Bubble Movement on Inclined Hydrophobic Surfaces - Langmuir (ACS

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Bubble Movement on Inclined Hydrophobic Surfaces Ali Kibar, Ridvan Ozbay, Mohammad Amin Sarshar, YongTae Kang, and Chang-Hwan Choi Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02831 • Publication Date (Web): 05 Oct 2017 Downloaded from http://pubs.acs.org on October 10, 2017

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Bubble Movement on Inclined Hydrophobic Surfaces Ali Kibar1, 2, Ridvan Ozbay2, Mohammad Amin Sarshar2, Yong Tae Kang3, Chang-Hwan Choi2 1

Department of Mechanical and Material Technologies, Kocaeli University, Arslanbey Campus, Kocaeli, 41285, Turkey 2

Department of Mechanical Engineering, Stevens Institute of Technology, Hoboken, New Jersey, 07030, USA 3

School of Mechanical Engineering, Korea University, Seoul, 02841, Korea

ABSTRACT The movement of a single air bubble on an inclined hydrophobic surface submerged in water, including both the upward- and the downward-facing sides of the surface, was investigated. A planar Teflon sheet with an apparent contact angle of a sessile water droplet of 106° was used as a hydrophobic surface. The volume of a bubble and the inclination angle of a Teflon sheet varied in the range of 5–40 µL and 0–45°, respectively. The effect of the bubble volume on the adhesion and dynamics of the bubble were studied experimentally on the facing-up and the facing-down surfaces of the submerged hydrophobic Teflon sheet respectively and compared. The result shows that the sliding angle has an inverse relationship with the bubble volume for both the upward- and the downward-facing surfaces. However, at the same given volume, the bubble on the downward-facing surface spreads over a larger area of the hydrophobic surface than the upward-facing surface due to the greater hydrostatic pressure acting on the bubble on the downward-facing surface. This makes the lateral adhesion force of the bubble greater and requires a larger inclination angle to result in sliding.

Keywords: Bubble, Hydrophobic, Adhesion, Sliding angle

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1. INTRODUCTION Bubbles dynamics are of great significance in many industrial processes and applications, which include boiling,1 nucleation,2 cavitation,3 electrolysis,4 water treatment,5 froth flotation,6 orifice plate,7 biomass energy,8 chemical reactors,9 nuclear reactors,10 and hydrodynamic drag reduction.11 In such applications, the substrates often stand as inclined or their angle changes over time. And the attachment/detachment of a bubble is highly important in the practical applications concerned. For example, the attachment of minerals to an air bubble is highly desired to efficient froth flotation process.12,13 In contrast, the presence (i.e., attachment) of air bubbles in microfluidic and thermal systems is undesirable.14,15 The adhesion and dynamics (i.e., sliding) of a bubble on solid surfaces are significantly affected by the wetting characteristics of the surfaces. Bubbles can detach and slide easily from hydrophilic surfaces, while they like to adhere and spread over hydrophobic surfaces.16 Previously, many studies were conducted on bubble motions on hydrophilic as well as hydrophobic surfaces, both experimentally and theoretically. For example, Perron et al.17 studied the influence of a bubble volume and the inclination angle of a substrate on the terminal velocity of a bubble on a hydrophilic surface. Ridvan et al.16 studied the volume effect on the sliding angle and the adhesion force of a captive bubble on micropillared superhydrophilic surfaces and compared to those on flat hydrophilic surfaces. Sonoyama and Iguchi18 studied the bubble motions at both the topside and the underside of a hydrophobic surface and determined the detachment condition. However, it should be noted that the bubble dynamic studies on hydrophobic surfaces in most of the previous works including attachment,19,20 formation,18 electroflotation,21 transportation22, growth of captive bubble on plastron,23 and pool boiling,24,25 have been discussed mainly in association with a buoyancy force. However, such a simple model


