Anal. Chem. 2007, 79, 3888-3893
Fundamental Equation of Dual-Mode Gradient Elution in Liquid Chromatography Involving Simultaneous Changes in Flow Rate and Mobile-Phase Composition A. Pappa-Louisi,* P. Nikitas, P. Balkatzopoulou, and G. Louizis
Laboratory of Physical Chemistry, Department of Chemistry, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
The rigorous derivation of the fundamental equation of the dual-mode gradient elution in liquid chromatography involving any type of simultaneous changes in flow rate and mobile-phase composition is developed following Drake’s approach. The equation is a generalization of the already known fundamental equations of single gradient when either the mobile-phase composition or the flow rate is constant. The theory was tested in the retention prediction from isocratic data 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. The average percentage error between experimental and predicted retention times ranged from 0.9 to 2.5%. Two approximations that simplify the calculations considerably without increasing the above error were also proposed. Gradient elution in liquid chromatography is based on programmed separation modes.1-4 The useful modes are the mobilephase composition, the flow rate, and the column temperature. At present, the majority of work on gradient elution is carried out using single gradients, and there are only a few reports concerning “dual-mode gradient”, i.e., the coupling of solvent and flow or temperature programming.5-12 A possible reason is that the development of dual-mode gradients is more difficult than that of the single gradient elution mode due to the lack of an appropriate theory for prediction of the solute elution times. The * To whom correspondence should be addressed. Tel.: +30 2310 997765. 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) Jandera, P.; Churacek, J. Gradient Elution in Liquid Chromatography. Theory and Practice; Elsevier: Amsterdam, 1985. (3) Snyder, L. R.; Dolan, J. W. Adv. Chromatogr. 1998, 38, 115. (4) Jandera, P. Adv. Chromatogr. 2005, 43, 1. (5) Lesins, V.; Ruckenstein, E. J. Chromatogr. 1989, 467, 1-14. (6) Hsieh, Y.; Wang, G.; Wang, Y.; Chackalamannil, S.; Korfmacher, W. A. Anal. Chem. 2003, 75, 1812-1818. (7) Paci, A.; Caire-Maurisier, A.-M.; Rieutord, A.; Brion, F.; Clair, P. J. Pharm. Biomed. Anal. 2001, 27, 1-7. (8) Yokoyama, Y.; Ozaki, O.; Sato, H. J. Chromatogr., A 1996, 739, 333-342. (9) Earley, R. L.; Miller, J. S.; Welch, L. E. Talanta 1998, 45, 1255-1266. (10) Warner, B. D.; Boehme, G.; Legg, J. I. J. Chromatogr. 1981, 210, 419-431. (11) Yokoyama, Y.; Tsuji, S.; Sato, H. J. Chromatogr., A 2005, 1085, 110-116. (12) Ricci, M. C.; Cross R. F. J. Liq. Chromatogr., Relat. Technol. 1996, 19 (4) 547-564.
3888 Analytical Chemistry, Vol. 79, No. 10, May 15, 2007
lack of this theory is mainly associated with the different properties that these three modes exhibit. Thus, a certain solvent gradient profile formed in the mixer of the chromatographic system meets the analyte inside the column after several minutes, whereas any change in flow rate or column temperature reaches the analyte almost instantaneously. In addition, changes in flow rate cause the deformation of the solvent gradient profile formed in the mixer when this profile reaches the inlet of the chromatographic column, because the various parts of this profile move with different velocities. In a recent paper,13 we have shown that, due to the above differences, the basic expressions for the movement of the analyte inside the column under solvent gradients at constant flow rate and under flow rate gradients at constant solvent composition are different, and this makes difficult the direct coupling of the two effects into a single and simple expression. Despite this difficulty, we succeeded in that paper to combine these two gradient modes in a single expression when a stepwise gradient profile of the volume fraction of the organic modifier in the water-organic mobile phase, φ, versus time, t, is coupled with flow rate variations. In the present paper, we attempt to extend this treatment to dualmode gradients composed of any type of φ versus t profiles and flow rate variations. THEORY Basic Expressions for the Solute Retention. Consider an arbitrary φ versus t gradient profile formed in the mixer. This profile is transformed to a new one, φ versus t*, at the inlet of the chromatographic column, since an event that takes place at the time tp in the mixer is transferred to the inlet of the column at tp*, where tp and tp* are interrelated through the following equation13
tD(F ) 1) )
∫
t*p
tp
F(t) dt
(1)
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). Following Drake’s approach,14,15 the curve of φ versus t* may be approximated by a stepwise curve composed of a large number of very small time (13) Nikitas, P.; Pappa-Louisi, A.; Balkatzopoulou, P. Anal. Chem. 2006, 78, 5774-5782. 10.1021/ac070090r CCC: $37.00
© 2007 American Chemical Society Published on Web 04/20/2007
Figure 1. Schematic representation of the movement of the fronts of the mobile phase when the φ versus t* profile (dotted line) at the inlet of the chromatographic column is approximated for simplicity by four steps under variable flow rate and the corresponding path of an analyte inside the column shown by the thick line 0abcd.
