446 GRADIENT ELUTION a change in k ∗ (Eq. 9.5c). For ‘irregular’ samples this can result in changes in selectivity. As selectivity for a gradient method should have been optimized (step 5 of Table 9.2) prior to a change in column conditions, it is important to maintain the same values of k* (and α*) when changing column conditions and N*.This can be achieved by maintaining (t G F/L) constant (Eq. 9.5c); for example, if column length is doubled, gradient time must also be doubled so that t G /L stays constant; if flow rate is doubled, gradient time must be decreased by half so as to keep t G F constant. For examples of this approach to optimizing N ∗ ,seeFigure9.6d–f.As long as values of k ∗ are maintained constant in this way, a change in column length or flow rate has the same effects on run time and resolution in either isocratic or gradient elution. A minor exception to this rule can occur for the resolution of early peaks in the chromatogram for larger values of V D —regardless of whether k ∗ is held constant (Section 9.2.2.4 [11]) [11]. When column conditions are changed for a segmented gradient, the time t seg for each segment must be adjusted so as to maintain t seg F/L constant. For example, consider the separation of Figure 9.11c, where the gradient is 0/23/42% B in 0/32/38 min. If column length were doubled, the length of each segment t seg would also require doubling, so that the new gradient would be 0/23/42% B in 0/64/76 min. 9.3.7 Determine Necessary Column-Equilibration Time (Step 8 of Table 9.2) After method development is complete, in most cases the resulting HPLC procedure will be used for routine sample analysis. During this application of the method the column must be washed between successive gradient runs with a sufficient volume of mobile phase whose composition matches that of the mobile phase at the start of the gradient (e.g., 5% B in the examples of Fig. 9.18). This column-equilibration step is intended to allow for (1) the holdup volume V D (or dwell-time t D = V D /F) of the gradient equipment, (2) gradient rounding (Section 3.10.1.2), and (3) slow equilibration of the stationary phase (removal of excess B-solvent) when switching from high %B at the end of one gradient to low %B at the beginning of the next gradient. Figure 9.19 illustrates the possible consequence and correction of the combined effects of dwell-volume, gradient rounding, and slow column equilibration, when sequential sample injections are made during routine analysis. In Figure 9.19a the solid lines describe a series of programmed gradients in terms of time, while the arrows mark the times when samples 1, 2, and so forth, are injected at the beginning of each gradient. These 5–100% B gradients in 10 minutes are followed by a between-run equilibration with 5% B for one minute (the equilibration time t eq = 1 min). The complete programmed gradient is therefore 5/100/5/5%B in 0/10/10/11 min. If the system dwell-volume and gradient rounding are negligible, and if column equilibration is fast, the actual gradient should be the same as the programmed gradient—and injection of each sample would then occur one minute after completion of the previous gradient. The same separation of each sample would then result. Figure 9.19b expands on the example of Figure 9.19a by introducing some additional features of an actual gradient (which are common in practice): a significant 9.3 METHOD DEVELOPMENT 447 t D + t x = t eq 0 5 10 15 20 25 30 35 (min) 1 2 3 Revised gradient with correct t eq : 5/100/5/5% B in 5/10/10/15 min t eq 0 5 10 15 20 25 30 (min) 123 t G 100% B 80% 60% 40% 20% 0% (a) (b) (c) Programmed gradient at column inlet 5/100/5/5% B in 0/10/10/11 min Actual gradient at column inlet t D t x 0 5 10 15 20 25 30 (min) 123 t x programmed gradient actual gradient actual gradient programmed gradient Figure 9.19 Illustration of different contributions to column nonequilibration in gradient elution. (a) Ideal gradient; (b) more realistic gradient; (c) addition of an adequate between-run equilibration time for the gradient of (b). (—) programmed gradient; (- - -) actual gradient at the column inlet; 1, 2, etc., refer to injections of sample 1, 2, etc. 448 GRADIENT ELUTION equipment dwell volume with gradient rounding and/or slow column equilibration. The dashed curves (- - -) in Figure 9.19b describe the actual (individual) gradients at the column inlet; there is a significant dwell-volume, resulting in a dwell-time t D = 2 min. Consequently the arrival of the