Chapter 7 Selection of The Optimum Gradient

157

Chapter 7

SELECTION OF THE OPTIMUM GRADIENT 7.1. ASPECTS OF OPTIMIZATION

The "ideal" chromatographic separation can probably be characterized as a separation t h a t provides perfect resolution of a l l sample compounds of i n t e r e s t i n the minimum time, and makes possible a very sensitive detection on the one hand and separations of large amounts of sample f o r preparative o r semi-preparat i v e purposes o r f o r further identification on the other. I t i s almost impossible t o obtain such an ideal chromatographic separation i n practice, because one feature of the separation can be improved only a t the cost of other parameters. Thus, the resolution or the amount of sample t o be separated can be increased only i f the time of analysis is also increased. Similarly, s e n s i t i v i t y can be improved a t the cost of resolution; the resolution can be improved, b u t the time of analysis, s e n s i t i v i t y of detection and the amount of sample t h a t can be loaded on the column are simultaneously adversely affected. Therefore , an "optimum" separation represents a compromise between these various aspects and such a compromise should be the best f i t w i t h respect t o the most important requirements, which may d i f f e r according t o the specific problem t o be solved and according t o the aim of the separation. We can optimize the separation t o obtain one of the following features: ( a ) maximum resolution i n the minimum time ( t h i s i s perhaps the most frequent requirement i n most routine applications); ( b ) maximum resolution, where a short analysis time is of i n t e r e s t b u t not of primary i n t e r e s t (this applies t o the analysis of very complex sample mixtures in natural, biological or environmental samples); ( c ) maximum s e n s i t i v i t y of detection ( t h i s i s required i n trace analysis); ( d ) maximum amount of sample t h a t can be loaded on the column (separations on a preparative or semi-preparative scale, required f o r the isolation o f sample components f o r further identification o r for other purposes). I t can be seen that the key t o optimization i n a l l of the above instances i s the resolution, which must be controlled s a t i s f a c t o r i l y . To this aim, we can optimize the three terms contributing t o resolution ( I , efficiency; 11, select i v i t y ; and 111, capacity). With given instrumentation, the optimization consists i n a suitable choice of the column and of the mobile phase. With some experience, References on p . 180.

158 TABLE 7.1 USEFUL COMBINATIONS OF MOBILE PHASE COMPONENTS (WEAKER ELUTION STRENGTH COMPONENT, a ; STRONGER ELUTION STRENGTH COMPONENT, b ) , THE SHAPE AND THE

STEEPNESS LIKELY TO YIELD GOOD SEPARATIONS OF UNKNOWN MIXTURES TO A FIRST APPROXIMATION (CONDITION FOR THE FIRST EXPERIMENT I N EMPIRICAL OPTIMIZATION OF GRADIENT ELUTION CONDITIONS) The i n i t i a l composition o f the mobile phase a t the s t a r t o f the gradient should be selected according t o the absolute r e t e n t i o n o f chromatographed compounds i n sample mixtures. Each gradient may be s t a r t e d from the mobile phase containing a c e r t a i n amount o f component b and use component b d i l u t e d by component a o r t o end the gradient before 100%b i s achieved. Gradients from 5 t o 95% o f b are recommended by Snyder. Column packing

Mobile phase components

a

Gradient shape

b

( I ) Normal-phase chromatography on p o l a r adsorbents: Silica, n-Hexane o r n-Propanol , n-heptane isopropanol , a1umi na , ethanol porous , 5-10 urn

Concave 15-25 (or linear - l e s s adequate) 10-20

' Chloroform, d i isopropyl ether, d i e t h y l ether*, methylene c h l o r i d e *

D i isopropyl ether

Methanol , acetonitri l e

I-Chlorobutane

Methanol , acetonitri l e

(I I ) Reversed-phase chromatography, aqueous : C18/si 1ica Water** Methanol , (C8/si 1i c a acetoni tri l e less frequently), Tetrahydrochemi ca 11y furan (dioxan) bonded; carbon adsorbents, 5-10 pm ( 111) Reversed-phase chromatography, non-aqueous : As i n (11) Acetonitri l e y Tetrahydrofuran methanol D i e t h y l ether*, d i isopropyl ether

Gradient steepness ('GlVm)

Concave

15-25 15-25

Linear , slightly convex

10-20

Linear

5-10

15-25

10-15

159 TABLE 7.1 ( c o n t i n u e d ) Column p a c k i n g

M o b i l e phase components

a

Gradient shape

b

Gradient steepness ('Glvm)

( I V ) Normal-phase chromatography on c h e m i c a l l y bonded phases o f medi urn p o l a r i t y : NH / s i l i c a , n-Hexane, n-Propanol , Linear 5-10 n-heptane isopropanol , o r concave CNYsi 1ica , ethanol, D i o l / s i 1i c a , chloroform, N02/si 1ica, diisopropyl ether, e t c . , 5-10 urn d i e t h y l ether: methylene c h l o ride* (V) Ion-exchange chromatography: Cation, anion Water exchanger*** Cation, a n i o n exchanger

PH 2 pH 6-8

0.1-0.5 M s a l t or buffers

Concave o r 1 in e a r

pH 6-8 PH 2

Linear 15-25 pH g r a d i e n t

( V I ) I o n - p a i r chromatography i n reversed-phase systems: As i n (11) I o n - p a i r i n g i o n s § , 0.0005-0.05 M I n water

10-15

Linear

10-20

Concave

5-10

I n CH30H o r CH3CN

0.1 M ion-pai ri n CH30H-H20§ 5 ion i n CHQOH-H~O%~~ o r i n CH$N-H205§§ CH3CN-H20s§5

* V o l a t i l e s o l v e n t , d i f f i c u l t i e s may o c c u r when r e c i p r o c a t i n g pumps a r e used. **A s a l t o r a b u f f e r a t a c o n c e n t r a t i o n o f 0.01-0.4 M may be added t o improve t h e s e p a r a t i o n o f s t r o n g l y p o l a r and i o n i z e d compounds ( a t t e n t i o n must be p a i d t o n o n - m i s c i b i l i t y o f s o l v e n t s a and b ! ) . ***Resinous exchangers, exchangers c h e m i c a l l y bonded on porous m a t e r i a l ( s i l i c a ) . 5 E. g , NaN03, Na2S04, KH2P04, K HPO4, CH$OONa, s o d i um c i t r a t e , b o r a t e , 1a c t a t e , formate, o x a l a t e , NH4C1, KC1 ( [ a l i d e i o n s a r e u s u a l l y n o t recommended owing t o c o r r o s i o n o f t h e equipment), a c e t a t e , phosphate o r c i t r a t e b u f f e r s . s§Tetrabutylammonium phosphate, s u l p h a t e o r p e r c h l o r a t e f o r chromatography of a c i d i c compounds; pentane- o r h e p t a n e s u l p h o n i c a c i d f o r chromatography o f b a s i c compounds, pH b u f f e r e d t o between 3-7 and/or i o n i c s t r e n g t h a d j u s t e d . § § § T h i sc o n c e n t r a t i o n i s k e p t c o n s t a n t d u r i n g t h e g r a d i e n t .

.

References on p . 180.

160 a suitable column packing material can be found relatively e a s i l y f o r a given separation problem, as there are f a r fewer commercially available useful column packing materials f o r l i q u i d chromatography than stationary phases for gas chromatography. On the other hand, with each column packing material, we can combine binary or more complex mobile phases. However, a few combinations of mobile phases f o r each packing material have proved useful f o r a number of practical separation problems and are l i s t e d in Table 7.1. Of course, numerous other comb i n a t i o n s may be more suitable f o r specific separation problems. A more detailed discussion of this aspect of separation would be beyond the scope of t h i s book, and can be found in other l i t e r a t u r e on liquid chromatography 75,76 Once the column packing material has been selected, the efficiency of separat i o n can be controlled by using appropriate p a r t i c l e s i z e of the material, column dimensions, mobile phase flow-rate a n d , t o a lesser extent, temperature. Columns w i t h an inner diameter between 3 and 6 mm and a length between 100 and 300 mm packed w i t h 5-10 pm particle diameter porous material have become standard for most routine analytical separations. Recently, more e f f i c i e n t columns packed w i t h 3 pm particles have become available commercially and may be used; there are also l e s s e f f i c i e n t columns packed w i t h 15-25 pm material. In ion-exchange chromatography and i n semi-preparative chromatography, columns of length 500-1000 mm and I.D. 8-10 mm are often used i n connection w i t h materia l s o f p a r t i c l e diameter 20-50 pm. Here, a large column loading and a lower cost of the packing material are of primary i n t e r e s t . Pellicular and controlled surface porosity packing materials containing a t h i n layer (1-5 pm) of the stationary phase coated, grafted or sintered on an impervious core, 30-80 pm i n diameter, were very popular i n the early 1970s, b u t they are used only occasionally now, as they combine low sample loadability w i t h a relatively lower efficiency of separation i n comparison w i t h Standard analytical columns. Their only advantage, the possibility of high-speed analysis a t a relatively low pressure, does not represent a significant benefit w i t h modern l i q u i d chromatographic instrumentation. In recent years, microbore columns of I.D. 0.5-2.0 mn, 1 t o several metres long, packed with 5-20 pm particle diameter material , have been introduced77 and have recently become commercially available. These columns o f f e r a very h i g h efficiency ( u p t o several hundred thousands of theoretical plates) , b u t the separat i o n times are longer than with commercial analytical columns. Microbore columns are useful in the chromatography of very complex samples and seem promising f o r d i r e c t coupling with mass spectrometers. Similar advantages would be offered by open-tubular columns, b u t t h e i r capacity i s low and extreme care must be devoted t o the miniaturization of other parts of the instrumentation (sample injectors, detector c e l l s ) .

