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: $\tilde{\zeta }$
, the ratio of the zeta potential ζ to the thermal voltage φT = kBT/ze; $\tilde{\lambda }$
, the ratio of the Debye length λ to the characteristic particle size l; and $\tilde{E}$
, the ratio of the characteristic potential drop l‖E0‖, associated with the applied field E0, to φT. (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 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 $\alpha = \epsilon \varphi _T^2/ \eta D$
appearing in the above is a common dimensionless group in electrokinetic analyses;7 since it is an inherently O(1) parameter8 the ratio β/l is of order $\tilde{\lambda } \tilde{\zeta }^2$
. In view of (1), it is exceedingly small. Thus, for all practical purposes, Eq. (8a) in Ref. 1 constitutes a homogenous Neumann condition; Morrison's argument9 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 $O(\tilde{\lambda } \tilde{\zeta }^2\tilde{E})$
electric-field corrections, implying an $O(\tilde{\lambda } \tilde{\zeta }^3\tilde{E})$
correction to Smoluchowski's velocity. The answer is no: the concentration-polarization effect, neglected in that paper, actually gives rise to an asymptotically larger mobility correction. To see why, consider the criterion for small concentration polarization provided by Orsini and Tricoli [after their Eq. (8b)]; in the above dimensionless notation, it reads $(2+1/\alpha ) \tilde{\lambda } |\tilde{\zeta }| \tilde{E}\ll 1$
. Since α is O(1), this simply implies
When valid, the salt perturbation is of $O(\tilde{\lambda } \tilde{\zeta } \tilde{E})$
relative magnitude compared with the uniform bulk concentration. That perturbation, in turn, gives rise to diffuso-osmotic slip, linear in its gradient.10 Since the proportionality coefficient in diffuso-osmosis scales as $\tilde{\zeta }^2$
for small $\tilde{\zeta }$
, that implies an $O(\tilde{\lambda } \tilde{\zeta }^3\tilde{E})$
correction to Smoluchowski's velocity, comparable to that animated by the electric-field perturbation. Moreover, because of the inherent coupling between salt and charge transport,5 the induced $O(\tilde{\lambda } \tilde{\zeta } \tilde{E})$
salt perturbation necessitates an electric-field perturbation of the same order. This will give rise, now through electro-osmosis, to an $O(\tilde{\lambda } \tilde{\zeta }^2\tilde{E})$
mobility correction (see Ref. 11), asymptotically 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 $|\tilde{\zeta }|$
, resulting13 in an effective surface conductivity scaling as $\tilde{\lambda } e^{|\tilde{\zeta }|/2}$
. Surface conduction, while concentrated in an $O(\tilde{\lambda })$
narrow layer, thus affects the leading-order bulk transport when $\tilde{\zeta }$
is logarithmically large.3–5 For O(1) values of $\tilde{\zeta }$
(let alone small ones, as in Ref. 1) surface conduction effects are $O(\tilde{\lambda })$
small and cannot be discerned from other corrections inevitable in the thin-layer approximation.
A final observation is warranted. In view of (1), condition (3) may be satisfied even for $\tilde{E}\gg 1$
. One may then claim that, at the very least, the analysis of Ref. 1 shows that Smoluchowski's formula may hold under strong fields if the zeta potential is sufficiently small. The persistence of Smoluchowski's formula under strong fields, however, was already demonstrated under more general conditions, where the zeta potential is not restricted to be small.14 For such moderate zeta potentials, Smoluchowski's formula actually remains a leading-order approximation up to very strong fields, scaling as $1/\tilde{\lambda }$
, where it eventually breaks down because of dielectric solid polarization.15 At these conditions, surface conduction is still subdominant.