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based on a buoyancy force is regarded improper for the precise analysis of the dynamics of bubbles which are partially attached on the hydrophobic surfaces. Furthermore, the bubble adhesion and dynamics on solid surfaces are significantly influenced by the orientation as well as the inclination of the surfaces. For example, Vachova et al.26 studied the bubble adhesion process on an inclined planar hydrophobic surface both experimentally and theoretically by examining the three-phase contact line expansion. Kim et al.27 investigated the effects of surface wettability on the sliding dynamics of bubbles and hence the heat transfer over heated surfaces in inclination, finding that the heat transfer coefficient increased significantly in the hydrophobic region of the surface where the bubble adhered. However, the effects of surface orientations (i.e., upward- or downward-facing) to the attached bubbles have not been studied much on the inclined surfaces. Depending on applications, bubbles are used with different surface orientations. For instance, bubbles are grown on upward-facing surfaces in boiling applications,1 while they are nucleated on downward-facing surfaces for nuclear engineering applications.10 Thus it is critical to understand the effects of surface orientations to the bubble dynamics on the inclined surfaces. Bubble volume is another factor which should affect their adhesion and dynamics (i.e., sliding) on the surfaces where the gravity (i.e., buoyancy or hydrostatic pressure) force plays an important role relative to the surface tension force.17 However, few studies have been done to systematically investigate the effects of the surface orientations and the bubble volume on the bubble adhesion and dynamics on hydrophobic surfaces in inclination. As for the adhesion, it should be noted that there are two different types of adhesion, including normal adhesion and lateral adhesion. The normal adhesion acts on a bubble in a perpendicular direction to a solid substrate, relating to the detachment of it.12, 28 On the other


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hand, the lateral adhesion acts on a bubble in a tangential direction against to the sliding (or depinning) motion of a bubble on a solid substrate.16 In this work, we study the lateral adhesion and sliding behaviors of an air bubble placed on a hydrophobic surface with different orientations (i.e., both facing-up and facing-down surfaces fully submerged in water) and systematically varied bubble volumes and inclination angles. We propose a theoretical model for the forces associated with the bubble behaviors and discuss the inefficiency of the simple models based on the buoyancy force that has occasionally been misused in the literature. Then, we compare our experimental results with the refined model to confirm the validity of the new model.

2. THEORETICAL MODELS

2.1. Bubble on a Horizontal Hydrophobic Surface Figure 1 illustrates the theoretical model of the forces acting on a bubble, including the cases on the upward-facing (Figure 1a) and the downward-facing sides (Figure 1b) of a horizontally-leveled hydrophobic surface. It is considered that the physical properties are constant so that there is no surface tension gradient present along the interface. In a quasi-steady manner, there is a force balance between the surface tension and external forces acting around a single air bubble in water.2 The internal pressure force acting on the inner surface of a bubble (FIP) is balanced by the hydrostatic pressure force (FHP) of water acting on the outer surface of the bubble, the weight of a bubble by gravity (FG) which is typically negligible, and the surface tension force acting at the bubble boundary (i.e., contact line). The pressure difference across the air-water interface of the bubble can be described by the Young-Laplace equation.2


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ଵ

ଵ

∆ܲ = ߛ ቀோ + ோ ቁ భ

మ

(1)

where ∆P is the pressure difference between the inside and the outside of a bubble (so-called Laplace pressure), γ is the coefficient of surface tension, and R1 and R2 are the principal radii of the curvatures. Atmospheric pressure (P0) prevails on the free surface of the water, as shown in Figure 1. We consider reference planes for the hydrostatic pressure at the top (apex) and the bottom (base) of the bubble in the cases of the upward- and the downward-facing surfaces, respectively. Then the excess of the hydrostatic pressure becomes zero at the apex of the bubble and maximum at the base of the bubble for the upward-facing surface (Figure 1a). On the contrary, the excess of hydrostatic pressure becomes maximum at the apex of the bubble and zero at the base of the bubble for the downward-facing surface (Figure 1b). Thus, in the case of the downward-facing horizontal surface (Figure 1b), the greater excess force toward to the apex of a bubble would press the bubble towards to the surface and the bubble would spread out more than the case of the upward facing. This excess of hydrostatic pressure can be called as a relative hydrostatic pressure when the highest level of a bubble is considered as a reference plane. The bold lines in Figures 1a and 1b represent the excesses of the hydrostatic pressure distributions around a bubble on the basis of the reference planes. The internal and external hydrostatic pressure forces (FIP and FHP) along the bubble meniscus act normal to the meniscus. Therefore, they have vertical and horizontal components (FIP,V and FIP,H for FIP; FHP,V and FHP,H for FHP) due to the spherical shape of a bubble. The vertical components effectively act on the projected area of the bubble to a normal plane, which equals to the spreading area on the hydrophobic surface. The difference between the vertical components of the internal pressure force (FIP,V) and the external hydrostatic pressure force


