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Laboratory of Physical Chemistry, Departm
Thessaloniki, 54124 Thessaloniki, Greece.
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† Electronic supplementary informa10.1039/c3an36425a
Cite this: Analyst, 2013, 138, 3771
Received 3rd October 2012Accepted 29th April 2013
DOI: 10.1039/c3an36425a
www.rsc.org/analyst
This journal is ª The Royal Society of
Retention modeling in combined pH/organic solventgradient reversed-phase HPLC†
Ch. Zisi, S. Fasoula, P. Nikitas and A. Pappa-Louisi*
An approach for retention modeling of double pH/organic solvent gradient data easily generated by
automatically mixing two mobile phases with different pH and organic content according to a linear
pump program is proposed. This approach is based on retention models arising from the evaluation of the
retention data of a set of 17 OPA derivatives of amino acids obtained in 27 combined pH/organic solvent
gradient runs performed between fixed initial pH/organic modifier values but different final ones and for
different gradient duration. The derived general model is a ninth parameter equation easily manageable
through a linear least-squares fitting but it requires eighteen initial pH/organic modifier gradient
experiments for a satisfactory retention prediction in various double gradients of the same kind with those
used in the fitting procedure. Two simplified versions of the general model, which were parameterized
based on six only initial pH/organic modifier gradients, were also proposed, when one of the final double
gradient conditions, pH or organic content was kept constant. The full and the simplified models allowed
us to predict the experimental retention data in simultaneous pH/organic solvent double gradient mode
very satisfactorily without the solution of the fundamental equation of gradient elution.
1 Introduction
Retention of ionogenic analytes in reversed-phase liquid chro-matography (RP-LC) is known to strongly depend on the pH ofthe eluent. A complete retention modeling in isocratic mode asa function of pH was developed and comprehensivelyreviewed.1,2 However, the performance of a pH-gradient or acombined pH/organic modier gradient during RP-LC separa-tions extends much more the selectivity and the resolution ofionizable compounds. Recent reports by Kaliszan's groupdemonstrated a comprehensive theory allowing the predictionof solute retention in the pH-gradient mode as well as the pKa
determination of monoprotic acids and bases.3–7 The theoreticalbasis of simultaneous pH/organic solvent double lineargradient mode was also presented8–10 and reviewed,11 whereasthe usefulness of this double gradient method was tested in theanalysis of biological samples.12 The current work on the linearpH-gradient mode was critically reviewed by our group recentlyin combination with the development of new expressions forthe retention time of ionizable analytes.13
All the above attempts to describe the retention of solutes inthe pH-gradient or simultaneous pH/organic solvent gradientmode theoretically are based on the solution of the fundamentalequation of gradient elution. Themain drawback, however, of the
ent of Chemistry, Aristotle University of
E-mail: [email protected]; Fax: +30
tion (ESI) available. See DOI:
Chemistry 2013
solute retention description based on the solution of the funda-mental equation of gradient elution is that all the expressionsderived for the solute elution time under pH or pH/organicsolvent gradient conditions are complicated and thismakes theirapplication rather difficult, since proper soware for nonlinearleast-squares tting is necessary to achieve convergence to theglobal optimum. For this reason, in a report published recentlyby our laboratory, a simple approach for retention modeling ofsolutes under single pH-gradient conditions at various constantorganic contents in the mobile phase was proposed, which iseasily manageable through a linear least-squares tting andallows a very satisfactory retention prediction for pH-changes ofthe same kind with those used in the tting procedure.14 Thisapproach was based on an empirical model arising from theexperimental properties of pH-gradient retention data obtainedbetween a given initial and nal pH value with different gradienttimes and at different but constant values of organic modiercontent in the eluent. The derived model expresses the soluteretention times in terms of the variables representing the abovegradient elution conditions, which are the programmed gradientduration and the organic modier content in the mobile phase.