.

161

When t h e e f f i c i e n c y o f separation has been established by the appropriate choice o f column geometry, p a r t i c l e s i z e o f the packing m a t e r i a l and flow-rate o f the mobile phase (which i s u s u a l l y between 0.5 and 4.0 ml/min i n work w i t h standard a n a l y t i c a l columns), the s e l e c t i v i t y and capacity c o n t r i b u t i o n s t o r e s o l u t i o n can be optimized ( t h e e f f i c i e n c y can be assumed n o t t o change s i g n i f i c a n t l y w i t h changing composition o f the e l u e n t ) . This can be done much more e a s i l y i n i s o c r a t i c e l u t i o n chromatography (as i s discussed i n Sections 2.1 and 2.2)

than i n gradient e l u t i o n chromatography. There, the optimum gradient p r o f i l e

has t o be selected, which means t h a t the shape and steepness o f t h e gradient and the i n i t i a l composition o f the mobile phase should be optimized ( t h i s a l s o holds t r u e f o r each step i n composed gradients). The o p t i m i z a t i o n i s complex and has u s u a l l y been done using a t r i a l - a n d - e r r o r method. However, the theory o f gradient e l u t i o n can be used t o increase the e f f i c i e n c y o f and t o speed-up the optimizat i o n procedure. The r a t i o n a l o p t i m i z a t i o n o f conditions f o r gradient e l u t i o n chromatography i s based on the o p t i m i z a t i o n o f r e s o l u t i o n . Two d i f f e r e n t s i t u a t i o n s should be distinguished, each o f which requires d i f f e r e n t methods f o r (1) s p e c i f i c case, where the sample compounds and the r e l a t i o n s h i p between

t h e i r r e t e n t i o n and the composition o f the mobile phase are known; ( 2 ) general case, where the parameters o f the k ' = f(c) r e l a t i o n s h i p f o r the

chromatographed compounds and o f t e n t h e i r exact number are n o t known.

7.2.

SPECIFIC CASE The optimized p r o f i l e of the gradient i d e a l l y should y i e l d a separation i n

which a l l sample compounds o f i n t e r e s t are s u f f i c i e n t l y w e l l resolved and t h e i r e l u t i o n i s achieved i n the minimum time. This means t h a t the peaks o f compounds should be spaced as r e g u l a r l y as possible w i t h the required r e s o l u t i o n (e.g., = 1.5) on the chromatogram from i t s beginning t o i t s end. A higher r e s o l u t i o n 9 than t h a t j u s t required would only increase the time o f analysis and t h e r e f o r e

R

i s undesirable. I t i s n a t u r a l l y only r a r e l y possible t o achieve such an i d e a l , e x a c t l y r e g u l a r

spacing o f the peaks o f a l l sample compounds i n p r a c t i c a l systems, where t h e capacity f a c t o r s u s u a l l y increase i r r e g u l a r l y between t h e i n d i v i d u a l sample compounds. This leads t o "bunching together" o f peaks a t c e r t a i n places on the chromatogram and t o "over-resol u t i o n " a t other places (homo1ogous, benzol ogous and oligomeric s e r i e s are exceptions). However, t h e gradient p r o f i l e can be optimized i n order t o improve the spacing o f peaks i n the chromatogram w h i l e maintaining the required r e s o l u t i o n and t o approximate the " i d e a l " separation. Gradient e l u t i o n o f f e r s more e f f i c i e n t means f o r t h i s purpose than e l u t i o n under i s o c r a t i c conditions. References on p . 180.

162

To optimize the p r o f i l e o f the gradient, the steepness

( B ) , shape

(K)

and the

i n i t i a l concentration o f the more e f f i c i e n t e l u t i n g component i n the mobile phase ( A ) should be chosen so as t o y i e l d optimum c o n t r i b u t i o n s o f t h e s e l e c t i v i t y and

capacity terms t o r e s o l u t i o n . [The e f f i c i e n c y term (number o f t h e o r e t i c a l p l a t e s ) i s assumed n o t t o depend s i g n i f i c a n t l y on the composition o f t h e mobile phase and, consequently, on the p r o f i l e o f the gradient.] Let us r e c a l l b r i e f l y the i n f l u e n c e o f the three parameters o f gradient p r o f i l e on separation, as i t appears on the chromatogram. I f the i n i t i a l concentration of the more e f f i c i e n t component i n the mobile phase, A, i s n o t too high and the steepness o f the gradient, B , n o t t o o low (otherwise the separation conditions are close t o i s o c r a t i c e l u t i o n ) , an increase i n A would have a much greater i n fluence on r e t e n t i o n volumes (decrease) than on t h e widths o f t h e peaks, which u s u a l l y decrease t o some e x t e n t also. Resolution and bunching together o f peaks are u s u a l l y n o t influenced much by minor changes i n A , w i t h the exception o f the e a r l y e l u t e d compounds. An increase i n B leads t o a s i g n i f i c a n t decrease i n the r e t e n t i o n volumes o f sample compounds. The differences i n the r e t e n t i o n volumes o f i n d i v i d u a l compounds and ( t o a l e s s e r e x t e n t ) t h e peak widths a l s o decrease w i t h increasing B. So does the r e s o l u t i o n , i f the s e l e c t i v i t y o f separation ( a ) does n o t depend s i g n i f i c a n t l y on the compos i t i o n o f the mobile phase. I n the opposite case, the p l o t s o f A versus B may 9 be complex (see Fig. 4.14 and the discussion i n Section 4.9). I f the shape o f gradient i s changed from convex t o l i n e a r and f u r t h e r t o concave ( w h i l e increasing the parameter 4.18) a

, the

K

o f the gradient function, eqns. 4.15-

r e t e n t i o n volumes and compression o f t h e chromatogram increase, and

decreases. A t a c e r t a i n curvature o f the gradient, maximum r e s o l u t i o n can be

9 observed. Generally, an increase i n

K

would lead t o an increase i n the resolu-

t i o n o f e a r l i e r e l u t e d compounds and, a t the same time, the r e s o l u t i o n o f l a t e r e l u t e d compounds may be impaired ( t h e tendency f o r bunching together o f peaks i s moved from the e a r l i e r t o the l a t e r p a r t s o f the gradient). The above general r u l e s apply i f o n l y one parameter of the g r a d i e n t f u n c t i o n i s changed w h i l e the remainder are kept constant and, f u r t h e r , i f are e l u t e d under t r u e gradient conditions, i.e.,

the compounds

i f the compound "senses" s u f -

f i c i e n t l y the change o f the mobile phase composition during the m i g r a t i o n along the column before the e l u t i o n . Otherwise, w i t h e a r l y e l u t e d compounds and w i t h " f l a t " gradients, the s o l u t e band i s e l u t e d under q u a s i - i s o c r a t i c conditions and the differences i n the gradient p r o f i l e can h a r d l y have a s i g n i f i c a n t i n f l u e n c e on separation. The best way o f o p t i m i z i n g a given gradient e l u t i o n separation seems t o begin I n most instances, we s e l e c t e i t h e r w i t h the s e l e c t i o n of the gradient l i n e a r concentration gradients ( i n reversed-phase systems) o r a c e r t a i n curvature

163 of the g r a d i e n t by analogy w i t h o t h e r experiments i n s i m i l a r systems (concave g r a d i e n t s are t o be p r e f e r r e d i n chromatography on p o l a r adsorbents and i n i o n exchange chromatography). The curvature may be c a l c u l a t e d w i t h respect t o t h e approximate compression o f the g r a d i e n t (E) r e q u i r e d (see Section 4.10). The second step i s t o choose a convenient steepness o f t h e g r a d i e n t i n order t o achieve t h e r e q u i r e d r e s o l u t i o n o f the p a i r o f sample compounds t h a t are most d i f f i c u l t t o separate. Of course, t h e separation o f such compounds i s f e a s i b l e i n g r a d i e n t e l u t i o n chromatography o n l y i f i t can be accomplished i n a given system under i s o c r a t i c c o n d i t i o n s w i t h t h e same components o f the mobile phase. F i n a l l y , t h e i n i t i a l concentration o f t h e more e f f i c i e n t e l u t i n g component i n the mobile phase can be c o n v e n i e n t l y adjusted w i t h t h e aim o f keeping t h e time o f separation as s h o r t as possible. The o p t i m i z a t i o n approach described may be performed u s i n g a t r i a l - a n d - e r r o r method, by c a l c u l a t i o n o r by combination o f c a l c u l a t i o n s and experiment, and t h e method s e l e c t e d w i l l depend on the nature o f the separation problem and on t h e experience o f t h e chromatographer. I n t h e f o l l o w i n g , a method w i l l be described f o r o p t i m i z a t i o n o f t h e steepness o f t h e g r a d i e n t ( B ) and o f t h e i n i t i a l concent r a t i o n o f t h e more e f f i c i e n t e l u t i n g component i n t h e mobile phase ( A ) by c a l culation5'.

