Anal. Chem. 2006, 78, 5774-5782
Theory of Stepwise Gradient Elution in Reversed-Phase Liquid Chromatography Coupled with Flow Rate Variations: Application to Retention Prediction and Separation Optimization of a Set of Amino Acids P. Nikitas,* A. Pappa-Louisi, and P. Balkatzopoulou
Laboratory of Physical Chemistry, Department of Chemistry, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
The coupling of stepwise mobile phase gradient elution and flow programming is proposed as an integrated approach to the general elution problem in reversed-phase liquid chromatography. A model is developed to describe the above separation process performed under simultaneous programming of two separation parameters by extending our previous work on the rigorous derivation of the fundamental equation governing the concentration gradient of organic modifier in the mobile phase, that is, a single gradient elution mode (Anal. Chem. 2005, 77, 5670-5677). The theory was tested in the retention prediction and separation optimization of 18 o-phthalaldehyde derivatives of amino acids in eluting systems modified by acetonitrile or methanol. The retention prediction obtained for all solutes under all dual-mode gradient conditions was excellent. In addition, it has been shown that the combination of mobile phase and flow rate programming modes is particularly favorable, whereas the separations among the analytes were considerably improved by using the acetonitrile eluting system, as compared to those obtained by the methanol system. At present, there are four possible techniques of programming gradient formation in liquid chromatography (LC): solvent programming, stationary phase programming (couple-column operation), temperature programming, and flow programming. In each of these techniques, only one parameter is varied during the analysis so that the resolution and widths of early- and lateeluting bands become more similar in a shorter analysis time. Although solvent programming is probably the best single gradient method in LC to control the separation where fast and highly efficient analysis is desirable, it can sometimes suffer from poor front-end resolution.1,2 Thus, the coupling of solvent and flow programming, that is, a “dual-mode gradient”, can become a powerful tool for altering separation quality in many situations. * To whom correspondence should be addressed. Phone: +30 2310 997773. Fax: +30 2310 997709. E-mail:
[email protected]. (1) Snyder, L. R.; Kirkland, J. J. Introduction to Modern Liquid Chromatography, 2nd ed.; Wiley-Interscience: New York, 1979. (2) Jadera, P.; Churacek, J. Gradient Elution in Column Liquid Chromatography. Theory and Practice; Journal of Chromatography Library; Elsevier: New York, 1985; Vol 31.
5774 Analytical Chemistry, Vol. 78, No. 16, August 15, 2006
However, there are only a few reports concerning such reversedphase, dual-mode gradient elution in the literature,3-10 since it seems that the development of this separation method is more difficult than that of a single-gradient elution mode. In general, the gradient technique exploits completely its potentialities only if we can make predictions of the retention time on the basis of the properties of the solute and the gradient profile. Until now, this could be done if we had only one gradient mode.11-17 The theory of combined gradients has not been advanced, possibly for the following reason. Consider a gradient profile of the volume fraction of the organic modifier in the water-organic mobile phase, φ, with time, t, and a corresponding profile of the flow rate, F vs t. Each of these two curves can be approximated by a stepwise curve composed of a large number, N, of infinitesimally small time steps, δt, each characterized by a constant instantaneous composition of the mobile phase or flow rate.12,17-21 A striking difference between the F-vs-t and φ-vs-t profiles is that any change in F reaches almost instantaneously the analyte inside the column, whereas this does not happen for the changes in φ. Due to this effect, if φ is constant, (3) Lesins, V.; Ruckenstein, E. J. Chromatogr. 1989, 467, 1-14. (4) Hsieh, Y.; Wang, G.; Wang, Y.; Chackalamannil, S.; Korfmacher, W. A. Anal. Chem. 2003, 75, 1812-1818. (5) Paci, A.; Caire-Maurisier, A.