2244 (3) (4) (5) (6) (7) (8)
(9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25)
Anal. Chem. 1991, 63, 2244-2252 Shiralwa, T.; Fu no, N. Jpn. J. Appl. m y . 1966, 5 , 886-899. Shkahva, 1.;Furh, N. X-Ray Speclrom. 1874, 3. 84-73. Sparks, C. A&. X-&y Anal. 1975, 79, 19-51, Van Dyck, P. M.; Tkok, S. B.; Van Qrieken, R. E. Anal. Chem. 1966. 58, 1761-1766. Nielson, K. K. Anal. Chem. 1977, 4 9 , 841-848. Nielson, K. K.; Sanders, R. W. The SAP3 Computer Program for Quantitatlve Multislement Analysis by Energy Dispersive X-Ray Fluorescence. Technlcal Report PNL-4173; Paclflc Northwest Laboratory: Rlchland, WA 99352, 1982. Rachlttl. A.; Wegschelder, W. A&. X-Ray Anal. 1986, 30, 143-151. Rachlttl, A.; Wegschelder, W. Anal. Chlm. Acta 1966, 788, 37-50. Sanders, R. W.; Olsen, K. B.; Welmer. W. C.: Nielson. K. K. Anal. Chem. 1963. 55, 1911-1914. Van Espen, P.; Van’t dack, L.; Adams. F.; Van Qrieken, R. Anal. Chem. 1979, 5 1 , 961-967. Van Dyck, P. M.; Van Qrleken, R. E. Anal. Chem. 1960. 52, 1859-1864. Araijjo, M. F.; Van Espen, P.; Van &kken, R. X-Ray Specfrom. 1990, 79, 29-33. Araijjo, M. F.; He, F.; Van Espen, P.; Van Qrieken. R. A&. X-Ray Anal. 1990, 33, 515-520. Bllbrey, D. 8.; Bogart, 0. R.; Leyden, D. E.: Hardlng. A. R. X-Ray Spectrom. 1966, 17, 63-73. Crlss, J. W.; Bifks. L. S. Anal. Chem. 1968, 4 0 , 1080-1088. Crlss, J. W.; M s , L. S.; GHfrich, J. V. Anal. Chem. 1976, 50, 33-37. Stephenson, D. A. Anal. Chem. 1971, 4 3 , 1781-1764. Dapple Systems, Sunnyvale, CA 94086. Crlss, J. W. A&. X-Ray Anal. 1980, 23, 93-97. Tao, (3. Y.; Pdla, P. A,; Rowseau, R. NBSQSC, A Fortran Program for Quantitative X-Ray spectrometrlc Analysis using Xaay Tube Excitetion. NIST Technlcal Note 1213; NIST Washington, DC, 1985. Abramowttz, M.; Segun, J. Handbook of MelhemeHCel Functbns; D e ver: New York, 1988. Vol. 5. Van Espen, P.; Adams, F. X-Ray Spectrom. 1961, 10, 84-68. QMrlch, J. V.; Blrks, L. S. Anal. Chem. 1966, 4 0 , 1077-1080.
(26) Gilkich, J. V.; Burkhalter, P. Q.; WhiUock, R. R.; Warden, E. S.; Birks, L. S. A M I . chem.1971, 43. 934-936. (27) bown. D. 6.; Qilfrkh, J. V. J. Appl. Pnys. 1971, 42, 4044-4048. (28) bown, D. B.; QlMich, J. V. J. Appl. Phys. 1975, 4 6 , 4537-4540. (29) Loomis, T. C.; Kelth, H. D. X-Ray Spectrom. 1976, 5 , 104-114. (30) Kelth, H. D.; LoOmlS. T. C. X-RBy SpeCtrOm. 1976, 5 , 95-103. (31) Tertian, R.; Broll. N. X-Ray Spect”. 1964, 73, 134-141. (32) Pella, P. A.; Feng, L. Y.; Small. J. A. X-Ray Spectrom. 1965. 14, 125- 135. (33) Ebel, H.; Ebel, M. F.; Wernlsch, J.; Poehn, Ch.; Wiaderschwlnger, H. X-Ray Spectrom. 1989, 78, 89-100. (34) Debertln. K.; Helmer, R. G. Qamme- and X-Ray Specbomstrv wtth ~e”C0ndUCtWDstectors ; Elsevler Science Publishers B.V.: Amsterdam, The Netherlands, 1988. (35) Vrebos. B. A. R.; Pella, P. A. X-Ray S F W . 1986. 17, 3-12. (38) Bambynek, W.; Crasemann, 6.; Fink, R. W.; Freund, H. U.; Mark, H.; Swift, C. D. Rev. Modbm Phys. 1972, 44, 716-748. (37) Krause, M. 0. J. Phys. C h m . Ref. Data 1979, 8, 307-329. (38) MCMaster, W. H.; Delgand, N. K.; Malbt, J. H.; Hubbell, J. H. Compllat i n of X-Ray Cross Sections. Lawrence Radiation Laboratory Report UCRL-50174, Sec. 11, Rev. 1; Universlty of Callfornla: Berkeley, CA, 1988. (39) Hubbell, J. H.; Velgele, Wm. J.; blggs: Brown, R. T.; Cromer, D. T.; Howerton, R. J. J. Phys. Chem. Ref. Data 1975, 4 , 471-538. (40) Pohen, C.: Wernlsch. J.: Hanke, W. X-Ray Specirom. 1965, 74, 120- 124. (41) SCOfield, J. H. PhJ’S. Rev. 1969, 179, 9-16. (42) Scofield, J. H. Phys. Rev. 1974, A9, 1041-1049. (43) SCOfleM. J. H. P h y ~Rev. . 1974, A10, 1507-1510. (44) MClller, R. 0. Spectrochemlcel Analysk by X-Rey fluofescencs; Pienum: New York. 1972. (45) Van Espen, P.; Janssens, K.; Nobels, J. Chemom. Intell. Lab. Syst. 1966, I , 109-114.
