Splitter Imperfections in Annular Split-Flow Thin Separation Channels

Abstract. The separation performance of split-flow thin (SPLITT) separation channels .... and the sample stream may occupy the full channel thickness ...
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Anal. Chem. 2003, 75, 1365-1373

Splitter Imperfections in Annular Split-Flow Thin Separation Channels: Effect on Nonspecific Crossover P. Stephen Williams,*,† Lee R. Moore,† Jeffrey J. Chalmers,‡ and Maciej Zborowski†

Department of Biomedical Engineering, The Cleveland Clinic Foundation, 9500 Euclid Avenue, Cleveland, Ohio 44195, and Department of Chemical Engineering, The Ohio State University, 140 West 19th Avenue, Columbus, Ohio 43210

The separation performance of split-flow thin (SPLITT) separation channels generally falls short of ideal behavior. There are many possible contributing factors to the loss of separation resolution, and these are discussed in the text. The possibility that small imperfections in the splitters play a significant role is examined in this study. Computational fluid dynamics is used to determine the flow pattern within an annular SPLITT channel having small imperfections in the inlet splitter. These results are used to calculate the nonspecific crossover of particles from the inner annular inlet to the outer annular outlet under various flow rate regimes. Nonspecific crossover, obtained through convective transport alone, and not the result of field-induced transport, is often used as a check of channel behavior. The results of a typical experimental determination of nonspecific crossover are included for comparison. It is concluded that geometrical imperfections can indeed play a significant role in the loss of resolution observed for these systems. Since the inception of the split-flow lateral-transport thin channel (SPLITT) separation cell,1 various forms of the device have been widely applied for the separation of particulate and macromolecular materials. Separation takes place within a flow of fluid that is driven through a thin enclosed channel of high cross-sectional aspect ratio. In the most common mode of operation, a separation of sample components is obtained by exploiting their different rates of transport across the thin channel dimension. This is known as the transport mode of operation. Lateral transport may be the result of diffusion from regions of higher concentration to regions of lower concentration.2,3 It may be due to the influence of hydrodynamic lift forces.4,5 Most commonly, however, it is due to interaction of the sample materials with a field that is applied across the thin channel dimension, perpendicular to the direction of flow. With the application of a * Corresponding author. Fax: 216-444-9198. E-mail: [email protected]. † The Cleveland Clinic Foundation. ‡ The Ohio State University. (1) Giddings, J. C. Sep. Sci. Technol. 1985, 20, 749-768. (2) Williams, P. S.; Levin, S.; Lenczycki, T.; Giddings, J. C. Ind. Eng. Chem. Res. 1992, 31, 2172-2181. (3) Levin, S.; Giddings, J. C. J. Chem. Technol. Biotechnol. 1991, 50, 43-56. (4) Giddings, J. C. Sep. Sci. Technol. 1988, 23, 119-131. (5) Zhang, J.; Williams, P. S.; Myers, M. N.; Giddings, J. C. Sep. Sci. Technol. 1994, 29, 2493-2522. 10.1021/ac020649h CCC: $25.00 Published on Web 02/15/2003

© 2003 American Chemical Society

transverse field, the SPLITT separation channels have much in common with those of field-flow fractionation, although the separation mechanisms for the two techniques are different. Fieldflow fractionation (FFF) also separates particulate and macromolecular materials in a flow of fluid through thin channels, across which a transverse field is applied. In FFF, however, the separation mechanism amplifies differences in transverse distribution in order to obtain a separation in the direction of fluid flow. An analytical separation is generally the objective, which takes the form of a sequential elution of sample components. On the other hand, the SPLITT separation channels may be run in continuous mode for preparative separation referred to as continuous SPLITT fractionation (CSF). The fields employed to date for SPLITT separation include gravitational,6,7 centrifugal,8,9 electrical,10,11 acoustic,12 and magnetic.13-15 Other field types that have been commonly employed in FFF may find future use in SPLITT separation. Flow FFF uses a cross-flow of fluid through semipermeable channel walls to drive suspended and dissolved materials across the channel. A cross-flow displaces all suspended materials at the same rate. It is not then, by itself, selective with regard to any sample component property and would not be useful for SPLITT separation. However, a nonuniform cross-flow combined with an opposing gravitational field has been proposed16 for the design of a highefficiency SPLITT elutriation device. Thermal FFF uses a temperature gradient across the channel thickness, and this induces transverse migration known as thermophoresis. Thermal gradients are not generally selective with regard to molecular weight of dissolved macromolecules in organic solvents.17 On the other hand, they have been found to be selective for the chemical (6) Springston, S. R.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1987, 59, 344-350. (7) Gao, Y.; Myers, M. N.; Barman, B. N.; Giddings, J. C. Part. Sci. Technol. 1991, 9, 105-118. (8) Fuh, C. B.; Myers, M. N.; Giddings, J. C. Ind. Eng. Chem. Res. 1994, 33, 355-362. (9) Fuh, C. B.; Giddings, J. C. Biotechnol. Prog. 1995, 11, 14-20. (10) Giddings, J. C. J. Chromatogr. 1989, 480, 21-33. (11) Levin, S.; Myers, M. N.; Giddings, J. C. Sep. Sci. Technol. 1989, 24, 12451259. (12) Mandralis, Z. I.; Feke, D. L. Chem. Eng. Sci. 1993, 23, 3897-3905. (13) Zborowski, M.; Williams, P. S.; Sun, L.; Moore, L. R.; Chalmers, J. J. J. Liq. Chromatogr., Relat. Technol. 1997, 20, 2887-2905. (14) Fuh, C. B.; Chen, S. Y. J. Chromatogr., A 1998, 813, 313-324. (15) Fuh, C. B.; Chen, S. Y. J. Chromatogr., A 1999, 857, 193-204. (16) Giddings, J. C. Sep. Sci. Technol. 1986, 21, 831-843. (17) Schimpf, M. E.; Giddings, J. C. Macromolecules 1987, 20, 1561-1563.

