Capillary imbibition of aqueous foams by miscible and nonmiscible liquids

When put in contact with a large liquid drop, dry foams wick owing to surface-tension-driven flows until reaching equilibrium. This work is devoted to the dynamics of this imbibition process. We consider imbibition of both wetting or nonwetting liquid, by putting the dry foam into contact either with the foaming solution that constitutes the foam or with organic oils. Indeed, with the appropriate choice of surfactants, oil spontaneously invades the liquid network of the foam without damaging it. Our experiments show an early-time dynamics in $t^{1/2}$ followed by a late-time dynamics in $t^{1/4}$. These features, which differ from theoretical works predicting a $t^{1/3}$ dynamics, are rationalized considering the influence of the initial liquid fraction of the foam in the driving capillary force and the impact of gravity through the capillary-gravity equilibrium.

Figure 1

(a) Experimental setup. (b) Geometrical elements of foams: Nodes, Plateau borders, and foam films. (c) Slender Plateau border with a radius of curvature $r_F$. (d) Capillary rise of oil in a dry aqueous foam. The olive oil drop appears white because of a fluorescent dye and invades the network of Plateau borders. Time interval between each frame is 67 s. The scale bar is 3 mm.

Figure 2

Height of the rising front for olive oil as a function of time for different initial liquid fractions and one mean bubble radius $R$ = 1.8 mm. Inset: log-log scale of the data corresponding to $\epsilon =0.7\times {10}^{-4}$. The solid line corresponds to the $t^{1/2}$ dynamics detailed in Ref. [13], and the dashed line illustrates the $t^{1/4}$ dynamics.

Figure 3

Typical evolution of the fluorescence intensity of a single node during sunflower oil imbibition as a function of the node's vertical dimension. Initially the node is located at $x$ = 1 mm and $z$ = 7.9 mm. The intensity of fluorescence is in arbitrary units, and the different curves correspond to fluorescence intensity measurements taken at different times. The black arrow indicates the evolution of time. Inset: Width of the peak of intensity (defined at half of the maximum intensity) as a function of time. The red arrow highlights the time at which the oil front has reached the node.

Figure 4

Normalized imbibition front $\alpha h$ as a function of normalized time $t^{*}$ in log-log plot. In the legend, AS1, AS10, and AS100 stand for aqueous solution with $\eta =1.4$, 10, and 100 mPa s, respectively, and OO and SO stand for olive oil and sunflower oil. The black line corresponds to ${(1+\alpha h)}^4+\frac{4}{1+\alpha h}-5=b\frac{2\delta }{15\sqrt{3}}{t}^{*}$ with $b\sim 0.3$, and the two dashed lines represent the power laws: $t^{1/2}$ at early times and $t^{1/4}$ at late times.