C O M P U T E R I Z E D OPTIMIZATION A N D S O L U T E R E C O G N I T I O N IN LIQUID C H R O M A T O G R A P H I C S E P A R A T I O N S
. ^ G H N / S c ,
P R O E F S C H R I F T
ter v e r k r i j g i n g van de graad van doctor in de technische wetenschappen aan de Technische Hogeschool D e l f t ,
op gezag van de Rector Magnificus, prof. dr. 3. M. D i r k e n , in het openbaar te verdedigen ten overstaan van
het college van Dekanen
op donderdag 28 maart 1985 te 16.00 uur door
Antorüus Christianus Johannes Hubertus Drouen,
geboren te Helmond, scheikundig ingenieur.
Druk: Dissertaliedrukkerij Wibro, Helmond,
TR diss
1430
Vit pioZ-ibdhüit a gozdge.ke.wid dooi de. piomotoi Vzoh. Vi. L. de Galan
aan ma.ige.ile, lenneke en maithe. aan mijn ou.de.16
Voorwoord
Aan de totstandkoming van dit proefschrift hebben velen een bijdrage geleverd.
Als eerste mijn promotor, professor ito dt Galan die door zijn waarde volle suggesties, aanwijzingen en discussies mede de loop van het onderzoek bepaald heeft. In het bijzonder wil ik verder danken Pttti Sthotnmaktib, die de basis voor dit werk heeft gelegd en aanleiding gaf tot vele waardevolle discussies en Hugo BiUitt voor alle praktische en morele steun tijdens mijn werk.
De studenten Albe.it Bot>mcw, Stt&aan BMie.t, Jolanda Cottltvci, Ttd Windkout, Bait Vandtnbwuckt en Joobt Stiabttib hebben elk hun prak tische medewerking verleend aan een gedeelte van dit proefschrift. De heren Htnk V. Vam, Fian6 Boudtwijn, Jaap Kitlmtti en Kttb van Ravtbttijn hebben ieder op hun gebied bijgedragen aan het installeren en in stand houden van de computer-apparatuur.
Het merendeel van het typewerk is gedaan door Bh van Uttn, een gedeelte van de tekeningen zijn op deskundige wijze door Fianb Solman en Ailt Schüti vervaardigd, de foto's zijn gemaakt door C. OJainaai en F. Hammtib.
Contents
Samenvatting
Sammauj
Chapte.1 }
An introduction on designs and criteria for optimizing liquid chromatographic separations
Chaptli 1
A simple procedure for the rapid optimization of reversed-phase separations with ternary mobile phase mixtures
ChcLpte.z 3
An improved optimization procedure for the selection of mixed mobile phases in reversed-phase liquid chromatography Chaptzi 4
A practical procedure for the optimization of reversed-phase separations with quaternary mobile phase mixtures
ChaptQ.i 5
Dual-wavelength absorbance ratio for solute recognition in liquid chromatography
Chaptzz 6
Multi-wavelength absorbance detection for solute recognition in liquid chromatography
Appendix I
Combined optimization of mobile phase pH and organic modifier content in the separation of some aromatic acids by reversed-phase high-performance liquid chromatography
Samenvatting
De optimalisering van de analytische scheiding van een monster heeft t o t doel om die condities te localiseren, waarbij de scheiding van d i t monster in al zijn samenstellende componenten te verwezenlijken is. Dit p r o e f s c h r i f t beschrijft de ontwikkeling van een optimaliseringsstrategie, welke toepasbaar is op alterlei gebieden van de hogedruk v l o e i s t o f c h r o m a -t o g r a f i e . Deze op-timalisering verloop-t snel en e f f i c i ë n -t en kan ges-tuurd worden met een computer.
Hoofdstuk 1 geeft een overzicht van 'de t o t op heden meest succesvol toegepaste optimaliseringsstrategieën. De voor- en nadelen worden aange geven ten opzichte van de ontwikkelde strategie.
In hoofdstuk 2 wordt het principe van de door ons ontwikkelde strategie uit de doeken gedaan. De methode berust op het volgende principe. De optimalisering wordt geïnitieerd met een aantal s t a r t c h r o m a t o g r a m m e n , minimaal gelijk aan het aantal te optimaliseren variabelen plus één. De in dit hoofdstuk beschreven optimalisering van de mobiele fase samenstel ling op het gebied van de reversed-phase chrornatografie gaat uit van 3 i n i t i ë l e chromatogrammen. Deze chromatogrammen zijn opgenomen in drie iso-eluotrope binaire mengsels van water gemengd met successieve lijk methanol, t e t r a h y d r o f u r a n en a c e t o n i t r i l . De mengverhouding (elutie-sterkte) van deze drie mobiele fasen is zodanig gekozen, dat zij een ongeveer gelijke totale analysetijd voor het monster geven (iso-eluotrope mengsels). Met behulp van de aanname dat de r e t e n t i e (uitgedrukt in de logarithme van de c a p a c i t e i t s f a c t o r ) lineair verandert als f u n c t i e van de mengverhouding van twee iso-eluotrope binaire mengsels, is het mogelijk om een ternaire mobiele fase te berekenen, waarbij de scheiding o p t i maal is. Deze berekening kan op een eenvoudige en duidelijke wijze weergegeven worden in een fase-selectiediagram.
De berekende mobiele fasesamenstelling kan daarna in de p r a k t i j k getest worden. Het verkregen resultaat wordt gebruikt om correcties in het vooropgestelde lineair verloop aan te brengen. Hierna kan opnieuw de optimale samenstelling berekend worden met behulp van het g e c o r r i geerde retentiegedrag. D i t nu wordt steeds herhaald, waardoor een i t e r a t i e v e procedure ontstaat, waarbij het werkelijke retentiegedrag steeds beter benaderd wordt en de condities voor de werkelijke optimale scheiding steeds beter benaderd kunnen worden. Deze i t e r a t i e gaat door tot er geen verbetering meer bereikt w o r d t .
Het begrip optimale scheiding wordt door ons gedefinieerd als een zo groot mogelijke spreiding van alle pieken over het t o t a l e c h r o m a t o g r a m . De mate van spreiding w o r d t uitgedrukt als het product van de resolutie factoren van alle opeenvolgende piekparen. Hierin is de resolutie een maat voor de scheiding van twee pieken, rekening houdend met de breedte van deze twee pieken. Bewezen kan worden, dat de hoogste verdelingsgraad bereikt is als het resolutieproduct maximaal is binnen een vaste totale analysetijd.
In hoofdstuk 3 worden enkele verbeteringen van de voorgaande o p t i m a l i seringsprocedure besproken. In eerste instantie werd steeds het voor spelde o p t i m u m gemeten en gebruikt om het retentiemodel te verbete ren. In sommige gevallen leidde dit t o t een groot aantal metingen waarbij de werkelijke optimale conditie slechts langzaam benaderd w e r d . In de nieuw gedefinieerde strategie wordt nu echter niet bij het
voor-spelde optimum gemeten, maar bij een verschoven samenstelling, waar door de procedure versneld wordt en een betere spreiding van de meetpunten wordt verkregen. Dit laatste levert een snellere beschrijving van het werkelijke retentiegedrag over het totale bereik van de te optimaliseren parameters.
Ook het criterium, geformuleerd in hoofdstuk 2, vertoonde enkele gebre ken. Een maximale waarde in het resolutieproduct geeft alleen dan de maximale spreiding aan van een verzameling chromatogrammen als de som van alle resolutiewaarden steeds constant Is. Dit houdt in dat over het totale parameterbereik de totale analysetijd nagenoeg constant moet zijn. Dit is zeker niet altijd het geval. Door nu steeds de werkelijke waarde van het resolutieproduct uit te drukken als de fractie van het maximaal haalbare product voor deze situatie, wordt deze fout gecorri geerd. De maximaal haalbare productwaarde wordt gegeven als de voor deze totale analysetijd realiseerbare optimale verdeling in resolutiewaar den (alle pieken gescheiden met gelijke resolutiefactor). Door de intro ductie van een fictieve startpiek kan bereikt worden dat het criterium naar een kortere analysetijd streeft.
