 615-8540, Japan b

a r t i c l e

i n f o

Article history: Received 7 July 2017 Received in revised form 21 September 2017 Accepted 21 December 2017

Keywords: Dynamic contact angle Low-surface-energy solid Sliding droplet Wettability

a b s t r a c t The timing of the onset of droplet sliding behaviour on an inclined solid surface is mainly characterised by the onset of the rear contact line movement. The critical inclined angle is predicted from contact angle hysteresis, which is the difference between the advancing and receding contact angles of the droplets. However, at present, it is difficult to explain the contact angle hysteresis using the properties of liquids and solids. In this study, the behaviours of water–ethanol mixture droplets sliding down a low-surfaceenergy solid, inclined with constant angular velocities, are experimentally investigated. The front and bottom views of the droplets are observed to understand the detailed variations of the droplet length, width, and contact area during the inclination of the solid. From the experimental results, it was found that the adhesion forces related to the onset of the advancing and receding contact lines exhibit proportional relation with respect to the surface energy density of the liquid. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The control of the motion of liquid droplets over a solid surface is important for various industrial applications such as cooling, inkjet printing, cleaning, coating, and nanoimprinting [1–5]. In these applications, the absence of wetting, as in a dry region, leads to defects in the products and systems. In heat removal systems, the heat transfer coefficient of dropwise condensation is much higher than that of film condensation [6]. In the dropwise condensation phenomenon, the four main processes such as the generation of small droplets, growth of the droplets, coalescence of the droplets, and movement of the droplets (sliding of droplets) are important for heat transfer enhancement. In particular, the sliding of droplets induces a new condensation surface on which new small droplets are generated [7]. This means that the condition required for droplets to slide on a solid surface is related to the heat transfer efficiency. However, the accurate prediction of the condition for the sliding of droplets (contact line motion) is very difficult. The onset of droplet motion on an inclined solid surface is mainly characterised by the onset of contact line motion of the droplet. In order to predict the onset of droplet sliding motion,

⇑ Corresponding author. E-mail address: [email protected] (Y. Yonemoto). https://doi.org/10.1016/j.ijheatmasstransfer.2017.12.099 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

many studies have been conducted based on theory, experiments, and numerical simulation. The most important parameter for predicting the onset of sliding motion is the difference between the advancing and receding contact angles, which is called contact angle hysteresis [8,9]. The relationship between the contact angle hysteresis and inclined critical angle is expressed by the Furmidge relation, which is as follows [10].

ql gV sin ac ¼ Lw rlg ðcos hre  cos had Þ

ð1Þ

In this equation, ql, g, V, and ac represent the density of liquid, gravitational acceleration, droplet volume, and inclined critical angle when the droplet starts to move, respectively. Lw, hre, and had are the width of the contact area, receding contact angle, and advancing contact angle, respectively. Lw is the width of the droplet contact area and is defined as a constant value. This value is perpendicular to the direction in which the droplet moves. The contact angle hysteresis (D cos h = cos hre  coshad) is determined by experimental measurement [10–14], because the theoretical prediction of this value is very difficult, although it is known that this value is influenced by the surface roughness and surface conditions such as hydrophilicity and hydrophobicity [15–21]. The dynamic sliding motion after the onset of contact line motion has been studied both experimentally and theoretically [22,23]. The contact angle hysteresis is an important factor for the prediction of sliding velocity. In the study of sliding velocity, contact angle hysteresis is used to express the energy dissipated by the motion of fluid molecules in

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the vicinity of the contact line. However, contact angle hysteresis is a kind of geometrical information. Wetting behaviour is mainly characterised by the interaction between solids and liquids. Therefore, in order to understand the sliding behaviour in detail, it is important to obtain the relationship between the onset of front and rear contact line movements and the properties of the liquid and solid. To the best of our knowledge, no such study has been conducted so far. In the present study, the sliding behaviour of water–ethanol mixture droplets on a low-surface-energy solid, whose inclination is varied at a constant angular velocity, is investigated using a high-speed video camera. Then, the variations of the geometrical parameters such as the droplet length, width, height, and receding and advancing contact angles are measured by observing the side and bottom views of the droplet. From the results, the timing of the onset of the front contact line in addition to the rear contact line is investigated in detail. In addition, adhesion forces related to the onset of the contact line motion are estimated using the existing model.

and glass is applied as shown in Fig. 3 [25]. If a transparent material, whose width and refractive index are Lg and n, respectively, is inserted in front of the camera (Fig. 3(b)), the optical length increases and the image point moves, compared to the situation in which there is no transparent material (Fig. 3(a)). Thus, the displacement of the focal point Dl is calculated as follows.

