Bubble−Solid Interactions in Water and Electrolyte Solutions

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Bubble-Solid Interactions in Water and Electrolyte Solutions Rada A. Pushkarova* and Roger G. Horn Ian Wark Research Institute, UniVersity of South Australia Mawson Lakes, Adelaide SA 5095, Australia ReceiVed March 6, 2008. ReVised Manuscript ReceiVed May 7, 2008 Surface forces between an air bubble and a flat mica surface immersed in aqueous electrolyte solutions have been investigated using a modified surface force apparatus. An analysis of the deformation of the air bubble with respect to the mutual position of the bubble and the mica surface, the capillary pressure, and the disjoining pressure allows the air-liquid surface electrical potential to be determined. The experiments show that a long-range, double-layer repulsion acts between the mica (which is negatively charged) and an air bubble in water and in various electrolyte solutions at low concentration, thereby indicating that the air bubble surface is negatively charged. However, there is clear evidence that charge regulation occurs at the air-water interface to maintain a constant surface potential, and as a result of this, the charge at this interface changes from negative to positive as the bubble approaches the mica surface. Because of the attraction that arises as a result of the charge reversal, a finite force is required to separate the bubble from the mica, though the mica remains wetted by the aqueous phase. At the low concentrations investigated, the potential on the gas-liquid interface is independent of the electrolyte type within experimental uncertainty.

1. Introduction Colloidal interactions between air bubbles and solid surfaces in water play a vital role in many technologies, including mineral processing, food processing, paper and plastics recycling, and biological systems. Such interactions are known to include electrical double-layer forces because there is ample evidence that air bubble surfaces are electrically charged. Understanding the origins of this charge and the solution conditions that affect it is of fundamental scientific interest. An improved understanding will also lead to a better ability to control the behavior of aqueous systems involving air bubbles. To this end, we report a method of determining the surface potential of air bubbles (from which surface charge is readily calculated) by measuring their interaction with a solid surface (mica) whose double-layer properties are well known. The measurement of interface potentials can encounter significant technical difficulties, especially if one or both of the surfaces in question are deformable, as is the case for an air bubble in water. Furthermore, the air-water interface is notoriously susceptible to the influence of surface-active impurities. There is a lack of agreement on the value of the air-water surface potential and the influence of various electrolytes on it. The literature reports a wide range of values ranging from -11001 mV through -80 to -108,2,3 -65,4,5 -45,6,7 -40,8 -34,9,10 -15 to -10,11 -10,12 and -0.6 mV.13 The large variation in * Corresponding author. E-mail: [email protected]. (1) Parfenyuk, V. I. Kolloidn. Zh. 2002, 64, 651–659. (2) Usui, S.; Sasaki, H.; Matsukawa, H. J. Colloid Interface Sci. 1981, 81, 80–84. (3) Usui, S.; Imamura, Y.; Sasaki, H. J. Colloid Interface Sci. 1987, 118, 335–342. (4) Graciaa, A.; Morel, G.; Saulner, P.; Lachaise, J.; Schechter, R. S. J. Colloid Interface Sci. 1995, 172, 131–136. (5) Creux, P.; Lachaise, J.; Graciaa, A.; Beattie, J. K. J. Phys. Chem. C 2007, 111, 3753–3755. (6) Zorin, Z. M.; Kolarov, T.; Esipova, N. E.; Platikanov, D.; Sergeeva, I. P. Kolloidn. Zh. 1990, 52, 666–673. (7) Hewitt, D.; Fornasiero, D.; Ralston, J. J. Chem. Soc., Faraday Trans. 1993, 89, 817–822. (8) Okada, K.; Akagi, Y.; Kogure, M.; Yoshioka, N. Can. J. Chem. Eng. 1990, 68, 393–399. (9) Bach, N.; Gilman, A. Acta Physicochim. URSS 1938, 9, 1–26. (10) Schulze, H. J.; Cichos, C. Z. Phys. Chem. (Leipzig) 1972, 251, 252–268. (11) Exerowa, D.; Churaev, N. V.; Kolarov, T.; Esipova, N. E.; Panchev, N.; Zorin, Z. M. AdV. Colloid Interface Sci. 2003, 104, 1–24.

the reported data for the air-water surface potential suggests that special attention should be paid to the selection and conduct of the measurement and data analysis techniques and that the system may be unusually prone to the effects of surface-active contamination. Experimental techniques that have been used to investigate the surface potential of bubbles in water include electrokinetics,2–4,8–10,12–18 the measurement of equilibrium wetting film thickness by interferometric methods,7,11,19–22 and AFM.23,24 Recently, we have used another technique based on the surface force apparatus (SFA).25 The SFA, originally developed by Tabor, Israelachvili, and others,26–28 was adapted in our laboratory by Connor29,30 for the measurement of interactions between deformable fluid drops and solid surfaces across an intervening liquid film. In particular, it has been demonstrated that this modification of the SFA allows the measurement of interactions between air bubbles and a mica surface immersed in aqueous (12) Samygin, V. D.; Derjaguin, B. V.; Dukhin, S. S. Kolloidn. Zh. 1964, 26, 493–501. (13) McShea, J. A.; Callaghan, I. C. Colloid Polym. Sci. 1983, 261, 757–766. (14) Collins, G. L.; Motarjemi, M.; Jameson, G. J. J. Colloid Interface Sci. 1978, 63, 69–75. (15) Huddleston, R. W.; Smith, A. L. Foams, Proc. Symp. 1976, 163–177. (16) Kelsall, G. H.; Tang, S.; Yurdakul, S.; Smith, A. L. J. Chem. Soc., Faraday Trans. 1996, 92, 3886–3893. (17) Kuznetsova, L. A.; Kovarskii, N. Ya. Kolloidn. Zh. 1995, 57, 657–660. (18) Usui, S.; Sasaki, H.; Hasegawa, F. Colloid Polym. Sci. 1983, 261, 757– 766. (19) Exerowa, D.; Kruglyakov, P. M. Foam and Foam Films: Theory, Experiment, Application; Elsevier: Amsterdam, 1998. (20) Karraker, K. A.; Radke, C. J. AdV. Colloid Interface Sci. 2002, 96, 231– 264. (21) Sedev, R.; Exerowa, D. AdV. Colloid Interface Sci. 1999, 83, 111–136. (22) Sheludko, A.; Exerowa, D. IzV. Inst. Fizikokhim. Bulg. Akad. Nauk. 1960, 1, 203–212. (23) Ducker, W. A.; Xu, Z.; Israelachvili, J. N. Langmuir 1994, 10, 3279– 3289. (24) Fielden, M. L.; Hayes, R. A.; Ralston, J. Langmuir 1996, 12, 3721–3727. (25) Pushkarova, R. A.; Horn, R. G. Colloids Surf., A 2005, 261, 147–152. (26) Tabor, D.; Winterton, R. H. Proc. R. Soc. London, Ser. A 1969, 312, 435–450. (27) Israelachvili, J. N.; Tabor, D. Proc. R. Soc. London, Ser. A 1972, 331, 19–38. (28) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1978, 74, 975–1001. (29) Connor, J. N. Measurement of Interactions between Solid and Fluid Surfaces: Deformability, Electrical Double Layer Forces, and Thin Film Drainage. Ph.D. Thesis, University of South Australia, Adelaide, Australia, 2001. (30) Connor, J. N.; Horn, R. G. Langmuir 2001, 17, 7194–7197.

