Mobile or Immobile? Rise Velocity of Air Bubbles in High Purity Water

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Mobile or Immobile? Rise Velocity of Air Bubbles in High Purity Water Piotr Pawliszak, Vamseekrishna Ulaganathan, Bronwyn H BradshawHajek, Rogerio Manica, David A Beattie, and Marta Krasowska J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 23 May 2019 Downloaded from http://pubs.acs.org on May 24, 2019

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The Journal of Physical Chemistry

Mobile or Immobile? Rise Velocity of Air Bubbles in High Purity Water

Piotr Pawliszak,1,2 Vamseekrishna Ulaganathan,1 Bronwyn H. Bradshaw-Hajek, 2 Rogerio Manica,3 David A. Beattie,1,2* Marta Krasowska1,2*

1 2

Future Industries Institute, University of South Australia, Mawson Lakes SA 5095, Australia School of Information Technology and Mathematical Sciences, University of South Australia, Mawson Lakes SA 5095, Australia

3 Department

of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta T6G 1H9, Canada

* Corresponding Authors: M.K. – [email protected]

1 ACS Paragon Plus Environment

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D.A.B. –[email protected]

ABSTRACT: The velocity of bubbles rising in pure liquids is affected by the boundary conditions at the liquid-air interface, with bubbles rising the fastest when the bubble surface is fully mobile. The presence of even very small amounts of surface active molecules causes tangential immobility at the liquid-air interface and subsequently results in slower bubble rise velocity. The existing literature on rise velocities of air bubbles in high purity water does not provide a conclusive picture on whether or not the water-air interface is immobile, with the most discrepancies reported for very small bubbles. This paper presents the first systematic study of bubble rise velocities in high purity water for a sufficiently wide range of bubble sizes (bubble diameter between 48 μm and 1.5 mm), and for the same experimental conditions, which will allow firm conclusions to be drawn on this issue. For bubbles of diameter 800 μm or larger the measured rise velocity is in a good agreement with the theoretical predictions for completely mobile water-air interface, but the velocity starts to deviate from that of mobile bubbles as the size becomes smaller. The smaller the bubble, the closer the rise velocity corresponds to immobile water-air interface, and for Re < 1 the bubble rise velocity agrees with the Stokes law. We use the rear stagnant cap model to explain why smaller bubbles are significantly more sensitive to any surface active impurities than larger ones.


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INTRODUCTION The phenomenon of a single bubble rising in water has intrigued scientists for decades. This is sometimes due to the interesting trajectories observed during the rise itself, but primarily due to its significance in natural (e.g. in oceans where aerosols are produced by rising bubbles, influencing the global climate1-3) and in industrial (including mineral flotation4, bioprocessing5 and food beverage manufacturing6) processes. Any object, rigid or soft, moving in a liquid experiences the hydrodynamic drag force, FD, which depends on the liquid viscosity, μ and also the velocity of the moving object, U, the object size (when spherical, defined by D – diameter), object shape/deformability, and the tangential mobility, i.e. boundary conditions, of the object interface. For an air bubble, the buoyancy force, FB, acting on it is given by 𝐹𝐵 =

𝜋𝐷3(𝜌 ― 𝜌𝑎)𝑔

(1)

6

where 𝜌 and 𝜌𝑎 are the densities of liquid and air, respectively, and g is the gravitational acceleration. The accelerating bubble reaches a steady state velocity once FB reaches equilibrium with FD (FB + FD = 0). For the most simplistic case, and for Reynolds number (𝑅𝑒 =

𝐷𝜌𝑈𝑡 𝜇

) ≪ 1, Stokes law predicts FD

to be7 𝐹𝐷_𝑆𝑡 = ―3𝜋𝐷𝜇𝑈𝑡

(2)

for objects moving at their terminal velocity, Ut. Equation (2) can be used for particles with a rigid/tangentially immobile interface. FD for a particle with a tangentially mobile interface (e.g. bubble in liquid devoid of any surface active impurities), for 𝑅𝑒 ≪ 1, was given by Hadamard and Rybczynski8-9. 𝐹𝐷_𝐻𝑅 = ―2𝜋𝐷𝜇𝑈𝑡

(3)



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By comparing equations (2) and (3) one can see that the drag force for smaller particles with a tangentially mobile interface is two thirds that of particles with an immobile interface. This results in a terminal velocity, that is 1.5 times larger. To satisfy 𝑅𝑒 < 1 for air bubbles rising in water, the bubble diameter must be smaller than 120 μm. This is experimentally challenging, as for most typical methods of bubble generation by a capillary/needle, the capillary inner diameter, DC, would need to be (as given by Tate’s law: 𝐷3 = 6𝐷𝑐𝛾/𝑔(𝜌 ― 𝜌𝑎), with γ being the water-air interfacial tension) as small as 0.039 μm to generate a 120 μm bubble. Such small inner diameters are difficult to fabricate and, in addition, a high pressure impulse is required to produce a single bubble using such a capillary. Due to these difficulties only very few studies have been carried out for this bubble size range 10-13. Parkinson et al. have produced such small bubbles by inducing detachment from a capillary with a mechanical pulse. They have shown that the velocity of bubbles rising in Milli-Q water obeyed the Hadamard-Rybczynski model10. Detsch et al. generated a bubble swarm and isolated a single bubble (with a diameter ranging from 20 μm to 1000 μm) rising in type I reagent grade pure water (equivalent to Milli-Q water). In contrast to Parkinson et al. the bubble rise velocity for the same bubble size range was in an agreement with the Stokes law11. Most experimental studies of bubble rise in pure water have focused on high Re (100 and higher). Unlike for very small bubbles (Re < 1), the experiments for larger bubbles show a good agreement for measured terminal velocities of bubbles rising in high purity water. In addition, for all the studies14-19 the bubble rise velocity agreed well with the model proposed by Manica et al.20 for fully mobile water-air interface. Recently, Liu et al.21 used high speed interferometry to study collisions between two bubbles in high purity water. The thin film thinned faster (indicating fully mobile water-air interface) for bubbles approaching at the highest speeds, while the thin film drainage was slowed down (indicating


