From Small Carbon Fragments to Self- From Small Carbon Fragments to Self- Assembled Fullerenes in Quantum Assembled Fullerenes in Quantum Chemical Molecular Dynamics Chemical Molecular Dynamics Guishan Zheng, Keiji Morokuma, and Stephan Irle Cherry L. Emerson Center for Scientific Computation and Department of Chemistry, Emory University, Atlanta, Georgia, U.S.A. International Congress of Nanotechnology, San Francisco,CA, November 2004 1
Overview: Proposed Fullerene Formation Mechanisms Overview: Proposed Fullerene Formation Mechanisms Hypothetical mechanisms • (C n ) x relying on more or less sound assumptions; no intermediate species confirmed so far. Underlying assumption of • structural order : Systematic “ construction ” from smaller fragments or collapse of highly pre-organized structures. C 60 No experimental or • theoretical verification ! 2 Scheme from: Yamaguchi, T.; Maruyama, S. JSME 1997 , 63-611B 2398
One Possible “ Designed ” Pathway to C 28 - Structures One Possible “ Designed ” Pathway to C 28 - Structures H=43.32260eV H=45.79176eV c28d2-12 c28d2-9 2 6 6 26 14 22 H=44.39414eV 2 c28d2-10 10 3 28 10 c28d2-11 H=42.06081eV 24 16 11 18 3 14 7 7 20 17 17 24 28 22 4 9 11 18 21 16 8 19 7 20 26 25 11 26 5 2 26 3 9 13 19 12 13 5 14 22 15 9 1 24 6 15 14 25 1 18 22 23 18 27 1 27 10 9 5 21 2 28 13 17 17 23 23 3 27 5 6 25 12 10 16 7 24 4 13 12 1 21 8 25 4 8 11 28 4 21 15 20 8 AM1 in G01B2+ 16 20 19 AM1 in G01B2+ AM1 in G01B2+ 19 12 15 23 27 AM1 in G01B2+ H=47.46386eV c28d2-8 27 23 6 c28d2-4 H=48.22512 c28d2-6 H=48.40815 27 15 2 2 H=45.93379eV 10 c28d2-7 23 15 8 12 12 4.2450 14 4 26 6 12 14 12 17 19 19 21 23 5.4282 23 18 8 8 8 26 13 20 25 27 10 20 18 9 4 27 4 16 22 1 22 17 2 6 28 15 16 26 5 21 16 9 6 3 14 19 2 11 10 28 24 22 20 4 21 3 21 25 3.5843 7 7 24 26 10 3 4.4414 11 14 28 18 17 9 AM1 in G01B2+ 15 13 3 11 13 22 25 5 1 24 7 5 25 18 17 AM1 in G01B2+ 16 7 9 13 19 1 1 20 5 11 28 AM1 in Gau01B2+ 24 c28d2-1 H=45.11716eV 4 16 8 28 21 12 20 2.9939 25 19 6 24 10 c28d2-2 11 c28d2 fullerene H=41.03134eV H=47.74820 23 13 19 17 2 15 1 13 17 27 10 7 20 18 H=48.59386eV c28d2-3 3.3966 25 3 5 14 18 21 15 20 11 1 16 19 26 11 9 9 14 16 27 22 6 28 24 12 3 7 5 15 28 24 2 4 7 28 3.3101 8 23 2.6356 3.5186 24 4 4 3 25 1 22 26 8 27 5 5 22 25 8 21 1 16 12 13 2.9288 7 2 23 6 11 26 22 23 9 13 9 12 3 10 20 27 14 10 6 17 17 2 26 18 18 19 15 14 AM1 in G01B2+ AM1 in g01b2+ AM1 calculation including all transition states and intermediates of a “ ring collapse mechanism ” 3 in the spirit of Mishra, R. K.;Lin, Y.-T.; Lee, S.-L. J. Chem.Phys. 2000 , 112, 6355-6364
