Anal. Chem. 1998, 70, 1610-1617
Calculation of the Hydrodynamic Contribution to Peak Asymmetry in High-Performance Liquid Chromatography Using the Equilibrium-Dispersive Model Brett J. Stanley,* Theresa L. Savage, and Jennifer J. Geraghty
Department of Chemistry, California State University San Bernardino, 5500 University Parkway, San Bernardino, California 92407-2397
The equilibrium-dispersive model of chromatography and measurements of sorption isotherms were applied to reversed-phase C18 systems of several solutes with 40: 60 methanol/water mobile phases to calculate the band shapes of eluites in the linear, analytical region of chromatography. These predicted profiles were compared to experimental profiles to determine the error associated with the equilibrium-dispersive model. This error is associated with the assumption of a uniform plug flow profile of mobile phase through the column. Analysis of the results of several systems and columns illustrates the asymmetry that exists in the actual mobile-phase flow profiles. The model error asymmetry decreases for higher efficiency columns as expected and increases in the form of peak tailing for lower efficiency columns. This behavior suggests that portions of the column are not swept by the mobile-phase flow at the same rate as the rest of the column, indicating an inhomogeneous packing density. The general method described is valuable for determining the relative contributions of thermodynamics and hydrodynamics to the peak asymmetry observed in liquid chromatography. The basic model of chromatography in the linear, analytical region of solute concentration, also termed the region of “infinite dilution”, assumes that retention times are independent of concentration and that the eluted peaks are Gaussian in shape.1 The definition of column efficiency, i.e., the number of theoretical plates or the plate height, is based on the standard deviation of the Gaussian profile.2 Band-broadening mechanisms are in turn summed up using this definition, thus assuming that these mechanisms result in symmetric broadening.2,3 The original van Deemter theory has been modified over the years into several working forms. Many of these equations imply the possibility of asymmetric band broadening through the mass-transfer term, but they are still grounded to the dependent plate height definition, which assumes a Gaussian profile.4-6 (1) Miller, J. M. Chromatography: Concepts and Contrasts; John Wiley & Sons: New York, 1988; Chapter 1. (2) Giddings, J. C. Dynamics of Chromatography; M. Dekker: New York, 1960. (3) van Deemter, J. J.; Zuiderweg, F. J.; Klinkenberg, A. Chem. Eng. Sci. 1956, 5, 271.
1610 Analytical Chemistry, Vol. 70, No. 8, April 15, 1998
The original theory of the Gaussian chromatographic profile can be linked to the assumption of a uniform, plug flow profile of mobile phase through the stationary phase at a constant velocity throughout the column.5,7 With this assumption, a column can be continuously divided into equal regions of plate height, H, from which the plate and rate theories are derived along with the assumption of a linear sorption isotherm. In practice, nonideal peak shapes are often incurred and it is generally recognized that the Gaussian shape is only a simplification, an approximate working model. Several studies have suggested alternative chromatographic profiles, such as the exponentially modified Gaussian, which better approximate the peak shapes typically obtained in chromatography.8 However, the fundamental underpinnings of why asymmetry typically exists has not progressed well in a theoretical or quantitative manner. Multitudes of studies have shown the nature and origin of peak asymmetry in a descriptive, qualitative manner. In reversed-phase liquid chromatography, a common contribution to asymmetry (typically peak tailing) is attributed to the active silanol site, in particular with the separation of basic solutes.9-15 The presence of the active silanol site can be directly ascertained by close examination of bonded-phase densities, which do not add up to the bare silica silanol site densities. Many studies have shown the nature and importance of adsorption onto these sites by various solutes in many different mobile-phase systems. Masstransfer considerations have also been implicated as possible contributors to peak asymmetry for several systems.6 (4) Hawkes, S. J. J. Chem. Educ. 1983, 60, 393-398. (5) Guiochon, G.; Golshan-Shirazi, S.