Li Zhigang, Wang Hai
Department of Mechanical Engineering, University of Delaware, Newark, Delaware 19716, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Dec;68(6 Pt 1):061207. doi: 10.1103/PhysRevE.68.061207. Epub 2003 Dec 31.
We propose a generalized treatment of the drag force of a spherical particle due to its motion in a laminar fluid media. The theory is equally applicable to analysis of particle diffusion and electric mobility. The focus of the current analysis is on the motion of spherical particles in low-density gases with Knudsen number Kn>>1. The treatment is based on the gas-kinetic theory analysis of drag force in the specular and diffuse scattering limits obtained in a preceding paper [Z. Li and H. Wang, Phys. Rev. E., 68, 061206 (2003)]. Our analysis considers the influence of van der Waals interactions on the momentum transfer upon collision of a gas molecule with the particle and expresses this influence in terms of an effective, reduced collision integral. This influence is shown to be significant for nanosized particles. In the present paper, the reduced collision integral values are obtained for specular and diffuse scattering, using a Lennard-Jones-type potential energy function suitable for the interactions of a gas molecule with a particle. An empirical formula for the momentum accommodation function, used to determine the effective, reduced collision integral, is obtained from available experimental data. The resulting treatment is shown to be accurate for interpreting the mobility experiments for particles as small as approximately 1 nm in radius. The treatment is subsequently extended to the entire range of the Knudsen number, following a semiempirical, gas-kinetic theory analysis. We demonstrate that the proposed formula predicts very well Millikan's oil-droplet experiments [R. A. Millikan, Philos. Mag. 34, 1 (1917); Phys. Rev. 22, 1 (1923)]. The rigorous theoretical foundation of the proposed formula in the Kn>>1 limit makes the current theory far more general than the semiempirical Stokes-Cunningham formula in terms of the particle size and condition of the fluid and, therefore, more attractive than the Stokes-Cunningham formula.
我们提出了一种对球形粒子在层流流体介质中运动所受曳力的广义处理方法。该理论同样适用于粒子扩散和电迁移率的分析。当前分析的重点是半径为1nm左右的球形粒子在克努森数Kn>>1的低密度气体中的运动。这种处理方法基于在前一篇论文[Z. Li和H. Wang，《物理评论E》，68，061206 (2003)]中得到的镜面反射和漫反射极限情况下曳力的气体动力学理论分析。我们的分析考虑了范德华相互作用对气体分子与粒子碰撞时动量传递的影响，并以有效约化碰撞积分的形式来表示这种影响。结果表明，这种影响对于纳米级粒子是显著的。在本文中，使用适合气体分子与粒子相互作用的伦纳德 - 琼斯型势能函数，得到了镜面反射和漫反射情况下的约化碰撞积分值。从现有实验数据中得到了用于确定有效约化碰撞积分的动量适应函数的经验公式。结果表明，这种处理方法对于解释半径小至约1nm的粒子的迁移率实验是准确的。随后，按照半经验气体动力学理论分析，将这种处理方法扩展到整个克努森数范围。我们证明，所提出的公式能很好地预测密立根油滴实验[R. A. Millikan，《哲学杂志》34，1 (1917)；《物理评论》22，1 (1923)]。所提出公式在Kn>>1极限下的严格理论基础使得当前理论在粒子尺寸和流体条件方面比半经验的斯托克斯 - 坎宁安公式更具普遍性，因此比斯托克斯 - 坎宁安公式更具吸引力。
