Sample Dispersion for Segmented Flow in Microchannels with

Jan 30, 2008 - Michiel T. Kreutzer,, Axel Günther, andKlavs F. Jensen* ... Neil Hawbaker , Eric Wittgrove , Bianca Christensen , Neal Sach , and Donn...
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Anal. Chem. 2008, 80, 1558-1567

Sample Dispersion for Segmented Flow in Microchannels with Rectangular Cross Section Michiel T. Kreutzer,† Axel Gu 1 nther,‡ and Klavs F. Jensen*

Department of Chemical Engineering, Massachusetts Institute of Technology, 25 Ames Street, Cambridge, Massachusetts 02139

Hydrodynamic dispersion in microchannels can be significantly reduced by segmentation with a second immiscible phase. We address the effect of microchannel cross section on the dispersion of analytes in a segmented gas-liquid flow of alternating bubbles and liquid segments. Channels of square or nearly square cross section are considered. A significant fraction of the liquid surrounds the bubbles and wets the channel walls in the form of films or menisci. This stagnant fraction of the liquid remains when gas and liquid segments flow by, and it is connected to the liquid within the liquid segments by diffusion only and it effectively increases dispersion. We design and fabricate a microchip with integrated analyte injection and detection to investigate the effects of the influence of the stagnant liquid in segmented flow through square microchannels on the analyte bandwidth. The measured data and a corresponding model confirm the experimental trends and suggest operating conditions at which the unwanted effect of dispersion in segmented microchannel flow is minimized. Dispersion is least when the liquid flow rate is greater than the gas flow rate, and the optimum ratio of the two flow rates slightly increases with increasing bubble velocity. Transport processes associated with the flow of monodisperse droplets or bubbles, that act as moving compartments that are isolated from each other in the flow, direction provide a key advantage of multiphase microfluidic systems.1,2 The reproducible formation of such streams of bubbles or droplets in microsystems can serve various applications: sample analysis,3,4 high-throughput screening,5 miniaturized chemical6 and material synthesis,7 and drug discovery.8 * Corresponding author. E-mail: [email protected]. † M.T.K. is currently with the Department of Chemical Engineering, DelftChemTech, Delft University of Technology, Julianalaan 136, 2628 BL Delft, the Netherlands. E-mail: [email protected]. ‡ A.G. is currently with the Department of Mechanical and Industrial Engineering and the Institute of Biomaterials and Biomedical Engineering, University of Toronto, 5 King’s College Road, Toronto, Ontario M5S 3G8, Canada. E-mail: [email protected]. (1) Gu ¨ nther, A.; Jensen, K. F. Lab Chip 2006, 6, 1487-1503. (2) Squires, T. M.; Quake, S. R. Rev. Mod. Phys. 2005, 77, 977-1026. (3) Hosokawa, K.; Fujii, T.; Endo, I. Anal. Chem. 1999, 71, 4781-4785. (4) Zhang, J. Z.; Kelbe, C.; Millero, F. J. Anal. Chim. Acta 2001, 438, 49-57. (5) Bartsch, J. W.; Tran, H. D.; Waller, A.; Mammoli, A. A.; Buranda, T.; Sklar, L. A.; Edwards, B. S. Anal. Chem. 2004, 76, 3810-3817. (6) Song, H.; Chen, D. L.; Ismagilov, R. F. Angew. Chem., Int. Ed. 2006, 45, 7336-7356.

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When fluid samples are analyzed in continuous microfluidic networks, an injected discrete analyte pulse spreads in the direction of flow as it traverses the microchannel network. When multiple sample plugs are periodically injected, dispersion limits the streamwise separation at which, for a given channel length, the injected samples can still be distinguished from one another. Flow segmentation is a well-known technique to reduce the unwanted axial dispersion.9 The separation of samples in continuous-flow analysis benefits from adding a second phase that confines the liquid into small, isolated segments. The second, dispersed phase acts as a barrier between two neighboring liquid volumes. The dispersed phase can be a gas or a liquid, but we only consider gases here. The interfacial tension between a gas and a liquid is often higher than that between two liquids, which stabilizes the flow to allow a wider range of fluid velocities. Also, gases are immiscible with liquids at conditions below the critical point, whereas it is particularly difficult to find two fluids that are immiscible, particularly for microchemical systems at high temperature. Inert gases that do not extract from the analyte phase or interact chemically with it are also easier to find. Segmented-flow continuous analyzers10 generally consist of cylindrical channels, in which inserted gas bubbles span most of the channel cross section, leaving only a very thin film of liquid that connects two subsequent liquid segments (slugs). The amount of cross-talk between the two segments is directly related to the film thickness. The present understanding of pulse broadening in circular cross section channels is well developed, including fluid mechanical effects that dictate the thickness of this film11-15 and the phenomena that describe diffusive transport from one slug to the next.9,16-23 (7) Yen, B. K. H.; Gunther, A.; Schmidt, M. A.; Jensen, K. F.; Bawendi, M. G. Angew. Chem., Int. Ed. 2005, 44, 5447-5451. (8) Dittrich, P.; Manz, A. Nat. Rev. Drug Discovery 2006, 5, 210-218. (9) Thiers, R. E.; Reed, A. H.; Delander, K. Clin. Chem. 1971, 17, 42-48. (10) Skeggs, L. T. Am. J. Clin. Pathol. 1957, 28, 311-322. (11) Ajaev, V. S.; Homsy, G. M. Annu. Rev. Fluid Mech. 2006, 38, 277-307. (12) Bretherton, F. P. J. Fluid Mech. 1961, 10, 166-188. (13) Clanet, C.; Heraud, P.; Searby, G. J. Fluid Mech. 2004, 519, 359-376. (14) Hazel, A. L.; Heil, M. J. Fluid Mech. 2002, 470, 91-114. (15) Kreutzer, M. T.; Kapteijn, F.; Moulijn, J. A.; Kleijn, C. R.; Heiszwolf, J. J. AIChE J. 2005, 51, 2428-2440. (16) El-Ali, J.; Gaudet, S.; Gunther, A.; Sorger, P. K.; Jensen, K. F. Anal. Chem. 2005, 77, 3629-3636. (17) Muradoglu, M.; Stone, H. A. Phys. Fluids 2005, 17. (18) Muradoglu, M.; Stone, H. A. J. Fluid Mech. 2007, 570, 455-466. (19) Snyder, L. R.; Adler, H. J. Anal. Chem. 1976, 48, 1017-1022. (20) Snyder, L. R.; Adler, H. J. Anal. Chem. 1976, 48, 1022-1027. (21) Trachsel, F.; Gunther, A.; Khan, S.; Jensen, K. F. Chem. Eng. Sci. 2005, 60, 5729-5737. 10.1021/ac702143r CCC: $40.75

