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the initial variation rate is very remarkable, the variation of the I-curve comes with less of a time lag compared to those with curve No. 10. It is understandable because the initial increase rate of the second component is sharper in the case of curve No. 2 compared to that in No. 10. There are some time lags between the curves of the M- and I-groups. A t the maximum about 4 min in Figure 6A is necessary for change, and 6 min in Figure 6B. For separation using the pH gradient, rapid variation of the pH gradient in the capillary column is preferred. Therefore, one must select an appropriate gradient curve number and a duration period for the pH gradient. The difference period between Figure 6A,B for changes in the curves might be due to the buffering action of the phosphate buffer used. Therefore one must be careful in selecting the medium. The medium having no buffering action is the best selection for pH gradient. But in some cases a separation problem of a complex mixture with such a medium in capillary zone electrophoresis may be encountered. The average pH estimated from current is based on the summed phenomena in the column. From average pH one can suppose the pH change along the column generated by the present method. As just the average pH in the column is known, it is better to get the local pH in the column. For further attempts a more direct way will be needed to measure the local pH. For the measurement of local zone development in isotachophoresis, Hirokawa et al." devised a scanningdetector system in which a UV detector is scanned along the column. Their device might be adapted to measurement of true pH variation in capillary zone electrophoresis. Electropherograms a-d in Figure 7 are obtained with different mixing percentage of Na2HP04in 5 mM phosphate buffer. The elution times of the four solutes are dependent on the mixing percentage. The elution order of dansylated L-serine and 5-bromouracil has been changed between 100 and 70% of Na2HP04in phosphate buffer. The elution times of all solutes become larger in acidic media compared to those in alkali media. Two electropherogramswith pH gradient are shown at G-1 and G-2 in Figure 7. As in the pH gradient electropherogram
of G-1, the elution time of dansylated L-serine is later than 5-bromouracil, and it is clear that some pH gradient is generated in the capillary column from the comparative study of electropherograms a-c and G-1. The solutes of aliphatic hydrocarbons with temperature programming in gas chromatographyare eluted generally with a regular time period between every two peaks; that is, the time period between two peaks of Cnfl and C, is always the same in a wide range (C, means the number of carbon atoms in the aliphatic hydrocarbon). The same pattern is obtained by pH gradient in capillary zone electrophoresis. One of the typical examples is shown at G-2 in Figure 7 . Although these solutes are not homologous, one can observe regular time periods between two peaks in the pH gradient electropherogram of G-2.
ACKNOWLEDGMENT I thank deeply Richard N. Zare and Kinio Doi for their very helpful discussions and John B. St. John for his technical assistance. Part of this paper was presented at the 15th International Symposium on Column Liquid Chromatography, June 3-7, 1991, Basel. REFERENCES BoEek. P.; Deml. M.; Pospkhal, J.; Sudor, J. J. Chromatogr. 1989, 470, 309-312. Pospichal, J.; Deml, M.; Gebauer, P.; BoEek, P. J. Chromatcgr. 1889, 470. 43-55. SustBEek, V.; Foret, F.; BoEek, P. J. Chromatogr. 1989, 480, 271-276. Balchunas, A. T.; Sepaniak, M. Anal. Chem. 1988. 6 0 , 617-621. Foret, F.; Fanall, S.; BoEek, P. J. Chromatogr. 1990, 516, 219-222. Sudor, J.; Stransky, Z.; Pospkhal, J.; Deml, M.; BoEek, P. Nectrophoresis 1989, 10, 802-605. Tsuda, T.; Sweedler. J. V.; Zare, R. N. Anal. Chem. 1990, 62, 2149-2 152. Tsuda, T.; Zare, R. N. J. Chromatcgr. submitted for publication. Everaerts, E. M.; Beckers, J. L.; Verheggen, Th. P. E. M. Isotachophoresis ; Journal of Chromatography Library; Elservler: Amsterdam, 1976; Vol. 6. BoEek, P.; Gebauer, P.; Deml, M. J. Chromatcgr. 1981, 219, 21-28. Hirokawa, T.; Yokota, Y.; Kiso, Y. J. Chromatogr. 1991, 538, 403411; 545, 267-261.
RECEIVED for review August 1,1991. Accepted November 13, 1991.
