Pattern Formation during the Impact of a Partially Frozen Binary Droplet on a Cold Surface

The impact of a droplet on an undercooled surface is a complex phenomenon as it simultaneously instigates several physical processes that cover a broad spectrum of transport phenomena and phase transition. Here, we report and explain an unexpected but highly relevant phenomenon of fingered growth of the solid phase. It emerges during the impact of a binary droplet that freezes from the outside prior to the impact on the undercooled surface. We establish that the presence of presolidified material at the advancing contact line fundamentally changes the resulting dynamics, namely, by modifying the local flow mobility that leads to an instability analogous to viscous fingering. Moreover, we delineate the interplay between the interfacial deformations of the impacting droplet and patterned growth of the solid phase as disconnected patterns emerge at faster impacts.

Figure 1

(a) Bottom-view snapshots, recorded via high-speed total-internal-reflection (TIR) imaging, highlighting the temporal growth of the solid phase in fingerlike patterns along the substrate after the impact of a partially frozen droplet ($R_{\mathrm{drop}}=0.85\mathrm{mm}$) of a binary mixture (80:20% vol) of hexadecane and diethyl ether. Here, $U=1.24\mathrm{m}/\mathrm{s}$ ($\mathrm{Re}\sim 440$) and $T_s=6°\mathrm{C}$ as substrate temperature. Both the contact line and the solidified phase (hexadecane) are seen as gray-shaded regions in the TIR images. (b) Side view high-speed recordings: Typical deformations experienced by the partially frozen interface as the droplet ($R_{\mathrm{drop}}=0.82$) impacts on the substrate $T_s=6°\mathrm{C}$ with $U=1.45\mathrm{m}/\mathrm{s}$ ($\mathrm{Re}\sim 517$). (c) Temporal evolution of the azimuthal undulations along the radius $R(\theta )$ of the contact line during the expansion of the wetted area shown in sequence in (a). (d) Scaled master curve of the spreading radius $R_f/R_f^{\infty }$ versus time $t/(2R_{\mathrm{drop}}/U)$ for droplet impacts on the undercooled substrate, $T_s=6°\mathrm{C}$, at different impact velocities $U$ = (Solid square) $\text{0.88\hspace{0.17em}\hspace{0.17em}}\mathrm{m}/\mathrm{s}$, (Star) $1.45\mathrm{m}/\mathrm{s}$, (Daimond) $1.90\mathrm{m}/\mathrm{s}$, (Filled circle) $2.25\mathrm{m}/\mathrm{s}$. Inset: Raw data for the temporal evolution of $R_f$ for these velocities.

Figure 2

(a) Sequence of events leading to the destabilization of the advancing contact line and fingered growth of the solid phase are outlined schematically. While a partially frozen droplet deforms on the substrate, presolidified material accumulates at the advancing contact line. This results in a mobility contrast between the unfrozen fluid away from the contact line and its vicinity, thus viscous-fingering occurs. (b) A liquid marble is used as a model system to mimic a droplet that is frozen from the outside. The bottom-view sequence recorded via TIR shows a typical outcome of the impact of the liquid marble ($R_{\mathrm{drop}}=1.06\mathrm{mm}$, $U=1.45\mathrm{m}/\mathrm{s}$). Notably the emergent features, namely, destabilization of the contact line and clustering of particles in radially growing fingerlike patterns, closely resemble the patterns observed for the impact of a partially frozen droplet. Small circles at the periphery of the droplet footprint highlight the fingered clusters of particles.

Figure 3

A comparison between the experimentally measured (red symbols) onset $t_{\ell }$ of the fingered growth of the solid phase and the theoretical predictions of the instant of lamella ejection, computed from Eq. (2) (solid black line). The experimental values correspond to the impacts of partially frozen droplets of 80:20% vol mixture of hexadecane and diethyl ether on the undercooled substrate at $T_s=9°\mathrm{C}$.

Figure 4

(a) Temporal growth of the solid-phase in disconnected patterns after the impact ($U=2.25\mathrm{m}/\mathrm{s}$, $\mathrm{Re}\sim 800$) of a binary droplet of a mixture (80:20% vol) of hexadecane and diethyl ether on an undercooled substrate, $T_s=9°\mathrm{C}$. (b) At higher impact velocities ($\mathrm{Re}>550$), the front of the ejected lamella initially lifts off from the substrate at the location $R_{\ell }$. Consequently no particle-induced viscous fingering takes place until the lamella rewets the substrate at $R_{\mathrm{rc}}$. The extent of uniformly solidified central patch $R_{\ell }$ marks the location of lamella ejection. For increasing impact velocities, $R_{\ell }$ slowly decreases as $R_{\ell }/2R_{\mathrm{drop}}\sim {\mathrm{Re}}^{-1/4}$ (solid black line). The width of the unfrozen annular gap $\mathcal{W}=R_{\mathrm{rc}}-R_{\ell }$ widens for increasing $U\sim \mathrm{Re}$. The dashed line is a guide to the eyes. The experimental data plotted here correspond to the impacts of partially frozen droplets of 80:20% vol mixture of hexadecane and diethyl ether on the undercooled substrate at $T_s=9°\mathrm{C}$.