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nature of the macromolecules.18 Also, recent studies have shown that, for aqueous and organic suspensions of colloidal particles, the induced thermophoretic motion is a function of both particle size and surface chemistry.19-21 In all implementations of SPLITT separation, the channel flow is divided at the outlet into two or more fractions that derive from different discrete lateral fractions of the channel thickness. Usually, only two fractions are obtained with the use of a single thin flow divider, or splitter, mounted between the channel walls. The divided fractions are carried to two outlets on opposite sides of the channel. Such a configuration yields a binary separation. Higher order separations can be obtained by taking successive binary cuts under appropriate conditions or by chaining binary separation channels.1,7 Ideally, the channel has perfectly uniform thickness across its breadth and the flow splitter is mounted at a constant fractional distance across the channel thickness. Under steady laminar flow conditions, a virtual surface inside the channel divides fluid streamlines that pass to each side of the outlet flow splitter and on to the two outlets. In an ideal system, this virtual surface is located at a constant fractional distance across the channel thickness, with the actual distance being a function of the ratio of the flow rates at the two outlets. In the case of an ideal parallel-plate system, this surface is planar and is known as the outlet splitting plane (OSP). For an ideal annular system, it is cylindrical and is known as the outlet splitting cylinder (OSC). Generally, it may be referred to as the outlet splitting surface (OSS). The fact that the ratio of flow rates determines the position of the OSS means that the lateral position of the splitter does not restrict or define the operational characteristics of the device. In fact, the lateral position of the OSS does not generally correspond to that of the splitter, which is often mounted at the midpoint of the channel thickness for ease of construction, if no other reason. Streamlines pass to each side of the splitter, and the OSS must therefore terminate at the splitter edge. As it approaches the splitter, it follows that the OSS must deviate from the lateral position that it assumes within the main body of the channel. This deviation occurs over a short distance, a distance comparable to the channel thickness and, therefore, a relatively small fraction of the channel length. This last point is important because the separation may then be considered to take place predominantly in the region of steady flow between the splitters where streamlines are parallel and the field is also generally arranged to be uniform. This simplifies the separation model considerably. If the device is to be operated over a fairly limited range of outlet flow rate ratios, then the splitter may be positioned close to the expected lateral range of the OSS in order to minimize this deviation of flow. When the high aspect ratio of typical channels is taken into account, the viscous drag of the sidewalls may be ignored and the model simplifies further. The assumptions of ideal infinite parallel-plate fluid flow and ideal particle migration allow for prediction of separation for given field and flow conditions to a high degree of accuracy. The selection of optimum flow conditions for a desired separation also becomes possible.6,22,23 (18) Gunderson, J. J.; Giddings, J. C. Macromolecules 1986, 19, 2618-2621. (19) Ratanathanawongs, S. K.; Shiundu, P. M.; Giddings, J. C. Colloids Surf., A 1995, 105, 243-250. (20) Jeon, S. J.; Schimpf, M. E.; Nyborg, A. Anal. Chem. 1997, 69, 3442-3450. (21) Mes, E. P. C.; Tijssen, R.; Kok, W. Th. J. Chromatogr., A 2001, 907, 201209.

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Figure 1. Schematic showing side view of a SPLITT separation channel. ISS and OSS represent the virtual inlet and outlet splitting surfaces described in the text.

In the transport mode, the sample must be introduced close to one of the channel walls (by convention, this is wall A) in order to exploit the different rates of transport across the channel thickness. The field acts to drive the sample components across the channel thickness toward the opposite wall (wall B) while at the same time they are carried along the length of the channel. The total channel flow rate therefore determines the time available for transverse transport. The sample components that have a lateral transportation rate insufficient to carry them beyond the OSS will exit at the outlet adjacent to wall A. Those having sufficiently high transportation rates that they are transported beyond the OSS will exit at the outlet adjacent to wall B. It was shown by Giddings22 that the efficiency of the separation is an inverse function of the initial thickness of the region occupied by the sample, known as the sample lamina thickness. The confinement of the sample to a thin region close to wall A is generally accomplished with the use of a second flow divider or splitter mounted at the inlet end of the channel. Two flow streams are introduced to the channel, one to each side of the inlet splitter. The sample or feed stream is introduced at the inlet adjacent to wall A, and a flow of the suspending fluid used to make up the sample is introduced at the other. The inlet splitter serves to guide these two contributions to channel flow to a smooth convergence at its downstream edge. By arranging for the sample flow rate to be much lower than that of the other, the merging of the streams confines the sample stream to a thin lamina adjacent to wall A. A second virtual surface, known appropriately as the inlet splitting plane, cylinder, or surface (ISP, ISC, or ISS, respectively), divides streamlines entering at the two inlets. In an ideal system, the ISS will be at a constant fractional distance across the channel thickness. The edge view of an ideal SPLITT channel, showing a binary separation of a bimodal sample, is shown in Figure 1. The channel thickness is not drawn to scale; it is expanded to illustrate the particle separation. Parallel-plate channels are typically a fraction of a millimeter in thickness, and between 5 and 30 cm in length. For continuous SPLITT fractionation (CSF), there is little advantage in the use of very thin channels, which is not the case for FFF channels. Parallel-plate systems require the channel to have a cross section of high aspect ratio in order to minimize the detrimental effects of the sidewalls on the separation, and limitations on channel breadth may put some constraint on channel thickness. Given this constraint, throughput in CSF has (22) Giddings, J. C. Sep. Sci. Technol. 1992, 27, 1489-1504. (23) Williams, P. S.; Zborowski, M.; Chalmers, J. J. Anal. Chem. 1999, 71, 37993807.