gradient at the column inlet is delayed by 2 minutes. Sample injection at the time the programmed gradient begins (the usual case) is still acceptable for the first sample, but later samples are now injected one minute before the previous gradient has been completed. Because samples 2, 3, and so forth, are injected into a mobile phase with much higher %B (just prior to the completion of the gradient), the result would be very small values of k ∗ for early bands—and an unacceptable loss in resolution for these peaks. Due to gradient rounding and/or slow column equilibration, there is a slow decrease in %B from 100% B to 5% B at the end of each gradient. A time t x = 3 min is required for the return of the gradient to the initial-%B value. Note that adjacent gradients (%B values) add in this example (not shown in Fig. 9.19b). The detrimental effects of dwell-volume plus the slow return of the gradient to baseline can be eliminated by the use of an equilibration time t eq that is made equal to t D + t x , as illustrated in Figure 9.19c. With this change in the gradient (5/100/5/5%B in 0/10/10/15 min), injection of each sample occurs at the start of its programmed gradient (with return of the preceding gradient to the initial-%B value); now the same (acceptable) separation is achieved for all samples. The required value of t eq can be determined by trial and error—where successive sample injections are made for a given value of t eq , then repeated for different values of t eq . The preferred value of t eq is the lowest value that gives an acceptable separation for successive samples. The trial-and-error approach also has the benefit of including any effect of autosampler delay in the equilibration process. For method-development experiments, 10 column-volumes or more of the starting mobile phase(φ = φ o ) should be passed through the column before start- ing the next gradient run (corresponding to 10V m /F,oratimeequalto10t 0 ). Otherwise, any change in the time between successive experiments (often the case in method development) may result in variable column equilibration and resulting changes in retention and separation. For a routine gradient assay as in Figure 9.19, however, the time devoted to column equilibration can be reduced to t eq ,with a corresponding shortening of run time (compared to 10t 0 ) and an increase in the number of samples that can be analyzed each day. Furthermore partial equi- libration of the column (i.e., incomplete return of the mobile phase to initial-%B) may be acceptable for routine analysis, with a reduction of the equilibration time to a value < t eq . Thus, if the resolution of early peaks is not compromised, and if the equilibration time is the same for each gradient run, each sample is treated the same and each separation will be the same—despite incomplete column equilibration. For HPLC systems that do not permit delayed injection, the preferred partial equilibration time t  eq for routine analysis will be given by t D ≤ t  eq ≤ (t D + t x ), where t  eq must be determined by trial and error (i.e., minimum allowable value of t  eq ). Values of t x are usually comparable to values of t D , which provides an initial estimate of t eq ≈ 2t D (followed by trial-and-error changes in equilibration time in order to determine t  eq ). For systems that allow injection of the sample at any time following the start of the gradient, sample injection can be programmed to occur at 9.3 METHOD DEVELOPMENT 449 atimet D after the start of each gradient. The corresponding equilibration time t  eq is then ≤ t x . However, methods based on sample injection after the gradient begins can only be carried out with gradient systems that allow delayed injection. Whatever column equilibration time is allowed for a routine assay procedure, it should be confirmed that repetitive sample injections yield the same (acceptable) chromatograms and reproducible data, except possibly for the first run (which can be discarded). The required column equilibration time will be less for systems with a smaller dwell-volume and reduced gradient rounding, and can be reduced further for the case of systems that permit delayed sample injection. Changes in the plumbing of the system can be used to minimize the effects of gradient rounding [43] and further reduce t eq , although most users will not make use of this option (but future equipment may eliminate the need for changes in system plumbing). For additional details on column equilibration in gradient elution, see [2, 44, 45]. 