Generation and Limitations of Peak Capacity in Online Two

Apr 21, 2009 - The different operating conditions of an online two-dimensional liquid chromatographic separation (2D-LC), such as the length of the co...
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Anal. Chem. 2009, 81, 3879–3888

Generation and Limitations of Peak Capacity in Online Two-Dimensional Liquid Chromatography Krisztia´n Horva´th,†,‡ Jacob N. Fairchild,† and Georges Guiochon*,† Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, and Department of Analytical Chemistry, University of Pannonia, P.O. Box 158, Veszpre´m, H-8200, Hungary The different operating conditions of an online twodimensional liquid chromatographic separation (2D-LC), such as the length of the column, the linear velocity and the composition of the mobile phase used in the second dimension, its initial organic content if this separation is carried out in gradient elution, the number of fractions of the first column eluent collected, and the analysis time of the first dimension all affect the achievable separation power of 2D-LC online systems. The influences of these factors on the separation performance were investigated, and an equation was derived for the calculation of the achievable peak capacity in online 2D-LC assuming (1) that the option of undersampling the first-dimension separation is acceptable, (2) that the solutes follow linearsolvent-strength behavior, and (3) that all the separations are made in gradient elution. This theoretical discussion shows that (1) highly efficient separations made with online 2D-LC require the second-dimension peaks to be very narrow, (2) the separation power of 2D-LC systems is maximum for an optimum number of fractions collected in the first dimension, (3) higher peak capacities can be achieved by using shorter second-dimension columns and collecting a relatively large number of fractions, (4) the achievable 2D peak capacity is maximum for a certain eluent flow rate and column length of the second-dimension column, and (5) the maximum achievable peak capacity increases with decreasing velocity and initial organic content of the second-dimension eluent. As a consequence, due to the time restriction of the seconddimension gradient time, online 2D-LC schemes cannot realistically afford peak capacities exceeding 10 000, even if they are implemented with exceptionally efficient columns and if long analysis times are accepted. One of the main goals of chromatographic analyses is to separate as many components as possible in the shortest possible time. Most of the important problems that analysts must now face involve the analysis of samples consisting of hundreds or even thousands of components present in a very wide range of concentrations. Such problems frequently arise in proteomic,1,2 * To whom correspondence should be addressed. E-mail: guiochon@ ion.chem.utk.edu. † University of Tennessee. ‡ University of Pannonia. (1) Link, A. J.; Eng, J.; Schieltz, D. M.; Carmack, E.; Mize, G. J.; Morris, D. R.; Garvik, B. M.; Yates, J. R. Nat. Biotechnol. 1999, 17, 676–682. 10.1021/ac802694c CCC: $40.75  2009 American Chemical Society Published on Web 04/21/2009

lipodomic, and metabolomic3 research. Separations by unidimensional chromatography are unable to provide acceptable analyses of these difficult samples.4 For this reason, two-dimensional liquid chromatographic separations have attracted intense interest over the past decade, due to the tremendous potential improvements in separation power that this method offers over that of onedimensional systems.5,6 The role of comprehensive two-dimensional liquid chromatography (2D-LC) is now important in the separation of proteins and of protein metabolites. This approach is fast becoming widely used in the analysis of other complex samples containing nonvolatile compounds, including low molecular weight extracts from plants, pharmaceuticals, and polymers as well as many environmental samples.5-7 Basically, comprehensive 2D-LC can be implemented through one of three schemes: online; stop-and-go; and off-line.5,6 Currently, the scheme most extensively used is the online scheme, in which the second-dimension analysis is carried out in real time while the first-dimension separation proceeds. It requires that the second analysis be completed during the time that has been allocated to the collection of a fraction from the first dimension. Most 2D-LC separations with analysis times shorter than two hours and having peak capacities between 500 and 1000 were obtained with this scheme.8-14 Wang et al.15 derived a simplified equation for optimizing the peak capacity of a 2D-LC system (nc,2D): (2) McDonald, W. H.; Ohi, R.; Miyamoto, D. T.; Mitchison, T. J.; Yates, J. R. Int. J. Mass Spectrom. 2002, 219, 245–251. (3) Bajad, S. U.; Lu, W.; Kimball, E. H.; Yuan, J.; Peterson, C.; Rabinowitz, J. D. J. Chromatogr., A 2006, 1125, 76–88. (4) Guiochon, G. J. Chromatogr., A 2006, 1126, 6–49. (5) Stoll, D. R.; Li, X.; Wang, X.; Carr, P. W.; Porter, S. E. G.; Rutan, S. C. J. Chromatogr., A 2007, 1168, 3–43. (6) Guiochon, G.; Marchetti, N.; Mriziq, K.; Shalliker, R. A. J. Chromatogr., A 2008, 1189, 109–168. (7) Marchetti, N.; Fairchild, J. N.; Guiochon, G. Anal. Chem. 2008, 80, 2756– 2767. (8) Stoll, D. R.; Carr, P. W. J. Am. Chem. Soc. 2005, 127, 5034–5035. (9) Stoll, D. R.; Wang, X.; Carr, P. W. Anal. Chem. 2008, 80, 268–278. (10) Dugo, P.; Favoino, O.; Luppino, R.; Dugo, G.; Mondello, L. Anal. Chem. 2004, 76, 2525–2530. (11) Dugo, P.; Skˆeı´kova´, V.; Kumm, T.; Trozzi, A.; Jandera, P.; Mondello, L. Anal. Chem. 2006, 78, 7743–7750. (12) Stoll, D. R.; Cohen, J. D.; Carr, P. W. J. Chromatogr., A 2006, 1122, 123– 137. (13) Jiang, X.; van der Horst, A.; Limac, V.; Schoenmakers, P. J. J. Chromatogr., A 2005, 1076, 51–61. (14) Huidobro, A.; Pruim, P.; Schoenmakers, P.; Barbas, C. J. Chromatogr., A 2008, 1190, 182–190. (15) Wang, X.; Stoll, D. R.; Carr, P. W.; Schoenmakers, P. J. J. Chromatogr., A 2006, 1125, 177–181.

