Anal. Chem. 2010, 82, 8525–8536
Comprehensive Study on the Optimization of Online Two-Dimensional Liquid Chromatographic Systems Considering Losses in Theoretical Peak Capacity in First- and Second-Dimensions: A Pareto-Optimality Approach G. Vivo´-Truyols,*,† Sj. van der Wal,†,‡ and P. J. Schoenmakers† Analytical-Chemistry Group, van’t Hoff Institute for Molecular Sciences, University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV-Amsterdam, The Netherlands, and DSM Resolve, P.O. Box 18, 6160 MD Geleen, The Netherlands A method to optimize different objectives (total analysis time, total peak capacity, and total dilution) has been applied to comprehensive two-dimensional liquid chromatography. The approach is based on Pareto-optimality, and it yields optimal parameters (column particle sizes, column diameters, and modulation times). Losses in the peak capacities in the first dimension (due to undersampling) and in the second dimension (due to high injection volumes) have been taken into account. The first effect (detection band broadening) reduces the original peak capacity by about a half, the second effect can reduce the total peak capacity by an additional half. Thus, the total loss in peak capacity may be 75% of its theoretical value. Analytical expressions to calculate these effects under gradient and isocratic conditions are derived. Of particular interest is the study of the optimal modulation times, which corresponded to between 2 and 3 two-dimensional runs per one-dimensional peak. The effects of using gradient or isocratic elution, conventional (40 MPa) or “ultra-high” (100 MPa) pressures, and focusing between the first and second dimensions were also studied. A trade-off between total peak capacity, total analysis time, and total dilution can be established. Separation scientists are fighting to satisfy mounting demands for analyzing and separating complex mixtures. In order to satisfy these challenging goals, chromatographers are designing systems with more and more separation power. More than 10 years ago Poppe wrote that “Resolving power is what it is all about in analytical separation science”.1 Since then our main goals have not changed. One way to express the resolving power of a chromatographic system is the peak capacity, defined as the maximum number of peaks that can be accommodated in a single analysis.2 It provides chromatographers with an objective measure of their potential for separating complex mixtures. * Corresponding author. Phone: +31 (0)20.525.6531. E-mail: g.vivotruyols@ uva.nl. † University of Amsterdam. ‡ DSM Resolve. (1) Poppe, H. J. Chromatogr., A 1997, 778, 3–21. (2) Giddings, J. C. Anal. Chem. 1967, 39, 1027–1028. 10.1021/ac101420f 2010 American Chemical Society Published on Web 09/23/2010
There are several ways to increase peak capacities. The most straightforward one is to enlarge the number of plates in onedimensional chromatography. This can be achieved by increasing the maximum pressure of the chromatographic system (i.e., ultrahigh pressure liquid chromatography, UHPLC), by decreasing the column resistance factor (e.g., by means of using monolithic columns), or by decreasing the mobile phase viscosity (e.g., using higher temperatures). All these improvements can be invested in longer columns, which is translated into higher resolving power of the system. However, although the improvements are substantial, they are not dramatic.3 Potentially, a much greater increase in peak capacity can be achieved by applying comprehensive two-dimensional chromatography. In this configuration (and assuming complete orthogonality of the two dimensions), the total peak capacity of the system is roughly equal to the product of peak capacities of each separation dimension. Especially for highly complex separations, comprehensive twodimensional liquid chromatography (LC × LC) offers substantial gains in peak capacity and/or analysis time. For example, Shen et al.4 demonstrated impressive one-dimensional LC separations of protein digests with peak capacities exceeding 1000 but analysis times around 2000 min. Thus, they could ideally separate one peak within 2 min. In contrast, Stoll and Carr5 demonstrated a peak capacity of 1024 for an LC × LC separation of metabolite samples in half an hour, which corresponds to one peak every 2 s. Some debate has arisen recently on the total peak capacity that is attainable in LC × LC practice. It has been argued that the total peak capacity is lower than what is ideally expected.5 A first concern is the orthogonality of the separation dimensions. The total peak capacity of the two-dimensional system (i.e., is the product of the peak capacities for each system independently) can be fully used only if both systems are orthogonal (i.e., the separation in one dimension is completely uncorrelated with the separation in the second). Other potential restrictions also have to be taken into account. The separation performance that would be obtained from each (3) Neue, U. D. J. Chromatogr., A 2008, 1184, 107–130. (4) Shen, Y.; Zhang, R.; Moore, R. J.; Kim, J.; Metz, Th. O.; Hixson, K. K.; Zhao, R.; Livesay, E. A.; Udseth, H. R.; Smith, R. D. Anal. Chem. 2005, 77, 3090–3100. (5) Stoll, D. R.; Wang, X.; Carr, P. W. Anal. Chem. 2008, 80, 268–278.
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system independently has to be maintained when a twodimensional setup is built. Otherwise, the observed peak capacity would be less than predicted by the product rule. The situation is different in spatial two-dimensional chromatography (e.g., twodimensional thin layer chromatography (TLC)) compared to time two-dimensional chromatography (e.g., LC × LC). The twodimensional version of column chromatography cannot conveniently be constructed “in space” but is usually performed by manipulating the time dimensions. This implies that the seconddimension separation is not performed simultaneously for the entire sample but sequentially for a (large) number of fractions of the first-dimension effluent. To realize this, the first-dimension separation is typically slow and the second one fast, allowing several fractions to be separated in the second dimension while a single peak is eluted from the first-dimension column. With this configuration, there is, however, a loss in the total peak capacity, because the peak capacity of each system independently is not maintained in the two-dimensional setup. This is mainly because of two reasons. First, the first dimension is undersampled. This becomes clear when the second-dimension chromatograph is considered as a detector for the first-dimension separation. Unless the second-dimension analysis time (i.e., the modulation time) is extremely short, the (undersampled) first-dimension chromatogram experiences an extra band broadening effect due to a lowfrequency detector. Thus, the practically observed first-dimension peak capacity is reduced. This effect was already measured by Blumberg6 and accounted for in two-dimensional separations by Murphy et al.7 and by Horie et al.8 The second reason is that the second-dimension chromatogram suffers from injection band broadening. Large volumes of sample (compared to the column volume) are injected in the second dimension, which lead to an increased band broadening. This extra band broadening has an impact on the observed peak capacity of the second-dimension separation. In this paper, the computation of peak capacities is reported in such a way that all these extra effects are properly accounted for. To this end, the effects of the additional band broadening in the first- and second-dimension are calculated. An analytical expression is derived that reflects the effects of extra band broadening in isocratic and gradient-elution LC on the total peak capacity. Once the theoretical equations that account for the loss of peak capacity are established, the impact of (first- and second-dimension) band broadening on the two-dimensional peak capacity can be investigated. It is possible to optimize the system in such a way that the maximum peak capacity is obtained in the minimum analysis time. This constitutes a clear case of multiobjective optimization. This family of optimization problems can be tackled in several ways.9 In this article, the concept of Pareto optimization10 is proposed. Pareto optimization has been applied already to several chromatographic problems, including one-dimensional
chromatography,11 selection of columns,12 and simulated moving bed chromatography.13 However, to the authors’ knowledge this is the first application of the concept for optimizing twodimensional chromatography. There is a similarity between the so-called Poppe plots1 or kinetic plots14 and the Pareto optimization presented here. The goal of this paper is to use the Pareto optimization approach to (i) measure the maximum system performance in terms of (maximum) peak capacity and (minimum) analysis time, (ii) compute the optimal chromatographic parameters (e.g., modulation times, column lengths, column diameters, and particle sizes) obtained along these optimal solutions, (iii) measure the impact of choosing different elution modes (gradient vs isocratic) or chromatographic systems (high pressure vs ultrahigh pressure) in two-dimensional chromatography, and (iv) apply the extension of this methodology to approach a tree-goal trade-off between (maximum) peak capacity, (minimum) time, and (minimum) sample dilution.