In hoofdstuk 2 en 3 worden steeds ternaire mengsels geoptimaliseerd, die samengesteld zijn uit twee van de drie binaire iso-eluotrope mobiele fasen. Dit resulteert dan in een scala quaternaire mobiele fasesamen-stellingen. Deze uitbreiding wordt besproken in hoofdstuk 4, waarbij de één-parameter optimalisering van de hoofdstukken 2 en 3 wordt uitge breid tot twee parameters- Natuurlijk worden nu in plaats van lineaire lijnsegmenten, platte vlakken gebruikt om het retentieverloop in In k als functie van de mobiele fasesamenstelling weer te geven. Deze platte vlakken worden steeds door drie datapunten getrokken. Ook dit is weer een iteratieve procedure, waarbij "verschoven" meetpunten gebruikt wor den om het werkelijke retentieverloop steeds beter te beschrijven. Ook hierdoor wordt een versnelling van de procedure bewerkstelligd. Natuur lijk is de berekening van het verschoven meetpunt ingewikkelder dan voor de ternaire procedure, maar nog steeds goed uitvoerbaar. Een stopcriterium is ingebouwd om aan te geven wanneer de procedure beëindigd kan worden. Deze aanpak leent zich bij uitstek voor automati sering, waarbij een computer zowel de berekening van de optimaliserings procedure, als de besturing van een chromatograaf verzorgt.
Niet alleen in deze optimaliseringsstrategie, maar ook in enkele andere optimaliseringsprocedures, is het steeds noodzakelijk om gedurende de gehele procedure alle componenten in elk chromatogram te volgen. Tot op heden werd dit gerealiseerd door aparte injecties van elke component in elke gemeten mobiele fasesamenstelling. Dit is een tijdrovende bezig heid en deze kan natuurlijk niet gebruikt worden voor monsters die onbekende stoffen bevatten.
In hoofdstuk 5 wordt de mogelijkheid besproken om de verhouding (ratio) van de UV-VIS absorptie, gemeten bij twee verschillende golflengten, te gebruiken als identificatiemogelijkheid. Modelstudies en enkele prakt ische voorbeelden tonen aan dat het gebruik van ratiogrammen in een optimaliseringsstrategie beperkt wordt door chromatografische en instru mentele factoren. De belangrijkste chromatografische factoren zijn de afhankelijkheid van de ratiowaarde van de mobiele fasesamenstelling en de vervorming van het ratiosignaal door het gedeeltelijk of geheel samenvallen van twee of meer componentpieken. Instrumentele factoren zijn onder andere de vervorming ontstaan door een electronische offset
U
op een of beide absorptiesignalen en door een niet gelijktijdig aftasten van de absorptiewaarden van de beide gemeten golflengten. Geconclu deerd kan worden, dat desondanks piekoverlap gedetecteerd kan worden tot op ongeveer 0.1 o. Daarentegen wordt de piekherkenning in verschil lende chromatogrammen te veel gestoord door de bovengenoemde facto ren. Door gebruik te maken van het gehele UV-VIS spectrum, waardoor meer informatie verkregen wordt moet het mogelijk zijn om in vele gevallen componenten op te sporen in een serie chromatogrammen, mits deze een redelijke divergentie vertonen in hun respectievelijke absorptie spectra.
In hoofdstuk 6 wordt dit aangetoond met gebruikmaking var, de in de laatste jaren ontwikkelde fotodiode-array detector en een door ons ontwikkelde stapsgewijze procedure. Met behulp van deze detector is het mogelijk om tijdens de elutie van een chromatogram continue spectra op te nemen met een maximale snelheid van 10 scans per seconde. Als eerste stap in de procedure wordt een maximum absorptie chromatogram getekend om all UV adsorberende componenten te detecteren. Hierna wordt met behulp van elutieprofielen, opgetekend bij verschillende ge schikte golflengten, een indruk verkregen van het minimaal aantal aanwezige componenten. Vervolgens worden spectra uit het gehele chro matogram geselecteerd en vergeleken met spectra, geselecteerd m de andere chromatogrammen, om zo de elutievolgorde van alle componenten in de chromatogrammen te achterhalen. De in hoofdstuk 6 besproken voorbeelden geven duidelijk aan dat deze methode zeer bruikbaar is, ook als meerdere componenten in hoge mate overlappen.
In appendix 1 wordt tenslotte aangetoond dat de ontwikkelde één en twee-parameter optimaliseringsmethoden ook toepasbaar zijn op andere scheidingsmethoden in de hoge druk vloeistofchromatografie. De optima lisering van de pH van de mobiele fase voor de scheiding van enkele aromatische zuren wordt eerst als een één-parameter optimalisering uitgevoerd. Hierna wordt een gelijktijdige optimalisering van zowel de pH als de modifier concentratie uitgevoerd. In beide voorbeelden wordt de optimale mobiele fasesamenstelling op een snelle en adequate wijze gelocaliseerd.
We mogen wel concluderen dat de door ons ontwikkelde procedure zeer bruikbaar is voor de optimalisering van scheidingen in de vloeistofchroma tografie. Dat deze methode niet alleen toepasbaar is voor de optimali sering van ternaire en quaternaire mobiele fasesamenstellingen in rever sed phase vloeistofchromatografie blijkt onder andere ook uit appendix I.
Summary
This thesis gives a description of a procedure for the optimization of separations in Reversed Phase Liquid Chromatography, which can be used as the basis for a fully automated system.
Chapter 1 gives a general introduction, in which the developed method is placed in the context of other optimization schemes.
Chapter 2 defines the general framework of the iterative regressive optimization. The method starts with a limited number of initial chroma-tograms with equal total analysis time but recorded under different conditions. For the separation of samples by reversed-phase liquid chro matography, these chromatograms are recorded in three iso-eluotropic binary mixtures of water with methanol, tetrahydrofuran and acetonitnl, respectively. The assumption that the logarithm of the capacity factor changes linearly with the mixing rate of these binary mixtures, makes it possible to construct a so-called phase selection diagram and to locate the ternary mobile phase composition for an optimal separation. This optimal separation is defined as the best possible equal spreading of all peaks in the total chromatogram and numerically defined as the maxi mum in the product of the resolution factors of all adjacent peak pairs. After recording a chromatogram under the optimal conditions, the retention data are used to refine the linear retention model and a new optimum is located. This procedure is repeated until no further improve ments are expected.
Chapter 3 gives some refinements of the procedure described in chapter 2. In the initial procedure each predicted optimum was checked and used to refine the retention model. This could result in small refinements and many additional measurements. In the newly defined strategy a more efficient use of previously measured data is incorporated. The chroma togram is not recorded at the optimal condition, but at a shifted composition, resulting in a more rapid approach of the final optimal conditions.
The resolution product criterium is refined and expressed as the fraction of the ideal separation, that theoretically could have been realized for that situation. This relative resolution product ranges from zero if two solutes coelute, up to unity for an ideal separation (equidistance in resolution). A stop criterium is added to decide after each refinement if the procedure should be continued or stopped.
Chapter 4 describes an extension of the one parameter, ternary mobile phase optimization to a two parameters, quaternary mobile phase optimi zation. Whereas for the ternary procedure the mixing ratio of two binary phases is optimized, for the quaternary phases a blend of three binary phases is possible. The retention is approximated by flat planes, fitted through three data points, instead of linear line segments between two successive points. As in the ternary concept, each additional measure ment is used to refine the retention planes and to predict a new optimum until no further improvement can be expected. The method also uses the concept of shifted compositions to speed up the procedure and to gather more information of the entire parameter space.
