Study of the Rise Time in Electroosmotic Flow within a Microcapillary

Anal. Chem. 2009, 81, 6590–6595

Articles Study of the Rise Time in Electroosmotic Flow within a Microcapillary Cuifang Kuang, Fang Yang, Wei Zhao, and Guiren Wang* Department of Mechanical Engineering & Biomedical Engineering Program, University of South Carolina, Columbia, South Carolina 29208 For the first time with a temporal resolution higher than 100 µs, we have successfully measured the risetime of electroosmotic flow (EOF) in a microcapillary using recently developed laser induced fluorescence photobleaching anemometer (LIFPA). Although there are some theoretical estimations in literature about the risetime in microcapillaries, to the best of our knowledge, this has never been experimentally validated in a microcapillary with inner diameter less than 100 µm. The LIFPA with high temporal and spatial resolution described in this paper is capable of measuring risetime distribution radially in a cylindrical microcapillary tube. The experimental results show that the risetime under a pulsed electric field decreases with the reduction of the inner diameter of a microcapillary, and the risetime also increases from the wall to the axis for a given microcapillary. These are qualitatively similar to the theoretical prediction. In addition, the initial pressure driven flow does not appear to affect risetime.

capillary electrophoresis,14,15 sample injection,16,17 and particle or cell manipulation by dielectrophoresis.18 Electrokinetics has also been used for active control to enhance mixing in microfluidics.19,20 Many electrokinetic phenomena involve transient processes, such as sample injection based on electrokinetics, AC EOF, traveling wave EOF and electrothermal flow, dielectrophoresis, and AC electrokinetic mixing during each period of charge relaxation time. EOF has been widely used in capillary electrophoresis (CE), microchip CE, sample injection, and micropump technology. EOF mostly coexists with electrophoresis once the electric field is established. Therefore, EOF is often used as a micropump to deliver fluid in lab-on-a-chip, with both DC21 and AC22 electric field. EOF is also used in sample injection, a critical step in electrophoresis based separation, partially because it can provide a plug sample due to its plug flow velocity property. There are steady and unsteady (or transient) EOFs. Although research has been carried out for steady EOF, transient EOF has attracted attention recently because of its association with microfluidics as aforementioned.23-36 For instance, in the starting phase

Recent progress in microfluidics or lab-on-a-chip has had a revolutionary impact in analytical chemistry and biological assays.1,2 In microfluidics, electrokinetics play an important role because it can provide actuation for pumping liquids through electroosmotic flow (EOF),3-6 AC EOF7-9 and traveling wave EOF10,11 or electrothermal flow,12,13 sample separation based on

(12) Wu, J.; Lian, M.; Yang, K. Appl. Phys. Lett. 2007, 90, 234103. (13) Chen, D. F.; Du, H. J. Micromech. Microeng. 2006, 2411. (14) Huang, Y. F.; Huang, C. C.; Hu, C. C.; Chang, H. T. Electrophoresis 2006, 27, 3503–3522. (15) Wey, A. B.; Thormann, W. J. Chromatogr., A 2001, 916, 225–238. (16) Zhu, L.; Lee, H. K. Anal. Chem. 2001, 73, 3065–3072. (17) Tang, L.; Huber, C. O. Talanta 1994, 41, 1791–1795. (18) Wang, X. B.; Huang, Y.; Gascoyne, P. R. C.; Becker, F. F. IEEE Trans. Ind. Appl. 1997, 33, 660–669. (19) Oddy, M. H.; Santiago, J. G.; Mikkelsen, J. C. Anal. Chem. 2001, 73, 5822– 5832. (20) Wang, S. C.; Lai, Y. W.; Ben, Y. x.; Chang, H. C. Ind. Eng. Chem. Res. 2004, 43, 2902–2911. (21) Chen, J. K.; Luo, W. J.; Yang, R. J. Jpn. J. Appl. Phys. 2006, 45, 7983–7990. (22) Loucaides, N.; Ramos, A.; Georghiou, G. Microfluid. Nanofluid. 2007, 3, 709–714. (23) Dose, E. V.; Guiochon, G. J. Chromatogr., A 1993, 652, 263–275. (24) Heiger, D. N.; Carson, S. M.; Cohen, A. S.; Karger, B. L. Anal. Chem. 1992, 64, 192–199. (25) So ¨derman, O.; Jo ¨nsson, B. J. Chem. Phys. 1996, 105, 10300–10311. (26) Dutta, P.; Beskok, A. Anal. Chem. 2001, 73, 5097–5102. (27) Green, N. G.; Ramos, A.; Gonza´lez, A.; Morgan, H.; Castellanos, A. Phys. Rev. E 2000, 61, 4011–4018. (28) Qiao, R.; Aluru, N. R. Int. J. Numer. Methods Eng. 2003, 56, 1023–1050. (29) Kang, Y.; Yang, C.; Huang, X. Int. J. Eng. Sci. 2002, 40, 2203–2221. (30) Yang, C.; Ng, C. B.; Chan, V. J. Colloid Interface Sci. 2002, 248, 524–527. (31) Chang, C. C.; Wang, C. Y. Electrophoresis 2008, 29, 2970–2979. (32) Manz, B.; Stilbs, P.; Joensson, B.; So ¨derman, O.; Callaghan, P. T. J. Phys. Chem. 1995, 99, 11297–11301.