2. Experiment

3. Results and discussion

The schematic of the experimental apparatus is depicted in Fig. 1(a). The droplet was initially deposited on the horizontal solid surface. Then, the solid substrate was inclined with a constant angular velocity x = da/dt; the values of x were 4, 7, and 10 degs1, and the solid surface was rotated in the anticlockwise direction as shown in Fig. 1(b). The droplet behaviour was analysed using a high-speed video camera (HX-5, NAC Image Technology, Ltd., Japan). Fig. 1(b) shows the schematic of the definition of the geometrical parameters. The geometrical parameters such as the contact area radius, width, height, and advancing and receding contact angles were measured. The optical resolution is 16.1 lm/ pixel. The contact angles were measured using a commercial software (FAMAS; Kyowa Interface Science Co., Ltd., Japan), for which the measurement accuracy is within 1°. In this study, water–ethanol binary mixtures were used for studying wettability on an inclined low-surface-energy solid. The four mixtures with different surface energy densities were labelled as follows: a: 0.0722 J/m2 (purified water), b: 0.0503 J/m2, c: 0.0380 J/m2, and d: 0.0303 J/m2. The viscosities were 1.002 [mPa s] for liquid a, 1.334 [mPa s] for b, 2.030 [mPa s] for liquid c, and 2.613 [mPa s] for liquid d. The surface energies of the liquids were measured using DM300 (Kyowa Interface Science Co., Ltd., Saitama, Japan). The volumes of the droplets were 20.0, 29.0, and 39.0 lL. The solid substrate (silicone rubber with a thickness of 0.05 mm) was laid on the slide glass. With respect to the silicone rubber substrate, the relationship among the contact angle, surface energy density of liquid, and critical surface tension can be evaluated as follows.

3.1. Variations of geometrical parameters during sliding

cos h0 ¼ 1 þ 2

rc : rlg

ð2Þ

The value of the critical surface tension of the silicone rubber was estimated to be 0.01956 J m2 [24]. In this experiment, the temperature and humidity were in the ranges of 20.0–25.0 °C and 47.0–55.0%, respectively. The image of the contact area was obtained from the mirror, as shown in Fig. 1(a). However, as shown in Fig. 2, it is difficult to capture the images of both the front and bottom shapes of the droplet, because the optical path A (front view of the droplet) is different from path B (bottom view of the droplet). Path B is longer than path A (Dl). Therefore, in order to capture both the images, the focal point in the optical path B is moved by changing the optical path length of B. The difference between the refractive indices of air

Dl ¼ Lg ðn  1Þ

ð3Þ

In this study, two kinds of borosilicate glass plates were used. One is a block type with dimensions 30  4015 mm and a refractive index of 1.515. The other one is a cover glass with dimensions 30  400.1 mm and a refractive index of 1.526. The latter was used for the fine adjustment of the focal point. In this study, Dl was 7.99 mm. Fig. 4 shows an example of the actual images of the droplets observed from the same camera position. Fig. 4(a) shows the droplet image before setting the glass plates. Fig. 4(b) shows the droplet image after setting the glass plates. It can be seen that the bottom image in Fig. 4(b) is clearer than that in Fig. 4(a).