(6) Blumberg, L. M. J. Sep. Sci. 2008, 31, 3358–3365. (7) Murphy, R. E.; Schure, M. R.; Foley, J. P. Anal. Chem. 1998, 70, 1585– 1594. (8) Horie, K.; Kimura, H.; Ikegami, T.; Iwatsuka, A.; Saad, N.; Fiehm, O.; Tanaka, N. Anal. Chem. 2007, 79, 3764–3770. (9) Bourguignon, B.; Massart, D. L. J. Chromatogr. 1991, 586, 11–20. (10) Massart, D. L.; Vandeginste, B. G. M.; Buydens, L. M. C.; de Jong, S.; Lewi, P. J.; Smeyers-Verbeke, J. Handbook of Chemometrics and Qualimetrics: Part A; Elsevier: Amsterdam, The Netherlands, 1997; p 790.
(11) Cela, R.; Martı´nez, J. A.; Gonza´lez-Barreiro, C.; Lores, M. Chemom. Intell. Lab. Syst. 2003, 69, 137–156. (12) van Gyseghem, E.; Jimidar, M.; Sneyers, R.; Redlich, D.; Verhoeven, E.; Massart, D. L.; Vander Heyden, Y. J. Chromatogr., A 2004, 1042, 69–80. (13) Paredes, G.; Mazzotti, M. J. Chromtogr., A 2007, 1142, 56–68. (14) Cabooter, D.; de Villiers, A.; Clicq, D.; Szucs, R.; Sandra, P.; Desmet, G. J. Chromatogr., A 2007, 1147, 183–91. (15) Tijssen, R. Handbook of HPLC; Katz, E., Eksteen, R., Schoenmakers, P., Eds.; Marcel Dekker: New York, 1998; p 57.
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THEORY The objective of this section is to provide the equations that relate several chromatographic parameters (first- and seconddimension particle sizes, column diameters and lengths, flow rates, and modulation time) to three “objective functions”, viz. the total peak capacity, the total analysis time, and the total dilution of the sample. The ultimate goal is to establish the impact of the chromatographic parameters (that can be modified by the user) on the objective functions. Once this link has been established, these equations constitute the basis to perform the Pareto optimization. Calculation of Total Peak Capacity and Total Analysis Time in LC × LC. Consider first the influence of particle size on the peak capacity in one-dimensional LC. The most important equations governing the peak capacity (irrespective of the chromatographic dimension considered) are the plate-height equation and the pressure drop. As for the plate-height equation, the Van Deemter equation in the dimensionless form is used:
h)a+
b + cυ υ
(1)
where h is the reduced plate height (h ) H/dp; H and dp being the plate height and the particle size, respectively), and ν is the reduced velocity (υ ) umdp/Dm; um, dp, and Dm being the linear velocity, particle size, and molecular diffusion of the analyte, respectively; see Table 1 for typical values of dp and Dm). In eq 1, a, b, and c are parameters related to the Eddydiffusion, molecular diffusion, and mass-transfer contribution, respectively. There is some ambiguity in the literature about the velocity (um) that should be used as “mobile-phase velocity”. Theoretical considerations15 suggest that the mobile-phase
Table 1. Values Adopted for Computing Peak Capacities, Analysis Times, and Dilution Factors Used in This Studya name
value
van-Deemter coefficient, a van-Deemter coefficient, b van-Deemter coefficient, c diffusion coefficient, D maximum pressure drop, ∆P column resistance factor, φ column diameter, dc mobile-phase viscosity, η column internal porosity, εi column external porosity, εe particle size, dp resolution, Rs retention time of the last eluted compound from the first dimension, 1tw retention time of the last eluted compound from the second dimension (equivalent to the modulation time), 2tw first-dimension injection volume, 1Vi focusing factor, FF ) (2k1e + 1)/(2k2e + 1) S∆φ tG/tm ka kw
1.5 1 0.15 10-9 4 × 107 1000 1, 2.5, 5, 7.5, 10 10-3 0.35 0.4 1.5, 2, 2.5, 3, 3.5, and 4 1 50-200
a
units
equation
elution mode
min
1 1 1 1 2 2 8 2 8 8 1 and 2 5, 6, 16 and 18 3 and 4
both both both both both both both both both both both both both
0.1-1
min
3 and 4
both
3 1 3.3 10 1 10
µL
20 13 7 4 and 7 5 3
both both gradient gradient isocratic isocratic
m2/s Pa mm Pa s µm
ref 26 26 26 1 27 27 27
17 17
When not specified otherwise, values of the parameters correspond to both first and second dimensions.
velocity corresponds to the interstitial velocity, considering that mobile-phase flow occurs only in the interstices between the packing particles. This velocity can be calculated by measuring the elution time of a solute that is totally excluded from the particle pores. In practice, however, this velocity is difficult to calculate. Chromatographers normally fit the Van Deemter equation using the velocity corresponding to an unretained solute that is able to fully penetrate the particle pores. The definition of the velocity does not affect the form of the equation, but it does affect the values of the a, b, and c parameters. In this paper, this practical approach is adopted, where um refers to the average velocity of an unretained compound able to penetrate the particle pore space, and the experimental values of a, b, and c found in the literature and based on the same convention (see the Experimental Section) are used. The pressure drop of the system (∆P) is governed by Darcy’s equation:
∆P )
umφηL dp2
(2)
where φ is the flow-resistance factor, η is the viscosity of the mobile phase, and L the column length. The same ambiguity as above surrounds the velocity in eq 2. In some studies, eq 2 is defined referring to the velocity of a (totally excluded) unretained compound, whereas in other cases um corresponds to the velocity of an unretained compound able to fully penetrate the particle pores. Again, ambiguity in the definition of um does not affect the form of the pressure-drop equation but only the value of φ. According to Cramers et al.,16 the velocity corresponding (16) Cramers, C. A.; Rijks, J. A.; Schutjes, C. P. M. Chromatographia 1981, 14, 439–444. (17) Schoenmakers, P. J.; Vivo´-Truyols, G.; Decrop, W. M. C. J. Chromatogr., A 2006, 1120, 282–290.
