多约束纳米结构的声子热导率模型研究

纳米技术的快速发展使得对微纳尺度导热机理的深入研究变得至关重要. 理论和实验都表明, 在纳米尺度下声子热导率将表现出尺寸效应. 基于声子玻尔兹曼方程和修正声子平均自由程的方法得到了多约束纳米结构的声子热导率模型, 可以描述多个几何约束共同作用下热导率的尺寸效应. 不同几何约束对声子输运的限制作用可以分开计算, 总体影响则通过马西森定则进行耦合. 对于热流方向的约束, 采用扩散近似的方法求解声子玻尔兹曼方程; 对于侧面边界约束, 采用修正平均自由程的方法计算边界散射对热导率的影响. 得到的模型能够预测纳米薄膜(法向和面向)及有限长度方形纳米线的热导率随相应特征尺寸的变化. 与蒙特卡罗模拟及硅纳米结构热导率实验值的对比验证了模型的正确性.

A model for phonon thermal conductivity of multi-constrained nanostructures

The rapid development of nanotechnology makes it possible to further understand nanoscale heat conduction. Theoretical analysis and experimental measurement have demonstrated the size-dependence of thermal conductivity on a nanoscale. As dielectric material (such as silicon), phonons are the predominant carriers of heat transport. Phonon ballistic transport and boundary scattering lead to the significant reduction of thermal conductivity. Various models, in which only one geometrical constraint of phonon transport is considered, have been proposed. In engineering situations the phonon transport can be influenced by multiple geometrical constraints, especially for material with long intrinsic phonon mean free path. However, at present a phonon thermal conductivity model in which the multiple geometrical constraints of phonon transport are taken into account, is still lacking. In the present paper, a multi-constrained phonon thermal conductivity model is obtained by using the phonon Boltzmann transport equation and modifying the phonon mean free path. The geometrical constraints are dealt with separately, and the effects of these constraints on thermal conductivity are then combined by the Matthiessen's rules. Different boundary conditions can lead to different influences on the phonon transport, so different methods should be used for different boundary constraints. The differential approximation method is utilized for the constraint in the direction of heat flux, while phonon scatterings on side surfaces are characterized by modifying the phonon mean free path. The model which characterizes various nanostructures including nanofilms(in-plane and cross-plane) and finite length rectangular nanowires, can well agree with the Monte Carlo simulations of different Knudsen numbers. The model with the Knudsen number Kn_x equal to 0 can well predict the experimental data for the in-plane thermal conductivity of nanofilm. When the Knudsen numbers Kn_y and Kn_z vanish, the model corresponds to the cross-plane thermal conductivity of nanofilm. Moreover, with Kn_x=0 and Kn_y=Kn_z, the model corresponds to the square nanowires of infinite length, and the similar slopes between the model and the experimental data of nanowires can be achieved.