426 GRADIENT ELUTION (a) (b) (c) 0 10203040 Time (min) 100% B 80% 60% 40% 20% 0% 0/23/42% B in 0/32/38 min R s = 2.1 0 1020304050 Time (min) 100% B 80% 60% 40% 20% 0% 0/40/100/100% B in 0/50/51/52 min R s = 2.1 0 10203040 Time (min) 100% B 80% 60% 40% 20% 0% 0-40% B in 50 min R s = 2.1 Figure 9.11 Gradient separations of a peptide digest of recombinant human growth hormone. Conditions: 150 × 4.6-mm C 18 column (5-μm); 45 ◦ C; 2.0 mL/min. (a )0–40%Bin50min; (b)sameasin(a), except a steep gradient segment is added in order to remove strongly reten- tive ‘‘junk’’ from the column; (c)sameasin(a), except a second gradient segment is added in order to accelerate elution of the last two peaks in the chromatogram. Gradient indicated by (- - -). Chromatograms recreated from data of [13]. by a short isocratic hold. Thus the final gradient in Figure 9.11b is 0/40/100/100%B in 0/50/51/52 min. Shortening run time is illustrated in Figure 9.11c for the sample of Figure 9.11a, without a final column-cleaning gradient step (which could be added, if needed). Because the last five peaks in the chromatogram are resolved with R s 2, it is possible to increase gradient steepness for these peaks, so as to reduce their retention times while maintaining R s ≥ 2 for all peaks. This way run time is shortened from 50 minutes in Figure 9.11a to 40 minutes in Figure 9.11c. Increasing resolution by adjusting selectivity for different parts of the chro- matogram can sometimes be achieved with a segmented gradient; gradient steepness (and values of k ∗ ) for different segments are optimized for different critical 9.2 EXPERIMENTAL CONDITIONS AND THEIR EFFECTS ON SEPARATION 427 024681012 Time (min) (a) (b) 40-100% B in 12.5 min R s = 1.4 * * 024681012 Time (min) 40/50/100% B in 0/7.5/11.5 min R s = 1.7 * * 100% B 80% 60% 40% 20% 0% 100% B 80% 60% 40% 20% 0% 3 4 14 15 Figure 9.12 Separation of a mixture of 16 polycyclic aromatic hydrocarbons, adapted from Figure 6.4 of [6]. Conditions: 150 × 4.6-mm (5-μm) C 18 column; 35 ◦ C; 2.0 mL/min. (a)Sep- aration with an optimized linear gradient; (b) separation with an optimized two-segment gradient. Gradient indicated by (- - -). See [6] for further details. peak-pairs. An example is shown in Figure 9.12 for the separation of a mix- ture of polycyclic aromatic hydrocarbons; peak-pairs 3–4 and 14–15 (marked by *) are critical. Whereas peak-pair 3–4 is better separated with a flatter gradient (larger values of k ∗ ), the separation of peaks 14 and 15 improves for a steeper gradient (smaller k ∗ ). In Figure 9.12a, the slope of a linear gradient has been selected for max- imum critical resolution of the sample. Maximum critical resolution corresponds to equal resolution for each of these two peak-pairs because a change in gradient steepness will increase resolution for one peak-pair while decreasing resolution for the other. However, the resolution of each peak-pair can be improved by the segmented gradient of Figure 9.12b, which combines a flatter gradient for peaks 3 and 4 with a steeper gradient for peaks 14 and 15. The small increase in R s shown in Figure 9.12b (+0.3R s -units vs. Fig. 9.12a) is typical of the effect of segmented gradients. It is rare to achieve an increase in resolution of more than ≈ 0.5 units with segmented gradients. In the absence of computer simulation (Section 10.2.3.4), the time required to develop such separations may not be worthwhile. Segmented gradients are not often used for improving resolution as in Figure 9.12 because their ability to enhance resolution without increasing run time is usually limited [14]. An increase in critical resolution as a result of the use of segmented gradients requires at least two critical pair-pairs that elute, respectively, early and late in the chromatogram (as in Fig. 9.12). Otherwise, the partial migra- tion of the second