to a fully penetrating unretained solute should be used in eq 2 and this convention is followed in this paper (although this can be a subject of discussion from a theoretical perspective). The appropriate values of φ and η are described in the Experimental Section. Theoretically, the maximum pressure delivered by the instrument constitutes the ultimate limitation to obtain a certain peak capacity within a certain analysis time. It is supposed in this work that the system is running all the time under the maximum pressure allowed by the instrument. At a given pressure drop and particle size and using a specified mobile phase (viscosity) and typical van-Deemter coefficients, it can be demonstrated1 that the combination of eqs 1 and 2 leaves only one degree of freedom. Therefore, a unique solution can be obtained if either L or um is fixed. Suppose that the analysis time, tw is fixed. Through eqs 1-7 (see below), a value for the peak capacity can be obtained. The first step is to relate tw with the dead time (tm). If the retention factor of the most retained compound (kw) is known, the equation is straightforward in the case of isocratic elution:
tm )
tw (kw + 1)
(3)
The equivalent equation for gradient elution is
tm )
(
tw tG +1 tm
)
(4)
where tG/tm is the ratio between the gradient time and the dead time, which has been fixed in this work at a value of 10 (which is a typical value in practice17). Equation 4 arises from considering tw ) tG + tm, which is normally a good approximation, especially for relatively hydrophobic solutes (the equation performs worse for highly water-soluble solutes). In this Analytical Chemistry, Vol. 82, No. 20, October 15, 2010
8527
equation, the dwell time and the equilibration time have been ignored. The ways to deal with these parameters either practically or theoretically are beyond the scope of this paper. At a given dead time (tm), a single value of flow velocity (um) and column length (L) can be obtained from eq 2, making use of the relationship L ) umtm. Knowing um and L, the application of eq 1 yields a value of the plate height (H) and thus the plate count (N ) L/H). The last step in the process is to relate the plate count with the peak capacity. In case of isocratic elution, Giddings’ equation is used: niso ) 1 +
(
1 + kw √N ln 4RS 1 + ka
)
(5)
where ka is the retention factor of the least retained compound and RS the desired resolution between consecutive peaks (normally a value of 1 is used). In the case of gradient elution, assuming that the peak bandwidth does not vary with the retention time,18 the following expression is obtained: ngrd )
tG tG √N ) 4RSσpeak tm 4RS(1 + ke)
(6)
where σpeak represents the peak dispersion under gradient conditions and ke is the retention factor at the moment of the elution, which is approximately equal to ke )
tG 1 tm S∆φ
(7)
In eq 7, S is the slope of the ln k vs φ (retention vs mobilephase composition) line and ∆φ is the range in composition covered by the gradient. In principle, the value of S is solutedependent, but it is assumed constant in the present treatment. Additionally, the flow rate can be calculated if the diameter (dc) and the total porosity εT of the column are known: F)
()
dc 2 πεTL 2 tm
(8)
Flow rates are needed to calculate the extra-column band broadening (see Influence of Extra Band Broadening in (Isocratic and Gradient) Peak Capacity) and dilution factors (see Dilution Factors in Comprehensive Two-Dimensional Chromatography). Equations 1-7 constitute a way to relate analysis time and peak capacity in one-dimensional chromatography. The same approach can be applied to each dimension in the case of two-dimensional chromatography. In online comprehensive chromatography, the condition is that the modulation time equals the analysis time in the second dimension. A pair of values of (first- and seconddimension) peak capacities (1n and 2n) are obtained from a pair of values of (first- and second-dimension) analysis time (1tw and 2 tw). When these two separation systems are linked in a comprehensive two-dimensional chromatographic device, the total analysis time (2Dtw) and the total peak capacity are calculated as follows: (18) Dolan, J. W.; Lommen, D. C.; Snyder, L. R. J. Chromatogr. 1989, 485, 91– 112.
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2D
tw ) 1tw + 2tw
(9)
2D
n ) 1n × 2n
(10)
Equation 9 arises from considering that the second-dimension separation is performed online in real time (during the separation in the first-dimension), so that only the last injection in the second dimension contributes to enlarge the total analysis time. Since many fractions must be taken from the first-dimension effluent, the first-dimension analysis time is significantly larger than the second-dimension analysis time and 2tw is negligible in eq 9, so that 2Dtw ≈ 1tw. Equation 10 arises from considering that a complete two-dimensional chromatogram can be obtained for each first-dimension peak. This equation is not affected by the orthogonality of the retention mechanisms in the first and the second dimension. However, the peak capacity obtained from eq 10 can only be used fully if the two mechanisms are completely independent (orthogonal). Equations 9 and 10 suggest a much higher separation power (at a marginal cost in analysis time) for comprehensive two-dimensional chromatography in comparison with the one-dimensional case. Peak capacities are multiplied in eq 10, but retention times are summed in eq 9. Influence of Modulation Time on Additional Band Broadening in the First- And Second-Dimensions. The modulation time has an obvious impact on the total peak capacity; a longer 2 tw increases 2n through eqs 1-6. However, this parameter has an additional impact on the total peak capacity, since it affects how much of the theoretical peak capacity (1n and 2n) can be maintained in eq 10. When eq 10 was written, it was assumed that the peak capacity in each dimension would not be affected by combining the two separation stages in an online (real-time) two-dimensional device. Strictly speaking, this is true only when 2 tw is extremely short. This is because of two reasons. First, the modulation time has to be sufficiently short enough to allow the first-dimension chromatogram to be sampled at a high enough frequency. Otherwise, additional band broadening occurs, leading to a decrease in 1n. This effect has previously been noticed by other authors.6-8,19,20 The second-dimension chromatograph can be considered as a detector for the first-dimension separation process, with a sampling frequency of 1/(2tw). This allows us to correct the peak width in the first dimension (1σtotal) as follows:21
1
σtotal )
(1σpeak)2 +
(2tw)2 δdet2
(11)
where 1σpeak is the peak width due to the chromatographic process. The value of δdet2 is still subject to discussion. A value of 12 represents the ideal case.22,8 However, Blumberg6 argued that the effect of low-sampling frequency of the first-dimension chromatogram can be approximated by the effect of a two consecutive box-car filters, and therefore the correct value of δdet2 should be decreased to 4. Recently, Davis et al.19 applied the (19) Davis, J. M.; Stoll, D. R.; Carr, P. W. Anal. Chem. 2008, 80, 461–473. (20) Li, X.; Stoll, D. R.; Carr, P. W. Anal. Chem. 2009, 81, 845–850. (21) Poole, C. F. The Essence of Chromatography; Elsevier: Amsterdam, The Netherlands, 2003. (22) Sternberg, J. C. Adv. Chromatogr. 1966, 2, 205.
