Dissipation of Film Drainage Resistance by Hydrophobic Surfaces in

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Dissipation of Film Drainage Resistance by Hydrophobic Surfaces in Aqueous Solutions Louxiang Wang,† Zhenghe Xu,*,†,‡ and Jacob H. Masliyah† †

Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2V4 Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084, China

‡

S Supporting Information *

ABSTRACT: Understanding and minimizing the film drainage resistance (forces) from a moving fluid are of great importance both scientifically and technologically. The direct and accurate measurement of film drainage resistance was made possible by integrating a speaker diaphragm of large displacement range and rapid responses with a sensitive bimorph force sensor and high resolution digital camera. Our study demonstrates that the liquid film drainage resistance can be greatly diminished or accurately controlled by increasing or controlling the hydrophobicity of solid surfaces. The results show that for a given solid surface hydrophobicity, the film drainage resistance at the point where film ruptures increases linearly with increasing bubble approach velocity. The dependence of the film drainage resistance on bubble approach velocity decreases linearly with increasing hydrophobicity of the solid surface. This finding has important implications for biological processes, microfluidic devices, and design of new materials.

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INTRODUCTION The efficient movement of sharks and fish shows a good example of drag force reduction. The reduction of the drag force is achieved by reducing the force along surfaces because of a unique surface structure or coating on the surface. Covered by small individual toothlike scales called dermal denticles (little skin teeth) and ribbed with longitudinal grooves, the low drag of shark skin can effectively inhabit the occurrence of turbulence and reduce wall frictions1,2 thereby making the shark one of the fastest swimming creatures in the ocean. The fish mucus on the skin, a naturally occurring polymer, on the other hand provides a reduction in drag when the fish moves through water. Inspired by the natural biological structure, efforts were made to modify the solid surface to reduce the hydrodynamic drag force or film drainage resistance. For example, by applying fish mucus on narrow pipe walls, a pressure drop in a pipe could be reduced by up to two-thirds.3,4 The drag force reduction can also be accomplished by solid hydrophobization. In 1999, Watanabe et al. noticed a drag reduction of water flowing in a 16 mm diameter pipe with a highly water repellent (superhydrophobic) wall. 5 Drag reduction was also reported at hydrophobic microchannel walls.6−8 In all the cases, the drag reduction is attributed to a slip boundary condition, that is, nonzero fluid velocity on the hydrophobic solid surface. The presence of slip velocity on a hydrophobic surface is further demonstrated by atomic force microscopy (AFM) measurements,9,10 surface force apparatus measurements,11,12 and relevant measurements using other techniques.7,8,13 A solidlike layer not participating in the flow © 2013 American Chemical Society

has been observed by SFA measurement for some organic liquids on solid surfaces.14−16 Furthermore, AFM measurements demonstrated a nonslip boundary condition for the water flowing over the hydrophilic glass surface.17−20 Generally, it is well accepted that the slip boundary condition should be applied for hydrophilic flows over the hydrophobic boundaries or vice versa.6,19,21,22 However, no direct force measurement was conducted at high approach velocities between two surfaces to illustrate reduction of drag resistance because of hydrophobizing solid surfaces. In this study, a custom-built integrated thin film drainage apparatus (ITFDA) was introduced, and the dynamic interactions between a spherical glass surface of controlled hydrophobicities and an air bubble in 1 mM KCl solutions at approach velocities up to 2.4 × 10−3 m/s were measured using this ITFDA.

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INSTRUMENTATION By integrating a speaker diaphragm of a large displacement range and rapid responses23 with a sensitive bimorph force sensor24,25 and a high resolution digital camera as shown schematically in Figure 1, the custom-built ITFDA allows direct and accurate measurements of forces on a stationary object from a moving air bubble, thin film rupture time, and advancing and receding contact angle in a single experiment. By simultaneously measuring the film drainage force and film Received: January 3, 2013 Revised: March 24, 2013 Published: March 25, 2013 8799