As both o f these parameters i n f l u e n c e t o a c e r t a i n e x t e n t b o t h t h e

r e s o l u t i o n and t h e r e t e n t i o n volumes o f sample solutes, which c o n t r o l t h e time o f separation, simultaneous o p t i m i z a t i o n o f A and B i s t h e most c o r r e c t t h e o r e t ically. F i r s t , i t i s necessary t o choose a p p r o p r i a t e l y two compounds j and k , w i t h adjacent peaks, t h e r e s o l u t i o n o f which should be kept a t a c e r t a i n r e q u i r e d These are u s u a l l y t h e two components o f a sample m i x t u r e t h a t are g,o' most d i f f i c u l t t o separate, which should guarantee adequate r e s o l u t i o n over t h e level, R

whole chromatogram. F u r t h e r , another sample compound, i, i s chosen, t h e r e t e n t i o n volume of which, Vii,

should be minimal, i n order t o keep t h e s e p a r a t i o n time as

s h o r t as possible. This i s u s u a l l y t h e most s t r o n g l y r e t a i n e d sample component. An a p p r o p r i a t e shape o f t h e g r a d i e n t (parameter

K)

i s pre-selected.

From the mathematical p o i n t o f view, t h e o p t i m i z a t i o n o f A and B c o n s i s t s i n the c a l c u l a t i o n o f the minimum o f t h e f u n c t i o n V ' = f(A,B) a t a given value gi o f K, f o r which A and B are f u r t h e r i n t e r r e l a t e d by t h e c o n d i t i o n t h a t the r e should be obtained. As t h e usual Lagrange method f o r the solu93 0 t i o n o f t h e problem would s u f f e r from severe d i f f i c u l t i e s , a m o d i f i e d method of

q u i r e d value R

s o l u t i o n was suggested, t h e d e t a i l s o f which, t o g e t h e r w i t h t h e c a l c u l a t i o n program, are given elsewhere5'.

The c a l c u l a t i o n r e q u i r e s a knowledge o f t h e

parameters o f the k' = f ( e ) dependences f o r t h e sample compounds i, j and k and i s f e a s i b l e f o r combinations of these f u n c t i o n s and g r a d i e n t f u n c t i o n s t h a t y i e l d an e x p l i c i t s o l u t i o n f o r r e t e n t i o n volume (Appendix 2 ) . A schematic diagram o f t h e o p t i m i z a t i o n program i s given i n Appendix 3. References on p . 180.

164

CH30H

6

- 50

25

15

20

0

Fig. 7.1. Optimized reversed-phase gradient e l u t i o n separation o f a mixture o f seven b a r b i t u r a t e s using a l i n e a r concentration gradient o f methanol i n water. Conditions and Nos. o f compounds as i n Fig. 4.7; compound 7 = amobarbital. Resol u t i o n optimized w i t h respect t o the compounds 6 and 7 and minimum r e t e n t i o n volume o f compound 1 required. Gradient function: c = 0.523 + 0.00846 v [c = %(v/v) methanol x 10-2; v i n ml]. (From r e f . 52.)

2

I

1

V(mi)

I

10

I

I

5

I

bo

Fig. 7.2. Reversed-phase separation o f a m i x t u r e o f seven b a r b i t u r a t e s using an e m p i r i c a l l y selected gradient ( l i n e a r gradient f u n c t i o n : c = 0.1 + 0.0172 V ) . Conditions and Nos. o f compounds as i n Figs. 4.7 and 7.1.

165 A practical example of the separation of a mixture of seven barbiturates on

an octadecylsilica column can i l l u s t r a t e this optimization procedure5'. A linear concentration gradient of methanol in water was optimized i n order t o achieve a resolution of 1.5 f o r the separation of the two most strongly retained compounds, 6 and 7 , while the retention volume of the e a r l i e s t eluted compound, 1 should be kept t o a minimum. The optimization calculation yielded a linear gradient function of the form c = 0.523 + 0.00846 V . The chromatogram obtained using the optimized gradient function i s shown in F i g . 7.1. For comparison, the chromatogram obtained originally u s i n g the gradient function selected by a trial-ande r r o r procedure i s shown in Fig. 7.2. A significant reduction in the time of separation i s apparent f o r the gradient optimized by calculation. Obviously, such a relatively simple separation could be achieved under isoc r a t i c conditions in an equal or even shorter analysis time i f we take i n t o account the time necessary t o re-equilibrate the column a f t e r the end o f the gradient. However, optimized gradient elution separations may r e s u l t i n s i g n i f i cant shortening of the time of analysis o f more complex mixtures. To elucidate the influence of the selected shape (curvature) of the gradient on the optimization procedure, the calculation approach was repeated f o r the same column and components of the mobile phase and the same optimization c r i t e r i a , b u t a logarithmic (convex) gradient function given by eqn. 4.20 (Table 4.1) was used. The c = f(V) plot according t o the calculated optimized logarithmic gradient function was almost identical with that in the previous instance of the calculated optimized linear gradient function. This suggests that the pre-selected form of the gradient function i s n o t very c r i t i c a l f o r the optimization procedure 52 To verify each calculated optimized gradient function, i t i s advisable t o calculate the retention volumes and resolution for a l l chromatographed compounds. Owing t o minor changes i n s e l e c t i v i t y w i t h changing composition of the mobile phase, which are d i f f i c u l t t o forecast without calculation, sometimes the pair of compounds expected t o be the most d i f f i c u l t t o separate i s well resolved under calculated optimized gradient conditions, b u t the resolution of another p a i r of sample compounds i s impaired. This is apparent from the calculated V' and R 9 9 values. In t h i s instance, the selection of the compounds i, j and k should be changed and the optimization calculation repeated. T h i s was the case w i t h the above example of the separation of barbiturates, where originally the gradient function was optimized f o r the resolution Rg,o=l.5 between compounds 1 and 2 and the minimum retention volume of the l a s t eluted compound, 7; under these conditions, however, the resolution of compounds 6 and 7 would be insufficient. Therefore, the c r i t e r i a of optimization were changed and the calculation was repeated as indicated above52

.

.

References on p . 180.

166 Further, we s h a l l bear i n mind t h a t i n each stationary-mobile phase system the r e s o l u t i o n o f each p a i r o f sample compounds i s l i m i t e d by c e r t a i n minimum and maximum values, i n both i s o c r a t i c and gradient e l u t i o n , which cannot be surpassed w i t h a given column ( t h e number o f p l a t e s i s assumed t o be constant and optimized) and a combination o f t h e mobile phase components (see Section 4.9)

*

Therefore, the r e q u i r e d value o f t h e r e s o l u t i o n i n the o p t i m i z a t i o n c a l c u l a t i o n s should be kept w i t h i n these p r a c t i c a l l i m i t s ( t h e e f f i c i e n c y o f the column could be changed t o y i e l d the r e s o l u t i o n required), otherwise t h e c a l c u l a t i o n would f a i l . The optimized gradient conditions can be simply transformed t o other columns o f d i f f e r e n t geometry and e f f i c i e n c y and t o other flow-rates o f the mobile phase f o l l o w i n g the r u l e s o u t l i n e d i n Section 4.5. Regular spacing o f t h e peaks of sample compounds on the chromatogram can u s u a l l y be b e t t e r approximated i n stepwise e l u t i o n chromatography and i n chromatography using composed gradients than i n simple continuous gradient e l u t i o n chromatography. Here, the conditions can be optimized step by step i n order t o achieve a "tailor-made" separation o f two ( o r more) neighbouring peaks i n each step. Three approaches have been suggested f o r t h e o p t i m i z a t i o n of stepwise e l u t i o n chromatography. The r e l a t i v e l y simple graphical method introduced by Golkiewicz and Socze~ i i i s k i is~used ~ f o r t h e construction of a stepwise gradient i n such a way t h a t each compound e l u t e d i n a given i s o c r a t i c step as the l a s t peak should have a c e r t a i n , pre-determined capacity f a c t o r i n t h i s step (e.g.,

k ' = 1). The composi-

t i o n o f t h e mobile phase i n t h i s step necessary f o r t h i s aim i s found from p l o t s o f k ' = f ( c ) f o r the i n d i v i d u a l compounds. On the other hand, t h e k ' = f(c) p l o t s are used t o determine the capacity f a c t o r s f o r compounds t h a t are s t i l l r e t a i n e d on the column a f t e r t h e end o f t h i s step and the corresponding RF values t h a t give the m i g r a t i o n o f these compounds along the column. However, t h i s approach does n o t take i n t o account the e f f i c i e n c y o f the column necessary t o achieve a certain resolution. The o p t i m i z a t i o n approach suggested by B o r b k o and co-workers 58*60 consists i n the c a l c u l a t i o n o f t h e volume and composition o f the mobile phase i n each i s o c r a t i c step, which i s necessary t o keep the r e s o l u t i o n o f t h e a d j o i n i n g peaks o f compounds w i t h i n c e r t a i n 1i m i t s . The approach o f Jandera and C h ~ r s E e ki ~s ~s i m i l a r , b u t much simpler owing t o another d e f i n i t i o n o f the column sequences corresponding t o t h e i n d i v i d u a l i s o c r a t i c e l u t i o n steps (see Section 5.1).

