Which fullerenes and why?

I somehow missed this recent PNAS piece by S Schein and T Friederich completely; it is a very interesting piece as you can see form the title and abstract:

A geometric constraint, the head-to-tail exclusion rule, may be the basis for the isolated-pentagon rule in fullerenes with more than 60 vertices

Carbon atoms self-assemble into the famous soccer-ball shaped Buckminsterfullerene (C60), the smallest fullerene cage that obeys the isolated-pentagon rule (IPR). Carbon atoms self-assemble into larger (n > 60 vertices) empty cages as well—but only the few that obey the IPR—and at least 1 small fullerene (n ≤ 60) with adjacent pentagons. Clathrin protein also self-assembles into small fullerene cages with adjacent pentagons, but just a few of those. We asked why carbon atoms and clathrin proteins self-assembled into just those IPR and small cage isomers. In answer, we described a geometric constraint—the head-to-tail exclusion rule—that permits self-assembly of just the following fullerene cages: among the 5,769 possible small cages (n ≤ 60 vertices) with adjacent pentagons, only 15; the soccer ball (n = 60); and among the 216,739 large cages with 60 < n ≤ 84 vertices, only the 50 IPR ones. The last finding was a complete surprise. Here, by showing that the largest permitted fullerene with adjacent pentagons is one with 60 vertices and a ring of interleaved hexagons and pentagon pairs, we prove that for all n > 60, the head-to-tail exclusion rule permits only (and all) fullerene cages and nanotubes that obey the IPR. We therefore suggest that self-assembly that obeys the IPR may be explained by the head-to-tail exclusion rule, a geometric constraint.

The two papers that (a) describe the geometric constraint and (b) the (plausible) reasons behind it are also pretty interesting; here is the abstract of (b):

In the companion article, we proposed that fullerene cages with head-to-tail dihedral angle discrepancies do not self-assemble. Here we show why. If an edge abuts a pentagon at one end and a hexagon at the other, the dihedral angle about the edge increases, producing a dihedral angle discrepancy (DAD) vector. The DADs about all five/six edges of a central pentagonal/hexagonal face are determined by the identities—pentagon or hexagon—of its five/six surrounding faces. Each “Ring”—central face plus specific surrounding faces—may have zero, two, or four edges with DAD. In most Rings, the nonplanarity induced by DADs is shared among surrounding faces. However, in a Ring that has DADs arranged head of one to tail of another, the nonplanarity cannot be shared, so some surrounding faces would be especially nonplanar. Because the head-to-tail exclusion rule is an implicit geometric constraint, the rule may operate either by imposing a kinetic barrier that prevents assembly of certain Rings or by imposing an energy cost that makes those Rings unlikely to last in an equilibrium circumstance. Since Rings with head-to-tail DADs would be unlikely to self-assemble or last, fullerene cages with those Rings would be unlikely to self-assemble.

Have fun!


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