76 BASIC CONCEPTS AND THE CONTROL OF SEPARATION Gradient elution refers to a continuous change in the mobile phase during separation, such that the retention of later peaks is continually reduced; that is, the mobile phase becomes steadily stronger (%B increases) as the separation proceeds. An illustration of the power of gradient elution is shown in Figure 2.25c,where all peaks for the sample of Figure 2.25a, b are separated to baseline in a total run time of slightly more than 7 minutes, with approximately constant peak widths and comparable detection sensitivity for each peak (assuming a similar detector response for each solute). The advantages of gradient elution for this sample are obvious. Gradient elution also can be used to deal with several other separation problems, as discussed in Sections 9.1.1 and 13.4.1.4. For a further discussion of gradient elution, see Chapter 9. 2.7.3 Peak Capacity and Two-dimensional Separation So far we have used critical resolution R s as the measure of a given separation. This criterion is appropriate when the peaks of interest in a chromatogram can all be resolved to some extent, and our goal is some minimum resolution for all peaks. Some samples contain so many components, however, that it is impractical to achieve a significant resolution for all peaks of interest. Then we need a different measure of ‘‘separation power’’ for various combinations of experimental conditions. The peak capacity of a separation refers to the total number of peaks that can be fit into a chromatogram, when every peak is separated from adjacent peaks with R s = 1. An example is shown in Figure 2.26a, for a retention range of 0 < k ≤ 20 and N = 100. For isocratic separation, peak capacity is given by [73] PC = 1 + N 0.5 4 ln t R,z t 0 = 1 + 0.575N 0.5 log t R,z t 0 (2.30) where t R,z refers to the retention time of the last peak in the chromatogram. For typical separations, with k ≤ 20 for the last peak and values of N as large as 20,000, PC = 108. If we exclude peaks with k < 0.5 so that 0.5 ≤ k ≤ 20, the peak capacity drops to PC = 93; if we require R s = 2 the number of peaks that fit between k = 0.5 and 20 drops to 47. Peak capacity is of much greater importance for separations of complex samples—those containing a very large number of components. It is seldom possible to separate such samples with an acceptable resolution of all peaks, so peak capacity becomes a better measure of overall separation than values of R s . Separations of complex samples are usually carried out by gradient elution, for which the concept of peak capacity is more relevant (Section 9.3.9.1). Peak capacity is of special interest for so-called two-dimensional (2D-LC) separation (Section 9.3.10), where fractions from a first separation are further resolved in a second separation, as illustrated in the example of Figure 1.4b,c. There it is seen that a group of overlapping peaks from the first separation (fraction 7) is spread out over the entire chromatogram of the second separation (orthogonal separation). Under these circumstances the combined peak capacity for the two separations will be equal to the product of peak capacities for each separation. For the example above of an isocratic peak capacity of PC 2.7 RELATED TOPICS 77 ≈ 100, the 2D-LC peak capacity would be PC = 100 × 100 = 10,000. Thus 2D-LC separation provides a lot more room in the combined chromatograms for sample peaks, so it is a powerful technique for separating complex mixtures that contain hundreds or thousands of individual components. The peak capacity of a separation should not be confused with the number of compounds separated at R s = 1, since it is rarely possible to achieve a regular spacing of peaks as in Figure 2.26a [73]. Figure 2.26b illustrates the required peak capacity PC req for the separation (where R s ≥ 1 for all peaks) of a sample with n components. Prior to the optimization of selectivity as in Section 2.5.2, a random arrangement of peaks within the chromatogram can be assumed. As seen in Figure 2.26b, a sample containing 10 components (‘‘random’’ curve, n = 10) would require a peak capacity of about 80 to achieve R s ≥ 1 for every peak. However, if separation selectivity has been optimized, critical peak-pairs will be separated to a greater extent, and the required peak capacity would decrease to about 40 (‘‘optimized’’ curve of Fig. 2.26b). See [74] for further details. 2.7.4 Peak Tracking The interpretation of separations obtained during method development requires peak tracking or peak matching. For each compound X in the sample, peaks in 0246810 Time (min) 0 ≤ k ≤ 20 peak capacity = 8 (a) (b) 010203040 50 n 500 400 300 200 100 PC req random ideal spacing (PC req = PC) Required PC (PC req ) for separation of n sample components with R s = 1.0 “optimized” Figure 2.26 Peak capacity. (a) Example of peak capacity (PC) for a separation where PC = 8; N = 100, and R s = 1 for every peak; (b) peak capacity required for the separation of a sample that contains n components [74]; ‘‘ideal spacing’’ is from Equation 2.30. 