Temperature-dependent divergence of thermal conductivity in momentum-conserving one-dimensional lattices with asymmetric potential

In this study we used a nonequilibrium simulation method to investigate the temperature dependent divergence of thermal conductivity in a one-dimensional momentum conserving system with an asymmetric double well nearest-neighbor interaction potential. We show that across all temperatures thermal conductivity exhibits power-law divergence with the chain length and the value of the divergence exponent ($\alpha$) depends on the temperature of the system. At low and high temperatures $\alpha$ reaches close to $\sim 0.5$ and $\sim 0.33$, respectively. Whereas in the intermediate temperature the divergence of thermal conductivity with the chain length saturates with $\alpha \sim 0.07$. Subsequent analysis showed that the estimated value of $\alpha$ in the intermediate temperature may not have reached its thermodynamic limit. Further calculations of local $\alpha$ revealed that its approach towards the thermodynamic limit is crucially dependent on the temperature of the system. At low and high temperatures local $\alpha$ reaches its thermodynamic limits in shorter chain lengths. On the contrary, in the case of intermediate temperature its progress towards the asymptotic limit is nonmonotonic.

Figure 1

Schematic representation of the double-well (DW) nearest-neighbor interaction potential, $V\left(x\right)$, with different values of asymmetric parameter $k_3$. $k_2=0.1$ and $k_4=0.002$.

Figure 2

Divergence of thermal conductivity, $\kappa$, as a function of chain length, $N$. Different colored symbols represent simulations with different average bath temperatures with fixed temperature difference, ($T, \mathrm{\Delta }T$); circle: (1.5,1), triangle: (4.5,1), square: (9.5,1), and star: (3.0,4). Solid lines are from power-law fitting ($\kappa \sim N^{\alpha }$). The values of $\alpha$ are indicated inside the plots for (a) $k3=0.003$ and (b) $k3=0.006$.

Figure 3

Divergence of $\kappa$ as a function of $N$ for different average $T$ and $\mathrm{\Delta }T$; circle: (1.5,1), triangle: (0.3,0.2), square: (0.15,0.1), and diamond: (0.075,0.05). The asymmetric parameter $k_3$ was 0.003. The $\alpha$ values are indicated inside the plot.

Figure 4

Divergence of $\kappa$ with $N$ at different temperatures with asymmetric parameter $k_3=0.006$. ($T, \mathrm{\Delta }T$) pairs are: circle (1.5,1), triangle (0.3,0.2), square (0.15,0.1).

Figure 5

Temperature dependence of $\alpha$ for two different values of asymmetric parameters $k_3$. The sizes of error bars on $\alpha$ are nearly the same as the sizes of the markers.

Figure 6

Comparison of the divergence of $\kappa$ with $N$ for different values of asymmetric parameter $k_3$ at various average bath temperatures, $T$. Solid line: $T=1.5$, dotted line: $T=0.3$, and dashed line: $T=0.15$.

Figure 7

Plot of local divergence coefficient ${\alpha }_N$ with $N$ for different values of asymmetric parameter $k_3$ and at different average bath temperature, $T$. The ${\alpha }_N$ was estimated by calculating the local slope of $\kappa$ vs. $N$ plots given in Figs. 3 and 4. Solid line: $T=1.5$, dotted line: $T=0.3$, dashed line: $T=0.15$, and dashed-dot line: $T=0.075$. The horizontal dashed line represents $\alpha =0.33$.

Figure 8

(a) Temperature dependence of the order parameter for a chain with $N=500$. The average displacement from the adjacent particle along the chain at $T=2.0$ (b) and $T=0.07$ (c).

Figure 9

Absolute average contributions of second, third, and fourth order terms in the potential for a chain with $N=1000$ and $k_3=0.003$. The values of temperature are indicated at the top of the figure. The system behaves similarly with $k_3=0.006$ as well.