Regulation of phonon localization on thermal transport in complex networks

The regulation of thermal transport is a challenging topic in complex networks. At present, the hidden physical mechanism behind thermal transport is poorly understood. This paper addresses this issue by proposing a complex network model that focuses on the thermal transport regulation through the manipulation of the network's degree distribution and clustering coefficient. Our findings indicate that increasing the degree distribution regulation parameter $\sigma$ leads to reduced phonon localization and improved thermal transport efficiency. Conversely, increasing the clustering coefficient $c$ results in enhanced phonon localization and reduced thermal transport efficiency. Meanwhile, by calculating the pseudodispersion relation of the network, we find that the maximum (or the second smallest) eigenfrequency decreases with increasing $\sigma$ (or $c$). Finally, we elucidate that phonon localization plays a pivotal role in the thermal transport of the network, as demonstrated through density of states and the participation ratio.

Figure 1

The influence of network topology on clustering coefficient. In the network with six nodes and eight edges, (a) and (b) are network structures of low clustering coefficient 0 and high clustering coefficient 0.83, respectively. (c) Kim's reconnection method: randomly select two edges $A\text{−}B$ and $C\text{−}D$ each time in the original network, then delete the two edges $A\text{−}B$ and $C\text{−}D$, and create two new edges $A\text{−}D$ and $B\text{−}C$.

Figure 2

Schematic diagram of heat flux distribution and node temperature distribution in a random network. The solid red circle represents the node, and its color depth represents the temper- ature. The node surrounded by the red (blue) circle is connected to the thermostat with the high temperature $T_h$ (low temperature $T_l$). The arrows represent the direction of heat flux at steady state, and their darker color means a larger heat flux. Here the size of the network $N=200$ and the average degree $\langle k\rangle =6$.

Figure 3

Effect of parameter $\sigma$ on the total heat flux $J$ of the network. (a) and (b) illustrate the variation of total network heat flux $J$ with parameter $\sigma$, where $T_h=0.9$ and $T_l=0.1$, at $\beta$ values of 0.0 and 1.0, respectively. (c) and (d) illustrate the variation of total network heat flux $J$ with parameter $\sigma$, where $T_h=0.09$ and $T_l=0.01$, at $\beta$ values of 0.0 and 1.0, respectively. Black squares, red circles and blue triangles represent the cases of clustering coefficient $c=0.1$, 0.4, and 0.7, respectively.

Figure 4

Effect of clustering coefficient $c$ on the total heat flux $J$ of the network. (a) and (b) illustrate the variation of the total network heat flux $J$ with the clustering coefficient $c$, where $T_h=0.9$ and $T_l=0.1$, at $\beta$ values of 0.0 and 1.0, respectively. (c) and (d) illustrate the variation of the total network heat flux $J$ with the clustering coefficient $c$, where $T_h=0.09$ and $T_l=0.01$, at $\beta$ values of 0.0 and 1.0, respectively. Black squares, red circles, and blue triangles represent the cases of $\sigma =0.0$, 0.5, and 1.0, respectively.

Figure 5

Effect on the total heat flux $J$ of the network after changing parameter $\sigma$ for different average degrees. (a)–(c) illustrate the variation of the total network heat flux $J$ with parameter $\sigma$ for $c=0.1, c=0.4$, and $c=0.7$ under different average degrees, respectively. Black squares and red circles represent the cases of $\langle k\rangle =6$ and 10, respectively.

Figure 6

Effect on the total heat flux $J$ of the network after changing clustering coefficient $c$ for different average degrees. (a)–(c) illustrate the variation of the total network heat flux $J$ with the clustering coefficient $c$ for $\sigma =0.0, \sigma =0.5,$ and $\sigma =1.0$ under different average degrees respectively. Black squares and red circles represent the cases of $\langle k\rangle =6$ and 10, respectively.

Figure 7

Nodes temperature distribution and thermal resistance of each edge. (a)–(d) are the distribution $D\left(T\right)$ of nodes temperature $T$ for different parameters $\sigma =0.0$, 0.3, 0.7, and 1.0 for clustering coefficient $c=0.1$, respectively. (e)–(h) are the values of thermal resistance corresponding to each edge for different parameters $\sigma =0.0$, 0.3, 0.7, and 1.0 for clustering coefficient $c=0.1$, respectively.

Figure 8

Heat flux and nodes temperature distribution in network. (a) and (b) are heat flux distribution with average degree $\langle k\rangle =6$ for $\sigma =0$ and 1, respectively. The difference in node color represents the difference in temperature. The nodes surrounded by red (blue) circles are connected to the thermostat with the high temperature $T_h$ (low temperature $T_l$). The arrows represent the direction of heat flux at steady state, and their darker color means a larger heat flux.

Figure 9

Nodes temperature distribution and thermal resistance of each edge. (a)–(d) are the distribution $D\left(T\right)$ of nodes temperature $T$ for different clustering coefficients $c=0.1$, 0.3, 0.5, and 0.7 for parameter $\sigma =1.0$. (e)–(h) are the values of thermal resistance corresponding to each edge for different clustering coefficients $c=0.1, c=0.3, c=0.5,$ and $c=0.7$ for parameter $\sigma =1.0$, respectively.

Figure 10

Heat flux and nodes temperature distribution in network. (a) and (b) are heat flux distribution with average degree $\langle k\rangle =6$ and $\sigma =1$ for clustering coefficient $c=0.1$ and 0.7, respectively. In these visualizations, variations in node color indicate differences in temperature. Nodes encircled by red (blue) circles are linked to the thermostat with high temperature $T_h$ (low temperature $T_l$). The arrows represent the direction of heat flux at steady state, and their darker color means a larger heat flux.

Figure 11

The influence of parameter $\sigma$ and $c$ on the pseudo-dispersion relation. (a) The eigenfrequency $\omega$ versus its eigenmode index $\lambda$ of random networks for $\sigma =0.0$, 0.2, 0.4, 0.6, 0.8, and 1.0, respectively. (b) The eigenfrequency $\omega$ versus its eigenmode index $\lambda$ of networks for $c=0.1$, 0.3, 0.5, and 0.7, respectively.

Figure 12

Phonon density of states for different parameters $\sigma$ of size $N=200$. (a)–(f) are the DOS of a network for $\sigma =0.0$, 0.2, 0.4, 0.6, 0.8, and 1.0, respectively.

Figure 13

Phonon density of states for different clustering coefficients $c$ of size $N=200$. (a)–(d) are the DOS of a network for $c=0.1$, 0.3, 0.5, and 0.7, respectively.

Figure 14

Eigenmode participation ratio of generalized networks vs. the eigenfrequency, $\omega$, for different $\sigma$: (a) $\sigma$ = 0.0, (b) $\sigma$ = 0.2, (c) $\sigma$ = 0.4, (d) $\sigma$ =0.6, (e) $\sigma$ = 0.8, and (f) $\sigma$ = 1.0. Simulation data for spring networks with $n=200$ and $M=1$.

Figure 15

Eigenmode participation ratio of generalized networks vs. the eigenfrequency, $\omega$, for different $c$: (a) $c$ = 0.1, (b) $c$ = 0.3, (c) $c$ = 0.5, and (d) $c$ =0.7. Simulation data for spring networks with $n=200$ and $M=1$.