78 BASIC CONCEPTS AND THE CONTROL OF SEPARATION the various method development chromatograms that correspond to X must be characterized or numbered (as in Fig. 2.20). Thus, if peak 1 in run 1 corresponds to compound A (whose chemical structure may or may not be known), it is necessary to know which peak in run 2 also corresponds to A. For many samples this may not be difficult. For example, in Figure 2.20b, d, the six peaks in each run can be matched on the basis of peak area and relative retention (which usually do not change drastically when separation conditions are varied). Peaks 3 and 4 change places in these two chromatograms, but the areas of these and other peaks are sufficiently different to allow unambiguous peak tracking between the two runs. Manual peak tracking can take advantage of peak area, peak shape, and the observation that retention order changes (when they occur) are usually minor (i.e., a peak for a given compound usually appears in the same region of the chromatogram). Peak tracking can be much more difficult in other cases, however, for example, when several peaks overlap as in the two separations of Figure 2.20a,c. While several workers have suggested ways to improve peak tracking with UV detection [75–79], no procedure has proved adequate for all samples. Method development is increasingly making use of mass spectrometer detection (LC-MS), which largely eliminates problems in peak tracking because of the ability of MS detection to (1) recognize each of two overlapped peaks and (2) assign a (usually unique) molecular mass to each peak in the chromatogram [75]. 2.7.5 Secondary Equilibria Chromatographic retention is based on a (primary) equilibrium between a solute molecule X in the mobile and stationary phases (as in Fig. 2.4 and Eq. 2.2): X (mobile phase) ⇔ X (stationary phase) (2.2) Solute molecules may undergo further (secondary) equilibria that involve the ioniza- tion of acids and bases, ion pairing, complex formation, or isomer interconversion. As a result it is possible for two forms of the solute to be in equilibrium during their migration through the column. A common example is the separation of a partially ionized carboxylic acid, which involves an equilibrium between the ionized and non-ionized forms: R−COOH ⇔ R−COO − + H + (2.31) The relative concentrations of each form of the molecule will be determined by compound acidity (its pK a value) and the pH of the mobile phase (Section 7.2), leading to some fraction F − of the molecules being in the ionized form and some fraction (1 − F − ) being in the neutral form. If the value of k for the ionized form is k − ,andifk 0 refers to k for the non-ionized acid, then a single peak will be observed for the two species, with its retention given by k = F − k − + (1 − F − )k 0 (2.32) As mobile-phase pH is varied, the ionization of an acidic solute and the value of F − will change, as will the value of k (Section 7.2). 2.7 RELATED TOPICS 79 For acid-base equilibria as in Equation (2.31) (for either acidic or basic solutes), it can be assumed that the ionization process will be quite fast, much faster than the time required for a solute molecule to move through the column. As a result each solute molecule will pass back and forth between the ionized and non-ionized states many times during its migration through the column, and its retention will be an average value as described by Equation (2.32). Peak width and shape are not adversely affected by secondary equilibria, despite frequent comments to the contrary. As noted by McCalley [80], ‘‘the popular assumption that a mixed-mode mechanism leads inevitably to (peak) tailing is shown to be unfounded.’’ On the other hand, peak tailing for both acids and bases is sometimes observed, primarily because of the properties of the column (Section 5.4.4.1) or inadequate buffering of the mobile phase (Section 7.2.1.1). When the rate of equilibration between two species is fast, only a single peak will be observed. This is the case for a partially ionized acid, where the two forms R–COOH and R–COO − rapidly equilibrate during their migration through the column. When the rate of equilibration between two species is slow, peak broadening, distortion, and/or the apperance of separate peaks can result. An example is the interconversion of cis and trans peptide isomers [81]. At higher temperatures, the interconversion is rapid, and a single, sharp peak is observed for the peptide where isomerization is possible. At lower temperatures, where the interconversion is much slower, two distinct peaks are observed. For intermediate temperatures, a single wide, distorted peak is seen. 2.7.6 Column Switching Column switching involves the use of two columns connected in a series via a switching valve (Section 3.6.4.1). A sample is injected into the first column, and one or more leaving fractions are transferred sequentially to the second column for further separation. Column switching can be used in each of the following applications: • sample preparation (Sections 3.6.4.1, 16.9) • two-dimensional liquid chromatography (2D-LC) (Sections 9.3.10, 13.4.5, 13.10.4) • increased sampling rate The use of column switching for sample preparation or 2D-LC usually involves the separation of one or more analytes from a complex sample where compounds of interest are completely overlapped in the first separation (with α ≈ 1.00). To