Mathematics says why ice turns round when it melts – 05/04/2023 – Fundamental Science

Every time I dropped an ice cube into a glass of water, I noticed the same thing: as it melted, not only did the cube change size (it got smaller, of course), but it became more rounded. Its edges and corners disappeared. And it was always like this: the boundary between the ice cube and the water seemed smoother and softer. Understanding what went into my glasses of ice water took at least 30 years. But I realized that a more accurate perception of these things involved learning more (and better) mathematics. After all, the behavior I observed in my glass of water didn’t just happen. Indeed, it was an important theorem.

Lucky for me, this same curiosity had also affected the Slovenian physicist Jošef Stefan (1835-93). Around 1890, Stefan, the first among his peers, focused on this problem that ended up receiving his name. His curiosity sounded less childish than mine, and was motivated by the formation of ice in marine waters and the freezing of soils.

We can describe this class of problems as follows. In a larger region (the glass) we have two sub-regions: the water and the ice cube. The laws that govern the temperature diffusion in the liquid are different from those that govern the temperature inside the ice cube. However, over time, the shape of these regions changes. And the border separating the cube from the liquid is not fixed, but free. Therefore, understanding the problem involves not only finding a temperature profile for the ice cube and the liquid, but also studying the interface properties between the two. That is, consider the geometry of the ice cube. Understand the free boundary of the problem.

Let’s look at another example. Imagine that someone covers Sugarloaf Mountain and Pedra da Urca with a huge sheet. In some areas, the sheet will touch the rocks; in others, no. The region where the sheet starts to separate from the stones is an interface between two worlds: the one where both structures are in contact and the one where they are separated. As in the case of the ice cube, the interface that forms depends on the processes that take place in both regions. It is also a free border.

Another example is smart materials – a coat that “notices” when you get on the subway and starts to heat up less; a wall that “hears” noise in a room and increases its acoustic insulation capacity; a shoe that “notices” if you’re walking faster and starts squeezing less. Heterogeneous materials, membranes detaching from an obstacle, and even the optimal moment to exercise a call and put option on a given financial asset are some more examples.

In common, these problems have diffusion laws that govern different processes in different subregions of the problem. These subregions are separated by a boundary that is determined by diffusion. It’s Tostines-type mathematics: the frontier depends on diffusion, but diffusion also depends on the frontier.

Mathematically, these problems are quite challenging. In addition to the diffusion laws in each sub-region (generally described by partial differential equations), the problem still depends on an interface that changes throughout the process. And these different elements must be studied together, which generally requires new techniques and strategies.

Immense advances of this theory – whose official name is free border problem theory – occurred in the second half of the 20th century. And even today, surprising results and new techniques appear in the specialized literature. And many problems remain without a complete or even satisfactory answer. In particular, an important theorem of the area guarantees that, however sharp the edges and corners of a cube are, they instantly smooth out when dipped in our favorite beverage.

Here, the fun lies in realizing that there is an interface that separates children’s curiosity from high-level scientific knowledge. As there is an interface that separates the stock of current knowledge and available techniques from what we want to know. With luck, it is noted that it is a free border.

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Edgard Pimentel is a researcher at the Mathematics Center of the University of Coimbra and professor at PUC-Rio.

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