Chapter 6 Programmed Analysis

CHAPTER 6

PROGRAMMED ANALYSIS Programmed analysis can be defined as a chromatographic elution during which the operation conditions are varied. The parameters that may be varied during the analysis include temperature, mobile phase composition and flow rate. In many respects programmed analysis does not differ from chromatography under constant conditions. Retention is still determined by the distribution of solute molecules over the two chromatographic phases and the selectivity of the system is still determined by differences between the distribution coefficients of the solutes. However, if the operation conditions are changed during the elution, then the distribution coefficients may change with time, thus affecting both retention and selectivity. In this chapter we will take a look at some aspects of programmed analysis, particularly those which bear relation to the chromatographic selectivity. The parameters involved in the optimization of programmed analysis will be divided into primary or program parameters and secondary or selectivity parameters. These parameters will be identified for different chromatographic techniques and procedures will be discussed for the optimization of both kinds of parameters.

6.1 THE APPLICATION OF PROGRAMMED ANALYSIS The general elution problem

In chapter 1 (section 1.6) we have seen that only a limited number of sample components can be eluted with optimum capacity factors in a chromatogram (see eqn.l.25). Real-life samples often confront us with the problem that some of the components are bunched together (and ill-resolved) early in the chromatogram, while some other components are eluted in the optimum range of k values (see figure 6.la). If we change the conditions so as to increase the capacity factors of the early eluting components, then the later eluting ones will tend to give rise to impractically high k values. This so-called “general elution problem” (see ref. [601], pp.54-55) is illustrated in figure 6.1 (chromatograms a and 6). The idea of programmed analysis is to vary the operating conditions during the analysis, so that all components of the sample may be eluted under optimum conditions. Such an ideal situation is illustrated in chromatogram c of figure 6.1. Although such an ideal situation may not always be realized, figure 6.lc provides a good illustration of the aim of programmed analysis. The analysis program may be defined as the function that describes the variation of the operating conditions (or elution parameters) with time. Most often, only one parameter is varied during the analysis. Many different programs may be used. The simplest program is a single step (figure 6.2a) in which the parameter x changes instantaneously at a certain time t. Other possible elution programs are illustrated in figure 6.2.

253

1

,

0

50

k-

100

Figure 6.1: Illustration of the general elution problem in chromatography. Chromatogramsa and b: constant elution conditions. Chromatogram c (opposite page): programmed analysis.

When to apply programmed analysis?

In general, programmed analysis may be applied to samples that give rise to the general elution problem, for example samples with a wide volatility (boiling point) range in GC or samples with a wide polarity range in LC. The different ways in which programmed analysis can be applied are summarized in figure 6.3. The first field of application involves the use of programmed analysis as a scanning or scouting technique for unknown samples. In this case the (volatility or polarity) range of 254

1

2

3

d

5

6

7

8

(C)

I

0

tlmin

10

the sample is not necessarily large, but the sample components may fall anywhere in a large range. This application of programmed analysis has been discussed extensively in section 5.4.

I

t-

t-

t-

t-

t-

Figure 6.2: Different shapes of elution programs in chromatography. Description of programs: (a) step; (b) linear; (c) convex; (d) concave; (e) multisegment.

255

The second field of application involves the occasional analysis of wide range samples. In this category we find samples which only occur in the laboratory occasionally and in small numbers, so that only a small number of chromatographic analyses have to be performed. For samples of this kind it is usually sufficient to realize a separation and it is not rewarding to try and optimize the selectivity, not even if the analysis time is rather long and the required number of plates high.. The third field of application in figure 6.3 concerns a routine situation, in which a large number of similar samples needs to be analyzed. It is in this field that it is usually worthwhile to optimize the program. The use of programmed analysis in a routine situation is not attractive. The application of programmed analysis 1. requires more complicated and therefore more vulnerable equipment, 2. leads to reduced analytical reproducibility, 3. leads to increased detection limits because of variations in the baseline*), and 4. will add to the analysis time because of the time required to return to the starting conditions.

Figure 6.3: Schematic illustration of the fields of application of programmed analysis in chromatography.

* Detection limits under programmed conditions compare unfavourably to those obtained with isocratic elution, provided that optimum k values can be obtained in the latter case. 256

Hence, ironically, the best possible result of the optimization of a programmed analysis is a non-programmed one, i.e. a set of conditions where an optimum separation (or at least optimum elution of all components) can be achieved without the need to change parameters during the analysis.

Multicolumn analysis One way to avoid the need for programmed analysis in a routine situation is to use of “multicolumn” or ”column-switching” methods. In these techniques more than one column is used to realize optimum capacity factors (and optimum separation) for all sample components. For example, if we look at the chromatogram of figure 6.1 a, we may use a short column to separate the later eluting components, but a column with a higher phase ratio ( VJ V,J is required to separate the early eluting components. Also, columns with different stationary phases may be used, as long as the columns are all compatible with the mobile phase. If different stationary phases are used, then the selectivity may be optimized using the fixed experimental designs described in section 5.5.1. Multicolumn analysis requires careful optimization. However, the effects of column length, phase ratio and particle size are all predictable, so that the separation that will be achieved on a multicolumn system can be predicted almost exactly. A different set of columns is usually required for every different analytical problem. The effort needed to develop and optimize a multicolumn method will become the more justified the larger the number of analyses that needs to be performed. More information on theoretical [602] and practical [603] aspects of multicolumn techniques in GC can be found in the literature. Ref. [604] contains a review of column-switching methods in LC.

6.2 PARAMETERS AFFECTING SELECTIVITY IN PROGRAMMED ANALYSIS The effects of changes in a parameter during a programmed elution will generally be the product of two independent factors: 1. the relationship between the parameter that is being programmed and the retention under non-programmed conditions, and 2. the variation of the parameter as a function of time. The relationships referred to in the first factor have been discussed extensively in chapter 3. Two important examples are the variation of retention with temperature in GC and with mobile phase composition in LC. If we use programmed analysis to separate wide range samples, then the parameters which are varied during the elution should have a large effect on retention. Hence, the most relevant parameters to be considered for programmed analysis are the primary parameters, which have been listed in table 3.10 for the various chromatographic techniques. Table 6.1 summarizes programmed analysis techniques for various forms of chromatography. An important characteristic of primary optimization parameters is that whereas they have a large effect on the capacity factors of the solutes, they have a relatively minor effect on the selectivity (a). This implies that the factors involved in optimizing an analysis program (the initial and final conditions, programming rate and shape of the program) do not affect the chromatographic selectivity to a large extent. This will be even more true 257

for parameters which have no effect whatsoever on the selectivity under non-programmed conditions. Hence, techniques which involve the programming of such parameters (e.g. .* flow rate programming) will not be discussed in this book. Table 6.1: Programmed analysis methods for various forms of chromatography. Method

Primary parameters (1)

Programmed analysis

GC

Temperature

Temperature programming

RPLC

Mobile phase polarity PH

Solvent programming (gradient elution) pH gradients

LSC

Eluotropic strength

Solvent programming

I EC

Ionic strength PH

Salt gradients pH gradients

I PC

Various

(2)

SFC

Mobile phase density

Density programming; pressure programming Solvent programming

Mobile phase composition ~

_

_

~

~~

~

(1) See table 3.10.

(2) Not compatible with programmed analysis owing to slow equilibration.

The second factor that determines the effects of programmed analysis, the variation of the elution parameter(s) with time, is usually referred to as the program, for example a temperature program in GC. In LC, the program is often referred to as a gradient. However, we will see below that a programmed analysis in LC involves more than just a gradient and therefore it is better to speak of a program or a gradient program. 6.2.1 Temperature programming in GC We have seen in section 3.1 that the primary parameter in both GLC and GSC is the temperature. We have also seen that retention in GC varies very strongly with the temperature. The followingequation was found to describe the relationship in quantitative terms: Ink=InT+A/T+ B,

(3.10)

where k is the capacity factor under isothermal conditions at an absolute temperature T and A and B are constants. 258

Figure 6.4 shows a schematic example of the variation of retention with temperature in GC for a number of solutes, which could, for example, form part of a homologous series. The vertical lines a and b correspond to temperatures at which chromatograms would be obtained which are similar to the chromatograms a and b in figure 6.1. Hence, we are confronted with the “general elution problem”. This is further illustrated by the two (almost) horizontal lines, which enclose the optimum elution range (1 < k < 10). Apparently, there is no single temperature at which all components can be eluted from the column under optimal conditions. Figure 6.5a shows a typical temperature program for GC. The relevant parameters of the program are also explained in this figure. Temperature programs in GC are almost exclusively linear programs, i.e. during the actual heating step in the program the temperature varies linearly with time. Occasionally a program may be comprised of several linear segments. Figure 6.5b shows the typical variation of the baseline with time during a programmed temperature run according to the program of figure 6.5a. The two main sources of baseline drift in programmed temperature GC are increased bleeding of the stationary phase at elevated temperatures and variations in the gas flow rate. The use of two identical columns and two detectors in a parallel configuration (baseline subtraction), of accurate flow controllers and, especially, the use of stable stationary phases are factors which may be used to reduce the blank signal. Harris and Habgood, in their standard work on programmed temperature GC [605] have shown that the retention time of a component under programmed temperature conditions is a function of the retention behaviour of the solute under isothermal conditions and the programming rate. The latter they defined as the heating rate ( r T :

-

10~1~

a

3

b

Figure 6.4: Schematic example of the variation of retention with temperature in gas chromatography. Retention lines are drawn for a group of 8 solutes (e.g. homologues). Vertical dashed lines (a and b) correspond to chromatograms (a and b) in figure 6.1. “Horizontal” dashed lines indicate the range of optimum capacity factors.

259

I

0 4

start ( injectI

,I,,

‘b’

ti

t-

-t

Figure 6.5: (a) Schematicillustration of a temperature program for gas chromatography. The relevant parameters of the program and the units in which they are typically expressed are as follows: Ti= initial temperature (“C); 7’’ = final temperature (“C); rT = heating rate (OC/min); ti = initial time (min); 9 = final time (min). (b) Typical variation of the baseline as a function of time in programmed temperature GC. “C/min) divided by the flow rate ( F ; ml/min). Because of this, it may be hard to reproduce retention data in temperature programmed G C exactly, because whereas it may be possible to accurately control the heating rate, it may be more difficult to reproduce the flow rate F within 0.5%. Resolution in programmed temperature GC is enhanced if the programming rate ( r T / F ) is decreased and if the initial temperature (Ti)is decreased. Giddings [606] suggested that the first peak in a programmed analysis should not appear within about five times the hold-up volume of the column. Since the temperature has little effect on the selectivity in GC (see section 3.1.1), the optimization of temperature programs is a process that may be seen as resolution optimization rather than as selectivity optimization.

