Short summary of zome-type superbuckyball part I: 1D Linear and Helical C60 Polymers

Recently I've been playing with all kinds of these superbuckyballs, based on the methodology of replacing balls and struts of zometool by Ih-symmetric fullerenes and straight CNTs. Taking C60 for example, blue struts correspond to removing two atoms next to some particular C2 rotation axes. And the connecting CNTs are of chiral vector (4,0). On the other hand, yellow and red struts correspond to C3 and C5 rotation axes, respectively. In addition, due to the property of the golden field, the algebraic field of zometool, superstructures using yellow or red struts only have the possibility of being polymers of C60. This means that the number of atoms is a multiple of 60, i.e. no atoms needed to be deleted or added when constructing the superbuckyball.

Here I will briefly summarize some of the cases I've done coding with. Hopefully I'd soon come up with a short paper ready to submit to the Bridges 2013 on this topic.

First let us start with the trivial C60 dimers. As mentioned, structures with red or yellow struts cases have the atom-preserving property. The C120 isomers corresponding to joining two C60s along their fivefold and threefold axes are shown below. I should mention that they were also discussed in Diudea and Nagy's book . In particular for the C3-joined case there are two possibilities of local atomic connectivity.

C3-fused C120, case 1 (with octagons and pentagons at the interface)

C3-fused C120, case 2 (with heptagons at the interface)

C5-fused C120

For the case of blue struts (twofold rotation axes)

C2-fused C116

C2-fused C132

Some of them were already made previously by us, see here for example. But we did not realize back then this particular connection with zometool. To my knowledge, there has not been any experimental characterization of such dimers. Synthetic chemists do make C60 dimers but those are of partial sp3 characteristics, i.e. some interfacial atoms have four neighbors instead of three. Please refer to Diudea and Nagy's book for further details if you are interested.

One can come up with the one dimensional C60 chains without too much effort by enforcing periodic boundary condition. So the structure repeats itself indefinitely along the direction of polymerization. See for example below.

Also, it is one step away from constructing the 2D analog of this kind of structure.

A little bit more sophisticated extension of the above scheme is to consider helical screw symmetry. A (discrete) helical curve is defined by the angle between adjacent unit cells and the dihedral angle between next-nearest neighbors. I recommend readers of interest to play with the awesome virtual zome program vZome developed by Scott Vorthmann. You have to write Scott an email for the license of the full version of vZome. Anyway, here are some examples of helical C60 polymers.

C3-fused fourfold C60 helix

C5-fused fivefold C60 helix

Notice that if you are looking along the axes of the helices, the C60s that are four/five unit cells away lie exactly on top of each other. Curiously, this result is actually symmetry-determined, since I've tested with the relaxation scheme that does not require such symmetry. In other words, even if I optimize the geometries with full degrees of freedom of a general helix, the screw angles will still be 2*pi/4 or 2*pi/5 in the above cases.