Coalescence Conditions for Bulk Nanobubbles
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cence Conditions for Bulk Nanobubbles S. I. Koshoridze Institute of Applied Mechanics, Russian Academy of Sciences, Moscow, 125040 Russia e-mail: [email protected] Received March 25, 2020; revised March 25, 2020; accepted April 13, 2020
Abstract—The coalescence of two nanobubbles with substantially different radii is considered. Studies of this problem have practical significance, because coalescence is an undesirable process (the nanobubbles lose their unique properties after enlargement). An energy-based approach is used with a simplified model of charged nanobubbles to demonstrate that bulk nanobubble coalescence can take place at specific values of surface charge density, concentration, and radius ratio, in contrast to the case of surface nanobubbles. Keywords: nanobubble in a bulk medium, coalescence, electrostatic pressure, Ostwald ripening. DOI: 10.1134/S1063785020070184
Nanobubbles (NBs) are widely used in various chemical, physical, and biological processes, and, therefore, it is important to increase the lifetime of NBs and prevent their dissolution and coalescence [1–3]. Surface NBs are resistant to diffusive dissolution if two necessary conditions are fulfilled: 1) the water is supersaturated with gas found inside the NBs and 2) the interface of the solid, liquid, and gaseous phases is fixed (this is termed “pinning”) [1]. If these two conditions are fulfilled, NBs with different curvature radii located close to each other on a substrate will not undergo “Ostwald ripening,” the diffusive dissolution of small NBs and growth of large NBs [2]. Mathematical modeling [3] shows that coalescence, the absorption of small surface NBs by large ones in particular, is also suppressed under these conditions. The situation for bulk NBs is not clear and the physical mechanism underlying the protection of these bubbles from rapid diffusive dissolution has not been fully elucidated. The connection between the stability of NBs in bulk medium and their surface charge has been proposed in [4]. Indeed, nanobubbles formed in media with pH > 4.5 possess a surface charge, presumably generated by OH– ions, and an associated electrokinetic potential of 25–40 mV [5, 6]. The charge generates a “negative” distending pressure to fully or partially compensate for the Laplacian pressure that compresses the bubbles: as a result, the rate of NB dissolution in water is reduced. Neither experimental nor theoretical studies of “Ostwald ripening” and coalescence of large and small NBs have been published to date. A simple model of charged NBs [7–10] is used in the present study to address coalescence—specifically,
the absorption of bulk NBs by larger NBs dependent on nanobubble concentration c, surface charge density σ, and the relative size of the NBs. Let us consider bulk NBs in water. Let us denote smaller NBs with radius R as S and the larger ones with radius nR (n > 1) as B. The gas pressure inside an NB in a state of dynamic equilibrium is 2 2 pS = 2α + pa − σ , pS = 2α + pa − σ , (1) R nR 2εε0 2εε0 for S and B, respectively
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