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(FHP,V), i.e., the vertical components of the Laplace pressure force (FLP,V), is balanced by the vertical component of the surface tension force and the weight of the bubble (FG). In addition to the vertical balancing, there is a horizontal component of the Laplace pressure force (FLP,H) for a bubble. This force effectively acts axisymmetrically on the horizontally-projected area around the bubble, which is πwH, where w is the contact width and H is the height of the bubble. According to the Archimedes principle, any object, wholly or partially submerged in a liquid, is buoyed up by a force equal to the weight of the liquid displaced by that object, which is equivalent to the integral of the partial hydrostatic pressure difference between the upper and the lower segments of infinitesimally small bubble columns consisting of the whole bubble.29 Thus, the buoyancy force for an unbounded bubble is the resultant force of the hydrostatic pressure distribution acting around the whole free bubble submerged in liquid and can be described as: ‫ = ܤܨ‬൫ߩ‫ ܮ‬− ߩ‫ ܩ‬൯ܸ‫݃ ܤ‬

(2)

where FB represents a buoyant force, ρL and ρG the densities of surrounding liquid and immersed gas (bubble), respectively, VB the volume of a bubble, and g a gravitational constant. However, when the bubble is attached and bounded firmly to a horizontal hydrophobic surface, the hydrostatic pressure force around the bubble acts only one-directional (i.e., only downward for the upward-facing surface, or only upward for the downward-facing surface) since there is no water to result in hydrostatic pressure between the bubble and the bounded solid surface. Therefore, this hydrostatic pressure force cannot strictly be called as a buoyancy force and estimated by Eq. (2). As already noted earlier, the terms of buoyancy and the equation of buoyancy force have been frequently used to represent the vertically-upward resultant force even when bubble attaches to a hydrophobic surface immersed in water.19,20,22,23,25,30,31 However, it should be noted again that there is no effective buoyancy force when a bubble attaches on a horizontal hydrophobic surface with the contact angle of a bubble (θb) less than 90°. Thus, the


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use of Eq. (2) for the force balance in such a case would be invalid.

2.2. Bubble on an Inclined Hydrophobic Surface Figures 2a and 2b illustrate the force balances on a bubble on an inclined hydrophobic surface for upward- and downward-facing sides, respectively. When a bubble stands on a horizontal surface, the horizontal component of the Laplace pressure force (FLP,H) acts around the bubble symmetrically. However, on the inclined surfaces, the symmetry is broken due to the asymmetric hydrostatic pressure distributions at the uphill and the downhill sides of the bubble. Due to the asymmetric hydrostatic pressure distributions and the different projected areas for the uphill and the downhill sides of the bubble, the Laplace pressure forces at the two sides are different. If the tilting of the surface for inclination is slow, the change of the bubble shape is also slow so that the forces depending on inertia effects can be neglected. Then, the lateral sliding motion of a bubble along the surface is driven by the tangential component of the net force (Ft) acting on the bubble. The tangential component of the net force for the case of an upward-facing surface (Figure 2a) can be formulated as:

Ft = ( FB , P + FLP,V − FG )sin α + ( FLP , H ,U − FLP , H , D ) cos α

(3)

where FB,P is an effective buoyancy force for the partial volume (VP) of the unbounded section of the bubble (see Figure 3a) which can be estimated as ρL gVP , FLP,V is the vertical component of the Laplace pressure force which can be estimated as