peak-pair under the influence of the initial gradient segment will result in little or no overall advantage from the use of the second gradient segment. However, this limitation of segmented gradients for an increase in sample resolution becomes less important for high-molecular-weight samples such as proteins [15, 16], 428 GRADIENT ELUTION since there is less migration of later peaks during an earlier gradient segment, and therefore less effect of the earlier segment on the resolution of later peaks. The use of segmented gradients for the purpose of increasing critical resolution is therefore somewhat more practical for the separation of mixtures of large biomolecules. How- ever, there are other—generally more useful—means for optimizing resolution by changing selectivity and relative retention (Section 9.3.3). Also separations that use segmented gradients to improve resolution are likely to be less reproducible when transferred to another piece of equipment. A more detailed examination of the use of segmented gradients in this way is offered in [17, 18]. Computer programs have also been reported for the auto- mated development of optimized segmented gradients [14, 19, 20]. Stepwise elution involving step gradients can be regarded as a simple (if less generally effective) kind of segmented gradient; a theory of such separations has been described [21]. 9.2.3 ‘‘Irregular Samples’’ The following section discusses gradient separations where relative retention changes for an ‘‘irregular’’ sample as a result of a change in some condition that affects k* (gradient time, flow rate, etc.). These examples are intended to supplement preceding examples in Figures 9.4 and 9.6 to 9.9 for ‘‘regular’’ samples, by illustrating changes in relative retention for ‘‘irregular’’ samples as a function of changes in conditions that affect k*. The reader may choose to skip to Section 9.3, and return to this section at a later time—or as needed. However, this treatment can add to the reader’s intuitive understanding of gradient elution, as well as find occasional practical application. Changes in k ∗ can result from a change in any of the experimental conditions included in Equation (9.5) (t G , F, V m or column length L, Δφ), as well as from a change in initial-%B, the introduction of a gradient delay, or a change in dwell volume. An increase in k ∗ will result in an average increase in retention time, resolution, and peak width for all samples, as illustrated by Figures 9.4 and 9.5 for changes in gradient time. In the case of ‘‘irregular’’ samples (Fig. 9.5) a change in k ∗ will also cause relative retention to change, which can result in a change in resolution for certain peaks. Any change in k ∗ for a given ‘‘irregular’’ sample will result in similar changes in relative retention and resolution, regardless of how k ∗ is caused to vary. This is illustrated in the remainder of this section for various changes in gradient or column conditions, using the examples of Figure 9.13 for selected peak-pairs (2–3 and 8–9) from the irregular sample of Figure 9.5. Because many real samples fall in the ‘‘irregular’’ sample category, the following discussion is expected to reflect the kind of changes most users will observe with changes in gradient elution conditions. A starting separation of peak-pairs 2–3 and 8–9 of the ‘‘irregular’’ sample of Figure 9.5 is shown in Figure 9.13a. These two peak-pairs have been chosen because their resolution responds in opposite fashion to a change in k ∗ (as a consequence of difference in S-values for these four solutes: S 3 > S 2 ; S 8 < S 9 ; see the similar examples of Fig. 6.7c). Consider first an increase in gradient time from5to20 minutes (Fig. 9.13b), corresponding to an increase in average k ∗ from 5 to 20. As a result the retention of peak 2 relative to that of peak 3 increases, and