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2 nc nc,2D ) 1tG 2 tcyc

(1)

where 1tG is the analysis time of the first-dimension separation (supposed to be conducted under gradient elution), 2nc is the peak capacity of the second-dimension separation, and 2tcyc is the cycle time of this second-dimension separation. The authors concluded that, at constant gradient time of the first separation, the total 2D-LC peak capacity can be maximized by maximizing the second-dimension conditional peak capacity production (i.e., 2 nc/2tcyc). Then, they focused on the effect of the particle size, the pressure, and the temperature on the one-dimensional gradient peak capacity of the second separation, by modifying the Poppe plot and assuming the applicability of the linearsolvent-strength model. They found that fast separations giving low conditional peak capacities but high conditional peak capacity production rates favor the use of small particles (∼2 µm) and short columns (∼3 cm). Schoenmakers et al.16 introduced a practical scheme based on the use of Poppe’s plots17 to optimize comprehensive, online 2D-LC. In this protocol, the analyst has only to define the maximum allowable pressure in both dimensions and the minimum diameter of the first-dimension column. Then, the protocol provides suitable column dimensions (lengths and diameters), particle sizes, and flow rates for the two columns, and the seconddimension injection volume (i.e., sampling loop size) corresponding to realistic practical conditions. The authors used their method to design a suitable online liquid chromatography-size exclusion chromatography (LC-SEC) system under realistic conditions. Recently, two-dimensional parallel-gradient systems were optimized18,19 for the separation of phenolic acids and of flavone natural antioxidants, using a bonded poly(ethylene glycol) (PEG) column in the first and a C18 column in the second dimension. During the optimization procedure, the authors used the linear free energy relationship (LFER) model for the selection of the most suitable (i.e., the most closely orthogonal) combination of separation columns. The gradients in the two separation dimensions were optimized to find the mobile phase composition providing the highest selectivity between the critical pairs of compounds by using the resolution mapping window diagram method.20 As a result of this optimization process, a significant improvement was achieved in the utilization of the available twodimensional retention space. One aim of our work is to investigate the effect of different operating conditions (the flow rate, the column length, etc.) on the maximum achievable separation power of online 2D-LC systems using a gradient separation in each dimension. Understanding the role and the significance of these parameters, and their relationship with each other, would allow further improvements of the separation power of online 2D-LC systems and would offer simple guidelines for their optimization. Another aim of this (16) Schoenmakers, P. J.; Vivo-Truyols, G.; Decrop, W. M. C. J. Chromatogr., A 2006, 1120, 21. (17) Poppe, H. J. Chromatogr., A 1997, 778, 3. ˆ esla, P.; Hjek, T.; Vohralk, G.; Vynuchalov, K.; Fischer, J. (18) Jandera, P.; C J. Chromatogr., A 1189, 1189, 207–220. ˆ esla, P.; Ha´jek, T.; Jandera, P. J. Chromatogr., A, in press. (19) C (20) Schoenmakers, P. J. Optimization of Chromatographic Selectivity: A Guide to Method Development; Elsevier: Amsterdam The Netherlands, 1986.

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paper is to find out whether there is a maximum limit to the practical performance of online 2D-LC. THEORY Achievable Peak Capacity in Online 2D-LC. The separation power of a one-dimensional chromatographic system is best qualified by its peak capacity. Originally, Giddings proposed to define this metric as the number of peaks separated with unit resolution that could be placed in a defined chromatographic time, e.g., between the peaks corresponding to the first and the last eluted components of interest in an analysis.21 This conditional peak capacity, sometimes referred to as the sample peak capacity, is used as a metric for the chromatographic space used in a separation. It is given by

n*c ) 1 +

tR,n - tR,1 w

(2)

where tR,1 and tR,n are the retention times of the first and last eluted compounds of interest, respectively, and w is the average width of the eluted peaks. For effective gradient separations, when the peaks of the sample compounds fill the whole possible retention window (i.e., the gradient time), the peak capacity, nc, can be approximated as

nc )

tG w

(3)

where tG is the gradient time. In two-dimensional chromatography, two separate, independent separation systems are combined to achieve a higher separation power. Thus, if both separation systems are used to their maximum potential, the overall peak capacity of the 2D-LC system considered would be equal to the product of their peak capacities. This relationship, originally suggested by Horva´th,22 was discussed further by Guiochon et al.23 and Giddings.24 The following equation, also called the “product rule”, was derived for the calculation of the total peak capacity of a 2D separation: nc,2D ) 1nc 2nc

(4)

where 1nc and 2nc stand for the one-dimensional capacities in the first and the second dimensions, respectively. This equation gives the peak capacity made available to the analyst by the two-dimensional chromatographic system considered. Its validity assumes that (1) the two retention mechanisms used are independent and the 2D separation space made available is entirely accessible to the components of the analyzed sample, and (2) that no separation is lost due to back-mixing during the transfer of the fractions of the first column eluent that are collected and transferred into the second column. There are three different schemes available to implement comprehensive, 2D-LC by coupling two column separations: (21) Giddings, J. C. Anal. Chem. 1967, 39, 1027–1028. (22) Karger, B. L.; Snyder, L. R.; Horva´th, Cs. An Introduction to Separation Science; Wiley-Interscience: New York, 1973. (23) Guiochon, G.; Gonnord, M. F.; Zakaria, M.; Beaver, L. A.; Siouffi, A. M. Chromatographia 1983, 17, 121–124. (24) Giddings, J. C. Anal. Chem. 1984, 56, 1258A–1270A.

