THE SlS I ATE UNIIVERSITY u I xv Irl'lllllilllililillll‘liil l 1293 01417 2112 This is to certify that the dissertation entitled PARAMETRIC MODULATION: A NOVEL COMPUTER-ASSISTED OPTIMIZATION METHOD FOR LIQUID CHROMATOGRAPHY presented by Patrick Hassan Sao Lukulay has been accepted towards fulfillment of the requirements for Doctorate degree in Analytical Chemistry ’Ucdma/ Ynglatiggw Major professor Date Arguojoa H95, M5 U i: an Affirmative Action/liq ual Opportunity Institution 0- 12771 LIERARY Michigan fitate 1 University PLACE N RETURN BOX to romavothb checkout from your board. TO AVOID FINES mum on or baton duo duo. usm AnMimm o A imam o Mon/Emu may PARAMETRIC MODULATION: A NOVEL COMPUTER-ASSISTED OPTIMIZATION METHOD FOR LIQUID CHROMATOGRAPHY BY Patrick Hassan Sao Lukulay A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1 995 Copyright by Patrick Hassan Sac Lukulay 1995 ABSTRACT PARAMETRIC MODULATION: A NOVEL COMPUTER-ASSISTED OPTIMIZATION METHOD FOR LIQUID CHROMATOGRAPHY by Patrick Hassan Sac Lukulay A novel computer-assisted optimization method called parametric modulation has been developed for univariate and multivariate optimization in liquid chromatography. The fundamental strategy of his approach is that chromatographic retention may be accurately predicted it the solute is constrained to undergo interaction independently within each mobile phase, stationary phase, and temperature environment. Thus, the overall solute retention may be calculated by summation of its retention within each individual environment. In this method, the predicted solute retention time is more accurate because it is calculated by summation of the known and measured behavior in discrete environments rather than by numerical regression in unknown or poorly characterized environments. , In this dissertation, the general concept and theory of solute retention and band dispersion under the condition of parametric modulation have been developed and a computer program written to implement the technique. Semi- empirical equations have been developed to predict solute retention and band broadening as a function of the length dimension of stationary phase, mobile phase and temperature environments. Experimental verification of this technique is demonstrated for both univariate and multivariate optimization. In the univariate optimization, mobile phase is optimized for the separation of eight common corticosteroids. The separation was compared to that obtained by conventional premixed solvents and the results were comparable with respect to accuracy, total analysis time, resolution of the least resolved solute pair, and the overall quality of the separation. The verification of the multivariate optimization is demonstrated in two independent studies for the separation of isomeric polynuclear aromatic hydrocarbons (PAH). In the first study, mobile and stationary phases are optimized simultaneously. By measuring solute retention times at various aqueous compositions of methanol and acetonitrile on both Octadecylsilica and B-cyclodextrinsilica stationary phases, the optimum combination of stationary phases and mobile phase was predicted by the computer program. Under these optimum conditions, all solutes were predicted to be baseline resolved . (Resolution . 1.5), except for two solutes with resolution of 0.88. The experimental retention time and peak width for each solute agreed well with the predicted values, with an average error of i3.5% for retention time and 121% for peak width. In the second study, temperature and mobile phase were optimized simultaneously. By measuring retention times of PAHs at 23, 35, 40, and 45°C on polymeric octadecylsilica stationary phase using mobile phases of pure methanol and acetonitrile, the optimum conditions of temperature and mobile phase were predicted. Under these optimum conditions, the experimental chromatogram shows excellent separation of all PAH standards with average relative errors of 0.9% and 156%, in predicted retention times and peak withs, respectively. Based on these results, the parametric modulation approach appears to be a powerful and versatile strategy for the multivariate optimization of chromatographic separations. This dissertation