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One of the CCD cameras records the entire displacement process in real time by the PC using a user-developed program interfaced with LabVIEW 8.0. By analyzing the recorded video and bimorph signals, the entire dynamic process of the upper bubble approaching and retracting from the lower surface can be investigated. Image analysis of the video, on other hand, allows accurate determination of the size of bubble and glass sphere, the contact area, and the receding (θr) and advancing (θa) contact angles in a single experiment, providing a practical means to link the surface properties with dynamic interaction forces and, hence, insights into these interactions. The selected frames from the recorded video of interest were analyzed using the image analysis program to determine geometry of the system. A toroidal approximation where the air−liquid interface on the axi-symmetrical plane was considered as a circular arc was employed. As shown in Figure 2 for both approaching (a) and retracting (b) of an air bubble

Figure 1. A schematic configuration of the integrated thin film drainage apparatus (ITFDA).

drainage time, the ITFDA bridges a gap between the modified thin film balance26−28 and the colloidal probe experiments using AFM.29−33 Using the thin film balance technique, only film thickness (film drainage time) can be measured, while in the AFM experiments interaction forces alone are obtained. In the ITFDA, the air bubble is generated using a microsyringe at the end of the glass capillary tube which is connected to a speaker diaphragm. A computer generates a desired waveform that controls the movement of the speaker diaphragm which in turn drives the air bubble to approach or retract from the lower (glass) surface in a well-controlled manner. With such an arrangement, the approach and retract velocity (Va and Vr, from μm/s to mm/s) as well as the range of displacement of the air bubble (up to 2 mm) can be accurately controlled. The extent (applied force) and the duration of the contact between the air bubble and the solid surface with or without an intervening liquid film can also be well controlled. The fused glass sphere on a glass rod is clamped at the free end of a bimorph beam. The bimorph is enclosed by a fluorinated ethylene propylene (FEP) sheath and is mounted on a small stainless steel chamber which is placed on a three-dimensional translation stage. Two cameras are placed perpendicular to each other near the sample chamber to align the interacting surfaces as well as to control the size of the air bubble and the gap between the bubble and solid surface. Details on sample preparation can be found in the Materials and Methods section. When the two surfaces approach each other, a deflection of the bimorph occurs because of the interaction forces between the surfaces. Bimorph is a piezoelectric device which generates electrical charge proportional to the applied force or the deflection along the central axis at the end of the bimorph.24 Hence, the force between the two interacting surfaces can be obtained by measuring the voltage generated from the deflection of the bimorph cantilever and its spring constant. The bimorph used in this study gives a force resolution of 0.05 mN/m or 0.1 μN, which can be varied by selecting bimorphs of different geometries. The bimorph force sensor shows an excellent reproducibility. The vertical displacement of the glass tube holding the air bubble, on the other hand, is independently measured using a displacement sensor with a sensitivity of 5 μm. The signals from the charge amplifier and the displacement sensor of the glass tube in response to the voltage applied to the speaker are recorded as a function of time. In the measurement, the initial separation distance between the air bubble and the glass surface is set at 120 μm, and the maximum glass tube displacement is set at 240 μm.

Figure 2. Still images and diagram of imaging analysis to determine the geometrical properties of the bubble−glass sphere interactions using a LabView 8.0 based image analysis program: (A) approaching and (B) retracting. The hydrophobic glass surface was obtained by octadecyltrichlorosilane treatment of hydrophilic glass.

on a hydrophobic glass surface with velocity of V, points were manually selected at the solid/water and air/water interfaces. These points were then fitted to a circular arc. Geometric properties of the system can then be determined. For example, the angle between the two fitted arcs at the intersection was considered as the contact angle. Because of the symmetry of the bubble and glass sphere, the difference between the contact angles at the left and right sides of the air bubble was determined to be less than 3°. The average value of the two was reported as the contact angle. The contact angles of the approaching and retracting air bubble on the glass surface in Figure 2 were determined to be 36° and 51°, respectively. Knowing the geometrical properties of the interfaces, the capillary force, FC, can be calculated by29,34 ⎛1 1⎞ FC = πR 22 sin 2 α ·γlv ⎜ + ⎟ l⎠ ⎝ R1 + [−2πR 2γlv sin α sin(θ − α)] =