Fig. 5 shows examples of the captured images for each droplet sliding on the inclined solid surface, when the solid substrate is rotated at an angular velocity of 4 degs1. The droplet volume is 20 lL. The initial contact area at a = 0° becomes large as the surface energy density of the liquid becomes small. As the solid substrate rotates, the droplet deforms and then slides in the downward direction (left direction). In the figure, ‘‘Front” and ‘‘Rear” indicate the timings for the onset of the front and rear contact line movements, respectively. It was found that the onset of the movement of the front and rear contact lines is different. For example, the critical inclined angle of the solid substrate is 12°, when the front contact line (CL) of case (a) starts to move. In this case, the critical inclined angle for the rear CL movement is 56°. The critical inclined angle for the movement of each contact line becomes small as the surface energy density of the liquid becomes small. In this study, there were no major differences in the critical inclined angles among the angular velocities of 4, 7, and 10 degs1. The droplet shape observed from the bottom view gradually changes from circular to oval shape in the case of purified water (case (a)). On the other hand, in case (d), the droplet shape changes from circular to slug shape. From the front views of the droplets, this may be related to the shape of the free surface of the droplets. The contact angle becomes small as the surface energy density becomes small. If the contact angle is small, the liquid moves to the forefront of the droplet (toward the front CL), when the solid substrate is inclined: the force acts on the front CL directly due to the movement of the gravity point. Therefore, the shape of the contact area changes to slug shape in the case of (d). On the other hand, in the case of (a), the contact angle is large. Therefore, the force mainly acts on the front surface of the droplet. This indicates that the force acting on the front CL is uniform, compared with the case of (d). Thus, the shape of the contact area changes to oval shape. Fig. 6 shows the variations of the geometric parameters such as length (D), height (h), and width (L) of the droplet for the cases of 0.0303, 0.038, and 0.0722 J m2. The angular velocities of the solid substrate are 4, 7, and 10 degs1. The value of D increases gradually as the value of a increases. As the droplet volume increases, the gradient dD/da becomes steep, because of the increase in mass. These tendencies are qualitatively the same for each droplet case. The value of h decreases gradually, as the value of a increases. Then, after reaching a certain value of a, h increases slightly for

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(a) Rotation stage LED light

Droplet Solid sample Slide glass High speed video

View A

camera Glass plate Mirror

Computer

Stage controller

(b) Solid sample

Droplet h

Slide glass

R

A

L A

D Fig. 1. Schematic of (a) experimental apparatus and (b) geometrical parameters of a droplet.

(a)

Camera lens l

Path A

l

Path B

Objective Camera lens

(b) l Fig. 2. Optical paths between the droplet and camera: Path A is between the front image of the droplet and the camera. Path B is between bottom image of the droplet and the camera.

all the cases. This behaviour can be understood by considering the variation of L. Fig. 6(c) shows the variation of L for the cases of 0.0303 and 0.038 J m2. Here, in the case of water, the contact angle exceeds 90°. Thus, it was difficult to measure L from the bottom image. From the results of L, it was found that the value of L remains constant up to a certain value of a. For example, the value of L in the case of rlg = 0.038 J m2 with 39 lL remains constant until a = 20°. In this region (a < 20°), the value of D increases. Therefore, in this region, the value of h decreases, because the droplet volume is constant. For a > 20°, the value of L decreases sud-

Objective

Lg Refraction index n

Fig. 3. Schematic of change in optical length: (a) before changing and (b) after changing.

denly. This may result in a slight increase of h. The same mechanism would be applicable to the case of water. Fig. 7 shows the relationship between the contact angle and the inclined angle for each liquid. The black dashed line represents the

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(a)

Front view

(b)

Bottom view

Front view

Bottom view

Fig. 4. Example of droplet images: (a) before setting the glass plates, (b) after setting the glass plates.

equilibrium contact angle. From the figures, it can be observed that the cosine component of the advancing contact angle decreases as the inclined angle increases whereas the cosine component of the receding contact angle increases as the inclined angle increases. This is because the solid substrate continuously rotates in the present study, thus inducing continuous deformation of the droplet shape. Similar to the results in Fig. 6, the behaviours of the contact angle for each liquid are qualitatively the same even if the angular velocity changes. Subsequently, the difference between the cosine component of the receding contact angle and the equilibrium contact angle becomes small as the value of the cosine component of the equilibrium contact angle approaches unity. This indicates that, in the hydrophilic case, the range of motion of the droplet surface toward the solid-liquid interface near the rear CL becomes small, whereas in the hydrophobic case, it becomes large. Furthermore, the difference between the cosine component of the advancing contact angle and the equilibrium contact angle becomes large as the value of the cosine component of the equilibrium contact angle approaches unity. A similar interpretation as above in the case of the receding contact angle can be applied to the case of the advancing contact angle by considering the range of motion of the droplet surface toward the solid-gas interface near the front CL. 3.2. Onset of movement of contact lines In this section, the onset of the movement of contact line (CL) is discussed. The relationship between the contact angle and critical inclined angle is expressed by the Furmidge relation (Eq. (1)). Fig. 8 shows the relationship between qlgV  sin a(t) and rlg(cos hre(t)  cos had(t)). The plots were evaluated from the experimental data. The plots exhibit linear relation for all the cases. The solid lines indicate the best fit to the experimental data. Although Eq. (1) is applied to solid surfaces with critical inclined angle, this result indicates that Eq. (1) may be applicable to a continuously inclined solid surface. For the cases of (a), (b), (c), and (d), the values of Lw in Eq. (1) were estimated as Lw_fit = 2.903  103, 3.683  103, 3.622  103, and 4.402  103 m, respectively, by the fitting procedure. Fig. 9 shows the relationship between the estimated values of Lw_fit and the initial contact area diameter. This result shows that Lw_fit is not simply equal to the initial contact diameter D0. At any rate, the characteristic behaviours that imply the onset of CL are not observed in Fig. 8, although Eq. (1) can fit the experimental data. However, the signs that imply the movement of CL can be observed from the relationship between the variations of D and L. Fig. 10 shows the variations of D and L against a for the 29 lL droplet with surface energy densities of 0.0303 and 0.038 J m2. In this case, the angular velocity of the solid surface is 10 degs1. In Fig. 10(a), the black and white circles represent the trajectories of D and L with