-M.; Rieutord, A.; Brion, F.; Clair, P. J. Pharm. Biomed. Anal. 2001, 27, 1-7. (6) Yokoyama, Y.; Ozaki, O.; Sato, H. J. Chromatogr., A 1996, 739, 333-342. (7) Earley, R. L.; Miller, J. S.; Welch, L. E. Talanta 1998, 45, 1255-1266. (8) Warner, B. D.; Boehme, G.; Legg, J. I. J. Chromatogr. 1981, 210, 419-431. (9) Yokoyama, Y.; Tsuji, S.; Sato, H. J. Chromatogr., A 2005, 1085, 110-116. (10) Ricci, M. C.; Cross, R. F. J. Liq. Chrom., Relat. Technol. 1996, 19, 547564. (11) Jandera, P. Adv. Chromatogr. 2005, 43, 1-108. (12) Nikitas, P.; Pappa-Louisi, A. Anal. Chem. 2005, 77, 5670-5677. (13) Nikitas, P.; Pappa-Louisi, A.; Papachristos, K. J. Chromatogr., A 2004, 1033, 283-289. (14) Nikitas, P.; Pappa-Louisi, A. J. Chromatogr., A 2005, 1068, 279-287. (15) Nikitas, P.; Pappa-Louisi, A.; Agrafiotou, P. J. Chromatogr., A 2006, 1120, 299-307. (16) Haber, P.; Baszek, T.; Kaliszan, R.; Snyder, L. R.; Dolan, J. W.; Wehr, C. T. J. Chromatogr. Sci. 2000, 38, 386-392. (17) Pappa-Louisi, A.; Nikitas, P.; Zitrou, A. Anal. Chim. Acta, in press. (18) Smith, R. D.; Chapman, E. G.; Wright, B. W. Anal. Chem. 1985, 57, 28292836. (19) Snijders, H.; Janssen, H. G.; Cramers, C. J. Chromatogr., A 1995, 718, 339355. (20) Chester, T. L. J. Chromatogr., A 2003, 1016, 181-193. (21) Chester, T. L.; Teremi, S. O. J. Chromatogr., A 2005, 1096, 16-27. 10.1021/ac0606655 CCC: $33.50
© 2006 American Chemical Society Published on Web 06/30/2006
then the distance δL traveled by the analyte inside the column under the influence of flow rate F at time δt is given by17
δL ) vδt )
Lδt LFδt ) tR t0(1 + k)
(1)
where L is the column length, v is the instantaneous velocity of the analyte, tR is its retention time when its velocity is constant and equal to v, t0 is the column hold-up time when F ) 1, and k is the solute retention factor. In contrast, if F is constant, the distance δL traveled by the analyte inside the column under the influence of a mobile phase with composition φ for a time period δt may be expressed as12-15
δL )
LFδt t0k
(2)
It is evident that F in eq 1 may change at the various δt, whereas in eq 2, the variable that has this property is φ. It is seen that these two expressions of δL are different, and this makes difficult the direct combination of the two effects into a single and simple expression, such as the already known expressions of the fundamental equations of gradient elution when the two effects act separately. At this point, we should clarify that theoretical treatments of combined gradients that do not exhibit different δL expressions have already been presented in the literature. For example, Kaliszan’s group has studied combined pH/organic modifier gradients in reversed-phase HPLC.22-24 In the present paper, we attempt to resolve the problem of dual gradients when their expressions of δL are different. In particular, we determine the relationships that allow for the calculation of the solute retention time when a stepwise gradient profile of φ vs t is coupled with flow rate variations. To do that we use the approach proposed by Drake25 and adopted in our previous work12 for a rigorous derivation of the fundamental equation of mobile-phase gradient elution. Finally, the validity of the developed theory governing the dual-mode gradient elution was tested in the retention prediction and separation optimization of 18 o-phthalaldehyde (OPA) derivatives of amino acids, since the analysis of amino acids is very important in biological and biomedical research,26 and as a result, any new analytical approach is worth being applied first to the analysis of these compounds. THEORY Basic Equations. Consider a q-steps variation pattern of φ vs t created in the mixer of the gradient chromatographic system under variable flow rate. Due to the variations in the flow rate, the fronts of the mobile phase are moving with variable velocity, and for this reason, their representation in the plot of x vs t, where x is the distance from the beginning of the chromatographic (22) Kaliszan, R.; Markuszewski, M. J.; Wiczling, P. Pol. J. Chem. 2004, 78, 1047-1056. (23) Wiczling, P.; Markuszewski, M. J.; Kaliszan, M.; Kaliszan, R. Anal. Chem. 2005, 77, 449-458. (24) Wiczling, P.; Markuszewski, M. J.; Kaliszan, M.; Galer, K.; Kaliszan, R. J. Pharm. Biomed. Anal. 2005, 37, 871-875. (25) Drake, B. Akriv. Kemi 1955, 8, 1. (26) Cooper, C.; Packer, N.; Williams, K. Amino Acid Analysis Protocols; Humana Press Inc.: Totowa, NJ, 2001.