statistical overlap theory to two-dimensional separations and found a experimental value of δdet2 of 4.76. In this work, we will follow the latest results and a value of 4.76 will be used. The second reason why the total peak capacity may be lower than predicted by eq 10 has to do with the injection band broadening in the second dimension. As relatively large fractions of the first dimension are injected in the second-dimension column, the peak width in the second dimension 2σtotal suffers from additional band broadening, i.e.,
2
σtotal )
() 1
(2σpeak)2 +
2 2
F
2
F
( tw)2
(12)
δinj2
where 1F and 2F are the flow rates in the first and second dimensions, respectively, and δinj2 is a parameter related to the injection system used that varies between 12 and 4 (usually, a value of 4 is accepted17). Equation 12 holds when there is no focusing effect. Focusing will occur at the inlet of the second-dimension column when the retention factor on that column with the first-dimension mobile phase (2k1e) is higher than the retention factor with the seconddimension mobile phase (2k2e). The injection volume is then contracted by a factor of (2k1e + 1)/(2k2e + 1).23,24 For convenience, this ratio will be called the “focusing factor” (FF). The modified equation for the total band broadening in the presence of focusing is
2
σtotal )
(2σpeak)2 +
( )( 1
F
2 2
2
F
2
k2e + 1
k1e + 1
) ( )(
(2σpeak)2 +
2 2
σtotal )
(2σpeak)2 +
niso
√N )1+ 4RS
2
( tw)
µ)
F
2
2
F
1 FF
2
)
(2tw)2 δinj2
(13)
(2tw′)2 δinj2
(14)
where the auxiliary variable 2tw′ can be considered as an “effective duration of the second-dimension injection” and is defined as
tw′ )
F (2k2e + 1) 2 tw 2 F (2k1e + 1)
(23) Lankelma, J.; Poppe, H. J. Chromatogr. 1978, 149, 587–598. (24) Jandera, P.; Cesla, P.; Hajek, T.; Vohralik, G.; Vynuchalova, K.; Fischer, J. J. Chromatogr., A 2008, 1189, 207–220.
( ) tR,1 +
(16)
d2µ
tR,1 + tR,n
2t√N
(17)
(18)
(ke + 1)2tm2 t2 + 2 N d
where we have been substituting eq 11 on the left-hand side of eq 6. Similar to the case of isocratic elution, we define t ) 2tw and d ) δdet for the first dimension, and t ) 2tw′ and d ) δinj for the first and second dimension. If we want to take the effects of additional band broadening into account in calculating the total two-dimensional peak capacity, eqs 16 and 18 should be used instead of eqs 5 and 6. First- and second-dimension peak capacities obtained by eq 16 or eq 18 are then used in eq 10 to calculate the total peak capacity. Dilution Factors in Comprehensive Two-Dimensional Chromatography. It is important to limit the dilution experienced by analytes during the two consecutive chromatographic processes. In elution chromatography, the DF (dilution factor) is17 DF ) √2π
(15)
Note that eq 14 has the same form as eq 11. This allows us to use universal expressions for additional band broadening, irrespective of whether this band broadening is due to a low frequency of detection (additional band broadening in the firstdimension) or injection effects (additional band broadening in the second-dimension). If the additional band broadening in the first dimension is to be calculated, 2tw and δdet are used; to calculate
ln
t√N d2µ t√N
tG
1
2
µ
tR,n +
The values of t and d depend on the dimension in which the isocratic elution is applied. If eqs 16 and 17 refer to the first dimension, then t ) 2tw and d ) δdet; for the second dimension t ) 2t′w and d ) δinj is used. In the case of gradient elution, the peak capacity becomes
δinj
1
1 d2
µ2 +
4RS
)
2
where
ngrd )
For convenience, eq 13 can be rearranged to be similar to eq 11:
2
the additional band broadening in the second dimension, 2tw′ and δinj are used instead. Influence of Extra Band Broadening in (Isocratic and Gradient) Peak Capacity. In the case of two-dimensional chromatography, eqs 5 and 6 should be modified if the extra band broadening effects, measured through eqs 11 and 14 for the first and second-dimension, respectively, are taken into account. In the case of isocratic elution, the peak capacity becomes, see theSupporting Information, SI1,
σF Vi
(19)
where σ is the standard deviation of the peak in time units. F is the flow rate and Vi is the injection volume. Thus, the DF in the first-dimension chromatographic process is σpeak1F
1
DF ) √2π
1
1
Vi
(20)
Note that in this equation, the term 1σpeak refers to the standard deviation of the peak at the end of the first-dimension column, which does not need to be corrected for the additional band Analytical Chemistry, Vol. 82, No. 20, October 15, 2010
8529
broadening due to the sampling frequency (eq 11). This is because this additional band broadening is not a physical process. The dilution experienced in the second dimension is DF ) √2π
2
2 2
2
2
) √2π 1
F σtotal Vi
F 2σtotal
F
2
tw
(21)
where in this case 2σtotal does include the injection band broadening, as calculated from eqs 12 and 13. Note that the equality 2Vi ) 1F2tw is implicit, which implies that the entire volume of eluent collected at the outlet of the first dimension during a modulation-period is injected in the second dimension and the modulation time is equal to the seconddimension analysis time. Finally, the dilution of the complete chromatographic process can be calculated from 2D
DF ) 1DF × 2DF
(22)
Pareto-Optimality Methodology. Equations 1-22 constitute the basis to link total peak capacity, analysis time, and dilution in two-dimensional chromatography. The chromatographer may control some of the parameters involved in eqs 1-22, i.e., maximum pressure drop, column diameter, gradient parameters, injection volume, column particle diameter, and modulation time. Other parameters are given by the system, i.e., coefficients for the plate-height equation, diffusion coefficient, viscosity, retention factors for first and last eluted compound, column porosity, column flow-resistance. With the application of eqs 1-22, values of the column lengths, flow rates, peak capacities, analysis times, and dilution factors are obtained for both dimensions. In practice, the objective is to study the effects of the most important chromatographic variables (column diameters, particle sizes, and modulation time), on the objective functions (peak capacity, analysis time, and dilution factor) and on the recommended column lengths and flow rates. This summarizes the optimization problem. The objective is to obtain the highest possible peak capacity within the shortest possible time and with the lowest possible sample dilution. As it will be shown later (Results and Discussion), these three objectives tend to be conflicting. For example, in order to obtain faster analysis, one has to sacrifice peak capacity and/or deal with higher dilution factors. The Pareto-optimality principle is a helpful technique to find the optimal conditions, or a collection of optimal conditions, when a trade-off between more than one objective needs to be made. This methodology is applied in many scientific disciplines. It was introduced in chromatography by Smilde et al.25 in the 80s. The Pareto-optimization approach is explained in detail in ref 10. The reader can find an explanation of the Paretooptimality approach applied to one-dimensional chromatography in the Supporting Information (SI2). To the authors’ knowledge, there is no report in the literature applying Pareto optimization to two-dimensional chromatography. (25) Smilde, A. K.; Knevelman, A.; Coenegracht, P. M. J. J. Chromatogr. 1986, 369, 1–10. (26) Gilar, M.; Daly, A.; Kele, M.; Neue, U. D.; Gebler, J. C. J. Chromatogr., A 2004, 1061, 183–192. (27) Cabooter, D.; Billen, J.; Terryn, H.; Lynen, F.; Sandra, P.; Desmet, G. J. Chromatogr., A 2008, 1178, 108–117.