respect to a, respectively. The red solid lines (a and b) represent the onset of the movement of the front and rear CLs, respectively. The red dashed line is the onset of the decrease in the value of L. In region 1 (before line a), both D and L takes the same value. This indicates that the contact area maintains the initial shape (circle). From line a, the front CL starts to move and the value of D increases gradually, while L remains constant. Then, after reaching line b, the rear CL starts to move. The value of D continues to increase. However, the gradients dD/da in regions 2 and 3 are different from each other. In region 3, both the front and rear CLs move, while in region 2, only the front CL moves. Therefore, the gradient dD/da in region 2 is greater than that in region 3. After line c, L starts to decrease and D continues to increase. In region 4, the gradient dD/da is between those of regions 2 and 3. In this case, the contact area can be measured as shown in Fig. 11. In region 1, the contact area remains constant. After line a, the contact area increases gradually until line c. After line c, the value of the contact area is almost constant, while the values of D and L change in the same region. From Figs. 10 and 11, it is found that some characteristics can be observed in the variations of D, L, and A, although the characteristic behaviour cannot be observed from the contact angles (Fig. 8). Fig. 12 shows the relationship between D(t) and L(t) for the cases of (a) 0.0303 and (b) 0.038 J m2. The black, blue1, and red triangles represent the experimental values for 20 lL, 29 lL and 39 lL, respectively. The solid and dashed arrows represent the onset of the front and rear CLs, respectively. This figure indicates that the value of Lw_fit obtained from the results of Fig. 8 differs from the measured value of L(t). Therefore, the value of Lw in Eq. (1) cannot be evaluated from the width of the droplet in the present case. There have been some previous studies in which Lw is not necessarily the width of the droplet contact area [26–28]. For example, a parameter k is added to the force balance (Eq. (1)). For parallel-sided or elliptical contact lines, the following relation is proposed [26,28]:

ql gV sin ac ¼ kD0 rlg ðcos hre  cos had Þ:

ð4Þ

In this equation, k is evaluated as follows:

k ¼ 0:115 þ 0:52

Dk ; D?

ð5Þ

where Dk and D? are the length and width of the contact area, respectively. By comparing Eq. (1) with Eq. (4), the relation Lw/D0 = k is obtained. In the present study, Dk and D? are evaluated using the values of D(t) and L(t) at rear CL movement in Fig. 12, respectively. From Fig. 12, L(t) almost corresponds to the initial contact diameter D0 until the movement of rear CL. The evaluated values of k for liquids 0.0722 J m2, 0.0503 J m2, 0.0380 J m2, and 0.0303 J m2 are 0.708, 0.678, 0.672, and 0.666, respectively. The order of values of k is the same as the estimated order in Fig. 9. However, the calculated values of k appear to change depending on the kinds of liquids. Eq. (5) is derived by considering the force balance between the surface force and external force (gravity). The change in the droplet shape is not considered, which may depend on the kinds of liquid and the change in the gravity point. Especially, in our case, the solid substrate changes continuously. Thus, the discrepancy between the value of the gradient Lw_fit/D0 and k may be related to such kind of complex geometrical change, which influences the effect of the external force on the contact line. 3.3. Adhesion forces for the onset of the movement of contact lines In the previous section, it was seen that there are two critical inclined angles where the contact line starts to move. One is the 1 For interpretation of color in Fig. 12, the reader is referred to the web version of this article.