Figure 1. Schematic representation of the movement of the fronts of the mobile phase due to a four-step variation pattern of φ vs t created in the mixer under a variable flow rate and the corresponding path of an analyte inside the chromatographic column, shown by the thick line.
column, is not linear. This is schematically shown in Figure 1, which depicts a four-step variation pattern of φ vs t under variable flow rate. The fronts of the mobile phase are indicated by the curved lines AA′, BB′, and CC′, whereas the analyte follows inside the column the path 0abcd, indicated by the thick line. It is evident that in Figure 1, a, b, and c are the points where the analyte meets the fronts of the mobile phase. In this example, t/A is the time duration of the first step, with mobile phase composition φ ) φ1; t/B - t/A is the duration of the second step with φ ) φ2; t/C - t/B is that of the third step with φ ) φ3; and the last step extends beyond t/C with φ ) φ4. More precisely, t/A, t/B and t/C are the times needed for the first, second, and third steps, respectively, to reach the column inlet. It is evident that these times depend on the F-vs-t profile and can be calculated as follows: Suppose that tA, tB, and tC are the programmed times that define the φ-vs-t profile formed in the mixer. That is, the first step in the mixer occurs at tA, the second step at tB, and so on. Then we have
t/A ) tA + tDA, t/B ) tB + tDB, t/C ) tC + tDC
(3)
where tDj, (j ) A, B, C) is the dwell time needed for step j to reach the beginning of the column. For the calculation of these dwell times, we can consider that at an infinitesimally small time, dt, any change in the mixer is moved by ds ) (F/S) dt along the tube connecting the mixer and the chromatographic column, which upon integration yields
tD(F ) 1) )
∫
tj+tDj
tj
F(t) dt, j ) A, B, C
(4)
Here, S is the cross section of the tube, and tD (F ) 1) is the dwell time when F ) 1. The solution of eq 4 determines the dwell times tDA, tDB, and tDC, which can be used for the calculation of t/A, Analytical Chemistry, Vol. 78, No. 16, August 15, 2006
5775
t/B, t/C, through eq 3, that is, the calculation of the arrival times of φ steps at the column inlet. Now if we consider that the F-vs-t curve is approximated by a stepwise curve composed of a large number of infinitesimal small time steps, δt, then for t/A, t/B, and t/C, we can write the following relationships
t/A ) m1δt, t/B ) m2δt, t/C ) m3δt
L1
)
L
l1
n1
δt
)
t0(1 + k1)
L
n1
n1
∑
(1 + k1)
Fi -
i)m1+1
t1 ) n1δt, t2 ) n2δt, t3 ) n3δt
(6)
n1
l1 )
∑
i)m1+1
Lδt t0i
)
Lδt t0
n1
∑
Fi
(7)
i)m1+1
L2
l2 - l1
)
t0
∑
∑
t0(1 + k2) i)n1+1
L
∑
n2
∑
Fi -
i)m2+1
Fi )
Fi
t0(1 + k2)l1/L
(12)
(13)
δt
i)n1+1
which can be used for the calculation of n2. The generalization is straightforward. The values of n1, n2, ... and those of l1, l2,... can be calculated from the recursive relationship np
l2 )
n2
δt
)
n2
(1 + k2)
(1 + kp)
n2
(11)
i
i)1
Equation 12, in combination with the first of eqs 8, results in
and similarly
Lδt
∑F )0
which allows for the calculation of n1 and, next, the calculation of l1/L from eq 10. Similarly, from the fact that F varies under constant φ ) φ2 for a time period equal to t2 - t1 ) (n2 - n1)δt, we obtain
L The coordinates l1, l2, l3 of points a, b, and c on the x-axis, where the column length is represented by L, may be derived by simple kinetic arguments: Along the arc Aa there exist n1 - m1 time steps, δt, where at each step the flow rate is assumed constant and equal to Fi. At each such step, the front of the mobile phase is moving with constant velocity equal to L/t0i ) LFi/t0, where t0i is the column hold-up time when the flow rate is Fi. Therefore,
(10)
i
i)1
