Negative thermal expansion

Some interesting science results recently, but I wanted to talk about one a little off the beaten path.  Most people have some exposure to the concept of thermal expansion, the idea that solids tend to increase in size as temperature is increased.  This is why people suggest running a stuck (metal) lid on a glass jar under hot water to make it easier to open - the idea is that the metal expands more with increasing temperature than the glass.  This is why there are flexible joints between sections of concrete road, rather than trying to cast the road in one giant section.  Thermal expansion of the pavement would otherwise buckle the roadway.  

Vibrating H2 molecule, electron density
from DFT, by Dr. Or Cohen.

Where does thermal expansion originate?  In a toy model, we can think of the bound atoms in a solid like balls and springs.  The springs in this case model forces between the atoms that result from the electrons involved in the chemical bonds that hold the solid together.  (We usually think of the nuclei as slow and the electrons as fast, so you can consider the nuclear positions, somehow solving for the electron density given those positions, and figuring out the net force on the nuclei.  There is a whole subfield now in shortcutting these calculations with machine learning.)  In an ideal harmonic oscillator, the potential energy is perfectly symmetric around its minimum position.  Giving the oscillator larger and larger amounts of kinetic energy therefore does not change the time average separation of the atoms. 

When dealing with interatomic potentials, though, the potential is anharmonic - the effective spring is softer in extension than compression.  Another way to put it:  at small separations, the "steric interactions" caused by the Pauli principle give the "hard core repulsion" that tends to keep atoms from overlapping.  As a result, the potential looks like the cartoon (red dashed parabola = harmonic approximation that is good near the equilibrium position).  Now, if you give the atoms more kinetic energy, their time-average separation gets larger.  This is the conventional origin of the usual positive thermal expansion.  (Fun historical note.  In 1910, Lindemann, Churchill's friend ("the prof") and science advisor during WWII, put forward what is now called the Lindemann melting criterion: monatomic solids melt roughly when the root mean square thermal vibration displacement is about 10% of the interatomic distance.  This paper is hard to find online, btw.  Lindemann, Frederick A. "Über die berechnung molekularer eigenfrequenzen" Phys. Z 11, 609-612 (1910).),

Interestingly, some materials have negative thermal expansion - as temperature is increased, the materials shrink!  How does that work?  It seems to fly directly counter to intuitive expectations.  Negative thermal expansion often involves materials with lots of open volume in their structure, built out of rigid subunits (e.g. tetrahedra or octahedra of atoms).  As temperature increases, the subunits can deform a bit and also can rotate in ways that allow them to pack more efficiently.  An example of a material like this is zirconium tungstate.   That brings me to this article in JACS, which reports colossal negative thermal expansion in a metal organic framework compound, with a fractional change in volume of around -0.0006 per Kelvin near around 50 degrees C.  This negative thermal expansion coefficient is six times larger than the previous record, and seems to result from distortion of Zr6/oxygen tetrahedra.  Pretty neat, and these kinds of motifs could lead to materials with more designer thermal structural properties.