Collective dissolution of microbubbles

TLDR: In this paper, asymptotic analysis is used to characterize collective shielding effects in bubble lattices, the resulting bubble lifetime and dissolution dynamics, which is driven by the diffusion of gas within the surrounding liquid.

Abstract: Microscopic bubble dissolution is driven by the diffusion of gas within the surrounding liquid. Asymptotic analysis is used to characterize collective shielding effects in bubble lattices, the resulting bubble lifetime and dissolution dynamics.

Figures

FIG. 4. Relative displacement of two identical bubbles due to their collective dissolution measured by the change in their center-to-center distance, δ, over the full duration of the dissolution process, as a function of their initial relative distance, d0c .

FIG. 3. Shielding effect on the lifetime of two identical bubbles of initial radius a0i = 1. (a) Relative dissolution time, Tf /T ∗f , of the bubbles (the value T ∗ f = 1/4 corresponds to the reference case of an isolated bubble) as a function of their initial relative distance. The inset displays the algebraic convergence rate in the far-field limit. In both cases, we show the results including hydrodynamics (i.e., induced bubble motion, blue symbols) and without hydrodynamics (i.e., bubbles whose centers are fixed, solid red line). (b) Instantaneous relative mass flux of two identical bubbles as a function of their instantaneous dimensionless relative distance, dc/a; q∗ = −2 is the reference of a single isolated bubble.

FIG. 14. Evolution of an hexagonal cluster of initially identical bubbles with unit radius (a0 = 1) and Nl = 10 layers (a total of N = 331 bubbles). The distance between closest neighbors is d = 4. The color shows the dissolved gas concentration (taken as uniform within the bubbles and equal to their surface concentration) and the initial positions of dissolved bubbles are shown as white dots. Corresponding videos of the dissolution process are also available as Supplemental Material [56].

FIG. 15. Influence of fluctuations in the bubble position on the dissolution time distribution, Tf (r), for circular lattices with Nl = 4 (left) and Nl = 10 layers (right). In each case, three different levels of noise (10%, 20%, and 50% bubble radius corresponding to decreasing shades of gray) are represented with 20 different configurations for each level. Each bubble’s lifetime is represented as a function of the bubble’s position within the lattice. The results for the regular circular lattice (green symbols) are shown as well as the predictions of the continuum model for the corresponding reduced density σ̃ (solid red line).

FIG. 1. Dissolution of a single bubble. Left: Relative change in the dissolution time, (Tf − T ∗f )/T ∗f , as a function of the relative bubble size (r0) and chemical saturation of the environment (ζ ). The case r0 ≪ 1 corresponds to the capillary-dominated regime (and serves as reference here with T ∗f = 1/4) while for r0 ≫ 1 the gas concentration in the bubble is independent of its size. Right: Temporal evolution of the radius of the bubble. The gray region corresponds to the envelope of all possible time dependence for undersaturated regimes, ζ ! 1. Two limit cases are highlighted: (i) Capillarity-dominated regime r0 ≪ 1 (solid blue line) and (ii) negligible capillarity r0 ≫ 1 (dashed red line).

FIG. 11. (a) Circular and (b) hexagonal lattices of bubbles with interbubble distance d = 4 and Nl = 4 layers. Initially, each bubble has unit radius. In the circular lattice, 6n bubbles are regularly spaced on a circle of radius nd away from the central bubble.

Frequently asked questions

Q1. What are the contributions in "Collective dissolution of microbubbles" ?

In this paper, the authors provide quantitative insight on the fundamental physics of diffusive shielding in the collective dissolution of microbubbles. 

Q2. What are the future works in "Collective dissolution of microbubbles" ?

These asymptotic frameworks could open the door to many different applications, e. g., as an attractive alternative to computationally expensive numerical simulations to study suspension dynamics. The present analysis was formulated to tackle bubble dissolution in an infinite liquid, but it could easily be extended to confined geometries ( e. g., bubbles near a wall ). Collective effects for such applications therefore deserve further analysis in particular for large lattices which were identified here as highly sensitive to fluctuations. 

Q3. What is the effect of diffusive shielding on bubbles?

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. 

Q4. What is the total dissolution time of the lattice?

