Anal. Chem. 2009, 81, 1217–1223
Multimode Gradient Elution in Reversed-Phase Liquid Chromatography: Application to Retention Prediction and Separation Optimization of a Set of Amino Acids in Gradient Runs Involving Simultaneous Variations of Mobile-Phase Composition, Flow Rate, and Temperature A. Pappa-Louisi,* P. Nikitas, K. Papachristos, and P. Balkatzopoulou Laboratory of Physical Chemistry, Department of Chemistry, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece The theory of multimode gradient elution in liquid chromatography involving combined gradients of the mobilephase composition with flow rate and column temperature is presented, and a very simple stepwise method that allows for the calculation of the elution time of a sample solute under all gradient conditions is proposed. The theory is successfully applied to the separation of 12 o-phthalaldehyde derivatives of amino acids in eluting systems modified by acetonitrile. Average errors below 2.9% have been found in the retention prediction using the above method, which is supported by adequate models and algorithms capable of describing the chromatographic behavior of solutes upon changes in the separation factors, such as the modifier content, flow rate, and temperature.
However, the performance of gradient elution is expected to enhance under multigradient conditions provided that the relevant theory is available. In multimode gradient elution, two or more separation variables are used for prediction and optimization. The multimode gradients may be divided into two categories: (a) Gradients related exclusively to the mobile-phase composition, that is, the separation variables are the volume fractions φ1, φ2, ..., of the constituents of the mobile phase and/or the pH of this phase, and (b) combined gradients of the mobile-phase composition with flow rate and/or column temperature. The fundamental equation for the gradients of the first category is the familiar relationship valid also for single φ and pH gradients
∫
tR-t0
0
Gradient elution in liquid chromatography is a powerful technique based on programmed separation modes and therefore the operation conditions are changed during the elution, according to a preset program.1-6 The main purpose of gradient elution is to accelerate the elution of strongly retained solutes, having the least retained component well resolved. It may operate in a single mode or multimodes. In single-mode gradient elution, only one separation variable, the mobile-phase composition, the column temperature, the flow rate, or the pH, varies with time. This is the most common operation of gradient elution since it is easier to control just one variable than many variables simultaneously. * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: +30 2310 997765. Fax: +30 2310 997709. (1) Snyder, L. R.; Kirkland, J. J. Introduction to Modern Liquid Chromatography, 2nd ed.; Wiley-Interscience: New York, 1979. (2) Jandera, P. Churacek, J. Gradient Elution in Column Liquid Chromatography: Theory and Practice; Journal of Chromatography Library 31; Elsevier: Amsterdam, 1985. (3) Snyder, L. R.; Glajch, J. L.; Kirkland, J. J. Practical HPLC Method Development; Wiley-Interscience: New York, 1997; Chapter 8. (4) Poole, C. F. The Essence of Chromatography; Elsevier: Amsterdam, 2003. (5) Schmidt-Traub, H. Preparative Chromatography of Fine Chemical and Pharmaceutical Agents; Wiley-VCH: New York, 2005. (6) Snyder, L. R.; Dolan, J. W. High Performance Gradient Elution; WileyInterscience: New York, 2007. 10.1021/ac801896n CCC: $40.75 2009 American Chemical Society Published on Web 01/05/2009
dt )1 t0kφ
(1)
where tR is the retention time of a sample analyte, k is its retention factor, and t0 is the column holdup time. This fundamental equation has been first proposed by Snyder et al.,2,7-11 whereas strict derivations of this equation may be found in refs 12 and 13. Equation 1 is inapplicable for the gradients of the second category, the theory of which is still under development in our laboratory. In particular, in previous publications,14-16 we studied two-mode gradient elution in liquid chromatography involving simultaneous changes in (a) flow rate and mobile-phase composition,14,15 and (b) temperature and mobile-phase composition.16 Here, we examine the more general case of gradients with (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)
Snyder, L. R. J. Chromatogr. 1964, 13, 415. Snyder, L. R. Chromatogr. Rev. 1965, 7, 1. Snyder, L. R.; Saunders, D. L. J. Chromatogr. Sci. 1969, 7, 145. Snyder, L. R. In High Performance Liquid Chromatography; Horvath, C., Ed.; Academic Press: New York, 1980; Vol. 1, p 207. Quarry, M. A.; Grob, R. L.; Snyder, L. R. Anal. Chem. 1986, 58, 907. Nikitas, P.; Pappa-Louisi, A. Anal. Chem. 2005, 77, 5670. Hao, W. Q.; Zhang, X. M.; Hou, K. Y. Anal. Chem. 2006, 78, 7828. Nikitas, P.; Pappa-Louisi, A.; Balkatzopoulou, P. Anal. Chem. 2006, 78, 5774. Pappa-Louisi, A.; Nikitas, P.; Balkatzopoulou, P.; Louizis, G. Anal. Chem. 2007, 79, 3888. Nikitas, P.; Pappa-Louisi, A.; Papachristos, K.; Zisi, C. Anal. Chem. 2008, 80, 5508.