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immobile water-air interface) for bubbles of the same size and in the same high purity water but approaching at lower speeds. Liu et al.21 pointed out that that the interfacial velocity can be determined by the balance of Marangoni stress and shear stress and hence the immobilization of water-air interface due to adsorption of traces of surface active impurities is less likely to happen when there is continuous flow sweeping surface active molecules off the water-air interface, i.e. at higher approach speeds. The same logic should apply for boundary conditions at the water-air interface for rising bubbles. However, there is no systematic study on bubble rise velocity for a wide range of bubble sizes (tens of microns to a few millimeters) in high purity water to confirm this. Our paper addresses this gap: we study the rise velocity of air bubbles of diameters between 48 μm and 1.5 mm in high purity water during the same experiment. In addition, the effect of residence time on bubble rise velocity is elucidated by looking at the change in rise velocity over a distance. Finally, we use the rear stagnant cap model to estimate the amount of surface active impurities sufficient to fully immobilize the water-air interface for bubbles of diameters < 120 μm (i.e. Re < 1).

MATERIALS AND METHODS Materials. The Milli-Q water used for experiments was provided by an Advantage system A10 (Millipore, USA). The Milli-Q water had a resistivity of 18.2 MΩ∙cm, interfacial tension of 72.4 mN∙m-1 at 22 °C, and less than 4 ppm of total organic carbon. For most experiments we have used Milli-Q water from a system with cartridges/UV lamps that were 3 months old. However, we have also used Milli-Q water from a system which had cartridges and UV lamps newly replaced, as well as being thoroughly cleaned 24 h before the experiment (resistivity of 18.2 MΩ∙cm, interfacial tension of 72.4 mN∙m-1 at 22 °C, less than 4 ppm of total organic carbon). We also distilled this Milli-Q water 4 times for the experiments with the smallest bubbles (interfacial tension of 72.4 mN∙m-1 at 22 °C).



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Cleanliness of glassware (as well as all other parts of the experimental set-up) is crucial, as any trace of surface active impurities will affect the bubble rise velocity. All the glassware and glass elements (column and cuvette) were soaked in 2% Extran® (Merck Millipore, USA) solution in demineralized water for 40 minutes and rinsed with demineralized water. Then, the components were soaked in 2M KOH aqueous solution for 40 minutes and then rinsed with Milli-Q water, and finally dried at 50,°C in an oven. All glass parts were air plasma cleaned (Harrick, PDC-OD2, USA) for 90 seconds prior to the experiments. The Teflon mounting holder was cleaned by soaking in 2% Extran® solution for 40 minutes followed by rinsing with Milli-Q water and sonicating in 100% undenatured ethanol (ChemSupply Pty Ltd., Australia) for 15 minutes. The holder was then rinsed with Milli-Q water again and dried in a nitrogen stream (99.999% purity, BOC, Australia). Gastight syringes supplied by Hamilton (U.S.A.) were sonicated in 100% undenatured ethanol for 15 minutes, followed by rising and sonicating in Milli-Q water for 15 minutes, and then dried in a nitrogen stream followed by exposure to air plasma for 90 seconds. The microfluidic chip, used for single bubble generation, was prepared for experiments by pumping 2% Mucasol™ (Sigma-Aldrich, Australia) solution in Milli-Q water for 30 minutes followed by flushing with Milli-Q water, pumping 2M KOH aqueous solution for a further 30 minutes, and rinsing with Milli-Q water until neutral pH. All solutions were pumped through the microfluidic chip at a constant flow rate of 1 ml∙h-1. The microfluidic chip was dried by injecting a nitrogen gas stream through the channels, and finally it was exposed to air plasma for 90 seconds. All the glassware used for storing Milli-Q and distilled water, as well as the glassware used for distillation, was soaked in 2% Extran® solution for 2 hours, rinsed with demineralized water, soaked in 2M KOH aqueous solution for 2 hours, and finally rinsed with Milli-Q water until neutral pH. Single Bubble Rise Experiment. The experiments were performed using a set-up similar to that reported previously22-24. The borosilicate glass column of a square 40 mm × 40 mm cross-section


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and a height of 15 cm was mounted onto a Teflon column holder hosting a microfluidic chip at the bottom. The column was completely filled with Milli-Q water. Unless stated otherwise the images of a rising air bubble were recorded at 12 cm above the microfluidic chip orifice and 3 cm from the waterair interface. All videos were recorded using a stereo-microscope (SZ-1145TR, Olympus, Japan) combined with a high-speed camera (SA4 Fastcam, Photron, USA) with a frame rate of 2000 fps and a resolution of 1024×768 pixels. Single and isolated air bubbles were created at a T-junction of a microfluidic chip. To allow for generation of air bubbles of different sizes, four microfluidic devices of different channel dimensions were used. Also, the flow rates of water and air were varied and fall within the range of 100 – 1000 μl∙h-1 for water and 20 – 200 μl∙h-1 for air. To ensure each of the generated air bubbles was isolated from the effect of previous or following bubbles, a new bubble was generated every 30 seconds. To minimise the effect of the temperature fluctuations on water viscosity and interfacial tension all experiments were carried out at 22 ± 1 °C for 30 minutes. The terminal bubble rise velocity was determined from the difference in air bubble position as a function of time. Recorded videos were converted to image sequences and analyzed using a custom MATLAB script. The bubble diameter and position as a function of time was determined for every bubble (for more details please see Supplementary Information). Interfacial tension of both types of Milli-Q water and distilled water was measured using the sessile drop technique (OCA 20, Data Physics, Germany) at 22 ± 1 °C.

