Article pubs.acs.org/Langmuir
New Interferometric Technique To Evaluate the Electric Charge of Gas Bubbles in Liquids Mario Corti,*,†,‡ Marco Bonomo,†,§ and Antonio Raudino∥ †
Dipartimento di Chimica, Biochimica e Biotecnologie per la Medicina, Università di Milano, LITA, Via Fratelli Cervi 93, 20090 Segrate, Milano (MI), Italy ‡ Instituto dei Processi Chimico-Fisici (IPCF), Consiglio Nazionale delle Ricerche (CNR), Viale Ferdinando Stagno d’Alcontres, 37, 98158 Messina, Italy ∥ Dipartimento di Chimica, Università di Catania, Viale Andrea Doria 6, 95125 Catania, Italy ABSTRACT: We report a new interferometric technique to measure the electric charge at the gas−liquid interface of a bubble in a liquid. The bubble rests by buoyancy against an electrode, and an alternating electric field excites its capillary oscillations. The oscillation amplitude of the quadrupolar mode frequency is measured by the interferometer, and it is used to evaluate the electric charge. The mode frequency scales with the square root of the interfacial tension and with a −3/2 power law as a function of the bubble radius. For bubbles in the millimeter diameter range in pure water, the measured negative charge scales with the square of the radius, hence, giving a constant surface charge density on the order of 1.8 × 10−5 C m−2, which is rather consistent with the electrophoretic values reported in the literature.
1. INTRODUCTION Electric charge effects at the gas−liquid interface of gas bubbles in a liquid are interesting for both fundamental physicochemical studies1−4 and practical applications in industrial separation processes.5 All measurements reported thus far rely on electrokinetic phenomena. Ordinary electrophoretic methods are difficult to apply because of gravity effects determined by the large density difference between the gas and the liquid. The rising velocity of macroscopic bubbles is generally much larger than their electromobility. Many different techniques have been described in the literature to measure the bubble ζ potential, each one with advantages and disadvantages.6 Here, we report an experimental study that shows the feasibility of measuring bubble charge by an optical interferometric method. The bubble remains still and rests by buoyancy against an electrode. A second electrode is placed below the bubble. In the presence of an electrical charge at the interface, an alternating electric field applies an alternating force to the bubble in the vertical direction. The deformation from the spherical shape is detected by the change of the optical path inside the bubble of a laser beam that traverses it in the horizontal direction. High sensitivity is reached because of both the interferometric nature of the measurement and also the resonant behavior of the bubble, which amplifies the effect of the applied force. In this paper, we give a description of the main features of the experimental setup, and as a demonstration of the validity of the proposed new experimental technique, we report the surface charge density evaluated for air bubbles on the order of 1 mm diameter in distilled water. © 2012 American Chemical Society
2. EXPERIMENTAL SETUP 2.1. Optical Interferometer. A laser beam is focused at the center of the bubble (Figure 1). Refractive index mismatch causes light
Figure 1. Interfering beams from the bubble. reflection at the gas−liquid interfaces. These two points can be considered as two coherent light sources S1 and S2 with a finite extension and divergence determined by the focusing optics of the Gaussian laser beam. For mild focusing inside the bubble, with waist on the order of 20−50 μm, the Rayleigh range has an extension that cannot be neglected in comparison to the bubble diameter (in the millimeter range). Therefore, Gaussian optics is used instead of simple geometrical optics to calculate the beam divergence at the interfaces. In the backward direction, a circular pattern of interference fringes is Received: January 24, 2012 Revised: March 8, 2012 Published: March 16, 2012 6060
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formed on any plane perpendicular to the beam axis, the envelope of which diverges with a much larger angle than the incoming beam. The crucial element, determining both the fringe pattern and their divergence, is the difference between the curvature of the wavefront and the bubble at S1 and S2. In actual experiments, the divergence of the beam reflected at the second (exiting) bubble interface, S2, can be twice as large as the divergence of the beam reflected at the first (entering) bubble interface, S1. A change in the distance between S1 and S2 gives rise to a change in the fringe pattern. A full cycle in the central fringe brightness that is from bright to dark to bright again is obtained for an optical path variation of λ that is for a bubble radius change of λ/4. The optical system is described in Figure 2. Designed