the resolution of peak-pair 2–3 therefore increases. At the same time the relative retention of peak 9relativetopeak8decreases when gradient time is increased, and the resolution 9.2 EXPERIMENTAL CONDITIONS AND THEIR EFFECTS ON SEPARATION 429 5-100% B in 5 min k* = 5 L = 50 mm F = 2.0 mL /min 5-100 % B in 20 min k* = 20 L = 50 mm F = 2.0 mL /min 5-100% B in 5 min k* = 2.5 L = 100 mm F = 2.0 mL/min 5-100 % B in 5 min k* = 20 L = 50 mm F = 8.0 mL /min 15-100% B in 4.5 min k* < 5 L = 50 mm F = 2.0 mL /min 5/5/100% B in k* > 5 0/5/10 min L = 50 mm F = 2.0 mL /min 3 2 8 9 2 + 3 3 (a) (b) (c) (d ) (e) (f ) Figure 9.13 Changes in peak spacing with changes in gradient conditions. Sample consists of peaks 2, 3, 8, and 9 of the irregular sample of Figure 9.5. Conditions: 28 ◦ C. The arrows in (b) indicate the relative movement of peaks 2 and 9 as a result of an increase in gradient time and k ∗ . Gradient indicated by (- - -). of this peak-pair decreases. Similar changes in relative retention and resolution for these two peak-pairs can be expected for changes in any other condition, which results in an increase in k ∗ . Opposite changes in relative retention will occur when k ∗ is decreased. In Figure 9.13c, column length L is increased from 50 to 100 mm, while other conditions remain the same as in Figure 9.13a; the value of k ∗ decreases by a factor 430 GRADIENT ELUTION of 2 to k ∗ = 2.5 (Eq. 9.5c below). As expected from this decrease in k ∗ (relative to the separation of Fig. 9.13a), the changes in relative retention seen in Figure 9.13b compared to Figure 9.13a are reversed in Figure 9.13c: peak 2 now moves toward peak 3 with a decrease in resolution, while peak 9 has moved away from peak 8, with an increase in resolution. The effect of an increase in flow rate (from 2.0 to 8.0 mL/min) is seen in Figure 9.13d. Because k ∗ has increased from 5 to 20 (Eq. 9.3), a similar change in relative retention is expected as for an increase in gradient time (Fig. 9.13b): again, peak 2 moves away from peak 3 with an increase in resolution, and peak 9 moves toward peak 8, with a decrease in resolution. When %B at the start of the gradient (φ o ) is increased while holding Δφ/t G constant (Fig. 9.13e), values of k ∗ calculated from Equation (9.5) remain the same. However, actual values of k ∗ for early-eluting peaks are decreased (Eq. 9.11), despite holding (t G /Δφ) constant. Thus Equation (9.5) no longer applies for early peaks in the chromatogram, resulting in the movement of peak 2 toward peak 3. The value of k ∗ for later peaks 8 and 9 is somewhat less affected by the increase in initial %B, so the relative retention and resolution of peaks 8 and 9 are less affected (compared to the separation of Fig. 9.13a). Finally, in Figure 9.13f ,agradient delay (or increase in dwell time t D )of 5 minutes is introduced into the separation of Figure 9.13a (other conditions the same). As in the preceding example (Fig. 9.13e), the value of k ∗ calculated from Equation (9.5) is unchanged (k ∗ = 5), but the effect of a gradient delay is to reduce the effect of the gradient on initial peaks in the chromatogram. This in turn means effectively higher values of k ∗ for these early peaks (Eq. 9.12). As a result a similar change in relative retention and resolution results as in Figure 9.13b, for an increase in gradient time—but to a somewhat lesser extent for later peaks 8 and 9 (whose values of k ∗ are less affected by either a gradient delay or a change in initial %B). A change in dwell-volume and dwell-time (due to a change in gradient system) would give the same result as this change in gradient delay in Figure 9.13f . Resolution is also affected by changes in k ∗ and N ∗ (see Eq. 9.15c below), apart from changes in relative retention. The former contributions to resolution may occasionally confuse the dependence of resolution on relative retention. 