406 GRADIENT ELUTION 9.1.1 Other Reasons for the Use of Gradient Elution Apart from the need for gradient elution in the case of wide-polarity-range samples like that of Figure 9.1, there are a number of other situations that favor or require the use of gradient elution: • high-molecular-weight samples • generic separations • efficient HPLC method development • sample preparation • peak tailing High-molecular-weight compounds, such as peptides, proteins, and oligonu- cleotides, are usually poor candidates for isocratic separation, because their retention can be extremely sensitive to small changes in mobile-phase composition (%B). For example, the retention factor k of a 50,000-Da protein can change by 3-fold as a result of a change in the mobile phase by only 1% B. This behavior can make it extremely difficult to obtain reproducible isocratic separations of macromolecules in different laboratories, or even within the same laboratory. Furthermore the isocratic separation of a mixture of macromolecules usually results in the immediate elution of some sample components (with k ≈ 0 and no separation), and such slow elution of other components (with k 100) that it appears that the sample never leaves the column; that is, the retention range of such samples is often extremely wide (isocratic k-values for different sample components that vary by several orders of magnitude). With gradient elution, on the other hand, irreproducible retention times for large molecules are seldom a problem, and resulting separations can be fast, effective, and convenient (Chapter 13). Generic separations are used for a series of samples, each of which is made up of different components; for example, compounds A, B,andC in sample 1, compounds D, E,andF in sample 2, and so forth. Typically each sample will be separated just once within a fixed separation time (run time), with no further method development for each new sample. In this way hundreds or thousands of related samples—each with a unique composition—can be processed in minimum time and with minimum cost. Generic separations by RPC (with fixed run times, for automated analysis) are only practical by means of gradient elution and are commonly used to assay combinatorial libraries [3], as well as other samples [4]. Generic separation is often combined with mass spectrometric detection [5], which allows both the separation and identification of the components of samples of previously unknown composition—without requiring the baseline resolution of peaks of interest. Efficient HPLC method development [6] is best begun with one or more gradient experiments (Section 9.3.1). A single gradient run at the start of method development can replace several trial-and-error isocratic runs as a means for estab- lishing the best solvent strength (value of %B) for isocratic separation. An initial gradient run can also establish whether isocratic or gradient elution is the best choice for a given sample. Sample preparation (Chapter 16) is required in many cases because some samples are unsuitable for direct injection followed by isocratic elution. Interfering 9.1 INTRODUCTION 407 peaks, strongly retained components, and particulates must first be removed. In some cases, however, gradient elution can minimize (or even eliminate) the need for sample preparation. For example, by spreading out peaks near the beginning of a gradient chromatogram (as in Fig. 9.1g vs.Fig.9.1a), interfering peaks (non-analytes) that commonly elute near t 0 can be separated from peaks of interest. Similarly strongly retained non-analytes at the end of an isocratic separation can result in excessive run times, because these peaks must clear the column before injection of the next sample. Gradient elution can usually remove these late-eluting compounds within a reasonable run time (Section 9.2.2.5). Peak tailing was a common problem in the early days of chromatography, and the reduction of tailing was an early goal of gradient elution [7]. Because of the increase in mobile-phase strength during the time a band moves through the column in gradient elution, the tail of the band moves faster than the peak front, with a resulting reduction in peak tailing and peak width (Section 9.2.4.3). However, peak tailing is today much less common, and other means are a better choice for addressing this problem when it occurs (Section 17.4.5.3). 