Study of the Relationship between Boundary Slip and Nanobubbles

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Study of the Relationship between Boundary Slip and Nanobubbles on a Smooth Hydrophobic Surface Dayong Li,†,‡ Dalei Jing,†,§ Yunlu Pan,*,† Bharat Bhushan,*,†,∥ and Xuezeng Zhao† †

School of Mechanical and Electrical Engineering, Harbin Institute of Technology, Harbin 150001, China School of Mechanical Engineering, Heilongjiang University of Science and Technology, Harbin 150022, China § School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China ∥ Nanoprobe Laboratory for Bio- & Nanotechnology and Biomimetics (NLB2), The Ohio State University, 201 W. 19th Avenue, Columbus, Ohio 43210-1142, United States ‡

S Supporting Information *

ABSTRACT: Surface nanobubbles, which are nanoscopic or microscopic gaseous domains forming at the solid/liquid interface, have a strong impact on the interface by changing the two-phase contact to a three-phase contact. Therefore, they are believed to affect the boundary condition and liquid flow. However, there are still disputes in the theoretical studies as to whether the nanobubbles can increase the slip length effectively. Furthermore, there are still no direct experimental studies to support either side. Therefore, an intensive study on the effective slip length for flows over bare surfaces with nanobubbles is essential for establishing the relation between nanobubbles and slip length. Here, we study the effect of nanobubbles on the slippage experimentally and theoretically. Our experimental results reveal an increase from 8 to 512 nm in slip length by increasing the surface coverage of nanobubbles from 1.7 to 50.8% and by decreasing the contact angle of nanobubbles from 42.8 to 16.6°. This is in good agreement with theoretical results. Our results indicate that nanobubbles could always act as a lubricant and significantly increase the slip length. The surface coverage, height, and contact angle are key factors for nanobubbles to reduce wall friction. observed by Zhang et al.5 and Simonsen et al.6 and concluded that nanobubbles do not significantly increase slip over bare, hydrophobic surfaces. In contrast, Wang et al.16 reported a slip length of 44 nm on a hydrophobic surface with observed nanobubbles, and a larger slip length of 257 nm on a superhydrophobic surface was also thought to be attributed to the formation of more nanobubbles, although they were not observed. Cottin-Bizonne et al.17 measured a typical slip length of about 100 nm on an octadecyltrichlorosilane (OTS) surface, and in their work,18 they thought that such a result might be affected by nanobubbles. To date, there is still no detailed experimental study to shed light on the effect of nanobubbles on slip length for liquid flow over smooth surfaces. Some studies investigated interfacial slip on structured surfaces with a trapped gaseous film,19−24 which are thought to be similar to the effect of nanobubbles. It was found that the change in orientation of the geometry on the surface with respect to the flow direction can cause a transition from an increased slippage state to a frictional state.25 Similarly, the protrusion angle (θp, as shown in Figure 1), quantifying the

1. INTRODUCTION Reducing the viscous friction of liquid flows in micro/nano devices has become an important issue with the development of micro/nano fluidic applications, especially because manipulating the hydrodynamic slippage at small scales remains a challenge.1−4 In order to achieve effective drag reduction of liquids at walls, surface nanobubbles have been suggested as a lubricant at liquid−solid interfaces. Surface nanobubbles are nanoscale gaseous domains with surprising properties including long lifetimes and small contact angles (gas side) on immersed substrates.5−8 The effect of surface nanobubbles on the boundary condition has been investigated intensively on various substrates in the past two decades.1−3,9−13 However, there are still disputes as to whether surface nanobubbles can increase the slip length effectively. The concept of slip length was first proposed by Navier to quantify the slippage of a liquid at a solid surface. Navier states that the velocity v of liquid is proportional to the velocity gradient, i.e., v = b∂v/∂z|z=0. The slip length b is the distance below the liquid/solid interface where the liquid velocity linearly extrapolates to zero.4,10 A larger slip length means a smaller drag of liquid flow at the wall. Some early theoretical results10−15 show that the increased slippage of liquid flow is ascribed to the presence of nanobubbles on substrates. Hendy and Lund9 estimated the value of the slip length on the basis of the nanobubbles © 2016 American Chemical Society

