Coalescence, Growth, and Stability of Surface-Attached Nanobubbles

Subscriber access provided by NEW YORK UNIV

Article

Coalescence, Growth and Stability of Surface Attached Nanobubbles Chon U Chan, Manish Arora, and Claus-Dieter Ohl Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b01599 • Publication Date (Web): 03 Jun 2015 Downloaded from http://pubs.acs.org on June 10, 2015

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 18

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

Langmuir

Coalescence, Growth and Stability of Surface Attached Nanobubbles Chon U Chan,†,‡ Manish Arora,†,‡ and Claus-Dieter Ohl∗,† Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371 Singapore. E-mail: [email protected]

Abstract Surface attached nanobubbles once formed and kept under constant conditions show remarkable stability against dissolution. When observing a large population of nanobubbles using a total internal reflection (TIRF) microscopy we find rare events of coalescence, i.e. the merging of two neighboring bubbles. The new bubble covers the convex hull of their “footprint”, with most of the three phase contact line remaining pinned. Interestingly, the newly formed bubble is not shape stable but grows in height within several 100 ms. This growth dynamic can be described with classical diffusion theory using contact line pinning and Henry’s law. This theory also shows that surface nanobubbles can attain a stable shape with contact angle larger than 90 degree in supersaturated liquid. ∗

To whom correspondence should be addressed Nanyang Technological University ‡ Contributed equally to this work †

1 ACS Paragon Plus Environment

Langmuir

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

1

INTRODUCTION

Surface attached nanobubbles are long-lived spherical-cap shaped objects populating surfaces in liquids. 1–3 Their content is assumed to be gas and vapor and they exhibit strong pinning of their three phase contact line. 4 Although the height is a few nanometers, their “footprint” may be several 100 nm wide. The large contact angle and their longevity challenge our current understanding. Some recent models suggested that contact line pinning and gas supersaturation provide diffusional stability to nanobubbles. 4–7 Yet, experimental studies revealing diffusion dominated growth or shrinkage of bubbles are lacking. In principle, gaseous objects should adjust their size to the surrounding gas concentration by gas in- or outflux. A rough estimate of the timescale for this process is given by the diffusion coefficient and the length scale, i.e. the growth of a nanobubble will be of the order of milliseconds if we assume a length scale of 1 µm and a diffusion constant of 10−9 m2 /s. Here we report on experiments of single nanobubble growth and test the observation with classical diffusion laws. Conventional atomic force microscopy is far too slow to achieve this temporal resolution while optical microscopy using total internal reflection microscopy (TIRF) 8 does not have a high lateral spatial resolution. However, TIRF is very sensitive to the growth of the bubbles in the direction normal to the substrate due to the exponentially decaying strength of the evanescent wave. Previous work has already explored the effect of changes to the liquid environment for the stability of nanobubbles: For example, chemical changes like change in pH and salt concentration, and the effect of the substrate’s electrostatic potential 9 have been shown to affect the number density and size of the nanobubble, while temperature and dissolved gas concentration 10,11 also may affect their stability. By exposing nanobubble to ultrasound, Brotchie and Zhang 12 showed they can grow in size, while coalescence of nanobubbles by interaction with a scanning probe microscope tip has been reported in number of studies. 10,13–15 In this work, we observed that neighbouring nanobubbles coalesce in a flow. After they coalesced, the merged bubble adjusts to a new equilibrium shape by growth in height. 2 ACS Paragon Plus Environment

Page 2 of 18

Page 3 of 18

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

Langmuir

2

EXPERIMENTAL SECTION

The nanobubbles are created with a standard Water-Ethanol-Water exchange process within a microchannel and are observed with a homebuilt total internal reflection microscopy, see Ref. 8. The visualization is achieved by coupling a green cw solid state laser with stable light output (up to 30 mW, 532 nm, Laser Quantum, UK) into an inverted microscope (Olympus IX71) for TIRF illumination. The images are captured with a cooled slow scan CCD camera (6.45 µm pixel size, Sensicam QE, PCO) at 50 frames per second. Using a 60X objective (NA 1.49, APON60X, Olympus) with this setup results in 108 nm/pixel and the diffraction limited resolution is about 230 nm. The Water-Ethanol-Water exchange is carried out in 20 µm high and 1 mm wide microchannels fabricated with standard soft lithography in PDMS. The channel is bonded through plasma activation to a cleaned glass coverslip. 16 Rhodamine 6G (5 µM conc., Sigma Aldrich) is dissolved in both DI water and ethanol; the dye adsorbs on the surface nanobubbles leading to strongly contrasted images of nanobubbles. 8 Plastic syringes are used, however subsequent experiments with the same equipment showed that the imaged objects are gaseous. 17

