The characteristics of phonons, i.e. linearized normal modes of vibration, provide important insights into many aspects of crystals, e.g. stability and thermodynamics. In this paper, we use the Objective Structures framework to make concrete analogies between crystalline phonons and normal modes of vibration in non-crystalline but highly symmetric nanostructures. Our strategy is to use an intermediate linear transformation from real-space to an intermediate space in which the Hessian matrix of second derivatives is block-circulant. The block-circulant nature of the Hessian enables us to then follow the procedure to obtain phonons in crystals: namely, we use the Discrete Fourier Transform from this intermediate space to obtain a block-diagonal matrix that is readily diagonalizable. We formulate this for general Objective Structures and then apply it to study carbon nanotubes of various chiralities that are subjected to axial elongation and torsional deformation. We compare the phonon spectra computed in the Objective Framework with spectra computed for armchair and zigzag nanotubes. We also demonstrate the approach by computing the Density of States. In addition to the computational efficiency afforded by Objective Structures in providing the transformations to almost-diagonalize the Hessian, the framework provides an important conceptual simplification to interpret the phonon curves. Our findings include that, first, not all non-optic long-wavelength modes are zero energy and conversely not all zero energy modes are long-wavelength; second, the phonon curves accurately predict both the onset as well as the soft modes for instabilities such as torsional buckling; and third, unlike crystals where phonon stability does not provide information on stability with respect to non-rank-one deformation modes, phonon stability in nanotubes is sufficient to guarantee stability with respect to all perturbations that do not involve structural modes. Our finding of characteristic oscillations in the phonon curves motivates a simple one-dimensional geometric nonlocal model of energy transport in generic Objective Structures. The model shows the interesting interplay between energy transport along axial and helical directions.
Bibliographical noteFunding Information:
Amin Aghaei and Kaushik Dayal thank AFOSR Computational Mathematics ( FA9550-09-1-0393 ) and AFOSR Young Investigator Program ( FA9550-12-1-0350 ) for financial support. Kaushik Dayal also acknowledges support from NSF Dynamical Systems ( 0926579 ), NSF Mechanics of Materials ( CAREER-1150002 ) and ARO Solid Mechanics ( W911NF-10-1-0140 ). Ryan S. Elliott acknowledges support from NSF CMMI-0746628 (Dr. George Hazelnut, Program Director) and The University of Minnesota Supercomputing Institute. This work was also supported in part by the NSF through TeraGrid resources provided by Pittsburgh Supercomputing Center. Kaushik Dayal thanks the Hausdorff Research Institute for Mathematics at the University of Bonn for hospitality. We thank Richard D. James for useful discussions.
Copyright 2013 Elsevier B.V., All rights reserved.
- Energy transport
- Objective structures
- Phonon analysis