© 2008 American Chemical Society Published on Web 01/30/2008

Significant residence times L/U between tens of seconds and several minutes are required by many on-chip analysis, chemical or materials synthesis, and catalysis applications of segmented flow. Symbols L and w describe the microchannel length and width, respectively, and U is the bubble velocity. After being made dimensionless with the time scale, w/U, the resulting dimensionless times are on the order of L/w ∼ 500-5000 for microchemical applications7,24 and 100-1000 in monoliths.25 It remains difficult to analyze this problem computationally, due to the threedimensionality of the problem, the long timescales, and the wide range of Peclet numbers Pe ) Uw/D and capillary numbers Ca ) µU/γ, where γ is the surface tension and µ is the continuousphase viscosity. Most numerical studies are limited to twodimensional axisymmetric channels. Typically, simulations are formulated for a single segment, e.g., Salman et al.26 analyzed dispersion in 2D axisymmetric channels for 10-6 < Ca < 10-3 and 103 < Pe < 106. Muradoglu et al.23 simulated domains containing two or six bubbles in channels of L/w ) 30 and L/w ) 60 for Ca ) 0.01. Dispersion was computed for Pe ) 10 - ∞ in a transient simulation in which the segments traveled a distance of up to 37 channel diameters. In a recent contribution, Wo¨rner27 was able to analyze segmented-flow dispersion in a periodic domain with a single bubble for a short traveling time of the segment (L/w ) 8) in a square channel using a nondiffusing tracer (Pe ) ∞) at Ca ≈ 0.2. The objective of this paper is to quantify cross-talk between liquid segments in channel geometries that are relevant to microfluidic and lab-on-chip applications. For many microfluidic channel networks, the layout of the microchannel network is lithographically defined in a 2D plane and then “extruded” into the third dimension, either by etching the bulk material (e.g., dryetching or laser ablation of silicon, glass, and quartz) or by patterning the channel into a layer of negative resist (e.g., SU-8). The resulting microchannels have (near-) rectangular cross sections. Recently, the notable effect of the shape of the microchannel cross section on hydrodynamic dispersion and pulse broadening has received due attention.28-32 For segmented flow, an even larger effect would be expected due to the significant amount of stagnant liquid contained in menisci of rectangular channels compared to the gas-liquid interface forming a film of uniform thickness around the circumference for circular channel cross sections. We will show that this difference leads to increased communication from one liquid segment to the next. The paper is organized as follows. First, we discuss chip design and fabrication. Next, we report experiments with single-phase flow used to validate our designs and experimental procedures. (22) Kreutzer, M. T.; Bakker, J. J. W.; Kapteijn, F.; Moulijn, J. A.; Verheijen, P. J. T. Ind. Eng. Chem. Res. 2005, 44, 4898-4913. (23) Muradoglu, M.; Gunther, A.; Stone, H. A. Phys. Fluids 2007, 19. (24) Khan, S. A.; Gunther, A.; Schmidt, M. A.; Jensen, K. F. Langmuir 2004, 20, 8604-8611. (25) Kreutzer, M. T.; Kapteijn, F.; Moulijn, J. A.; Heiszwolf, J. J. Chem. Eng. Sci. 2005, 60, 5895-5916. (26) Salman, W.; Gavriilidis, A.; Angeli, P. AIChE J. 2007, 53, 1413-1428. (27) Wo ¨rner, M.; Ghidersa, B.; Onea, A. Int. J. Heat Fluid Flow 2007, 28, 8394. (28) Ahn, H.; Brandani, S. AIChE J. 2005, 51, 1980-1990. (29) Ajdari, A.; Bontoux, N.; Stone, H. A. Anal. Chem. 2006, 78, 387-392. (30) Chatwin, P. C.; Sullivan, P. J. J. Fluid Mech. 1982, 120, 347-358. (31) Doshi, M. R.; Daiya, P. M.; Gill, W. N. Chem. Eng. Sci. 1978, 33, 795-804. (32) Dutta, D.; Ramachandran, A.; Leighton, D. T. Microfluidics Nanofluidics 2006, 2, 275-290.