2γ

A where A is the horizontally projected

r0

area and r0 is a radius of the curvature of the bubble at apex, FLP,H,U and FLP,H,D are the horizontal components of the Laplace pressure force at the uphill and the downhill sides, respectively,


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which can be estimated as

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2γ 2γ AU and AD where r1 and r2 are the radii of the curvatures of r1 r2

the bubble at the uphill and the downhill sides, respectively, and AU and AD are the horizontally projected areas of the bubble at the uphill and the downhill sides, respectively (see Figure 4a), and α is an inclination angle of the surface. Then, Eq. (3) can further be reduced to:

   2γ  2γ π w2 2γ Ft =  ρ L gVP + AU − AD  cos α cos α − ρG gVB  sin α +  r0 4 r2  r1   

(4)

As illustrated in Figure 3, in the cases of inclined surfaces, the contact angle of a bubble (θb) can become greater than 90° due to the deformation of the bubble with the inclination so that the partial buoyancy force (FB,P) for the unbounded sections should be considered, as summarized in Table 1. Similarly, in the case of a downward-facing surface (Figure 2b), the tangential component of the net force can be formulated as:

Ft = ( FB, P − FLP ,V − FG )sin α + (FLP , H ,U − FLP , H , D ) cos α

(5)

Then, Eq. (5) can further be reduced to:

   2γ  2γ π w2 2γ Ft =  ρ L gVP − cos α − ρG gVB  sin α +  AU − AD  cos α r0 4 r2  r1   

(6)

It should be noted that, compared to the case of an upward-facing surface, the direction of the vertical component of the Laplace pressure force (FLP,V) is opposite in the case of a downward-facing surface. Figures 3b and 4b illustrate the effective buoyancy force applicable for the partial volume of the unbounded section of the bubble and the horizontally projected areas of the bubble at the uphill and the downhill sides (AU and AD) for the case of a downward-facing surface, respectively. When the net tangential force (Ft) overcomes the lateral adhesion force of a bubble


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(FLA) on the surface, the contact line of a bubble depins from the surface and starts to slide up along the surface at the critical angle called as a sliding angle (α). Thus, the lateral adhesion force of a bubble on an inclined hydrophobic surface can be estimated as:

FLA = Ft

(7)

On the other hand, the lateral adhesion force for a sliding bubble can also be estimated on the basis of the Furmidge equation,32 as follows:

FLA−Furmidge = kwγ (cosθmin − cosθmax )

(8)

where k is a retentive force factor which depends on the morphology of a contact line (shape and length) as well as the contact angle distribution along the contact line.33,34 θmin and θmax represent the receding (or minimum) and the advancing (or maximum) contact angles of a bubble that starts to slide at the downhill and the uphill sides, respectively. Alternatively, the lateral adhesion force of a bubble on an inclined hydrophobic surface can also be estimated on the basis of Tadmor’s equation,35 which is associated with the solid-liquid interaction, as follows:

FLA−Tadmor

 cos θmin cos θ max  − 2γ 2 w   r2 r1   = Gs

(9)

where Gs is the interface modulus which is associated with the reorientation of the solid molecules on the outermost layer of the surface and dependent on the normal force acting on the bubble. In this study, we use Eq. (7) to estimate the lateral adhesion force of a bubble at sliding on an inclined hydrophobic surface by experimentally measuring the net tangential force (Ft) based on Eqs. (4) and (6) for the upward-facing and the downward facing surfaces, respectively.


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Then, the retentive force factor (k) and the interface modulus (Gs) of the hydrophobic surface are evaluated based on Eqs. (8) and (9), respectively. Then, we also examine the compatibility of the proposed theoretical model by comparing the lateral adhesion force obtained by Eq. (7) to those by Eqs. (8) and (9) with the estimated retentive force factor (k) and the interface modulus (Gs).