RECEIVED for review February 20,1991. Accepted July 1,1991.
Gradient Elution Chromatography at Very High Column Loading: Effect of the Deviation from the Langmuir Model on the Band Profile of a Single Component M. Zoubair El Fallah and Georges Guiochon* Department of Chemistry, The University of Tennessee, Knoxville, Tennessee 37996- 1501, and Division of Analytical Chemistry, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 -6120
The elution proflks of very large samples of 2-phenylethanol obtalned with steep concentration gradients are compared with those predicted by theory. The main difflcuiths encountored in this modeling stem from the needs of an accurate detmlnatlon of the ackorption data at high concentrations, an accurate modeling of those data and the determination of the exact InJoctionprofile. I n gradknt elution the high-concentration part of the irothorm influences the band profiles more than In isocratic chromatography. The import a m of an accurate .qullibrknn isotherm k much higher for the accurato dmulatbn of profiles obtained In gradhl eiutlon than for those provlded by isocratic elution, because In the latter case, the bands dilute steadliy durlng their migratlon whlk In the former caw they concentrate. shrllarly, “pies InJectodIn gradknt elution have typlcalty a much larger voim e and are mare dilute. Unwuai profilea are predicted and obsorved In agreement with these predlctions, whon the sample amount le of the order of the column saturatlon capacHy and/or the sample volume is of the order of the retontion volume at the mobile phase composition of the gradient start.
*Towhom correspondence should be sent a t the University of Tennessee. 0003-2700/91/0363-2244$02.50/0
INTRODUCTION In a recent paper (1) the profiles of high-concentration bands of a single component (2-phenylethanol)were recorded under a linear gradient elution mode, using a reversed-phase chromatographic system. These experimental profiles were compared to the band profiles calculated from the measured equilibrium isotherm, using a numerical simulation based on the Craig distribution ( 2 4 ) . In these calculations, a Langmuir adsorption isotherm was fit to the adsorption data and the variation of the two coefficients with the fraction of the organic modifier was assumed to be an exponential decay function. This is the classical assumption of the linear solvent strength model (5). In general, a good agreement was observed between experimental and calculated band profiles, especially when the gradient program was not very steep and for moderate column loading, up to 10% of the column saturation capacity. In this case, the experimental band profiles exhibited shapes very similar to those calculated by other authors who also had assumed a Langmuir adsorption model (6, 7). From these limited results, it could seem that, besides some computational difficulties (I), the study of band profiles in gradient elution would not require more ingenuity nor provide more insight into the phenomena associated with column overloading than previous studies made in isocratic elution (8, 9). @ 1991 Amerlcan Chemical Socbty
ANALYTICAL CHEMISTRY, VOL. 63,
For higher loading factors, however, strange experimental band profiles were recorded. These profiles could not be accounted for by using a Langmuir model, even when performing the computations with the true injection profile, which is different from a narrow rectangular plug (10). Significant deviations of the Langmuir model from the experimental data are certainly expected at high concentrations, and these deviations should increase with increasing eluate concentration. They are due to the failure a t high concentrations of the unrealistic assumption made in deriving the Langmuir isotherm that both the solution of eluate in the mobile phase and the adsorbed solution have an ideal behavior. The influence of marked deviations from the ideal adsorption behavior expressed by the Langmuir model is expected to be more severe in the gradient elution mode than under isocratic conditions. In the latter case, chromatographic bands dilute rapidly during their migration. In the former case, on the contrary, either the bands do not dilute much or, more often and especially for steep gradient programs, they concentrate. Accordingly, the injection profile and particularly ita volume width, as well as the exact isotherm model, play a key role in the determination of the band profile at very high loading factors. The aim of thispaper is to study the influence of these factors on the band profiles, at very high loading factors, up to nearly half the column saturation capacity. The calculated profiles have been derived by using the method previously described (1, 4, 11). EXPERIMENTAL SECTION Equipment. Experiments involving overloaded elution in either the gradient or the isocratic modes were performed on a Hewlett-Packard (Palo Alto, CA) HP-1090 liquid chromatograph with a solvent delivery system, a UV-diode m a y detector operated at 275 nm, and a data station. Although designed to perform gradient elution, the solvent delivery system was also used to carry out the frontal analysis experiments needed to measure the solute adsorption isotherm at various mobile-phase compositions (8). Column and Chemicals. A 25 cm long, 4.6 mm i.d. column was homepacked with YMC 120 A octadecylsilica (YMC, Morris Plains, NJ). The physical characteristics of the lot supplied are as follows. The average particle size is 4.6 Fm, the uniformity coefficient ( D m / D ~is) 1.30. The average pore size is 124 A, the pore volume is 0.99 mL/g, the specific surface area is 319 mz/g, the carbon content is 17.8%, and the hydrogen content is 3.5%. This corresponds to a bonding density of