Special Issue: Nanobubbles Received: August 2, 2016 Revised: September 21, 2016 Published: September 22, 2016 11287

DOI: 10.1021/acs.langmuir.6b02877 Langmuir 2016, 32, 11287−11294

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Langmuir

2. MATERIALS AND METHODS 2.1. Sample/Water Preparation. Samples of polystyrene (PS) coated on 1 cm × 1 cm silicon (100) substrates were prepared by spincoating PS (molecular weight 350 000, Sigma-Aldrich) solution at a speed of 2000 rpm. To obtain PS surfaces of different roughnesses, two different concentrations of PS solution of about 0.30 and 0.50% (by weight) are used. The root-mean-square (RMS) values of surface roughness for the substrates over a scanned area of 5 μm × 5 μm were measured using tapping mode atomic force microscopy (TM-AFM) in air. The thickness of the PS films was determined by AFM nanoshaving,31 and the contact angle (water side) was measured using the sessile drop method. These parameters can be seen in Table 1. Before spin-coating, the silicon (100) wafers were boiled in piranha solution (1:3 mixture of 30% H2O2 and 98% H2SO4) for 30 min and then acetone for 30 min and then were rinsed using ethanol and pure water (three times each). After spin-coating, the wafer pieces were blown dry with nitrogen, put in culture dishes, and then left in an oven for 5 h at 50 °C. In our experiments, water was purified with a MilliQA10 system (18.2 MΩ·cm); air-equilibrated water and partially degassed water were used. To ensure gas-saturated water, about 50 mL of water in a glass container was allowed to equilibrate at atmospheric pressure for about 10 h. Two kinds of degassed water were prepared by degassing purified water at 20 mbar for 30 min and 1 h, respectively. The experimental process is usually performed in about 40−50 min. Consider that exposing water to air will increase the gas concentration, which might affect the formation of nanobubbles and thus affect the slip length. To reduce this effect on the slip length measurements, all experiments were performed as soon as possible (the whole process was usually completed in about 40 min). In addition, the incubation experiments of nanobubbles in degassed water were carried out, which can be seen in Figure S3 and Table S1 in the Supporting Information. There is no obvious increase in the number and volume of nanobubbles after about 1 h of incubation time. Therefore, the effect of gas dissolution on slip length measurements in degassed water was neglected in this article. 2.2. Nanobubble Measurements. TM-AFM (NTEGRA platform, NT-MDT Company, Zelenograd, Moscow) was used to image nanobubbles on PS films in purified water. Rectangular cantilevers (CSG30, NT-MDT company, typical tip curvature radius of 10 nm, spring constant of k = 0.13−2 N m−1, actual spring constant of 0.68 ± 0.02 N m−1 determined by the resonance frequency method32) were treated with a plasma cleaner (Harrick Plasma, PDC-32G-2) for about 1 min before use. The resonance frequency of the cantilever in air and water was measured with a lock-in amplifier (SRS 830), and the corresponding values were about 65 and 21 kHz, respectively. The scanning frequency used was 1 Hz at a set-point ratio of 95% with a typical free amplitude value of 3.6 ± 0.2 nm. To avoid contamination, the fluid cell and clip spring need to be rinsed with ethanol and pure water before use. All of these experiments were performed in a clean room (∼25 °C, relative humidity (RH) ∼47%). The NT-MDT SPM software (Nova), grain analysis, was used to analyze the height, diameter, and surface coverage of surface bubbles. 2.3. Hydrodynamic Force and Slip Length Measurements. Contact-mode AFM (CM-AFM, NTEGRA platform, NT-MDT Company, Zelenograd, Moscow) was used to measure the hydrodynamic force. The colloidal AFM probe was prepared by attaching a borosilicate sphere (GL018B/45-33, MO-Sci Corporation) to a rectangular cantilever (ORC8, Bruker) using epoxy (Araldite, Bostik,

