Biomolecular Sensing Processing and Analysis  Rashid Bashir and Steve Wereley
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midplane concentration proﬁles for the three ﬂow arrangements shown in Figure 14.2. In all these ﬁgures a neutral mixing species (i.e. µep = 0, thereby ignoring any electrophoretic transport) with a diffusion coefﬁcient D = 3 × 10−10 m2/s is considered. As expected both symmetric ﬂow ﬁelds discussed above have yielded symmetric concentration proﬁles as shown in Figure 14.3a and 14.3b. While mixing in the homogeneous case is purely diffusive in nature, the presence of the symmetric circulation regions, Figure 14.3b, enables enhanced mixing by two mechanisms, ﬁrstly through convective means by circulating a portion of the mixed downstream ﬂuid to the unmixed upstream region, and secondly by forcing the bulk ﬂow through a signiﬁcantly narrower region, as shown by the convergence of the streamlines in Figure 14.2b. In Figure 14.3c the convective effects on the local species concentration is apparent from the concentration contours generated for the nonsymmetrical, offset patch arrangement.
14.5. CASE STUDY II: OSCILLATING AC ELECTROOSMOTIC FLOWS IN A RECTANGULAR MICROCHANNEL
In the previous example we simulated the electroosmotically driven transport of a dilute species. As discussed in Section 14.2.1, the ﬂow transients in such a situation need not be considered since they occur on timescales much shorter than other onchip processes (i.e. species transport). A situation where this condition is not met is when the applied electric ﬁeld is transient or time periodic. Recently the development of a series of related applications has led to enhanced interest in such time periodic electroosmotic ﬂows or AC electroosmosis. Oddy et al. [46] for example proposed and experimentally demonstrated a series of schemes for enhanced species mixing in microﬂuidic devices using AC electric ﬁelds. Green et al. [28] experimentally observed peak ﬂow velocities on the order of hundreds of micrometers per second near a set of parallel electrodes subject to two AC ﬁelds, 180◦ out of phase with each other. Brown et al. [4] and Studer et al. [57] presented microﬂuidic devices that incorporated arrays of nonuniformly sized embedded electrodes which, when subject to an AC ﬁeld, were able to generate a bulk ﬂuid motion. Other prominent examples include the works by Selvaganapathy et al. [55], Hughes [32] and Barrag´an [1].
In this section we will demonstrate the modeling of an AC electroosmotic ﬂow in a rectangular channel geometry, shown in Figure 14.4. AC electroosmotic ﬂows are in general a good candidate for demonstrative simulation as they represent a case where the appropriate time scale (i.e. the period of oscillation) is signiﬁcantly shorter than that required for the ﬂow ﬁeld to reach a steady state. For example if the AC ﬁeld is applied at a frequency of 1000Hz the period of oscillation is 0.001s compared with 0.1s for the conservative case of Re = 1, discussed in Section 14.2.1. As such the transient term in Eq. (14.1a) must be considered (note that this is a simpliﬁed analysis for estimating when ﬂow ﬁeld transients should be considered, for a more detailed discussion of the relevant timescales in AC electroosmosis see Erickson and Li [15]). Further complicating matters is the work by Dutta and Beskok [12] who demonstrated that depending on the frequency and the double layer thickness there may be insufﬁcient time for the liquid within the double layer to fully respond. As such the use of a slip condition at the edge of the double layer was shown to produce largely inaccurate results.
Thus to model this situation we consider the transient form of Eq. (14.1a), replacing the steady state , with φappsin(2π ft), where f is the frequency of oscillation. Since the ﬂow ﬁeld is uniaxial and symmetric about the center axes the computational domain is reduced
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Microchannel
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FIGURE 14.4. System geometry and computational domain for AC electroosmosis simulation.
to 2D in a single quadrant as shown in Figure 14.4. The double layer and applied potential ﬁelds can be decoupled, as is implied above, since the gradient of EDL ﬁeld is perpendicular to the wall and the applied potential ﬁeld parallel with it. The uniaxial condition implies that the derivative of the ﬂow velocity in the lengthwise direction is zero, thus the momentum convection term in Eq. (14.1a) disappears, independent of the ﬂow Reynolds number. Here we used the DebyeHuckel¨ linearization to determine the EDL ﬁeld and net charge density, Eq. (14.9) and Eq. (14.10).
