1.Beijing Key Laboratory of Heat Transfer and Energy Conversion, MOE Key Laboratory of Enhanced Heat Transfer and Energy Conservation, College of Energy and Power Engineering, Beijing University of Technology, Beijing 100124, China 2.Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Hong Kong, China
Fund Project:Project supported by the National Natural Science Foundation of China (Grant No. 51776007), the Beijing Nova Program of Science and Technology, China (Grant No. Z191100001119033), and the Young Talent Project of Beijing Municipal Education Committee, China (Grant No. CIT&TCD201904015)
Received Date:30 September 2020
Accepted Date:05 November 2020
Available Online:25 February 2021
Published Online:05 March 2021
Abstract:Thermophoresis refers to the motion of small particles suspending in a fluid with non-uniform temperature distribution due to the temperature gradient around the particle. Usually, the fluid molecules coming from the hot side carry more kinetic energy than those from the cold side, which results in a net thermophoretic force in the direction opposite to the temperature gradient. Since it was discovered more than 100 years ago, thermophoresis has been of major importance in a variety of applications, where it can play either beneficial role or adverse role, including material synthesis, micro- and nano-fabrication, and environmental science. Therefore, it is necessary to accurately evaluate the thermophoretic force. In the present work, the thermophoretic force on nanoparticles is examined in the free molecule regime by using non-equilibrium molecule dynamics (MD) simulation. It has been widely accepted that the thermophoretic force conforms with the Waldmann equation for large Knudsen numbers. However, due to the effect of the nonrigid-body interactions between the particle and gas molecules, the thermophoretic force on nanoparticles might deviate greatly from the classical theory. In our MD model, a single nanoparticle with a diameter of several nanometers suspends in a diluted gas. The Lennard-Jones (L-J) potential is employed to simulate the intermolecular interactions. To avoid deforming the nanoparticle, the solid molecules within the nanoparticles are linked to their nearest neighbors through a finite extensible nonlinear elastic bonding potential. The thermophoretic force on a nanoparticle is calculated by imposing a harmonic potential on the nanoparticle, which eliminates the effect of the Brownian motion of the nanoparticle on the thermophoresis. The effective thermal conductivity of the ambient gas is employed in Waldmann equation for the thermophoretic force due to the finite volume effect. It is found that the Waldmann theory for thermophoresis is still valid for nanoparticles in the case of weak gas-particle interaction or high gas temperature. With the increase of the gas-particle interaction strength or the decrease of the gas temperature, the Waldmann theory is invalid due to the effect of gas-particle nonrigid-body collisions and the adsorption of gas molecules on the particle surface. By considering the gas-particle nonrigid-body interaction and the modified particle size, the theoretical results for thermophoretic force accord with the MD simulations quite well. Keywords:nanoparticle/ thermophoretic force/ molecular dynamics/ free molecule regime
表1分子动力学模拟系统的几何特征参数 Table1.Geometric and characteristic parameters of the simulation systems.
分别采用Phillips (11)式和Waldmann (1)式进行热泳力理论计算, 并与分子动力学模拟计算结果进行对比分析. Phillips公式的计算结果表示为FT,Phillips, 分子动力学模拟结果为FT,MD. Waldmann公式将分别使用气体的宏观热导率κ和有效热导率κ'计算, 计算结果分别用FT,B和FT,E表示. 图3(a)与图3(b)分别表示在kij = 5.26 (强气-固相互作用下)和kij = 1.0 (弱气-固相互作用下)情况下FT/FT,B随l的变化图. 由图3可以看出, 模拟系统中“壁面”间距是影响颗粒所受热泳力的重要因素之一, 这说明有限空间效应确实存在. FT,E与FT,Phillips吻合较好, 表明Phillips对Waldmann热泳力计算式的修正是可以通过对气体热导率的修正来实现的. 图 3FT, MD/FT, B, FT, E/FT, B 和FT, Phillips/FT, B随模拟空间尺寸的变化 (a) kij = 5.26; (b) kij =1 Figure3. Influence of the size of simulation box on FT, MD/FT, B, FT, E/FT, B and FT, Phillips/FT, B: (a) kij =5.26, (b) kij =1.
对于强气-固相互作用(图3(a)), 分子动力学结果FT,MD与理论结果FT,E和FT,Phillips存在较大误差, 这种偏差是由于气体分子与纳米颗粒之间的非刚体碰撞所引起的. 对于弱气-固相互作用(图3(b)), 气体分子与纳米颗粒之间的非刚体碰撞效应较弱, 此时, 基于刚体碰撞模型假设的Waldmann理论仍然适用于纳米颗粒, FT,E和FT,Phillips都与分子动力学结果FT,MD吻合较好. 图4为热泳力随气体有效热导率κ' 的变化曲线. 与前文结果类似, 对于弱气固耦合的情况(kij = 1.0), FT,E和与分子动力学结果FT,MD吻合较好; 对于强气固耦合(kij = 5.26), 分子动力学模拟得到的热泳力则明显高于基于有效热导率的理论计算结果. 图 4 热泳力FT,MD和FT,E随气体有效热导率κ' 的变化 Figure4. Influence of the effective thermal conductivity of the media gas on the thermophoretic forces.
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4.1.系统温度(气-固结合强度)影响
由前文所述可知, 将气体的有效热导率引入到Waldmann公式中, 可得到有限空间中气体的Waldmann自由分子极限. 但是, 对于纳米颗粒来说, 特别是当气-固相互作用较强时, Waldmann热泳力计算式的误差较大. 基于分子动力学模拟结果, 热泳力随系统平均温度$T_{0}^*$, 气-固结合强度kij和温度梯度$ \nabla T $的变化情况如图5所示(图5(a)中的插图为在相同系统参数下本文分子动力学结果与文献[28]中结果的对比). 由图5可以看出, 对于纳米颗粒来说, Waldmann热泳力计算式的适用性受系统平均温度$ T_{0}^* $与气-固结合强度kij的影响, 当系统温度较低或气-固结合强度较大时, Waldmann公式给出的热泳力计算结果误差较大; 随着系统温度的升高或气-固结合强度的减小, 纳米颗粒所受热泳力的分子动力学计算结果逐渐收敛于Wald-mann公式的预测值. 这是由气体分子与纳米颗粒间非刚体碰撞效应的强弱受温度或气-固结合强度影响所导致的. 当结合强度较大或温度较低时, 势能在气体分子与纳米颗粒进行动量交换的运动轨迹中占主导地位; 而当结合强度较小或温度较高时, 气-固相互作用势能的影响十分微弱, 刚体碰撞模型近似成立. 图 5 不同参数下热泳力的分子动力学结果FT,MD与Waldmann公式结果FT,E比较图 (a) 环境温度T*; (b) 气-固结合强度kij; (c) 温度梯度$ \nabla T^* $ Figure5. The variation of thermophoretic force FT between present MD simulation result FT, MD and Waldmann equation result FT, E under different parameters: (a) The environ-ment temperature $ T_{0}^* $; (b) the intensity of gas-particle interaction kij; (c) temperature gradient $ \nabla T^* $.