9.1.2 Gradient Shape By gradient shape, we mean the way in which mobile-phase composition (%B) changes with time during a gradient run. Gradient elution can be carried out with different gradient shapes, as illustrated in Figure 9.2a–f. Most gradient separations use linear gradients (Fig. 9.2a), which are strongly recommended during the initial stages of method development. Curved gradients (Fig. 9.2b,c) have been used in the past for certain kinds of samples, but for various reasons such gradients have been largely replaced by segmented gradients (Fig. 9.2d). Segmented gradients can provide most of the advantages of curved gradients, are easier to design for different samples, and can be replicated by most gradient systems. The use of segmented gradients for various purposes is examined in Section 9.2.2.5. Gradient delay or ‘‘isocratic hold’’ (Section 9.2.2.3) is illustrated by Figure 9.2e; an isocratic hold can also be used at the end of the gradient. Step gradients (Fig. 9.2f), where an instantaneous change in %B is made during the separation, are a special kind of segmented gradient. They are used infrequently—except at the end of a gradient separation for cleaning late-eluting compounds from the column; a sudden increase in %B (as in i of Fig. 9.2f) achieves this purpose. A step gradient that provides a sudden decrease in %B (as in ii of Fig. 9.2f) can return the gradient to its starting value for the next separation. In the past, step gradients were sometimes avoided because of a concern for column stability; with today’s well-packed silica-base columns, however, step gradients can be used without worry about column damage. A linear gradient can be described (Fig. 9.2g) by the initial and final mobile-phase compositions, and gradient time t G (the time from start-to-finish for the gradient). We can define the initial and final mobile-phase compositions in terms of %B, or we can use the volume-fraction φ of solvent B in the mobile phase (equal to 0.01%B): values φ o and φ f , respectively. The change in %B or φ during the gradient is defined as the gradient range andisdesignatedbyΔφ = φ f − φ 0 (or the equivalent Δ%B = [final%B] − [initial%B]). In the present book, values of %B and φ will be used interchangeably; that is, φ always equals 0.01%B, and 100% B (φ = 1.00) signifies pure organic solvent in RPC. For reasons discussed in Section 408 GRADIENT ELUTION %B %B %B %B %B %B time time Concave Segmented (a) (c) (e) (g) time time Linear Convex (b) (d) (f ) (h) time time Gradient delay Step gradients i ii 100 80 60 40 20 0 100 80 60 40 20 0 0 5 10 15 2 0 (min) 5/25/40/100% B at 0/5/15/20min t G = 20 min = (0.80 – 0.10) = 0.70 %B 0 5 10 15 20 (min) 10-80% B in 20 min o = 10% B ≡ 0.10 f = 80% B ≡ 0.80 = f – o Figure 9.2 Illustration of different gradient shapes (plots of %B vs. time). 17.2.5.3, it is sometimes desirable for the A- and/or B-solvent reservoirs to contain mixtures of the A- and B-solvents, rather than pure water and organic, respectively; for example, 5% acetonitrile/water in the A-reservoir and 95% acetonitrile/water in the B-reservoir. For the latter example, a (nominal) 0–100% B gradient would then correspond to 5–95% acetonitrile, with Δφ = (0.95 – 0.05) = 0.90. 9.1 INTRODUCTION 409 By a gradient program, we refer to a description of how the mobile-phase composition changes with time during a gradient. Linear gradients represent the simplest program, for example, a gradient from 10–80% B in 20 minutes (Fig. 9.2g), which can also be described as 10/80% B in 0/20 min (10% B at 0 min to 80% B at 20 min). Segmented programs are usually represented by values of %B and time for each linear segment in the gradient, for example, 5/25/40/100%B at 0/5/15/20 min for Figure 9.2h. 9.1.3 Similarity of Isocratic and Gradient Elution A peak moves through the column during gradient elution in a series of small steps, in each of which there is a small change in mobile-phase composition (%B). That is, gradient separation can be regarded as the result of a large number of small, isocratic steps. Separations by isocratic and gradient elution can be designed to give similar results. The resolution achieved for selected peaks in either isocratic or gradient elution will be about the same, when average values of k in gradient elution (during migration of each peak through the column) are similar to values of k in isocratic elution, and other conditions (column, temperature, A- and B-solvents, etc.) are the same. Isocratic and gradient separations where the latter conditions apply are referred to as corresponding. Thus the isocratic examples of Figure 9.1b–f can be compared with the ‘‘corresponding’’ gradient separation of Figure 9.1g.The isocratic separations of individual groups of peaks in Figure 9.1b–f each occur with k ≈ 3, while in the gradient separation of Figure 9.1g the equivalent value of k for each peak is also ≈ 3. We see in this example that the peak resolutions of Figure 9.1b–f are similar to those of Figure 9.1g. In isocratic elution we can change values of k by varying the mobile-phase strength (%B). In gradient elution, average values of k can be varied by changing other experimental conditions—as described below in Section 9.2. 