Size effects in thermal conduction by phonons

Abstract

Heat transport in nanoscale systems is both hard to measure microscopically, and hard to interpret. Ballistic and diffusive heat flow coexist, adding confusion. This paper looks at a very simple case: a nanoscale crystal repeated periodically. This is a popular model for simulation of bulk heat transport using classical molecular dynamics (MD), and is related to transient thermal grating experiments. Nanoscale effects are seen in perhaps their simplest form. The model is solved by an extension of standard quasiparticle gas theory of bulk solids. Both structure and heat flow are constrained by periodic boundary conditions. Diffusive transport is fully included, while ballistic transport by phonons of long mean free path is diminished in a specific way. Heat current J(x) and temperature gradient ∇T (x ) have a non-local relationship, via κ(x − x ), over a distance |x − x | determined by phonon mean free paths. In MD modeling of bulk conductivity, finite computer resources limit system size. Long mean free paths, comparable to the scale of heating and cooling, cause undesired finite-size effects that have to be removed by extrapolation. The present model allows this extrapolation to be quantified. Calculations based on the Peierls-Boltzmann equation, using a generalized Debye model, show that extrapolation involves fractional powers of 1/L. It is also argued that heating and cooling should be distributed sinusoidally (ė ∝ cos(2πx/L)) to improve convergence of numerics.

Removing artificial Kapitza effects from bulk thermal conductivity calculations in direct molecular dynamics

“…4(b) in this work and Fig. 3 in Allen, 30 we note some similarities. This suggests that the nonlocal model may be a promising approach, in particular, in case where the temperature gradients vary significantly.…”

“…Nevertheless, it is of interest to explore the change in temperature gradient and heat current along the beam to make contact with the nonlocal model of Allen. 30 Figure 4 presents the absolute values of the temperature gradient and thermal conductivity for bins away from thermostats for the case of ¼ 1.0 ps shown in Fig. 3.…”

“…(17) a higher-order dependence on 1/L m . Allen 30 concluded that the extrapolation procedure should include fractional powers of 1/L m based on a model that includes a nonlocal dependence of the heat current on the temperature gradient. In our simulations of solid argon, the heat current appears nearly constant away from the thermostats (see Figure 3) suggesting that nonlocal effects would be small.…”

“…This allows them to enter the periodic system multiple times and interfere with the system randomly, which is another reason for the size dependent reduction in the values of computed thermal conductivities. 30 The dependence of k on beam length is also affected by the choice of thermostat and its parameters. A separate issue is whether the size dependence is linear or nonlinear.…”

Influence of thermostatting on nonequilibrium molecular dynamics simulations of heat conduction in solids

“…It has been usually argued that nonlinearities are artifacts related to e.g., the local thermostats in the NEMD simulation setup and should be excluded. However, recent studies suggest that such nonlinearities have a physical origin, related to transport in finite size systems [16,17]. If the nonlinear parts need to be excluded, then |∇T | = ∆T /L, leading to an inconsistency among Eqs.…”