second order by the coupling of the overall bubble translational motion to the bubble diameter determined by hydrodynamic effects. It is important to notice that such sensitivity can be exploited at best along measurements of vibration amplitudes. Indeed, such sensitivity is still available also in quasi-static measurements of the change in the bubble volume, provided that the thermodynamic condition of the bubble is carefully controlled to avoid drifts because of changes in the gas solubility. 2.2. Measurement Cell. The measurement cell has an internal cubic shape with approximately 1 cm3 volume. It is made of poly(methyl methacrylate) (PMMA) with three polished windows. A 316 stainless-steel electrode of 3 mm in diameter is hanging from the top. A second electrode, 1.6 mm in diameter, comes up from the bottom. The gap between the two electrodes is 3 mm. The lower electrode has a central hole, 0.3 mm in diameter, which allows for the gas to go through to form the bubble that, by buoyancy, rises up toward the upper electrode. The upper electrode is screwed in the cell in a way that it is easily extractable, so that it can be treated externally with a 5 M HNO3 solution prior to each set of measurements. This is performed to ensure a rather reproducible hydrophilic surface of the electrode. Inlet and outlet tubes serve to fill the cell. An extra tube at the top connects the cell with a piezoelectric transducer used for acoustic excitation of the bubble. The acoustic impedance is kept low by making this tube short and with a larger internal diameter. The cell is slightly tilted in the vertical plane to avoid the back reflections of the windows to interfere with the fringe pattern. A miniature chargecoupled device (CCD) camera is placed laterally to view the bubble in the cell. The conductivity properties of the cell have been tested by measuring real and imaginary parts of its impedance as a function of salinity, frequency, and amplitude of the voltage applied to the electrodes. Perfect linearity for salinity was found, in the test range between no added salt and 5 mM NaCl, an undetectable difference of conductivity in the applied voltage range of 0.1−5 V, and a maximum deviation of 2% in the conductance measurements changing the applied voltage frequency from 300 to 8000 Hz for the 2 mM NaCl solution. Impedance data are obtained with a Stanford SR780 spectrum analyzer, which supplies the variable frequency voltage to the cell and measures the amplitude and phase of the current through it.
Figure 2. Bubble interferometer. to be compact and simple, it is mounted on a rigid metal plate inside a sound isolating box. A 5 mW He−Ne laser beam operating at λ = 633 nm is focused at the center of the bubble by the lens L of 15 cm focal length. The fringe pattern is displaced from the laser beam axis by the beamsplitter (BS). A photodetector (Hamamatsu R2949) placed on the central fringe axis, with an aperture much smaller than the central fringe width, measures a signal of the following form:
I(x) = A(I1 + I2 + 2(I1I2)1/2 cos(2πx /λ))
(1)
where A is a constant that depends upon the photodetection circuitry, I1 and I2 are the light intensities coming from S1 and S2, respectively, and x is the optical path difference. The fringe visibility is excellent because, from the Snell law of reflection, 2(I1I2)1/2/(I1 + I2) = 0.9998 for a gas bubble in a liquid of refractive index n = 1.33. For central fringe operation, the interferometer is strictly differential because x depends upon the optical path inside the bubble only (x = 4R, with R being the bubble radius) and not, at first order, the overall movement of the bubble relative to the laser source or the detector. The form of the response function, eq 1, suggests that the bubble vibrations are easily measurable for very small amplitudes, in the 10 nm range. In fact, the interferometer can operate in a region where the response function I(x) is nearly linear with x, that is, around a point where 2πx/ λ is equal to an odd multiple of π/2. In this case, a 1% deviation from linearity is reached with a peak−peak vibration amplitude ΔR = 13.4 nm for the 633 nm wavelength of the laser. With the experimental setup of Figure 2 and