Taking crystals to a higher dimension

Crystallization is a phase transition, the solidification process that transforms a liquid (or gas) into a translationally ordered solid state that is a tightly packed and highly organized structure of atoms. The process occurs as liquid water turns into ice, whether as raindrops becoming snowflakes in a cloud, or as a tray of water becomes cubes of ice in the freezer. Crystallization is ubiquitous in our daily lives, but there is still much we don’t understand about it.

Caroline Gorham and David Laughlin, of the Department of Materials Science and Engineering, have recently published a paper that considers a novel framework to understand crystallization, specifically crystalline solids that contain permanent structural defects. The paper is part of a detailed series of articles they are developing to better understand crystallization in solids beyond glass formation and perfect crystals, which have no defects and are not readily observed in nature.

While crystalline and glassy solids are ubiquitous in the world around us, a detailed theoretical description of solidification processes remains rather rudimentary. With this article and continued research, Gorham, a postdoctoral research fellow, and Laughlin, professor of MSE, are developing a detailed approach to crystallization that they hope can impact how we think about and use solids in everyday life.

In this project, Gorham and Laughlin looked at crystallization from new perspectives and applied concepts of topology, a field in mathematical physics concerned with the properties of materials that remain unaffected by deformations such as stretching, bending, and twisting, but not cutting or gluing. Topology is crucial in discerning the characteristics of defects in materials. The particular crystals they focused on are known as topologically close-packed structures (also known as Frank-Kasper systems), which require the presence of permanent structural topological defects in order to exist in the solid state.

The topological order of phase transitions has become a popular topic in physics and materials science since John Kosterlitz and David Thouless earned a Nobel Prize in 2016 for their description of topological superfluidity—a quantum phenomenon in which a fluid has zero viscosity and flows without losing kinetic energy. They described a novel phase transition to the superfluid state, in one- and two-dimensional systems, by considering a binding mechanism that applies to topological defects. Gorham and Laughlin have built on the Kosterlitz and Thouless description and have generalized its concepts to approach a unified description of crystallization and glass formation.

By moving to quaternions, we were able to make so many analogies with the physics of complex ordered systems in two- and one-dimensions that have provided breakthroughs in topological materials over the last half century.

Caroline Gorham, Postdoctoral research fellow, MSE

This new framework puts crystallization in four-dimensional space. In particular, they have applied a quaternion order parameter—a higher-dimensional complex number system—to describe the orientational order that develops upon crystallization and glass formation. This works because quaternions, which form a four-dimensional vector space, are useful in representing rotations in three-dimensions, which form the basis for orientational order in three-dimensional solid states.

“This is the first time that anyone has provided a full ordering field theory to the solid state by applying a quaternion order parameter,” said Gorham. “By moving to quaternions, which are four-dimensional, we were able to make so many analogies with the physics of complex ordered systems in two- and one-dimensions that have provided breakthroughs in topological materials over the last half century.”

One of the four number systems of modern algebra, quaternion numbers exist in a system with three axes over the real number line, supporting rotations of three-dimensional objects, like solid states. Think imaginary numbers, but a level more complex. Though common in fields such as computer science, aviation, and graphic design, this application of quaternions to solidification processes is novel. Like the topological ordering of superfluids in one- and two-dimensions, Gorham and Laughlin make use of topological ordering of quaternion ordered systems in three- and four-dimensions to describe crystallization.

“We’re having to reach upwards in dimensionality,” said Gorham. “It’s quite a challenge because we live in a three-dimensional universe and we’re trying to understand solid states when identified with four-dimensional quaternion numbers.”

With the transition to quaternions, they are generalizing physics to ultimately understand the different types of solid states. This particular paper on crystallization making use of a quaternion order parameter is just one part of an ongoing series of articles by Gorham and Laughlin, which can be found at Gorham’s website.