Second Law Considerations in Fourier Heat Conduction of a Lattice Chain in Relation to Intermolecular Potentials

TL;DR

This paper investigates the validity of Fourier's law in a 1-D lattice chain with harmonic potentials, revealing discrepancies with standard models and proposing a stationary principle for certain regimes, with implications for thermal circuit design.

Contribution

It challenges standard Fourier-based heat conduction models in lattice chains with harmonic potentials and introduces a new stationary principle for non-Fourier regimes.

Findings

Standard models do not hold for harmonic potentials in certain regimes.

Simulation results show different temperature profiles with more realistic thermostats.

Sinusoidal temperature profiles suggest potential for thermal circuit applications.

Abstract

Two aspects of conductive heat are focused here (i) the nature of conductive heat, defined as that form of energy that is transferred as a result of a temperature difference and (ii) the nature of the intermolecular potentials that induces both thermal energy flow and the temperature profile at the steady state for a 1-D lattice chain. It is found that the standard presuppositions of people like Benofy and Quay (BQ) following Joseph Fourier do not obtain for at least a certain specified regime of intermolecular potential parameters related to harmonic (quadratic) potentials for nearest neighbor interactions. For these harmonic potentials, it appears from the simulation results that steady state solutions exist utilizing non-synthetic thermostats that couple not just the two particles at the extreme ends of the lattice chain, but to a control volume of $N$ particles located at either ends of the chain that does not accord with the unique analytical solutions that obtains for single particle thermostatting at the ends of the lattice with a different thermostatting algorithm that utilizes coupling coefficients. If the method used here is considered a more "realistic" or feasible model of the physical reality, then a re-evaluation of some aspects of the standard theoretical methodology is warranted since the standard model solution profile does not accord with the simulation temperature profile determined here for this related model. We also note that the sinusoidal temperature profile generated suggests that thermal integrated circuits with several thermal P-N junctions may be constructed, opening a way to create more complex thermal transistor circuits. A stationary principle is proposed for regions that violate the Fourier principle $\mathbf{J_q.}\nabla T \le 0 $, where $\mathbf{J_q}$ is the heat current vector and $T$ the temperature.