where k1 is the solute retention factor at the first step of the φ-vs-t curve. If we equate this equation to eq 7, we obtain
(5)
which, in fact, define the integer numbers m1, m2, and m3, because t/A, t/B, and t/C are calculated through eqs 3 and 4, and the time step δt is selected arbitrarily, provided it fulfills the condition δt , t/A, t/B, t/C. For the time coordinates of points a, b, and c, we can write similarly
∑F
∑
np
∑
Fi -
i)mp+1
Fi )
t0(1 + kp)lp-1/L
i)np-1+1
(14)
δt
Fi
i)m2+1
in combination with
lp-1
and
l3 )
Lδt t0
L
n3
∑
)
Fi
t0
np-1
∑
Fi
(15)
i)mp-1+1
(8)
i)m3+1
Note that the integer numbers n1, n2, and n3 appearing in the above relationships are not known and may be calculated as follows: It is evident that during the time period t1, the analyte is under the influence of a constant mobile-phase composition, φ ) φ1. Similarly, the analyte is under the influence of φ ) φ2 for time t2 - t1 and under the step φ ) φ3 for a time period equal to t3 t2. However, when F varies under constant φ, eq 1 is valid. This equation for each time step δt of the time period t1 ) n1δt may be written as
δLi δt Fiδt δt ) ) ) L tRi t0i(1 + k) t0(1 + k)
δt
Conditions for Solute Elution. The solute may be eluted during the first step, at an arbitrary intermediate step, or during the last step. Elution during the First Step. The condition for elution at the first step is
l1/L g 1
(16)
In this case, from eq 10, we obtain
δt t0(1 + k1)
m
∑F )1 i
(17)
i)1
(9) and the solute retention time is calculated from
and yields 5776
Analytical Chemistry, Vol. 78, No. 16, August 15, 2006
tR ) mδt
(18)
Elution during the Intermediate p Step. If the solute is eluted during the intermediate step, p, we have
lp-1/L < 1 and
lp/L g1
(19)
and therefore,
lp-1
+
m
δt
∑
t0(1 + kp) i)np-1+1
L
Fi ) 1
(20)
Evaluating m from this equation, the retention time is calculated from eq 18. Elution during the Last Step, q. The condition for elution at the last step, q, can be expressed as
lq-1/L < 1
(21)
Again, the retention time, tR, is calculated from eq 18, with m determined from the solution of the following equation.
lq-1 L
+
m
δt
∑
t0(1 + kq) i)nq-1+1
Fi ) 1
(22)
Limiting Cases. The treatment presented above is a generalization of previous treatments of gradient elution of variable φ at constant F13 or variable F at constant φ17. Therefore, these cases should arise as limiting cases of the present treatment. Constant F and Variable φ. Let us assume for simplicity that F ) 1. Then from eq 11, we readily obtain that
n1 )
1 + k1 m1 k1
(23)
which in combination with eq 10 yields
tA L1 m1δt ) ) L t 0k 1 t0k1
(24)
1 + kp (mp - mp-1) kp
(25)
If this equation is introduced into the extension of eq 12 for Lp/L, we obtain
Lp (mp - mp-1)δt tp ) ) L tokp t0kp
constitute the fundamental equations of stepwise gradient elution under constant flow rate and yield directly the final expression for the elution time, tR, when the analyte is eluted during the p step:2,13
k1 - kp k2 - kp + t2 +‚‚‚+ k1 k2
tR ) t0(1 + kp) + t1
tp-1
(26)
where tp is the time duration of the p step. Equations 24 and 26
kp-1 - kp (27) kp-1
Constant φ and Variable F. In this case, the fronts of the mobile phase AA′, BB′, and CC′ in Figure 1 do not exist. Therefore, it is absolutely equivalent to elution with variable φ and F but when the solute is eluted during the first step. Thus, the elution is described by eq 17, which may take the following integral expression.2,17
1 t0(1 + k)
Similarly, from eqs 14 and 15, we have
np - np - 1 )
Figure 2. Seven different dual-mode gradient programs (I, II, III, IV, V, VI, and VII) in which a two-step variation in φMeCN()) from φ1 ) 0.3 to φ2 ) 0.5 (A) or from φ1 ) 0.3 to φ2 ) 0.4 (B) in addition to a flow rate variation I, II, III (A) or IV, V, VI, VII (B) were simultaneously time-programmed.