9.2.4 Quantitative Relationships The LSS model allows the derivation of a number of exact relationships for retention and peak width; these equations form the basis of computer simulation for gradient elution (Section 10.2). Apart from computer simulation and the dependence of k ∗ on experimental conditions (Eq. 9.4), following Equations (9.5a) to (9.15) have somewhat limited practical application. For this reason the reader may wish to skip to Equation (9.16) at the end of this section, and return to the remainder of this section as needed. For the derivation of the various equations contained in this section, and for details on their application, see Chapter 9 of [2]. Linear RPC gradients are assumed for each of the following equations. Values of k ∗ can be described by a relationship that corresponds to Equation (9.1) for isocratic elution: log k ∗ = log k w − Sφ ∗ (9.5a) 9.2 EXPERIMENTAL CONDITIONS AND THEIR EFFECTS ON SEPARATION 431 where φ* refers to the value of φ for mobile phase in contact with the solute band when it has reached the column midpoint. Values of k w and S are the same for either isocratic or gradient elution. V m can also be estimated (Eq. 2.7a, which assumes a total column porosity ε T = 0.65) from column length L and internal diameter d c : V m ≈ 5 × 10 −4 Ld c 2 (units of L and d c in mm) (9.5b) For the usual column diameter of 4.6 mm, it is convenient to approximate V m by 0.01 times the column length in mm; for example, V m ≈ 1.5 for a 150 × 4.6-mm column. Combining Equations (9.5) and (9.5b), we have k ∗ = 1740t G F Ld c 2 ΔφS (9.5c) or for S ≈ 4 for small solute molecules, k ∗ ≈ 450t G F Ld c 2 Δφ (for solutes < 500 Da, S ≈ 4) (9.5c) Thus k ∗ will increase for larger values of t G and F or smaller values of column length L, column diameter d c or gradient range Δφ. From Equations (9.4) and (9.5), we see also that k ∗ is related to the gradient-steepness parameter b: k ∗ = 0.87 b (9.6) That is, the value of k ∗ decreases for steeper gradients with larger values of b. 9.2.4.1 Retention Time The calculation of retention time t R of a solute in gradient elution takes different forms, depending on (1) whether a significant dwell volume is assumed (V D > 0) and (2) whether the initial value of k at the start of the gradient (k 0 ) is small. The value of k 0 is given by log k 0 = log k w − Sφ 0 (9.7) If k 0 is large, and if V D = 0, t R = t 0 b log(2.3k 0 b + 1) + t 0 (9.8) ≈ t 0 b log(2.3k 0 ) + t 0 (9.8a) If k 0 is large, and if V D > 0, t R = t 0 b log(2.3k 0 b + 1) + t 0 + t D (9.9) 432 GRADIENT ELUTION ≈ t 0 b log(2.3k 0 b) + t 0 + t D (9.9a) Here t D = V D /F is the column dwell-time. If k 0 is small, and if V D > 0, t R = t 0 b log{2.3k 0 b[1 − t D t 0 k 0 ] + 1}+t 0 + t D (9.10) Equation (9.10) is valid, regardless of the values of k 0 or V D . Equations (9.8) to (9.10) assume that the peak does not elute before or after the gradient. For equations that cover the latter cases, see [22]. Equation (9.9) is often a reasonable approximation for gradient separations and is frequently cited in the literature (although different symbols are sometimes used; see pp. xxv–xxvi of [2]). Values of the gradient retention factor k ∗ can also vary with values of V D and k 0 . For small values of k 0, and V D = 0, k ∗ = 1 1.15b + (1/k 0 ) (9.11) For small k 0 and V D > 0 (or any values of k 0 and V D ), k ∗ = k 0 2.3b[(k 0 /2) − (V D /V m )] + 1 = k 0 2.3b[(k 0 /2) − (t D /t 0 )] + 1 (9.12) Thus a small value of k 0 leads to smaller values of k ∗ , compared to values from Equation (9.4) or (9.6). Likewise, for larger values of t D (or a gradient-delay time t delay ), the value of k ∗ will be larger, compared to values from Equation (9.4) or (9.6). 