the photomultiplier gain of 104, a typical variation of 1 V on a load resistance of 100 kΩ is measured when the central fringe goes from maximum light to dark, that is, with ΔR = λ/8 = 84.25 nm. Background light noise is reduced to a minimum by means of an interference filter placed in front of photomultiplier aperture. The electrical noise is mainly due to the shot noise term related with the granular nature of the photodetection process. Practically, this gives the theoretical limit for the bubble interferometer sensitivity. In a bandwidth of a few hundred hertz, the root-meansquare noise amplitude is evaluated to be on the order of a few tens of millivolts, which, in terms of radius change, corresponds to a few hundredths of a nanometer. Such theoretical sensitivity is very high and can be even improved further, if necessary, using a polarizing interferometer scheme (a polarizing beam splitter and a λ/4 waveplate), more laser power, and an avalanche photodiode, which has a higher quantum efficiency in the red. In our experiments, the actual sensitivity was not so good, about 0.1 nm. In fact, because of external acoustic noise, the differential nature of the interferometer can be reduced at
3. BUBBLE OSCILLATIONS At first order, bubble oscillations can be distinguished in two types: volume and surface oscillations. Volume oscillations are easily excited by modulating an uniform pressure field. For small-amplitude oscillations, the bubble behaves as an harmonic oscillator,7 with a characteristic frequency νv determined by the thermodynamic conditions of the bubble and its radius. For diameters in the millimeter range, the gas in the bubble behaves not completely adiabatically nor isothermally. Its characteristic frequency, in the order of kilohertz, is inversely proportional to the radius with the expression νv = (3kP /(4π 2ρ))1/2 /R
(2)
A surface term has been neglected because it is very small compared to the pressure term of eq 2. The external pressure is P, R the bubble radius, and ρ is the liquid density. The polytropic coefficient k has a value that is in between the isothermal exponent k = 1 and the adiabatic exponent k = 1.4. It can be taken to be 1.14 from the data of ref 8. At the surface, the “periodic” boundary conditions naturally provided by the bubble closed geometry allow for only a discrete spectrum of bubble stationary oscillation modes.9 A bubble deformation can, in general, be expressed as a superposition of spherical harmonic contributions, which, at different orders, can be considered as a bubble mode, because according to a linearized hydrodynamic description, it obeys an independent equation of motion with its own characteristic frequency. The i ≥ 2 order 6061
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modes refer to oscillations of the bubble interface, which keep the internal volume constant. Their characteristic frequency νS,i is given by10 νS, i = [(i − 1)(i + 1)(i + 2)σ/(ρR3)]1/2 /(2π)
(3)
where σ is the gas−liquid surface tension. A major difference between surface and volume modes is the leading elastic force acting to restore the spherical shape, while the restoring force for the bubble deformed according to i ≥ 2 modes is the surface tension. In the case of the volume mode, the restoring force is mainly due to the difference between the pressure inside and outside the bubble. For small deformations, where the linear response dominates, all of the bubble modes are expected to show a simple Lorentzian frequency dispersion.9 The i = 2 quadrupolar mode transforms the bubble into a spheroid, and it is simple to excite. Uniform acoustic fields are unable to excite surface modes. In fact, shape deformation is obtained only if the excitation force is applied on some preferential axis. Such deformation can be achieved by an electric field, which is directional. We have already shown that an electric field may couple selectively to surface modes.8 This may happen even in the absence of surface charge because of the difference in the dielectric constant between the gas within the bubble and the surrounding liquid. The coupling is quadratic in the electric field. In turn, a more effective coupling can be obtained if a net surface charge is present at the gas−liquid interface of the bubble. If the bubble is constrained onto a surface, for instance, by buoyancy onto the upper electrode (as in Figure 3), an
Figure 4. Oscillation amplitude (dots) of a bubble with a radius of 0.75 mm as a function of frequency with a sine wave excitation of 1 V. The full line is a Lorentzian fit.
