Probing the Mechanical Strength of an Armored Bubble and Its Implication to Particle-Stabilized Foams

Bubbles are dynamic objects that grow and rise or shrink and disappear, often on the scale of seconds. This conflicts with their uses in foams where they serve to modify the properties of the material in which they are embedded. Coating the bubble surface with solid particles has been demonstrated to strongly enhance the foam stability, although the mechanisms for such stabilization remain mysterious. In this paper, we reduce the problem of foam stability to the study of the behavior of a single spherical bubble coated with a monolayer of solid particles. The behavior of this armored bubble is monitored while the ambient pressure around it is varied, in order to simulate the dissolution stress resulting from the surrounding foam. We find that above a critical stress, localized dislocations appear on the armor and lead to a global loss of the mechanical stability. Once these dislocations appear, the armor is unable to prevent the dissolution of the gas into the surrounding liquid, which translates into a continued reduction of the bubble volume, even for a fixed overpressure. The observed route to the armor failure therefore begins from localized dislocations that lead to large-scale deformations of the shell until the bubble completely dissolves. The critical value of the ambient pressure that leads to the failure depends on the bubble radius, with a scaling of $\mathrm{\Delta }{P}_{\text{collapse}}\propto R^{-1}$, but does not depend on the particle diameter. These results disagree with the generally used elastic models to describe particle-covered interfaces. Instead, the experimental measurements are accounted for by an original theoretical description that equilibrates the energy gained from the gas dissolution with the capillary energy cost of displacing the individual particles. The model recovers the short-wavelength instability, the scaling of the collapse pressure with bubble radius, and the insensitivity to particle diameter. Finally, we use this new microscopic understanding to predict the aging of particle-stabilized foams, by applying classical Ostwald ripening models. We find that the smallest armored bubbles should fail, as the dissolution stress on these bubbles increases more rapidly than the armor strength. Both the experimental and theoretical results can readily be generalized to more complex particle interactions and shell structures.

Popular Summary

Foams, which consist of bubbles dispersed in a liquid or solid medium, are useful in many applications such as food, cosmetics, and construction to control the physical properties of the materials in which they are embedded. However, foams suffer from aging problems; bubbles grow or disappear over time, which leads to a degradation of desired properties. One way to stabilize a foam is to add tiny solid particles to it that attach to the interface between the liquid and the gas. These particles form a type of armor that allows the foam to remain stable for months. Nevertheless, the physical mechanisms underlying this stabilization remain mysterious. Here, we study this problem by measuring the behavior of a single particle-coated bubble in conditions that mimic the influence of the surrounding foam.

We focus on a single bubble trapped in a chamber with a conical ceiling, and we apply a monolayer of polystyrene microbeads with diameters of between 0.5 and $\text{4.5\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$. We vary the ambient pressure around this bubble, and we study how the bubble stability depends on parameters such as bubble size and particle diameter. The failure, by buckling, depends on the bubble radius but not on the diameter of the coating microbeads. We quantify the resistance of the particle shell and observe the way in which it fails when it becomes unstable. We compare our unique microfluidics experiments with a theoretical model that we develop in order to capture the relevant physics. We find that the armored bubbles obtain their stability from the geometrical arrangement of the particles, like bricks in an arch, and that the armor failure arises from dislocations of individual particles. This new physical understanding allows us to predict the fate of any bubble in a particle-stabilized foam.

We expect that our results can be generalized to more complex situations such as sticky or rough particles that appear in real-life applications.

Figure 1

(a) General view of the microfluidic device which consists of a bubble formation zone, a coating channel, and an observation chamber. (b)–(d) Composite bright-field or fluorescence images where the particles appear in green. (b) First, a slopped injector ($\text{height}=30\mu \mathrm{m}$ to $80\mu \mathrm{m}$, angle 2.3°) produces a single air bubble. (c,d) The bubble is then squeezed through the coating channel by the flow of a particle dispersion where it progressively collects particles. Scale bars are $200\mu \mathrm{m}$. The armored bubble is then released in the observation chamber, and the excess of unattached particles in suspension is washed away, yielding a bubble coated with a single layer of particles with loosely packed crystalline domains. (e) Phase contrast micrograph of such a bubble with $4.5\text{−}\mu \mathrm{m}$ diameter particles. The scale bar is $30\mu \mathrm{m}$. (f) Cross-section schematic of the conical observation chamber (not to scale). The air bubble is trapped even if the liquid around it is flowed.

