Scaling state of dry two-dimensional froths: Universal angle-deviations and structure

We characterize the late-time scaling state of dry, coarsening, two-dimensional froths using a detailed force-based dynamical model. We find that the slow evolution of bubbles leads to small and decreasing deviations from ${120}^{\circ }$ angles at threefold vertices in the froth, but with a side-number dependence that is independent of time and apparently universal. We also find that a significant number of $T1$ side-switching processes occur for macroscopic bubbles in the scaling state, though most bubble annihilations involve four-sided bubbles at microscopic scales.

Figure 1

Illustrations of the fast topological processes involved in froth evolution. From top to bottom are a side-switching $T1$ event, and three-, four-, and five-sided bubble deletions ($T{2}_3$, $T{2}_4$, and $T{2}_5$, respectively). From left to right are examples of the bubble walls and vertex positions before, during, and after the corresponding topological process. The details of resolution of $T{2}_4$ and $T{2}_5$ events will, in general, be determined by the forces acting along the bubble walls, as described in Sec. 2.

Figure 2

The average bubble area of a coarsening froth, $⟨A⟩$, vs the natural time variable, $\tilde{t}\equiv tD\sigma$. We see that linear growth is obeyed at late times, with a best-fit $⟨A⟩=0.{94}^{\pm 0.01}(\tilde{t}-{\tilde{t}}_0)$ indicated by a solid line [21]. The amplitude is expected to universally apply to coarsening froths in their scaling state, with the effective time origin ${\tilde{t}}_0$ determined by initial conditions.

Figure 3

Film energy of a coarsening froth, $E$, equal to the surface tension $\sigma$ times the total film length, normalized by the energy of a uniform hexagonal froth $E_{\mathrm{hex}}\equiv N\sigma \sqrt{2\sqrt{3}{A}_0}$, with the same number of bubbles, $N$, and average bubble area, $A_0$, plotted vs $\tilde{t}=tD\sigma$. The asymptotic value in the scaling state is $E∕E_{\mathrm{hex}}=0.938\pm 0.001$ [21] indicated by the horizontal line. The symbols “$\times$” and “$+$” indicate data from $D=0.2$ and $D=1$, respectively, both with $N_0={10}^4$.

Figure 4

The second moment of the side-number distribution, ${\mu }_2$ vs $\tilde{t}$. We extract [21] an asymptotic value of ${\mu }_2=1.24\pm 0.01$, indicated by a horizontal line. Significant transients are seen for the larger $D$ data (“$+$”), including a distinctive preasymptotic peak. The symbols “$\times$” and “$+$” indicate data from $D=0.2$ and $D=1$, respectively, both with $N_0={10}^4$.

Figure 5

Average angle deviations of $n$-sided bubbles scaled by the $n=5$ sided deviations, $⟨\mathrm{\Delta }{\theta }_n⟩∕\mid ⟨\mathrm{\Delta }{\theta }_5⟩\mid$, vs $n$. The data are for $\tilde{t}=27969$ (triangles and squares) and $\tilde{t}=158217$ (“$\times$” and “$+$”) for $D=0.2$ and $D=1$, respectively. Also shown (circles) are the scaled data from experimental undrained froths [2], which illustrates the universality of the scaled angle deviations. Inset: time dependence of angle deviations of $n=5$ sided bubble vertices, $\mid ⟨\mathrm{\Delta }{\theta }_5⟩\mid$, scaled by $\gamma D^{-1}$ (see text) vs $\tilde{t}\equiv tD\sigma$. The solid line shows the expected ${\tilde{t}}^{-1∕2}$ behavior. All data are with $N_0={10}^4$.

Figure 6

The proportion of $T{2}_3$, $T{2}_4$, and $T{2}_5$ processes, for three-sided, four-sided, and five-sided bubble annihilation, denoted by triangles, squares, and pentagons, respectively, vs $\tilde{t}$. The filled symbols are for $D=0.2$ while the open symbols are for $D=1$, both with $N_0={10}^4$. The number of sides are counted as the bubble area approaches microscopic scales, at $\ell \equiv \sqrt{r_{c}L\left(t\right)}$. The proportions approach constant values of $9^{\pm 2}\%$, ${84}^{\pm 2}\%$, and $7^{\pm 2}\%$, respectively, for $D=0.2$ [21], indicated by horizontal lines. Significant transients are apparent even at late times for $D=1$, though the trend is towards agreement with $D=0.2$.

Figure 7

The proportion $R_{T1}∕R_{T2}$ of macroscopic $T1$ events to microscopic $T2$ events, where only side shedding involving bubbles larger than $\ell \equiv \sqrt{r_{c}L\left(t\right)}$ are counted as $T1$ events. We find a ratio of $R_{T1}∕R_{T2}=0.53\pm 0.05$ in the scaling regime [21], indicated by the horizontal line. Very long transients are found for $D=1$ (“$+$”), consistent with the previous figure.