Chern Chuang and Paul Hildebrandt with G and D surfaces

A photo of Chern (with G surface) and Paul (with D surface) in the Bridges conference held in the Towson university, Baltimore, USA:

(Photo by Helen Yu)

這一年我們一起串的巴克球!! (The buckyballs we bead together this year)

I got an announcement (in Chinese) about an activity or more exactly a competition to make beaded buckyballs for local high-school students.

workshop for ICCE/ECRICE

I had a workshop for joint meeting of the 22nd International Conference on Chemistry Education and the European Conference for Research in Chemical Education yesterday in Rome, Italy. My collaborator and I prepared materials for 40 people. But only around 10 people participated the event.

Before participants to make their bead models in this hands-on workshop, I gave a 30-min introductory talk about what can done with beads and particularly the connection of the bead model to the valence sphere model (VSM). I tried to emphasize that bead model is the best method to realize the VSM for both fullerenes and other molecules with sp3, dsp3 and d2sp3 hybridization.

Hyperbolic "buckyball" with octagons

I played with carbon tetrapods described in the Diudea and Nagy's book last weekend and discovered quite accidentally that it is possible to make another type of hyperbolic "buckyball" with neighbored nonhexagons separated by only one CC bond. Particularly, all nonhexagons in this structure are octagons. So this one is different from another hyperbolic "buckyball" made of heptagons and hexagons, or simply heptagonal buckyball. The skeleton of this structure is the same as the heptagonal "buckyball". But each unit cell, which consists of two connected tetrapods, has 96 carbon atoms. I am not sure if people have mentioned this kind of hyperbolic "buckyball" or not.

Heptagonal "buckyball", C168 (This model was given to Prof. Sonoda as a gift when I was invited to Japan for a series of talks and workshops.):

(photo by Dancecology)

Octagonal "buckyball", C96:

Truncated octahedron

C60 and extended C168 are unique because neighbored nonhexagons in them are separated by exactly one carbon-carbon bond. Is there any other graphitic structure with the similar property? The answer is yes. Chern and I have written an article on the carbon nanotori and nanohelices with this property a few years ago.

Chuang, C; Jin, B.-Y.* “Hypothetical Toroidal, Cylindrical, Helical Analogs of C60.” J. Mol. Graph. Model. 2009, 28, 220-225.

Of course, it is easy to see that there are another four Archimedean solids with this property if we allow nonhexagons to be squares or triangles. They are truncated octahedron (see the following photo), truncated cube, truncated tetrahedron, and truncated dodecahedron. Note that C60 is the truncated icosahedron. So all five truncated Platonic solids belong to this class.

Coalescence of two or more C60s

Previously, I constructed many large graphitic structures by fusing C60 along a particular 5-fold axis. But in addition to the 5-fold axes, one can also fuse two C60s along a particular 3-fold axis to get different kind of fused structures. One can easily see that the topological charges of necks in these two structures are the same, i.e. (-1)x6=0x3+(-2)*3=-6. The total topological charges are, of course, equal to 18x1+(-6)=12. However the relative arrangement for the two C60s are different, staggered for the pure-heptagon neck, while eclipsed for the hexagon-octagon neck. In fact, Diudea and Nagy have already done a lot on this possibility. Read the book for more information.

The following two pictures show two possible ways of coalescence: the neck of the first one consists of a ring of octagons and hexagons alternatively; the second one is a ring of six heptagons, instead.

In principle, one can create a one-dimensional tube by coalescence of many C60s repeatedly.

From carbon cube to Schoen's I-WP surface

As I mentioned in the previous post, we can create necks by replacing every pentagon around corners by a heptagon. The structure thus obtained is a single unit cell for the Schoen's I-WP surface! The eight necks in this particular case are, however, too thin to be chemically stable.

Carbon cube

Using the same combination of an octagon together with four pentagons for a face as described in the carbon cuboctahedron, one can make the following bead model of the carbon cube. Each corner of a cube is mimicked by three fused pentagons. In principle, we can use the same strategy to create a whole series of fullerenes with the shape of a truncated cube.

More carbon tetrapods

In the appendix of "Periodic Nanostructures", M. V. Diudea and C. L. Nagy gave an extensive list of tetrapod-like structures of carbon nanotubes. Here are some bead models based on the structures listed in the book.

Carbon cuboctahedron

Most of cage-like fullerenes belong to either icosahedral or tetrahedral groups. But if we remove the restriction of using only pentagons and hexagons, we can create cage-like fullerenes with octahedral symmetry (or cubical shape) too. Of course, all P-type TPMSs, which are extended systems, posted before have the same symmetry. But I have never made cage-like fullerenes with cubical shape before. I found a few examples of cage-like fullerenes with cubical shape in an interesting book with the title, Periodic Nanostructures, by M. V. Diudea and C. L. Nagy recently. In this structure, each face contains an octahedron surrounded by four pentagons, which give a topological charge of 2. There are six faces in a cube, so the total topological charge is 12 as required by the Euler theorem. It is also easy to see that the eight vertices of this molecule are covered by flat coronenes. Therefore, the molecule looks like a cuboctahedron (立方八面體).

It is not hard to see that one can grow eight carbon nanotubes along eight vertices of the cube. The result will be a Schoen's I-WP surface I described before. If one inserts six tubes along the six faces, one get a single unit cell of the P-surface. Or one can also terminate the CNTs to get a dendritic fullerene with a cubic-shape core.