Interfacial thermal conduction and negative temperature jump in one-dimensional lattices

We study the thermal boundary conduction in one-dimensional harmonic and ${\varphi }^4$ lattices, both of which consist of two segments coupled by a harmonic interaction. For the ballistic interfacial heat transport through the harmonic lattice, we use both theoretical calculation and molecular dynamics simulation to study the heat flux and temperature jump at the interface as to gain insights into the Kapitza resistance at the atomic scale. In the weak coupling regime, the heat current is proportional to the square of the coupling strength for the harmonic model as well as anharmonic models. Interestingly, there exists a negative temperature jump between the interfacial particles in particular parameter regimes. A nonlinear response of the boundary temperature jump to the externally applied temperature difference in the ${\varphi }^4$ lattice is observed. To understand the anomalous result, we then extend our studies to a model in which the interface is represented by a relatively small segment with gradually changing spring constants and find that the negative temperature jump still exists. Finally, we show that the local velocity distribution at the interface is so close to the Gaussian distribution that the existence or absence of a local equilibrium state is unable to be determined by numerics in this way.

Figure 1

Heat flux as a function of interfacial coupling $k_c$ via the LEGF approach as $k_c$ approaches to zero. Symmetry: $k_L=k_R=1,f_L=f_R=0$; asymmetry: $k_L=1,k_R=2,f_L=f_R=0$; symmetry with on-site potential: $k_L=1,k_R=1,f_L=f_R=2$; and asymmetry with on-site potential: $k_L=1,k_R=2,f_L=f_R=2$. For all cases, we set $T_L=2,T_R=1$ and $N=64$.

Figure 2

The heat flux (a) and temperature jump between the $N/2$-th and $(N/2+1)$-th particles (b) as a function of $k_c$ via both the LEGF approach and MD simulations. Here $f_L=f_R=0,T_L=2,T_R=1,{\lambda }_L={\lambda }_R=0$, and $N=64$. In panel (b), the horizontal dotted line is drawn as a reference line for $\mathrm{\Delta }{T}_b=0$.

Figure 3

The contribution of normal modes to local temperature at the interface for (a) $k_c=0.5$, (b) $k_c=1$, (c) $k_c=1.5$, and (d) $k_c=2$, respectively. Here $k_L=k_R=1, f_L=f_R=0, {\lambda }_L={\lambda }_R=0, T_L=2,T_R=1$, and $N=64$.

Figure 4

(a) Temperature profile in the ${\varphi }^4$ lattice for $k_c=0.1, 0.5, 0.9, 1.1, 1.5$, and $1.9$, respectively. (b) $\mathrm{\Delta }{T}_b$ as a function of $\mathrm{\Delta }T$ for $k_c<1$. (c) $\mathrm{\Delta }{T}_b$ as a function of $\mathrm{\Delta }T$ for $k_c>1$. The lines in panels (a), (b), and (c) are drawn to guide the eyes. Here we set $T_L=T_R+\mathrm{\Delta }T, T_R=0.5, {\lambda }_L={\lambda }_R=1, k_L=k_R=1$, and $N=64$.

Figure 5

The temperature profile of the harmonic chain with an interfacial junction, whose spring constants are smoothly varied. The distribution of spring constants for the whole system is given as follows: $k_i=k_L=1$ for $1\le i\le 7N/16$; $k_i=exp[-{(i-N/2)}^2/50]+1$ for $7N/16<i\le 9N/16$; and $k_i=k_R=1$ for $9N/16<i\le (N-1)$, where $i$ is the index of particle number and $N=256$. The first three even moments of velocity are given by $T_i^{\left(2\right)}=m\langle {\dot{x}}_i^2\rangle$ for the second moment, $T_i^{\left(4\right)}=m{\left(\langle {\dot{x_i}}^4\rangle /3\right)}^{1/2}$ for the fourth moment and $T_i^{\left(6\right)}=m{\left(\langle {\dot{x_i}}^6\rangle /15\right)}^{1/3}$ for the sixth moment, respectively. Here $T_L=2,T_R=1$, and $N=256$.

Figure 6

The temperature profile of the ${\varphi }^4$ lattice with an interfacial junction, whose spring constants are smoothly varied. The distribution of the spring constants in the interfacial junction is the same as that for Fig. 5. Here ${\lambda }_L={\lambda }_R=1,T_L=2,T_R=1$, and $N=256$.