6-22 Gradient elution in LC We have seen in chapter 3 (table 3.10 b-d) that the composition of the mobile phase is a primary parameter in various forms of LC (LLC, RPLG, LSC). Gradient elution is only

relevant for the latter two techniques, because the LLC system is not compatible with mobile phase gradients. Figure 6.6a shows a typical gradient program for LC. The complete program can be divided into a number of segments. The program starts and ends at the purge segment (P). The reason for this is related to the typical baseline observed in a gradient elution LC experiment (figure 6.6b). Unlike the situation in GC, the main cause of the blank signal in programmed solvent LC* is formed

* By analogy with the term “programmed temperature GC” [605]we will use the term “programmed solvent LC”, although “solvent programmed LC” is also commonly used. 260

t

Ip

f/

t

I

t-

#

I

t-

Figure 6.6:(a) Schematic illustration of a solvent program (or gradient program) for LC. (p = mobile phase composition: P = purge: R = reverse: E = equilibrate; 1 = inject; G = gradient. (b) Typical variation of the baseline as a function of time in programmed solvent LC.

by impurities in the mobile phase, especially in the weaker solvent. Because of the high capacity factors in this solvent, impurities tend to be concentrated at the top of the column when the weak solvent is run through the column in the equilibrate segment (E). If a gradient is subsequently applied, then the impurities will be washed from the column and appear as peaks in the chromatogram. In order to minimize the background signal, the equilibrate segment (E) should be kept as short as possible. A second factor that contributes to the baseline variation is the difference in the background signal (absorption; fluorescence) between the two solvents. This effect causes the difference in the baseline level between the left and the centre in figure 6.6b. A more extensive discussion on baseline variations in programmed solvent LC can be found in ref. [607]. The actual gradient is denoted by G in figure 6.6a. Because large instantaneous variations in the composition may reduce both the reproducibility of the analysis and the lifetime of the column, a reverse segment (R) is also necessary in a gradient program. A reproducible blank signal can only be obtained if the duration of the reverse, equilibrate and gradient segments, as well as the time of injection (I) and the flow rate, are accurately controlled. The duration of the purge segment is not relevant in this respect. Therefore, it is to be recommended that a solvent program in LC is built up from a minimum of four segments, starting and ending at the purge level. In RPLC retention varies exponentially with the composition of the mobile phase, i.e. approximately straight lines are obtained in a plot of In k vs. cp (see section 3.2.2). If we look at the retention behaviour of each individual solute, then the optimum conditions (LSS gradient, see section 5.4) correspond to a linear gradient (figure 6.2b). Linear gradients will indeed be optimal when acetonitrile-water mixtures are used as the mobile 26 1

t

t I(c'

-3

log 9

-

-2

0.8

cp-.

0.85

Figure 6.7: Schematic illustration of the variation of retention with mobile phase composition in LC. (a) RPLC with acetonitile-water mixtures; (b) RPLC of small molecules with methanol-water mixtures; (c) LSC; (d) RPLC of large molecules with methanol-water mixtures.

phase. The typical variation of retention with mobile phase composition for some low molecular weight solutes in this system is illustrated in figure 6.7a. Linear, roughly parallel lines are obtained in a plot of In k vs. 'p (see also section 3.2.2). However, if methanol-water mixtures are used as the mobile phase, the retention lines for individual solutes tend to diverge towards 'p= 0, as is schematically illustrated in figure 6.7b (see also figure 3.14). In this system, a linear composition gradient would result in a series of peaks with decreasing intervals. This can easily be understood by following the horizontal dashed line for which In k = 2.3 (k= 10) from left to right in figure 6.7b. As a consequence, slightly convex gradients are optimal for RPLC with methanol-water (and THF-water) mixtures [608]. Nevertheless, for most practical purposes linear gradients are acceptable for RPLC. In LSC, an approximately linear behaviour is observed if In k is plotted against In 9. This is schematically illustrated in figure 6.7~.Hence, in order to obtain the same effect of the gradient program as in RPLC (figure 6.7a for the simplest case of a linear gradient), we should aim at a linear variation of In 'p with time, i.e.*

* Eqn.(6.1) arises from the definition equation of LSS gradients (eqn.54, if a linear relationship is assumed to exist between In k and In cp. 262

ln(cp+d)=ar+ b or cp = c exp ( a t ) - d

(6.la)

In eqm(6.1) and (6.la) a, 6, c and d are constants. Because cp should increase with time, a is a positive constant in both equations and hence a concave gradient (figure 6.2d) is

optimal for LSC. Figure 6.7d shows the variation of retention with composition for some large molecular weight solutes in RPLC. In this case, the mobile phase composition has a very strong effect on the retention of the solutes (note the tenfold expansion of the horizontal axis). For low molecular weight solutes, the slope in the retention vs. composition plots is typically around 7 (see section 3.2.2) and therefore a typical solute can be eluted with a capacity factor in the optimum range (1 < k < 10) at mobile phase compositions which span a range of about 30% (2.3 x 100/7). Hence, for a limited number of low molecular weight solutes, there is a good chance that an isocratic composition can be established that is within the optimum range of each individual sample component. An example is given by the vertical line in figure 6.7a. The situation is quite different for high molecular weight solutes, as is illustrated in figure 6.7d. For large, polar molecules that may be eluted with RPLC (e.g. proteins), retention may be expected to be an exceptionally strong function of mobile phase composition [609]. In this case, every individual sample component will have a very narrow composition range over which optimum capacity factors will occur. If a number of different large molecular weight components are present in the sample it may be almost impossible to find a constant (isocratic) composition that will give rise to optimum capacity factors for all sample components, and hence the use of gradient elution may be hard to avoid. It turns out [609] that the slope in the In k vs. cp plots is mainly determined by the molecular weight of the solute. Solutes with very large molecular weight show very steep lines. The lines denoted by 1 and 2 in figure 6.7d form two examples. Retention, however, is mainly determined by the polarity of the solute. Therefore, a component with a much lower molecular weight but also a lower polarity than solutes 1 and 2 in figure 6.7d may have a similar retention time, but show a much shallower retention vs. composition line (solute 3 in figure 6.7). Consequently, the regular picture of figure 6.7b is disturbed and a much less structured pattern remains. It will be clear from figure 6.7 that the nature of the mobile phase (compare figures 6.7a and 6.7b) and the stationary phase (compare figure 6 . 7 ~with figures 6.7a and 6.7b) have a great effect on the character of the retention vs. composition plots and hence on the shape of the required (optimum) gradient. It will also be clear that, unlike the situation in GC, the selectivity may be greatly influenced by variations in the mobile phase. The situation becomes more complex if we realize that figure 6.7 only provides schematic illustrations of the typical retention behaviour in different forms of LC. Examples of anomalous behaviour will not be hard to find. For example, figure 6.8 shows the variation of retention with composition for 23 phenylthiohydantoin (PTH) derivatives 263

k

0

0.10

0.20 0.30 0.U

0.50

0.60

9 -

Figure 6.8: Experimental variation of the retention of 23 phenylthiohydantoin(PTH) derivatives of amino acids with mobile phase composition in RPLC. Mobile phase: mixtures of acetonitrile and 0.05M aqueous sodium nitrate buffer (pH = 5.81). All mobile phases contain 3’/0 THF. Stationary phase: ODS silica. Solutes: D = aspartic acid; C-OH = cysteic acid; E = glutamic acid; N = asparagine; S = serine; T = threonine; G = glycine; H = histidine; Q = glutamine; R = arginine; A = alanine; METS = methionine sulphone; ABA = a-aminobutyric acid; Y = tyrosine; P = proline; V = valine; M = methionine; NV = norvaline; I = isoleucine; F = phenylalanine; L = leucine; W = tryptophan; K = lysine. Figure taken from ref. [610]. Reprinted with permission. of amino acids in RPLC using acetonitrile-water mixtures that contain a small amount (3’10) of THF as the mobile phase. The retention behaviour of these solutes under isocratic and gradient conditions was studied by Cohen et al. [610]. In figure 6.8 we recognize a rough pattern of parallel In k vs. cp lines, but we also see that many lines intersect (“cross-over”) due to variations in the slopes for individual solutes. Another complication may arise if we choose to vary the selectivity of a gradient program in LC by varying more than one parameter at the same time. For example, the concentration of two organic modifiers may be varied independently in RPLC (so-called ternary gradients) or both the mobile phase composition and the temperature may be programmed. In figure 6.9 we take a closer look at some ternary gradients in which the composition (cp) of two solvent components ( B and C) is varied with time*. For simplicity, figure 6.9 has been limited to linear gradients. In figure 6.9a the concentration of both organic modifiers is seen to increase with time.

* The third and weakest solvent A is assumed to make up the solvent to loO%, i.e. p,, + p 264

~ pc= + 1.

t-

t-

t-

t-

t

cp

Figure 6.9: Examples of linear ternary gradients in which the concentration of two modifiers (Band C) is varied simultaneously.The concentration of the base solvent ( A ) is not indicated in the figure.