∫
tR
0
F dt ) 1
(28)
Equation 28 is the fundamental equation for gradient elution under variable flow rate when all the other column parameters are constant. EXPERIMENTAL SECTION To test the above theory, different types of stepwise gradient elution coupled with flow rate variations depicted in Figures 2 and 3 as well as in Table 1 were implemented in the reversedphase HPLC separation of 18 o-phthalaldehyde derivatives of amino acids: L-arginine (Arg), taurine (Tau), L-asparagine (Asn), L-glutamine (Gln), L-serine (Ser), L-aspartic acid (Asp), L-glutamic Analytical Chemistry, Vol. 78, No. 16, August 15, 2006
5777
Table 2. Experimental Retention Times (in min) of Amino Acid Derivatives under Isocratic Conditions at a 1 mL/min Flow Rate in Mobile Phases Modified with Different φMeCN Values
Figure 3. Three different dual-mode gradient programs, A, B, and C, in which a three-step variation of φMeCN(-) in addition to a linear variation of F (- - -) were simultaneously time-programmed. Table 1. Three Different Dual-Mode Gradient Programs (times in min) in Which a Three-Step Variation in φMeOH (φ1 ) 0.55, φ2 ) 0.6, and φ3 ) 0.7) in Addition to a Linear Variation of F between F1 ) 0.5 and F2 ) 1.5 mL/min Were Simultaneously Time-Programmed gradient
1
2
3
tφ1 tφ2 tF1 tF2
4 12 9 17
5 13 6 13
5 13 8 16
acid (Glu), L-threonine (Thr), glycine (Gly), β-(3,4-dihydroxyphenyl)-L-alanine (Dopa), L-alanine (Ala), 4-aminobutyric acid (GABA), L-methionine (Met), L-valine (Val), L-tryptophan (Trp), L-phenylalanine (Phe), L-isoleucine (Ile), and L-leucine (Leu). The derivatives were formed by the reaction of OPA with amino acids in the presence of 2-mercaptoethanol (2-ME) according to the previously published nonautomated, manual, precolumn derivatization procedure27 with minor modifications. The detection of derivatized amino acids was performed using a spectrofluorometric detector (Shimadzu, model RF-10AXL) at 455 nm after excitation at 340 nm coupled in series with a Gilson electrochemical detector (model 141) equipped with a glassy carbon electrode. The electrochemical detection of the analytes was performed at 0.8 V vs the Ag/AgCl reference electrode. By using fluorometric and electrochemical detection in series, positive peak identification can be obtained in a single chromatographic run; however, only results (27) Zunin, P.; Evangelisti, F. Int. Dairy J. 1999, 9, 653-656.
5778 Analytical Chemistry, Vol. 78, No. 16, August 15, 2006
solute
φ ) 0.3
φ ) 0.35
φ ) 0.4
φ ) 0.5
Arg Tau Asn Gln Ser Asp Glu Thr Gly Dopa Ala GABA Met Val Trp Phe Ile Leu
4.434 5.478 6.014 6.700 7.693 9.352 11.33 11.64 13.55 15.79 22.92 26.06 63.06 81.42 107.2 116.5 150.4 160.6
3.353 4.123 4.520 4.828 5.654 6.326 7.242 7.724 9.268 9.351 14.03 15.27 31.51 39.78 46.78 51.75 67.27 70.85
2.980 3.365 3.700 3.844 4.500 4.764 5.246 5.671 6.733 6.366 9.422 9.905 17.79 21.85 23.57 26.19 34.04 35.32
2.543 2.739 2.949 2.986 3.407 3.422 3.614 3.925 4.490 3.925 5.583 5.664 8.281 9.711 9.322 10.35 13.27 13.45
obtained by the fluorescence detector installed at constant sensitivity were chosen to assess the theory for predicting elution in gradient flow programming. Appropriate working concentrations of underivatized amino acids were used in the derivatization procedure by OPA/2-ME reagent (Tau, GABA ) 0.25 µg/mL; Ser, Asp, Thr, Gly ) 0.5 µg/mL; Trp, Phe, Ile, Leu ) 2 µg/mL; others ) 1 µg/mL) so that the peak heights of the OPA derivatives recorded by the fluorescence detector do not differ significantly. The liquid chromatography system used for separations of the test compounds consisted of a Shimadzu LC-20AD pump and a model 7125 syringe loading sample injector fitted with a 20-µL loop (Rheodyne, Cotati, CA). The mobile phases were aqueous phosphate buffers (pH 2.5) modified with different volume fractions, φ, of either acetonitrile (MeCN) or methanol (MeOH). Their total ionic strength was held constant at I ) 0.02 M. A MZAnalyzentechnik column, PerfectSil Target ODS-3 (5 µm, 250 × 4.6 mm) thermostated by a CTO-10AS Shimadzu column oven at 30 °C, was used for the separations of derivatized amino acids. The dwell time, tD (F ) 1), was 1.19 min, whereas the hold-up times, t0, were estimated as 2.14 and 2.58 min for the MeCN and MeOH eluting systems, respectively, at a constant flow rate, F ) 1 mL/min. Note that the hold-up times were measured by injection of water under different isocratic and isorheic conditions used in this study, as well as from the corresponding disturbance of the baseline at the beginning of the chromatogram. Finally, an average, constant t0 value at F ) 1 mL/min was estimated for each eluent system. The reference flow rate, F ) 1 mL/min, was measured at the start and at the end of the experiments to check the correctness of the pump settings. The obtained experimental retention data for the mixture of amino acids under isocratic conditions in MeCN- or MeOHmodified mobile phases are shown in Tables 2 and 3. Note that the 18 amino acids are listed in Tables 2 and 3 according to their elution times in φMeCN ) 0.3 and φMeOH ) 0.55, respectively. Finally, the retention data of the solutes tested under different dual-mode gradients are given in Tables 4 and 5 for MeCN gradients and in Table 6 for MeOH gradients. All retention data have been properly corrected for the extra-column volume.