RESULTS AND DISCUSSION Velocity Comparison for Bubbles Generated by a Microfluidic Chip and a Capillary. Since the channels of the microfluidic device have a large surface area and are made of Pyrex glass, i.e. a material of high surface energy, they are prone to contamination due to adsorption of surface



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active impurities. Our first target for the bubble rise experiment was to compare whether or not the terminal rise velocity of air bubbles generated by a microfluidic chip were the same as the terminal rise velocities for the air bubbles generated with a glass capillary. The terminal velocity of air bubbles of a diameter ranging from 740 to 1500 µm generated by a microfluidic chip rising in Milli-Q water, measured 12 cm above the microfluidic chip orifice, is presented in Figure 1 (black open circles). The terminal velocity is the lowest (166 mm∙s-1) for the smallest bubble, and increases with the bubble size, reaching 353 mm∙s-1 for the largest air bubbles. The data are in very good agreement with the data obtained by Zawala and Niecikowska for air bubbles generated in distilled water with a glass capillary18 (red open triangles in Figure 1). Moreover, the theoretical model that is valid for bubbles of a diameters between 400 – 2000 μm for a contamination-free system, based on the equations described by Manica et al.20, 25 shows very good agreement with our experimental data (compare black solid line and open black circles in Figure 1). Such an agreement with the literature data for both the experiment and theory confirms that the presence of any potential surface active impurities in microfluidic device, as well as the effect of the confinement on the bubble formation/initial stages of bubble life, on the air bubble terminal velocity can be excluded.


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Figure 1. Terminal velocity of air bubbles of different diameters rising in Milli-Q water. ○ – experimental results for bubbles generated with a microfluidic chip; △ – experimental results of Zawala and Niecikowska for bubbles generated with a glass capillary in distilled water18; solid black line – a theoretical model by Manica et al20

The biggest advantage of using a microfluidic chip for bubble generation is the ability to generate a wide range of bubble sizes, including very small ones (down to a few tens of m), in a wellcontrolled and repeatable manner during the same experiment. However, for such small bubbles, it is important to exclude any wall effects (i.e. increased hydrodynamic resistance due to the presence of an interface)26-28. The bubble local velocity as a function of bubble distance from the water-air interface for bubbles of different diameters is presented in Figure 2. As can be seen, even for the smallest bubble



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(D = 215 μm; green closed squares in Figure 2) the rise velocity is constant when the bubble is away from water-air interface (at distances > 1 bubble diameter), but it starts to decrease rapidly when the bubble approaches the interface (e.g. ~ 1 bubble diameter or less). The bubble velocity decreases to half at a distance ~ 0.2 bubble diameter from the water-air interface. For larger bubbles (D = 335 μm; red closed triangles in Figure 2), the velocity starts to deviate from its terminal value at a distance ~ 0.5-0.75 bubble diameter from the water-air interface, and it drops to half at a distance ~ 0.13 bubble diameter from the water-air interface. For the largest bubbles (D = 415 μm; black closed circles in Figure 2), the bubble velocity starts to decrease at the closest distance from the water-air interface (~ 0.5-0.65 bubble diameter away) and the bubble local velocity decreases to half at the smallest distance (~ 0.05 diameter away from water-air interface). This indicates clearly that the measurements of bubble terminal velocity 3 cm from water-air interface are not affected by wall effects, which is in a good agreement with our previous studies29.


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Figure 2. Local velocity of air bubbles of different diameter in Milli-Q water as a function of the distance from water-air interface normalized by the bubble diameter, xi/D. The average local velocity for each of the bubble sizes calculated from 20 independent measurements.

Figure 3A presents the terminal velocity (measured 12 cm above the microfluidic chip orifice and 3 cm below water-air interface) of air bubbles rising in Milli-Q water as a function of bubble diameter. Each data point in Figure 3A is the average of at least 20 independent measurements with a relative standard deviation, RDS, for the terminal velocity lower than 1.0%, and RDS for the bubble size lower than 0.5%.



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Figure 3. Panel A: terminal velocity of air bubbles of different diameter in Milli-Q water. ● – experimental results (each point is the average of at least 20 independent measurements), dashed red line – a theoretical model by Schiller and Naumann30, dotted black line – a theoretical model by Mei et al31; Panel B: terminal velocity of 215 μm bubble rising in different types of high purity water (constant value of interfacial tension – 72.4 mN∙m-1 – during 30 minutes of experiment at 22 °C).

