Chapter 4 Optimization Criteria

CHAPTER 4

OPTIMIZATION CRITERIA Before any optimization process can be started, the goals of the process should be defined unambiguously. For chromatographic separations this is not always a straightforward matter. A chromatogram is more than a simple unique number in one dimension. Nevertheless, we want to reduce the information contained in the chromatogram to a single number during the course of the optimization process. An additional complicating factor is that the goals of the chromatographic optimization may vary considerably from one case to another. For example, all peaks may need to be separated, or just some relevant peaks in a complex chromatogram; large series of presumably identical samples may have to be run in a quality control situation, or a screening method may need to be developed for a relatively large number of potentially present drugs or pollutants. The aim of this chapter is to translate such different analytical goals into objective functions, i.e. into different criteria which can be the objective goals of an optimization process. In the literature many different terms are used for such criteria: (chromatographic) response functions, objective functions or (chromatographic) optimization functions. Throughout the rest of this chapter, the neutral term optimization criteria will be used. 4.1 INTRODUCTION

In this introduction the possible goals of an optimization process will be investigated. In the following sections we will then try to translate these goals into simple mathematical algorithms. At the end of this chapter the different goals and the recommended optimization criteria will then be summarized. 4.1.1 Separation of two peaks

The resolution of two chromatographic peaks has been defined in terms of retention times and bandwidths in chapter 1: At R, = '/2( w, + w2) and it was shown that for symmetrical (Gaussian) peaks eqn.(4.l) can be transformed into a very useful equation (see section 1.5):

The fact that resolution as defined by eqm(4.1) can be related so elegantly to the fundamental parameters of the separation process (i.e. a, and N ) is a great advantage for the use of resolution (R,) to quantify the extent of separation of a pair of chromatographic peaks. However, in opting for R, we need to accept all characteristics 116

of this quantity, which may not always be advantageous. The following three characteristics appear to be relevant: 1. The resolution (R,) is independent of the (relative) height(s) of chromatographic peaks. This is a fair proposition only if we work in the linear range of chromatographic operation, i.e. if the peak height increases linearly with the injected quantity and the peak width remains constant. However, even then differences in the ratio of the two peak heights will lead to differences in the resolution (subjectively) observed by the chromatographer. A detailed description can be found in ref. [401], pp.34-48. 2. R , may be expressed in terms of fundamental chromatographic quantities (eqn.4.2) for symmetrical (Gaussian) peaks, but for such peaks only. Ideally, all chromatographic peaks fall within the above qualification. However, in the practice of GC and in particular LC, this is usually not true. 3. R, cannot easily be estimated from a chromatogram using eqn.(4.1), since this requires knowledge of both retention times and peak widths. Establishing the latter from the chromatogram requires tedious manual measurements or complex mathematical algorithms for integrators or computers. In either case, the resulting estimates for the peak widths are not usually very reliable. The use of eqn.(4.2) with a given number of theoretical plates yields more reproducible results, but again it assumes the peaks to be Gaussian. In this case only retention times need to be established from the chromatogram. N may be obtained from independent measurements or from a “test chromatogram”. It may be a fixed number, but also a function of the capacity factor (k). A disadvantage of the use of eqn.(4.2) may be a variation of N with time, for instance due to a gradual deterioration of the column. Such a process is not accounted for if the resolution is characterized without obtaining an up-to-date measurement for the peak width. In section 4.2. we will define other criteria that may be used to characterize the separation between a pair of adjacent peaks in a chromatogram (so-called elemental criteria). From the above it is obvious that the following three questions need to be addressed in this chapter: 1. Should the criteria used to characterize the extent ofseparation of apair of adjacentpeaks in a chromatogram be affected by the relative peak heights ? 2. Should the shape of the peaks be reflected in the value of the criterion ? 3. Can a criterion be defined which bears relation to fundamental chromatographic parameters, but is yet conveniently obtained from the chromatogram ? 4.1.2 Separation in a chromatogram

To some extent the quantification of the amount of separation in a chromatogram can be seen as an expansion of the characterization of the separation achieved for each pair of successive peaks. However, a straightforward expansion of a criterion for a pair of peaks, for instance a summation of individual R, values, may easily yield numbers that do not at all correspond to the chromatographer’s own (subjective) opinion of what constitutes a good chromatogram. This is easily illustrated by the two chromatograms shown in figure 4.1. These two 117

(a1

1

(bl 1

0

1

5

k-

Figure 4.1: Two schematic chromatograms,constructed with N = 10,000. The capacity factors of the peaks in the chromatograms are listed in table 4.1. Table 4.1: Resolution data for the chromatograms of figure 4.1 Chromatogram

4.1.a

4.1.b

Peak no.

k

1

1

2

1.I

3

1.25

1

1

2

1.1

3

5

RS

1.22 1.72

1.22

24.07

= R S

2.9

25.3

chromatograms are identical, apart from the position of the last peak. Table 4.1 lists the k values and the resolution factors (calculated from eqn.4.2) which correspond to the two chromatograms of figure 4.1. Also given in the table is the sum of all resolution factors in each chromatogram. Due to the improved resolution of the last two peaks, the sum of the resolution values is much higher for the bottom chromatogram, suggesting this one to 118

be vastly superior. However, as long as all peaks are of equal importance, every experienced chromatographer would prefer the top chromatogram to the bottom one, because it would yield a much shorter analysis time. The condition that all peaks must be of equal importance is relevant in this context. For example, an analyst who is only interested in the quantization of the last peak in the chromatogram might well prefer the bottom one. On a very much shorter column this would yield a very fast resolution of the last peak from the (unresolved) rest of the chromatogram. In the case that all peaks are (or must be) considered to be of equal importance, we will speak of the general case. If only a few peaks are of interest, or if some peaks are of more importance than others, we will speak of specific cases. The above discussion has led to the following three questions: 1 . How can we expand criteria that measure the resolution between apair of succesivepeaks to criteria that measure the quality of separation achieved in an entire chromatogram? 2. How should analysis time be reflected in such criteria ? 3. May we use the same criteria for the general case and for specific cases, or how should we adapt the criteria to serve these dzfferent purposes?

4.2 ELEMENTAL CRITERIA The resolution between two peaks has been defined in chapter 1 and this definition has been reviewed in section 4.1.1. In this section we will define and investigate various other criteria that may be used to quantify the extent of separation between a pair of adjacent peaks in a chromatogram. We will refer to these criteria as “elemental criteria”. Later in this chapter the elemental criteria will serve as the basis of criteria for judging the extent of separation in entire chromatograms.

4.2.1 Peak-valley ratios Three definitions of peak-valley ratios are illustrated in figure 4.2. All of them express the extent of separation as some measure of the depth of the valley between two peaks divided by some measure of the peak height. The first criterion (P) measures the depth of the valley relative to the interpolated peak height as shown in figure 4.2.a. The corresponding expression is:

P = f/g

(4.3)

where g is the interpolated peak height, i.e. the height to the baseline of the line connecting two peak tops at the location of the valley, and f i s the depth of the valley relative to the interpolated baseline. This criterion was suggested by Kaiser [402]. The second peak-valley ratio was suggested by Schupp [403] and is illustrated in figure 4.2.b. In this criterion, which we will refer to as the median peak-valley ratio (Pm),the depth is measured relative to of the valley at a point midway between two successive peaks the average peak height (g,) which equals the interpolated peak height at that point. The corresponding equation is:

urn)

119

\

P=flg

Figure 4.2: Three definitions for peak-valley ratios as elemental criteria to quantify the extent of separation between a pair of adjacent peaks in a chromatogram. (a) Peak-valley ratio ( P ; eqn.4.3) accordingto Kaiser, (b) median peak-valleyratio (P,; eqn.4.4)according to Schupp and (c) (opposite page) the valley-to-top ratio ( P ; eqn.4.5) according to Christophe.

120

p, = f , / g , .

(4.4)

The third criterion, the valley-to-top ratio ( Pv), was introduced by Christophe [404]. It measures the height of the valley relative to the height of either of the peak tops. Hence, for a cluster of two peaks as in figure 4.2, two values of P, can be obtained, one for each peak. If the ratio of the height of the valley (v) to the peak height (h) is subtracted from unity, the resulting definition is very similar to the previous two. It is illustrated in figure 4.2.c and the appropriate equation for P, is

P, =

1

-

v / h

(4.5)

The parameters used in eqns.(4.3), (4.4) and (4.5) are all illustrated in figure 4.2. All three definitions for peak-valley ratios are very similar. According to the last definition, a value for the valley-to-top ratio (P,)can be assigned to each peak rather than to each pair of peaks. However, in the case of two Gaussian peaks of equal heights all three definitions yield exactly the same results. Even if the relative peak heights vary, the first two definitions will still yield comparable results. The definition for P, implies that the value will be higher for the larger peak and lower for the smaller peak (proportionally to the relative height). The peak-valley ratios vary from zero for separations where no valley can be detected, to unity for complete separation. It ought to be noticed that a P value equal to zero does not necessarily imply that two solutes elute with exactly the same retention time (or k value). There is a threshold separation below which the presence of two individual bands in one peak only leads to peak broadening or deformation, without the occurrence of a valley. In these cases R , values are indeed not equal to zero, because by definition (eqn.l.14) R, is proportional to the difference in retention times. 121

Three characteristics of peak-valley ratios are: P can readily be estimated from a chromatogram. 2. In theory P will vary with varying relative peak heights (or areas) of the two peaks involved. A simulation for Gaussianpeaks reveals that this variation is small for both P and P,, but it is substantial for P, [405]. 3. Because of its pragmatic definition P automatically reflects peak asymmetry and it can be applied to peaks of all shapes, not exclusively to Gaussian peaks. 1.

'

For Gaussian peaks of equal height the value of the peak-valley ratio (then the same according to all three definitions) can readily be expressed in terms of R,. This can be done by relating the parameters f, g and v (see figure 4.2) to the parameters that describe a Gaussian peak (ISand h). For the first of a pair of Gaussian peaks (peak A) we can write (eqn.1.15): A ( t ) = h, exp

-

-'I2

(4.6)

while a similar expression holds for peak B. If we now substitute t = 1/2 ( t , + tB) and assume the values of IS for close peaks to be approximately equal (cr, z 0, FS 5)we find for the combined signal v (see figure 4.2.c): ' A +'B

v =g-f=A(*)+B(.) t + t , = h, exp

(t;i:) - + 2

-'/2

w (hA

+ h,)

= (h,

+

exp

h,exp

2 -'/2

2

-'I2

h,)exp -(2 R:)

and given that g = (h,

+

h,) / 2

eqns.(4.7) and (4.8) can be combined to yield

(4.10) 122

Eqn.(4.10) gives the relationship between P and R , for two Gaussian peaks, assuming that oAw a, and assuming that the position of the peak tops is not significantly altered because of peak overlap. For Gaussian peaks of equal height P= P,= P, and eqn.(dlO) applies to all three criteria. Calculations performed on simulated Gaussian peaks [405] confirm that both P and P, closely follow the theoretical curve described by eqn.(4.10), even when the relative peak heights vary. Clearly, for non-symmetrical peaks (e.g. typical solvent peaks) the value of P will be affected by the peak heights. However, if the first of a pair of peaks is a “solvent peak” it may be well-nigh impossible to use any of the definitions in figure 4.2 to establish either the peak-valley ratio (P) or the median peak-valley ratio ( P A from the chromatogram. Only a pragmatic ratio between the top of the observed peak and the height of the valley on the solvent front preceding the peak may be established from the chromatogram. Eqm(4.10) also provides insight into the threshold value for P for symmetrical peaks. According to eqn(4.10) P will be estimated as zero for

or R,y < 0.59

.

(4.1 1)

This figure applies to Gaussian peaks, but clearly, for peaks of other shapes there will also be some threshold value below which changes in the extent of separation will not be reflected in P. Wegscheider et al. [406] have modified P so that it will also reflect baseline noise:

P’

=f

/(g+2n)

(4.12)

where n is the (peak-to-peak) noise level on the baseline. According to eqn.(4.12), P‘ will decrease when the noise level increases, as well as when the absolute peak heights (reflected in f and g ) decrease. If noise is a significant factor, eqn.(4.12) may provide a more realistic evaluation of the merits of the actual separation than does eq~(4.3).Eqm(4.4) and (4.5) can be modified analogously to account for the influence of baseline noise on P, and P,. Because of the great similarity between the definitions for P and P,, we will not try to establish superiority of one over the other. To a large extent, the choice for one of them will depend on the software that is available to obtain P values from a chromatogram. The choice between P or P, on the one hand and P, on the other will be determined by whether or not an influence of the (relative) peak heights is wanted (see discussion below). 4.2.2 Fractional peak overlap An obvious criterion by which to judge the extent of separation of chromatographic peaks, especially for the optimization of a quantitative analysis, is the fraction of the peak that is free of overlap from adjacent peaks. The definition for this so-called fractional overlap criterion is illustrated in figure 4.3. An equation to describe the fractional overlap is 123

(4.13) where A, is the area of the nth peak and A,,,.-, and A,.,, the preceding and the following peaks, respectively.