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67.2

60.8

12

56

Rear

(a)

48

38.4

Front

-2

lg=0.0722Jm

27.4

24

10.7

Rear

(b)

33.6

26.4

Front

lg=0.0503Jm

16.8

8

Rear

28.8

19.8

=0

Front -2

lg=0.038Jm

14.4

11.2

8.6

Rear

(d)

=0

-2

19.2

(c)

=0

=0

Front -2

lg=0.0303Jm

Fig. 5. Images of droplet shape (20 lL): (a) rlg = 0.0722, (b) 0.0503, (c) 0.038, and (d) 0.0303 J m2.

critical angle aad, where the front CL starts to move. The other one is the critical angle are, where the rear CL starts to move. The rear CL moves after the front CL moves. Therefore, the droplet sliding motion occurs after the rear CL moves. In this study, the adhesion forces for the onset of the advancing and receding contact line movements were estimated. Based on the Von Buzágh and Wolfram equation [29], the relationship between ac and E can be expressed as follows:

sin ac ¼ E

pD0 ; ql gV

ð6Þ

where E is a constant value [N/m]. This value is generally evaluated by fitting Eq. (6) to the experimental data of the relationship between sin ac and D0/V. In this study, two values of E (for the cases re of aad c and ac ) were estimated based on Eq. (6). The rearrangement of Eq. (6) yields,

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0.01

0.008

0.008

0.008

0.006

0.006

D

29 L

D

39 L

0.006

D

0.01

0.004

0.004

0.004

20 L 0.002 0

0.002

-2 lg=0.0722Jm

0

20

40

60

0

80

0

20

40

60

0

80

-2

=0.0303Jm

lg

0

10

20

30

(t) (a) Length of droplet (D) [m]

(t)

40

50

=0.0303Jm-2

lg=0.038Jm

lg

0.002

0.002

h

h

20 L

50

0.003 -2

39 L 29 L 0.002

40

(t)

0.003

0.003

h

0.002

-2 lg=0.038Jm

-2

=0.0722Jm

0.001

lg

0

0.001

20

40

60

80

0

20

40

60

0.001

80

0

10

20

30

(t)

(t)

(t)

0.01

0.01

0.008

0.008

0.006

0.006

L

L

(b) Height (h) [m]

0.004

0.004 -2

0.002 0

=0.038Jm

0.002

lg

0

=0.0303Jm-2

lg

20

40

60

0

80

0

10

20

(t)

30

40

50

(t) (c) Width (L) [m]

Fig. 6. Variation of geometrical parameters (left rlg = 0.0722 J m2, centre rlg = 0.038 J m2, right rlg = 0.0303 J m2): (a) length of droplet, (b) height, and (c) width.

sin ac ¼ k k¼

pE : ql g

D0 : V

ð7Þ ð8Þ

Fig. 13(a) and (b) shows the relationships between sin aad c and D0/V, and between sin are c and D0/V, respectively. In these figures, the solid lines indicate the best fit of Eq. (7) to the experimental data. Fig. 14 shows the relationship between E (Ead and Ere) and rlg  rc. The plots show the experimental values of Ead and Ere that were evaluated from the results in Fig. 13. These results indicate that the adhesion force, which determines the onset of the contact line, exhibits linear relation with respect to the liquid and solid surface properties such as the surface energy density of the liquid (rlg) and the critical surface tension (rc). This result implies that the contact angle hysteresis (cos hre  cos had) in Eq. (1) may be related to rlg and rc. Then, as the value of rlg  rc increases, the value of E also increases. From this result, it is clear that the onset of motion of the droplet under the condition of small rlg  rc is faster than that under large rlg  rc. In the case of small rlg  rc, the droplet exhibits hydrophilicity against the solid surface; the contact angle is acute. Thus, due to the movement of the gravity point, the liquid flows into the acute meniscus, which may have a precursor film [30], and the force concentrates on the contact line. On the other hand, in the case of large rlg  rc, the contact angle is obtuse. This means that the force diverges at the front of the droplet surface and does not concentrate on the

advancing contact line. Therefore, the onset of the droplet motion for the hydrophilic case is faster than that for the hydrophobic case; the value of E in the case of small rlg  rc is smaller than that in the case of large rlg  rc.