* To whom correspondence should be addressed: (phone) 01 (803)-777-8013; (fax) 01 (803)-777-0106; (e-mail) [email protected]. (1) Beebe, D. J.; Mensing, G. A.; Walker, G. M. Annu. Rev. Biomed. Engineering. 2002, 4, 261–286. (2) Reyes, D. R.; Iossifidis, D.; Auroux, P. A.; Manz, A. Anal. Chem. 2002, 74, 2623–2636. (3) Laser, D. J.; Santiago, J. G. J. Micromech. Microeng. 2004, 14, R35-R64. (4) Chen, C. H.; Santiago, J. G. J. Microelectromech. Syst. 2002, 11, 672–683. (5) Zeng, S.; Chen, C. H.; Santiago, J. G.; Chen, J. R.; Zare, R. N.; Tripp, J. A.; Svec, F.; Frechet, J. M. J. Sens. Actuators, B 2002, 82, 209–212. (6) Yao, S.; Myers, A. M.; Posner, J. D.; Rose, K. A.; Santiago, J. G. J. Microelectromech. Syst. 2006, 15, 717–728. (7) Ramos, A.; Morgan, H.; Green, N. G.; Castellanos, A. J. Colloid Interface Sci. 1999, 217, 420–422. (8) Bazant, M. Z.; Ben, Y. Lab Chip 2006, 6, 1455–1461. (9) Studer, V.; Pepin, A.; Chen, Y.; Ajdari, A. Analyst 2004, 129, 944–949. (10) Ramos, A.; Gonza´lez, A.; Castellanos, A.; Green, N. G.; Morgan, H. Phys. Rev. E 2003, 67, 056302. (11) Gonza´lez, A.; Ramos, A.; Castellanos, A. Microfluid. Nanofluid. 2008, 5, 507–515.

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10.1021/ac901017a CCC: $40.75  2009 American Chemical Society Published on Web 07/17/2009

