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, Japan2Kirensky 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

Acknowledgements

Prof. Keiji MorokumaCREST “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

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

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)

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

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

Simultaneous Growth and Shrinking

Background Growth vs Shrinking

7

Jin et al, ACS Nano 2, 1275 (2008)

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)

Growth vs Shrinking

C2 C2 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

Fullerenes are like clouds

Fullerene cages are made in a dynamic process!

Saha, SI, Morokuma, J. Phys. Chem. A 112, 11951 (2008)

Formation Mechanism

C2C2

Entropy

“dissipative structure” gravity

Warm

humid air

rises

r(C/C2)

shrinking growth

“Lego philosophy”

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

New Method to estimate kinetic stabilityA. 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:

 

X k =

2kBT

mkw k

2

“high T” “low T”

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

New Method to estimate kinetic stabilityA. 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

New Method to estimate kinetic stabilityA. 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

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

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

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)

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!

Thank you for your attention!

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