(28) Pappa-Louisi, A.; Nikitas, P.; Balkatzopoulou, P.; Malliakas, C. J. Chromatogr., A 2004, 1033, 29-41.
1.4 1.9 2.1 1.8 1.7 1.8 4.4 4.0 2.9 3.2 1.8 1.8 0.9 0.5 0.6 0.8 0.1 0.3 1.8 7.10 8.57 9.21 10.04 11.20 13.03 14.06 14.24 15.10 15.42 18.05 18.68 25.53 28.60 29.95 31.67 37.11 38.03 7.002 8.411 9.024 9.862 11.01 12.80 14.70 14.84 15.55 15.94 18.38 19.01 25.76 28.76 30.14 31.92 37.16 38.14 0.9 1.4 1.5 1.2 1.1 1.1 1.2 2.6 2.5 2.6 1.2 1.1 0.3 0.1 0.1 0.1 0.8 0.7 1.1 6.60 7.89 8.46 9.18 10.19 11.77 13.25 13.36 14.06 14.41 16.56 17.10 22.93 25.81 27.17 28.88 34.32 35.25 6.543 7.778 8.336 9.072 10.08 11.64 13.41 13.72 14.42 14.79 16.76 17.29 23.00 25.79 27.15 28.90 34.06 35.00 3.4 0.7 0.7 0.7 0.7 0.8 1.0 0.9 0.6 0.6 0.4 0.6 1.4 1.4 2.1 2.3 1.8 2.1 1.2 8.78 10.34 10.68 11.13 11.80 12.72 13.16 13.30 13.88 14.03 15.92 16.37 21.98 24.85 26.15 27.87 33.32 34.23 8.490 10.27 10.60 11.06 11.72 12.82 13.30 13.42 13.96 14.12 15.99 16.47 22.30 25.20 26.72 28.53 33.93 34.98 0.0 0.5 0.5 0.3 0.0 0.1 0.5 0.7 0.3 1.8 1.0 0.9 0.0 0.4 0.5 0.4 1.1 1.0 0.6 5.62 6.62 7.06 7.61 8.37 9.56 10.86 11.08 12.38 13.02 14.74 15.27 21.11 24.00 25.46 27.16 32.58 33.51 5.622 6.588 7.022 7.590 8.367 9.549 10.92 11.16 12.42 13.26 14.89 15.42 21.11 23.90 25.33 27.06 32.22 33.18 1.9 1.0 0.8 0.5 0.7 0.7 0.2 0.2 0.1 0.0 0.3 0.2 0.1 0.5 0.1 0.5 0.8 1.1 0.5 8.78 10.34 10.68 11.13 11.80 12.67 12.87 12.96 13.25 13.23 14.15 14.29 16.25 17.19 17.08 17.76 19.63 19.77 8.619 10.24 10.60 11.07 11.72 12.75 12.90 12.99 13.24 13.23 14.11 14.26 16.23 17.27 17.10 17.85 19.78 19.99 0.8 1.0 0.4 0.2 0.1 0.1 0.5 0.7 0.2 1.2 0.8 0.8 0.6 1.0 0.8 1.1 1.3 1.6 0.7 5.62 6.62 7.06 7.61 8.37 9.56 10.86 11.08 12.38 12.79 13.61 13.80 15.91 16.88 16.85 17.52 19.38 19.53 5.664 6.555 7.033 7.593 8.378 9.551 10.92 11.16 12.41 12.95 13.72 13.91 16.00 17.06 16.98 17.71 19.63 19.85
and the adjustable parameters a, b and c were determined for each eluting system using the Solver of Excel. The obtained isocratic retention parameters were used in an algorithm constructed on the model developed in the theoretical section to describe the process in which the organic solvent concentration was stepwise changed in addition to a simple linear or stepwise flow rate gradient. This algorithm predicts the retention time of solutes under conditions of a combination of these two gradient modes using eq 18 by evaluating m from eq 17, 20, or 22. The values of n1, n2, ... and those of l1, l2, ... are calculated from the recursive eq 14 in combination with eq 15. Tables 4 and 5 summarize the comparison between the calculations and the experimental results obtained in the mobile phases containing MeCN as the organic modifier. The agreement is excellent in all the different types of dual-mode gradient programs tested, that is, in concurrently time-programmed two- or threestep variations of φMeCN in addition to stepwise or different steepness linear variations of the flow rate, shown in Figures 2