approximately 2 pmol/m2. The column void volume was 2.71 mL (determined by uracil injections). This column was operated at 40 OC, with a mobile-phase flow rate of 1 mL/min. 2-Phenylethanol from Aldrich (Milwaukee, WI) was used as solute, as in our previous study and for the same reasons (I). Water for HPLC was distilled in the laboratory. The acetonitrile (ACN)was from J. T. Baker (Phillipsburg, NJ) and was used as received. Procedures. The solute was dissolved in a 10% ACN solution in water, and the sample solution was injected by using one of the pumps of the solvent delivery system. This method permits the recording of the injection band profies. All gradient programs were linear in concentration (which in reversed-phase chromatography is close to linear in elution strength) and started with the aame ACN concentration (10%). Because of holdup volumes in the line between the solvent delivery system and the column, the concentration gradient does not actually begin at the set time. This delay (38 s) is measured by using uracil and corrected for in the calculations. All the chromatogramsrecorded (absorbance profiles) were transferred to the VAX computer of the University of Tennessee Computer Center and converted into concentration profiles, using the same procedure as described in our previous study (I). Solute adsorption isotherms were determined at different mobile-phase compositions by frontal analysis (I, 4 1 2 ) . These determinations were performed over a range of concentrations wider than that done in our previous study. Furthermore, this
NO. 20, OCTOBER 15, 1991 2245
range increases with increasing ACN fraction, from 0 to 0.15 M with 10% ACN to 0 to 0.50 M with 30% ACN and above. The solubility of 2-phenylethanolin a 10% aqueous solution of ACN is slightly above 0.15 M it increases rapidly with the ACN content of the aqueous solution and exceeds 0.5 M in 30% ACN. Units. Since in this work the Langmuir isotherm model is not satisfactory,the corresponding column saturation capacity is not meaningful. It would not make much sense to refer the sample size to the saturation capacity of an incorrect Langmuir isotherm. For the lack of a better reference,it appeared convenientto report the sample size as the average number of molecules per chemically bonded octadecyl group in the whole column. The highest concentrations in the stationary phase we have reached in the measurement of adsorption data, close to saturation of the mobile phase, are about 2.5 molecules of 2-phenylethanol per bonded octadecyl group. This value is nearly independent of the ACN concentrationof the mobile phase. The absolute value of the sample size is also given in all cases. The column saturation capacity of ACN is 5.8 molecules per bonded group. Most of the experiments were made by using two sample sizes corresponding to loading ratios of 29 and 43 molecules per 100 bonded groups in the whole column, respectively. Calculations. Band profiles were calculated as previously described (I). A correct simulation of a gradient experiment includes a first step during which the sample injection is performed and the amount injected elutes under isocratic conditions (0.38 9). Then, the gradient begins, immediately after the injection is finished. As the same pump is used to carry out the injection and the gradient, there is no significant delay between the end of the injection and the beginning of the gradient.
RESULTS AND DISCUSSION I. Band Profile in the Isocratic Elution Mode. In the isocratic elution mode, an excellent agreement has been reported in many previous studies between the calculated band profiles of pure compounds and the experimental profiles. This agreement has been observed with pure mobile phases (8), with a solution of a weak organic solvent in water (9,13) and also with a solution of a strong organic modifier in a weak organic solvent (14). In all these studies, the band profiles were computed by using the semiideal model of chromatography, taking into account the finite column efficiency (4,11, 15), and using a Langmuir isotherm model fitted to the adsorption data. An excellent agreement has also been found between calculated and experimental band profiles when a bi-Langmuir isotherm model had to be used for a better fit of the adsorption data (16). In all these studies, however, the column loading did not exceed 10% of the column saturation capacity, based on the Langmuir fit of the isotherm data (1). We know that the band profiles are quite sensitive to the shape of the isotherm, and there is no experimental evidence that the Langmuir isotherm will account for adsorption data at high concentrations. On the contrary, there are good theoretical reasons to think that it will falter. A good test of that prediction is that, for the lack of accuracy of the Langmuir isotherm a t high concentrations, the band profiles calculated with the semiideal model, using this isotherm, will not agree with experimental profiles at very high column loadings. A better isotherm model will be needed. Obviously, it is hopeless to attempt to predict band profiles in gradient elution for very high loadings if we cannot at least predict them accurately in the isocratic elution mode. This is because in isocratic elution, the band dilutes steadily during its migration. In gradient elution, on the contrary, the band dilutes only slightly or, especially for steep gradients, it actually concentrates. So, our first series of results (Figure la-c) shows the band profiles of 2-phenylethanol recorded for three different injection volumes, corresponding to loading ratios of about 9.5, 19, and 28.5 molecules of 2-phenylethanol for 100 bonded octadecyl groups with a solution of 10% ACN in water as the mobile phase.