1⎞ πD 2 ⎛ 1 γlv ⎜ + ⎟ + [−πDγlv sin(θ − α)] l⎠ 4 ⎝ R1

(eq 1)

where R2 is the radius of the spherical glass surface, α is the half-filling angle (measured at the center of the glass sphere) of the sphere−capillary bridge contact as shown in Figure 2, γlv is the liquid−vapor interfacial tension, R1 and l are the principal radii of the capillary bridge, θ is the contact angle of the bubble on the glass surface measured through the aqueous phase, and D is the diameter of the bubble−glass contact area. In this 8800

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RESULTS AND DISCUSSION Force Profiles. The bimorph response as a function of measurement time between an air bubble and a hydrophobized spherical glass surface of θa = 52° is shown in Figure 3.

communication, the capillary forces at various states during TPC were calculated on the basis of eq 1 and were compared with the measured forces.

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MATERIALS AND METHODS A piezo ceramic actuator, purchased from Fuji Ceramics Corp. with dimensions of 20 × 3 × 0.3 mm and capacity of 20 nF, was used to fabricate the bimorph force sensor. The glass-sphere surface was prepared by melting a 1.5 ± 0.1 mm diameter Pyrex rod under a butane-oxygen flame until the surface tension of the melting Pyrex produced a spherical surface with a diameter of roughly 4.5 mm. The surface prepared as such was found to be molecularly smooth for force measurements. A glass capillary tube with an inner diameter of 1.1 ± 0.05 mm (Fisher Scientific) was used to generate and displace the air bubble. One end of the glass tube was placed under a butane flame to create a smooth end to facilitate reproducible bubble generation. Extreme caution was taken to avoid overheating which would result in an asymmetric glass end because of deformation. The glass sphere and capillary tube were cleaned in freshly prepared piranha solutions (3 H2SO4:1 H2O2, by volume) at 80−90 °C for 30 min and were rinsed with Milli-Q water prior to their use in the experiments. The surfaces prepared in this manner were free of contamination and were completely water wettable (i.e., having a contact angle of zero degree). Octadecyltrichlorosilane (OTS), received from SigmaAldrich, was used to prepare 1 mM OTS in toluene (Fisher Scientific) solutions which were used to hydrophobize the hydrophilic glass spheres by surface silanation reactions. Different degrees of surface hydrophobicity were obtained by varying the soaking time of the hydrophilic glass sphere in 1 mM OTS solutions.30,35,36 The treated glass sphere was rinsed with toluene followed by anhydrous ethyl alcohol (Commercial Alcohols Inc.) and was blow-dried with ultrapure nitrogen to remove residual OTS from the glass surface, avoiding the deposition of OTS precipitates from residual OTS solutions and hence the formation of rough surfaces. Potassium chloride (KCl), purchased from Sigma-Aldrich, was used as the supporting electrolyte, and concentrated NaOH and HCl solutions were used as pH modifiers. To obtain the degassed solution, 1 L of 1 mM KCl solution was boiled at 100 °C for 90 min to eliminate dissolved gases after which the solution was rapidly cooled to room temperature in an ice−water bath. The solution was then diluted with degassed Milli-Q water (treated the same way) to 1 L volume to maintain the desired initial concentration. The degassed 1 mM KCl solution was used for experiments immediately to minimize introduction of air from the atmosphere. At the beginning of each experiment, the glass sphere was clamped at the end of the bimorph beam as shown in Figure 1. The chamber was then filled with a test solution and was placed on a three-dimensional translation stage. The glass capillary tube was filled with fresh air before being brought into the solution, and the system was then left for two hours to stabilize the bimorph signals. Before the measurement of interaction forces, a fresh air bubble was generated with the diameter of 1.5 ± 0.05 mm and was conditioned in desired solutions for 5 min to equilibrate the air−water interface. At least 10 measurements were conducted for each condition, and a single representative force profile was reported in this communication (the signal was smoothed for noise reduction).