Figure 1. Sketch of a concave trapped bubble and a convex trapped bubble and the definition of protrusion angle θp.

morphology of liquid−gas menisci, has also been demonstrated to lead to such a transition from slippage to friction.2,3,19,22,25 Simulation studies2,3 and numerical studies15,22 have predicted a negative slip length when the protrusion angle is larger than a critical value. These critical values have been identified as about 65,22,26 69,25 and 62 or 68° for bubble coverages of 54 and 38%, respectively.3 A maximum slip length will be obtained when the protrusion angle is 1022 and 11°.3 The experimental results of a decreased slip length with increasing protrusion angle were recently verified in the study by Karatay et al.3 The effective slip length for trapped bubbles with a protrusion angle of 43° is smaller than that for those with a protrusion angle of 21°. A fact in these studies is that the bubble mattress on the surfaces has been built up of equally shaped and evenly sized bubbles. However, for nanoubbbles, the assumption of such a situation of evenly sized and equally shaped bubbles is not realistic. Hyväluoma et al.27 simulated a liquid flow over structured surfaces with two different sizes of attracted bubbles and found that the slip length depends on the pattern geometry and the maximum slippage can be obtained by varying the position and size of the trapped gas bubbles. Actually, the surface used by Hyväluoma et al. is close to, but still not in accordance with, the nanobubbles’ situation. Furthermore, surface nanobubbles, unlike the artificially trapped bubbles, are very special gaseous domains with distinctive properties, including an extremely high inner pressure,7,28 which may lead to different viscosity, uncertain movability, a long lifetime,7,28,29 and a smaller contact angle (gas side).7,30 These properties are different from those of macroscopic bubbles and might have a different influence on slip length. Therefore, carrying out an experimental study on unstructured surfaces with spontaneously formed nanobubbles is appropriate and significant. Also, such research can contribute to the understanding of the interfacial behavior and surprising properties of surface nanobubbles. In this article, we report on slip length measurement experiments on polystyrene (PS) substrates with different surface nanobubbles. We further investigate in detail the effects of surface coverage, height, and contact angle of surface nanobubbles on the effective slippage. Table 1. Parameters of Four Different PS Substrates

a

substrate

1

2

3

4

RMS (nm) contact angle (deg) film thickness (nm) concentration of PS solution (weight)

0.86 ± 0.02 93 ± 2 27 ± 2 0.30%

1.05 ± 0.03 93 ± 2 29 ± 2 0.30%

1.87 ± 0.05 94 ± 2 56 ± 3 0.50%

2.16 ± 0.06 95 ± 2 71 ± 4 0.50%

The contact angle was measured through the water side of a water drop. 11288

DOI: 10.1021/acs.langmuir.6b02877 Langmuir 2016, 32, 11287−11294

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Figure 2. AFM images of nanobubbles on PS films. Height images a and b show the nanobubbles imaged in partially degassed water at 20 mbar for 1 and 0.5 h, respectively. Height image c shows the nanobubbles formed on a PS film with a roughness of RMS = 1.87 nm imaged in air-equilibrated water, image d shows a cross-sectional profile of bubble b1 in image c, and the inset is the 3D image of bubble b1. Image e shows a sketch of a nanobubble (not to scale). Height image f shows nanobubbles formed on a PS film with a roughness of RMS = 2.16 nm imaged in air-equilibrated water, and image g is the magnified section of the rectanglar region in image f. Coubert). The diameter of the borosilicate sphere was about 44.4 μm as measured with an optical microscope. The hydrodynamic force is measured by driving the substrate to approach the sphere with a constant velocity of Vd = 65.6 μm/s. A liquid is squeezed out of the space between a sample surface and a colloid probe. The forces, including the hydrodynamic force, the drag force, the van der Waals force, and the electrostatic force, acting on the colloidal probe can be obtained from the deflection of the cantilever. The diameter of the sphere used was about 44.4 μm. This means that there is a large separation distance between the sample and the AFM cantilever, thus leading to a small hydrodynamic force exerted on the cantilever that could be neglected.10,33 The van der Waals force was assumed to be small compared to the hydrodynamic force and was also neglected in this article. The hydrodynamic force on a silicon surface was first measured for a hydrophilic flat surface on which the liquid flow is under the no-slip boundary condition and the deflection of the cantilever Def can be written as10

