Introduction to Modern Liquid Chromatography, Third Edition part 48 pps

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 diffe

(b)

(c)

Time (min)

100% B 80%

60%

40%

20%

0%

0/23/42% B in 0/32/38 min

Rs= 2.1

Time (min)

100% B 80%

60%

40%

20%

0%

0/40/100/100% B in 0/50/51/52 min

R s= 2.1

Time (min)

100% B 80%

60%

40%

20%

0%

0-40% B in 50 min

R s= 2.1

Figure9.11 Gradient separations of a peptide digest of recombinant human growth hormone Conditions: 150× 4.6-mm C18column (5-μm); 45◦C; 2.0 mL/min (a) 0–40% B in 50 min; (b) same as in (a), except a steep gradient segment is added in order to remove strongly reten-tive ‘‘junk’’ from the column; (c) same as in (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

0 2 4 6 8 10 12

Time (min)

(a)

(b)

40-100% B in

12.5 min

R s= 1.4

*

*

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

Figure9.12 Separation of a mixture of 16 polycyclic aromatic hydrocarbons, adapted from Figure 6.4 of [6] Conditions: 150× 4.6-mm (5-μm) C18column; 35◦C; 2.0 mL/min (a) Sep-aration with an optimized linear gradient; (b) sepSep-aration 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 Maxmax-imum 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 sshown

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],

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: S3> S2; S8< S9; see the similar

examples of Fig 6.7c) Consider first an increase in gradient time from 5 to 20 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

9 relative to peak 8 decreases when gradient time is increased, and the resolution

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 )

Figure9.13 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

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 holdingΔφ/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 , a gradient 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:

whereφ* 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 × 10−4Ld c2 (units of L and d cin 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

or for S≈ 4 for small solute molecules,

k∗≈ 450t G F

Ld c2Δφ (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 rangeΔφ From Equations (9.4) and (9.5), we see also that k∗is related to the gradient-steepness parameter b:

k∗= 0.87

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 (k0) is small The value

of k0is given by

If k0is large, and if V D= 0,

t R=



t0

b



≈



t0

b



If k0is large, and if V D > 0,

t R=



t0 b

 log(2.3k0b + 1) + t0+ t D (9.9)



t0

b



Here t D = V D /F is the column dwell-time.

If k0is small, and if V D > 0,

t R=



t0

b

 log{2.3k0b[1−



t D

t0k0

 ]+ 1} + t0+ t D (9.10)

Equation (9.10) is valid, regardless of the values of k0 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 Dand

k0 For small values of k0,and V D= 0,

For small k0and V D > 0 (or any values of k0and V D),

2.3b[(k0/2) − (V D /V m)]+ 1

Thus a small value of k0 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 kw

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 k0 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 ratioβ = 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 b1can be calculated as

t R1 − (t R2 /β) − (t0+ t D)(β − 1)/β (9.13)

log k0=



b1(t R1 − t0− t D)

t0



Insertion of b1into Equation (9.4a) allows the calculation of a value of S, while log

k w is then calculable as log k0+ Sφ0

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 )Gt0



1+ 1

2.3b



(9.14)

≡ (4N ∗−0.5 )Gt0



1+k∗ 2



(9.14a)

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= 2.3k0b

k0+ 1

for large k0 G is then given in terms of p as

 (1+ p + [p2/3])

(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∗

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

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) Values of W 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 ∗1/2logα



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∗



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

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)

Table 9.2

Outline for the Development of a Routine Gradient Separation (Compare with Fig 9.14)

1 Review information on sample a Molecular weight>5, 000 Da? (see

Chapter 13)

b Mobile phase buffering required?

c Sample pretreatment required?

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 selectivityα* 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 Rs

or if very short run times are required, vary

conditions further in order to optimize

peak spacing (for maximum Rsor

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 Rs

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)