FO =

, are the areas it shares with

An -An,n-1 -An.n+l An

Figure4.3: Illustration of the definition of the fractional overlap (FO) as a criterion for the separation of a pair of adjacent peaks in a chromatogram. Clearly, FO gives a good indication of the accuracy with which a peak can be quantitatively determined in a chromatogram. However, it is not the same as the error involved in quantitative analysis. The latter is affected not only by the extent of separation (reflected in FO), but also by the algorithms or programs used to establish the peak area. If the peaks are assumed to be Gaussian and if the exact peak positions and peak widths are known (the latter are very difficult to obtain accurately from a chromatogram), then FO can be calculated. But even then, the calculation is fairly complicated and simple equations relating FO to the difference in retention times and the standard deviations of the two peaks cannot be derived. For non-Gaussian, non-symmetrical peaks FO can only be estimated if the profiles of each of the individual peaks in the chromatogram can be established. This can be done in a purely mathematical way by “deconvolution”. This requires some mathematical function that describes the shape of the real peak with some degree of accuracy, and preferably also knowledge of the number of peaks actually present in the part of the chromatogram. It also requires complex computer programs. A more practical way to obtain the profiles of the individual peaks may be a sensible application of modem multichannel detection techniques (see section 5.6.3). It should be noted that neither mathematical deconvolution nor multichannel detection can be a

124

substitute for chromatographic separation. They only serve to illustrate that sophisticated techniques are required if FO is to be used as a criterion by which to judge the separation between adjacent peaks in a chromatogram. At present, therefore, the FO criterion seems to be a merely theoretical proposition. 4.2.3 Separation factor

In chapter 1 (eqn.l.20) we have seen that the resolution (R,) can be described as the product of two factors, one covering the chemical and physical characteristics of the separation and one reflecting the column efficiency: k,-k, .-fi (4.14) R s = k,+k,+2 2 The first factor in eqn.(4.14) combines the effects of the capacity factor ( k ) and the selectivity (a)on the resolution. To some extent, k and a can be varied independently for the purpose of optimization. Notably, k varies with the phase ratio (eqn.l.10) while adoes not. Hence, if the largest value for a is observed in conditions where k values are either too high or too low, variations in the phase ratio may be used to realize an optimum separation. The parameters that can be used for this purpose have been classified as “capacity parameters” in section 3.5. These parameters and the ways in which they affect the capacity factors are summarized in table 4.2. Table 4.2: Summary of parameters which affect retention (k), but do not affect selectivity (a). The proportionalities given assume all other parameters to be constant. Stationary phase Solid; CBP

Liquid

Column diameter ( k cc dc-’)

Open columns

Surface area ( k a S,)

Packed columns

I

(k a

E-’)

For various reasons the “capacity parameters” listed in table 4.2 will not often be used to optimize k values. 125

In the first place column characteristics will often be determined by practical conditions, such as availability of columns and materials and instrumental considerations. In the second place, the parameters listed in table 4.2 cannot always be varied independently and, moreover, will have side effects on yet other parameters. All the capacity parameters affect the phase ratio (V,/V,,J. If all other parameters are kept constant, then the film thickness and the surface area will affect V,, the porosity will affect V,,, and the diameter of open columns will affect both V,,, and V,. However, it is often impossible to keep all other parameters constant. For instance, it would be very difficult to vary the porosity without changing the surface area. An example of the effect of variations in the capacity parameters on other parameters is the decrease in the number of theoretical plates in the column that usually accompanies an increase in the stationary phase film thickness in GLC. Thirdly, the parameters in table 4.2 turn out to be proportional or inverselyproportional to k,whereas other parameters which affect both k and a, such as temperature in GC and mobile phase composition in LC, have an exponential effect on k (see table 3.10). Hence, even if higher a values can be obtained at some temperature or composition outside the range where k is optimal, chances are that the parameters listed in table 4.2 do not offer sufficient flexibility to move k values back into the optimum range. For all these reasons, it is usually realistic to treat R, as the product of only two independent factors according to eqn.(4.14). The first factor will depend on the retention ( k ) and the second factor will reflect the efficiency of the chromatographic system (N). Since we are most interested in small values for R, (i.e. k , M k J , the variation of N with k can be neglected as a first approximation*, and the two factors can be treated as independent. Therefore, we can define a separation factor independent of the column efficiency: S =

h-k, k,+k,+2

(4.15)

This separation factor was first suggested by Ober [407]an it has been used more recently by Jones and Wellington [408] and by Schoenmakers and Drouen [409,410]. For a given value of S, the number of plates required to realize a given value of R, (Nne)can easily be obtained from N,, = 4 ( R, / S

)2.

(4.16)

S has the advantagethat it is obtained from the chromatogram much more readily than is the case for R,. To establish the value of S no estimate of the peak width is required. Moreover, if we substitute k = (t- @ / t o in eqn.(4.15), we find that

(4.17) Hence, S can be obtained directly from the retention times of two successive peaks,

* While this may be true for a pair of adjacent peaks in the chromatogram, it may not be quite as valid an approximation if an entire chromatogram is considered (section 4.3). 126

without the use of an estimate for the hold-up time (to).Conversely, if S is calculated from k values, the to value used to obtain the latter will not affect the value of S. We may summarize the advantages of the separation factor S as follows: 1. S is directly related to chromatographic theory ( a s is RJ. 2. S can readily be obtained from the chromatogram. 3. N o estimate for the hold-up time to is required to establish S . The disadvantages are: 1. Use of S implies the assumption of Gaussian peaks. 2. The plate count N is assumed to be constant throughout (parts ofl the chromatogram (i.e. N is independent of k), as well as constant in time. The second disadvantage can largely be removed by expressing the plate count (N) as a function of the capacity factor (k). This has been demonstrated by Svoboda [41 I]. If N is not assumed to be a constant, but some function f(k) of k, then we assume that the peak-broadening process is determined by the properties of the column and the phase system and not by the properties of the solute (e.g. its diffusion coefficient). In other words, if two very different solutes elute with the same capacity factor, we would expect the widths of the two peaks to be the same. While this may not always be completely true, it appears that a useful refinement of the elemental criterion is possible in this way. The function f(k) will add an extra factor to eqn.(4.15): S’ =

k2 - k, k,+k,+2

*

f(k)

.

(4.1 5a)

However, the main reason for preferring S over R, (i.e. that no estimate is required for N) is now no longer relevant, and therefore, it is more appropriate to introduce the function f(k) into eqn.(4.14):

R, =

k, - k, k ,+k ,+2

.-f( k)1’2 2

(4.14a)

In order to estimate k (and hence f( k)), an estimate for the hold-up time to is required. However, this can be avoided if N is expressed as a function f( V,) of the retention volume. 4.2.4 Discussion

A comparison of various elemental criteria has been reported by Knoll and Midgett [412] and by Debets et al. [413]. Figure 4.4 shows the variation of some of the criteria for the separation of pairs of chromatographic peaks as a function of the time difference between the peak tops (At = t2 - t , ) . By definition, R, (and hence S) varies linearly with At. The peak-valley ratios (P)and the fractional overlap both increase rapidly with increasing AZat first, but level off towards At =: 4 (T to reach the limiting value of 1. At high values of At, R , and S will keep increasing, while the other criteria will not. Figure 4.5 shows the variation of the fractional overlap criterion with At for three different values of the ratio of peak heights (A). These data were calculated for Gaussian peaks. It is clear that FO will be lower for peak ratios different from unity. Similar 127

calculations reveal that P and P, are virtually independent of A (for Gaussian peaks), but that P, varies with A as expected. FO accurately describes the real extent of quantitative separation obtained in a chromatogram. If an analysis is optimized on the same column on which it will later be run as a routine separation, then this is a fair criterion. If however, the analysis will eventually be run on a different column of a potentially different length (or diameter, etc.), then it will often be hard to predict the value of FO on that other column. In the case where

2

L

AL 0

.-*

6

8

Figure 4.4: Variation of some elemental criteria as a function of the difference in retention times between the two solutes. Data calculated for Gaussian peaks of equal height. Courtesy of Anton Drouen [405].

'I

-0

2

6

8

Figure 4.5: Variation of the fractional overlap criterion (FO) and the resolution ( R J as a function of the difference in retention times between the two solutes. FO data calculated for Gaussian peaks of varying peak height ratios ( A = h,/h,). Courtesy of Anton Drouen [405].

128

FO FOequals equalsunity unitythis thisbecomes becomesquite quiteimpossible. impossible.InInother otherwords, words,given giventhe thefinal finalcolumn columnfor for routine routineanalysis, analysis,very verylarge largevalues valuesofofAt Atare areunattractive, unattractive,since sincethey theydo donot notincrease increasethe the value valueofofFO, FO,but butdo dolead leadtotoan anincrease increaseininanalysis analysistime. time.If, If,however, however,we wecan cantailor tailorour our column columntotothe theresult resultofofthe theoptimization optimizationprocedure procedure(i.e. (i.e.totothe thenumber numberofofplates platesrequired), required), then thenlarge largevalues valuesofofAt Atleading leadingtotovery verylarge largevalues valuesofofR,R,are areindeed indeedsignificant. significant.Hence, Hence, inin the the case case where where the the column column dimensions dimensionscan can be be chosen chosen after after completion completion ofof the the optimization optimizationofofselectivity, selectivity,the theuse useofofRR, ,ororSSisispreferred, preferred,because becauseofofthe theclear clearand andsimple simple relationship relationshipbetween betweenthese thesecriteria criteriaand andthe therequired requirednumber numberofoftheoretical theoreticalplates. plates. The Thesame sameargument argumentholds holdseven evenmore morestrongly stronglywith withrespect respecttotopeak-valley peak-valleyratios. ratios.Not Not only onlyisisthere thereaarange rangeofoflarge largeAt Atvalues valuesfor forwhich whichPP==1,1,there thereisisalso alsoaaconsiderable considerablethreshold threshold range rangeininwhich whichAt At>>0,0,but butthere thereisisno nodiscernible discerniblevalley valleybetween betweenthe thepeaks peaksand andhence hencePP==0.0. For ForGaussian Gaussianpeaks peaksofofequal equalheight heightthis thisthreshold thresholdrange rangewas wasshown showntotoequal equalaaresolution resolution ofof0.59 0.59ororless less(eqn.4.11). (eqn.4.11). IfIfthe theanalysis analysistotobe beoptimized optimizedinvolves involvesaasample sampleininwhich whichthe therelative relativepeak peakareas areasare are expected expectedtotobe beconstant constant(for (forinstance instanceininaaquality qualitycontrol controlsituation), situation),then thenaacriterion criterionmay may be beused usedthat thatisisaffected affectedby bythe therelative relativepeak peakheight height(A), (A), i.e. i.e.FO FOororPP, ,may maybe beused. used.IfIfthis this isisnot notthe thecase, case,then thenaacriterion criterionshould shouldbe beselected selectedthat thatdoes doesnot notvary varywith withAA(R,$ (R,$ ororS;S;PP ororPPJJ. .This Thiswill willavoid avoidthe thevery veryunattractive unattractivesituation situationininwhich whichthe thelocation locationofofthe theoptimum optimum isisaafunction functionofofthe the(quantitative) (quantitative)composition compositionofofthe thesample, sample,so sothat thatinintheory theorythere theremay may be bedifferent differentoptimum optimumconditions conditionsfor forevery everysingle singlesample! sample! This Thiseffect effectwill willbe bemost mostpronounced pronouncedininthe thecase casewhere whereaasolvent solventpeak peakdominates dominatesthe the chromatogram chromatogramand andsolutes solutesneed needtotobe beanalyzed analyzedon onthe thetail tailofofthis thispeak peak(see (seesection section4.6.3). 4.6.3). AAsimilar similarargument argumentholds holdsfor forthe theinfluence influenceofofthe thepeak peakshape shapeon onthe theseparation separationcriterion. criterion. InInthe thenon-linear non-linearpart partofofthe thedistribution distributionisotherm, isotherm,the theshape shapeofofthe thepeak peakwill willbe beaafunction function ofofthe theinjected injectedquantity. quantity.Hence, Hence,once onceagain, again,the thelocation locationofofthe theoptimum optimummay maybe beaffected affected by bythe thecomposition compositionofofthe thesample. sample.Also, Also,the theeffect effectofofcolumn columndimensions dimensionson onthe thepeak peakshape shape may maybe behard hardtotopredict, predict,and andthe thepeak peakshape shapemay maytotoaalarge largeextent extentbe bedetermined determinedby bythe the characteristics characteristicsofofthe theinstrument, instrument,rather ratherthan thanofofthe thecolumn. column.Therefore, Therefore,ififthe thecomposition composition (or (orthe theconcentration) concentration)ofofthe thesample samplecan canbe beexpected expectedtotovary varyconsiderably, considerably,and andififititisis desirable desirablethat that the theresult resultofof an anoptimization optimization process processcan can be beextrapolated extrapolatedtotodifferent different columns columns(of (ofthe thesame sametype) type)and andtotodifferent differentinstruments, instruments,then thenititisisadvisable advisabletotouse usecriteria criteria that thatare arenot notaffected affectedby bythe therelative relativepeak peakareas, areas,nor norby bythe theshape shapeofofthe thepeaks. peaks. For Forpractical practicalevaluation evaluationFO FOisisaavery veryunattractive unattractivecriterion. criterion.Its Itsvariation variationwith withAt Atand and with withthe thepeak peakarea arearatio ratioAAisissimilar similartotothat thatofofthe thepeak-valley peak-valleyratio ratioP,. P,.Pand PandPP, ,are aresimilar similar totoeach eachother otherininall allrespects. respects.PP, , may maybe beobtained obtainedfrom fromthe thechromatogram chromatogramslightly slightlymore more easily easilythan thanP,P,because becauseititonly onlyrequires requireslocation locationofofthe thepeak peaktops, tops,and andnot notofofthe thevalleys. valleys.To To calculate calculateRR, ,from fromthe thechromatogram chromatograman anestimate estimateofofNNisisrequired. required.Scan Scanbe beestimated estimatedvery very easily, easily,using usingonly onlythe theretention retentiontimes timesofofindividual individualpeaks. peaks. Below Belowaacertain certainthreshold thresholdresolution, resolution,no novalley valleycan canbe beobserved observedbetween betweentwo twoadjacent adjacent peaks peaksininaachromatogram. chromatogram.InInthat thatcase casethe thevalue valuefor forany anyofofthe thepeak-valley peak-valleyratios ratioswould would equal equalzero. zero.InIntheory, theory,the thevalue valuefor forR,R,and andSSwould wouldexceed exceedzero zerofor forany anytwo twopeaks peaksthat that have havedifferent differentretention retentiontimes times(At (At>>0). 0).InInpractice, practice,this thisdifference differencevanishes vanishesififthe thepresence presence ofoftwo twopeaks peakscannot cannotbe bediscerned discernedfrom fromthe thechromatogram. chromatogram.However, However,the theoccurrence occurrenceofof ill-resolved ill-resolvedpeaks peaksininaachromatogram chromatogrammay maybe berecognized recognizedvisually visuallyatatresolutions resolutionswell wellbelow below 0.6 0.6(the (thethreshold thresholdvalue valuebelow belowwhich whichPPequals equalszero zerofor forGaussian Gaussianpeaks peaksofofequal equalheight) height) (see (seeref. ref.[401], [401],figure figure2.1 2.11,1,p.38). p.38).Moreover, Moreover,there thereare areseveral severaltechniques techniqueswhich whichmay maybe beofof 129 129