The single-phase experiments are carried out in square and circular cross section channels and are interpreted using classical Taylor dispersion theory. We then report dispersion measurements for segmented flow in square channels for a wide range of gas and liquid flow rates. In our experiments, 2 × 10-5 < Ca < 2 × 10-4 and 3 × 103 < Pe < 3 × 104 in channels of L/w on the order of 1000. We adapt the models for cylindrical channel dispersion to square duct geometries by combining the theory for sample carry-over with scaling arguments for the lengths of the continuous liquid segments and the dispersed gas bubbles in microfluidic systems. Finally, we express the dependencies of the number of theoretical plates on various operating parameters in explicit relationships. EXPERIMENTAL SECTION Device Fabrication. Pulse broadening experiments were performed in poly(dimethylsiloxane) (PDMS) microfluidic devices that were fabricated using standard soft-lithographic techniques33 for defining masters by patterning the negative resist SU-8 on silicon wafers. The cross section of the flow channel was 300 µm (width) × 300 µm (depth), with some variation in depth from master to master. Particularly for single-phase dispersion experiments, uniformity of the channel cross section is of vital importance. The thickness of the patterned SU-8 layer was controlled by spin-coating successive layers of SU-8 2050 (Microchem, Newton, MA) at 2200 rpm with a thickness of ∼100 µm per layer, that were successively spun and prebaked for 10 min. In this way, the thickness of the final SU-8 layer varied only within 2% across the entire wafer, where thickness measurements were determined by an interferometer at 10 different locations across the wafer. Spin-coating at lower speeds yielded thicker layers with much more variation in height. The accuracy in channel width was determined by the printing resolution of the transparency mask (20 000 dpi) that was lithographically transferred to the SU-8 master. Thin (∼2 mm) PDMS layers were cast on top of the SU-8 master and cured. Subsequently, a piezoelectric bending disc element (0.5 in. diameter, Piezo Systems, Cambridge, MA) was glued (epoxy) on top of the membrane covering the fluid tracer reservoir. On top of the thin PDMS and piezoelectric element, a second, thicker (∼6 mm) layer of PDMS was then cast and cured. The molded PDMS pattern was peeled off, cut to size, holes were punched for the fluidic connections, and the device and a cover glass slide (Corning, Corning, NY, 50 mm × 75 mm × 1 mm) were exposed to an oxygen plasma (Harrick, Ithaca, NY, model PDC-32G) and subsequently bonded. Microchip Layout. Figure 1 shows the design of the microfluidic device. Three fluidic connections in the center of the chip allow for feeding of the gas, liquid, and tracer streams. The carrier liquid and gas first pass through 50 µm × 300 µm channels that ensure that most of the pressure drop is over the single phase feeding section before the gas and liquid meet.34 The tracer is fed into a circular well in the middle of the device that connects to the carrier liquid channel, 12 channel diameters upstream of the T-junction where the gas is introduced. (33) Duffy, D. C.; McDonald, J. C.; Schueller, O. J. A.; Whitesides, G. M. Anal. Chem. 1998, 70, 4974-4984. (34) de Mas, N.; Gunther, A.; Kraus, T.; Schmidt, M. A.; Jensen, K. F. Ind. Eng. Chem. Res. 2005, 44, 8997-9013.

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Figure 1. Microfluidic device with fluid inlets/outlets, gas-liquid separator, fiber optical ports, and integrated piezoelectric bending disc actuator.

The injected tracer flows with the carrier liquid through the racetrack shaped channel. Different devices were fabricated, in which the racetrack shaped microchannel was either 1.2 or 1.77 m long. Sharp bends would induce increased dispersion inside the microchannel.35 The chosen channel layout therefore minimizes the curvature of the channel. The radius of the innermost track of the microchannel is 12.5 mm. For single-phase flow at sufficiently high flow rates, the channel curvature could lead to an unwanted recirculatory motion in the cross sectional plane (Dean vortices). These vortices only have influences on dispersion if a tracer molecule takes less time to diffuse across the channel width than it takes to migrate convectively across the channel in such a vortex. The criterion for this condition36 is (d/a)Re2Sc < 20, or

d/2 2Ud 2 ν < 20 a ν D

( )

(1)

in which d is the channel diameter, a is the track radius, ν is the kinematic viscosity. Symbol U is the fluid velocity and D is the diffusion coefficient. For our design, working liquids and fluorescent markers, the maximum allowable velocity is on the order of 5 mm/s. (35) Culbertson, C. T.; Jacobson, S. C.; Ramsey, J. M. Anal. Chem. 1998, 70, 3781-3789. (36) Janssen, L. A. M. Chem. Eng. Sci. 1976, 31, 215-218.