I t works as follows. The composition and

the volume o f the mobile phase i n each i s o c r a t i c e l u t i o n step are c a l c u l a t e d provided a s i n g l e sample compound i s e l u t e d i n each step and t h e r e s o l u t i o n

167 The width o f between the neighbouring peaks i s kept a t a required value R 9.0' the peak o f the compound e l u t e d i n step n i s c o n t r o l l e d by the capacity f a c t o r o f the compound i n t h i s step ( a t the time o f e l u t i o n ) (eqn. 5.6)

( t h i s i s some-

what s i m p l i f i e d ) . Then, the r e t e n t i o n volumes o f the p a i r o f compounds w i t h adj a c e n t peaks t h a t are e l u t e d i n two neighbouring steps,

n-1 and n , are i n t e r -

r e l a t e d as f o l l o w s :

where the subscripts n-1 and n r e l a t e t o V ' k ' and w o f compounds e l u t e d i n g' 9 steps n-1 and n , respectively. The capacity f a c t o r s , k;, o f the s o l u t e e l u t e d i n step n , and the volumes o f the mobile phase passed through the column, Ve,i

i n each step ifrom 1 t o n-1

are known. Then, i t i s possible t o c a l c u l a t e t h e capacity f a c t o r , k;, necessary between t h i s t o e l u t e the s o l u t e i n step n so t h a t the required r e s o l u t i o n 8 93 0 compound and the s o l u t e e l u t e d i n step n-1 i s achieved. For t h i s purpose, eqn.

7.1 i s combined w i t h eqn. 5.4 as f o l l o w s :

n

A f t e r i n t r o d u c i n g k; c a l c u l a t e d from eqn. 7.2 i n t o the appropriate k' = f ( e ) f u n c t i o n t h a t f i t s a given chromatographic system (Table 2.1), the necessary concentration o f the more e f f i c i e n t e l u t i n g component i n the mobile phase, e n , i s calculated. F i n a l l y , the volume o f the mobile phase i n step n, Ven,

i s de-

i f step n i s f i n i s h e d j u s t a t the time when the e l u t i o n o f the

termined, e.g.,

sample compound i s accomplished:

'en

= V'

gn

+

w

R

gn Q Y O 2

n- 1

c vei

i=l

= V'

gn

This sequence o f c a l c u l a t i o n s o f k;,

+

"mRg

V G

o

. (1 + k;)

-

n-1 i=l 'ei

(7.3)

en and Ven i s consequently repeated f o r

each step and compound eluted, beginning w i t h step 1 u n t i l the step i n which the l a s t sample compound i s eluted. I t i s convenient t o consider t h e e l u t i o n o f two compounds i n step 1 w h i l e t h e composition o f the mobile phase i n t h i s step i s calculated as i n chromatography under i s o c r a t i c conditions a f t e r i n t r o d u c i n g the

References on p . 180.

168 corresponding k' = f ( e ) functions f o r the two compounds i n t o the definition equation f o r resolution (see Appendix 1). This calculation approach may be e a s i l y adapted t o the construction o f stepwise gradients where two compounds instead of a single one are eluted in each i s o c r a t i c step. Fig. 7.3 shows an example of the separation of a mixture of barbiturates on an octadecylsilica column using a stepwise concentration gradient of methanol i n water optimized for the resolution R = 1.75 between the ad49 QYO jacent bands

.

2

6

10

v(ml)

5

-

0

Fig. 7.3. Reversed-phase separation of a mixture o f barbiturates using a stepwise gradient optimized by calculation t o achieve a resolution R =1.75 between neighbouring peaks. Step 1: 2.37 ml , 52% ( v / v ) methanol, compoun8s 1 and 2 eluted. Step 2: 1.78 m l , 55% ( v / v ) methanol , compounds 3 and 4 eluted. Step 3: 64% ( v / v ) methanol, compounds 5 and 6 eluted. Conditions and Nos. of compounds as i n Fig. 4.7. (From ref. 52.) Of course, the optimization calculation approach described i s subject t o certain limitations. I t i s possible t o obtain the resolution within certain practical limits, as under i s o c r a t i c conditions and as i n chromatography u s i n g simple continuous gradients (see above). Sometimes the concentration of the more e f f i c i e n t eluting component in the mobile phase f o r step n, en, calculated u s i n g the above optimization approach

169 may be lower than the corresponding concentration i n step n-1. I f t h i s c a l c u l a t e d concentration were r e a l y used i n step n , d i f f i c u l t i e s might occur i n t h e form o f broad and sometimes even unsymmetrical peaks. I n o t h e r instances, t h e c a l c u l a t i o n may y i e l d a r e t e n t i o n volume o f t h e compound e l u t e d i n step n lower than t h e sum o f Vei

f o r steps 1 t o n-1, which means t h a t t h e concentration o f t h e more

e f f i c i e n t e l u t i n g component i n the mobile phase i n step n-1 i s t o o h i g h t o a l l o w the required resolution R between t h e adjacent peaks e l u t e d i n steps n-1 and gY0 n. I f one o r the o t h e r s i t u a t i o n a r i s e s , i t i s necessary t o r e - c a l c u l a t e t h e c o n d i t i o n s f o r step n-1 so t h a t t h e e l u t i o n o f t h e two compounds i s accomplished This c o n d i t i o n i s easy t o e s t a b l i s h i n step n-1 w i t h the r e q u i r e d r e s o l u t i o n R i n a program o f o p t i m i z a t i o n c a l c u l a t i o n

.

499 $0'

should be used 9 I n p r i n c i p l e i t would

For more p r e c i s e c a l c u l a t i o n s , eqn. 5.6a f o r peak widths w i n s t e a d o f eqn. 5.6 i n combination w i t h eqns. 7.1

-

7.3.

be p o s s i b l e t o optimize g r a d i e n t e l u t i o n using composed g r a d i e n t s c o n s i s t i n g o f several subsequent continuous steps i n an analogous way t o stepwise e l u t i o n . Of course, t h e c a l c u l a t i o n procedure would be much more complex. I n such c a l c u l a t i o n s , l i n e a r concentration changes i n t h e i n d i v i d u a l steps are s u f f i c i e n t , because the curvature o f t h e g r a d i e n t i n a r e l a t i v e l y s h o r t p a r t o f t h e g r a d i e n t has an almost n e g l i g i b l e i n f l u e n c e on separation. I t i s p o s s i b l e t o c a l c u l a t e t h e steepness o f t h e g r a d i e n t necessary f o r t h e e l u t i o n o f one o r two sample compounds i n each step i n order t o achieve t h e r e s o l u t i o n required. The c o n t r i b u t i o n o f each step t o V ' and t o Vm should be c a l c u l a t e d u s i n g t h e a p p r o p r i a t e g equations from Appendix 2 (see Section 5 . 2 ) . The s o l u t i o n would be p o s s i b l e o n l y w i t h the a i d of a computer o r a t l e a s t a programmable c a l c u l a t o r .

A s i m i l a r o p t i m i z a t i o n approach was a p p l i e d by Svoboda6' t o c a l c u l a t e optimum composed g r a d i e n t s f o r anion-exchange separations o f mixtures o f n u c l e o t i d e s .

7.3.

GENERAL CASE For most p r a c t i s i n g chromatographers, i t i s o f g r e a t i n t e r e s t t o have the

p o s s i b i l i t y o f o p t i m i z i n g g r a d i n e t e l u t i o n w i t h o u t t h e i s o c r a t i c experiments necessary t o determine the constants o f the k' = f ( c ) f u n c t i o n s f o r t h e i n d i v i d u a l sample compounds i n a given system. Such a g e n e r a l l y a p p l i c a b l e g r a d i e n t o p t i m i z a t i o n i s u s u a l l y performed u s i n g a t r i a l - a n d - e r r o r method, u n t i l t h e chromatographer i s s a t i s f i e d w i t h the r e s u l t o f t h e separation. Some commercial g r a d i e n t e l u t i o n l i q u i d chromatographs make i t p o s s i b l e t o pre-programme a sequence o f t r i a l - a n d - e r r o r experiments t o be performed subsequently by t h e i n ~ t r u m e n t ~ ' A~f.t e r t h e end o f t h i s unattended s e r i e s o f experiments, t h e chromatographer s e l e c t s the "best l o o k i n g " s e p a r a t i o n and the corresponding p r o f i l e

o f the g r a d i e n t f o r r o u t i n e repeated analyses.