9.3.8 Method Reproducibility Some causes of irreproducible results or poor method precision are the same for both isocratic and gradient separations (Section Section 11.2). Other sources of irreproducibility are either unique to gradient elution, or more likely for this technique: 1. poor control of experimental conditions from run to run 2. malfunctioning or poorly designed equipment 3. insufficient column equilibration between gradient runs 4. differences in equipment dwell volume The contributions above to separation variability can impact both method devel- opment (following Section 9.3.8.1), and the subsequent routine use of a gradient method (Section 9.3.8.2). Additionally the accuracy and precision of gradient assays (and the interpretation of method development experiments) can be compromised by drifting baselines during a gradient run, as well as artifact peaks that are inde- pendent of the sample (Section 17.4.5.2). To rule out such problems, we strongly recommend that every series of gradient runs be preceded by a blank gradient:a gradient run without sample injection, or (better) with injection of only the sample solvent (Section 3.10.1.2). 9.3.8.1 Method Development Consider first the need for repeatable data during method development, where it is advisable to replicate each experiment so as to verify that the data obtained are reproducible from run to run; this is especially important for gradient elution experiments. Retention times in duplicate, back-to-back runs should not vary by more than some set amount; for example, ±0.02 min or ±0.1%, whichever is larger. Poorly controlled experimental conditions and malfunctioning equipment (items 1 and 2) fall largely under the heading of good laboratory technique. For purposes of the present discussion, we will assume that all experimental conditions are controlled within limits necessary for repeatable separation. We will also assume that the equipment is operating properly, and that column performance meets the 450 GRADIENT ELUTION manufacturer’s specifications (Section 3.10.1.2). Apart from operator and equip- ment issues, however, a major objective of method development should be a final method that can tolerate small, largely unavoidable changes in gradient conditions, temperature, and mobile phase composition (pH, buffer concentration, etc.), from day to day and from system to system. If a method appears not to be robust, efforts should be made to reduce the dependence of the method on experimental conditions, by examining both method robustness and resolution as a function of conditions (Section 12.2.6). Insufficient column equilibration (item 3) is a major source of variable retention in gradient elution, so a column-equilibration step between each run or experiment is necessary (Section 9.3.7). Retention-time repeatability should be checked initially for two replicate, successive runs that use the selected minimum equilibration time (e.g., 10t 0 ) between the two runs, but with a 2-fold longer equilibration time prior to the first run. An equilibration time > 10t 0 min may be required for some samples and/or separation conditions, and very slow changes in retention may occur over a longer time period [43] (but have little effect on method development). To ensure reproducibility of method-development runs, it is prudent to allow more than the minimum required equilibration time between runs; this can be trimmed to reduce the run time when the method is finalized for routine use. Differences in equipment dwell volume (item 4) can significantly affect exper- imental results (Section 9.2.2.4). For this reason it is strongly recommended to carry out all method-development experiments for a given sample on the same (or equivalent) equipment, in order to avoid changes in dwell-volume among different experiments. 