help in confirming the purity of the peaks obtained in a chromatogram during an optimization process (see section 5.6). In some optimization procedures the capacity factor is known as a function of the parameters. These so-called “interpretive optimization methods” will be described in section 4.5. From known capacity factors R, and S can be calculated much more easily than peak-valley ratios and, moreover, from known capacity factors the R, or S values can be calculated, no matter how small the difference between the two capacity factors is. In other words, the resolution of a pair of peaks can be calculated in a range in which it would be very difficult to obtain an estimate for the resolution from an actual chromatogram. Therefore, the use of R, or S as a criterion to judge the separation in combination with interpretive optimization methods enables us to appreciate variations in the resolution in the range of 0 < R,< 0.6. Such variations are very significant because (i) on a different (more efficient) column the separation with the highest value for R, is most easily realized and (ii) on the same column, improvements in resolution in the range 0 < R, < 0.6 will help to send the optimization process in the right direction. Hence, in combination with interpretive methods the use of R, or S as the resolution criterion appears to be always advantageous. A further refinement may be sought by incorporating a function f(k) to describe the dependence of the plate count on the capacity factor (see eqns. 4.14a and 4.15a). The characteristics of the different criteria are summarized in table 4.3. Table 4.4 lists the recommendations formulated above for the use of different criteria. Table 4.3: Characteristics of different elemental criteria for measuring the extent of separation of a pair of chromatographic peaks. Criterion

Affected by Peak area ratio

Reflects actual separation

Transfer towards other columns

Ease of calculation

+/+/-

++ ++

+/-

-

+ + +

+ + +

+/+/-

+

++

-

Peak shape

P pm P“

FO

-

+ +

-

++

(I) (1)

+/-

+/+/--

(1) Indirectly via eqn.(4.10), but only in the range where (approximately) 0.05 < P<0.95.

130

Table 4.4 Recommendations for the use of different elemental criteria for measuring the extent of separation of a pair of chromatographic peaks. The preferred criteria are given, while possible alternatives appear in brackets. Optimization on final analytical column YES Interpretive method (1) YES

NO

NO

(1) See section 5.5. (2) The noise level can be incorporated in analogy to eqn. (4.12).

4.3 CHROMATOGRAMS We will base our discussion about criteria by which to judge the quality of an entire chromatogram on the elemental criteria for pairs of chromatographic peaks, which have been defined in chapter 1 (R, and a) and in the previous section . We will look at several ways of combining the numbers for all individual pairs of peaks into a single number. We will then discuss the influence of other parameters, such as the analysis time and the number of peaks on the proposed criteria. Initially, the discussion will be focussed on the general case (see section 4.1), in which all peaks in the chromatogram are considered to be of equal importance and all peaks have to be separated. At the end of this chapter, we will discuss some specific cases, for which the requirements are different. 4.3.1 Sum criteria

Summation of resolution values has been used by Berridge [414] and summation of separation factors has been suggested by Jones and Wellington [408]. In the introductory section of this chapter it was shown that a straightforward summation of resolution values does not yield a satisfactory criterion for the evaluation of complete chromatograms (see figure 4.1 and table 4.1). A problem that can readily be appreciated from the example given there, is that the sum of R, values will be determined mainly by the largest values of R, that occur in the chromatogram. For example, in the 131

chromatogram shown in figure 4.lb, the sum of resolutions turns out to be 25.3. Of this, 24.07 or 95% is due to the excessive separation of the last two peaks. However, in judging the separation, the chromatographer will immediately refer to the separation of the first two peaks, rather than to that of the last pair. This is correct, because the first two peaks determine the efficiency of the chromatographic system that is required to realize the separation of all three peaks (see also section 4.3.3 below). Apparently, it is the occurrence of very large R, values that causes problems. Obviously, this problem can then be avoided by substituting for R, one of the criteria which level off for very large time differences between the two peaks (see figure 4.4). In this way, the contribution of the abundantly separated pairs of peaks in a chromatogram is limited. The resulting sum of FO or P values is to a much smaller extent determined by the largest values, although in some extreme cases large contributions may still obscure important changes in small ones. For example, if 20 pairs of peaks were to occur in a chromatogram and each of these pairs were almost separated to the baseline (P=O.9), then the sum ZP would equal 18. This is less than if 19 out of the 20 pairs of peaks were amply separated (P= l), but two peaks showed complete overlap (P= 0), giving rise to a value for ZP of 19. Assuming that all peaks are of equal importance, the latter chromatogram is obviously inferior. Of course, this is a hypothetical example, but it illustrates a potential limitation of the criterion ZP. Figure 4.6 illustrates the dependence of ZP and ZR, on the number of plates in the

50

fi-

100

Figure 4.6: Variation of the sum of peak-valley ratios as a function of the number of plates for the two chromatograms( a and b) shown in figure 4.1, and for a third chromatogram(c), shown in figure 4.8. P was calculated from eqn.(4.10). Negative values for P were set equal to zero. The sum of resolution values is shown as a dashed line for chromatogram a only.

132

column. This figure has been calculated from the data in table 4.1, assuming that the chromatograms of figure 4.1 have been run on a series of columns with different Nvalues. Obviously (eqn.1.22), ZR, increases linearly with V N . ZS (not indicated in figure 4.6) is independent of N (see eqn.4.15) The behaviour of ZP is vastly different. When there are no plates ( N = 0), there is complete overlap of all peaks (ZP= 0). For chromatogram b this soon changes, since only a handful of plates is necessary to achieve baseline separation of the last two peaks. For a while, the ZP value for chromatogram b is then larger than that for chromatogram a, because two peaks are distinguishable instead of one. However, at still higher values of N the last two peaks in chromatogram a start to be resolved, soon followed by the first two peaks in both chromatograms. Eventually, abundant resolution will be achieved for all pairs of peaks, all values of P will equal unity and the ZP values for both chromatograms will be the same (ZP= 2). From this point on the use of ZP no longer enables us to quantify the quality of the chromatogram, because it does not differentiate between chromatograms a and 6. Moreover, above a certain threshold number of plates (around 10,000 in the example of figure 4.6) the ZPcriterion becomes very insensitive to the number of plates and to changes in the relative peak positions, unless these changes have a significant effect on R , values between about 0.6 and 1.5. From the above discussion the following five conclusions can be drawn:

1. ZR, is not a useful criterionfor judging the quality of a chromatogram, since its value

2.

3.

4.

5.

is determined largely by the largest values of R, that occur in the chromatogram. i.e. by thepairs ofpeaks which are the least relevantfor the realization ofa separation. Thesame is true for the sum of separation factors (ZS). ZP gives a better representation of the actual separation achieved on a given column, since there is a limit to the contribution of amply resolvedpairs ofpeaks. Above a certain threshold value for the number of plates, a pair of peaks will become irrelevantfor the determination of ZP. When this threshold is reached for all pairs of peaks. Z P will have reached its limiting value. Changes in N will no longer be reflected in ZP, whereas changes in the (relative) retention times become increasingly irrelevant as N increases. Below a certain threshold value for the number ofplates, all P values will be zero and again ZP becomes insensitive to changes and provides no information about the chromatogram. The values of Z R , and ZS on another column with a different plate count can easily be predicted, since ZR, is proportional to V N and since ZS is independent of the plate count. For ZP, the situation is less straightforward. The value of ZP is relevant for the separation on a given column, but cannot be extrapolated from there.

As the main conclusion from this section it appears that ZR,(and hence ZS)is not a useful criterion, and that ZP may be used for a comparison of chromatograms on a single column. A problem that remains is the fact that ZP becomes a very insensitive criterion once the limiting value (equal to n - 1 if n is the number of peaks in the chromatogram) is approached, as well as in the range in which one or more of the P values become equal to zero. In section 4.3.4 we will investigate whether composite criteria (involving for instance the analysis time) can be used to avoid this problem.

133

4.3.2 Product criteria ress the extent of separation in an A second major category of proposed criteria to entire chromatogram is that in which the product is taken f the values for all pairs of peaks of one of the elemental criteria defined before. Taking the products of these criteria is equivalent to taking the sum of the logarithms, for instance

\

lIR,=exp(ZInR,).

(4.18)

The use of the sum of logarithms may have a slight disadvantage in the case where a value of zero occurs for one of the pairs of peaks. If any of the peak-valley ratios (P,P, or Pv) is used, then this problem is aggravated because these criteria take on a value of zero below a certain threshold resolution. The obvious way around this problem, however, is to set the sum of logarithms equal to minus infinity or to a large negative number once a value of zero occurs. A summary of the use of product functions in the literature is given in table 4.5. In some cases, the products were part of composite criteria involving other factors or terms. We will come back to these criteria in section 4.3.4. It is clear from table 4.5 that product criteria have been used more extensively than have sum criteria. Table 4.5: Summary of product criteria proposed in the literature to express the extent of separation achieved in a chromatogram. Elemental criterion

Product criterion proposed by

Ref.

RS

Glajch et al., JC 199, (1980),57

415

S

Schoenmakers et al., Chr 15, (1 983),688

409

FO

Smits et al. ZAC 273, (1975),1

416

P

Morgan and Deming, JC 112, (1975),267 Watson and Carr, AC51, (1979),1835 Spencer and Rogers, AC 52, (1980),950

417 418 419

P'

Wegscheider et al., Chr 15, (1982),498

406

Explanation of abbreviations: A C = Analytical Chemistry Chr = Chromatographia JC = Journal of Chromatography ZAC = Fresenius Zeitschrift fur Analytische Chemie One obvious advantage of product criteria is that the result will be mainly determined by the smallest values for the elemental criterion, i.e. by the least resolved pairs of peaks. 134

All the criteria that have been discussed in section 4.2 will equal zero if one pair of peaks shows complete overlap and therefore once such a situation occurs the resulting product will be zero as well.

I

/

5

/

-

~ 1 1 0 ~

10

Figure 4.7: Variation of the product of peak-valley ratios as a function of the number of plates for the two chromatograms ( a and b) shown in figure 4.1, and for a third chromatogram (c), shown in figure 4.8. P was calculated from eqm(4.10). Negative values for P were set equal to zero. The product of resolution values is shown as a dashed line for chromatogram a only.

Figure 4.7 shows the variation of the products of R, and P values as a function of the number of theoretical plates for the chromatograms of figure 4.1 and an additional chromatogram shown in figure 4.8. Plotting the products against N yields a straight line for nR,.In general, l7R, will be proportional to N, where p is the number of pairs of peaks in the chromatogram. Therefore, the differences between l7R, and l7P will be more pronounced for chromatograms with large numbers of peaks. As with the sum criteria, the use of P instead of Rs does not yield a simple relationship for the variation with N. However, it is very clear from figure 4.7 that the differences between the l7P values for the chromatograms a and b are much smaller than the differences in the ZP values (figure 4.6). This illustrates that the value of l 7 P is mainly determined by the least separated pair of peaks, i.e. the first two peaks, which are the same in both chromatograms. All product criteria will be zero if any single pair of peaks is completely unresolved. For FO, R, and S this situation theoretically only occurs if the retention times of two peaks are equal. For peak-valley ratios a value of zero is estimated from the chromatogram below a certain threshold sewration, which for Gaussian peaks corresponds to R, < 0.59 135

(eqn.4.11). Thus, l7P equals zero once any pair of peaks shows no discernable valley. Of course, there is an important increase in separation if R, is increased from 0 to 0.6. This is a serious drawback to the use of l7P as an optimization criterion, since it does not acknowledgedefinite improvements below a certain threshold and it illustrates once more that the elemental criterion P may be used only if the optimization process is carried out on the final analytical column. Even more than ZP, l7Pis a threshold criterion. Its value is zero or one, with only a small range over which intermediate values occur. Threshold criteria can be used to allocate areas in which a certain condition is fulfilled. They divide the parameter space (i.e. the space formed by all the parameters considered in the optimization process, see section 5.1.3) into areas where a certain condition is met and areas where this is not the case. In the case of l7P there is a diffuse boundary in between the different areas. So far, all the criteria that have been discussed have suggested either that chromatogram b is superior to chromatogram a, or that both chromatograms have equal credentials. The

0

k -

Figure 4.8 Three schematic chromatograms.Constructed for N = 10,000. Capacity factors are listed in table 4.6.