Effects of Analyte Velocity Modulation on the Electroosmotic Flow in Capillary Electrophoresis Tshenge Demana, Urmi Guhathakurta, and Michael D. Morris* University of Michigan, Department of Chemistry, Ann Arbor, Michigan 48109-1055 Modulation of the electroosmotlc flow In capillary zone electrophoresls by modulatlon of the drlvlng voltage gives rlse to a flow proflle that changes between lamlnar and flat profiles. The changlng flow profile Induces a radial movement of Sample specles to and from the capillary surface. The Induced sample concentration gradlent can be monitored by carefully problng the capillary surface. The resuitlng slgnal Is a derlvative of the normal-shaped peak. Derlvatlve-shaped peaks can be observed wlth catlons, but not with anlons. Anions are unable to approach the doublelayer reglon and therefore are unaffected by the modulatlon process.
INTRODUCTION Modern capillary electrophoresis (CE), introduced first in 1979 by Mikkers' and then Jorgenson? is a powerful analytical
technique for the separation of charged specie^.^-^ The advantages of short analysis times, high resolution, and low buffer and sample consumption continue to generate great interest in the application of CE to bioseparati~ns.~-~ Capillaries of less than 100 pm i.d. are used in order to eliminate convection. The tiny column volumes are ideal for the separation of nanoliter samples. However, the ultrasmd samples pose a severe challenge to detector d e ~ i g n . ~ The nanoliter samples of capillary electrophoresis make the concentrationdetection limits of most detectors relatively high. Laser fluorometry1°and perhaps on-line voltammetry'l are the major exceptions. However few compounds fluoresce12 or are electrochemically active without prior derivatization. Thus, it is worthwhile to consider improving the performance of CE detectors which are not operating at fundamental shot noise limits. Analyte velocity modulation13J4was developed
0003-2700/92/0364-0390$03.00/00 1992 American Chemical Society
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toward this end. It is an optimization technique which should be useful with excess noise limited detectors such as the refractive index detector or the indirect fluorescence detector. Analyte velocity modulation is performed by imposing an ac voltage on the driving dc voltage. The modulation technique moves the detection region into a narrow frequency band (0.1-1 Hz) centered around the modulation frequency, providing opportunities for SIN enhancement. Synchronous demodulation with a lock-in amplifier rejecta all signals not within the passband. The technique is especially effective in systems subject to flicker noise, or other excess low-frequency noise.13 We have recently shown that analyte velocity modulation is also a useful form of pulsed field electrophoresis for separation of nucleic acids in gel-fied capillariea.15 Here, the goals are to decrease migration times and achieve improved resolution. Although the modulation technique is unchanged, demodulation of the detector signal is not necessary, because detector performance improvements are not sought. In an early communication we reported a simplified theoretical treatment of the modulation te~hnique.'~ This treatment assumed that bclbx, the axial concentration gradient, was constant in a spatially f i e d coordinate system. A consequence of that mumption was the prediction of an axial modulation of the concentration gradient and generation of a signal which is the derivative of the normal peak-shaped response. Subsequent work cast doubt on the proposed theory but presented no rigorous explanation for derivative-shaped ~u17res.l~In thiswork we r e f i e the theory of modulation and describe conditions under which normal-shaped and derivative-shaped peaks are obtained. We show that derivative shapes are observed by modulation of the electroosmotic flow in and near the electric double layer. We confirm experimentally that the modulation affects radial concentration gradient and does not affect the axial concentration gradient directly.