online, stop-and-go, and off-line.6 The online coupling consists of the second dimension being carried out in real time with the first dimension, during the time that it takes (1) to collect a sufficiently narrow fraction of the first column eluent, (2) to transfer this fraction and inject it into the second column, (3) to analyze it, and (4) and to restore the second column into its initial conditions for the next analysis. Only a fraction of the time during which fractions of the first column effluent are collected is available to carry out their analysis on the second column, and the cycle time of the second dimension, 2tcyc, can be divided into two parts: (1) the gradient time, tG, that is required for the separation and (2) the additional time, tadd, that is required to transfer the sample from the first- to the second-dimension column and rinse and equilibrate the second-dimension column. Let λ be the fraction of the cycle time devoted to the separation (2tG ) 2 tcycλ, 0 e λ e 1). Then (1 - λ) is the fraction of the cycle time used to transfer the sample fraction and restore the column to its initial conditions. From the principle of comprehensive online 2D-LC, it results that, in order to limit back-mixing, the analysis time in the second dimension, 2tcyc, of the fractions collected at the exit of the first column must be equal to a small fraction of the width, 1w, of a peak eluted from this first column. This sets a rather drastic limit to the implementation of the second-dimension separation in online 2D-LC. The other two schemes, the off-line and the stop-and-go ones, do not suffer from the same restriction. The condition that must be satisfied by the second-dimension separation in an online scheme is given by the following relationship between the total analysis time in the second dimension and the bandwidth of the peaks eluted from the first column: 1

w r

2

tcyc )

(5)

where r is the fraction collection ratio, that is, the ratio of the average width of the eluted peaks and the sampling time, i.e., the number of fractions collected per average peak width. Murphy et al. showed25 that a significant loss in resolution in the first dimension takes place if fewer than three samples of the first column eluent are analyzed on the second column per peak width of the first column eluent and that little resolution is gained if this collection frequency exceeds five (i.e., they recommended that 3 < r < 5). These authors showed that the optimum width of the transferred fractions should be close to a fourth of the width of the peaks eluted from the first column (r = 4). This means that, if the first column has a peak capacity 1nc, the number of fractions that should be transferred from the first- to the second-dimension column should be 41nc. Accordingly, by combining eqs 3-5, the peak capacity of the online 2D-LC separation is given by 1

nc,2D =

tGλ

2

w4

(6)

Although numerous authors seem now to consider the MurphySchure-Foley condition as a requirement, this rule is not applied in certain cases. So far, the fastest second-dimension gradients

used in 2D-LC separations are between 12 and 20 s,8,26 which suggests that either the number of fractions collected per peak is usually less than four, given that first-dimension peaks narrower than 1 min could easily be achieved in HPLC or that the analysis time in the first dimension is purposefully selected to be larger than could possibly be achieved with the first column in order to broaden the first-dimension peaks. Equation 6 can be generalized as 1

nc,2D =

tGλ

2

wr

(7)

Equations 6 and 7 are similar to eq 1, although the latter was derived on the basis of different assumptions. Equation 7 shows that highly efficient online 2D-LC separations require very small values of 2w and that the peak capacity of a two-dimensional separation does not directly depend on the efficiency of the first separation nor on the duration of the analysis made in the second dimension. This conclusion justifies the relative lack of concern that practitioners of online two-dimensional chromatography have for the performance of the first column of their separation process. Recently, Li et al.27 derived a simplified equation for the calculation of the effective peak capacity in online 2D-LC. They also concluded that, for relatively short 2D-LC separations, the first-dimension peak capacity is not really important and need not to be fully optimized. Equations 6 and 7 have another important consequence. They suggest that it is impossible to develop an online 2D-LC separation that is fast and in the same time has a high peak capacity. This surprising conclusion suggests that the achievable separation power of online 2D-LC is limited by the consequences of the time restriction imposed on the second separation. However, eq 7 also explains why on-line 2D-LC systems generate higher peak capacities than unidimensional ones. Since 2w and 1tG are independent, the peak widths of the second dimension can be decreased without affecting the first-dimension analysis time, 1tG, and/or this analysis time can be increased without increasing the width of the peaks eluted from the second-dimension column, and these modifications can be made far more flexibly than in the case of unidimensional separations. This constitutes the real advantage of 2D-LC over unidimensional separations for attaining high peak capacities. Effect of Undersampling on the Achievable Peak Capacity. The derivation of eq 7 assumed the validity of the “product rule”, eq 4. However, two phenomena lower the actual value of the total peak capacity: (1) the undersampling of the first dimension25,28-30 and (2) the degree of correlation between the retention mechanisms involved in the two separations. Only the first factor can easily be changed by the analyst, so the problem of the lack of orthogonality between the two retention mechanisms used will not be discussed in this work. (25) Murphy, R. E.; Schure, M. R.; Foley, J. P. Anal. Chem. 1998, 70, 1585– 1594. (26) Fairchild, J. N.; Horva´th, K.; Guiochon, G. J. Chromatogr., A 2009, 1216, 1363–1371. (27) Li, X.; Stoll, D. R.; Carr, P. W. Anal. Chem. 2009, 81, 845–850. (28) Seeley, J. V. J. Chromatogr., A 2002, 962, 21–27. (29) Horie, K.; Kimura, H.; Ikegami, T.; Iwatsuka, A.; Saad, N.; Fiehn, O.; Tanaka, N. Anal. Chem. 2007, 79, 3764–3770. (30) Davis, J. M.; Stoll, D. R.; Carr, P. W. Anal. Chem. 2008, 80, 461–473.