In figure 6.9b we see that one organic modifier is gradually being replaced by another. In this particular kind of gradient, the two limiting compositions (in figure 6.9b 60% B in A and 40% C in A ) may be of equal eluotropic strength, so that only the selectivity and not the eluotropic strength of the eluent is varied during the elution. Glajch and Kirkland [611] refer to this kind of gradient as “isocratic multi-solvent programming”, because the elution pattern of the solutes and the resulting chromatogram will represent isocratic elution much more closely than typical gradient elution. Although it may be possible to vary the selectivity in different parts of the chromatogram by “isocratic multi-solvent programming”, it should be noted that this technique features all the disadvantages of programmed analysis described in section 6.1. Hence, if “simple isocratic mixtures” (mixtures of constant composition [61 I]) can be used, the use of “isocratic multi-solvent programming” should be avoided. a ternary gradient is shown in which a small concentration of the second In figure 6 . 9 ~ modifier C i s present throughout the elution. Figure 6.9d shows a gradient that runs from 100% A to 100% B and subsequently from 100% B to 100% C. This may be a sensible program if C is a considerably stronger solvent than B. Although all the gradients in figure 6.9 are ternary gradients in that two parameters (the concentrations of two modifiers) are varied at the same time, we may interpret three of the four gradients as special forms of binary gradients (solvent A‘ -solvent W ) ,in which A’ and B’ are mixtures of the pure components A, B and C. We may refer to A‘ and B‘ as pseudo-solvents and to the ternary gradients as pseudo-binary gradients. The last gradient in figure 6.7 (figure d) can be seen as a combination of two binary gradients 265

-

(solvent A' solvent B' -, solvent C'). The compositions of the different pseudo-solvents for the gradients shown in figure 6.7 are listed in table 6.2. Table 6.2: Compositions of the solventsin pseudo-binary gradients (A'-B' are equivalent to the ternary gradients of figure 6.7. Gradient Figure 6.7

Solvent A' '/o B

0 60 0 0

Solvent B' 90

c

0 0 10

0

'/o B 60 0 90

100

or A'+B'+C'

)which

Solvent C' 'lo c

40 40 10 0

'/o B

O /'

c

N/A N/A

0

N/A 100

From the above we may conclude that many of the ternary gradients which may be used in LC can be seen as special forms of binary gradients. Of course, this conclusion is no longer correct if we do not restrict the discussion to linear gradients and allow the shape of the gradient for one solvent to be different from that for another. However, it may be difficult to find applications for which such complicated ternary gradients can be proved to yield better results than the simpler (pseudo-) binary ones. Summary In this section we have come to the following conclusions. I . A solvent program (or gradient program) in LC should be comprised of at least four segments (purge, reverse, equilibrate and gradient; seefigure 6.6). The program should begin and end at the purge stage. 2. The pattern of the variation of retention with composition in LC is aflected by the choice of both the stationary and the mobile phase. The optimum shape of the gradient for unknown wide range samples is dictated by the phase system. Linear or slightly convex gradients are optimal for RPLC. Concave gradients are optimal for LSC. 3. For specijk samples the optimum shape may deviatefrom this general rule. The retention and selectivity under gradient conditions may not follow the expected pattern because of anomalies in the isocratic retention vs. composition relationships. 4. The selectivity in programmed solvent LC may be varied by varying the solvents used or by the application of ternary or even more complicated gradients. However, most ternary gradients can in fact be reduced to binary ones using mixed (pseudo-) solvents. 6.3 OPTIMIZATION OF PROGRAMMED ANALYSIS

There are two aspects involved in the optimization of programmed analysis. The first one is the optimization of the parameters of the program. These parameters include the initial and final conditions, the shape of the program (see figure 6.2) and the duration of 266

the program segments, for instance the heating rate (in programmed temperature GC), or the slope of the gradient in programmed solvent LC. Programmed analysis almost always involves the variation of primary parameters during the analysis. These parameters (and others such as the flow rate and the length of the column) will affect the separation, but the selectivity (a) is only slightly (or not at all) affected. Nevertheless, the program parameters form a relevant part of the optimization of programmed analysis. For instance, the choice of the initial conditions will affect the resolution for peaks that appear early in the chromatogram and the shape of the gradient will determine the overall distribution of the peaks over the chromatogram and may affect the selectivity for some pairs of peaks. Multisegment programs (see figure 6.2e) may allow the optimization of the resolution throughout the entire chromatogram. Examples of this will be given below. The second aspect of optimization in programmed analysis involves adapting the selectivity by variation of secondary parameters. The various secondary parameters listed in table 3.10 may be used to vary the selectivity of a chromatographic system without affecting retention to a great extent (see the discussion in section 3.6.1). The situation in programmed analysis is similar to the one described above for chromatographic elution under constant conditions, in that retention and selectivity may be optimized more or less independently. However, under constant elution conditions the optimization of the retention only involves adapting the primary parameters such that all capacity factors fall into the optimum range (1 < k < 10). In programmed analysis the optimization of the retention involves optimizing the characteristics of the program (initial and final composition, slope and shape) in conjunction with the physical parameters (e.g. flow rate and column dimensions, see section 3.6). If the program is optimized so that all sample components are eluted under optimal conditions, then other (secondary) parameters may be used for the optimization of the selectivity. However, changes in the secondary parameters may imply that the parameters of the program need to be re-optimized. For example, if the selectivity in a temperature programmed GC analysis is insufficient, then another stationary phase may be used to enhance the separation. However, the optimum program parameters obtained with one stationary phase cannot be transferred to another column that contains another stationary phase. The re-optimization of the temperature program for the other column will require at least one additional experiment to be performed. The primary and secondary parameters that may be used for the optimization of the program and the selectivity in programmed analysis, respectively, are listed in table 6.3 for the various chromatographic techniques. It can be seen in table 6.3 that the optimization of selectivity in programmed temperature GC involves variation of the (nature or composition) of the stationary phase. To vary this parameter, a different column and re-optimization of the (primary) program parameters will be required. This is clearly not a very attractive proposition and therefore the optimization of programmed temperature GC is usually restricted to optimizing the program. In programmed solvent LC the nature of the modifier(s) in the mobile phase is the most common secondary parameter that may be used for the optimization of the selectivity. This is an attractive parameter, because different modifiers may be selected and programmed automatically on various commercial instruments. Therefore, the possibilities for selectivi267

Table 6.3: Parameters for the optimization of programmed analyses in various chromatographic techniques. Primary parameters may be used to optimize the program parameters (initial and final conditions, slope and shape). Secondary parameters may be used to optimize the selectivity. Method

Primary parameter(s) (program)

Secondary parameter(s) (selectivity)

GC

Temperature

Stationary phase

RPLC

Mobile phase polarity; pH

Nature of modifier(s); stationary phase

LSC

Eluotropic strength

Nature of modifier(s); stationary phase

I EC

Concentration of counterion; pH

Nature of modifier(s), counterion or buffer

SFC

Mobile phase density

Nature of mobile phase; stationary phase Nature of modifier(s)

Mobile phase composition

ty optimization in programmed analysis are much greater in programmed solvent LC than they are in programmed temperature GC. Selectivity optimization vs. multisegment programs Two ways are open that may lead to the optimization of the resolution of all pairs of peaks in the chromatogram. The first is to use the primary (program) parameters in designing a multisegment gradient, the second relies on the optimization of secondary (selectivity) parameters. In the first case, the resulting programs will be generally complex and consist of many segments. In the second case, relatively simple, continuous programs will be obtained. The latter is generally to be preferred, for the following reasons: 1. With simple, continuous elution programs the elution conditions for the individual peaks (in terms of peak width and detector sensitivity) will either be constant throughout the chromatogram, or will vary in a continuous way. 2. Simpler instrumentation may be used and the effect of the quality of instrumentation on the resulting chromatogram is reduced. 3. The reproducibility of the analysis will be enhanced. 4. Column lifetime will be increased. 5. Selectivity optimization of simple, continuous gradients will be easier than the optimization of complex multisegment programs, because there are bound to be serious 268

discrepancies between theory and practice, which will prohibit the exact calculation of programs comprised of many “subtle” segments. 6. Optimization of the primary parameters of the gradient program may only lead to sufficient separation if the selectivity is sufficiently large. If the a values between one or more pairs of solutes are low, resolution may be enhanced by a reduction of the programming rate and by increasing the number of plates. However, this will be at the expense of increased analysis times and the resolution will never be better than under constant elution conditions. Therefore, selectivity optimization, is in principle to be preferred over multisegment gradients. In GC, where selectivity optimization is not attractive because of the requirement to use different columns, one may wish to resort to multisegment gradients for practical reasons. In LC, where selectivity optimization is readily possible by using different modifiers in the mobile phase, multisegment gradients are of little practical interest. We have seen in section 6.1 that a programmed analysis in chromatography generally requires more time than an experiment under constant elution conditions. Therefore, optimization procedures that require large numbers of experiments are the least attractive for the optimization of programmed analysis. The procedures that were found to require the largest numbers of experiments under constant elution conditions in chapter 5 were the simultaneous (“grid search”) optimization methods (see section 5.2). For this reason, such procedures have not been contemplated for the optimization of programmed analysis and they will not be discussed in this section. Attention in this section will be focussed on the choice between sequential methods as described in section 5.3 and interpretive methods as described in section 5.5. 6.3.1 Optimization of programmed temperature GC 6.3.1.1 Sequential methods

Simplex optimization

Walters and Deming [612] have used a Simplex procedure for the optimization of the program parameters in programmed temperature GC. We have seen in chapter 5 (section 5.3) that one of the main advantages of Simplex optimization procedures is that no knowledge is required about the relationships between the parameters considered on the one hand and the retention and selectivity on the other. Hence, a Simplex program that can be used for the optimization of chromatographic separations under constant elution conditions may be used equally well for the optimization of programmed analysis. All that is necessary to adapt the Simplex program for this purpose is to select an appropriate optimization criterion for programmed analysis. This subject has been discussed in section 4.6.2. The two parameters considered by Walters and Deming [612]were the initial temperature and the heating rate. They used a composite optimization criterion (see section 4.4.2) and imposed a time constraint of 5 minutes on the system by assigning a very unfavourable 269

(infinite) value to the criterion when the analysis time was longer*. The procedure required 13 experiments, two of which could not be performed because negative heating rates were suggested by the optimization program. Because this optimization only concerned program parameters and not selectivity parameters, the response surface will have been relatively simple. Therefore, the probability that the Simplex procedure would arrive at the global optimum rather than at a local one was greater than it was in section 5.3, where wedescribed the use of the Simplex method for selectivity optimization. Systematic sequential optimization