THEORY The internal surface of an unmodified fused-silica electrophoresis capillary is considered to be a negatively charged polysilicate ion. Because of the presence of this stationary polyanion, electrophoretic movement of hydrated cations toward the cathode gives rise to electroosmotic flow directed toward the cathode. The electroosmotic flow is proportional to the applied electric field and is a function of the {potential a t the surface.16 As the applied electric field is modulated, both the electrophoretic and electroosmotic velocities are modulated. The electric field varies from a minimum to a maximum at each half-cycle. The variation in the magnitude of the electric field causes the flow to vary from the conventional plug flow profile of capillary electrophoresis. For instance, a t a modulation depth of 100%, where modulation depth is defined as the ratio of the ac voltage to the driving dc voltage, the flow profile is expected to vary from laminar profile a t the zero minimum voltage to plug flow at the maximum voltage. In the case of 50% modulation depth the flow profie will change in a similar way but to a lesser extent. In other words modulation of electroosmosis c a w s the radial velocity distribution to systematically move between the nearly-uniform or plug profile characteristic of CZE and the parabolic profile characteristic of laminar flow. In turn, this velocity profile fluctuation induces a radial concentration gradient. If the peaks were stationary, this concentration profile would simply follow the average concentration. Because the peak center moves, the profile tracks 6c16.z. Therefore the pertinent concentration profile equations are radial equations. For analyte velocity modulation, the electric field is the instantaneous voltage E given by eq 1,where V,, is the dc
991
voltage, V,, cos w t is the instantaneous ac voltage, and L is the total length of the capillary column. According to Tay-
E=
v,, + v,, cos ut L
(1)
lor,l7 eq 2 is the sample diffusion equation under flowing conditions in a cylindrical tube, where u is the flow velocity, assumed to be a combination of laminar and plug flow in the case of analyte velocity modulation, c is the sample concentration, x is the axial position which is assumed to advance with the flow, D is the diffusion coefficient, and r is the radial position in the capillary. u 6c =
6x
D - -6( r r 6r
E)
It has been shown in refls following Taylor's assumption^'^ that the radial concentration profile in the case of laminar flow is given by eq 3, where u is the mean flow velocity and a is the radius of the capillary. (3) The radial concentration profile with plug flow can be derived in a similar manner. We assume that without modulation the electroosmotic flow has a flat velocity profie. The electroosmotic flow velocity as derived by Rice and Whiteheadlg is given by eq 4. u, is the electroosmoticflow, e is the (4)
dielectric constant of the buffer, coo is the potential at the capillary wall, and q is the viscosity of the medium. Io is the zero-order-modified Bessel function of the first kind, and k is the reciprocal of the double-layer thickness. In case of electroosmotic flow, eq 2 can then be expressed as eq 5.
(5)
Equation 5 can be solved to give 7 below. Equation 7 satisfies the boundary conditionz0at r = 0, 6c/6r = 0. Here c, is the bulk concentrationat the center of the capiuaTy where r = 0.
Equation 7 can be combined with eq 3 to give eq 8 below. In eq 8, we have expressed the radial concentration profile as a combination of the laminar and the plug flow profile terms. Substitution of eq 1 into eq 8 gives the sinusoidal
radial concentration lag, eq 9. In eq 9, cac(x,r,t)is the component of concentration which varies synchronously with modulation voltage.
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Cac(x,r,t) =
This time-dependent concentration can be observed most strongly near the walls of the capillary, where r is largest and the alternation between plug flow and laminar flow is strongest. Because this modulation is directly related to the existence of the capillary double layer, we expect to find it at unmodified silica capillaries only for cations and solvated neutral molecules. Anions, which are repelled from the capillary wall, should show little or no radial concentration modulation. Similarly, if the capillary walls are coated to minimize electroosmosis, little or no radial modulation should be observed for either cations or neutrals.
EXPERIMENTAL SECTION The experimental apparatus used in this work has been described previ~usly.'~ Briefly, a high-voltagepower supply (Series EH, Glassman High Voltage Inc.) provided the driving dc voltage. The modulating ac voltage was provided by an 881 step-up transformer connected between the capillary exit end and the system ground. A sinusoidal signal derived from a Wavetek 112 signal generator and amplified through a solid-state operational amplifier (Apex PA85),was used to drive the transformer. The modulation frequency typically employed was 160 Hz, with 50% depth of modulation. The driving dc voltage typically used was 20 kV with a current of 20 pA. The analyte signal was demodulated through a lock-in amplifier (EG&GPARC 5209) operating with a 0.1-s output time constant and 12 dB/octave rolloff. The dc signal was recovered through a low-pass filter/amplifier (Ithaco 1201) operating with a 3-Hz cutoff frequency and digitized with the auxiliary A/D converter of the lock-in amplifier. Data were recorded at 10 samples/s on an IBM personal computer. The separation capillary was 55 cm long, 50-cm effective length, with an i.d. of 75 pm (Polymicro Technologies, TSP/075/375). For certain experiments the procedure of HjertenZ1was used to coat the capillary walls with polyacrylamide in order to reduce electroosmosis. The laser-induced fluorescence detector used consisted of a 442-nm He-Cd laser (Liconix Model 4230NB) attenuated to 10 mW with neutral density fdters, a 5X microecope objective (Rolyn 80.3045) to focus the laser light into the capillary, a 100 mm focal length achromat to collect the fluorescence, a sharp cut orange glass fiter (SchottGG475) to reject the laser line, and a photodiode (EG&G,HUV 40oO). The laser deflection refractive index detector has been described previ~usly.'~It consisted of a 3-mW 750-nm diode laser (Oriel 79401) focused onto the capillary through the 5X objective. Half the beam from the capillary was blocked with a razor blade and the remaining light was focused with a 100 mm focal length lens onto a position-sensing photodiode (United Detector Technology, UDT-LSC/5D). The typical buffer was 10 mM phosphate at a pH of 7. In the case of refractive index detection, Tris-HC1 at a pH of 8 was the running buffer. All sample solutions were prepared in the running buffer. Type I deionized water was used throughout. Sample injection was by electromigration for 3 s at a potential of a 100 V unless indicated otherwise. The sample injection current was 0.2 PA.