3.4. Sliding motion after the onset of movement of CL In this section, the relationship between the contact angle and the velocity of the CLs after the onset of movement of CLs is discussed. In the present study, to understand the relationship between the dynamic motion of the contact line and the dynamic contact angle, two approaches are employed [31,32]. One is the molecular-kinetic (M-K) theory, which is expressed as follows:

" U ¼ 2j0 k sinh

rlg ðcos heq  cos hD Þk2 2kB T

# :

ð9Þ

In this equation, kB [m2 kg s2 K1] and T [K] represent the Boltzmann constant and the absolute temperature, respectively. j0 [s1] is the characteristic frequency and is related to the frequency of the random molecular displacements within the threephase zone characterised by the activation free energy of wetting. k [m] is the average distance of the displacement and the scale of this value is usually of molecular dimension. The values of l and k are mainly determined by fitting Eq. (9) with the experimental data for the relationship between the dynamic contact angle hD

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1

1

1

Recede

20

0

Advance

= 4 deg·s–1 -1 0

cos (t)

0

cos (t)

cos (t)

39 L 29 L 20 L

40

60

= 10 deg·s–1

= 7 deg·s–1 -1 0

80

20

40

(a)

0

20

lg =

0

40

60

80

0

20

(b)

40

60

80

lg =

20

cos (t)

cos (t) 60

-1 0

80

80

60

80

1

0

= 10 deg·s–1

= 7 deg·s–1 40

40

0.038 Jm-2

0

= 4 deg·s–1 20

0

(t)

1

0

60

= 10 deg·s–1 -1

(t)

1

80

0

= 7 deg·s–1 -1

60

0.0722 Jm-2

(t)

-1 0

40

1

= 4 deg·s–1 0

20

(t)

cos (t)

cos (t)

cos (t)

80

1

1

cos (t)

60

-1 0

(t)

(t)

-1

0

20

40

60

-1 0

80

20

40

(t)

(t)

(t) (c)

lg =

0.0303 Jm-2

Fig. 7. Variation of contact angle (left x = 4 degs1, centre x = 7 degs1, right x = 10 degs1): (a) rlg = 0.0722 J m2, (b) rlg = 0.038 J m2, and (c) rlg = 0.0303 J m2.

-2

0.0002

L=2.903x10 |r|=0.980 0.02

0.04

(cos lg

b

0.06

-cos re

L=3.622x10 |r|=0.980 0

0.0003

-2

0.02

0.04

(cos lg

-cos re

0.02

0.06

) ad

0.04

(cos lg

=0.0503Jm

L=3.683x10 |r|=0.984 0

-3

0.0001

d

lg

0.0001

-2

=0.038Jm

lg

) ad

0.0002

0

c

0.0002

0

0.08

Vgsin (t)

Vgsin (t)

0.0003

-3

lg

0.0001 0 0

lg

0.0003

=0.0722Jm

lg

Vgsin (t)

a

-3

lg

lg

Vgsin (t)

0.0003

0.08

0.06

-cos re

0.08

) ad

-2

=0.0303Jm

lg

0.0002 0.0001

0

L=4.402x10 |r|=0.973 0

0.02

(cos lg

Fig. 8. Relationship between qlVgsin a and rlg(cos hre  cos had).

0.04

-cos re

0.06

) ad

-3

0.08

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L(t) [m]

0.008

Onset of CL movement Front CL Rear CL

0.006 39 L 29 L 20 L

0.004

-2

(a) 0.002 0.002

0.004

=0.0303 Jm

lg

0.006

0.008

D(t) [m] Fig. 9. Relationship between D0 and Lw_fit for each droplet.

0.008 Onset of CL movement Front CL Rear CL

0.008

D or L

L(t) [m]

(a) 0.007

4

1 a 2 b3 c

0.006 39 L 29 L 20 L

0.004

0.006

-2

(b) Front Rear

0

0.002 0.002

L D

0.005 10

20

30

0.004

=0.038 Jm

lg

0.006

0.008

D(t) [m] 40

Fig. 12. Relationship between L(t) and D(t) with Lw_fit for (a) rlg = 0.0303 J m2 and (b) rlg = 0.038 J m2.