of EOF when potential is applied, several dynamic processes occur including propagation of the electric field, capacitive charging of the double layer, resistive heating of the electrolyte, and subsequent development of thermal gradients in both electrolyte and substrate.23 In sample injection using EOF, the startup interval, in which electric potential is applied, is relatively short and the process of sample injection is transient. In AC EOF pumping, the process during each period of charge relaxation is also transient as it is in the process of AC electrokinetic mixing. Studying transient EOF helps to understand the chemical and physical features of many transient electrokinetic processes and thus promote optimal design of microfluidic devices. An important property of transient EOF is its risetime Tr, a brief time interval that begins at the switch-on of an electric current and ends at the establishment of the steady EOF. Tr depends on several variables: channel cross section size, length, transverse position, electric field intensity, buffer concentration, and pH etc. Normally, Tr of EOF in a microcapillary is short, on the order of 1-500 µs.23 Understanding the factors that control Tr facilitates development of pulsed-field capillary systems and ultimately provides useful separation of large DNA molecules.24 Some studies of transient EOF are based on theoretical analysis.23,25-31,36 Dose and Guiochon23 applied numerical analysis based on fundamental physical properties to study the development of EOF after a potential was applied to a capillary filled with solution. They concluded that Tr was directly proportional to the square of the inner diameter (ID) of the microcapillary. Kang and Yang et al.29 obtained the same result by solving the Poisson-Boltzmann equation as it relates to the dynamics of EOF in a cylindrical microcapillary. Using a finite element method, Yang et al. studied the transient characteristics of EOF in hydrophobic and hydrophilic microchannels.36 They also found that the steady time of EOF was proportional to the square of microchannel height and a scale of microseconds. So ¨derman and Jo ¨nsson25 studied Tr in a 2D flow, where a detailed theoretical framework was developed describing the phenomenon of transient capillary EOF. Their work described the temporal and spatial evolution of the velocity of EOF when electric field pulses are applied parallel to the capillary. Finally, the results of their theoretical prediction agree well with experimental measurement results obtained from nuclear magnetic resonance (NMR) imaging techniques applied to a millimeter scale tube. Qiao and Aluru28 studied theoretically the Tr in several typical electrokinetic geometries using semiimplicit-step technique. They found that Tr varied across the width of the channel, specifically Tr near the wall was much less than Tr at the center line of the channel. Yang et al.30 and Chang et al.31 studied the time evolution of a slit microchannel and an annulus channel after electric potential was applied. The former theoretical results show that electroosmotic flow reaches steady state more quickly in smaller sized channels. Also, the time for the electroosmotic flow to reach steady

state is nearly the same regardless of differences of the electric double layer (EDL) parameters for a fixed size microchannel. The latter theoretical results show that total transient flow rate of several smaller pores or channels may largely exceed that of a single large pore or channel with the same total cross-section on the transient time scale of the smaller channels. Few publications reported experimental results for measuring Tr in microchannels. Manz et al.32 studied experimentally the risetime of EOF in a tube in millimeter scale (where the risetime was in the order of 100 ms). They obtained the time evolution of EOF in the tube with 4 mm ID by NMR imaging method. Time evolution of the EOF profiles was compared to the prediction of a theoretical model. The two-dimensional image of the tube cross section was obtained with a spatial resolution of 78 µm, which is relatively low for a microcapillary. The nuclear magnetic resonance microscopy (NMR) technique was also used to measure the risetime of EOF in a capillary tube with 2.07 mm ID. The experimental results showed that the temporal resolution was about 4 ms,33 which is also not sufficient to resolve the risetime of EOF in a microcapillary. To investigate transient phenomena in a microfluidic device, Shinohara et al. developed a high-speed micro-PIV technique and obtained 500 µs temporal resolution.34 Yan and Nguyen et al.35 used the micro-PIV technique to measure Tr in a 300 µm channel and also reached a 500 µs temporal resolution. However, to the best of our knowledge, the temporal resolution of NMR and micro-PIV has not been high enough to measure the risetime of EOF in a microcapillary with an ID of less than 100 µm. In analytical chemistry and lab-on-a-chip, however, the capillary ID used in microcapillary electrophoresis is in many cases less than 100 µm. Heiger et al.,24 on the other hand, studied the RC electrical characteristics of a CE system. Their experimental results show that the risetime of the applied voltage increases with increasing resistance or parasitic capacitance of the capillary. The lack of suitable instrumentation may explain the absence of studies involving the risetime of EOF in microcapillary with ID less than 100 µm. Currently there is no velocimetry that can measure Tr with sufficiently high temporal resolution, even though velocimeters and flow meters exist for microfluidics measurement.37 Consequently simple and convenient methods capable of measuring Tr for transient EOF with temporal resolution in the order of 1-100 µs are highly desirable. In this study, an optical technique was developed based on laserinduced fluorescence photobleaching anemometer (LIFPA) to measure Tr in microcapillaries. LIFPA, developed recently,38-40 provides an easy, fast, instantaneous, accurate, and potentially in-line velocity (or flow rate) measurement. The current LIFPA method employs a single-point measurement technique based on photobleaching. As flow velocity drops, dye in the flow has a longer residence time in the laser beam at the detection point. Increased exposure to the laser