For these iterative regressive designs, it is necessary to keep track of all solutes in each successively recorded chromatogram. Without this knowledge it is impossible to fit the retention model. So far this tracing 7
is done by injecting all solutes separately for each measured mobile phase condition. This is a time consuming method and it can, obviously, not be applied to samples containing unknown solutes.
Chapter 5 discusses the possibility of using the ratio of the absorbance measured at two different wavelengths for solute tracing. Model calcula tions and practical examples show that the applicability of the absor bance ratio for solute tracing in optimization procedures is restricted by chromatographic and instrumental limitations. Although peak overlap can be detected down to a peak separation of only 0.1 o, solute tracing is hampered by instrumental limitations and by the influence of the composition of the mobile phase on the value of the absorbance ratio. Chapter 6 describes the possibilities of fast scanning photodiode array detectors for tracing UV-active solutes in a series of successive chroma-tograms. Instead of recording only two wavelengths, as in chapter 5, the full spectrum is recorded to gain more information. An operator inter active strategy is developed. First a maximum absorbance plot is used to detect all absorbing solutes. Secondly, elution profiles at selected wave lengths are constructed to get an indication of the total number of solutes. Then representative spectra are selected for the final matching of the solutes in different chromatograms.
Appendix 1 describes an extension of the two parameter, quaternary optimization in RPLC to the simultaneous optimization of pH and modifier content in aqueous-organic phases. This appendix shows, that the concept of the optimization scheme, which was originally developed for the optimization of the mobile phase composition of mixtures of three organic modifiers with water, can easily be extended to include other parameters. The only restriction is that the retention behaviour, as a function of the parameters, varies continuously and smoothly between the initial data points, so that it can be described by a linear segmented model.
CHAPTER I
AN INTRODUCTION ON DESIGNS AND CRITERIA FOR OPTIMIZING LIQUID CHROMATOGRAPHIC SEPARATIONS
1-1 Introduction
High Performance Liquid Chromatography (HPLC) is an analytical sepa ration technique which has grown enormously in versatility an applica bility over the past fifteen years. This growth was initiated by the availability of high quality equipment. Also, the understanding of the many physico-chemical processes that control and influence the separa tions has grown. Nevertheless, when confronted with a novel, reasonably complex separation problem, many chromatographers still rely to a large extent on experience and intuition in developing a suitable HPLC-separa-tion for their particular problem. At best this is time-consuming and inefficient, since many factors influencing the separation interact and are, therefore, difficult to optimize by an empirical approach. The decisions, that confront a chromatographer, are first the suitable separa tion mode (straight-phase, reversed-phase, ion-exchange, ion-pairing, size-exclusion), then the appropriate instrument variables {gradient or isocra-tic, flow rate, column dimensions, temperature, detector) and finaly, an adequate mobile phase composition (solvent polarity, nature and concen tration of the constituents, pH, ionic strength, ion-pair reagent). Indeed, a major portion of time is spent in fine-tuning the instrumental variables and the composition of the mobile phase. It is, therefore, not surprising that efforts have been made to delegate this step to computer algo rithms and automation, especially since micro-processors are built into almost every modern instrument.
During the last four years many studies dealing with this type of automated optimization (whether or not built into a commercially avail able apparatus) have been published. Most apply to re versed-phase separations (RP), because this is the most versatile and popular mode of HPLC- Here the sample is separated on a column packed with a support material consisting of silica particles with a diameter of 3 to 10 jjm chemically coated with Ca or Cia hydrocarbons. After having selected the column dimensions (diameter from 2 to 7 mm and length 5-25 cm) the sample must be eluted with a polar mobile phase. This mobile phase consists usually of water mixed with one or more organic modifiers. The mixing ratio determines the elution strength and thus the total analysis time. Separation power (selectivity) is offered by the type of organic modifier. In addition to the organic modifiers, the pH and minor additions of salt or ion-pair reagent also influence the selectivity. Thus, in reversed-phase liquid chromatography, the separation can be optimi zed by adjustment of the mobile phase composition.
The optimization schemes proposed in the literature can be broadly divided into two catagories. Before we discuss the catagories (paragraph
1.5), some aspects common to both categories will be explained first.
1.2 Parameter space
The chrornatographer must define the number and the ranges of the parameters that are to be optimized. These parameters must all be continuously variable such as temperature, pH, flow rate, concentrations, etc. However, the more parameters to be optimized and the more they interact, the more difficult and time consuming the optimization proce dure will be. It is, therefore, prudent to restrict the optimization to those key parameters that exert the greatest influence upon the separa tion. In HPLC these are the concentration of the constituents of the mobile phase. Indeed, a parameter often encountered in RP-HPLC is the blending of two or three binary phases (H20 with an organic modifier) yielding respectively a ternary or quaternary mobile phase composition. This is a one-dimensional optimization for the ternary and a two-dimen sional optimization for the quaternary compositions. Other parameters that have been included in RP-HPLC optimization are flow-rate and temperature.
1.3 Response surface
The computer algorithm must be able to express the quality of the chromatographic separation for each chromatogram in a single number, often called the separation criterion. The variation of this criterion over the n-dimensional parameter space is called the response surface. This response surface must be searched for the highest value of the criterion, which defines the optimal chromatogram and its chromatographical conditions. It is in this search that difference is found between the two categories of optimization procedures (1). They can broadly be divided in the brute force or black box approaches and in the analytical or mathematical approaches as will be discussed in paragraph 1.5.
10
IA Characterization and evaluation of chromatograms
In all automated optimization schemes, the computer algorithm must be able to express the separation achieved in a chromatogram in a single number: the criterion value. This value must be related to the desired quality of separation. What constitutes a good chromatographic separa tion, is decided by the chrornatographer and the separation he tries to achieve. Sometimes he may desire baseline separation of all components in the sample. In other cases, he may be satisfied with the separation of only a few solutes. An additional restriction, sometimes imposed by the chrornatographer, can be that the total analysis time may not exceed a certain limit.
The selection of a proper criterion is very important, because its highest value directs the optimization procedure within the constraints of the parameter space and determines the final result (2-5). Unfortunately, in most published optimization schemes, the chrornatographer has little or no freedom in its formulation. Up to now, nearly ail procedures aim at baseline separation of as many solutes as possible within a reasonable analysis time. They are all based on a chromatographic separation criterion of adjacent peaks, sometimes in relation with the total analysis time. t 2 - to t l - to At 2(ol + o2) e / g
F*g. / Vabitdtion oh ^activity factoi a [A], solution lactox Ri (A) peafc tepawtwn factoi SF (6) and vedfay to pzak mtio UP (CJ.
[AA Two-peak separation criteria
Many c r i t e r i a are described in the l i t e r a t u r e for the quality of the separation of two peaks. The most simple one is the selectivity factor a defined by the ratio of capacity factors of t w o adjacent peaks (Fig. 1A). It does not take into acount the overlap of the peak areas and thus fails to discriminate between narrow and broad peaks. The resolution f a c t o r , Rs, defined by the ratio of the difference in elution t i m e and t w i c e the sum of the peakwidths, o (Fig. 1A), accounts for peak broadening. However, experimental measurement of a is d i f f i c u l t for strongly overlap ping peaks. A more readily measured c r i t e r i o n , introduced by Kaiser in 1960 (6), is the peak separation factor SF (Fig. IB). A closely related measure is the valley to peak r a t i o VP (Fig. 1C), defined by Christophe in 1970 (7).
To compare the valley to peak ratio w i t h the other c r i t e r i a , which approach zero for poor separations, it is convinient to consider 1-VP. Whereas Rs and a change almost linearly w i t h the extent of separation, the behaviour of 1-VP and SF is curved. This is shown in F i g . 2, where the c r i t e r i o n values are plotted for two gaussian peaks of equal height as function of the separation t i m e , dt, expressed in the peakwidth a. If separation improves SF and 1-VP change quickly f r o m 0.0 up to 0.8, and level off towards 1.0 when baseline separation is approached.