Analytical Chemistry, Vol. 81, No. 3, February 1, 2009
1217
simultaneous changes in mobile-phase composition, i.e., organic modifier content, φ, flow rate, F, and temperature, T, and present a very simple stepwise method that allows for the calculation of the elution time of a sample solute under all gradient conditions. The validity of the developed theory was tested in the retention prediction and separation optimization of 12 o-phthalaldehyde (OPA) derivatives of amino acids.
step, t0,i is the holdup time under the same conditions, ki ) (tR,i t0,i)/t0,i is the corresponding retention factor, and L is the column length. Equation 3 upon integration results in the fundamental equation of gradient elution valid under any single mode or multimode gradient profile
∫
tR
0
THEORETICAL SECTION Basic Relationships. As pointed out above, the theory of twomode gradient elution has been developed in refs 14–16. This theory can be readily extended to three-mode gradients involving simultaneous variations in φ, T, and F. However, here we present an alternative approach, which is based on the numerical solution of the fundamental equation of multimode gradient elution. This approach has the advantage that it is computationally much simpler than the method developed in refs 14–16. In multimode gradient elution, a composite gradient (x1, x2, ..., xm) versus t is formed in the mixer of the chromatographic system, where t is the time variable and the coordinates x1, x2, ..., may be volume fractions of the mobile-phase constituents, φ1, φ2, ..., the pH of this phase, the flow rate F, or the column temperature T. Since the temperature that the analyte feels inside the column may be not the column temperature T but an effective one, Tef, we should consider Tef in place of T.16,17 The (x1, x2, ..., xm) versus ′ t profile is transformed to a new one, (x1′ , x2′ , ..., xm ) versus t, at the inlet of the chromatographic column, because any variation in F or T in the mixer reaches the column almost instantaneously, whereas any change in φi or pH needs some time to cover the same distance. Moreover, if F varies, there is a distortion of the φi or pH versus t profiles when they reach the inlet of the chromatographic column since any change in φi or pH that takes place at the time tp* in the mixer is transferred to the beginning of the column at tp, where tp* and tp are interrelated through the following equation14,15 tD(F ) 1) )
∫
tp
tp*
F(t) dt
(2)
Here, F(t) is the function of flow rate upon t and tD(F)1) is the product of the dwell time when the flow rate F is equal to 1, in arbitrary units, by the unit flow rate (F ) 1). Consider that we approximate the φi versus t (or pH vs t) profiles in the mixer by stepwise curves composed of a large number of infinitesimally small time steps ∆t. Due to the fact that the analyte and the mobile phase are moved with different velocities, the analyte feels each ∆t step for a time period equal to δtc different from ∆t.12 Since δtc is also an infinitesimally small time step, during δtc all separation variables may be considered constant. Therefore, the analyte moves at the ith time step δtc with a constant velocity va ) L/tR,i and covers a distance equal to δLi δtc δtc ) ) L tR,i t0,i(1 + ki)
(3)
where tR,i is the analyte retention time under the constant flow rate, temperature and composition of the mobile phase of the ith (17) Pappa-Louisi, A.; Nikitas, P.; Zisi, C.; Papachristos, K. J. Sep. Sci. 2008, 31, 2953.
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Analytical Chemistry, Vol. 81, No. 3, February 1, 2009
dtc )1 t0(1 + k)
(4)
The derivation of eq 4 leads almost directly to a very simple stepwise method that solves this equation under all gradient conditions. Note that the use of stepwise methods for the estimation of the elution time of a sample analyte has a long tradition in chromatography. Usually the fundamental equation is solved numerically by dividing the time axis into small segments,12,18-22 which is equivalent to approximate a gradient profile by a stepwise one.23-29 Consider that in the mixer the following profiles are formed: φi ) φi(t), pH ) pH(t), F ) F(t), and T ) T(t). From the latter we may know the effective profile Tef ) Tef(t). We divide the time axis into small segments, δt, and consider the ith time step. During this time step, the analyte covers a distance equal to δLi and its position from the inlet of the chromatographic column, li, can be calculated from li ) li-1 + δLi