9.2.4.2 Measurement of Values of S and k w Values of S and k w can be obtained from isocratic values of k as a function of φ from Equation (9.7), or from two gradient runs where only gradient time is varied. When values of k 0 are large for gradient elution, Equation (9.9a) accurately describes linear-gradient retention in RP-LC. For this case it is possible to calculate values of log k w and S for each compound in any sample, based on two experimental gradient runs where only gradient time is varied. Thus suppose gradient times for the two experiments of t G1 and t G2 (t G1 < t G2 ), with a ratio β = t G2 /t G1 . Given values of t R for a given solute in run-1 (t R1 ) and run-2 (t R2 ), a value of b 1 can be calculated as b 1 = t 0 log β t R1 − (t R2 /β) − (t 0 + t D )(β − 1)/β (9.13) 9.2 EXPERIMENTAL CONDITIONS AND THEIR EFFECTS ON SEPARATION 433 Similarly log k 0 = b 1 (t R1 − t 0 − t D ) t 0 − log(2.3b 1 ) (9.13a) Insertion of b 1 into Equation (9.4a) allows the calculation of a value of S, while log k w is then calculable as log k 0 + Sφ 0 . 9.2.4.3 Peak Width Peak width W in gradient elution is defined in the same way as for isocratic separation (Section 2.3) and is given by any of the following equivalent equations: W = (4N ∗−0.5 )Gt 0 1 + 1 2.3b (9.14) ≡ (4N ∗−0.5 )Gt 0 1 + k ∗ 2 (9.14a) ≡ (4N ∗−0.5 )Gt 0 (1 + k e ) (9.14b) That is, W can be related to gradient steepness b, a value of k ∗ , or the value of k when the peak leaves the column (k e ); as noted in Section 9.1.3.2, k e = k ∗ /2. The peak compression factor G describes the narrowing of a peak in gradient elution, due to the faster migration of the band tail (in a higher%B mobile phase) compared to the band front (in a weaker%B mobile phase) [23, 24]. G can be related to gradient steepness b [25]. First define the quantity p as p = 2.3k 0 b k 0 + 1 ≈ 2.3b (9.15) for large k 0 . G is then given in terms of p as G = (1 + p + [p 2 /3]) (1 + p) 2 0.5 (9.15a) Values of G vary with gradient steepness b as follows: for 0.05 < b < 2 (correspond- ing to 17 > k ∗ > 0.4), 1 > G > 0.6; that is, large b or small k ∗ corresponds to smaller G. Thus the value of G varies from 0.6 for very steep gradients to 1.0 for very flat gradients. A more convenient equation for G can be derived from the similarity of equations for isocratic and gradient elution (Eqs. 2.24 and 9.15 below): G ≈ 1 + k ∗ 1 + 2k ∗ (9.15b) For values of k ∗ ≥ 1, Equation (9.15b) is accurate within a few percent. The theory of peak compression in gradient elution was well developed by 1981, but subsequent experimental studies failed to confirm this phenomenon until 434 GRADIENT ELUTION 2006 [24]. It is now believed that this past uncertainty concerning peak compression was mainly the result of a moderate failure of Equation (9.1), combined with the use of Equation (9.14) instead of Equation (9.14b); the latter relationship is more accurate when plots of log k against φ are slightly curved (i.e., failure of Eq. 9.1). 9.2.4.4 Resolution An equation analogous to Equation (2.24) for isocratic elution can be derived for gradient elution [26]. Starting with Equation (2.23), Equation (9.8a) can be substituted for values of t R(j) and t R(i) .ValuesofW i and W j can be replaced by a single peak width W (Eq. 9.14), and the quantity G can be approximated by Equation (9.15b). to give R s = 2.3 4 N ∗1/2 log α k ∗ 1 + k ∗ (9.15c) With the final approximation 2.3 log(α) ≈ (α − 1), for small values of α, we have R s = 1 4 k ∗ 1 + k ∗ (α ∗ − 1)N ∗0.5 (9.16) Here α* is the value of the separation factor α when the band-pair reaches the middle of the column (at which time k ≡ k ∗ ), and N ∗ is the value of N when the band reaches the middle of the column. Values of N ∗ in gradient elution are the same as N in isocratic elution, when k = k ∗ . Equation (9.16) is primarily of conceptual value; it describes how resolution depends on k ∗ , the separation factor or selectivity, and the column plate number. We will find this relationship useful in our following discussion of gradient method development (Section 9.3). Equation (2.23), which defines resolution for both isocratic and gradient elution, is more accurate than Equation (9.16) and is used in this book for all calculations of resolution—but Equation (2.23) is of little use as a guide for method development. 