The SF and 1-VP values can only be derived f r o m experimental data. If there is no valley in the elution p r o f i l e , these two c r i t e r i a are unde fined. The resolution and selectivity factor remain defined, although they are d i f f i c u l t to measure for strongly overlapping peaks. At. a l t e r n a tive method is to measure retention t i m e and estimate the peakwidth f r o m chrornatographic column paramieters.
2 w K I 3 0 0 d t , o 8
Tig. 1 Variation oi the selectivity ^actoi a [plate numbei 5000], isola-tion dantoi TU, peak be.pa.ia.tion iaetot SF, and valley to peak mtio 1-VP a6 a function o{) the. sepamtion time, dt, expiehbed in the. peakwidth, a, iot two adjacent gaaiiian peafc4 o& equal height. l(, both peaki have equal height then 1-VP equah ST. The dashed line tepieienti l-2expt-2R42).
For the regressive designs (paragraph 1.5.2), the c r i t e r i o n must be calculated f r o m fundamental chrornatographic parameters. The c a l c u l a tion of a is s t r a i g h t f o r w a r d f r o m the predicted retention times. As o is d i r e c t l y related to the retention t i m e , t r , as a function of the plate number, N (o = tr / / N ) , RS can be calculated f r o m the retention times and the plate number of the column. SF and 1-VP cannot d i r e c t l y be calculated f r o m chrornatographic paramieters. From their d e f i n i t i o n i t can be derived t h a t , for two adjacent gaussian peak shapes of equal height,
SF = 1 - VP = 1 - 2e 2 R s ( 1 )
This function is plotted in Fig. 2 and gives a good approximation of SF and 1-VP.
l.^f.2 Multi-peak separation c r i t e r i a
Many multi-component o p t i m i z a t i o n c r i t e r i a are based on an increasing c r i t e r i o n value (SF, 1-VP, Rs or a) of the worst separated peak pair (8-12). In this case all i n f o r m a t i o n about the separation of other peak pairs is ignored, and minor improvement of the worst separation can lead to a decreased separation of other peak pairs. Moreover, and much more serious, unresolved peaks are not taken into account.
What is clearly needed, then, is a measure including separation of all adjacent peaks. Such a multi-peak c r i t e r i o n was f i r s t o f f e r e d by Giddings (13), called the total overlap f u n c t i o n ,
PO = I e "2 R s , i 1=1
and extended by many authors (1*0 to the Chrornatographic Response Function, C R F :
n-1
CRF = Z w. l n ( S F . ) ( 3 ) 1=1 1
in which n is the number of peaks, wi a weighting f a c t o r for the i-th component and SF the peak separation factor of Kaiser. The separation c r i t e r i a for each individual adjacent peak pair are simply added and the sum is optimized towards a higher value. For e f f i c i e n t use of this c r i t e r i o n , the number of peaks should either be known in advance or kept track of during o p t i m i z a t i o n , as there is no indication f r o m this type of c r i t e r i o n , whether many substances are poorly separated or a few are f a i r l y well separated (15). The t o t a l analysis t i m e or, more
i m p o r t a n t , the t o t a l number of peaks in the chroma tog ram can be included. This is given by Berridge (16) as:
n - i
CRF i ( I R s , i ) + nX - a ( T - T ) - b (T - T j (4) . , m n 0 1
i = 1
which includes the t o t a l number of peaks detected, n, the difference between an acceptable (Tm) and actual total analysis t i m e (Tn) and a factor to prevent that the first peak ( T i ) elutes before a minimum retention time (To). Factors x, a and b are applied to give more or less weighting to each contributing t e r m .
The CRF c r i t e r i a which include separation, analysis t i m e and number of peaks are d i f f i c u l t to interpret. They simply sum t o t a l l y d i f f e r e n t concepts such as separation, time and number of peaks. One can hardly say how much analysis t i m e equals a certain separation or an extra peak. For this reasons, 1 t h i n k , this type of c r i t e r i a often f a i l , especially for d i f f i c u l t separations.
Another type of multi-peak c r i t e r i u m is defined by the present author as the product of the two-peak c r i t e r i o n values in a chromatogram (chapter 2 and 3) instead of the sum. This c r i t e r i u m aims at an even spreading of the peaks over the t o t a l chromatogram, resulting in as much i n f o r m a t i o n as possible. To correct for changes in t o t a l analysis t i m e , the c r i t e r i o n can be expressed as the f r a c t i o n of the ideal separation (= equal spacing in Rs) as given below:
n-1 n-1 n-1
r = II R si + 1 / [ ( I R si + 1 ) / ( n - 1 ) ] (5)
1.5 Optimization designs
1.5-1 Brute force or black box approaches
These techniques observe only the e f f e c t of changing the parameters on the response and apply no chromatographical insight. The total response surface is covered w i t h a fine maze of measured response values and the optimal condition is simply defined by the highest value measured. L a t t i c e design
In this type of procedures the total response surface is estimated by a latice of discrete measured data points, which cover the t o t a l parameter space ( f i g . 3A). In its most simple f o r m the analyst visually selects the best separation f r o m all recorded chrornatograms, thus defining the optimal chromatographical conditions. If the o p t i m i z a t i o n is controlled by a computer the separation resulting f r o m each step can be expressed as a single c r i t e r i o n value. When stepping through the entire o p t i m i z a tion space, the computer algorithm remembers the highest c r i t e r i o n value and its chromatographical conditions, determining the best separa tion possible.
A B
THF/Ha0
Fig. 5 Fiarnz A >>howh a lattice, dtiign Ojj 2 / iuni> foot a quate.inaztj mobile. phdhZ optimization. Fiame. 8 bhowt» a two-paiame.tei Umptex pwctduze. in which the timplzx mavzè tiowty to the. highe.U value..
It may be clear that this method uses many test runs, c e r t a i n l y when more than one parameter is selected. This can be i l l u s t r a t e d w i t h a simple example of a reversed phase o p t i m i z a t i o n . 11 runs are needed to cover a f u l l range of w a t e r / m e t h a n o l (MeOH) binaries in steps of 10%. This is repeated for two other organic modifiers ( t e t r a h y d r o f u r a n , T H F , and a c e t o n i t r i l e , ACN) which brings the t o t a l up to 33 runs. If also ternary mixtures of the previous binaries have to be covered ( M e O H -THF, M e O H - A C N and T H F - A C N ) the t o t a l w i l l be 63 runs or even 96 runs including the quaternary mixtures. A f t e r all runs, one may still have missed the optimum by stepping over i t . However, the procedure is easily automated for unattended execution and indead i t is c o m m e r c i a l l y available f r o m at least one company (Perkin-Elmer).
Simplex design
A much more e f f i c i e n t approach is offered by the self-searching simplex procedure (17). It has been used in a wide variety of a n a l y t i c a l situations and offers a rapid approach to the highest value provided t h a t the response surface is rather smooth. The method starts w i t h a geometric figure called simplex, defined by a number of points equal t o one more than the number of dimensions of the parameter space. A simplex i n a two-parameter space is a t r i a n g l e , in a three-parameter space a t e t r a hedron, etc. Each of the points in the simplex results f r o m a c h r o m a t o -graphic experiment of which the separation is translated i n t o the corresponding c r i t e r i o n value. The objective of this method is to force the simplex t o move to the region of the highest response value. This is achieved by r e j e c t i n g the point giving the worst result and r e f l e c t i n g i t in the direction of the remaining points. This is i l l u s t r a t e d for t w o
parameters in Fig. 3B. Without refinements the procedure is rather slow but improvements have been made to gain speed by a continuous expansion and contraction of the step size of the simplex (18).
