Initial solidification dynamics of spreading droplets

When a droplet is brought in contact with an undercooled surface, it wets the substrate and solidifies at the same time. The interplay between the phase transition effects and the contact-line motion, leading to its arrest, remains poorly understood. Here we reveal the early solidification patterns and dynamics of spreading hexadecane droplets. Total internal reflection imaging is employed to temporally and spatially resolve the early solidification behavior. With this, we determine the conditions leading to the contact-line arrest. We quantify the overall nucleation behavior, i.e., the nucleation rate and the crystal growth speed and show its sensitivity to the applied undercooling of the substrate. We also show that for strong enough undercooling it is the rapid growth of the crystals which determines the eventual arrest of the spreading contact line. By combining the Johnson-Mehl-Avrami-Kolmogorov nucleation theory and scaling relations for the spreading, we calculate the temporal evolution of the solid area fraction, which is in good agreement with our observations.

Figure 1

Schematic of the experimental setup for total internal reflection imaging.

Figure 2

Characteristic sequences of hexadecane drops spreading on a sapphire prism of varying temperature, the red bar indicates a length of 1 mm: (a) $\mathrm{\Delta }T=1.2\mathrm{K}$: Random nucleation with subsequent dendritic growth. (b) $\mathrm{\Delta }T=2\mathrm{K}$: Continuous nucleation with subsequent radial crystal growth. Note the different timescales in (a) and (b).

Figure 3

Number of distinct crystals as function of dimensionless time. Separate experiments are denoted with different symbols. The dashed line ($---$) indicates the scaling [Eq. (1)]. The capillary timescale ${\tau }_c={(\rho R_0^3/\sigma )}^{1/2}$. The color bar shows the temperature difference $\mathrm{\Delta }T=T_f-T_c$. The inset shows the same data rescaled with the probability of the formation of a stable crystal nuclei at different $\mathrm{\Delta }T\mathrm{s}, exp\left(E_a/k_{b}T\right)$.

Figure 4

Contact-line arrest and crystal growth: (a) Nucleation near the contact line, leading to local contact-line arrest (red circles), the red bar indicates a length of 1 mm, $\mathrm{\Delta }T=1.0\mathrm{K}$. (Left panel) Nucleation far away from the contact line. (Center panel) Nucleation close to the contact line leads to contact-line arrest. (Right panel) Crystals growing towards the contact line lead to contact line arrest. (b) Growth velocity as a function of the undercooling $\mathrm{\Delta }T$. The error bars indicate the minimal and maximal growth velocity at a certain temperature. The dashed line shows the growth velocity $U_{\mathrm{g}}=\beta \mathrm{\Delta }T$ with $\beta \approx 4.5\mathrm{mm}{\mathrm{s}}^{-1}{\mathrm{K}}^{-1}$. (c) Radius of arrest versus temperature. $---$ is the model of Eq. (2). The error bars show the minimum and maximum radii of a single arrest event.

Figure 5

Time dependence of the fraction of solidified material at the substrate for various temperatures. The solid lines show Eq. (3) for $\mathrm{\Delta }T=2.1,\mathrm{\Delta }T=2.7,\mathrm{\Delta }T=3.2,\mathrm{\Delta }T=5.5,\mathrm{\Delta }T=6.5,\mathrm{\Delta }T=8.5\mathrm{K}$. A large change in solidification behavior is seen for the model between $2.7\mathrm{K}<\mathrm{\Delta }T<3.2\mathrm{K}$.