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 particle^{2} 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:

_{T}=

*k*

_{B}

*T*/

*ze*;

*l*; and

*l*‖

**E**

_{0}‖, associated with the applied field

**E**

_{0}, to φ

_{T}. (We use the notation of Ref. 1 throughout.) The approximations used in Ref. 1 thus, respectively, read

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 λ^{2} = εφ_{T}/2*zec*_{∞} of the Debye length λ and the resulting expression κ = ε*D*/λ^{2} 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

^{7}since it is an inherently

*O*(1) parameter

^{8}the ratio β/

*l*is of order

^{9}then implies Smoluchowski's velocity. The subsequent analysis in Ref. 1 of various geometries with finite values of β/

*l*is a mere mathematical exercise, devoid of any physical significance.

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

*O*(1), this simply implies

When valid, the salt perturbation is of

^{10}Since the proportionality coefficient in diffuso-osmosis scales as

^{5}the induced

*larger*than that related to the (obviously incomplete) surface-conduction mechanism represented in condition (8a) of Ref. 1. (Since the authors of Ref. 1 were apparently unaware that their surface-conduction effect is asymptotically small, it is hardly surprising that they overlooked this inconsistency.

^{12})

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

^{13}in an effective surface conductivity scaling as

^{3–5}For

*O*(1) values of

A final observation is warranted. In view of (1), condition (3) may be satisfied even for

^{14}For such moderate zeta potentials, Smoluchowski's formula actually remains a leading-order approximation up to very strong fields, scaling as

^{15}At these conditions, surface conduction is still subdominant.

## REFERENCES

*l*and the salt concentration

*c*

_{∞}; for typical diffusivities in aqueous solutions (

*D*is simply proportional to ion size, independent of the electrolyte viscosity η.