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: New York, 1994; Chapter 6. (6) Fornstedt, T.; Zhong, G.; Bensetiti, Z.; Guiochon, G. Anal. Chem. 1996, 68, 2370-2378. (7) Giddings, J. C. Unified Separation Science; John Wiley & Sons: New York, 1991; Chapter 4. (8) Foley, J. P.; Dorsey, J. G. J. Chromatogr. Sci. 1984, 22, 40. (9) Fairbank, R. W. P.; Wirth, M. J. Anal. Chem. 1997, 69, 2258-2261. (10) Ko ¨hler, J.; Chase, D. B.; Farlee, R. D.; Vega, A. Kirkland, J. J. J. Chromatogr. 1986, 352, 275. (11) Shapiro, I.; Kolthoff, I. M. J. Am. Chem. Soc. 1950, 72, 776. (12) Unger, K. K.; Becker, N.; Roumeliotis, P. J. J. Chromatogr. 1976, 125, 115. (13) Sadek, P.; Carr, P. W. J. Chromatogr. Sci. 1983, 21, 314. (14) Landy, J. S.; Ward, J. L.; Dorsey, J. G. J. Chromatogr. Sci. 1983, 21, 314. (15) Trushin, S.; Kever, J. J.; Vinogradova, L. V.; Belenkii, B. G. J. Microcolumn Sep. 1991, 3, 185. S0003-2700(97)01096-2 CCC: $15.00
© 1998 American Chemical Society Published on Web 03/06/1998
The above causes of peak asymmetry are all thermodynamic or kinetic in origin and deal with the chemical interactions of the solute between the mobile and stationary phases. Hydrodynamic contributions that have nothing to do with the interfacial surface chemistry typically have been attributed to extracolumn band broadening.16,17 Recently the possibility of an inhomogeneous stationary-phase packing or mobile-phase flow profile has been reexamined as a significant contribution to peak asymmetry.18-22 It is likely that several contributions cause the observed result of peak asymmetry in any given system. For optimally packed columns, the mobile-phase flow profile should be close to uniform and thus not a significant contributor; for systems where the extracolumn volume is minimized, this contribution is negligible especially for longer columns and retention times; for stationaryphase/solute interactions which are nearly constant everywhere on the stationary-phase surface, heterogeneous adsorption is minimized; for many stationary-phase/solute interactions, masstransfer limitations have been shown to be negligible as the kinetics are very fast. Of course, the best-shaped chromatograms exist when all these factors have been optimized simultaneously for a given system. Oftentimes, this across-the-board optimization is not possible or realized. In the development of new stationary phases and separations, it may not be obvious which contribution(s) are operative or most important and thus require attention. This can add to the development cost and hinder the production of new columns. In this paper, we demonstrate how to quantitatively determine the relative contributions of interfacial or “thermodynamic” interactions and “hydrodynamic” considerations to overall peak asymmetry using a few simple experiments. Reversed-phase, C18 systems were studied. These systems have been shown to result in fast kinetics, so that mass-transfer limitations should be negligible and any thermodynamic contribution to peak shape is observable in an appropriately sampled sorption isotherm. THEORETICAL SECTION Chromatographic Model. The equilibrium-dispersive model (E-D model) of chromatography is based on the mass balance equations which describe the movement of a zone of component concentration through a stationary phase in which sorption can occur.23 Solutions of this model have been extensively studied by Guiochon and co-workers for nonlinear systems in which the curvature of the isotherm at high concentrations is a large contributor to the peak shapes observed in these systems.24 This model is based on the system of partial differential equations that describe the migration of component i through a column: (16) Katti, A. M.; Czok, M.; Guiochon, G. J. Chromatogr. 1991, 556, 205. (17) Dolan, J. W. LC-GC 1996, 14, 562. (18) Guiochon, G.; Farkas, T.; Guan-Sajonz, H.; Koh, J.-H.; Sarker, M.; Stanley, B. J.; Yun, T. J. Chromatogr., A 1997, 762, 83-88. (19) Farkas, T.; Chambers, J. Q.; Guiochon, G. J. Chromatogr., A 1994, 679, 231-245. (20) Yun, T.; Guiochon, G. J. Chromatogr., A 1994, 672, 1-10. (21) Tallerek, U.; Baumeister, E.; Albert, K.; Bayer, E.; Guiochon, G. J. Chromatogr., A 1995, 696, 1-18. (22) Bayer, E.; Baumeister, E.; Tallarek, U.; Albert, K.; Guiochon, G. J. Chromatogr., A 1995, 704, 37-44. (23) Haarhoff, P. C.; Van der Linde, H. J. Anal. Chem. 1966, 38, 573. (24) Guiochon, G.; Golshan-Shirazi, S.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: New York, 1994; Chapter 10.