The aim of the present study is to extend our previous workin ref. 14 to simultaneous pH/organic solvent double gradients,since the simultaneous changes in mobile phase pH andorganic modier content during a chromatographic experimentis the most general and challenged gradient mode for theseparation of ionogenic analytes in complex mixtures. In moredetail, it is attempted to describe retention in double gradientsby expressing the solute retention in terms of variables
Analyst, 2013, 138, 3771–3777 | 3771
Analyst Paper
representing double gradient elution conditions. This theoret-ical treatment is based exclusively on combined pH/organicsolvent gradients data and the accuracy of the proposedapproach is tested by the experimental data obtained for 17 o-phthalaldehyde (OPA) derivatives of amino acids in various pH/organic solvent gradient experiments.
2 An empirical model arising from theexperimental properties of combined pH/4-gradient retention data
In the present work, we restricted our study to linear pro-grammed pH/organic modier gradients, where the pH andorganic modier content, 4, both change between initial, pH0/40, and nal, pHf/4f, values with the same starting time, ts, andwith the same gradient duration, tG.
In an effort to develop a model describing the retention ofsolutes in the above described double gradient mode we followedthe approach adopted in ref. 14. In this reference, a simple modeldescribing the retention of solutes in single pH-gradient runscarried out between a given initial, pH0, and nal, pHf, pH valuewith different gradient duration, tG, and with different butconstant organicmodier content, 4, in the eluent was proposed.This model was based on the experimental dependence of thesolute retention time, tR, upon each of the two variables gov-erning the retention, which were tG and 4 in that case. As far asthe equations that express separately the dependence of tR uponeach of these factors are known, the combined effect of thesefactors on the solute retention, tR (tG,4), may be written as15
tR(tG,4) ¼ tR(tG)tR(4) (1)
Taking combined pH/4 linear gradient runs into consider-ation, the solute retention should depend on the followinggradient elution conditions: tG, pH0, pHf, 40, 4f, and ts. Bykeeping 40, pH0 and ts constant, the retention is governed bythree variables, i.e. tG, 4f and pHf, and consequently thecombined effect of these factors on the solute retention,tR(tG,4f,pHf), may be written as
tR(tG,4f,pHf) ¼ tR(tG)tR(4f)tR(pHf) (2)
Supposing a linear dependence of tR upon tG (with constant4f and pHf) and a quadratic variation of tR against 4f (withconstant tG and pHf) or against pHf (with constant tG and 4f),eqn (2) yields
tR(tG,4f,pHf) ¼ c0 + c1tG + c24f + c3pHf + c44f2 + c5pHf
2
+ c6tG4f + c7tGpHf + c84fpHf (3)
provided that we keep terms of order up to 2. In eqn (3), coef-cients c0.c8 are adjustable parameters determined by tting aseries of 3D pH/4-gradient retention data obtained in pro-grammed linear changes of both mobile phase pH and 4
between xed pH0 and 40 values but different pHf and 4f values,respectively, and for different programmed gradient time tG.
The evaluation of experimental data obtained in this studywill verify the validity of the above assumptions.
3772 | Analyst, 2013, 138, 3771–3777
3 Experimental3.1 Instrumentation and solutes
The liquid chromatography system consisted of a Shimadzu LC-20AD pump, a Shimadzu DGU-20A3 degasser, a model 7125syringe loading sample injector tted with a 20 mL loop, a 250 �4.6 mm MZ-Analytical column (PerfectSil Target ODS-3 HD 5 mm)thermostatted at 25 �C by a CTO-10AS Shimadzu column oven anda Shimadzu RF-10AXL spectrouorometric detector (Shimadzu,Model) working at 455 nm aer excitation at 340 nm. The soluteswere the following 17OPA/2-mercaptoethanol derivatives of aminoacids: L-arginine (Arg), L-asparagine (Asn), L-glutamine (Gln),L-serine (Ser), L-aspartic acid (Asp), L-glutamic acid (Glu), L-threo-nine (Thr), beta-(3,4-dihydroxyphenyl)-L-alanine (Dopa), L-alanine(Ala), L-tyrosine (Tyr), 4-aminobutyric acid (GABA), L-methionine(Met), L-valine (Val), L-tryptophan (Trp), L-phenylanine (Phe),L-isoleucine (Ile) and L-leucine (Leu). The working concentration ofunderivatized amino acids used in the derivatization procedure byOPA/2-mercaptoethanol reagent was 5 mg mL�1.