10.1021/la8007156 CCC: $40.75  2008 American Chemical Society Published on Web 07/26/2008

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electrolyte.25,29,31 The SFA avoids some of the problems of electrokinetic methods, which are difficult to interpret when the moving “particles” are deformable.32 Electrokinetic methods can also suffer from the influence of turbulence and the possible charging of air bubble surfaces, as reported by Kuznetsova and Kovarskii.17 To some extent, the SFA technique resembles disjoining pressure isotherm methods in which the aqueous film thickness between air and a solid surface is measured by using optical methods (see, for example, reports of Exerowa et al.11 and Pashley and Kitchener33), but it allows a wider range of film thicknesses to be explored. The SFA also has the advantage, in common with AFM force measurements,23,24 that it can probe attractive forces as well as repulsive ones. Moreover, the SFA has significant advantages over AFM because it allows the measurement of the absolute bubble-surface separation as well as the measurement of the deformations of the bubble surface when it interacts with another body. Surface charging mechanisms can be explored with the SFA by measuring the surface potentials under various electrolyte conditions, as has been done previously for mica34 and silica35 surfaces. In this article, we present the results of SFA measurements of air bubble-mica interactions across dilute solutions of a number of aqueous electrolytes. Accurate measurements of the deformation of the bubble as it approaches the mica can be fitted by using a model based on well-established theories (the Young-Laplace equation and the Poisson-Boltzmann (PB) equation), and from this fit, it is possible to determine the magnitude and the sign of the surface potential of the bubble. It is demonstrated that the potential is small and negative on an isolated bubble, but the bubble surface undergoes charge reversal when it is close to the more highly charged negative mica surface. This results in an interaction that is weakly repulsive at long range but attractive at shorter range, leading to an adhesion between the bubble and the mica surfaces even though a thin wetting film of water remains between them.

2. Experimental Section In the modified SFA, an air bubble was formed in a blind cylindrical capillary of Teflon (internal diameter of 2.8 mm and depth of 1.5 mm) fixed to the bottom of a chamber that is filled with an aqueous electrolyte solution. A flat, horizontal sheet of mica of about 5 to 8 µm in thickness and ∼1 cm2 in area with a thin (10 nm) layer of silver forming a partial mirror on its upper surface was glued to a silica disk and mounted above the capillary. The disk carrying the mica could be moved vertically with subnanometer precision by using a differential spring mechanism, similar to that described by Israelachvili and Adams,28 to press down on the air bubble. White light interferometry in reflection36 was used to measure the distance between the air bubble-electrolyte interface and the mica surface, that is, the thickness of the aqueous film. This method measures the film thickness across a vertical section through the apex of the bubble, allowing an analysis of the curvature of the bubble and detailed measurements of how it was deformed by surface and hydrodynamic forces acting on the bubble surface. Special attention was paid to the purity of the solution in order to create and maintain a clean bubble surface. The water used in the experiments was passed through multistage reverse osmosis, deionization, and organic adsorption cartridges, followed by subboiling distillation in an all-quartz apparatus (Quartz et Silice, France).29,37 Electrolyte solutions were prepared from LiCl, KCl, (31) Connor, J. N.; Horn, R. G. Australian Institute of Physics 17th National Congress, Brisbane, December 2006. (32) Sotskova, T. Z.; Bazhenov, Yu. F.; Kul’skii, L. A. Kolloidn. Zh. 1982, 44, 989–994. (33) Pashley, R. M.; Kitchener, J. A. J. Colloid Interface Sci. 1979, 71, 491– 500. (34) Pashley, R. M. J. Colloid Interface Sci. 1981, 83, 531–546.

Langmuir, Vol. 24, No. 16, 2008 8727 NaCl, CsCl, KBr, KI, and KNO3 (AnalaR, BDH), and KF and KCH3COO (ACS reagents, Sigma-Aldrich). Solutions of the required concentration were prepared by dilution from a fresh stock solution of 1 M concentration and used within several hours of their preparation. The pH was measured before and after each experiment and remained at 5.6 ( 0.1 because of equilibration with CO2 in the air. All glassware for preparing and storing the solutions was cleaned immediately before the experiments with a glassware cleaning solution containing 2% chromic acid (Ajax Finechem, Australia) then KOH (10% w/w) before it was rinsed thoroughly with water (purified as described above). All cleaning, assembling, water distillation, and solution preparation procedures were conducted in laminar flow cabinets with HEPA filters removing >99% of airborne dust particles greater than 1 µm in size. The routine procedure for assembling the SFA for an experiment included washing all parts with ethanol (BDH AnalaR grade 99.7-100%; Merck Pty Ltd., Australia), wiping with low-lint tissues (Kimwipes, Kimberly Clark) while avoiding all contact with hands by wearing double gloves and using tweezers as much as possible, washing again with ethanol, then drying by a jet of highpurity nitrogen gas (BOC, Australia) from a gas gun (Wafergard Gas Filter Gun; Millipore). The SFA chamber was filled with water or with an electrolyte solution in a laminar flow cabinet. In the flow cabinet, before the chamber was sealed, bubbles were introduced by using a gastight syringe (Hamilton) with a stainless steel needle. First, a stream of bubbles was passed through the aqueous phase for 3-5 min to purge it. About 20-30 bubbles were then attached to a Teflon circle fixed at the bottom of the chamber to act as a sink for any remaining contaminants from the water or the solution. Only after that was the bubble to be investigated affixed to the blind Teflon capillary. The chamber was then sealed, and the apparatus was moved from the laminar flow cabinet to an optical table for white light interferometry measurements based on fringes of equal chromatic order observed in reflected light.36 In conventional SFA measurements, the force acting between two solid objects is determined by measuring the deflection of a cantilever spring to which one of the solids is attached. With the present setup there is no spring. However, the bubble deforms in response to an applied force, so in a sense, it provides its own spring. Attard38 proposed that it could be treated in a first approximation as a linear spring with a stiffness equal to the interfacial tension. A more accurate relationship between the force acting on a fluid drop and the amount by which its surface is compressed has been derived recently by Carnie, Chan, and coworkers,39,40 who showed that the “spring” is nonlinear. The instantaneous spring stiffness (gradient of the force-displacement curve) has the same order of magnitude as the surface tension, so we will continue to use the surface tension as an effective spring constant of the bubble to make approximate estimates of forces later in the article. The simple concept of the bubble as a spring allows a qualitative understanding of the behavior observed in the experiments and a semiquantitative determination of the total force exerted on the bubble’s surface. However, more detailed information can be obtained by modeling the experiment using the well-established YoungLaplace and PB equations. The method was presented by Horn et al.41 and is outlined in the following section. Similar methods have (35) Grabbe, A.; Horn, R. G. J. Colloid Interface Sci. 1993, 157, 375–383. (36) Connor, J. N.; Horn, R. G. ReV. Sci. Instrum. 2003, 74, 4601–4606. (37) Pushkarova, R. A. Investigation of Bubble-Solid Interactions Using Surface Force Apparatus. Ph.D. Thesis, University of South Australia, Adelaide, Australia, 2006. (38) Attard, P.; Miklavcic, S. J. Langmuir 2001, 17, 8217–8223. (39) Manica, R.; Connor, J. N.; Dagastine, R. R.; Carnie, S. L.; Horn, R. G.; Chan, D. Y. C. Phys. Fluids 2008, 20, 032101-1–032101-12. (40) Chan, D. Y. C.; Manor, O.; Connor, J. N.; Horn, R. G. Soft Matter 2008, 4, 471–474. (41) Horn, R. G.; Bachmann, D. J.; Connor, J. N.; Miklavcic, S. J. J. Phys.: Condens. Matter 1996, 8, 9483–9490. (42) Carnie, S. L.; Chan, D. Y. C.; Lewis, C.; Manica, R.; Dagastine, R. R. Langmuir 2005, 21, 2912–2922. (43) Dagastine, R. R.; White, L. R. J. Colloid Interface Sci. 2003, 247, 310– 320.