3. EXPERIMENTAL Figure 5a shows the goniometer system with an automated tilting stage (Model 590, Rame-hart) that was used to measure the volume, width, contact angle, and sliding angle of an air bubble on a hydrophobic surface whose inclination angle was gradually increased. A Teflon sheet with an apparent contact angle of 106° for a sessile droplet of water was used as a hydrophobic surface. A custom-made rectangular acrylic tank (11 cm long, 8 cm wide, and 8 cm high) was attached on the stage of the goniometer and filled with distilled water up to around a half of the tank. The Teflon sheet was fixed to an acrylic plate and the acrylic plate was mounted on the bottom of the tank in the case of the experiment for an upward-facing surface (Figure 5b). In the case of the experiment for a downward-facing surface, the Teflon sheet was mounted on the underside of the acrylic plate (Figure 5c). Then the acrylic plate was attached on the bottom of the tank with a gap (2.5 cm) for the loading of a bubble from the underneath. After the Teflon sheet was loaded into the tank filled with water, a single air bubble was dispensed on the surface. When dispensing a bubble on the upward-facing surface of the Teflon sheet, a single air bubble was carefully injected from above by using a micropipette until it touched the surface and became stable. In the case of the experiments for the downward-facing surface, a bubble was injected under the Teflon sheet by using an inverted microneedle. After a bubble was placed on either surface horizontally, the stage of a goniometer was gradually tilted


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at the rate of 0.5° per second for all experiments until it reached the point when the bubble started to slide up. While the stage was tilted, the images of the bubble were captured at ten frames per second (10 fps). These pictures were analyzed to determine the volume, height, width, surface area, and maximum/minimum/sliding angles of the bubble, by using image processing software (DROPimage advanced v2.4, Rame-hart). Maximum and minimum contact angles were determined from the image of the bubble just before it started to slide. The obtained experimental data were then used for the theoretical models [Eqs. (4) and (6)]. To study the effects of a bubble volume on the dynamics, the volume of the bubble was varied in the range of 5-40 µL (0.5˂Bo˂1.6, where Bo is the Bond number and estimated as Bo=(ρL- ρG)gVB2/3/γ). Generally, a gravity effect can be neglected for the small bubbles (Bo1). This is mainly due to the distinctive distributions of the hydrostatic pressure around the bubble depending on the surface orientation. In the case of the downward-facing surface, the compression force is applied at the apex of the bubble with the highest hydrostatic pressure, increasing the spreading area (i.e., contact width) and the contact angle hysteresis of the bubble. While the lateral adhesion force generally increases with the increase of the bubble volume due to the increase of the contact width (i.e., the effective length of the contact line), the sliding angle of a bubble on both upward- and downward-facing surfaces decreases with the increase of


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bubble volume, which is more pronounced on the upward-facing surface. This is mainly due to the greater increase of the Laplace pressure force at the uphill side of the bubble with the increase of the bubble volume, compared to that of the lateral adhesion force. While our theoretical models based on the force balance generally agree well with the other models such as Furmidge’s and Tadmor’s approaches, the results suggest that the retentive force factors or the interface moduli employed in their models should depend on the surface orientation as well as the bubble volume. In summary, this study shows that both the surface orientation and the bubble volume significantly affect the lateral adhesion and the sliding motion of a bubble on an inclined hydrophobic surface. Such new understanding will be of great significance in many engineering and scientific applications related bubbles, such as boiling, flotation, microfluidics, hydrodynamic drag reduction.

ACKNOWLEDGEMENTS This work was supported by the Scientific and Technical Research Council of Turkey (TUBITAK-BIDEB-2219-1059B191000491) and the NSF of the United State of America (Award Numbers 1462499 and 1537474).


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Váchová, T.; Basařová, P.; Brabcová, Z. Three-Phase Contact Expansion During the Bubble Adhesion on an Inclined Plane. Procedia Eng. 2012, 42, 1897–1907.

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Kim, J.; Lee, J.S. Surface-Wettability-Induced Sliding Bubble Dynamics and Its Effects on Convective Heat Transfer. Appl. Thermal Eng. 2017, 113, 639-652.

