Auxiliary Material for the Paper by A. N. Volkov and L. v. Zhigilei " Scaling Laws and Mesoscopic Modeling of Thermal Conductivity in Carbon Nanotube Materials " Derivation of Eqs. (1,2, and 4)

The derivation of the theoretical equations for thermal conductivity is performed for a model material composed of straight nanofibers. Each nanofiber has a shape of soft-core spherocylinder (SC), i.e. a circular cylinder of length T L and external radius T R capped on its both ends by two hemispheres. The “soft-core” assumption implies that SCs can freely intersect with each other and the intersections are treated as thermal contacts between nanotubes. In real materials, the intersections would be accommodated by local bending of the interacting nanofibers. The analysis is performed under assumption that the intrinsic thermal conductivity of SCs is infinitely large and the conductivity of the material is governed by the inter-tube thermal contact resistance. As a consequence, every SC i in the sample is characterized by a single value of temperature i T and the heat flux at a contact between SCs i and j is equal to ) ( i j c ij T T Q − σ = , where the inter-tube contact conductance, c σ , is assumed to be the same for all contacts, 0 c c σ = σ . To ensure transparent connection between the analytical equations and the results of the numerical calculations presented in the paper, we consider finite-size square (in 2D case) or cubic (in 3D case) systems with a size of S L . The distribution of SCs within the systems is homogeneous and isotropic, with the total number of centers of SCs defined by the surface number density, S n , and volume number density, V n , in the 2D and 3D cases, respectively. It is assumed that a constant gradient of averaged temperature, x T ∇ , is maintained along the x-axis and periodic boundary conditions are applied in the other direction(s) in the system. The heat flux through a cross sections of the systems at 0 = x can be calculated as ∑∑ + δ − = ij ij x Q Q ) ( , where 1 ) ( = δ + ij if SC i intersects axis 0 = x and the point of contact between SCs i and j is located to the right of the cross section, otherwise 0 ) ( = δ + ij . The derivation of Eqs. (1-2) is based on representation of the ensemble averaged heat flux as