radius, for an air bubble of radius 0.75 mm, excited by a swept sine wave of 1 V amplitude. The bubble is in water at a temperature of 25 °C. The diameter of the bubble is measured by means of a cathetometer with micrometric movements. The spectrum has a Lorentzian shape with a peak frequency ν = 225 Hz. The peak is a true resonance peak because the correct 180° phase shift between the excitation and bubble response is observed when the sine wave sweeps through resonance. The bubble behaves as a damped harmonic oscillator forced by the external electric field. The resonance width is related to the damping connected to water viscosity.11 The measured peak frequency copes very well with the i = 2 quadrupolar mode frequency predicted by eq 3, with R = 0.000 75 m, σ = 0.071 95 N/m for the surface tension of water at 25 °C, and ρ = 1000 kg/m3 for the water density. The same bubble can be excited also by a pressure field obtained with the piezoelectric transducer driven by a swept sine wave. The resonance is found to be at ν = 4720 Hz. This value is very similar to the volume mode frequency νv = 4840 Hz, which eq 2 predicts for the 0.000 75 m diameter bubble, with an external atmospheric pressure of 0.989 × 105 Pa, k = 1.14, and water density ρ = 1000 kg/m3. The validity of eq 2 is seen to be very good, so that, in principle, one could avoid measuring the bubble diameter optically. It would be enough to measure its volume mode frequency and use eq 2 to calculate the bubble radius with no risk of confusion because volume and surface modes are well-separated in frequency. The peak amplitude of the surface mode of Figure 4 linearly follows the amplitude of the exciting electric field. This is direct evidence that a net electric charge exists at the bubble interface. If it were electric excitation by dielectric constant mismatch, we would have a factor of 2 difference between the frequency of the exciting electric field at resonance and the resonant frequency itself. Besides, in the spectrum of Figure 4, there is no evidence of a second peak at twice the resonance frequency, which confirms operation of the interferometer in the linear region and also that electric excitation by mismatch of the dielectric constant is negligible at such small exciting voltages. Figure 5 shows a log−log plot of the measured resonance frequency for bubbles of different radii. Data follow a power law dependence with a slope of −3/2 , as predicted by eq 3. The resonance frequency of eq 3 depends upon the square root of the surface tension. This is verified experimentally in Figure 6.
Figure 3. Bubble (0.4 mm radius) standing below the upper electrode.
applied electric field may induce deformation of the bubble shape. In particular, the surface mode with i = 2 can be excited by applying an electric field at its characteristic frequency.
4. RESULTS The cell is filled with freshly distilled water. Care is taken to keep a low contamination of salts and CO2. Typical resistivity and pH are 18 MΩ/cm and 6.5, respectively. The bubble is generated by injecting a fixed amount of air through the internal hole of the lower electrode. A Stanford Research spectrum analyzer SR780 is used to generate a sine wave sweeping through the frequency range, in which we want to excite bubble vibrations and also to register the frequency spectrum of the interferometer signal. Measurements are enabled only when the interferometer operates within its linear region. Figure 4 shows the frequency spectrum of ΔR, the vibration amplitude of the 6062
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the optical path of the light inside the bubble. The spectrum reported in Figure 4 is a measure of such deformations around the equilibrium diameter, which the bubble has in the absence of the alternating electric field. Indeed, the bubble is not a sphere because buoyancy deforms the bubble into an oblate spheroid and deformations as a result of the applied electric field superimposed on it. Static deformation is quite small and should not appreciably affect the theoretical prediction for the sphere. In fact, if we call the vertical (short) and horizontal (long) semi-axes of the spheroid b and a, respectively, the equilibrium deformation of the bubble can be easily computed from the air−water surface tension and results are about b/a = 0.98 for a bubble having a diameter d = 1 mm. With the data of Figure 4, which represents the vibration amplitude of the horizontal semi-axis a, it is possible to evaluate the work performed by the electric field to deform the bubble and, hence, the surface charge. At a constant bubble volume, the semiaxes are coupled by the relation a2b = R03, where R0 is the radius of the equivalent sphere. Hence, the change in the semi-axes a and b and the surface area S for an oblate ellipsoid can be computed in terms of the variation of its eccentricity e.
