Marginal regeneration-induced drainage of surface bubbles

The prediction of the lifetime of surface bubbles necessitates a better understanding of the thinning dynamics of the bubble cap. In 1959, Mysels et al. [Soap Films: Studies of Their Thinning and a Bibliography (Pergamon, New York, 1959)], proposed that marginal regeneration, i.e., the rise of patches thinner than the film should be taken into account to describe the film drainage. Nevertheless, an accurate description of these buoyant patches and of their dynamics as well as a quantification of their contribution to the thinning dynamics is still lacking. In this paper, we visualize the patches, and show that their rising velocities and sizes are in good agreement with models respectively based on the balance of gravitational and surface viscous forces and on a Rayleigh-Taylor-like instability. Our results suggest that, in an environment saturated in humidity, the drainage induced by their dynamics correctly describes the film drainage at the apex of the bubble within the experimental error bars. We conclude that the film thinning of soap bubbles is indeed controlled, to a large extent, by marginal regeneration in the absence of evaporation.

Figure 1

Photograph of a surface bubble with a radius of curvature of 1.13 cm (Bo = 25) illuminated in white light for a TTAB concentration of 0.5 cmc; credits: Serge Guichard.

Figure 2

(a) Sketch of the experimental setup with the notations used. (b) Example of an experimental image centered on the bottom of the bubble on which we can measure the characteristic size of the patches ${\ell }_0$ (m) and the number of patches per unit area $n_0$ (${\mathrm{m}}^{-2}$).

Figure 3

(a) Rising velocity of the patches $V^{+}$ as a function of time $t$. The Bond number is ranging from 15 to 35, which corresponds to bubbles with a spherical cap radius between 9 and 13 mm. Error bars correspond to the standard deviation measured by averaging over ten patches. (b) Rising velocity of the patches $V^{+}$ as a function of the film thickness $h$. The error for $h$ is taken to be 50 nm (the choice of this value has no noticeable impact on the errors calculated for ${\eta }_s$ and $V_d$; see next figures).

Figure 4

(a) Evolution of the characteristic size of the patches ${\ell }_0$ ($\mu \mathrm{m}$) as a function of the film thickness $h$ (nm) for 387 measurements performed on all systems. The orange circles correspond to direct measurements. The blue triangles correspond to the average values of ${\ell }_0$ over an interval of 50 nm for $h$. The green curve represent an adjustment of Eq. (4) with $\delta =36\mu \mathrm{m}$. Vertical error bars correspond to the standard deviation measured. (b) Surface shear viscosity ${\eta }_s$ as a function of time $t$. The solid line corresponds to the mean value measured for the surface shear viscosity: ${\eta }_s=(2.0\pm 0.1)\times {10}^{-8}$ Pa m s. The error bars take into account errors in the measured parameters and are calculated using the error propagation formula.

Figure 5

(a) Time evolution of the film thickness $h$ at the bubble apex for different bubble sizes and humidities. The dashed and dotted black lines correspond respectively to second-order and power law adjustment of the film thickness $h$ as a function of time $t$ around the instant $t$ = 45 s for Bo = 35 obtained at RH = $50\%$ and $t$ = 130 s for bubbles with Bo = 25 obtained at RH = $100\%$. (b) Thinning rate $V_d$ predicted by the model as a function of the measured thinning rate $dh/dt$. The black line corresponds to a line of equation $y$ = $x$. The black dashed line corresponds to the equation $y=-11.1+x$. The error bars take into account errors in the measured parameters and are calculated using the error propagation formula.