The terminal velocity increases with increasing bubble size, from 18.3 ± 0.3 mm∙s-1 up to 80.7 ± 0.5 mm∙s-1. The terminal velocity is attained when there is a balance between the drag force (for Re 𝐹𝐷_𝑀 =

𝜋𝜌𝐷2𝑈2𝑡 8

𝐶𝑑

(4)

> 50)32 and the buoyancy force (equation (1)), as

𝑈𝑡 =

4𝐷2𝜌 𝑔 3𝐶𝑑𝑅𝑒𝜇

(5)

where Cd is the drag coefficient, and we have neglected the density of air.


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For Re between 0.1 and 200, a model proposed by Schiller and Naumann30 for a bubble with an immobile surface rising in water gives the Cd as (6) 𝐶𝑑𝑅𝑒 = 24(1 + 0.15𝑅𝑒0.687) The terminal velocity predicted from this model is plotted as the dashed red line in Figures 3A and B. In contrast, for intermediate Re (1 < Re < 50), a theoretical model proposed by Mei et al.31 for a bubble with a fully mobile water-air interface rising in water gives Cd as

{ [ ( )] } 8

1 3.315 𝐶𝑑𝑅𝑒 = 16 1 + + 1+ 1 𝑅𝑒 2 𝑅𝑒2

―1

(7)

The terminal velocity predicted by Mei et al. is plotted as dotted black line in Figure 3A. Unlike for larger bubbles, the terminal velocities measured for bubble diameter from 215 μm to 505 μm deviate from the predictions given by Mei et al. for a fully mobile water-air interface. In addition, as the bubble diameter decreases, the measured terminal velocities get closer to the Schiller-Naumann model, suggesting the mobility of the water-air interface is decreasing with the decrease of the bubble size. The terminal velocity measured for the smallest bubble (D = 215 μm) indicates that the surface of the bubble is almost completely immobile. Figure 3B shows the terminal velocity of these smallest bubbles in three different types of high purity water: (i) Milli-Q water from the system with 3 months old cartridges/UV lamps – black closed circles, (ii) Milli-Q water from the system which had cartridges and UV lamps replaced, as well as when thoroughly cleaned, just 24 h before the experiment – red closed triangles, and (iii) 4 times distilled Milli-Q water – grey closed diamonds. For all three types of high purity water the measured interfacial tension was 72.4 mN∙m- 1 and did not decrease during 30 minutes of the experiment at 22 °C. Despite the different origin of high purity water used in this experiment, the terminal velocity of 215 μm bubbles was, within the experimental error, the same (18.3 13 ACS Paragon Plus Environment


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mm∙s-1) and far from the values predicted by Mei et al. (26.1 mm∙s-1) for a fully mobile water-air interface. To quantify the degree of deviation from full mobility at the water-air interface we introduce a mobility percentage parameter, Mp, which is defined as 𝑈𝑒𝑥𝑝 ― 𝑈𝑆𝑁

𝑀𝑝 = 𝑈𝑀𝑒𝑖 ― 𝑈𝑆𝑁 × 100

(8)

where Uexp is the measured terminal velocity, USN is the terminal velocity accounting for an immobile water-air interface given by Schiller and Naumann model, and UMei is the terminal velocity accounting for full mobility at the water-air interface given by Mei et al. model. The calculated values of Mp for all bubble sizes are listed in Table 1. None of these values reaches 100%, indicating that full mobility at water-air interface has not occurred.

Table 1. Diameters, Terminal Velocities, Reynolds Numbers, and Mobility Percentage Values for the Air Bubble Rising in Milli-Q Water D (μm) Ut (mm∙s-1) Re Mp (%) 215 ± 2 18.3 ± 0.3 3.72 4.62 250 ± 2 23.8 ± 0.4 5.63 17.23 335 ± 2 40.5 ± 0.6 12.81 42.11 415 ± 2 65.0 ± 0.9 25.29 80.73 430 ± 2 66.1 ± 0.6 26.88 73.65 490 ± 2 78.2 ± 0.3 36.21 68.13 505 ± 2 80.7 ± 0.5 38.53 65.52 The Mp is the highest (~ 65 - 81%) for the largest bubbles (D ~ 400 - 500 μm), and it is decreasing with the decrease in bubble diameter, reaching the smallest value of 4.62% for 215 μm bubbles. The terminal velocities measured here are still higher than these reported by Okazaki33 for bubbles rising in distilled water. In their study the measured velocity for bubbles of D ~ 315 μm was ~ 30 mm∙s-1 (USN for bubbles of such a size is 30.4 mm∙s-1), while for bubbles of D ~ 440 μm the terminal velocity was ~ 52 mm∙s-1 (USN = 46.5 mm∙s-1), about ~ 14 mm∙s-1 lower than the terminal velocity measured by us for slightly smaller (D = 430 μm) bubbles. 14 ACS Paragon Plus Environment

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For larger air bubbles (D > 600 μm) rising in high purity water, it has been determined that after the initial acceleration stage the bubble reaches its terminal velocity at approximately 40 mm from the point of bubble formation14, 16-17, 34. For smaller bubbles this distance is smaller10. Figure 4 presents local velocities of bubbles of three different diameters as a function of the distance from the point of bubble release. For the largest bubbles (D = 415 μm, see panel A of Figure 4), the velocity is, within experimental error, the same for all the distances. However, for smaller bubbles (D = 335 μm and D = 215 μm, see Figure 4 panels B and C, respectively) there is a very different trend, with higher local velocities measured at a shorter distance from the point of bubble release. Due to the design of our experimental set-up we are unable to measure local velocities closer to the point of bubble formation, hence we can only speculate that after the initial acceleration stage there will be a region in which the air bubble starts to decelerate. Such velocity profiles (with a velocity maximum proceeding the decrease in local velocity) are characteristic for bubble rise in solutions of surfactants/surface active impurities, which would preferentially adsorb at a water-air interface16, 35-36.