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Tadmor, R.; Das, R.; Gulec, S.; Liu, J.; N’guessan, H. E.; Shah, M.; Wasnik, P. S.; Yadav, S. B. Solid-Liquid Work of Adhesion. Langmuir 2017, 33, 3594-3600.


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Lin, J. N.; Banerji, S. K.; Yasuda, H. Role of Interfacial Tension in the Formation and the Detachment of Air Bubbles. 1. A Single Hole on a Horizontal Plane Immersed in Water. Langmuir 1994, 10 (1), 936–942.

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Yasuda, H. Luminous Chemical Vapor Deposition and Interface Engineering; C. R. C. Press: New York, 2004.

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Furmidge, C. G. Studies at Phase Interfaces. I. The Sliding of Liquid Drops on Solid Surfaces and a Theory for Spray Retention. J. Colloid Sci. 1962, 17 (4), 309–324.

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Antonini, C.; Carmona, F. J.; Pierce, E.; Marengo, M.; Amirfazli, A. General Methodology for Evaluating the Adhesion Force of Drops and Bubbles on Solid Surfaces. Langmuir 2009, 25 (11), 6143–6154.

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Extrand, C. W.; Kumagai, Y. Liquid Drops on an Inclined Plane: The Relation between Contact Angles, Drop Shape, and Retentive Force. J. Colloid Interface Sci. 1995, 170 (2), 515–521.

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Tadmor, R. Approaches in Wetting Phenomena. Soft Matter 2011, 7 (5), 1577–1580.

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Das, S.; Morsi, Y. S.; Brooks, G.; Chen, J. J. J.; Yang, W. Principal Characteristics of a Bubble Formation on a Horizontal Downward Facing Surface. Colloids Surfaces A Physicochem. Eng. Asp. 2012, 411, 94–104.

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De-pinning on Superhydrophobic Surfaces. Phys. Rev. Lett. 2012, 109, (024504). (39)

ElSherbini, A.I.; Jacobi, A.M. Liquid Drops on Vertical and Inclined Surfaces I. An Experimental Study of Droplet Geometry. J. Colloid Interface Sci. 2003, 273, 556-565.

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Tadmor, R.; Bahadur, P.; Leh, A.; N’Guessan, H. E.; Jaini, R.; Dang, L. Measurement of Lateral Adhesion Forces at the Interface between a Liquid Drop and a Substrate. Phys. Rev. Lett. 2009, 103 (26).


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Table 1. Effective conditions for partial buoyancy Upward-facing surface

Downward-facing surface

Side

Downhill side

Uphill side

Downhill side

Uphill side

Condition

ߠ௠௜௡ + ߙ > 90°

ߠ௠௔௫ − ߙ > 90°

ߠ௠௜௡ − ߙ > 90°

ߠ௠௔௫ + ߙ > 90°


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(a)

(b)

Figure 1. Force balance for an air bubble on the (a) upward-facing and the (b) downward-facing sides of a horizontal hydrophobic surface.


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(a)

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(c)

(d) (b)

Figure 2. Force balance for an air bubble on the (a) upward-facing and the (b) downward-facing sides of an inclined hydrophobic surface. The radii of curvatures at the apexes at uphill and downhill sides of the bubble for the (c) upward-facing and the (d) downward-facing surfaces.


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(b)

(a)

Figure 3. Partial buoyancy forces effective at the (a) upward-facing and the (b) downward-facing sides of the inclined hydrophobic surfaces.


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(a)

(b)

Figure 4. Projected areas of the bubble from the downhill (AD) and the uphill (AU) sides for the (a) upward-facing and the (b) downward-facing surfaces.


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(a)

(b)

(c)

Figure 5. Experimental setup. (a) Goniometer system and a liquid chamber. A Teflon sheet mounted on an acrylic plate for the (b) upward-facing and the (c) downward-facing experiments.


