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Kinetic Stability Governs Relative Fullerene Isomer Abundance

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Stephan Irle
Stephan Irle

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. Maolepsza 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).

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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 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
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 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
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
Curls 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 Curls 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 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
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 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
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
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! 20

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Kinetic Stability Governs Relative Fullerene Isomer Abundance

  • 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. Curls 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 Curls 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