Stan and Steinbach [613] have described a sequential optimization procedure for programmed temperature GC that searches for an optimum multisegment program in a systematic way. This procedure can be divided into three different stages: 1. Separation of a maximum number of peaks by adapting the programming rate of each segment, as well as the length of the preceding isothermal period**. 2. Increasing the resolution (R,; eqn.l.14) values to exceed 1.5 for each pair of peaks by reducing segment slopes and inserting isothermal periods. 3. Reducing the resolution values to be less than 1.5 for each pair of peaks by increasing segment slopes and shortening isothermal periods. The first stage is the actual sequential optimization procedure. It involves the optimization of the heating rate of each segment followed by the optimization of the duration of the preceding isothermal period. As an example, a program was described in ref. [613] that started with (splitless) injection of the sample at 100 O C . The injection period was followed by a rapid (30 OC/min) heating to the initial program temperature (150 "C). The total span of the program from 150 to 250 OC was divided in five non-isothermal segments, each spanning 20 OC. Isothermal segments could be inserted before each of these, so that a total of ten segments was considered during the first stage of the process. The procedure starts by recording a first chromatogram in which the maximum heating rate (30 OC/min) is applied throughout the program. After the injection period, the temperature is raised from 100 OC (injection temperature) to 250 "C. The resulting chromatogram is shown in figure 6.10a. In this example, 28 peaks can be registered from the chromatogram. The sequential optimization procedure now starts by optimizing the last segment of the program (230-250 "C)aiming to increase the number of peaks observed in the chromatogram. To this end, the slope of this segment is successively reduced from 30 OC/min to 8, 4 and 2.67 OC/min. If reducing the slope does not result in an increase in the observed number of peaks, then the value is rejected and the previous one is retained. A similar procedure is followed for the next segment (isothermal at 230 "C). The duration of this

* This time constraint is required because, as we have seen in section 4.4, the incorporation of a (weighted) time term into the optimization criterion is not an effective way to constrain the analysis time (see eqn.4.29 and subsequent discussion). ** In this optimization procedure a segment usually refers to a part of the temperature program at which heating takes place. Such segments may be separated by isothermal periods, during which the temperature is kept constant. In the present discussion we will refer to the two kinds of segments as non-isothermal and isothermal, respectively. 270

period is increased in steps from 0 to 1 , 2 or 3 minutes, until there is no further increase in the observed number of peaks. For the optimization of each segment a minimum of one and a maximum of three experiments needs to be performed. Every experiment involves a complete temperature program from 100 O C (injection) to 250 O C . For the optimization of the entire ten-segment program, 11 to 31 experiments (including the initial run) are required. Figure 6.10b shows the resulting chromatogram after 16 experiments were performed following the procedure described above. The temperature program is shown underneath the chromatogram. During the procedure, the number of registered peaks has been increased from 28 to 36, During the second stage of the procedure, each pair of peaks is checked for sufficient resolution. If R,< 1.5 (eqn.l.l4), then depending on the elution temperature observed for the peak pair, either an isothermal segment may be inserted in the program, or the slope of a non-isothermal segment may be reduced. This procedure may be followed simultaneously for every ill-resolved pair of peaks. Therefore, few additional experiments are required*. Figure 6.1 Oc shows the resulting chromatogram and temperature program after two more injections. It is seen that the program is now much more complicated and that the analysis time has been increased from about 25 to about 43 minutes. During the second stage of the optimization process the number of registered peaks was increased from 36 to 38. Whereas it was claimed in ref. [613] that 38 is the actual number of peaks present in the sample, it seems that at this stage of the procedure additional peaks may only be found accidentally. This will be the case if, in striving for sufficient resolution of one particular pair of peaks somewhere in the chromatogram, a hidden peak is suddenly revealed. In a second optimization cycle these peaks may subsequently be resolved with R,> 1.5. However, if peaks are hidden in parts of the chromatogram in which the resolution appears to be sufficient for all registered peaks, they will not be found during stage 2 of the optimization process. The fact that the presence of two more peaks was revealed during the this stage suggests that additional peaks may be “hidden” in the chromatogram. Therefore, it may illustrate one of the shortcomings rather than one of the advantages of the method. Finally, in the third stage of the process, a procedure similar to that of the second stage may be followed to reduce the resolution of abundantly resolved pairs of peaks (R,>2). During this stage, slopes may be increased and isothermal periods shortened, leading to a reduction of the required analysis time. Figure 6.10d shows the result obtained after an additional two chromatograms. It is seen that the analysis time has been reduced from about 43 to about 37 minutes. The entire procedure illustrated in figure 6.10 involved 21 (1 + 16 + 2 + 2) chromatograms and took about 10 hours. Because of the sequential nature and because of the selection of the criteria, automation is relatively easy. A serious disadvantage of the method, besides the large number of required experiments and the complexity of the resulting program, is the dependence of the result on the column used. Possibly, a different

* It may not be possible to achieve sufficient resolution for all the pairs of peaks in the chromatogram on the particular column. Therefore, a stop criterion is needed in the optimization procedure, for instance a maximum of two attemptsto separatea particular pair of peaks. If it is difficult to recognize (pairs of) peaks, then a maximum of two or three optimization cycles each for stage 2 and stage 3 of the optimization procedure may be considered. 27 1

t lmin -c

250 -

0

tlmin-

I

-

5 10 15 20 25 30 35 LO L5 50 55 60 tlmin

Figure 6.10: Application of the systematic sequential optimization procedure of Stan and Steinbach [613] for the optimization of a temperature program in capillary GC. Column: 25 m x 0.2 mm I.D. coated with dimethylsilicone bonded phase BP-1 (S.G.E.); Carrier gas: helium; Detector: electron capture; Sample: halogenated pesticides (residue analysis). (a) Initial chromatogram; (b) Resulting chromatogram after stage 1 (maximum number of peaks); (c) Resulting chromatogram after stage 2 (increased resolution); (d) (opposite page) Final chromatogram after stage 3 (reduced resolution). For explanation see text. Figure taken from ref. [613]. Reprinted with permission.

272

100,

0

-

5 10 15 20 25 30 35 LO 15 50 tlmin

result may even be obtained on the same column under different operating conditions (e.g. flow rate). This is due to the use of the column-dependent R, criterion during the second and third stages of the optimization process (see discussion in section 4.3.3). Finally, a reasonable estimate for the initial and final program temperatures should either be made on beforehand, or established from the first chromatogram. 6.3.1.2 Interpretive methods

The obvious alternative to the sequential optimization methods is the use of an interpretive optimization method. In such a method a limited number of experiments is performed and the results are used to estimate (predict) the retention behaviour of all individual solutes as a function of the parameters considered during the optimization (retention surfaces). Knowledge of the retention surfaces is then used to calculate the response surface, which in turn is searched for the global optimum (see the description of interpretive methods in section 5.5). For programmed temperature GC the framework of such an interpretive method has been described by Grant and Hollis [614] and by Bartd [615]. All interpretive optimization methods are by definition required to obtain the retention data of all sample components at each experimental location. If the sample components are known and available they may be injected separately (at the cost of a large increase in the required number of experiments). For unknown samples, for samples of which the individual components are not available, and in those situations in which we are not prepared to perform a very large number of experiments (as will usually be the case in the optimization of programmed analysis) we need to rely on the recognition of all the individual sample components in each chromatogram (see section 5.6). 273

If many peaks occur in a chromatogram this appears to be a very difficult proposition. However, the requirement of solute recognition may not give rise to insurmountable problems in the optimization of programmed temperature GC for the following two reasons: 1. The optimization procedure may be carried out on the basis of a limited number of (major) components in the sample [614]. 2. Changes in elution order (”component cross-overs”) are unlikely to occur. For the interpretive optimization of the primary (program) parameters in the programmed analysis of complex sample mixtures it may well be sufficient to optimize for the major sample components. This may be done if it is assumed that the primary parameters do not have a considerable effect on the selectivity, so that if the major sample components are well spread out over the chromatogram, the minor components in between these peaks will follow suit automatically, and if it is assumed that the minor peaks are randomly distributed over the chromatogram. The major chromatographic peaks can be separated to any desired degree if optimization criteria are selected which allow a transfer of the result to another column. Changes in elution order are unlikely to occur, because temperature is not a truly selective parameter (see section 3.1). To a first approximation, the elution order of the peaks, and certainly the elution order of the major components, may therefore be assumed constant. The retention behaviour of solute moleculesunder programmed temperature conditions is completely characterized by 1. the parameters of the program, and 2. the variation of the retention with the temperature under isothermal conditions. If we leave out of account the delay that both the column and the packing material may cause in the temperature program inside the column relative to the program followed in the column oven [615], then the program parameters are naturally known. In principle, the description of the retention vs. temperature relationships requires two experiments, because a straight line can be obtained by plotting In (k/T) vs. 1 / T (eqn. 3.10). Grant and Hollis [614] assume a linear relationship between In k and 1/T: Ink=A+B/T

(6.2)

where A and Bare solute-dependent coefficients.They assume that the intercept A remains “sensibly constant”, and that the slope B is proportional to the (absolute) boiling point for solutes within a given class. Therefore, the isothermal retention data for some “typical” solutes from a class at a minimum of two different temperatures is thought to be sufficient to describe the retention behaviour of all solutes within that class under programmed temperature conditions. Unlike eqn.(3.10), eqn.(6.2) is not a fundamentally linear relationship. Since both equations require two experimental data points two establish the coefficients, the use of the former is to be preferred. BartQ [615] uses a different relationship to describe the retention vs. temperature relationship. His equation is also not fundamentally linear and requires a minimum of three parameters: 274

In(t,-C')

=A

+ B/T.

