The peculiar shapes of air bubbles trapped in ice

Open the freezer and look at an ice cube. Chances are it is not perfectly transparent; maybe it is white; if so, it likely contains air bubbles. Water usually contains dissolved gases, and because freezing is a purifying process these gases are expelled as ice forms. Bubbles appear at the freezing front and are then entrapped into the ice. Such bubbles come in a large range of sizes from microns to millimeters and their shapes are peculiar; never spherical but elongated, and usually fore-aft asymmetric. They result of the interplay between freezing, capillarity and diffusion. More generally, gas-laden liquids yield porous materials when they freeze.

We conduct experiments to characterize the bubbles obtained at different freezing rates; we obtain empirical laws for the dimensions and aspect ratio. Then, we show that the problem can be reduced to a simple yet non-linear ordinary differential equation; it involves only three parameters, related to the freezing rate and the gas saturation. This equation predicts asymptotical regimes which describe the top and the bottom of the bubble; our experiments confirm both regimes. Finally, we provide a method for matching solutions of the differential equation to given experiments. Most bubble shapes correspond to solutions starting with a straightforward initial condition; others require a more subtle one but are still well matched by theory. This matching enables a indirect measurement of the gas concentration and freezing rate under which the bubble is formed.