136

main reason for this is that the capacity factors of the last peaks in the chromatograms in figure 4.1 are vastly different, i.e. we are comparing two chromatograms which on the same column under identical conditions would show vastly different analysis times. Clearly, some correction is required once this is the case. Figure 4.8 shows the two chromatograms of figure 4.1, together with a third chromatogram (c), in which the capacity factors of the first and the last peak equal those observed in chromatogram b, but the separation of the first two peaks has been improved dramatically. Table 4.6 lists the data for all three chromatograms. The capacity factors are Table 4.6: Data for capacity factors, elemental criteria and for criteria judging the extent of separation in the entire chromatograms. Chromatograms are shown in figure 4.8. Criteria for pairs of peaks: separation factor (S, eqn.4.15), resolution ( R , eqn.4.14) and peak-valley ratio ( P , eqn.4.10). Criteria for entire chromatograms: sum criteria (section 4.3.1), product criteria (section 4.3.2), normalized resolution product (r, eqn.4.19), calibrated normalized resolution product (r*, eqn.4.21) and minimum resolution (section 4.3.3). For discussion see text. Chromatogram

a

Peak number

k

1

1

2

1.1

3

1.25 2

n b

1

1

2

1.1

3

5

z:

n C

1

1

2

2.5

3

5

z:

n

S

R S

P

0.0244

1.22

0.898

0.0345

1.72

0,995

0.059 8.4.10-4

2.9 2.1

1.893 0.893

0.0244

1.22

0.898

0.481

24.1

1

0.51 1.2.10

25.3 29.4

1.898 0.898

0.273

13.6

1

0.263

13.2

1

0.54 7.2.10

26.8 180

2

1

r = 0.97 r* = 0.13 Rs,min = 1.2

r = 0.18 r* = 0.18 Rs,min = 1.2

r = 1.00 r* = 0.98 Rs,min = 13.2

137

,

used to calculate S, R, and P. The latter is estimated from eqn.(4.7). Also, the various sum and product criteria are shown in the table. Since there is a constant factor of 50 (vN12) between S and R , we will focus on R, and P only. In all three chromatograms the threshold number of plates for separation appears to have been approached, so that CP and are only slightly affected by the differences between the chromatograms. No distinction can be made between the sum criterion CP and the product criterion nP, The difference between the two is 1 for all three chromatograms. Both criteria yield marginally higher values for chromatogram b in comparison to.chromatogram a. Chromatogram c yields the maximum values of 2 for ZP and 1 for nP.In fact, it is well into the region in which the peak-valley ratio is completely insensitive to variations in the capacity factors. If the resolution (R,) equals 2, then eqn.(AlO) yields a Pvalue of 0.999. In chromatogram c of figure 4.8 the resolution between each pair of peaks is about 13. ZR, is much higher for chromatograms band c than it is for chromatogram a. However, it is about equal for the bottom two chromatograms. Hence, ZR, is more sensitive to changes in the capacity factor of the last peak than it is to changes in the extent of separation. n R , does yield a much higher value for chromatogram c than it does for chromatogram b. Hence, nR, can be used for a quantitative comparison of chromatograms of similar length (capacity factor of the last peak). When the length of the chromatogram changes (for instance in going from chromatogram a to chromatogram b), nR, is not a useful criterion. Normalized resolution product

Drouen et al. [410] have recognized this problem and proposed a solution by using a product of normalized resolution values. They divide every value of R, by the average R, value where the average is taken over all the pairs of peaks in the chromatogram:

(x,),

(4.19)

where n is the number of peaks and (4.20) The average S value (3 is defined analogously. Because both R,sand are proportional to V N , the normalized product of R, values is equal to that of the S values, and both are independent of the number of plates. The normalized resolution product ( r ) will vary from zero, in the case where one or more pairs of peaks show no resolution, to one, if the resolution is equal for all the pairs of peaks in the chromatogram. Therefore, in choosing r as the optimization criterion the aim is to achieve an equal distribution of the peaks over the chromatogram. This aim is realized much better in chromatogram c in figure 4.8 than it is in chromatogram b. The r values given in table 4.6 illustrate that fact. For chromatogram c the r value is (almost) equal to 1, while for chromatogram b it is not larger than 0.18.

x,$

138

Chromatogram a is very different from chromatogram c, but it also shows good spacing of the peaks over the chromatogram and hence an r value of about 1. Clearly, r values do reflect the distribution of the peaks over the chromatogram and are not seriously affected by the absolute k values. Chromatogram c appears to show a much longer analysis time than does chromatogram a. However, if we are free to define the column dimensions after the selectivity optimization process, chromatogram c can be the basis for a very quick separation on a very short column.

Calibrated normalized resolution product In theory, all peak pairs may show equal R , values, but the peaks may occur very late (with very high capacity factors). A bunch of peaks may move about through the chromatogram and will yield the same value for any of the criteria discussed so far as long as the mutual resolution factors between all the different pairs of peaks remain unaltered. To avoid bunching of the peaks at some high value of k, Drouen et al. have suggested the inclusion of a hypothetical peak at t = to in the calculation of r [410]*. This yields the calibrated normalized resolution product (r*),defined as n-I

r* = ll (R,i,i+,/ i=O

n-1

R,) = ll

i=O

(Si,i+l1s)

(4.21)

where (4.22) The r* values for the three chromatograms in figure 4.8. are also shown in table 4.6. It appears from these values that r* is very high (close to the maximum value of 1) for chromatogram c, but that the bunching of peaks around k = l in chromatogram a is effectively penalized. The introduction of the hypothetical peak at t = to has the effect of “calibrating” or “anchoring” the real peaks in the chromatogram to a starting point. The optimum value of r (or r*) does not correspond to a unique chromatogram, but rather to a series of chromatograms, each of which has the peaks spread out at constant resolution intervals in the chromatogram. In other words, the absolute value for R, or S may vary, but all the normalized values are equal to 1. r* is higher for chromatogram b than it is for chromatogram a. If we include a peak at t = to, then the value for S between this peak and the first real peak in the chromatograms is 0.33 (R, = 16.7; P = 1). This means that for chromatogram a one high value for S occurs (0.33) in combination with two low values (0.02 and 0.03), while in chromatogram b two high values (0.33 and 0.48) are combined with one low value (0.02). If the goal is to make all S values equal, both situations are equally bad. Nevertheless, because of the much smaller k value for the last peak (and given the equal resolution for the first two real peaks) chromatogram a may well be preferred to chromatogram b.

* A hypothetical peak assumed at any time after t = to [410] will give rise t o considerable problems, because peaks may be distributed over the chromatogram before as well as after this imaginary peak. 139

Although r* appears to be very useful as a criterion that strives towards a clear and objective goal in selectivity optimization, it is still not perfect once two chromatograms are compared which show very different capacity factors for the last peak, especially when both r* values are low. The use of P , in sum and product criteria

The use of P, in sum or product criteria creates a special problem, because two values for P, can be calculated for each peak in the chromatogram. This applies to isolated as well as to ill-resolved peaks and it also applies to the first and the last peaks observed in the chromatogram. The number of P, P,, R,, or S values in an entire chromatogram usually equals n-1 (where n is the number of peaks). If an imaginary peak is assumed to be present at t = to, the number of values for the elemental criteria becomes n. We can deal with this problem in three ways. 1. Use all P, values without correction. This is a useful approach if no other criteria are to be considered, so that the criteria based on P, will not have to be compared with other criteria. 2. Use of the lowest value for P, that occurs on either side of a peak. A considerable disadvantage of this approach is that large improvements in the resolution may go by unnoticed. 3. Use of an average value for P,. If two values for P, for each peak are obtained, then this strategy corresponds to the use of (112) ZP,or VnP,as the optimization criterion. This third approach appears to be the most correct one, since it creates a common basis for all sum and product criteria (i.e. those based on P, P , P, R , and S), which may allow a comparison between the different propositions. Another peculiar aspect of the use of P, is that its value is by definition proportional to the height of the peak (see also section 4.2.4). Hence, for a pair of peaks with a certain valley between them, the P, value will be largest for the largest peak and proportionally smaller for the smallest one. Since sum criteria are mainly determined by the largest values for the elemental criteria (see section 4.3.1), we may expect that the value of ZP,will be affected most by the largest peaks (major components) in the sample. On the other hand, product criteria are affected most by the smallest values for the elementa1 criteria (see section 4.3.2) and hence n P , will be determined mainly by the minor components in the sample, i.e. the smallest peaks which are detected and considered relevant in the optimization process. Because one very small value (close to zero) for P, will largely determine the value for the entire product, this emphasis on small peaks is much greater than the emphasis put on the large peaks by the use of ZP,,. We might say that if P, is used as the elemental criterion, a weighing factor is automatically built into the (sum or product) criterion for the entire chromatogram, which puts the emphasis either on the major or on the minor components in the sample.

4.3.3 Minimum criteria The lowest value for a which occurs in a chromatogram has been used extensively in GC [420] and LC [421-4231 (see also section 5.5) as a criterion to quantify the extent of 140

separation achieved in a chromatogram. This so-called minimum a (amin) criterion is set equal to the lowest value for (I that occurs for any pair of peaks in the chromatogram. However, the value of a is not a good indication for the separation of a pair of peaks. For example, if an a value of 1.05 occurs somewhere early in the chromatogram, say around k = 0.5, then the corresponding S value is 0.008 and 60,000 plates are required to achieve adequate resolution (R,= 1) of the two peaks. If the same value for awere to occur around k = 3 , then S would equal 0.018 and to realize an R, value of 1 about 12,000 theoretical plates would be sufficient. Clearly, it is advisable to substitute R, (a quantity which depends on the plate count) or S (independent of N) for a. In judging a chromatogram on the basis of the*minimum it becomes very easy to estimate the number of plates that value for R, ( Rs.min)or S (Smin), is required to realize the separation with sufficient but not excessive resolution. For instance, if the final result of a selectivity optimization process is a chromatogram with an Rs.minvalue of 0.5 on a column with 2,500 theoretical plates, then a column with 10,000 plates will yield an Rs,minvalue of 1 under identical conditions. criterion can be used in two During a selectivity optimization process the Rs.minor Smin ways: 1. By setting a threshold value (e.g. Rs.min= I), above which the result is acceptable. If x is the threshold value, we can describe this criterion as (4.23) This criterion may be used during a sequential optimization process (see chapter 5), leading to an acceptable result and to completion of the optimization process once the threshold value has been reached. Alternatively, it may be used to establish ranges of conditions in the parameter space for which the result will be acceptable. This latter approach has been followed by Glajch et al. 14151, by Haddad et al. [424] and by Weyland et al. [425] and was referred to as resolution mapping by the former. Within the permitted area(s) secondary criteria are then required to select the optimum conditions. For example, the conditions at which the k value of the last peak ( k J is minimal while the minimum value for R,sexceeds1may be chosen as the optimum. Such a composite criterion can be described as min k ,

f l

Rs,min> x .