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Due to undersampling, the achievable peak capacity in the firstdimension separation, 1n′, c is smaller than the maximum peak capacity that could be given by the column. The actual peak capacity achieved in the first dimension is30 1 1

nc′ )

nc γ

(8)

where γ represents the loss of first-dimension peak capacity due to undersampling. It can be calculated as30,31

γ)

√r2 + 3.424

1

nc′ )

ncr

√r

2

+ 3.424

)

f

√r

2

(10)

+ 3.424

In order to take into account the effect of undersampling on the achievable peak capacity, eq 7 should be divided by γ (see eq 9), which gives 1

nc,2D =

tGλ 2

1 w √r2 + 3.424

(31) Horva´th, K.; Fairchild, J. N.; Guiochon, G. J. Chromatogr., A 2009, 1216, 2511–2518.

Analytical Chemistry, Vol. 81, No. 10, May 15, 2009

tR-t0

0

dt ) t0 k(t)

(12)

where t0 is the hold-up time of the column. The integration of this equation requires prior knowledge of the relationship between the local retention factor, k, and the local composition of the mobile phase, φ. In practice, the relationship between the logarithm of the isocratic retention factor, ln k, and the fraction of organic solvent in the mobile phase, φ, is almost always accounted for in RPLC by an empirical quadratic and/or cubic equation.32,33 However, for the sake of simplicity, a linear relationship will be considered here as valid. Accordingly, k[φ] ) k0 exp(Sφ)

(13)

where k0 is the retention factor of the compounds for φ ) 0, and S is the slope of the log k[φ] versus φ plot. S is a practical measure of the strength of the strong solvent for the given analyte. The value of S is negative (S < 0) and usually decreases (i.e., increases in absolute value) with increasing molecular size of the solute in reversed-phase HPLC. In most theoretical studies, linear gradients are considered, consistent with their practical importance. This allows the analytical solution of eq 12. The actual composition of the eluent at any time, t, at the inlet of the column is given by

(11)

A close examination of eq 11 shows that there are three practical ways to increase the separation power of an online 2D-LC system: (1) to increase the analysis time of the first dimension, 1tG, (2) to minimize the product of 2w and (r2 + 3.424)1/2 (note that it is not possible to minimize separately 2w and (r2 + 3.424)1/2 because the average peak width in the second dimension depends on the number of fractions collected from the firstdimension eluent), and/or (3) to maximize the value of λ or, in other words, to minimize the of equilibration time of the second-dimension column. Finally, eq 11 provides another interesting conclusion. In order to achieve a peak capacity with a 2D-LC system that is larger than that yielded by the first-dimension column, the average peak width in the second dimension must be smaller than 1wλ/[(r2 + 3.424)1/2].

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∫

(9)

r

where r is the number of fractions collected per peak (see eq 5). Note that eq 9 is valid only in the range of 0.25 j r j 20. It is obvious that if two fractions only are collected from the eluent of a first column having a peak capacity of 100 (r ) 1/50), the actual peak capacity for the purpose of estimating the peak capacity of the column, 1n′, c would be 2. In contrast, however, eqs 8 and 9 suggest that 1n′c would be equal to ∼1.07. Most probably, eqs 8 and 9 are too strict for small values of r’s, which is why we limited the value of r to between 0.25 and 20, which still involves most cases of practical interest. Considering that the number of fractions collected from the first-dimension eluent, f, is the product of r1nc, eq 8 can be rewritten as

1

Calculation of Retention Times and Variances in Gradient Chromatography. The prediction of the retention time, tR, of a compound in gradient elution has attracted much attention and generated a large number of publications.32,33 We consider here only retention times in reversed-phase HPLC (RPLC). Similar approaches, using analogous relationships, can be used in other modes of chromatography. In addition, the contribution of the dwell volume of the system will be neglected during the derivation of the equations. Retention times in gradient elution can be calculated by solving the following integral for tR:

φ[t] ) φ0 + βt

(14)

where β is the gradient slope and φ0 the initial eluent composition. In order to allow the utilization of the whole retention window, β should be set knowing the retention behavior of the sample components separated (see eq 24 later in this section). By combining eqs 13 and 14 with eq 12, we derive the retention time of a compound eluted with a linear gradient:

(

tR ) t 0 1 -

ln(1 - k[0]St0 β) St0 β

)

(15)

where k[0] is the retention factor of the compounds in the initial eluent at the beginning of the elution (φ ) φ0, see eq 13). (32) Jandera, P.; Chura´cˇek, J. Gradient Elution in Column Liquid ChromatographysTheory and Practice; Elsevier: Amsterdam, The Netherlands, 1985. (33) Snyder, L. R.; Dolan, J. W. High-Performance Gradient Elution; WileyInterscience: Hoboken, NJ, 2007.