9.1.3.1 The Linear-Solvent-Strength (LSS) Model This section provides a quantitative basis for the treatment of gradient elution in this chapter. However, the derivations presented here are of limited practical utility per se (although necessary for a quantitative treatment of gradient elution). The reader may wish to skip to Section 9.1.3.2 and return to this section as needed. Isocratic retention in RPC is given as a function of %B (Section 2.5.1) by log k = log k w − Sφ (9.1) For a given solute, the quantity k w is the (extrapolated) value of k for φ = 0(water or buffer as mobile phase), and S ≈ 4 for small molecules (<500 Da). A linear gradient can be described by %B = (%B) 0 + t t G [(%B) f − (%B) 0 ] (9.2) Here %B refers to the mobile-phase composition at the column inlet, (%B) 0 is the value of %B at the start of the gradient (time zero), (%B) f is the value of %B at the 410 GRADIENT ELUTION finish of the gradient, t is any time during the gradient, and t G is the gradient time. We can restate Equation (9.2) in terms of φ, the volume-fraction of B: φ = φ 0 + t t G (φ f − φ 0 ) = φ 0 + Δφ t G t (9.2a) where φ 0 is the value of φ at the start of the gradient, φ f is the value of φ at the end of the gradient, and Δφ = (φ f − φ 0 ) is the change in φ during the gradient (the gradient range); see Figure 9.2g. The quantity φ refers to values at the column inlet, measured at different times t during the gradient. Thus the mobile-phase composition at time t = 0 (the start of the gradient) is φ = φ 0 , provided that no delay occurs between the gradient mixer and the column inlet (Section 9.2.2.3). Equations (9.1) and (9.2a) can be combined to give log k = log k w − Sφ 0 − ΔφS t G t = C 1 − C 2 t (9.3) For a linear gradient, a given solute, and specified experimental conditions C 1 and C 2 are constants, so log k varies linearly with time t during the gradient (the value of k in Eq. 9.3 refers to the value of k measured at the column inlet at any given time t). Gradients for which Equation (9.3) applies are called linear-solvent-strength (LSS) gradients; linear RPC gradients are therefore (approximately) LSS gradients. Exact equations for retention and peak width can be derived for LSS gradients (Section 9.2.4). LSS separations are much easier to understand and to control, compared to the use of other gradient shapes. Finally, LSS gradients provide a better separation of most samples that require gradient elution. A fundamental definition of gradient steepness b for a given solute is b = V m ΔφS t G F (9.4) or as t 0 = V m /F, b = t 0 ΔφS t G (9.4a) This definition of gradient steepness follows from Equation (9.3), which can be written as log k = log k w − Sφ 0 − t 0 ΔφS t G t t 0 or log k = log k w − Sφ 0 − b t t 0 (9.4b) 9.1 INTRODUCTION 411 where (log k w − Sφ 0 ) for a given gradient and solute is equal to log k at the start of the gradient (and therefore varies with φ 0 ; see later Eq. 9.7). A larger value of b corresponds to a faster decrease in k with time, or a steeper gradient. Retention times and peak widths in gradient elution can be derived from the relationships above (see Section 9.2.4). 9.1.3.2 Band Migration in Gradient Elution Consider next how individual solute bands move through the column during gradient elution (Fig. 9.3). For an initially eluted compound i in Figure 9.3a, the solid curve (x[i]) marks the fractional migration x of band i through the column as a function of time (note that y = 1onthey-axis represents elution of the band from the column; y = 0 represents the band at the column inlet). Band migration is seen to accelerate with time, resulting in an upward-curved plot of x versus t. Also plotted in Figure 9.3a is the instantaneous value of k for band-i (dashed curve, k[i]) as it migrates through the column. The quantity k(i) is the value of k at time t for an isocratic mobile phase whose composition (%B) is the same as that of the mobile phase in contact with the band at time t.Peakwidthandresolutioningradient elution depend on the median value of k: the instantaneous value of k when the k(i) k(j ) x(i) x(j) 1.0 0.0 5 3 1 k Band migration x k* k e 024681012141618 Time (min) i j 5 10 15 (min) (a) (b) Figure 9.3 Peak migration during gradient elution. (a) Band-migration x and instantaneous values of k related to time, showing average (k ∗ ) and final values of k (at elution, k e ); (b)result- ing chromatogram. 