Note that the underivatized common amino acids arediprotic ampholytes of the typical structure H2NCH(R)COOHwith widely different aqueous pKa values of their carboxylgroups (z2.3) and of their ammonium groups (z9.5). Conse-quently, throughout the entire eluent pH range used in thepresent study (3.21–7.86), the solute retention should be onlyaffected by the carboxyl group pKa. This seems to be true if it istaken into account that the formation of the amino acid OPAderivatives and the presence of acetonitrile in eluents, in aconcentration range of 4¼ 0.25 to 0.5 in this study, increase theaqueous pKa values of the carboxyl groups of free amino acids toa value close to z 4.2.14,16
The pH of the mobile phases used in different pH-gradientswas measured aer mixing the aqueous buffers and the organicmodier, whereas the electrode system was calibrated with theusual aqueous standards.17 The measurements were done witha Mettler Toledo Seven Easy pH-meter.
3.2 Chromatographic experiments
In order to investigate the effects brought on retention of testsolutes by pH/organic modier gradients as well as the validityof the proposed theoretical treatment, we performed a series of27 double organic solvent and pH gradient runs. All the chro-matographic measurements were done at 25 �C with an eluentow rate of 1.0 mL min�1 and were generated by automaticallymixing two mobile phases according to a linear pump programwith the same starting time (0 min) but with different gradientdurations, tG ¼ 10, 20 or 30 min. The mobile phases used forgeneration gradient runs consisted of aqueous phosphatebuffers with a total ionic strength of 0.02 M with xed pH0 and40 values but different pHf and 4f values. In more detail, threeseries of pH/4 gradient experiments were conducted in thisstudy. In the rst one, gradient experiment no. 1–9 in Table 1,the pH was changed from pH0 ¼ 3.21 to pHf ¼ 7.86 simulta-neously with acetonitrile (MeCN) as an organic modier with 4
between 40 ¼ 0.25 and 4f ¼ 0.35, 0.40 or 0.50. The above pH/4gradients were formed by mixing according to a linear HPLC
This journal is ª The Royal Society of Chemistry 2013
Table
1Experim
entalreten
tiondata(in
min)oftest
solutesobtained
indifferen
tpH/4
double
gradientruns
Exp
erim
ent
no.
t G40
4f
pH0
pHf
Arg
Asn
Gln
Ser
Asp
Glu
Thr
Dop
aAla
Tyr
Gab
aMet
Val
Trp
Phe
Ile
Leu
110
0.25
0.35
3.21
7.86
6.65
7.60
8.30
8.60
9.10
9.37
9.63
10.14
10.66
11.16
11.68
13.54
13.54
15.04
15.56
15.56
15.98
220
0.25
0.35
3.21
7.86
6.97
8.35
9.44
10.04
11.68
13.40
13.40
14.82
15.59
16.17
17.32
19.27
19.54
21.69
22.34
22.34
22.95
330
0.25
0.35
3.21
7.86
7.12
8.66
9.93
10.61
12.88
15.37
15.37
18.65
20.53
21.17
22.76
24.66