steps, δt*. Similarly, the F versus t curve may be approximated by a stepwise curve of a large number of infinitesimally small time steps, δt, where δt , δt* for reasons explained in ref 13. That is, both curves of φ versus t* and F versus t are approximated by stepwise curves, which have different time steps, δt* and δt, respectively. Under these conditions, the problem we are treating becomes similar to that examined in ref 13, where a stepwise gradient profile of φ versus t is coupled with flow rate variations. Figure 1 depicts schematically the approximation of a φ versus t* profile (broken line) by only four steps in φ for shake of simplicity. 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 versus t, where x is the distance from the beginning of the chromatographic column, is not linear. 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. In this figure, a, b, and c are the points where the analyte meets the fronts of the mobile phase. If the φ versus t* curve is approximated by q steps and choose the steps δt* and δt such that their ratio is to be an integer number, say m ) δt*/δt, the steps of the φ versus t* curve occur at the following times
t*1 ) δt* ) mδt,
Figure 2. Effect of flow rate variation on a gradient profile (- - -) programmed and formed in the mixer. The flow rate (in mL/min) changes linearly from 0.5 at t ) 0 to 1.5 at t ) 5 min. Lines (...) and (s) represent the gradient profile transformed at the inlet of the chromatographic column when F is constant and equal to 1 mL/min, and if we correct only the times tφ1, tφ2 by means of eq 1, respectively. Points represent the gradient profile at the inlet of the chromatographic column when it is calculated from eq 1. tD ) 1.19 min for F ) 1 mL min-1.
in the mixer is transformed to a new one, φ versus t*, at the beginning of the column, whereas inside the column, the analyte is subject to the φ versus tc profile. When the φ versus t* curve is approximated by a stepwise curve with steps at t1*, t2*, t3*, ..., the φ versus tc profile is also stepwise with steps at tc1, tc2, tc3, ..., as is shown in Figure 1, whereas the φ values of the steps, φ1, φ2, φ3, ..., remain the same. Thus, the present treatment corrects the mobile-phase composition not only at the column inlet but also inside the column, where all relative equations utilize the value of φ at the peak location. Note that the integer numbers n1, n2, ... appearing in eq 3 are not known. According to the treatment presented in ref 13, these numbers and therefore the times tc1, tc2, ..., can be calculated from the following equation that determines the solute movement inside the column np
(1 + kp)
∑
i)pm+1
np
∑
Fi -
Fi )
i)np-1+1
to(1 + kp)Ip-1/L , δt
p ) 1, 2, 3, ... (4)
where
lp-1 L
np-1
) (δt/to)
∑
Fi
(5)
i)(p-1)m+1
t*2 ) 2δt* ) 2 mδt, ..., t*q) qδt* ) qmδt (2)
In addition, the solute meets the fronts of the mobile phase at
tc1 ) n1δt,
tc2 ) n2δt, ...,
tcq ) nqδt
(3)
where the integers n1, n2, ..., nq are, in fact, defined from eq 3. It is seen that the initial φ versus t profile programmed and formed
Here, L is the column length, lp-1 is the distance from the inlet of the chromatographic column traveled by the mobile phase for tcp-1 time (see Figure 1), to is the column holdup time when F ) 1 in arbitrary units, Fi is the flow rate during the ith step of the F versus t curve, and kp is the solute retention factor at the pth step of the φ versus t* curve. Note that indices i and p denote (14) Drake, B. Akriv. Kemi 1955, 8, 1. (15) Nikitas, P.; Pappa-Louisi, A. Anal. Chem. 2005, 77, 5670-5677.