Def =

Fhydro k

=

6πηR2V kD

For the hydrophobic surface on which the liquid flow is under the slip boundary condition, the hydrodynamic forces can be given at a large separation distance (D ≫ b) as34

V 1 = (D + b) Fhydro 6πηR2

(2)

The slip length b, based on eq 2, can be described as the position of the linear extrapolation of V/Fhydro intercepts at the axis of the separation distance.

3. RESULTS AND DISCUSSION Here, we investigate the effect of surface nanobubbles on the slippage at the solid−gas−liquid interface. To obtain surfaces with different bubble coverage, polystyrene (PS) surfaces with different surface roughness were immersed in partial degassed water or air-equilibrated water. Experimental studies35,36 and theoretical studies37 have shown that the gas concentration in water has an important effect on the formation of surface nanobubbles and that nanobubbles can survive in slightly degassed water. Figure 2a−c,f,g shows the topography of surface nanobubbles imaged by using TM-AFM. The parameters of four different PS substrates can be seen in Table 1, the water used in the experiments for Figure 2a,b is degassed water at 20 mbar for 1 and 0.5 h, respectively, and the water used in the experiments for Figure 2c,f is air-equilibrated water. From the height images of Figure 2a−c, an increase in size and surface coverage of nanobubbles can be found, which

(1)

where Fhydro is the hydrodynamic force, k is the cantilever stiffness, R and V are the radius and velocity of the sphere, respectively, D is the separation distance between the bottom of sphere and the substrate surface, and η is the dynamic viscosity of the liquid. By fitting the experimental data of Def as a function of D using eq 1, a value of about 26.4 nm s was obtained for k/6πηR2, and then with the values of η = 0.98 × 10−3 Pa s and R = 22.2 μm, a value of k = 0.24 N m−1 was calculated. 11289

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that on a smoother surface. Noticeably, in our experiments, the rough surface usually had a thicker PS film, which should be more stable than the thinner PS film on a smoother surface. In addition, our recent study has shown that nanobubbles have little effect on the topography of PS film for about 1 h of incubation time.29 Therefore, although the pinning force of nanobubbles has been found to change the topography of a PS film after a long incubation time, the effect of the variation of the nanobubble pinning force on the slip length was neglected. (The incubation time of nanobubbles in our experiments is about 40 min.) After imaging surface nanobubbles, slip length measurements were carried out immediately on the measured nanobubbles at the same PS substrate in the same water. In our experiments, the sphere is regarded as a plane; however, unlike a plane, the spherical surface will intensify the effect of the uneven distribution of nanobubbles and will lead to a different slip length at different locations. Therefore, an average value of slip length obtained at three positions at the location or close to the location of measured nanobubbles is used as the slip length value. The ratios of the approaching velocity and the hydrodynamic force applied to the colloidal probe V/Fhydro as a function of separation distance D are shown in Figure 3, and the slip length b can be described as the intercept on the axis of separation distance for the linear fitting line of V/Fhydro at a larger separation distance.18 On the PS film with a 15.5% surface coverage of nanobubbles (Figure 2c), we show in Figure 3a the results of slip length measured at three different positions. The linear extrapolation of the hydrodynamic force intersects the D axis at distances of about 130, 78, and 47 nm from the origin, which correspond to three effective slip lengths b of 130, 78, and 47 nm, respectively. The values of the slip length measured at different points on one substrate are unequal. This should be due to the randomness in the distribution density and size of nanobubbles, which can be shown in Figure 2c and the three-dimensional image of the inset in Figure 3a. In this case, the average value of 85 nm is regarded as the slip length. Similarly, the average values of slip length for the surfaces in Figure 2a,b,f are calculated to be 8 nm (the slip length values obtained at three positions are 7, 8, and 9