The total dissolution time of the lattice is an increasing function of the mean lattice radius, R̄, and increases with the number of layers. 

Q5. What is the effect of hydrodynamics on the bubbles?

Hydrodynamics tends to increase the lifetime of the bubbles by bringing them closer and thus increasing their instantaneous diffusive shielding. 

Q6. What is the role of surface pinning in the dissolution of gas bubbles?

Surface pinning was further identified to play a key role in the coarsening process surface nanobubbles and droplets, in particular stabilizing them against Ostwald ripening [43,44]. 

Q7. How does the first reflection provide an estimate of the diffusive mass flux?

Turning now to the case of many bubbles, and neglecting the role of hydrodynamics, the first reflection provides an estimate of the diffusive mass flux valid up to an O(ε4) error by superimposing the influence of each bubble as a simple source of intensity qj . 

Q8. What is the purpose of the present analysis?

The present analysis was formulated to tackle bubble dissolution in an infinite liquid, but it could easily be extended to confined geometries (e.g., bubbles near a wall). 

Q9. What is the typical hydrodynamic pressure in the quasisteady framework?

In the quasisteady framework described above, the typical hydrodynamic pressure is also small in comparison with capillary pressures so that the bubble remains spherical. 

Q10. What is the translation velocity of a bubble whose center is located at Xj?

the translation velocity Ẋj of bubble j whose center is located instantaneously at Xj (t) follows from solving for the Stokes flow forced by the shrinking motion of the bubbles under the conditionsnj · u|rj =aj = 

Q11. How long does the dissolution time of large bubbles last?

While all bubbles still dissolve in finite time, the final dissolution times of large bubble lattices may be orders of magnitude larger than the typical lifetime of an isolated bubble. 

Q12. What are the main points of view in the dynamics of small bubbles?

From a fluid mechanics standpoint, two main points of view, or types of questions, have been considered in the dynamics of small bubbles. 

Q13. How does the shielding effect of the second bubble change as the bubble size decreases?

as the bubble decreases in size, the instantaneous ratio a/d decreases and the shielding effect of the second bubble becomes negligible [see Fig. 3(b)]. 

Q14. How do the authors solve the case of two bubbles?

The authors first use an exact semianalytical solution to capture the case of two bubbles and analyze in detail the shielding effect as a function of the distance between the bubbles and their size ratio. 

Q15. What is the simplest explanation for the complex behavior of bubbles?

This complex behavior arises together with a critical sensitivity to fluctuations in the spatial arrangement of the bubbles, or their size distribution, leading to chaotic dissolution patterns of the lattice. 

Q16. How many Legendre modes are used to solve the hydrodynamic and diffusion problems?

For both the hydrodynamic and diffusion problems, a sufficiently large number of Legendre modes is chosen to ensure the convergence of the results. 

Q17. What is the effect of the large pressure on the dissolution of small bubbles?

This increased dissolution of small bubbles stems from the large capillary pressure inside them that translates into a large dissolved gas concentration contrast between their surface and their environment. 

Q18. What is the limiting assumption for the gas?

The limiting assumption is therefore on ( ∼ RT/KH = Cs/ρg which depends only on the ambient temperature and the nature of the gas considered (and not on the bubble size). 

Q19. How do the authors solve the case of multiple bubbles?

In order to tackle the case of multiple bubbles, the authors 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. 

Q20. What is the dissolution time of the bubbles?

In Fig. 13, the authors further show the final dissolution time of the bubbles as a function of their radial position which characterizes the inward propagation of the dissolution front. 

Q21. What is the dissolution process of a lattice?

The dissolution process follows the same qualitative pattern as for one- and two-dimensional lattices, with the outermost bubbles dissolving first, thereby shielding the central ones which experience a much extended lifetime (the corresponding video of the dissolution process is available as Supplemental Material [56]). 

Q22. What is the dissolution time of a bubble of the same initial radius?

Here the authors use T ∗ f,i = (a0i )2/4 to denote the dissolution time of an isolated bubble of same initial radius a0i .is sufficiently large or the bubbles are initially far apart, or accelerated when the neighboring bubble is much larger and the contact distance is small [Fig. 5(b)].