25.13
27.70
28.61
28.61
29.40
410
0.25
0.40
3.21
7.86
6.51
7.35
7.95
8.28
8.73
9.24
9.24
9.64
10.05
10.36
10.93
12.03
12.03
12.97
13.26
13.26
13.45
520
0.25
0.40
3.21
7.86
6.85
8.12
9.04
9.64
11.01
12.57
12.57
14.10
14.95
15.29
16.40
17.35
17.64
19.00
19.49
19.49
19.87
630
0.25
0.40
3.21
7.86
6.88
8.33
9.42
10.07
11.93
14.08
14.08
16.97
19.39
19.96
21.44
22.37
22.89
24.39
24.84
25.18
25.62
710
0.25
0.50
3.21
7.86
6.28
7.05
7.53
7.88
8.29
8.90
8.90
9.30
9.64
9.70
10.31
10.66
10.79
11.26
11.59
11.59
11.59
820
0.25
0.50
3.21
7.86
6.63
7.77
8.52
9.07
10.14
11.44
11.44
12.89
14.37
14.53
15.57
15.77
16.09
16.70
16.84
17.12
17.24
930
0.25
0.50
3.21
7.86
6.85
8.14
9.05
9.68
11.19
12.93
12.93
15.20
17.91
18.50
20.23
20.70
21.15
21.95
21.95
22.38
22.54
1010
0.25
0.35
3.21
5.86
6.91
8.11
8.96
9.49
10.64
11.96
11.96
13.20
14.03
14.28
16.57
16.90
17.20
19.10
19.44
20.28
20.99
1120
0.25
0.35
3.21
5.86
6.89
8.43
9.60
10.31
12.29
14.52
14.52
17.80
21.16
22.17
24.28
25.02
25.88
27.72
27.94
28.99
29.71
1230
0.25
0.35
3.21
5.86
7.56
9.26
10.62
11.36
13.99
16.75
16.75
21.24
25.95
28.47
30.48
34.08
35.14
37.08
37.08
38.36
39.09
1310
0.25
0.40
3.21
5.86
6.73
7.80
8.48
8.99
9.88
11.03
11.03
12.22
13.43
13.52
14.88
15.23
15.36
16.27
16.27
16.96
17.25
1420
0.25
0.40
3.21
5.86
6.82
8.16
9.14
9.80
11.46
13.28
13.28
15.89
19.02
19.95
21.66
23.18
23.83
24.63
24.63
25.41
25.72
1530
0.25
0.40
3.21
5.86
6.98
8.45
9.59
10.29
12.40
14.66
14.66
18.18
22.39
24.25
25.33
31.28
32.74
33.57
33.57
34.62
34.93
1610
0.25
0.50
3.21
5.86
6.26
7.11
7.60
8.06
8.67
9.42
9.59
10.44
11.93
11.93
12.93
13.45
13.84
14.00
14.00
14.57
14.57
1720
0.25
0.50
3.21
5.86
6.74
7.90
8.66
9.27
10.49
11.83
11.83
13.59
16.15
16.58
17.47
20.59
21.75
22.18
22.18
23.26
23.26
1830
0.25
0.50
3.21
5.86
6.86
8.18
9.09
9.78
11.41
13.19
13.19
15.63
18.98
19.97
20.64
26.14
28.26
29.06
29.39
31.43
31.74
1910
0.25
0.35
3.21
4.68
6.80
7.99
8.82
9.39
10.62
12.02
12.02
13.59
15.39
16.00
20.93
21.80
25.57
29.26
30.36
36.47
39.28
2020
0.25
0.35
3.21
4.68
6.96
8.42
9.60
10.28
13.37
14.64
14.64
18.08
21.75
23.10
25.67
29.25
33.02
36.93
37.96
44.14
46.94
2130
0.25
0.35
3.21
4.68
7.32
8.94
10.23
11.02
13.76
16.44
16.44
21.07
25.95
28.81
30.25
37.03
40.75
44.84
45.59
51.92
54.65
2210
0.25
0.40
3.21
4.68
6.60
7.67
8.37
8.90
9.87
11.04
11.04
12.38
13.96
14.18
16.62
17.49
19.62
20.99
21.38
24.97
26.24
2320
0.25
0.40
3.21
4.68
6.93
8.26
9.27
9.89
11.60
13.43
13.43
16.06
19.32
20.32
21.70
25.02
27.14
28.72
28.72
32.57
33.72
2430
0.25
0.40
3.21
4.68
7.07
8.54
9.70
10.41
12.59
14.86
14.86
18.42
22.64
24.57
25.43
32.46
34.79
36.56
36.56
40.39
41.53
2510
0.25
0.50
3.21
4.68
6.32
7.18
7.68
8.15
8.82
9.74
9.74
10.64
12.21