(5)
where l0 ) 0. In order to calculate δLi we may use eq 3 with δt in place of δtc, because eq 3 is valid for every infinitesimally small time step δt irrespective of its physical meaning. In addition, for the calculation of δLi we need to know the values of the separation variables that affect the analyte during the ith step. That is, the values of φ1i, φ2i, ..., pHi, Fi, and Tefi. Since any change in F and T is transferred almost immediately from the mixer to the analyte, we have the following: Fi ) F(t) and Tefi ) Tef(t), where t ) iδt. For the values of φ1i, φ2i, ..., and pHi, we have to take into account the delay in any change in φi or pH created in the mixer to reach the analyte. This delay can be calculated by means of eq 2, which in the present case is extended to tD + t0li ⁄ L )
∫ F(t) dt t
t*
(6)
where t0 is the hold up time when F ) 1, t* is the time that a change in φi or pH takes place in the mixer and t is the time needed for this change to reach the analyte inside the column. (18) Vivo-Truyols, G.; Torres-Lapasio, J. R.; Garcia-Alvarez-Coque, M. C. J. Chromatogr., A 2003, 1018, 169. (19) Smith, R. D.; Chapman, E. G.; Wright, B. W. Anal. Chem. 1985, 57, 2829. (20) Snijders, H.; Janssen, H. G.; Cramers, C. J. Chromatogr., A 1995, 718, 339. (21) Chester, T. L. J. Chromatogr. A 2003, 1016, 181. (22) Chester, T. L.; Teremi, S. O. J. Chromatogr. A 2005, 1096, 16. (23) Cela, R.; Barroso, C. G.; Viseras, C.; Perez-Bustamante, J. A. Anal. Chim. Acta 1986, 191, 283. (24) Cela, R.; Barroso, C. G.; Perez-Bustamante, J. A. J. Chromatogr. 1989, 485, 477. (25) Cela, R.; Lores, M. Comp. Chem. 1996, 20, 175. (26) Cela, R.; Lores, M. Comp. Chem. 1996, 20, 193. (27) Cela, R.; Leira, E.; Cabaleiro, O.; Lores, M. Comp. Chem. 1996, 20, 285. (28) Cela, R.; Leira, E.; Cabaleiro, O.; Lores, M. Comp. Chem. 1996, 20, 315. (29) Cela, R.; Martinez, J. A.; Gonzalez-Barreiro, C.; Lores, M. Chemom. Intell. Lab. Syst. 2003, 69, 137.
Table 1. Experimental Retention Data (in min) of Amino Acid Derivatives Obtained in Different Isothermal/Isocratic Chromatographic Runs at F ) 1 mL/min and Used for Fitting Eq 10 temp, °C 15 φ solutes Arg Tau Asn Gln Ser Thr Dopa Met Val Phe Ile Leu
37
0.26
0.3
0.4
3.222 3.735 4.067 4.691 5.146 8.553 13.16
2.310 2.673 2.920 3.196 3.675 5.382 7.244 31.83 40.19 60.89 77.23 83.48
1.732 1.752 1.943 1.993 2.275 2.753 3.027 8.397 10.13 12.35 15.88 16.62
0.5
75
0.26
0.3
0.4
2.747 3.282 3.383 3.870 4.247 6.827 9.342
2.134 2.467 2.597 2.854 3.201 4.611 5.634 22.86 30.21 42.13 56.52 60.40
1.641 1.727 1.842 1.913 2.114 2.532 2.638 6.938 8.609 10.11 13.35 13.92
4.022 4.656 4.983 6.313 6.414
0.5
0.26
0.3
0.4
0.5
2.225 2.668 2.678 2.980 3.252 4.859 5.855
1.832 2.124 2.157 2.322 2.551 3.465 3.866 13.61 18.65 23.55 33.14 34.61
1.535 1.621 1.681 1.725 1.866 2.191 2.195 4.983 6.333 7.047 9.153 9.685
2.789 3.317 3.423 4.344 4.359
3.503 4.117 4.359 5.552 5.636
Table 2. Coefficients of the Sixth-Order Polynomials Outline the Actual Oven Temperature (Toven in K) Gradients T-gradient programmed
R0
R1
R2
R3 × 102
R4 × 103
R5 × 105
R6 × 107
20 f 80 °C at 2 min 15 f 75 °C at 30 min
290.19 287.67
8.54 0.55
-0.96 0.30
8.51 -2.69
-4.43 1.20
11.86 -2.52
-12.70 1.95
Therefore, in order to calculate the values of φ1i, φ2i, ..., and pHi we put in the upper limit of eq 6 t ) iδt and determine from this equation the value of the lower limit t*. If t* < 0, we put t* = 0. Using this value of time, the effective separation variables are calculated from φ1i ) φ1(t*), φ2i ) φ2(t*), ..., and pHi ) pH(t*). Now δLi is determined by means of eq 3 using t0,i ) t0/Fi and ki ) k(φ1i, φ2i, ..., pHi, Fi, Tefi), where to is the value of the holdup time that corresponds to F ) 1. The position of the analyte inside the column is estimated from the recursive eq 5. The analyte is eluted when lp-1 ⁄ L < 1 and lp ⁄ L g 1
(7)
and therefore the retention time may be calculated from tR ) pδt
(8)