For the simplex procedure no chromatographical insight is needed and many parameters can be optimized simultaneously (of course more parameters will need more additional measurements). However, the simplex must be applied to a smooth response surface, preferably with one single valued maximum. If the response surface is very erratic with many peaks and valleys, there is a danger that the simplex foccusses onto one of the secondary peak crests. This is the more serious as the simplex provides no information of the entire response surface, but only indicates a maximum. To reduce this risk, the simplex may be started from different regions in the parameter space, leading to a large number of runs to complete the whole procedure (16,19).
This method has been applied with success by Berridge (16) for the automated optimization of two and three component mobile phases in RP-HPLC. For the optimization of the separation of four 2-substituted pyridines 25 runs were neccessary to optimize the flow-rate and the concentration of acetonitril in water. The separation of four substituted phenols with a ternary mobile phase of methanol, acetonitril and water was optimized with a total of 2U chromatograms.
The number of runs, though less then in the lattice design, is still appreciable. Again, the simplex can be easily automated for running overnight. No chromatographical insight is needed and many parameters can be optimized simultaneously. The elution order of the solutes needs not be known, so that the optimization of a sample consisting of unknown solutes is no problem.
1.5.2 Analytical or mathematical approaches
The second catagory can be described as the analytical or mathematical approach. They are based on some knowledge of the underlying physico-chemical principles and effects of the parameters on the response. In the regressive designs a regression technique is applied to fit a model through a minimal number of pre-planned data points. This model can either describe directly the response surface, or the behaviour of another variable from which the response surface can be derived. Regessive designs
The procedures discussed in paragraph 1.5.1 are doomed for failure when the response surface varies highly erratically over the parameter space (20). In such cases it is mandatory to derive the full response surface of the criterion and determine the global maximum. It should be rembered that the criterion value can be calculated from retention data (eq. 5), so that the response surface can be derived from the retention behaviour of the solutes as a function of the parameters. In HPLC the retention varies much more regularly than the optimization criterion. In most HPLC modes the variation of retention can be described with a simple relation (linear or quadratic).
This type of optimization was introduced by Laub and Purnell for a single parameter space in gaschromatography under the name of window diagram (8-12). In its simplest form retention data for all solutes are
16
parameter
VÏQ. 4 Example, of an one-paiametei window diagram for the optimiza-tion of the. reparaoptimiza-tion of, Ux sofatei. Top half, ihowi the retenoptimiza-tion model Uine.at line, ie.gme.nti), bottom half, ihowh the. calculated reiponie surface baled on the. relative resolution product r (eq. 5).
measured at a few selected points spread regularly over the parameter space. For each solute a surface is fitted through the retention data (usually linear or quadratic). The criterion is then calculated over the whole parameter space and the global optimum is immediately located. Fig. ^ shows a window diagram for six imaginary solutes. The top half shows the retention model (for this sample linear line segments) and the bottom half shows the calculated response surface of the relative resolution product r, eq. (5). This optimization scheme has been adapted to other types of chromatography.
Single-parameter systems for which the window diagram technique has been used successfully include variation of stationary phase composition in GC (S), variation of modifier concentration in RP-HPLC (21), vari ation of pH in liquid chromatography (22-24).
An approach for two interrelated variables, based on the window dia gram, is offered by Glajch et. al., under the name: mixture-design statistical technique (25). The system aims at the optimization of the mobile phase composition in RP-HPLC (26) and liquid-solid chromato graphy (LSC) (27,28). Seven experiments are employed to fit experimen tal retention data to a second order polynomial equation with respect to mixtures of three binary mobile phases. For the RP-HPLC mode (26) the three compositions are binary mixtures of H20 with THF, ACN and MeOH, selected using the Snyder Solvent Selectivity Triangle (29). In the LSC mode the binary mixtures consist of n-Hexane (or 1,1,2-trifluoro-trichloroethane) with methyl tert-butyi ether, chloroform and methylene chloride, respectively. As response surface, they construct an overlay of resolution maps (called ORM) of all peaks. This results in an area with a minimal resolution for the least separated peak pair.
Related two-parameter optimizations are also described by Deming et. al. (30). They use the window diagram approach of Laub and Purnell to optimize both pH and concentration of ion-interaction reagents for a nine component mixture.
We conclude that the regressive mapping design offers some important advantages compared to the brute force techniques. Only a minimal nuTiber of preliminary test runs is required to describe the retention behaviour of each solute (usually quadratic, i.e. 3-10 initial runs). The optimal separation condition is rapidly calculated from the response surface based on the retention behaviour. Because the whole parameter space is searched for the highest criterion value, the procedure does not focuss on a local optimum.
However, this procedure has some restrictions. Although, a few measure ments are neccessary to fit the retention model, many calculations are required to obtain the criterion values to form the response surface. To avoid lengthy calculations, the lattice in the parameter space for calculating the response surface is restricted to about 10,000 points, which represents for instance a lattice for two parameters with incre ments of 1% of each parameter. Another restriction is imposed by the need to known the elution order of all solutes in the recorded chromato-grams. If no change in the order occurs, interpolation between the data points is easy, but in this case optimization is usually pointless. When peak reversals occur, elution orders must be established in all chromato-graphic runs. In the literature so far, this is done by injection of each solute separately, which lengthens the procedure.
Another critical point in regressive designs is the equation describing the retention and, hence the predicted optimum relies on the correct shape chosen for the retention surface. If it fails to describe the real retention behaviour, a wrong prediction will result. A more sophisticated design is offered, in which the retention curvature is adapted during the acquisi tion of each new collected data point.
Iterative regressive designs
In this thesis a procedure was developed during the last four years by the author and will be explained in detail in chapter 2, 3, k and appendix I. Essentially, this procedure formulates first a linear relation between retention and optimization parameters on the basis of a few initial runs. After locating the highest response value this predicted optimum is verified. An observed difference between predicted and measured reten tion time is used to refine the retention surface. These iterative refinements are continued until no further improvement can be expec ted. This is in contrast with the regressive mapping design where no feed-back is incorporated between the retention model and measured retentention data of the optimal condition.
The iterative technique has been applied with success to one- and two-parameter optimizations in HPLC. The one-parameter procedure starts with retention data taken at the extreme limits of the concen tration range considered, e.g. modifier concentration (chapter 2 and 3), pH {appendix I) and ion-pair concentration (20). The true retention behaviour is first approximated by linear interpolation (straight lines for In k versus concentration) between these extremes. As long as the procedure continues, each added data point devides the segment in which
it is situated into two new linear segments. This segmentation is continued until no further improvements of the retention behaviour can be expected. Fig. It shows the final step of an optimization in which the retention shows a curved behaviour.
Chapter 4 describes an extention of the iterative regressive design for two interrelated parameters. Instead of linear line segments as in the one-parameter scheme, the retention behaviour is described by flat planes fitted through 3 data points. Thus, the procedure is initiated with a minimum of three initial runs, after which the response surface is calculated based on the constructed retention planes. Each new added data point devides the triangle in which it is situated into two or three new triangles. Again, the procedure is continued until no improvement in the retention behaviour can be expected. The procedure is described for the optimization of the mixing ratio of three binary mixtures, yielding quaternary mobile phase compositions. The retention behaviour is descri bed as a function of the mixing ratio.
Another two-parameter optimization based on the theory of chapter h is described for the optimization of pH and modifier concentration simulte-niously (Appendix I). This optimization starts with 5 initial runs, 4 on the limits of the parameter space and one in the center, yielding an initial number of k- triangles to approximate the retention behaviour. It may be concluded that a rapid selection of the mobile phase compo sition is facilitated by this iterative regressive design. Because no retention model is assumed, this scheme can be applied to any type of parameter as long as the retention behaviour can be approximated by linear segmentation. Also irregularities in retention surfaces are easily recognized during the optimization and can be corrected for. However, the need to recognize the elution order of the solutes in successive chromatograms remains.