Figure 5. Resonance frequency versus bubble radius (■) and the −3/2 power law in the log−log plot ().
5. CALCULATION OF THE BUBBLE RESPONSE TO THE ELECTRIC FIELD AND DISCUSSION With the ellipticity being defined as e = (1 − b2/a2)1/2, the surface of the oblate ellipsoid is S(e) = 2πa2 + {πb2 ln[(1 + e)/ (1 − e)]}/e. With the constant volume condition a2b = R03, where R0 is the radius of the equivalent sphere, a(e) = R0(1 − e2)−1/6, b(e) = R0(1 − e2)1/3, and S(e) = 2πR02(1 − e2)−1/3{1 + [(1 − e2)/(2e)]ln[(1 + e)/(1 − e)]}. In the limit e ≪ 1 S(e) = 4πR 02 + (8/45)πR 02e 4 + O(e5)
(4a)
b(e) = R 0(1 − (1/3)e 2 − (1/9)e 4 + O(e6))
(4b)
The total energy associated with the bubble is Figure 6. Square of the resonance frequency versus the log of the molar concentration of the non-ionic surfactant C12E8 for a 0.67 mm radius bubble. The frequency squared is proportional to the surface tension. The interpolating lines cross at the cmc, at about 8 × 10−5 M.
W = γS(e) + ΔρgVR 0 + WELECTR (e)
(5)
where γ is the air−water interfacial tension, Δρ is the water−air density difference, V is the bubble volume, and g is the gravity acceleration. The last term in eq 5, WELECTR = ∫ SσΦw(E)dS, describes the interaction between the electric potential Φw(E) at the bubble−water interface (related to the applied alternating current electric field E) and the surface charge density σ. Let us represent the bubble surface charge density as uniformly smeared over the ellipsoid surface. In the absence of free moving charges in water, the potential generated at the electrode obeys the Laplace equation ∇2Φw = 0. It linearly decays along the z axis from Φw = Φwo at z = 0 to Φw = −Φwo at z = D. This is a correct approximation when one is investigating the interaction of the electric field with small-molecular-sized objects. In the present case, the bubble radius is on order of 10−3 m; therefore, the effect of the ions coming from the spontaneous water autodissociation H2O ⇔ H+ + OH− plays an important role. The effect of free moving ions has been taken into account by a linearized Poisson−Boltzmann equation (LPBE): (∂2Φw)/(∂z2) = κ2Φw, where κ is the inverse of the Debye length. In pure water, at T = 25 °C, we measured pH 6.5 (because of partial solubilization of the atmospheric carbon dioxide) and the calculated screening is κ ≈ 2 × 106 m−1. The solution to the LPBE can be described as the sum of the unperturbed potentials stemming from the two oppositely
The square of the resonance frequency of a 0.75 mm radius bubble is plotted as a function of the molar concentration of a non-ionic surfactant of the alkyl polyoxyethylene glycol monoether family, the C12E8,12 which self-aggregates into micelles above a critical micelle concentration (cmc). It is wellknown that surface tension drops from the value of pure water down to a stable value after the cmc. This is exactly what it is observed in Figure 6. Experimentally, it is simple to know if the interferometer output corresponds to an increase or a decrease of the bubble radius. It is sufficient to slightly increase the external pressure while monitoring the interferometer output. The observed change corresponds to a decrease in the optical path inside the bubble. This is important to allow