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....)

Monday, June 18, 2018

Scientific American - what the heck is this?

Today, Scientific American ran this on their blogs page.  This article calls to mind weird mysticism stuff like crystal energy, homeopathy, and tree waves (a reference that attendees of mid-1990s APS meetings might get), and would not be out of place in Omni Magazine in about 1979.

I’ve written before about SciAm and their blogs.  My offer still stands, if they ever want a condensed matter/nano blog that I promise won’t verge into hype or pseudoscience.

Saturday, June 16, 2018

Water at the nanoscale

One reason the nanoscale is home to some interesting physics and chemistry is that the nanometer is a typical scale for molecules.   When the size of your system becomes comparable to the molecular scale, you can reasonably expect something to happen, in the sense that it should no longer be possible to ignore the fact that your system is actually built out of molecules.

Consider water as an example.  Water molecules have finite size (on the order of 0.2 nm between the hydrogens), a definite angled shape, and have a bit of an electric dipole moment (the oxygen has a slight excess of electron density and the hydrogens have a slight deficit).  In the liquid state, the water molecules are basically jostling around and have a typical intermolecular distance comparable to the size of the molecule.  If you confine water down to a nanoscale volume, you know at some point the finite size and interactions (steric and otherwise) between the water molecules have to matter.  For example, squeeze water down to a few molecular layers between solid boundaries, and it starts to act more like an elastic solid than a viscous fluid.  

Another consequence of this confinement in water can be seen in measurements of its dielectric properties - how charge inside rearranges itself in response to an external electric field.  In bulk liquid water, there are two components to the dielectric response.  The electronic clouds in the individual molecules can polarize a bit, and the molecules themselves (with their electric dipole moments) can reorient.  This latter contribution ends up being very important for dc electric fields, and as a result the dc relative dielectric permittivity of water, \(\kappa\), is about 80 (compared with 1 for the vacuum, and around 3.9 for SiO2).   At the nanoscale, however, the motion of the water molecules should be hindered, especially near a surface.  That should depress \(\kappa\) for nanoconfined water.

In a preprint on the arxiv this week, that is exactly what is found.  Using a clever design, water is confined in nanoscale channels defined by a graphite floor, hexagonal boron nitride (hBN) walls, and a hBN roof.  A conductive atomic force microscope tip is used as a top electrode, the graphite is used as a bottom electrode, and the investigators are able to see results consistent with \(\kappa\) falling to roughly 2.1 for layers about 0.6-0.9 nm thick adjacent to the channel floor and ceiling.  The result is neat, and it should provide a very interesting test case for attempts to model these confinement effects computationally.

Friday, June 08, 2018

What are steric interactions?

When first was reading chemistry papers, one piece of jargon jumped out at me:  "steric hindrance", which is an abstruse way of saying that you can't force pieces of molecules (atoms or groups of atoms) to pass through each other.  In physics jargon, they have a "hard core repulsion".  If you want to describe the potential energy of two atoms as you try to squeeze one into the volume of the other, you get a term that blows up very rapidly, like \(1/r^{12}\), where \(r\) is the distance between the nuclei.  Basically, you can do pretty well treating atoms like impenetrable spheres with diameters given by their outer electronic orbitals.  Indeed, Robert Hooke went so far as to infer, from the existence of faceted crystals, that matter is built from effectively impenetrable little spherical atoms.

It's a common thing in popular treatments of physics to point out that atoms are "mostly empty space".  With hydrogen, for example, if you said that the proton was the size of a pea, then the 1s orbital (describing the spatial probability distribution for finding the point-like electron) would be around 250 m in radius.  So, if atoms are such big, puffy objects, then why can't two atoms overlap in real space?  It's not just the electrostatic repulsion, since each atom is overall neutral.

The answer is (once again) the Pauli exclusion principle (PEP) and the fact that electrons obey Fermi statistics.  Sometimes the PEP is stated in a mathematically formal way that can obscure its profound consequences.  For our purposes, the bottom line is:  It is apparently a fundamental property of the universe that you can't stick two identical fermions (including having the same spin) in the same quantum state.    At the risk of getting technical, this can mean a particular atomic orbital, or more generally it can be argued to mean the same little "cell" of volume \(h^{3}\) in r-p phase space.  It just can't happen

If you try to force it, what happens instead?  In practice, to get two carbon atoms, say, to overlap in real space, you would have to make the electrons in one of the atoms leave their ordinary orbitals and make transitions to states with higher kinetic energies.  That energy has to come from somewhere - you have to do work and supply that energy to squeeze two atoms into the volume of one.  Books have been written about this.

Leaving aside for a moment the question of why rigid solids are rigid, it's pretty neat to realize that the physics principle that keeps you from falling through your chair or the floor is really the same principle that holds up white dwarf stars.