is dedicated to my family for their unconditional love and support ACKNOWLEDGEMENT i would like to express my sincere thanks and gratitude to my advisor, Dr. Victoria L. McGuffin for her guidance and direction through out the course of my graduate study. I would also like to thank the members of my committee; Dr. Stan Crouch, Dr. Ned Jackson, Dr. Carl Lira and for their critical evaluation and helpful feedback about my work This work would not have been fun without the many discussions and words of encouragement from my colleagues in the laboratory including, Dr. Jon Wahl, Dr. Marina Tavares, Daniel Hopkins, Dr. Chris Evans, Dr. Shu Hui Chen, John Jugde, Dr. Peiru Wu, and Chomin Lee. My thanks go especially to Dr. Jon Wahl and Dr. Thomas Atkinson for making the transition to computer optimization less tasking. I wish. to send a special message of thanks to Helen, for her hospitality toward me and my family. To my friends, at Fourah Bay College, especially Maada King, Munir and Leo for the wonderful companionship during rough times. I owe my accomplishments to my lovely wife and son; Adjoa and Aruna, for their encouragement and for being there for me during the course of this-work. To the rest of the Lukulay family, in Potoru and Freetown, especially by parents and Bra sheku, I say Be Seah. ' Finally, I thank the almighty God for his blessing and protection through out the course of my academic and social development. vi CHAPTER 1 TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES FUNDAMENTAL CONSIDERATION OF LIQUID CHROMATOGRAPHIC SEPARATIONS AND THEIR OPTIMIZATION 1.1 Introduction 1.2 Modes of Liquid Chromatography Adsorption Panmon Ion Exchange Size Exclusron 1.3 Partition Theory of Solute Retention Solubility Parameter Model Solvophobic Model Lattice Model 1.4 Optimization Procedures Simultaneous Method Sequential Method Regression Method Theoretical Method 1.5 Univariate O timization Mobile P ase Composition Stationary Phase Composition Temperature Other Parameters 1.6 Multivariate Optimization Multiple Mobile Phase Parameters Mobile and Stationary Phases Mobile Phase Com osition and Temperature Temperature and low Rate 1.7' Conclusion 1.8 References vii xi xiii 11 25 31 38 42 CHAPTER 2 CHAPTER 3 CHAPTER 4 THEORY OF PARAMET RIC MODULATION 2.1 Theoretical Development Theory of Solute Retention Theory of Solute Band Broadening Theory of Solvent Zone Broadening 2.2 Optimization Strategy - 2.3 Conclusion 2.4 References UNIVARIATE OPTIMIZATION OF MOBILE PHASE BY SOLVENT MODULATION 3.1 Introduction 3.2 Theoretical Concept and Optimization Strategies Premixed MDDIIB Phases Solvent Modulation 3.3 Experimental Methods Experimental System Materials and Methods Computer-Assisted Optimization Programs 3.4 Results and Discussion Premixed Solvents Solvent Modulation 3.5 Conclusion 3.6 References MULTIVARIATE OPTIMIZATION OF MOBILE AND STATIONARY PHASES BY SOLVENT MODULATION ON SERIALLY COUPLED COLUMNS 4.1 Introduction 4.2 Theoretical Development 4.3 Experimental Methods Materials and Methods Experimental System Computer-Assisted Optimization Program 4.4 Results and Discussion viii 48 55 62 64 65 65 69 73 95 96 97 97 99 102 CHAPTER 5 CHAPTER 6 CHAPTER 7 4.4 Results and Discussion 4.5 Conclusion 4.6 References MULTIVARIATE OPTIMIZATION OF MOBILE PHASE AND TEMPERATURE BY PARAMETRIC MODULATION 5.1 Introduction 5.2 Theoretical Development 5.3 Experimental Methods Materials and Methods Experimetal System Computer-Assisted Optimization Program 5.4 Results and Discussion 5.5 Conclusion 5.6 References EXPERIMENTAL AND COMPUTER SIMULATION STUDIES OF SOLUTE-SOLUTE INTERACTIONS IN LIQUID CHROMATOGRAPHY 6.1 Introduction 6.2 Experimental Methods Materials and Methods Experimental System Molecular Mechanics and Dynamics Simulations 6.3 Results Experimental Studies Computer Simulation Studies 6.4 Discussion 6.5 Conclusion 6.6 References SUMMARY AND FUTURE WORK 7.1 Summary 7.2 Future Work 102 136 137 138 138 142 145 170 171 172 175 180 208 210 211 214 217 APPENDIX 1 7.3 References COMPUTER PROGRAM A1.1 Program Description A1 .