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Figure 4. Local velocity of air bubbles of different diameter in Milli-Q water as a function of the distance from the point of bubble release. Panel A: D = 415 μm; Panel B: D = 335 μm; Panel C: D = 215 μm. The average local velocity for each of the bubble sizes is calculated from 20 independent measurements. Please note – only approximate timescale can be given as we do not see first 50 mm, hence the bubble velocity for distances smaller than 50 mm can be only approximated. Taking into account that for very low concentrations of such impurities (note that the interfacial measurement performed using the emerging bubble technique did not indicate any decrease in waterair interfacial tension even for measuring times significantly longer (30 minutes) than the time (a few 16 ACS Paragon Plus Environment

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seconds) of bubble residing in the column during the velocity measurement experiments), we should observe a maximum local velocity close to the terminal velocity for a fully mobile water-air interface (UMei = 26 mm∙s-1 for 215 μm bubble and UMei = 50 mm∙s-1 for 335 μm bubble). Since the highest local velocities in Figure 4 are quite far (19.7 mm∙s-1 for 215 μm bubble and 42.0 mm∙s-1 for 335 μm bubble) from values predicted by Mei et al. we suspect the maximum occurs at shorter distances, at which we cannot perform the measurements. The mobility percentage, calculated from equation (8), as a function of the distance from the point of bubble release in Milli-Q water for bubble diameters of 215 μm, 335 μm, and 415 μm is presented in Figure 5 (green closed squares, red closed triangles, and black closed circles, respectively). Whilst for the largest (D = 415 μm) bubbles the mobility percentage is not dependent on the distance from point of bubble release (at least not within the distance range in this study), it is not 100%, indicating there is only partial mobility at the water-air interface. For smaller bubbles the Mp is the largest at smaller distances and its value decreases for larger distances from the point of bubble release. Mp decreases with the bubble size, and it reaches 42.3% and 3.38% for 335 μm, and 415 μm bubbles, respectively.



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Figure 5. Mobility percentage, Mp, as a function of the distance from the point of bubble release in Milli-Q water for air bubbles of different diameter. The experimental results of the few existing studies dedicated to the rise velocity of air bubbles in high purity water for Re < 1 contradict each other. Parkinson et al.10 have shown that the velocity of bubbles rising in Milli-Q water obeyed Hadamard-Rybczynski model, i.e. they had a full mobility at the water-air interface. In contrast, in studies by Detsch et al11 and Takahashi37, the bubble rise velocity measurements were in an agreement with Stokes law, indicating a fully immobile water-air interface. Parkinson et al. used Milli-Q water of interfacial tension of 72.7 mN∙m-1 at 22.1 °C, while Detsch et al.used type I reagent grade pure water (equivalent of Milli-Q water). Both groups were rigorous with glassware cleaning (Parkinson et al. used warm 2 M KOH solution, Detsch et al.used chromic acid). The major difference between these studies was the point at which the bubble velocity was measured. Parkinson et al. measured the velocity 6 mm from the capillary orifice, i.e. the equivalent of 0.5 – 3 s bubble residence time in Milli-Q water, while Detsch et al.needed at least 10 – 20 s to record one 18 ACS Paragon Plus Environment

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bubble. It is likely that the small air bubbles close to the point of their formation, i.e. 6 mm, still had a contamination-free surface hence their rise velocity was in agreement with Hadamard-Rybczynski model. For longer residence times (i.e. in the case of rise velocity measured further from the point of the bubble formation) the bubble accumulated enough surface active impurities to change the boundary conditions at water-air interface from mobile to immobile, resulting in lower rise velocities in agreement with Stokes law. The effect of residence time/point at which the bubble velocity is measured is manifested in two later studies by Parkinson and Ralston12 and Manica et al.13 where the rise velocity of air bubbles Re < 1 could be only described by Stokes law (indicating immobile water-air interface) when the bubble was approximately 25 mm above the point of bubble formation. Also Detsch et al, who measured velocities of bubbles residing in high purity water for significantly longer times, observed a good agreement with Stokes law. In this study the effect of the residence time on bubble rise velocity is elucidated by looking at the change in velocity over the distance (see Figure 4). Figure 6 presents the rising velocity of very small bubbles (diameter of 48 – 180 μm) in Milli-Q water. The velocity was measured at 12 cm above the point of bubble release, i.e. above the point at which such small bubbles achieved their terminal velocity. The experimental data are in good agreement with Stokes law, i.e. immobile water-air interface, and with studies by Detsch et al11 and by Takahashi37. This clearly demonstrates that smaller bubbles are significantly more sensitive to any surface active impurities, even in the same system in which larger bubbles were not affected (as indicated by their rise velocity agreeing with that for fully mobile water-air interface). There are several reasons for this: (i) smaller bubbles have a lower rise velocity than larger ones, hence they reside longer in the solution, increasing the chance for adsorption of the surface active impurities at water-air interface, (ii) they have greater surface to volume ratio, and even minute



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amounts of impurities adsorbed at the water-air interface will have a significant effect on the hydrodynamic drag force for a bubble that has an opposing buoyancy force that is low. Such a force balance will result in a significant drop in the bubble rise velocity. In the case of bigger bubbles, which have a lower surface to volume ratio, a higher concentration of adsorbed surface active molecules is required for the hydrodynamic drag force to counteract the much higher buoyancy force.