Figure 3. Bimorph response as a function of time as an air bubble approaches and retracts from a hydrophobized glass sphere of θa = 52° and Va = Vr = 120 μm/s in a 1 mM KCl solution at pH 7.7. The inset shows bimorph response during the drainage of intervening liquid film, and the still images at various key locations of force profile are shown to help interpret the force profile. The red circular symbols are the calculated FC during the TPC using eq 1. The diameters of air bubble and glass surfaces are 1.46 mm and 4.31 mm, respectively. (See SI_Movie_1 in the Supporting Information.)

Typically, there are four regions in a complete force profile: (I) negligible net force between the two surfaces at large separation distances; (II) a rapid increase in repulsive forces from points a to b; (III) a sudden drop in the measured repulsive forces to become attractive from b to c followed by a continuous decrease in attractive forces and an increase in repulsive forces from points c to c′ and a slight increase in the repulsive forces from points c′ to d; and (IV) a continuous decrease of the repulsive forces and reverse to increase in attractive forces followed by a slight decrease in the attractive forces from points f and g, and a sudden decrease to zero of the force at point h. Points o to c′ represent approaching process where the air bubble is driven down by the glass capillary tube at the velocity of 120 μm/s toward the lower glass surface, while points d−k show the process where the air bubble is retracted from the glass surface at the same velocity. Between points c′ and d, the capillary tube is held stationary. A video clip showing the bimorph signals synchronized with the movie recorded during the experiment is provided in the Supporting Information (SI_Movie_1). When the air bubble approaches to or retracts from the hydrophobic glass surface at a constant velocity and large separation distances, there is no net (measurable) interaction force between the two surfaces, shown by a zero bimorph displacement as a flat baseline with electrical noises (region I). As the two surfaces come into a distance where the hydrodynamic or surface forces begin to dominate, a repulsive force (film drainage resistance) is detected at point a, represented by an increase in bimorph signal, and the lower 8801

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Figure 4. Bimorph response as a function of measurement time with Va and Vr of 240 μm/s and T = 20 ± 0.5 °C: (A) between an air bubble and a hydrophobized glass sphere of θa = 103° in a 1 mM KCl solution at pH 7.7 where at point h an orphan air bubble was left behind; (B) followed with A, at Va of 240 μm/s but between the air bubble and the orphan bubble on the solid surface; (C) repeat of A at Va of 240 μm/s but in the degassed 1 mM KCl solution. The inset in A shows a close view of the bimorph signal during intervening liquid film thinning process in comparison with the partial force profiles shown in B and C for better interpretation of the force profiles. The red circular symbols are the calculated FC during the TPC using eq 1. The diameters of air bubble and glass surfaces are 1.51 mm and 4.37 mm, respectively. (See SI_Movie_2 of the Supporting Information.)

surface deflects downward while the aqueous film between the bubble and the glass surfaces starts to drain (region II in Figure 3). During the film drainage as the bubble continues to move down, the repulsive force between the two surfaces increases and reaches a maximum at the critical film thickness (Tc), point b where the liquid film becomes unstable and ruptures, forming a TPC where the air bubble attaches on the glass surface. In this case, the lower hydrophobic glass surface is pulled up to the bubble surface by interfacial tension force, and a jump into the bubble is observed. This is shown by a dramatic drop of bimorph signal in the profile from points b to c as a result of the TPC line expansion on the glass surface. The maximum applied force, which is used to overcome the force barrier between the two surfaces before TPC, is determined as force barrier (Fbar). Right after point c in region III, the bubble is pinned on the glass surface without any movement of the TPC line. The attached bubble continues to deform against the lower glass surface as the capillary tube continues to drive the bubble downward, thereby increasing the force pushing down on the glass sphere and reducing the contact angle. When the contact angle reaches θr, the TPC line starts to recede, that is, the air bubble advances on the glass surface to increase the contact area of the bubble on the solid surface and, hence, the capillary force. The pinning and advancing of the TPC line on the glass surface from points c to c′ can be clearly observed in the SI_Movie_1 of the Supporting Information. The bubble is then held on the glass surface with little displacement of the capillary tube from point c′ to d during which the bimorph response increases slightly and then remains constant as a stable TPC is established. In region IV, as the bubble retracts from the solid surface at point d, the bubble exerts an increasing upward pull force on the glass surface as a result of bubble pinning from θr toward θa (points d−e). Once θa is reached at point e, the TPC line (bubble) starts to move inward on the glass surface, exhibiting a less significant further increase in the upward lift force as the bubble continues to be stretched. From points f to g, the average pulling force decreases only slightly with further

