On the shape of air bubbles trapped in ice

Water usually contains dissolved gases, and because freezing is a purifying process these gases must be expelled for ice to form. Bubbles appear at the freezing front and are then trapped in ice, making pores. These pores come in a range of sizes from microns to millimeters and their shapes are peculiar; never spherical but elongated, and usually fore-aft asymmetric. We show that these remarkable shapes result of a delicate balance between freezing, capillarity, and mass diffusion. A non-linear ordinary differential equation suffices to describe the bubbles, which features two non-dimensional numbers representing the supersaturation and the freezing rate, and two additional parameters representing simultaneous freezing and nucleation treated as the initial condition. Our experiments provide us with a large variety of pictures of bubble shapes. We show that all of these bubbles have their rounded tip well described by an asymptotic regime of the differential equation, and that most bubbles can have their full shape quantitatively matched by a full solution. This method enables the measurement of the freezing conditions of ice samples, and the design of freeze-cast porous materials. Furthermore, the equation exhibits a bifurcation that explains why some bubbles grow indefinitely and make long cylindrical ``ice worms'', well known to glaciologists.