X-ray Microscopy Study of Freezing Sessile Droplets

Abstract

A sessile water droplet on a cold substrate freezes into a shape with a sharp apex because of water's expansion upon freezing, yielding a universal tip angle across various conditions. Using \textit{in situ} X-ray imaging, we report that this angle changes with substrate temperature, and the deviation originates from bubble formation during freezing. Three-dimensional tomography enables direct quantification of the effective ice-water density ratio, accounting for trapped bubbles. Incorporating this effective density ratio reconciles the temperature-dependent tip angles. We also confirm that a bubble-free frozen droplet in a vacuum chamber exhibits the universal tip angle. Furthermore, X-ray imaging allows us to measure the three-phase boundary angles \textit{in situ}, thereby validating the geometric theory behind tip formation. These findings advance our understanding of the freezing dynamics associated with multiphase systems and highlight the capabilities of high-resolution X-ray imaging in ice research.

Figures (3)

• Figure 1: X-ray imaging of the freezing process of a sessile water droplet placed on the cold polycarbonate plate. After placing a droplet of 2µL in volume on the substrate at room temperature, we decrease the substrate temperature to -5. (a) The sketch of the experimental setup. Four distinct stages of the droplet freezing are shown. (b) Liquid cooling. A droplet remains in the supercooled liquid phase despite sub-zero substrate temperature. The black scale bar is 500µm. (c) Recalescence. Dendritic ice nucleates; the inset with the white 100 µm scale bar displays its magnified view. (d) and (e) Freezing. Curved water(top)-ice(bottom) interface, water-ice-air boundary, and the trapped air bubbles are visible. (f) Solid cooling. The completely frozen droplet forms the pointy tip at the top. Elapsed time after the recalescence is shown in the top-right corner.
• Figure 2: Effect of trapped bubbles on the tip angle according to the substrate temperature. The representative 2D X-ray images of the frozen water droplets of 1µL in volume on a cover glass at (a) -5, (b) -10, and (c) -20, respectively, in the ambient condition. The representative images of the bubble-free frozen water droplet on a cover glass at (d) -5 and (e) -20, respectively, in a vacuum chamber. All scale bars are 200µm. (f) Measured tip angles depending on substrate temperature and presence of bubbles. The black-filled circles and red-empty triangles represent the tip angles of individual frozen droplets with bubbles and without bubbles, respectively. The center line of the box plot corresponds to the median of each dataset, and the top and bottom lines are the 75th and 25th percentiles, respectively. Blue solid line at 139 deg and the band around it indicate the average and standard deviation of the tip angles reported in Marín et al. Marin2014: $139\pm8$°.
• Figure 3: X-ray imaging-enabled characterization of droplet freezing. (a) The schematic of the three-phase boundary. The advancing ice front has the height $h$ and the angle $\gamma$ to the air-water interface at the three-phase boundary. $H$ indicates the final height of the frozen droplet. (b) $\gamma$ as a function of the normalized height $h/H$ according to the substrate temperature. The data points are acquired from multiple droplets of the same volume in ambient conditions. (c) Theoretical curves predicting the tip angle according to $\gamma$ and the effective ice-to-water density ratio $\nu_{\text{effective}}$. The black solid is for $\gamma$ = 90°, and the blue dashed line is for $\gamma$ = 72.2°, which is the estimated $\gamma$ at $h/H = 1$ by the linear fit in (b). The representative X-ray tomographic cross-sections of the frozen water droplets of 1µL in volume on a cover glass at (d) -5 and (e) -20, respectively, in the ambient condition. The scale bar is 200µm. (f) Estimated $\nu_{\text{effective}}$ in the bubble-laden ice volume truncated at a distance $z$ from the end of the tip. The data points represent the average value of 20, 18, and 11 specimens at -5, -10, and -20, respectively, and the standard deviations are shown as bands.