been shown to be independent of channel thickness.22,23 In annular SPLITT channels, sidewalls and their effects are absent. The annular channels used in our laboratory are typically ∼2 mm in thickness. The use of thicker channels allows some relaxation in construction tolerances without compromise on throughput. The convention for labeling walls, inlets, and outlets is included in Figure 1. The sample stream enters at inlet a′ adjacent to wall A, and the higher flow of pure suspending fluid enters at inlet b′ adjacent to wall B. The product streams exit at outlets a and b adjacent to walls A and B, respectively. The region between the ISS and OSS is known as the transport lamina. A particle must traverse this fluid lamina of thickness wt in order to be carried to the outlet adjacent to the far wall. The transport lamina thickness and the total channel flow rate together determine the division between those particles that exit the a outlet and those that exit the b outlet. The physical position of the inlet splitter, like that of the outlet, does not define the position of the relevant splitting surface. The position of the ISS is determined by the ratio of the two inlet flow rates. The inlet splitter need only be mounted at some fixed lateral position across the channel thickness in order for the ISS to fall at a constant lateral position. Like the outlet splitter, it may be mounted at a position close to the relevant splitting surface in order to minimize the lateral deviation of flow streams. An equilibrium mode of SPLITT separation is possible when separands migrate in opposite directions across the channel thickness. This can occur in a density-based separation using a suspending fluid of intermediate density in a gravitational or centrifugal SPLITT channel. Positively and negatively charged materials will migrate in opposite directions in an electrical SPLITT channel, and in a magnetic SPLITT channel, paramagnetic and diamagnetic materials will move in opposite directions. In all of these cases, an inlet splitter is not required, and the sample stream may occupy the full channel thickness at the inlet.

DEVIATION FROM PREDICTED BEHAVIOR It was mentioned above that under ideal conditions (ignoring any small effect of sidewalls) it is possible to predict the separation obtainable with a SPLITT channel under any given flow rate regime. It is also possible to select the optimum flow rate conditions to achieve some desired separation. In practice, however, the critical particle cut between populations exiting at the two outlets or the sharpness of the cut (resolution) may not exactly correspond to prediction. There are many possible causes for this deviation from predicted behavior. Most lead to some degradation of separation, but one effect that can improve separation is the lateral transport of sample materials onto the surface of the inlet splitter before the merging of the inlet streams.6,22 This has the effect of reducing the sample lamina thickness, thereby improving resolution. The simple SPLITT separation model ignores the disturbance to simple parabolic flow profile produced by the sidewalls of parallel-plate SPLITT channels. The drag produced by the sidewalls reduces the fluid velocity close to the channel edges. This allows more time for lateral migration for those particles entrained by the fluid flow in these regions and can lead to “leakage” of some particles into the fluid exiting at outlet b that should otherwise exit at outlet a. This effect was said to be apparent for

a small analytical SPLITT channel that had an aspect ratio of only 26.24 This effect is not a concern for annular SPLITT channels where sidewalls are absent.23 The small transient regions where the flow streams merge and then divide around the splitters have already been mentioned. Particle migration in these regions can contribute to deviation from the predictions of the ideal model, and the deviation may be expected to increase with the ratios of flow rates V˙ (b′)/V˙ (a′) and V˙ (b)/V˙ (a), and with total flow rate V˙ ) V˙ (a′) + V˙ (b′) ) V˙ (a) + V˙ (b). Under extreme conditions, vortexes can form close to the splitters and these have the potential to disturb the separation.25 The presence of hydrodynamic lift forces has also been mentioned in relation to their exploitation for particle separation. However, if the primary selective transport is induced by the application of a field, then the action of lift forces may bring about a perturbation to the expected separation. Lift forces appear to attenuate rapidly as a particle migrates away from a bounding wall. Far from the walls and under typical flow conditions, lift forces are predicted to have little influence on transverse migration, except in the case of vertical flow with significant particle sedimentation opposite to the direction of flow.26 In this case, lift forces can act to drive particles to the midpoint of the channel thickness. Too high a particle concentration in the sample stream can result in a degraded separation. A high concentration may correspond to a sample stream density that is higher than that of the pure fluid entering at inlet b′. In gravitational or centrifugal operation, this would correspond to an unstable stratification of density and could lead to mixing of the two inlet substreams with consequent degradation of separation.2,27-30 In the rotating channel of a centrifugal SPLITT system, Coriolis forces can induce fluid mixing and degraded separation, even in the presence of a stable density gradient.31 Mixing is exacerbated when the density gradient is unstable.32 Unstable density gradients may be regarded as essentially an overloading effect because they can be reduced by reducing sample concentration. They can also be eliminated by adjusting the density of the fluid stream entering at inlet b′ with the use of a suitable additive such as sucrose. The Coriolis forces are specific to centrifugal systems, but their detrimental effect on separation can be reduced by lowering total flow rate.32 High particle concentration may also degrade separation through a mechanism known as shear-induced diffusion. If the ISS is located in a region of relatively high shear rate, as it tends to be for conditions conducive to thin sample lamina streams, then this effect may contribute to enhanced lateral transport for all (24) Fuh, C. B.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1992, 64, 31253132. (25) Fuh, C. B.; Trujillo, E. M.; Giddings, J. C. Sep. Sci. Technol. 1995, 30, 38613876. (26) Williams, P. S. Sep. Sci. Technol. 1994, 29, 11-45. (27) Gupta, S.; Ligrani, P. M.; Giddings, J. C. Sep. Sci. Technol. 1997, 32, 16291655. (28) Jiang, Y.; Kummerow, A.; Hansen, M. J. Microcolumn Sep. 1997, 9, 261273. (29) Ligrani, P. M.; Gupta, S.; Giddings, J. C. Int. J. Heat Mass Transfer 1998, 41, 1667-1679. (30) Gupta, S.; Ligrani, P. M.; Giddings, J. C. Int. J. Heat Mass Transfer 1999, 42, 1023-1036. (31) Gupta, S.; Ligrani, P. M.; Myers, M. N.; Giddings, J. C. J. Microcolumn Sep. 1997, 9, 213-223. Errata: J. Microcolumn Sep. 1997, 9, 521. (32) Gupta, S.; Ligrani, P. M.; Myers, M. N.; Giddings, J. C. J. Microcolumn Sep. 1997, 9, 307-319. Errata: J. Microcolumn Sep. 1997, 9, 521.