(4.24)

2. A second way to use the Rs,mincriterion is to try and maximize the value for Rs.min. In other words, one may strive towards conditions at which the lowest value for R, observed in the chromatogram is as high as possible. We can describe this criterion as (4.25) This criterion aims at a chromatogram that can be realized with the lowest possible number of theoretical plates. Indeed, if the highest possible value of Rs.minhas been reached, this automatically corresponds to the lowest number of ptates required (see section 4.4.3). Although the goat of achieving the separation with a minimum number of plates appears 141

to be clear and unambiguous, the resulting chromatogram is not well-defined.The fact that Rs,min has the highest possible value reveals nothing about the remainder of the chromatogram. A very simple example can be found from the two chromatograms in figure 4.1, which yield the same value for Rs,min. Moreover, merely because of the direct relationship between R, and k (see section 1S), R, values will tend to be larger for larger values of k. Hence, in striving towards a maximum value for one may be striving implicitly towards very high values for k. Therefore, the criterion described by eqn.(4.25) should only beconsidered as a criterion for a selectivity optimization process if the overall capacity factors are not expected to change considerably. In those cases in which the overall capacity factors do vary, it is more realistic to use the Rs,mincriterion in the way as described under 1. above. Eqn.(4.23) can be used to define the boundaries within which acceptable resolution can be achieved. It has the advantage that no implicit aim towards high capacity factors is present in the criterion. A disadvantage of the use of eqn.(4.23) is that the boundaries defined by the threshold criterion will change when the acceptance level is changed, i.e. they will be different if x in eqn.(4.23) is set equal to 1 from when x= 1.5. Along the same line, the boundaries will change when the number of plates is changed, i.e. when another column is used, or even when the flow rate is changed on a given column. If subsequently secondary criteria are used to define a unique set of optimum parameters rather than an acceptable range (eg. the hierarchic criterion of eqn.4.24), then the location of the optimum may very well depend on the threshold value (x) selected by the user, or on the column used during the process of selectivity optimization. For example, if the following criterion had been used min k, n Rs,min > 1

(4.26)

and the resulting chromatogram had an Rs,minvalue of 1.3, then necessarily a different optimum would have been found than if the criterion min k, n Rs,min > 1.5

(4.26a)

had been used, or equivalently if the original optimization process were repeated on a column having a factor of 2.25 (1.5*) fewer theoretical plates than the original one. In theory, the problem of the result being dependent on the number of plates used during the selectivity optimization process can be circumvented by setting a limiting value for S, rather than for R,. Nevertheless, the problem that the resulting location of the optimum will depend on the arbitrarily selected value for the threshold x still remains. There seems to be no logical way to define a single unambiguous value for Sminwhich can be used in all cases. It is interesting at this point to notice the similarity between the use of l 7 P and the threshold criterion of eqn. (4.23). In figure 4.9 this is illustrated by plotting IlP,n R , and Rs,min > 1 as a function of N for the two chromatograms of figure 4.1. Clearly, both l7P and Rs,minserve as threshold criteria, with the boundaries of the acceptable areas being abrupt when the minimum resolution criterion is used and diffuse for n P . This invites the use of IlP in a similar way as i.e. analogous to eqn.(4.24). The combination of UP with other factors to form composite criteria will be discussed in section 4.4. 142

Figure 4.9: Variation of the product of peak-valley ratios, (IIP), the product of resolution values (IIR,) and the minimum resolution (R,,in) criteria as a function of the number of plates for the chromatograms shown in figure 4.1. P was calculated from e q ~ ( 4 . 1 0 )Negative . values for P were set equal to zero.

Although the value of R , for a given pair of peaks can be quickly transferred from one column to another by using the proportionality of R, and V N , this is not the case for the threshold criterion of eqn.(4.23). The problem is that if we know the boundaries of the area for which Rs.min > 1 using a column of 10,000 plates, we only know the boundaries of the area for which Rs,min> 0.5 for a column with 2,500 plates. We d o not know what the boundaries for Rs.min > 1 are in the latter case, because we have no information on how the value of Rs.minchanges with variations in the parameter settings. Only if the variation of the capacity factors as a function of the relevant parameters is known, can the boundaries of the area in which the resolution is adequate be calculated for different columns with different numbers of theoretical plates. Optimization methods in which this is the case (so-called “interpretive methods”) will be discussed in section 5.5. The following conclusions can be drawn from the discussion in this section: 1. The threshold criterion of eqa(4.23) can be used to establish acceptable areas for adequate separation on a given column. 143

2. The use of this criterion gives results that are similar to the use of IlP. 3. The areas defined by the threshold criterion are not transferable from one column to another, unless this is done indirectly by means of known capacityfactors.

4. Maximization of the minimum resolution value observed in the chromatogram(eqn.4.25) corresponds to aiming at the minimum number of plates required to eflectuate the separation. 5. This criterion can only be used ifthe overall capacityfactors are roughly constant during the optimization process. 6. This criterion can readily be transferredfrom one column to another. 4.3.4 Other criteria

The first criterion to be suggested for the evaluation of the quality of an entire chromatogram was defined as the “total overlap” by Giddings in 1960[426]. The definition reads: cp =

Z: exp -(2 R , ) .

(4.27)

In figure 4.10 the function exp -(2 R,) for a single pair of peaks has been plotted as a function of R,. Also shown in this figure is the theoretical value for P (eqn.4.7). Figure 4.10 shows that the two functions are roughly complementary, although the variation of P with R, is more abrupt. This is logical when we consider the theoretical

Figure 4.10: Variation of Giddings’ peak overlap criterion (p) for one pair of peaks and the peak-valley ratio (P) as a function of the resolution ( R J of the pair. P values were calculated from eqn.(dlO). Negative values for P were set equal to zero.

144

relationship between P and R, (eqn.4.10) in which the term 2 exp -(2 R:) appears. In comparison with eqn(4.27) the change of P with R, will be more abrupt than that of cp (for one pair of peaks) because of the ocurrance of the square of R, in eqn.(AlO). Roughly speaking, for a complete chromatogram, the criterion Q behaves similarly to ZP. It functions as a threshold criterion with diffuse (and stepwise) boundaries, establishing areas for which adequate separation is obtained (cp=O). Because it is based on R, rather than on P, cp cannot easily be determined from the chromatogram. On the other hand, cp may more easily be calculated if the capacity factors and the plate number are known. Both cp and ZP should only be used for optimization processes run on the final analytical column. In the following discussions cp will not be considered as a separate criterion. Its merits correspond to those of the ZP criterion. 4.3.5 Summary

In the preceding four sections we have discussed various “sum”, “product” and “minimum” criteria. A schematic summary of these criteria and an indication of their applicability are given in table 4.7.

Table 4.7: Summary of sum, product and minimum criteria. Glossary: Continuous criterion, transferable to other columns tha,thf Threshold criteria, which locate boundaries at arbitrary (a) or fixed (f) degrees of separation Not useful as an optimization criterion. X C

Criterion

Sum

Product

Minimum eqn. 4.25

eqn. 4.23

Rs S

P (1) Normalized R , or S values should be used if the capacity factor of the last peak is expected to vary. (2) Only to be used if the capacity factor of the last peak is expected to be constant. (3) Use of l 7 P is to be preferred rather than ZP.

It is seen in the table that three groups of criteria are readily discernible: 1. The useless criteria. This category comprises ZR, and ZS and needs no further comments.

145

2. The criteria which vary continuously between a low value of zero and a maximum

value located at the optimum conditions. All these criteria allow a transfer of the resulting optimum from one column to another. 3. The threshold criteria. which may be used to define the boundaries of areas in which an acceptable result can be achieved. The boundaries can be at some arbitrary value for R , or S (or at an arbitrary valuefor P< I ) , or at a fued value. The latter arises naturallyfrom tht. use of ZP or (preferably) l 7 P , or by setting Pmin= 1. The use of l7P (or ZP)leads to di#ke boundaries.

4.4 COMPOSLTE CRITERIA

In many cases we have seen that the criteria in table 4.7 are adequate for judging the extent of chromatographic separation, but insufficient to account for the effects of chromatographic parameters on the separation. Two additional factors readily come to mind, and both have been used extensively in the literature. These factors are the number of peaks in the chromatogram and the analysis time. Table 4.8 summarizes the requirements of the different criteria for these two additional factors. Table 4.8: Requirements for additional parameters in the optimization criteria listed in table 4.7. The number of peaks present in the sample is asumed to be known. Glossary: n Requires additional parameter to account for the number of peaks appearing in the chromatogram. t Requires additional time parameter x Not useful as an optimization criterion Criterion

Sum

Product

Minimum eqn. 4.25

eqn.4.23

-

t

R S

t

P (1)

(n)

t

Time parameter is less necessary when normalized R,yor S values are used.

4.4.1 Number of peaks

One obvious conclusion from table 4.8 is that a provision for the number of peaks in the chromatogram is only required for the criteria which are not recommended for use in 146

selectivity optimization. All product and minimum criteria become equal to zero once a peak disappears completely, while CP only approaches its limiting value of n-1 ( n being the number of peaks) if all pairs of peaks are adequately resolved. If CP is used instead of nP,then a provision for the number of peaks in the chromatogram can easily be made by dividing ZP by the number of pairs of peaks in the chromatogram:

c,=-.ZP n-I

(4.28)

In eqn.(4.28) C, is the optimization criterion corrected for the number of peaks in the chromatogram*. However, the above is only true if the number of solutes in the sample (i.e. the number of peaks that should appear in the chromatogram) is known, which is assumed to be the case in table 4.8. Obviously, if the number of peaks present in the sample is not known, complete overlap of two peaks may go unnoticed. This problem will affect all criteria, although not all to the same extent. Product criteria will often be affected more than will sum criteria. As soon as it is known that the number of peaks in a chromatogram equals at least the number n, then all the useful criteria in table 4.8 (groups 2 and 3) will automatically penalize chromatograms which show fewer than n peaks. Hence, if the number of peaks decreases during the optimization process, there is n o need to correct any of the useful criteria in table 4.8 for the number of peaks present in the chromatogram. A different situation may arise if the number of peaks increases during the optimization process, which will be more frequently the case if the process guides us in the right direction. In that case the situation may arise in which the calculated value for the criterion C is lower in a newly obtained chromatogram than it was in previous ones, while the number of peaks has actually increased. In that case we have clearly interpreted the previous chromatograms incorrectly. In many cases (for instance simultaneous methods, section 5.2 o r interpretive methods, section 5.5) it is not necessary to introduce a separate correction for the criterion values if the observed number of peaks increases during the optimization process. This is because the calculation of the criterion (response) values is the final step in such a procedure. In each calculation step the number of peaks can be taken equal to the largest number of peaks observed in any of the chromatograms. If this number increases, then the results of the calculations are automatically adapted. The situation is different if a sequential method of optimization (e.g. Simplex optimization, section 5.3) is used. In this case a criterion value is assigned to every chromatogram and the result is compared with previously obtained values. Hence, if the number of observed peaks increases, this may lead to incorrect comparisons. For example, if in one chromatogram three fully separated peaks were observed, the value of l7P for that chromatogram would equal one. However, if in the next chromatogram four peaks were observed which were not completely resolved (e.g. Pvalues between each pair of successive peaks of 0.9), then the resulting value for l7P would only be 0.73. However, the second chromatogram is clearly to be preferred to the first one.

* In this section we will generally use C for some function of the elemental criteria ( R , S or P), for instance one of the optimization criteria in table 4.8. C , refers to a criterion which has been corrected for the number of peaks in the chromatogram, while C, refers to a time-corrected criterion. 147

To deal with this problem, it appears to be more correct to update the previously found criteria values than it is to increase the value of the new one. To do so, it is not only required to keep track of the criterion values of previous chromatograms, but also of the number of observed peaks. In the case of Simplex optimization this is especially easy, since only the criterion values of three chromatograms need to be remembered (see section 5.3). Hence, for the above example the previous result might be nP= 1 with n = 3. If the new result is nP=0.73 with n = 4, then the previous result needs to be updated to nP=0 with n=4. It is extremely easy to update the old values for the criterion, because all product criteria become zero for chromatograms which contain less than the highest number of peaks, whereas all sum criteria remain unaltered. If a composite criterion is used, in which a time factor occurs, then the previous values for the optimization criterion (C,) may usually only be updated if the values for the individual contributions (the value of the criterion C and a time factor) are stored separately. 4.4.2 Analysis time

Analysis time has been incorporated into optimization criteria as a separate term [414,415,418] or as a separate factor [406,416]. Typically, a separate term for the influence of the retention time appears as follows: (4.29) where I, is the retention. time of the last peak (“analysis time”), t,,, is the maximum allowable analysis time and a and 6 are (positive) weighting factors*. The last three parameters can all be chosen arbitrarily by the user. However, the actual influence of the user on the optimization process is limited to one parameter only, i.e. the ratio between the weighting factors a and 6. This can easily be understood from eqn.(4.29), once it is rewritten as C, = a . { C

+ ( W a ) t,,,

= a . { C - (b/a)t,

- ( b / a )t , }

+

C}

(4.29a)

where c is a constant. Variations in c cause all values of C , to be increased by the same amount, and hence the location of the optimum and the course of the optimization process are by no means affected. The same is true for variations in the weighting factor a, which cause all values of the criterion to be changed by a constant factor. Only variations in the 6 / a ratio that change the weighting of t, vs. C will affect the selectivity optimization process and the

* Note that for a meaningful summation in eqm(4.29) reciprocal time units (e.g. min- ’) are required for the parameter b. In this and subsequent equations we will tacitly assume that the correct

dimensions have been assigned to the parameters. 148

location of the optimum. Although users can easily be made to believe that the optimization process can be influenced by demands with regard to the maximum allowable analysis time, the fact is that a criterion that corresponds to eqm(4.29) completely ignores the maximum analysis time selected by the user. Berridge [414] used a different term to incorporate the analysis time in the optmization criterion:

C, = ZR, - b

It,

-

tm,,l

(4.30)

- d (tmin - t,)

where d is another constant, tmin the required “minimum retention time” and t, the retention time of the first peak. The last term is added in order to avoid overlap of the peaks of interest with solvent peaks and other signals around t = 1,. For the same reason as before, the value of tmin is completely irrelevant to the optimization process. However, t,,, has now become relevant. Eq~(4.30)can be divided into two equations. For t , < t,,, C, = ZR,

+ b.t, + d.t, + c’

where c’ is a constant equal to - (b.t,,, C, = ZR, - b.t,

+

d.t,

+ C”

(4.30a)

