2. Growing graphene

Our examination of graphene structures begins with a Mathematica program designed to examine the dendritic crystallization of unlayered graphene from a non-equilibrium melt, as one possible mechanism to explain the unlayered-graphene cores (Bernatowicz1996, Fraundorf2002, Mandell2007) found in a subset of the pre-solar graphite onions (obtained from primitive meteorites) whose isotopic composition indicates condensation in the atmosphere of asymptotic giant branch stars (which nucleosynthesize the lion's share of carbon in the universe). Although the program is designed to add carbon atoms also to sheets with defects, in this context we've simply added carbon atoms to a flat "graphene seed". We start with 24 atoms bounded by one pair of external atoms on each of the 6 sheet-sides for 2×6=12 surface atoms with only two bonds and 12 internal atoms with 3 bonds each. This beginning state has 6 + 6 + 12 = 24 atoms.

The first growth step adds three external atoms to each pair, leaving us now with 24 internal atoms and 3×6 = 18 surface atoms for 42 atoms total. The sequence of additions now looks like 6 + 6 + 12 + 18 = 42.

The second growth step adds a singly-bonded edge atom and two doubly-bonded edge atoms for each of the 6 previous surface pairs. Now there are 42 triply-bonded internal atoms, 6 × 2 = 12 doubly-bonded surface atoms and 6 × 1 = 6 singly-bonded surface atoms for a total of 60 atoms. The sequence here is 6 + 6 + 12 + 18 + 18 = 60.

Finally we add 4 doubly-bonded surface atoms on each side. Now there are 60 triply-bonded internal atoms, and 6 × 4 = 24 doubly-bonded surface atoms with a total of 84 atoms in the same closed-shell arrangement of the starting sheet. The sequence is now 6 + 6 + 12 + 18 + 18 + 24 = 84.

One possible pattern of atom-increments is (0) + 6 + 6 + (12) + 18 + 18 + (24) + 30 + 30 + (36) + 42 + 42 + (48) etc. The totals would then read (0), 6, 12, (24), 42, 60, (84), 114, 144, (180), 222, 264, (312) etc. Jumping only between the closed shells in parentheses, the total number of carbon atoms goes from 0, 24, 84, 180, 312, 480 etc. with 0, 6, 12, 24, 36, 48, 60 etc. "sheet-surface" or edge-atoms respectively.

The closed-shell recurrence relation therefore looks like Nn+1 = Nn + 3Sn + 24, where Sn+1 = Sn + 12. From this, it looks like closed shells with n atom-pairs along each of 6 sides (2n "surface atoms") have a volume of Nhex = 6n(1+3n) = 3m+9m2/2 carbon atoms with Shex = 12n = 6m atoms on the 2D surface (the hexagonal sheet perimeter), in terms of the number of atom-pairs n = 0, 1, 2, etc. and the number of atoms m ≡ 2n = 0, 2, 4, etc. on each side. 

related references

• T. Bernatowicz, R. Cowsik, P. C. Gibbons, K. Lodders, B. F. Jr., S. Amari, and R. S. Lewis (1996) "Constraints on stellar grain formation from presolar graphite in the Murchison meteorite", Astrophysical Journal 472, 760. (pdf).

• P. Fraundorf and Martin Wackenhut, "The core structure of pre-solar graphite onions" (2002) Ap. J. Lett. 578(2) L153-156.

• Eric Mandell, "Electron beam characterization of carbon nanostructures" (2007) is the topic of a dissertation in partial fulfillment of the requirements for a Ph.D. in Physics and Astronomy at UM-StL and UM-Rolla.