Heat conduction in a chain of dissociating particles: Effect of dimensionality

The paper considers heat conduction in a model chain of composite particles with hard core and elastic external shell. Such model mimics three main features of realistic interatomic potentials—hard repulsive core, quasilinear behavior in a ground state, and possibility of dissociation. It has become clear recently that this latter feature has crucial effect on convergence of the heat conduction coefficient in thermodynamic limit. We demonstrate that in one-dimensional chain of elastic particles with hard core the heat conduction coefficient also converges, as one could expect. Then we explore effect of dimensionality on the heat transport in this model. For this sake, longitudinal and transversal motions of the particles are allowed in a long narrow channel. With varying width of the channel, we observe sharp transition from “one-dimensional” to “two-dimensional” behavior. Namely, the heat conduction coefficient drops by about order of magnitude for relatively small widening of the channel. This transition is not unique for the considered system. Similar phenomenon of transition to quasi-1D behavior with growth of aspect ratio of the channel is observed also in a gas of densely packed hard (billiard) particles, both for two- and three-dimensional cases. It is the case despite the fact that the character of transition in these two systems is not similar, due to different convergence properties of the heat conductivity. In the billiard model, the divergence pattern of the heat conduction coefficient smoothly changes from logarithmic to power-like law with increase of the length.

Figure 1

Distribution of (a) the local heat flux $J_n$ and (b) of temperature $T\left(x\right)$ along the chain of length $L=340$. The density of the disks packing is $p=1$ (average distance between disk centers is $a=1/p=1$), the temperature of the thermostats $T_{+}=0.000105,T_{-}=0.000095$. The red horizontal dashed curve in part (a) represents the value of the heat flux $J$. The red dashed line in part (b) represents the linear temperature gradient.

Figure 2

Dependence of the heat conduction coefficient $\kappa$ on the length of chain $L$ for (a) temperatures $T=0.0001$, 0.001, 0.01 (curves 1, 2, 3) with packing density $p=1$, and (b) for packing densities $p=1$, 10/11, 5/6 (curves 4, 5, 6) with temperature $T=0.001$. Black straight lines are calculated using Green-Kubo relation (10).

Figure 3

Exponential decay of the autocorrelation function $c\left(\tau \right)$ for a chain of elastic disks with packing density $p=10/11$ and temperature $T=0.001$.

Figure 4

Dependence of the heat conduction coefficient $\kappa$ on the modeling temperature of the chain $T$ (the packing density is $p=1$). The dashed horizontal line represents the heat conductivity at the limit $T\to 0$.

Figure 5

Dependence of the heat conduction coefficient $\kappa$ on the packing density of the chain, $p$ (the modeling temperature is $T=0.001$).

Figure 6

Model of quasi-1D chain with length $L_x=30$, width $L_y=1.6$, and with left end (red disks) attached to $T=T_{+}$ thermostat and right end (blue disks) attached to $T=T_{-}$ thermostat (density $p=1$).

Figure 7

Exponential decay of the autocorrelation function $c\left(\tau \right)$ for a system of elastic disks. The packing density is $p=1$ and the modeling temperature $T=0.001$, width of the channel $L_y$=1 and 1.5 (curves 1 and 2).

Figure 8

Dependence of the heat conduction coefficient $\kappa$ on the length of the quasi-1D channel $L_x$ for temperature $T=0.001$, packing density $p=1$, and width of the channel $L_y=1$ and 1.5 (curves 1 and 2). Black straight lines are calculated using Green-Kubo relation (10).

Figure 9

Dependence of the normalized heat conduction coefficient $\kappa /{\kappa }_m$ on the width of the quasi-1D channel $L_y$ for temperature $T=0.0001$ (${\kappa }_m=47$), $T=0.001$ (${\kappa }_m=156$) and $T=0.01$ (${\kappa }_m=3728$)–curves 1, 2 and 3.

Figure 10

Local distribution of temperature $T_n$ in the chain of hard disks in a 2D channel with length $L_x=41$ and width $L_y=1.5$ (packing density $p=1$, number of disks $N=(L_x-1)p=40$).

Figure 11

Dependence of the heat conduction coefficient $\kappa$ on length of the channel $L_x$ for chain of hard disks (curve 1 and 2) and spheres (curve 3). The width of the channel $L_y=1.5$ (curves 1 and 3) and 1.8 (curves 2). Black straight lines correspond to the logarithmic relations $\kappa =\alpha ln{L}_x$ for $\alpha =0.25$, 0.37, and 0.72.

Figure 12

Power law decay of the autocorrelation function $c\left(t\right)$ for chain of hard disks in 2D channel with width $L_y=1.8$ and temperature $T=0.001$. The straight line corresponds to the power function $t^{-0.44}$.

Figure 13

Dependence of the heat flux $J$ on the width of channel $L_y$ in a rectangular channel of size $L_x\times L_y$, filled with hard disks (number of disks corresponds to the length of the channel) for lengths $L_x=43$, 83, 163 (curves 1, 2, 3). Temperature of boundary walls $T_{+}=0.00105$ and $T_{-}=0.00095$. The straight line 4 corresponds to the limit value of heat flux when $L_y\to \infty$.

Figure 14

Repulsive interaction potential $V\left(r\right)$, Eq. (4) (curve 1, parameter $d=0.8$), and smoothened potential $V_h\left(r\right)$, Eq. (A2), for $h=0.00001$, 0.0001, 0.001, 0.01 (curves 2, 3, 4,5).

Figure 15

Interaction potential with the walls $U\left(u\right)$, Eq. (12) (curve 1) and smoothened potential $U_h\left(u\right)$ for the length between walls $L=4$ and smoothening parameter $h=0.00001$, 0.0001, 0.001, 0.01 (curves 2, 3, 4, 5).