Figure 6. Terminal velocity of air bubbles of different diameter in Milli-Q water. ● – experimental results, dashed red line – a theoretical model by Stokes7 for Re < 1, solid black line – a theoretical model of Hadamard-Rybczynski8-9 for Re < 1. For a bubble rising in a surfactant solution there is an adsorption flux on to the top pole of the bubble. The hydrodynamic flow around the bubble induces a tangential shear stress, pushing the surfactant molecules towards the rear pole of the bubble. Such non-uniform surfactant distribution creates an interfacial tension gradient opposing the tangential hydrodynamic viscous stress and causes 20 ACS Paragon Plus Environment

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immobilization of the bubble surface. The degree of mobility depends on the amount of the surfactant molecules adsorbed at the bubble surface38. For bubbles of 𝑅𝑒 < 1 and for slow adsorption kinetics Zholkovskij et al.39 elaborated the rear stagnant cap, RSC, model. The adsorbed amount of surfactant per bubble, M, can be approximated from the shear stresses involved during the bubble rise using 𝑀(𝜓) =

𝜋𝐷4𝜌𝑔 2𝜓 ― 4𝜓cos (𝜓) ― sin (2𝜓) + 4sin (𝜓)

(9)

24𝑅𝑇 2𝜓 + sin (𝜓) ― sin (2𝜓) ― 1sin (3𝜓) + 4𝜋 3

where R is the gas constant and T is the temperature. The M is a function of the angle, 𝜓, that encompasses the RSC (see insert in Figure 7). For a bubble with no surfactant, i.e. fully mobile waterair interface, 𝜓 = 0 , while for bubble surface completely covered with surfactant molecules 𝜓 = 180°. Dukhin et al. predicted that for 𝜓 = 1200 the bubble surface starts to behave as completely immobile40. Figure 7 presents the bubble surface coverage, θ, with surfactant molecules necessary to completely immobilize the bubble surface as a function of bubble diameter. The quantity is expressed as a percentage and defined as 𝑀

𝜃 = 𝜋𝐷2Γ × 100

(10)

∞

where 𝛤∞ is the maximum surface concentration) necessary to completely immobilize the bubble surface as a function of bubble diameter. 𝛤∞= 8.42×10-6 mol∙m-2, which is maximum surface concentration for sodium dodecyl sulphate, a commonly used surfactant41, was used to estimate the bubble surface coverage.



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Figure 7. The surface coverage (in %) of adsorbed surface active molecules at the bubble surface as a function of bubble diameter at 𝜓 = 120° for Re < 1. Insert: Schematic of the rear stagnant cap formation at the surface of rising bubble. In Figure 7 the surface coverage necessary for complete immobilization of the bubble surface is plotted as a function of bubble diameter using equation (10). It is clear that the amount of surfactant required to change a fully mobile water-air interface to a completely immobile water-air interface increases with bubble size. However, even for larger bubbles, i.e. D = 100 μm, the surface coverage required for a complete immobilization of bubble surface is very low (0.013%). The adsorbed amount of surfactant per bubble, M, corresponding to such surface coverage is ~ 3.53 × 10-17 mol, and the 𝑀

surface concentration, 𝛤 = 𝜋𝐷2, to ~ 1.1×10-9 mol∙m-2. Analyzing Г vs bulk concentration for sodium dodecyl sulphate in a study by Fainerman et al42 Г~ 1×10-8 mol∙m-2, i.e. value 10 times higher, already corresponds to the surfactant bulk concentration which is too low to be detected by classic tensiometry. The relationship in Figure 7 offers an explanation why the smallest bubbles are the most prone to even minute amount of surface active impurities. We can extrapolate this trend to low (but larger than 1) Re 22 ACS Paragon Plus Environment

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to explain why the smallest bubbles (D = 215 μm) were the closest to having fully immobile water-air interface, while for larger bubbles the water-air interface became more mobile.

CONCLUSION The results presented in this paper demonstrate that bubbles of diameter 800 μm and larger rise in high purity water with a velocity that indicates fully mobile water-air interface, in good agreement with previous studies and with the theoretical predictions of Manica et al.20 However, in the same high purity water, the velocity of smaller bubbles (diameter of 215 – 550 μm) starts to deviate from the one predicted for a fully mobile water-air interface. The smaller the bubble, the closer its velocity gets to that of an immobile water-air interface predicted by the Schiller-Naumann model30. The smallest bubbles (diameter below 180 μm) exhibit tangentially immobile water-air interface, as predicted by the Stokes model for Re < 17. This shift from full mobility to immobile water-air interface is attributed to minute traces of surface active impurities in water. The RSC model estimates the amount of surface active impurities per bubble required to immobilise the water-air interface to be of the order of 10-17 moles per bubble of diameter 100 μm. Such small amounts cannot be detected with classic tensiometry. The RSC model and our data demonstrate that smaller bubbles are significantly more sensitive to any surface active molecules (even for the same system in which the rise velocities of larger bubbles indicated no presence of such impurities) opening opportunities to use rise velocity of small bubbles as a method to detect even the smallest traces of surface active impurities. As a final caveat, it should be noted that in almost any practical situation, impurity free water does not exist.