ANALYTICAL CHEMISTRY, VOL. 63,NO. 20, OCTOBER
2246
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(1-tp-to) min. (1-tp-to) min. Flgum 1. Comparison between experimental (symbols) and CaWted band profiles using either the Langmuir isothenn (dotted line) or the isatherm model given by eq 1 (dashed line), In the isocratfc elution mode. Experimental conditions: sample, solution of 2phenylethanol In 10% ACN In water; MobHe phase flow rate, 1 mL/mln; mobile phase composition, 10% ACN In water; column temperature, 40 O C ; cdumn dead volume, V, = 2.71 mL. (a, lett) Conditions: injection volume, VI = OSV,; sample size, 0.58 mmol, corresponding to a loading factor L , = 6.7%. (The loading factor, L, is calculeted on the basis of the satwatkn capachy 0, = 2.1 mol4 of the best Langmut isdhenn.) Inset: symbols,experimemel data as In main figure; lines, profiles calculated with the Langmuh Isotherm and 10 000 plates (solid line) or 1000 plates (dotted line). (b, middle) Conditions: injection volume, VI = V,; sample size, 1.17 mmoi, corresponding to a loading factor, L , = 13.4%. (c, right) Conditions: Injection volume, V, = 1.5V0; sample size, 1.75 mmol, corresponding to a loading factor L , = 20.0%.
Table I. Isotherm Coefficients
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The coefficients of the isotherms were obtained by a least-squares fit of the equilibrium adsorption data obtained by frontal analysis to the Langmuir model or to eq 1 (Table I). It is obvious that the experimental band profiles in Figure la-c are in excellent agreement with the second set of calculated profiles, those obtained with the isotherm given by eq 1. On the contrary, the profiles calculated with the best Langmuir isotherm account poorly for the experimental band profiles, especially for the retention time of the front, which is too long, for the profile of the tail,and for the limit retention time at zero concentration,which corresponds to a calculated value of k ,' that is well below the experimental one (seeFigure 2, inset). Figure 2 shows the adsorption data for 2-phenylethanol derived by frontal analysis in a 10% ACN solution in water (symbols),along with the best isotherms obtained by fitting the two models to these data. The fit is excellent with the isotherm model of eq 1 (sum of a linear and a Langmuir term). The Langmuir model, on the contrary, gives a poor fit with systematic deviations, particularly pronounced at low and high concentrations. Obviously, the Langmuir model is unable to
Figure 2. Adsorption isotherm of 2-phenylethanol (squaresymbols). Mobile phase: 10% ACN in water. Curve 1 is a Langmuk regresskn of these data (q = &/[I b c ] )with a = 24.88 and b = 11.79 mL/mml. Curve 2 is the best fit of the adsorption data to the other model (eq I), with a , = 4.69, 8 2 = 23.31, and b2 = 26.01 mL/mmd. Inset: blowup of the isotherm in the iow concentration range.
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account correctly for the adsorption data, having too high a curvature for the initial slope and the amount adsorbed at high concentrations. Thus, the best Langmuir isotherm (Figure 2) predicts an initial slope that is too low (hence, too short retention times in Figure la-c) and an amount adsorbed at high concentrations that is too high (hence, too long retention times for the band fronts). This explains the deviations observed in Figure la-c between the experimental band profiles and the profiles calculated with the Langmuir isotherm. That the use of an incorrect column efficiency cannot explain the difference between the two bands is demonstrated by the inset of Figure la, where band profiles calculated with the Langmuir isotherm and efficiencies of lo00 and loo00 plates are compared with the experimental profile. The agreement is no better than in the main figure. We note that when a 10% ACN mobile phase containing 0.15 M 2-phenylethanolis in equilibrium with the stationary
ANALYTICAL CHEMISTRY, VOL. 63, NO. 20, OCTOBER 15, 1991
phase, the contributions of the two terms of the isotherm (eq 2) to the stationary-phase content are nearly equal (0.70M for the linear term and 0.71 M for the Langmuir term). A t low concentrations, on the other hand, the contribution of the latter is 5 times larger than that of the former. This explains why we can no longer use the loading factor as a measure of the sample size to characterize column overloading at very high concentrations. Attempts at fitting the data to a bi-Langmuir isotherm (16) or to a quadratic isotherm (111,using appropriate curvefitting programs, failed. The fitting of the experimental data to the bi-Langmuir model was unsuccessful, giving very small values of the b coefficient and, depending on the ACN concentration, sometimes negative ones. This result confirms the experimental validity of eq 1. Success with a quadratic isotherm representing an S-shaped curve would have required data points at concentrations much higher than 0.15M. Such data cannot be acquired, however, as the most concentrated solution used (0.15M) is nearly (more than 90%) saturated. It is difficult to give a physical explanation for this new adsorption isotherm model which accounts very well for all the adsorption data determined, in the entire range of composition of the 2-phenylethanol/ACN/watermobile phase that is accessible to experiment. A similar behavior has been reported recently with cholesterol in a nonaqueous reversedphase system (17). Empirically, this model is the simplest possible extension of the failing Langmuir isotherm. However, one can hardly imagine a column with an infinite saturation capacity. This behavior could be explained by assuming that, above a certain mobile-phase concentration, the solute distribution between stationary and mobile phase proceeds by more of a partition phenomenon than of an adsorption one. More probably, however, this is the result of the progressive deviation with increasing concentrations of the behavior of the liquid and adsorbed solutions from the one predicted by the ideal solution model. A degree of adsorbate-adsorbate interactions could contribute to explain these deviations. Further physicochemical measurements are needed to elucidate the adsorption mechanism at high concentrations. 