EXPERIMENTAL SECTION In order to test the above theory, different isocratic, isothermal, and isorheic chromatographic runs as well as different types of single-mode, dual-mode, and three-mode gradient conditions were implemented in the reversed-phase HPLC separation of 12 OPA derivatives of amino acids: L-arginine (Arg), taurine (Tau), Lasparagine (Asn), L-glutamine (Gln), L-serine (Ser), L-threonine (Thr), β-(3,4-dihydroxyphenyl)-L-alanine (Dopa), L-methionine (Met), L-valine (Val), L-phenylanine (Phe), L-isoleucine (Ile), and L-leucine (Leu). The derivatives formed by the reaction of OPA with amino acids in the presence of 2-mercaptoethanol (2-ME). The detection of derivatized amino acids was performed at a spectrofluorometric detector (Shimadzu, model RF-10AXL) at 455 nm after excitation at 340 nm. The fluorescence detector was installed at constant sensitivity. Appropriate working concentrations of underivatized amino acids were used in the derivatization procedure by OPA/2-ME reagent (Tau, 0.2 µg/mL; Dopa, 2 µg/ mL; Met, Val, 4 µg/mL; Phe, Ile, Leu, 8 µg/mL; others, 1 µg/
mL) so that the peak heights of the OPA derivatives recorded by 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 (with a pH 2.5 and a total ionic strength, I ) 0.02 M) modified with different volume fractions, φ, of acetonitrile. Under the gradient process, the total ionic strength was also held constant at I ) 0.02 M, since all the mobile phases involved in this process had the same ionic strength. A Zorbax SB-C18 column (3.5 µm, 150 × 4.6 mm) thermostated by a CTO-10AS Shimadzu column oven was used for the separations of derivatized amino acids. The CTO-10AS oven is a rather fast-temperature change oven. It accommodates the injector and the gradient mixer and provides preheating flow lines to regulate mobile-phase temperature. At a constant flow rate, F ) 1 mL/min, the dwell time, tD, was found at 1.1 min, whereas the holdup time, t0, was estimated to be 1.37 min. Note that, this value of t0 is an averaged value of holdup times determined previously in different mobile phases and column temperatures by using water as a marker.31 The obtained experimental retention data for the mixture of amino acids under different isocratic and isothermal conditions (varying from φ ) 0.26 to φ ) 0.5 at different temperatures from 15 to 75 °C) at F ) 1 mL/min are shown in Tables 1 and 4. The retention data of the solutes tested under different single-mode gradients, dual-mode gradients, and three-mode gradients are given in Tables 5, 6, and 7, respectively. In Table 5, the one-mode gradients are denoted by g(φ), g(F), or g(T), when φ, F, or T is gradually varied during the chromatographic (30) Pappa-Louisi, A.; Nikitas, P.; Zitrou, A. Anal. Chim. Acta 2006, 573-574, 305. (31) Pappa-Louisi, A.; Nikitas, P.; Papachristos, K.; Zisi, Ch. J. Chromatogr., A 2008, 1201, 27.
Analytical Chemistry, Vol. 81, No. 3, February 1, 2009
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Table 3. Values of Adjustable Parameters Obtained by Fitting the Data of Table 1 to Eq 10 solutes
c1
c2
c3
c4
c5
c6
Arg Tau Asn Gln Ser Thr Dopa Met Val Phe Ile Leu
-31.39 -11.58 -11.70 -15.32 -10.11 -11.74 -7.91 -12.62 -13.80 -15.50 -21.16 -17.55
11799 5949 7182 8328 5877 7237 6693 8702 9612 10939 14267 11824
-10.44 -3.31 -4.99 -8.81 -3.44 -6.92 -2.81 -8.05 -10.07 -12.26 -18.29 -14.48
6738 4838 6132 7364 4698 6552 6030 7898 8978 10507 13784 11333
-7.312 -0.931 0.471 0.039 -0.103 0.432 1.204 0.403 0.453 0.821 0.880 0.561
2708 725.6 515.8 621.1 604.3 480.1 236.5 442.8 461.0 360.7 435.8 448.6
run. When φ and F, φ and T, F and T, or φ, F, and T, programming conditions are used simultaneously, the multimode gradients are denoted by g(φ,F), g(φ,T), g(F,T), or g(φ,F,T), respectively; see Tables 6 and 7. As shown in these tables, the following types of one-mode gradients were used in different multimode gradient runs: (a) two types of linear φ-gradients in which the organic content was increased from φ ) 0.34 or 0.3 to φ ) 0.4 and this increase was started at 3 or 0 min and completed at 4 or 30 min, respectively; (b) two types of F-gradients, in which F was increased linearly between 0.5 and 1.5 mL/min from 6 to 6.1 min or from 0 to 30 min; and (c) two types of linear T-gradients shown in Figure 1. Note that, due to a limitation of the oven we used, the programmed T-gradient profiles were stepwise. T-Gradients were generated