Acknowledgement



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The authors thank Professors Kazimierz Malysa, Roger Horn, Reinhard Miller and Volodymyr Kovalchuk for helpful and fruitful discussions. The financial support of Joint Research Centre for In-Line Chemical and Mineral Sensing for Sustainable Mineral Processing Australia-China Fund (project JRCAF-55), administered by The Department of Industry, Innovation and Science is acknowledged. The microfluidic chips for bubble generation were fabricated at South Australian node of the Australian National Fabrication Facility, a company established under the National Collaborative Research Infrastructure Strategy to provide nano- and micro-fabrication facilities for Australia’s researchers.

SUPPORTING INFORMATION The details of terminal rise velocity determination and the discussion regarding dissolution of very small air bubbles are given in the Supporting Information file.

REFERENCES 1.

Andreas, E. L.; Edson, J. B.; Monahan, E. C.; Rouault, M. P.; Smith, S. D. The Spray

Contribution to Net Evaporation from the Sea: A Review of Recent Progress. Boundary-Layer Meteorology 1995, 72, 3-52. 2.

Lewis, E. R.; Lewis, E. R.; Lewis, R.; Schwartz, S. E. Sea Salt Aerosol Production:

Mechanisms, Methods, Measurements, and Models; American Geophysical Union, 2004; Vol. 152. 3.

Blanchard, D. C. The Electrification of the Atmosphere by Particles from Bubbles in the Sea.

Progress in Oceanography 1963, 1, 73-202.


Page 25 of 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

4.


Kracht, W.; Finch, J. A. Effect of Frother on Initial Bubble Shape and Velocity. International

Journal of Mineral Processing 2010, 94, 115-120. 5.

Bavarian, F.; Fan, L.; Chalmers, J. J. Microscopic Visualization of Insect Cell-Bubble

Interactions. I: Rising Bubbles, Air-Medium Interface, and the Foam Layer. Biotechnology Progress 1991, 7, 140-150. 6.

Liger-Belair, G.; Marchal, R.; Robillard, B.; Dambrouck, T.; Maujean, A.; Vignes-Adler, M.;

Jeandet, P. On the Velocity of Expanding Spherical Gas Bubbles Rising in Line in Supersaturated Hydroalcoholic Solutions: Application to Bubble Trains in Carbonated Beverages. Langmuir 2000, 16, 1889-1895. 7.

Stokes, G. G., On the Effect of the Internal Friction of Fluids on the Motion of Pendulums; Pitt

Press Cambridge, 1851; Vol. 9. 8.

Hadamard, J. Mouvement Permanent Lent D'une Sphere Liquid Et Visqueuse Dans Un Liquide

Visqueux. CR Hebd. Seances Acad. Sci. Paris 1911, 152, 1735-1738. 9.

Rybczynski, W. Uber Die Fortschreitende Bewegung Einer Flussigen Kugel in Einem Zahen

Medium. Bull. Acad. Sci. Cracovie A 1911, 1, 40-46. 10.

Parkinson, L.; Sedev, R.; Fornasiero, D.; Ralston, J. The Terminal Rise Velocity of 10-100 Mu

M Diameter Bubbles in Water. J. Colloid Interface Sci. 2008, 322, 168-172. 11.

Detsch, R. M. Small Air Bubbles in Reagent Grade Water and Seawater: 1. Rise Velocities of

20‐to 1000‐Μm‐Diameter Bubbles. Journal of Geophysical Research: Oceans 1991, 96, 8901-8906. 12.

Parkinson, L.; Ralston, J. The Interaction between a Very Small Rising Bubble and a

Hydrophilic Titania Surface. J. Phys. Chem. C 2010, 114, 2273-2281.



13.

Page 26 of 30

Manica, R.; Parkinson, L.; Ralston, J.; Chan, D. Y. Interpreting the Dynamic Interaction

between a Very Small Rising Bubble and a Hydrophilic Titania Surface. J. Phys. Chem. C 2009, 114, 1942-1946. 14.

Duineveld, P. The Rise Velocity and Shape of Bubbles in Pure Water at High Reynolds

Number. J. Fluid Mech. 1995, 292, 325-332. 15.

Wu, M.; Gharib, M. Experimental Studies on the Shape and Path of Small Air Bubbles Rising

in Clean Water. Phys. Fluids 2002, 14, L49-L52. 16.

Malysa, K.; Krasowska, M.; Krzan, M. Influence of Surface Active Substances on Bubble

Motion and Collision with Various Interfaces. Adv. Colloid Interface Sci. 2005, 114, 205-225. 17.

Malysa, K.; Zawala, J.; Krzan, M.; Krasowska, M., Bubbles Rising in Solutions; Local and

Terminal Velocities, Shape Variations and Collisions with Free Surface. In Bubble and Drop Interfaces, Miller, R.; Liggieri, L., Eds. CRC Press: 2011; pp 251-300. 18.

Zawala, J.; Niecikowska, A. “Bubble-on-Demand” Generator with Precise Adsorption Time

Control. Review of Scientific Instruments 2017, 88, 095106. 19.

Yang, B.; Prosperetti, A.; Takagi, S. The Transient Rise of a Bubble Subject to Shape or

Volume Changes. Phys. Fluids 2003, 15, 2640-2648. 20.

Manica, R.; Klaseboer, E.; Chan, D. Y. C. The Impact and Bounce of Air Bubbles at a Flat

Fluid Interface. Soft Matter 2016, 12, 3271-3282. 21.

Liu, B.; Manica, R.; Liu, Q.; Klaseboer, E.; Xu, Z.; Xie, G. The Coalescence of Bubbles with

Mobile Interfaces in Water. Phys. Rev. Lett. 2019, 122, 194501. 22.