0.6 1.0 0.6 0.6 0.7 0.9 0.0 0.1 0.3 0.7 0.4 0.3 0.3 0.5 0.7 0.7 1.0 1.2 0.6
(29)
4.40 5.51 6.02 6.70 7.70 9.35 10.87 11.09 12.39 12.79 13.61 13.80 15.91 16.88 16.85 17.52 19.38 19.53
ln kφ ) a - cφ/(1+ bφ)
4.372 5.568 5.982 6.657 7.649 9.269 10.87 11.10 12.36 12.88 13.66 13.85 15.95 16.96 16.96 17.65 19.57 19.77
RESULTS AND DISCUSSION For predicting elution under simultaneously varied flow and solvent conditions, it is first necessary to know the isocratic behavior of the compounds under investigation. Thus, the isocratic retention data shown in Tables 2 and 3 for the MeCN and MeOH eluting systems, respectively, were fitted to the following expression of the dependence of kφ on φ28
Arg Tau Asn Gln Ser Asp Glu Thr Gly Dopa Ala GABA Met Val Trp Phe Ile Leu av
The algorithms used for prediction, according to the theory presented above, were written in-house in C++. All other calculations were performed using Excel spreadsheets on a 3-GHz Pentium CPU running under windows XP.
tR tR tR tR tR tR tR tR tR tR tR tR tR tR (exptl) (theor) % error (exptl) (theor) % error (exptl) (theor) % error (exptl) (theor) % error (exptl) (theor) % error (exptl) (theor) % error (exptl) (theor) % error
3.266 3.450 3.639 3.948 4.007 4.007 4.393 4.605 4.791 4.396 6.465 6.797 9.524 9.505 11.45 12.47 18.03 18.28
solute
4.132 4.402 4.878 5.411 5.707 5.964 7.001 7.192 7.401 7.419 12.09 13.20 22.97 25.76 32.01 33.15 55.66 56.78
VII
4.928 5.280 6.071 6.762 7.279 7.938 9.737 9.739 9.850 10.75 17.77 20.04 38.92 47.71 58.85 58.92 107.0 108.1
VI
5.659 6.017 7.063 7.865 8.644 9.645 12.08 12.02 12.01 14.03 22.65 26.48 54.20 71.78 85.92 85.77 159.6 160.0
V
Arg Asn Gln Ser Tau Asp Glu Thr Gly Dopa Ala GABA Met Trp Phe Val Ile Leu
IV
φ ) 0.7
III
φ ) 0.6
II
φ ) 0.55
I
φ ) 0.52
gradient
solute
Table 4. Comparison of Experimental and Predicted Retention Times (in min) of Amino Acid Derivatives Obtained under Dual-Mode MeCN Gradients Depicted in Figure 2
Table 3. Experimental Retention Times (in min) of Amino Acids Derivatives under Isocratic Conditions at a 1 mL/min Flow Rate in Mobile Phases Modified with Different φMeOH Values
Analytical Chemistry, Vol. 78, No. 16, August 15, 2006
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Table 5. Comparison of Experimental and Predicted Retention Times (in min) of Amino Acid Derivatives Obtained under Dual-Mode MeCN Gradients Depicted in Figure 3 gradient A
B
C
solutes
tR (exptl)
tR (theor)
% error
tR (exptl)
tR (theor)
% error
tR (exptl)
tR (theor)
