A methodology to evaluate the kinetic stability of molecular nanostructures is presented based on the assumption of the independent and random nature of thermal vibrations, calculated at the density functional theory (DFT) level of theory using the harmonic approximation [1]. The kinetic stability (KS) is directly correlated to the cleavage probability for the weakest bond of a given molecular geometry. The application of the presented method to a selection of fullerenes (see Fig. 1) and carbon nanotubes yields clear correlation to their experimentally observed relative isomer abundances. Moreover, we present good agreement of harmonic vibrational eigenmodes between DFT and the computationally more efficient density-functional tight-binding (DFTB) method [2-4]. Thus, DFTB-based KS calculations allow the estimation of kinetic stability for more than 100,000 isomers of the fullerenes C20-C100. We found that the experimentally observed isomer abundances, as recorded for instance by mass spectroscopic investigations, are reasonably well reproduced by the Boltzmann-weighted kinetic stabilities of the cage isomers. This result suggests a mechanism of fullerene formation involving cage destruction, such as recently predicted by quantum chemical molecular dynamics (QM/MD) simulations [5-6]. Rerefences: [1] A. S. Fedorov et al., Phys. Rev. Lett., 107, 175506 (2011). [2] H. A. Witek et al., J. Chem. Phys., 121, 5163 (2004). [3] E. Małolepsza et al., Chem. Phys. Lett., 412, 237 (2005). [4] H. A. Witek et al., J. Chem. Phys., 125, 214706 (2006). [5] S. Irle et al., J. Phys. Chem. B, 110, 14531 (2006). [6] B. Saha et al., J. Phys. Chem. A, 115, 22707 (2011).

1. Kinetic Stability Governs Relative Fullerene Isomer
Abundance
Stephan Irle,1 Yoshifumi Nishimura,1 Alexander S. Fedorov,2 Henryk A. Witek3
1WPI-Institute of Transformative Bio-Molecules (ITbM) & Department of
Chemistry, Nagoya University, Nagoya, Japan
2Kirensky Institute of Physics, Russian Academy of Science, Krasnoyarsk, Russia
3Department of Applied Chemistry, National Chiao Tung University, Taiwan
National Chiao Tung U Nagoya University
http://qc.chem.nagoya-u.ac.jp
223rd ECS Meeting
H3 Symposium “Endofullerenes and
Metallofullerenes”, No. 1101
Toronto, Ontario, Canada
May 15, 2013
C60
Russian Academy of Science (RAS)

2. 2
Acknowledgements
Prof. Keiji Morokuma
CREST “Multiscale Physics”
JSPS-RFBR Bilateral Researcher Exchange Program
Prof. Henryk A. Witek
Dr. Yoshifumi Nishimura
National Chiao Tung U Nagoya University
Earlier DFTB/MD Simulations:
Russian Academy of Science (RAS)
Prof. Alexander S. Fedorov

3. Background Fullerene Abundances
3
C60
Becker et al., 31st Lunar and Planetary Science
Conference, Houston, TX, 1000, 1803 (2000)
C60
O2-lean petroleum
combustion
Johnson et al., Carbon 40, 189 (2002)
C60
PMCS
(Cn expansion
into cold He)
Milani et al., New Journal of
Physics, 7, 81 (2005).
Heat
&
Carbon

4. 4
Hypothetical mechanisms relying
on more or less sound
assumptions; no large intermediate
species experimentally identified.
No experimental or theoretical
verification !
C60
(Cn)x
Scheme from: Yamaguchi, T.; Maruyama, S. JSME 1997, 63-611B 2398
“Centrally managed” C60 formation models
Buckminster Fuller 1895-1983
“Lego philosophy”
Background C60 Formation Models
“Closed Network
Growth” (Kroto et
al. Nature
Commun. 2012)

5. Dunlap et al. J. Phys. B. 29,
4907 (1996)
“Bucky” C60
not most
stable!
Giant fullerenes thermodynamically more stable than C60
5
Stability of FullerenesBackground
C∞ Graphite
is most
stable!!

6. 6
0.0 ps 0.1 ps 1.6 ps 8.5 ps 14.5 ps
40.2 ps 56.8 ps 81.1 ps 94.7 ps 104.1 ps
158.1 ps 320.1 ps 320.4 ps 360.0 ps 361.5 ps
Morokuma/Irle et al: Nano Lett. 3, 1657 (2003), J. Chem. Phys. 122, 14708 (2005)
J. Chem. Phys. B 110, 14531 (2006), J. Nanosci. Nanotechnol. 7, 1662 (2007); Nano 2, 21 (2007)
“octopus on a rock”
Our Shrinking Hot Giant RoadBackground

7. Simultaneous Growth and Shrinking
Background Growth vs Shrinking
7
Jin et al, ACS Nano 2, 1275 (2008)

8. 8
Growth vs Shrinking
Johnson et al., Carbon 40, 189 (2002)
Background Growth vs Shrinking
Shrinking Hot Giant road
Fullerene shrinking:
observed when environmental
C/C2 concentration is low
Fullerene road Brinkmann et al., CPL 428, 386 (2006)
Endo-Kroto insertion patch
Fullerene road (CNG):
observed when C/C2
concentration near cage is high
Huang et al. Phys.
Rev. Lett. 99,
175503 (2007)
Ogata et al., Carbon 47, 683
(2009)
C60 C70
Kroto et al. Nature
Commun. (2012)