in the oven by a sequence of small equal isothermal steps of 0.1 min for the T-gradient between 20 and 80 °C and of 1 min for the T-gradient between 15 and 75 °C, whereas the variation in the temperature was 3 and 2 °C at each step, respectively. The actual oven temperature displayed on the corresponding panel of the oven and recorded using a 14-bit ADDA card may present a delay with respect to the programmed T-gradients, the extent of which depends on the slope of the programmed gradient. The actual oven temperature, Toven, gradients that correspond to the stepwise T-programmed gradients used in this study are also depicted in Figure 1 by broken lines. The values of the oven temperature were fitted to a sixth-order polynomial, Toven ) R0 + R1t + · · · + R6t6, the coefficients of which are given in Table 2. However, due to hysteresis phenomena that appeared when we used conventional chromatographic columns, the oven temperature is not the effective temperature experienced by the analyte. This temperature is estimated by means of Newton’s law, which yields16,17 1 (a + 2a2t + ··· + nantn-1) + h 1 1 n 1 (2a2 + ··· + n(n - 1)antn-2) ··· (n ! an) + ce-ht (9) 2 h h
Tef ) (a0 + a1t + a2t2 + ··· + antn) -
( )
when Toven ) R0 + R1t + · · · + Rntn. Here, h is a constant characteristic of the system and c is an integration constant. Following the procedure explained in refs 16 and 17, we found that h ) 0.12 ± 0.02 min-1. Then the integration constant c is calculated from the initial condition Tef(t ) 0) ) Toven(t ) 0) ) a0. All algorithms used for fitting retention data obtained under isocratic, isothermal, and isorheic conditions, and for testing the 1220
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prediction ability of the theory of multimode gradient elution presented above, were homemade written in C++. The exe files of the basic programs are free available upon request from the authors. RESULTS AND DISCUSSION For the application of eqs 3, 5, 7, and 8 to predict tR values under any single mode or multimode gradient profile conditions, the first step is to describe the retention as a function of the experimental factors (φ, F, and T) examined in this work. The accuracy of this description by an appropriate retention model is decisive for the reliability of retention prediction under any new, both isocratic/isorheic/isothermal and gradient, elution conditions. As the solute retention factor is independent of F,14,15,30 a retention model considering simultaneously φ and T should be built with data obtained for sets of carefully designed experiments. In the present study, we adopted the following retention relationship31
(
)[ ( )] [ ( ) ]
c4 c6 c3 + exp c5 + φ c2 T T ln kφT ) c1 + T c6 1 + exp c5 + -1 φ T
(10)
where c1, c2, ..., c6 are adjustable parameters. Note that when eq 10 is applied under gradient conditions, T is the effective temperature, Tef, whereas under isocratic conditions, T ) Toven. Equation 10 is a straightforward extension of the following equation32-34
ln kφ ) a +
bβφ 1 + (β - 1)φ
(11)
if we assume the constants a and b depend linearly upon 1/T, i.e., a ) c1 + c2/T, b ) c3 + c4/T, and a similar expression is valid for the equilibrium constant β of the retention process
ln β ) c5 +
c6 T
(12)
Note that, according to the thermodynamic derivation of eq 11,32–34 this model is strictly valid when adsorption predominates in the retention mechanism. In this case, constant β is closely related to the equilibrium constant of the coadsorption process of modifier and water molecules on the alkyl chains of the stationary phase. The dependence of this constant upon T is discussed extensively in ref 31. For the determination of the adjustable parameters c1, c2, ..., c6 of eq 10, we used the RND_LM algorithm described in ref 35 and the isocratic/isorheic/isothermal data presented in Table 1. Note that a 3 × 3 φ and T experimental design was used in this (32) Pappa-Louisi, A.; Nikitas, P.; Balkatzopoulou, P.; Malliakas, C. J. Chromatogr., A 2004, 1033, 29. (33) Nikitas, P.; Pappa-Louisi, A.; Agrafiotou, P. J. Chromatogr., A 2002, 946, 9. (34) Nikitas, P.; Pappa-Louisi, A.; Agrafiotou, P. J. Chromatogr., A 2002, 946, 33. (35) Nikitas, P.; Pappa-Louisi, A.; Papageorgiou, A. J. Chromatogr., A 2007, 1157, 178.
