A contribution to the amaranthine quarrel between true and average electrical mobility in the free molecular regime

Abstract

Landau and Lipschitz's approach—termed here H&B due to the use of Happel and Brenner's slow rotation approximation—for calculating the average electrical mobility over all orientations of an ion in the free molecular regime is shown in this manuscript to be an invalid assumption for non-globular ions when a fixed electrical field is present. The reason behind the invalidity seems to be the confusion between average “settling” velocity (the calculation intended by H&B) and the average mobility (drag) in the direction of the field. When a missing orientation is taken into account by rotating the drag tensor, the average mobility obtained through Landau's approach coincides with well-known orientationally averaged Kinetic Theory Methods such as those of Mason and McDaniel (M&M). H&B's averaging approach, however, can be related to the true mobility displacement of the ion or, in other words, the displacement occurring in the direction of the velocity. This true mobility displacement only agrees with the average mobility displacement if ion velocity and electrical field have always the same direction, which only happens under special cases. Analytical and numerical calculations of collision cross-sections of linear and planar structures using a momentum transfer kinetic theory approach are chosen here as a means to prove that a single rotation of the drag tensor is sufficient to show agreement between both methods. A projected area approach is also used to prove the inadequacy of the H&B method.