indicates that the size and coverage of these bubblelike objects imaged in Figure 2 are affected by the gas saturation of water. These nanobubbles formed on four substrates are generally spherically cap-shaped and with height and diameter on the order of ∼150 nm and ∼2 μm, respectively. As shown in Figure 2c, a typical nanobubble with a diameter of about 600 nm and a height of about 55 nm is labeled as b1, and the section analysis and 3D image of b1 are shown in Figure 2d. The section analysis in five different directions of b1 is shown in Figure S2 in the Supporting Information, which shows a spherical capshaped nanobubble. Figure 1e shows the definition of the parameter of nanobubbles: H is the height, r is the three-phase contact line radius, θ is the contact angle, and R is the radius of curvature of the nanobubble. A detailed statistical analysis of the bubble parameters is shown in Table 2. To ensure the Table 2. Parameters of Nanobubbles on PS Films Immersed in Water and the Slip Length Measured on Corresponding Substrates substrate

average diameter (nm)

average height (nm)

coverage rate (%)

average contact angle (deg)

1 2 3 4

78 157 332 451

15.3 21.6 35.6 44.8

1.7 4.8 15.5 50.8

42.8 30.8 24.2 16.6

a

average slip length (nm) 8 21 85 512

± ± ± ±

2.4 3.2 6.3 27.6

The contact angle was obtained through gas side of nanobubbles.

statistical significance of the nanobubbles, the data of all of the parameters were obtained at five different locations (5 μm × 5 μm) in about a 30 μm × 30 μm scanning area, which is close to some marked surface morphology, and the average values are used as the characteristic parameters of nanobubbles in this study. While calculating the contact angle of nanobubbles, the irregularly shaped nanobubbles were not considered in order to ensure accuracy. Given that the pinning force of nanobubbles on a smooth surface is different from that on a rough surface, the pinning force of nanobubbles on a rougher surface should be larger than

Figure 3. Slip length measured by using CM-AFM on PS surfaces with different sizees and surface coverages of nanobubbles. (a) Slip measurements at three different positions on one PS surface with a nanobubble coverage ratio of about 15.5% and V/Fhydro with a high driven velocity (Vd = 65.6 μm/s) as a function of the separation distance; the inset is the 3D image of nanobubbles formed on the corresponding PS film. (b) Typical values of slip length measured on different PS surfaces; V/Fhydro with a high driven velocity (Vd = 65.6 μm/s) as a function of separation distance on PS surfaces with different surface nanobubble coverage ratios of about 1.7, 4.8, 15.5, and 50.8%, respectively. 11290

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Figure 4. Effective slip length b as a function of surface coverage of nanobubbles ϕ, height and contact angle obtained by AFM measurements, and theoretical calculations. (a−c) Relationships between slip length and surface coverage, slip length and height of nanobubbles, and slip length and contact angle, respectively. All of the theoretical results are obtained on the basis fo the parameters of nanobubbles in the experiments.