Analytical Chemistry, Vol. 79, No. 10, May 15, 2007
3889
Table 1. Dual-Mode Gradients Used in Mobile Phases Modified with MeCNa gradient φ1 φ2 tφ1 tφ2 F1 F2 tF1 tF2 aTime
I
II
III
IV
V
VI
0.3 0.5 0 10 0.5 1.5 0 10
0.3 0.5 0 25 0.5 1.5 0 10
0.3 0.5 10 16 0.5 1.5 2 9
0.3 0.4 0 20 0.5 1.5 0 10
0.3 0.4 0 20 1 1.5 0 10
0.3 0.4 0 20 1.0 1.5 10 10.01
Table 2. Initial and Final Times (in min) for the Linear Variation of φ from 0.55 to 0.7 and F from 0.5 to 1.5 mL/min in Mobile Phases Modified with MeOH
(a) the ith and pth step of the F versus t and φ versus t* curves, respectively, throughout this paper. After rearrangement eq 4 is simplified to
kp
∑
pm
∑
Fi ) (1 + kp)
i)np-1+1
Fi,
p ) 1, 2, 3, 4, ...
which allow for the determination of np and through eq 5 for the calculation of the distance lp. Equation 6 is a recursive equation, which starts with p ) 1. Then we have np-1 ) n0 ) 0. Fundamental Equation. The distance Lp traveled by the analyte during the time period tcp - tcp-1 is given by13
)
lp - lp-1
L
np
δt
)
∑
to(1 + kp)i)np-1+1
L
Fi
(7)
Lp
r
p)1
{
δt
np
∑
to(1 + kp)i)np-1+1
}
Fi g 1
(8)
where r is the lowest number of terms of the sum that makes the above inequality valid. If we consider that the quantity δtc ) tcp - tcp-1 is an infinitesimal small quantity and use the first mean value theorem of integrals, we obtain np
∑
Fiδt )
i)np-1+1
∫
tcp
tcp-1
F(t) dt ) (tcp - tcp-1)F(tcp) )δtcF(tcp) (9)
Then eq 8 may be readily expressed as
∫
tR
0
F dtc to(1 + k)
)1
(10)
which is the fundamental equation of the dual-mode gradient elution involving any simultaneous changes in flow rate and mobile-phase composition. In eq 10, dtc is equal to δtc. From this 3890
3
tφ1 tφ2 tF1 tF2
4 12 9 17
5 13 6 13
5 13 8 16
Analytical Chemistry, Vol. 79, No. 10, May 15, 2007
F
∫
tR
0
dtc to(1 + k)
)F
∫
tR-to
0
dt ) 1, tok when F is constant (11)
(b)
1 to(1 + k)
∫
tR
0
F(t) dt ) 1,
when φ is constant
(12)
It is evident that the application of eq 10 to the calculation of tR requires the expression of tc in terms of t or t* or equivalently the determination of k as a function of tc. Alternatively, a more direct calculation of tR can be performed by means of eqs 5 and 6 as follows. The analyte is eluted when
lp-1/L < 1
and
lp/L g 1
(13)
Therefore, the retention time may be calculated from
tR ) npδt
From this equation we obtain that the analyte is eluted when
∑ L )∑
2
(6)
i)(p-1)m+1
Lp
1
equation, we obtain directly the fundamental equations of single gradient when either φ or F is constant:2,3,10,16
in min; F in mL/min.
