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of on-line detection, the image system is also a universal detector and therefore allows detection of all analytes without tedious derivatization procedures. The detection limits of this system of proteins can be estimated to be at the lo4 M level from the signal to noise ratio in Figure 5c. Background correction methods will significantly improve the signal to noise ratio since the light scatter from refractive index defecta inside the capillary c a w s the baseline fluctuations in the present system, as shown in Figure 5. In addition, the second-derivative GauRsian correlation filter could be used to more effectively differentiate random noise from the signal. This imaging system can also be developed into a system for two-dimensional image detection, when a camera is used instead of a one-dimensional diode array. This arrangement will allow simultaneous monitoring of the separations in several capillaries. The future capillary electrophoresis instruments based on this principle will consist of a bundle of capillaries arranged side by side and detected, just as in the electrophoresis performed on gel slabs, by the two-dimensional image system. Such a system would allow analysis of many samples a t the same time.
REFERENCES (1) Righeni. P. G. Isoelectric Focusing: Theory, Methodokgy and AppUcations; Eisevier Press: Amsterdam, 1983. (2) Jorgenson, J. W. Anal. Chem. 1088, 5 8 , 743A-760A. (3) Zhu, M.; Hansen, D. L.; Burd, S.; Gannon, F. J . Chromatogr. 1989, 480, 311-319.
Kilar, F.; Hjerten, S. El8ctmphwssls 1080, 70, 23-29. Wehr, T.; Zhu, M.;Rodriguez, R.; Burke, D.; Duncan, K. Am. Blotechno/. Lab. 1990, 8 , 22-29. 746-751l K. C.; Koutny, L. B.; Yeung, E. S. Anal. Chem. 1091, 63, Chan,
. _- ."_.
Thormann, W.; Tsai, A.; Michaud. J.; Mosher, R. A.; A,; Bier, M. J . Chromatogr. 1087, 389, 75-86. Thormann, W.; Mosher, R. A.; Bier, M. J . Chromatogr. 1088. 351, 17-29. Righetti, P. G.; Pagani, M.; Glanazza, E. J . Chromatogr. 1075, 109, 341-356. Pawiiszyn, J. Spectrochim. Acta Rev. 1000, 73, 311-354. Pawliszyn, J. Anal. Chem. 1088. 60, 2796-2801. McDonnell, T.; Pawiiszyn, J. Anal. Chem. 1901, 63, 1884-1889. Pawliszyn, J.; Wu, J. J . Chromatogr. 1991, 559, 111-118. Wu, J.; Pawiiszyn, J. Anal. Chem., preceding article in this issue. Merzkirch, W. Flow Visualization;Academic Press: New York, 1987. Grushka, E.; Israeli, D. Anal. Chem. 1990, 62, 717-721. Kiiar, F. J . Chromatogr. 1901, 545, 403-406.
Jiaqi Wu Janusz Pawliszyn* Department of Chemistry University of Waterloo Waterloo, Ontario N2L 3G1, Canada
RECEIVED for review July 25,1991. Accepted October 18,1991. This work was supported by the Natural Sciences and Engineering Research Council of Canada. Donation of the high-voltage power supply by Beckman Instruments Inc. is appreciated.
Analytical Solution for Dispersion in Capillary Liquid Chromatography with Electroosmotic Flow Sir: The relatively recent techniques of open-tubular capillary liquid chromatography (LC) (1) and capillary electrophoresis (CE), (2,3)have both demonstrated a considerable potential for very efficient separation of highly complex mixtures, although some practical problems still remain to be resolved. The eluant flow in capillary LC is pressure driven, whereas that in CE is electroosmotic. Electroosmosis has also been proposed for capillary LC since it can provide substantially lower zone dispersion as compared with pressure-driven flows ( 4 , 5 ) . This is due to the considerably flatter velocity profile of electroosmotic flow as compared with the parabolic laminar velocity profile of pressure-driven flow. Thus, Pretorius et al. ( 4 ) and Tsuda et al. (5) obtained plate heights for unretained solutes that were from 10 to about 30 times smaller than the corresponding values for pressuredriven flow. Martin and Guiochon (6) and Martin et al. (7)have developed the theory of axial dispersion in open-tubular capiUary liquid chromatography with electroosmotic flow for the case of low {-potential, based on the approach of Golay (8)and Aris (9). However, for mathematical simplicity, they approximated the theoretical electroosmotic velocity profie (10)by empirical expressions and then obtained approximate analytical expressions for the plate height. Our objective here is to provide a completely analytical solution for the plate height in capiuary liquid chromatography with electroosmotic flow that is based on the exact analytical expression for the electroosmotic velocity profile for the case of small {-potential (10).The result is also applicable to the case of zone-spreading in CE for a neutral solute undergoing adsorption onto the capillary walls. The expression agrees
well with the numerical results of Martin and Guiochon (6) and Martin et al. (7) and reduces appropriately to the expression for plate height in CE for a neutral solute without any wall interaction (I1,12).