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EXPERIMENTAL SECTION Parameter values. The parameter values used in eqs 1-20 are listed in Table 1. When relevant, the table includes the reference from which the value of the parameter is taken. Pareto-Front Computation. The Pareto-optimization algorithm (constructed in house, see Software) can be applied to study a chromatographic optimization problem. The first step of the process is to generate all possible combinations of the factors that are being optimized. Suppose, for example, that the user selects possible particle sizes for the first- and second-dimension columns to be 2, 3, and 4 µm, fixing the remainder of the variables at a single value. This will result in the software automatically generating 3 × 3 ) 9 cases, with particle sizes of 2-2, 2-3, 2-4, 3-2, 3-3, 3-4, 4-2, 4-3, and 4-4 (the first number corresponding to the firstdimension particle size and the second number to the seconddimension particle size). In a second step, the software calculates the output parameters (dilution factors, peak capacities, analysis times, column lengths, etc.) for each case (in this context, a “case” is a particular combination of parameters). In a third step (once the objectives are defined), the Pareto-optimization algorithm is applied, discarding all the cases that are not Pareto-optimal. The Pareto front is then constructed, and the optimal values along the Pareto line can be inspected. The number of cases that are generated grows exponentially with the number of factors that are being optimized. For example, in the experiments described in the Results and Discussion, millions of cases are generated. In complex optimization problems (requiring examination of more than 2 million combinations), the Pareto-optimality computations were performed stepwise. In the first steps, the ranges of the factors were set relatively broad. For example, if the column diameters were to be optimized (in the range of 1 and 10), the column diameters considered were 1, 2.5, 5, 7.5, and 10 mm. After the first Pareto-optimization computation, the ranges of the factors were restricted to select only the area in which optimal parameters were found and the computational power was used to examine this area in more detail. Suppose that in the previous example, all the Pareto-optimal experiments were found to have column diameters of 1, 2.5, or 5 mm. In a second step the column diameters could then be set to 1, 2, 3, 4, or 5 mm. After a few repetitions of the Pareto optimization (with stepwise restriction of the ranges of the factors), a precise solution could have been found. Software. Home-built routines, written in MATLAB 7.7 (The Mathworks, Natick, MA), were used for all computations. In order to provide the reader with the possibility to apply the equations and the methods described in this paper, a free, compiled version of the software can be requested directly via an e-mail to the corresponding author. RESULTS AND DISCUSSION Influence of Additional Band Broadening on Total Peak Capacity. Figure 1 depicts the impact of additional band broadening on the total peak capacity as a function of the number of second-dimensional analyses per first-dimension peak (i.e., number of cuts per peak). The y-axis depicts the ratio of the corrected total peak capacity, 〈2Dnc〉 (considering the different bandbroadening effects) and the theoretical total peak capacity, 2Dnc (without additional band broadening). A value of 1 implies that
Figure 1. Fraction of theoretical peak capacity (2Dnc) that is actually realized as a function of the number of second-dimension runs per one-dimensional peak considering: (solid line) band broadening due to low detection frequency in the first dimension and (dashed line) band broadening due to low detection frequency in the first dimension and injection band broadening in the second dimension. For the second case, a variety of situations have been averaged (different flow-rates, see range in Table 1). Error bars represent the standard deviation of these averages. The case considering band broadening only due to low detection frequency in the first dimension and δdet2 ) 12 is overlaid (solid line with the “+” symbol). For the other cases, δdet2 ) 4.76 has been used.
the theoretical total peak capacity is maintained. Computations were performed considering a two-dimensional system with the parameters listed in Table 1 in each dimension. For simplicity, the gradient-elution mode was considered in both dimensions, with values of S∆φ and tG/tm as indicated in the table. Particle sizes were varied between 1.5 and 5 µm, but column diameters for the first and the second dimensions were kept constant at 4 mm. The two solid lines show the impact of additional band broadening in the first dimension on the total peak capacity, with values of δdet2 of 12 (solid line with “+” symbol) and 4.76. The dashed line shows the impact of additional band broadening in both the first and second dimensions on the total peak capacity (considering δdet2 ) 4.76 and δinj2 ) 4). The dashed line is in fact an average of a number of different cases (different analysis times, particle sizes, and flow rates), which result in different relative contributions of the injection volume to band broadening in the second dimension. Therefore, different values for the total peak capacity are obtained for the same number of cuts per peak. Error bars represent the standard deviation of the average values. When the effect of injection band broadening in the second dimension is neglected, four or more cuts per peak preserves most of the total peak capacity (see solid line). This was noticed by Seeley28 in earlier work. In fact, the solid line of this plot with “+” symbols coincides with the results published by Horie et al.8 (and it is included here for reference). The plain solid line corresponds to the same study but using the corrected value of δdet2 of 4.76 published by Davis et al.19 However, injection band broadening in the second dimension may diminish the obtained peak capacity even further. As can be seen by comparing the (28) Seeley, J. V. J. Chromatogr., A 2002, 962, 21–27.
Figure 2. Pareto fronts for the optimization of time and peak capacity of an LC × LC system using the parameters of Table 1 (gradient elution is considered in both dimensions), taking into account different equations to calculate the total peak capacity. The solid line represents the total peak capacity without any external band broadening. The dashed line represents the total peak capacity considering detection band broadening in the first dimension. The dashed-dotted line represent the total peak capacity considering detection band broadening in the first dimension and injection band broadening in the second. For all computations, column diameters were fixed at 4 mm.