np
gradient
(14)
with an accuracy of the order of δt*. The application of the above equations to the calculation of the prediction retention time tR from eq 14 necessarily requires the functional dependence of the retention factor k upon the mobile-phase composition. In the present study, we have adopted the following expression17
ln kφ ) a - cφ/(1 + bφ)
(15)
where the adjustable parameters a, b, and c may be determined for each eluting system from isocratic data. In more detail, the calculation of tR by means of eqs 5, 6, 13, and 14 involves the following procedure: We first define the duration δt* of each step of the φ versus t* profile at the inlet of the chromatographic column as well as the number m ) δt*/δt. In this paper, we used δt* ) 0.1 min and m ) 1000, since δt should be much smaller than δt*, δt , δt*. Then, at each time t*p) pδt* (p ) 1, 2, 3, ...) of the φ versus t* profile, the corresponding time tp in the mixer is calculated from eq 1. Thus, from the known φ versus t gradient formed in the mixer, the mobile-phase composition φ ) φp at t*p and the corresponding kp value are determined. Next, the recursive eq 6 is used for the calculation of n1, n2, n3, ... The value of Fi at the ith step, necessary for the application of eq (16) Pappa-Louisi, A.; Nikitas, P.; Zitrou, A. Anal. Chim. Acta 2006, 573-574, 305-310. (17) Pappa-Louisi, A.; Nikitas, P.; Balkatzopoulou, P.; Malliakas, C. J. Chromatogr., A 2004, 1033, 29-41.
Table 3. Comparison of Experimental and Predicted Retention Times (in min) of Amino Acids Derivatives Obtained under Dual-Mode MeCN Gradients Depicted in Table 1a gradients I
II
III
IV
V
VI
solutes
tR (exp)
tR (th)
% error
tR (exp)
tR (th)
% error
tR (exp)
tR (th)
% error
tR (exp)
tR (th)
% error
tR (exp)
tR (th)
% error
tR (exp)
tR (th)
% error
Arg Asn Gln Ser Tau Asp Glu Thr Gly Dopa Ala GABA Met Trp Phe Val Ile Leu
5.65 6.43 6.74 7.01 7.48 7.89 8.32 8.51 9.12 9.07 10.50 10.70 12.98 14.00 13.98 14.67 16.59 16.79
5.46 6.05 6.41 6.63 7.28 7.70 8.11 8.39 8.99 8.90 10.32 10.49 12.77 13.77 13.77 14.48 16.39 16.47
3.3 5.8 4.9 5.5 2.7 2.4 2.6 1.4 1.4 1.9 1.7 1.9 1.7 1.6 1.5 1.3 1.2 1.9
5.60 6.56 6.90 7.31 7.94 8.65 9.41 9.64 10.48 10.78 13.18 13.77 19.05 20.95 21.64 22.63 25.10 25.45
5.53 6.30 6.75 7.19 7.76 8.44 9.26 9.42 10.33 10.49 12.88 13.49 18.58 20.48 21.19 22.08 24.58 24.88
1.3 3.9 2.1 1.7 2.3 2.4 1.6 2.4 1.4 2.7 2.3 2.0 2.4 2.3 2.1 2.4 2.1 2.3
6.16 7.19 7.54 8.05 8.78 9.87 11.21 11.44 12.72 13.99 16.09 16.57 19.34 20.43 20.43 21.24 23.13 23.37
6.23 7.21 7.60 8.07 8.84 9.90 11.18 11.37 12.68 13.68 15.68 16.19 18.98 19.97 20.19 20.78 22.69 22.77
-1.2 -0.2 -0.8 -0.1 -0.7 -0.3 0.3 0.6 0.3 2.3 2.6 2.3 1.9 2.2 1.2 2.2 1.9 2.6
5.59 6.56 6.91 7.37 8.03 8.88 9.78 10.03 10.97 11.49 14.39 15.23 22.67 25.60 27.37 29.14 34.40 35.45
5.50 6.39 6.82 7.23 7.92 8.81 9.70 9.86 10.89 11.38 14.18 14.98 22.28 25.18 26.87 28.58 33.98 34.98
1.7 2.7 1.3 1.9 1.4 0.8 0.8 1.7 0.7 1.0 1.5 1.6 1.7 1.7 1.8 1.9 1.2 1.3
4.05 4.98 5.34 5.80 6.50 7.44 8.50 8.72 9.78 10.42 13.52 14.40 22.36 25.24 27.21 28.89 34.24 35.23
4.05 4.87 5.23 5.70 6.34 7.28 8.21 8.51 9.54 10.22 13.07 13.98 21.68 24.68 26.48 28.08 33.48 34.48
0.0 2.4 2.0 1.7 2.5 2.1 3.5 2.5 2.4 2.0 3.3 2.9 3.0 2.2 2.7 2.8 2.2 2.1
4.41 5.55 5.98 6.56 7.46 8.66 10.13 10.29 11.32 11.84 14.80 15.56 23.07 25.94 27.82 29.53 34.92 35.88
4.43 5.41 5.76 6.38 7.30 8.49 9.78 10.14 11.06 11.69 14.37 15.17 22.37 25.38 27.08 28.78 34.08 35.08
-0.6 2.6 3.6 2.7 2.2 2.0 3.4 1.4 2.3 1.3 2.9 2.5 3.0 2.1 2.7 2.5 2.4 2.2
absolute average a
2.5
2.2
1.3
1.5
2.4
2.4
The predicted retention times, tR(th), were calculated from eqs 5, 6, 13, and 14.