1.6 Multi-wavelength detection
In the regressive design methods, one has to keep track of the elution orders of each solute in each recorded chromatogram. This can be done by simply injecting each solute separately at each formulated composi tion. Thus, the regressive design procedures cannot be used for samples containing unknown solutes, unless the detector offers a tool for recog nition of solutes in a series of chromatograms. In HPLC, such a tool is provided by the multi-wavelength detection for UV-VIS absorbing compo nents. This is discussed in chapter 5 and 6. After we analyse the potential of dual-wavelength absorbance ratios for this purpose we draw the conclusion that, although peak-overlap can be detected down to a peak separation of only 0.1 a, recognition was hampered by instrumental limitations and by the influence of the mobile phase composition on the ratio. In chapter 6 we use the complete absorption spectra to establish, in a four-step operator-interactive strategy, the elution order in a series of chromatograms containing 5 and 13 solutes.
References
(1) D.L. Massart, A. Dijkstra, L. Kaufman, Evaluation and optimization of laboratory methods and analytical procedures, Elsevier Scientific Publishing Company, Amsterdam, 1978 Chapter 11.
(2) S.L. Morgan, S.N. Deming, Sep. and Pur. Methods 5, 333 (1976).
(3) W. Wegscheider, E.P. Lankmayr, M. Otto, Anal. Chim. Acta 150, 87 (1983).
(I) H.J.G. Debets, B.L. Bajerna, D.A. Doornbos, Anal. Chim. Acta, 151, 131 (1983).
(5) J.W. Weijlands, C.H.P. Bruins, H.J.G. Debets, B.L. Bajema, D.A. Doornbos, Anal. Chim. Acta 153, 93 C1983)■
(6) R. Kaiser, Gas-chromatographie, Portig, Leibzig, p.33, 1960.
(7) A. B. Christophe, Chromatographia t, 155 (1971). (8) R.J. Laub, J.H. Purnell, J. Chromatogr. 112, 71 (1975). (9) R.J. Laub, J.H. Purnell, Anal. Chem. 48, 799 (1976). (10) R.J. Laub, J.H. Purnell, Anal. Chem. 18, 1720 (1976). (II) R.J. Laub, J.H. Purnell, P.S. Wiliams, J. Chromatogr. -\y\,
219 (1975).
(12) R.J. Laub, A. Pelter, J.H. Purnell, Anal. Chem. 51, 1878 (1979).
(13) J.C. Giddings, Anal. Chem. 32, 1707 (I960).
(11) S.L. Morgan, S.N. Deming, J. Chromatogr. 112, 267 (1975). (15) W.A. Spencer, L.B. Rogers, Anal. Chem. 52, 950 (1980). (16) J.C. Berridge, J. Chromatogr. 214, 1 (1982).
(17) S.N. Deming, S.L. Morgan, Anal. Chem. 15, 278A (1973). (18) J.A. Nelder, R. Mead, Computer J. 7, 308, (1965). (19) J.C. Berridge, Chromatographia 16, 172 (1982). (20) H.A.H. Billiet, A.C.J.H. Drouen, L. de Galan, J.
Chromatogr. 316, 231 (1981)
(21) H. Colin, A. Krstulovio, G. Guiochon, J.P. Bounine, Chromatographia 17, 209 (1983).
(22) S.N. Deming, M.L.H. Turoff, Anal. Chem. 50, 516 (1978). (23) W.P. Price Jr, R. Edens, D.L. Hendrix, S.N. Deming, Anal.
Biochem. 93, 233 (1979).
(21) W.P. Price Jr, S.N. Deming, Anal. Chim. Acta 108, 227 (1979).
(25) J.L. Glajch, J.J. Kirkland, Anal. Chem. 55, 319A (9183).
20
(26) J.L. Glajch, J.J. Kirkland, K.M. Squire, J.M. Minor, J. Chromatogr. 199, 57 (1980).
(27) J.L. Glajch, J.J. Kirkland, L.R. Snyder, J. Chromatogr. 238, 269 (1982).
(28) P.E. Antle, Chromatographia 15, 277 (1982). (29) L.R. Snyder, J. Chrom. Sci. 16, 223 (1978).
(30) B. Sachok, R.C. Kong, S.N. Deming, J. Chromatogr. 199, 317 (1980).
CHAPTER 2
A SIMPLE PROCEDURE FOR THE RAPID OPTIMIZATION OF REVERSED-PHA5E SEPARATIONS WITH TERNARY MOBILE PHASE MIXTURES
Summary
A simple rapid procedure is described f o r estimating optimum composi tions of ternary mobile phase mixtures for the separation of samples by reversed- phase liquid chromatography (RPLC). Retention data in two iso-eluotropic binary mobile phase mixtures (mixtures w i t h equal r e t e n t i o n times) are required to i n i t i a t e the procedure. The logarithm of the capacity f a c t o r is assumed to vary linearly w i t h the composition of iso-eluotropic ternary mixtures formed by mixing the t w o l i m i t i n g binaries. Using the product of resolution f a c t o r s of adjacent peaks as the c r i t e r i o n , an optimum ternary composition is then c a l c u l a t e d . A f t e r a chromatogram has been obtained with the predicted optimum ternary mobile phase, the procedure is repeated until no f u r t h e r improvement can be achieved. Examples of the application of the present procedure are described to i l l u s t r a t e its effectiveness.
Introduction
Liquid chromatographic separations may be pursued for quite divergent purposes. Frequently it is sufficient to determine one or two known com ponents in a complex matrix, in which case specific action can be taken, such as specific detection, a specific stationary phase or pH ad]ustment. It may be desirable to check the purity of a sample, in which case high efficiency columns are preferred to separate, but not necessarily to resolve, completely as many peaks as possible.
Frequently, however, the sample contains a limited number of unknown components and the question then is to resolve as many peaks as possible, preferably all of them. In that case specific action is not possible and a general strategy to achieve this aim is desirable. This problem has been addressed in a previous paper (1) using binary mobile phases and in the present paper it will be extended to ternary mobile phases. It is our intention to develop a strategy with the aim of separating as many components as possible in an unknown sample within an acceptable analysis time.
The stipulation that the sample is unknown removes two degrees of free dom: there is no possibility of relying on specific detection or specific stationary phases. Instead we shall use a general detector and a general purpose, non-polar stationary phase, namely, octadecylsilylated silica. Separation must then be achieved by a judicious choice of the mobile phase composition. With respect to the constituents of the mobile phase our choice is again rather limited. Although non-aqueous RPLC can be applied to the analysis of low-polarity samples (2,3), a generally selec tive mobile phase requires the presence of water in the mobile phase (k). This restricts the choice of so-called organic modifiers to those that are miscibie with water in all proportions. Some possible modifiers must be discarded for practical reasons, such as high viscosity (glycol and glycerol), high volatility (acetone) or high cost (trifluoroethanol). The remaining modifiers are listed in Table I classified according to Snyders group number (5) and their overall solubility parameter (6). From Table I it appears that the number of different organic modifiers that may be considered for RPLC is not more than three or, perhaps, four, it is argued that DMSO is not entirely similar to the other two solvents in Snyder's group III in LC practice (see for example ref. (7)). The three prefered organic modifiers are printed in capitals in Table I: methanol (MeOH), acetonitril (ACN), and tetrahydrofuran (THF). Consequently, only a few binary mixtures of water with one organic modifier need be investigated to determine whether a well-separated chromatogram can be achieved with a binary mobile phase. In principle this still leaves an infinite number of mobile phases as a result of the variable proportion of water and organic modifier.