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sample particles. Again, this would constitute an overloading effect, which could be avoided by suitable dilution of the sample.33,34 High particle concentration may give rise to nonspecific crossover by yet another mechanism. The transport of a population of particles under the influence of the field could draw with it, by viscous interaction, particles that would otherwise not migrate. Again, this effect could be eliminated by sufficient dilution of the sample. Finally, geometrical imperfections must be considered. Any variation in channel thickness across the breadth will influence the local fluid velocity. Regions of reduced thickness will offer higher resistance to flow resulting in reduced fluid velocity. Particles entrained in flow through these regions will have a greater time available for lateral transport. The overall picture may be more complicated, depending on the nature and distribution of these nonuniformities in thickness. Well-constructed SPLITT channels are formed from highly polished wall materials that are rigidly fixed around very uniform spacers, and so nonuniformities in thickness may be assumed to be relatively small, if not negligible. Any variation in channel breadth is also likely to be insignificant, although care is required in assembling centrifugal systems. The integrity of the splitters remains to be considered. This aspect of SPLITT channel construction has always been of concern. In the case of parallel-plate channels, the edges of the laminate layer forming the splitters may be put under tension in order to hold the splitters parallel to the channel walls.5 The use of high inlet and outlet flow rate ratios will result in significant pressure difference on the two sides of a splitter. The tendency for the splitter to bow under the influence of this pressure difference is reduced by this practice. Centrifugal forces in centrifugal SPLITT operation may also contribute to this tendency to bow. An alternative remedy is to lay narrow strips of spacer material along the length of each side of the splitters.8 These must stop short of the splitter edges so that the flow pattern is undisturbed at the point of merging or division of flow streams. The determination of nonspecific crossover under conditions of thin transport lamina thickness has been used, on a routine basis, to establish acceptable behavior for a given SPLITT system. A gravitational SPLITT system would be mounted vertically for this procedure so that the field acts in the direction of flow, rather than perpendicular to flow. The crossover of micrometer-sized polystyrene beads from the sample stream to outlet b would be taken to indicate some misalignment of the splitters with the channel walls or distortion of the splitters. The method has also been used to check the behavior of the quadrupole magnetic cell separation channels. The practice has an added significance for quadrupole magnetic cell separation. It is often required to selectively remove a magnetically labeled population of biological cells from those that are unlabeled. If it can be established that an acceptably small number of unlabeled cells cross over to the b outlet at small transport lamina thickness, then the SPLITT system can be operated most efficiently. The procedure tends to be carried out at fixed inlet flow rate ratio. Samples of polystyrene bead standards (typically 8-15 µm in diameter) are introduced at sample inlet a′ and detected at the two outlets using HPLC (33) Leighton, D.; Acrivos, A. J. Fluid Mech. 1987, 177, 100-131. (34) Leighton, D.; Acrivos, A. J. Fluid Mech. 1987, 181, 415-439.

