[Paper Review] Conical tip in frozen water drops

Michael Nauenberg|arXiv (Cornell University)|Apr 17, 2014

Arctic and Antarctic ice dynamics3 citations

TL;DR

This paper proposes a theoretical model explaining the formation of a conical tip in water drops frozen on a cold surface, attributing it to volume expansion upon freezing and the resulting concave ice-water interface. By modeling the remaining liquid as a spherical drop in contact with a concave front and applying volume conservation with ice's density ratio (r = 0.917), the theory predicts a cone aperture angle of θ = 33.55°, matching experimental observations.

ABSTRACT

A theory is presented for the formation of a conical tip in water drops that are frozen on a flat surface below freezing temperature. For the known ice to water density ratio r = .917, the angle of aperture of this cone is found to be θ = 33.550, consistent with observations.

Motivation & Objective

• To explain the origin of the conical tip observed in water drops frozen on a cold surface.
• To resolve the discrepancy between prior theories assuming a planar freezing front and experimental observations showing a strongly concave front.
• To develop a quantitative model that accounts for the volume expansion upon freezing and its geometric consequences.
• To predict the cone angle of the tip based on the known ice-to-water density ratio (r = 0.917).
• To validate the model by showing consistency with experimental observations of the tip forming only at the final stage of freezing.

Proposed method

• Model the remaining liquid phase just before complete freezing as a spherical drop of radius ρ in contact with a concave ice-water interface.
• Assume the freezing front curvature is such that the liquid remains confined in a spherical shape due to surface tension.
• Use geometric analysis of two intersecting spheres (initial and partially frozen drop) to compute the volume difference δV between the liquid segments.
• Apply the volume expansion factor ξ = 1/r = 1.0905 for ice formation from water to relate the initial liquid volume to the final frozen cone volume.
• Derive a transcendental equation (Eq. 3) linking the cone angle θ to the volume expansion ratio ξ.
• Solve Eq. 3 numerically for ξ = 1.0905 to obtain the predicted cone aperture angle.

Experimental results

Research questions

• RQ1 Why does a conical tip form at the apex of a water drop during freezing on a cold surface?
• RQ2 How does the volume expansion upon freezing (from water to ice) influence the shape of the freezing front and the final tip morphology?
• RQ3 Why do previous theories assuming a planar freezing front fail to predict the conical tip, despite experimental evidence of a concave front?
• RQ4 What is the theoretical cone angle of the tip formed under the known ice-to-water density ratio (r = 0.917)?
• RQ5 At what stage of freezing does the conical tip emerge, and what physical conditions enable its formation?

Key findings

• The theory predicts a cone aperture angle of θ = 33.55° for the frozen water drop tip, which matches experimental observations.
• The conical tip forms only at the very end of the freezing process, when the remaining liquid volume becomes very small.
• The formation of the tip is driven by the volume expansion upon freezing and the resulting concave shape of the ice-water interface.
• The model assumes the remaining liquid is confined into a spherical drop by surface tension, just before complete solidification.
• The derived equation (Eq. 3) relates the cone angle θ to the volume expansion ratio ξ = 1/r, with ξ = 1.0905 for r = 0.917.
• When ξ = 1 (equal densities), the model predicts θ = 90°, confirming that no tip forms in the absence of volume expansion.