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When I put ice cubes in a glass of water, I find that sometimes they will stick together and form a sort of "bridge" between them as they melt. There is usually a visible line where one ends and the other begins, and they break apart if pushed (but for the most part they stick together if they aren't interfered with). I'm wondering how they form this bridge in the first place if they're melting, and why it stays together.

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When you put the ice cubes in, the temperature of the cubes is (much) below freezing temperature. The drink in the glass is above freezing temperature. The interface between the ice and the liquid (the surface of the ice cube) is cooled by the ice cube, but heated by the liquid. The ice cube heats up in this process and the liquid cools down in this process (which is the main reason why you put the cubes in your drinks anyway).

Usually the liquid wins this competition and the ice cube starts shrinking. However, at places where two ice cubes are very close together, the liquid in between the cubes cools very fast (as it is cooled from two sides and cannot easily be replaced by new, warmer, liquid since it is contained between two walls of ice). At these places, the ice cube is able to win and starts expanding. The liquid close to the cubes therefore freezes, forming a bridge.

Note that bridges have the highest probability to form if the ice cubes come from a very cold freezer and if your drink is not too warm.

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