for access to the sign of the electrical charge at the interface. The bubble rests in place below the upper electrode by buoyancy. The hydrophilic nature of the upper electrode avoids a finite contact angle of the bubble with the metal. A finite contact angle would drastically change the theoretical prediction of eq 3, which is not the case in our experiments. Therefore, the constraint is such that an electric attraction of the bubble toward the upper electrode gives rise to a surface deformation, which increases 6063
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such as (κR0e)2n that are by no means small. In the following formulas, the integral I(e 2 ) is calculated numerically. Furthermore, in the adopted geometrical setup, the bubble is contacting the upper electrode: Dmin ≈ 0, D ≈ 3 × 10−3 m, and κ ≈ 106 m−1; hence, in eq 9, we may set: exp(−κDmin) − exp(κD) ≈ 1. Within this approximation, we find
charged electrodes set at a distance D apart (weak overlap approximation). The potential shows the typical decay from the interfacial value Φwo . Φw = Φow [exp(−κz) − exp(−κ(D − z))]
(6)
Simple geometry (see Figure 7) enables us to calculate the electric potential acting on the ellipsoid surface. Adopting polar
WELECTR (e) = σΦow I(e 2)
(11)
The numerically calculated behavior of the electrostatic contribution is correct: if the bubble and the contacting electrode bear opposite charges, the bubble becomes more oblate to favor the electrostatic attraction by decreasing the average relative distance. Just the opposite happens when the bubble and electrode are identically charged. Inserting eq 11 into the expression for the total energy W, minimizing W with respect to e, and solving dW/de = 0, we find ⎡ ∂I(e 2) ⎤ ⎥=0 2e⎢ −A1 + 2A2e 2 + σΦow ⎢⎣ ∂e 2 ⎥⎦
where A1 ≡ 1/3ρgVR0 and A2 ≡ − 1/9ρgVR0 > 0. When the applied field is zero, the last term in eq 12 vanishes and the resulting eccentricity because of buoyancy is e*(0) = (A1/2A2)1/2, with the asterisk denoting the optimized value. The interferometer measures the larger semi-axis a(e) = R0(1 − e2)−1/6. The measured variation of a(e) with and without the applied electric field E reads
Figure 7. Schematic drawing of an oblate bubble near a planar charged interface. Dmin is the shortest bubble−electrode distance.
spherical coordinates r(Θ), Θ, and ϕ, the distance z from a generic point of the surface of an oblate ellipsoid to the upper electrode reads z = Dmin + b(e) − r(Θ)cos Θ ⎡ ⎤ cos Θ ⎥ = Dmin + b(e)⎢1 − ⎢⎣ (1 − e 2 sin 2 Θ)1/2 ⎥⎦
Δa /R 0 = (1 − e*2 (E))−1/6 − (1 − e*2 (0))−1/6 1 ≈ (e*2 (E) − e*2 (0)) 6
(7)
Inserting eq 7 into eq 6, we may calculate the bubble−electrode interaction WELECTR =
12A2
= σΦow (exp( −κDmin) − exp(−κD))
∫
(8)
Because dS = (r (Θ) + (∂r(Θ)/(∂Θ)) ) obtain after simple algebra
sin ΘdΘdϕ, we
WELECTR (e) = 2πσΦow (exp( −κDmin) − exp(−κD)) I(e 2)
(9)
where I(e 2) ≡ 2πb2(e)exp( −κb(e)) 1 ⎡ (1 + e 2(e 2 − 2)(1 − X 2))1/2 ⎤ ⎢ ⎥ ⎥⎦ −1 ⎢⎣ (1 − e 2(1 − X 2))2
∫
⎡ ⎤ κb(e)X ⎥d X exp⎢ ⎢⎣ (1 − e 2(1 − X 2))1/2 ⎥⎦
∂I(e 2) Δa + σΦow R0 ∂e 2
=0 e = e * (E)
(14)
