Fast Dynamics of Water Droplets Freezing from the Outside In

A drop of water that freezes from the outside in presents an intriguing problem: the expansion of water upon freezing is incompatible with the self-confinement by a rigid ice shell. Using high-speed imaging we show that this conundrum is resolved through an intermittent fracturing of the brittle ice shell and cavitation in the enclosed liquid, culminating in an explosion of the partially frozen droplet. We propose a basic model to elucidate the interplay between a steady buildup of stresses and their fast release. The model reveals that for millimetric droplets the fragment velocities upon explosion are independent of the droplet size and only depend on material properties (such as the tensile stress of the ice and the bulk modulus of water). For small (submillimetric) droplets, on the other hand, surface tension starts to play a role. In this regime we predict that water droplets with radii below $50\mu \mathrm{m}$ are unlikely to explode at all. We expect our findings to be relevant in the modeling of freezing cloud and rain droplets.

Figure 1

Experimental method. (Supplemental Material [29], video 1) (a)–(b) A droplet of liquid water rests on a superhydrophobic soot surface in the center of a small vacuum chamber. Here it becomes supercooled by evaporative cooling. An iced floor plate controls the final supercooling of the drop by providing a vapor buffer. (c) Trace of chamber pressure $P$ and droplet temperature $T$ (measured by inserting a thermocouple in the drop in one case). The moment of ice nucleation and the three phase equilibrium temperatures involved (ice-liquid, ice-vapor, and liquid-vapor) are indicated. (d) Once the droplet has reached a steady temperature, ice nucleation is induced by touching the drop with a tip of silver iodide (AgI). (Scale bars: 1 mm).

Figure 2

High speed recording of an inwards freezing water droplet. (Supplemental Material [29], video 2) (a) Ice nucleation at the drop surface. (b) Formation of an ice shell, causing the droplet to roll over slightly. (c) An ice spicule is pushed out from the shell. (d) Spicule ceases to grow, the ice shell cracks and vapor cavities appear below surface (dashed circles). (e) Cavities have completely healed. (f) Spicule tip explodes with splinter velocities of $3.5\mathrm{m}/\mathrm{s}$. (g) More cracks and cavities appear, accompanied by thrown out ice splinters. (h) Droplet appearance just before the final explosion. (i) Final explosion with fragment velocities of $1.5\mathrm{m}/\mathrm{s}$. (Scale bar: 1 mm)

Figure 3

Basic picture of the dynamics of an inwards freezing water drop. (a) High speed recording of a droplet bursting into two equal halves. (Scale bar: 1 mm). (b) Cross-sectional views of the halves of a burst droplet, showing a clear core region surrounded by a strong ice shell. A healed crack emerges from the core and runs over the length of the droplet. (Scale bars: 1 mm). (c)–(e) Cartoon of the freezing and bursting process. (c) A temperature difference $T_{il}-T_{iv}$ over the ice shell makes the freezing front $R_i$ gradually proceed inward [see (f)]. (d) Expansion of the freezing water leads to a buildup of pressure $P_i$ in the inclusion, which in turn leads to an azimuthal stress ${\sigma }_{\theta \theta }$ in the ice shell. (e) Once the stress in the ice shell exceeds ${\sigma }_c$ the shell cracks open. (f) Theoretical time evolution of the freezing front, scaled by ${\tau }_f=(\rho L_m{R}_o^2)/(\kappa \mathrm{\Delta }T)$, where $\kappa$ is the thermal conductivity of the ice shell and $L_m$ is the specific latent heat of melting (see Supplemental Material [29], Sec. I). (g) Time evolution of pressure in the inclusion during the opening of a crack for three different ratios $R_i/R_o\in 0.1$, 0.5, 0.8. The pressure rapidly varies from its positive initial value to a negative value (tension) of the same magnitude. The peak-to-peak pressure increases as the relative inclusion size shrinks. (h) Explosion energy (solid black curve) per unit volume $e=\rho v^2/2$ (scaled by ${\sigma }_c^2/K$) as a function of $R_i/R_o$. A maximum in energy occurs for $R_i/R_o\approx 0.57$. The dashed red lines indicate the minimum surface energy (per unit volume) required to tear apart the liquid in the inclusion for different droplet sizes. Below $R_o\approx 50\mu \mathrm{m}$ the required surface energy is higher for all $R_i/R_o$.