References on p . 180.

170

P = f/g w

v)

Z 0 0, v)

w d

5IYI-

;

i TIME

Fig. 7.4. Method f o r calculation of the peak separation function P. (Adapted from ref. 79.)

This empirical optimization method may be aided and controlled using a computerized simplex algorithm method suggested by Watson and Carr78. In this method, the peak separation function, P , as originally defined by Kaiser” (see Fig. 7 . 4 ) , i s used as a measure of the quality of separation and i s integrated into the chromatographic response function ( C R F ) . This function involves the experimental peak separation f o r each pair of adjacent peaks (pi), the desired peak separation ( P o ) , the actual analysis time (tL) and the acceptable analysis time (t,,,): pi

CRF = c l n - +

*(tM -

tL)

(7.4)

where a i s an arbitrary weighting factor. p0 and tMshould be consistent w i t h the achievable signal-to-noise r a t i o or with the desired resolution. Then, the optimization consists i n the search of a separation that yields the CRF optimization function closest t o zero. In subsequent optimization experiments, the peak separation functions, Pi, f o r each pair of peaks Must be evaluated as in-

171 dicated i n Fig. 7.4.

This o p t i m i z a t i o n procedure was v e r i f i e d i n the p r a c t i c a l

separation o f a five-component phenylthiohydantoin-amino a c i d mixtureir8 and, i n a f u r t h e r r e f i n e d form, on the separation o f

antioxidant^^^

by reversed-phase

chromatography. A s i m i l a r simple s t a t i s t i c a l approach was suggested f o r optimizat i o n o f gradient e l u t i o n conditions both i n reversed-phase and i n normal-phase chromatography, b u t t h i s approach took i n t o consideration a r a t i o n a l choice o f 84 solvents from the p o i n t o f view o f possible e f f e c t s on s e l e c t i v i t y I t i s possible t o give r u l e s f o r "optimum" gradients w i t h general a p p l i c a b i l -

.

i t y w i t h considerably s i m p l i f i e d assumptions o n l y as " h i n t s " t h a t cannot be ex-

pected t o provide the best separation o f a given sample mixture, b u t r a t h e r t o o f f e r the general p r o b a b i l i t y o f y i e l d i n g acceptable separations i n a wide v a r i e t y o f d i f f e r e n t separation problems. According t o

the " l i n e a r sol vent strength" (LSS) gradients are

e s p e c i a l l y s u i t a b l e f o r p r o v i d i n g the optimum separation conditons i n the above sense. Snyder and co-workers 3 7 y 3 8 suggested an o p t i m i z a t i o n approach f o r determining the optimum steepness o f LSS gradients t h a t would y i e l d maximum r e s o l u t i o n per u n i t time over the whole chromatogram. Let us r e c a l l t h a t the peak widths and r e s o l u t i o n are approximately equal i n d i f f e r e n t p a r t s o f the chromatogram f o r solutes w i t h equal r e l a t i v e r e t e n t i o n s ( i s o c r a t i c ) , provided t h a t the separat i o n factors, a. o f compounds i n the sample mixture do n o t change s i g n i f i c a n t l y w i t h the composition o f the mobile phase. I n t h i s instance, an optimum steepness, 6, o f an LSS gradient can be derived by analogy w i t h the optimum capacity f a c t o r

under i s o c r a t i c conditions (eqn. 4.9):

B = 0.1-0.3

f o r d i f f e r e n t column packing

materials, provided there i s a constant column l e n g t h and a v a r i a b l e separation time. For example, t h i s corresponds i n p r a c t i c e t o an increase o f 1-3% i n the amount o f organic solvent i n the mobile phase per m i l l i l i t r e o f t h e e l u a t e from the column i n reversed-phase chromatography on an o c t a d e c y l s i l i c a column w i t h

vm = 3 m l . This i s i n good agreement w i t h the e m p i r i c a l l y suggested optimum gra-

61 d i e n t steepness i n reversed-phase chromatography

.

I n reversed-phase chromatography, i t i s possible t o p r e d i c t t h e steepness o f a l i n e a r concentration gradient o f the organic solvent necessary t o achieve approximately the r e s o l u t i o n required f o r such mixtures, where the s e l e c t i v i t y o f separation between the i n d i v i d u a l compounds (under i s o c r a t i c c o n d i t i o n s ) i s approximately constant and e i t h e r i s known o r can be estimated. For example, the s e l e c t i v i t y i s approximately constant between t h e neighbouring members of a homologous o r o f a benzologous s e r i e s i n reversed-phase chromatography ( a constant). = constant) 9 under the gradient conditions, so t h a t the d i f f e r e n c e s o f the r e t e n t i o n volumes,

Further, the widths o f a l l peaks are approximately constant ( w

AV' and the r e s o l u t i o n between the adjacent peaks are approximately constant '52. also . References on p . 180.

172

Provided that the elution is started a t a low concentration of the organic solvent i n the mobile phase, the term l0"l i n eqn. 7.5 can be neglected t o a f i r s t approximation. If a = constant, the constants rn in the k' = f ( c ) functions (eqn. 1.28) are approximately equal f o r different sample compounds. From eqn. 7.5, the steepness of the linear concentration gradient, B , necessary t o yield the required resolution, R i s given as follows: g' B=--

& log

log a

t m ~R

h V R

9 9

mg

a

(1 + kk)

If we use k; = 1, we obtain B=-

&log a (7.7) &'mRg

or, introducing the more correct eqn. 4 . 3 0 ~f o r k; in gradient elution using LSS gradients : B =

&log a

-

1.73 Rg

4rnV R mg

The calculation of optimum gradient steepness f o r separation i n a homologous series i s i l l u s t r a t e d by Table 7.2. Fig. 4.5(8) shows the separation of a homologous s e r i e s o f fluorescent derivatives o f aliphatic amines using a linear gradient, w i t h near t o optimum conditions ( B = 0.015) 70 In practice, the parameters rn increase w i t h increasing retention and the separation factors, a ( i s o c r a t i c ) , of the neighbouring members i n a homologous series decrease t o a certain extent w i t h increasing content of the organic solvent i n the mobile phase. To compensate f o r t h i s moderate influence of concentration on a, the mean values o f rn and o f the separation factors (corresponding t o the composition of mobile phase i n the middle part of the gradient) s h o u l d be considered in calculations according t o eqns. 7.6-7.8. A practical consequence of the dependence of a on mobile phase composition i s that s l i g h t l y convex gradients sometimes approximate better a linear solvent strength change than linear gradients of concentration of the organic solvent in the mobile

.

173

TABLE 7.2

OPTIMUM GRADIENT STEEPNESS, B

OP'

OF A LINEAR CONCENTRATION GRADIENT OF AN

ORGANIC SOLVENT AS A FUNCTION OF RESOLUTION, R GRAPHY OF HOMOLOGOUS SERIES

g'

I N REVERSED-PHASE CHROMATO-

Column, C18/LiChrosorb S i 100 (10 pm), 300 x 4.2 mm 1.0.; V = 3.1 m l ; YZ = 2000. C a l c u l a t i o n based on separation o f a homo1ogous s e r i e s o f ff uorescent d e r i v a t i v e s of C 1 - C ~ Qn-alkylamines using ( a ) eqn. 7.7 and ( b ) eqn. 7.8 ( B i n g r a d i e n t f u n c t i o n given by eqn. 4.17; VG = volume o f the g r a d i e n t ( m l ) i f t h e e l u t i o n i s s t a r t e d w i t h zero c o n c e n t r a t i o n o f t h e organic solvent; A = 0 ) . Organic s o l v e n t

m*

l o g a* Parameter

Rg

1.0 a Methanol

5.25

0.12

Bop

Acetoni tri l e

2.95

0.13

vG Bop

Tetrahydrofuran

5.44

0.13

Bop

I%

vG

1.5 b

a

2.0 b

a

b

0.041 0.056 24 18

0.027 0.028 37 36

0.021 0.015 48 67

0.079 0.111 13 9

0.053 0.059 19 17

0.040 0.032 31 25

0.043 0.061 23 16

0.029 0.032 34 32

0.022 0.017 45 59

*Mean values. 7.4.

GRADIENT ELUTION AS A "SCOUTING" TECHNIQUE FOR ISOCRATIC ELUTION LIQUID CHROMATOGRAPHY I n c e r t a i n chromatographic systems, t h e g r a d i e n t e l u t i o n technique can be used

f o r t h e r a p i d s e l e c t i o n o f a s u i t a b l e composition o f t h e mobile phase i n l i q u i d column chromatography under i s o c r a t i c c o n d i t i o n s . According t o Snyder and c o - ~ o r k e r s ~ ~optimized '~~, e l u t i o n using " l i n e a r s o l v e n t strength'' (LSS) g r a d i e n t s can be used w i t h advantage f o r t h i s purpose. I f optimized LSS g r a d i e n t s (where 6 = 0.2 i n eqn. 4.4) are used, a l l sample compounds should e l u t e w i t h approximately equal instantaneous c a p a c i t y factors,

k;

2.2 ( c f . ,

eqn. 4 . 3 0 ~ ) .