+

d.tmin),and for t,> I,,,

(4.30b)

where the constant c ” equals (b.t,,, - d.tmin). According to eqn.(4.30a) the value of the time-dependent criterion C , will increase with increasing analysis time (tdif the selected maximum analysis time (r,,,,,) has not yet been reached. According to eqn.(4.30b), an increase of t , above t,,, will result in a negative contribution to C,. Hence, ,t serves as a desired value for the analysis time, rather than as a maximum value. The importance of the aim to realize the separation in a time that equals t,,, can be varied by varying the weighting factor 6. The criterion C, is always increased if t , increases. In other words, a large value for t , will be one of the goals set by the selection of an optimization criterion according to eqm(4.30). Hence, the two time terms in eq~(4.30)join forces to bunch the peaks between a retention time of the first peak, which is as high as possible, and the desired maximum analysis time. It may be expected that the two terms will try to direct the chromatographic parameters included in the optimization process in opposite directions. The sum of the resolution factors contributes to this conflict of interest, as ZR, will tend to increase if the peaks are spread out over larger time intervals. The balance between the different factors is in principle decided upon by the user in choosing the values of the parameters b and din eqm(4.30). However, it seems impossible to establish an objective balance between the importance of resolution on the one hand and retention time on the other. This situation is disturbing, especially because the course and the result of a selectivity optimization process will depend on the arbitrarily selected weighting factors. Smits et al. [416] and Wegscheider et al. [406] incorporated a time correction factor into their optimization criterion as follows:

c, = c / t,

(4.31) 149

where t, again represents the retention time of the last peak*. Essentially, in this way the obtained separation (expressed in the criterion C ) per unit time becomes the optimization criterion. Again, a weighting factor may be added, i.e.

c, = c /

(4.32)

t:

where r is an arbitrary weighting factor. Nevertheless, the choice of r= 1 does seem to be a natural one. Unlike the situation with the contribution of a time term (see the discussion above), C, can readily be expressed in inverse time units (e.g. min-’), so that a dimension problem will not arise. Hence, both from the point of view of weighting factors and from that of dimensions a time factor as in eqn.(4.31) appears to be more appropriate than a time term as in eqn.(4.28). A more fundamental advantage of the use of time factors may be that we no longer find ourselves in a position in which a compromise has to be established between two conflicting contributions to the optimization criterion. A proper balance between longer analysis times yielding higher resolution and shorter analysis times yielding lower resolution may be hard to find. The effect of a change in the chromatographic parameters will usually be such as to increase one term in eqn.(4.28), but to decrease the other. This may easily lead to oscillation effects in which the conditions are pushed back and forth, while the optimum is approached only very slowly. An example where such a problem seems to occur can be found in ref. [414]. An increase in the retention time accompanied by an increase in the resolution has the effect of increasing both the numerator and the denominator in eqn.(4.31), so that oscillations between high and low values are less likely to occur. Nevertheless, the use of eqn.(4.32) may result in a slower optimization process than if a simple sum or product criterion were used. It is also unclear at the outset how the process would respond to chromatograms with the same value for C , but widely different values for C. In other words, the criterion cannot differentiate between a bad resolution in a short time and a good resolution in a long time. This problem can most easily be circumvented by using a threshold criterion for C. If Cequalled one in acceptable regions of the parameter space and zero outside these regions, then the result of the use of eqn.(4.32) would correspond to the shortest possible chromatogram for which the resolution is acceptable (C= 1). A similar situation would occur if we were to use a threshold criterion with a diffuse change between zero and one,

* In fact, the time needed to elute 95% of the last peak was taken, in which case eqn.(4.31)would read for Gaussian peaks

c , = - =C

t , + 2 a,

C t,(l+2/VN)‘

(4.31 a)

Clearly, the difference between eqm(4.31) and (4.31a) is small. If we assume that genuine chromatographycolumns have a minimum of 2500 theoretical plates, then the differencebetween the two equations is always less than 4’/0. Even for non-symmetrical (“tailing”) peaks eqn.(4.31) will almost always be an adequate approximation of eqn.(4.31a).

150

such as DP. In fact, the criterion

c, = D P /

(4.3 1b)

t,

is similar to the criterion of eqa(4.24). Because Pdecreases very rapidly once the resolution of a pair of peaks becomes less than 1, the use of eqn.(4.31b) will not usually result in chromatograms in which a poor resolution is compensated by a very short analysis time. The difference between eqns.(4.24) and (4.3 1b) is that in the first case R, and k, are used, which are useful characteristics if the result of the optimization needs to be transferred t o another column, while the use of P and t, and eqn.(4.31b) makes this criterion more appropriate for optimization on the final analytical column. E q ~ ( 4 . 31b) has the advantage over eqn(4.24) of being a continuous criterion rather than a combination of two separate ones used hierarchically. We have seen before that the use of P suffers from the insensitivity of this criterion to changes in the range of high P values (P = 1)and in the range of badly resolved peaks (P= 0). The use of eqn.(4.31b) will eliminate the first problem, but the latter problem will remain. 4.4.3 Column-independent time factors The retention time is determined by three factors:

1. The capacity factor of the solute

2. The column dimensions (hold-up volume) 3. The flow rate. Only the first factor is influenced by the physico-chemical separation process (the selectivity), while the other two factors are determined by the column and the operating conditions, respectively. If C is a continuous criterion (see table 4.7), then both C and C, can be transferred from one column to another. Both column dimensions and flow rate have trivial effects on the analysis time t,. However, if the final analysis is to be run on a different (optimized) column, then it is more logical to use the dimensionless, column-independent factor (1 + k,) in eqn.(4.31) instead of t,:

c,= C / ( l +

kJ.

(4.33)

In the case, where the dimensions of the column are to be optimized after completion of the selectivity optimization process, another time factor may be even more appropriate. The first step to be taken after the completion of the optimization process is to establish the required number of plates (N,,). If the lowest resolution value encountered in the chromatogram is R,,,inr and if the required resolution is R,,,,, then (4.34) where N , is the number of plates available on the column used during the optimization process. If Smin is the lowest value for S in the chromatogram, then (4.35) 151

Since all continuous criteria (see table 4.7) require knowledge of the R, or S values, eqn.(4.34) or (4.35) can be used immediately to calculate the required number of plates. In many cases the required number of plates will only be used to estimate the length of the final analytical column, while all other parameters are being kept constant. For example, in GC a capillary column with a given diameter may be used, operated at a given flow rate, with N being directly proprotional to the column length (L). In GC or in LC, packed columns with a given particle size may be used at given flow rates. In all these common cases, the following proportionality series applies: t , cc to cc Vb cc L cc N,,

(4.36)

and hence the final analysis time will be proportional to N,,. Of course, t, is also proportional to (1 + kd (eqn.l.6), and therefore an appropriate time corrected criterion is (4.37) where f and d denote constant flow rate and diameter (of open columns or of particles in packed columns), respectively. The above simple proportionality between t , and N,, (eqn.4.36) is not always obeyed. For instance, at constant flow rate and particle size, the number of plates that can be achieved is limited by the maximum allowable column pressure. In that case, we are forced to vary the flow rate, the particle size, or both. If we do so, the analysis time ( t J will no longer be proportional to the required number of plates ( N,,). In chapter 7 (section 7.2.3) we will see that in the case where the pressure drop over a packed column is kept constant and both the flow rate and the particle size are allowed to vary in order to realize optimum operating conditions, the retention time will be proportional to the square of the required number of plates, i.e. t, =

p . N,: . (1 + kJ

(4.38)

where pis a constant, the value of which depends on the viscosity of the mobile phase, the pressure drop and the quality of the packing. According to eqn.(4.38) a time-corrected optimization criterion under constant pressure conditions (denoted by the subscript p) may be defined as (4.39) In the reality of LC practice, eqn.(4.38), which is based on a different optimum particle size for a different required number of plates, will not usually be realistic. For LC the truth may be somewhere in between the two extremes described by eqn.(4.36) (constant flow rate and particle size) and eqn.(4.38) (constant pressure drop). Relatively long columns with 10 pm particles may be used for difficult separations, requiring relatively large numbers of plates. For more convenient separations, somewhat shorter 5 pm particle columns may be used, and for relatively simple separations requiring modest numbers of plates 3 pm particles packed in very short columns may provide fast analysis. 152

If three or more different particle sizes are to be considered after the completion of the selectivity optimization process, then this may be an argument for the use of eqn.(4.39) instead of eqn.(4.37). The required analysis time itself appears to be both a logical and an elegant choice for an optimization criterion, Either tnecan be minimized, or, for reasons of consistency, 1I t n e can be maximized. The criterion of minimum required analysis time then corresponds to a constant value for C in eqn.(4.37): 4

(4.40)

and with eqn~(4.34)and (4.39, neglecting the constant factors which are irrelevant for optimization purposes, (4.41) An analogous expression can be found for the case of a constant pressure drop over a packed column with variable particle size from eqn.(4.39): ‘&in

CIlP = ( l + k J N ;

a- g i n (I+kJ’

(4.42)

are especially attractive, since no estimate of the column The equations in terms of Smin plate count is required. If R,is used, an estimate of the peak width or plate count is required twice, but since R, a VN the plate count cancels in eqm(4.41) and (4.42). This becomes clear when the two equations are expressed in terms of S. 4.4.4

Time-corrected resolution products

In section 4.3.2 we have seen that the normalized resolution product criterion r aims at achieving a chromatogram in which all peaks appear at constant resolution intervals from the first one. If r* is used instead of r, then the regular intervals start at an imaginary peak at t = c., A chromatogram for which r* = 1 is one of a series for which the constant intervals can be found. Once the absolute value of S, the number of peaks and the plate number are known, the chromatogram is defined unambiguously. The question we can now ask ourselves is whether all.members of the series are equally good chromatograms. In other words, is the criterion r* on its own sufficient for judging the quality of a chromatogram. To simplify the discussion, we will investigate the necessity of time correction factors for chromatograms for which r* = 1, and then extend the result to include the more realistic chromatograms for which r* < 1. In the case of a chromatogram for which r* = 1, the value of k, can be calculated if the absolute value of S (Shas the same value for all pairs of peaks in the chromatogram!) and the number of peaks is known. The capacity factor for the first peak can be found from

S =

k, - ko - (k, + l)-(ko+ 1) k0+k,+2 (k,+l)+(k,+l)

(4.43)

and this leads for ko=O to 153

1+s

2s

k, = -( k o + l ) - 1 = -. I-s 1-s

(4.44)

For the second peak we find in a similar way 1+s 1+s * (k,+l) - 1 = - l

(GI

k“==

(4.45)

and in general terms (4.46) Eqm(4.46) allows us to calculate the k value for the nth peak in an “optimal” chromatogram (r* = I). Ifwesubstitute S = 2 R,/VNandassume S < 1, then theequation for the peak capacity (eqn.l.25) follows directly from eqm(4.46). Of course, in practice k values may also be obtained from the chromatogram instead of from eqr(4.46).

-2

log

s-

-1

0

Figure 4.1 1: Calculated characteristics for optimum chromatograms (r* = 1) containing 10 equally resolved peaks as a function of the separation factor S.Plotted on a logarithmic scale are the capacity factor of last peak (1 + k, eqn.4.46), the required number of plates (Nne;eqn.4.47), the required analysis time under conditions of constant flow rate and particle diameter (tnelr.,,; eqn.4.48), and required analysis time under conditions of constant pressure drop t,&; eqn.4.49). For explanation see text,

154

In figure 4.11 the function log (1 + k,,) is plotted as a function of log S for chromatograms with r* = 1. k,, denotes the capacity factor of the 10th peak. Clearly, for very low values of S all k values are very low. At an S value of 0.03 (log S z - 1.5) all 10 peaks still elute before a k value of 1. Around an S value of 0.12 the 10th peak elutes at k=10, while at S=O.25 the capacity factor becomes equal to about 100. Roughly, optimum capacity factors are found around S= 0.1. The number of plates required for realizing adequate resolution can be found from eqn.(4.35). If the required resolution (RS.,,) is unity, then

N,, = 4 /

smin*.

(4.47)

Again, choosing any other value for R,,, is totally irrelevant for the optimization of selectivity. In figure 4.11 log N,, is also plotted against log S. In accordance with eqn.(4.47) a straight line results that has a slope of - 2. The number of required plates quickly decreases with increasing S. Two other lines are drawn in figure 4.1 1. These correspond to the time correction factors in eqm(4.37) and (4.39). Under conditions of constant flow rate and constant diameter (of particles or open columns) the analysis time, neglecting constant factors, can be expressed by (4.48) while for packed columns under conditions of constant pressure drop and optimal particle size (4.49) Log t,,l,,, and log tnelp are illustrated in figure 4.11. It can be seen that under the conditions of eqa(4.48) a broad optimum exists around S=O.l. Over the range 0.03 < S < 0.2 the required analysis time varies by about a factor of 2. This range in Scovers a very large range in k values. For S = 0.03 the capacity factors for ten equally resolved peaks range from 0.06 to 0.82. For S = 0.2 the capacity factor ranges from 0.50 to 56.7. Hence, there is a broad optimum range around S = 0.1 in which the required analysis time does not vary considerably. In this range the criterion r* can be used to try and locate the best possible chromatogram. Outside the optimum range this is no longer true. If ten peaks are equally resolved (r* = 1) with S values of 0.001, then according to eqn.(4.47), four million plates are required for adequate resolution. Moreover, we can see from figure 4.1 1 that the required analysis time is a factor of about 600 larger (under constant flow and diameter conditions) than it would be if S equalled 0.1. If S was 0.5, the analysis time would be a factor of about 200 larger than in the optimum. Hence, we may conclude that for optimization processes during which the capacity factors may be expected to vary dramatically, a time correction factor is required even when r* is used as the optimization criterion. If we consider packed columns under constant pressure conditions, i.e. if we use eqn.(4.49) instead of eqn.(4.48), then the optimum that corresponds to the shortest analysis 155

time will be observed around S = 0.2. A variation in tneby a factor of two allows operation in the range 0.1 < S < 0.3. The corresponding ranges in k are 0.22 to 6.44 for ten peaks at S=O.1 and 0.86 to 487 for ten peaks at S=0.3. Again we see that the optimum range is quite broad. The range of k values usually considered as optimal, i.e. 1 < k < 10, is well encompassed in the optimum working ranges of both eqns.(4.48) and (4.49). It should be noted that in the optimum ranges the number of plates required for “ideal” chromatograms that show constant resolution intervals throughout is always very small. The limiting values of S for the optimum ranges correspond to plate numbers of around 4500 (S=0.03) to 45 (S=0.3). When the number of peaks increases, the (1 + k d factor increases and the optimum shifts towards lower S values (to the left in figure 4.1 1). For instance, for 15 peaks the optimum S value shifts down from S = 0.1 (400 plates required) to S=O.O7 (800 plates) if eqn.(4.48) is used, and from S=O.2 (100 plates) to about 0.13 (250 plates) using eqn.(4.49). Unfortunately, in practice “ideal”.chromatograms showing r* = 1 will be hard to find.