8728 Langmuir, Vol. 24, No. 16, 2008 subsequently been used to analyze AFM measurements on deformable surfaces.42–44 Experimentally what is required is to move the mica toward (or away from) the fixed capillary holding the bubble and to measure the resultant minimum film thickness, that is, the gap between the apex of the air bubble surface and the mica surface. For most of the measurements presented below, the mica was moved in a series of ∼100 nm steps repeated every 10 s while optical interference fringes were recorded with a sensitive video camera (Dage MTI VE 1000 SIT camera and controller unit). After each drive step there was a pause of 2 s, followed by data collection of 3-5 video frames during the following 8 s (the rate being limited by the time taken to digitize and store the video signal using a frame grabber) before repeating the process. An estimate from the Taylor equation of the hydrodynamic force45 acting on the bubble at this rate of approach (∼10 nm/s on average) shows that if the bubble remains undeformed then the hydrodynamic force remains less than the minimum detectable force down to separations of a few tens of nanometers. Hence, any force detected at larger separations than this can be attributed to nonhydrodynamic effects (e.g., a double-layer force). However, when it does occur, double-layer repulsion tends to flatten the bubble, increasing the radius of curvature near its apex. This effect, combined with the decreasing film thickness as the surfaces are pressed closer, increases the hydrodynamic force so that it cannot be ignored at small separations. Non-negligible hydrodynamics complicates the analysis of the results to be presented below, but it is not practicable to drive the surfaces more slowly during the experiment because of concerns about thermal drifts and possible contamination of the air-water interface that could occur if the experiment were to be prolonged. An important parameter in the experiment is the Laplace pressure of the bubble (i.e., the excess internal pressure with respect to the ambient pressure). This was found for each particular experiment from measurements of the curvature of the undeformed bubble apex when it was far from the mica so that the two surfaces were not interacting. Typical values of the radius were ∼2 mm, corresponding to Laplace pressures of ∼70 Pa. Measurements of surface tension were made by the maximum bubble-pressure method using a Lauda MPT2 tensiometer. The measurements were conducted for randomly chosen solutions at the beginning and at the end of the experimental runs. The value of the surface tension was in the range 0.072 ( 0.002 N/m for all of the tested solutions (compared with the tabulated value of 0.0728 N/m46), which shows that the bubble surface remained clean throughout the experiments.

3. Theoretical Analysis The Young-Laplace equation relates the local curvature of a fluid interface to the pressure difference across that interface. In the SFA setup, contributions to the pressure come from the internal pressure within the bubble (also called Laplace or capillary pressure); the variation of hydrostatic pressure with height due to gravity; the disjoining pressure in the aqueous film resulting from surface forces acting between the mica and bubble surfaces; and possibly the hydrodynamic pressure if the surfaces are in relative motion. Including the first three of these gives the augmented Young-Laplace (AYL) equation, which can be solved numerically under the appropriate boundary conditions: the bubble surface is horizontal at its apex and the bubble is anchored with a fixed radius at the top of the capillary.41 If the disjoining pressure Π is known as a function of the film thickness, then an integration of the AYL equation allows the shape of the entire bubble surface to be calculated for a given mica-capillary distance, which we denote as X. The film thickness is denoted as h(r), where r ) |y| is the horizontal distance from the symmetry axis of the capillary (44) Nespolo, S. A.; Chan, D. Y. C.; Grieser, F.; Patrick, G.; Stevens, G. W. Langmuir 2003, 19, 2124–2133. (45) Chan, D. Y. C.; Horn, R. G. J. Chem. Phys. 1985, 83, 5311–5324. (46) Handbook of Chemistry and Physics, 78th ed.; Lide, D. R., Ed.; CRC Press LLC: Boca Raton, FL, 1997.