Table 4. Comparison of Experimental and Predicted Retention Times (in min) of Amino Acid Derivatives Obtained Under Dual-Mode MeOH Gradients Depicted in Table 2a gradient 1 solutes Arg Asn Gln Ser Tau Asp Glu Thr Gly Dopa Ala GABA Met Trp Phe Val Ile Leu absolute average a
2
3
tR (exp)
tR (th)
% error
tR (exp)
tR (th)
% error
tR (exp)
tR (th)
% error
9.50 10.05 11.16 11.93 12.35 12.81 13.68 13.75 13.87 13.97 15.82 16.15 18.43 18.56 19.98 20.57 24.74 24.74
9.86 10.28 11.41 12.13 12.48 12.87 13.67 13.74 13.91 14.07 15.79 16.12 18.20 18.27 19.56 20.17 24.08 24.17
-3.8 -2.3 -2.3 -1.6 -1.1 -0.4 0.1 0.1 -0.2 -0.7 0.2 0.2 1.3 1.6 2.1 2.0 2.7 2.3 1.4
8.60 8.96 9.75 10.40 10.79 11.19 12.03
8.82 9.11 9.93 10.57 10.89 11.24 12.09 12.06 12.13 12.41 14.29 14.55 16.89 17.08 18.38 18.87 22.77 22.98
-2.5 -1.6 -1.8 -1.7 -0.8 -0.4 -0.5
9.28 9.75 10.74 11.50 11.98 12.43 13.33
-3.7 -2.4 -2.4 -1.7 -1.7 -1.0 -0.4
0.3 -0.6 0.0 0.8 0.4 0.4 0.4 0.9 1.1 0.2 0.9
13.49 13.63 15.57 15.89 18.07 18.07 19.55 20.14 24.13 24.13
9.62 9.99 11.00 11.70 12.19 12.56 13.38 13.46 13.53 13.79 15.61 15.83 17.97 18.17 19.48 20.09 23.87 24.08
12.17 12.34 14.29 14.66 16.96 17.15 18.46 19.03 23.03 23.03
-0.2 -1.2 -0.2 0.4 0.6 -0.6 0.4 0.3 1.1 0.2 1.1
The predicted retention times, tR(th), were calculated from eqs 5, 6, 13, and 14.
6, is readily calculated from the known F versus t profile formed in the mixer. The values of n1, n2, ... are further used for the calculation of l1/L, l2/L, ... from eq 5 until eq 13 is fulfilled. Then the retention time is calculated from eq 14. It is evident that this approach is an extension of that proposed in ref 15 as well as of the time-segmented approaches for predicting gradients reported in refs 18-22. (18) Zunin, P.; Evangelisti, F. Int. Dairy J. 1999, 9, 653-656.
The above approach can be also used for the calculation of tR by means of eq 10, since it establishes the correspondence between np and kp and, therefore, the dependence of kp upon tcp (19) Smith, R. D.; Chapman, E. G.; Wright, B. W. Anal. Chem. 1985, 57, 28292836. (20) Snijders, H.; Janssen, H. G.; Cramers, C. J. Chromatogr., A 1995, 718, 339355. (21) Chester, T. L. J. Chromatogr., A 2003, 1016, 181-193. (22) Chester, T. L.; Teremi, S. O. J. Chromatogr., A 2005, 1096, 16-27.