3 3.1

RESULTS AND DISCUSSION Nanobubble Merger Under Flow

After the water-ethanol-water exchange, the nanobubbles appear quickly on the glass coverslip (approx. 1.1 bubbles/µm2 ). Once a stable nanobubble populations has formed, we maintain the water flow rate at 200 µL/min (equivalent to an averaged velocity of 0.17 m/s). Interestingly, in rare events a sudden increase of the TIRF intensity occurs. There two nanobubbles have merged into one with a greatly increased intensity. Fig. 1a depicts such a merging event; the first frame marks two neighboring bubbles (B1 and B2) of nominal radius of 209 nm and 313 nm, respectively, which merged in the successive frame into a considerably


Langmuir

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

brighter object of 603 nm in radius. Measurement uncertainty of the lateral size is estimated to be of the order of one pixel (≈ 108 nm). Note that the area of the merged nanobubble is larger than the original areas and it spans the convex hull of the previous nanobubbles. Interestingly, the triple contact line during the merging of the initial and final bubble remains fixed, except for the central area connecting both bubbles. After merging a gradual decrease in the brightness of the merged bubble is observed while the surrounding bubbles remain at their brightness and contrast, see Fig. 1c. The temporal change of intensity is documented in Fig. 3b; there the averaged pixel intensity of the merged bubble area is plotted as a function of time (time of merger t = 0). The intensity increases to a higher brightness within the camera’s inter-frame time of 20 ms and is followed by a slow decrease on a time scale of several hundred milliseconds. What is the cause of this slowly reducing brightness? We can rule out photo bleaching as the intensity of the surrounding bubbles remain unchanged. The diffusion limited transport of dye molecules from the bulk onto the new surface of the bubble is considerably faster than the temporal resolution. Advection of the dye if present will further reduce this re-equilibration time. Hence both diffusion and advection effect for the florescent dye can be neglected. Therefore, the height of the merged bubble must be changing with time: initially after the merger, the bubble interface must be very close to the surface and over time the bubble increases in height, see Fig. 1b. In order to understand the changes in the height of surface nanobubbles, the kinetics of gas diffusion from the medium into the bubble must be considered. In the following section, we model this diffusion of gas for a surface attached nanobubble with its contact line being pinned.

3.2

Nanobubble Diffusion Model

We consider a surface attached spherical-cap shaped bubble of height h and lateral radius a (Fig. 2). A similar model has been independently developed by Lohse and Zhang. 7 The


Page 4 of 18

Page 5 of 18

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

Langmuir

number of gas molecules inside the nanobubble, N , is given by

N = Vb ρg =

π (3a2 + h2 )hρg 6

,

(1)

where, Vb is the volume of the bubble and ρg the number density of the gas phase. A change of the number of gas molecules leads to changes in the volume and the density of the gas. Relating the density to the Laplace pressure and assuming an ideal gas law we obtain dN π(3a2 + h2 )h(a2 − h2 ) 4σ π dh =[ + (a2 + h2 )ρg ] 2 2 2 dt 6(a + h ) RT 2 dt

.

(2)

The lateral radius a of the merged bubble is assumed constant due to contact line pinning. Using Fick’s 1st law of diffusion, the net flux of dissolved gas into the bubble is given by the product of concentration gradient at the bubble wall interface, ∇Cs , surface area, Ab and the diffusion coefficient D: dN = −Ab D∇Cs dt

.

(3)

Thus we can solve for (a2 + h2 )D dh = − (3a2 +h2 )h(a2 −h2 ) 4σ ∇Cs dt + 1 (a2 + h2 )ρg 2 2 2 6(a +h )

RT

.

(4)

2

To solve Eq.4 we approximate the concentration gradient as

∇Cs =

Cs (t) − Cl L

.

