# Interdisciplinary Applied Mathematics

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the local velocity field, the sample also experiences electrophoretic transport, as shown by equation (7.44). In Figure (d), we present the sample distribution along the channel center at various times at which diffusion ef-

FIGURE 7.23. Numerical simulation results of electrophoretic transport in a microchannel under mixed electroosmotic/pressure-driven flow. Figures (a-c) show concentration contours at various times, while (d) shows the sample distribution along the channel center, and (e) shows the streamwise velocity distribution. The velocity is normalized with the Helmholtz-Smoluchowski velocity of equation (7.18), and simulations are performed for Re = 0.03, Pe = 500, and X/h = 0.01 conditions.

fects are visible. Figure (e) shows the velocity distribution across the channel for mixed electroosmotic/pressure-driven flows. The velocity profile is merely a superposition of pure electroosmotic flow with pluglike velocity distribution and a parabolic velocity profile of pressure-driven flow. Sharp velocity variation across the EDL is also visible in the figure.

The initial sample shape plays a key role in determining the species type using capillary electrophoresis. If we can prescribe the sample shape at the entry of the capillary, it is possible to determine the sample shape at any time (and place) in the channel. For example, an injection with a Gaussian distribution (with ao initial standard deviation) can be written as

n(x,0) =-y?=exp

^o 2n

X

2

Solution of equation (7.45) using this initial distribution results in (Bharad-

waj et al., 2002)

n(x, t)

o

exp

(x — Ut)2

2^ )

where, a2 = ao2 + 2tDe. Therefore, knowing the initial distribution of the injection, we can easily deduce the electrophoretic mobility of the species using j^ep = U/E, where U = x/t is found by measuring the location (x) of the sample at a given time (t). Noise associated with the initial sample shape or satellite sample droplets creates difficulties in identification of the sample type using this technique. To this end, there have been numerous experimental and numerical studies on initial sample focusing. For example, (Ermakov et al., 1999) utilized electroosmotic flow in a cross-channel to focus and pinch the sample for electrophoretic detection in a straight channel. Due to the convective-diffusive transport nature of electrophoretic detection, channel length plays an important role. One way to fit a very long channel in a microchip is to use spiral or serpentine channels, which result in severe dispersion effects due to the channel curvature. Several studies were conducted to understand and eliminate this effect by modification of the channel geometry (Molho et al., 2001; Dutta and Leighton, 2002), as well as local modifications of the channel zeta potential (Qiao and Aluru, 2003a). In    addition,    there    have    been    efforts    to    modify    the    hydrodynamic

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