Molecular theory of translational diffusion: Microscopic generalization of the normal velocity boundary condition

Abstract

A simple molecular theory is presented for the diffusion constant D for a test hard sphere translating in a hard sphere solvent. It is argued that there is a breakdown of the applicability of hydrodynamics in the neighborhood of the test particle due to collisional effects. It is shown that, as a consequence, the traditional hydrodynamic boundary condition (BC) on the particle–solvent normal relative velocity is incorrect for molecular motion. An approximate replacement for this BC is constructed from collisional considerations. With this new BC and the usual hydrodynamic equations, D is found to have two additive contributions. The first is the microscopic, collisional Enskog diffusion constant; the second is of the hydrodynamic Stokes–Einstein form. It is shown how the standard hydrodynamic Stokes–Einstein relation for D can hold numerically to a good approximation despite the dominance of (or significant contribution to) the motion by microscopic collisional contributions. Observed trends of D with size and mass ratios which contradict the analytic Stokes–Einstein relation are reproduced. The predicted D values are compared with available results of renormalized kinetic theory and Boltzmann-level kinetic theory. High density deficiencies of the new BC are discussed.

Kinetic equation and transport properties of large, heavy ions in a gas in any regime in external fields

“…In fact, the general result (27): (i) has a form which recalls the result obtained by Hynes et al [15] (by a quite different method and in the special case of the hard-sphere interaction model) for particles moving in a gas under ''slip'' boundary conditions, and (ii) explicitly expresses the simple dependence of D both on k R and on k S . It must be noted, however, that, while for D (and for the l.h.-ion mobility K) the contributions due to the Stokes' force and to the single collisions are only additive (and therefore separable), for the mean energy hi 1 this does not happen.…”

A mode coupling theory analysis of microscopic friction in the macroscopic limit

“…First we analytically examine the contribution of each mode (binary, density and current) to the total friction/diffusion in the limit of large solute size. This analysis shows that the final expression for the zero frequency friction reduces to a form similar to that given by Hynes et al [18] and Mehaffey and Cukier [17]. The contribution from the density mode becomes zero as the microscopic structure of the solvent becomes unimportant to a large solute.…”

“…It was analytically shown by Gaskell and coworkers [13], that once the current mode makes the dominant contribution to diffusion the SE relation with the slip boundary condition can be recovered. Other theories, such as renormalized kinetic theory (ring theory) [14][15][16], repeated ring theory [17] and a semi-phenomenological theory by Hynes et al [18] have also demonstrated this cross-over to the hydrodynamics for large solutes.…”

Translational and rotational friction on a colloidal rod near a wall

“…For a heavy sphere with diffusing boundaries (which randomly scatter colliding particles according to the Maxwell-Boltzmann law) the translational Enskog friction is given by ξ e = 2 3 √ 2πmk B T γD 2 (1 + 2χ)/(1 + χ), where χ = 2/5 is the gyration ratio. It has been confirmed [25,26,[31][32][33] that the two sources of friction act in parallel, i.e. that the total friction ξ is given by…”