(t) 0.008

0.3

20 L

(a)

0.007

1

2

a

3

b

c

4

29 L

0.2

[-]

0.006 0.005

Front

0

sin

ad

D or L

(b)

Rear

10

20

30

39 L

0.1

40

(t) Fig. 10. Variation of length of droplet (D) and width (L) of 30 lL (w = 10 degs (a) rlg = 0.0303 J m2 and (b) rlg = 0.038 J m2.

0 0

1

):

1

2

3 [ 105]

2

3 [ 105]

-2

D0/V [m ] 1

(b)

0.00004

2

b

3

4

c

[-]

a

re

0.00003

sin

2

A[m ]

1

0.00003

20 L

0.8 29 L

0.6 39 L

0.4

Front Rear

0.2 0.00002 0

10

20

30

0 0

40

(t)

-2

D0/V [m ]

Fig. 11. Variation of contact area for case (a) in Fig. 9.

Fig. 13. Relationship between the critical angle (sin ac) and D0/V: (a) sin aad c and D0/ V and (b) sin are c and D0/V.

[rad] and the contact line velocity U [m s1]. hD corresponds to the advancing and receding contact angles. The other approach is the hydrodynamic theory, which is expressed as follows:



h3D  h3m ¼ 9Ca ln

 L : Lm

1

ð10Þ

In this equation, L [m] and Lm [m] are the macroscopic and microscopic length scales, respectively. From a practical point of view, the value of ln(L/Lm) is usually treated as an adjustable parameter. Ca is the capillary number expressed as llU/rlg where ll [Pa s] is the liquid viscosity. hm is the microscopic contact angle.

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Y. Yonemoto et al. / International Journal of Heat and Mass Transfer 120 (2018) 1315–1324

-1

Ere, Ead [Nm ]

0.015 Ere Ead

0.01

0.005

0 0

0.01 0.02

0.03 0.04 0.05 0.06

– lg

-2

c

[Jm ]

Fig. 14. Relationship between E (Ead, and Ere) and rlg  rc.

In the present study, hst is treated as the static contact angle owing to the difficulty of measurement. In practice, hst is considered as the value obtained immediately before the CLs starts to move. Fig. 15 shows the result of the comparison of the experimental data with the solutions of Eqs. (9) and (10) for the relationship between the contact line velocity and the dynamic contact angle. The velocities of the front and rear CL are expressed as positive and negative values, respectively. Fig. 15(a) –(d) represents the cases for rlg = 0.0722, 0.0503, 0.038, and 0.0303 J m2, respectively. In each case, the characters are the experimental data and all the experimental conditions are illustrated. The solid and dashed lines are the analytical results for Eqs. (9) and (10), respectively. The red and blue lines represent the receding and advancing cases, respectively. The red point is the equilibrium contact angle. From the figure, it can be observed that the results of the M-K theory exhibit superior consistency with the experimental data as compared to those of the hydrodynamic theory. Especially, the transition zone where the contact line starts to move is well fitted by the M-K theory. From the experimental observation, the veloc-

ities of the CLs gradually increase after the onset of each CL. Especially, the transition zone of the receding contact angle exhibits a more gradual curve than that of the advancing contact angle. It may be difficult for the hydrodynamic theory to express this kind of gradual change in the CL velocity and the contact angle. Further, the parameters of j0 and k used in the M-K theory for each liquid are obtained by fitting: (a) j0 = 3.46  105 [s1] and k = 1.20  109 [m] for the blue solid line and j0 = 1.50  104 [s1] and k = 0.95  109 [m] for the red solid line; (b) j0 = 2.27  105 [s1] and k = 1.31  109 [m] for the blue solid line and j0 = 3.78  105 [s1] and k = 0.75  109 [m] for the red solid line; (c) j0 = 14.2  105 [s1] and k = 0.89  109 [m] for the blue solid line and j0 = 5.47  105 [s1] and k = 0.76  109 [m] for the red solid line; (d) j0 = 5.59  105 [s1] and k = 1.1  109 [m] for the blue solid line and j0 = 1.84  105 [s1] and k = 1.08  109 [m] for the red solid line. Subsequently, the parameters of ln(L/Lm) used in the hydrodynamic theory are as follows: (a) 1.593  103 for the blue dashed line and 1.066  103 for the red dashed line; (b) 0.727  103 for the blue dashed line and 0.493  103 for the red dashed line; (c) 0.278  103 for the blue dashed line and 0.165  103 for the red dashed line; (d) 0.138  103 for the blue dashed line and 0.077  103 for the red dashed line. 4. Conclusions The wettability of water–ethanol mixture droplets on a lowsurface-energy solid inclined with a constant angular velocity was experimentally observed. From the captured images, the geometrical parameters of the droplets such as length, width, height, contact area, and receding and advancing contact angles were measured. Then, the onset of the movement of the front and rear contact lines was investigated. Although there are no characteristic tendencies between the contact angles of the droplets and the onset of CLs, the behaviours