Figure 2

Free energy of a single gas bubble of radius $R$ in a weak gas-liquid solution. If $\mathrm{\Delta }P=0$, the bubble radius is stationary. The bubble shrinks if $\mathrm{\Delta }P>0$ and it swells if $\mathrm{\Delta }P<0$ because of gas exchange between the liquid and the bubble.

Figure 3

(a) Typical evolution of the bubble projected area (plus) with the imposed overpressure $\mathrm{\Delta }P$ (minus) over time. The insets are fluorescence micrographs from the corresponding experiment. The figure shows the blocked armor (square, $R=100\mu \mathrm{m}$), the appearance of fractures during the bubble swelling step (triangle), and finally the buckled armor (circle). (b) Plot of $\mathrm{\Delta }{P}_{\text{collapse}}$ as a function of $1/R$, for three particle diameters. The error bars are estimated from the uncertainty in the measurement of $P_{\mathrm{eq}}$ and the size of the $\mathrm{\Delta }P$ steps. The black line shows the value of collapse pressure predicted by our model $\mathrm{\Delta }{P}^{*}\approx 1.57\gamma /R$. (c) Detail of the buckling event for a bubble armored with $4.5\mu \mathrm{m}$ beads dissolving at $\mathrm{\Delta }P=50\mathrm{mbar}$. Scale bars are $30\mu \mathrm{m}$.

Figure 4

(a) Sketch of the colloidal armor with an arbitrary 1D fiber highlighted in red. (b) Sketch of the buckling of the fiber. The red line shows the undeformed initial state of the fiber of length $L$; the blue lines are the deflected sinusoidal states. Light blue is the lowest frequency mode; dark blue is the highest frequency mode with alternating beads. (c) Detail from fluorescence micrographs of the buckling event (movie 4 in Ref. [26]), where the beads overlaid in red and yellow overlap and come out of the plane of the monolayer. The direction of the compression is indicated by the white arrows. Scale bars are $5\mu \mathrm{m}$.

Figure 5

(a) The blue-green curves display the overpressure felt by a bubble as a function of its inverse radius $R$ in a foam with a log-normal bubble-size distribution. The mean of the distribution is kept constant at $100\mu \mathrm{m}$, while the standard deviation is varied from $3.5\mu \mathrm{m}$ (blue) to $84\mu \mathrm{m}$ (green), as shown in the lower inset. The black line represents the resistance $\mathrm{\Delta }{P}^{*}$ of a colloidal shell as defined by our model and our experiments. Bubbles subjected to a pressure below this critical value will collapse and disappear, while the others will remain stable. The case of unarmored bubbles ($\mathrm{\Delta }P=0$) is displayed in gray for comparison. The inset is a close-up of the region where $\mathrm{\Delta }P=\mathrm{\Delta }{P}^{*}$ ($\circ$), which defines the cutoff radii $R_{\text{cut}}$. (b) The percentage of bubbles that collapse or dissolve under the influence of the whole foam, in the armored case (black) and the unarmored case (gray). The polydispersity is calculated as the standard deviation of the distribution divided by the mean. The buckled percentage is calculated by integrating the probability density function on $[0,R_{\text{cut}}]$, as shown in the right inset. The left inset shows the probability density functions $F(R)$ used for the calculations, with increasing polydispersity.

Figure 6

Three-dimensional view of the brass mold used for the casting of the microchips. It was obtained with an optical profilometer (Zygo) by stitching images from multiple acquisitions. The height of the coating channel is $80\mu \mathrm{m}$, and its width is $300\mu \mathrm{m}$. The slope of the bubble injector is 4%.

Figure 7

Measurement of the time evolution of the normalized radius of an air bubble in water, for different values of $\mathrm{\Delta }P$.

Figure 8

Armored bubble collapse pressure as a function of the inverse radius. The black line is our model prediction, the gray symbols correspond to our experimental points, and the colored symbols correspond to measurements from Ref. [17] for two different particle diameters: $16\mu \mathrm{m}$ and $40\mu \mathrm{m}$. The inset is a close-up for small values of $1/R$.