A crystal-shaped riddle is finally solved

A crystal’s shape is determined by its inherent chemistry, a characteristic that ultimately determines its final form from the most basic details. But sometimes the lack of symmetry in a crystal renders the surface energies of its facets unknowable, confounding any theoretical prediction of its shape.

Rice University theorists say they have found a way around this conundrum by assigning arbitrary latent energies to its surfaces or, in the case of two-dimensional materials, to its edges.

Yes, it sounds like cheating, but in the same way that a magician finds a selected card in a deck by narrowing down the possibilities, a little algebraic sleight of hand solves the problem of predicting the shape of a a crystal.

The method described in Computational science of nature shows that using what they call auxiliary edge energies can align predictions with Wulff’s construction, a geometric recipe that has been used for over a century to determine how crystals arrive at their final equilibrium shapes.

The open access paper by materials physicist Boris Yakobson, lead author and alumnus Luqing Wang, and their colleagues at Rice’s George R. Brown School of Engineering presents algorithms that use arbitrary numbers for the factors of right in the equations and always provide the correct unique form-solution.

“The shape question is compelling, but researchers have tried and failed for years to calculate surface energies for asymmetric crystals,” Yakobson said. “Turns out we were falling down a rabbit hole, but we knew that if nature can find a solution through a gazillion atomic moves, there should be a way for us to figure that out as well. “

He said the surge of interest in 2D materials in recent times prompted the new study. “We had a ‘eureka’ moment: After changing our thinking from geometry to algebra, we added closure equations containing arbitrary parameters,” Yakobson said. “These seem useless, but we ran everything through the computer and watched a well-defined shape come out of it,” he said.

“The hardest part was convincing our reviewers that peak energy really is indefinable, but a solution can always be found,” Wang said.

The work could provide a valuable tool for researchers growing bottom-up crystals for catalytic, light-emitting, sensing, magnetic and plasmonic applications, especially when their shapes and active edges are of particular importance.

The researchers pointed out that natural crystals enjoy the luxury of geological time. They arrive at their forms by “relentlessly carrying out an experiment of trial and error” as they seek balance, the minimum energy of all their constituent atoms.

But computational and theoretical approaches simply can’t deal with billions of atoms at once, so they typically rely on the energies of outward-facing atoms. For many crystals that have equivalent facets or edges, this works very well.

In 2D materials, virtually all atoms are “outward facing”. When their edges are equivalent by symmetry – in rectangles, for example – completing a Wulff construction is simple after calculating the energies of the edges via density functional theory.

But in the absence of symmetry, when all the edges are different, the calculated average energy is meaningless, Yakobson said.

“Nature has the answer for shaping a crystal, regardless of whether or not it ‘knows’ about edge energies,” he said. “So there is an answer. Our challenge was to emulate it with theory.”

The first step toward a solution was to consciously forgo finding the unknowable absolute edge energies and instead deal with their well-defined calculable combinations, Yakobson said. Geometrically it was quite an enigma, and for asymmetrical bulk materials it was hopelessly complicated.

“But 2D materials and their plane polygons made solving the problem easier to think about than having to deal with multi-faceted polyhedra,” he said.

Finding and establishing average energies was only the first step, followed by “closing equations” which used arbitrary material latent energy for the right side of the equation. Even though these latter figures were intentionally incorrect, applying all of them to the Wulff construction of the manual resulted in the correct crystal form.

The group tested their theory on several 2D crystals and compared the results to the observed final shapes of the crystals. Their versatile equations successfully predicted the shapes, shown experimentally, of the truncated rectangle formed by 2D tin selenide, a promising thermo- and piezoelectric material, and the asymmetric needles formed by silver nitrite.