Sunday, June 24, 2018

There is no such thing as a rigid solid.

How's that for a provocative, click-bait headline?

More than any other branch of physics, condensed matter physics highlights universality, the idea that some properties crop up repeatedly, in many physical systems, independent of and despite the differences in the microscopic building blocks of the system.  One example that affects you pretty much all the time is emergence of rigid solids from the microscopic building blocks that are atoms and molecules.  You may never think about it consciously, but mechanically rigid solids make up much of our environment - our buildings, our furniture, our roads, even ourselves.

A quartz crystal is an example of a rigid solid. By solid, I mean that the material maintains its own shape without confining walls, and by rigid, I mean that it “resists deformation”. Deforming the crystal – stretching it, squeezing it, bending it – involves trying to move some piece of the crystal relative to some other piece of the crystal. If you try to do this, it might flex a little bit, but the crystal pushes back on you. The ratio between the pressure (say) that you apply and the percentage change in the crystal’s size is called an elastic modulus, and it’s a measure of rigidity. Diamond has a big elastic modulus, as does steel. Rubber has a comparatively small elastic modulus – it’s squishier. Rigidity implies solidity. If a hunk of material has rigidity, it can withstand forces acting on it, like gravity.  (Note that I'm already assuming that atoms can't pass through each other, which turns out to be a macroscopic manifestation of quantum mechanics, even though people rarely think of it that way.  I've discussed this recently here.)

Take away the walls of an aquarium, and the rectangular “block” of water in there can’t resist gravity and splooshes all over the table. In free fall as in the International Space Station, a blob of water will pull itself into a sphere, as it doesn’t have the rigidity to resist surface tension, the tendency of a material to minimize its surface area.

Rigidity is an emergent property. One silicon or oxygen atom isn’t rigid, but somehow, when you put enough of them together under the right conditions, you get a mechanically solid object. A glass, in contrast to a crystal, looks very different if you zoom in to the atomic scale. In the case of silicon dioxide, while the detailed bonding of each silicon to two oxygens looks similar to the case of quartz, there is no long-range pattern to how the atoms are arranged. Indeed, while it would be incredibly difficult to do experimentally, if you could take a snapshot of molten silica glass at the atomic scale, from the positions of the atoms alone, you wouldn’t be able to tell whether it was molten or solidified.   However, despite the structural similarities to a liquid, solid glass is mechanically rigid. In fact, some glasses are actually far more stiff than crystalline solids – metallic glasses are highly prized for exactly this property – despite having a microscopic structure that looks like a liquid. 

Somehow, these two systems (quartz and silica glass), with very different detailed structures, have very similar mechanical properties on large scales. Maybe this example isn't too convincing. After all, the basic building blocks in both of those materials are really the same. However, mechanical rigidity shows up all the time in materials with comparatively high densities. Water ice is rigid. The bumper on your car is rigid. The interior of a hard-boiled egg is rigid. Concrete is rigid. A block of wood is rigid. A vacuum-packed bag of ground espresso-roasted coffee is rigid. Somehow, mechanical rigidity is a common collective fate of many-particle systems. So where does it originate? What conditions are necessary to have rigidity?

Interestingly, this question remains one that is a subject of research.  Despite my click-bait headline, it sure looks like there are materials that are mechanically rigid.  However, it can be shown mathematically (!) that "equilibrium states of matter that break spontaneously translational invariance...flow if even an infinitesimal stress is applied".   That is, take some crystal or glass, where the constituent particles are sitting in well-defined locations (thus "breaking translational invariance"), and apply even a tiny bit of shear, and the material will flow.  It can be shown mathematically that the particles in the bulk of such a material can always rearrange a tiny amount that should end up propagating out to displace the surface of the material, which really is what we mean by "flow".   How do we reconcile this statement with what we see every day, for example that you touching your kitchen table really does not cause its surface to flow like a liquid?

Some of this is the kind of hair-splitting/no-true-Scotsman definitional stuff that shows up sometimes in theoretical physics.  A true equilibrium state would last forever.   To say that "equilibrium states of matter that break spontaneously translational invariance" are unstable under stress just means that the final, flowed rearrangement of atoms is energetically favored once stress is applied, but doesn't say anything on how long it takes the system to get there.

We see other examples of this kind of thing in condensed matter and statistical physics.  It is possible to superheat liquid water above its boiling point.  Under those conditions, the gas phase is thermodynamically favored, but to get from the homogeneous liquid to the gas requires creating a blob of gas, with an accompanying liquid/gas interface that is energetically expensive.  The result is an "activation barrier".

Turns out, that appears to be the right way to think about solids.  Solids only appear rigid on any useful timescale because the timescale to create defects and reach the flowed state is very very long.  A recent discussion of this is here, with some really good references, in a paper that only appeared this spring in the Proceedings of the National Academy of Sciences of the US.  An earlier work (a PRL) trying to quantify how this all works is here, if you're interested.

One could say that this is a bit silly - obviously we know empirically that there are rigid materials, and any analysis saying they don't exist has to be off the mark somehow.  However, in science, particularly physics, this kind of study, where observation and some fairly well-defined model seem to contradict each other, is precisely where we tend to gain a lot of insight.  (This is something we have to be better at explaining to non-scientists....)


Surajit Sengupta said...

Thanks for the wonderful post. I would like to, however, point out the following. All of this could indeed merely be a mathematical "hair-splitting" definitional thing unless this viewpoint also leads one to new ways for calculating quantities that can be checked by experiment. This is what the PNAS achieves. By using this picture, one can compute the maximum deformation a solid can take before beginning to flow, a quantity that depends on the rate at which deformation is applied. This is probably the first time such a computation has become possible, at least for an ideal solid i.e. one without any pre existing lattice defect.

Douglas Natelson said...

Oh absolutely! There is no doubt that your paper adds real new quantitative understanding. It’s also a very cool way to think about this.

I guess a better example of hair-splitting is when people point to Mermin-Wagner as an argument that there can be no stable 2D crystals, and then it turns out that basically arbitrarily small ripples are enough to let graphene, e.g., evade that restriction.

Unknown said...
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Surajit Sengupta said...

Thanks Doug. Yes, I also get irritated when in every paper that I try to publish on two dimensional crystals, at least one referee raises the Mermin Wagner issue and I have to write the same thing all over again for the nth time!