Poppe et al.34 derived a simplified equation for the calculation of the peak variance in the case of linear-solvent-strength (LSS) gradients. Later Gritti and Guiochon35 generalized this equation for non-LSS cases. According to Poppe and assuming a Dirac pulse injection, the variance of the peak can be calculated as 1 1 + p + p2 3 1 + k[L] σt ) HL u0 (1 + p)2

(

2

)

2

(16)

et al. confirmed this result.39,40 Thus, we will use 22t0 as an estimate for the minimum equilibration time of the second column. To minimize the product of 2w and (r2 + 3.424)1/2, we first need to investigate their relationship in detail. The cycle time of the separation done in the second dimension must be equal to the ratio of the analysis time in the first dimension, 1tG, and the number of fractions collected from its eluent, f. 1

where H is the height of a theoretical plate of the compound on the column considered (under isocratic conditions), L is the column length, u0 is the linear velocity of the mobile phase, u0 ) L/t0, and k[L] is the retention factor of the compound at the time of its elution (when the peak maximum leaves the column), with

k[L] )

k[0]u0 u0 - k[0]SLβ

(17)

p)-

SLβ k[0] u0 1 + k[0]

(18)

and p is given by

The first fraction in eq 16 represents the peak compression factor,36,37 which is due to the rear part of the band moving in a stronger solvent, hence at a higher velocity than its front part. The last term of the equation converts the variance from length scale into time scale. From eq 16, the width of the peak in linear gradient elution is

w ) 4σt ) 4

1 + k[L] u0(1 + p)

 (

1 HL 1 + p + p2 3

)

(19)

If the retention factor of the compound at the beginning of the elution, k[0], is not too small (that is the case in practice) eq 19 simplifies to 4 w) βu0



H (3u02 + SLβ(SLβ - 3u0)) 3LS2

2

tcyc )

1 2

tG )

tG 2 - t02 f

(22)

and the fraction of the second-dimension cycle time that is devoted to the separation, λ, is 2 tG 2f ) 1 - 2t0 1 λ) 1 tG ⁄ f tG

(23)

The gradient time, 2tG, is the time when the last eluting component has just finished to completely leave the seconddimension column. By rearranging eq 15, we find that the gradient slope of the separation made in the second dimension, β, can be calculated as

(20)

(34) Poppe, H.; Paanakker, J.; Bronckhorst, M. J. Chromatogr. 1981, 204, 77– 84. (35) Gritti, F.; Guiochon, G. J. Chromatogr., A 2008, 1212, 35–40. (36) Snyder, L.; Saunders, D. J. Chromatogr. Sci. 1969, 7, 195. (37) Snyder, L. R.; Dolan, J. W.; Gant, J. R. J. Chromatogr., A 1979, 165, 3. (38) Cole, L.; Dorsey, J. Anal. Chem. 1990, 62, 16–21.

(21)

This statement is the same as eq 5, considering that the number of collected fractions is equal to the product of the peak capacity, 1 nc, and the fraction collection ratio, r (f ) 1ncr). Equation 21 shows that the higher the number of fractions collected from the first-dimension eluent, the shorter the cycle time in the second dimension. Since the minimal equilibration time is twice the hold-up time of the second-dimension column, it is obvious that 2tcyc must be higher than 22t0, so the effective gradient time of the second dimension is

β)

Separation Power of the Online Scheme. Equation 11 shows that the separation power of a 2D-LC system can be increased either by increasing the analysis time of the firstdimension separation, by minimizing the product of 2w and (r2 + 3.424)1/2, and/or by minimizing the equilibration time of the column used in the second dimension. Since the consequences of an increase of 1tG is straightforward, it will not be discussed in detail. Cole and Dorsey have shown that the equilibration of an RPLC column after the end of a gradient run needs two column volumes of the initial eluent.38 Recently, Schellinger

tG f

u0 u0 ρ fu0 ρ u0 ) + 2 + 1 LSlkl[0] S ( t u - L) LSlkl[0] S ( t u - 3Lf ) l G 0 l G 0 (24)

where

[

ρ ) W-1

(

3Lf - 1tGu0 3Lf - 1tGu0 exp Lfkl[0] Lfkl[0]

)]

(25)

where W-1(x) is the Lambert W-function41 that gives the principal solution for ϑ in x ) ϑeϑ for the cases in which ϑ e (39) Schellinger, A. P.; Stoll, D. R.; Carr, P. W. J. Chromatogr., A 2008, 1192, 41–53. (40) Schellinger, A. P.; Stoll, D. R.; Carr, P. W. J. Chromatogr., A 2008, 1192, 54–61. (41) Weisstein, E. W. Lambert W-function. MathWorldsA Wolfram Web Resource. http://mathworld.wolfram.com/LambertW-Function.html. (42) Gilar, M.; Daly, A. E.; Kele, M.; Neue, U. D.; Gebler, J. C. J. Chromatogr., A 2004, 1061, 183.

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-1. In eq 24 the parameters Sl and kl[0] are S and k[0] of the last eluting compound; L and u0 refer to the column length and eluent velocity of the second dimension, respectively. By substituting eq 24 into eq 20, the peak width of the sample compound can be derived from

2

w)

4 u0z

LH z(z +33) + 3

(26)

where

z)

Lf(kl[0]ρ - 3) + 1tGu0 kl[0](3Lf - 1tGu0)

(27)

Accordingly, the achievable 2D peak capacity in the case of LSS behavior can be calculated by combining eqs 11, 23, and 26: 1

nczu0(1tG- 2t02f )

nc,2D =

 (

4

HL

1 2 z + z + 1 (f 2 + 3.4241nc2) 3

(28)

)

It should be noted that during the derivation of eq 28, the slopes, S, of the plots of the logarithms of the retention factors of all the compounds studied versus the concentration fraction of the strong eluent were assumed to be equal to Sl. This way of estimating peak capacities in gradient elution is not original (e.g., see refs 16 and 42). METHOD OF CALCULATIONS The derivation of the equations and all other calculations were carried out by using Mathematica 6.0 (Wolfram Research) software. Figures 1-7 were calculated on the basis of eq 28 by changing one or two parameters at a time and keeping the other ones constant. During the calculations, and unless differently indicated in the figure captions, the value of the different parameters were the following (see Table 1): 1tG ) 120 min (1nc ) 165), f ) 165, φ0 ) 0.05, L ) 5 cm, u0 ) 60 cm/min, dp ) 0.0005 cm (5 µm), Dm ) 10-4 cm2/min, Sl ) -43, and k0,l ) 100 000 (note that all but the first two parameters are those of the second dimension). In addition, there are two restraints in