1

1 lg=0.0722

Jm-2

b

cos (t)

cos (t)

a

Advance 0

Recede

lg=0.0503

Jm-2

0

Equilibrium point Equilibrium point -1 -0.03 -0.02 -0.01

0

0.01

0.02

-1 -0.03 -0.02 -0.01

0.03

U (t) 1

0.01

0.02

0.03

1 lg=0.038

Jm-2

d

cos (t)

c

cos (t)

0

U (t)

0

0

lg=0.0303

Jm-2

Equilibrium point

Equilibrium point -1 -0.03 -0.02 -0.01

0

U (t)

0.01

0.02

0.03

-1 -0.03 -0.02 -0.01

0

0.01

0.02

0.03

U (t)

Fig. 15. Relationship between dynamic contact angle and contact line velocity: (a) rlg = 0.0722 J m2, (b) rlg = 0.0503 J m2, (c) rlg = 0.038 J m2, and (d) rlg = 0.0303 J m2. The solid red and blue lines represent the results by M-K theory. The dashed red and blue lines represent the results by hydrodynamic theory. The red point is the equilibrium point (initial condition of droplet). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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among the droplet length, width, and contact area exhibit some tendencies with respect to the onset of CLs. In the present experiment, it was found that the front CL starts to move prior to the rear CL. Eq. (1) is mainly used to explain the relationship between the critical inclined angle (ac) and the contact angle hysteresis (coshre  coshad) via droplet width (Lw). However, this equation cannot predict the CL behaviours observed in the present experiment. Our experimental observations revealed that the fitting value of Lw in Eq. (1) is quite small compared to the observed values. This indicates that the sliding behaviour on the solid substrate inclined with a constant angular velocity cannot be explained only with Eq. (1). The relationship between the gravitational force and the adhesion force of the contact area was evaluated on the basis of the existing model. The adhesion force that determines the onset of the advancing and receding CL movements exhibits linear relations with respect to the surface energy density of the liquid and the critical surface tension. The actual sliding behaviour may be related to the change in the gravity point of the droplet. Thus, the effect of the movement of the gravity point on the onsets of the CL may be included in the values of Ead and Ere. From the consideration of the droplet sliding motion, it was observed that the relationship between the dynamic contact angle and the contact line velocity after the onset of the contact line movement could be well fitted by the molecular-kinetic theory. The sliding behaviour may change if the combination of solid and liquid is changed. Therefore, in order to understand the sliding behaviour of droplets in detail, further investigation using various kinds of liquids and solids needs to be performed, which will be our next step. Conflict of interest The authors have no potential COI to disclose. References [1] P.S.R. Krishna Prasad, A. Venumadhav Reddy, P.K. Rajesh, P. Ponnambalam, K. Prakasan, Studies on rheology of ceramic inks and spread of ink droplets for direct ceramic ink jet printing, Mater. Process. Technol. 176 (2006) 222–229. [2] Y. Kosuke, S. Munetoshi, I. Toshihiro, M. Sachiko, N. Akira, Investigation of droplet jumping on superhydrophobic coatings during dew condensation by the observation from two directions, Appl. Surf. Sci. 315 (2014) 212–221. [3] J. Daehwan, K. Dongjo, M. Jooho, Influence of fluid physical properties on inkjet printability, Langmuir 25 (2009) 2629–2635. [4] H. Jiang, L. Lu, K. Sun, Experimental investigation of the impact of airborne dust deposition on the performance of solar photovoltaic (PV) modules, Atm. Environ. 45 (2011) 4299–4304. [5] Y.-Y. Quan, L.Z. Zhang, R.-H. Qi, R.-R. Cai, Self-cleaning of surfaces: the role of surface wettability and dust types, Sci. Rep. 6 (2016) 38239. [6] S.G. Kandlikar, V.K. Dhir, M. Shoji, Hnadbook of Phase Change – Boiling and Condensation, Taylor & Francis, Philadelphia, PA, 1999.

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