Analytical Chemistry, Vol. 79, No. 10, May 15, 2007
3891
) npδt. Knowing this dependence, eq 10 is solved numerically using Simpson’s rule. An Approximation. Although the above equations can be easily applied for the calculation of the solute elution time, the following approximation may simplify the calculations considerably. If we assume that Fi in the sum of the left-hand side of eq 6 takes a mean value Fp, then np is calculated directly from
np ) (A + Fpnp-1kp)/(Fpkp)
(16)
where pm
A ) (1 + kp)
∑
Fi
(17)
i)(p-1)m+1
For the application of eqs 16 and 17, we calculated Fp at the mean point
) tcp-1 + (tcp-1 - tcp-2)/2
EXPERIMENTAL SECTION In order to test the theory, we used mainly linear φ versus t and F versus t gradients. The dual-mode gradients tested in the present study are given in Tables 1 and 2. In these tables, each gradient is determined by the initial φ or F value indicated by subscript 1, the corresponding final value denoted by subscript 2, the time at which φ or F starts to increase, tφ1 or tF1, and, finally, the time at which this increase is completed, tφ2 or tF2. The chromatographic and the eluent system as well as the solutes used in the present study are the same as that in ref 13. The retention data of the solutes, which are amino acids derivatives with o-phthalaldehyde, under the dual-mode gradients applied are given in Tables 3 and 4, for acetonitrile (MeCN) and methanol (MeOH), respectively. All retention data have been properly corrected for the extracolumn volume. The algorithms used for prediction, according to the theory presented above, were homemade written in C++. The exe files of the basic programs used for the calculation of tR by means of either eq 10 or eqs 5, 6, 13, and 14 are free available upon request from the authors. Note that if we use m ) δt*/δt ) 1000 and δt* ) 0.1 min these two approaches give practically identical results for tR. For this reason, the above values of m and δt* were used in all calculations. All other calculations were performed in Excel spreadsheets on a 3-GHz Pentium CPU running under windows XP. Finally, the adjustable parameters a, b, and c of eq 15 were determined by fitting this equation to the isocratic retention data for MeCN and MeOH eluting systems appeared in Tables 2 and 3 in ref 13. RESULTS AND DISCUSSION The use of linear or multilinear F versus t gradients has the advantage that the integral of eq 1 has an analytical solution allowing for the direct determination of t*p from tp and vice versa. In addition, all sums in eqs 5-7 are transformed to integrals, which are again easily calculated and allow for the subsequent determination of n1, n2, ... and l1/L, l2/L,... until eq 13 is fulfilled. 3892 Analytical Chemistry, Vol. 79, No. 10, May 15, 2007
Table 5. Percentage Error between Experimental and Predicted Retention Times from Eqs 13, 14, and 16 (in min) of Amino Acid Derivatives Obtained under Dual-Mode MeCN (I, II, III, IV, V, VI) and MeOH (1, 2, 3) Gradients Depicted in Tables 1 and 2 solutes
I
II
III
IV
V
VI
1
2
3
Arg Asn Gln Ser Tau Asp Glu Thr Gly Dopa Ala GABA Met Trp Phe Val Ile Leu absolute average
3.3 5.8 4.9 5.5 2.7 2.4 2.6 1.4 1.4 1.9 1.8 1.9 1.7 0.8 0.6 1.3 1.3 1.2 2.4
1.3 3.9 2.2 1.7 2.3 2.4 1.6 2.4 1.4 1.5 2.3 2.1 2.5 2.3 2.1 2.5 1.7 1.9 2.1
-1.2 -0.2 -0.7 -0.1 -0.6 -0.3 0.3 0.6 0.3 2.3 2.6 2.4 1.9 2.3 1.2 1.6 2.0 2.1 1.3
1.6 2.6 1.3 1.9 1.4 0.8 0.8 1.7 0.7 1.0 1.5 1.7 1.7 1.2 1.4 1.6 0.9 1.0 1.4
-0.1 2.4 2.1 1.7 2.5 2.1 3.5 2.5 2.4 2.0 3.3 3.0 3.0 2.2 2.7 2.4 1.9 1.8 2.3
-0.6 2.6 0.9 2.7 2.2 2.0 3.4 1.4 2.3 2.3 2.8 2.5 3.0 2.2 2.3 1.9 2.1 1.7 2.2
-1.3 0.6 1.4 2.0 1.3 2.0 2.2 2.1 2.5 1.2 1.3 2.0 2.5 2.1 2.5 3.0 3.5 2.7 2.0
-2.5 -1.6 -1.8 -1.7 -0.8 -0.4 -0.5
-3.7 -2.4 -2.4 -1.7 -1.7 -0.9 -0.3
0.3 -0.6 -0.1 -0.2 0.4 0.5 0.5 0.9 1.1 0.2 0.8
-0.2 -1.2 -0.2 0.4 0.6 -0.5 0.4 0.9 1.1 0.2 1.1