Table 1. Value of Parameters Used during the Calculationsa parameter

value

1

unit

120 165 5 60 5 × 10-4 10-4 -43 105 5 × 10-2

tG f L u0 dp Dm Sl k0,l φ0

min cm cm/min cm cm2/min

a All but the first two parameters are referring to the second dimension.

the calculation algorithm. Since the organic content of the eluent cannot be higher than 1 (100%), the algorithm gave results only for the cases in which the product of 2tG and β is smaller than or equal to 1. Besides this, the number of collected fractions per peak, r, was limited to values between 0.25 and 20. Although many parameters can affect the achievable peak capacity in the first dimension, all their effects were not investigated deeply in this work. In the case of Figure 1, the peak capacity of the first dimension was an independent variable, whereas in the case of all the other figures, it was calculated according to the following equation:

1

nc )

1751tG 7.8 min + 1tG

(29)

Equation 29 is the simplified form of the equation derived by Neue et al.43 for the calculation of peak capacities in reversed-phase gradient elution. The original equation considers the efficiency and dead time of the column, the gradient slope, and the slope of the logarithm of the retention factor as a function of the fraction of the stronger eluent in the mobile phase. Since we did not consider the first dimension in details, in eq 29, these parameters were lumped together into two constants. In a previous paper,31 it was shown that eq 29 fitted well to the measured peak capacity data and was used in the optimization of off-line 2D systems31 and in the comparison of different 2D-LC schemes.26 In eq 29 1tG has the units of minute. The height equivalent to a theoretical plate, H, was estimated using the following van Deemter equation:

H ) 1.5dp + Dm ⁄ u0 +

0.3dp2u0 Dm

(30)

where dp is the particle size of the stationary phase and Dm is the diffusion coefficient of the sample compound. The coefficients of 30 were chosen as those of a well-packed column. Similar values can be found in the chromatographic practice (see, e.g., ref 42). Figure 1. Achievable 2D peak capacity as a function of the peak capacity of the first dimension. The number of collected fractions are 50 (dot-dashed), 100 (dashed), 150 (dotted), and 200 (solid line). For all the other parameters see Table 1. 3884

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(43) Neue, U. D.; Carmody, J. L.; Cheng, Y.-F.; Lu, Z.; Phoebe, C. H.; Wheat, T. E. Design of Rapid Gradient Methods for the Analysis of Combinatorial Chemistry Libraries and the Preparation of Pure Compounds. Advances in Chromatography, Vol. 41; Brown, P. R., Grushka, E., Eds.; Marcel Dekker Inc.: New York and Basel, Switzerland, 2001; pp 93-137.

The contribution of the extracolumn band broadening was not included in the calculations; the dwell volume of the instrument was also neglected. RESULTS AND DISCUSSION Effect of the Operating Parameters on the Achievable 2D peak Capacity. The peak capacity of an online 2D-LC system can be calculated using eq 28. Several independent parameters affect the value of nc,2D: L, u0, 1nc, f, 1tG, φ0, dp, Sl, and k0,l. Aside from the last two ones, all the parameters can be adjusted by the analyst. In the following, the values of Sl, and k0,l are set to be -43 and 1 × 105, respectively. Figure 1 illustrates the effect of the first-dimension peak capacity, 1nc, on the achievable 2D peak capacity. The numbers of collected fractions are 50, 100, 150, and 200 while the other parameters, including the gradient time of the first dimension, were kept constant (see Table 1). Since 1tG was constant, eq 29 was not used for the calculation of the peak capacity in the first dimension and 1nc was used as the independent variable. Figure 1 shows that there is a limit for the first-dimension peak capacity beyond which the total achievable 2D peak capacity does not increase significantly. When 50 fractions are collected (lower curve), it would make no sense to increase the separation power of the first dimension beyond ca. 50. In contrast, when 100 fractions are collected (second lower curve), a significant improvement would be achieved by increasing 1nc from 50 to ca. 110. This conclusion confirms the recent results of Li et al.27 Figure 1 shows that the number of collected fractions has quite a significant effect on the achievable 2D peak capacity. Figure 2 illustrates the influence of the number of fractions of the first column eluent that are collected and analyzed online during 2D-LC runs made with 3, 5, 10, and 15 cm long seconddimension columns. This figure shows that the separation power of the 2D-LC system has an optimum depending on the number of collected fractions from the first-dimension eluent. The cycle time (1tG/f) of the second-dimension separation must decrease when more fractions from the first-dimension eluent are collected. Since the eluent linear velocity, u0, and the initial eluent strength, φ0, are kept constant, the only possibilities to reduce the analysis time to the value determined by the number of collected fraction (see eq 22) are either to increase the gradient slope, β, and/or to decrease the column length. In the first case, increasing the gradient slope increases also the peak compression and more importantly decreases the retention factor of the compound at the time of its elution, k[L], which results into thinner peaks (see eq 16). Furthermore, increasing the number of collected fractions decreases the degree of undersampling of the separation made on the first column, thus decreases the loss of first-dimension peak capacity. On the other hand, however, the shorter the cycle time in the second dimension, the lower its contribution to the peak capacity of the 2D-LC system since the fraction of the cycle time of the second dimension devoted to the separation, λ, decreases rapidly (see eq 23), which explains why 2nc goes through a maximum, then decreases at high numbers of collected fractions. Figure 2 illustrates also the importance of using shorter second-dimension columns and of collecting a relatively large number of fractions on the achievable online 2D-LC peak capacity.

