Classification and Construction of Topologically Nontrivial

Graphitic MaterialsBih-Yaw Jin

Department of ChemistryNational Taiwan University

byjin@ntu.edu.tw

ACS meeting, Washington DC, Aug.20, 2009

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Classification

Polygonal TorusTamura, R. et al. Phys. Rev. B. 2005, 71, 045418.

Chuang, Fan & Jin, J.Chem.Inf & Model 2009, 49, 361

A family of achiral TCNTs with Dnh symmetry can be systematically constructed by polygonal prisms defined by four parameters, (n75, n77, n55, s).

Series I : the increasing of n75 while keeping other parameters, n77, n55, and s, constant. Series II : varying (n77, n55). Series III : the increasing of s.

Evolution of TCNTs as a function of certain indices among the four parameters, (n75, n77, n55, s)

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Construction of Fullerenes

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Beads and beadworks

1-D

2-D

3-D

Beaded Buckyball

• Bead Representation = Bond Representation

• 90 beads are needed for C60

• Beaded fullerene represent the pi-bond network.

60× 3/2 = 90

Weaving code �Spiral Code

9•C60: [1 7 9 11 13 15 18 20 22 24 26 32 ]

566666565... planar graph

7 Isomers of C80

• [1 7 9 11 13 15 28 30 32 34 36 42]

•[1 7 9 11 13 18 25 30 32 34 36 42]

• [1 7 9 11 14 22 27 30 34 36 38 40]

•[1 7 9 11 14 23 28 30 33 35 37 39]

•[1 7 9 12 14 20 26 28 32 34 39 42]

•[1 7 10 12 14 19 26 28 32 34 39 42]

•[1 8 10 12 14 16 28 30 32 34 36 42]

P. W. Fowler, An Atlas of FullerenesSpiral Codes

Structures of Fullerenes

P. W. Fowler, An Atlas of Fullerenes Point group: D5d�D2�C2v�D3�C2v�D5h�Ih

Advantages of Beaded Models

• The shapes of the beaded fullerenes are in good agreement with the optimized geometries based on more sophisticated molecular force fields since the mechanical interactions among three neighbored beads can effectively mimic the steric repulsion in the sp2-hybridized fullerenes.

• Beaded fullerenes can be constructed systematically based on the spiral codes together with simple construction rules.

Minimal Length of String Required

•What is the minimal length of string to weave

a particular fullerene?

•Hamiltonian Circuits or Paths

Lmin = 2sNd + Lext

• d : diameter of bead.

• s = 1.1

• Lex: extra length

Truncated Icosahedron

C60 C80 C140 C180 C240 C280

Truncated Tetrahedron

End-capped CNT

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Classification

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Beaded Models

TCNT 240 (T240) Construction Procedure of T240

Helically Coiled CNTsHorizontal shifting

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Beaded Model for HCCNT

More beaded models for HCCNT Triple Stranded Carbon Helix

L.P. Hwang, Phys. Chem. Comm.(2002)

Helicity of 2nd Kind: Vertical Shifting

Combination of Two Kinds of Helicities: CNT Space Curves

Carbon Trefoil Knots

Carbon Nanotube Torus Knot

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Classification

Neck Structure: Construction of High Genus Fullerenes

Chuang & Jin, J.Chem.Inf & Model 2009, 49, 1664

D5h Necks

High Genus Fullerenes

Chuang & Jin, J.Chem.Inf & Model 2009, 49, 1664

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Chuang & Jin, J.Chem.Inf & Model 2009, 49, 1664

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Beaded models for High-Genus Fullerenes

Superbuckyball

C60

C70

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Chuang & Jin, submitted

Doubly periodic graphitic structures

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P-Type Minimal Surface

D-Type TPMS Gyroidal Minimal Surfaces

C168 Pseudo P-Type surface

H-Type Surface Sierpinski buckyball

D=log 90/log scaling factor = log(90)/log(7) ~ log(90)/log(6.5) = 2.3 ~ 2.4

Hausdorff (fractal) dimension

Sierpinski's pyramid

http://commons.wikimedia.org/wiki/Image:Sierpinski_pyramid.png

Conclusion

• The shapes of the beaded fullerenes are in good agreement with the optimized geometries based on more sophisticated molecular force fields since the mechanical interactions among three neighbored beads can effectively mimic the steric repulsion in the sp2-hybridized fullerenes.

• Beaded fullerenes can be constructed systematically based on the spiral codes together with simple construction rules.

• Bead is a versatile low-cost high quality material for creating various complicated 3D structures of sp2-hybridized carbon nanostructures.

• Beaded molecules are aesthetically pleasing artworks.

Advantages of Beaded Models

KM

Blog for Beaded Fullereneshttp://thebeadedmolecules.blogspot.com/

C. Chuang

NSC, TaiwanCTS and CQSE, NTU