Bubble dynamics inside an outgassing hydrogel confined in a Hele-Shaw cell

We report an experimental study of bubble dynamics in a non-Newtonian fluid subjected to a pressure decrease. The fluid is a hydrogel, composed of water and a synthetic clay, prepared and sandwiched between two glass plates in a Hele-Shaw geometry. The rheological properties of the material can be tuned by the clay concentration. As the imposed pressure decreases, the gas initially dissolved in the hydrogel triggers bubble formation. Different stages of the process are observed: bubble nucleation, growth, interaction, and creation of domains by bubble contact or coalescence. Initially bubble behave independently. They are trapped and advected by the mean deformation of the hydrogel, and the bubble growth is mainly driven by the diffusion of the dissolved gas through the hydrogel and its outgassing at the reactive-advected hydrogel-bubble interface. In this regime, the rheology of the fluid does not play a significant role on the bubble growth. A model is proposed and gives a simple scaling that relates the bubble growth rate and the imposed pressure. Carbon dioxide is shown to be the gas at play, and the hydrogel is degassing at the millimeter scale as a water solution does at a smaller scale. Later, bubbles are not independent anymore. The growth rate decreases, and the morphology becomes more anisotropic as bubbles interact because they are separated by a distance smaller than the individual stress field extension. Our measurements show that the interaction distance scales with the bubbles' size.

Figure 1

Representation of the experimental setup.

Figure 2

Image sequence of bubbles growth inside a 4% hydrogel with 100 s between two images and a pressure rate $-0.67$ mbar ${\mathrm{s}}^{-1}$. Bubbles are advected toward the pumping direction, here on the right-hand side, due to the global expansion of the hydrogel triggered by the nucleation and growth of bubbles.

Figure 3

Image sequence of bubble growth: (a–d) inside a 3% hydrogel with 130 s between two images and a pressure rate $-0.30$ mbar ${\mathrm{s}}^{-1}$, (e–h) inside a 5% hydrogel with 300 s between two images and a pressure rate $-0.45$ mbar ${\mathrm{s}}^{-1}$. Pumping from the right side.

Figure 4

(a) Imposed pressure for three different pressure decrease rates: $-30$, $-1.4$, and $-0.7$ mbar ${\mathrm{s}}^{-1}$, respectively. (b) Experimental snapshots obtained at times $t_1$, $t_2$,..., $t_6$. The hydrogel concentration is 4%. The observation field is $2.6\text{cm}\times 3.3$ cm.

Figure 5

Diluted regime, Set 1. Constant pressure ($\square$) and decreasing pressure ($○$) experiments with the same hydrogel composition 3%. For both experiments, several bubbles are tracked, and their respective data points are connected by solid lines. Pressure as a function of time (a), area as a function of time (b), area as a function of pressure (c), and area growth rate as a function of pressure (d). For the constant pressure experiment, all bubbles grow with a quasiconstant and homogeneous growth rate. Its value is centered around $0.8\times {10}^{-9}$ ${\mathrm{m}}^2$ ${\mathrm{s}}^{-1}$. For the decreasing pressure experiment, the bubbles do not nucleate simultaneously, and each of them follows its own area-time or area-pressure trajectory. For a given time, the growth rate is the same for all bubbles. It increases as the pressure decreases, and when plotting the growth rate as a function of the imposed pressure, all the data points collapse onto the same trend.

Figure 6

Diluted regime, Sets 2 and 3. (a–b) Three decreasing pressure experiments at different pressure rates $[-0.33$ ($▵$), $-0.45$ ($○$), and $-0.81$ ($\diamond$) mbar ${\mathrm{s}}^{-1}]$ with the same hydrogel composition 3%: pressure as a function of time (a) and area rate as a function of pressure (b). (c–d) Three hydrogel compositions [3% ($○$), 4% ($*$), and 5% ($+$)] with the same pressure rate $-0.45$ mbar ${\mathrm{s}}^{-1}$: pressure as a function of time (c) and area growth rate as a function of pressure (d). For a given experiment, all data points collapse onto a single trend when plotting the growth rate as a function of the imposed pressure. By varying the pressure rate or the hydrogel composition, trends are maintained. The small differences observed do not enable to discriminate the impact of these two parameters on the evolution of the growth rate (see text).

Figure 7

Area rate $dA/dt$ vs $(P_0-P)/P$. For simplicity, only four sets of data are plotted (Table 2). All time steps for every bubble of each experiment are represented. Measurements are interpolated by straight lines modeled by Eq. (6) with $P_0=1000$ mbar and the best fits for the parameter $\mathcal{D}$.

Figure 8

Model parameter $\mathcal{D}$ for gases in the air composition (black disks and Table 3) and measured in the experiments (see Table 2 for symbols).

Figure 9

Experimental snapshots of two close-neighbor growing bubbles: (a) $t=50$ s, (b) $t=100$ s, (c) $t=150$ s, (d) $t=200$ s, (e) $t=250$ s, (f) $t=300$ s, (g) $t=350$ s, and (h) $t=400$ s. Outer contours, bubble radii for given angles, and the center-to-center distance ${\ell }_{cc}$ are indicated. Very curved parts begin to grow on the contour of the bubbles between $t=150$ and 200 s; this marks the onset of the bubble interaction. The hydrogel concentration is 4% and the pressure is constant, 100 mbar. The observation field is $2.5\text{mm}\times 4.6\text{mm}$.

Figure 10

Morphology correlation measurements for the two bubbles of Fig. 9. (a) Angle-time diagram of the correlation function $\mathcal{C}(\alpha ,t)$.( b) $\mathcal{C}$ vs time for two specific values of the angle $\alpha$: $0^{\circ }$ (red circles) and ${180}^{\circ }$ (black triangles) respectively. The crossover is observed around 185 s (vertical dashed line).

Figure 11

Center-to-center distance ${\ell }_{cc}$ as a function of the bubble radius $R$ for pairs of bubbles at the crossover between the diluted and dense regimes. The criterion is based either on the bubble morphology for hydrogels of composition 3% ($○$), 4% ($*$), and 5% ($+$). Error bars are plotted in light gray and are inferred from the area difference between the two interacting bubbles. The dashed line represents ${\ell }_{cc}=4.3R$.