12.2 1-DBand broadening [microchip separations top]

Many separation modes exist, each of which uses a different approach to give large or unique Δu. Much of the phenomena that predict w(t) are general, and so we begin by describing these band broadening phenomena, which compete with the separation technique and limit the resolution. Since these broadening phenomena involve dispersion and diffusion in the transport of the analyte, we consider these mass transport phenomena. For a model problem, namely a microchip electrophoretic separation, we consider in turn below the effects of analyte diffusion, diffusion with electrophoresis, and finally diffusion and electrophoretic separation with dispersive effects like pressure-driven flow or geometric turns.

12.2.1 Analyte transport: quiescent flow, no electric field

We start by considering quiescent flow to describe how analyte bands diffuse in the absence of separation or dispersion. Consider a 1-D distribution of the concentration c of a sample bolus of width w₀ and concentration c₀ centered at x = 0 at time t = 0. The total number of moles of analyte per unit area Γ is given by Γ = c₀w₀.

The governing equation for the evolution of the species concentration is the Nernst-Planck equation:

(12.2)

Noting that _i is zero given no flow and no electric field, and simplifying to one dimension, we get

(12.3)

which is just a 1-D passive scalar diffusion equationas discussed in Chapter 4. The solution for the distribution of c at large t is given by

If we take the limit w₀ → 0 as Γ is held constant, this solution is valid for all t. This solution is obtained by performing a similarity transform on the PDE to obtain an integrable ODE. From this solution, we can conclude that the peak concentration diminishes as and the width widens as . Specifically, the FWHM (full width at half-maximum) w is given by w = 4 and the peak is given by . Thus diffusion affects the separation in two ways. First, because of the peak concentration dependence, our ability to experimentally detect the analytes decreases as . Second, the separation modality must separate species faster than or diffusion obscures the separation.

12.2.2 Transport of analytes: electroosmoticflow andelectrophoresis

Given the results above for pure diffusion, we now add separation to the mix, giving us an idealized result for electrophoretic separations. Here we consider the action of applied electric fields in thin double layer devices, causing both a separation owing to electrophoresis and uniform velocity owing to electroosmosis.

If electroosmosis in thethin-double-layer limit leads to a uniform velocity u_EO and the convective transport of an analyte is given by the netelectrokinetic velocity u_EK = u_EO +u_EP, the solution is the same as Equation 12.4 except for a spatial transformation:

Note that the peak decrease and spreading are unaffected by electroosmotic flow because electroosmotic flow is uniform in the thin-double-layer limit. Thus electroosmotic flow is not dispersive in the thin double layer limit.

From the above relation, we see that applying an electric field in the thin double layer limit leads to electrokinetic motion of all analytes, leading to separation based on net electrokinetic mobility. Separation of the centers of each bolus is given by Δμ_EPEt. The width of each bolus expands with time owing to diffusion, and for large t is given by 4. Thus the band width scales with and the resolution increases with as well. This is the ideal result, an example of which is shown in Figure 12.6.

Effect of pressure-driven flow As discussed in Chapter 4, pressure-driven flow is dispersive, and pressure driven flow with a mean velocity u(x) leads to both a net motion of the analyte and an increased effective diffusion. Since these effects often vary with space and time, we write the solution as

where D_eff = D per Equation 4.12. Section 12.4.1 discusses sources of pressure-driven flow in microsystems.

Related work on chemical separations from our research group can be found here.

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