14.5.1. Flow Simulation
Figure 14.5 compares the timeperiodic velocity proﬁles in the upper left hand quadrant of a 100 µm square channel for two cases (a) f = 500 hz and (b) f = 10 khz with φapp = 250 V/cm. To illustrate the essential features of the velocity proﬁle a relatively large double layer thickness has been used, κ = 3 × 106 m−1 (corresponding to a bulk ionic concentration no = 10−6 M), and a uniform surface potential of ζ = −25 mV was selected. For a discussion on the effects of double layer thickness the reader is referred to Dutta and Beskok [12]. From Figure 14.5, it is apparent that the application of the electrical body force results in a rapid acceleration of the ﬂuid within the double layer. In the case where the momentum diffusion time scale is much greater than the oscillation period (high f, Figure 14.2b) there is insufﬁcient time for ﬂuid momentum to diffuse far into the bulk ﬂow and thus while the ﬂuid within the double layer oscillates rapidly the bulk ﬂuid
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FIGURE 14.5. Steady state time periodic electroosmotic velocity proﬁles in upper quadrant of a 100 µm square microchannel at (a) f = 500 hz and (b) f = 10 khz with φapp = 250V/cm. In each case n represents a “sufﬁciently large” whole number such that the initial transients have died out.
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remains nearly stationary. At f = 500 hz there is more time for momentum diffusion from the double layer, however the bulk ﬂuid still lags behind the ﬂow in the double layer.
Another interesting feature of the velocity proﬁles shown in Figure 14.5 is the local velocity maximum observed near the corner (most clearly visible in the f = 10 khz case at t = n/4 f and t = n/2 f ). The intersection of the two walls results in a region of double layer overlap and thus an increased net charge density. This peak in the net charge density increases the ratio of the electrical body force to the viscous retardation allowing it to respond more rapidly to changes in the applied electric ﬁeld and thus resulting in a local maximum in the transient electroosmotic ﬂow velocity. Such an effect would not be observed had the slip condition approach used in Section 14.4 been implemented here.
14.6. CASE STUDY III: PRESSURE DRIVEN FLOW OVER HETEROGENEOUS SURFACES FOR ELECTROKINETIC CHARACTERIZATION
In the above two examples we limited ourselves to low Reynolds number cases where the ﬂow ﬁeld had no presumed effect on the EDL ﬁeld. Here we will examine a case where such a condition is not met, speciﬁcally pressure driven slit ﬂow over heterogeneous surfaces. Such a situation arises in a variety of electrokinetic characterization and microﬂuidics based applications. For example the streaming potential technique has been used to monitor the dynamic or static adsorption of proteins [13, 45, 60, 62], surface active substances [25] and other colloidal and nanosized particles [29, 63] onto a number of surfaces. In this technique the surface’s electrokinetic properties are altered by introducing a heterogeneous region, for example due to protein adsorption, which has a different ζpotential than the original surface. The introduction of this heterogeneous region induces a change in the surface’s average ζ potential, which is monitored via a streaming potential measurement, and related back to the degree of surface coverage.
In this ﬁnal example we demonstrate the simulation of pressure driven ﬂow through a slit microchannel with an arbitrary but periodic patchwise heterogeneous surface pattern on both the upper and lower surfaces (such that the ﬂow ﬁeld is symmetric about the midplane). Since the pattern is repeating, the computational domain is reduced to that over a single periodic cell, as shown in Figure 14.6. As mentioned above, in this example we are interested in examining the detailed ﬂow structure and coupled effect of the ﬂow ﬁeld on the electrical double layer. As such we require a very general solution to the Poisson, Eq. (14.4), NernstPlanck, Eq. (14.5), and NavierStokes, Eq. (14.1), equations in order to determine the local ionic concentration, double layer distribution and the overall ﬂow ﬁeld. Note that here in this theoretical treatment we limit ourselves to steady state ﬂow, but we do wish to consider cases of higher Reynolds number. Thus the transient term in Eq. (14.1a) can be ignored however the higher momentum convection term should not.