Figure 2. Effect of the number of analyzed fractions on the separation power of an online 2D-LC system. The parameters are given in Table 1, except the column lengths of the second dimension that are 3 (dot-dashed), 5 (dashed), 10 (dotted), and 15 cm (solid line).

Figure 3. 2D-LC peak capacity of an online system as a function of the length of the second-dimension column at different eluent velocities (15, solid; 30, dotted; 60, dashed; 90 cm/min, dot-dashed line). For the remaining parameters, see Table 1.

Figure 3 illustrates the achievable 2D peak capacity as a function of the length of the second column, for different eluent velocities: 15, 30, 60, and 90 cm/min. It shows that the maximum achievable 2D-LC peak capacity offered by the system under given conditions can be reached either by using shorter column/lower eluent flow rate or longer column/higher eluent flow rate combinations in the second dimension if all other parameters are kept constant. When a longer second column is used and everything else is kept constant, the gradient slope must be increased in order to achieve the elution of all the components within the available retention window, 2tG. Similarly as in the previous case, an increase of the second gradient slope can result in thinner peaks, due to the smaller retention factor of the solute at the time of elution and the higher degree of peak compression even if the length of the second-dimension column is higher. However, an increase of the column length in the second dimension causes an increase of the time needed for its equilibration (a time assumed to be 2t02 in our case) and a decrease of the fraction of the cycle time of the second dimension that is devoted to the separation, λ. As a result of these processes, the achievable 2D-LC peak capacity goes through a maximum suggesting that the eluent flow rate and the length of the column in the second dimension have to be Analytical Chemistry, Vol. 81, No. 10, May 15, 2009

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Figure 4. Effect of the linear velocity of the eluent on the achievable peak capacity of an online 2D-LC system in case of a number of collected fractions equal to 0.51nc (solid), 1nc (dotted), 21nc (dashed), and 41nc (dot-dashed line). The values of the other parameters are in Table 1.

changed and optimized together in order to reach the maximum achievable 2D peak capacity. Figure 4 shows the influence of the linear velocity of the eluent in the second-dimension column on the separation power of the 2D-LC system for different numbers of collected fractions (r ) 0.5, 1.0, 2.0, 4.0; f ) 82, 165, 330, 660 fractions). As could be expected from previous conclusions, nc,2D goes through a maximum at a certain linear velocity of the eluent. Since the retention times of the sample components and the eluent velocity are inversely proportional, an increase of the latter causes a decrease of the width of the retention window. In order to keep 2tG constant, the gradient slope must be decreased. As a result, the peaks broaden resulting in a decrease of the peak capacity in the second dimension. However, at the same time, the increase of the eluent flow rate decreases the time needed for the column equilibration as well. As a result, λ increases, and that increases the achievable peak capacity in the second dimension. These two processes act in opposition, resulting in a maximum of the achievable 2D peak capacity. This means that using too high eluent flow rates in the second dimension is not advantageous. Figure 4 illustrates this important conclusion. Figure 5 illustrates the influence of the initial organic content of the eluent in the second-dimension separation, φ0, on the online 2D-LC peak capacity. The observed decrease of the separation power of the 2D system with increasing φ0 was expected since the higher the initial organic content of the initial eluent, the smaller the slope of the gradient needed in order to utilize the full width of the retention window, which gives broader elution peaks due to the reasons described above. Figure 6 illustrates the influence of the duration of the firstdimension gradient on the 2D-LC peak capacity that can be achieved with different numbers of collected fractions. In Figure 6a, the number of collected fractions depends on the peak capacity of the first dimension (with r ) 0.5, 1.0, 2.0, 4.0). In this case, the degree of undersampling remains constant. The figure shows that the peak capacity tends toward a limit and that, beyond a certain range, 1tG has only a small influence on the separation power of the 2D-LC online system. This observation could seem surprising since eq 21 showed that 1tG and nc,2D are directly proportional. However, the peak capacity of the first dimension does not increase significantly beyond ∼100 min in our case 3886

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Figure 5. Achievable peak capacity of an online system as a function of the initial organic content of the eluent in the second dimension. The other operating parameters can be found in Table 1.

(see value of the first-dimension peak capacity in Method of Calculations section, eq 29). Any further increase merely causes peak broadening (w ) tG/nc, see eq 3) without increasing either the separation power of the first dimension or the number of collected fractions. However, according to eq 22, the cycle time of the second dimension still increases beyond 1tG ∼ 100 min. In order to exploit all the width of the retention window (2tG), either the eluent velocity or the gradient slope must be decreased. In Figure 6a, the eluent velocity was kept constant and the gradient slope changed according to eq 24. This increases k[L], decreases the degree of peak compression, and gives broader peaks. The consequences of the increases of 1tG and 2w are opposed, and they tend to compensate. In Figure 6b, the number of collected fractions depends on the analysis time of the first dimension (f ) 0.5, 1, 2, and 3 times the numerical value of 1tG that has unit of minutes). In this case, the gradient times of the second-dimension separation were constant, with 2

tG )

