Glass

The Window That Did Not Flow

The story is that glass is a liquid, only a very slow one, and that the proof is in medieval cathedral windows, which are measurably thicker at the bottom than the top. Centuries of sagging, caught in the act. It is a lovely piece of evidence and it is wrong in every part.

The panes are thicker at the bottom because of how they were made. Crown glass was produced by gathering a blob on a rod and spinning it into a flat disc, which comes out thin in the middle and thick toward the rim, and panes cut from such a disc are wedges. A glazier fitting a wedge into a frame puts the heavy edge down, as anyone would. Some surviving medieval windows are thicker at the top, which the sagging story cannot account for and a careless glazier can.

And the arithmetic finishes it. Estimates for the viscosity of window glass at room temperature vary enormously, but take a middling one and work out how far a pane a meter tall creeps under its own weight in a thousand years. The answer is about a micrometer: a fiftieth of the width of a human hair, over a millennium. Roman glass two thousand years old shows no measurable sag either. The windows are not flowing, and nothing you own is.

And Yet It Is Not a Crystal Either

Having taken the myth away, here is the fact underneath it that made the myth plausible. Point an X-ray beam at molten glass and at the cold window in your wall, and the two pictures are almost the same: a couple of broad smears rather than the sharp spots a crystal gives. Whatever happened on the way down, it did not rearrange the atoms into a pattern.

That is what makes glass different from everything else that freezes. When water becomes ice, the molecules snap into a lattice, the volume jumps, heat is given out, and there is a single temperature at which it happens. All of that is missing here. Cool a glass-forming liquid and it simply gets thicker, with no jump, no latent heat and no temperature you can point to and say the change occurred there.

So a window is not a slow liquid, and it is not a crystal. The interesting question is what it is instead. The honest first answer is that it is a liquid whose atoms have stopped being able to move past one another, and the rest of this page is about how strange that turns out to be.

Twelve Orders of Magnitude, and Nothing Moves

Follow the thickness down. At the temperature a furnace works glass, around a thousand degrees, it pours like treacle. Four hundred degrees cooler it is stiffer than pitch. A hundred and fifty degrees below that it is not pourable by any definition anybody would accept. That is twelve powers of ten in the resistance to flow, across a few hundred degrees, with the arrangement of the atoms essentially unchanged throughout.

No ordinary phase transition behaves like this. Boiling and freezing are events: a structure gives way at a temperature and something discontinuous happens. Here nothing is discontinuous. The liquid is simply slower at every step, and at some point it becomes slower than the experiment.

The picture below is that climb, on a ruler of thickness marked with things you can pour. The dial is the temperature, and it stops at the point where the number stops being measurable. Watch the last row of the readout rather than the bar: how long a pane of this glass would take to sag a millimeter under its own weight. It runs from a fraction of a millisecond to three months.

Which Makes Solid a Statement About Your Patience

Since nothing discontinuous happens, somebody had to choose where to draw the line, and the choice is frankly arbitrary. The glass transition temperature is defined as the point where the material takes about a hundred seconds to rearrange itself, which corresponds to a particular round number of a viscosity. There is no physics in the hundred seconds. It is a human patience, written into a material constant.

And it behaves exactly as an arbitrary line would. Cool the same glass ten times more slowly and its transition temperature comes out a few degrees lower, because the atoms had ten times as long to keep rearranging before you stopped watching. Cool it faster and it comes out higher. No other transition temperature does this. Water freezes at zero however slowly you approach it.

So the sentence a window is a solid is not quite a statement about the window. It is a statement about the window and the length of a human life, taken together, and that is a strange thing to find inside a material property.

The Drop That Falls Once a Decade

There is an experiment that sits exactly on that line, and it has been running since 1927. A physicist at the University of Queensland melted a lump of pitch, poured it into a funnel, let it settle for three years, and then cut the stem. Nine drops have fallen since. The wait between them is around a decade.

Pitch at room temperature is a glass by every structural test, and it is also unambiguously flowing, which is the point of keeping it in a cupboard for a century. Its thickness is around a hundred million times water's. That is enough to make a drop take ten years and enough to make the flow undeniable, and it is the boundary case that shows why the cathedral story fails.

Window glass is not a hundred million times thicker than water. It is somewhere upward of a hundred million million million times. Pitch is to water roughly as water is to a thin gas; window glass is another twelve powers of ten past pitch. At that remove a drop does not take longer than a lifetime. It takes longer than there has been life.

The Paradox That Says Something Must Happen

So far the story has been that nothing happens and the liquid merely runs out of time. In 1948 Walter Kauzmann noticed that this cannot be the whole account, and the argument is short enough to follow.

Take a liquid that has been cooled below its freezing point without crystallizing, and measure its entropy as it goes down: a count of the ways its molecules can be arranged and share their energy. It falls, and it falls faster than the crystal’s does. Now extrapolate the two lines below the point where the glass takes over. They cross. At a definite temperature, a few tens of degrees below the transition, the extrapolated liquid would have fewer ways open to it than the crystal.

That is hard to believe of a jumbled liquid set beside an ordered crystal, and further down it becomes impossible: the liquid would run out of ways altogether while still being a liquid. So one of two things has to be true. Either the extrapolation is wrong and the curve bends before it gets there, or something real happens on the way, a genuine transition to a uniquely well-packed arrangement that nobody has ever reached. Both remain live.

The Deepest Unsolved Problem in Solid State

Philip Anderson won a Nobel Prize for work on disordered solids. In 1995 he wrote that the nature of glass and the glass transition was probably the deepest and most interesting unsolved problem in the theory of solids. Three decades later it is still open, and the shape of the disagreement is worth stating, because it is unusually clean.

On one side, the transition is a real thermodynamic event hidden below where anyone can look. There is a particular arrangement the liquid is heading for, and the reason nobody has seen it is that reaching it needs infinite patience: the closer you get, the longer the material takes to rearrange, and the time runs away faster than the temperature comes down. On the other side, nothing thermodynamic is happening at all. It is a traffic jam, and the arbitrary line in the fourth section is arbitrary all the way down.

The two accounts make almost identical predictions everywhere an experiment can be done, which is exactly why the question has survived. The interesting temperature is unreachable by construction, so the disagreement is about the behavior of a material in a region that cannot be visited by cooling.

Two lines of attack get closer than cooling can. Amber that has sat undisturbed for twenty million years has had time to settle further than any laboratory could arrange, and it can be measured. And since 2007 there has been a way to build a glass rather than cool one, by depositing it a molecule at a time onto a cold surface so that each arrival can find a good position before the next lands. Those come out denser and more stable than anything cooling produces, and by some measures correspond to millennia of settling. Neither has reached the place the paradox points at, and both have made it a thing you can approach rather than only argue about.

And It Is Not a Rare State

It would be easy to file all of this under a curiosity about windows. It is not. Almost every plastic is a glass. So are most gels, many ceramics, a great deal of what is in a pharmacy, and the sugar in a boiled sweet. Even metals can be made glassy, by cooling them fast enough that the atoms have no time to find a lattice, and the resulting alloys are used where springs must not lose their bounce.

The same mathematics turns up further from home than that. A pile of sand that will not pour, a traffic jam that has not cleared, a crowd that cannot move: all of them are systems whose parts are stuck because their neighbors are stuck, with no pattern anywhere and no single moment when it happened. Whether those are the same problem in a useful sense is itself debated, which is the sort of thing that happens when a question stays open for a century.