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Figure 6. Comparison of the profiles of the bubbles on the upward- and the downward-facing surfaces of a Teflon sheet in a horizontal position. The bubbles have the same volume (VB≅29 µL). The red lines represent the excess of the hydrostatic pressure distribution over the bubble.


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(a)

(b)

4

3

Height (mm)

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2

1 Upward facing surface Downward facing surface 0 0

10

20

30

40

Volume ( L)

Figure 7. Variation of the (a) height and the (b) width of a bubble with respect to the bubble volume on the upward- and the downward-facing surfaces. (The solid lines are drawn to guide the trend). The inset in (b) shows the change of the bubble with respect to the Bond (Bo) number.


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(a)

(b)

Figure 8. Comparison of the profiles of the bubbles (VB≅29 µL) on the horizontal and the inclined hydrophobic surfaces. (a) Upward-facing surface. (b) Downward-facing surface. Red lines illustrate the excess of the hydrostatic pressure distribution over the bubble.


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(a)

(b)

Figure 9. Changes of the forces acting on the bubble on the (a) upward-facing and the (b) downward-facing surfaces at the sliding angle with respect to the bubble volume.


()

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( )

Langmuir

Figure 10. Sliding angles of a bubble on the inclined surface with respect to the bubble volume. The inset shows the change of the sliding angle with respect to the Bond (Bo) number.


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(a)

(b)

140 120 100 80 60 max_upward

40

facing surface

min_upward facing surface

20

max_downward

facing surface

min_downward facing surface

0 0

10

20

30

40

Volume ( L)

Figure 11. (a) Change of the maximum and the minimum contact angles of a bubble with respect to the bubble volume on the upward- and the downward-facing surfaces. (b) Change of the difference of the cosine values of the minimum and the maximum contact angles and the width of a bubble. The solid lines in the graphs are drawn to guide the trends.


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max+

160 140 min+

120 100

effective upward facing

80

ineffective

60

,

max-

min+

,

max+

180

min-

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Page 40 of 43

downward facing

max-

40

min-

20 0 0

10

20 (µL) Volume (mL)

30

40

Figure 12. Critical bubble volume effective for the partial buoyancy force. The region above the red line represents effective buoyancy force, while that below corresponds to the conditions when the buoyancy force is ineffective.


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(a)

(b)

Figure 13. (a) Estimated retentive force factors (k) based on Eqs. (7) and (8). (b) Estimated interface moduli (Gs) based on Eqs. (7) and (9).


Langmuir

(a) 0.3

(b) 0.3

Downward facing with (k =1.05)

Downward facing with (G s=63 Pa) Upward facing with (Gs=53 Pa)

FLA-Tadmor (mN)

FLA-Furmidge (mN)

Upward facing with (k =1.21)

0.2

0.1

0.0

0.2

0.1

0.0 0

10

20 Volume ( L)

30

40

0

(c)

10

20 Volume ( L)

30

40

(d) 0.35

0.35 Ft FLA-Furmidge with k=1.21

0.30

0.30

FLA-Tadmor with Gs = 53 Pa

0.25 Force (mN)

0.25 Force (mN)

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0.20 0.15

0.20 0.15

0.10

0.10

0.05

0.05

0.00

0.00

Ft FLA-Furmidge with k=1.05 FLA-Tadmor with Gs = 63Pa

0

10

20 Volume ( L)

30

40

0

10

20 Volume ( L)

30

Figure 14. (a-b) Lateral adhesion forces of a bubble of varying volume on the upward- and the downward-facing surfaces, estimated based on the Furmidge equation32 and the Tadmor’s model35 with the best fitted retentive force factors and the interface moduli, respectively, for the varied bubble volume. (c-d) Comparisons of the lateral adhesion forces estimated by the theoretical models [Eqs. (4) and (6)] to those by the Furmidge equation32 and the Tadmor’s model35 for the upward- (c) and the downward-facing (d) surfaces, respectively.


40

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Figure for Table of Content

Upward Facing

Downward Facing


Bubble Movement on Inclined Hydrophobic Surfaces - Langmuir (ACS

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