12.21
13.13
14.16
15.07
15.32
15.32
16.91
17.24
2620
0.25
0.50
3.21
4.68
6.97
7.95
8.71
9.32
10.57
11.95
11.95
13.72
16.34
16.88
17.58
21.16
22.59
23.20
23.20
24.86
25.18
2730
0.25
0.50
3.21
4.68
7.73
8.38
9.38
10.02
11.74
13.49
13.49
16.02
19.34
20.49
20.97
26.78
28.96
29.87
30.29
32.71
33.11
This journal is ª The Royal Society of Chemistry 2013 Analyst, 2013, 138, 3771–3777 | 3773
Paper Analyst
Fig. 2 Variation of tR as a function of the duration, tG, of pH/4-gradients for Ile(:) and Val (-) obtained in gradient experiment no. 13, 14 and 15 of Table 1.Points are experimental data; solid lines are obtained by linear fitting of experi-mental data, whereas dotted lines are derived from eqn (3) with the corre-sponding adjustable parameters of Table S1 in the ESI.†
Fig. 1 The actual pH changes correspond to gradient experiment no. 4 (—), 13(/) and 22 (----) of Table 1.
Analyst Paper
pump program two mobile phases with different pH values(3.21 or 7.86), which contained a xed concentration of MeCN.At the start of the gradient the content of MeCN in the eluentwith pH0 ¼ 3.21 was 40 ¼ 0.25, whereas 4f ¼ 0.35, 0.4 or 0.5 wasused as an organic modier in the eluent with pHf ¼ 7.86 at theend of each gradient. In the other two series of pH/4 gradients,gradient experiment no. 10–18 and 19–27 in Table 1, the onlydifference was that the nal pH values of double gradients were5.86 or 4.68, respectively, instead of 7.86, which was the nal pHvalue in the rst series of gradients. The experimental retentiondata obtained under all the above described double pH and 4
gradient runs are shown in Table 1. These 27 chromatographicruns were selected according to a 3� 3� 3 experimental designrelated to the three variables, i.e. tG, 4f and pHf, governing theretention under study. Note that in the above gradient runs theactual 4-proles are linear but the actual pH-changes are curvedwith a shape that depends on the value of pHf and not on thevalue of 4f at least in the range of 4f tested. See for example inFig. 1 the real pH changes that correspond to gradient experi-ment no. 4, 13 and 22 in Table 1, which in fact are almostidentical to all the corresponding gradient runs with the sametG and pHf values but with different 4f values.
3.3 Fitting and prediction algorithms
The algorithms used for tting and testing the prediction abilityof the models derived in this study were written in C++ andbased on the theory of linear least-squares. However, thedetermination of the tting parameters of the proposed simpleretention models additionally with the prediction derived bythese retention equations could also be easily done on Excelspreadsheets using the Regression tool.