9.3.8.2 Routine Analysis During method development it is necessary to anticipate possible changes in the separation that might inadvertently occur when the method is transferred to another laboratory for routine analysis. Variation in experimental conditions and malfunc- tioning equipment (items 1 and 2 above) can be recognized by system suitability tests (Section 12.3.2.9). Column equilibration (item 3) should be handled differently in routine analysis than in method development. During method development, a between-run equilibration time of at least 10t 0 min is usually acceptable. For routine analysis, where the equilibration time between runs is generally fixed, it is desirable to shorten the equilibration time as much as possible, in order to minimize the time between sample injections (Section 9.3.7)—as well as the overall run time. Differences in equipment dwell volume (item 4) are a common reason for the failure of a gradient method during method transfer or routine application on a different HPLC system. The dwell-volume V D can vary significantly between different gradient systems; older equipment usually has larger values of V D . A different gradient system will often be used to carry out routine assays, compared to the system used to develop the method. If the second system has a different dwell-volume (V D ) compared to the original system, unacceptable changes in separation can result (Section 9.2.2.4), especially for irregular samples. When the value of V D for the second system is smaller, this difference in dwell-volumes can be compensated by adding a gradient-delay time t delay for the separation carried out on the second system, as this is equivalent to an increase in dwell-volume. The length of this 9.3 METHOD DEVELOPMENT 451 gradient delay in minutes should be made equal to the difference in dwell-times t D for the two systems (t D = V D /F). A second gradient system with a larger dwell-volume (the more likely case) presents a more difficult problem. For this reason an effort should be made in method development to anticipate the maximum dwell-volume likely to be encountered in other laboratories that will use a given procedure. The original method can be developed with a total gradient delay (equal t D + t delay ) that effectively increases dwell-time to the maximum value of t D expected in other labs to which the method will be transferred; a value of t delay can then be selected in each transfer lab, so as to compensate for differences in dwell-volume relative to the original equipment (so that t D + t delay remains constant for each system). Alternatively, a delayed injection of the sample (if the system allows this option) can be used to effectively reduce the dwell volume of the second system. See the more detailed discussion of [2]—as well as [46], where other options for dealing with varying dwell-volume are discussed. 9.3.9 Peak Capacity and Fast Separation The peak capacity (PC) of an isocratic separation was discussed in Section 2.7.3. The definition of peak capacity is the same for both isocratic and gradient elution; PC equals the maximum number of peaks that can be inserted into a given chromatographic space (e.g., a gradient chromatogram) with a resolution R s = 1.0 for all adjacent peaks (a defined run time is assumed). In a gradient separation, where every peak has approximately the same peak width W, peak capacity can be approximated in terms of the gradient time t G as PC = 1 +  t G W  ≈ t G W (9.20) Figure 9.20 illustrates this definition of peak capacity for a (hypothetical) separation in a gradient time of 10 minutes. For the example of Figure 9.20a, the average peak width W = 0.2 min. Therefore PC for this example equals t G /W = 10/0.2 = 50, as illustrated in Figure 9.20b, where 50 peaks, each with W = 0.2 min, can be fit into the 10-minute chromatogram with R s = 1 for each adjacent peak-pair. The concept of peak capacity has been used to evaluate the relative performance of separations by gradient elution, in place of measurements of the column plate number N (which are possible, but less convenient; see p. 38 of [2]). As a measure of separation effectiveness, values of PC are especially useful for samples that generate more peaks than can be individually separated to baseline, that is, ‘‘complex’’ samples as in Figure 9.17c or Figure 9.20a (e.g., peptide digests, plant extracts, etc). ‘‘Peak capacity’’ is a hypothetical (if measurable) quantity that generally overestimates the separation power of an actual gradient chromatogram. Thus, in the separation of Figure 9.20a, peaks appear only between 2 and 9 minutes, so that only a fraction of the gradient chromatogram is actually used: (9–2)/10 = 70%. When sample peaks are confined within part of the chromatogram (as in Fig. 9.20a), rather than being distributed over the entire chromatogram, the effective peak capacity of the separation is less than the value of PC defined in Figure 9.20b for 452 GRADIENT ELUTION 0 2 4 6 8 10 (min) 100% B 80% 60% 40% 20% 0% t o + t D 0 2 4 6 8 10 ( min ) Peak capacity PC = t G /W 100% B 80% 60% 40% 20% 0% (a) (b) 10 20 30 40 50 Figure 9.20 Peak capacity in