One Possible “ Designed ” Pathway to C 28 - Energetics One Possible “ Designed ” Pathway to C 28 - Energetics 27 23 c28d2-6 H=48.40815 15 c28d2-3 H=48.59386eV 12 20 19 19 11 16 8 28 24 15 20 3.3101 7 4 3.5186 4 16 3 25 1 2 6 27 21 5 8 26 10 12 14 28 13 2.9288 22 3 11 25 23 22 24 7 17 9 18 10 9 13 6 17 26 5 1 2 18 14 AM1 in Gau01B2+ eV AM1 in g01b2+ 8 Energetics Diagram for the ring-collapse mechanism of C28-D2 fullerene c28d2-2 H=47.74820 19 20 3.3966 15 d2-3 16 11 27 12 3 7 28 24 d2-6 4 8 23 2.6356 H=47.46386eV c28d2-8 7 6 d2-4 2 25 5 22 10 21 1 8 12 4 2 6 26 14 17 9 21 23 13 18 c28d2-4 H=48.22512 10 14 26 13 2 25 27 17 18 9 14 4.2450 26 6 22 1 12 5 28 15 5.4282 23 d2-2 Large barrier associated 16 19 8 24 20 3 10 7 18 11 27 22 17 AM1 in G01B2+ 9 4 21 d2-8 3.5843 3 4.4414 15 13 5 25 with ring strain. 6 7 16 19 1 20 11 28 24 H=45.79176eV c28d2-9 6 2 10 14 17 18 21 4 8 26 25 13 12 9 22 23 1 5 5 3 27 Energy stabilization in 7 24 11 28 15 16 20 19 AM1 in G01B2+ d2-7 d2-9 c28d2-1 H=45.11716eV 27 c28d2-7 H=45.93379 eV 4 16 23 15 8 28 21 final steps through 3D- - 12 12 20 2.9939 25 19 19 24 8 6 10 20 11 23 13 4 16 2 15 1 2 17 26 6 27 21 7 3 14 22 11 10 28 3 25 14 5 H=44.39414eV 7 24 18 c28d2-10 4 18 17 26 9 28 9 24 16 13 22 3 11 5 1 aromaticity 7 20 22 AM1 in G01B2+ 19 d2-1 26 9 5 15 14 18 1 27 2 17 13 23 6 10 25 12 21 4 8 AM1 in G01B2+ H=43.32260eV d2-10 c28d2-12 2 6 26 3 14 22 3 10 18 7 17 9 11 13 5 1 24 25 21 28 4 16 20 8 19 12 15 23 27 AM1 in G01B2+ d2-12 2 Very high reaction barrier: 5.09 eV ~ 117 kcal/mol 1 c28d2 fullerene H=41.03134eV 17 d2-11 13 10 18 25 21 1 9 14 6 5 c28d2-11 H=42.06081eV 2 28 11 24 28 16 24 4 7 20 2 26 3 d2-n : n represents the number of broken bonds from d2-fullerene 19 22 26 8 14 22 6 15 16 18 27 7 10 9 23 11 17 23 12 5 12 3 13 20 27 1 8 25 4 0 21 These energies are from AM1 calculation in G01B2+ and the 19 15 AM1 in G01B2+ AM1 in G01B2+ d2-fullerene structures of these molecules can be found in related files. 4
Need for Reactive Molecular Dynamics Simulations Need for Reactive Molecular Dynamics Simulations High temperature (1000 - 5000 K) reduced relevance of 1. thermodynamically favorable pathways. Can sample structures of high potential energies. 2. High-dimensionality prohibits systematic determination of structures and energies of intermediates and transition states. Need high temperature molecular dynamics (MD) approach. Need inexpensive method for calculating potential energy function which allows bond breaking/formation: Semiclassical Brenner REBO (Reactive empirical bond-order) 1. molecular force field potential, (e.g., Brenner et al, Phys. Rev. B 1990 , 42 , 9458, for simulation od diamond) 5 2. Semiempirical quantum chemical methods (AM1, PM3, DFTB)
Brenner-Potential MD Simulation of the Fullerene Brenner-Potential MD Simulation of the Fullerene Formation Process - Time Scale Formation Process - Time Scale Yamaguchi, Y.; Maruyama, S. Chem.Phys.Lett., 1998 , 286 , 336-342 T=1500 K Cluster size Time scale: nanoseconds 6 time [ps]
