In a recent paper,1 Orsini and Tricoli considered particle electrophoresis assuming thin double layers and small zeta potentials. Extending Levich's analysis of surface conduction about a spherical particle2 to arbitrary geometries, they obtained a boundary condition governing the bulk electric potential, incorporating a surface-Laplacian term representing surface conduction. As in Levich's analysis, the role of concentration polarization is neglected altogether. Using that effective condition, the authors predict non-Smoluchowski mobilities for various particle shapes.
If correct, these results are quite remarkable. In the classical weak-field régime,3,4 surface conduction leads to the breakdown of Smoluchowski's formula only at sufficiently large zeta potentials; in that approximation, moreover, the contribution (through diffuso-osmosis) of concentration polarization is comparable to the contribution (through electro-osmosis) of the modified electric-field distribution.5 Naïvely, then, the paper of Orsini and Tricoli identifies an overlooked small-zeta-potential régime (where the applied field is not necessarily weak) in which surface conduction affects the electrophoretic mobility while concentration polarization is negligible. As we demonstrate here, the surface-conduction effect addressed in Ref. 1 is actually negligible in the very limit considered in that paper. Furthermore, this effect cannot even be interpreted as a small perturbation, as it is dominated by concentration-polarization effects, neglected at the outset.
The dimensional notation used in Ref. 1 tends to obscure these oversights. We accordingly define the following dimensionless parameters:
Consider first Eq. (8a) in Ref. 1. This boundary condition implies that the relative ratio of surface-to-bulk conduction is of order β/l. (This is the “electrophoretic number” defined by Orsini and Tricoli.) The length β is provided two lines after Eq. (8a) as a function of various physicochemical parameters. Using the definition λ2 = εφT/2zec∞ of the Debye length λ and the resulting expression κ = εD/λ2 for the bulk conductivity, the electrophoretic number β/l is simply
namely, the small-zeta-potential limit of the familiar Dukhin number [see Eq. (2.18) in Ref. 6]. The ion-drag coefficient
A question then arises whether condition (8a) in Ref. 1 can at least be used in a perturbative approach, where the small surface-conduction term in that condition yields
When valid, the salt perturbation is of
The importance of surface conduction actually lies in the limit of large zeta potentials, rather than the one considered in Ref. 1. In view of the Boltzmann distribution within the thin diffuse layer, the counterion concentration scaling becomes exponentially large in
A final observation is warranted. In view of (1), condition (3) may be satisfied even for