Direct Self-Sustained Fragmentation Cascade of Reactive Droplets

A traditional hand-held firework generates light streaks similar to branched pine needles, with ever smaller ramifications. These streaks are the trajectories of incandescent reactive liquid droplets bursting from a melted powder. We have uncovered the detailed sequence of events, which involve a chemical reaction with the oxygen of air, thermal decomposition of metastable compounds in the melt, gas bubble nucleation and bursting, liquid ligaments and droplets formation, all occurring in a sequential fashion. We have also evidenced a rare instance in nature of a spontaneous fragmentation process involving a direct cascade from big to smaller droplets. Here, the self-sustained direct cascade is shown to proceed over up to eight generations, with well-defined time and length scales, thus answering a century old question, and enriching, with a new example, the phenomenology of comminution.

Figure 1

Traces of the droplets paths illustrating the sustained cascading fragmentation process in a traditional firework [10]. Top: Exposure time $1/60\mathrm{s}$, the 2 mm diameter paper string holds the mother drop. Bottom: A longer exposure time of $1/25\mathrm{s}$ reveals the cascade of successive breakups with up to four visible generations on the in-focus arborescence.

Figure 2

(a) Mother drop surface bubbling, drop ejection from a bursting surface cavity, and subsequent breakup. (b) Bursting cascade shown every $30\mu \mathrm{s}$. Arrows indicate ruptures producing the next generation. (c) Temperature trajectories of 40 different droplets, from their formation to bursting. Time $t$ is normalized by the lifetime $t_n$ for several cascade steps $0\le n\le 1$. The dashed line indicates the melting temperature of ${\mathrm{K}}_2{\mathrm{SO}}_4$ (1342 K). (d) Evolution of a droplet with initial radius $R$ as it bursts, inflated up to $R_{\star }$ by a gas cavity with radius $b$.

Figure 3

Statistics of the cascade for $0\le n\le 8$: (a) The number of daughter droplets is variable from $p=2$ to $p=8$. (b) The fragmentation ratio $\beta =⟨R_{n+1}/R_n⟩\approx 0.5$. (c) The distribution (pdf) of the daughter sizes follow a Gamma distribution of order 4. (d) The droplets lifetime is such that $t_n\approx R_n^2/\kappa$.