Coalescence in Two-Dimensional Foams: A Purely Statistical Process Dependent on Film Area

While coalescence is ultimately the most drastic destabilization process in foams, its underlying processes are still unclear. To better understand them, we track individual coalescence events in two-dimensional foams at controlled capillary pressure. We obtain statistical information revealing the influence of the different parameters which have been previously proposed to explain coalescence. Our main conclusion is that coalescence probability is simply proportional to the area of the thin film separating two bubbles, suggesting that coalescence is mostly stochastic.

Figure 1

(a) Experimental setup. The liquid height $h$ allows us to fix the capillary pressure in the 2D foam confined by the Plexiglas plates in a Hele-Shaw cell. (b),(c) Top view of the bubbles for two different capillary pressures with an enlargement at the scale of a few bubbles in the insets. (d) Evolution of the radius $R_{\mathrm{PS}}$ of the pseudo-Plateau borders with the liquid height. The lines are the fits used to extract $R_{\mathrm{PS}}$ for each liquid height. For a spacer of 1 mm (dashed blue line), the equation is $3.3335\times {10}^{-3}h-8.0883\times {10}^{-5}{h}^2+6.739\times {10}^{-7}{h}^3$, whereas for a spacer of 2 mm (solid green line), we used $7.09\times {10}^{-3}h-4.7568\times {10}^{-4}{h}^2+1.3484\times {10}^{-5}{h}^3-1.6367\times {10}^{-7}{h}^4+7.2203\times {10}^{-10}{h}^5$

Figure 2

(a) Section of a 2D foam confined in a Hele-Shaw cell of thickness $e$. (b) Sketch of the different steps along a $T_1$ event with the extraction of a film of length $\ell$ from a Plateau border of radius of curvature $R_{\mathrm{PB}}$.

Figure 3

Detection of two mother bubbles and of their common daughter.

Figure 4

For each experiment, at fixed $R_{\mathrm{PS}}$, plot of (a) the number of observed bubbles and the number of coalescence events, and (b) the probability of a bubble to coalesce.

Figure 5

Plot of (a) the number of observed bubbles and of coalescence events and (b) the probability of a bubble to coalesce, as functions of $R_{\mathrm{PS}}/\ell$. Note that the $x$ axis is different in both plots.

Figure 6

Plot of (a) the number of observed films and of coalescence events and (b) the probability of a film to break, as functions of the film area. The black solid line is a guide for the eye.