Table 4. Experimental Retention Data (in min) of Amino Acid Derivatives Obtained in Different Isocratic/Isorheic/ Isothermal Chromatographic Runs and Absolute Percentage Errors between Them and Calculated Retention Data by Eq 10 Using the Parameters of Table 3 temp, °C φ solutes
15
37
40
0.34
0.34
0.36
65 0.3
75
0.34
0.4
0.34
tR (exp) % error tR (exp) % error tR (exp) % error tR (exp) % error tR (exp) % error tR (exp) % error tR (exp) % error
Arg Tau Asn Gln Ser Thr Dopa Met Val Phe Ile Leu
1.903 2.144 2.356 2.502 2.899 3.856 4.701 17.20 21.32 29.32 37.54 40.04
average
2.7 0.5 0.4 0.3 0.2 0.3 0.3 0.1 0.2 0.3 1.0 0.4
1.842 2.064 2.205 2.341 2.638 3.463 3.921 13.22 17.02 21.97 29.38 31.08
0.5
0.4 1.7 2.3 2.0 2.2 1.9 2.3 0.0 0.2 0.1 1.2 0.3
1.731 1.887 2.025 2.104 2.369 3.004 3.278 10.02 12.83 15.96 21.40 22.48
1.2
1.1 0.4 1.4 0.2 0.9 0.9 1.3 0.5 0.0 0.3 1.8 0.6
1.953 2.245 2.305 2.492 2.753 3.800 4.329 15.42 21.03 27.17 37.78 39.71
0.8
1.7 1.4 2.1 1.8 1.9 1.8 2.5 1.5 1.3 1.5 1.4 1.8
1.737 1.933 1.993 2.124 2.315 2.970 3.176 9.535 12.60 15.32 21.07 21.99
1.7
1.1 2.1 2.1 3.0 1.9 2.2 2.7 0.4 0.4 0.7 2.1 0.7
1.591 1.701 1.752 1.792 1.943 2.275 2.305 5.457 6.887 7.762 10.39 10.69
1.6
1.0 3.2 2.0 1.3 1.1 0.2 0.7 1.7 1.3 1.6 2.8 1.3
1.655 1.836 1.867 1.980 2.164 2.756 2.894 8.278 11.07 13.22 18.23 18.93
1.1 0.5 1.4 0.5 0.8 0.1 0.0 2.2 1.6 1.1 0.1 1.4
1.5
0.9
Table 5. Experimental Retention Data (in min) of Amino Acid Derivatives Obtained in Different Single-Mode Gradient Runs and Absolute Percentage Errors between Them and Calculated Retention Data by the Predictive Approach Explained in the Text gradient
g(φ)1
g(φ)2
g(φ)3
g(F)1
g(T)1
g(T)2
g(T)3
φ
0.34 f 0.4
0.34 f 0.4
0.34 f 0.4
0.36
0.36
0.36
0.36
t,min
3f4
3f4
3f4
F, mL/min
1
0.5
1.5
0.5 f 1.5
1
0.5
1.5
20 f 80
20 f 80
20 f 80
0f2
0f2
0f2
t, min
6 f 6.1
temp, °C
40
40
40
40
t, min
solutes tR (exp) % error tR (exp) % error tR (exp) % error tR (exp) % error tR (exp) % error tR (exp) % error tR (exp) % error Arg Tau Asn Gln Ser Thr Dopa Met Val Phe Ile Leu aver.
1.807 2.018 2.133 2.275 2.545 3.352 3.766 9.105 10.78 12.31 15.42 15.96
1.5 0.2 0.2 0.4 0.1 0.4 0.8 0.9 0.8 1.2 0.4 1.2 0.7
3.516 3.954 4.218 4.483 5.056 6.699 7.570 16.46 19.78 22.85 29.12 30.23
4.3 1.8 1.0 1.1 0.5 0.4 1.3 0.3 0.3 0.1 0.7 0.1 1.0
1.223 1.324 1.424 1.544 1.730 2.277 2.556 6.788 7.924 9.021 11.13 11.50
0.1 1.4 0.3 2.2 2.1 2.3 2.6 0.5 0.6 0.4 1.2 0.5 1.2
study for fitting eq 10, in order to avoid “overfitting” problems appearing when only two values of temperature are used,31 but a 3 × 2 φ and T experimental design could also be successfully used in some experimental systems. The values of the c1, c2, ..., c6 parameters found by the above fitting procedure are listed in Table 3. Once the dependence of ln k upon φ and T is known, the retention, tR, can be calculated in any isocratic/isorheic/isothermal runs as well as in gradient runs involving variations of φ, F, or T. Tables 4-7 summarize the comparison between calculated and experimental retention times of solutes under isocratic/isorheic/ isothermal conditions (Table 4) and under gradients where one
3.375 3.709 3.962 4.146 4.657 5.902 6.231 10.72 12.57 14.59 18.15 18.86
3.7 1.4 0.8 1.3 0.8 0.9 0.7 0.3 0.2 0.3 0.6 0.3 0.9
1.792 1.943 2.144 2.235 2.567 3.292 3.765 11.41 13.90 16.78 21.01 21.82
2.6 1.0 1.0 1.6 1.3 1.2 1.8 1.4 1.2 1.4 0.5 1.5 1.4
3.534 3.866 4.238 4.46 5.115 6.565 7.491 21.53 26.26 31.00 38.97 40.35
3.8 1.3 1.6 1.2 0.6 0.2 0.7 2.1 1.4 2.1 0.6 1.7
1.258 1.359 1.510 1.570 1.792 2.315 2.658 8.322 10.12 12.37 15.41 16.04
1.4
2.5 3.7 4.2 3.5 3.1 3.8 3.0 2.9 2.6 1.3 1.1 0.0 2.6
(Table 5) or more (Tables 6 and 7) of the experimental factors examined in this studied are varied simultaneously. It seen that the agreement is very satisfactory in all-different types of elution conditions. Average errors below 2.9% have been found in the retention prediction for all types of gradient programs tested using the prediction procedure described above; see Tables 4-7. The high accuracy of retention prediction achieved may permit the determination of the best simultaneous variation patterns of φ, F, and T versus t that leads to the optimum separation of the sample of interest. However, optimization of multimode gradients increases the complexity of the computations. For this reason, the optimizaAnalytical Chemistry, Vol. 81, No. 3, February 1, 2009