9. Growth vs Shrinking
C2C2 ejection C2 capture
QM/MD Simulations: Fullerenes can Eject and
Capture C2 Molecules!
Saha, SI, Morokuma, J. Phys. Chem. C 115, 22707 (2011)
Formation Mechanism
Observed C2
insertion events:
Endo-Kroto insertion patch

10. Fullerenes are like clouds
Fullerene cages are made in a dynamic process!
Saha, SI, Morokuma, J. Phys. Chem. A 112, 11951 (2008)
Formation Mechanism
C2
C2
Entropy
“dissipative structure” gravity
Warm
humid air
rises
r(C/C2)
shrinking growth
“Lego philosophy”

11. Curl’s hypothesis
What determines fullerene isomer abundance?
Fullerene Abundance
11
If not thermodynamic stability, then “the suprising abundance of
C60 and C70 must be of kinetic origin”.
Curl et al. Phil. Trans. R. Soc. A 343, 19 (1993)
Combination of growth and shrinking
Curl’s Spreading the Distribution Mechanism
Curl et al., J. Phys. Chem. A 112, 11951 (2008)
Initial Population: C154,
followed by Kinetic
Monte Carlo

12. New Method to estimate kinetic stability
A. S. Fedorov, D. A. Fedorov, A. A. Kuzbov, Y. Nishimura, SI, H. A. Witek, Phys. Rev.
Lett. 107, 175506 (2011), Erratum: 108, 249902 (2012)
Kinetic stabilityFullerene Abundance
12
Alexander S. Fedorov
Wish to derive a “cleavage probability” Pcleav on the basis of
molecular vibrations at given temperature T
C20 “Cage breathing mode”, 795 cm-1 (SCC-DFTB)
Assumption 1): Thermal equilibrium:
Etotal = Ekin + Epot = kBT (for each vibration)
Assumption 2): Harmonic approximation; vibrational amplitude:

Xk =
2kBT
mkwk
2
“high T” “low T”

13. 3N-6=54 displacement vectors of C20 (Ih) at SCC-DFTB*
374.97 374.97 374.97 374.97 449.61 449.61 449.61 449.61 449.61
467.30 467.30 467.30 495.90 495.90 495.90 495.90 540.00 540.00
540.00 540.00 540.00 754.49 754.49 754.49 795.02 850.03 850.03
850.03 969.98 969.98 969.98 969.98 969.98 1018.97 1018.97 1018.97
1051.60 1051.60 1051.60 1051.60 1051.60 1057.20 1057.20 1057.20 1057.20
1097.04 1097.04 1097.04 1097.04 1253.04 1253.04 1253.04 1253.04 1253.04
*harmonic vibrational frequency [cm-1] with slkoopt parameter

14. New Method to estimate kinetic stability
A. S. Fedorov, D. A. Fedorov, A. A. Kuzbov, Y. Nishimura, SI, H. A. Witek, Phys. Rev.
Lett. 107, 175506 (2011), Erratum: 108, 249902 (2012)
Kinetic stabilityFullerene Abundance
14
Alexander S. Fedorov
•Now calculate time-dependent displacements of atoms n and m:
•Project this quantity on the direction of the original bond:
•Assumption 3): We assume that a bond is broken when:
In our case, Xmax = 1.95 A

15. New Method to estimate kinetic stability
A. S. Fedorov, D. A. Fedorov, A. A. Kuzbov, Y. Nishimura, SI, H. A. Witek, Phys. Rev.
Lett. 107, 175506 (2011), Erratum: 108, 249902 (2012)
Kinetic stabilityFullerene Abundance
Alexander S. Fedorov
•Assumption 3): Assuming that bond is broken for:
•Assumption 4): Probability for this condition to occur
is approximated by help of central limit theorem as:
variance of Xi
= Pcleav (n,m)
(cleavage probability for n,m bond)
15Assumption 5):
Winner takes all: weakest bond determines cleavage probability

16. Application of Kinetic Stability to Fullerene Isomers
Kinetic stabilityFullerene Abundance
T=1500 K
16
A. S. Fedorov, D. A. Fedorov, A. A. Kuzbov, Y. Nishimura, SI, H. A. Witek, Phys. Rev.
Lett. 107, 175506 (2011), Erratum: 108, 249902 (2012)
Frequency calculation:
PBE DFT, VASP 4.6
PW basis set
UPP
287 eV kinetic energy
cutoff

17. Visualization of “weakest bonds”
Kinetic stabilityFullerene Abundance
17
A. S. Fedorov, D. A. Fedorov, A. A. Kuzbov, Y. Nishimura, SI, H. A. Witek, Phys. Rev.
Lett. 107, 175506 (2011), Erratum: 108, 249902 (2012)
Pcleav
/bond

18. What about carbon nanotubes?
Kinetic stabilityCNT abundance
18Kinetic and thermodynamic stability is correlated!
A. S. Fedorov, D. A. Fedorov, A. A. Kuzbov, Y. Nishimura, SI, H. A. Witek, Phys. Rev.
Lett. 107, 175506 (2011), Erratum: 108, 249902 (2012)

19. Summary
Kinetic stabilityFullerene Abundance
19
•Method to estimate kinetic stability developed and applied
•Using DFTB, harmonic normal mode calculation is easy for
~100 atom systems, 100,000 calculations!
•Fullerene isomer abundance can be correlated with kinetic
stability, not with thermodynamic stability
•Carbon nanotubes are produced under conditions closer to
thermodynamic equilibrium
•Fullerene isomers show “flatter” kinetic stability distributions
at higher temperatures; cooling is important!

20. Thank you for your attention!
20