14.6.1. System Geometry, Basic Assumptions and Modeling Details
In general boundary conditions on the respective equations are as described in section 2 with the exception of the inlet and outlet boundaries where periodic conditions are applied (for a general discussion on simulation in periodic domains the reader is referred to the work by Patankar et al. [50]). As mentioned above, in the absence of externally applied
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y 
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FIGURE 14.6. 3D spatially periodic computational domain for pressure driven ﬂow through a slit microchannel with “closepacked” heterogeneous surface pattern.
electric ﬁelds the convective, streaming current components of the current conservation equation, Eq. (14.11b), is no longer dwarfed by the conduction component and thus must be considered. Additionally since we have rapid changes in the surface charge pattern, charge diffusion cannot be neglected. The highly coupled system consisting of all the general equations is difﬁcult to solve numerically, thus several simplifying assumptions were made. Firstly it was assumed that the gradient of the induced potential ﬁeld, φ, was only signiﬁcant along the ﬂow axis. Also, we assumed that Eq. (14.11b) was satisﬁed by applying a zero net current ﬂux at every cross section along the ﬂow axis, consistent with what has been done by others [7, 42]. In general we are interested in the steady state behavior of the system, thus the transient term in Eq. (14.1a) could be ignored, however we do wish to examine higher Reynolds number cases thus the convective terms should be included.
As discussed above, the resulting system of equations was solved using the ﬁnite element method. The computational domain was discretized using 27noded 3D elements,
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which were reﬁned within the double layer region near the surface and further reﬁned in locations where discontinuities in the surface ζpotential were present (i.e. the boundaries between patches). In all cases periodic conditions were imposed using the technique described by S´aez and Carbonell [53].
The solution procedure here began with a semiimplicit, nonlinear technique in which the ionic species concentrations and the double layer potential equations were solved for simultaneously. In general it was found that a direct solver was required as the resulting matrix was neither symmetric nor well conditioned. Once a solution was obtained the forcing term in the NavierStokes equations was be evaluated the ﬂow ﬁeld solved using a penalty method [30] which eliminated the pressure term from the formulation. While this reduced the number of equations that had to be solved, the resulting poorly conditioned matrix dictated the use of a direct solver over potentially faster iterative solver, such as that described in Section 14.4. Further details on the solution procedure and model veriﬁcation stages are available elsewhere [19].
14.6.2. Flow Simulation
This example considers the ﬂow over the periodic surface pattern described above with a periodic length of lx = 50 µm and width ly = 25 µm, a channel height of 50 µm (therefore ly = 25 µm in Figure 14.6). The liquid is considered as an aqueous solution of a monovalent species (physical properties of KCl from Vanysek [58] are used here) with a bulk ionic concentration of 1 × 10−5 M. Note that the selection of this reasonably low ionic concentration is equivalent to the double layer inﬂation techniques discussed earlier in order to bring the double layer length scale closed to that of the ﬂow system. The Reynolds number Re = 1 and a homogeneous ζ potential of −60 mV was selected. Figure 14.7 shows the ﬂow streamlines for different heterogeneous ζpotentials (a) −40 mV, (b) −20 mV and
(c) 0 mV. As can be seen the streamlines along the main ﬂow axis remain relatively straight, as would be expected for typical pressure driven ﬂow; however, perpendicular to the main ﬂow axis a distinct circular ﬂow pattern can be observed. In all cases the circulation in this plane is such that the ﬂow near the surface is always directed from the lower ζpotential region (in these cases the heterogeneous patch) to the higher ζpotential region. As a result the direction of circulation is constantly changing from clockwise to counterclockwise along the xaxis as the heterogeneous patch location switches from the right to the left side. It can also be observed that the center of circulation region tends to shift towards the high ζpotential region as the difference in the magnitude of the homogeneous and heterogeneous ζpotential is increased. The observed ﬂow circulation perpendicular to the pressure driven ﬂow axis is the result of an electroosmotic body force applied to the ﬂuid continua caused by the differences in electrostatic potential between the homogeneous surface and the heterogeneous patch [19]. In general it was found that the strength of this circulation was proportional to both Reynolds number and double layer thickness.