I

0

(C)

5

k-

10

Figure 4.12: Three schematicchromatogramswith equal values for the lowest separation factor (Smin. determined by the first pair of peaks) as well as for the capacity factor of the last peak (kd.

156

Therefore, for a chromatogram with ten peaks and an Smin value of 0.1, the capacity factor of the tenth peak is bound to be much higher then the value of 6.4 predicted by eqn.(4.46). In general therefore, the value for Smin in a real chromatogram will be shifted to the left (lower S values), the number of required plates will be higher and so will the analysis time. The time correction factors of eqn~(4.48)and (4.49) can readily be used with the lowest S value (Smi,,)and the largest k value ( k d observed in the chromatogram. The use of the required analysis time as the optimization criterion (eqns.4.41-42 for criteria to be maximized, or eqns.4.48-49 for criteria to be minimized) yields a balanced comparison between the minimum resolution on the one hand and the retentjon on the other. However, in using the criterion the main disadvantage of the use of Sminas the function describing the resolution in the entire chromatogram remains. If Smin is used, no attention is paid to all but one pair of peaks in the chromatogram. By using the required analysis time as the optimization criterion care is taken that other pairs of peaks do not extend the length of the chromatogram with impunity. However, even when the Smin values and the k, values of different chromatograms are the same, these chromatograms can still be very different. This is illustrated by the three chromatograms shown in figure 4.12. The Sminvalue, determined by the first two peaks is the same in all three chromatograms, and so is the k, value. The top chromatogram shows two peaks early in the chromatogram and a bunch of peaks between k = 5 and k = 10. The middle chromatogram shows four pairs of peaks and the bottom chromatogram shows good spacing of the peaks after the first pair. Table 4.10 lists the required analysis times for the three chromatograms of figure 4.12, as well as the r* values. Constant flow rate and diameter are chosen as the conditions for table 4.10. Clearly, the required analysis times are the same for all three chromatograms. However, r* reveals large differences between the different chromatograms. These changes in r* are relatively large in comparison with possible changes in the time factor tne. We concluded from figure 4.1 1 that there were large ranges over which the required analysis time varied by less than a factor of two. The variation in relative peak positions in the chromatograms of figure 4.12 gives rise to changes in r* which amount to a factor of 50 between the top and the bottom chromatogram. Table 4.10: Required analysis times and time-corrected resolution products for the three chromatograms of figure 4.12. Constant flow rate and diameter (of open columns or particles) (i.e. constant f,d conditions) have been assumed. Mult. factor 103 10-4 10-2 10-6 10-4

Criterion

tne1f.d

1’tne1f.d r* rr ‘“1

Eqn.

4.48 4.41 4.21 4.50 4.52

Chromatogram top

middle

bottom

4.4 2.21 1.09 2.47 1.30

4.4 2.27 7.27 16.5 1.64

4.4 2.27 56.4 128 2.12 157

The ratio *

*

r, = r / t,,

(4.50)

could be used as an optimization criterion to try and accommodate both peak distribution and analysis time in one criterion. In that case an increase in rf by a factor of 50 would compensate for an increase in the required analysis time by the same factor. The bottom chromatogram in figure 4.12 may be more attractive than the top one, but it is quite obvious that a factor of 50 increase in analysis time is too high a price to pay for the improved peak distribution. The reason for this is that changes in resolution are over-emphasizedin the criterion r*, because n separate resolution factors for n pairs of peaks ( n - 1 if r is used) occur in r*. In this way, resolution to the nth power is balanced vs. tne. Therefore, a more sensible criterion would be rit =

V7 / t,,

(4.51)

which for the conditions of eqn.(4.48) (constant f,d) becomes (4.52) and for the conditions of eqn.(4.49) (constant p) (4.53) The values of r&d and r:rlf,d for the three chromatograms of figure 4.12 are included in table 4.10. The values of r: show an increase of a factor of 50 as discussed before. The values of rif increase by about 60% in going from the top to the bottom chromatogram. Hence, the improved peak distribution observed in the bottom chromatogram may outweigh a 60% increase in analysis time.

4.5 RECOMMENDED CRITERIA FOR THE GENERAL CASE The final recommendations for optimization criteria to be used in the general case (i.e. when all peaks are considered to be of equal importance) are summarized in table 4.1 1. The table shows recommended criteria for four different cases. Below the dashed line, an alternative criterion (second best choice) is given for each case. 4.6 SPECIFIC PROBLEMS

4.6.1 Limited number of peaks of interest

So far, we have only considered chromatograms in which all peaks were treated as being of equal importance. Now we will look at a chromatogram in which a number (n)of peaks appears, but only some peaks @; p < n) are of interest. An example is shown in figure 4.13. In this chromatogram, eight peaks occur, but only two of these (peaks 3 and 5 ) are of interest. Seven or eight (if a peak is assumed at t = to) S values can be obtained from the chromatogram. Four of these involve one of the peaks of interest. 158

Table 4.11: Recommended criteria for use in selectivity optimization processes in the general case (all peaks equally important).

U Optimization on final analytical column?

ves

I

I

sample

I

YFS

max W P , / t, eqn.(4.31b)

I

overall

no

YFS

1

max l 7 P / t, or max

LIP,

no

/ t"

I

no I *

max r*

max rn,

eqm(4.2 1) and (4.22)

eqn.(4.52) or (4.53)

max Smin

max l/tne

eqn.(4.25)

eqn.(4.41) or (4.42)

. . . . . . . . eqn.(4.31 . . . .b). . . . . . . . . . . . . . . . . . . . . min t , Rs.min

n > x

(1) eqn.(4.24)

min t , Rs.min

(1)

n > x

eqn.(4.24)

(1) Suggested value for x: 1.5.

The k and S values corresponding to the chromatogram of figure 4.1 3 are given in table 4.12. The relevant values are underlined in this table*. The chromatogram in figure4.12 can be improved dramatically by using a multicolumn technique (see also section 6.1). The entire sample can be eluted from a short column to obtain a rough separation, and only the fraction that contains the relevant peaks (3 and 5) can be passed on to a longer column to realize the entire separation. The first column may be referred to as a clean-up column or pre-column and the second one as the analytical column. However, with or without the use of multicolumn techniques, irrelevant peaks (e.g. peak no.4) will usually appear in real-life chromatograms and optimization criteria have to be developed to cope with them. Most of the criteria used so far can readily be applied to cases where a limited number of peaks is of interest. The parameters Rs,min, Smin and Pminretain their value, but now the lowest value should be selected from the relevant pairs of peaks (a pair of peaks is

* The number of relevant S values is never higher than 2p. If two peaks of interest are adjacent in the chromatogram, then the number of relevant S values is decreased by one. 159

3 N

2

Figure 4.13: A chromatogramwith eight peaks ( n = 8), of which only two peaks are of interest ( p = 2). The relevant peaks (3 and 5) are indicated by an asterisk.

Table 4.12: Retention and resolution data corresponding to the chromatogram of figure 4.13.The relevant peaks and the relevant values for k and S are underlined.*! was calculated from eqns.(4.55a) and (4.56a). f,$ was calculated from eqn.(4.57). Peak no.

k

0

0

1

0.5

5

0.20

Smin= 0.067

0.14

Smin = 0.077

0.20 0.25 0.091 0.077

0.067 0.16

160

relevant if either of the two peaks is relevant). We distinguish the appropriate criteria for the specificcase again by underlining, hence13,,in,Smin andLmi,refer to the specificcase. Even if 0.067 is the lowest separation factor observed in the chromatogram of fig 4.13 (Smin), then 0.077 is still the lowest relevant value (Smin). The analysis time is still determined by the capacity factor of the last peak (k,= lo), no matter whether or not this is a relevant peak. However, the lowest relevant value for Rs or S should now be used in eqn.(4.48) or eqn(4.49). The criteria which may be used for the specific case in which only a limited number of peaks is of interest are listed in table 4.13. Table 4.1 3: Recommended criteria for use inselectivity optimization processes for the specific case in which only a limited number of peaks is of interest (see also figure 4.13).

Constant sample composition? Yes

no

Constant overall k value? Yes

no

( 1 ) Suggested value for x: 1.5. (2) ine can be calculated from eqn.(4.48) or eqn.(4.49), using the lowest relevant value for S

emin).

The square root should now be incorporated in all product criteria, i.e. also in the case in which lTP or I7P, is used instead of nP, This is because it is now now a sensible convention to incorporate two values for a peak-valley ratio (be it P, P , or P J into the criterion for every peak of interest. If we did not follow this convention, then a different situation would exist if two relevant peaks were adjacent (yielding one combined value for the peak-valley ratio) or separated by an irrelevant peak (which would lead to two different 161

P or P, values). Hence, two values may best be used for every peak of interest and the resulting product may be “normalized” by taking the square root. For example, we can write for the product of P values i7P -=

P

n i=

1

(4.54)

vPi,i-,.Pi.i+l

where Pi,i-, refers to the separation factor between the ith relevant peak and the preceding one and Pi,i+ that between the ith relevant peak and the following one and where the number of factors in this product equals the number of relevant peaks @). If peak-valley ratios are used as elemental criteria, then the separation between the first peak and the (imaginary) preceding one, as well as the separation between the last peak and the (imaginary) following one, may readily be characterized by a P value of one. The retention time of the last peak, which may be used in combination with a product of P values (see table 4.13) refers to the last appearing peak in the chromatogram, no matter whether or not this is a relevant peak. The calculation of (normalized) resolution products if only a limited number of peaks is of interest requires some additional thought. In calculating normalized resolution products either R, or S may be used. We will use S in the following discussion. The ideal situation with regard to the normalized resolution product will be that all relevant S values are equal, while the irrelevant peaks contribute nothing to the overall length of the chromatogram (i.e. for the relevant values and and for all other S=O). In that case the following product should equal unity:

s=s

whereSi.i- refers to the separation factor between the ith relevant peak and the preceding one and that between the ith relevant peak and the following one and where

&+,

-

S = l/@-I)

n-1

c

i= 1

Si*i+l .

(4.56)

When a hypothetical peak is assumed at t = to the calibrated normalized resolution product becomes:

with

-

n-I

s = l / p i =zO si,i+l .

(4.56a)

In these equations, p is again the number of peaks of interest. Hence, the product includes only the relevant S values, while the sum is taken over all pairs of peaks. If the sum is divided by the number of relevant peaks (p - 1 or p ) , then a value of r= 1 or r* = 1 can only be reached if all irrelevant peaks appear “nowhere” in the chromatogram, i.e. coincide with the imaginary peak at t = r,,. If we divided the sum by the total number of peaks (n), then the resulting values for r would not be restricted to the range O < r< 1, and we would no longer be able to refer to r as a normalized resolution product. 162

The separation factor between the first peak and the (imaginary) preceding peak, as well as that between the last peak and the (imaginary) following one, do not take on a natural value, as was the case for peak-valley ratios (which in both cases could be said to equal one). In the case of the normalized resolution product the most logical choice is to take the optimum value, i.e. the average value 3, as the limiting separation factor of the chromatogram. The time-corrected normalized resolution product now becomes: :;t

=

(4.57)

/ t,,

where t,, can be found from eqm(4.48) or eqn.(4.49), using the lowest relevant value for S (Smin) in the calculations. Weighting factors The special case described above is in fact a particular example of the use of weighting factors in the optimization criteria. If some peaks are of interest whereas others are not, we could use weighting factors of one and zero, respectively, as we have in fact done above. However, we have seen that even the introduction of these simple weighting factors has caused some problems regarding the calculation of the various recommended criteria. These problems will be aggravated if weigthing factors other than one and zero are allowed. If a product of peak-valley ratios is used as the optimization criterion, then two values would need to be used for every peak, one describing its resolution from the preceding peak in the chromatogram and the other one describing its resolution from the following peak. Because a product criterion is used, the weighting factors (8) will appear as powers in the product. If we assume the weighting factors to be positive, we may write n

nP,

=

n (Pi.i-, . Pi.i+l)gi’* i= 1

(4.58)

or, equivalently,

nP, = exp

n

{ Z ( g / 2 ) In i= 1

. P i , i + , ) }.