dashed line with the solid line, roughly about half the peak capacity is lost because of this effect. One should note that the average is computed for an arbitrary range of situations that may be far from optimal. For example, the diameters of both column were set equal at 4 mm. This may have a detrimental effect on the estimated values of total peak capacity. Better results may be obtained using a Pareto-optimality approach. The Pareto-optimality method was applied to the system described above. Two objectives were considered, i.e., minimizing the analysis time and maximizing the peak capacity. Parameters were taken from Table 1 (gradient elution was considered in both dimensions). Particle sizes and analysis times of the first- and second dimensions were allowed to vary within the ranges listed in Table 1. Column diameters were again kept constant to 4 mm for both dimensions. The procedure described in Pareto-Front Computation was applied. Figure 2 depicts the Pareto-front of the optimization. The solid line depicts the theoretical values of the peak capacity, without considering any loss in either the first or second dimensions due to sampling or injection band-broadening effects, respectively. The dashed line is obtained when the additional band-broadening is only considered in the first dimension. The dotted-dashed line considers both band-broadening effects. As can be seen, the reduction in peak capacity due to additional band-broadening contributions is a multiplicative effect when only the optimal experiments are considered. The additional bandbroadening effect due to the low sampling rates in the first dimension reduces the total (maximum) peak capacity to approximately 1/2 (compare solid line with dashed line). This effect is different that what was be observed in Figure 1, since the experimental situations depicted in the Pareto fronts are not the same. When the total peak capacity does not reflect any additional band-broadening effects (solid line), the optimal modulation times Analytical Chemistry, Vol. 82, No. 20, October 15, 2010
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tend to be as large as possible, so as to accommodate the maximum number of peaks in the second dimension. This is because this (simple) model does not have any restrictions on the number of cuts per peak, nor on the total peak capacity when the first dimension is undersampled. This effect was noticed recently by Stoll et al.5 Injection band broadening in the second dimension reduces the total peak capacity by a further factor of 2. In the present case, with the diameters of the two columns equal, this is a major effect which certainly cannot be neglected. Equation 13 reveals that there are two ways to reduce this second-dimension injection band broadening, i.e., to increase the flow-rate ratio 2F/1F (which can be achieved by increasing the ratio of column diameters, 2dc/ 1 dc) or by focusing the sample at the top of the second dimension column (increasing the value of the focusing factor, FF). These effects were not considered in this section, since the objective was to explore the possible effect of the additional second-dimension band broadening on the total peak capacity. In this section the column diameters were fixed at 4 mm. Dual-Objective Pareto Optimization in Two-Dimensional Chromatography: Optimal Particle Sizes, Column Dimensions, and Modulation Time. Gradient × Gradient Case. The Pareto optimization method described in Influence of Additional Band Broadening on Total Peak Capacity was extended to optimize the column diameters, the values of which were varied between 1 and 10 mm for both dimensions. Particle sizes and analysis times were optimized as in Influence of Additional Band Broadening on Total Peak Capacity (using the ranges listed in Table 1), keeping the other factors constant. Computations in which additional factors were optimized (e.g., tG/tm ratio) did not change the results significantly. The gradient-elution mode was selected for both the first and the second dimensions. No focusing effect was taken into account (FF ) 1). Because the band broadening effects were found to be significant in both dimensions (see Influence of Additional Band Broadening on Total Peak Capacity), the total peak capacities were calculated using eqs 11 and 13 for the first and second dimensions, respectively. The complete Pareto analysis is depicted in Figure 3. The optimal values of parameters along the Pareto front (as a function of the total peak capacity) are depicted in subgraphs a-e. The Pareto front is depicted in Figure 3f. Some discontinuity can be seen in the optimal values of flow-rates, column lengths, and modulation time along the Pareto optimum. This is due to the discrete levels of the input parameters (column particle sizes, column lengths, and analysis times). Allowing intermediate values would decrease the oscillation of the parameters but increase the computation time and the demand on RAM memory (see ParetoFront Computation). Also, particle sizes, column diameters, etc. cannot be varied freely in practice. For these reasons, we have kept the number of discrete levels of the parameters fairly low. It is instructive to analyze the optimal values of the various parameters along the Pareto front. For example, the optimal column diameters (results not depicted in the figure) were found to be independent of the peak capacity. The selected first- and second-dimension column diameters were always the minimum and the maximum values (i.e., 1 and 10 mm for the first- and second-dimension, respectively). This is in agreement with what 8532
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Figure 3. Pareto-optimal results for an optimization of peak-capacity and analysis time of an LC × LC system using the parameters from Table 1 (with gradient elution in both dimensions). In parts a-f, the X axis is the total peak capacity (in parts a-e the X axis has been omitted). Y axis: (a) optimal flow rates for the first (solid line, left axis) and second (thick line, right axis) dimensions; (b) optimal column length for the first (solid line, left axis) and second (thick line, right axis) dimensions; (c) optimal particle sizes for the first (solid line) and second (thick line) dimensions; (d) optimal number of cuts per peak (i.e., number of second-dimension runs for a first-dimension peak); (e) modulation time; (f) total analysis time.
was anticipated in Influence of Additional Band Broadening on Total Peak Capacity. The additional band broadening in the second-dimension is minimized by maximizing the flow-rate ratio 2 F/1F, see eq 13 and Figure 3a, through increasing the ratio of column diameters (2dc/1dc). Not surprisingly, with a column diameter ratio (2dc/1dc) equal to 10, the Pareto-front values of analysis time versus peak capacity are almost coincident with those obtained when the injection band broadening in the second-dimension is neglected (dashed line in Figure 2). However, while high values of 2dc/1dc have a beneficial impact on the total peak capacity, they have a detrimental effect on dilution, especially in the second dimension (eq 21). Dilution effects have not been taken into account in this section (they will be discussed in Dilution Factors: The Trade-Off between Analysis Time, Peak Capacity, and Dilution). The optimal flow-rate in the second dimension may be too high to be operative with current instrumentation. In cases like these, it is possible to operate at
suboptimal conditions. The impact of departing from optimal conditions on the peak capacity vs time curve is variable. As it will be discussed in Dilution Factors: The Trade-Off between Analysis Time, Peak Capacity, and Dilution, a large decrease of 2 F (via decreasing 2dc) has a small impact in the peak capacity vs time curve. The optimal column lengths in both dimensions increase with the total peak capacity (Figure 3b), i.e., longer analysis times are needed to obtain higher peak capacities. This is achieved by increasing the total number of plates through using longer columns. This forces a decrease in the flow rates (Figure 3a) to keep the pressure at the prescribed (maximum) value. These results are in accordance with previous research.17 Optimal particle sizes are greater in the first dimension than in the second dimension. Much of the total peak capacity is obtained in the (slow) first dimension, using longer columns and larger particles to keep the pressure at the specified (maximum) level. The optimal particle size in the second dimension is always the smallest value allowed (1.5 µm), suggesting further benefits of even smaller particles when applied in the second dimension. The optimal number of cuts per peak oscillates between 2 and 3 (Figure 3d), which is somewhat similar to the optimal values ranging from 2.2 to 4 reported by Horie et al.8 The results should be comparable, because the injection band broadening in the second dimension has been almost eliminated by increasing 2F/1F. Any change in one (or several) of the optimized factors affects the optimal values of the other parameters. For example, forcing the column diameters to be 4 mm in both dimensions not only changes the Pareto front (as described in Figure 2) but also makes the optimal number of cuts per peak higher (optimal values are between 6 and 10). This is because a low number of cuts per peak has a greater negative impact on the total peak capacity, since the peak capacities in both dimensions are reduced. With both column diameters equal it is not possible to maintain high flowrate ratios (2F/1F). In fact, the ratio is altered because the optimal flow rates and the optimal particle size in the second dimension are affected. In other words, the optimal peak width in the second dimension is obtained by decreasing the second term in eq 13 (by increasing the flow rate in the second dimension) and increasing the first term (by increasing particle sizes). Indeed, the optimal particle sizes in the second dimension are no longer 1.5 µm (the lower limit) but between 1.5 and 2 µm. Gradient vs Isocratic. The Pareto method can be used to compare the performance in different situations. This can simply be done by comparing different Pareto fronts. For example, suppose that we want to study the effect of changing elution modes (from gradient to isocratic) in one or both dimensions. This results in four possibilities: gradient in both dimensions (gradient × gradient), isocratic in the first dimension and gradient in the second (isocratic × gradient), gradient in the first dimension and isocratic in the second (gradient × isocratic), or isocratic elution in both dimensions (isocratic × isocratic). Note that the equilibration times for those cases in which gradient elution is operating in the second dimension have been ignored. This may be realistic if two second-dimension columns are operating alternatingly.29 In the other cases, re-equilibration times should be taken into (29) Stoll, D. R.; Carr, P. W. J. Am. Chem. Soc. 2005, 127, 5034–5035.