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Figure 1. Schematic illustration of the difference between the virtual and actual minimum film thicknesses. (a) Bubble and mica far apart and not interacting and (b) the effect of a repulsive interaction (e.g., electrical double-layer force) with mica closer to the bubble surface.

with y being the horizontal axis shown in Figure 1a. From this numerical solution, it is possible to predict how the minimum film thickness h(0) ) h0 varies as X varies. Recall that X is controlled in the experiment, and h is measured, so these are appropriate independent and dependent variables for the comparison with data. The general procedure is to adjust the disjoining pressure function Π(h) in the AYL model until a good fit between the model and the data is obtained. It is convenient to subtract the height above the capillary of the “noninteracting” bubble, z0, from X. The noninteracting bubble is one that feels the effects of gravity and its own internal pressure but not of disjoining or hydrodynamic pressure. This is the shape that the bubble adopts when the mica is far away from it. Setting

∆X ) X - z0

(1)

resets the scale so that ∆X ) 0 at the (virtual) position where the mica would just touch the top of the bubble if the latter did not deform as the mica approached. It is simple to convert the known displacement of the mica to the ∆X scale from the experimental data because ∆X ) h0 when the mica and bubble are far apart. Note that ∆X can go to negative values if the bubble is flattened by repulsive interactions with the mica, and the mica is pushed down past the original apex position of the noninteracting bubble. Figure 1 gives a graphical depiction of this notation. For simplicity, we will refer to ∆X as “mica displacement”. The inclusion of hydrodynamic effects makes for a more complicated problem because the hydrodynamic pressures themselves depend on the shape of the bubble (or fluid drop). However, the problem can be solved numerically by combining the Reynolds lubrication equations with the AYL equation and by including the appropriate boundary conditions. This has recently been done for a constant-volume liquid drop in the SFA setup,47 and the model has been shown to fit well to experimental data.48 Figure 2 shows a number of possibilities for the variation of the minimum film thickness h0 as a function of the mica position ∆X for different disjoining pressure isotherms Π(h). The isotherms are calculated from DLVO theory, combining electrical doublelayer and van der Waals (vdW) interactions. In this system, the mica-aqueous solution-air vdW force is repulsive, meaning that water is attracted to mica more strongly than air, promoting a wetting film. An accurate calculation of the (positive) vdW (47) Manica, R.; Connor, J. N.; Carnie, S. L.; Horn, R. G.; Chan, D. Y. C. Langmuir 2007, 23, 626–637. (48) Connor, J. N.; Horn, R. G. Faraday Discuss. 2003, 123, 193–206.

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a wetting film on mica when the electrolyte concentration is above a threshold that is specific to the cation species. The double-layer disjoining pressure can be positive or negative depending on the potentials on the bounding mica and air surfaces. The double-layer pressure was calculated from the nonlinear PB equation by using the Chan-Pashley-White (CPW) algorithm,52 assuming constant-potential boundary conditions. Some experimental observations will be presented later that show that this boundary condition is appropriate for the bubble surface. The exact nature of the boundary condition at the mica does not have a strong bearing on the results. We found that the Hogg-Healy-Fuerstenau (HHF) linear approximation for the disjoining pressure between two surfaces of different constant potential53,54

Πedl ) -

Figure 2. (a) An example set of disjoining pressure isotherms calculated for a constant-potential double-layer (edl) interaction plus the van der Waals (vdW) component. The Laplace pressure inside the bubble is shown by a horizontal dotted-dashed line for P0L ) 100 Pa. Conditions: 1 mM 1:1 electrolyte, surface potentials +100/-100 mV (dotted line, curve 1); -4.5/-100 mV (dashed line, curve 2a, long-range part due to edl forces, 2b, short-range repulsion due to vdW forces); and -100/ -100 mV (solid line, curve 3). (b) Schematic picture of the h0 vs ∆X curve. The solid line represents typical experimental data. The dashed line shows the theoretical calculation corresponding to curve 2 in part a when dissimilar-magnitude potential edl and vdW forces are acting in the liquid film. The dotted-dashed line shows purely repulsive (curve 3) interactions and the dotted line shows purely attractive (curve 1) interactions. Regions A-G are discussed in the text.

contribution to the disjoining pressure is made by using Lifshitz theory with an algorithm written by Grabbe49 using the Parsegian50 representation of the dielectric response function of water and that of Horn and Smith51 for mica. The results of this calculation are fitted by the following empirical expressions for the vdW pressure in water

log ΠvdW(h) ) -0.4627(log h)2 - 11.5582 log h - 60.3763 (2) and for 1 mM electrolyte

log ΠvdW(h) ) -0.1602(log h)2 - 5.986 log h - 34.863 (3) In these expressions, Π is given in pascals, h is given in nanometers, and log means base 10. Another force that may contribute to the short-range repulsion between the bubble and the mica is the hydration force. It is known from the literature34 that hydration forces exist between two mica surfaces immersed in electrolyte solution at concentrations above 0.1-10 mM, depending on the cation. It is not clear whether this force would exist between the mica and the air bubble. If it does occur, then the hydration effect could stabilize (49) Grabbe, A. Langmuir 1993, 9, 797–801. (50) Parsegian, V. A. J. Colloid Interface Sci. 1981, 81, 285–289. (51) Horn, R. G.; Smith, D. T. Appl. Opt. 1991, 30, 59–65.

εκ2 [(ψ 2 + ψ022) cosech(κh) 2 sinh(κh) 01 2ψ01ψ02 coth(κh)] (4)

in which ε is the dielectric permittivity for water, κ is the Debye-Huckel parameter, and ψ01 and ψ02 are the surface potentials on the two surfaces that are separated by a distance h, gives a remarkably accurate representation of the full nonlinear PB solution if a simple shift in film thickness is applied. This is described in the Appendix. Because the HHF expression is more computationally convenient than the CPW algorithm, it was used, with the appropriate shift in h, in our numerical solution of the AYL equation. The potential on the mica was measured in separate mica-mica SFA experiments and found to be -120 ( 5 mV in distilled water, -110 ( 5 mV in 0.1 mM KCl, and -100 ( 5 mV in 1 mM KCl solutions.37 These values are consistent with the SFA results of Pashley34 and also with results of electrokinetic measurements by Hartley,56 Scales,57 Lyons58 and Debacher.59 The potential on the air-electrolyte interface was treated as an adjustable parameter, with different values used in eq 4 until the best fit to each experimental h(∆X) curve was obtained. Surfaces of different potentials can interact in a number of ways, as illustrated in Figure 2a. If the two surface potentials have opposite signs, then the force is attractive at long range, and the disjoining pressure is negative (curve 1). If the potentials are of unequal magnitudes and opposite signs, then the force becomes repulsive at shorter range (not shown). When the potentials are of the same sign, the force is repulsive at long range (more than 1 to 2 debye lengths); however, if the potentials are unequal in magnitude, then there is a phenomenon known as charge reversal in which the higher-magnitude potential surface compels the charge on the lower-magnitude potential surface to change sign, and the force switches from repulsive to attractive as the surfaces approach.54,60–63 This behavior is illustrated by (52) Chan, D. Y. C.; Pashley, R. M.; White, L. R. J. Colloid Interface Sci. 1980, 77, 283–285. (53) Hogg, R.; Healy, T. W.; Fuerstenau, D. W. Trans. Faraday Soc. 1966, 62, 1638–1651. (54) Hunter, R. J. Foundations of Colloid Science; Claredon Press: Oxford, U.K., 1987; Vol. 1. (55) Derjaguin, B. V. Theory of Colloids and Thin Film Stability; Nauka: Moscow, 1986. (56) Hartley, P. G.; Larson, I.; Scales, P. J. Langmuir 1997, 13, 2207–2214. (57) Scales, P. J. Langmuir 1990, 6, 582–589. (58) Lyons, J. S.; Furlong, D. N.; Healy, T. W. Aust. J. Chem. 1981, 34, 1177–1187. (59) Debacher, N.; Ottewill, R. H. Colloids Surf. 1992, 65, 51–59. (60) Derjaguin, B. V. Faraday Discuss. 1954, 18, 85–98. (61) Derjaguin, B. V.; Churaev, N. V. Wetting Films; Nauka: Moscow, 1984. (62) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Surface Forces; Consultants Bureau: New York, 1987. (63) Mishchuk, N. A.; Koopal, L. K.; Dukhin, S. S. J. Colloid Interface Sci. 2001, 237, 208–223.