∂qi ∂Ci ∂Ci ∂2Ci +F +u ) Da,i 2 ∂t ∂t ∂z ∂z
(1)
where C is the component concentration in the mobile phase, q is the concentration in the stationary phase, F is the column phase ratio, u is the linear mobile-phase velocity, Da,i is the apparent dispersion coefficient, t is time, and z is distance along the column. These equations apply in chromatography when the mass-transfer kinetics are fast. Mobile-phase components can be neglected in the description of eluite band profiles if they are much less strongly sorbed in the stationary phase than the eluite component of interest. The apparent dispersion coefficient is dependent on the plate height. Although no analytical solution exists for eq 1, a numerical solution exists in which the numerical dispersion of the algorithm matches the apparent axial dispersion when a forward-backward finite difference scheme is used.24 This finite difference method solves eq 1 for the band profile C(t), at z ) L (the length of the column), by dropping the dispersion term (right-hand side of eq 1), as j C n+1 ) C jn + (h/T)(G jn - G j-1 n )
(2)
G(C) ) (1/u)(C + Fq)
(3)
where
h and τ are the space and time increments, respectively, and n and j are the space and time indexes, respectively. The space increment, h, is set equal to the estimated, experimental plate height, H. The time increment, τ, is then chosen such that
ui‚T/h ) 2
(4)
where ui is the linear velocity of the eluite component at infinite dilution. This is called the Courant number and has been shown to provide stability to the algorithm and equate the numerical dispersion of the calculation with the required axial dispersion in eq 1 for the case of linear chromatography.25 Knowledge of the isotherm, q ) f (C), and the initial and boundary conditions are required for the solution of eqs 2 and 3. If these conditions are approximated with a rectangular injection profile, any observed peak asymmetry must be modeled by a nonlinear sorption isotherm. In nonlinear chromatography, h should be set greater than H for a convex-upward isotherm (Courant number >2). In this study, we interpret linear chromatography as possibly nonlinear. Any asymmetric band tailing (as occurs with a convex-upward isotherm) would cause an overestimation of H; thus, the value of h used in the algorithm does indeed exceed the true experimental plate height, H. Experimental deviation from a rectangular injection profile can cause peak asymmetry when retention factors are low and the injection volume is relatively high.16 Under all other conditions, peak asymmetry must be attributed to the sorption behavior. The possibility of a heterogeneous flow profile through the column is (25) Czok, M.; Guiochon, G. Anal. Chem. 1990, 62, 189.
Analytical Chemistry, Vol. 70, No. 8, April 15, 1998
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Table 1. Packing Characteristics of Columns Studied
b
column
silica
solventa
efficiencyb
asymmetryb
A B C
Zorbax Pro 10-150 Zorbax Pro 10-150 Vydac 218TP-10410TS
70:30 CHA/CCl4, MA 70:30 IPA/CH2Cl2, IPA c
2110 3290 2290
1.3 1.03 0.93
a First solvent is used to prepare the slurry; second solvent is the packing or pushing solvent. CHA, cyclohexanol; IPA, 2-propanol; MA, methanol. Values listed for p-cresol. Number of theoretical plates for efficiency, tailing factor for asymmetry. c Packed by Vydac, conditions proprietary.