(33) Wu, D. H.; Chen, A.; Johnson, C. S. J. Magn. Reson., Ser. A 1995, 115, 123–126. (34) Shinohara, K.; Sugii, Y.; Aota, A.; Hibara, A.; Tokeshi, M.; Kitamori, T.; Okamoto, K. Meas. Sci. Technol. 2004, 15, 1965–1970. (35) Yan, D. G.; Nguyen, N. T.; Yang, C.; Huang, X. Y. J. Chem. Phys. 2006, 124, 021103. (36) Yang, D.; Liu, Y. Sci. China, Ser. E: Technol. Sci. 2009, 1–6.

(37) Devasenathipathy, S.; Santiago, J. G.; Takehara, K. Anal. Chem. 2002, 74, 3704–3713. (38) Wang, G. R. Lab Chip 2005, 5, 450–456. (39) Wang, G. R.; Sas, I.; Jiang, H.; Janzen, W. P.; Hodge, C. N. Electrophoresis 2008, 29, 1253–1263. (40) Kuang, C.; Zhao, W.; Yang, F.; Wang, G. Microfluid. Nanofluid. 2009, in press.

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Figure 1. Schematic diagram of experimental setup.

extends photobleaching. Residence time within the detection region is approximately equal to the photobleaching time. Because photobleaching diminishes fluorescence, fluorescence intensity increases with flow velocity because of the shorter residence time. As a result, the relationship between fluorescence and velocity can be determined. The higher the velocity, the higher the fluorescence signal. At very low flow velocities, fluorescence intensity and velocity are linearly related. From the calibration curve, velocity is easily derived from fluorescence intensity alone. Temporal resolution can be enhanced by optimizing some parameters such as laser power, objective lens, dye concentration, etc.39 Spatial resolution depends on the numerical aperture of the objective lens and laser wavelength, i.e., dictated by the Rayleigh diffraction limit.40 For the first time, temporal resolution greater than 100 µs has been achieved based on LIFPA technique described in this paper. It is the highest temporal resolution for noninvasive velocimeter and 5 times higher than that from a µPIV. In this study, we focus on measuring the Tr of EOF in a 50 µm ID capillary, which is on the order of a typical size used in; ab-on-a-chip. EXPERIMENTAL SECTION Materials. Most electrolyte solutions have charged ions. Consequently, velocity driven by EOF could be affected by electrophoresis. To eliminate this effect on EOF, we used a neutrally charged, small molecular dye coumarin 102 (SigmaAldrich Corp., MO). This dye has a relatively high quantum efficiency of photobleaching, with a high absorption coefficient at around 400 nm wavelength and high emission coefficient around 460 nm wavelength. The dye was diluted with pure methanol to a neutral dye solution with concentration of 100 µM. Because the molecule size of the dye coumarin 102 is about 0.6 nm, it is easy to keep the dye concentration uniform in the entire microcapillary. Configuration of the Measurement System. All experiments were performed using a custom built system that was designed and constructed in house. The system is shown schematically in Figure 1. A violet laser, 405 nm wavelength (Crystalaser Corp., NV), was used to excite the fluorescent dye. The laser beam passed through a beam expander to enlarge the beam diameter to 5 mm, and was then collimated to a dichroic mirror. The laser was reflected to a 40×, NA of 0.6 objective lens (Olympus Corp., PA) by the dichroic mirror to illuminate the test section of a microcapillary. The fluorescent signal from the microcapillary was 6592