(6.3)

In this equation t , is the retention time under isothermal conditions at the temperature T and A, B and Care constants. An analogous expression is used to describe the variation of the peak width at the location of half the peak height ( w , , ~with ) temperature. The two experiments required to describe the isothermal retention vs. temperature relationships through eqn.O.10) or eqn.(6.2), or the three required by eqn.(6.3), may either be performed isothermally or under programmed conditions. However, in the latter case the calculations to obtain the coefficients A, B, and, if eqn.(6.3) is used, C, will be more complicated and more than two or three experiments may be required to estimate the coefficients with sufficient accuracy. The latter aspect suggests the use of an iterative interpretive method, in which the values of the coefficients are updated after each new experiment until the accuracy of the predicted optimum turns out to be sufficient. Neither the procedure described by Grant and Hollis [614], nor that of Bartd [615] is a complete optimization procedure. They do not provide a generally useful strategy for unknown or ill-known samples. The application of either approach in practice has not been described. 6.3.1.3 Discussion Simplex optimization of the primary (program) parameters in programmed temperature GC analysis has been demonstrated [612]. A systematic sequential search [613]may be used as an alternative. The Simplex method may be used to optimize a limited number of program parameters, whereas the latter approach was developed for the optimization of multisegment gradients. The use of interpretive methods has so far only been suggested [614, 6151. As was the case in its application to the optimization of chromatographic selectivity under constant conditions, the Simplex algorithm appears to require a rather large number of experiments. This is also true for a systematic sequential procedure. If interpretive methods are used, the calculations involved may be complicated and it is necessary to recognize the individual solutes in each chromatogram. Because the optimization procedure may be carried out for a limited number of major sample components and because the elution order is not likely to vary, this will not usually be a serious problem. It certainly would not have been a problem in the example for which the Simplex program was demonstrated in ref. [612]. In this sample only four components were present. The selection of this particular example to demonstrate the applicability of Simplex optimization for programmed temperature GC was unfortunate in any case, because a straightforward isothermal separation of the sample at 70 OC also appeared to be possible. The example shown in figure 6.10 (ref. [613]) for the optimization of a multisegment temperature program was more impressive. Unfortunately, the required number of experiments was large (21). The selection of simple criteria (e.g. maximum number of peaks) may greatly enhance the possibilities for fully automatic optimization. If an interpretive method is used, then the number of experiments required to allow an accurate prediction of the global optimum may be somewhat larger than the theoretical minimum of two experiments. However, this still compares favourably with the 21 275

experiments performed by Stan and Steinbach [613b who used a systematic sequential procedure, and to the 11 experiments performed by Walters and Deming [612]to locate the optimum with a Simplex method. Moreaver, Wdters and Deming performed 8 additional experimentsin the vicinity of the qtimurn to enhance the accuracy oithe result. 6.3.1.4 Selectivity optimization

In order to optimize the selectivity in programmed temperature GC, the parameter to be varied is the nature or composition of the stationary phase. If this kind of optimization is to be pursued, then the Simplex procedure will be especially unattractive, because it will require large numbers of experiments using different stationary phases and, consequently, different columns. Therefore, interpretive methods are to be preferred for optimizing the selectivityin programmed temperature GC. Because of the experimentallyobserved linear relationship between retention and composition in isothermal GC using mixed stationary phases (eqn.3.14), fixed experimental designs may be used, similar to those employed for optimizing the stationary phase composition in isothermal GC (window diagrams, see section 5.5.1). 6.3.1.5 Summary 1. Simplex optimization of the primary parameters in programmed temperature GC analysis is possible.

2. As with other applications of the Simplex algorithm in chromatography (see section 5.3), a large number of experiments is required. 3. The response surface for the optimization of the primary (program) parameters in programmed temperature GC is less convoluted than a typical response surface obtained in selectivity optimization procedures (see section 5.1). This will increase the possibility of a Simplex procedure locating the global optimum. 4. A systematic sequential optimization procedure may be used to establish an optimum multisegment temperature program. 5. Such a procedure requires a large number of experiments, but may readily be automated. 6. For simple samples, in which the individual components can be recognized, interpretive methods should ailow the prediction of the (global) optimum from a small number of experiments. 7. For complex samples the separation of the major components may be optimized by an interpretive method. The resulting optimum program presumably corresponds to the optimum for the entire sample. 8. Optimization of the selectivity in programmed temperature GC requires the application of diflerent stationary phases or stationary phase mixtures. 9. In that case, interpretive methods based on f u e d experimental designs (window diagrams) may be used. 6.3.2 Optimization of programmed solvent LC

The (primary) program parameters may be used to optimize the separation in programmed solvent LC in a non-selective way. Since this involves optimization of the 276

retention rather than the selectivity, this kind of optimization will only be adressed briefly in this section. The optimization of the program parameters has been discussed extensively by Snyder [616,6171 and more recently in an excellent book by Jandera and ChurhEek [618]. The most useful secondary parameter for the optimization of the selectivity in programmed solvent LC is the nature of the modifier(s) in the mobile phase. The selectivity can be varied by selecting various solvents (pure solvents for binary or ternary gradients; mixed solvents for pseudo-binary gradients). Analogous to the situation in isocratic LC, it is possible to use different modifiers (and hence to obtain different selectivity),. while optimum retention conditions are maintained for all solutes. This possibility to optimize the selectivity in programmed solvent LC will be discussed below. 6.3.2.1 Simplex optimization

As with programmed temperature GC, the application of the Simplex optimization procedure to programmed solvent LC is relatively straightforward. The same procedure can be used both for isocratic and for gradient optimization, as long as an appropriate criterion is selected for each case*. After earlier applications of the Simplex algorithm for the optimization of programmed solvent LC by Watson and Carr [619] and by Fast et al. [6201, the possibility of applying (slightly) different versions of a single Simplex program for the optimization of isocratic and programmed solvent analysis in LC was demonstrated by Berridge [621]. He used the Simplex procedure to optimize three program parameters: the initial and the final composition and the duration of a linear gradient. The convergence of the Simplex algorithm to the final optimum was said to be rapid, but still 15 experiments were required to arrive at the optimum. A reason for such a “ r a p i d convergence was suggested to be the location of the resulting optimum on the edge of the parameter space (final composition: 100 %B). Another reason may be the relative simplicity of the response surface in comparison to isocratic optimization in which the selectivity (secondary parameter: nature and concentration of modifiers) is varied. An indication of this latter effect can be found in figure 6.1 1, which shows the result of the Simplex optimization procedure applied to the programmed solvent LC separation of three antioxidants [621]. The sum of peak-valley ratios was used as the resolution term in a composite optimization criterion, which otherwise corresponds to eq~(4.30).Berridge also added a term to describe the contribution of the number of peaks (n). With this, the complete optimization criterion became

The desired analysis time (t,,,) was set equal to 4 min., whereas the value of the minimum time ( tmin,which is irrelevant for the optimization process; see section 4.4.2) was taken to be 1.5 min.

* For criteria based on the peak-valley ratio ( P ) no modification of the criterion used for isocratic optimization may be necessary (see section 4.6.2).

277

0

1

2 tlmin-

3

Figure 6.1 1: Resulting chromatogram from a Simplex optimization procedure applied to the separation of three antioxidants. Solvents: 5’h acetonitrile in water (A) and 5Oh water in acetonitrile (B). Linear gradient 44to 100% B in A in 1.5 min. Column: 10 cm x 5 mm I.D. 5 pm Lichrosorb C-18. Flow rate: 2.0 mL/min. Solutes: 1 = propyl gallate; 2 = 2-t-butyl-p-methoxyphenol (BHA); 3 = unknown; 4 = 2,6-di-t-bytul-p-cresol(BHT).Figure taken from ref. [621].Reprinted with permission. It can be seen in the chromatogram of figure 6.1 1 that four peaks (the three antioxidants plus an unknown impurity) are amply resolved to the baseline. This implies that all values for the peak-valley ratio Pare equal to 1 and that the criterion has become very insensitive to (minor) variations in the resolution between the different peak pairs. In the area of the parameter space in which four well-resolved peaks are observed, the only remaining aim of the optimization procedure is to approach the desired analysis time of 4 minutes. The irrelevance of the “minimum time” tmin is illustrated by the occurrence of the first peak in figure 4.9 well within the value of 1.5 min chosen for this parameter. The application of the Simplex procedure for the optimization of the selectivity in programmed solvent LC (e.g. for the application of ternary gradients) has not yet been reported. However, there is no apparent obstacle to the applicability of the Simplex procedure for this purpose. Of course, the simultaneous optimization of different (primary) program parameters (initial and final composition, slope and shape of the gradient) and secondary parameters (nature and relative concentration of modifiers) may involve too many parameters, so that an excessive number of experiments will be required to locate the optimum. This problem may be solved by a separate optimization of the program (primary parameters) and the selectivity (secondary parameters) based on the concept of iso-eluotropic mixtures (see section 3.2.2). This will be demonstrated below (section 6.3.2.2). However, the transfer of

278

the program parameters optimized with one modifier to an analysis program using another modifier (or a combination of two modifiers in a ternary gradient) requires more knowledge and understanding of the relationships between chromatographic retention and the parameters considered in the optimization procedure than is usual for Simplex optimization.

6.3.2.2 Systematic optimization of program parameters Optimization without solute recognition The concept of linear solvent strength (LSS) gradients developed by Snyder (see also sections 5.4.2 and 6.2.2) incorporates optimization of both the shape and the slope of gradient programs. The shape of an LSS gradient is determined by 1. the definition equation of LSS gradients, i.e. log kin = log k , - b ( t / t o ) , where kinis the capacity factor of the solute under the isocratic conditions at the column inlet at time r, k , the capacity factor under isocratic conditions corresponding to the initial composition of the gradient program, b the gradient steepness parameter, t the time elapsed since the start of the gradient (or, more precisely, the time elapsed since the arrival of the gradient at the inlet of the column) and to the hold-up time of the column. 2. The relationship between retention and composition under isocratic conditions, i.e. the function k = f(q).

(6-5)

The combination of these two factors determines the required shape of an LSS gradient. Linear gradients were shown to result for RPLC in section 5.4, whereas a concave gradient was found to be optimal for LSC in section 6.2.2. The optimal slope of the gradient also follows from the LSS concept, since it was shown by Snyder et al. I6161 that optimum values for the gradient steepness parameter b are in the range 0.2 < b < 0.4. If the function f(q) is known, then the optimum slope of the gradient can be calculated. For example, in RPLC the relationship between retention and composition over the range 1 < k < 10 can be described by I n k = Ink,

-

Sq.

(3.45)

In RPLC an LSS gradient is a linear gradient that can be described by

(5.6)

q = A + B t .

Combination of eq~(3.45)with eqa(5.5) (see also section 5.4) yields b = S B to / 2.303

.