RESULTS AND DISCUSSION As described above, electric field modulation induces a radial movement of analyte material to and from the capillary wall synchronous with the modulation frequency. The variation in velocity profile should be strongest close to the capillary w&, in the double layer region, and should disappear at the capillary center. Focusing the laser beam at the capillary wall should maximize the derivative signal, while focusing it a t the center should produce no derivative signal. Figure 1 is an electrophoretogram of Rhodamine 6G, a cationic dye, showing both the demodulated ac and the dc fluorescence signals. In Figure 1A the region close to the wall
where the laser enters the capillary is probed. Figure 1B shows the signals obtained in the center of the capillary. In Figure 1C the signals are obtained at the laser exit wall. The demodulated ac signals obtained a t the laser walls closely resemble the derivative of the normal dc signal obtained anywhere in the capillary. To show this we have numerically differentiated the dc signal of Figure lA, which is plotted together with the demodulated ac signal. The same behavior is obtained a t the opposite wall of the capillary. These observations are entirely independent of the modulation frequency. Over the available range of our apparatus, 50-500 Hz, neither the shape nor the intensity of the ac signals depends on modulation frequency. The resulb of Figure 1can be explained by eq 8. According to this equation the sample concentration at the center of the capillary (r = 0) is unaffected by the voltage modulation. Indeed the peak shape of figure 1B is Gaussian in shape, indicating that it is independent of the modulation voltage. However as the probe moves away from the center (r > 0), the signals obtained depend on the modulation; parts A and C of Figure 1 are both derivatives. A derivative response can be observed even when a large fraction of the axial concentration gradient is probed. In Figure 2 we show the ac and dc response for Rhodamine 6G obtained with a 50 mm focal length cylindrical lens focusing the beam to a 1 mm long line oriented along the capillary length. A derivative is also observed as the ac response in this experiment. The ac response is the normal peak shape when the center of the capillary is probed with the cylinder lens. This behavior is inconsistent with an axial modulation. We define modulation length, ml, as the axial distance covered by the analyte as a result of the superimposed ac field. By definition, ml = L/(2rfrt), where L is the column length, f is the modulation frequency, and rt is the migration time. For the 50-cm effective column length, 160-Hz modulation frequency, and 240-9 migration time of our experiments, ml = 10 pm. Therefore, probing a length of 1 mm should integrate the signal to a normal-shaped peak. Because a derivative is observed, the effect of the ac field cannot be to generate an axial concentration modulation. For anions we can find no conditions under which derivative signals are generated in an unmodified silica capillary. Figure 3 shows the dc and demodulated ac signals from 2,7-dichlorofluorescein, an anionic dye. As in Figure lA, the laser is focused at the entrance wall of the capillary. In the capillary center and at the exit wall the same peak-shaped ac responses are observed. The lack of a derivative signal here suggests that for a derivative to exist, the sample species must approach the double layer. Negatively charged dichlorofluorescein is repelled by the negatively charged capillary wall. It is clear from either eq 8 or 9 that the modulation effect is highest a t larger r, near the walls. If the sample species cannot approach the walls then neither of these equations apply. The results of this experiment are consistent with a radial concentration gradient. In order to examine the effects of electroosmotic flow, experiments were performed in a capillary with the inner wall silanized and coated with a monomolecular layer of noncross-linked polyacrylamide. Coating the capillary wall reduces the magnitude of the electroosmotic flow because silanization eliminates hydroxy groups and creates a thin surface layer of high viscosity. Because there is little or no electroosmosis in this capillary, we shortened the capillary and increased the driving dc field to keep the transit times short. The reduction of the electroosmotic flow in turn reduces the magnitude of the electroosmotic term in eq 8. Therefore, as shown in Figure 4, derivative signals are not observed with a coated capillary even for cations when the laser is focused
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i
Dc
DC
h
I
I 1w
1 9
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I
I
m
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-.
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Figure 2. Electrophoretograms obtained with a 50 mm focal length cylindrical lens. The laser beam is focused to a 1-mm line along the capillary column. The unmarked middle curve is the numerical differentiation of the dc response. The sample and running condltions are the same as in Figure 1.
u ) ( :
. .