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collected by the same objective lens, reflected by the dichroic mirror, and focused on a detector by a convex lens. The detector window for the fluorescence signal wavelength was set between 435 and 485 nm. The fluorescence signal was detected confocally through a confocal pinhole, whose diameter is 10 µm. The pinhole and band fluorescence filter was situated in front of the detector. The pinhole rejects stray and ambient light to reduce noise. The signal from the detector was amplified by a preamplifier RS570 (Stanford Research System, CA), then acquired by an A/D converter USB-6259 (National Instruments, TX). The microcapillary was a fused silica microcapillary with 50 µm ID, 150 µm OD coated with linear polyacrylamide (Polymicro Technologies Corp., AZ), and was placed on a manual three-dimensional translation stage 17ANC005/MD (MellesGriot Corp., CA). A Harvard syringe pump PHD 2000 (Harvard Apparatus, MA) was used to drive a syringe containing fluorescence dye solution. Two Tee connections (Upchurch Corp., TX) were used to connect the plastic tubing and platinum electrodes respectively. A fast relay was used to control and switch the circuit loop. A high voltage power supply Keithley 248 (Keithley Instruments Inc., OH) was used to provide 0-5 kV DC signal. Resolution Analysis of Measurement System. The temporal resolution of the measurement system based on photobleaching is qualitatively related to, but does not correspond quantitatively the half decay time of the fluorescence.38-41 The shorter the half decay time, the higher the temporal resolution. Several factors influence photobleaching half decay time including; photobleaching quantum yield, laser power, dye concentration, solution pH value, buffer concentration, etc. These factors have been discussed in detail in previous works.38,39,41,42 In addition, the A/D convertor was set at the rate of 20 kHz for the data acquisition in this experiment. Rise Time with High Voltage Turned on. Most commercial high voltage power supplies have risetime considerably longer than 1 ms,24 which cannot meet the requirement of experimental measurement of Tr. For instance, the risetime of the Keithley 248 power supply used here is about 300 ms for a square wave from 0 to full range under full load, which is too slow for our experimental tests. Hence, we have used a fast switch circuit using a fast relay. The controlling schematic is shown in Figure 2a. The relay was driven by a power supply. A function generator AFG3102 (Tektronix Corp. TX) provided a square wave signal to control the relay. To validate the performance of this fast switch subsystem, we used an oscilloscope LeCroy 9384CL1Gs/s (LeCroy Corp., NY) to monitor the voltage wave forms by adding a resistance in series with the microcapillary. The results are shown in Figure 2b, from which we can see the risetime of turning-on 2000 V high voltage is less than 0.3 µs, which is sufficiently short to satisfy the requirement of our experiments and has almost no effect on the measurement of Tr. Measuring Rise Time of EOF in the Microcapillary. Rise time, Tr, is defined herein as the time required for the velocity of EOF to increase from 10 to 90% of its maximum steady value. Tr measurements were performed at room temperature using the setup shown in Figure 1. The distance between the ends of the two platinum electrodes was about 90 mm. There was an (41) Wang, G. R.; Fiedler, H. E. Exp. Fluids 2000, 29, 265–274. (42) Britt, A.; Moniz, W. IEEE J. Quantum Electron. 1972, 8, 913–914.

Figure 4. Effect of different initial flow velocities on Tr of the 50um ID, 90 mm length microcapillary. Table 1. Rise Time of Microcapillay ID: 50 and 75 µm

Figure 2. (a) Schematic diagram of the relay used as a fast switch; (b) rise time of the relay turned on at 2000 V high voltage.

Figure 3. Rise time of the 50 µm ID capillary tubing at 2000 V high voltage; built-in curve is the local enlargement of the original signal.

optical detection window in the center of the tubing. Laser power was set at 40 mw and focused on the center of the microcapillary by the objective lens. The high voltage power supply was set to 2000 V. The positive and negative electrodes were connected at the inlet and the outlet T-connector respectively. An initially steady pressure driven flow with very low flow rate was maintained by the syringe pump. In this case, the detector received a steadygoing fluorescence signal under the steady flow velocity.39 When a pulsed potential is applied, the velocity of EOF changes, which in turn causes the fluorescence signal to rise rapidly. The experimental result is shown in Figure 3, from which we see a Tr of 0.62 ms as the signal increases from 10 to 90% of the steady EOF velocity. The time interval of data acquisition is 50 µs. The results show that temporal resolution of the measurement

diameter (µm)

experimental (ms)

theoretical (ms)

deviation (%)