(5.8)

279

Typical S values for small solutes using methanol-water mixtures as the mobile phase are in the range 5 < Sc 10 [608]. The value of to is determined by the column and the flow rate. For example, if a column of 15 cm length with an internal diameter of 4.6 mm is used, the hold-up volume ( Vo)is around 1.5 ml, so at a flow rate of 1.5 ml/min the hold-up time ( f a ) is about 1 min. An optimal gradient with a b value of 0.3 then leads to a range of B values in eqm(5.6) given by 0.069 < B

-= 0.138,

where B is expressed in min-I. The optimum programming rate is seen to be between about 7 and 14 %/min. For a 0-100% gradient this corresponds to gradient durations (t,) in the range

where f , is expressed in minutes. Snyder et al. [616,622] recommend a simple trial-and-error approach for the optimization of the remaining two parameters of the program, i.e. the initial and the final composition. These parameters should be adapted such that solute bands are eluted neither too early, nor too late in the chromatogram. If larger solute molecules (e.g. proteins) are to be separated by programmed solvent LC, then much higher S values may be expected and consequently (eqn.5.8) a lower B value (shallower gradient) will be required [609]. The Snyder procedure would have led to a quick solution of the separation problem shown in figure 6.1 1. However, the answer would have been different from that obtained with the Simplex optimization program. If we assume an S value of about 7 for the solutes involved and estimate the hold-up volume of the column to be around 1.18 mL (60% of the volume of the empty column), then we can estimate the b value for the gradient used in figure 6.1 1

6 = S B to / 2.303 = S 19 V, / (2.303 F) = 7 x 0.373 x 1.18 / (2.303 x 2) = 0.67. This shows that the very fast gradient (t,= 1.5 min.) used in figure 6.1 1 was indeed two or three times steeper than the optimum conditions suggested by Snyder. Following Snyder’s approach, the first experiment could have been a gradient of 0 100 %B in A in 6 minutes ( b = 0.30). As a result of this gradient, the initial concentration could then have been increased to yield (after one or two experiments) an optimum program with a gradient from about 50 to 100% B in 3 minutes. The overall analysis time (retention time of the last peak) would not have been much longer than the 3 minutes observed in figure 6.11, whereas all peaks would have been eluted under optimum conditions with roughly equal peak widths. The last peak in figure 6.1 1 is considerably broader than the other ones, because it is eluted after the completion of the gradient program. However, the most important difference between the Simplex procedure and a systematic approach such as the one suggested by Snyder is not in the quality of the

280

resulting chromatogram but is the number of experiments required. For the optimization of the primary (program) parameters the former required 15 experiments, whereas the latter would not have required more than 2 or 3.

Optimization with limited solute recognition In the Snyder approach to gradient optimization the characteristics of the individual solutesare largely neglected. The optimum shape of the gradient is determined by the phase system and the optimum slope is usually estimated from simple rules for the retention behaviour of the solutes (e.g. assuming S = 7 for small solute molecules as we did above). Only the initial and the final conditions are adapted to the requirements of the sample. A strategy for the optimization of gradient programs based on the actual retention behaviour of some sample components has been described by Jandera and Churaeek [623, 6241. This approach relies on the possibility to calculate retention and resolution under gradient conditions from known retention vs. composition relationships and plate numbers. Both typical RPLC (eqn.3.45) and LSC (eqn.3.74) relationships can be accommodated in the calculations and linear, convex and concave gradients are all possible because of the use of a flexible equation to describe the gradient function. This equation reads Q

=

+ B V)

where A is the initial concentration, B the slope of the gradient and V is the volume of eluent delivered since the start of the gradient. Vis related to the elapsed time t and the flow rate F by V = Ft. K characterizes the shape of the gradient. If K = 1 the gradient is linear. K < 1 corresponds to a convex gradient and K > 1 to a concave one. Optimum gradients were defined by Jandera and ChuraEek [624] to yield 1. a preset (required) value for the resolution (R,) between two arbitrary solutes, and 2. a minimum retention volume for another arbitrary solute. We can summarize this optimization goal in a way that is consistent with the criteria described in chapter 4 (section 4.3.3) as follows:

In eqn.(6.7) the pair of solutes for which a minimum resolution of x is required is denoted by i and i 1. j denotes the sample component for which the retention volume under gradient conditions ( Vg)is to be minimized. If the retention vs. composition relationships for the solutes i, i + 1 andjare known, then the gradient parameters A, B and K can readily be calculated for the optimum gradient according to equation 6.6. Not unexpectedly, the value of the shape parameter K turns out to be of little significance for an optimization procedure in which only three solutes affect the result [624]. Therefore, it may be sufficient to optimize the parameters A and B for a linear gradient ( K = 1). Figure 6.12a shows the resulting optimal chromatogram for the separation of a mixture of seven barbiturates by programmed solvent RPLC. This figure was obtained with the following optimization criterion:

+

281

Vlml

10

-

0 V/mllO

5

Vlml

-

5

0

-

10

Figure 6.12 Resulting chromatograms from the Jandera and ChuraEek gradient optimization method. (a) requiring a minimum resolution ( R 3 between solutes 1 and 2 of 1.7 and minimizing the retention volume ( Vg)of solute 7 (eqn.6.7a); (b) requiring a minimum resolution between solutes 6 and 7 of 1.75 and minimizing the retention volume of solute 1 (eqn.6.7b); (c) linear gradients used to obtain the chromatograms a and b (gradient a: 9 = 0.368 + 0.061 V; gradient b (p = 0.523 + 0.0082 V). Mobile phase components: water (A) and methanol (B). Stationary phase: Lichrosorb ODs. Solutes: 1. barbital; 2. heptobarbital; 3. allobarbital; 4. aprobarbital; 5. butobarbital; 6. hexobarbital; 7. amobarbital. Figure taken from ref. (6241. Reprinted with permission. (6.7a) Figure 6.1 2b shows the resulting chromatogram obtained under the conditions RJ7,6 > 1.75

n min Vg,,.

(6.7b)

The two different linear gradients are shown in figure 6.12~. It can be seen in figure 6.12 that the two different criteria described by eqns.(6.7a) and (6.7b) result in different gradient profiles and different chromatograms. In figure 6.12a the resolution between the last two peaks is clearly insufficient. In figure 6.12b the resolution of these last two peaks has increased, but at the expense of a decreased resolution of the first two peaks. In the first chromatogram the gradient is too steep to obtain sufficient resolution. In the latter chromatogram the initial concentration may be slightly too high. Clearly, neither in chromatogram a nor in chromatogram b is the resolution optimized 282

throughout the chromatogram. This is a disadvantage of the procedure. Another disadvantage is that a choice needs to be made as to which two components will be the most difficult to separate (“critical pair”) and for which solute the retention volume should be minimized. The two above chromatograms illustrate that a different choice for the solutes involved in the optimization criterion will lead to a different result. Apparently, in order to improve the method other optimization criteria need to be considered. For example, the resolution could be optimized for both the first two and the last two peaks in the chromatogram. Advantages of the procedure are that the calculations are relatively simple and that only the retention vs. composition relationships of the three solutes involved in the optimization criterion need to be known. Complete mathematical optimization If the retention vs. composition plots of all solutes are known, then it is in principle possible to calculate the optimum program parameters for a simple, continuous gradient (figure 6.2a-d). In such a procedure an appropriate optimization criterion can be selected such that the distribution of all the peaks over the chromatogram, as well as the required analysis time, can be taken into account (see chapter 4). However, the calculations required for such an optimization are quite involved. This is caused by the requirement to calculate the retention times of each solute (and the resolutions of each pair of adjacent peaks) from the isocratic retention vs. composition relationships. In order to characterize the response surface, these calculations need to be performed a number of times. Finally, the optimum needs to be found on the response surface. If all four program parameters (initial and final concentration, slope and shape) are considered, the number of calculations would be large, even though the response surface may be simple compared with those encountered in selectivity optimization (see the discussion in section 6.3.2.1). Multisegment gradients

A procedure that avoids the lengthy calculation procedure mentioned above is the one described by Noyes [625]. She designed a multisegment gradient program on the basis of visual interpretation of the isocratic retention vs. composition relationships for a number of phenylthiohydantoin (PTH) amino acids. It was claimed that the mixture could not be separated by a continuous linear gradient, but no further details on the design of the multisegment gradient were given. Issaq et al. [626] have reported on a method for the optimization of a multisegment gradient program for the optimum resolution of all pairs of peaks in a programmed solvent LC chromatogram. In their procedure a number of programmed solvent experiments are performed, either a series of linear gradients between two solvents A and B of variable duration ( t G) , or a series of linear gradients with constant tG, but a variable final concentration of B in A. For each pair of adjacent solutes the gradient which yields the best resolution is then selected and the different linear segments are combined into a multisegment program. The exact procedure in which the multisegment gradient is built up from the optimum 283

gradients for the individual pairs of peaks is not clarified, however, and it remains to be seen whether the calculated program wifl indeed result in optimum separationfor all pairs of peaks in the chrumatogram. It appears that this goal can only be achieved if the elution pattern of a pair of peaks through the column is only affected by the particular segment designed for the optimum resolution of this pair. Unless the different solute pairs are very far apart in the chromatogram (in which case the overall distribution of the peaks over the chromatogram would be far from optimal!), the resolution of a pair of peaks is likely to be much affected by the preceding segments of the program. No examples to demonstrate the applicability of the method are given in ref. €6261. 6.3.2.3 Znterpretive methodsfor selectivity optimization

Glajch and Kirkland [627]have extended the Sentinel optimization method (seesection

5.5.1) to include the optimization of the selectivity in programmed solvent LC. This

optimization procedure allows the use of linear gradients in RPLC using one or more organic modifiers in water. The relative concentration of the modifiers does not change during the analysis (so-called iso-selectivemuiti-solvent gradients (6111, see figure 6.7a). This allows a straightforward extension of the Sentinel method. For the optimization of programmed solvent LC the Sentinel method starts by establishinga suitablebinary methanol-water gradient. The appraa& of Snyderdescribed above may be used for this purpose. For example 16271, a gradient from 20 to IO@h methanol (in water) in 20 minutes may be the result.

THF

ACN

Figure 6.13: Figure illustrating the 7 linear gradients used in the Sentinel optimization method for programmed solvent U=. Initial and final compositions of the gradients are listed in table 6.4. 284

Next, the concept of iso-eluotropic mobile phases is used to determine the binary acetonitrile-water and THF-water mixtures that correspond to the initial and the final composition. For example, 20% methanol corresponds [627] to 17% acetonitrile and to 12% THF, whereas 100% methanol corresponds to 84% acetonitrile and to 59% THF. By analogy with the isocratic Sentinel optimization procedure a series of 7 gradients (all of the same duration time) can now be defined. These gradients are shown in figure 6.13 and the initial and final compositions are listed in table 6.4. The individual retention times of all solutes in a 14-component sample mixture were measured and used to calculate resolution values (eqn.1.14, because eqn.l.22 is invalid) between each pair of peaks in the chromatogram. The largest value for the limiting resolution (max Rs.min;eqn.4.25) was used as the optimization criterion. Table 6.4 Initial and final compositions of the 7 linear gradients shown in figure 6.13. All 7 gradients have the same duration. Gradient number

Mobile phase composition (% v/v) Water

MeOH

ACN

THF

1

Initial Final

80 0

20 100

0

0

0 0

2

Initial Final

83 16

0 0

17 84

0 0

3

Initial Final

88 41

0 0

0 0

12 59

4

Initial

81

10

9

0

Final

8

50

42

0

5

Initial Final

85 28

0 0

9 42

6 30

6

Initial Final

20

84

10 50

0

6 30

7

Initial Final

83 19

7

33

6 28

8 (1)

Initial Final

83 16

2 10

14 69

0

4

20

1

5

(1) Predicted optimum gradient.