-2 Optimization Program Program name: paramod1.for A1.3 References 219 220 224 236 Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 LIST OF TABLES Capacity Factors for Corticosteroids on Octad Isilica Stationary Phase using Methanol and Acetonitrile obile ases. Comparison of Experimental and Theoretical Capacity Factors under the Predicted Optimum Conditions for Premixed Mobile Phases. Evaluation of the Permutations for a Two-Solvent Modulation Sequence. Comparison of Experimental and Theoretical Capacity Factors under the Predicted Optimum Conditions for Solvent Modulation. Capacity factors (--) for polynuclear aromatic hydrocarbons on octa Isilica stationary phase using aqueous methanol and acetonitrile mobile phases. Capacity factors (ki- ) for polynuclear aromatic hydrocarbons on B-cyclogextrinsilica stationary phase using aqueous methanol and acetonitrile mobile phases. Evaluation of the permutations of a two-column/two—solvent chromatog|raphic system for the separation of polynuclear aromatic Jdrocarbons. The columns are octadecylsilica (ODS) an p-cyclodextrinsilica (BCEB; the solvents are aqueous mixtures of methanol CH H) and acetonitrile CH3CN) as given in Tables 4.1 and 3.2. The analysis time u), minimum resolution ((Ri 1+1)min) and minimum hromatographic Resolution Statistic (CRSmin) function corresponding to the optimum conditions are given. Evaluation of the permutations of a two-column/two-solvent chromatographic system for the separation of polynuclear aromatic J'drocarbons. The columns are octadecylsilica (ODS) an B-cyclodextrinsilica 8-00; the solvents are aqueous mixtures of methanol CH H) and acetonitrile CH30N) as given in Tables 4.1 and .2. The analysis time 1%, minimum resolution ((Rii+1)minl and minimum romatographic Resolution Statistic (CRSmm) function corresponding to the optimum conditions are given, when analysis time is not considered to be a goal of the optimization. Com rison of experimental (EXPT) and theoretical (TH ORY) retention time and peak width under the prgdicted optimum conditions illustrated in Figures 4.7 and .xi 74 82 94 105 106 108 110 134 Table 5.1 Table 5.2 Table 5.3 Table 5.4 Table 6.1 Table 6.2 Table 6.3 Capacity factors (lq- ) for pol nuclear aromatic hyprocarbons on polyme’iic octadecylsi ica stationary phase 0 ained at different temperatures using 100% methanol as mobile phase. Capacity factors (k. ) for pol nuclear aromatic hyprocarbons on polymech octadecylsi ica stationary phase 0 ained at different temperatures using 100% acetonitrile as mobile phase. Evaluation of the selected permutations of a two- temperature/one-solvent and Mo-temperature/two-solvent chromatographic system for the separation of polynuclear aromtic hydrocarbons. The- column is polymeric octadecylsillca; the solvents are 100% methanol and acetonitrile as given in Tables 5.1 and 5.2. The analysis time (9), minimum resolution ((R-M) and minimum hromatogrphic Resolution Stati c 8min) function ‘ correpondlng to the optimum conditions are given. Comparison of experimental (EXPT) and theoretical (Theory) retention time and peak width under the predicted optimum conditions illustrated in Figures 5.7 and 5.8. Comparison of the chromatographic figures of merit for corticosteroids analyzed individually and in mixtures. Experimental conditions as given in Flgure 6.2. Total energy (E) and the components of van der Waals energy (Em), electrostatic energy (E0), and hydrogen bondlng energy (E...) at infinite separation distance (co) and at the optimum separation distance (opt) for the pairwise gogggurations of corticosteroids shown in Figures 6.4A to Total energy (E) and the components of van der Waals energy (Em), electrostatic energy (E0), and hydrogen bondlng energy (Ebb) at infinite separation distance (co) and at the optimum separation distance (opt) for the pairwise gogggurations of corticosteroids shown in Figures 6.5A to xii 152 153 159 169 185 199 207 Figure 1.1 Figure 1.2 Figure 1a Figure 1.4 Figure 2.1 Figure 22 Figure 3.1 Figure 3.2 LIST OF FIGURES The seqpential method of optimization for two parameters using t e simplex approach. The initial sim lex is comprised of vertices 1, 2, and 3. The vertex wit worst response is identified, then replaced by projection through the midpoint of the remaining vertices. The simplex eventually converges at the optimum, which is located in the vicinity of vertex 6 in this