11. Variation of the Isotherm Parameters with the ACN Concentration. In order to simulate band profiles in overloaded gradient elution chromatography one must have a model allowing an accurate description of the variation of the isotherm coefficients with the mobile-phase composition. These functional dependences, together with the isotherm model, are entered into the computer program previously described (I), based on the Craig distribution ( 2 , 4 ) ,and the band profiles are calculated for different input settings, i.e. injection volume, input concentration, and gradient program. Figure 3 shows the variation of the coefficient al of the linear term of the isotherm (eq 1)with the fraction of ACN in the mobile phase. A linear regression describes well the variation of al with increasing ACN fraction. This result is surprising in the case of a phenomenon related to liquid-solid adsorption. It is consistent, however, with either one of the two possible explanations for a linear term in the equilibrium isotherm: (i) the progressive deviation (with increasing 2-phenylethanol concentrations) of the behavior of the liquid and adsorbed solutions from the ideal solution behavior or (ii) the existence of a partition mechanism that becomes important at high concentrations. On the other hand, the variation of both az and b2 with the ACN concentration in the mobile phase was found to follow a more classical pattern, fitting well the conventional model of a linear solvent strength where the logarithm of the retention factor (proportional to az) decreases linearly with
2247
35
ACN fraction Flgure 3. Plot of the coefficient a , of the linear term of the isotherm (eq 1) versus the fraction of ACN in the mobile phase. Experimental values (symbols) obtained by regresslon to eq 1 of the adsorptkn data obtained by frontal analysis for varlous ACN concentrations. The sdid line is the linear regression of the data points. Slope S, , = 14.606 = 6.136. and intercept a
increasing fraction of ACN in the mobile phase. The variations of the coefficients az and b2 with the ACN fraction are shown in Figure 4. The two linear regressions are nearly parallel. As a consequence, the dependence of the column saturation capacity of the Langmuir term of the isotherm on the mobile-phasecomposition is minimal. A similar behavior was observed when the isotherm data acquired in a lower concentration range were fit to the Langmuir model (1). Taking into account the variations of the three isotherm coefficients with the mobile-phase composition just described, one can now write a general equation for the adsorption isotherm:
where al,oand Sa, are the intercept and the slope of the plot of al versus the fraction of ACN, 9,respectively; az,oand Sa, are the intercept and the slope of the plot of In (az) vs 9, respectively; and bz,Oand S, are the intercept and the slope of the plot of In (b,) vs @, respectively. 111. Comparison between the Langmuir and the Composite Isotherm. Samples of 2-phenylethanol corresponding to a loading ratio of 29.4 molecules per 100 bonded groups were injected and eluted by using two different gradient programs. The recorded band profiles (symbols)are compared in Figure 5a (1%ACN/min) and 5b (2% ACN/min) to two different calculated profiles, the first one obtained by using the Langmuir model (dashed line), the second one by (dotted line) using the new composite isotherm (eq 2). For both gradient programs, the band profiles calculated with the composite isotherm are in much closer agreement
2248
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with the experimental profile than those calculated with the Langmuir model. The agreement is not as good as for the isocratic profdes (Figure la-c), but still the calculated profiles do predict the inflection observed on the rear of the experimental profiles, even if the towering look of the experimental band tops and their rear inflection point are somewhat imperfectly rendered. The 20-s delay observed between the experimental and calculated profiles could be explained by our neglect of the ACN retention in the model, an effect that is discussed below. Other experimental band profiles were recorded a t lower loading ratios and with different gradient programs (1,2, and 4% ACN/min). Again the Langmuir model failed to describe correctly these experimental profiles. However, the composite isotherm does not account well for the experimental band profiles obtained with very steep gradients (4% ACN/min), although it performs much better than the Langmuir model. The same discrepancy has been reported previously at moderate loading factors for steep gradient programs (I). Comparing these earlier results with those reported here, we understand that there are at this stage, two possible and nonexclusive explanations for the disagreement, either deviations at high concentrations between the actual isotherm and the model used (eq 2) or the effect of a retention of the organic modifier that is likely to distort the front of the gradient profiles (18). We have already explained that it is not possible to acquire adsorption data at higher concentrations. The effect of the nature of the regression method used for the calculation of the isotherm coefficients (al, a2, and b2 versus the ACN fraction) on the shape of the calculated band profiles has been