285

tlmin I

(b)

al

In C

:: 0

a,

K

0.0

I

:e

LO

-

-

8.0

tlmin

12.0

160

Figure 6.14 Result of a Sentinel optimization of programmed solvent LC. Experimental design according to figure 6.13 and table 6.4. (a) Predicted optimum linear gradient and (b) chromatogram obtained with the optimum linear gradient. Stationary phase: Zorbax alkylsilica. Flow rate: 3.0 ml/min. Solutes: A = resorcinol; B = theophylline;C = phenol; D = benzyl alcohol; E = caffeine; F = methyl paraben; G = benzonitrile; H = nitrobenzene; I = cortisone;J = propyl paraben; K = ramrod L = butyl paraben; M = chloro-isopropyl-N-(3-chlorophenyl) carbamate (CIPC); N = progesterone. Figure taken from ref. [627]. Reprinted with permission. Figure 6.14 shows the resulting quaternary gradient and the resulting chromatogram for the 14-component mixture to which the 7 gradients described in figure 6.13 and table 6.4 have been applied. It will be clear that the interpretive procedure described here allows the recalculation of the resolution surfaces (and the response surface) after the retention times of the individual solutes have been obtained from the chromatogram at the predicted optimum (figure 6.14), so that a n iterative optimization procedure, in which the accuracy of the resulting optimum is improved, is also possible. The Sentinel gradient optimization method, by analogy with the isocratic Sentinel method, requires a minimum of 7 chromatograms to be recorded before the optimum conditions can be predicted and it requires the retention data of all solute components to be established at each experimental location.

286

la1

Step 100

I

I

3

tlrnin-

t

t b

i:I 8.0

tlmin

-

Figure 6.1 5: (a) Step-selectivity gradient program designed after “visual interpretation” of the 7 chromatograms obtained during a Sentinel gradient optimization procedure (figure 6.13). (b) Chromatogram obtained with the step-selectivitygradient of figure 6.15a. Sample and conditions as in figure 6.14. Figure taken from ref. 16271. Reprinted with permission.

Advantagesare that the selectivity is optimized (secondary parameters) so that optimum resolution can be obtained and that all components of the sample are considered in the optimization procedure. Unlike the result of the gradient optimization procedure suggested by Jandera and ChuraEek, (section 6.3.2.2) the lowest value for the resolution in the chromatogram is maximized and not the resolution of an arbitrary pair of solutes. However, because of the selection of the max Rs,min criterion the distribution of the peaks over the rest of the chromatogram (other than the critical pair of peaks) is not optimized (see discussion in section 4.3.3). This was realized by Glajch and Kirkland [627] who therefore tried to optimize a “selective multi-solvent’’ gradient, in which a series of segments is allowed in order to try and optimize the resolution in various parts of the chromatogram. They did not describe a formal procedure for the optimization of such step-selectivity gradients, but used “visual interpretation” of the seven chromatograms obtained during the optimization procedure described above to design the gradient shown in figure 6.15a. The chromatogram obtained with this gradient is shown in figure 6.15b. 287

The chromatogram in figure 6.1 5 is only marginally (if at all) better than the one shown in figure 6.14. However, Glajch and Kirkland correctly state that very few of the possibilitiesof exploiting various selectivegradients have yet been explored. If the relative concentrations of the organic modifiers are allowed to vary and if the variation of composition with time is not restricted to linear relationships, then the distribution of the peaks over the chromatogram may still be greatly improved. However, the use of simple continuous gradients is to be preferred to the use of complex multisegment gradients for a number of reasons outlined in the introduction of section 6.3. predictive optimization method The Sentinel method of GIajch and Kirkland described above involves the measurement of retention data under gradient conditions and the direct optimization of the selectivity, i.e. the differences between these retention times for different solutes. Jandera et al. [628] have described a predictive optimization method in which 1. retention vs. composition relationships are obtained under isocratic conditions using several modifiers, 2. the retention data using ternary gradients are predicted from the isocratic data, and 3. an adequate ternary gradient is selected based on the predicted retention times. According to Jandera et al. [628], the isocratic retention behaviour of solutes in ternary solvents in RPLC may be predicted from data obtained with binary mixtures. However, such predictions are only accurate within about 5%. This accuracy is insufficient for the purpose of selectivity optimization, where small differences in retention times between adjacent peaks are of critical importance. Therefore, ideally, binary as well as ternary mixtures should be used in the isocratic experiments. The selection of an adequate ternary gradient takes place largely on a trial-and-error basis. However, instead of trial experiments, trial calculations are performed until a satisfactory result is predicted. Only then will a trial experiment be performed. Figure 6.16 illustrates the application of the method of Jandera et al. for the selection of a satisfactory linear gradient for the separation of a mixture of 9 phenolic solutes. It is seen in figure 6.16a and figure 6.16b that the mixture is not completely separated using either a binary methanol-water (chromatogram a) or a binary acetonitrile-water gradient (chromatogram 6). Also,an ”iso-selective”linear gradient, in which the ratio between the concentrations of methanol and acetonitrile is kept constant, provides insufficient resolution. Figure 6.16d shows the chromatogram obtained with a linear ternary gradient which was predicted to provide a satisfactory separation. Indeed, the resolution is better than in any of the previous chromatograms (a, b and c) and is sufficient with the column and conditions used in figure 6.16. Figures 6.16e and 6.16f show two chromatograms using gradients which were predicted to yield insufficient separation. Using the optimization procedure of Jandera et al., a number of gradient programs can be tested by calculating the resulting chromatograms, so that the number of experiments required can be greatly reduced. It is interesting to note that the gradient predicted by Jandera et al. could not have been arrived at using the Sentinel method described in figure 6.13. The predictive optimization method of Jandera et al. is designed to yield an “adequate” result. In other words, a threshold optimization criterion is used (eqn.4.23). Once a certain

288

8

60 V/ml

LOV/ml

30

30

1

20

20

I

I

10 - 0

10

-

I

0

V/ml

Vlml

6

I

3

I

30

1

0

10

I

I

20

30

-

I

20

10

-

I

0

8.9

(fl

II

3

8.9

7

LOVlml

30

I

20

10-

I

0

Vlml30

20

10 - 0

I

Figure 6.16: Illustration of the predictive optimization method for ternary gradients in RPLC of Jandera et al. [628]. All figures were recorded with linear gradients from 100°h solvent A to l0O0/o solvent B in 60 min. Stationary phase: Lichrosorb C18. Flow rate: 1.0 mllmin. Solutes: 1 = 4cyanophenol; 2 = 2-methoxyphenol; 3 = 4-fluorophenol; 4 = 3-fluorophenol; 5 = m-cresol; 6 = 4-chlorophenol; 7 = 4-iodophenol; 8 = 2-phenylphenol; 9 = 3-t-butylphenol. Mobile phase components: (a) solvent A: 20% methanol (in water), solvent B: 100°h MeOH; (b) A: 100% water, B: 100°/o acetonitrile (ACN); (c) A: 100°h water, B 60°h MeOH + 40°/o ACN; (d) A: 20°/0 ACN, B: 100°h MeOH; (e) A: lOoh ACN, B lOOohMeOH; (f) A: 30°h ACN, B lOOohMeOH. Figure adapted from ref. [628]. Reprinted with permission.

289

minimum resolution is predicted for all the pairs of peaks in the chromatogram this is said to be an adequate or sufficient result, provided that it can be verified experimentally. A disadvantage of the method of Jandera et al. is the requirement to know the isocratic retention vs. composition relationships. If these data are not already known, which will most likely be the case in the optimization of real-life samples, the experimental effort needed to obtain sufficient data of sufficient accuracy will be very large. 6.3.2.4 Discussion

We have seen that the primary (program) parameters can be optimized in one of several ways. If the actual gradient consists of a single segment, four parameters may be considered, of which two (the slope and the initial composition of the gradient) are most relevant for the result in terms of resolution. The final composition may affect the required analysis time (the program should not extend beyond the chromatogram), whereas the shape of the gradient will have an effect on the overall distribution of the peaks over the chromatogram. The Simplex optimization procedure allows different optimization criteria to be used, so that a good distribution of all the peaks over the chromatogram may be aimed at. However, the Simplex method does require a large number of experiments, and therefore seems to be very inefficient for optimization of the primary parameters alone. Without knowing much about the sample, the Snyder approach may also be used to optimize the program parameters. This is an empirical approach in which the sample properties are largely disregarded, but it does lead to the formulation of reasonable working conditions after only one or two chromatograms have been obtained. The approach of Jandera and ChuraEek allows the optimization of the resolution of one given (arbitrary) pair of sample components and the minimization of the retention volume of another (arbitrary) solute. It requires knowledge of the isocratic retention vs. composition relationships of these three solutes. The information needed may be acquired from gradient elution experiments performed as part of the optimization procedure, or from separate isocratic experiments.The selection of the three arbitrary solutes considered during the optimization process appears to have a large effect on the result and the resolution cannot be optimized throughout the chromatogram. In principle, the retention behaviour of all sample components under gradient conditions can be calculated once two experimental retention times have been obtained (either under isocratic or under gradient conditions) [616,618,623,629]. Therefore, in principle, it ought to be possible to calculate optimum gradient parameters from two solvent programmed experiments [630]. However, to account for inaccuracies in the gradient elution data [630,631,632], a few more experiments may be required. Procedures to obtain isocratic retention vs. composition relationships from a series of gradient experiments have also been described by Jandera and ChuraEek [633,634]. Determination of the optimum program parameters based on the retention vs. composition relationships for all (or all major) sample components will require quite complicated and extensive calculations. It is the charm of the methods described in section 6.3.2.3 that the required computational effort is either minimal (i.e. a few computations, which can easily be performed on a pocket calculator for the Snyder method) or small (i.e. a limited number 290

