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Floppy vibration modes explain negative thermal expansion in solids

By Jennifer Huber January 2, 2020
ScF3animation
Animation showing how solid crystals of ScF3 shrink upon heating. While the bonds between scandium (green) and fluorine (blue) remain relatively rigid, the fluorine atoms along the sides of the cubic crystals oscillate independently, resulting in a wide range of distances between neighboring fluorine atoms. The higher the temperature, the greater the buckling in the sides of the crystals leading to the overall contraction (negative thermal expansion) effect. Credit: Brookhaven National Laboratory

Matching the thermal expansion values of materials in contact is essential when manufacturing precision tools, engines, and medical devices. For example, a dental filling would cause a toothache if it expanded a different amount than the surrounding tooth when drinking a hot beverage. Fillings are therefore comprised of a composite of materials with positive and negative thermal expansion, creating an overall expansion tailored to the tooth enamel.

The underlying mechanisms of why crystalline materials with negative thermal expansion (NTE) shrink when heated have been a matter of scientific debate. Now, a multi-institutional research team led by Igor Zaliznyak, a physicist at Brookhaven National Laboratory, believes it has the answer.

As recently reported in Science Advances, the scientists measured the distance between atoms in scandium trifluoride powder, a cubic NTE material—at temperatures ranging from 2 K to 1099 K—using total neutron diffraction. The research team determined the probability that two particular atomic species would be found at a given distance. They studied scandium trifluoride because it has a simple atomic structure in which each scandium atom is surrounded by an octahedron of fluorine atoms. According to the prevailing rigid-unit-mode (RUM) theory, each fluorine octahedron should vibrate and move as a rigid unit when heated—but that is not what they observed.

“We found that the distances between scandium and fluorine were pretty rigidly-defined until a temperature of about 700 K, but the distances between the nearest-neighbor fluorines became ill-defined at temperatures above 300 K,” says Zaliznyak. “Their probability distributions became very broad, which is basically a direct manifestation of the fact that the shape of the octahedron is not preserved. If the fluorine octahedral had been rigid, the fluorine-fluorine distance would have been as well defined as scandium-fluorine.”

With the help of high school researcher David Wendt and condensed matter theorist Alexei Tkachenko, Zaliznyak developed a simple model to explain these experimental results. The team went back to the basics—the fundamental laws of physics.

“When we removed the ill-controlled constraint that there must be these rigid units, then we could explain the fundamental interactions that govern the atomic positions in the [ScF3] solid using just Coulomb interactions.”

The team developed a negative thermal expansion model that treats each Sc-F bond as a rigid monomer link and the entire ScF3 crystal structure as a floppy, unconstrained network of freely jointed monomers. Each scandium ion is constrained by rigid bonds in all three directions, whereas each fluorine ion is free to vibrate and displace orthogonally to its Sc-F bonds. This is a direct three-dimensional analogy of the well-established behavior of chainlike polymers. And their simple theory agreed remarkably well with their experiments, accurately predicting the distribution of distances between the nearest-neighbor fluorine pairs for all temperatures where NTE was observed.   

“Basically we figured out how these ceramic materials contract on warming and how to make a simple calculation that describes this phenomenon,” Zaliznyak says.

Angus Wilkinson, an expert on negative thermal expansion materials at the Georgia Institute of Technology who is not involved in the project, agrees that Zaliznyak’s work will change the way people think about negative thermal expansion in solids.

“While the RUM picture of NTE has been questioned for some time, the experimental data in this paper, along with the floppy network (FN) analysis, provide a compelling alternative view,” says Wilkinson. “I very much like the way the FN approach is applicable to both soft matter systems and crystalline materials. The floppy network analysis is novel and gives gratifyingly good agreement with a wide variety of experimental data.”

According to Zaliznyak, the next major step of their work will be to study more complex materials that exhibit NTE behavior now that they know what to look for.

Read the article in Science Advances.