gradient elution. (a) Hypothetical separation; (b) illustration of peak capacity PC for separation of (a). the full gradient. In actual separations as in Figure 9.20a, separation performance is better defined by the number of resolved peaks that can be fit between the first and last peaks in the chromatogram (not necessarily the beginning and end of the gradient). The latter quantity will be referred to as the equivalent peak capacity n c = PC (t Z − t A )/t G ; n c has also been called conditional peak capacity or sample peak capacity [47]. In practice, equivalent peak capacities will be smaller than values of PC given by Equation (9.20) (n c = 35 for the example of Fig. 9.20a). Note that the selection of conditions that maximize PC will also tend to maximize n c . Because a single gradient run can be inadequate for the separation of complex samples, two-dimensional (2D) separation is often employed (Sections 9.10, 13.4.5, 13.10.4). In 2D separation, fractions from a first column are transferred to a second column for further separation (as in the example of Fig. 1.4b,c). If the two separations are orthogonal (i.e., uncorrelated retention times; Section 6.3.6.2), the peak capacity of the combined separations equals the product of the peak capacities for each separation. A common goal is to maximize the peak capacity of each separation—but in minimum overall run time [47–58]. In this section we will approach an understanding (and control) of peak capacity on the basis of similar considerations as for isocratic elution (Section 2.4.1). It is also relevant to note that detection by mass spectrometry (MS) and multi-wavelength absorbance effectively 9.3 METHOD DEVELOPMENT 453 add an extra dimension to one- or two-dimensional separations [53], hence further increasing the overall peak capacity. 9.3.9.1 Optimized Peak Capacities The primary value of the present section is limited to just two applications: very fast gradient separations (Section 9.3.9.2) and two-dimensional HPLC (2D-LC) (Section 9.3.10). Much of the additional detail of this section had not appeared in the literature as of 2009. Unless the reader has an immediate interest in fast or 2D-LC separation, it may be advisable to skip to Section 9.4. It is useful to express peak capacity in terms of experimental conditions that can be optimized for maximum values of PC and/or minimum run times. For a full-range gradient (Δφ = 1), Equation (9.20) can be expressed as PC = t Z − t A W (9.21) where t A and t Z refer to the retention times for two peaks that elute, respectively, at the start and finish of the gradient (e.g., peaks 1 and 50 in Fig. 9.20b). Values of W can be assumed to be approximately equal for peaks A and Z, so Equation (9.21) is equivalent to the resolution of these two peaks (Eq. 2.23). Resolution R s in Equation (9.15c) can therefore be replaced by PC to give PC =  2.3 4  N ∗0.5 log α ∗  k ∗ 1 + k ∗  (9.21a) If we assume equal values of S for peaks A and Z (although S is usually larger for the later peaks in the chromatogram), then α* will equal the ratio of k w -values for peaks Z and A (Eq. 9.5a), which in turn is equal to ΔφS. Equation 9.21a then takes the form PC =  2.3 4  SΔφ N ∗0.5  k ∗ 1 + k ∗  (9.22) (i)(ii)(iii) This expression for peak capacity PC assumes a full-range gradient (Δφ = 1), and the value of S is determined by the molecular weight M of the sample (Section 13.4.1.4): S ≈ 0.25M 0.5 (9.23) For samples with M ≤ 500, a value of S ≈ 4 can be assumed. Term i of Equation (9.22) is therefore constant for a sample of defined molecular weight. Term ii varies with column length, particle size, flow rate, temperature, and sample molecular weight (Section 2.3.1). Finally, term iii varies with gradient time, flow rate, column length, and S or sample molecular weight (Eq. 9.5c). The elaboration of Equation (9.22) in terms of the latter experimental variables can lead to fairly complex relationships that are challenging to both interpret and apply. 