Size effects on thermal rectification in mass-graded anharmonic lattices

In this work we study the thermal rectification efficiency of a one-dimensional mass-graded anharmonic oscillator lattice at large system sizes. A modest increase in rectification is observed. When the magnitude of the mass gradient scales with the system size, the aforementioned effect increases substantially. This result can be unmistakeably attributed to an asymmetry in the local temperature profile obtained for the employed parameter values.

Figure 1

Thermal rectification $r=r(J_{+},J_{-})$ as a function of the system size $N$ for a mass gradient of $\Delta M=20$ (circles), 30 (squares), and 40 (diamonds), with $T_{{}_0}=0.1,\Delta T=0.16$, and $n_{{}_L}=n_{{}_R}=3$.

Figure 2

Temperature profiles for the $\Delta M=20$ case depicted in Fig. 1 using the same $T_0,\Delta T$, and $n_{{}_{L,R}}$ values as in the previous figure. In panel (a) the case corresponding to $T_{{}_L}>T_{{}_R}$ is shown, whereas in panel (b) is the case when the two reservoirs at the ends are swapped, i.e., $T_{{}_L}<T_{{}_R}$. Arrows in both figures indicate the direction in which system size increases.

Figure 3

(a) Temperature profile for the $N=30000$ case depicted in Fig. 2; same $T_0,\Delta T$, and $\Delta M$ values as in previous figure. (b) Corresponding local heat flux $J_i$ to the temperature profile in panel (a). Arrow in panel (b) indicates the direction of the total heat flux $J_{+}$.

Figure 4

(a) Temperature profile for the $N=30000$ case depicted in Fig. 2. (b) Corresponding local heat flux $J_i$ to the temperature profile in (a). Thick arrow in (b) indicate the direction of the total heat flux $J_{-}$ and thin arrows those of the “local” heat fluxes $j_{+}$ and $j_{-}$ (see text for more details). In both figures the vertical dashed line indicates the position along the lattice length wherein the TP attains its maximum value.

Figure 5

Thermal rectification $r$ vs system size $N$ for $\rho =\Delta M/N$ values of $0.025$ (circles), $0.05$ (squares), $0.1$ (diamonds), and $0.25$ (triangle up). Same $T_0$ and $\Delta T$ values as in Fig. 1. Dashed lines indicate the corresponding exponential fit (see text for details).

Figure 6

Temperature profiles for cases (a) $J_{+}$ and (b) $J_{-}$, both with $\rho =0.25$; the same $n_{{}_{R,L}},T_0$, and $\Delta T$ values are used as in Fig. 1.

Figure 7

Thermal rectification for three system sizes $(N=1000,10000$, and $20000)$, all with $\rho =0.1$, vs percentage of thermostated oscillators (view text for more details). Arrows indicate previously obtained values for the case $n_{{}_{L,R}}=3$.

Figure 8

Temperature profiles for cases (a) $J_{+}$ and (b) $J_{-}$, both with $\rho =0.1$ and $N=20000$ of previous figure. Arrows indicate the direction in which the percentage of thermostated oscillators increases; the same $T_0$ and $\Delta T$ values are used as in Fig. 1.