Equation 14 is easily solved for σ. Using A2 ≡ 8/45πγR02 − 1/ 9ρgVR0 > 0.2 × 10−7 Nm, Δa ≈ 10−9 m, R0 ≈ 0.75 × 10−3 m, Φwo ≈ 0.5 V, and (∂I(e2)/∂e2)|e = e*(E) ≈ −0.2 × 10−8 m2 as numerically calculated, we derive the following estimate: σ ≈ 1.8 × 10−5 C m−2. The peak oscillation amplitude of the 0.75 mm bubble of Figure 4 is 8.5 nm. The Δa used in the calculation of the surface charge has been reduced by a factor of 8, which is the Q factor of the resonance, just to take care of the oscillation amplitude, which would have been present in the absence of the resonance because of the driving electric force. The charge at the interface is negative, as revealed by the relative phase between the measured Δa and the applied electric field. Figure 8 shows a log−log plot of the peak oscillation amplitude at resonance versus the bubble radius. The straight line represents a slope of 2, which means that the oscillation amplitude and, hence, the surface charge density σ are radius-independent for bubbles in the millimeter range. The novel interferometric technique presented in this paper gives a surface charge density of air bubbles in pure water, which compares rather well with electrophoretic data. Indeed, the literature reports a wide range of the air−water surface ζ potential, −65 mV,13 −35 mV,14 −25 mV,15 and −10 mV,4,16 which corresponds to charge densities in the range from 10−5 to 10−4 C m−2. In the present experiment, the bubble does not drift under the applied electric field,
⎤ ⎡ κb(e)cos Θ ⎥d S exp( −κb(e)) exp⎢ ⎢⎣ (1 − e 2 sin 2 Θ)1/2 ⎥⎦ S 2 1/2
(13)
Inserting these relationships for e*(E) and e*(0) into eq 12, eventually we obtain
∫S σΦw (E)dS
2
(12)
8/45πγR02
(10)
with X ≡ cos Θ. Unfortunately, κb(e) ≈ κR0 ≈ 10 . Such a large value does not permit us an expansion of the exponential terms in the power series of the small eccentricity parameter e2n (n = 1, 2, ...). This is because the resulting expressions contain terms 3
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Figure 8. Oscillation amplitude at resonance versus bubble radius (squares) and the linear fit of the log−log plot (straight line with a slope of 2).
Figure 10. Resonant behavior of a bubble 1.6 mm in diameter attached to the upper electrode excited by a swept sine voltage of 0.025 V amplitude.
such as for electrophoresis, so that it could not be appropriate to assign the estimated effective charge to the surface of shear. Figure 9 shows a plot of the peak oscillation amplitude at
attached to the upper electrode excited by a swept sine voltage of 0.025 V amplitude. The peak oscillation amplitude is only 4.5 Å. The reported spectrum is obtained with six frequency sweeps of 10 s each. Unfortunately, the theory does not exist yet to treat this particular case. Hopefully, this work will stimulate future developments with finite-element calculations of the dynamic behavior of bubbles attached to a wall.
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AUTHOR INFORMATION
Corresponding Author
*Telephone: +39-3470387099. Fax: +39-0250330365. E-mail:
[email protected]. Present Address §
BEL Engineering, Via Carlo Carrà 5, 20900 Monza, Italy.
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS The authors thank Franco Aliotta and Norberto Micali for interesting discussions and Stefano Motta for help in setting up the experiments.
Figure 9. Oscillation amplitude at resonance versus added salt (bubble radius = 0.74 mm).
