Entropy and Glasses

Having spent much of the last few weeks concentrating non-stop on exams, I have learned to appreciate this subject a little more. So, as a celebration of finishing yesterday, I wanted to tell you all why it's quite so cool.

The two most important statements in physics are Second Laws. Newton's Second Law of Motion states "Rate of change of momentum is proportional to the force producing it". This leads to the equation F=ma. That's pretty cool, almost as cool as the Second Law of Thermodynamics, which states that Entropy Always Increases.

Entropy is disorder. If atoms are in a mess, entropy is high. Entropy means that when you add a nice square lump of sugar into tea, it will stop being nice and square and orderly, and spread its atoms all through the tea by dissolving (making your tea taste all sweet and gross). Entropy also means that if you heat up your room, the heat spreads out of the windows, doors and through the walls, and in the end your room is almost as cold as it ever was. Hot things have high entropy, because the atoms jiggle a lot and swap places and move around. And there's no way that they will stay in a nice ordered way like that. There is order involved in having all the heat in one place (your room), and it's much more disordered to share it out randomly (everywhere else), hence your room getting cold.

This law has never been proven wrong. If something creates order in one place, you will create that much disorder (and probably some more) in another place to compensate. Entropy always increases. We are doomed to become disordered.

The law leads to an equation that goes like this:
G = H - TS
This is one of the most useful equations ever. The second law of thermodynamics says entropy must go up. This equation does the same thing. G is called a free energy and in all processes, free energy G must go down. G is related to entropy of the universe somehow. S is the entropy of what you're looking at. If you line up a load of atoms in a row, then entropy of those atoms, S, goes down. But although you'd think that wouldn't be allowed from our law, it is, because the entropy of the whole universe could still go up, even if the entropy of this little bit goes down.

It happens all the time. You freeze some water to make an ice cube, and it has quite a low entropy. But freezing gives out heat. And it heats up the rest of the universe, so the rest of the universe has a higher entropy. This process of giving out heat (or even taking heat in) is explained in H. H is called the enthalpy. Giving out heat gives H a minus sign, and taking in heat gives H a positive value. The other letter in this equation is T, for Temperature.

Now, imagine that you have a load of atoms just sitting around. You decide to cool them down until the temperature, T, is zero. Absolute zero. It's minus 270-odd degrees C. Now, according to the equation, G = H - TS. If T is zero, then T * S = 0, no matter what S is. So G will try to get to a minimum value, and the only way it can do that is by lining up all the atoms so that H is as low as possible. From the paragraph above, we see that we need to find a way of giving out as much heat as possible, since this makes H as low as possible and so makes G as low as possible.

For any pair of atoms, you can work out what energy there is between them. They interact in some way, and have an energy. If there is more energy available than they need to interact, they give it out as heat. We can make a guess that there will be some optimal distance between the atoms that will make them give out the maximum amount of heat.

If you have a load of atoms at a really low temperature, they will all move to that distance away from each other, because that gives the lowest H and therefore the lowest G. That's what a crystal structure is. It's an ordered layout of atoms. Most solids have a nice ordered layout of atoms.

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Just let me make it clear... the entropy of this crystal is very low, but in making a low entropy system, the rest of the universe is getting hotter, and entropy is increasing. So from the second law of thermodynamics we can work out that crystals are the most stable structure of anything if you get cold enough.

Glass has atoms that are all over the place though. And cooling glass down to absolute zero won't change that. That goes precisely against what I have been saying. Glass should not exist. It is unstable, according to the second law of thermodynamics, which is NEVER wrong. In fact it CAN exist, but only exist because at low temperatures, atoms don't have enough energy to move around, so when they try to arrange themselves into a crystal structure as they cool, they can't arrange themselves quickly enough so a disordered structure is left over.

The fact that science can describe so much using the law "Entropy always increases" (and I've only just scratched the surface of one area), and the things it predicts are correct is the reason why I love science. It's a triumph of logic, and just so, so powerful. That's why I'm sad I won't go to any more undergraduate lectures about it.