412 GRADIENT ELUTION band has migrated halfway through the column. This median value of k in gradient elution is defined as the gradient retention factor k ∗ . Peak width is determined by the value of k when the peak leaves the column (defined as k e ,equaltok ∗ /2). A similar plot for a second band j (with values of x = x[j], and k = k[j]) is also shown in Figure 9.3a. The resulting chromatogram for the separation of Figure 9.3a is shown in Figure 9.3b. A comparison of band migration in Figure 9.3a for the two compounds i and j shows a generally similar behavior, apart from a delayed start in the migration of band j because of its stronger initial retention (larger value of k w ). Specifically, values of k ∗ and k e for both early and late peaks in the chromatogram are approximately the same for solutes i and j, suggesting that resolution and peak spacing need not decrease for earlier peaks, as in isocratic elution for small values of k (compare the gradient separation of peaks 1–6 in Fig. 9.1e with their isocratic separation in Fig. 9.1a). Values of k e are also usually similar for early and late peaks in gradient elution, meaning that peak widths (and heights) will be similar for both early and late peaks in the chromatogram (contrast the peak heights for peaks 1–14 in the gradient separation of Fig. 9.1e with these same peaks in the isocratic separation of Fig. 9.1a). The relative constancy of values of k* and k e for a linear-gradient separation are responsible for the pronounced advantages of gradient over isocratic elution for the separation of wide-range samples such as that of Figure 9.1. 9.2 EXPERIMENTAL CONDITIONS AND THEIR EFFECTS ON SEPARATION The gradient retention factor k ∗ of Figure 9.3 has a similar significance in gradient elution as the retention factor k has in isocratic elution. Values of k in isocratic elution are important for the understanding and control of separation, and we will see that values of k ∗ play the same role in gradient elution. The value of k ∗ depends on the solute (its value of S in Eq. 9.1) and experimental conditions: gradient time t G , flow rate F, column dimensions, and the gradient range Δφ [2]: k ∗ = 0.87t G F V m ΔφS (9.5) Here V m is the column dead-volume (mL), which can be determined from an experimental value of t 0 and the flow rate F (Section 2.3.1; V m = t 0 F). Values of S for different samples with molecular weights in the 100 to 500 Da range can be assumed equal to about 4. This means that values of k* for different solutes in a given linear-gradient separation (with constant values of t G , F, V m , and Δφ) will all be about the same. Let us next compare isocratic and gradient separation in terms of values of k and k ∗ , for the same sample and similar conditions (same A- and B-solvents, column, flow rate, and temperature). The isocratic separations of Figure 9.4a–c illustrate the effect of a change in %B (and k), for mobile phases of 70, 55, and 40% B. Similar values of k ∗ in gradient elution can be achieved by varying gradient time t G (Eq. 9.5 with S = 4); see Figure 9.4d–f ,wherek ∗ = 1, 3, and 9 for t G = 3, 10, 9.2 EXPERIMENTAL CONDITIONS AND THEIR EFFECTS ON SEPARATION 413 and 30 minutes, respectively. Isocratic and gradient separations will be referred to as ‘‘corresponding’’ when the average value of k in the isocratic separation equals the value of k ∗ for the gradient separation (for the same sample and experimental conditions, except that %B and k ∗ are allowed to vary). In the example of Figure 9.4, separations (a)and(d) are ‘‘corresponding,’’ as are separations (b)and(e), and (c)and(f ). ‘‘Corresponding’’ separations as in these examples should be similar in terms of resolution and average peak heights—except that peaks in the gradient separation can be taller by as much as 2-fold. For either isocratic or gradient elution, an increase in k or k ∗ corresponds to an increase in run time (other conditions the same). In isocratic elution, resolution increases for larger values of k (Eq. 2.24), as observed in Figure 9.4a–c (R s = 0.4, 1.7, and 3.4). For similar values of k ∗ in gradient elution (Fig. 9.4d–f), the observed resolution is seen to be about the same for each ‘‘corresponding’’ separation (R s = 0.4, 1.7, and 3.6). Finally, peak widths in isocratic elution increase with k (decrease in %B), resulting in decreased peak heights. Again, similar changes in peak width and height are observed in gradient elution as k ∗ is varied in Figure 9.4d–f . Thus changes in %B for isocratic elution, or gradient time in gradient elution, lead to similar changes in run time, resolution, and peak heights. 