Our second aim, to achieve the separation within a reasonable analysis time, however, puts a restriction on the water content of the mobile phase. Although a higher water content generaliy improves the resolution (8), it also lengthens the analysis time significantly. In an earlier publication (1), we have proposed the use of a gradient-run to determine the isocratic composition of a binary mobile phase to analyse a sample within a range of capacity factors between 1 and 10. For many samples the results indicate a binary composition with only a very small margin for the water content (such as k0~h5% water with 60-65% MeOH). 2k
Table. I, Oiganic modifczu fioi RPLC
solvent METHANOL (MeOH) Ethanol n-Propanol i-Propanol ACETONITRIL (ACN) TETRAHYDROFURAN (THF) 1,1-Dioxane Dimethylsulphoxide (DMSO) Group II II II II vib III III III 6 ,1/2 -3/2 cal era 15.85 13-65 12.27 12.37 13. 11 9.88 10.65 13-15
Separation may then be achieved through a change in nature of organic modifier (transfer to ACN or THF). Again, however, if we wish to retain an acceptable analysis time (kmax about 10), the transfer from one binary mobile phase to another is straightforward (I). In other words, the number of binary mobile phases of a given eluotropic strength is equal to the number of organic modifiers considered, that is, three or four. Such mixtures that are expected to yield the same capacity factor for a hypothetical "average" solute, have been designated as iso-eluotropic mixtures (1). For real solutes, the transfer wiJl usually cause the peak to shift either forward or backward in the chromatogram. We will refer to these relative changes in retention as specific effects or specifity (1,9), In some cases, complete separation of all sample components may be achieved with an appropriate binary composition derived from the initial run (for RPLC usualy a methanol-water mixture). In some other cases, the transfer to an iso-eluotropic binary mixture with another modifier may lead to the desired result as a consequence of specific effects (1). In many cases, however, the separation will be insufficient in all binary iso-eluotropic mixtures considered. If, however, the incomplete separa tion is due to the overlap of different pairs of solutes in two different binary mixtures, then the use of ternary solvent mixtures may be advantageous. In a previous publication (10), we have studied the reten tion behaviour in two ternary systems, namely, mixtures of MeOH, ACN and water and of MeOH, THF and water. The main conclusion was that ternary mixtures form a smooth transition between two iso-eluotropic binary compositions. It is thus possible to make an infinite number of iso-eluotropic ternary mixtures from the two limiting binary mixtures, whereby the specific effects may be expected to change gradually over the range from one modifier to another.
White the potential of ternary mobile phases for RPLC has been recognized by others (ref. (7)), little has been done to rationalize their selection. Almost always, trial and error procedures have been applied to determine a ternary composition of suitable selectivity. A notable
exception is the rational approach of Glajch et al. (11), which starts with seven initial experiments with well defined mobile phases contai ning one or more of three modifiers (MeOH, ACN, THF). Using either a Simplex optimization or chromatographic insight, Glajch et al. then establish the optimum composition on the basis of the seven initial data sets.
In the present paper we offer a more efficient alternative procedure for the rapid determination of the optimum ternary mobile phase composi tion. It starts with one gradient run and as many isocratic runs as the number of organic modifiers considered - in our case three. Simple chromatographic rules are used to predict a first ternary mixture. The optimum ternary mixture is then determined by an iterative process that usually involves only a few chromatographic runs. The entire procedure can easily be completed within a working day.
Theory
Optimization Criterion
The selection of an optimum mobile phase composition to be described in this section is based on successive decisions on superior separation between various chromatograms. For such a comparison we need an objective criterion to express the quality of a complete chromatogram. For the present purpose the criterion must be a quantitative number that is easily calculated and readily derived from an experimental chromatogram. For the separation of two adjacent peaks in a chroma togram several expressions have been suggested in the literature. Only a few are readily extended to describe the separation achieved in a complete chromatogram (chapter 1).
The present problem is very similar to that of mixed stationary phases in gas chrornatography, for which an optimization scheme has been developed by Laub and Purnell (13-15). These authors use the lowest value of the relative retention between two adjacent peaks as the optimization criterion. Jones and Wellington (16) recently showed that a better and more logical choice would be the lowest value for the resolution (Rs). With either criterion a so-called window diagram is obtained from which the phase system yielding the highest value for the worst resolved peak pair is readily derived. A similar approach is used by Glajch et al. (11) for RPLC in their 'resolution mapping'. Rs values for several adjacent peaks are plotted in phase diagrams to locate areas of ternary or quaternary mobile phase mixtures that provide an Rs value larger than an accepted threshold (such as Rs>1.5) for all peak pairs. Obviously, a criterion that uses only one pair of peaks is inadequate to compare complete chromatograms. It is easy to visualize two quite different chromatograms that show the same value for the lowest resolution. In our opinion, it is not correct to state that the chromato-grapher's eye judges chromatograms on the basis of the least separated pair of peaks. Rather, it ignores all those peak pairs that are amply separated. Also, the optimum mobile phase need not necessarily provide adequate resolution (Rs> 1.5) for all peak pairs. Since the aim of our optimization is the separation of as many components of the sample as possible, we want the peaks to be distributed regularly over the
chroma-togram. However, a constant distance between adjacent peaks ignores the continuously increasing peak width. Therefore we propose as the optimization criterion the product of all Rs values between adjacent peaks in the chromatogram. Now,
(Tr1 - Tr2) (kg - k ^
R s1,2 z (2.a„ + 2.öJ z 2(kg + k. + 2) v
where tr is gross retention time, a standard deviation and k capacity factor of a peak; N is the plate number of the column. Hence, neglecting constant factors, our optimization criterion can be defined as:
n _ 1 n i
( k . - k.
-n—-—
L
. , (k, , + k. + 1=1 ï + i i
where n is the number of peaks in the chromatogram.
Since we use iso-eluo tropic mobile phase mixtures in the proposed strategy, we can expect the analysis time to be roughly constant, in which case the highest value for ÏÏRs corresponds to an even spreading of the peaks over the chromatogram taking into account their increasing band width. The lowest Rs value has a strong influence on the product IIRs, which becomes zero if any one pair of peaks coincides completely (ki+i = ki), however, two chromatograms with an equally low value for the resolution factor of one peak pair can still yield different values for the criterion, since it appreciates also changes in resolution of other pairs.
As the optimization procedure is executed on a given column, the plate number is not retained in the criterion (cf eqs. (1) and (2)). If we assume the plate number to be independent of the solute, any change in it changes all Rs values in the same way and, therefore, improves or worsens the entire chromatogram. The criterion expressed by eq. (2) is a relative number designated to select an optimum mobile phase composi tion, but not necessarily a satisfactory chromatogram. However, if the result of the procedure is a chromatogram in which the sample compo nents are insufficiently or abundantly separated, the plate number may be increased or reduced by, for example, changing the column length or the flow rate. As long as the capacity factors remain unaltered, such changes do not affect the previously established optimum mobile phase composition.
The criterion expressed by eq. (2) is easily calculated for an experimen tal chromatogram. If we want to use the expression, however to predict changes in the criterion with changes in mobile phase composition, two conditions must be met. First, we must be able to recognize a given solute in different chromatograms. In the present discussion this will be
done on the rather primitive basis of peak areas. In the case of complex samples more sophisticated means will be required, such as dual wave length UV detection. Secondly, we must be able to predict variations of capacity factors with a change of organic modifier content in iso-eluo-tropic mixtures. This will be discussed in the next section.
Iso-Eluotropic Binary Mixtures
In a previous paper (1) we have discussed the selection of iso-eluotropic binary mobile phases suitable for the analysis of an unknown sample. Because these chromatograms form the basis of the present optimization scheme, the procedure will be briefly reviewed.
In the first step the sample is subjected to a gradient, usually running from pure water to pure organic modifier, preferably methanol. The resulting chromatogram allows us to decide whether all components in the sample can be eluted isocratically with capacity factors roughly between 1 and 10 and, if so, what the isocratic binary composition of a water-methanol mixture should be. If the result is negative, the optimi zation scheme is disrupted.