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ANALYTICAL CHEMISTRY, VOL. 63,NO. 20, OCTOBER 15, 1991
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Flguro 6. Adsorption of acetonlrlle in pure water. Detection was at 190 nm. Square symbols represent experimental data points and the solid line glves the best fk to the Langmuir Isotherm. Coefficlents are given in Table I. studied by using a quadratic regression of the data shown in Figures 3 and 4. No significant improvement has been observed. IV. Effect of t h e Adsorption of Acetonitrile on t h e Band Profiles. Most theoretical work carried out in reversed-phase chromatographyassumes that the strong organic solvent (e.g., ACN or even methanol) is not adsorbed by the stationary phase but acts only by increasing the solubility of the eluites in the mobile phase. As shown recently by Velayudhan and Ladisch (181, this is only approximate. The effect of the adsorption of ACN could also be neglected if the column were practically saturated a t the initial concentration (10% in the present work). This is not true either, as demonstrated by the ACN isotherm in Figure 6. The capacity factor of ACN in pure water was found to be around 0.6, and the column saturation capacity, estimated by using a Langmuir regression of the adsorption data, was found equal to 7.5 mmol/mL. This value corresponds to an average of 5.8 ACN molecules per CI8bonded group. However, at the beginning of the gradient program (10% ACN), the stationary-phase concentration is only about 1.5 mmol/mL and the slope of the isotherm tangent is far from negligible. Thus, we cannot neglect the adsorption of the organic modifier in this work. In analytical gradient chromatography under the linear solvent strength mode (5) and the assumption of a linear isotherm of the organic modifier, Martin (19) has shown that the retention of this modifier leads to an increase in the solute retention time almost equal to k’*to/2, where k k is the (constant) capacity fador of the modifier and to is the holdup time of the column. In the case of a nonlinear isotherm of the organic modifier, however, the gradient profile is not only delayed but also changed (18). Because of the self-sharpening effect associated with a convex upward isotherm, the elution profile of a linear gradient has a steep front followed by a curved ramp, the deviation from a linear profile decreasing slowly (see Figure 7). Using the parameters of the ACN isotherm, it is easy to calculate the outlet profiles of different ACN gradient programs. The calculation procedure is the same as for the calculation of isocratic band profiles (11). The only difference is in the boundary condition. Instead of a rectangular pulse, the input is the linear concentration ramp of the gradient program. The initial condition includes the column equilibration with 10% ACN. Figure 7 compares the linear input gradients, the experimental elution profiles and the calculated output profiles obtained in the case of the three input gradients used (1, 2, and 4% ACN/min).
( t - t d min.
Flgure 7. Comparison between the Meal ACN output profiles (solid lines) and simulated profiles (dashed lines) using the Langmulr model or experimental ACN outlet profiles (dotted lines): (series 1) gradient 1 % ACN/min; (series 2) gradient 2% ACNlmh.1; (series 3) gradient 4 % ACNlmin.
The agreement between calculated and experimental profiles is good, except at very high concentrations, near the end of the gradient. This discrepancy is probably due to the fact that the Langmuir isotherm does not account very well for the adsorption isotherm of ACN at high concentrations. We have also compared the calculated profiles and those obtained by using the analytical solution derived by Velayudhan and Ladisch (18). They agree perfectly. Using these results, we can now take the retention of ACN into account in the calculation of the elution profiles of large samples of 2phenylethanol. V. Comparison between Experimental and Calculated Band Profiles. Parts a and b of Figure 8 compare the experimental band profiles (symbols) obtained with loading ratios of 29 (Figure 8a) and 43 (Figure 8b) molecules of 2phenylethanol for 100 bonded chains using a 1%ACN/min gradient with two calculated band profiles. The first profile (dotted line) was calculated with the procedure described above (Figure 5a), which ignores the retention of ACN. The second profile (solid line) takes into account the adsorption of ACN. Both calculated profiles used the composite isotherm (eq 2). This second profile was calculated with a modification of our previous Craig computer program. This improved version consists of propagating the ACN input profile, taking into account the adsorption of ACN on the stationary phase (Figure 6). Using a cell propagation scheme, one cannot predict the local concentration of ACN a t any time and location, unless its concentration in the previous nodes of the propagation grid are known (4). This makes the computer program more complicated and slower than when the retention of the strong solvent is neglected. An alternative, faster and more convenient procedure could be based on the analytical solution giving the ACN concentration profile along the column at any time (18). This method would be valid if the
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ANALYTICAL CHEMISTRY, VOL. 63,NO. 20, OCTOBER 15, 1991
4
6
8
10
12
(t-tp-to)
14
16
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11
13
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"I