For example, t o achieve k ' * 4 under i s o c r a t i c c o n d i t i o n s , i t would be necessary t o s e l e c t a composition o f t h e mobile phase corresponding t o k ; = 4 i n eqn. 4.4.

Eqn. 4.4 can then be used t o c a l c u l a t e the time, tC,from t h e s t a r t o f

g r a d i e n t e l u t i o n , when t h e compound e l u t e d i n time t had t h e c a p a c i t y f a c t o r g

k ' = k ' = 4 a t the i n l e t o f t h e column, i.e., C

References on p . 1 8 0 .

174

= t

9

- 2.5

-

t,

tZ

(7.9)

i s the d i f f e r e n c e between the times corresponding t o the i n s t a n t a c/f neous capacity f a c t o r s k' = k) and k' k; a t the o u t l e t o f t h e column and tZ i s

where A t

the delay time corresponding t o the v o i d (dead) volumes between the o u t l e t from the gradient-forming device and the top o f the column. The composition o f t h e mobile phase corresponding t o the time tc can be e a s i l y found from the g r a d i e n t f u n c t i o n used t o c o n t r o l the mobile phase composition programme. Snyder and co-workers 3 7 y 3 8 found good agreement between the experimental and expected chromatographic data i n chromatography on chemically bonded non-pol a r s t a t i o n a r y phases using l i n e a r concentration gradients o f a c e t o n i t r i l e i n water. A more extensive study by Elgass61 showed t h a t nine compounds on an octadecyls i l i c a column had instantaneous capacity f a c t o r s a t the time o f e l u t i o n o f

k; = 2.4-3.2 k; = 0.9-2.5

f o r l i n e a r concentration gradients o f a c e t o n i t r i l e i n water and f o r l i n e a r concentration gradients o f methanol i n water ( 6 * 0.2 i n

these examples)

, which

i s i n s a t i s f a c t o r y agreement w i t h Snyder's assumptions

( k b = 2.2 f o r B = 0.2). Based on these experiments, the f o l l o w i n g empirical r u l e s were derived6 1

.

Capacity f a c t o r s k ' = 1 can be expected under i s o c r a t i c conditions when worki n g w i t h a mobile phase w i t h a composition corresponding t o t h e time o f e l u t i o n o f peak maxima f o r compounds t h a t y i e l d the f o l l o w i n g r e t e n t i o n volumes i n grad i e n t e l u t i o n when e l u t e d w i t h l i n e a r concentration gradients beginning a t zero concentration o f the mori? e f f i c i e n t e l u t i n g component i n the mobile phase: (a) V; = 12 Vm f o r l i n e a r gradients o f methanol concentration i n water, s t a r t i n g w i t h a zero concentration o f methanol, on a C18 column; (b) V' = 9 Vm f o r l i n e a r gradients o f a c e t o n i t r i l e concentration i n water, 9 s t a r t i n g w i t h a zero concentration o f a c e t o n i t r i l e , on a C18 column; ( c ) V' 2 11 t o 12 vm f o r l i n e a r gradients o f methylene c h l o r i d e concentration 9 i n n-heptane, s t a r t i n g w i t h a zero concentration o f methylene chloride, on a s i l i c a column. The p r e c i s i o n o f these estimates should be w i t h i n f 20-40% r e l a t i v e . Another p o s s i b i l i t y f o r o p t i m i z a t i o n o f i s o c r a t i c conditions i s t o c a l c u l a t e the constants o f the k' = f ( c ) functions i n a given system from the data obtained i n gradient e l u t i o n chromatography. Under gradient conditions, we can deternine these constants f o r a l a r g e r number of compounds w i t h s i g n i f i c a n t d i f f e r e n c e s i n r e t e n t i o n than i s possible under i s o c r a t i c conditions. These constants can be used t o c a l c u l a t e the composition o f the mobile phase necessary t o achieve t h e required r e s o l u t i o n f o r given p a i r s o f compounds, as i n d i c a t e d i n P a r t I.

175 Jandera and C h u r f ~ c ' e k ~showed ' ~ ~ t h i s p o s s i b i l i t y f o r systems where eqn. 1.23 describes t h e k' = f ( c ) r e l a t i o n s h i p s (such as chromatography on p o l a r adsorbents o r on i o n exchangers) and where t h e general g r a d i e n t f u n c t i o n g i v e n by eqn. 4.18 c o n t r o l s t h e c o n c e n t r a t i o n programme o f t h e more e f f i c i e n t e l u t i n g component i n t h e mobile phase. Here, t h e constants o f t h e two-parameter eqn. 1.23,

kl, and m y

may be c a l c u l a t e d from t h e regression l i n e s o f l o g V ' versus l o g B p l o t s obtained 9 from t h e data under g r a d i e n t c o n d i t i o n s (eqn. 4.34). The f o l l o w i n g r e l a t i o n s h i p s between t h e constants o f t h e r e g r e s s i o n l i n e s ,

a

and 5 , and t h e parameters kl,

and rn apply: (7.10) and (7.11)

where

K

c h a r a c t e r i z e s t h e shape ( c u r v a t u r e ) o f t h e g r a d i e n t (eqn. 4.18).

The

accuracies o f t h e constants k i and m c a l c u l a t e d i n t h i s way are comparable t o those determined under i s o c r a t i c c o n d i t i o n s .

A s i m i l a r method was suggested by Schoenmakers e t al.41 f o r o p t i m i z a t i o n o f i s o c r a t i c c o n d i t i o n s i n reversed-phase l i q u i d chromatography. They observed an e m p i r i c a l c o r r e l a t i o n between t h e parameters m and kl, o f t h e k ' = f ( c ) r e l a t i o n s h i p a p p l y i n g i n reversed-phase systems (eqn. 1.28) i f aqueous m e t h a n o l i c mobile phases are used: (7.12)

m = p ' t q ' l o g k:

This c o r r e l a t i o n may be i n t r o d u c e d i n t o eqn. 4.27 f o r r e t e n t i o n volmnes i n r e versed-phase g r a d i e n t e l u t i o n l i q u i d chromatography t o y i e l d

m( 1-q ' A ) - p 10 9' mB

'

+

-

(7.13)

With t h e known column dead volume, Vm, and t h e parameters o f t h e l i n e a r conc e n t r a t i o n g r a d i e n t f u n c t i o n , A and B y t h e o n l y unknown q u a n t i t y on t h e r i g h t hand s i d e o f eqn. 7.13 i s t h e constant rn o f eqn. 1.28, which can be determined from t h e experimental r e t e n t i o n volume u s i n g eqn. 7.13 and i t e r a t i o n c a l c u l a t i o n s , provided t h e parameters p'and q ' o f eqn. 7.12 a r e known ( f r o m t h e d a t a o f Schoenmakers e t al.41, References on p . 180.

p r = 2.86 and 9' = 1.77 f o r an o c t a d e c y l s i l i c a column and

176 aqueous methanolic mobile phases). The correlation i s poorer w i t h mobile phases t h a t contain other organic solvents. To calculate the constant k ; , the value of rn determined from eqn. 7.13 i s introduced into eqn. 7.12 and, once the constants of the k' = f ( c ) function according t o eqn. 1.28 are known, the composition of the mobile phase necessary t o achieve the required resolution i n a given mixture may be calculated as i n the previous optimization method for normal-phase chromatography. The method has been used t o find optimum isocratic conditions f o r the separation of mixtures of phenolic compounds and of phenylthiohydantoin amino acids4'. However, one must be careful when using the correlation according t o eqn. 7.12, as the constants p'and q' were evaluated using only a limited number of compounds and exceptions are likely t o occur ( i t i s interesting t h a t the correlat i o n given by eqn. 7.12 can be predicted theoretically using the interaction indices model for reversed-phase separation mechanism, as mentioned i n Section 1.1 83). 7.5. EMPIRICAL APPROACH FOR SELECTION OF OPTIMUM GRADIENT CONDITIONS W i t h a basic knowledge of the theory of gradient elution, i t i s possible t o place on a rational basis an empirical design and optimization of gradient elution conditions. This empirical optimization approach may be understood i n terms of a subsequent approach t o the conditions t h a t would yield a resolution i n the chromatogram as near t o the required "best" values as possible in a reasonably short separation time. For this purpose, the three contributions t o resolution (efficiency, s e l e c t i v i t y and capacity) should be suitahly adjusted. The efficiency, i . e . , the number of plates i n the column used, can be i n creased and the resolution improved i n much the same way as under i s o c r a t i c conditions, i . e . , by ( a ) increasing the column length, ( b ) decreasing the p a r t i c l e diameter of column packing material, ( c ) decreasing the flow-rate of the mobile phase and ( d ) increasing the temperature or choosing components of the mobile phase of lower viscosity. However, changes ( a ) and ( c ) o r t h e i r simultaneous combination are connected with an increase in the time of analysis; ( a ) and (6) lead t o an increase in the operating pressure, which is increased most i f the column length and the flow-rate of the mobile phase are increased simultaneously t o hold the analysis time constant. The change ( d ) usually produces minor improvements. As b o t h the time of analysis and the operating pressure have practical upper limits, the optimization of efficiency i s limited in practice. Currently used standard analytical columns packed w i t h materials with p a r t i c l e diameters of 5-10 pm are usually s u f f i c i e n t l y e f f i c i e n t f o r most practical separation problems.

