Non-stationary elastic wave scattering and energy transport in a one-dimensional harmonic chain with an isotopic defect

TL;DR

This work analyzes non-stationary elastic wave scattering and energy transport in a $1$-D harmonic chain containing an isotopic defect. By deriving the Green function in the frequency domain and performing large-time asymptotics, it decomposes the wave field into pass-band and stop-band contributions and validates these results against numerics. In the stochastic loading setting, it separates slow and fast thermal motions to reveal a thermal shadow behind the defect and a jump in slow temperature at the defect, analogous to Kapitza resistance, along with a non-stationary transmission function for the kinetic temperature. The findings offer a tractable, non-stationary model of interfacial heat transfer and anti-localization phenomena with implications for nanoscale thermal transport and design of defect-engineered materials.

Abstract

The fundamental solution describing non-stationary elastic wave scattering on an isotopic defect in a one-dimensional harmonic chain is obtained in an asymptotic form. The chain is subjected to unit impulse point loading applied to a particle far enough from the defect. The solution is a large time asymptotics at a moving point of observation, and it is in excellent agreement with the corresponding numerical calculations. At the next step, we assume that the applied point impulse excitation has random amplitude. This allows one to model the heat transport in the chain and across the defect as the transport of the mathematical expectation for the kinetic energy and to use the conception of the kinetic temperature. To provide a simplified continuum description for this process, we separate the slow in time component of the kinetic temperature. This quantity can be calculated using the asymptotics of the fundamental solution for the deterministic problem. We demonstrate that there is a thermal shadow behind the defect: the order of vanishing for the slow temperature is larger for the particles behind the defect than for the particles between the loading and the defect. The presence of the thermal shadow is related to a non-stationary wave phenomenon, which we call the anti-localization of non-stationary waves. Due to the presence of the shadow, the continuum slow kinetic temperature has a jump discontinuity at the defect. Thus, the system under consideration can be a simple model for the non-stationary phenomenon, analogous to one characterized by the Kapitza thermal resistance. Finally, we analytically calculate the non-stationary transmission function, which describes the distortion (caused by the defect) of the slow kinetic temperature profile at a far zone behind the defect.

Figures (7)

• Figure 1: The particle velocity ${\mathcal{V}}_n^N$ versus the spatial variable $n$. The source position is indicated by the vertical magenta solid line. The right leading scattered wave-front is indicated by the vertical magenta dashed line. (a) The case of a heavy defect, (b) the case of a light defect
• Figure 2: Frequencies $\omega_-$ and $\omega_+$ versus $n$. The source position is indicated by the vertical magenta solid line. The right leading reflected wave front is indicated by the vertical magenta dashed line
• Figure 3: Kinetic temperature $\mathcal{T}_n^N$ versus spatial variable $n$. The source position is indicated by the vertical magenta solid line. The right leading reflected wave-front is indicated by the vertical magenta dashed line
• Figure 4: Kinetic temperature $\mathcal{T}_n^N$ versus time $t$ for various positive $n$. The instant of coming of the leading reflected wave-front is indicated by the vertical magenta dashed line
• Figure 5: Kinetic temperature $\mathcal{T}_n^N$ versus time $t$ ($n=0$)