seen in Figure 4a−c, respectively. This indicates that the surface coverage, height, and contact angle of nanobubbles are key factors for nanobubbles to increase the slippage of liquid flow. However, a phenomenon can be found in Table 2. A larger contact angle is often accompanied by a lower surface coverage and a smaller bubble height. Considering that the three parameters have a coupling effect on slip length, we thus cannot demonstrate here which parameter will contribute to a larger slip length more than others. Surface nanobubbles forming at PS films immersed in water are spontaneous; the size and surface coverage of nanobubbles are usually uncontrollable. We change the size and surface coverage of nanobubbles in this study by changing the gas concentration of water and surface roughness of substrates. Neither of these methods can modify only one parameter while the others remain constant. Significantly, the contact angle of nanobubbles is calculated on the basis of the parameters of nanobubbles, including bubble height and base radius, which means that the size, height, and contact angle of nanobubbles are always coupled in this study. However, it is still the first time that the effect of nanobubbles on boundary slip is decoupled from other factors such as the surface potential, surface materials, and surface structure. Instead of surface nanobubbles, some studies investigated the slippage of liquid over structured surfaces with trapped bubbles, which is thought to be similar to the effect of nanobubbles. For the effect of surface coverage on the slip length, our result is in line with the qualitative behavior reported in the case of liquid flow parallel to bubbles trapped at grooves,25 i.e., the slip length increases with an increase in gas fraction as liquid flow over the trapped bubble mattress. For the effect of bubble morphology on the slip length, the experimental results22 and simulative results2,22,25 obtained on the trapped bubble mattress show an increase in slip length as the protrusion angle changes from 45 to 12°. This is in good agreement with our results. However, a notable feature of trapped bubbles is that the meniscus of trapped bubbles is not always convex (a sketch of a concave bubble and a convex bubble can be seen in Figure 1) and trapped bubbles have a protrusion angle of ∼90°,2,22 which is a much larger scope than that (0° ≈ 45°) of nanobubbles.8 Simulation studies2,22,25 also show that the trapped bubbles will lead to a transition from slippage to friction for the fluids when the protrusion angle is larger than 60°. This is unlike nanobubbles, which will never provide friction for the liquids and will always increase the slip length because of their smaller contact angle. Karatay et al.3 investigated the effect of trapped bubbles on the slip length at a wide range of protrusion angles

nm), 21 nm (the slip length values obtained at three positions are 15, 20, and 28 nm), and 512 nm (the slip length values obtained at three positions are 233, 582, and 721 nm), respectively. Figure 3b shows the typical values of slip length measured on these four surfaces, which are 8, 28, 130, and 582 nm, respectively. It is obvious that the boundary slip was affected by the nanobubbles on the surface. As a comparison, Wang et al.16 measured a slip length of 44 nm, which is about half of our result of 85 nm, on a hydrophobic film with a comparable surface fraction covered by nanobubbles of about 20%. However, the height of nanobubbles in their study is around 6 nm, which is much smaller than that of 21.6 nm in this study. This should be the main reason for the difference in slip length. Furthermore, we found that the hydrophobic surface they used has an RMS surface roughness of 11 nm and a peak−valley distance of 66 nm, which are much larger than 1.87 and 8 nm, respectively, for the PS substrate used in our experiment. (The corresponding height image of the PS substrate in air is shown in Figure S1 in Supporting Information.) For a rough surface, on one hand, the system error derived from AFM measurements of the boundary slip might be large, which may be close to the peak−valley distance. On the other hand, the large surface roughness itself might have a nonignorable influence on the effective slip length. Bonaccurso et al.38 investigated the effect of surface roughness on the boundary condition by using AFM and found that there is an increase in slip length with an increase in roughness. This is in line with a simulated (lattice Boltzmann, LB) study presented by Kunert and Harting.39 Surface roughness is believed to be helpful in the slippage of fluids, and the effect of surface roughness is therefore not the reason for the smaller slip length in the study in ref 16. Kunert and Harting39 also found that the slip length will be less than 10 nm because the average roughness (Ra) is less than 10 nm, and the effect of roughness on the slip length can be neglected for surfaces with a small Ra. The largest roughness of the surfaces we used is 2.16 nm; the effect of surface roughness on slip length in this study is therefore not considered. By cross-checking the parameters of surface nanobubbles formed on corresponding substrates in Table 2, the slip length increases from several nanometers to more than 0.5 μm with an increase in the surface coverage of nanobubbles from 1.7 to 50.8%, an increase in bubble height from 15.3 to 44.8 nm, and a decrease in the contact angle (gas side) from 42.8 to 16.6°. The relationships of slip length and surface coverage, slip length and bubble height, and slip length and contact angle can also be 11291