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The ability to reliably measure diffusion coefficients requires the radial diffusion time to be less than the time it takes to flow through the channel:

d2 L . U D

(2)

This expression defines an upper limit of approximately 5 mm/s, which is similar to the velocities considered here. Figure 2 shows a detail of the outlet flow section. At the end of the dispersion channel, the liquid passes a previously described gas-liquid separator.37 The liquid is removed from the dispersion channel by applying a small vacuum, corresponding to an approximately 10 cm liquid column, over an array of slits1 with openings of dslit ∼ 10 µm. The pressure difference across the slits draws liquid through them but is well below the pressure difference, of the order of γ/dslit, required to draw a gas-liquid meniscus through the slits. The gas is allowed to leave the chip through an outlet that is open to the atmosphere. Tracer Injection. The working liquid was ethanol in all experiments, and the tracer was Rhodamine-B. Several techniques exist to introduce sharp pulses of tracer into a flow channel. Commonly, a simple cross38 or a staggered double T-junction39 is used to fill a section of the main liquid channel with tracer, (37) Gunther, A.; Jhunjhunwala, M.; Thalmann, M.; Schmidt, M. A.; Jensen, K. F. Langmuir 2005, 21, 1547-1555.

Figure 2. Separation of gas from liquid and detection of tracer at the outlet of the chip. The comblike structure consists of a row of narrow slits that prevents the gas from being sucked in. Downstream of the separator in the liquid only channel, fibers are guided and self-aligned on opposite sides of the channel, creating a transmission flow cell.

preferably without flow of the carrier liquid. When a plug of tracer or analyte has filled the main channel, the analyte inflow is stopped and carrier liquid starts to flow. Such a cross-flow technique is not suitable for multiphase systems, because of the dynamic behavior of the flow. Stopping, or even just changing, the liquid feed disturbs the two-phase flow in the channel and leads to strong fluctuations that take so long to decay that constant flow cannot be assumed in the experiment. Here, we have used the deformability of PDMS by fabricating a liquid reservoir that can eject tracer into the carrier liquid under continuous flow of the liquid. The fluid injection is caused by a deflecting reservoir top wall, due to the deformation of a piezoelectric bending disc element. Applying a voltage of 200 V DC across the bending disc element caused a controllable amount of tracer liquid, approximately 100 nL, to be injected into the carrier liquid.21 The introduction of gas causes pressure pulses at the moment of pinch-off of the bubble. These pressure fluctuations cause bleeding of tracer into the carrier liquid after analyte injection. To prevent this bleeding, a distance of 10 channel diameters was required between the injection port and the T-junction where the gas is introduced. Even without gas bubbles, bleeding of analyte into the channel is caused by diffusion from the reservoir. We eliminated such bleeding by reducing the applied voltage from +200 V to -200 V over the course of 45 s, which draws a minimal amount of clear carrier liquid into the thin channel that connects the reservoir and the main channel. Tracer Detection. The concentration of tracer flowing out of the device was measured by transmission spectroscopy at 10 Hz. Two polished fiber ends were introduced such that they were aligned and each faced one side of the liquid-filled channel (Figure 2). One fiber was connected to a visible-light source (DH2000-SDUV, Ocean Optics, Dunedin, FL), the other one to a spectrometer (HR2000CG, Ocean Optics, Dunedin, FL). The intensity was recorded at a wavelength of 560 nm. At the inlet of the device, a camera was used to record time-lapsed fluorescence images of the tracer injection at a rate of 10 fps. The field of view (FOV) included the injection of the tracer, the T-junction where the gas was mixed in, and the first two tracks of the long channel, as (38) Culbertson, C. T.; Jacobson, S. C.; Ramsey, J. M. Talanta 2002, 56, 365373. (39) Chmela, E.; Blom, M. T.; Gardeniers, J. G. E.; van den Berg, A.; Tijssen, R. Lab Chip 2002, 2, 235-241.

Figure 3. The injection of a tracer, as recorded from fluorescence micrographs at 10 Hz. The field of view of the camera is shown on the left, and inverted fluorescence micrographs are shown on the right. Liquid flow rate 11 µL/min, gas flow rate ∼7 µL/min.

indicated in Figure 3. From subsequent frames, portions of the image were cut out and overlapped, where the offset between two frames was determined by cross-correlation in a Matlab routine (Figure 4a). The temporal evolution of the tracer pulse was calculated from the image using data the cross sectional intensity c(t) ) ∫w c(x,t) dx, as shown in Figure 4b,c. The recorded or reconstructed peaks were normalized, such that the area under the peak equals unity. The moments of the distribution of elution times c(t) are given by

∫t c(t) dt ∫c(t) dt n

〈z 〉 ) n

(3)

