Short story: a thin layer of water on the surface of the ice melts in contact with your skin, and then freezes in all the little grooves of your skin, “gripping” it and sticking. Have you ever seen two very polished plates or other flat things that are hard to separate once you press them against each other? That’s because they’re both flat and a lot of the surface is in contact with the other one. When the ice freezes it makes a surface that mates with your skin perfectly.
Longer story: it may seem counterintuitive that ice can melt and re-freeze again… is it warm enough to melt or cold enough to stay frozen? The key is that ice can be colder than it needs to be in order to stay frozen, and heat doesn’t transfer through ice instantly.
So when you first touch an ice cube, your warm skin beats the outer layer of ice and it melts. However the ice can transfer heat away better than your blood can warm your finger skin, so before too long that heat that melted the outer layer is absorbed by the rest of the ice cube, allowing that layer to freeze again. The whole ice cube is a little warmer now because of all this, but it’s still frozen.
If the ice cube is warm enough, the environment is warm enough and the part of your body touching it is warm enough, it won’t be cold enough for the ice to re-freeze, and it will simply melt and get wet. This is why if you fish a ice cube out of your drink on a warm day, it won’t stick to your hand like we’re talking about.
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Water doesn’t “stick”, but it exhibits capillary action: it gets “sucked” into little grooves by itself. This is how a paper towel dipped into water can soak it up the towel against gravity.
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