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curve 2 in Figure 2a. If the potentials are equal, then the doublelayer force is repulsive at all separations (curve 3), or if they are nearly equal, then the repulsive maximum is high (greater than the Laplace pressure in the bubble), which prevents the charge reversal region from being accessed. Inserting disjoining pressure isotherms of these types in the AYL equation leads to predictions of different types of behavior in the h0 versus ∆X plots41 (Figure 2b). With surfaces beyond the range of double-layer forces, the predicted behavior follows the line h0 ) ∆X (region A). When the disjoining pressure is positive, the repulsive force pushes the bubble surface away from the mica, which makes h0 > ∆X, and the curve lies above the line h0 ) ∆X (region B). The surfaces would continue to approach until they reached a separation at which Π(h0) equaled the internal pressure in the bubble. In this situation, there is no pressure difference across the air-water interface, so it becomes flat. If the maximum in Π(h) exceeds the internal bubble pressure P0L, then pressing the mica down further increases only the extent of the flat region without causing the surfaces to come any closer (region E). If the surface potentials were opposite, then the force would be attractive at long range, which would pull the top of the bubble toward the mica, making h0 < ∆X and deviating the curve below the line and then back to the right (larger ∆X) (region F). Note that regions of negative gradient are mechanically unstable, and the two surfaces would jump to the next stable part of the curve at a smaller separation, in this case, a short-range repulsive region where a wetting film is stabilized by vdW forces (curve 2b). Potentials of the same sign but different magnitudes repel at long range, so the behavior on approach initially follows region B. However, at a certain separation (typically 1 to 2 debye lengths54) charge reversal causes the force to turn attractive, and the h0-∆X curve bends down (region G). The short-range vdW repulsion would cause the curve to turn back again to the left, so the prediction is that on approach there would be a jump from a thick aqueous film to a thin wetting film. These curves are reminiscent of Derjaguin’s55 discussion of wetting films of type I (corresponding to curve 3 in Figure 2a and region E in Figure 2b) or type II (corresponding to curve 2 in Figure 2a and region G in Figure 2b). Regions C and D will be discussed below.

4. Results Minimum Separation versus Mica Displacement. Following the protocol described in the Experimental Section (moving the mica toward the bubble apex in a regular series of equal steps of ∼100 nm repeated at 10 s intervals), measurements of the minimum separation h0 as a function of the distance from the undeformed bubble position ∆X have been conducted in water and in a series of electrolytes with different cations and a constant anion (LiCl, NaCl, KCl, CsCl) and with different anions and a constant cation (KF, KCl, KBr, KI, KCH3COO). The dependence of the minimum separation on the undeformed bubble position has similar character for all electrolytes and water, which is represented schematically by regions A-D in Figure 2b. Two representative data sets are shown in Figure 3. At large separations, the bubble remains undeformed, and the apex position is constant relative to the end of the capillary. This stage corresponds to the region labeled A in Figure 2b. The exact step size is obtained by calibrating the drive against the change in h0 in this linear region. As the separation decreases, the apex of the bubble surface deflects from the undeformed position in a direction that indicates a repulsion between the mica and the bubble surfaces (region B in Figure 2b). In distilled water, the deflection starts from a separation of ∼150-200 nm (Figure 3a). For all electrolyte solutions with a concentration of 1 mM, the

PushkaroVa and Horn

Figure 3. Change in minimum separation as a function of mica displacement (a) in water and (b) in 1 mM KI solution in the range of separations where the deflection from an undeformed shape begins. The solid lines represent the baseline h0 ) ∆X corresponding to the undeformed bubble shape. Letters label the regions of behavior corresponding to Figure 2b and are discussed in the text.

deviation from the baseline starts from a smaller separation of 50 to 70 nm. The dependence of the range of force on the electrolyte strength is good evidence that the repulsion arises from electrical double-layer interactions. As the mica is pressed further down on the bubble, the flattening of the bubble increases, and hydrodynamic effects resulting from the finite drainage rate of the thin aqueous film come into play. The observed behavior shows a quasi-linear decrease in h0 as ∆X becomes increasingly negative (region C in Figure 2b). This “ramp” persists over a wide range of ∆X. Region C is observed in all experimental solutions, and the ramp is quasi-linear for all solutions except KCl and CsCl, in which region C is slightly curved. Note that region E (a thick wetting film stabilized by a double-layer repulsion) is never observed in our experiments. When the minimum film thickness is reduced further, it reaches a constant value on the order of nanometers. In this regime, further movement of the mica results in increased deflection of the bubble but no further decrease in the separation (region D). A thin wetting film of a measurable thickness in the nm range is formed in 1 mM solutions of NaCl (7 ( 1 nm), KCl, KI, KCH3COO, and KNO3 (3 ( 1 nm). In other solutions, including water, the measured film thickness is small and is comparable to the measurement uncertainty in h0 (1 ( 1 nm), so the resolution of the optical interference method does not allow the proof or disproof of the existence of a thin film less than ∼2 nm thick between the air bubble and the mica. However, observations before and after each experiment show that the advancing contact angle of the electrolyte solution on the mica surface is small and finite (∼50) and independent of the electrolyte type; therefore, it is likely that a thin film is formed in all of the experiments. Shape of the Bubble. When mica is driven against the bubble, the bubble shape shows four qualitatively different stages of deformation. A typical set of data is shown in Figure 4. The first stage occurs when the bubble and the mica are at a large separation,

Bubble-Solid Interactions

Langmuir, Vol. 24, No. 16, 2008 8731 Table 1. Estimation of the Adhesion Between 2-mm-Radius Bubbles and Mica