The mean residence time of the tracer is then given by tavg ) 〈z〉 and the tracer band variance σ2 is given by σ2 ) 〈z2〉 - t2avg. Experimental Procedure. The clear liquid stream and the tracer liquid were fed from a syringe using a syringe pump (Harvard, Holliston, MA, model PHD 2000), and the gas was supplied from a nitrogen bottle through a manual pressure reducer (PR50A15Z1 from Valco Instruments, Houston, TX). After fixing the set points for gas feed pressure and liquid flow rates, the system was allowed to stabilize (∼20 min). Once stabilized, a regular succession of gas bubbles and liquid segments flowed through the microchannel. For liquid-only experiments, the gas feed was capped after the gas feeding channels were filled with liquid. Analytical Chemistry, Vol. 80, No. 5, March 1, 2008

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Figure 4. Tracer peaks as constructed from consecutive micrographs: (a) Images as recorded by the camera, aligned such that the displacement in pixels can be seen, (b) peak for the tracer shortly after injection, (c) peak after completing one round in the racetrack microchannel.

We determined the linear velocity U of the tracer from the residence time and length of the channel. The superficial velocity of the liquid uL was calculated from the flow rate of carrier liquid FL and the cross sectional area of the microchannel. For segmented flow experiments, the flow rate of gas, FG, was not measured directly. We estimated FG either by using the ratio of bubble length to segment lengths, which equals the ratio of flow rates, or from the linear tracer velocity and the liquid superficial velocity using uG ≈ U - uL. Both methods delivered comparable values. The plate height was calculated from the variance of the elution time distribution and the linear velocity by H ) σ2U2/L and the number of theoretical plates was calculated as NTP ) (L/σU)2. RESULTS AND DISCUSSION Tracer Injection. Figure 3 shows a time series of four consecutive images that capture the release of the tracer solution into the channel. We are interested in characterizing dispersion at large times (L/U > w2/D), where inlet and outlet effects can be ignored with respect to the extent of hydrodynamic dispersion due to the length of the channel. For the tracer injection depicted in Figure 4, the variance of the inlet pulse is σin2 < 0.25 s2. In the analysis of contributions to dispersion, the variances of the pulses are additive, so for a peak width at the outlet of the devices σout > 5 s, the contribution of the inlet pulse width is less than one percent. Single-Phase Experiments. The measurement of the diffusion coefficient D of Rhodamine-B in ethanol served as a reference experiment for the device. For Rhodamine-B in water, which has a viscosity that is 20% lower than ethanol at room temperature, 1562

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diffusion coefficients between 2.7 × 10-10 m2/s and 1.3 × 10-9 m2/s have been reported.35,40-42 For Rhodamine-B in ethanol, we did not find a value for the diffusion coefficient reported in the literature. We measured D using the Taylor dispersion method with a conventional circular capillary (L ) 1.44 m, inner diameter d ) 250 µm, acoil ) 15 cm) to obtain a value of D ) (2.89 ( 0.05) × 10-10 m2/s at U ) 3.40 mm/s and D ) (2.81 ( 0.05) × 10-10 m2/s at U ) 6.80 mm/s. In these experiments, the residence time calculated from the elution curve agreed within 1% with the residence time based on flow rate and capillary volume, indicating that no significant adsorption occurred. In determining the diffusion coefficient from single-phase pulse broadening in the square channel device, the cross section of the channel, i.e., the channel depth h and channel width w, needs to be known with sufficient accuracy. We used established correlations28,30,31,43,44 for determining the dispersion coefficient K in

K Ud 2 w g2 ) 210 D D h

( ) ()

(4)

where w/h is the channel aspect ratio. Function g2 approaches the value of 7.95 at large w/h and varies significantly with w/h at small deviations from unity, which is why we determined h (40) Jacobson, S. C.; Hergenro¨der, R.; Koutny, L. B.; Ramsey, J. M. Anal. Chem. 1994, 66, 1114-1118. (41) Jacobson, S. C.; Hergenro¨der, R.; Koutny, L. B.; Warmack, R. J.; Ramsey, J. M. Anal. Chem. 1994, 66, 1107-1113. (42) Munson, M. S.; Hawkins, K. R.; Hasenbank, M. S.; Yager, P. Lab Chip 2005, 5, 856-862. (43) Dutta, D.; Leighton, D. T. Anal. Chem. 2001, 73, 504-513. (44) Dutta, D.; Leighton, D. T. Anal. Chem. 2003, 75, 3352-3359.