(5)

Here, Cs is the gas concentration just outside the nanobubble surface and Cl is the gas concentration in the liquid far away from the nanobubble. Though the value of Cl might change over time due to changes in experimental conditions, for the timescale and conditions involved in these experiments Cl can be treated as constant. An appropriate choice for the


Langmuir

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

Page 6 of 18

mass diffusion boundary layer thickness L, is

L = (√

1 −1 1 + ) πDt rb (t)

.

(6)

This choice of boundary layer thickness is motivated by the fact that after a step change (such as merger of two nanobubbles or abrupt change in curvature) the boundary layer √ needs to rebuild for the new surface concentration. Its thickness scales as πDt. After some growth, the boundary layer develops and for later times, assuming quasi-static growth of the nanobubble, the diffusion layer is given by the radius of curvature, rb , of the bubble. This expression for the boundary layer thickness can be shown to be exact for the case of a quasistatic spherical bubble. 18 With the above approximations, Eq. 4 can be solved numerically using a standard RungeKutta scheme. To compare with experimental observations, we start with a flattened surface nanobubble, i.e. lateral radius a(t = 0) = 603 nm, initial height h(t = 0) = 2 nm. This initial height of the merged bubble is chosen by applying mass conservation i.e. the merged bubble is formed by distributing the gas present in bubbles B1 and B2 over the footprint of the merged bubble. 19 For these numerical calculations, we further assume that the concentration of dissolved gas on the liquid side of the nanobubble interface is given by Henry’s law, i.e.

Cs (t) =

2σ 1 (P∞ + ) . kH rb (t)

(7)

with kH = 1465 L·atm/mol as the Henry’s law constant for air 20, 21 and P∞ = 1 atm as the ambient pressure. Since the dissolved gas concentration in the vicinity of the nanobubbles is not known, the numerical simulations are carried out at various concentrations, see Fig 3a. For undersaturated liquids, i.e. gas concentrations below 100% here, the nanobubbles dissolve but above this value the nanobubbles reach a stable height. The effect of the dissolved gas concentration on the resulting height of a stable surface attached nanobubble is analyzed in the next section. 6 ACS Paragon Plus Environment

Page 7 of 18

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

Langmuir

As a first approximation the intensity of the nanobubble images is proportional to the intensity of the illuminating evanescent wave. Because the experimental TIRF intensity I(h) is a function of height and drops as I(h) = I0 exp(−h/λair ) 22 we can compare the observed change in intensity with the model prediction. 23 The penetration depth of the evanescent wave in air, λair , is about 40 nm for our experimental setup. Figure 3b compares the experimental TIRF signal (dashed line) with the expected intensity for a variety of dissolved gas concentrations. The simulated TIRF intensities show similar reduction over time as seen in experiments for mild super-saturations in the range of about 116%. Here, both the time scale and magnitude of the observed TIRF intensity agrees with simulation results. Yet, to obtain this good agreement we had to reduce the surface area, Ab , by 8 times. We speculate that the gas transport is hindered by Rhodamine dye absorbed on the surface, effectively reducing the surface area. The reduction of the surface area only affects the time scale of the dynamics but does not change the final height of the simulated nanobubbles. In the five merger events that we recorded, the growth dynamics agrees nicely with the model with moderate oversaturation levels between 110% and 130% in the liquid, see Fig. 4. We therefore can describe the growth of nanobubbles with a classical diffusion theory under the assumption of contact line pinning. The second interesting observation is that the bubbles grow to a limited final size. This is in contrast to free gas bubbles in the bulk, which either grow or dissolve indefinitely by gas diffusion. 24,25 We now discuss the stability of the nanobubble shape.

3.3

Equilibrium shape of surface attached nanobubbles

Considering surface nanobubbles as a dynamical system governed by Eq. (3), its stable and unstable equilibrium shape can be found by finding the zeros of the time derivative, i.e. Ab D dN =− dt LkH

2σ P∞ + − kH C l = 0 . rb


(8)

Langmuir

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

Page 8 of 18

It is convenient to define the supersaturation pressure Pss ≡ kH Cl − P∞ as a measure of the supersaturation of the liquid. It follows that

Cl =

P∞ + Pss kH

(9)

and the equilibrium condition now reads

Pss =

2σ rb

.

(10)

Pss is positive if the liquid is supersaturated and negative if the liquid is undersaturated.

