# What is the maximum amount of carbon atoms that can comprise a fullerene?

As the fullerene sum formula can be defined as $$\ce{C_n}$$ with $$n$$ being any even integer number, is there any fundamental theoretical limit as to how big $$n$$ can get?

If there is no limit to the chemistry, but there are difficulties in synthesizing large fullerenes, what has been the largest fullerene ever made? Would there be any differences in behaviour?

• $n$ is not just any integer. It can't be odd, to begin with. Now to the point: just as you expected, there is no fundamental upper bound. Oct 7 at 6:18
• At some point, the molecule would be so big that it cannot exist at a given temperature, because it is also stiff (has a 1D long range order). I will then just snap in two with a certain half-life time. I don't think a controlled synthesis can ever get close to that value.
– Karl
Oct 7 at 10:51
• Btw.: it is extremely hard to produce macroscopic amounts of very large molecules (a substance) with a defined, uniform molecular weight, i.e. number of carbon atoms in this case. For polyethylene, the record is afaik $\ce{C390H782}$, far, far below the average size of a UHMWPE molecule (easily a million carbon atoms).
– Karl
Oct 7 at 10:59
• One can consider a capped nanotube a fullerene. Oct 7 at 19:05
• To be honest I don't think you will get an answer to the first question 'is there any fundamental theoretical limit as to how big "n" can get?' as I suspect large fullerenes will be thermodynamically unstable WRT decomposition to both smaller fullerenes and graphite, and so the question becomes why is this kinetically infeasible, which is very hard. So you are left with a literature search for the biggest fullerene synthesised or detected. Oct 20 at 8:19

Not a full answer, but possibly the start of one. I came across a paper[1] discussing some mathematical properties of fullerene graphs. In section 4.2, they give a formula for a lower bound on the diameter based on the number of vertices $$n$$, assuming there are only pentagonal and hexagonal face. $$\text{diam}(F)\ge\frac{\sqrt{24n-15}}{6}-\frac{1}{2}$$

Given a lower bound on the diameter, it may be possible to prove that a ball of a certain size would be mechanically unstable. I suspect this would give a fairly high upper bound on the size and it wouldn't account for the wide variety of shapes that are possible for fullerenes[2], but this would at least be a starting point.

This could hopefully be refined somewhat by using the largest synthesized/reported fullerene size as a lower bound. Thus far $$\ce{C_{500}}$$ is the largest fullerene I can find experimental evidence for [3,4]. I suspect this is still pretty far from the largest possible fullerene, even considering just spherical/icosahedral structures.

Based on [2], you will need to look for some kind of mechanical/kinetic stability limit for the upper bound, as the paper suggests that fullerenes should increase in thermodynamic stability as $$n$$ increases, approaching the structure of graphene or a infinite nanotube.

1. Vesna Andova, František Kardoš, Riste Škrekovski. Mathematical aspects of fullerenes. Ars Mathematica Contemporanea, DMFA Slovenije, 2016, 11, pp.353 - 379. ffhal-01416354f
2. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207
3. Shinohara, H., Sato, H., Saito, Y., Izuoka, A., Sugawara, T., Ito, H., Sakurai, T. and Matsuo, T. (1992), Extraction and mass spectroscopic characterization of giant fullerences up to C500. Rapid Commun. Mass Spectrom., 6: 413-416. https://doi.org/10.1002/rcm.1290060702
4. Front. Chem., 03 December 2020 | https://doi.org/10.3389/fchem.2020.607712
• really cool paper! I've just asked What does 3-regularity mean in the context of this diagram of a fullerene? in Math SE.
– uhoh
Oct 26 at 10:57
• While this is only a start, I found the paper really interesting and salient, and it has got me thinking, and I'm loath to waste a good bounty. +200 :-)
– uhoh
Oct 27 at 1:08
• @uhoh Thank you! I'll try to keep expanding the answer as I run into anything that might fit. I can at least now give a (probably very loose) lower bound with experimental evidence of at least $\ce{C_{500}}$. Oct 27 at 15:01