Brenner-Potential MD vs. Quantum Chemical Potential Brenner-Potential MD vs. Quantum Chemical Potential REBO Force Field is several orders of magnitude faster than semiempirical quantum chemical methods, in addition: scaling ~ N 2 vs. N 3 REBO Force Field was developed for vapor decomposition of graphite under high pressure to form diamond; can only describe bond formation/breaking processes. Quantum chemical all valence electron approaches include naturally directionality, i.e. bond formation/breaking. Quantum chemical potential includes naturally aromaticity , conjugational stabilization, C sp C sp 3 hybridization 7
Density Functional Tight Binding (DFTB) Density Functional Tight Binding (DFTB) Seifert et al., Int. J. Quant, Chem. 1996 , 92 , 185 Extended Hückel type method using atomic parameters from DFT (PBE), diatomic repulsive potentials from B3LYP orb atom E n i i V AB • Seifert, Eschrig (1980-86): A B STO-LCAO; 2-center approximation k • Porezag et al. (1995): efficient parameterization scheme • Elstner et al. (1998): charge self-consistency: SCC-DFTB “ approximate DFT ” Only time consuming step: Matrix diagonalization 8
Order out of chaos? Ensembles of C 2 molecules as Order out of chaos? Ensembles of C 2 molecules as starting structures for DFTB/MD simulations starting structures for DFTB/MD simulations Experimental conditions of fullerene formation: Many carbon clusters • available in great abundance under great heat and normal pressure Non-equilibrium dynamics with large kinetic energy and carbon cluster • material fluctuations: Monomolecular approach may not be valid. More realistic starting point for DFTB/MD simulations: Ensembles of • randomly oriented C 2 molecules under ~ 2000 K, providing steady supply of additional C 2 molecules: Open exchange of energy and carbon material , NO SINGLE POTENTIAL ENERGY SURFACE Example for order created dynamically out of chaos: • Dissipative structures (e.g. Rayleigh-Benard convection cells) without associated single potential energy function 9
Adding More C2 ’ s is a key to formation of fullerenes Adding More C2 ’ s is a key to formation of fullerenes “ S2 ” to “ S5 ” 60 C 2 s 6ps 6ps in 30 Å add 10 more C 2 add 10 more C 2 2000K 2000K cubic box 0.7g/cm 6ps 2000K preferred for earlier stage 2000K 6ps 6ps add 10 more C 2 add 10 more C 2 add 10 more C 2 2000K 2000K 6ps 3000K 10-48ps add 10 more C 2 3000K 10
DFTB/MD on n C 2 in 30Å periodic boundary box (I) DFTB/MD on n C 2 in 30Å periodic boundary box (I) 0.00ps 0.24ps 0.29ps S1 First big rings Long entangled chains Initial state 12.1ps 3.86ps 6.05ps Big rings Many long chains More smaller rings collapse into at the edges created by ring collapse smaller rings 11
DFTB/MD on n C 2 in 30Å periodic boundary box (II) DFTB/MD on n C 2 in 30Å periodic boundary box (II) 39.78ps 22.07ps 14.54ps Growth by collapse Short chain of chains on edges connect with another long chain 49.72ps 43.27ps 43.26ps One more hexagon Cycloaddition between Fullerene with 26 penta Created by reaction Adjacent chains 42 hexa, and 15 heptagons, Between wobbling On border similar to CNT 146 carbons in the cage C2 and C3 12
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