177 S e l e c t i o n of an a p p r o p r i a t e column p a c k i n g m a t e r i a l and c o m b i n a t i o n o f s o l v e n t s f o r a g i v e n s e p a r a t i o n i s a l s o c o n t r o l l e d by t h e same r u l e s as i n i s o c r a t i c e l u tion

Some u s e f u l combinations o f column p a c k i n g m a t e r i a l s and

components o f t h e m o b i l e phase a r e g i v e n i n Table 7.1 2-4,61

Once h a v i n g s e l e c t e d t h e a p p r o p r i a t e column and t h e components o f t h e m o b i l e phase, t h e o p t i m i z a t i o n of g r a d i e n t e l u t i o n s e p a r a t i o n c o n s i s t s i n a search f o r the best gradient p r o f i l e . F o r t h i s purpose, we r u n a g r a d i e n t c o r r e s p o n d i n g t o a change f r o m 0 t o 100% of t h e more e f f i c i e n t e l u t i n g component i n t h e m o b i l e phase i n a volume o f grad i e n t VG = 10-15 Vm ( g e n e r a l l y , a 15-45 min g r a d i e n t a t a f l o w - r a t e o f 1 ml/min, when u s i n g common a n a l y t i c a l columns). A l i n e a r c o n c e n t r a t i o n g r a d i e n t i s s e l e c t e d f o r work i n reversed-phase systems, whereas a concave g r a d i e n t would be g e n e r a l l y p r e f e r r e d i n ion-exchange chromatography and i n chromatoqraDhy on p o l a r adsorb e n t ~ The ~ ~ h~i n. t s c o n c e r n i n g t h e shape and steepness o f t h e i n i t i a l g r a d i e n t 2-4,61 p r o f i l e are a l s o given i n Table 7.1 O f course, f o r c e r t a i n s e p a r a t i o n s , t h e s e l e c t i v i t y may be m o d i f i e d by a d d i t i o n

o f a ligand-forming,

i o n - p a i r i n g compound o r a n o t h e r s u i t a b l e compound i n t o t h e

m o b i l e phase which o f f e r s s p e c i f i c i n t e r a c t i o n s w i t h t h e sample compounds.

Ift h e peaks of a l l sample compounds a r e e l u t e d i n t h e p a r t o f t h e g r a d i e n t volume c o r r e s p o n d i n g t o a ca. 20-25% change i n t h e c o n c e n t r a t i o n o f t h e more e f f i c i e n t e l u t i n g component b i n t h e m o b i l e phase, i t i s l i k e l y t h a t i s o c r a t i c c o n d i t i o n s c o u l d be found f o r a s u c c e s s f u l s e p a r a t i o n o f t h e sample m i x t u r e . I f t h e compounds a r e e l u t e d o v e r a l a r g e r p a r t o f t h e g r a d i e n t , g r a d i e n t e l u t i o n u s u a l l y should be used t o speed up t h e e l u t i o n . O f course, t h i s r u l e i s o n l y approximate and t h e above i n d i c a t e d range o f c o m p o s i t i o n i s i n f l u e n c e d b y t h e system used and b y t h e d i f f e r e n c e s i n t h e e l u t i o n s t r e n g t h s o f t h e s o l v e n t s a and b . I t s h o u l d a p p r o x i m a t e l y f i t reversed-phase systems w i t h methanol o r aceton i t r i l e as t h e s o l v e n t b . I n normal-phase chromatography on p o l a r adsorbents, t h i s r u l e can be used i f t h e compounds a r e n o t e l u t e d i n t h e e a r l y p a r t of t h e g r a d i e n t , where t h e changes i n c o n c e n t r a t i o n o f t h e more p o l a r s o l v e n t have a much more d r a m a t i c i n f l u e n c e on r e t e n t i o n . ( a ) The peaks a r e r e g u l a r l y spaced on t h e chromatogram, b u t t h e y a r e n o t s u f f i c i e n t l y r e s o l v e d and o v e r l a p : The steepness o f t h e g r a d i e n t , B , s h o u l d be decreased o r t h e number o f p l a t e s i n c r e a s e d t o improve r e s o l u t i o n . The t i m e o f a n a l y s i s and t h e peak w i d t h s a r e t h u s i n c r e a s e d and t h e peak h e i g h t s lowered. A decrease i n t h e i n i t i a l c o n c e n t r a t i o n o f t h e more e f f i c i e n t e l u t i n g agent i n t h e m o b i l e phase, A , a l s o l e a d s t o a s l i g h t increase i n resolution, b u t a l s o t o a l a r g e increase i n t h e analysis time, and i t s h o u l d be t r i e d o n l y i f t h e f i r s t peaks e l u t e d a r e c l o s e t o Vm and p o o r l y r e s o l v e d , o t h e r w i s e a decrease i n B i s u s u a l l y a more e f f i c i e n t remedy. References on p . 180.

178

-

( b ) The peaks are r e g u l a r l y spaced a t l a r g e distances on t h e chromatogram they are "over-resolved",

which leads t o unnecessarily l o n g a n a l y s i s times:

The steepness o f t h e g r a d i e n t should be increased t o y i e l d t h e r e s o l u t i o n required; t h e time o f a n a l y s i s i s thus shortened and t h e peaks become h i g h e r and narrower. A s h o r t e r column o r a h i g h e r f l o w - r a t e o f t h e mobile phase may be used. ( c ) The peaks a r e r e s o l v e d s a t i s f a c t o r i l y b u t t h e f i r s t compound has a l a r g e e l u t i o n volume: The c o n c e n t r a t i o n o f t h e more e f f i c i e n t e l u t i n g component b i n t h e mobile phase a t t h e beginning o f t h e g r a d i e n t , A , should be increased. As an e m p i r i c a l r u l e , a s t a r t i n g c o n c e n t r a t i o n A s e l e c t e d as ca. 50% o f t h e c o n c e n t r a t i o n o f t h e component b a t t h e i n s t a n t o f e l u t i o n o f t h e f i r s t sample band i n t h e i n i t i a l g r a d i e n t run i s l i k e l y t o p r o v i d e a s a t i s f a c t o r y improvement. T h i s u s u a l l y s l i g h t l y i m p a i r s t h e r e s o l u t i o n ( t o compensate f o r t h i s e f f e c t , B can be lowered), b u t decreases s i g n i f i c a n t l y the t i m e o f a n a l y s i s . ( d ) Sample compounds are e l u t e d t o o e a r l y and are b a d l y resolved: Both A and B should be decreased; i n c e r t a i n i n s t a n c e s t h e s o l v e n t b i s t o o s t r o n g and should be changed f o r a weaker one. I f t h e s l o p e o f t h e g r a d i e n t i s low and t h e r e s o l u t i o n i s s t i l l poor and t h e peaks are r e l a t i v e l y e a r l y e l u t e d a t A = 0, t h e l e s s e f f i c i e n t e l u t i n g component i n t h e m o b i l e phase ( s o l v e n t a) has t o o h i g h an e l u t i o n s t r e n g t h and should be changed f o r a weaker one. I f t h i s i s n o t p o s s i b l e (e.g., i f a i s water i n reversed-phase chromatography o r an n-alkane i n normal-phase chromatography), a change o f t h e whole chromatographic system would be necessary. The r e t e n t i o n may be increased u s i n g a c e r t a i n form o f a d d i t i o n a l i n t e r a c t i o n s i n t h e mobile phase, such as a change i n t h e i o n i c s t r e n g t h o f pH o r t h e a d d i t i o n o f i o n - p a i r - f o r m i n g o r complex-forming agents i n reversed-phase chromatography o f s t r o n g l y p o l a r o r i o n i z e d s o l u t e s . ( e ) A number o f compounds can be e l u t e d o n l y a f t e r t h e end o f t h e g r a d i e n t , i n pure component b o f t h e mobile phase: I f t h e r e s o l u t i o n i s adequate and a l l sample s o l u t e s are e l u t e d i n a reasona b l e time, i t i s n o t necessary t o change t h e p r o f i l e o f t h e g r a d i e n t , which i s a two-step g r a d i e n t w i t h a f i n a l hold-up ( S e c t i o n 5.4). However, i f sample peaks are "over-resolved"

and t h e e l u t i o n volumes o f t h e most s t r o n g l y r e t a i n e d com-

pounds a r e excessive, t h e i n i t i a l l y chosen s o l v e n t b has t o o l o w an e l u t i o n s t r e n g t h and should be changed f o r a s t r o n g e r one. ( f ) The l a t e r e l u t e d peaks a r e s u f f i c i e n t l y r e s o l v e d o r "over-resolved",

but

the e a r l y e l u t e d compounds are p o o r l y resolved; t h e peaks a r e "bunched t o g e t h e r " towards t h e beginning o f t h e chromatogram; t h e peak w i d t h s i n c r e a s e towards t h e end o f t h e chromatogram: The shape ( c u r v a t u r e ) o f t h e g r a d i e n t should be changed i n such a way t h a t t h e g r a d i e n t i s l e s s steep a t t h e beginning and steeper towards t h e end, i.e.,