resonance of a 0.62 mm radius bubble versus added NaCl. Increasing the ionic strength of the solution gradually diminished the oscillation amplitude. This may be due to counterion shielding, as already observed in the literature for the ζ potential.15 The most acceptable explaination of the negatively charged interface is a specific adsorption of OH− ions coming from water dissociation.17 Specific adsorption could be connected to a reduced mobility of OH− ions at the interface.1 The negative charge can also be explained as a natural consequence of the spontaneous polarization of the water dipoles.18 As a final remark, authors hope to have shown that the proposed interferometric technique can be a valuable way to obtain access to surface properties with minimum perturbation of the interface itself. The present work refers to bubbles that are not attached to the upper electrode in the cell. For eventual technological developments, it would be preferable to work with bubbles having a finite contact angle with the upper electrode. In fact, it would have been much simpler to prepare bubbles with a finite contact angle at the upper electrode, and measurements would have been even better, because secondorder hydrodynamic effects are reduced. As an example, Figure 10 shows the resonant behavior of a bubble 1.6 mm in diameter
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REFERENCES
(1) Marinova, K. G.; Alargova, R. G.; Denkov, N. D.; Velev, O. D.; Petsev, D. N.; Ivanov, I. B.; Borwankar, R. P. Charging of oil−water interfaces due to spontaneous adsorption of hydroxyl ions. Langmuir 1996, 12, 2045−2051. (2) Creux, P.; Lachaise, J.; Gracia, A.; Beattie, J. K. Specific cation effects at the hydroxide-charged air/water interface. J. Phys. Chem. C 2007, 111, 3753−3755. (3) Lützenkirchen, J.; Preočanin, T.; Kallay, N. A macroscopic water structure based model for describing charging phenomena at inert hydrophobic surfaces in aqueous electrolyte solutions. Phys. Chem. Chem. Phys. 2008, 10, 4946−4955. (4) Puskarova, R.; Horn, R. G. Bubble−solid interactions in water and electrolyte solutions. Langmuir 2008, 24, 8726−8734. (5) Li, C.; Somasundaran, P. Reversal of bubble charge on multivalent inorganic salt solutions. J. Colloid Interface Sci. 1992, 148, 587. (6) Gracia, A.; Creux, P.; Lachaise, J. Electrokinetics of gas bubbles. In Interfacial Electrokinetics and Electrophoresis; Delgado, A. V., Ed.; Marcel Dekker: New York, 2001; pp 825−835. (7) Plesset, M. S.; Prosperetti, A Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 1977, 9, 145−185.
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(8) Bellini, T.; Corti, M.; Gelmetti, A.; Lago, P. Interferometric study of selectively excited bubble capillary modes. Europhys. Lett. 1997, 38, 521−526. (9) Leigthon, T. G. The Acoustic Bubble; Academic Press: London, U.K., 1994. (10) Prosperetti, A. Normal-mode analysis for the oscillations of a viscous liquid drop in an immiscible liquid. J. Mec. 1980, 19, 149−182. (11) Asaki, T.; Marton, P. L. Free decay of shape oscillations of bubbles acoustically trapped in water and sea water. J. Fluid Mech. 1995, 300, 149−167. (12) Degiorgio, V. Non-ionic micelles. In Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; Degiorgio, V., Corti, M., Eds.; North Holland: Amsterdam, The Netherlands, 1985; pp 303−335. (13) Graciaa, A.; Morel, G.; Saulner, P.; Lachaise, J.; Schechter, R. S. The ξ potential of gas bubbles. J. Colloid Interface Sci. 1995, 172, 131− 136. (14) Takahashi, M. ξ potential of microbubbles in aqueous solutions: Electrical properties of the gas−water interface. J. Phys. Chem. B 2005, 109, 21858−21864. (15) Yang, C.; Dabros, T.; Li, D.; Czarnecki, J.; Masliyah, H. Measurement of ξ potential of gas bubbles in aqueous solutions by microelectrophoresis method. J. Colloid Interface Sci. 2001, 243, 128− 135. (16) Exerowa, D.; Churaev, N. V.; Kolarov, T.; Esipova, N. E.; Pauchev, N.; Zorin, Z. M. Foam and wetting films: Electrostatic and steric stabilization. Adv. Colloid Interface Sci. 2003, 104, 1−24. (17) Stubenrauch, C.; von Klitzing, R. Disjoining pressure in thin liquid foam and emulsion filmsNew concepts and perspectives. J. Phys.: Condens. Matter 2003, 15, R1197−R1232. (18) Ciunel, K.; Armelin, M.; Findenegg, G. H.; von Klitzing, R. Evidence of surface charge at the air/water interface from thin-film studies on polyelectrolyte-coated substrates. Langmuir 2005, 21, 4790− 4793.
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dx.doi.org/10.1021/la3003542 | Langmuir 2012, 28, 6060−6066
