Heat conduction in nanoscale materials: A statistical-mechanics derivation of the local heat flux

We derive a coarse-grained model for heat conduction in nanoscale mechanical systems. Starting with an all-atom description, this approach yields a reduced model, in the form of conservation laws of momentum and energy. The model closure is accomplished by introducing a quasilocal thermodynamic equilibrium, followed by a linear response approximation. Of particular interest is the constitutive relation for the heat flux, which is expressed nonlocally in terms of the spatial and temporal variation of the temperature. Nanowires made of copper and silicon are presented as examples.

Figure 1

Partition of the system: The atoms are grouped into different cells. The interface between the neighboring cells ${\Omega }_I$ and ${\Omega }_J$ is highlighted. This partition is generated from a Voronoi tessellation.

Figure 2

Coarse graining the copper nanowire. The system is aligned in the $\langle 100\rangle$ direction with dimension $46.3\times 14.5\times 14.5{\mathrm{nm}}^3$. For the CG procedure, it is divided into blocks, in each of which the total energy (momentum) is chosen as CG variables. The flux $j_{I+1/2}=q_{I,I+1}$ is defined locally at the interface of two blocks and will be computed from (33).

Figure 3

Comparison of the correlation functions of the heat flux: the kernel function ${\kappa }_{I,I}\left(t\right)$ (top) and the kernel function ${\kappa }_{I,I+1}\left(t\right)$ (bottom). The unit for time is $0.081155$ ps and the unit for the kernel function $\kappa$ is $5.9179\times {10}^{-6}{\mathrm{W}}^2$.

Figure 4

Comparison of the correlation function of the heat flux in the nanowire and in the bulk: the kernel function ${\kappa }_{I,I}\left(t\right)$ (top) and the kernel function ${\kappa }_{I,I+1}\left(t\right)$ (bottom). The unit for time is $0.081155$ ps and the unit for the kernel function $\kappa$ is $5.9179\times {10}^{-6}{\mathrm{W}}^2$.

Figure 5

Plot of the correlation functions ${\kappa }_{I,J}\left(t\right)$. The unit for time is $0.081155$ ps and the unit for the kernel function $\kappa$ is $5.9179\times {10}^{-6}{\mathrm{W}}^2$.

Figure 6

Comparison of the correlation functions of the heat flux for two nanowires of different length: the kernel function ${\kappa }_{I,I}\left(t\right)$ (top) and the kernel function ${\kappa }_{I,I+1}\left(t\right)$ (bottom). The time scale is $0.081155$ ps and the unit for the kernel function $\kappa$ is $5.9179\times {10}^{-6}{\mathrm{W}}^2$.

Figure 7

Plot of the kernel functions ${\kappa }_{I,J}\left(t\right)$. The time scale is $0.081155$ ps and the unit for the kernel function $\kappa$ is $5.9179\times {10}^{-6}{\mathrm{W}}^2$.

Figure 8

Correlation functions ${\kappa }_{I,I}\left(t\right)$ (top) and ${\kappa }_{I,I+1}\left(t\right)$ (bottom) with respect to ${\rho }^{\mathrm{eq}}$ (solid lines) and ${\rho }_{\mathrm{loc}}^{\mathrm{eq}}$ (dashed lines). The time scale is $0.081155$ ps and the unit for the kernel function $\kappa$ is $5.9179\times {10}^{-6}{\mathrm{W}}^2$.

Figure 9

Correlation functions ${\kappa }_{I,I}\left(t\right)$ (top) and ${\kappa }_{I,I+1}\left(t\right)$ (bottom) with respect to ${\rho }^{\mathrm{eq}}$ (solid lines) and ${\rho }_{\mathrm{loc}}^{\mathrm{eq}}$ (dashed lines). The time scale is $0.081155$ ps and the unit for the kernel function $\kappa$ is $5.9179\times {10}^{-6}{\mathrm{W}}^2$.