THEORY Some of the assumptions involved in the derivation are enumerated by Datta and Kotamarthi (11). The plate height, H,takes the following form with the assumption of negligible resistance to mass transfer in the stationary phase:
where (v,) is the cross-sectional average velocity in the axial direction, z, Di is the solute diffusion coefficient in the mobile phase, a is the capillary radius, and C, is a dimensionless coefficient, dependent on the form of the velocity profile of the eluant, for the term representing the resistance to mass transfer in the mobile phase. For the parabolic velocity profile of Poiseuille flow resulting from a pressure-gradient, C, is given by the well-known Golay equation (8). In general, for a given velocity profiie, C , may be obtained by evaluating the following expression (6),based on the generalized dispersion theory of Aria (9):
where k { is the column capacity factor for the solute i and the definite integrals AI and A2 are given respectively by
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 2, JANUARY 15, 1992
(3)
(4) where
Pb) = ]y2~’4b’) 0 dy’
(5)
Here y = r / a is the dimensionless radial distance. In eq 5 , Q(y) is the dimensionless velocity profile defined by Qb) = u,b)/(u*) (6) where u,(y) is the axial velocity profile. The electroosmotic velocity profiie, for the case of relatively low [-potential when the Debye-Huckel approximation can be invoked to linearize the Poisson-Boltzmann equation, is given by (10, 11)
(7) where Io is the modified Bessel function of the first kind of order zero, 4 is the electrokinetic capillary radius, made dimensionless with the Debye length, A, i.e., 4 = a / A , and the characteristic electroosmotic velocity ueo is given by the Helmholtz-Smoluchowski equation
where dV/dz is the applied voltage gradient or the electric field strength and ueois the electroosmotic mobility given in terms of the surface [-potential, permittivity of the eluant, e, and the eluant viscosity, F , by ueo = ~ C / P (9) The area-average electroosmotic eluant velocity is obtained by integrating the velocity given by eq 7 over the capillary cross section (10) (uz) = ueo(1 - a) where the function 7 is 2 Il(4)
v=--
4 Io(4)
where I, is the modified Bessel function of the first kind and first order. The function 9 has the following limiting forms: 7 1.0 as 4 0, and 7 214 as 4 m. For 4 1 3, which includes practically all cases of interest, 7 may be estimated accurately (within 1%) by the following approximate expression (6): 2 1 1 1 ... (12) 7%------4 $2 443 444
-
-
-
-
+
Equations 7 and 10 are next substituted into eq 6 to obtain Q(y). Upon substitution of Q ( y ) into eq 5 and by carrying out the indicated integrations in eqs 3-5 in which use is made of Lommel’s integral (13),the following expression finally results for the coefficient C, in the expression for the plate height: 6
1
(13)
where the integrals A , and A , are given, respectively, by the terms within the first and second set of curly braces on the right-hand side of eq 13.
RESULTS AND DISCUSSION Comparison with Previous Results. The analytical expression for C, given by eq 13 is compared below with the corresponding semianalytical results of Martin and Guiochon (6) and Martin et al. (7). Martin and Guiochon (6) divided the capillary cross-section into two regions separated by y = 1 - p , where p is the dimensionless distance of the dividing boundary from the capillary wall. They then approximated the theoretical velocity profile, eq 7, empirically by u,b) = u,, U,O
u,(y) = - - [ ( 2 p P2
y I 1- p
- 1) + 2 0 - p)y - y2]
(14) 1- p I y 5 1 (15)
These expressions were substituted into eq 6, and the integrals AI and A , were then evaluated. The value of the fitted parameter p, for a given $, was estimated by equating the average velocity obtained from these approximate expressions to the theoretical average velocity given by eq 10. The empirical velocity profile used to Martin et al. ( 7 ) , on the other hand, is uzb) = ueo(1
- Y”)
(16)
where the fitted exponent n was evaluated in terms of 4 by equating the average velocity obtained by eq 16 to the actual one. Equation 16 resulted in an expression for C, that was both simpler and more accurate ( 7 ) than that of Martin and Guiochon (6). These authors further assumed that u,,, = oeo,where u, is the maximum velocity at the capillary axis. As seen from eq 7, this is accurate only for relatively large 4 ( 7 ) . A comparison of the analytical result for C,, eq 13, with the semianalytical expressions obtained by Martin and Guiochon (6) and Martin et al. ( 7 ) , as well as with the values evaluated by numerical integration of A , and A2 based on the true velocity profile (6, 3,was done for 4 = 10 and 4 = 50 and with k‘i varying from 0 to w . It was found that the C, values provided by eq 13 agreed very well with the numerical solution, with a maximum relative error of 0.4%. The maximum relative error in C, values given by the approximate expression of Martin and Guiochon (6) was 16.5%, and in the expression of Martin et al. (7) was 7.0%, both at k’i = 0. For higher values of kti, their expressions are more accurate. The limiting form of eq 13 for the case of no adsorption, i.e., for k’i = 0, may also be compared with the corresponding expression obtained by Datta and Kotamarthi (11)for electrokinetic dispersion in CE, wherein typically the eluant flow is purely electroosmotic and generally there is no wall adsorption. Thus, for k : = 0, eq 13 reduces, after some rearrangement, to
This result is the same as that obtained by Datta and Kotamarthi (11) for electrokinetic dispersion in CE with electroosmotic flow. For the sake of completeness, the other limiting case of large kfi may also be considered. For k\ a,eq 13 reduces to
-
Equation 17 was also used by Datta and Kotamarthi (11) to obtain the dimensionless electrokinetic radius, 4, by comparison of the theory with the data of Pretorius et al. (4) and Tsuda et al. (5)for plate heights of unretained solutes with electroosmotic flow. The values of 4 thus calculated agreed
ANALYTICAL CHEMISTRY, VOL. 64, NO. 2, JANUARY 15, 1992
0
A
10
0
229
ReSc 7n
U
Figure 1. Variation of the coefficient C, with capillary electrokinetic radius, $, and the capacity factor, k',.
well with the corresponding values calculated by Martin and Guiochon (6) for this data. Variation of C, with kfand $. It is seen from eq 13 that C, is a function only of k i' and $. The two limiting cases of eq 13 for kfi = 0 and k f a have already been considered above in eqs 17 and 18. Figure 1is a three-dimensional plot of C, as a function of $ and k
