Microscopic determination of the interlayer binding energy in graphite
Introduction
Literature values for the energy of attraction between graphite layers differ by a factor of a hundred. Empirical interatomic interactions fit to a database of hydrocarbon binding energies, a two-parameter interatomic potential fit to the c-axis compressibility and the interlayer spacing of graphite, and application of intermolecular N2 interaction potentials to carbon atoms all yield weak interlayer attractions of εattr≈0.002, 0.002, and 0.003 eV/atom respectively 1, 2, 3. In contrast, local density approximation (LDA) calculations yield εattr≈0.02 eV/atom 4, 5, 6and εattr≈0.03 eV/atom (M. Côté, private communication). Bear in mind that the LDA can have difficulty with the correlation-born Van der Waals energy. Alternative theoretical calculations and semiempirical estimates yield a clutch of larger values, εattr≈0.04, 0.1, 0.17, 0.2 eV/atom 7, 8, 9, 10. A single heat of wetting experiment yields εattr≈0.04 eV/atom [11]. Although this value is in reasonable agreement with the difference between a bond-energy sum and the graphite heat of vaporization [12], this estimate is the difference of large, imprecisely known numbers.
Our measurement exploits the well-known mean curvature modulus of graphite in a unique experimental geometry which balances curvature energy with the interlayer attraction 13, 14. A carbon nanotube [15]can collapse into a flat strip with bulbs on the edges [16], the large radial deformations being stabilized by the intersheet attraction2 between opposing sides of the inner wall. Collapse is favored in large tubes: as tube circumference increases, the excess curvature energy in the bulbs asymptotes to a constant while the energetic advantage of the intersheet attraction increases linearly. The ratio of the interlayer attraction to the graphitic mean curvature modulus determines the size of the bulbs and the radius above which collapse is favored. Careful analysis of transmission electron microscopy (TEM) images of flattened, twisted tubes reveals the size of the bulbs, allowing us to extract the strength of the interlayer binding, 0.035+0.015−0.01 eV/atom, while simultaneously exploring the detailed morphology.
Section snippets
Experiment
Carbon nanotube samples were synthesized by the standard arc method [17]. Nanotubes were transferred to holey carbon grids by rubbing the material between glass slides. The samples were then imaged in a JEM JEOL 200CX TEM operating at 200 keV. Of three examples of twisted flattened nanotubes that proved suitable for further analysis, we will concentrate on Fig. 1. Previous sample rotation [16]confirms that the tube is uniformly collapsed along its length. The flattening persists in untwisted
Elastic model
Opposing walls of a flattened multiwall nanotube collapse to an optimum distance d apart. If the graphite sheets do not tear, then the two flat sections must join at the edges. If εattrd2∼k, the graphite sheet mean curvature modulus [19], then the edges bulge smoothly to decrease the local curvature energy, yielding the cross-section shown in Fig. 4. We model the bulbs as semicircles joined smoothly to the flat region by identical curves αsin(βx+γ)+δ. The interwall distance is fixed at d in the
Summary
Continuum elasticity theory with a Lennard–Jones description of intersheet attraction yields the equilibrium cross-sections of collapsed carbon nanotubes. Tubes with few walls and large radii favor collapse over the more familiar circular cross-section. The size of the bulbs on the edges of collapsed tubes depends on the ratio of the intersheet attraction to the mean curvature modulus. Detailed analysis of TEM images of twisted collapsed tubes allows us to extract the number of walls, tube
Acknowledgements
This research was supported by the National Science Foundation Grant No. DMR-9520554 and by the Office of Energy Research, Office of Basic Energy Sciences, Materials Sciences Division of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098. NGC acknowledges support from the Department of Education. AZ and SGL acknowledge support from the Miller Institute for Basic Research in Science.
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Present address: Optical Technology Div., Physics Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20878, USA.