stretching the bubbles. However, a zigzag response appears during this state, indicating a stick−slip motion of the TPC line as the bubble approaches instability of pulling away from the solid surface. At point g, the restoring force of the bimorph beam overcomes the capillary (adhesion) force, and the bubble detaches from the glass surface as shown by a sudden change in bimorph signal. After detachment of the bubble from the solid surface, the bimorph cantilever returns to its free position as shown by a zero signal response at point h, while the glass tube keeps moving until it returns to its original position. As shown in Figure 3, the measured forces (solid line) are in an excellent agreement with the calculated capillary force (open circles) over the TPC region (points c−g), demonstrating the accurate measurements of force and geometrical properties. Reduction of Film Drainage Resistance. There are two forces exerted on the two interacting surfaces during the film drainage period: hydrodynamic and surface forces. When the bubble approaches the glass surface, the repulsive hydrodynamic force dominates the total force at large separation distances. At close separation distance, as the air bubble continues to deform prior to liquid film rupture, the external force applied by the glass capillary through the air bubble increases to overcome the repulsive film drainage resistance and further thins the intervening liquid film. Eventually, the film reaches a critical thickness and ruptures at point b in Figure 3. Interestingly, when the hydrophobicity of the glass sphere was increased to a θa of 103° at Va = 240 μm/s, a very different force profile was obtained as shown in Figure 4A. Even though the bubble approach velocity was increased to 240 μm/s, the film drainage resistance detected before TPC in Figure 3 from points a to b disappeared as shown in Figure 4A, indicating diminished resistance by the strongly hydrophobic glass surface. Instead, a sudden jump of glass surface into the air bubble is observed at points b−c, suggesting a strong and long-range attractive force which depresses all repulsive forces between the two surfaces even at this high bubble approach velocity. This attractive force is attributed to the change in the wettability of 8802

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surface is highly hydrophobic (i.e., θa = 103°) but only up to 48 μm/s for a moderately hydrophobic surface (i.e., θa = 84°). The motion of the glass tube is speculated to cause additional flow around the glass sphere and, hence, to add additional force on the measured force by the bimorph. To quantify this effect if present, experiments were conducted to determine the force from the capillary tube with a bubble cap on the glass sphere fixed on the bimorph when the glass tube approached the glass sphere. As shown in Figure 6A, the initial separation distance between the glass tube and the glass sphere and the maximum

the glass surface from low hydrophobicity of θa = 52° to strong hydrophobicity of θa = 103°. As shown in the inset photo of Figure 4A at point h, a visible orphan bubble was left on the hydrophobic glass surface after the detachment of the air bubble from the glass surface. A strong repulsive force before TPC formation as shown in Figure 4B was detected while moving the air bubble toward the orphan bubble on the hydrophobic glass surface. This observation is in great contrast to the case of an air bubble approaching the strongly hydrophobic solid surface as in case A, where there was no noticeable repulsive force. More detailed information is available in SI_Movie_2 of the Supporting Information. Dissipation of film drainage resistance was also observed in a degassed KCl solution with an air bubble approaching the same hydrophobic glass surface (Figure 4C). Again, a good agreement between the calculated and measured FC was reached in Figure 4 during the bubble approaching to, holding on, and initially retracting from the glass surface. However, the magnitude of the calculated FC was much smaller than the measured FC with the further retraction (stretching) of the bubble. The deviation could be attributed to the breakdown of the toroidal approximation at large bubble deformation (stretching). On the basis of the toroidal approximation, the Laplace pressure (Δp) across the air/ water interface at the three phase contact region can be expressed as Δp = γlv[(1/R1) + (1/l)]. When the bubble was stretched extensively, as shown in the inset photo at point e in Figure 4, the radius of the air/water interface (R1) becomes very large. As a result, the actual Laplace pressure of the air bubble becomes much smaller than the calculated Δp where Δp ∼ γlv/l. Therefore, the measured forces are much larger than the calculated FC over this region. Effect of Solid Hydrophobicity. As shown in Figure 5, increasing the air bubble approach velocity increases the Fbar between the air bubble and the hydrophobic glass surfaces. Fbar remains undetectable up to Va of 240 μm/s when the glass