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UV-visible detectors at a range of different outlet flow rate ratios. COMPUTER SIMMULATIONS The computational fluid dynamics software package, CFX-4 (CFX International, AEA Technology plc, Harwell, Didcot, U.K.) was used to predict the fluid flow pattern through a perfectly annular channel, with fluid entering via an imperfect splitter, the outlet splitter being assumed perfectly concentric. The geometry assumed for the simulations was based on that of a prototype quadrupole magnetic cell separation channel (Mark II, in-house classification). Splitter lengths were set at 0.500 cm, the channel length between splitter edges was 9.500 cm, the radius of the inner wall of the channel ri (corresponding to that of the core rod) was 0.239 cm, the internal radius of the outer wall ro was 0.439 cm, the inner radius of the splitters was 0.321 cm, and the outer radius of the splitters was 0.357 cm. The splitters therefore had a thickness of 0.036 cm, and they were assumed to be cut squarely and cleanly. Two cases of splitter imperfection were considered: (1) a lateral offset of a splitter by 100 µm, with its axis remaining parallel to that of the annular channel, and (2) an elliptical distortion of the splitter such that the difference between its major axis and minor axis was 200 µm, its axis remaining in line with that of the channel. Both the offset splitter and the elliptical splitter were placed only at the inlet, the other splitter being concentric and undistorted. In all cases, the total flow rate was fixed at 2.50 mL/min, and flow rate ratios at the inlets V˙ (a′)/V˙ (b′) equated to those at the outlets V˙ (a)/V˙ (b), and were set to 1/4, 1/9, and 1/19, making a total of six simulations. In every case, the plane of symmetry was taken into account to reduce the size of the calculation. A temperature of 298 K was assumed, with fluid properties of water at this temperature (density of 0.9966 g/mL, and viscosity of 0.009 00 P), giving a channel Reynolds number of 2.18 in all cases. A mesh was constructed over each model geometry that was finely and uniformly developed close to the edges of the splitters. Briefly, 20 elements were placed along the length of each splitter, 100 elements were placed along the channel length between the splitters, 30 elements were placed across the thickness of the inner inlet and outlet (a′ and a), 15 elements were placed across the thickness of the outer inlet and outlet (b′ and b), 12 elements were placed across the thickness of the splitter, and 18 elements were placed around the half-circumference (only half the channel was modeled with consideration of the plane of symmetry). Elements were progressively varied in length along all edges. For example, elements along the distance between the splitters were 100 times longer at the center than at the ends, close to the splitters. The fluid elements entering at the two inlets were visualized by consideration of an arbitrarily low diffusing solute (diffusion coefficient, D ) 10-6 cm2/s) of unit concentration in the sample stream entering at inlet a′. RESULTS Offset Inlet Splitter. Figure 2 shows a cross sectional view of the channel inlet and outlet regions in the plane of symmetry, with the flow running from left to right. The inlet splitter is offset downward by 100 µm with respect to the channel axis. The color spectrum corresponds to red for the unit solute concentration entering at the inner inlet a′ and blue for zero concentration

Figure 2. Cross sections through inlet and outlet regions of annular SPLITT channel, with flow from left to right. The inlet splitter is offset by 100 µm (downward), and the cross section is in the plane of symmetry. The color represents local sample concentration from unity (red) to zero (blue). Inlet flow rate ratio V˙ (a′)/V˙ ) 0.050 and is equal to outlet flow rate ratio V˙ (a)/V˙ , with total flow rate V˙ ) 2.5 mL/min.

Figure 3. Cross sections of annular SPLITT channel perpendicular to the direction of flow, at the inlet plane, at the midpoint of channel length, and at the outlet plane, from left to right. Flow rates correspond to those of Figure 2.

entering at inlet b′. The flow rate ratios V˙ (a′)/V˙ and V˙ (a′)/V˙ both correspond to 0.050. The negligible diffusional transport is indicated by the maintenance of the sharp boundary between regions of unit and zero concentration along the length of the channel. The regions of red and blue therefore distinguish the fluid elements that enter at inlets a′ and b′, respectively. The annular thickness of inlet b′ varies from 920 µm on the upper side, as it appears in the figure, to 720 µm on the lower side. The reverse is true for the annular thickness of inlet a′. Resistance to flow is directly related to the local annular thickness. Local volumetric flow rate will therefore have a higher order dependence on local annular thickness. The actual dependence is complicated by the merging of the two flow streams and the circumferential components of flow. However, the result is clear. At the top where the thicker b′ annulus allows higher flow rate of pure solvent, and the thinner a′ annulus restricts the flow of sample, the ISS is driven closer to the inner channel wall. The opposite is true at the bottom of the figure. At the outlets, it may be seen that some of the fluid that enters at inlet a′ at the bottom of the figure spills over into the b outlet. This fluid would carry entrained, nonmigrating particles to outlet b, and the figure shows, quite clearly, the nonspecific crossover expected at equal inlet and outlet flow rate ratios. Figure 3 shows the solute distribution over the cross section of the channel at the plane of the inlets, at the midpoint of the channel length, and at the outlet plane. The flow rate ratios are the same as in Figure 2. The distortion of the ISS is plainly visible in the plane of the channel midpoint, as is the crossover of some of the fluid and entrained sample to the b outlet. Careful examination of the calculated fluid velocity profiles revealed that the flow was fully developed within a relatively short distance of the splitter edge for every ratio of inlet and outlet flows. This is evident from Figure 2, where the position of the ISS is seen to be established within a short distance of the inlet splitter, and is maintained along the length of the channel to within a short

distance of the outlet splitter. The maintenance of the step functional form for the concentration profile suggested that the position of the ISS could be assumed to correspond closely to that of a concentration isosurface corresponding to 0.5 of the sample concentration. The calculations to be described below depend on the flow being fully developed in a region sufficiently removed from the splitters. The channel is assumed to be a perfect annulus, and the fluid should have no radial velocity component in this region. The positions of the inlet splitting surfaces around the annular circumference could therefore be safely determined at the midpoint of the channel length. This was accomplished by extracting concentration profiles for the slowly diffusing solute at 10° intervals and interpolating to the point where concentration was predicted to be 0.5 that of the sample. Due to the small degree of diffusion, the concentration profiles still retained, to a large extent, their step functional form and the interpolation was easily obtained. The calculations are extremely repetitive, so only one example of inlet flow rate ratio V˙ (a′)/V˙ of 0.050 is given. Figure 4 shows the interpolated points giving the position of the ISS in terms of reduced radius F () r/ro) versus angle. The simulation was carried out for the range of 0°-180°, but taking into account the plane of symmetry, the plot shows the complete range from -90° to 270°. The range was plotted from -90° to 270° to better show some small disturbance apparent near the plane of symmetry at 0° and 180°. A smoothing was applied in the form of a leastsquares best-fit cubic in the cosine of the angle (resulting in FISS ) 0.603922 - 0.023478 cos(θ) + 0.010833 cos2(θ) - 0.003041 cos3(θ)), and the plotted curve represents this smooth function backtransformed to a function of angle. Also plotted in the figure is the calculated position of the OSS for V˙ (a′)/V˙ ) 0.050 (see below), at F ) 0.6121. Figure 5 shows a polar plot of the ISS for the 100-µm offset inlet splitter for V˙ (a′)/V˙ ) 0.050, with the radial axis being the Analytical Chemistry, Vol. 75, No. 6, March 15, 2003

