Collective dissolution of microbubbles

Abstract

A microscopic bubble of soluble gas always dissolves in finite time in an under-saturated fluid. This diffusive process is driven by the difference between the gas concentration near the bubble, whose value is governed by the internal pressure through Henry's law, and the concentration in the far field. The presence of neighbouring bubbles can significantly slow down this process by increasing the effective background concentration and reducing the diffusing flux of dissolved gas experienced by each bubble. We develop theoretical modelling of such diffusive shielding process in the case of small microbubbles whose internal pressure is dominated by Laplace pressure. We first use an exact semi-analytical solution to capture the case of two bubbles and analyse in detail the shielding effect as a function of the distance between the bubbles and their size ratio. While we also solve exactly for the Stokes flow around the bubble, we show that hydrodynamic effects are mostly negligible except in the case of almost-touching bubbles. In order to tackle the case of multiple bubbles, we then derive and validate two analytical approximate yet generic frameworks, first using the method of reflections and then by proposing a self-consistent continuum description. Using both modelling frameworks, we examine the dissolution of regular one-, two-and three-dimensional bubble lattices. Bubbles located at the edge of the lattices dissolve first, while innermost bubbles benefit from the diffusive shielding effect, leading to the inward propagation of a dissolution front within the lattice. We show that diffusive shielding leads to severalfold increases in the dissolution time which grows logarithmically with the number of bubbles in one dimensional lattices and algebraically in two and three dimensions, scaling respectively as its square root and 2/3-power. We further illustrate the sensitivity of the dissolution patterns to initial fluctuations in bubble size or arrangement in the case of large and dense lattices, as well as non-intuitive oscillatory effects.

Collisions and rebounds of chemically active droplets

“…Yet, it may also prove particularly useful to analyse a variety of other time-dependent problems such as the unsteady mass transfer and viscous growth/dissolution of gas bubbles (e.g. near catalytic surfaces or during boiling), or the collective dynamics of such bubbles or droplets (Michelin, Guérin & Lauga 2018).…”

Physicochemical hydrodynamics of droplets out of equilibrium

“…The situation becomes more interesting once multiple dissolving or evaporating sessile droplets interact, as in general they shield each other: The reason is that dissolving neighboring droplets reduce the concentration gradient at the interface and thus the outflux from the droplet, leading to longer (and heterogeneous -depending on the position of the droplet) life-times [42,43,[116][117][118][119][120], even when the solvent is flowing over the sessile droplets [42]. Such a situation has recently been explored both experimentally and numerically, leading to reasonable agreement, as seen in figure 3.…”

The shielding effect extends the lifetimes of two-dimensional sessile droplets

“…In the CIO mode, the inner and outer contact lines of both droplets are pinned, I ≡ I 0 and Ω ≡ Ω 0 = I 0 + 2. We may then immediately integrate (47) to obtain…”

“…Similar approaches have since been applied to elastic punches [40] and flow through pores [41], and have been put on a more rigorous asymptotic basis [42][43][44]. All these studies essentially considered the equivalent of thin circular droplets in three dimensions; recent work has used a variety of approaches to investigate the closely related problem of the dissolution of immersed nanobubbles and nanodroplets [45][46][47][48].…”