Averaged model for probabilistic coalescence avalanches in two-dimensional emulsions: Insights into uncertainty propagation

A two-dimensional concentrated emulsion exhibits spontaneous rapid destabilization through an avalanche of coalescence events which propagate through the assembly stochastically. We propose a deterministic model to explain the average dynamics of the avalanching process. The dynamics of the avalanche phenomenon is studied as a function of a composite parameter, the decay time ratio, which characterizes the ratio of the propensity of coalescence to cease propagation to that of propagation. When this ratio is small, the avalanche grows autocatalytically to destabilize the emulsion. Using a scaling analysis, we unravel the relation between a local characteristic of the system and a global system wide effect. The anisotropic nature of local coalescence results in a system size dependent transition from nonautocatalytic to autocatalytic behavior. By incorporating uncertainty into the parameters in the model, several possible realizations of the coalescence avalanche are generated. The results are compared with the Monte Carlo simulations to derive insights into how the uncertainty propagates in the system.

Figure 1

avalanches in a 2D emulsion, experiments [2], and simulation [3]: (a) and (d) trigger events where a pair of drops is allowed to coalesce. (b) Avalanching process stops with just one more coalescence event which results in a small sized avalanche. (c) Experiments of Bremond and co-workers [2] where coalescence avalanches both small and large in size are observed [courtesy: N. Bremond (ESPCI, France)]; (d)–(g) dynamic growth of a large sized avalanche [time (in generations): 1, 5, 15, and 25].

Figure 2

Modeling schemes to understand the collective averaged behavior of coalescence avalanches in 2D emulsions. There are two ways to model—a stochastic approach and an averaged model (deterministic) approach.

Figure 3

(a) A configuration of drops that contains two active sites with ten neighbors in total. The angles made by the neighbor drops are marked in the figure. (b) A configuration with ten neighbor drops and four active sites with angles marked in the figure. (c) The probability of propagation as measured by Bremond et al. in their experiments [2], which shows the anisotropic nature of propagation. [Note: the average angle made by the configuration in (b) is greater than that in (a). Hence the propagation rate is expected to be greater for (b)].

Figure 4

The number of neighbors per active drop $\beta$, plotted as a function of avalanche size from the Monte Carlo simulations (points) and the averaged model [Eq. (3), solid lines] for different values of $\alpha$ (tunable parameter that accounts for the propensity to propagate). (a) $\alpha =1.2$ higher propensity for coalescence. (b) $\alpha =1$ (corresponds to the propensity for coalescence as observed by Bremond et al. [2] in their experiments). (c) $\alpha =0.7$ (lower propensity for coalescence).

Figure 5

A comparison of the predictions of Monte Carlo simulations of the stochastic model [3] and the average model for coalescence avalanches. The time evolution of avalanche size and active sites is plotted for different values of the parameters $\alpha ,\tau$ which quantify the propensity to propagate. The scales used to normalize the avalanche size, active site number, and time are based on the growth kinetics: Eqs. (11) and (13). (a) Averaged model with ${\tau }_D=0.28,\gamma =0.1$ exhibit autocatalytic propagation characterized by the sigmoidal rise in the time evolution of the avalanche size; Monte Carlo simulations are carried out on a $14\times 14$ lattice with $\alpha =1.9$. (b) Averaged model with ${\tau }_{D,\mathrm{cr}}=0.57,\gamma =0.1$, a critically autocatalytic condition characterized by a maximum in the number of active sites at $T_s=0$; Monte Carlo simulations: $14\times 14$ lattice with $\alpha =0.84$. (c) Averaged model with ${\tau }_D=0.68,\gamma =0.1$, and nonautocatalytic propagation characterized by the pseudofirst order response (nonsigmoidal); Monte Carlo simulations: $14\times 14$ lattice with $\alpha =0.5$.

Figure 6

Plot of ${\alpha }_{\mathrm{cr}}\mathrm{vs}N$ that divides the parameter space into two regions: the region above the solid line corresponds to the autocatalytic propagation, whereas that region below it corresponds to the nonautocatalytic propagation regime. The solid line with the circular bullets corresponds to the case where propagation is anisotropic, and the solid line with the square bullets corresponds to the case where the propagation is isotropic. The results of Bremond et al. [2] (anisotropic propagation) lie in the autocatalytic region. The solid line for the anisotropic propagation has been approximated as a power law with a constant. However, for the isotropic case ${\alpha }_{\mathrm{cr}}$ is not a function of the $N$ size of the system $\left(\alpha ={\alpha }_{\mathrm{cr}}\right)$. The probability of propagation at the drop level for the isotropic case is $P\left(\theta \right)=\alpha$. When $P\left(\theta \right)\approx 0.28$, there is a sharp transition to autocatalytic behavior.

Figure 7

Equilibrium avalanche size $A_{\mathrm{eq}}$ as a function of the decay time ratio ${\tau }_D$ for different values of $\gamma$. There is a sharp transition from autocatalytic propagation, which results in large values of $A_{\mathrm{eq}}$, to nonautocatalytic propagation, which results in hardly any propagation (low values of $A_{\mathrm{eq}}$).

Figure 8

(a) A plot of $A_{\mathrm{eq}}$ as a function of ${\tau }_D$. The parameter ${\tau }_D$ is sampled in three different regions. $P\left({\tau }_D\right)-\mathrm{I}$ corresponds to ${\tau }_D$ sampled on the rhs of the critical region, $P\left({\tau }_D\right)-\mathrm{II}$ corresponds to ${\tau }_D$ sampled in the critical region, and $P\left({\tau }_D\right)-\mathrm{III}$ corresponds to ${\tau }_D$ sampled on the lhs of the critical region (probability distributions with arbitrary mean, variance, and bounds were used to sample ${\tau }_D$ in the regions of interest). (b) The probability of an avalanche as a function of its size corresponding to the distributions for $P\left({\tau }_D\right)-\mathrm{I}$, which contains only avalanches of smaller sizes. (c) The probability of an avalanche as a function of its size corresponding to the distributions for $P\left({\tau }_D\right)-\mathrm{II}$. The probability distribution captures the emergence of a hump as observed in the Monte Carlo simulations [3]. (d) The probability of an avalanche as a function of its size corresponding to the distributions for $P\left({\tau }_D\right)-\mathrm{III}$, which consists of avalanches of large sizes.

Figure 9

(a) A plot of $A_{\mathrm{eq}}\mathrm{vs}{\tau }_D$ from the averaged model under conditions $\gamma =0.3,A\left(0\right)=0.1,X\left(0\right)=0.1$. The uncertainty that has to be incorporated in the parameter ${\tau }_D$ to explain the results from the Monte Carlo simulations as shown in (b). (b) Monte Carlo simulation under conditions $N=8^2-{16}^2,\alpha =1,\mathrm{aspect}\mathrm{ratio}\approx 1$ (for the algorithm in Ref. [3]). The probability of an avalanche as a function of its size $A$.