Krasowska, M.; Kor, M.; Pawliszak, P.; Bernardis, F. L.; Bradshaw-Hajek, B. H.; Beattie, D.

A. Controlling Bubble-Solid Surface Interactions with Environmentally Benign Interfacial Modifiers. J. Phys. Chem. C 2019.


Page 27 of 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

23.


Kor, M.; Korczyk, P. M.; Addai-Mensah, J.; Krasowska, M.; Beattie, D. A.

Carboxymethylcellulose Adsorption on Molybdenite: The Effect of Electrolyte Composition on Adsorption, Bubble–Surface Collisions, and Flotation. Langmuir 2014, 30, 11975-11984. 24.

Wu, J.; Delcheva, I.; Ngothai, Y.; Krasowska, M.; Beattie, D. A. Bubble–Surface Interactions

with Graphite in the Presence of Adsorbed Carboxymethylcellulose. Soft matter 2015, 11, 587-599. 25.

Manica, R.; Klaseboer, E.; Chan, D. Y. The Hydrodynamics of Bubble Rise and Impact with

Solid Surfaces. Adv. Colloid Interface Sci. 2016, 235, 214-232. 26.

Happel, J.; Brenner, H., Low Reynolds Number Hydrodynamics with Special Applications to

Particulate Media; Springer: Dordrecht, 1983. 27.

Brenner, H., Hydrodynamic Resistance of Particles at Small Reynolds Numbers. In Advances

in Chemical Engineering, Elsevier: 1966; Vol. 6, pp 287-438. 28.

Małysa, K. Wet Foams: Formation, Properties and Mechanism of Stability. Adv. Colloid

Interface Sci. 1992, 40, 37-83. 29.

Krasowska, M.; Sellapperumage, P. M.; Ralston, J.; Beattie, D. A. Influence of Dissolved Air

on Bubble Attachment to Highly Oriented Pyrolytic Graphite. Physicochemical Problems of Mineral Processing 2018, 54. 30.

Schiller, L.; Naumann, A. Fundamental Calculations in Gravitational Processing. Zeitschrift

Des Vereines Deutscher Ingenieure 1933, 77, 318-320. 31.

Mei, R.; Klausner, J. F.; Lawrence, C. J. A Note on the History Force on a Spherical Bubble at

Finite Reynolds Number. Phys. Fluids 1994, 6, 418-420. 32.

Vakarelski, I. U.; Manica, R.; Li, E. Q.; Basheva, E. S.; Chan, D. Y. C.; Thoroddsen, S. T.

Coalescence Dynamics of Mobile and Immobile Fluid Interfaces. Langmuir 2018, 34, 2096-2108.



33.

Page 28 of 30

Okazaki, S. The Velocity of Ascending Air Bubbles in Aqueous Solutions of a Surface Active

Substance and the Life of the Bubble on the Same Solution. Bulletin of the Chemical Society of Japan 1964, 37, 144-150. 34.

Sanada, T.; Sugihara, K.; Shirota, M.; Watanabe, M. Motion and Drag of a Single Bubble in

Super-Purified Water. Fluid Dynamics Research 2008, 40, 534. 35.

Krzan, M.; Malysa, K. Profiles of Local Velocities of Bubbles in N-Butanol, N-Hexanol and

N-Nonanol Solutions. Colloids and Surfaces A: Physicochemical and Engineering Aspects 2002, 207, 279-291. 36.

Krzan, M.; Lunkenheimer, K.; Malysa, K. On the Influence of the Surfactant's Polar Group on

the Local and Terminal Velocities of Bubbles. Colloids and Surfaces A: Physicochemical and Engineering Aspects 2004, 250, 431-441. 37.

Takahashi, M. Ζ Potential of Microbubbles in Aqueous Solutions: Electrical Properties of the

Gas− Water Interface. J. Phys. Chem. B 2005, 109, 21858-21864. 38.

Dukhin, S.; Miller, R.; Loglio, G., Physico-Chemical Hydrodynamics of Rising Bubble. In

Studies in Interface Science, Elsevier: 1998; Vol. 6, pp 367-432. 39.

Zholkovskij, E.; Koval'chuk, V.; Dukhin, S.; Miller, R. Dynamics of Rear Stagnant Cap

Formation at Low Reynolds Numbers: 1. Slow Sorption Kinetics. J. Colloid Interface Sci. 2000, 226, 51-59. 40.

Dukhin, S. S.; Lotfi, M.; Kovalchuk, V. I.; Bastani, D.; Miller, R. Dynamics of Rear Stagnant

Cap Formation at the Surface of Rising Bubbles in Surfactant Solutions at Large Reynolds and Marangoni Numbers and for Slow Sorption Kinetics. Colloids and Surfaces A: Physicochemical and Engineering Aspects 2016, 492, 127-137.


Page 29 of 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

41.


Wüstneck, R.; Miller, R. The Adsorption Behavior of Solutions Containing Sodium Dodecyl

Sulfate and Different N-Alkanols. Colloids and Surfaces 1990, 47, 15-21. 42.

Fainerman, V.; Aksenenko, E.; Petkov, J.; Miller, R. Equilibrium Adsorption Layer

Characteristics of Mixed Sodium Dodecyl Sulphate/Triton Solutions. Colloids and Surfaces A: Physicochemical and Engineering Aspects 2011, 385, 139-143.

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Mobile or Immobile? Rise Velocity of Air Bubbles in High Purity Water

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