achieve the separation of compounds with very similar retention, a change in selectivity for the second separation is usually employed—this is generally achieved by the use of both a different column and a different mobile phase. An example of such an application of column switching was illustrated in Figure 1.4b,c. Another application of column switching for routine analysis can provide an increase in sampling rate, after conditions have been optimized for the fastest possible separation. A hypothetical example is illustrated in Figure 2.27a for the routine assay of peak c or d (or both peaks). The overall run time is 52 minutes, meaning an assay rate of only slightly more than one sample an hour. Sample pretreatment in this example might be able to remove late-eluting compounds 80 BASIC CONCEPTS AND THE CONTROL OF SEPARATION e and f, in which case the separation time could be reduced to about 25 minutes (a sampling rate of 2.4/hr). If a large number of samples are to be analyzed on a given day, however, it is possible to significantly increase sampling rate for assays such as this by means of a column-switching technique called boxcar chromatography [82]. Because the two peaks c and d in Figure 2.27a are well separated from other peaks in the chromatogram, these two peaks can be segregated from other sample components with a shorter column and a faster flow rate—as illustrated in Figure 2.27b for a total run time of < 2 minutes (and a potential assay rate of > 30 samples/hr). If samples are injected every 2 minutes, a fraction that contains peaks c and d can be diverted via a switching valve to the column of Figure 2.27a.For this way of column switching (Fig. 2.27c), a separate pump would deliver the same mobile phase to the second column at 0.5 mL/min, so as to achieve an equivalent separation of peaks c and d as in Figure 2.27a (i.e., with adequate resolution). Because bands c and d occupy only a small fraction of the second column during their migration through the column, it is possible to simultaneously separate several samples at the same time, as illustrated in Figure 2.27d. Here 12 fractions from the first separation can be separated simultaneously, as illustrated by an inside view of column 2 for fractions 1, 6, 10, and 12 at the beginning of this column-switching separation (other peaks not shown). The final separation by the second column is shown in Figure 2.27e; after a delay of about 25 minutes, separated peaks c and d begin to leave the second column at a rate of 30 samples per hour. Boxcar chromatography relies on the simultaneous separation of different samples within column 2, which requires that two successive samples not overlap during their movement through column 2. To avoid such sample overlap, the rate of sample injections into column 1 must be coordinated with the time required for the peaks of interest (e.g., c and d) in a given sample to leave column 2. The use of boxcar chromatography has rarely been reported in the literature [83], and today the availability of mass spectrometric detection might seem to further reduce the potential advantage of this technique for most samples. Where extremely large values of N are required for resolution—as in the preparative separation of compounds differing only in isotopic substitution—boxcar chromatography offers the possibility of achieving a much higher throughput rate than by any other technique. 2.7.7 Retention Predictions Based on Solute Structure Obviously predictions of retention times from experimental conditions and the molecular structures of sample compounds would be useful for selecting the best conditions for a separation. Unfortunately, sufficiently accurate predictions of this kind were generally not possible at the time this book was published. Where predictions of retention may be useful, however, is for confirmation of the identity of an unknown peak in the chromatogram. The retention k of a compound is determined by its molecular structure and separation conditions. For a given set of conditions, log k can be approximated by log k = A + R M(i) (2.33) 2.7 RELATED TOPICS 81 (a) (b) (c) (d ) (e) 02040 Time (min) 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Time (min) a b c d e f a b c + d e f column-2 400 × 4.6-mm 3-μm 0.5 mL/min column-1 50 × 4.6-mm 3-μm 2.0 m/min waste detector column-1 switching valve column-2 sample valve S S D Column switching 1 5 10 15 20 20 30 40 50 60 (min)70 1 6 12 10 Migration through column-2 inlet outlet Sequential analysis pump-2 pump-1 Figure 2.27 Illustration of boxcar chromatography for a hypothetical sample. (a) Optimized separation of the sample for acceptable resolution (column 2); (b) fast separation of the sample with a shorter column and faster flow rate (column 1); (c) equipment setup for separations of the present sample by boxcar chromatography; (d) migration of selected sample fractions (1, 6, 10, 12) within column 2, viewed just prior to elution of the fraction for sample 1; (e)early part of the chromatogram from column 2. 