Figure 4. Changing bubble profiles as the mica is driven in a series of steps toward the bubble in a 0.1 mM KCl solution. The initial radius of the noninteracting bubble is 2.0 mm. ∆X values (in nm) and the corresponding regions discussed in the text for the illustrated bubble shapes are (from bottom to top): 177 (region A); 64, -49, -613, -5240 (region B); -11 680, -18 230, -23 200, -29 300, -33 810, -38 330, -42 280 (region C); and -45 670 (region D). Note that the scale of the figure (nm vs µm) greatly exaggerates the apparent curvature of the bubble surface.

which corresponds to region A in Figure 2b. The bubble has a spherical shape near its apex and remains undeformed. The second stage occurs when the separation comes within the range of surface forces, and the bubble surface begins to flatten. The flattening is a typical deformation for all bubbles investigated and begins at a distance in the range of 50-200 nm, depending on the electrolyte strength. This corresponds to region B in Figure 2b. The deflection from the undeformed shape is observed in water at a separation of between 150 and 200 nm. In a dilute electrolyte solution (0.1 mM KCl), the flattening of the bubble surface begins at a separation of about 100 nm. In all of the 1 mM electrolyte experiments, the flattening is observed beginning at separations of between 50 and 70 nm. Because it is difficult to identify precisely where the deformation starts, the differences within this latter range are not considered to be significant. In the third stage (corresponding to region C in Figure 2b), as the mica is pressed further toward it, the bubble becomes more rounded again after the previous flattening in region B. This is a general observation that is made in all of the electrolyte solutions. Finally the fourth stage (region D) corresponds to complete flattening of the bubble surface, which occurs without further decrease in the separation. Thin films of a few nanometers in thickness were formed, as described in the previous subsection. The features shown in Figure 4 are qualitatively consistent with similar measurements reported by Connor.29,31,48 Adhesion Between Bubble and Mica. After pressing the mica down on the bubble until a nm-scale film has been formed (region D), the mica is then retracted. This is done in a series of 500 nm steps repeated at 10 s intervals. In every case, the system demonstrates adhesive behavior, manifested in a sudden jump apart from a very small separation (on the order of nanometers) to a much larger one (>10 µm), in the same way that adhering solids jump apart because of the mechanical instability of the spring on which one of them is mounted.28 In the present configuration, the air bubble also acts like a spring. As an order-of-magnitude estimate of the pull-off force that is required to separate the mica from the bubble, the jump-out distance was multiplied by the surface tension.38 The results for different electrolytes are listed in Table 1. It is seen that there is a variation in the magnitudes of the pull-off force, but it should be stressed that all experiments show adhesion between the mica and the bubble. The bubble has become attached to the mica through a stable thin film, as predicted by Derjaguin60 in DLVO

solution

concentration (mM)

film thickness from which jump out occurs (nm)

jump out (µm)

estimated pull-off force (µN)

water LiCl KCl KCl CsCl KF KBr KI

∼5.6 × 10-3 1 1 0.1 1 1 1 1

1(1 1(1 3(1 1(1 1(1 1(1 1(1 3(1

25 48 25 34 85 58 75 27

1.8 3.5 1.8 2.4 6.1 4.2 5.4 1.9

theory for the interaction between two surfaces of the same sign but significantly different magnitude potentials. To be sure that this is a result of an attractive surface force acting between the bubble and the mica surfaces, we must examine the possibility that such a behavior could result from hydrodynamic effects. It is known that the viscosity of a liquid medium can create an apparent adhesion between two bodies, such as a sphere and a flat surface brought close together in that medium,64 and indeed, this is the basis for everyday pressure-sensitive adhesives. Francis65 analyzed this effect in detail and found a universal relationship between the peak adhesive force and the initial surface separation when a spherical surface is pulled away from a flat one, using a spring (or a compliant machine) being driven at constant speed. This is close to the experimental situation here (taking the average speed of separation as the step size divided by the step interval and assuming a spring constant of the bubble equal to its surface tension), except that the deformation of the bubble is not considered. As a first estimate, we consider the bubble to have its initial undistorted radius. Of course, the bubble is significantly flattened in region D, but as the mica is retracted, the bubble becomes rounded again, even somewhat elongated before the abrupt jump out. The elongation reduces the hydrodynamic drag so that the calculation assuming constant radius probably gives an overestimate. Using eq 15b of ref 65, we can estimate that the hydrodynamic adhesion of a 2-mmradius sphere brought to within 1 nm of a flat surface in water and then separated at a rate of 50 nm/s would be about 40 nN. Because the adhesive force observed in the experiment is 2 orders of magnitude greater than this value, the observed jump out must be attributed to effects other than hydrodynamics. The observation that a finite force is required to pull the mica away from the bubble provides clear evidence that there is an attractive force between them for at least some range of the surface separation.

5. Discussion Three qualitative conclusions can immediately be drawn from the observed behavior: (1) When the air bubble is separated from the mica by a few debye lengths, flattening of the bubble indicates a repulsive interaction between the surfaces. Because the mica is negatively charged at the low electrolyte concentrations investigated here, the air bubble surface must also be negatively charged. The surface potential of the air bubble will be quantified below. (2) The observation of an adhesive force between the bubble and the mica indicates that the disjoining pressure must be negative over at least some range of separation. From the preceding observation, and from the fact that the vdW contribution to the disjoining pressure is positive for the mica–electrolyte–air system, the only way to account for this is by a region of electrostatic attraction. (Hydrophobic attraction is ruled out (64) Matthewson, M. J. Philos. Mag. 1988, 57, 207. (65) Francis, B. A.; Horn, R. G. J. Appl. Phys. 2001, 89, 4167–4174.