Figure 6. (A) A schematic configuration of the experimental setup to determine hydrodynamic effect of moving glass capillary tube on force measurement by bimorph; (B) effect of glass tube displacement velocity (48−4800 μm/s) on the force measurement of the bimorph. The initial separation distance between the end of the glass tube and the glass sphere and the maximum glass tube displacement was 1290 and 240 μm, respectively.

glass tube displacement were set at 1290 and 240 μm, respectively (the same value as the bubble−glass sphere interactions where an air bubble was attached at the end of the glass tube). Different drive velocities were tested to determine the effect of moving glass tube on the measured force because of the displacement of water around the capillary tube. The results in Figure 6B show a negligible force on the bimorph during the motion of the glass tube. The results demonstrate that over the velocity range of the capillary tube displacement in our study (up to 2400 μm/s) we can safely neglect the effect of the flow disturbance caused by the capillary tube on the measured forces by the bimorph sensor. Therefore, the change on the measured force by the bimorph sensor is due to the air bubble. It is recognized that the bubble deforms when it reaches the boundary proximity of a solid surface during the approach. The bubble deformation will change the area of interaction and, hence, the total boundary force. It is therefore important to account for the effect of bubble deformation on the measured total force to elucidate the effect of surface hydrophobicity on dissipation of film drainage resistance. For this reason, Fbar determined as shown in Figure 5 is normalized with the interaction area to obtain an average pressure (p)̅ of the drainage film, that is, p̅ = (Fbar)/(πr2), which is referred to as film drainage resistance. Here, r is the radius of the projected area of interaction at the point where the film ruptures. As it can be seen from Figure 7, for a given solid wettability (θa), the film drainage resistance increases linearly with approach velocity, Va. At a given bubble approach velocity, the film

Figure 5. Effect of glass sphere hydrophobicity on force barrier between an approaching air bubble and a hydrophobized glass surface prior to three phase contact formation in a 1 mM KCl solution at pH 5.6 and 20 ± 0.5 °C as a function of bubble approach velocity. Solid lines show the general trend between the two parameters; the diameters of the air bubble and the glass surfaces are 1.5 ± 0.05 mm and 4.3 ± 0.1 mm, respectively. 8803

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98% with the increasing contact angle from 38° to 103°. Therefore, the reduction of slope is most likely due to a dramatic decrease in S with the increasing θa, that is, increasing slippage boundary layer thickness with increasing θa, manifested by an increased reduction in film drainage resistance. By extrapolation, solids with θa larger than 106° would lead to a zero slope with a negligible film drainage resistance over the bubble approach velocities studied, indicating a complete slippage of the liquid film on this type of solids. Interestingly, this contact angle value is close to the lower limit of contact angle value for smooth superhydrophobic surfaces,39 providing another scientific insight of superhydrophobic surfaces. Nevertheless, the importance of Tc can never be overestimated. According to eq 2, the film drainage resistance can be a strong function of Tc, and the slope k scales with the Tc to the second power. Small variations in Tc can have a dramatic effect on the film drainage and rupture. Therefore, determination of the film thickness during the film drainage process is critical to understand the whole process. A further improvement of the current ITFDA which enables the direct measurement of film profile using interferometry method is in progress.