1 min 2 - t02 y

(31)

where the value of y is 0.5, 1, 2, or 3. Figure 6b shows that the achievable online 2D-LC peak capacity increases steadily with increasing analysis time in the first dimension because, although the first-dimension peak capacity does not increase beyond a certain limit, the number of collected fractions increases, which reduces the effects of the undersampling of the first-dimension separation. Thus, the achievable 2D-LC peak capacity keeps increasing even when the peak capacity of the first dimension has become constant. A comparison of parts a and b of Figure 6 shows that, under the experimental conditions listed in Table 1, the largest 2D peak capacity can be achieved if the cycle time of the separation in the second dimension is constant, as it is in Figure 6b, and is ∼20-30 s (f = 2(1/min) or 3(1/min)1tG). This is in relatively good agreement with the conclusions of Li et al.27 Under their experimental conditions, the peak capacity production rate reached a maximum at a gradient time of about 15-20 s. Limitations of Online 2D-LC. Parts a and b of Figure 6 also show the potential limit of online 2D-LC chromatography. Never does the achievable peak capacity ever exceed significantly 2500 within a reasonable time. The same observation can be made in

min, and the number of collected fractions was 3(1/min)1tG. The remaining parameters remained the same as in the Table 1 section. These conditions are consistent with the use of a very high-pressure liquid chromatography (VHPLC) column in the second dimension. Even in this case, the peak capacities remained below 5000. Considering eq 28 and Figure 7 we may safely conclude that, in all probability, the online scheme is limited to peak capacities below 8-10 000 even if the method is welloptimized and long analysis times are accepted. This is due to the time restriction of the second-dimension analysis (see eq 22). In order to reach peak capacities significantly larger than 10 000, one of the other two schemes (stop-and-go or off-line) should be used. Being exempt from any restrictions regarding the gradient time of the second-dimension separation, these schemes are more flexible than the online scheme. Finally, Figures 1-6 show how the complete optimization of an online 2D-LC separation is difficult due to the relatively large number of parameters affecting its performance and the many different ways in which they should be considered simultaneously.

Figure 6. Effect of the analysis time of the first dimension, 1tG, on the achievable 2D-LC peak capacity. The number of collected fractions are (a) 0.51nc (solid), 1nc (dotted), 21nc (dashed), 41nc (dotdashed line) and (b) (0.5/min)1tG (solid), (1/min)1tG (dotted), (2/min)1tG (dashed), (3/min)1tG (dot-dashed line).

Figure 7. Achievable 2D-LC peak capacity as a function of the firstdimension analysis time (1tG). The particle diameter of the stationary phase of the column used in the second dimension is 1.7 µm, the initial organic content of the eluent, φ0, is 1%, the linear velocity of the eluent is 60 cm/min, and the number of collected fractions is (3/ min)1tG. The remaining parameters are the same as in Table 1.

CONCLUSIONS The calculation of the influence of different operating conditions (flow rate, column length, etc.) on the achievable separation power of 2D-LC online systems was made on the basis of the LSS model of gradient chromatography. We considered the effects of an undersampling of the first dimension and of the time restriction of the second-dimension gradient time. The slope of the seconddimension gradient was not an operating parameter; it was derived from the other operating parameters and from the solute characteristics, in order to utilize the whole width of separation space. The equations that were derived should help in understanding and optimizing online 2D-LC methods. We conclude that many parameters must be considered simultaneously during this optimization and that the separations in the first and the second dimensions must be optimized together. The optimization process must be carefully thought out because the method leaves little room for empirical adjustments. Finally, although it can be stated that the online scheme of 2D-LC will probably never permit the achievement of peak capacities in excess of 10 000, this limit still leaves considerable room to practitioners since it exceeds by about 1 order of magnitude the current level of performance of the online 2D-LC separations published. ACKNOWLEDGMENT This work was supported in part by Grant DE-FG05-88-ER13869 of the U.S. Department of Energy and by the cooperative agreement between the University of Tennessee and Oak Ridge National Laboratory. GLOSSARY Glossary of Symbols

Figures 1-5. However, as we noted earlier, the calculations leading to the figures do not correspond to the best possible conditions, given the present state of column technology. The plot in Figure 7 corresponds to the use of 1.7 µm instead of 5 µm particles in the second dimension. The initial organic content of the eluent, φ0, was 1%, the linear velocity of the eluent, 60 cm/

Dm dp f H k

molecular diffusion coefficient or molecular diffusivity in the mobile phase (cm2 min-1) particle size of the stationary phase (cm) number of fractions collected from the first dimension height equivalent to a theoretical plate (cm) retention factor of a compound Analytical Chemistry, Vol. 81, No. 10, May 15, 2009

3887

k0

kl, k0,l L i nc n*c 1 nc,1 ′ nc,2D r S

Sl

t0 2 tcyc i tG

3888

retention factor of a compound in the weak solvent of the eluent (pure water); intercept of the ln(k) vs φ function k and k0 of the last eluting compound column length (cm) peak capacity of the ith dimension of a multidimensional chromatographic system conditional peak capacity corrected peak capacity of the first dimension, considering the effects of undersampling two-dimensional peak capacity ratio of the number of collected fractions and the peak capacity of the first dimension sensitivity of the retention factor to the variation of the organic modifier concentration; slope of the plot of ln(k) vs φ sensitivity of the retention factor to the variation of the organic modifier concentration in case of the last eluting compound hold-up time of the column cycle time of the second dimension gradient time of the ith dimension (min)

Analytical Chemistry, Vol. 81, No. 10, May 15, 2009

tR,1 tR,n u0 w

retention time of the first eluted compound of a mixture (min) retention time of the last eluted compound of a mixture (min) linear velocity of the eluent (cm/min) width of the peak (cm)

Greek Letters

β γ λ σt φ φ0

slope of the gradient (1/min) first-dimension broadening factor due to undersampling the fraction of the cycle time that is devoted for separation standard deviation of a peak in time scale (min) volumetric fraction of the strong solvent in the eluent initial composition of the eluent during a gradient run

Received for review December 19, 2008. Accepted March 25, 2009. AC802694C