179 use a more concave g r a d i e n t . I f t h e peaks o f t h e e a r l y e l u t e d compounds s t i l l remain b a d l y r e s o l v e d and a r e e l u t e d c l o s e t o V,,,, t r y t h e use o f a two-step e l u t i o n w i t h an i n i t i a l g r a d i e n t d e l a y u s i n g p u r e component a ( w i t h a l o w c o n t e n t o f t h e component b i n t h e f i r s t , i s o c r a t i c s t e p ) . ( 9 ) The e a r l i e r e l u t e d peaks a r e s u f f i c i e n t l y r e s o l v e d o r " o v e r - r e s o l v e d " , b u t t h e l a t e r e l u t e d compounds a r e p o o r l y r e s o l v e d ; t h e peaks a r e "bunched t o g e t h e r " towards t h e end o f t h e chromatogram; t h e peak w i d t h s decrease towards t h e end o f t h e chromatogram: The shape ( c u r v a t u r e ) o f t h e g r a d i e n t s h o u l d be changed t o become s t e e p e r a t t h e b e g i n n i n g and l e s s s t e e p towards t h e end, i . e . ,

use a more convex g r a d i e n t .

A two-step g r a d i e n t w i t h a f i n a l hold-up, e i t h e r a t 100% o r a t a l o w e r c o n t e n t o f t h e s o l v e n t b i n t h e m o b i l e phase, may improve t h e s e p a r a t i o n . ( h ) The r e s o l u t i o n and s p a c i n g o f most peaks on t h e chromatogram a r e s a t i s f a c t o r y , b u t t h e r e a r e one o r more p a i r s o f compounds i n s u f f i c i e n t l y r e s o l v e d a t d i f f e r e n t p a r t s o f t h e chromatogram: I f an i n c r e a s e i n column e f f i c i e n c y does n o t h e l p , i t i s necessary t o a d j u s t t h e s e l e c t i v i t y o f s e p a r a t i o n f o r t h e s e " c r i t i c a l " p a i r s o f compounds. T h i s may be a t t e m p t e d by changing t h e steepness o f t h e g r a d i e n t (changes i n t h e shape o f t h e g r a d i e n t o r i n t h e i n i t i a l c o n c e n t r a t i o n o f t h e more e f f i c i e n t e l u t i n g component i n t h e m o b i l e phase may sometimes a l s o improve t h e s e l e c t i v i t y , b u t t h e i n f l u e n c e o f B on s e l e c t i v i t y i s more s i g n i f i c a n t ) , which can be s u c c e s s f u l f o r p a i r s o f compounds w i t h d i f f e r e n t c o e f f i c i e n t s rn o f t h e two-parameter k ' = f ( c ) f u n c t i o n s . A change i n B may e i t h e r improve o r i m p a i r t h e s e l e c t i v i t y ( f o r more d e t a i l s , see S e c t i o n 4.8).

I f a change i n B i s n o t s u c c e s s f u l , we can t r y a com-

posed g r a d i e n t w i t h g r a d i e n t d e l a y s a t p o s i t i o n s o f e l u t i o n o f t h e u n r e s o l v e d p a i r s o f peaks. I f t h e s e p a r a t i o n i s s t i l l n o t s a t i s f a c t o r y , i t i s necessary t o change t h e more e f f i c i e n t e l u t i n g component i n t h e m o b i l e phase ( s u c h as methanol f o r a c e t o n i t r i l e o r t e t r a h y d r o f u r a n i n reversed-phase chromatography);

a change

o f t h e l e s s e f f i c i e n t e l u t i n g component i s u s u a l l y n o t o f much h e l p . T h i s means 75 t h a t t h e s o l v e n t b f r o m a n o t h e r s e l e c t i v i t y group ( a s d e f i n e d by Snyder ) i s t o be p r e f e r r e d . The use o f a t e r n a r y g r a d i e n t may o f t e n s o l v e t h e p r o b l e m e f f i c i e n t l y ( s e e Chapter 6 ) . Another p o s s i b i l i t y f o r i m p r o v i n g t h e s e l e c t i v i t y i n v o l v e s t h e a d d i t i o n o f a f u r t h e r component t o t h e m o b i l e phase t h a t can undergo s p e c i f i c i n t e r a c t i o n s w i t h some o f t h e b a d l y r e s o l v e d compounds ( a change i n pH, a d d i t i o n o f a complex-forming compound, e t c . ) . I f t h e s e p a r a t i o n problem i s n o t s o l v e d by any o f t h e s e means o r b y i n c r e a s i n g t h e e f f i c i e n c y , t h e chromatographic system (column p a c k i n g m a t e r i a l ) s h o u l d be changed. Some f u r t h e r d e t a i l s o f p o i n t s ( a ) - ( h ) s h o u l d be considered. F o r c e r t a i n p a i r s o f compounds, where t h e s e l e c t i v i t y depends s i g n i f i c a n t l y on t h e c o m p o s i t i o n of t h e m o b i l e phase, a change i n B o r A may l e a d t o o p p o s i t e r e s u l t s t o t h o s e p r e References on p . 1 8 0 .

180 dicted above owing t o changes in s e l e c t i v i t y , b u t this does not ocurr very f r e quently. Further, we should r e c a l l t h a t the steepness of the g r a d i e n t , B , i s defined as the concentration change o f component b in the mobile phase per u n i t voZme of t h e e l u a t e from the column. Consequently, i f we a r e changing the flowr a t e of t h e mobile phase, we should not f o r g e t t o change correspondingly the time of the gradient in order t o keep B constant. I f we decrease the flow-rate of t h e mobile phase t o improve the e f f i c i e n c y of separation (and r e s o l u t i o n ) , b u t we do not change t h e time of the gradient, an increase i n the gradient steepness, B , would r e s u l t , which usually would g r e a t l y predominate over the influence of the mobile phase flow-rate and t h e resolution would not be increased, b u t lowered! Further d e t a i l s concerning the p o s s i b i l i t i e s of u s i n g t h e optimum gradient p r o f i l e , once determined, with columns of other geometries o r w i t h o t h e r mobile phase flow-rates a r e given i n Section 4.5. Not only the time of the g r a d i e n t , b u t a l s o t h e time necessary f o r re-equil i b r a t i o n of t h e column back t o the i n i t i a l gradient conditions should be included in the analysis time. The s p e c i f i c requirements f o r separation, such as t h e detection s e n s i t i v i t y o r semi-preparative o r preparative separation purposes, should be considered when developing the optimum separation (they would l a r g e l y determine the resolution necessary). With c e r t a i n gradient p r o f i l e s , additional d i f f i c u l t i e s may occasionally occur, such as band t a i l i n g and solvent demixing e f f e c t s . With c e r t a i n gradient e l u t i o n instruments, the gradients a r e poorly reproducible i n the i n i t i a l and f i n a l p a r t s . Here, i t i s recommended t o s t a r t the gradient a t ca. 5% of component b and t o end the gradient a t a maximum of 95% o f component b38. The compositions of the two l i q u i d s mixed during thk gradient e l u t i o n should be adjusted correspondingly. REFERENCES TO PART I1 1 L.R. Snyder, Chromatogr. Rev., 7 (1965) 1. 2 L.R. Snyder and J.J. Kirkland, Introduction to Modern Liquid Chromatography, 2nd ed., Wiley-Interscience, New York, 1979, Ch. 16. 3 L.R. Snyder, i n Cs. Horvath ( E d i t o r ) , High-Performance Liquid Chromatography, Advances and perspectives, Vol. 1, Academic Press, New York, 1980, p. 208. 4 P. Jandera and J. ChurlEek, Advan. Chromatogr. , 19 (1981) 125. 5 L.B. Sybrandt and E.F. Montoya, Int. Lab., J u l y / A 5 u s t (1977) 51. 6 S.R. Bakalyar, R. McIlwrick and E. Roggendorf, J . Chromatogr., 142 (19771 353. 7 D.J. Popovich, J.B. Dixon and W.A. McKinley, h e r . Lab., May (1981). 8 P. Jandera and J. ChurBEek, J. Chrmatogr., 91 (1974) 223. 9 R.P.W. S c o t t , J . Chromatogr. S c i . , 9 (1971) 3755. 10 R.P.W. Scott and P. Kucera, J . Chrozatogr. Sci., 11 (1973) 83. 11 R.P.W. Scott and P. Kucera, J . Chromatogr., 83 ( 1 v 3 ) 257. 12 R.P.W. S c o t t and P. Kucera, AnaZ. Chem., s p 9 7 3 ) 749.