not present in the E-D chromatographic model, because everywhere the local velocity is assumed equal to u. The difference between the calculated and experimental band profiles can be attributed to errors in the E-D model, the isotherm model, or the injection profile. In this study, it is assumed that the injection profile estimate is sufficiently accurate. We show empirically below that the isotherm model chosen is sufficiently accurate. Thus the difference between the experimental and predicted band profiles, which we shall refer to as “difference” profiles (∆ profiles), is assigned to the error present in the E-D model. Some of this error can be attributed to incorrectly modeled axial dispersion, because the numerical algorithm outlined above is only approximate. However, both numerical dispersion and axial dispersion in a homogeneously packed column should be symmetrical. Since we are concerned with the contribution to peak asymmetry, only the unsymmetrical nature of the ∆ profiles can be proposed to originate from a distribution of flow rates through the column. If peak tailing is observed and not predicted by the E-D model (solved with an accurately modeled isotherm), it is due to a portion of the column possessing a local flow rate somewhat lower than the average flow rate through the rest of the column. Isotherm Determination. The sorption isotherms were measured using the frontal analysis (FA) method.26 This method elutes a series of increasing step concentrations of the sample and calculates the amount sorbed at the ith step as
qi ) qi-1 + (Ci - Ci-1)(VF,i - V0)/Va
(5)
where VF,i is the breakthrough volume of the ith step, V0 is the breakthrough volume of an unretained compound, and Va is the volume of the stationary phase. The accuracy of this method is independent of the efficiency of the chromatographic process or its symmetry in the elution mode. The concentration range in which the isotherm is sampled is important. The isotherm must be adequately sampled across the range of concentrations that elute in the experimental band profile, especially at the lowest concentrations. However, it is also important to sample concentrations above that eluted in the injection experiments. This “oversampling” has been shown to increase the accuracy of the determined coefficients of the sorption isotherm model, assuming the chosen model is the correct one.27 The isotherm model chosen for this study is the Langmuir model.28 This model has been illustrated repeatedly to agree well (26) Eltekov, Y. A.; Kazakevich, Y. V.; Kiselev, A. V.; Zhuchkov, A. A. Chromatographia 1985, 20, 525. (27) Guan, H.; Stanley, B. J.; Guiochon, G. J. Chromatogr. 1994, 659, 27-41. (28) Everett, D. H. Trans. Faraday Soc. 1965, 61, 2478.
1612 Analytical Chemistry, Vol. 70, No. 8, April 15, 1998
with experimental results in liquid chromatography, especially reversed-phase HPLC.29 It can conveniently be expressed in the form
q)
aC 1 + bC
(6)
where a and b are numerical coefficients. The a parameter corresponds to the initial slope of the sorption isotherm, i.e., when C is very small. EXPERIMENTAL SECTION Equipment and Materials. All experiments were carried out with a Shimadzu LC-10AS, dual-pump chromatograph equipped with a model SPD-10AV UV-visible detector (Cole-Scientific, Moorpark, CA). The columns studied were 10 × 0.46 cm and were packed with C18-modified silica particles having a diameter of 10 µm. The column packing was performed with a conventional downward-fill system using an air compressor, Haskel amplifier pump, and 35-mL slurry chamber (YMC, Wilmington, NC). Three columns were studied and are described in Table 1. The solutes phenol, p-cresol, and benzene were of ACS reagent grade and the methanol was of HPLC grade (Spectrum, Gardena, CA). The water was in-house distilled and deionized. All experiments were performed in 40:60 CH3OH/H2O mobile phase at a flow rate of 1.0 mL/min. The mobile phase was filtered through 0.2-µm filter paper before use. The columns were maintained at 30 °C for all experiments with an Eldex heater (Cole-Scientific). Isotherm Measurements. The sorption isotherms were determined over several orders of magnitude in eluite concentration. One FA experiment encompassed 10 steps over one factor of 10 in concentration. The results from the first step were discarded. The time between steps was 3.33 min. Four average isotherms corresponding to four concentration ranges were spliced together to construct the overall isotherm. Thus, for each sample, 36 isotherm data points ranging from approximately 1 × 10-6 to 1 × 10-2 M were obtained, for example, at mobile-phase concentrations 2 × 10-6, 3 × 10-6, ... , 1 × 10-5; 2 × 10-5, 3 × 10-5, ... , 1 × 10-4; 2 × 10-4, 3 × 10-4, ... , 1 × 10-3; and 2 × 10-3, 3 × 10-3, ... , 1 × 10-2 M. Three experiments were performed for each range to provide a sound average of the isotherm data. However, it was determined that the isotherm is precise to within 1% RSD when the column is well conditioned and equilibrated, so that in general, three trials for each range is not necessary for adequate results. The isotherms were fit to the Langmuir (29) Guiochon, G.; Golshan-Shirazi, S.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: New York, 1994; Chapter 3.
