Structural Models of Negatively Curved Graphitic Carbon
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STRUCTURAL MODELS OF NEGATIVELY CURVED GRAPHITIC CARBON
S. J. Townsend*, T. J. Lenosky*, D. A. Muller*, C. S. Nichols**, and V. Elser* *Cornell University, Department of Physics, Ithaca, New York 14853 "**Cornell University, Department of Materials Science and Engineering, Ithaca, New York 14853 ABSTRACT We have developed an automatic computational method for generating graphitic carbon structures. We propose models for crystalline and amorphous forms of carbon consisting of a single negatively curved graphitic sheet forming an extended structure. Stability and elastic properties are computed using an energy estimate fit to local density approximation(LDA) calculations. These forms of carbon are found to be more stable than C 60. The radial distribution function for the random structures closely matches that of films of amorphous carbon grown on NaCI substrates from sublimated graphite.
Introduction The invention of a method for the production of macroscopic quantities of fullerenes C6 0 and C7 0 [1] and the discovery of curved graphite tubules[2] have led to speculation about other new forms of carbon. Fullerenes and carbon tubules have positive and zero Gaussian curvature, respectively. In contrast, new possibilities involve graphitic sheets with negative Gaussian curvature[3, 4, 5]. The sheets form an extended structure that may be either periodic, as in previous models, or random. The random structures would, if synthesized, be classified as a form of amorphous carbon. We call both groups of structures schwarzite, after the mathematician H. A. Schwarz, who first studied periodic curved surfaces with zero mean curvature[6, 7].
Calculations Early crystalline schwarzite models were developed on an individual basis by assuming symmetries and using ball-and-stick models as guides(Figure la). We have replaced this process with an automatic method for generating carbon structures whose sheets conform to a given surface (details in [8]). The method introduces a cost function that is the sum of a purely repulsive pair potential between points representing the centers of carbon rings. Minimization of this cost function by simulated annealing yields a triangular array of nearly evenly spaced points. The full carbon structure is deduced from this array of ring centers. Figure lb shows a structure produced by this method using a P periodic minimal surface. A similar method was developed independently and has been applied to the study of fullerenes[9]. It is thought that a random pore structure would be more easily formed than crystalline schwarzite.[3, 5] To model such structures we produced random surfaces with many Mat. Res. Soc. Symp. Proc. Vol. 270. @1992 Materials Research Society
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Figure 1. Views of two crystalline schwarzites. Each has 216 carbon atoms per primitive unit cell with 80 6-membered rings and 24 7-membered rings. The structure in (a) lies in a cubic cell 16.2 A on a side. It was hand-generated to resemble the P minimal surface. The structure in (b) was automatically generated on a P minimal surfa
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