454 GRADIENT ELUTION A simpler, more convenient use of Equation (9.22) is as follows, assuming conditions that provide maximum values of PC for a given run time and some maximum allowable column pressure P. In Section 2.4.1.1 it was argued that the best use of a column (for some required value of N in minimum time) occurs when the value of the reduced plate height h is a minimum, corresponding to a value of the reduced mobile-phase velocity ν ≡ hd p /D m ≈ 3 (case ‘‘C’’ in Section 2.4.1.1, which assumes a wide choice of column lengths and particle sizes). Optimum values of N (isocratic elution) are then defined (as in Figure 2.15) as a function of particle size d p , pressure, and separation time (for given values of mobile-phase viscosity η and solute diffusion coefficient D m ). Specific values of the column length, flow rate, and particle size then result for each separation time and pressure (Eqs. 2.13a and 2.21b), such that ν = 3 (optimum value). As separation time increases, column length and particle size must increase, and flow rate decrease—so as to maintain constant pressure (while also maintaining ν = 3 by varying d p ). From Equation (9.22), it can be shown that values of PC are proportional to k ∗3/4 /(1 + k ∗ ), which then results in an optimum value of k ∗ = 3. A summary of optimal values of PC as a function of experimental conditions is shown in Figure 9.21a for a small-molecule sample (note the essential similarity of this plot, based on gradient elution, and the corresponding isocratic plot of Figure 2.15 for values of N). For these optimized conditions, we see that peak capacity increases with gradient time and pressure, while the required particle diameter also increases with gradient time. For very short runs, especially at higher pressures, column packings that are presently unavailable are required (d p < 1.5 μm). In these cases, various (sub-optimum) expedients are available to maximize peak capacity for very short gradients (e.g., t G ≤ 1 min). The data of Figure 9.21a also assume ideal, equipment-related conditions, which are difficult to attain for very fast separations (Section 9.3.9.2) When the optimized conditions of Figure 9.21a require particle sizes < 1.5 μm (i.e., for very short gradients), larger particles can be used, with some loss in peak capacity. Also, as long as ν (and N ∗ ) is optimized, the selection of other values of k ∗ within the range 1 ≤ k ∗ ≤ 10 (i.e., by varying gradient time only) leads to a ≤ 10% reduction in the optimum value of PC for a given run time (value of t G ). This then provides a simple means for varying gradient time (over a 10-fold range) while maintaining near-optimum values of PC over the gradient time.The achievement of sub-optimum, shorter gradients is illustrated in Figure 9.21b,for columns of different (discrete or noncontinuous) lengths packed with 3-μm particles; flow rate is also varied to maintain P = 6000 psi. The solid line represents optimal values of PC taken from Figure 9.21a for P = 6000 psi (where particle size and column length were allowed to vary continuously). Each of the other curves shown in this figure correspond to values of k ∗ that vary from 1 to 10—but with particle diameter fixed at 3 μm, and P = 6000 psi. The use of these sub-optimum conditions is seen to result in some loss in peak capacity, compared to optimum values for a given gradient time—but this loss in PC is usually less than half. Note that the optimum column length in this example for d p = 3 μmisL = 1500 mm. Table 9.4 summarizes conditions for the optimized separations of Figure 9.21b (6000 psi). 9.3 METHOD DEVELOPMENT 455 (a) (b) 1000 500 200 100 PC 15,000 psi 6,000 2,000 1 μm 1.5 μm 2 μm 3 μm 4 μm 1 2 5 10 20 50 100 200 500 1000 t G (min) 1000 500 200 100 PC 150-mm column 300-mm 600-mm 1500-mm optimized (d p varies) 1 μm 1.5 μm 2 μm 3 μm 1 2 3 10 20 50 200 200 500 1000 t G (min) k* = 1 P = 6000 psi (d p = 3 μ m for individual columns) k* = 3 k* = 10 Figure 9.21 Optimized peak capacity as a function of particle diameter, gradient time and column pressure. Data for a small-molecule sample (S = 4, D m = 10 −5 ; 0–100% B acetonitrile-buffer gradient (η = 0.75); calculations based on Equation (2.17) with A = 1, B = 2, and C = 0.05. (a) Optimized values (ν = 3, k ∗ = 3) for P = 2000, 6000, and 15,000 psi; (b) sub-optimized values for a pressure of 6000 psi and 3-μm particles (k ∗ allowed to vary for each column length). . μm 4 μm 1 2 5 10 20 50 100 200 500 1000 t G (min) 1000 500 200 100 PC 150- mm column 300-mm 600-mm 1500 -mm optimized (d p varies) 1 μm 1.5 μm 2 μm 3 μm 1 2 3 10 20 50 200 200 500 1000 t G (min) k*. method development to anticipate the maximum dwell-volume likely to be encountered in other laboratories that will use a given procedure. The original method can be developed with a total gradient. for this example equals t G /W = 10/0.2 = 50, as illustrated in Figure 9.20b, where 50 peaks, each with W = 0.2 min, can be fit into the 10-minute chromatogram with R s = 1 for each adjacent peak-pair.