DOI: 10.1021/acs.langmuir.6b02877 Langmuir 2016, 32, 11287−11294

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Langmuir ⎛η ⎞ b = ⎜⎜ l − 1⎟⎟Hbϕ ⎝ ηg ⎠

by using microparticle image velocimetry (μPIV). The slip length they obtained for a surface coverage of ϕ = 54% is larger than that for a surface coverage of ϕ = 38%. At the same time, a decrease in slip length is found as the protrusion angle increased from 12 to 45°. These are in good agreement with the results in this study. However, our result (512 nm, surface coverage ϕ = 50.8%, contact angle θ = 16.6°) is 9 times smaller than their maximum slip length of 4.8 ± 0.1 μm, which was obtained on a trapped bubble mattress with a comparable surface coverage of ϕ = 54% and a protrusion angle of θ = 12 ± 2°. Furthermore, Maali et al.40 measured a slip length of 356 nm on a structured surface with a surface coverage of trapped bubbles of ϕ = 63% by using CM-AFM. This result is comparable to the slip length of 512 nm in this study. Nevertheless, the meniscus of the trapped bubbles they captured, unlike that of nanobubbles, is not convex and spherically cap-shaped but concave and irregular. (A sketch of a concave bubble and a convex bubble can be seen in Figure 1.) On the basis of the above analysis, although nanobubbles are similar to trapped bubbles, the values of slip length obtained on the two kinds of bubble mattresses might be different. Except for the different experimental techniques and experimental operations, the difference between nanobubbles and trapped bubbles should be an important factor in leading to the difference in slip length values. First, the inner pressure of nanobubbles is different from that of trapped bubbles. The inner pressure of nanobubbles is much higher than that of macrobubbles.7,28 For example, for a nanobubble with a threephase contact line radius of 300 nm and a height of 28 nm, its inner pressure is about 2 atm.7 The trapped bubbles, however, might be thought of as having an inner pressure that is the same as that of macrobubbles because of their extra volume; i.e., the manufactured cavity under the protruding meniscus of the captured bubbles neutralizes the Laplace pressure, which is caused by the capillary meniscus. As a consequence, the trapped bubbles should be softer than similarly sized nanobubbles, which is in favor of the slippage of liquids and thus produces a larger slip length. Another difference between the two kinds of bubbles is their distribution and size, which are random for nanobubbles but uniform for trapped bubbles. The randomness for the distribution and size of nanobubbles should also be favorable to an increase in slip length. A recent simulation study has demonstrated that a maximum slippage can be obtained by varying the position and size of trapped bubbles on patterned surfaces.27 Furthermore, the contact angle of nanobubbles is usually in the range of ∼45°. However, the nominal protrusion angle for the meniscus of trapped bubbles can change from concave to convex over a large range. In view of the effect of the contact angle of nanobubbles on the slip length analyzed above, the difference in the meniscus of the two kinds of bubbles will also lead to different slip lengths. Therefore, our results show that although nanobubbles and trapped bubbles are two kinds of similar air bubbles, they have different effects on the slip length of liquid flow.

(3)

where ηg = 18.6 μPa s and ηl = 851.5 μPa s at a temperature of 300 K are the density of gas and the viscosity of water, respectively.16 In this model, the effect of the morphology of nanobubbles is neglected, which is, however, proven to be very important in the theoretical and experimental studies of trapped bubbles.2,3,22,25,27 To improve this model, supposing the nanobubbles are spherically cap-shaped, parameter Hb in eq 3 can be replaced by a bubble morphology function κ, considering both the height and the contact angle of nanobubbles, given as κ=

Vb (π /3)(3R − h)h2 = Sb πr 2

(4)

Here, Vb and Sb are the volume and projected area of nanobubbles. R, r, and h are the radius of curvature, three-phase contact line radius, and height of nanobubbles, respectively. r 2 + h2 2h h 2 θ . tan 6 2