Fig. 3 Variation of tR as a function of 4f for Ile (:) and Val (-) obtained ingradient experiment no. 11, 14 and 17 of Table 1. Points are experimental data;solid lines are obtained by a second order polynomial fitting of experimental data,whereas dotted lines are derived from eqn (3) with the corresponding adjustableparameters of Table S1 in the ESI.†
4 Results and discussion
In order to verify the validity of the assumptions adopted inSection 2 for the derivation of eqn (3), we evaluate the experi-mental data obtained in this study. In more detail, based, forexample, on the retention data obtained in gradient experimentno. 13, 14 and 15 in Table 1, we found that a linear dependencetR upon tG describes quite satisfactorily the experimental data
3774 | Analyst, 2013, 138, 3771–3777
obtained by pH/4-gradient runs performed with differentgradient durations (10, 20 and 30min, respectively) but betweenxed initial and nal pH/4 values, which in this case is frompH0/40 ¼ 3.21/0.25 to pHf/4f ¼ 5.86/0.4. Indicative examples forthe above linear variation of tR against tG are depicted in Fig. 2for two solutes, Val and Ile. Moreover, the evaluation of exper-imental data in Table 1, such as those recorded in gradientexperiment no. 11, 14 and 17, reveals a quadratic expression forthe dependence of tR upon 4f in pH/4-gradient runs from pH0/40 ¼ 3.21/0.25 to pHf/4f ¼ 5.86/(0.35, 0.4 or 0.5) with a xedgradient duration, tG ¼ 20 min. Examples for the quadraticvariation of tR against 4f are depicted in Fig. 3 for the samesolutes. A similar quadratic dependence seems to be valid forthe dependence of tR upon pHf in pH/4-gradient runs 4, 13 and22 of Table 1, which were recorded from pH0/40 ¼ 3.21/0.25 topHf/4f ¼ (4.68, 5.86 or 7.86)/0.4 with a xed gradient duration,tG ¼ 10 min. Such a quadratic variation of tR against pHf isdepicted in Fig. 4 for the same representative analytes used.
This journal is ª The Royal Society of Chemistry 2013
Fig. 4 Variation of tR as a function of pHf for Ile (:) and Val (-) obtained ingradient experiment no. 4, 13 and 22 of Table 1. Points are experimental data;solid lines are obtained by a second order polynomial fitting of experimental data,whereas dotted lines are derived from eqn (3) with the corresponding adjustableparameters of Table S1 in the ESI.†
Table 2 An overview of the retention description of solutes by different equa-tions. Average and maximum absolute percentage error between experimentaland calculated tR values of test solutes in different gradients runs of Table 1
Equation Experiment no. Mean Max
Eqn (3) 1–27 2.9 18.9Eqn (4) 1–9 1.3 5.1Eqn (4) 10–18 1.4 4.8Eqn (4) 19–27 1.0 5.9Eqn (5) 1–3, 10–12, 19–21 1.9 9.2Eqn (5) 4–6, 13–15, 22–24 1.9 7.4Eqn (5) 7–9, 16–18, 25–27 1.7 7.0
Fig. 5 Mean absolute percentage errors between experimental and calculatedretention data of each solute obtained from eqn (3) (C), eqn (4) (B, O, ,) andeqn (5) (+, �, *), respectively, in each group of gradients runs in Table 1 shown inthe figure.
Paper Analyst
Having experimentally veried that the assumptions adoptedfor the derivation of eqn (3) are valid at least in the above limitedgradient data, we tted all data depicted in Table 1 to eqn (3)(values of tting parameters c0.c8 are listed in Table S1 in theESI†) in order to evaluate the performance of eqn (3) on theretention description of these data and to ensure that eqn (3)fullls the properties of a linear dependence of tR upon tG (withconstant 4f and pHf) and a quadratic variation of tR against 4f
(with constant tG and pHf) or against pHf (with constant tG and4f). Indeed, using the values of tting parameters c0.c8 listed inTable S1,† the dotted lines in Fig. 2–4 were drawn by means ofeqn (3). It is seen that these lines are almost identical to the solidlines in the same gures obtained by a linear (in Fig. 2) or asecond order polynomial tting (in Fig. 2 and 3) of experimentaldata. Therefore, the assumptions adopted for the derivation ofeqn (3) should be valid. Moreover, the ability of eqn (3) todescribe the retention data in pH/4 gradients with constantinitial conditions, pH0/40, but different nal conditions, pHf/4f
and gradient duration, tG, is rather satisfactory since the overallaverage percentage error between calculated and experimentalretention data was only 2.9% whereas the maximum absolutepercentage error in tR values was 18.9%.
