Frozen Impacted Drop: From Fragmentation to Hierarchical Crack Patterns

We investigate experimentally the quenching of a liquid pancake, obtained through the impact of a water drop on a cold solid substrate ($0°\mathrm{C}$ to $-60°\mathrm{C}$). We show that, below a certain substrate temperature, fractures appear on the frozen pancake and the crack patterns change from a 2D fragmentation regime to a hierarchical fracture regime as the thermal shock increases. The different regimes are discussed and the transition temperatures are estimated through classical fracture scaling arguments. Finally, a phase diagram presents how these regimes can be controlled by the drop impact parameters.

Figure 1

Scheme of the frozen pancake obtained after a liquid drop impacted a cold substrate. The pancake has a radius $R$ and a typical thickness $h_0$. The substrate, at a temperature $T_s$, cools the pancake, so that a layer of thickness $h(t)$ is frozen, above which the liquid is at the freezing temperature $T_0$.

Figure 2

(a) Frieze presenting snapshots of the frozen pancakes formed after a water drop impacted, from a falling height $H=36\mathrm{cm}$, a cold substrate at various temperatures, $T_s=-20.0°\mathrm{C}$, $-31.1°\mathrm{C}$, $-41.2°\mathrm{C}$, $-50.3°\mathrm{C}$, and $-59.6°\mathrm{C}$ from left to right (with $\mathrm{\Delta }T=-T_s$). Depending on $\mathrm{\Delta }T$, the frozen pancake presents different crack patterns that can be gathered into three different regimes: (I) no cracks, (II) fragmentation regime, (III) hierarchical fracture regime. The transition temperatures are $\mathrm{\Delta }{T}_{\mathrm{I}\text{−}\mathrm{II}}^{(\exp )}\sim 27°\mathrm{C}$ and $\mathrm{\Delta }{T}_{\mathrm{II}-\mathrm{III}}^{(\exp )}\sim 42°\mathrm{C}$. (b) Sequence showing the drop impact and solidification dynamics preceding the fracture pattern observed on the second image of (a): $T_s=-31.1°\mathrm{C}$. (c) Sequence preceding the fracture pattern observed on the fifth image of (a): $T_s=-59.6°\mathrm{C}$. On these two sequences, the time and the scale bar are on the images.

Figure 3

Appearance time of the first crack, $t_c$, plotted as a function of $\mathrm{\Delta }T=T_0-T_s=-T_s$, with $T_s$ being the substrate temperature, for five different falling heights of the impacting drop. $t_c$ is determined considering the initial time when the drop reached its maximum spreading diameter after impact. The open diamonds correspond to the fragmentation regime (II), while the closed triangles correspond to the hierarchical fracture regime (III). The dashed line, representing $t_c\propto \mathrm{\Delta }{T}^{-5}$, follows reasonably well the points in regime III.

Figure 4

Phase diagram for the cracks pattern as the substrate temperature ($-\mathrm{\Delta }T$) and the drop impact velocity (here noted by $H$, the height of the drop fall) vary. The three regimes observed are represented with the same symbols and the same color as in Fig. 2: white squares for regime I, red diamonds for regime II, and blue triangles for regime III.