Designability of Graphitic Cones
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Designability of Graphitic Cones M. M. J. Treacy and J. Kilian NEC Research Institute, Inc. 4 Independence Way, Princeton, NJ 08540-6685, USA ABSTRACT We show that, with topologically flexible seeds which are allowed to explore different growth modes, graphitic cones are inherently more “designable” than flat graphitic disks. The designability of a structure is the number of seed topologies encoding that structure. We illustrate designability with a simple model, where graphite grows onto Cn (5n ULQJ seeds. For a wide range of ring sizes, cones are the most likely topological outcome. Results from the model agree well with data from special cone-rich carbon black samples. The concept of designability allows entropy to be incorporated into the “pentagon road” model of the formation of curved graphitic structures. INTRODUCTION Recently, Krishnan et al. (1997) [1] reported a new carbon black material, made by Kvaerner Engineering a.s. Norway in a proprietary industrial process that involved the pyrolysis of methane in a plasma torch. Transmission electron microscopy (TEM) showed that the sample (designated KVR) contained high concentrations of multilayer graphitic disks, cones and tubes. Cones had been sighted before in carbon black [2], but not in the quantity and variety exhibited by KVR. Five distinct cone angles were observed, each corresponding to the inclusion of a disclination of order 1m7KHPHDVXUHGFRQHDQJOHVθ, were found to correspond closely to the predicted values θ=2sin–1(1–m/6). Although graphitic disks dominated the sample, among the cones the medium-angle 60° (m=3) and 38.9° (m=4) cones occurred most frequently. From an energetics viewpoint, the small-angled cones (those with the sharpest cone points and largest m value) should have the highest elastic strain energy per carbon atom, particularly near the tip area where cone wall curvature is high. Furthermore, there is an additional energy penalty for forming the tip structure, which must enclose one or more disclinations. Although pentagon inclusion probably costs the least energy on forming the core of the disclination, defect structures with dangling bonds and different ring sizes are also possible. From consideration of the enthalpy of formation alone, the disk topology with no strain and no pentagons, must be the lowest energy form, and the 19.2° (m=5) cone and the cylinder (m=6) must be the highest energy forms, and therefore the rarest. An enthalpy argument would predict, in terms of θ values, the order of likelihood to be 180° > 112.9° > 83.6° > 60° > 38.9° > 19.2° > 0°. TEM shows, however, that the 180° disks, and the 60° and 38.9° cones dominate the KVR sample. The reason must lie with the differences in Gibb’s free energy, ∆Gij = ∆Hij – T∆Sij. As usual, ∆Hij is the enthalpy of formation when a chemical system transforms from state i to state j, and T is the temperature at which the transformation occurs. The entropy change ∆Sij is a measure of the number of reaction pathways available to system i that can lead to product j. If a
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