Figure 4. Pareto fronts in the optimization of the total peak capacity and total analysis time for an LC × LC system using the parameters from Table 1. Gradient (“Grd”) or isocratic (“Iso”) elution modes (at 40 MPa) have been considered in the first and the second dimension, as indicated. UHPLC (maximum pressure drop of 100 MPa) and HPLC (maximum pressure drop of 40 MPa) have also been considered in the first and the second dimension for the gradient × gradient case.
account in eq 4. We want to investigate whether a change in the elution modes will affect the optimal values of the parameters and study the impact of changing elution modes on the total peak capacity/time trade off. To perform the computations, the gradient parameters, S∆φ and tG/tm should be converted to the equivalent isocratic ones (ka and kw). From the approximations made in Theory, the equivalence between kw and tG/tm can be deduced. Therefore, kw is set to 10. However, the value of ka depends on the value of mobile-phase composition φ selected for the isocratic elution. The value of ka ) 1 has been arbitrarily chosen. Figure 4 shows a comparison of the results obtained in the four situations. As expected, choosing isocratic elution diminishes the potential separation power of the system. Around 30% of the peak capacity is lost when only one dimension is isocratic. The use of isocratic elution in the second dimension is a common choice in practice, since it does not require (fast) column re-equilibrations and may alleviate detection problems. The isocratic × gradient choice has been added here for coherence, but it is quite uncommon and will not be discussed. Selecting isocratic elution in both dimensions reduces the peak capacity by around 50% in comparison with the gradient × gradient case. The use of isocratic elution slightly increases the optimal number of cuts per peak. The optimal number of cuts per peak in the gradient × isocratic system is between 2 and 3, whereas in the isocratic × isocratic case it is between 2.5 and 4. In this latter case, the number of cuts per peak is calculated as an average value, since the peak width in the first dimension is not constant. A modification of the mode of operation in which the modulation time is variable (adaptable to the peak width resulting from the first dimension) is beyond the scope of the present paper. Analytical Chemistry, Vol. 82, No. 20, October 15, 2010
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Benefit of High Pressure. The benefits of high pressure have been already reported.30 The use of ultrahigh-pressure LC (UHPLC) systems in comprehensive two-dimensional chromatography is studied in this paper with the aid of the Paretooptimization approach. We have been applying the equations described in Theory, assuming that they are still valid under pressures of 108 Pa (1000 bar). However, the results of this exercise should be interpreted with some caution, since additional band-broadening effects due to thermal gradients have not been taken into account. Similarly to the previous section, four situations are taken into account: UHPLC × UHPLC, HPLC × UHPLC, UHPLC × HPLC, and HPLC × HPLC. In all cases and in both dimensions, gradient elution (using the same conditions as in Gradient × Gradient Case) has been selected as the elution mode. The maximum pressures of the UHPLC and HPLC systems were set at 108 Pa (100 bar) and 4 × 107 Pa (400 bar), respectively. The results are shown in Figure 4. As can be seen, the use of UHPLC in one of the dimensions increases the peak capacity by 15-20%, whereas the use of UHPLC in both dimensions improves the peak capacity by close to 25-30% in comparison with the original HPLC × HPLC system. In terms of analysis time, the use of UHPLC in one of the dimensions leads to a reduction by 25%, whereas the use of UHPLC in both dimensions allows obtaining the same separation power in around 35% of the analysis time. If UHPLC is available in the first dimension, the optimal particle sizes are slightly smaller (between 2 and 3 µm) and the columns longer (between 30 and 100 cm) in the first dimension. The optimal number of cuts per peak is still between 2 and 3. Apparently, the performance of an UHPLC × HPLC system is slightly lower than that of an HPLC × UHPLC system. In the second case, higher peak capacities are obtained by lengthening the second-dimension column (between 15 and 25 mm), while slightly reducing the number of cuts per peaks to between 1.8 and 2.8. This is understandable, since there is more to gain in the second dimension due to the higher pressures available. Also, the optimal flow rates in the second dimension are higher, reducing the loss of peak capacity due to injection band broadening in the second dimension. However, in these situations the dilution factors (that have not yet been taken into account in the optimization) become very high. Other factors (such as the use of different retention-factor windows, the use of monolithic columns, or the use of slower gradients) are not discussed in this paper, but they may easily be studied by the reader using the free software available on request. Dilution Factors: The Trade-Off between Analysis Time, Peak Capacity, and Dilution. From eq 13, one obvious way to decrease the additional (injection) band broadening in the second dimension is to increase the flow-rate ratio (2F/1F). This is achieved via increasing the ratio of column diameters, i.e., 2d/ 1 d, as has been explained in Dual-Objective Pareto Optimization in Two-Dimensional Chromatography: Optimal Particle Sizes, Column Dimensions, and Modulation Time. However, the ratio 2 F/1F also appears in eq 21. Increasing the flow-rate ratio is seen to result in higher dilution factors in the second dimension. In other words, the maximization of peak capacity in the second (30) Fallas, M. M.; Neue, U. D.; Hadley, M. R.; McCalley, D. V. J. Chromatogr., A 2010, 1217, 276–284.