Figure 7. Effect of glass hydrophobicity on film drainage resistance (p̅) between an approaching air bubble and a hydrophobized glass surface prior to three phase contact formation in a 1 mM KCl solution at pH 5.6 and 20 ± 0.5 °C. Solid lines show a linear fitting of the pressure as a function of bubble approaching velocity. The inset illustrates the effect of glass hydrophobicity on the slope, k, as given by eq 2.

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drainage resistance decreases with increasing θa. As shown in the inset of Figure 7, the dependence (slope) of the film drainage resistance on bubble approach velocity decreases almost linearly with increasing θa, illustrating progressive dissipation of film drainage resistance by increased surface hydrophobicity. Since the surface forces are independent of approach velocity for a given solid-aqueous system, the measured increase in film drainage resistance with increasing bubble approach velocity is most likely linked with the hydrodynamic resistance of liquid film drainage. This hydrodynamic resistance at the point of film rupture, p, between two surfaces approaching each other, can be described by37 ⎡ 3μRS ⎤ p = ⎢ 2 ⎥V = kV ⎣ Tc ⎦

CONCLUSIONS The interaction forces between an air bubble and a hydrophobic spherical glass surface with different hydrophobicities were measured using the ITFDA under a wide range of air bubble approaching velocities. It has been found that the force barrier before three-phase contact can be greatly diminished or accurately controlled by increasing or controlling the hydrophobicity of solid surfaces. The film drainage resistance or the normalized force barrier was found to increase linearly with increasing bubble approach velocity. The dependence (slope) of film drainage resistance on bubble approach velocity decreases linearly with increasing advancing contact angle of solid, indicating that hydrophobization of solid surface appears to be a practical approach to reduce film drainage resistance. Using a high-resolution camera and image analysis program, the interactions between the air bubble and the spherical glass surface were analyzed. The obtained geometrical properties of the system were used to calculate FC. The calculated and measured FC are shown to agree well with each other for the low bubble deformation, demonstrating but not for large bubble deformation demonstrating a limitation of using toroidal approximation to calculate the Laplace pressure of the bubble.

(eq 2)

where μ is the viscosity of the solution, Tc is the separation distance at the point of film rupture, V is the surface approach velocity, S is the dimensionless function used to consider the slip boundary condition, and R is the reduced radius of the system. Here, R is given by R = (R1R2)/(R1 + R2), where R1 and R2 are the radius of the upper and lower surfaces, respectively. For two hydrophilic surfaces, S = 1. When the slip boundary condition is applicable at the surface, S becomes less than unity.18,19 For a given glass surface, Tc and S remain the same at the rupture point. One would therefore obtain a linear relationship between p and V, that is, p = k·V, which can be clearly seen in Figure 7. As illustrated in the inset of Figure 7, the slope k decreases with increasing surface hydrophobicity (θa) and has an intercept of 106° at the x-axis. Tc is a function of solid hydrophobicity and can be calculated by38 Tc = 23.3[γ(1 − cos θa)]0.16

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ASSOCIATED CONTENT

S Supporting Information *

Two video clips showing the recorded bimorph signals synchronized with the movies recorded during the experiments between air bubble and hydrophobic glass sphere with θa = 52° and 103° in aqueous solutions are presented. This material is available free of charge via the Internet at http://pubs.acs.org.

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(eq 3)

According to eq 3, increasing θa would lead to an increase in Tc; the observed decrease in the slope as given in eq 2 with increasing θa is not unexpected. On the basis of eq 3, Tc ranges from 36 to 47 nm in the studied contact angle range. Such a slight increase of Tc would only account for 40% decrease of the slope. However, as shown in Figure 7, the slope decreased by

AUTHOR INFORMATION

Corresponding Author

*Tel.: +1 780 492 7667; fax: +1 780 492 2881; e-mail: zhenghe. [email protected]. Notes

The authors declare no competing financial interest. 8804

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ACKNOWLEDGMENTS Financial support for this work from Natural Sciences and Engineering Research Council (NSERC)-Industrial Research Chair Program in Oil Sands Engineering and Discovery Grants is greatly appreciated.

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dx.doi.org/10.1021/jp4000945 | J. Phys. Chem. C 2013, 117, 8799−8805