In an attempt to improve the capability of eqn (3) to describethe solute retention in combined pH/4 gradient mode, theretention data depicted in Table 1 were divided into two groups.The rst one consisted of retention data obtained under pH/4gradient conditions where the only experimental gradientvariables were tG and 4f since the change of pH, pH0 / pHf, inthese double gradients, was xed. Gradient experiment no. 1–9,10–18 or 19–27 in Table 1 represent examples of these restric-tions. In this case, eqn (3) may be simplied to
tR(tG,4f) ¼ a0 + a1tG + a24f + a34f2 + a4tG4f (4)
Values of parameters a0.a5 obtained from the tting ofretention data of gradients 1–9, 10–18 or 19–27 are listed inTables S2–S4, respectively, in the ESI.† The ability of eqn (4) todescribe the retention data in such pH/4 gradients ismuch better
This journal is ª The Royal Society of Chemistry 2013
than the general one (eqn (3)). The accurate description of theabove retention data obtained from eqn (4) is shown in Table 2.
A similar expression with eqn (4) is easily derived for theother group of gradients, like those consisting of gradientexperiment no.: (1–3, 10–12, 19–21), (4–6, 13–15, 22–24) or (7–9,16–18, 25–27). Under the above pH/4 gradient conditions, thevariation of 4, 40 / 4f remains constant and consequently thesolute retention is governed only by tG and pHf. Thus, in thesedouble gradients, eqn (3) may be written as
tR(tG,pHf) ¼ b0 + b1tG + b2pHf + b3pHf2 + b4tGpHf (5)
Values of adjustable parameters b0.b5 obtained from thetting of retention data of gradients (1–3, 10–12, 19–21), (4–6,13–15, 22–24) or (7–9, 16–18, 25–27) are listed in Tables S5–S7,respectively, in the ESI.† The ability of eqn (5) to describe theretention data in such pH/4 gradients is much better than thegeneral one, eqn (3), but a little worse than that of eqn (4). Thedescription of the above retention data obtained by eqn (5) isalso shown in Table 2. The worse performance of eqn (5) thaneqn (4) should be due to the fact that eqn (5) treats retentiondata obtained by pH-gradient proles with different shapes likethose depicted in Fig. 1, whereas all the gradient data tted toeqn (4) correspond to the same pattern of pH vs. time gradient.
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Table 3 An overview of the retention prediction of solutes by different equations. Average and maximum absolute percentage error in fitting and prediction of tRvalues of test solutes in different gradients runs in Table 1
Equation Experiment no. used for tting Mean Max Experiment no. used for prediction Mean Max
Eqn (3) All except those predicted 2.7 19.2 2, 5, 8, 11, 14, 17, 20, 23 and 26 3.5 11.8Eqn (4) 1, 3, 4, 6, 7 and 9 0.5 1.7 2, 5 and 8 2.7 7.7Eqn (4) 10, 12, 13, 15, 16 and 18 1.0 3.4 11, 14 and 17 2.6 7.2Eqn (4) 19, 21, 22, 24, 25 and 27 0.5 2.1 20, 23 and 26 2.1 8.9Eqn (5) 1, 3, 10, 12, 19 and 21 1.7 7.8 2, 11 and 20 2.2 8.9Eqn (5) 4, 6, 13, 15, 22 and 24 1.7 8.1 5, 14 and 23 2.6 7.2Eqn (5) 7, 9, 16, 18, 25 and 27 1.3 7.0 8, 17 and 26 2.6 5.0
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A general picture of the retention description of each solutetested by empirical models arising from the properties of theexperimental systems is given in Fig. 5, where it is clear thatthese models describe equally well the retention behavior of allsolutes, particularly in the case where limited pH/4 gradientconditions are used, such as those referred to in eqn (4) and (5),respectively. However, in the case of using the general eqn (3),which should describe all the retention data, in pH/4 gradients,with constant initial conditions but with different nal onesand different gradient durations, there is a rather signicantdeviation between the experimental and theoretical values ofretention of the two last eluted solutes Ile and Leu. This may bedue to the fact that these solutes are most affected by thecomplicated double pH/4 gradients used in this study, where,although 4 changes linearly with time, the variation of pHexhibits a curved form.