0246 Time (min) 70% B R s = 0.4 1 2 3 4 5 1 2 3 4 5 01020 Time (min) (a) (b) (c) 02 Time (min) 1 + 2 3 4 5 t 0 1 ≤ k ≤ 2 55% B R s = 1.7 2 ≤ k ≤ 8 40% B R s = 3.4 6 ≤ k ≤ 23 Figure 9.4 Isocratic (a–c) and gradient (d–f) separations compared for a regular sample and change in either %B or gradient time. Sample: 1, simazine, 2, monolinuron; 3, metobro- muron; 4, diuron; 5, propazine. Conditions: 150 × 4.6-mm C 18 column (5-μm particles); methanol-water mobile phase (%B or gradient conditions indicated in figure); ambient tem- perature; 2.0 mL/min. Note that actual peak heights are shown (not normalized to 100% for tallest peak). Chromatograms recreated from data of [8]. 414 GRADIENT ELUTION 2.2 2.4 2.6 2.8 3.0 3.2 3.4 Time (min) 1 + 2 4 5 3 6.0 7.0 8.0 Time (min) 1 2 3 4 5 10 12 14 16 18 Time (min) 1 2 3 4 5 0-100% B in 3 min k* = 1 R s = 0.4 0-100% B in 10 min k* = 3 R s = 1.7 0-100% B in 30 min k* = 9 R s = 3.6 (d) (e) (f ) Figure 9.4 (Continued) The sample of Figure 9.4 can be described as ‘‘regular’’ (Section 2.5.2.1) because there are no changes in relative retention when k or k ∗ are varied by varying isocratic %B or gradient time, respectively (holding other conditions constant). Consequently critical resolution increases continuously in Figure 9.4d–f as gradient time (and k ∗ ) is increased. A similar series of experiments are shown in Figure 9.5 for an ‘‘irregular’’ sample (Section 2.5.2.1), composed of a mixture of substituted anilines and benzoic acids. Relative retention for an ‘‘irregular’’ sample changes as either isocratic %B or gradient time is varied. As in Figure 9.4, the same trends in average resolution, peak heights, and run time result in Figure 9.5 when gradient time is increased. However, changes in relative retention also occur for the sample of Figure 9.5 when gradient time is changed (note the changes in relative retention of shaded peak 3 and—to a lesser extent—peaks 7–10). As a result of these changes in relative retention with t G , maximum (‘‘critical’’) resolution for this sample occurs for an intermediate gradient time of 10 minutes (Fig. 9.5b; R s = 0.9), whereas the resolution of the ‘‘regular’’ sample in Figure 9.4 continues to increase as gradient time (and k ∗ ) increases. For ‘‘irregular’’ samples a change in either k (isocratic) or k ∗ (gradient) will result in similar changes in relative retention; consequently maximum sample resolution may not correspond to the largest possible value of k or k ∗ for such samples. 9.2 EXPERIMENTAL CONDITIONS AND THEIR EFFECTS ON SEPARATION 415 0 246 Time (min) 02 Time (min) 3-min gradient k* = 1.5, R s = 0.4 10-min gradient k* = 5, R s = 0.9 30-min gradient k*= 15, R s = 0.1 (a) (b) (c) 1 4 5 + 6 1 2 3 4 5 − 7 9 8 10 11 2 7 + 8 9 11 10 024681012 Time ( min ) 2 + 3 4 8 10 11 1 9 5 - 7 3 Figure 9.5 Separations of an irregular sample as a function of gradient time t G . Sample: a mix- ture of substituted anilines and benzoic acids. Conditions: 100 × 4.6-mm C 18 column (3-μm particles), 2.0 mL/min, 42 ◦ C, 5–100% acetonitrile-pH-2.6 phosphate buffer in (a)5min- utes, (b) 15 minutes, and (c) 30 minutes. Peak 3 is cross-hatched to better illustrate changes in relative retention for this sample as gradient time is varied. Note that actual peak heights are shown (not normalized to 100% for tallest peak). Chromatograms recreated from data of [9]. 9.2.1 Effects of a Change in Column Conditions Column conditions—column length and diameter, flow rate, and particle size—affect the column plate number N (Section 2.4.1) and run time. Column conditions are chosen at the start of method development, then sometimes changed after other separation conditions have been selected—in order to either improve resolution or reduce run time (Section 2.5.3). In isocratic elution, a change in column conditions has no effect on values of k or relative retention. Resolution and run time usually increase for an increase in column length or a decrease in flow rate, while peak heights decrease for longer columns and faster flow. These changes in isocratic separation, when only column length or flow rate is changed, are illustrated in Figure 9.6a–c for the ‘‘regular’’ sample of Figure 9.4. Figure 9.6a represents a starting separation, while Figures 9.6b and 9.6c show the results of an increase in either column length or flow rate, respectively. Note the resulting . (nominal) 0–100% B gradient would then correspond to 5–95% acetonitrile, with Δφ = (0.95 – 0.05) = 0.90. 9.1 INTRODUCTION 409 By a gradient program, we refer to a description of how the mobile-phase composition. separations by RPC (with fixed run times, for automated analysis) are only practical by means of gradient elution and are commonly used to assay combinatorial libraries [3], as well as other samples. B-solvent reservoirs to contain mixtures of the A- and B-solvents, rather than pure water and organic, respectively; for example, 5% acetonitrile/water in the A-reservoir and 95% acetonitrile/water