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the fully developed flow in an annular channel was given previously.23 The volumetric flow rate at outlet a is given by

V˙ (a) )

∫

rOSS

2πr ν(r) dr

ri

(4)

where v(r) describes the fully developed annular flow velocity profile as a function of radius r. The integration of eq 4 yields

V˙ (a) ) (πro2〈ν〉/A1)I2[Fi, FOSS] Figure 4. Reduced radius of ISS and OSS as functions of angle for 100-µm offset inlet splitter. The points along the ISS curve were obtained by interpolation to a concentration of 0.50. Flow rates correspond to those of Figure 2.

(5)

in which I2[Fi, FOSS] represents the result of an integration and is given by

I2[Fi, FOSS] ) [f2(F)] FFiOSS ) [2F2 - F4 + 2A2F2 ln F A2 F2] FFiOSS (6) It follows that

V˙ (a) I2[Fi, FOSS] ) V˙ A (1 - F 2) 1

(7)

i

which can be solved numerically for FOSS, the reduced radius of the OSS. The value for FOSS corresponding to V(a′)/V˙ ) 0.050 was calculated to be 0.6121, and this is the value plotted in Figure 4. Mass conservation requires that the integration of fluid velocity over the cross section of the annulus within the ISS be equal to the volumetric flow rate at inlet a′. This can be used as a check of internal consistency for our approach to determining the position of the ISS. This can be written as Figure 5. Polar plots of ISS and OSS for 100-µm offset inlet splitter. Flow rates correspond to those of Figure 2.

reduced radius F. The smoothed ISS curve of Figure 4 is simply replotted in polar coordinates. Also shown is the OSS for a perfectly concentric outlet splitter for flow rate ratio V˙ (a′)/V˙ ) 0.050. The position of the OSS is obtained from a consideration of the fully developed fluid velocity profile in an annular channel,35 given as a function of reduced radius F by 2

ν(F) ) (2〈ν〉/A1)(1 - F + A2 ln F)

(1)

where is the mean fluid velocity along the length of the annulus and A1 and A2 are functions of the ratio of inner to outer radii, ri/ro ) Fi of the annulus: 2

A1 ) (1 + Fi - A2)

(2)

A2 ) (1 - Fi2)/ln(1/Fi)

(3)

The method of calculating the position of a splitting cylinder within (35) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 1960; pp 51-54.

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V˙ (a′) )

∫ ∫ 2π

0

rISS (θ)

ri

ν(r) r dr dθ ) ro2

∫ ∫ 2π

0

FISS (θ)

Fi

ν(F) F dF dθ (8)

in which the angle θ has units of radians. Substituting for v(F) using eq 1, and integrating over F results in

V˙ (a′) )

ro2〈ν〉 { 2A1

∫

2π

0

f2(FISS(θ)) dθ -

∫

2π

0

f2(Fi) dθ} (9)

and, with further reduction and division by V˙ ) πro2(1 - Fi2), we obtain, with consideration of symmetry,

{

V˙ (a′) 1 1 ) 2 V˙ π A1(1 - Fi )

∫

π

0

}

f2(FISS(θ)) dθ - f2(Fi)

(10)

Knowing the form of the smoothed function for FISS(θ), this is easily calculated numerically. For the specific case of 100-µm offset inlet splitter and V˙ (a′)/V˙ ) 0.050, the result of the calculation given by eq 10 was 0.049 44. This is an error of only 1.1%, which may be attributable to discretization error. This excellent agreement therefore serves to justify the approach taken in determining the

position of the ISS. The possibility that discretization is a contributing factor is supported by the fact that the corresponding errors for V˙ (a′)/V˙ ) 0.10 and 0.20 were calculated to be only 0.54% and 0.22%, respectively. The error was therefore found to decrease with increase of V˙ (a′)/V˙ . The consistency with discretization error may be explained as follows. As V˙ (a′)/V˙ increases, a larger fraction of the annular cross section is occupied by the fluid entering at inlet a′, and a larger number of discrete mesh points is associated with this fraction of the total flow. Integration over the larger number of mesh points would give rise to a smaller discretization error. All nonspecific transport is assumed to be convective. Therefore, the fractional mass or number crossover of particulate material that enters at inlet a′ and exits at outlet b will be equal to the fraction of volumetric flow that enters at inlet a′ and exits at outlet b. This crossover volumetric flow rate can be referred to as V˙ cross. This is obtained by integration of fluid velocity over the region where (FISS(θ) - FOSS) is positive:

∫ ∫

V˙ cross ) ro2

2π

0

FISS(θ)

FOSS

ν(F) F dF dθ

for FISS(θ) > FOSS (11)

The fractional crossover of both fluid and entrained material is given by the ratio V˙ cross/V˙ (a′):

V˙ cross V˙ (a′)

)

ro2〈ν〉

∫ πA V˙ (a′) 1

π

0

(f2(FISS(θ)) - f2(FOSS)) dθ for FISS(θ) > FOSS (12)

