Conduction through a damaged honeycomb lattice

The temperature distribution and rate of heat transfer across an infinite periodic strip of a honeycomb lattice consisting of conductive segments or links joined at nodes or junctions is discussed. A pristine honeycomb behaves like an isotropic medium whose effective conductivity is independent of the orientation of an applied macroscopic temperature gradient. Monte Carlo simulations are performed to determine the effect of link damage or disruption and lattice deformation due to junction displacement. In the simulations, a specified percentage of randomly distributed links are assigned a conductivity that is lower than that of the undamaged links. The balance equations governing the nodal temperatures at the junctions are solved by iteration subject to a periodicity condition along the strip and the Dirichlet condition along the two infinite edges of the strip. The results illustrate the effect of imperfections on the temperature distribution over the network and document the dependence of the effective conductivity on the percentage and conductivity of the defective links. In the case of nonconductive damaged links, the effective conductivity becomes nearly zero when a critical percentage of links are clipped, in agreement with bond percolation theory. However, the functional form of the number density of possible pathways connecting the lower to the upper edge of the strip predicted by percolation theory differs from that of the effective conductivity. Lattice deformation due to random node displacement has a small effect on the effective conductivity of the network.