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Table 6. Experimental Retention Data (in min) of Amino Acid Derivatives Obtained in Different Dual-Mode Gradient Runs and Absolute Percentage Errors between Them and Calculated Retention Data by the Predictive Approach Explained in the Text gradient
g(φ,F)1
g(φ,F)2
g(φ,F)3
g(φ,F)4
g(φ,T)1
g(F,T)1
φ
0.34 f 0.4
0.34 f 0.4
0.34 f 0.4
0.3 f 0.4
0.34 f 0.4
0.36
t, min
3f4
3f4
3f4
0 f 30
3f4
F, mL/min
0.5 f 1.5
0.5 f 1.5
0.5 f 1.5
0.5 f 1.5
1
t, min
6 f 6.1
6 f 6.1
6 f 6.1
0 f 30
temp, °C
40
20
60
45
0.5 f 1.5 6 f 6.1
20 f 80
20 f 80
0f2
0f2
t, min
solutes
tR (exp)
% error
tR (exp)
% error
tR (exp)
% error
tR (exp)
% error
tR (exp)
% error
tR (exp)
% error
Arg Tau Asn Gln Ser Thr Dopa Met Val Phe Ile Leu
3.559 3.987 4.260 4.514 5.093 6.314 6.600 9.583 10.67 11.69 13.77 14.14
3.1 1.0 0.0 0.4 0.2 0.9 1.2 0.5 0.2 0.2 0.7 0.1
3.653 4.077 4.495 4.764 5.502 6.499 6.923 10.45 11.65 13.13 15.51 16.02
5.7 3.5 2.9 3.2 2.3 0.5 0.2 0.6 0.2 0.2 0.6 0.3
3.443 3.867 4.002 4.266 4.704 6.090 6.265 8.891 9.888 10.61 12.38 12.63
1.1 1.0 0.9 1.6 1.4 2.0 1.6 1.6 1.4 1.6 1.8 1.1
3.685 4.158 4.339 4.711 5.195 6.937 7.883 17.90 20.57 23.06 26.34 26.89
0.4 0.9 0.1 0.2 0.7 1.5 1.7 1.0 1.4 1.6 2.1 1.7
1.893 2.094 2.305 2.436 2.799 3.715 4.410 9.958 11.51 13.11 15.85 16.35
2.6 1.3 1.3 1.7 1.6 1.7 2.2 1.5 1.6 1.9 1.4 2.1
3.387 3.667 4.046 4.236 4.871 6.127 6.431 11.18 12.81 14.81 17.47 17.99
5.9 4.6 2.1 2.3 0.2 1.5 1.3 2.7 3.0 2.6 3.6 4.7
aver.
0.7
1.7
Figure 1. Two stepwise programmed in the oven T- gradients (s), the corresponding actual oven temperature profiles (- -), and the Tef-gradients (s) that the analytes experience.
tion process proposed in this paper is based on the selection of the best combination of predefined sets of single-mode gradient profiles carefully chosen for the analysis under consideration. Thus, the first step is a rough estimation of possible single-mode gradient programs, which will be the input parameters in the optimization algorithm used in this paper. The number of φ-, F-, or T-gradients that can be input are virtually unlimited, but in practice, the selection of optimum multimode gradient conditions is carried out using a certain set of predefined single-mode gradient profiles. From the preliminary experimental data shown in Table 1, it is clear that our sample consists of two groups of analytes, the weakly retained Arg, ..., Dopa and the strongly retained Met, ..., Leu. Having this in mind, steep single-mode programming conditions should be input in the optimization algorithm in order to balance a satisfactory 1222
Analytical Chemistry, Vol. 81, No. 3, February 1, 2009
1.4
1.1
1.7
2.9
separation for the least retained solutes and a reasonable elution time for the most retained ones. Using the input φ-, F-, or T-single gradients, the optimization algorithm forms all possible (φ, F, T) three-mode gradients and at each multimode gradient it computes the minor distance of adjacent peaks, δt, and the analysis time, tg, determined by the elution time of the most distant solute. Thus, the output of the optimization algorithm is a table that contains the values δt and tg of each (φ, F, T) three-mode gradient. It is evident that the best three-mode gradient is that it yields the maximum δt value in the shorter analysis time. Finally, the corresponding chromatogram that is expected to give maximal resolution in the shorter run time is calculated. The optimum three-mode gradient profile involving simultaneous variations of φ, F, and T, which is selected for the separation of the mixture of 12 amino acid derivatives following the above-described optimization procedure, is that denoted by g(φ,F,T)1 in Table 7. Figure 2C shows the chromatogram selected as optimum. A regular peak distribution is shown in the optimized gradient chromatogram leaving a relatively large void in the initial part of chromatogram, which is very useful for the analysis of real samples. For comparison, Figure 2A shows the chromatogram obtained under isocratic conditions, and Figure 2B the chromatogram recorded under one-mode gradient conditions. Thus, this combination of three single-mode gradients results in narrower peaks, more regularly spaced in the chromatogram and in the minimum analysis time, ∼14 min, although the benefits of T-gradient are mostly lost because of the two delay phenomena, one between programmed and actual oven temperature and the other between the actual and effective temperature depicted in Figure 1. For this reason, Tef does not change significantly during the optimal multimode gradient run