8732 Langmuir, Vol. 24, No. 16, 2008

because the mica surface remains hydrophilic before and after the experiments.) The electrostatic repulsion at long range and the attraction at short range indicates that the phenomenon of charge reversal occurs when two surfaces approach with their surface potentials kept constant. Because air is the weaker of the two surfaces (the magnitude of its surface potential is less than that of the mica), the sign of the charge at the air-water interface changes from negative when the bubble is isolated to positive when it is within 1 to 2 debye lengths of a strong negative surface such as mica. (3) When an air bubble is pressed against the mica, a very thin (on the order of nanometers) film of water remains between them. This is consistent with a positive disjoining pressure arising from the vdW force and is also consistent with the observed low contact angle of water on mica. The possibility of a hydration effect on mica was mentioned in the theory section, but we find no evidence of a corresponding variation in film thickness with electrolyte type. Such a variation would have been expected from the results of ref 34, which showed that a hydration repulsion exists in 1 mM KCl but not in 1 mM NaCl. If hydration was affecting the film thickness, then it would be expected to be greater for the former than the latter, but we observe the opposite (measured film thickness 3 ( 1 nm in KCl compared with 7 ( 1 nm in NaCl). One feature of the experimental data does not fit the qualitative picture for the equilibrium forces embodied in Figure 2b, and it currently lacks a quantitative analysis. Figure 2b has regions labeled A, B, D, E, F, G, all of which correspond to possible behaviors based on DLVO interactions. However, the data show the behavior represented by curve C: a ramp region in which h decreases quasi-linearly as ∆X decreases. Over the same range, the deformation of the bubble is observed to become less flattened and slightly more elongated (though still considerably flatter than the original undeformed curvature) as the surfaces approach (Figure 4). The likely explanation for this behavior lies in hydrodynamic interactions between the surfaces associated with the drainage of the thin aqueous film that separates them. Although several seconds were allowed for equilibration between each step in the drive and the subsequent measurement of the bubble shape and its separation from the mica, it is probable that this waiting period was not long enough in region C, and some residual hydrodynamic effects (thin-film drainage) influenced the measurements. In the absence of hydrodynamics, we would expect an instability with a jump in from the repulsive region (a metastable thick film, discussed below) through the attractive region of the disjoining pressure to a stable thin film (region G of Figure 2b); it appears that hydrodynamic effects have prevented this jump from being observed. Recently, Chan and coworkers47 analyzed the thin-film drainage behavior of a comparable experiment in which the air bubble is replaced by a drop of mercury held at a fixed electrical potential, and a flat mica surface is driven toward it. The theoretical model, incorporating both hydrodynamic and electrical double-layer interactions between dissimilar constant potential surfaces, is very successful at fitting experimental data for the mercury drop.48 However, the same model has not been able to fit the data presented here for the mica approaching an air bubble (R. Manica, personal communication). Although the model does include the possibility of charge reversal affecting the disjoining pressure, it does not include the possible effects of gas compressibility, dissolution, nonuniform interfacial tension due to Marangoni effects, or more-subtle electrokinetic effects coupling surface

PushkaroVa and Horn

Figure 5. Comparison of the data measured in region B (symbols) and the theoretically predicted (-) deformation at the bubble’s apex for different potentials (-3 (- · · -), -4 (---), -4.5 ( · · · ), and -5 mV ( · - · -) at the air-liquid interface in (a) 1 mM LiCl and (b) 1 mM KBr solutions. Different symbols show repeat measurements. Table 2. Potential and Charge at the Air Bubble-Aqueous Solution Interface electrolyte

concentration (mM)

potential (mV)

charge density (mC/m2)

water LiCl NaCl KCl KCl CsCl KF KBr KI KCH3COO

∼5.6 × 10-3 1 1 1 0.1 1 1 1 1 1

-9.0 ( 1.0 -4.5 ( 0.5 -4.0 ( 0.5 -3.5 ( 0.5 -5.0 ( 0.5 -3.5 ( 0.5 -3.7 ( 0.5 -3.0 ( 0.5 -4.0 ( 0.5 -4.0 ( 0.5

-0.073 ( 0.007 -0.36 ( 0.04 -0.29 ( 0.04 -0.25 ( 0.04 -0.11 ( 0.01 -0.25 ( 0.04 -0.27 ( 0.04 -0.22 ( 0.04 -0.29 ( 0.04 -0.28 ( 0.04

tension and surface charge to the electrolyte flow.66,67 The failure of the model to describe the air bubble behavior is presumably associated with one or more of these phenomena. We now return to discussing the surface potential of the air bubble. Although the region C behavior is not fully understood, the behavior at slightly larger separations (region B) can be explained and modeled according to DLVO theory. To do this, we calculate curves in the form shown in Figure 2b by using the known value of the mica surface potential, and various values of the air surface potential, until the best fit to the data is obtained. Typical results are shown in Figure 5, and the surface potentials obtained from a range of experiments in different electrolyte solutions are summarized in Table 2. Several features are apparent: (a) All of the surface potential values are low in magnitude, ranging from -3.0 to -4.5 mV in 1 mM electrolyte solutions and -9 mV in water. These results are in good agreement with the results of Samygin et al.12 obtained (66) Marrucci, G. Chem. Eng. Sci. 1969, 24, 979–985. (67) Yaminsky, V. V., Ohnishi, S., Vogler, E. A., Horn, R. G. In preparation. (68) Sherwood, J. D. J. Fluid Mech. 1986, 162, 129–137.

Bubble-Solid Interactions

Figure 6. Example of the solution of the PB equation in the linear HHF approximation (O), the nonlinear method (solid line), and the linear approximation with shift (+). Conditions: water (∼5.6 × 10-3 mM), potentials ψ01 ) -12 mV, ψ02 ) -120 mV, shift in κh ) 0.3503.

by the Dorn effect, McShea and Callaghan13 obtained by electrophoresis, and Exerowa et al.11 and Hewitt et al. (for electrolyte)7 obtained by the thin-film balance technique. However, the electrokinetic measurements of Bach and Gilman,9 Usui et al.,2,3 Okada et al.,8 Graciaa et al.,4,5 Schulze and Cichos,10 and Huddleston (cited in ref 68) and the thin-film measurements of Zorin et al.6 and Yaminsky et al.67 report higher-magnitude potentials than those measured in the current work. (b) The magnitude of the potential decreases with concentration, whereas the magnitude of the negative charge increases from -0.07 mC/ m2 in water to -0.36 mC/m2 in 1 mM LiCl. The results indicate that the presence of an electrolyte, even in small concentrations, reduces the potential. For 0.1 mM KCl, the potential is -5 ( 0.5 mV, approximately half of the magnitude that it has in pure water. A further increase in the concentration does not change the potential as much: for a 1 mM solution of KCl, the potential is -3.5 ( 0.5 mV. These changes are consistent with compression of the electrical double layer.5 (c) There is no clear dependence of surface potential on the electrolyte type at these low concentrations. It is known that there are electrolyte-specific effects on the surface tension of water69,70 and on bubble coalescence,71,72 but it is also known that these effects become significant only at higher concentrations, on the order of 100 mM. Creux et al.5 observed slight effects of the cation and no effects of the anion on the bubble surface potential. Note that the low magnitude of surface potential on the air bubble is consistent with the observation of an attractive region. If the potential were more negative, then double-layer repulsion between the mica and the bubble would be sufficiently strong to prevent the surfaces from approaching closer than the point at which repulsive disjoining pressure (at long range) equals the internal Laplace pressure of the bubble, and the surfaces would not reach the charge-reversal region. Instead, a stable thick film, tens of nanometers thick, would have been observed (region E). Figure 2 shows that a potential of -4.5 mV on the air bubble (with -100 mV on the mica) is enough for the repulsive maximum to be comparable to a typical Laplace pressure of 100 Pa in a 1.4-mm-diameter bubble at a film thickness of some 70 nm. The height of the maximum is sensitive to the air bubble potential, and higher (magnitude) potentials than this would mean that no (69) Pugh, R. J.; Weissenborn, P. K.; Paulson, O. Int. J. Miner. Process. 1997, 51, 125–138. (70) Weissenborn, P. K.; Pugh, R. J. J. Colloid Interface Sci. 1996, 184, 550– 563. (71) Craig, V. S.; Ninham, B. W.; Pashley, R. M. J. Phys. Chem. 1993, 97, 10192–10197. (72) Craig, V. S.; Ninham, B. W.; Pashley, R. M. Nature 1993, 364, 317–319.