In order to test the accuracy of retention predictionsobtained by each of the proposed equations, eqn (3), (4) or (5),for each series of pH/4 gradient experiments, in which thesemodels are valid, the following procedure was adopted: all theexperimental retention data in Table 1, except those used fortting, were tested. In more detail, eqn (3) requires 18 chro-matographic runs for tting, i.e. a 2 � 3 � 3 table of retentiondata for each solute, whereas only 6 preliminary pH/4 gradientexperiments, corresponding to a 2 � 3 table of retention data,are required for tting each of eqn (4) and (5). In Table 3, thegradient data used for tting as well as those used for testingthe prediction ability of each of the proposed model in thisstudy are shown, which in fact are the centre points of 27chromatographic runs performed according to a 3 � 3 � 3experimental design. It is seen that both eqn (4) and (5) enablean equivalent predictive ability for all tested sets of retentiondata, which is quite satisfactory. Consequently, it seems thatstarting from six pH/4-gradient runs conducted with twodifferent gradient durations between the given initial pH0/40
values and three different nal pH/4 values, pHf/(4f1, 4f2 or 4f3)in the case of eqn (4) or (pHf1, pHf2 or pHf3)/4f in the case of eqn(5), both the above models enable an accurate prediction for anyother pH/4-gradients with different tG and 4f or pHf values,respectively, but within the ranges used in the tting procedure.Moreover, eqn (3) requiring 18 preliminary pH/4-gradientscarried out with two different gradient times between xedinitial pH0/40 values but different nal ones of the followingnine combinations (pHf1, pHf2 or pHf3)/(4f1, 4f2 or 4f3) allows
3776 | Analyst, 2013, 138, 3771–3777
also an acceptable retention prediction for various pH/4changes of the same kind with those used in the tting proce-dure (see Table 3).
5 Conclusion
A 9th parameter general equation, eqn (3), was proposed todescribe the retention behavior of solutes in pH/4 double gradi-ents carried out between xed initial pH/4 values but differentnal ones and for different gradient durations. Two simpliedversions of the general model, 5th parameter eqn (4) and (5), werealso proposed when one of the nal double gradient conditions,pHf or 4f, is kept constant. All the proposedmodels, although theyare simple, empirical in nature and easily manageable through alinear least-squares tting, allow us to predict analyte retention incombined pH/4 gradient mode very satisfactorily, particularlywhen the most restricted gradient conditions are used.
However, the most interesting point of the use of the abovemodels is their ability to describe the solute retention in doublepH/4 gradients, where, although 4 changes linearly with time,the variation of pH exhibits a curved form. This is accomplishedbecause of the fact that all the theoretical treatment is basedexclusively on similarly recorded gradient data. The performanceof the derived models is expected to considerably improve indouble gradients where linear changes of both 4 and pH withtime are produced by using appropriate buffers mixed in variousproportions, an experimental procedure, however, certainly morecomplex than that adopted in this study. Consequently, theresults of this papermay encourage chromatographers to use thiscombined gradient technique in order to improve the separationof ionogenic analytes in complex mixtures, since a very simpletheory is now available in the analysis of double pH/organicsolvent gradient data easily generated by automatically mixingtwo mobile phases with different pH and organic contentaccording to a linear pump program.
Acknowledgements
This research has been co-nanced by the European Union(European Social Fund – ESF) and Greek national funds throughthe Operational Program "Education and Lifelong Learning" ofthe National Strategic Reference Framework (NSRF) – ResearchFunding Program: Heracleitus II. Investing in knowledgesociety through the European Social Fund.
This journal is ª The Royal Society of Chemistry 2013
Paper Analyst
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