14.6.3. Double Layer Simulation
Figure 14.8 shows the inﬂuence of the ﬂow ﬁeld on the double layer distribution along the indicated plane in the computational domain at Reynolds numbers of (a) Re = 0.1,
(b) Re = 1 and (c) Re = 10 respectively. In all cases the heterogeneous ζ potential is
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(a)
(b)
(c)
FIGURE 14.7. Inﬂuence of heterogeneous patches on ﬂow ﬁeld streamlines for close packed pattern. Homogeneous ζpotential = −60 mV (white region), heterogeneous patches (dark patches) have ζpotential (a) −40 mV,
(b) −20 mV, (c) 0 mV. In all cases Re = 1 and no = 1 × 10−5 M KCl.
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FIGURE 14.8. Electrostatic potential contours in the double layer region parallel with the ﬂow axis on z = 0 µm (see Figure 14.6) for (a) Re = 0.1, (b) Re = 1.0 and (c) Re = 10.0. In each case the ζpotential of the homogeneous
region and heterogeneous patch are ζ0 = −60 mV and ζp = −40 mV respectively and the bulk ionic concentration is no = 1 × 10−5 M.
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−40 mV and the bulk ionic concentration is 1 × 10−5 M. As can be seen the double layer ﬁeld becomes signiﬁcantly distorted at higher Reynolds numbers, when compared to the diffusiondominated ﬁeld at Re = 0.1 and Re = 1. The diffusiondominated ﬁeld in these cases represents a classical PoissonBoltzmann distribution. This distortion of the double layer ﬁeld is the result of a convective effect observed by Cohen and Radke [7] in their 2D numerical work and comes about as a result of an induced ydirection velocity in the transition region between the two heterogeneous surfaces. This velocity perpendicular to the surface is a result of a weaker electroviscous effect over the lower ζ potential heterogeneous patch (see Li [38] for a discussion on the electroviscous effect) and thus the velocity in the double layer is locally faster. To maintain continuity, a negative ydirection velocity is induced when ﬂow is directed from a higher ζpotential region (i.e. more negative) to a lower and a positive ydirection velocity is induced for the opposite case.
14.7. SUMMARY AND OUTLOOK
In this chapter we have described the relevant equations, assumptions and numerical techniques involved in performing efﬁcient microscale ﬂow and transport simulation for microﬂuidic and labonchip systems. There are very few “standard” problems in such systems and thus through a series of illustrative examples we have endeavored to demonstrate the range of complexity which may be required depending on the information of interest.
In this work we have limited our discussion to ﬂow and species transport systems. These two aspects form the fundamental basis for the operation of most LabonChip devices and cover the majority of what many researchers are interested in simulating at present. In general however they do not present a complete picture of what is required to engineer a true lab onchip. The majority of the developmental work in LabonChip simulation is currently directed towards either the examination of more complex ﬂow systems, particularly those with nonuniform electrical conductivity [52], those involving thermal analysis [16], or those which couple reaction kinetics to the transport problem [17]. The eventual goal is towards the integration of all these models into a single “numerical prototyping” platform enabling the coupled simulation of microﬂuid dynamics, microtransport, microthermal, micromechanics, microelectronics and optics with chemical and biological thermodynamics and reaction kinetics.
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