The Math of Glass

This is what I thought I knew about glass: if you wait long enough, it finally turns into something else. By “long enough,” I mean centuries at least, and by “something else,” I mean either an opaque solid or a gooey liquid. Glass is, or so I thought, already close to the latter—well below its freezing point but very, very slow to crystallize. On the presumably inevitable way to that state, it is as we customarily see it. Yet it is ever changing.

Is this true? At last, I know the answer: No! Glass is stable. While really old glass may be discolored or hazy, this may merely reflect imperfections in the fabrication process. (Think I’ll believe MIT over Wikipedia.) No, wait, my Inorganic Chemistry textbook says there is a long-term change too. “Because glasses are metastable solids, they may eventually crystallize or devitrify, especially if held close to their melting points for long periods (as novice glassblowers learn to their dismay).” So, have I learned anything? Or have I simply self-enstupidated? Could be the latter, as I pick up on the Kauzmann paradox – or as it may now be called, the discredited Kauzmann paradox.

The Kauzmann paradox is, or was, I think, this: under certain circumstances in a supercooled liquid, such as might be found during the glass transition, a negative entropy can be calculated. But that can’t be. And perhaps it isn’t. I found one non-Wikipedia reference which declared: “After nearly eighty years of research, there is no conclusive evidence that supports the concepts proposed by Kauzmann. As such, it can be concluded that future work should be focused elsewhere.” OK! Will do!

My feeling is that if whatever happens conflicts with theory, then that’s the theory’s problem...or the theorists’ problem. This could be like the bees-can’t-fly idea. Oh yes, they can! Kind of. Depends on how you observe 'em.

The only reason I got on this subject at all was that, while researching for my recent chromium post, I consulted the abovementioned textbook. An inquiry into the atomic latticework of stainless steels – which are 10-18% chromium – drifted into crystals of metals and then solids in general, and that brushed against the topic of glasses. In this book, glasses are said to have some repeating structure we might call crystalline. But unlike the case with, say, big hunks of quartz, the repeats do not go on infinitely or even very far. Soooo...how is that described formally? What is the math of glass? Sine waves that are damped, or just quit after a while? Or maybe it’s a composite of several Boltzmann distributions, which seems to appear in that paradox-bustin’ article above in its section titled Statistical Mechanics.

I am not doing this subject justice; it really is far more rigorous than I am making it sound. I cannot claim I comprehend it, but noodling is seldom unrepaying. I came across, and not for the first time, some great lecture notes online. By “great,” I mean of course “helpful to me.” I can’t guarantee the same for you. As it happens, this reference does not mention the Kauzmann paradox, but it does speak of other researchers I’d found cited alongside Kauzmann. It does seem to be talking about the same subject. And in ways I somehow find more instructive. I had, for example, jumped to the conclusion that “fictive temperature” was, well, dishonest; but no, it has a graphical meaning and is calculable. (Though the former may not be precise and the latter may not be unambiguous. Thanks a lot!)

Probably all the references I looked at spoke of the same things, but only – or at last – in this one I glommed onto the distinction between vibration (of individual molecules) and structural relaxation (of groups of molecules); how the sum of these helps describe what happens as a liquid is cooled; how they differ in their sensitivity to temperature...and also how fast that temperature is changing and how long you’re watching all this. If you set this up right, the liquid may veer from mere supercooling to glass formation. Where that veers is – or is very close to – the fictive temperature.

For some reason, it didn’t bother me this time to read, “The most general and most generous definition of the fictive temperature is to assume that there are an infinite number of fictive temperatures.” Maybe because a little before that, as the professor described the ratio of vibration and relaxation contributions, the concept of an ergodic process, and how this might depend on observation time, he mentioned the Deborah constant and gave for that name its Biblical source as well as noted its descriptive limitations:

NOTE: The same experiment may be ergodic for one observer and nonergodic for a different observer. e.g. For God, D <<<<<< 1

Which would suggest that at supreme-being time scales, glass does indeed flow. And maybe man was not meant to know! Sorry, couldn’t resist. I will, however, resist categorizing this post as Religion and Philosophy. Wherever you are now, Dr. Kauzmann, let it all hang out. You've earned it!