Figure 5. Elution curves for (a) single phase flow and (b) segmented flow at comparable residence time. Linear velocity, liquid-to-gas ratio, mean elution time, variance, and number of theoretical plates (NTP) are given in the inset. Microchannel dimensions: L ) 1.77 m, w ) 300 µm, h ) 281 µm.

accurately in the following manner. First, the volume of liquid contained in the flow channel was calculated from the residence time t and the liquid flow rate FL. Then, this volume, divided by the area of the channel footprint gives the channel height h. The height thus obtained agreed with the direct depth measurement using interferometry. For a chip with a microchannel cross section of 281 µm (h) × 300 µm (w), we obtained D ) (3.09 ( 0.42) × 10-10 m2/s at U ) 2.9 mm/s and D ) (3.07 ( 0.31) × 10-10 m2/s at U ) 9.8 mm/s. These values are within the limits of experimental uncertainty of the values determined in the capillary device. These measurements were performed at 293 K, and it was important to switch off all light sources in the proximity of the device; performing the experiments under a microscope resulted in less reproducible and higher values of D. We attribute this deviation to the change in viscosity due to heating of the device. A thermocouple taped to the device revealed a local temperature up to 308 K when a nearby light source was switched on. Dispersion in Segmented Flow. Figure 5 quantifies dispersion in square channels for segmented flow and single phase flow. The segmentation of the liquid by gas reduces the peak width by a factor of 5, or a 25-fold increase in the number of theoretical plates. Over the length of 1.77 m, the liquid that was initially injected into approximately five segments (cf. Figure 4b) has spread out to more than 50 segments. Note that the segments that were filled with tracer at the injection had transferred the analyte to upstream segments. In several experiments we tracked the movement of the slug that, right after the injection of gas, had the highest amount of tracer. This slug eluted about 20% faster than the slug that had the highest concentration at the detection, i.e., at the peak maximum. The peak width strongly depended on the flow rates of the gas and the carrier liquid. We describe these variations using the linear tracer velocity U and the ratio of liquid flow rate to gas flow rate FL/FG. This volumetric feed ratio is equal to the slugto-bubble length ratio LS/LB, which is easily measured visually. The width of the peak increased more than linearly with residence

Figure 6. The effect of linear velocity on peak width: (a) U ) 5.3 mm/s, FL/FG ) 1.2, (b) U ) 8.4 mm/s, FL/FG ) 1.4, (c) U ) 16.9 mm/s, FL/FG ) 1.4, (d) U ) 24.3 mm/s, FL/FG ) 1.5. The small inset shows the number of theoretical plates (NTP) as a function of the capillary number Ca ) µU/γ. Microchannel dimensions: L ) 1.2 m, w ) 300 µm, h ) 288 µm.

Figure 7. The effect of gas-to-liquid ratio on peak width: (a) U ) 6.3 mm/s, FL/FG ) 0.46, (b) U ) 6.7 mm/s, FL/FG ) 2.8, (c) U ) 7.8 mm/s, FL/FG ) 7.2. The small inset shows the number of theoretical plates (NTP) as a function of the gas-to-liquid ratio FL/FG. Microchannel dimensions: L ) 1.2 m, w ) 300 µm, h ) 288 µm.

time. Figure 6 shows how the peak width changes with velocity, at constant liquid-to-gas ratio. The number of theoretical plates (NTP) decreases with increasing velocity, but the effect is less than linear: a 5-fold increase of the velocity results in a 2-fold reduction in NTP. Changing the liquid-to-gas ratio has a more profound effect on the peak width. Figure 7 shows that for large liquid-to-gas ratios and at constant linear velocity, dispersion is reduced significantly. Model for Peak Width in Square Channels. While the effect of segmentation is not as pronounced as in capillaries, there is still a significant effect of adding gas bubbles. It is noteworthy that the single-phase peak has a perfectly Gaussian shape, as theoretically predicted, whereas the segmented-flow peak exhibits a tail. The tail can be explained as a fingerprint of exchange with stagnant liquid. The volumes of stagnant liquid that account for this tailing are the menisci in the corners of the square capillary, which are much thicker, on the order of one-tenth of the channel Analytical Chemistry, Vol. 80, No. 5, March 1, 2008

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diameter, than the thin lubricating films of thickness on the order that separate bubbles from the wall in circular capillaries. The simplest analysis for dispersion in segmented flow is obtained by regarding each liquid segment as a well mixed volume that mixes with a stream from its nearest downstream segment and that feeds a stream to its nearest upstream segment.19 If only the zeroth segment contained tracer at t ) 0, then for the kth segment

Am L Ck qk ) e-q , with q ) C0 k! A LS

(5)

in which q is a volumetric ratio of the volume of wetting film that has been deposited by the segment (i.e., AmL with Am the cross sectional area of the menisci and L the length of the channel), divided by the volume of an individual slug (i.e., the product ALs of the length of a slug Ls and the channel cross section A). Equation 5 is the Poisson distribution, which for large values of q, or for long channels, approaches the Gaussian distribution with variance σs2 ) q, where q1/2 is expressed in number of liquid segments. This expression can easily be expressed in time using the segment frequency f ) (LB + LS)/U, 2 L Am (LS + LB) σt ) LS A U2 2

(6)

It is often convenient to normalize the width of the pulse with respect to the mean time required to travel through the dispersion channel, L/U, to obtain 2

σθ2 )

(LS + LB) 1 ) NTP LSL

( ) Am A

Figure 8. Relevant geometric parameters for segmented flow dispersion: (a) the length of bubbles and liquid segments, as can be measured directly from bright-light micrographs and can be predicted from scaling arguments. The geometry of the T-junction, where the phases meet, determines the segment length. (b) The cross section of an elongated bubble shows the area of liquid near the menisci Am and the bubble area Ab. (c) The fraction of the channel cross section that is occupied by the liquid in the menisci, as a function of the capillary number. The dashed line represents values for circular cross section channels, the full line is for square channels.