The second step is the suggested isocratic run in MeOH-water. Provided we know the number of components in the sample, we can judge whether or not adequate separation has been obtained. In the more usual case that the number of components is unknown or the separation inadequate, the sample is subjected to isocratic runs with organic modifiers different from methanol. In order to keep the overall analysis time (the capacity factors) roughly constant, such alternative isocratic binary mixtures must have the same eluotropic strength. For the organic modifiers presently under consideration (ACN and THF) simple transfer rules have been derived (1). For other organic modifiers such transfer rules can be determined in the same way, or predicted from simplified solubility parameter theory, as described eiswhere (4).
The argument for a change of the organic modifier is that the transfer rules describes an average behaviour of solutes, but take no account of specific effects towards a particular solute. For example, if the result of the gradient run suggests an isocratic elution with 50% MeOH in H20, the transfer rules predict similar retention with 40% ACN in water and 32% THF in water. The capacity factors of individual solutes, however, may show variations of up to a factor of two between these three iso-eluotropic mixtures.
In consequence, isocratic runs with different organic modifiers may lead to substantial changes in overall separation of the sample components or reveal the presence of novel components. The quality criterion expressed by eq. (2) may be used to determine the best chromatogram and visual inspection may show this to be an adequate separation. In an examle of 8 PTH-aminoacids described in ref. (1), incomplete separation was observed with the organic modifiers methanol and acetonitrile, whereas the use of tetrahydrofuran led to an excellent separation. In that case, the optimization may be considered complete and there is no need for a ternary solvent to be used.
It may well occur, however, that the separation is insufficient in all three binary mobile phases. If in all cases the same two (or more) sample components overlap, there is little reason to expect vast impro
28
vements when ternary solvents are used. Here, we shall address the more interesting situation that the insufficient separation in two chroma tograms run with binary solvents is due to the overlap of different solutes. In that case, ternary solvents may well provide superior and adequate separation.
As a result of our initial experiments we have at our disposal the retention data of all components in three iso-eluotropic binary solvents (or more if we elect to use more modifiers). These data form the basis of our first prediction of an optimum ternary solvent. To explain the procedure two steps will be distinguished: the preparation of iso-eluo tropic ternary mobile phases and the construction of what we will refer to as phase selection diagrams.
Iso-eluotropic Ternary Mixtures
An intensive experimental survey of ternary solvent behaviour in RPLC (10) has earlier enabled us to obtain so-called iso-eluotropic diagrams. These diagrams consist of iso-eluotropic lines that connect series of compositions which can be expected to yield approximately constant analysis times for a given sample. The iso-eluotropic diagrams for the systems MeOH-THF-water (a) and MeOH-ACN-water (b) are reproduced in Fig. 1. It is seen from these diagrams, that the iso-eluotropic lines for mixtures with more than 40% water follow the predictions from simplified solubility parameter theory (4) and can be approximated by straight lines. This means that to a good approximation two iso-eluo tropic binary mixtures can be mixed in different proportions to yield a series of iso-eluotropic ternary solvents. This confirms the usefulness of Snyder's polarity-approach (5) as used by Glajch et al. (11) and the
Ftg.7 Ho-tluotiopic diagicm^ hoi two ttinaiij mobile, phait 4£/i-tomb. Thtoittitat [dabhtd] and tx.ptiirre.nt a I [boiid] ii>o-tluo-tiopic. lint>> ait bhoWft contbponding to binaiy mixtaiti, otf rmthanol and uxitti at 10% inttivaU. Thtoitticat iintb ait pitdlcttd by bolubility paiartitti thtoiy and txptiiwzntal lintb itpititnt avtiagt ibo-tlaotiopic corrpof>ition>> hoi laigt numbti otf botixttb. Vlgme itpilnttd iioir\ it&.[10}[with ptimibbion).
simplified solubility parameter model (4) as previously applied to ternary mixtures by us (10) for the formulation of ternary solvent compositions for RPLC Hence, for ternary phase systems other than those in Fig. 1, the composition of iso-eluotropic mixtures can be predicted, as long as the solubility parameters of the three constituents are known.
It should also be realized that the iso-eluotropic lines in Fig. 1 repre sents averages over a large number of solutes. Due to specific effects, the retention of a particular solute can deviate from the average behaviour by as much as a factor of two. Therefore, minor deviations of the linear approximation from the true (experimental) iso-eluotropic line are irrelevant.
Hence, for the purpose of the present work, we decided to approximate the iso-eluotropic lines in Fig. 1 by straight lines, connecting the two binary compositions.
Specific Effects
Although iso-eluotropic mobile phase mixtures are designed to keep retention times constant when a change of solvents is made, changes in retention times will occur for individual solutes. We refer to these changes as specific effects or specificity- It is, of course, thanks to specific effects only, that a chromatographic separation may be altered significantly by applying different iso-eluotropic mobile phase mixtures. Previously (10), we showed that specific effects could be classified towards the functional groups present in the solute molecule. Hence i1; may be expected that specific effects are most likely to be of assis tance in the separation of samples with a wide divergence in chemical compounds, for example, samples that contain esters, alcohols, aldehy des. Hence the separation of a homologous series should not be approa ched using multicomponent mobile phases, while the separation of PTH-aminoacids, in which a variety of functional groups is present, offers a good example to demonstrate the potential of ternary solvents (17). Knowing the retention times for each solute in the sample in various iso-eluotropic binaries, the obvious question now becomes how we can use these binary data to predict retention behaviour in ternary solvents. Jandera et al. (18) have suggested that an empirical relationship, rela ting the logarithm of the capacity factor (In k) linearly to the volume fraction of organic modifier, may serve as a good approximation. Of course, along the linear iso-eluotropic line the volume fractions of the two organic modifiers vary proportionally, inspection of our data on ternary solvent behaviour published before (10) reveals that some devia tions from linearity should be expected - about 10% deviation in terms of k for the methanol-THF-water system and 20% for the rnethanol-ace-tonitrile-water system. However, as long as we have no other retention data available than those obtained with binary mobile phases, the linear interpolation of In k seems to be the most sensible approach.
We therefore decided to use the straight line approximation to estimate capacity factors in ternary systems from experimental data in the two corresponding iso-eluotropic binaries. As soon as more data are availa ble, linear interpolation between adjacent data points can be used to improve the accuracy of the predictions. Linear interpolation is prefered over a polynomial fit to the data points, since experimental errors can cause large errors in the latter case.
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Fie Vaüation ofa In k with composition along Utneti.il iboztuotzopic Und. —) actual itttntion bzhaviowi, [*)ixptümzntai datapolntb deviating ^iom the. actual cuivz deu to zxpziimdntat zitoib, [ ) Uncaz intzipolatlon between the £&& two datapointh, I J Unzai Intctpo-lation between th-ree datapoints and [-.-.-.-.] quadiatic cuivz tkiough tkico. axp<ziim£fttal datapolntb.
The procedure is illustrated in Fig. 2. Here In k is plotted vs compo sition for an imaginary solute along a linear iso-eluotropic line running from 50% methanol (50% water) to 32% THF (68% water). The thick solid line in this figure represents the actual retention behaviour as a function of composition. Three experimental data points are indicated in this figure. These data points include a purposely exaggerated experimen tal error, so that they do not fall on the thick line.
It is seen that a reasonable description is given by the straight line between the two binary data points (thin line) and that linear interpola tion between the three data points (interrupted thin line) descsribes the actual retention behaviour in this (fictious) case within experimental error. The quadratic curve (dashed/dotted line) through the three data points, however, shows a completely different behaviour and introduces errors much larger than the experimental ones. Since it is easy to imagine that similar problems will arise in practice, we prefer to use linear interpolation between individual data points.