I
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Time, sec
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Flgure 3. Electrophoretograms of 0.002 mM 2,7-Dichlorofluorescein in the phosphate buffer. The region probed is the same as that of Figure 1A. The separation voltage is 20 kV with a l 0 k V ac modulating voltage.
I
I 150
MO
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3w
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Flgure 1. Electrophoretograms of 0.01 mM Rhodamine 6G in the phosphate buffer. (A) Signals shown here were obtained by probing the region close to the cpillary wall on the laser entrance side to the capillary column. The unmarked signal in the middle is the numerical differentiationof the dc signal obtained through the SPECTRACALC program. (B) Signals were obtained at the center of the capillary. (C) Signals were obtained at the laser exit wail. The total column length is 55 cm, and the separation voltage Is 15 kV with a 7 k V modulating ac voltage.
Figure 4. Electrophoretograms of 0.01 mM Rhodamlne 6 0 in the phosphate buffer. The capillary column used in this case is coated with IxKIcTos&HnkBd polyacrylamide and the region probed is the same as that of Figure 1A. Sample Injection is through electromlgration for 3 s at 1 kV with a current of 1 MA. The column length is 40 cm, and the driving dc voltage is 20 kV with a 10-kV modulating ac voltage.
near the capillary wall. The demodulated ac signal has the same shape as its dc counterpart. Therefore, elimination of the conditions for generating a radial concentration modulation eliminates derivative-shaped signals. This experiment further confirms the source of these signals as a radial concentration gradient.
We also observe that methanol, dimethyl sulfoxide, and acetonitrile, each at a concentration of 1%in the running buffer, give derivative signals when the wall region is probed. For these measurements, we employ laser deflection refractive index detection. Figure 5 shows electrophoretograms of 1% v/v dimethyl sulfoxide and 1%v/v acetonitrile. As before,
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Electrophoretograms of 1 % dimethyl sulfoxide and 1 % acetonitrile each in 0.01 M Tris-HCI buffer at a pH of 8. The top two curves are the dc and ac responses of dimethyl sulfoxide. The bottom two curves are the dc and ac responses of acetonblle. Samples were electroinjected for 5 s at 3 k V with a current of 2 MA. The capillary length is 30 cm with a l 0 k V separation voltage and a 5-kV modulating voltage. Figure 5.
the derivative ac signals are obtained only at the capillary walls. Probing the capillary center gives ac and dc responses with the same band shapes. McIntire et al. have suggested that methanol and dimethyl sulfoxide interact with the capillary wall much more strongly than acetonitrile.22If it is assumed that the derivative signals that we observe with cations are due to adsorption and desorption which lag the instantaneous concentrations, then we would expect to see normal bands for the AC signals of a weakly adsorbed ion or molecule. That we see derivatives with acetonitrile, a neutral reported to have no adsorption to the capillary surface under CE conditions, indicates that derivative shapes cannot be explained in terms of adsorption. Taken together, our experiments demonstrate that derivative-shaped signals observed in driving voltage modulation result from a radial concentration gradient generated as a result of alternation between plug and laminar flow induced by the superimposed ac voltage. The strongest variation occurs near the capiUary walls, and so derivative-shaped signals are observed when the wall region is probed. A t the exact center of the capillary, there is no radial modulation, and it is weak on either side of the center. Therefore probing the center of the capillary yields normal signals as the ac response. By classical Debye-Huckel calculation^,^^ the electrical double layer is about 3 nm thick at the 0.01 M buffer concentration we employ. Since the depth of field (d = X/8(NA)2, where NA is the numerical aperture of the objective) of our microscope objective is about 2.6 pm and its position can be set to within 3 4 pm, we can probe a region close to the surface but we cannot actually probe nanometer depths. Yet we see only normal-shaped curves for anions, suggesting that the double layer in CZE might extend further out than the Debye-Huckel theory suggests. With our current instrumental system, it is not possible to accurately determine the double-layer thickness. We note however that other worke r ~ ~ have ~ , proposed ~ ~ , ~ that~ the , ~double ~ layer actually extends out to 1pm or further under CZE conditions.