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8
3
10 12 14 16 18
(t-tp-to)
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Figun 8. Comparlson between experlmental and calculated band proflles In gradlent elutlon. Experimental conditions are as In Figure 5a with concentration of the sample sdutlon cg = 132.5 mmol/L and a gradlent program of 1% ACNImln. Experimental band proflles (square symbols) and profiles calcutated uslng the composite Isotherm, taking Into account (sol# #ne)or not (dotted#ne) the admptkm of ACN
on the stationaty phase. (a, top) Conditions: Injection volume, Vi = 4 mL; sample slze, 0.53 mmol; loading ratlo, 29 molecules per 100 bonded groups. (b, bottom) Condlnons: in)ectkn volume, Vi = 6 ml, sample size, 0.80 mmol; loading ratlo, 43 molecuks per 100 bonded groups. adsorption isotherm of the modifier is Langmuirian and the column efficiency is high. In both cases, improved results are obtained if we take into account the adsorption of ACN on the stationary phase. The improvement of the fit is especially significant on the lower part of the front and rear profiles, which are now exactly on time and accurately accounted for. However, the fit of the pinnacle of the band is hardly improved at all and what appears to be a shock layer just at the rear of the band maximum (Figure 8b) is not explained by the model. Parts a and b of Figure 9 compare the calculated and experimental (symbols) band profiles obtained with the same sample sizes as above, but with a steeper gradient, 2% ACN/min. The two calculated profiles are as above: the first
5
7
9
(t-tp-to) Figure 9.
111315l7
(mid
Same as Figure 8, except with a gradient program of 2%
ACN/min.
one (dotted line) does not take the adsorption of ACN on the stationary phase into account (as in Figure 5b); the second one (solid line) does. As in the previous example (Figure 8a,b), the second calculation procedure, which takes into account the adsorption of ACN, gives an excellent fit of the band profile for the lower size sample and accounts well for the profile of the larger size sample, except for the band pinnacle. Although the results obtained with the first two gradient programs and large-size samples are quite good, the agreement at very steep gradient programs is not as satisfactory. We compare in Figure 10a,b the experimental and calculated profiles for the same two samples eluted with a very steep gradient program, 4% ACN/min. In this case, there is a profound disagreement between the experimental results and the profiles calculated with the best available approach. The new procedure, using a composite isotherm and taking the adsorption of ACN into account, improves the fit. It still fails, however, to describe correctly the experimental band profiles. VI. Effect of the Injection Volume on the Band Profiles. The amount of sample injected is proportional to the
ANALYTICAL CHEMISTRY, VOL. 63, NO. 20, OCTOBER 15, 1991
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0 ln
2-
! ‘(r
L x
n
0
2
4 6 8 10 . (t-tp-to) Flgvo 10. Same as Figure 8, except with a gradknt program of 4 % ACNlmin.
x
(mid
sample volume and to the solute concentration. In the case of overloaded isocratic elution, it has been shown that, if a solute isotherm follows the Langmuir model, the band profile can be normalized and described with only one parameter, the loading factor, Lf(20,21). This assumes, however, a nearly instantaneous injection. If the injection volume is not negligible, the bandwidth is increased and ita height decreased. In gradient elution chromatography, a concentration effect takes place. The front of the injection profiles moves at the velocity d a t e d with its concentration in the injection plug, Le., a shock velocity. During the gradient elution, the tail of the band tends to move faster than the front. This concentrates the band, which becomes narrower and taller. A taller front shock moves faster and the band profile reaches a dyanmic equilibrium. The intensity of this effect depends, however, on the retention of the solute in the mobile phase at the initial concentration of the gradient. The effect is strong if the solute is highly adsorbed at the beginning of the gradient program, i.e., if ita associated velocity is nearly zero. In general, however, and especially if the solute is not strongly adsorbed at the initial concentration, the shape of the band profile depends on the injection volume. It depends
Figure 11. Influence of the sample vdume at a constant k d n g factor
on the elution profile. Experimental chromatograms: conditions as In Figure 9. (a, top) Comparison of the chromatogram (X) shown in Figure 9a (injection volume, 4 mL; sample concentration, 132.5 mmoilL) and of a chromatogram (0)obtained for an injection of the same amount, but with a volume of 8 mL of a 66.25 mmd/L sdutlon. (b, bottom) Comparison of the chromatograms obtained for a 2-mL injection of a co = 132.5 mmoi/L solution (O),for a 4 m L injection of a c, = c0/2 = 66.25 mmoilL solution (X), and for an 8-mL injectlon of a c p = c,/4 = 33.1 mmoilL solution (V). also on the sample concentration that controls the associated velocity, inversely proportional to the local derivative of the isotherm. In the case where the isotherm is given by eq 1, this derivative is
It is obvious that the contribution of the linear term is more important at high concentrations than that of the Langmuir term, which tends toward zero with increasing concentrations. On the contrary, the Langmuir contribution to the velocity associated with a concentration prevails at low concentrations. However, the degree to which one or the other contribution dominates depends also on the mobile-phase composition.
