Understanding the mobility of nonspherical particles in the free molecular regime

Abstract

An approach to obtain the mobility of nonspherical particles is proposed by averaging the drag force orientationally, and two other widely used approaches in the literature, the averaged-collision-integral and averaged-drift-velocity methods, are summarized and extended. The concept of orientationally averaged collision integrals based on Chapman-Enskog theory for small gas-phase ions is re-examined for macromolecular ions whose surface cannot be treated as specular, but with inelastic interactions. A well accepted collision model considering inelastic collisions is Epstein's theory, which has been extended to include long-range potential forces by Li and Wang [Phys. Rev. E 68, 061206 (2003)] for spherical particles. This work extends Li and Wang's spherical particle theory to convex nonspherical particles considering long-range potential, and simplifies this collision integral to a product of the averaged projection area and an enhancement factor for short-range interactions (hard collisions), which is independent of convex particle shape and is identical to the value for a sphere that people are using. We also show that the averaged projection area of a convex particle in free molecular regime for hard collisions is equal to its mobility diameter. The second approach is the averaged-drift-velocity approach using the friction coefficient in a tensor form, which is often employed in aerosol science. We extend this approach in our previous work for axisymmetric particles to develop an expression for the mobility of nonspherical particles in a general form. Furthermore, it is pointed out that this approach is only valid when the particle Brownian rotation is slow compared with the particle translational relaxation time. If the particle Brownian rotation is fast, usually so in the case of very small ions and particles, we propose an "averaged-drag-force" approach. The three approaches are then compared for a randomly oriented rod and the protein GroEL. We show that for a cylinder rod in the free molecular regime at random orientation, the averaged-drag-force approach is identical to the averaged-collision-integral approach for short-range interactions (hard collisions). We then summarize the relationship between collision-integral based approach and tensor based approaches. For readers only interested in implementation of the theory, we provide useful expressions in Tables I and II.

Drag force and transport property of a small cylinder in free molecule flow: A gas-kinetic theory analysis

“…Li et al [30] for slow Brownian rotation [12,30,55] as a function of the CNT length (symbols). The error bars for 2R = 15 and 22 nm data are taken from the reference sources directly, while those for 2R = 10 nm are estimated to be 10%, based on a TEM resolution of 0.5 nm for CNT diameter determination.…”

“…The above equation can be written equivalently following Li et al [30,46] as (45) where c x , c y and c z are the drag-coefficient components in along the x, y and z-axes,…”

“…Several studies [12,14,16,20,30,[54][55][56][57] have reported data of CNT mobility measured by a DMA, among them the relationship between the mobility diameter and CNT length is provided for a given tube diameter. For some of the CNTs, the length and diameter were determined directly by transmission electronic microscopy (TEM).…”

“…FIG 4 shows the length-versus-mobility diameter data of CNTs, all of which are 15 nm in diameter and taken from several sources [12,30,54,55]. In this set of data, the CNT diameter was determined by TEM, while the length ( ) was estimated from the TEM projected area ( A p ) as .…”

“…They showed that the mobility is a strong function of the wire orientation, which in turn, are dependent on the electric field strength and wire aspect ratio. Li et al [30] considered the effect of cylinder rotation and obtained an expression for the orientation averaged drag force from Dahneke's expressions. TABLE 1 presents several drag force formulations relevant to the current discussion.…”