CONCLUSIONS In this work we have demonstrated that analyte velocity modulation generates a radial concentration gradient rather
than an axial concentration gradient. A principal motivation for our work has been exploration of voltage modulation as a technique for improving the signal/noise ratios of existing detectors. Improvements are possible only where there is excess detector noise to be circumvented and do not depend on whether the demodulated signal has the form of the derivative of the normal detector response. We have used laser fluorescence here as a convenient and sensitive detection technique. We do not expect to obtain improved performance, and indeed the signal to noise ratio of the demodulated lock-in signal at the center of the capillary is the same or as much as a factor of 2 smaller than that of the corresponding dc signal. This is to be expected since in fluorescence detection there is negligible excess noise to reject. On the other hand, we have previously that analyte velocity modulation can improve the performance of a laser beam deflection refractive index detector. Our results also indicate that the double-layer thickness in our electrophoresis capillary could be larger than suggested by the Debye-Huckel theory. If the technique of analyte velocity modulation is used together with confocal fluorescence microscopy, which can achieve axial resolutions of 0.3 pm or less,26the double-layerthickness might be probed simply and directly. In principle the resolution will be limited only by the focal depth of the confocal microscope. Experiments toward that goal are in progress.
ACKNOWLEDGMENT We wish to thank Maureen Lanan for valuable discussions and insights. This work has been supported in part by the National Institutes of Health through Grant GM-37006. REFERENCES
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(1) . . Mlkkers. F. E. P.:Everaets. F. M.: Verheaoen. Th. P. E. M. J. Chromatogr. 1979, 169, 11-20. (2) Jorgenson, J. W.; Lukacs, K. D. Anal. Chem. 1981, 53, 1298-1302. (3) Karger, B. L.; Cohen, A. S.:Guttman. A. J. Chromafogr. 1989, 492, 585-612. .-... -.
(4) Ewlng, A. G.; Wallingford, R. A,; Olefirowicz, T. M. AM/. Chem. 1989, 6 1. 292A-303A. (5) Knox, J. H.; McCormack, K. A. J. Li9. Chromafogr. 1989, 12, 2435-2470. (6) Lauer, H. H.; McManiglll, D. Anal. Chem. 1988, 58, 166-170. (7) Jorgenson, J. W.; Lukacs, K. D. Science 1983, 222, 266-272. (8) Oleflrowlcz, T. M.; Ewing, A. G. Anal. Chem. 1990, 6 2 , 1872-1876. (9) Huang, X.; Shear, J. B.; Zare, R. N. AM/. Chem. 1990, 6 2 , 2049-2051. (10) Cheng. Y.-F.; Dovichi, N. J. Science 1988, 242, 562-564. (11) Waliingford, R. A.; Ewlng, A. G. Anal. Chem. 1987, 59. 1761-1766. (12) Kuhr, W. G.; Yeung, E. S. Anal. Chem. 1988, 6 0 , 1832-1834. (13) Chen, C.-Y.; Demana, T.; Huang, S.-D.; Morris, M. D. Anal. Chem. 1989, 6 1 , 1590-1593. (14) Demana, T.; Chen, C.-Y.; Morris, M. D. J. High Resolut. Chrometogr. 1QQO. ...-, 13. . ., 587-589 .- . .. .. (15) Demana, T.; Lanan. M.; Morris, M. D. Anal. Chem., 1991, 63, 2795-2797. (16) Pretorius. V.; Hopklns, B. J.; Schieke, J. D. J. Chromafogr. 1974, 9 0 , 23-30. (17) Taylor, G. R o c . R . SOC.London 1953. A219, 166-203. (18) Probstein, R. F. Physicochemlcal Hydrodynamics; Butterwotths: Boston, 1989; pp 85-88. (19) Rice, R. L.; Whitehead, R. J. Phys. Chem. 1985, 6 9 , 4017-4024. (20) Datta, R.; Kotamarthi, V. R. AIChE J . 1990, 3 6 , 916-926. (21) Hjerten, S. J. Chromatogr. 1985, 347, 191-198. (22) VanOrman. B. 8.; LhrersMge, 0. 0.; Oleflrowlcz. T. M.; Ewing, A. G.; McIntire, G. L. J. Mlcrocolumn Sep. 1980, 2 , 176-180. (23) Hunter, R. J. Founu'afbns of Colb& Science; Oxford University Press: New York, 1989; p 332. (24) Stevens, T. S.;Cortes, H. J. Anal. Chem. 1983, 55, 1365-1370. (25) Martin, M.; Gulochon, G. Anal. Chem. 1984, 56, 614-620. (26) Wilson, T. I n Confocal Mlcroscopy; Wilson T., Ed.; Academic Press: New York, 1990; p 58.
RECEIVED for review August 12,1991. Accepted November 11, 1991.