Table 7. Experimental Retention Data (in min) of Amino Acid Derivatives Obtained in Different Three-Mode Gradient Runs and Absolute Percentage Errors between Them and Calculated Retention Data by the Predictive Approach Explained in the Text gradient g(φ,F,T)1
g(φ,F,T)2
g(φ,F,T)3
φ
0.34 f 0.4
0.3 f 0.4
0.3 f 0.4
t, min
3f4
0 f 30
0 f 30
F, mL/min
0.5 f 1.5
0.5 f 1.5
0.5 f 1.5
t, min
6 f 6.1
0 f 30
0 f 30
temp, °C
20 f 80
20 f 80
15 f 75
t, min
0f2
0f2
0 f 30
solutes
Figure 2. Fluorescence detection chromatograms of a 12-component mixture of derivatized amino acids obtained (A) under isocratic/ isorheic/isothermal conditions at φ ) 0.36, F ) 1 mL/min, and 40 °C, (B) by using the single-mode gradient g(φ)1 shown in Table 5, and (C) by using the selected as optimum three-mode gradient g(φ,F,T)1 shown in Table 7. The elution order of the amino acid derivatives is the following: Arg, Tau, Asn, Gln Ser, Thr, Dopa, Met, Val, Phe, Ile, Leu.
and to first approximation when running an identical gradient program to the optimum, but at fixed temperature of 40 °C, i.e., g(φ,F)1 gradient profile shown in Table 6, the produced chromatogram is practically the same as that obtained under optimum threemode programming conditions. CONCLUSIONS In this paper, an algorithm capable of predicting the separation of any gradient runs involving simultaneous variations of φ, F, and T was developed, based on the substitution of linear single-mode gradients by several subsequent isocratic/isorheic/isothermal steps with various values of φ, F, and T. Data coming from isocratic/ isothermal experiments of a mixture of 12 OPA amino derivatives at a constant flow rate (1 mL/min) were used to anticipate the retention at any multimode gradient conditions. Average errors below 2.9% have been found in the retention prediction for all types of gradient programs tested provided that all delay phenomena considering variations of temperature were properly embodied into the treatment. Additionally, since the optimization of two or more separation factors complicates the optimization treatment notably, a simple optimization approach was developed, which permits the selection of the optimal combination of predefined single-mode gradient profiles. This optimization strategy was successfully applied to the separation of these 12 OPA amino derivatives, even though the variation of temperature or flow rate is generally less effective than the variation of mobile-phase composition to affect the retention at least to the conventional reversed-phase column used in this paper.
Arg Tau Asn Gln Ser Thr Dopa Met Val Phe Ile Leu
tR (exp)
% error
tR (exp)
% error
tR (exp)
% error
3.604 4.007 4.420 4.692 5.387 6.161 6.877 9.978 11.02 12.03 13.92 14.24
4.6 2.8 0.2 0.2 2.4 2.3 0.1 0.3 0.6 1.0 1.0 1.6
3.987 4.450 4.842 5.246 5.850 7.773 9.112 18.27 20.41 22.31 24.93 25.30
0.8 2.2 0.8 0.5 0.1 0.9 1.0 0.1 0.1 0.3 0.1 0.8
4.057 4.510 4.937 5.397 6.031 8.075 9.726 20.16 22.31 24.68 27.33 27.80
0.2 2.2 1.3 0.0 0.1 0.8 0.9 1.1 1.1 0.9 1.2 0.8
average
1.4
0.6
0.9
The use of three-mode gradients has been shown to have a big potential for producing well-balanced optimized chromatograms in the minimum analysis time. However, in order to get full advantage from the benefits of multimode gradient runs involving simultaneous variations of T or F, there are some equipment-related factors that have been kept in mind. Temperature can be a strongly complementary variation factor in gradient elution only if a relatively rapid response of the temperature inside the column to the temperature setting in the oven is enabled. On the other hand, flow programming should not be limited by the maximum pressure allowed by instrumentation. These drawbacks may be improved in future by using more sophisticated heating equipment for liquid chromatography and micro- or capillary-separation columns, stable enough at high temperatures, with higher permeability and lower back pressure than conventional particle-packed columns. In any case, this paper is the first step to encourage chromatographers to use and optimize multimode gradient elution conditions in order to separate complex samples with a maximal resolution in as short time as possible. ACKNOWLEDGMENT IKY, Greece for a PhD Fellowship to P. Balkatzopoulou.
Received for review September 8, 2008. Accepted October 10, 2008. AC801896N
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