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dimension (by avoiding injection band broadening) is achieved at the expense of high dilution factors. Dilution factors may indeed become unacceptably high in the situations that were considered optimal in Dilution Factors: The Trade-Off between Analysis Time, Peak Capacity, and Dilution. To minimize the dilution, the total dilution of the system (2DDF; eq 22) can be introduced as a separate objective in the Pareto optimization. In this way the tradeoff between peak capacity, analysis time, and dilution can explicitly be considered. The first objective (peak capacity) should be maximized, whereas the second and the third objectives (analysis time and dilution) should be minimized. Results of the optimization for the gradient × gradient case are shown in Figure 5a. Because this is a three-objective optimization, the Pareto front is not a line but a surface, representing the limiting possibilities of the current system. In order to simplify and quantify the representation, Figure 5b represents the optimal values of peak capacity and time along “iso-dilution lines”. Each iso-dilution line represents the (classical) trade-off between analysis time and peak capacity at a fixed value of the total dilution factor. One should take into account, however, that the iso-dilution line is calculated by interpolation. Therefore, some spurious effects may occur, especially at low dilution, where the number of Paretooptimal cases is very low, so that the surface is poorly defined. The surface is quite steep at high dilution values (which can be observed from the close proximity of iso-dilution lines at high dilution values). This means that dilution factors can be decreased from, for example, 50 to 20 with little loss in peak capacity and analysis time. If the objective is to reduce the dilution even further (to values of 10 or 7.5), the costs in terms of peak capacity and analysis time become more significant (see Figure 5b), especially at higher peak capacities. The cost of reducing dilution is also more significant at longer analysis times, where iso-dilution lines are found further apart. For sample-limited analyses and in situations in which detection forms a major bottleneck, the reduction of the dilution factors is of paramount importance. It is instructive to see how this decrease in dilution factors can be achieved. To this end, the optimal values of the various parameters along the Pareto surface can be analyzed in a similar way as was done in the dual-objective case (results can be inspected in the Supporting Information, Figure SI3). To reduce dilution, smaller second-dimension column diameters are recommended (Figure SI3b in the Supporting Information). This is because a decrease in the flow-rate in the second dimension is obtained by decreasing the column diameter (at the expense of lower peak capacities and higher analysis times). On the other hand, smaller first-dimension particle sizes (see Figure SI3a in the Supporting Information) are beneficial. Hence, shorter columns in the first dimension (Figure SI3c in the Supporting Information) become optimal to maintain the pressure without decreasing flow rates. In fact, the optimal particle sizes are different from those found when dilution is not taken into account. Along the surface, however, the optimal particle sizes still depend on the analysis time. Higher analysis times still require larger particle sizes (as in Figure 3c). The optimal number of cuts per peak is maintained between 2 and 3 along the surface (Figure SI3d in the Supporting Information), except in extreme situations (low peak capacities, low analysis times, and low dilution factor) where more cuts are possible.
Figure 5. (a) Pareto-optimal surface resulting from optimizing total peak capacity, total analysis time, and total dilution for an LC × LC system using the parameters from Table 1 (gradient elution was considered in both dimensions). Overlaid dots represent actual Pareto experiments, whereas the surface represents the linear interpolation of the dots. Part b depicts the iso-dilution map corresponding to part a (iso-dilution lines at dilution factors of 7.5, 10, or 20 are interpolated, whereas iso-dilution lines at 30 and 40 correspond to actual Pareto experiments). Parts c and d represent the iso-dilution maps as in part b but considering different values of the focusing factor, FF (eq 13). Plot c corresponds to FF ) 2 and plot d corresponds to FF ) 5. The iso-dilution lines in parts c and d are not interpolated but represent actual computations.
Effect of Focusing. Focusing can play a major role in the trade-off between peak capacity, analysis time, and dilution. Focusing occurs when the mobile phase from the first dimension (at the time of elution) is a weak eluent in the second dimension, and it is reflected in eq 13. When focusing occurs, the injection band broadening in the second dimension is reduced. This decreases the need to have high flow rates values in the second dimension. Thus, narrower second-dimension columns can be used, which in turn reduce the dilution in the second dimension. Figure 5c,d depicts the iso-dilution maps for optimizing total analysis time, total peak capacity, and total dilution for cases in which FF is 2 (Figure 5c) and 5 (Figure 5d). These maps should be compared with Figure 5b (where FF was 1). As can be seen, focusing constitutes an excellent way to decrease dilution without a significant sacrifice in terms of analysis time or peak capacity. This follows from the steep surfaces for FF ) 2 and FF ) 5. When FF ) 2, there is almost no loss in peak capacity nor an increase in analysis times. Restricting the dilution to 10 requires only a light decrease in the peak capacity (or a slight increase in analysis times). Even more extreme is the case in which FF ) 5. In this case, the surface is so steep that a dilution factor of 10 can be easily achieved without any penalty in terms of peak capacity or analysis time. CONCLUSIONS When constructing an online, comprehensive two-dimensional separation system with (valve-based) modulation, two extracolumn band-broadening effects should be taken into account, viz., the “detection band-broadening” in the first dimension and
injection band-broadening in the second dimension. The seconddimension can be seen as a detector for the first dimension; extracolumn band-broadening is observed in the first dimension due to low sampling rates (i.e., relatively long modulation times). Additionally, after modulated fractions of the first dimension are injected in the second dimension, extra-column band broadening is observed due to the injection volumes, especially with high modulation times. This latter effect is treated and optimized for the first time in this paper. The first effect (detection band broadening) reduces the original peak capacity by about a half, the second effect can reduce the total peak capacity by an additional half. Thus, the total loss in peak capacity may be around 75%. The Pareto-optimality method constitutes an excellent way to analyze the trade-off between different objectives in chromatography (such as achieving low dilution factors and high peak capacities in the shortest analysis time possible). The Paretomethodology is flexible enough to incorporate other chromatographic objectives. The analysis of optimal chromatographic parameters along the Pareto front allows the chromatographer to select the best possible system given the common trade-off of separation power and time. At present, computers allow one to perform these computations in a couple of seconds and to analyze the optimal solutions. Computations are facilitated for the reader through freeware (on request). The Pareto analysis can be applied to a variety of situations, analyzing the impact of the use of gradient elution (instead of isocratic elution) or the use of high pressure systems in either Analytical Chemistry, Vol. 82, No. 20, October 15, 2010
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(or both) dimensions. The method yields optimal values for particle sizes, column diameters, column lengths, and modulation times along the Pareto-optimal front that represents the trade-off front between peak capacity and time. Consistently, the optimal number of cuts per first-dimension peak was found to be between 2 and 3. A balance should be found between the losses in peak capacity in the first and second dimensions (due to detection and injection band broadening) and the increase in peak capacity observed in the second dimension with high modulation times. One way to reduce the second-dimension injection band broadening is to increase the flow-rate (and the column diameter) in the second dimension. This will, however, result in higher dilution of the analytes prior to detection. Pareto analysis can be applied to the multiobjective trade-off between peak capacity, time, and dilution. In normal conditions, dilution factors could be reduced from 60 to 30 without a significant loss in peak capacity or a significant increase in analysis times. A very effective way to gain with respect to all three objectives is to introduce focusing. This will, for example, occur when the mobile phase used in the first dimension is a weak eluent in the second. If the retention time on the second-dimension column is twice as high in the first-dimension eluent as in the second-dimension eluent, a dilution factor of 20 can be reached without a significant sacrifice in terms of analysis time and peak
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capacity. A ratio of retention times of 5 makes a dilution factor of 10 achievable without significant penalties. An improved equation for the peak capacity in isocratic conditions has been derived, which incorporates the possibility of external band broadening. An approximate analytical solution was established, allowing the external band broadening effects (“detection” band broadening in the first dimension and/or “injection” band broadening in the second dimension) to be taken into account. This has made it possible to calculate an analytical solution for the peak capacity when one (or both) of the two dimensions is operated under isocratic conditions. ACKNOWLEDGMENT This work was presented at the symposium HPLC-07 (Ghent, June, 2007). SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.
Received for review June 8, 2010. Accepted August 10, 2010. AC101420F
