6.5 Electrokinetic pumps [electroosmosis top]

Unlike Poiseuille flow (Chapter 2), for which the mean velocity for a given pressure gradient decreases as the channel dimension decreases, the mean electroosmotic velocity is independent of length scale as long as the thin double layer approximation applies. This means that as length scales are reduced, electric fields become much more effective at generating fluid flow than pressure gradients are. This property is central to the performance of an electroosmotic or electrokinetic pump, in which an electric field is applied along a capillary to generate flow and pressure. This is a simple flow case, but one that shows compellingly how length scale can dictate the relative importance of flow phenomena.

6.5.1 A planar electrokinetic pump

Consider a system in which an electric field E = ΔV∕L is applied across an open rectangular microchannel withdepth 2d, width w, length L, and cross sectional area A = wd. Assume that λ_D ≪ d ≪ w ≪ L, and no pressure gradient is applied. As we have discussed previously, the boundary condition can be modeled with the equation _wall = μ_EO. If the microchannel area is uniform, the electroosmotic velocity in the channel is uniform (this is a degenerate case of a Couette flow, in which both surfaces move with the same velocity). Summarizing the relations for the pressure drop across the capillary (Δp), the velocity distribution across the capillary cross-section, and the total flow rate in this system (Q_tot), we have:

(6.20)

(6.21)

and

(6.22)

Now suppose the downstream port of this capillary is closed, and we attach a pressure transducer to this port. For simplicity (and because L ≫ w) we consider only the region of the capillary that is far from the ends, and we assume all flow is strictly in the x-direction. In this case, the net flow through any cross-section must be zero, because the port is closed. The electroosmotic flow equation still holds, but the assumption that the pressure gradient is zero does not. In fact, an adverse pressure gradient and a reverse Poiseuille flow must coincide with the electroosmotic flow to satisfy mass conservation. For a 2-D system (plates of width w separated by 2d, where w ≫ d), the electroosmotic and pressure-driven contributions to the flow are:

(6.23)

and

(6.24)

Integrating and setting the net flowrate equal to zero, the downstream pressure generated by the flow is given by

In the thin double layer limit, an electric field applied across a closed microchannel generates a downstream pressure proportional to the applied voltage and inversely proportional to the channel depth squared. A similar relation can be derived for a circular capillary of radius r (see Exercise 6.26):

We have thus derived two limiting cases: an open capillary, for which the generated flowrate is maximum, but the pressure generated is zero, and a closed microchannel, for which the generated flowrate is zero, but the pressure generated is maximum. In the general case for which Q_tot is nonzero and Δp is nonzero, the solution is a linear combination of these two solutions, with an example flowfield shown in Figure 6.7. We can write for the planar electrokinetic pump:

(6.35)

and this can be rearranged as:

(6.36)

where Q_max is the maximum flow rate (at Δp = 0), given by

(6.37)

and Δp-max is given by

(6.38)

Equation 6.36 is shown graphically in Figure 6.8. Since the power generated by a pump is given by ΔpQ, Equation 6.36 can be used to derive the maximum power point. Since the power input to an electrokinetic pump is given by the current-voltage product VI, the thermodynamic efficiency can be defined as

where the current I is given by I = V∕R and the resistance R is given by R = L∕σA in the bulk conductivity limit. Efficiencies of electrokinetic pumps range approximately from 0.01%–5%.

6.5.2 Types of electrokinetic pumps

The analysis above is for a single capillary, but these results can be extended analytically to other geometries. Geometries used have included single capillaries [44], arrays of capillaries, capillaries packed with microparticles [45, 46, 47], planar microfabricated structures, and porous polymer monoliths. Two examples of these are shown in Figure 6.9. These devices have been used for chemical separations [47] and many other applications.

Experimentally, these pumps can be evaluated by monitoring flowrates and pressures as applied voltages are varied.

Related work on electrokinetic pumps from our research group can be found here.

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