50 75

0.69 1.39

0.43 1.01

60 37

system is better than 100 µs, because values can be easily discriminated between 100 µs time intervals. To understand the relationship between risetime and the microcapillary diameter, which is important for capillary electrophoresis, the Tr of another microcapillary with ID of 75 µm was investigated. In the 75 µm ID microcapillary, the risetime of EOF was measured to be about 1.39 ms. The maximal relative deviation error of this measurement system was about 18.1%. The experimental results for the different ID of the microcapillaries are shown in Table 1. The relative errors between experimental and theoretical values for 50 and 75 µm ID microcapillary are, 60 and 37%, respectively. This could indicate that at relatively large ID, the measured Tr is generally consistent with the theoretical prediction that Tr is directly proportional to the square of the ID of the microcapillary.23,25,29 In analytical chemistry, it is also interesting to know whether hydrodynamically pressure driven flow influences the risetime of EOF. For this purpose, a series of Tr measurements at the center of the microcapillary were recorded for various initial pressure driven flow velocities. The results are shown in Figure 4, which shows that the initial velocity of pressure driven flow has vitually no effect on Tr within the initial velocity range of 0.85-8.5 mm/s. To study risetime distribution in the radial direction, we measured Tr at various radial positions at the same cross-section using a translation stage by leveraging the high spatial resolution of the measurement system. The results, shown in Figure 5, indicate that Tr increases from the vicinity of wall to the axis. Average standard deviation error is about 100 µs. The Tr profile of the microcapillary resembles a parabolic curve, which was validated by theoretical analysis as discussed in the next section. Figure 5 indicates that risetime increases from the wall to the axis of the microcapillary. The spatial resolution of the measuring system is determined by equation of Airy spot. The diameter of the focused laser beam is about 0.8 µm in the Analytical Chemistry, Vol. 81, No. 16, August 15, 2009

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Figure 5. Rise time of the 50 µm ID, 90 mm length capillary tubing at different position. The time consumed of velocity up to 90% of maximum at different position after potential applied (a comparison of theoretical data and experimental results).

6.5 ms. However, when the laser power is increased to about 28 mW and higher, Tr is about 0.7 ms. This indicates that at laser power lower than 28 mW, Tr is affected by the temporal resolution of the measuring system. However, when laser power is above 28 mW, temporal resolution is sufficiently high that the measured Tr is no more limited by the temporal resolution of the measuring system. Furthermore, the results indicate that the measurement uncertainty of this system is about 100 µs from Figure 6. Theoretical Analysis. To compare the experimental results, we describe a theoretical analysis of Tr in this section. This analysis is derived from theoretical framework of So ¨derman and Jo ¨nsson.25 Here, we focus only on the pulsed EOF. In the analysis of electroosmosis presented below, assumptions similar to those in the reference works have been employed.23,25 The solution to the momentum equation of incompressible 1D flow can be written in cylindrical coordinates as follows (these equations are reproduced from refs 25 and 32). ∞

v(r, t) )

∑ [M ber(T r) + N bei(T r)]cos(nωt) + n

n

n

n

n)0

(1)

[Mnber(Tnr) - Nnbei(Tnr)]sin(nωt)] where ω is angular frequency; Mn and Nn are given by M0 ) Rvmax Mn ) vmax

(2)

ber(TnR) 4 nπ nπ cos sin k nπ 2 2 ber(T R)2 + bei(T R)2 n n

( ) ( )

(3)

Figure 6. Rise time of the 50 µm ID, 90 mm length capillary tubing at different laser power.

present study (the diameter is constricted by the diffraction limit, D ) (1.22λ)/(NA)43 where λ is wavelength of excitation laser and NA is objective numerical aperture). However, the spatial resolution, which is based on the Reyleigh criterion R ) 0.5λ/NA due to the insertion of a pinhole in front of the detector,44 could be better than 0.4 µm for measuring the Tr distribution in the transverse direction of the microcapillary. It is clear from these experimental results that the risetime is on the order of 1 × 10-4 s for a 50 µm ID microcapillary. To ensure that the temporal resolution of the system is sufficiently high, Tr was measured at various temporal resolutions. Since the half decay time decreases with increasing of the laser power,38-40 the smaller the half decay time, the higher the resolution. Therefore, various laser power settings can produce various temporal resolutions. Under the aforementioned conditions, a series of rise times were measured at the center of the microcapillary with increasing laser powers. In other words, Tr was measured with various temporal resolution conditions. Tr at various laser power settings is shown in Figure 6, where Tr tends to be almost constant with increasing laser power settings, indicating that the measurement system has sufficient temporal resolution in the 50 µm ID microcapillaries. For instance, at laser power of about 18 mW, the Tr measured is (43) Speidel, M.; Jona´sˇ, A.; Florin, E. L. Opt. Lett. 2003, 28, 69–71. (44) Heintzmann, R.; Ficz, G. Briefings Funct. Genomics Proteomics 2006, 5, 289–301.