Therefore, for a given simulation, yielding a function for FISS(θ), the value for fractional crossover may be calculated as a function of fractional outlet flow rate V˙ (a)/V˙ . The results for these calculations for the particular example of 100-µm offset inlet splitter and V˙ (a′)/V˙ ) 0.050 are shown in Figure 6a as the full curve. Also shown in Figure 6a as the dashed line is the ideal result for perfectly concentric inlet and outlet splitters. This shows no crossover of fluid or entrained particles for all V˙ (a)/V˙ > V˙ (a′)/V˙ ) 0.050 and a linear increase in V˙ cross/V˙ (a′) as V˙ (a)/V˙ is reduced from 0.050 to zero. When V˙ (a)/V˙ becomes zero, then all fluid exits outlet b, and therefore, all fluid entering at inlet a′ necessarily crosses over to outlet b. When V˙ (a)/V˙ ) 0.050, corresponding to zero nominal transport lamina thickness, the nonspecific crossover is calculated to be almost 24%. The minimum reduced radius of the ISS is calculated to correspond to F of 0.5882 (see Figures 4 and 5). For V˙ (a)/V˙ < 0.0214, the imperfect system behaves indistinguishably from a perfect system. The maximum reduced radius of the ISS is calculated to correspond to F of 0.6413. The OSS must therefore be raised to this level in order to reduce nonspecific crossover to zero. This corresponds to V˙ (a)/V˙ ) 0.0996, corresponding to an increment over V˙ (a′)/V˙ of 0.0496. We can also calculate that the transport lamina thickness wt must nominally be set to (0.6413 - 0.6121) × 0.439 cm ) 128 µm to eliminate nonspecific crossover. For convenience, these results, together with the results discussed below, are collected in Table 1. The results of the corresponding simulations with V˙ (a′)/V˙ set to 0.10 and 0.20 are shown in Figure 6b and c, respectively.

Figure 6. Fractional nonspecific crossover as a function of V(a)/V˙ for V(a′)/V˙ set to (a) 0.050, (b) 0.10, and (c) 0.20. The full lines are the result of calculations based on fluid dynamics modeling of 100µm offset inlet splitter, and the dashed lines correspond to the ideal result.

Nonspecific crossover at zero nominal transport lamina thickness is calculated to be 19 and 15%, respectively. It might appear that the problem of nonspecific crossover is reduced with increase of V˙ (a′)/V˙ . However, the calculation of necessary nominal transport lamina thickness to eliminate nonspecific crossover suggests the opposite. In the case of V˙ (a′)/V˙ ) 0.10, the maximum in FISS ) 0.6747, and at the same outlet flow ratio, FOSS ) 0.6415. A nominal transport lamina thickness of 146 µm is therefore required. This is obtained by setting V˙ (a)/V˙ to 0.1738, corresponding to an increment of 0.0738 over V˙ (a′)/V˙ . When V˙ (a′)/V˙ ) 0.20, the maximum in FISS ) 0.7225 while at the same outlet flow ratio FOSS ) 0.6851. This requires a nominal transport lamina thickness of 164 µm, obtained by setting V˙ (a)/V˙ to 0.3054, corresponding to an increment of 0.1054 over V˙ (a′)/V˙ . (Due to the parabolic fluid velocity profile, the nominal transport lamina thickness wt does not increase in direct proportion to the required increment over V˙ (a′)/V˙ .) There appears to be an advantage in operating at lower V˙ (a′)/V˙ when there is a requirement to compensate for splitter imperfection. The effect of splitter imperfection on the separation of materials that migrate across the channel thickness under the Analytical Chemistry, Vol. 75, No. 6, March 15, 2003

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Table 1. Results of Numerical Calculations

V˙ (a′)/V˙

ideal FISS

minimum in FISS

0.050 0.10 0.20

0.6121 0.6415 0.6851

0.5882 0.6135 0.6515

0.050 0.10 0.20

0.6121 0.6415 0.6851

0.5954 0.6220 0.6615

V˙ cross/V˙ (a′) (%)

V˙ (a)/V˙ req for zero crossover

nominal wt req for zero crossover (µm)

Offset Splitter 0.6413 0.6747 0.7225

24 19 15

0.09961 0.1738 0.3054

128 146 164

Elliptical Splitter 0.6289 0.6608 0.7073

14 11 8

0.07677 0.1409 0.2608

74 85 97

maximum in FISS

Figure 7. Reduced radius of ISS and OSS as functions of angle for elliptical inlet splitter. The points along the ISS curve were obtained by interpolation to a concentration of 0.50. Flow rates correspond to those of Figure 2.

influence of an applied field from those that do not is beyond the scope of the present work. This will be the subject of a later study. Elliptical Inlet Splitter. The inlet splitter was modeled with an elliptical cross section in line with the channel axis. The inner major and minor axes were set to 0.652 and 0.632 cm, and the outer major and minor axes were set to 0.724 and 0.704 cm. The difference between major and minor axes was 200 µm. The circumference of the elliptical splitter was therefore close to that of one of the undistorted circular splitters having an outer diameter of 0.714 cm. The flow pattern was established using the fluid dynamics software as in the previous case. The crossover calculations were then performed in the same way. Figure 7 shows the results of the ISS determination for V˙ (a′)/V˙ ) 0.050. The points correspond to interpolated concentrations equal to 0.5 that of the sample entering at inlet a′. As in the previous case, this is assumed to correspond to the position of the ISS. It is seen to be bimodal in form, which is to be expected for the elliptical inlet splitter. This bimodal form lent itself to a cubic fitting in the cosine of 2θ. The least-squares best fit is given by FISS ) 0.607125 + 0.015677 cos(2θ) + 0.005056 cos2(2θ) + 0.001064 cos3(2θ).) The transformed function is shown by the curve drawn through the points in Figure 7. The difference between major and minor splitter axes was 200 µm. The splitter therefore dipped 100 µm below its mean radius and protruded 100 µm above. These extremes correspond exactly to those of the offset circular splitter. The frequency of the oscillation around the mean is higher (being bimodal as opposed to a single oscillation around the circumference), and it is not surprising to see that the ISS varies in radius a little less than for the offset circular splitter under the same flow rate regime. Figure 8 shows the ISS and OSS (for the same flow rate ratio) plotted in polar 1372 Analytical Chemistry, Vol. 75, No. 6, March 15, 2003