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Parts a and b of Figure 11 illustrate the effect of the injection volume, at a fixed injected amount. Figure l l a compares the profiles obtained in a case where the effect of the injection volume is important, because the loading ratio is relatively high (29 molecules per 100 bonded groups) and the concentrationof the plug is in the range where the isotherm deviates significantly from the Langmuir model. Doubling the sample volume at constant amount changes dramatically the shape of the elution band top. This change occurs because the retention of the solute by the stationary phase at the initial mobile-phase concentration is finite and even relatively small compared to what happens in many cases when gradient elution is applied. So the front of the injection zone moves at a velocity that depends quite significantly on its concentration and the retention time of the band front in the gradient elution experiment depends on the concentration of the sample. On the contrary, for a lower value of the loading ratio (14.5 molecules per 100 bonded groups), an increase in the sample volume by a factor of 4 has almost no effect on the shape of the profile. In all cases, the use of the composite isotherm permits the calculation of band profiles that account well for the experimental results. CONCLUSION At very high concentrations, the equilibrium isotherm of 2-phenylethanoldoes not follow the Langmuir model. We are of the opinion that this is a general result, at least for reversed-phase chromatographic systems (I 7). As a consequence, the band profiles observed in gradient elution chromatography at high loading factors cannot be correctly described by the profiles calculated with the use of a Langmuir isotherm. On the contrary, an accurate description of these profiles is achieved at high concentrations with a composite isotherm, which is the sum of a Langmuir and a linear term. However, the quality of the agreement between calculated and experimental band profiles depends on the steepness of the gradient program. The agreement is excellent for shallow gradients, good for moderate gradients, and only fair for very steep gradients. This satisfactory agreement is obtained only if the retention of the organic modifier on the stationary phase is accounted for as suggested previously (18) in the case of compounds that are less retained than the modifier. Because high concentrations are usually less retained than low ones in nonlinear chromatography, an interaction of the type re-
ported here will often tend to take place when the column is overloaded in gradient elution. The sample volume also plays an important role in determining the band profile, especially at very high column loadings. This effect is easy to account for, and the agreement between calculated and experimental band profiles is excellent at high sample volumes, because a large sample volume corresponds to a low solute concentration in spite of the concentrating effect of the gradient.
ACKNOWLEDGMENT We acknowledge the gift of the stationary phase and the supply of its physicochemical characteristics by R. Cooley (YMC, Morris Plains, NJ). LITERATURE CITED El Fallah, M. 2.; Guiochon. G. AM/. Chem. 1091, 63, 859. Craig, L. C. J . Bld. Chem. 1044, 755, 519. Eble, J. E.; Grob, R. L.; Antle, P. E.; Snyder, L. R. J . Chromatogr. 1087, 405, 51. Czok, M.; Guiochon. G. AM/. Chem. 1000, 6 2 , 189. Snyder, L. R. In Hlgh Performance LlquM Chromatography-Advances and Persmctlves; How4th. Cs.. Ed.: Academic Press: New York.
1980 Voi. 1, p 280. Snyder, L. R.; Cox: G. 0.; Antle, P.E. J . Chrometog. 1988, 444, 303. Antla, F. D.; Horvath, Cs. J . Chromatogr. 1080, 484, 1. Golshan-Shirazi, S.; Ghodbane, S.; Guiochon, 0. AM/. Chem. 1088, 60.. _... 2830. Golshan-Shirazi, S.; Guiochon. 0.Anal. Chem. 1088, 60, 2634. Katti, A. M.; Ma, 2.; Guiochon. 0.A I C M J . 1000, 38, 1722. Golshan-Shkazi, S.; Jauimes, A. AM/. chem.1088, 60, Guiochon, 0.; 1856. Jacobson, J.; Frenz. J. M.; Horvith, Cs. J . Chromatogr. 1084, 376, 53. El Fallah, M. 2.; Guiochon, G. J . Chrcmatogr. 1990, 522, 1. Golshan-Shirazi, S.; Guiochon, G. AM/. Chem. 1080. 67. 462. Lin, 8.; Guiochon, G. Sep. Scl. Techno/. 1088, 2 4 , 32. Jacobson. S.; Golshan-Shirazi, S.; Guiochon, G. J . Am. Chem. Soc. 1900, 772, 8492. Jandera, P.; Guiochon. 0. Submltted for publication in J . Chromatcgr. Velayudhan, A.; Ladisch. M. R. AM/. Chem., in press. Martin, M. J . Liq. ChrOmatogr. 1088, 7 1 , 1809. Golshan-shirazi, S.; Gukchon, 0.Anal. Chem. 1008, 60, 2364. Golshan-Shirazi, S.; Guiochon, G. J . Chromatogr. 1000, 506, 495.
RECEIVED for review April 1, 1991. Accepted July 9, 1991. This work was supported in part by Grant CHE8901382 from the National Science Foundation and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory. We acknowledge support of our computational effort by the University of Tennessee Computing Center.
