Buckminsterfullerene is a molecule shaped like a soccer ball, made of 60 carbon atoms. If one of the bonds between two hexagons rotates, we get a weird mutant version of this molecule:

This is an example of a Stone-Wales transformation: a 90° rotation in a so-called ‘π bond’ between carbon atoms. Here’s how it works in graphene:

Graphene is a sheet of carbon molecules arranged in hexagons. When they undergo a Stone–Wales transformation, we get a Stone–Wales defect with two pentagons and two heptagons, like this:

This picture is from Jacopo Bertolotti, based on a picture by Torbjörn Björkman which appeared in this paper:

• Torbjörn Björkman, Simon Kurasch, Ossi Lehtinen, Jani Kotakoski, Oleg V. Yazyev, Anchal Srivastava, Viera Skakalova, Jurgen H. Smet, Ute Kaiser and Arkady V. Krasheninnikov, Defects in bilayer silica and graphene: common trends in diverse hexagonal two-dimensional systems, Scientific Reports 3 (2013), 3482.

This paper also shows other interesting defects, and electron microscope pictures of how they actually look in graphene and hexagonal bilayer silica.

You’ll notice that in buckminsterfullerne the Stone–Wales transformation turned 2 hexagons and 2 pentagons into two pentagons and 2 hexagons, while in graphene it turned 4 hexagons into 2 pentagons and 2 heptagons. The general pattern is this:

I think it’s cool how a simple topological transformation akin to the Pachner moves shows up in chemistry!

I got the above image from here:

• Wei-Wei Wang, Jing-Shuang Dang, Jia Zheng, Xiang Zhao, Shigeru Nagase, Selective growth of fullerenes from C₆₀ to C₇₀: inherent geometrical connectivity hidden in discrete experimental evidence, The Journal of Physical Chemistry C 117 (2013), 2349–2357.

I got the image of a Stone–Wales transformation in buckminsterfullerene from here:

• L. A. Openov and Mikhail Maslov, On the vineyard formula for the pre-exponential factor in the Arrhenius law, Physics of the Solid State 56 6 (2014), 1239–1244.

The Arrhenius equation is a simple rule of thumb for the rates of chemical reactions. A lot of statistical mechanics gives simple laws for equilibrium behavior. For example, a state of energy will show up with probability proportional to in equilibrium at temperature where is Boltzmann’s constant. But dynamics is much harder, so the Arrhenius equation for the rates of transitions between states is precious, even though only approximate. I would like to understand this law better, and its range of approximate validity. The above paper digs into that. As an example, it studies the rate at which Stone–Wales transformations happen in buckminsterfullerene!

There should be a nice theory of topological transformations like Stone–Wales transformations or Pachner moves occurring randomly, following the laws of statistical mechanics. I guess the study of matrix models moves in this direction, but there’s no Arrhenius equation there, I don’t think!

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