26

Hence a equilibrium radius only exists in supersaturated liquid. Note that Eq. (8) has no geometrical constrain and Eq. (10) is applicable to bulk bubbles. In the case of bulk bubbles, it can be easily shown that dN/dt has the same sign as the radius perturbation and thus the equilibrium radius is always unstable. 25 In the case of surface attached nanobubbles, there are additional constrains due to contact line pinning and spherical-cap geometry. By expressing rb explicitly in terms of lateral contact radius a, we obtain a quadratic equation about h:

h2 −

4σ h + a2 = 0 , Pss

(11)

it becomes apparent that a surface attached nanobubble is in equilibrium with the surrounding liquid for two heights, h1 and h2 :

h1 =

2σ −

p 4σ 2 − (Pss a)2 Pss

, h2 =

2σ +

p 4σ 2 − (Pss a)2 Pss

.

(12)

To analyse the stability of these two equilibrium heights, Eq. (8) can be written as: dN Ab D Pss = (h − h1 )(h − h2 ) . dt LkH (a2 + h2 )


(13)

Page 9 of 18

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

Langmuir

Note that, since dN/dt has same sign as dh/dt for a surface attached nanobubbles (i.e. a fixed, see Eq. (2)), the smaller of the two equilibrium heights, h1 , is stable and h2 is unstable. As illustrated in Fig. 5, for a given supersaturation, nanobubbles smaller than h1 in height and between h1 and h2 will change their height by gaining or losing gas respectively till their height reaches h1 (see Supplementary Information for comparison with our experiment). While the bubbles larger than h2 in height will grow indefinitely. Even though there is no equilibrium solution for undersaturated liquid (i.e. Pss is negative in Eq. (10)), as long as the liquid is slightly supersaturated the equilibrium nanobubble height is stable (see Fig. 3a). The smaller the supersaturation, the smaller the stable equilibrium height. An alternative form of the equilibrium shape is in terms of the liquid side contact angle, θc , which can be obtained from the geometry: sin θc =

a Pss 2σ

.

(14)

The equilibrium contact angle also imposes a maximal lateral size. As the left hand side of Eq. (14) is positive and smaller than 1 it follows that the lateral radius of the nanobubble a < 2σ/Pss . Bubbles beyond this lateral radius grow unbounded as there is no solution in Eq. (11) and dN/dt > 0 for all bubble heights. In ambient pressure (Pss = 1.0 atm), inserting the above estimated saturation of 116% (Pss = 0.16 atm) we obtain a maximum lateral nanobubble radius of a = 8 µm, and a = 1.2 µm for 200% saturation. These predictions give a reasonable size estimate, i.e. we have not observed bubbles larger than a few micrometers in the experiments reported here. The predicted contact angle of stable nanobubbles and maximum feasible lateral size are consistent with numerical results obtained by Liu and Zhang. 5

4

CONCLUSIONS

Our experimental results show that most nanobubbles are stable even in a strong flow yet some may depin partly and coalesce. After coalescence the area of the merged bubble spans 9 ACS Paragon Plus Environment

Langmuir

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

the convex hull of both initial bubbles. The bubble remains pinned yet the nanobubble height changes rapidly. A classical gas diffusion model for the surface attached nanobubble agrees with the experimental findings that the new bubble grows up to a unique height given by the gas supersaturation and the lateral size. The timescale of this growth can be modelled with a reduced surface area participating in the gas diffusion. This is expected as the dye molecules necessary for the observation adsorb at the interface thus effectively reduce the surface area. This model supports that, due to contact line pinning, surface attached nanobubbles can attain a stable shape, which is in contrast to bubbles in bulk liquid. In supersaturated liquid, a surface nanobubble with lateral size smaller than ∼ 10 µm and initial height smaller than a critical height will approach an equilibrium shape within one second. If either one of these dimensions is larger than the critical size, it will grow continuously. Our model further predicts that the equilibrium contact angle of surface nanobubbles is always larger than 90◦ and independent of the macroscopic equilibrium contact angle of the substrate. Hence it explains the anomalous large contact angle reported in the literature as a consequence of contact line pinning and gas diffusion. In contrast, Young’s relation is violated, which leads to an capillary force pointing towards the nanobubble’s center 27 that must be balanced by a finite pinning force. Once this pinning force is overcome, the contact line starts moving and the nanobubble shrinks. 17 Although our work sheds little light on the origin of this pinning force, we think that both hydrophobic and hydrophilic 8,11,17 surfaces support nanobubbles; yet the pinning force is higher on a hydrophobic surface. Hydrophilic surfaces with some surface heterogeneity (in our case likely through adsorption of the dye onto the glass) or chemical etching, e.g. Piranha cleaning 28 provide the necessary pinning force. In summary, a classical model is sufficient to describe the dynamics of nanobubble growth to a stable size. The model is based on a constant surface tension coefficient, Laplace’s equation, and Henry’s law applied to the mildly curved gas-liquid interface. Yet, gas super-


