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In this paper we have used equation (4) only to define temperatures in sections that are "bulk like" and do not contain any grain boundaries. The Stillinger-Weber phonon dispersion for bulk crystalline silicon was, therefore, used with equation (4). From the physical continuity in the temperature profile, one could estimate an average steady-state temperature of a section containing an isolated grain boundary simply from the average of the temperatures of the two neighboring bulk like sections. The assumption is that each section is wide enough for the local phonon modes not to be significantly affected by the presence of a grain boundary in a neighboring section. For a more general (disordered) case, it is necessary to define an appropriate phonon dispersion local to each section.
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In order to calculate the thermal conductivity using classical MD, one must define the energy carried by each atom in the simulation box. Such a definition is unambiguous for pair potentials but not in the case of the Stillinger-Weber potential, which contains three-body terms.
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