Langmuir, Vol. 24, No. 16, 2008 8733

attractive force would be experienced, and no adhesion would be observed. As discussed above, the double-layer force changes from repulsion to attraction when two constant-potential surfaces of the same sign but unequal magnitudes of surface potential approach each other.54,60–63 AFM measurements by Fielden et al.24 of the force between a hydrophilic glass sphere and an air bubble indicated a weak adhesion that may also have been due to charge reversal. Derjaguin and Churaev61 and Mishchuk et al.63 suggested that particle-bubble attachment may occur because of the presence of a force minimum at a finite separation, allowing the possibility of “contactless flotation” in mineral or other particle separation processes. If the local maximum of the disjoining pressure is less than the internal bubble pressure, then the bubble surface is not prevented from coming closer to the solid surface to a distance where an attractive force is balanced by the shorterrange vdW repulsion, and a stable thin film is formed. This is what Derjaguin55 calls a “type II” wetting film. Many authors have discussed the origins of the negative surface potential at an air-water (or oil-water) interface. The present results at low electrolyte concentration and a single pH (5.6) do not enable us to add any new information to the debate. A common view is that the negative charge at an air-water (or oil-water) interface arises from the adsorption of excess OH- at the water surface. We remark that OH- is nothing other than a water molecule that is missing a proton, and so it makes just as much, or perhaps more, sense to talk about the depletion of H+ from the interface. A driving force for such a depletion could arise from the fact that the hydrogen bond network cannot adopt its normal 3-D icelike structure when interrupted by an interface with a nonpolar medium.73 Hydrogen bonding is thus frustrated at the water surface (accounting for the comparatively high surface tension of water); the frustration could be partially alleviated (at the expense of electrostatic energy) if a small fraction of the water molecules were to lose a proton. From the values of the surface charge given above, the number of ionized water molecules per unit area of surface is 1 per 2200 nm2, which is only about 1 in 2 × 104 surface molecules.

6. Conclusions The experiments show that a long-range double-layer repulsion is present between mica and an air bubble in water and electrolyte solutions of different concentrations. However, this repulsion is not strong enough to stabilize a thick wetting film. The doublelayer repulsion indicates that the air bubble surface is negatively charged. Our data show that the charge is very weak, with surface potentials of only a few millivolts in the low electrolyte concentrations investigated here. There is clear evidence that charge regulation occurs at the air-water interface to maintain a constant surface potential, and as a result of this, the charge at this interface changes from negative to positive as the bubble approaches the negatively charged mica surface. This results in the double-layer force turning from a repulsion to an attraction at smaller separations, and a thin film is stabilized by vdW pressure. Because of the attraction, a definite pull-off force is required to separate the bubble from the mica, though the mica remains wetted by the aqueous phase. This form of disjoining pressure corresponds to water having a finite contact angle atop a thin wetting film, a situation known as pseudopartial wetting in the wetting literature. The charge reversal should have led to an unstable region of film thickness when the bubble was approached by a mica surface, (73) Kudin, K. N., Car, R. J. Am. Chem. Soc., 2008, DOI 10.1021/ja077205t, published online March 1, 2008.

8734 Langmuir, Vol. 24, No. 16, 2008

PushkaroVa and Horn

Table 3. Shift in Kh: 2-D Interpolation in Water (∼5.6 × 10-3 mM)

Table 4. Shift in Kh: 2-D Interpolation in 1 mM Electrolyte Solution

ψair (mV)

ψair (mV)

ψmica (mV)

-1

-10

-20

-30

-40

-50

-60

ψmica (mV)

-1

-10

-20

-30

-40

-50

-60

-120 -100 -80 -60

0.3503 0.2725 0.1946 0.1168

0.3503 0.2725 0.1946 0.1168

0.3503 0.2725 0.1946 0.1168

0.3892 0.3114 0.2102 0.1557

0.4281 0.3503 0.2335 0.1868

0.4671 0.3892 0.2491 0.2180

0.5060 0.4281 0.2958 0.2569

-120 -100 -80 -60

0.3542 0.2708 0.1771 0.1042

0.3646 0.2708 0.1875 0.1146

0.3854 0.2917 0.2083 0.1354

0.4063 0.3229 0.2292 0.1563

0.4375 0.3542 0.2708 0.1875

0.4690 0.3958 0.3021 0.2292

0.5040 0.4271 0.3333 0.2708

but in practice, under the conditions of our experiment, the instability was suppressed by hydrodynamic drag, and the film thickness decreased steadily. Detailed information on the bubble deformation was obtained through that region, but we have not yet succeeded in analyzing the dynamic behavior quantitatively. Acknowledgment. This work was supported by the Australian Research Council Special Research Centre for Particle and Material Interfaces. We thank Jason Connor for experimental advice and Derek Chan, Steve Carnie, Rogerio Manica, Vassili Yaminsky, James Beattie, Tom Healy, and John Ralston for helpful discussions.

Appendix Comparison between the linear HHF approximation53 (eq 2) and the accurate CPW solution52 of the nonlinear PB equation

assuming constant-potential boundary conditions is shown in Figure 6. It appears that a small shift along the separation axis brings the two curves into close agreement. The analysis of the offset between the linear and nonlinear calculations has been done for water and 1 mM solutions for a range of potentials on the mica and air surfaces. To bring the simplified linear calculation into agreement with the CPW algorithm, it turns out to be enough to introduce a shift in κh. The distance correction has been calculated for mica potentials from -60 to -120 mV and air surface potentials from -1 to -60 mV. For arbitrary values of the mica and air potentials within the predefined ranges, the shift can then be calculated by a 2-D interpolation from the predefined array of values listed in Table 3 or Table 4. LA8007156