Thermal rectification in three-dimensional mass-graded anharmonic oscillator lattices

In this work we study the thermal rectification efficiency, i.e., asymmetric heat flow, of a three-dimensional mass-graded anharmonic lattice of length $N$ and width $W$ by means of nonequilibrium molecular dynamics simulations. The obtained rectification, which is of the same order of magnitude as that of the corresponding one-dimensional lattice, saturates at low values of the aspect ratio $W/N$, consistent with the already known behavior of the corresponding heat fluxes of the homogeneous system under analogous conditions. The maximum rectification is obtained in the temperature range wherein no rectification could be obtained in other one-dimensional systems, as well as in the corresponding one-dimensional instance of the model studied herein.

Figure 1

Thermal rectification $r$ vs system width $W$ for lateral lengths of $N=16$ (circles), 32 (triangles), 64 (diamonds), and 128 (squares) with $M_{\max }=10$. Open symbols correspond to $T_{{}_0}=0.1$ and $\mathrm{\Delta }T=0.16$; filled symbols, to $T_{{}_0}=5$ and $\mathrm{\Delta }T=9$. Solid lines are a guide for the eye.

Figure 2

(a) Temperature profiles of a 3D mass-graded lattice with $W=16$, $T_{{}_0}=0.1$, and $\mathrm{\Delta }T=0.16$; the $M_{\max }$ value is the same as in Fig. 1. Results for $T_{{}_L}>T_{{}_R}$ (circles) and $T_{{}_L}<T_{{}_R}$ (triangles) are shown. Solid, dashed, and dotted lines correspond to $N=32$, 64, and 128, respectively. (b) Same as (a), but for $T_{{}_0}=5$ and $\mathrm{\Delta }T=9$. In both instances the dashed horizontal lines represent the corresponding $T_{{}_0}$ values.

Figure 3

(a) Power spectra for an oscillator on the left side, $i=4$ (red), and one on the right side, $i=60$ (blue), for $W=16$ with $N=64$, $M_{\max }=10$, $T_{{}_0}=0.1$, $\mathrm{\Delta }T=0.16$, and $T_{{}_L}>T_{{}_R}$. (b) Same as (a), but for $T_{{}_L}<T_{{}_R}$.

Figure 4

Same as described in the caption to Fig. 3, but for $T_{{}_0}=5$ and $\mathrm{\Delta }T=9$. (a) $T_{{}_L}>T_{{}_R}$ and (b) $T_{{}_L}<T_{{}_R}$.

Figure 5

(a) Dependence of thermal rectification $r$ on temperature difference $\mathrm{\Delta }T$ for two lateral lengths $N$, 32 (circles) and 64 (triangles), both with $W=16$, $M_{\max }=10$, and $T_{{}_0}=0.1$. (b) Same as (a), but for $T_{{}_0}=5$. Solid lines are a guide for the eye.

Figure 6

(a) Thermal rectification $r$ as a function of the largest mass $M_{\max }$ for $N=32$ (circles) and $N=64$ (triangles), both with $W=16$. Open symbols correspond to $T_{{}_0}=0.1$ and $\mathrm{\Delta }T=0.16$, whereas filled ones correspond to $T_{{}_0}=5$ and $\mathrm{\Delta }T=9$. Solid lines are a guide for the eye.

Figure 7

Thermal rectification $r$ as a function of the lateral length $N$ for $W=16$ and $M_{\max }=10$. Circles correspond to $T_{{}_0}=0.1$ and $\mathrm{\Delta }T=0.16$; triangles, to $T_{{}_0}=5$ and $\mathrm{\Delta }T=9$. Solid lines are a guide for the eye.