Tables 3 and 4 summarize the comparison between calculations and experimental results obtained in mobile phases containing MeCN and MeOH, respectively, as organic modifiers. The agreement is excellent in all-different types of dual-mode gradient programs tested. The average percentage error between experimental and predicted retention times ranges from 1.3 to 2.5% in MeCN and from 0.9 to 1.4% in MeOH depending on the gradient profile. Although the differences between experimental and calculated retention times are very small, there is a kind of bias and the calculated retention times are systematically lower than the experimental ones for acetonitrile mobile phases. According to a recent study,23 this bias depends on the organic modifier usedsit is small for mobile phases modified by acetonitrile and practically negligible in methanol solutionss and it is attributed to two conflicting processes, a solvent “demixing” process and a slow change in stationary-phase conformation as a result of change in mobile-phase composition during gradient elution. It should be pointed out that a similar good prediction could be achieved if we make the following simplification in the prediction algorithm. Instead of applying eq 1 at each step p ) 1, 2, 3, ... for the determination of n1, n2, ..., we may correct only the times tφ1, tφ2 by means of eq 1 assuming a linear dependence of the φ versus t* in the interval [tφ1, tφ2]. Note that an arbitrary linear φ versus t gradient profile formed in the mixer is transformed to a nonlinear φ versus t* gradient; see in Figure 2 such a transformation of a programmed gradient profile. In fact, the portion of the φ versus t* curve exhibits some curvature in the interval between tφ1 and tφ2. We have found that this approximation simplifies considerably the calculations and has a small or negligible effect on the calculated retention times at least in our experimental systems, where the range of different flow rates used (0.5-1.5 mL min-1) is rather small as well as the value (23) Pappa-Louisi, A.; Nikitas, P.; Agrafiotou, P. J. Chromatogr., A 2006, 1127, 97-107.
of the dwell time (1.19 min). For example, the average percentage error for dual-mode gradients 1, 2, and 3 with the above simplification was found to be 1.5, 0.9, and 1% instead of 1.4, 0.9, and 1.1% previously found; see Table 4. Another very useful approximation that also simplifies the calculations considerably is the determination of the sequence n1, n2, ... by means of eqs 16 and 17. Table 5 shows the percentage error between experimental and predicted retention times calculated from this approximation. It is seen that the performance of eqs 16 and 17 is excellent. In fact, eqs 16 and 17 give results indistinguishable from that of the accurate eq 6, although in same cases, they seem to give slightly better predictions than eq 6. This might be due to column equilibration effects, which are likely to have a small impact on the experimental retention times.23 However, the use of the approximate eqs 16 and 17 should be handled carefully, because there is no guarantee that these equations perform equally well for nonlinear φ gradients or when there is a greater variation in F values. To sum up, this paper is the first report of development of a mathematical model describing the peak position within a
chromatogram when both the flow rate and mobile-phase composition change simultaneously with time. No restriction is imposed on the φ versus t and F versus t profiles. The fundamental equation of dual-mode gradient elution in liquid chromatography involving simultaneous changes in flow rate and mobilephase composition is determined if we take into account that the initial φ versus t profile programmed and formed in the mixer is transformed to a new one, φ versus t*, at the beginning of the column, and inside the column, the analyte is subject to the φ versus tc profile. The retention prediction obtained for our experimental system of 18 o-phthalaldehyde derivatives of amino acids in eluting systems modified by acetonitrile or methanol under all dual-mode gradient conditions was excellent.
Received for review January 15, 2007. Accepted March 21, 2007. AC070090R
Analytical Chemistry, Vol. 79, No. 10, May 15, 2007
3893