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N0 ) 0 Nn ) vmax

(4)

bei(TnR) 4 nπ nπ cos sin k nπ 2 2 ber(T R)2 + bei(T R)2 n n

( ) ( )

(5) With Tn ) √Fωn/η ber(x) ) 1 +

(6) 4l

l

(x/2) ∑ (-1) [(2l)!] 2

(7)

lg1

bei(x) )

l

4l+2

(x/2) ∑ (-1) [(2l + 1)!]

2

(8)

lg0

where ber(x) and bei(x) are the real and imaginary parts of the Bessel function, respectively; R is the duty cycle (the ratio between the pulse duration and the period of a rectangular waveform) of the rectangular wave, whose value is 0.5 here; vmax is set to be a constant of 1. For a microcapillary filled with a solution similar to pure methanol at room temperature, the quantity
F/η is approximately equal to 1452, when the density Fand viscosity η of pure methanol are 792 kg/m3 and 0.544 × 10-3 kg m-1 s-1, respectively. r and R represent distance from the centerline

Figure 7. Axial velocity profiles at difference time after potential applied using theoretical analysis.

Figure 8. Time development of the axial velocity at different position using theoretical analysis.

and the radius of the capillary, respectively. By combining eqs 1-8, the time evolution of the radial velocity profile was obtained in a cylindrical microcapillary, whose ID is 50 µm, as is shown in Figures 7 and 8. Figure 7 shows that it takes about 0.7 ms for the plug flow to develop in such a microcapillary. The time evolution of the velocity profile differs somewhat from the results of Dose et al.,23 due in part to the different solutions used. Figure 8 shows the time evolution of velocity at various positions on the cross section. The plots show the variation in risetime in radial direction of the microcapillary. Tr becomes shorter as proximity to the wall is

reduced.28 Figure 6 compares the theoretical and experimental results. The diagram shows that risetime is on the order of 1 × 10-4 s for the microcapillary. Also, the risetime in the center is longer than it is near the wall. Tr near the wall might be shorter because the charged surface attains its maximum velocity shortly (typically a few nanoseconds25) after the electric field is turned on. Subsequently, this liquid layer close to surface exerts frictional shear stress on the adjacent layer of liquid. The liquid is then driven by friction stress layer by layer. The difference between the theoretical and experimental data was observed to be about 0.2 ms. The difference might be explained as follows: (1) The measurement system has about 100 µs uncertainty, which can be seen from Figure 5 when the laser power is sufficiently high. (2) Assumptions in the aforementioned theoretical analysis are a slightly different from the actual situation. For instance, the risetime on the wall is assumed to be zero, which may not be true. (3) The risetime of the high voltage supply may influence the measurement of the EOF risetime as well. CONCLUSIONS A simple method based on LIFPA technique has been successfully applied to measure the EOF risetime in microcapillaries with inner diameter less than 100 µm. The dependence of risetime on the capillary diameter, radial position, and pressure driven flow rate has been studied. Temporal and spatial evolutions of the EOF risetime were experimentally investigated and compared with theoretical analysis. For the risetime of EOF, the relative error between experimental and theoretical values increases as the ID of the microcapillary decreases. Our experimental results divulge a simple way of gaining and monitoring the time evolution of EOF. The method can also be performed in common microfluidic devices. ACKNOWLEDGMENT This work has been financially supported by the NSF RII funding (EPS-0447660). The authors thank our colleague, Mr. David Westbury and Muhammad Yakut Ali for revising this manuscript. Received for review January 11, 2009. Accepted June 28, 2009. AC901017A

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