13.2 Electrophoretic velocity dependence on particle size [particle electrophoresis top]

In the previous section, the analysis was simple because the flow was unidirectional (and thus _ext was uniform) or the electrical double layers were thin (and thus _ext was uniform within the double layer). For the particles we manipulate in microfluidic devices (usually spheres or particles of roughly spherical shape), the flow around the particle is not unidirectional. Further, we routinely work with particles with small radius, as low as 5 nm.

In the general case, in which the particle radius cannot be assumed large as compared to the electrical double layer thickness, the assumption (first implemented in Chapter 6) that _ext is uniform throughout the electrical double layer cannot be employed. This leads to a considerable increase in the complexity of the analysis. As discussed above, electrophoretic velocity in the large particle, thin DL limit (Figure 13.2) is analogous to the result for electroosmosis:

However, to describe particles whose radius is not necessarily much greater than λ_D (Figure 13.3), we rewrite Equation 13.10 with a correction factor:

where f is a multiplicative factorof order 1. This factor accounts for the variation of the local electric field throughout the electrical double layer. The sections below discuss these results in detail.

13.2.1 Smoluchowski velocity:large particles, small zeta

Recall that, for electroosmosis, the effect of the double layer can be modeled as a slip velocity as long as the local radius of curvature of the surface is large as compared to the double layer thickness, in which case the surface can be treated as locally flat. Similar arguments can be used to model the flow around a particle. If we consider a motionless particle in an infinite fluid, electroosmosis generates fluid flow relative to the particle with the electroosmotic velocity. As discussed above, making a coordinate transform to make the fluid motionless and and the particle mobile, we can show that the velocity of the particle is given by

(13.12)

This is the Smoluchowski equation. It requires two assumptions, namely that (1) the surface radius of curvature is much larger than theDebye length, and (2) that the double layer potential φ₀ is small, i.e., that ion distributions (and therefore the local conductivity and therefore the local electric field solution) are only slightly perturbed. In the next section, we discuss what happens when these assumptions are relaxed.

13.2.2 Henry’s function: effect of finite double layers for small φ₀

Smoluchowski’s equationassumes that the double layer is thin, and the electric field seen by the ions in the double layer is precisely that at the surface of the particle. When this is not the case, the electrophoretic mobility changes.

Now assume that the double layer has a finite thickness, but assume that φ₀ is small. Since φ₀ is small, we can neglect any changes in conductivity in the double layer—thus, the electric field solution around the particle is unaffected by the double layer. The key difference is that the excess ions in the double layer now experience a variety of local electric fields. This means we cannot treat E_ext as uniform along the normal coordinate of the electrical double layer. The net effect of this is a retardation of the particle velocity since the component of ρ_E in the direction of the applied field is lower as one moves away from the particle surface along the normal coordinate. We write the result as

where f₀ is Henry’sfunction. For a spherical particle of radius a with a^* defined as a∕λ_D, Henry’s function is

where E_n is the exponential integral of order n. This is reasonably well-approximated by the relation

(13.15)

which is more straightforward to implement.

In the small-particle limit (a^*→ 0), f₀ = 2 3 and the resultingelectrophoretic mobility relation is termed Hückel’s equation:

We can write similar relations for an infinite cylinder aligned perpendicular to the electric field— this is essentially the two-dimensional version of above. The exact equation is rather complicated, but again an approximate relation is available:

(13.17)

13.2.3 Large surface potential—effect of counterion distribution

The previous relations hold for small zeta potentials, in which case it can be assumed that the electrical double layer exhibits one-way coupling—the electric field applies a force to the double layer, but the presence of double layer does not perturb the electric field. For large surface potentials, this is no longer the case, and the f₀ from Henry’s equation(Equation 13.14) no longer accurately describes particle electrophoresis. In particular, if the double layer thickness λ_D is on the same order as the particle radius, the motion of ions through the double layer leads to a distorted double layer that counteracts much of the applied electric field. A schematic of double layer asymmetry caused by relaxation is shown in Figure 13.4.

The effect of this asymmetric counterion cloud is to make the correction factor f lower in the region near a^* = 1, as shown in Figure 13.5. Chapter 15 will recapitulate the conclusion that most electrokinetic phenomena are most complicated when the double layer thickness λ_D is on the order of the particle or channel size.

In general, a value forHenry’s function for large φ₀ can only be solved numerically. Sample results from these numerical calculations are shown in Figure 13.6.

For those wanting to avoid solutions of differential equations, an analytical approximation exists for symmetric electrolytes of valence z and a^*≫ 10, derived in [135]:

(13.24)

ζ_EP^co and ζ_EP^ctr are nondimensional ion mobilities of the coions and counterions respectively:

(13.25)

and t, F , G, H, and K are abbreviations to simplify Equation 13.23:

(13.26)

(13.27)

(13.28)

(13.29)

(13.30)

Using this analytical approximation, the electrophoretic mobility as a function of zeta potential is shown in Figure 13.7 as a function of a^*.

Similarly, the correction factor as a function of zeta potential is shown in Figure 13.8 as a function of a^*.

Finally, the correction factor as a function of a^* is shown in Figure 13.9 as a function of a^*.

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