82 BASIC CONCEPTS AND THE CONTROL OF SEPARATION Here A refers to log k for a parent molecule (e.g., benzene) and R M(i) is the increase in log k that results from the substitution of group i into the molecule (e.g., insertion of a nitro group i into benzene to form nitrobenzene). Smith [84] has reported values of R M(i) for a number of common substituent groups and different RPC mobile-phase conditions, allowing estimates of retention as a function of solute molecular composition (for a very limited number of possible solutes and separation conditions). For the case of a homologous series, Equation (2.33) assumes the form log k = A + nα CH2 (2.34) Here n is the number of methylene groups (–CH 2 –) within the molecule, and α CH2 is the increase in log k due to the addition of one –CH 2 – group to the molecule. As a consequence of Equation (2.34), plots of log k for a homologous series versus n are generally observed to be linear (but note the exception of Section 6.2.2 and Fig. 6.5). Relationships similar to Equation (2.34) apply for other compound series based on the presence of some number n equivalent groups in the molecule (e.g., oligomers of polyvinylalcohol [–CH 2 CH 2 O– repeating groups], polystyrene [–CH 2 (C 6 H 5 )CH 2 – repeating groups], etc.) Equations (2.33) and (2.34) are each referred to as the Martin equation, in recognition of A. J. P. Martin’s first use of these relationships. In the case of gradient elution, Equation (2.33) becomes t R ≈ A + t R(i) (2.35) where A is the retention time of the parent compound, and t R(i)) is a constant for a given group i that is substituted into the parent compound. Equation (2.35) has been used for the prediction of gradient retention times for a wide variety of solute molecules; for example, triacylglycerols [85], peptides [86], and polysacchrides [87]. In each case these predictions apply only for a specific set of separation conditions. While Equation (2.33) or (2.35) can prove occasionally useful in estimating where a compound peak should be found within a chromatogram, other factors than the number and kind of substituent groups can have a significant effect on retention, especially for the complex polar molecules that are commonly present in samples for HPLC separation. Since the 1950s a large number of workers have investigated the relationship of sample retention to structure, with the hope of eventually being able to predict retention and separation in the absence of experiments (the ‘‘Holy Grail’’ of chromatography). In general, it has not proved possible to predict chromatographic retention in HPLC with an accuracy that is anywhere near sufficient to support method development (see [88] for a failed example). An interesting exception to these past failures of predictions of retention as a function of solute molecular structure was reported in 2007 [89], where mass spectrometric detection was combined with retention predictions to permit the identification of individual peptides in protein-digest mixtures. 2.7.7.1 Solvation-Parameter Model A well-documented and widely applied solvent-parameter approach has been used to rationalize RPC retention as a function of the sample, column, and separation REFERENCES 83 conditions (see [90, 91] and especially [14]). A non-ionized sample is assumed, in which case retention can be approximated as a result of hydrophobic and hydrogen-bonding interactions among sample, mobile phase, and column. The solvent-parameter model takes the form log k = C 1 + νV x + rR 2 + sπ H 2 + aα H 2 + bβ 2 (2.36) (i)(ii)(iii)(iv)(v) A solute retention factor k is related to (1) a constant C 1 that is a function of column and conditions, (2) solute-dependent quantities ν, r, s, a,andb,and(3) solute-independent quantities V x , R 2 , π H 2 , α H 2 ,andβ 2 .Termsi to iii of Equation (2.36) together account for hydrophobic interactions, while terms iv and v are the result of hydrogen bonding between solute and either the column or the mobile phase. Values of ν, r, s, a,andb for a large number of different solutes have been tabulated, and values of C i , V x , R 2 , π H 2 , α H 2 ,andβ 2 can be determined for a column and given conditions by the use of appropriate tests solutes. Equation (2.36) can provide insight into the factors that determine RPC separation, but the errors in predictions of values of k (about ±20%) are too large to be useful for method development. Equation (2.36) is further limited by the fact that it cannot be applied to ionized solutes, and it neglects a number of additional interactions that can affect retention (see the related discussion of Section 5.4). REFERENCES 1. M. M. Fallas, U. D. Neue, M. R. Hadley, and D. V. McCalley, J. Chromatogr. A, 1209 (2008) 195. 2. C. A. Rimmer, C. R. Simmons, and J. G. Dorsey, J. Chromatogr. A, 965 (2002) 219. 3. J. M. Berm ´ udez-Salda ˜ na, L. Escuder-Gilabert, R. M. Villanueva-Cama ˜ nas, M. J. Medina- Hern ´ andez, and S. Sagrado, J. Chromatogr. A, 1094 (2005) 24. 4. M. Shibukama, Y. Takazawa, and K. Saitoh, Anal. Chem., 79 (2007) 6279. 5. M. A. Quarry, R. L. Grob, and L. R. Snyder, J. Chromatogr., 285 (1984) 19. 6. L. R. Snyder, unreported data. For a single mobile phase (50% acetonitrile/buffer) and several hundered different columns. 7. T. Braumann, G. Weber, and L. H. Grimme, J. Chromatogr., 261 (1983) 329. 8. L. C. Tan, P. W. Carr, and M. H. Abraham, J. Chromatogr. 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