less of the channel length, allowing more segments to fit into the channel, and (2) the length of the liquid segments changes. Garstecki et al.46 reported a scaling law for the length of bubbles, LB, as a function of the gas feed rate FG and liquid feed rate FL,

FG LB )1+R d FL

(8)

which can readily be rewritten for liquid segment lengths using LS/LB ) FL/FG as

FL LS )R+ d FG

(9)

(7)

Equation 6 shows that the peak width depends on various geometric parameters, i.e., A, Am, Ls, and Lb. For circular cross section channels, the film thickness varies with the capillary number Ca ) µU/γ, and therefore q depends on the velocity U and fluid properties γ and µ. For square channels, the thickness of the film far away from the corners also depends on Ca but this portion of the wetting film does not contribute significantly to Am. Rather, the volume of the wetting liquid is dominated by the liquid in the corners, and this amount of liquid is hardly affected by the value of Ca. Figure 5c shows the parameter (Am/ A) as a function of the capillary number, based on correlations for the meniscus shape.45 For the range of Ca in our experiments (2 × 10-5 < Ca < 2 × 10-4), the value of (Am/A) ranges from 0.075 to 0.095. The fractional liquid area for round channels is also plotted in Figure 8c. For cylindrical channels, this parameter is more than an order of magnitude smaller than for square channels. The liquid volume that connects the liquid segments is hardly affected by operating parameters or fluid properties. This leaves the gas-to-liquid feed ratio as the most suitable parameter to adjust q. Decreasing the gas flow has two effects: (1) bubbles will occupy

For the T-junction geometry that we have used in our device, we determined from micrographs at different flow ratios that R ≈ 1.5, where we lumped inaccuracies in determining LS and LB and a small systematic error because LS/LB is slightly smaller than FL/FG into this parameter R. Combining eqs 6-9 and using a constant meniscus shape, this model predicts that the number of theoretical plates is fully determined by the liquid-to-gas ratio. The plate height should also be independent of velocity, in contrast to the experimental findings. Figure 9 shows that eq 6 underestimates the peak variance σ2 significantly, by almost an order of magnitude on average. The failure of the simple segments-in-series model lies in the assumption of equilibrium between the liquid segments and the stagnant liquid in the corners. It is important to realize that this circulating flow in the liquid segments, which is sometimes referred to as bolus flow, never comes in direct contact with the wall of the microchannel.14 Rather, the lubricating layer that separates the bubble from the wall remains when the segment passes by. The diffusion time, associated with this thick layer of ∼0.1d, amounts to 3 s, which is much longer than the contact time LS/U between the segment and the stagnant layer. Figure 9c shows that the discrepancy between the simple model depends on the contact time, and that a short contact time results in a

(45) Kreutzer, M. T.; Du, P.; Heiszwolf, J. J.; Kapteijn, F.; Moulijn, J. A. Chem. Eng. Sci. 2001, 56, 6015-6023.

(46) Garstecki, P.; Fuerstman, M. J.; Stone, H. A.; Whitesides, G. M. Lab Chip 2006, 6, 437-446.

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Figure 9. (a) Schematic of fluid phase distribution and recirculation in the liquid phase in a simplified situation under the assumption of full equilibrium between a segment and the film it leaves behind, (b) comparison of the observed peak width with the ideal-model value, and (c) the deviation from the ideal model, as a function of the contact time of a segment with the stagnant liquid.

Figure 10. (a) Slow mass exchange leads to added dispersion. This schematic figure shows a 2D analogon. (b) Comparison of the observed peak width with the complete model value.

stronger deviation. Indeed, the data suggest that agreement with the segments-in-series model will be obtained when LS/U > 1 s. The slow exchange between segments and stagnant liquid gives rise to the asymmetry in the peak, and the additional dispersion can be accounted for using an additional term for the peak width20

σt2 )

(

)

q 1 Ud Am d L 1+ 2 36 D A L LS f

(10)

This additional term has the same geometric parameters, i.e., Am/A, d/L, L/LS, and includes a Peclet number, Ud/36D, for radial exchange between the bolus flow and the stagnant liquid. Equation 10 is an analytical result for round channels, based on the reasonable assumption that the relevant distance for diffusion is one-third of the channel radius. The cross section of

the moving segment, i.e., the area indicated as Ab in Figure 8b, is almost circular. For a square cross section, using the same estimate of distance for diffusion is appropriate. Figure 10a shows the recirculation pattern in bolus flow and depicts the added dispersion graphically. Equation 10, combined with the correlations for segment length and meniscus shape, predicts the experimental peak width very accurately, as shown in Figure 10b. The mass-exchange contribution scales with (Am/A)2, note that q is also proportional to Am/A. The significant amount of stagnant liquid in the corners of the channel makes a significant contribution to the more pronounced effect of analyte band broadening in square channels, as opposed to round channels. In our system, the second term between brackets in eq 10 is on the order of 10, in fact, it agrees roughly with the variance ratio shown in Figure 9c. For the same-sized circular channels, (Am/A) would be at least 30 times smaller and this second term would be insignificant with respect to the segments-in-series term. Analytical Chemistry, Vol. 80, No. 5, March 1, 2008

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Figure 11. Number of theoretical plates, NTP, (a) as a function of the liquid-to-gas feed ratio and (b) as a function of the linear velocity, U, calculated from eq 11 with L ) 1.2 m, d ) 300 µm, D ) 3 × 10-10 m2/s.

