Saturday, February 23, 2013

Short summary of zome-type superbuckyball part II: Superfullerenes

We will move on to our main and original goal of devising this technique: constructing superfullerenes. Readers familiar with zometool would know that C60 can be built with solely blue struts (two fold rotation axes). So here it is.

C60⊗C60 with g=0: C3240

I've borrowed the notation of Kronecker product (⊗) since these two mathematical operations are in some sense similar: each entry (atom) of the matrix (fullerene) before the ⊗ sign is "expanded" into the second matrix times the original entry (the spatial location of that atom). The meaning of g will become clear once you see an example of g=1 as shown below.

C60⊗C60 with g=1: C4680

It is obvious that g indicates the length of the struts (straight CNTs). In the first case the length is essentially zero, so pairs of heptagons "merge" into octagons at the interface. For clarity the rotatable models of four connected superatoms of the above two superbuckyballs are presented below as well. Since all of these superatoms are identical and can be related through mirror symmetries, readers of interest can start with them to build your own models.

For convenience I also show the rotatable models for the superbuckyball proposed and its beaded model constructed by Bih-Yaw in the previous posts.

C60⊗C60 with g=0: C2700

C60⊗C60 with g=1: C4500

Upon close inspection, notice that there is still local threefold symmetry at each of the node in this case. While on the contrary, there are only mirror symmetries in the zome-type superbuckyball. This asymmetry leads to the fact that there is almost no strain when constructing the beaded model of these structures, or even the actual microscopic realization. This is not an issue concerning the 1D structures in the previous post, since they are all simply connected and there is no such thing as commensurability among multiple struts that join at the same node. However, the above two superstructures seem to be pretty stable and beadable, which surprise me a lot in this regard. According to Bih-Yaw, the tension of the five-member rings and the stress in the six-member rings magically balance each other. This is not so for the dodecahedron case, where tension is built everywhere in the model without being compensated by stress.

Having demonstrated the above mentioned, there is nothing so different in constructing other types of superfullerenes. Below I listed a few that I have done coding with.

C20⊗C60 with g=1: C1560

Although I have not tried to build this one with beads yet, I believe that it is quite doable in the sense of stability as mentioned above.

C80⊗C60 with g=1: C6720

C180⊗C60 with g=1: C15480

I'd like to point out in the last two cases you need red struts as well as blue ones. It can be shown that all (n,n) or (n,0) type icosahedral fullerenes are constructable from zometool (with blue and red struts). For now I just manually find out what are the atoms needing to be deleted/connected, well, in an efficient way. I hope one day I can come up with a general automatic routine that does all these for me, which should be taking account of different orbits in a symmetry group. C180 (a (3,0) Ih fullerene) is the largest one I've ever played with.

I plan to talk about other types of regular polyhedra in the next post.

No comments: