Anal. Chem. 1991, 63,859-867 (28) Langer, P. R.; WaMrop, A. A.; Ward, D. C. Roc. Natl. Acad. Sci. U . S . A . 1981, 78, 6633-6637. (29) Jabbnski, E.; Moomaw, E. W.; Tullis, R.; Ruth, J. L. Nucleic Acids Res. 1988. ..- -, 14. . . , 6115-8128 - . .- - .- -. Auron, P. E.; Sullivan, D.; Fenton, M. J.; Clark, B. D.; Cole, E. S . ; Galson, D. L.; Peters, L.: Teller, D. BioTechniques 1988, 6, 347-353. Mullis, K. B.; Faloona. F. A. Methods Enzymol. 1987, 755, 335-350. Glbbs. R . A. Anal. Chem. 1990, 62, 1202-1214. (33) Wang, A. M.: Doyle, M. V.; Mark, D. F. Roc. Natl. Acad. S d . U . S . A . 1989, 86, 9717-9721. (34) Oakley, B. R.; Kirsch, D. R.; Morris, N. R. Anal. Biochem. 1980, 705, 36 1-363
(35) Towbin, H.; Staeheiin, T.; Gordon, J. Proc. Natl. Acad. Sci. U . S . A . 1979, 76, 4350-4354. (36) Lee, C.; Hu, S.-E.; Lok. M. S.;Chen, Y . C.; Tseng, C. C. BioTechniuues 1988. 6.216-224. (37) dighetti, P. G.; Gianazza, E.; Gelfi. C.; Chiari. M.; Singa, P. K. Anal. Chem. 1989, 61, 1602-1612. (38) Scopes, R . K. Protein Purification Principles and Practice, 2nd ed.; Springer-Veriag: New York, 1987. (39) Lucas, C.; Nelson, C.;Peterson. M. L.; Frie, S.; Vetterlein, D.: Gregory, T.; Chen. A. B. J . Immunol. Methods 1988, 713, 113-122. (40) Bloom, J. W.: Wong, M. F.: Mitra. G. J . Immunol. Methods 1989. 777, 83-89.
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(41) Baker, R. S.;Ross, J. W.: Schmidtke, J. R.; Smith, W. C. Lancet 1981, 2 , 1139-1141. (42) Anicetti. V. R.; Fehskens, E. F.; Reed, B. R.; Chen, A. 8.; Moore, P.; Geier, M. D.; Jones, A. J. S. J . Immunol. Methods 1988, 9 1 , 213-224. (43) Anicetti, V. R.; Simonetti, M. A,; Blackwood, L. L.: Jones, A. J. S.; Chen, A. B. Applied Biochem. Biotechnol. 1989, 22, 151-168. (44) Olson, J. D.; Panfili, P. R.: Armenta, R.; Femmel. M. B.; Merrick, H.; Gumperz, J.; Goltz, M.; Zuk, R. F. J . Immunol. Methods 1990, 734, 71-79. (45) Hafeman, D. G.; Parce, J. W.; McConnell, H. M. Science 1988, 240, 1 182- 1 185. (46) Bousse, L.; Kirk, G.; Sigal, G. Sensors Actuators 1990, 67, 555-560. (47) Briggs, J.; Kung, V. T.; Gomez, B.; Kasper. K. C.; Nagainis. P. A,; Masino, R. S.; Rice, L. S.; Zuk, R. F.; Ghazarossian, V. E. BioTechniques 1990, 9 , 598-606. (48) Kung. V. T.; Panfili. P. R.; Sheldon, E. L.: King, R. S.; Nagainis, P.; Gomez, B.; Ross, D. A.; Briggs, J.: Zuk. R. F. Anal. Biochem. 1990, 787, 220-227. (49) King, R. S.;Panfili, P. R. J . Biochem. Biophys. Methods, in press. (50) All inquiries about unpublished comparative studies should be directed to P. R . Panfili. (51) Olson, J. D.; Panfili, P. R.; Zuk, R. F.; Sheldon, E. L. Unpublished results.
ARTICLES
Comparison of Experimental and Calculated Results in Overloaded Gradient Elution Chromatography for a Single-Component Band M. Zoubair El Fallah and Georges Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, and Division of Analytical Chemistry, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6120
The elutlon profiles of hightoncentrationbands of pure compounds In gradient elution chromatography are Investigated. Emphasls Is placed on the Influences of the amount of sample injected and of the gradlent program. At moderate gradient rates, the band profiles recorded for 2-phenylethanol agree very well wlth those predlcted by numerlcal integratlon of the mass balance equation, followlng a procedure widely used in isocratlc chromatography which is now conventional. For steep gradients, the agreement is still good but differences appear, probably related to the higher Importance of the equlllbrlum Isotherm data at high solute concentrations.
INTRODUCTION Since its introduction nearly four decades ago ( 1 , 2 ) ,gradient elution liquid chromatography has become one of the most widely used techniques in analytical chromatography. A large number of theoretical studies have investigated the relationships between the retention times or volumes of analytes or their resolution and the gradient profile used for different chromatographic systems (3-7). Such relations allow the prediction of the chromatograms obtained under different experimental conditions and facilitate their optimization and
* To w h o m correspondence should be sent a t t h e U n i v e r s i t y of Tennessee.
the development of new analytical procedures. Furthermore, rapid developments in the field of analytical instrumentation and the design of advanced programmable solvent delivery systems have led to the commercial availability of numerous chromatographic instruments that provide the precision and the reproducibility required for quantitative analytical procedures. The technique has become a popular analytical method. Gradient elution liquid chromatography has found one of its most current areas of application in the analysis of the complex mixtures of clinical, biochemical, or environmental origins (8,9). Some of the most important advantages of this mode of chromatography are a reduction of the analysis time, an enhancement of the detection sensitivity, and an increase in the useful column peak capacity. Recently, with the increase in the need of high-purity bioactive compounds and of other high-value-added chemicals, chromatography has established itself as a method of choice in extracting or preparing high-purity products (10-14). This is especially due to the high degree of flexibility provided by the many different modes available and by the numerous implementations available for each mode. Whereas gradient elution does not seem attractive in the field of large-scale industrial separations and purifications by preparative chromatography, its use for laboratory scale applications has drawn much interest and has become an important area of investigation (15-18). In industrial applications, the cost of regenerating the chemicals used to prepare a mobile phase of constantly changing composition is higher than for an
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isocratic mode, when either one or a few closely eluted components have to be recovered. The long-retained impurities can be eliminated less expensively through the use of a preseparation stage, a precolumn (backflushed or not) or a step gradient. In laboratory scale applications, on the other hand, the production time is more important than the production cost. The mobile phase is frequently wasted while more complex mixtures are often separated and the fractions collected for the identification of unknown components are eluted over a wider range of retention volumes. In opposition to the significant contributions made toward the understanding of the band profiles in nonlinear isocratic elution chromatography over the last 45 years (19-29), few theoretical studies of nonlinear gradient elution chromatography have been made, even in the simplest case of a single-component problem. There is no closed-form solution of the mass balance equation in the case of gradient elution, even in the framework of the ideal model (column of infinite efficiency). It is not possible to relate simply the characteristics of the profile to the thermodynamic functions of the retention data and to the gradient parameters. Among the significant contributions made to this problem, Frey has recently developed an asymptotic solution of the elution band profile in the case when the solute has a linear isotherm (30). Besides this theoretical attempt, computational methods have provided the only workable approach to the calculation of elution band profiles in the single-component problem, and of course in the more important two-component problem, which permits the investigation of the extent of separation afforded by the chromatographic column. Snyder et al. (31-33) have used the Craig model to simulate band profiles in overloaded gradient elution chromatography, whereas Antia and Horvkth (34)have developed a more sophisticated, robust, and fast solution of the system of mass balance equations using the orthogonal collocation method to solve the mass balance equation of a single component or of the components of a binary mixture. So far, however, no significant experimental results have been reported regarding the elution band profiles in overloaded gradient elution chromatography and the validity of predictions based on the current theory of nonlinear chromatography. The main goal of the present work is to provide a detailed comparison between the experimental band profiles obtained at various column loadings and with different gradient programs and the profiles calculated from the relevant thermodynamic data (i.e., the equilibrium isotherms measured under isocratic conditions covering most of the useful range) and from the average column efficiency. In this work we assume that the strong solvent used to make the concentration gradient is not adsorbed by the stationary phase. The strong solvent does not compete with the solute for adsorption, and no system peak phenomena are observed. The concentration gradient injected in the column is transported to the exit with a delay but without any changes in its profile. This simplifying assumption permits the use of a single-component Langmuir isotherm and a single mass balance equation.
THEORY Ideal Model of Chromatography. In this model, we assume that the kinetics of mass transfer in the column is infinitely fast and that the axial dispersion does not affect the band profile (19-29). In such a case, the column efficiency is infinite and the solute band profile is determined solely by (i) its mass balance equation
where C, is the solute concentration in the mobile phase, C, is the solute concentration in the stationary phase at equilibrium, F is the column phase ratio (ratio of the volumes available to the mobile phase between and within the particles), uo is the mobile-phase linear velocity, and z and t are the abscissa along the column and the time, respectively, (ii) its equilibrium isotherm
and (iii) the initial and boundary conditions that describe the experiment in mathematical terms. In gradient elution, the initial condition states that the column is filled only with the pure weak mobile phase at time origin:
C,(O,Z) = 0
cP(0,Z) = 0
0cz