Page 10 of 18

Page 11 of 18

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

Langmuir

saturation and pinning of the contact line are required for stable bubbles, in agreement with most experiments reported.

Acknowledgement We are very thankful for fruitful discussions with Derek Chan, Xuehua Zhang, Detlef Lohse and Beng Hau Tan. This work was supported by the Singapore National Research Foundations Competitive Research Program funding (NRF-CRP9-2011-04).

Supporting Information Available The equilibrium shape predicted in Eq. 12 is compared against experimental observation. This material is available free of charge via the Internet at http://pubs.acs.org/.

Notes and References (1) Seddon, J. R. T.; Lohse, D. Nanobubbles and micropancakes: gaseous domains on immersed substrates. J. Phys.: Cond. Matter 2011, 23, 133001. (2) Craig, V. S. J. Very small bubbles at surfacesthe nanobubble puzzle. Soft Matter 2011, 7, 40–48. (3) Zhang, X.; Lohse, D. Perspectives on surface nanobubbles. Biomicrofluidics 2014, 8, 041301. (4) Zhang, X.; Chan, D. Y. C.; Wang, D.; Maeda, N. Stability of interfacial nanobubbles. Langmuir 2013, 29, 1017–1023. (5) Liu, Y.; Zhang, X. Nanobubble stability induced by contact line pinning. J. Chem. Phys. 2013, 138, 014706. (6) Liu, Y.; Zhang, X. A unified mechanism for the stability of surface nanobubbles: Contact line pinning and supersaturation. J. Chem. Phys. 2014, 141, 134702. 11 ACS Paragon Plus Environment

Langmuir

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

(7) Lohse, D.; Zhang, X. Pinning and gas oversaturation imply stable single surface nanobubbles. Phys. Rev. E 2015, 91, 031003. (8) Chan, C. U.; Ohl, C.-D. Total-internal-reflection-fluorescence microscopy for the study of nanobubble dynamics. Phys. Rev. Lett. 2012, 109, 174501. (9) Mazumder, M.; Bhushan, B. Propensity and geometrical distribution of surface nanobubbles: effect of electrolyte, roughness, pH, and substrate bias. Soft Matter 2011, 7, 9184–9196. (10) Yang, S.; Dammer, S. M.; Bremond, N.; Zandvliet, H. J. W.; Kooij, E. S.; Lohse, D. Characterization of Nanobubbles on Hydrophobic Surfaces in Water. Langmuir 2007, 23, 7072–7077. (11) Zhang, X. H.; Zhang, X. D.; Lou, S. T.; Zhang, Z. X.; Sun, J. L.; Hu, J. Degassing and Temperature Effects on the Formation of Nanobubbles at the Mica/Water Interface. Langmuir 2004, 20, 3813–3815. (12) Brotchie, A.; Zhang, X. H. Response of interfacial nanobubbles to ultrasound irradiation. Soft Matter 2011, 7, 265. (13) Yang, J.; Duan, J.; Fornasiero, D.; Ralston, J. Very Small Bubble Formation at the SolidWater Interface. J. Phys. Chem. B 2003, 107, 6139–6147. (14) Simonsen, A. C.; Hansen, P. L.; Kl¨osgen, B. Nanobubbles give evidence of incomplete wetting at a hydrophobic interface. J. Colloid Interface Sci. 2004, 273, 291299. (15) Bhushan, B.; Wang, Y.; Maali, A. Coalescence and movement of nanobubbles studied with tapping mode AFM and tipbubble interaction analysis. J. Phys.: Cond. Matter 2008, 20, 485004. (16) Glass coverslips were cleaned with acetone, isopropyl alcohol and deionized water in ultrasonic bath for 15min each before bonding to the PDMS structures. 12 ACS Paragon Plus Environment