Using the independently determined values Am ≈ A/12 and R ≈ 3/2, the following practical formula is found by combining eqs 6-10 2 1 (1 + FL/FG) dL 1 Ud σt ) 2(FL/FG) + 3 + 24 U2(F /F )2 216 D

[

2

L

G

] (11)

This equation has no additional fitted parameters and supports our results (L ) 1.2 m, d ) 300 µm, 0.4 < FL/FG < 8, 3 mm/s < U < 25 mm/s) with an average relative error of 17%. A parity plot of the experimental values versus eq 11 is indistinguishable from Figure 10b. Figure 11 shows the dispersion characteristics, expressed in number of theoretical plates, based on the complete model, i.e., including finite mass exchange rate, for square channels as given in eq 11. The number of theoretical plates has a maximum with respect to the liquid-to-gas ratio. At very low velocities, the largest number of plates is observed when the liquid flow rate is slightly higher than the gas flow rate. The location of the maximum increases with velocity as

(FL/FG) )

1 + 2

x134 + 2161 UdD

(12)

The highest value of NTP, corresponding to the narrowest peaks, is obtained at a liquid-to-gas ratio between 5 and 15 for typical values of the velocity. If the liquid-to-gas ratio is lower than the optimal value, the segments are short and the incomplete mass-exchange with the stagnant liquid causes excessive pulse broadening. However, if the liquid-to-gas ratio is increased, the segments become very long and this reduces the number of segments inside a given length of channel. The dependence on velocity is more straightforward. The massexchange term is proportional to U through the Peclet number. Decreasing the velocity reduces dispersion. For very small velocities, the segments have enough time to equilibrate with the surrounding stagnant liquid, and the effect of the nonidealities vanish and the simple segments-in-series model applies. For all the conditions analyzed herein, the square channels exhibit more pronounced analyte spread than circular channels. 1566

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We have limited ourselves to channels that are nearly square, i.e., that have aspect rations of w/h ≈ 1. When the channel is much wider than deep, the fraction of the channel that is taken up by stagnant liquid is reduced. Limited data is available on bubble shapes in such shallow channel, but it is reasonable to assume that the thickness of the liquid layer behind the meniscus is governed by h if h , w. For w/h ) 20, the wetting film can be estimated to have just twice the volume of the wetting film in a corresponding circular cross section channel. Experimental data and computational modeling that describes how the parameters Am/A and R depend on channel geometry and on inlet geometry, respectively, would be very welcome to improve the design of segmented-flow devices. Provided the velocity is small enough to allow sufficient time for the segments to exchange with the film, it should be possible to microfabricate devices for segmented gasliquid flow with very favorable dispersion characteristics. Shallow channels will have much smaller plate heights than square ones, in contrast to single-phase dispersion, where shallow channels are characterized by greater dispersion. CONCLUSIONS This work describes the reduction of dispersion by flow segmentation in microfluidic networks having square microchannel shapes that are common in many microfluidics and lab-on-achip applications. As in circular channels, dispersion is greatly reduced, but the noncircular channel shape impacts dispersion in two ways: increased cross-talk between subsequent liquid segments and slower diffusive exchange between the bulk fluid of a liquid segment and the stagnant liquid layer close to the wall. In square channels, a film of finite thickness interconnects the liquid segments, especially in the liquid-filled menisci in channel corners. The effect leads to increased cross-talk from segment to segment, and band broadening is more extensive in square channels than circular channels. The amount of analyte exchanged due to cross-talk hardly depends on segment velocity, in contrast to channels of circular cross section. Having such nonuniform menisci surrounding the bubbles and segments has minimal impact when the contact time of the moving segments is long enough to even out concentration differences between a segment and the surrounding liquid. Conversely, short and fast-moving segments do not equilibrate completely with the bulk fluid contained in the liquid segments, and the slow diffusive exchange contributes significantly to band broadening, a contribu-

tion that vanishes for bubble velocities approaching zero. At higher bubble velocities, an optimal liquid-to-gas ratio exists that is dependent on the lengths of the segments and bubbles. ACKNOWLEDGMENT This work was supported by the U.S. Army through the Institute for Soldier Nanotechnologies (under Contract DAAD19-02-0002 with the U.S. Army Research Office). M.T.K. has been supported by Grant DPC.7172 from the Dutch Foundation for

Applied Sciences STW. We acknowledge the help of MIT Microsystems Technology Laboratories staff and Brian K. H. Yen for assistance with the fluorescence images and fruitful discussions.

Received for review November 26, 2007.

October

17,

2007.

Accepted

AC702143R

Analytical Chemistry, Vol. 80, No. 5, March 1, 2008

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