Universality of Tip Singularity Formation in Freezing Water Drops

Abstract

A drop of water deposited on a cold plate freezes into an ice drop with a pointy tip. While this phenomenon clearly finds its origin in the expansion of water upon freezing, a quantitative description of the tip singularity has remained elusive. Here we demonstrate how the geometry of the freezing front, determined by heat transfer considerations, is crucial for the tip formation. We perform systematic measurements of the angles of the conical tip, and reveal the dynamics of the solidification front in a Hele-Shaw geometry. It is found that the cone angle is independent of substrate temperature and wetting angle, suggesting a universal, self-similar mechanism that does not depend on the rate of solidification. We propose a model for the freezing front and derive resulting tip angles analytically, in good agreement with the experiments.

Theory and experiments on the ice–water front propagation in droplets freezing on a subzero surface

“…This calculation is based on the approximation that the propagation of this front can be split into two stages: a) the front is planar until it reaches a critical height when the shape of the droplet evolves into a cone due to the expansion of the volume of water when it freezes into ice, and b) afterwards, the front takes the shape of a spherical . cap centered at the apex of this cone 7,9 , illustrated in Fig. 3.…”

“…3). The angle θ is obtained by the requirement that upon freezing, the remaining volume of water expands into this cone and into the spherical cap 7,9 . The ratio ν = .917 of ice to water density determines θ ≈ 65 o , and corresponds to z 0 ≈ .64.…”

“…Recently, there has been some renewed interest in the properties of water droplets that are frozen on a planar surface at subzero temperatures [1][2][3][4]6,7,9 . A theoretical analysis, and numerical solutions of the heat diffusion differential equations for the propagation of the ice-water front in spherical droplets have been carried out by W.W. Schultz, M. G. Worster, and D.M.…”

Fast Dynamics of Water Droplets Freezing from the Outside In

“…From daily experience we know that water freezing up inside a closed container can exert extreme pressures on its container walls, as is exemplified by the fracturing of rocks by freezing water inclusions [3] and by the playful "ice bomb", in which freezing water bursts out of a thickwalled metal flask [4]. For unenclosed water drops, it has been found that the combination of expansion and geometrical confinement by surface tension leads to the formation of a singular tip in the final stage of solidification [5]. However, in these experiments the spherical symmetry was purposefully broken by cooling the droplets only from one side.…”

Growing bubbles and freezing drops: depletion effects and tip singularities

“…To minimize image distortion due to this meniscus, the glass is treated such that the wetting contact angle ≈ 90 • . This gives a clear view on the quasi-two-dimensional freezing process -typical videos can be found in the supplementary materials of [104]. Figure 6.3a shows that the solidification front in the Hele-Shaw cell grows towards the top of the drop in a similar fashion as the unconfined experiment, although the timescale of the process is a bit faster.…”