9.3 METHOD DEVELOPMENT Method development for a gradient separation (Table 9.2, Fig. 9.14) is conceptually similar to the development of an isocratic procedure (Section 6.4, Fig. 6.21). The composition of the sample must first be considered (step 1 of Table 9.2 and Fig. 9.14), in order to establish appropriate starting conditions. Defining the goals of separation comes next (step 2), for example, as baseline resolution (R s ≥ 2.0), the shortest possible run time, and conditions that favor (or do not hinder) the detection and measurement of individual peaks of interest. Other aspects of method development that are similar for isocratic or gradient separation include: • a possible need for sample pretreatment prior to injection (Chapter 16) • checking that all experiments are reproducible (replicate runs) • verifying column reproducibility (two or more columns from different lots; Section 9.3.8) 9.3 METHOD DEVELOPMENT 435 Table 9.2 Outline for the Development of a Routine Gradient Separation (Compare with Fig. 9.14) Step Comment 1. Review information on sample a. Molecular weight > 5, 000 Da? (see Chapter 13) b. Mobile phase buffering required? c. Sample pretreatment required? 2. Define separation goals Section 6.4 3. Carry out initial separation (run 1) a. Conditions of Table 9.3; 10-min gradient b. Any problems? (Section 9.3.1.1, Fig. 9.17) c. Isocratic separation possible? (Fig. 9.15) 4. Optimize gradient retention k ∗ Conditions of Table 9.3 should yield an acceptable value of k ∗ ≈ 5 5. Optimize separation selectivity α* Increase gradient time by 3-fold (run 2, 30 min); increase temperature by 20 ◦ C (runs 3 and 4); see examples of Figure 9.18 5a. If best resolution from step 5 is R s 2, or if very short run times are required, vary conditions further in order to optimize peak spacing (for maximum R s or minimum run time) a. Replace acetonitrile by methanol and repeat runs 1–4 b. Replace column and repeat runs 1–4 c. Change pH and repeat runs 1–4 d. Consider use of segmented-gradients (Section 9.3.5; least promising) 6. Adjust gradient range and shape a. Select best initial and final values of %B for minimum run time with acceptable R s b. Add a steep gradient segment to 100%B for ‘‘dirty’’ samples (e.g., Fig. 9.11b) c. Add a steep gradient segment to speed up separation of later, widely spaced peaks (Fig. 9.11c) d. Add an isocratic hold to improve separation of peaks eluting at start of gradient (Fig. 9.9d) 7. With best separation from step 5 or 6, choose best compromise between resolution and run time Vary column conditions (Section 9.3.6) 8. Determine necessary column equilibration between successive sample injections Using the conditions selected above, carry out successive, identical separations while varying the equilibration time between runs; select a minimum equilibration time that provides acceptable separation (Section 9.3.7) . conditions that affect k*. The reader may choose to skip to Section 9.3, and return to this section at a later time—or as needed. However, this treatment can add to the reader’s intuitive understanding. increase in gradient time from 5to2 0 minutes (Fig. 9.13b), corresponding to an increase in average k ∗ from 5 to 20. As a result the retention of peak 2 relative to that of peak 3 increases, and. shortened from 50 minutes in Figure 9.11a to 40 minutes in Figure 9.11c. Increasing resolution by adjusting selectivity for different parts of the chro- matogram can sometimes be achieved with a