With R =

κ=

h 2

+

and r =

h , tan θ /2

κ can be written as

Therefore, eq 3 can be rewritten as

⎛η ⎞⎛ h h θ⎞ l ⎜ b = ⎜ − 1⎟⎟⎜ + tan 2 ⎟ϕ ⎝ 2 6 2⎠ η ⎝ g ⎠

(5)

On the basis of this model, the effective slip length b is evaluated by using the bubble parameters obtained in the experiments, with calculated values of 7, 26, 136, and 561 nm, which are close to the experimental values of 8, 21, 85, and 512 nm, respectively. Likewise, the calculated values of slip length using eq 3, without considering the effect of bubble morphology, are 13 nm, 57 nm, 270 nm, and 1.1 μm, which are much larger and more than twice the experimental values. This indicates that the morphology of nanobubbles in the experiments may reduce the slippage of fluids, and the effect of bubble morphology on slip length cannot be neglected. In addition, a two-dimensional model based on trapped bubbles derived by Sbragaglia and Prosperetti15 is used to examine the experimental results, where the effective slip length b is described as a function of bubble radius r and bubble coverage ϕ, given as b = (8/9π )ϕr

(6)

With the experimental data of nanobubbles, the values of slip length are 0.4, 2.1, 17.2, and 64.8 nm, respectively, based on eq 6; these are much smaller than our measurements of the boundary slip, probably because the bubble height was omitted. The relationship between the slip length and bubble height is shown in Figure 4c; both the experimental and theoretical results show an increase in slip length with an increase in bubble height. This, together with the experimental results of Wang et al.16 discussed above, indicates that the height of surface nanobubbles is also an important element in the slippage of liquids.

4. COMPARISON WITH THEORY There are a few theoretical models that describe the relationship between nanobubbles and the boundary slip. Wang et al.16 considered the nanobubbles effectively equal to a continuous gas layer at a solid−liquid interface and obtained a model based on the model provided by Vinogradova.10 The effective boundary slip can be expressed by the average bubble height Hb and the surface coverage of nanobubbles ϕ as

5. CONCLUSIONS Our experiments quantitatively demonstrate the effect of surface nanobubbles on slip length. The slip length changes from several nanometers to about 1 μm with the change in size and surface coverage of nanobubbles, which can be controlled 11292

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by changing the surface roughness of PS substrates and the gas concentration of water. The result provides the first experimental evidence of the fact that nanobubbles forming at smooth, hydrophobic surfaces can increase the slip length effectively. The slip length increased from 8 to 512 nm with an increase in the surface coverage of nanobubbles from 1.7 to 50.8% and a decrease in the contact angle of nanobubbles from 42.8 to 16.6°. A simple model considering the morphology of nanobubbles is presented, on the basis of which the calculated values of slip length are in good agreement with our experimental results. At the same time, the numerical and theoretical results of slip length obtained on trapped bubbles are compared and discussed. Our results indicate that the slip length can be affected by the surface coverage, height, and contact angle of nanobubbles, and unlike the trapped bubbles, surface nanobubbles will be always promote the slippage of liquid flow. Furthermore, our results suggest agreement with the result in the study of Hyväluoma et al.;27 i.e., maximum slippage can be obtained on designed surfaces by varying the position of the bubbles as well as by changing the bubble size. Our results also can help us to analyze the gas state of nanobubbles and contribute to the understanding of the open questions of nanobubbles such as the abnormally long lifetime and the small contact angle.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b02877. AFM image of the height of a PS-coated silicon wafer, section analysis of a bubble, and changes in the surface bubbles during degassing experiments and time incubation experiments (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge the financial support of the National Natural Science Foundation of China (nos. 51475118 and 51505108) and the Heilongjiang Province Natural Science Foundation of China (no. E2016059).

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REFERENCES

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DOI: 10.1021/acs.langmuir.6b02877 Langmuir 2016, 32, 11287−11294

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DOI: 10.1021/acs.langmuir.6b02877 Langmuir 2016, 32, 11287−11294