Rough boundary effect in thermal transport: A Lorentz gas model

A Lorentz gas model with rough periodic boundary conditions is proposed to investigate the boundary effect on thermal transport of noninteracting particles. The normal thermal transport is confirmed as the linear temperature gradient, and length independent thermal conductivity is obtained. We demonstrate that the thermal conductivity indeed decreases as the degree of roughness increases, which is consistent with previous works. Furthermore, we find that the thermal conductivity is oscillating with the degree of the boundary roughness. This interesting phenomenon is related to the successful transport probability of particles, which results from the interference between transporting particles and the periodic boundary conditions.

Figure 1

Schematic setup of the models and their temperature distributions. (a) The ballistic transport model. The outline of this model is a rectangle with a length of $L$ (200 in this model) and a width of $D$ (200 in this model). The left and right ends are isothermal heat reservoirs in which temperatures are $T_1=1.1$ and $T_2=1.0$, respectively. (b) The temperature distribution along the transmission direction in the ballistic transport model. There is no gradient of temperature elsewhere except a temperature jump near the heat source. (c) Sinusoidal rough boundary model. We employ two sinusoidal curves with an amplitude of $A$ and a period of ${\lambda }_{\sin }$ to replace the parallel border. (d) The temperature distribution in the sinusoidal rough boundary model. ${\lambda }_{\sin }=2$ and $A$ are equal to 0.3 (dash line), 0.5 (dot line), 0.7 (dash dot line), and 0.9 (solid line), respectively. Linear gradients in temperature are discovered. The larger the amplitude, the more obvious the linear gradient.

Figure 2

(a) The flow $I$ is proportional to $L{}^{-1}$; the red line is a reference line of which the slope is $-1$. (b) The thermal conductivity keeps constant as the length of the sinusoidal rough boundary model changes. We keep the value of $D$, ${\lambda }_{\text{sin}}$ (equal to 10) and A (equal to one-thousandth of $D$) constant in each line, calculating the thermal conductivity of different $L$. The black squares are data for $D=100$, the red rounds are for $D=150$, and the blue triangles are for $D=200$.

Figure 3

(a) The thermal conductivity $\kappa$ does not change when the roughness $u$ is constant although the amplitude $A$ and period of boundary ${\lambda }_{\text{sin}}$ changes. We keep the value of $u$ constant and calculate the thermal conductivity of different $A({\lambda }_{\text{sin}}$ changes simultaneously within the range of ${\lambda }_{\text{sin}}\in [2,10]$). The black squares represent $u=0.0625$, the red rounds represent $u=0.09375$, and the blue triangles represent $u=0.125$. (b) We keep ${\lambda }_{\text{sin}}$ as 4 and change $A$. The conductivity decreases as the value of $u$ gets larger.

Figure 4

Both the successful transport probability (red open circles, left $y$ axis) and the thermal conductivity $\kappa$ (black solid circles, right $y$ axis) vibrate with the degree of roughness $u$. The two curves are highly consistent in frequency (both have the same oscillation period corresponding to a characteristic frequency $\nu =1.75$), which confirms the relationship between the successful transport probability of the particle and the roughness of the sinusoidal boundary, which leads to the oscillation of the $\kappa \text{−}u$ curve.

Figure 5

(a) The comparison between the thermal conductivity with superimposed boundary conditions and the average thermal conductivity with two pure boundary conditions as the function of amplitude $A$. The red open circles represent the thermal conductivities for the superimposed model with ${\lambda }_{{\text{sin}}_1}=2$ and ${\lambda }_{{\text{sin}}_2}=4$ for upper and lower boundaries, respectively. The black solid circles are the average thermal conductivities of the two pure models with ${\lambda }_{\text{sin}}=2$ or 4, respectively. It can be seen that the two curves are almost consistent with each other. This verifies that the complex boundary conditions can be replaced by many simple boundary conditions. (b) The power spectrum of the $\kappa \text{−}A$ curve. The two peaks represent two values of the frequencies of ${\nu }_1=0.88$ (corresponding to ${\lambda }_{\text{sin}}=4$; here the ${\nu }_1=0.88$ of $\kappa -A$ is equal to ${\nu }_1=1.76$ of $\kappa -u$, which is consistent with ${\nu }_1=1.75$ in Fig. 4) and ${\nu }_1=1.76$ (corresponding to ${\lambda }_{\text{sin}}=2$), respectively. This means we can get all periodic compositions of any rough boundary through Fourier analysis of $\kappa \text{−}A$ data. More deeply, from the double relationship between two frequency peaks, we can get other information that the oscillation cycle of the $\kappa \text{−}u$ curve has a linear relationship with the boundary period.

Figure 6

The $\kappa \text{−}A$ curve for random boundary model. We take the adjustment parameter $r_{\text{ratio}}=1$. When the boundary periods are random, the regular concussion of the $\kappa \text{−}A$ (or $u$) curve disappears.