Page 12 of 18

Page 13 of 18

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

Langmuir

(17) Chan, C. U.; Chen, L.; Arora, M.; Ohl, C.-D. Collapse of Surface Nanobubbles. Phys. Rev. Lett. 2015, 114, 114505. (18) Enr´ıquez, O. R.; Sun, C.; Lohse, D.; Prosperetti, A.; van der Meer, D. The quasi-static growth of CO2 bubbles. J. Fluid Mech. 2014, 741, R1. (19) The volume of bubbles B1 and B2 are estimated from equilibrium heights corresponding to an iteratively estimated supersaturation of 116%. (20) Henry’s law constant for air is obtained by a weighted average of Henry’s law constants for Nitrogen and Oxygen. (21) Sander, R. In NIST Standard Reference Database; Linstrom, P., Mallard, W., Eds.; 2014. (22) Gell, C.; Berndt, M.; Enderlein, J.; Diez, S. TIRF microscopy evanescent field calibration using tilted fluorescent microtubules. J. Microscopy 2009, 234, 38–46. (23) This relationship is a stark simplification of the complex scattering near the three phase contact lines and therefore may only hold for sufficiently large bubbles. (24) Epstein, c. e. P.; Plesset, M. On the Stability of Gas Bubbles in Liquid-Gas Solutions. J. Chem. Phys. 1950, 18, 1505–1509. (25) Brennen, C. E. Cavitation and bubble dynamics; Cambridge University Press, 2013; Chapter 2.4 Stability of Vapor/Gas Bubble, pp 39–41. (26) In the literature, gas concentration in the liquid is often expressed in the unit of % supersaturation, Cl,% . The corresponding supersaturation pressure is Pss = (Cl,% − 100%)P∞ . (27) De Gennes, P.-G. Wetting: statics and dynamics. Rev. Mod. Phys. 1985, 57, 827. (28) Private communication with Holger Sch¨onherr,. 13 ACS Paragon Plus Environment

Langmuir

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

Figure 1: a) Rapid merging event of two individual bubbles B1 and B2 within less than 20 ms. Thereafter the combined bubble is considerably brighter but slowly decreases in intensity within 400 ms. b) Sketch of the side view of the nanobubble merger event, followed by slow vertical growth c) Consecutive images of the merged bubble as a function of time, please note the decrease in the intensity of the merged bubble while the surrounding nanobubbles remain unaffected.


Page 14 of 18

Page 15 of 18

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

Langmuir

Figure 2: Sketch of the surface attached nanobubble system modeled in the present paper. The nanobubble is in spherical-cap shape with height h, lateral radius a and radius of curvature rb . The system is kept at ambient pressure P∞ . The gas concentration in the bulk liquid and outside the bubble surface are Cl and Cs , respectively. The difference in these two concentrations drives gas diffusion over a boundary layer thickness L.


Langmuir

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

Figure 3: a) Temporal evolution of the height of simulated nanobubble at various dissolved gas concentration (solid and dash-dotted lines are from the model, diamond experiments) b) Simulated TIRF intensity for the growing nanobubble (solid line) in comparison to the experimentally observed change in TIRF intensity (dashed line).


Page 16 of 18

Page 17 of 18

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

Langmuir

Figure 4: Simulated (lines) and experimental (dots) TIRF intensity change is plotted as a function of the nanobubbles lateral size for 110%, 120% and 130% dissolved gas concentration. The horizontal error bar is estimated to be one pixel. These curves are compared against 5 experimental observations of bubble merging. All observations are in agreement with a mild super-saturation of 10%-30%.

Figure 5: Diagram using Eq. (13) to illustrate the stability of the two equilibrium heights. An initial height smaller than h2 converge to a stable height h1 , otherwise it grows unbounded and eventually detaches. The bubble shapes corresponding to the two equilibrium heights are shown, too. 17 ACS Paragon Plus Environment

Langmuir

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

For Table of Contents Only.


Page 18 of 18

Coalescence, Growth, and Stability of Surface-Attached Nanobubbles

Recommend Documents