On the link between mechanics and thermal properties: mechanothermics

TL;DR

This paper develops a theoretical framework linking mechanical mode Brownian motions to macroscopic thermal properties in insulating rods, emphasizing modal energy mixing and its relation to phonon mean-free-path, with implications for nanostructure thermal transport.

Contribution

It introduces an analytic theory connecting mechanical mode correlations to thermal transport, extending fluctuation-dissipation concepts to nanomechanics.

Findings

Temperature gradient linked to mode cross-correlations.

Energy transport mediated by modal wave mixing.

Theory applicable to nanostructures and quantum regimes.

Abstract

We report on the theoretical derivation of macroscopic thermal properties (specific heat, thermal conductivity) of an electrically insulating rod connected to two reservoirs, from the linear superposition of its mechanical mode Brownian motions. The calculation is performed for a weak thermal gradient, in the classical limit (high temperature). The development is kept basic as far as geometry and experimental conditions are concerned, enabling an almost fully analytic treatment. In the modeling, each of the modes is subject to a specific Langevin force, which enables to produce the required temperature profile along the rod. The theory is predictive: the temperature gradient (and therefore energy transport) is linked to motion amplitude cross-correlations between nearby mechanical modes. This arises because energy transport is actually mediated by mixing between the modal waves, and not by the modes themselves. This result can be tested on experiments, and shall extend the concepts underlying equipartition and fluctuation-dissipation theorems. The theory links intimately the macroscopic size of the clamping region where the mixing occurs to the microscopic lengthscale of the problem at hand: the phonon mean-free-path. This clamping region, which is key, has received recently a renewed attention in the field of nanomechanics with topical works on "phonon shields" and "soft clamping". We believe that our work should impact the domain of thermal transport in nanostructures, with future developments of the theory toward the quantum regime.