of computations involving more complex but analytical expressions for the method of Jandera and ChuraEek). A second reason not to become involved in extensive calculations for the complete mathematical optimization of the (primary) program parameters is that a more powerful way to optimizethe separation of all sample components in the mixture may be to optimize the selectivity of the gradient by varying the nature of the mobile phase components (secondary parameters). Three methods appear to be available for optimizing the selectivity in programmed solvent LC: 1. the Simplex procedure, 2. interpretive methods, and 3. the predictive optimization method. The Sentinel method is the outstanding exponent of the group of interpretive methods, as it has already been applied successfullyfor selectivity optimization in programmed solvent LC. However, other interpretive methods, based either on fixed experimental designs or on iterative procedures, can be applied along the same lines. It was seen in section 6.3.2.3 that the extension of the Sentinel method to incorporate gradient optimization was fairly straightforward. For the Simplex optimization procedure the common disadvantage of the large number of required experimentsweighs more heavily for programmed analysis,because more time is required for each experiment (see section 6.1). Also, the response surfaces encountered in the optimization of selectivity in programmed solvent LC appear to be no less convoluted than the ones encountered in isocratic selectivity optimization [627], so that there is again a large chance that the Simplex algorithm will arrive at a local rather than the global optimum. The advantages and disadvantages of interpretive methods are also fully analogous to those listed in chapter 5 (section 5.5). Fewer experiments are needed, but the recognition of the different sample components is required in each experiment. Contrary to the complete optimization of the (primary) program parameters, interpretive methods for the optimization of the selectivity under programmed conditions do not require more complicated calculations than do their isocratic analogs. This was amply demonstrated by Glajch and Kirkland [627], who used the same computer program for the two optimization processes. The predictive method of Jandera et al. [628] requires knowledge of the isocratic retention data of all solute components in binary and (preferably) ternary mobile phase mixtures. Once these data are available, the method may be very helpful in obtaining an “adequate” (but not an optimum) separation with a ternary gradient. Unfortunately, the data required for the application of this predictive method are almost never available, and hence a large number of experiments need to be performed before any predictions can take place. When this is the case the method is of very little practical use. The final question we need to address in this discussion is the general need for gradient optimization procedures, both for optimizing the program parameters and for optimizing the selectivity. In section 6.1 several disadvantages of programmed analysis were described and it was concluded that its application should be avoided if possible. Especially for large 29 I

series of samples, the use of alternative (multicolumn) techniques should be considered. In isocratic analysis, the general motivation is that the larger the supply of a particular kind of sample, the more optimization effort is warranted. In programmed analysis this is not true. In that case, the larger the supply of samples, the larger the urge to look for alternativemethods. Therefore, gradient optimization procedures are only relevant if they represent a limited effort. It yet remains to be established just how far the word "limited" will reach. 6.3.2.5 Summary

The characteristics of the different methods for gradient optimization are summarized in table 6.5. In table 6Sa, the different methods for the optimization of the program parameters are compared. Bearing in mind that a large effort is generally not warranted for the optimization of programmed analysis (seesection 6.3.2.41, we shouldconclude that the Simplex method is not suitable because of the large experimental effort required, and Table 6.5: Summary of the characteristics of gradient optimization methods. a. Optimization of primary (program) parameters Simplex method

Snyder method

Jandera method

Complete mathematical optimization

No-experiments Large

1 or2

few

few

Computational effort

Moderate

Minimal

Small

Large

Resolution optimization

YeS (1)

No (2)

One pair

All solutes

Time optimization

Yes (3)

YeS

(4)

One solute

YeS (3)

Recognition requirements

None

None

Three solutes

All (major) solutes

Ea4y

Easy

Difficult

Difficult

Complete automation

(1) Any optimization criterion can be selected that assigns a single criterion value to each

chromatogram. (2) Optimum slope is selected to provide optimum elution conditions for "average" solutes. (3) Time factor may be incorporated in optimization criterion. (4) Initial and final conditions may be adapted to first and last peaks to minimize analysis time.

292

that a complete mathematical optimization is unattractive because of the large computational effort involved. The method proposed by Jandera and ChuraEek requires somewhat more effort than that of Snyder. It requires some calculations, the recognition of three solutes, and knowledge of the isocratic retention vs. composition relationships for these solutes, obtained either during the optimization procedure or from independent (isocratic) experiments. Table 6.5: Summary of the characteristics of gradient optimization methods. b. Selectivity optimization. Simplex

Interpretive methods

method

Fixed design (Sentinel)

Iterative design

Predictive optimization method

No.experiments

Large

7

5-10

Large (1)

Computational effort

Small

Moderate

Moderate

Moderate

Optimum found

Local

Global

Global(2)

“Adequate” (3)

Accuracy of optimum

High

Low

High

-

Impression of response surface

Poor

Good

Moderate

Poor

Optimization criterion

Single value

Any

Any

Recognition required

No

Yes

Yes

Yes (4)

Easy

Partly easy (5)

Difficult

-

Complete automation

Rs.min

’

x

(eqn.4.23)

(1) Large number of isocratic experiments required. (2) Global optimum may be overlooked if large areas remain unsearched. (3) This method aims at achieving an adequate rather than an optimum result. (4) Recognition of the peaks is required during the isocratic experiments to establish the retention vs. composition relationships. ( 5 ) Experimental part may easily be automated.

293

On the other hand, this method does take into account the resolution of the most critical pair of solutes. If this pair can easily and unambiguously be identified, then the method of Jandera and ChuraiSek may be worth the extra effort. In table 6.5b the methods for selectivity optimization are compared. Again, the Simplex method turns out to be unattractive, because of the large number of experiments required. Also, the resulting optimum may well be a local one. Interpretive methods will generally arrive at the global optimum after a limited number of experiments. However, (by definition) the recognition of the individual solutes is required in each experimental chromatogram. Also, the computational requirements are relatively high, especially if the simultaneous optimization of several parameters is considered. For example, (linear) ternary gradients (one parameter) will be much easier to optimize than quaternary gradients (two parameters). Interpretive methods may possibly be used for the complete optimization of selectivity in solvent programmed LC. If any gradient program (multisegment gradients, see figure 6.2e) is allowed, then it may be possible to optimize the resolution of each pair of peaks in the chromatogram. This possibility has been largely unexploited. However, it also appears to be of limited practical interest, because of the disadvantages of multisegment gradients compared with simple, continuous gradients (see introduction section 6.3).

REFERENCES 601. L.R.Snyder and J.J.Kirkland, Introduction to Modern Liquid Chromatography,2nd edition, Wiley, New York, 1979. 602. J.F.K.Huber, E.Kenndler, W.Nyiri and M.Oreans, J.Chromatogr. 247 (1982) 21 1. 603. W.Blass, K.Riegner and H.Hulpke, J.Chromatogr. 172 (1979) 67. 604. C.J.Little, D.J.Tompkins, O.Stahel, R.W.Frei and C.E.Goewie, J.Chromatogr. 264 (1983) 183. 605. W.E.Harris and H.W.Habgood, Programmed Temperature Gas Chromatography, Wiley, New York, 1966. 606. JCGiddings in: N.Brenner, J.E.Callen and M.D.Weiss (eds.), Gas chromatography, Academic Press, New York, 1962, pp.57-77. 607. V.V.Berry, J.Chromatogr. 236 (1982) 279. 608. P.J.Schoenmakers, H.A.H.Billiet and L.de Galan J.Chromatogr. 185 (1979) 179. 609. L.R.Snyder, MStadalius and M.A.Quarry, AnaLChem. 55 (1983) 1421A. 610. K.A.Cohen, J.W.Dolan and S.A.Grillo, J.Chromatogr. 316 (1984) 359. 61 1. J.L.Glajch and J.J.Kirkland, AnaLChern. 54 (1982) 2593. 612. F.H.Walters and S.N.Deming, AnaLLett. 17 (1984) 2197. 613. H.-J.Stan and B.Steinbach, J.Chromatogr. 290 (1984) 31 1. 614. D.W.Grant and M.G.Hollis, J.Chromatogr. 158 (1978) 319. 615. V.BartO, J.Chromatogr. 260 (1983) 255. 616. L.R.Snyder, J.W.Dolan and J.R.Gant, J.Chromatogr. 165 (1979) 3. 61 7. L.R.Snyder in: Cs.Horvath (ed.), HPLC, Advances and Perspectives, Vol.1, Academic Press, New York, 1980, p.207. 61 8. P.Jandera and J.ChurhEek, Gradient Elution in Column Liquid Chromatography. Theory and practice, Elsevier, Amsterdam, 1985. 294

619. 620. 621. 622. 623. 624. 625. 626. 627. 628. 629. 630. 631. 632. 633. 634.

M.W.Watson and P.W.Carr, Anal.Chem. 51 (1979) 1835. D.M.Fast, P.H.Culbreth and E.J.Sampson, Clin.Chem. 27 (1981) 1055. J.C.Berridge, J.Chrornatogr.244 (1982) 1. J.W.Dolan, J.R.Gant and L.R.Snyder, J.Chromatogr. 165 (1979) 31. P.Jandera and J.ChuraEek, J.Chromatogr. 192 (1980) 1. P.Jandera and J.ChuraEek, J.Chromatogr. 192 (1980) 19. C.M.Noyes, J.Chrornatogr. 266 (1983) 451. H.J.Issaq, K.L.McNitt and N.Goldgaber, J.Liq.Chromatogr. 7 (1984) 2535. J.L.Glajch and J.J.Kirkland, J.Chromatogr. 255 (1983) 27. P.Jandera, J.ChuraEek and H.Colin, J.Chrornatogr.214 (1981) 35. P.J.Schoenmakers, H.A.H.Billiet, R.Tijssen and L.de Galan, J.Chrornatogr. 149 (1978) 519. M.A.Quarry, L.R.Grob and L.R.Snyder, J.Chrornatogr. 285 (1984) 1. M.A.Quarry, L.R.Grob and L.R.Snyder, J.Chromatogr. 285 (1984) 19. P.Jandera and J.ChuraEek, J.Chromatogr. 192 (1980) 37. P.Jandera and J.ChuraEek, J.Chrornatogr.91 (1974) 223. P.Jandera and J.ChuraEek, J.Chrornatogr.93 (1974) 17.

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