% error
Arg Tau Asn Gln Ser Asp Glu Thr Gly Dopa Ala GABA Met Val Trp Phe Ile Leu av
5.616 6.673 7.068 7.635 8.423 9.648 11.04 11.27 12.57 13.34 15.06 15.57 19.42 20.24 20.80 21.12 23.41 23.28
5.62 6.62 7.06 7.61 8.37 9.56 10.86 11.08 12.38 13.02 14.74 15.27 18.95 19.99 20.18 20.82 22.65 22.83
0.1 0.8 0.1 0.3 0.6 0.9 1.6 1.6 1.5 2.4 2.1 1.9 2.4 1.3 3.0 1.4 3.2 1.9 1.5
6.774 8.218 8.752 9.563 10.39 10.91 11.47 11.75 12.64 12.70 15.37 15.92 20.04 21.27 21.38 22.15 24.17 24.42
6.89 8.27 8.87 9.17 9.69 10.25 10.89 11.21 12.20 12.13 14.99 15.53 19.63 20.81 20.91 21.62 23.63 23.80
1.7 0.6 1.3 4.1 6.7 6.1 5.1 4.6 3.4 4.5 2.4 2.4 2.0 2.1 2.2 2.4 2.2 2.6 3.1
6.145 7.188 7.521 8.049 8.753 9.852 11.20 11.45 12.69 13.28 15.00 15.56 19.75 20.85 21.05 21.74 23.64 23.89
6.23 7.18 7.58 8.09 8.80 9.90 11.20 11.42 12.66 13.12 14.85 15.38 19.61 20.64 20.88 21.51 23.34 23.52
1.4 0.1 0.8 0.5 0.5 0.5 0.0 0.2 0.3 1.2 1.0 1.2 0.7 1.0 0.8 1.0 1.3 1.6 0.8
Table 6. Comparison of Experimental and Predicted Retention Times (in min) of Amino Acids Derivatives Obtained under Dual-mode MeOH Gradients Shown in Table 1 gradient 1
2
3
solutes
tR (exptl)
tR (theor)
% error
tR (exptl)
tR (theor)
% error
tR (exptl)
tR (theor)
% error
Arg Asn Gln Ser Tau Asp Glu Thr Gly Dopa Ala GABA Met Trp Phe Val Ile Leu av
9.504 10.04 11.12 11.75 12.12 12.55 13.73 13.73 13.73 14.08 16.62 16.94 19.28 19.44 20.85 21.49 25.86 25.86
9.82 10.29 11.02 11.63 12.02 12.40 13.49 13.60 13.75 13.94 16.25 16.55 18.65 18.84 20.12 20.69 24.57 24.72
3.3 2.5 0.9 1.0 0.8 1.2 1.8 1.0 0.1 1.0 2.2 2.3 3.3 3.1 3.5 3.7 5.0 4.4 2.3
8.607 8.970 9.757 10.36 10.67 11.01 11.88
8.79 9.15 9.80 10.25 10.55 10.85 11.72 11.80 11.92 12.10 15.18 15.92 18.18 18.47 19.73 20.26 24.18 24.37
2.1 2.0 0.4 1.0 1.1 1.4 1.3
9.314 9.790 10.77 11.50 11.87 12.27 13.28
3.0 2.7 0.7 0.9 1.0 1.4 1.2
1.3 1.3 1.5 0.3 0.6 0.5 0.6 0.7
13.51 13.73 16.55 16.86 19.12 19.35 20.68 21.24
0.3 1.0
25.34
9.59 10.05 10.85 11.39 11.75 12.10 13.12 13.21 13.34 13.55 16.37 16.67 18.83 19.07 20.35 20.88 24.80 24.95
12.08 12.26 15.41 15.97 18.28 18.56 19.86 20.40 24.45
and 3. The average percentage error between experimental and predicted retention times was ∼1.2% for these gradient profiles. Here, it should be pointed out that a similarly good prediction could be achieved if in the prediction algorithm we use for simplicity a constant tD value instead of a variable one, tDj, derived from eq 4. Indeed the accuracy of such predictions was examined by using the value tD (F ) 1) ) 1.19 min because F ) 1 mL/ min is the average flow rate of the total flow range used in this study. The average error between the predictions obtained using variable and constant tD values for all dual-mode predictions was