isotherm using nonlinear regression with the JMP/SAS software (SciTech, Chicago, IL). The total void volume, V0, in eq 5 was determined with an FA experiment utilizing uracil as an unretained component. Four steps were measured and the average taken from the results of the latter three. The adsorbent volume, Va, was determined as the difference between the empty, geometrical column volume and the column void volume. The column void volume was determined as the difference between the total void volume and the extracolumn void volume. The extracolumn void volume was determined by replacing the column with a low dead volume union and repeating the uracil FA experiment. This latter experiment need only be determined once if the system and tubing remain constant. The column and total void volumes must be measured for each new column studied and should remain constant for each as a function of time. Elution Experiments. The elution band profile data were obtained with 10-µL injections of the solutions of the eluites studied (solution molarities: 1 × 10-4 M for phenol and p-cresol and 1 × 10-2 M for benzene). Retention times, efficiencies, and asymmetry factors were determined with an in-house computer program. Efficiency was calculated according to the standard half-width definition
N ) 5.54(tR/W1/2)2
(7)
and asymmetry as the ratio of the area to the right of the peak maximum to the area to the left of the peak maximum at 10% peak height (tailing factor, TF)
TF ) Ar/Ai
(8)
Calculation of Band Profiles. The predicted band profiles were determined with an in-house program based on eqs 2 and 3, initial and boundary conditions based on the concentration and volume injected (rectangular pulse) and the column dimensions, the flow rate, the isotherm coefficients of the determined Langmuir isotherm, the phase ratio, and the number of plates observed for the eluted component. The extracolumn retention time was determined with a union replacing the column for the injection experiments and added to the retention times output by the E-D model. Agreement with experiment was determined by comparing the retention times of the eluite as well as the hold-up times of the nonretained compound (uracil). A 1% relative error was used as a quality control measure. If the retention times differed by more than 1%, further comparisons were not attempted. Peak areas were also compared to ensure the concentrations were correct (detector calibration can be determined directly from the FA experiments). This was estimated only visually with a comparison criterion of less than ∼10% difference being satisfactory, because peak shape and retention time were not variable over a concentration range significantly larger than this. Observed inaccuracies in retention times between predicted and experimental were due primarily to two causes: (1) systematic systemderived errors in the isotherm or peak measurement or (2) inaccurate isotherm model. With careful measurement, the retention times could be predicted to within 1% with the Langmuir isotherm for phenol, p-cresol, and benzene on all three columns.
Figure 1. Frontal analysis experiment for p-cresol in 40:60 CH3OH/H2O on column B. Concentration range corresponds to 0.00010.009 M. Breakthrough volumes, VF,i, are determined from the inflection points of each step. The first step, 0.0001 M, is discarded.
Calculation of ∆ Profiles. If the predicted vs experimental retention times (and peak areas) were determined to be in agreement with each other, the retention times of the experimental profiles were adjusted (shifted) so that the retention times of the peak maximums matched exactly. The predicted and experimental profiles were also divided by their respective peak areas. This two-step normalization procedure assured that random fluctuations in the amount injected, measurement of eluite retention time, or small inaccuracies in the isotherm model did not contribute to the differences observed between the predicted and the experimental profiles. If the isotherm was measured and modeled accurately, the “thermodynamic” contribution to the peak shape was accounted for in the predicted band profile. The difference between the predicted band profile and the experimental band profile is illustrated by calculating the difference between the normalized signals as a function of time:
∆i ) Sexp,i - Spred,i
(9)
where Sexp,i and Spred,i are the experimental and predicted signals, respectively at time i. To calculate the differences at the same times, a cubic spline was placed through the profiles (MathCad, Cambridge, MA) and interpolated at 0.01-min intervals. RESULTS AND DISCUSSION The series of experiments described in the Experimental Section is illustrated in Figures 1-4 for p-cresol on a Zorbax column in 40:60 CH3OH/H2O mobile phase (column B). The column efficiency for this particular column was 3300 plates (for p-cresol), yielding a reduced plate height of 3.0 particle diameters. The agreement between the predicted and the experimental band profiles is observed to be good in Figure 3. Subtle differences can be noted. The experimental profiles possess small “wings” that deviate from the predicted profiles at the very beginning and end of the peak. The predicted peak is very close to a perfect Gaussian profile. This is because the measured isotherm is very nearly linear (see Figure 2), especially in the range of concentrations that correspond to the eluted concentrations in the experimental peaks, which are