Figure 8. Polar plots of ISS and OSS for elliptical inlet splitter. Flow rates correspond to those of Figure 2.

coordinates. Finally, Panels a-c of Figure 9 show the calculated nonspecific fractional crossover as a function of V˙ (a)/V˙ for the three cases of V˙ (a′)/V˙ equal to 0.050, 0.10, and 0.20, respectively. The respective maximums in FISS are 0.6289, 0.6608, and 0.7073, which correspond to V˙ (a)/V˙ of 0.07677, 0.1409, and 0.2608. The required nominal transport lamina thicknesses are calculated to be 74, 85, and 97 µm, respectively. Again, these are a little smaller than in the case of the 100-µm offset splitter. Incidentally, it might appear from Figure 9a, and to a lesser extent from Figure 9b and c, that, at low V˙ (a)/V˙ , crossover in the ideal case is slightly higher than with the elliptical inlet splitter. This is an artifact of discretization error. The minimum values for FISS in Figure 9a-c are 0.5954, 0.6220, and 0.6615, respectively. These values correspond to V˙ (a)/V˙ of 0.02885, 0.06513, and 0.1423, respectively. For V˙ (a)/V˙ below these limits, the crossover behavior of the undistorted and distorted splitters should be indistinguishable in each case. The dashed lines corresponding to undistorted behavior were drawn without regard to discretization error. EXPERIMENTAL MEASUREMENTS The results of a typical experimental determination of nonspecific crossover for an annular SPLITT channel are shown in Figure 10. The channel used (Mark IV, in-house classification) was not the same as that upon which the flow simulations were

Figure 10. Example of experimental determination of nonspecific crossover using 15.8-µm polystyrene standard particles in the Mark IV annular SPLITT channel having very similar dimensions to the Mark II model channel used for flow simulations.

imperfections in the real channel may be comparable to those considered for the simulations.

Figure 9. Fractional nonspecific crossover as a function of V˙ (a)/V˙ for V˙ (a′)/V˙ set to (a) 0.050, (b) 0.10, and (c) 0.20. The full lines are the result of calculations based on fluid dynamics modeling of elliptical inlet splitter, and the dashed lines correspond to the ideal result.

based (the Mark II channel). The Mark IV channel differs from the Mark II only in the thickness of the channel wall and splitters. The radius of the outer wall ro is 0.453 cm, compared to 0.439 cm for the Mark II. The inner radius of the annular channel remained at 0.238 cm. The inner and outer radii of the splitters were 0.347 and 0.362 cm, compared to 0.321 and 0.357 cm, respectively. The splitters were therefore considerably thinner. Polystyrene standard particles of 15.8-µm diameter (Duke Scientific Co., Palo Alto, CA) were used at a concentration of 2 × 106/mL. The total flow rate was set to 4.00 mL/min and V˙ (a′)/V˙ was fixed at 0.20. The carrier fluid contained 0.1% Triton X-100 (Sigma-Aldrich) and 0.02% sodium azide as a bactericide. Crossover was calculated via detector peak area integration, taking into account outlet flow rates, detector gain, and relative detector response. The measured crossover at equal inlet and outlet flow rate ratios was ∼14%, which is very close to the 15% predicted for the 100-µm offset inlet splitter. This agreement is coincidental, but it indicates that the

CONCLUSIONS The fluid dynamics simulations have demonstrated the importance of constructing SPLITT channels to close tolerances. The relatively small imperfections that were considered for the inlet splitter predict a nonspecific crossover similar in magnitude to that observed for real systems. In real systems, imperfections are just as likely to be present in the outlet splitter, and they may be present in both. For the special case of a displacement of splitters in the same direction, we might expect some compensation in the effects on nonspecific crossover, as the ISS and OSS are distorted in the same direction. The same might be expected for splitters that have the same orientation of elliptical distortion. Imperfections in the splitters are likely to be randomly orientated with respect to one another, however, and the effects are just as likely to exacerbate crossover as to reduce it. It was pointed out that nonspecific crossover has special significance for biological cell separations using an annular SPLITT channel with a quadrupole magnetic field. The present study was concerned only with nonspecific crossover. However, the establishment of the influence of imperfections in splitters on both the ISS and OSS will allow the quantitative prediction of degradation of separation resolution. As mentioned earlier, this will be the subject of a future report. ACKNOWLEDGMENT This work was supported by Grants R01 CA62349 (M.Z.) and R33 CA81662 (J.J.C.) from the National Institutes of Health, and Grants BES-9731059 (J.J.C. and M.Z.) and CTS-0125657 (P.S.W., J.J.C., and M.Z.) from the National Science Foundation. The authors also acknowledge expert technical help in channel construction by J. Proudfit (Prototype Laboratory, Lerner Research Institute, The Cleveland Clinic Foundation). Received for review October 18, 2002. Accepted January 7, 2003. AC020649H

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