(4.59)

This product of peak-valley ratios can be normalized to the sum of weighting factors, so that the “true” value of the resolution product is less obscured by the arbitrarily selected values for g: (4.60) Eqn.(4.60) is equivalent to an equation suggested by Morgan and Deming [417], apart from the requirement to use two P values for each peak. Eqns.(4.58) to (4.60) can be used with P , (if the sample composition is expected to be constant) or with P or P , if the sample composition is expected to vary. 163

The normalized resolution product (r or r*) can be obtained in an analogous way. In terms of S the product reads: (4.61)

-

s

=

n-1

n

i= 1

i= 1

x q i + /, x

(4.62)

gi

or, with the inclusion of a (hypothetical peak at t = to) (4.61a) with

-

n

n-1

s = i =xO si,i+l / x i=O

(4.62a)

gi

Table 4.14 Recommended criteria for use in selectivity optimization processes for the specific case in which weighting factors are used to indicate the importance of each individual peak. Optimization on final analytical column?

I

"p:.g 1

no

Constant sample composition ?

Constant overall k value?

I

Yes max

Y e

*"

eqn.(4.60)

Rs.min

Yes

max

max rg

max

eqn.(4.61a) and (4.62a)

eqm(4.63)

*

"Pi/

t,

W n . g

1t ,

or max

(1)

min x

eqn.(4.24)

t,

Rs.min

n 2 x

(1) eqn.(4.24)

(1) Suggested value for x: 1.5.

164

no I

no 1

eqn(4.60)

min t, n

I

i

r*,,.g

Finally, the time-corrected normalized resolution product can be found from I

t

= exp In ri /

n

z: ;=o

\

g;

(4.63)

where t,, can again be found from eqn.(4.48) or eqn.(4.49), using the lowest value for S (Smin)in the calculations. Table 4.14 lists the criteria that may be used in combination with weighting factors for each peak and refers to the appropriate equations. The use of Sminor 1/rne as alternative criteria when the optimization does not take place on the final analytical column (see table 4.1 1 and table 4.13) is not recommended in table (4.14), because these criteria are not compatible with the use of weighting fqctors. 4.6.2 Programmed analysis The important aspect of programmed elution techniques with respect to optimization criteria is that the peak width does not increase with the retention time in a manner corresponding to eqm(l.16). In programmed analysis a constant peak width is wanted throughout the chromatogram (see section 6.1). Because of their pragmatic definitions, the different P values are not at all affected by changes in the elution mode, i.e. they may be applied under programmed elution conditions in exactly the same manner as under constant conditions. Resolution (R,) factors are not fundamentally affected, i.e. the definition given in eqn.(l.l4) can still be applied. However, the relationship between R, and fundamental chromatographic parameters given by eqm(l.22) is no longer valid. The separation factor S loses its usual meaning, since its definition originates from the above eqn.(l.22). A simple solution to this problem is to use the difference in retention time (At) between two peaks as the sole criterion for resolution. This is justified by the fact that for ideal elution programmes the nominator in eqn.(l.l4) has a constant value. Hence, Atmin can be used instead of Rs,min or Smin.Also, a normalized resolution product may be defined as

r,, =

n-I

n

i= 1

n-l

=

n

i= 1

- ti) / Z i }

(Ar/z)

(4.64)

where (4.65)

It should be noted that a constant peak width will only be achieved by approximation in most programmed elution chromatograms. Early eluting peaks (not subjected to gradient conditions during their migration through the column) may be considerably narrowed, while late eluting peaks (eluting after completion of the program) may be considerably broadened. Harris and Habgood (ref. [427], p.123) have suggested a different definition for a 165

separation factor in programmed temperature gas chromatography. Their definition is based on the assumption that the width of a peak that is eluted from a gas chromatograph at a temperature T, during a programmed analysis is the same as it would have been if the same component had been eluted from the column under isothermal conditions at the temperature T,, Therefore, the isothermal retention volume at T = T, ( VT) may be used to characterize the peak width in the denominator of the separation factor, i.e.

- -F S = VRj- VR,i‘TPi+

‘T,J

‘T

Trj- Tr,i ‘Tri+

(4.66)

‘T,J

where V, is the retention volume under programmed elution conditions, Fis the flow rate (for example expressed in mL/min) and rT the heating rate (e.g. OC/min). Harris and Habgood proceed to suggest that eqn.(4.66) can be used to explain the influence of the “programming rate” (r7/ F OC/mL) on the resolution in programmed temperature GC. However, this is a somewhat simplified picture because the retention temperature T, is affected by the programming rate. Nevertheless, their conclusion that the resolution in programmed temperature GC increases with a decrease in the programming rate is essentially correct. Snyder [428,429] has suggested a way to relate resolution in solvent programmed (gradient elution) liquid chromatography to the fundamental parameters of the chromatographic process by defining a median capacity factor under gradient conditions which is characteristic for the average speed of migration of the solute molecules through the column. In terms of we can use an equation identical to eqn.(1.22) to describe the resolution R,:

(K),

%

(4.67) It is important to realize two things in using eqn.(4.67). 1. The median capacity factor is not directly related to the retention time under gradient conditions. In fact, it can be shown that under someconditionsqhas the same value for all the peaks in a chromatogram obtained under programmed conditions*. 2. In deriving eqn.(4.67), the relative retention a is assumed to be independent of the composition. In other words, plots of retention (In k) vs. composition (p)obtained under isocratic conditions are assumed to yield parallel lines. For components which are eluted under “ideal” gradient conditions (i.e. those components that appear neither at the very beginning nor after the end of the actual gradient in linear solvent strength gradients**, it can be shown that the median capacity factor is inversely proportional to the gradient steepness parameter, defined as [428]

5

(4.68) where S is the slope of the retention (In k) vs. composition (9)plots (see eqn.3.45), V, the

* The suggestion given before that in an “ideal” programmed analysis the p z k width should be the same for all solutes (see also figure 6.1~)corresponds to the assumption that k, is equal for all peaks. ** For a definition of linear solvent strength (LSS) gradients see section 5.4. 166

hold-up volume of the column and d@dt the programming rate. The latter can be related directly to the span (Ap) and the duration (1,) of a linear gradient. It appears from eqn.(4.68) that if the flow rate and the span of the gradient are kept constant, the gradient steepness parameter ( b ) is inversely proportional to the duration time (t,) of the gradient, and, hence, that the median capacity factor is directly proportional to t,. Therefore, under these conditions, in gradient elution t, may take the place of the capacity factor in the resolution equation and eqn.(4.67) may be rewritten as

(k,)

(4.67a) where c is a constant. Cohen et al. [430] have demonstrated the validity of eqn.(4.67a) in practice for cases in which a is constant. However, they have also shown that the equation is no longer valid if a varies with composition under isocratic conditions. Nevertheless, eqn.(4.67a) may serve as a good rule of thumb for the optimization of gradient duration times (see chapter 6). Another aspect of programmed elution that will affect the quality of the chromatogram is the variation (“drift”) of the baseline during the program. Methods to reduce the baseline drift (or blank signal) and other aspects of programmed analysis will be discussed further in chapter 6. 4.6.3 Dealing with solvent peaks

In many chromatograms a “solvent peak” will appear. This is typically a large signal that appears early in the chromatogram. In GC, solvent peaks are usually highly non-symmetrical (tailing) peaks. In LC they may occur in many different ways, for instance as large negative and positive signals early in the chromatogram. In some forms of LC, especially the ionic separation methods (section 3.3) peaks induced by the mobile phase may occur as genuine (or negative) peaks much later in the chromatogram. This latter kind of solvent-induced signal can be dealt with as an additional (irrelevant) peak in the chromatogram, from which the (relevant) peaks need to be separated. In this section we will discuss some aspects of the more common type of solvent peaks, i.e. large signals which appear early in the chromatogram. Figure 4.14 shows a typical chromatogram in which three solute peaks are preceded by a large solvent peak. There are two fundamentally different ways in which we can deal with solvent peaks. I. Reduce the solvent signal. This can be done chromatographically, for instance by using a column switching technique, that prevents the first part of the chromatogram from entering the analytical column (see also section 6.1). Reduction (or elimination) of the solvent signal may also be achieved mathematically by subtracting the signal of the pure solvent (“blank”) from the chromatogram. 2. By modifying the optimization criteria such that an optimum separation of the solutes from the solvent signal is achieved. This latter method, which is the relevant one in the context of this chapter, has not received much interest in the literature so far. Solvent peaks are usually highly non-symmetrical, so that neither R , nor Scan be used. 167

Moreover, neither P nor P, can be established from the chromatogram. The only criterion which maintains a realistic value is P, As is shown for the first peak in figure 4.14, this parameter may be estimated from the chromatogram in the usual way (cf. figure 4.2). Hence, criteria based on P,may be used for chromatograms that resemble figure 4.14. A great disadvantage of this is that the use of P , has been recommended only for optimization of samples of constant composition on the final analytical column (tables 4.4 and 4.1 1).

0

k-

5

Figure 4.14 A typical chromatogramcontaining a solvent peak followed by three small peaks. h, and v, can be used to estimate the peak-valley ratio of the first peak (see figure 4.2.c).

A possible way to deal with solvent peaks if the resolution ( R , ) or the separation factor (S) is opted for as the elementary criterion (which is to be recommended if the optimization

process does not take place on the final analytical column), is the introduction of (large) weighting factors for the solvent peak, using for example the criteria described by eqns.(4.61) to (4.63). For example, if a large weighting factor were assigned to the solvent peak in figure 4.14 (e.g. g= lo), while small factors were assigned to the (relevant) solute peaks (e.g. g= l), then the resulting criterion would have the effect of trying to enhance the separation between the solvent peak and the first peak in the chromatogram. From there on, a normalized resolution product criterion would again aim at a regular spacing of peaks in the chromatogram. 168

REFERENCES 401. L.R.Snyder and J.J.Kirkland, An lntroduction to Modern Liquid Chromatography, Second edition, Wiley, New York, 1979. 402. R.Kaiser, Gas-chromatographie, Geest und Portig, Leibzig, 1960, p.33. 403. 0.E.Schupp 111, Gas Chromatography, Wiley, New York, 1968, p.22. 404. A.B.Cristophe, Chromatographia 4 (1971) 455. 405. A.C.J.H.Drouen, Unpublished results, 1981. 406. W.Wegscheider, E.P.Lankmayr and K.W.Budna, Chromatographia 15 (2982) 498. 407. S.S.Ober in: V.J.Coates, H.J.Noebels and 1.S.Fagerson (eds.), Gas Chromatography, Academic Press, New York, 1958, pp.41-50. 408. P.Jones and C.A.Wellington, J.Chromatogr. 213 (1981) 357. 409. P.J.Schoenmakers, A.C.J.H.Drouen, H.A.H.Billiet and L.de Galan, Chromarographia 15 (1982) 688. 410. A.C.J.H.Drouen, P.J.Schoenmakers, H.A.H.Billiet and L.de Galan, Chromatographia 16 (1982) 48. 411. VSvoboda, J.Chromatogr. 201 (1980) 241. 412. J.E.Knoll and M.R.Midgett, J.Chromatogr.Sci. 20 (1982) 221. 413. H.J.G.Debets, B.L.Bajema and D.A.Doornbos, Anal.Chim. Acra 151 (1983) 131. 414. J.C.Berridge, J.Chromatogr. 244 (1982) 1. 41 5. J.L.Glajch, J.J.Kirkland, K.M.Squire and J.M.Minor, J.Chromatogr. 199 (1980) 57. 416. R.Smits, C.Vanroelen and D.L.Massart, Fresenius Zeitschr.Anal.Chem.273 (1975) 1. 417. S.L.Morgan and S.N.Deming, J.Chromatogr. 112 (1975) 267. 418. M.W.Watson and P.W.Carr, AnaLChem. 51 (1979) 1835. 419. W.A.Spencer and L.B.Rogers, Anal.Chem. 52 (1980) 950. 420. R.J.Laub in: Th.Kuwana (ed.), Physical Methods in Modern Chemical Analysis, Vo1.3, Academic Press, New York, 1983, pp.249-341. 421. S.N.Deming and M.L.H.Turoff, AnaLChem. 50 (1978) 546. 422. B.Sachok, J.J.Stranahan and S.N.Deming, Anal.Chern. 53 (1981) 70. 423. J.W.Weyland, H.Rolink and D.A.Doornbos, J.Chromatogr. 247 (1982) 221. 424. P.R.Haddad, A.C.J.H.Drouen, H.A.H.Billiet and L.de Galan, J.Chromatogr. 282 (1983) 71. 425. J.W.Weyland, C.H.P.Bruins and D.A.Doornbos, J.Chrornatogr. Sci. 22 (1984) 31. 426. J.C.Giddings, Anal.Chem. 32 (1960) 1707. 427. W.E.Harris and H. W.Habgood, Programmed Temperature Gas Chromatography, Wiley, New York, 1966. 428. L.R.Snyder, J.W.Dolan and J.R.Gant, J.Chrornatogr. 165 (1979) 3. 429. L.R.Snyder in: Cs.Horvath (ed.), HPLC, Advances and Perspectives, Vol.1, Academic Press, New York, 1980, p.207. 430. K.A.Cohen, J.W.Dolan and S.A.Grillo, J.Chromatogr. 316 (1984) 359.

169