example. The re ression method of optimization for one parameter using t 9 window diagram approach. The resolution or other quality criterion is calculated by regression for each solute pair. the shaded window regions represent the limiting resolution of the least-resolved solute pair. The, optimum is located at the highest point in the highest window, which is approximately 0.8 for parameter 1 in this example. ' The re ression method of optimization for two parameters usin t e overlapping resolution mapping approach. The reso ution or other quality criterion is calculated by regression for each solute pair. The regions with resolution less than a threshold value (9.9., Rm = 1.5) are shaded. The optimum is located by overlapping the maps for all solute pairs, which is approximately 0.7 for parameter 1 and 0.6 for parameter 2 in this example. Schematic illustration of the parametric modulation concept for n solvent zones in q serially coupled columns. cS)r:9h5e)matic illustration of solvent zone broadening (1 — g = Effect of solvent zone length on zone purity according to Equations (3.12) to (3.14). (A) Constant particle diameter 5 pm) with varying column lengths of 10 cm (—), 25 cm -), and 100 cm (-). (B) Constant column length 25 cm) and varying particle diameters of 3 pm (—), 5 pm -), and 10 um {-)- Structure of corticosteroids. Critical resolution as a function of the mobile phase composition for aqueous acetonitrile mixtures. Column: 47 x 0.46 cm i.d., packed with octadecylsilica material. Solutes: (1) prednisone, (2) cortisone, (3) prednisolone, (4) hydrocortisone, (5) dehydrocorticosterone, (6) methyl- prednisolone, (7) tetrahydrocortisol, (8) tetrahydrocortisone. xiii 17 21 24 54 57 71 77 Flgure 3.3 Figure 3.4 Figure 3.5 ' Figure 3.6 Figure 3.7 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Critical resolution as a function of the mobile phase composition for aqueous methanol mixtures. Experimental conditions as given in Figure 3.2. . Experimental chromatogram of corticosteroids obtained under the predicted optimum conditions for premixed mobile phases. Column: 47 x 0.46 cm i.d., packed with octadecylsilica material. Mobile phase: 56% methanol, 0.5 mUmin. Detector: UV-visible absorbance detector, 240 nm‘, 0.005 AUFS. Solutes: (1) prednisone, (2) cortisone, ' (3) prednisolone, (4) hydrocortisone, (5,) dehydrocorticosterone, (6) methylprednisolone, tetrahydrocortisol, (8) tetrahydrocortisone. Separation of corticosteroids in individual mobile phases for solvent modulation. Mobile phase: (A) 60% methanol, (B) 35% acetonitrile, 0.5 mUmin. Al other experimental conditions as given in Figure 3.4. Topographic (A) and contour (8) maps of the CRS response surface as a function of the fractional zone lengths‘for 35% acetonitrile and 60% methanol. Experimental chromatogram of corticosteroids obtained under the predicted optimum conditions for solvent modulation. Mobile phase: solvent modulation se uence of 35% acetonitrile and 60% methanol in fractiona zone lengths of 0.4 and 4.8, respectively, 0.5 mUmin. All other experimental conditions as given in Figure 3.4. Structures of isomeric four- and five-ring polynuclear aromatic hydrocarbons. Topographic (A) and contour (B) maps of the CRS response surface as a function 0 the column length (cm) for octadecylsilica and p-cyclodextrinsilica. Topographic (A) and Contour (B)1 maps of the CRS response surface as a function 0 t 9 so vent zone length (cm) for 80% methanol and 50% acetonitrile. Predicted chromatogram for the isomeric PAH with a 75 cm octadecylsilica column usin the optimal solvent modulation sequence of 80% methano and 50% acetonitrile in zones of 167 and 318 cm length, respectively. Solutes: 1 pxrene, (2) triphenylene, (3) benzo[c]phenanthrene, 4 c rysene, (5) benz[a]anthracene, (6) benzo[e]pyrene, 7 pa lane, (8) benzo[alpyrene, (9) dibenz[a,c)anthracene, (107dibenz[a,h]anthracene. - Topographic (A) and contour B) maps of the CRS response surface as a function 0 the column length (cm) for octadecylsilica and B-cyclodextrinsilica. xiv 79 81 86 93 104 112 116 121 123 Figure 4.6 Figure 4.7 Figure 4.8 Figure 5.1 Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 benz[a anthracene, (6) benzo[e]pyrene, Topographic (A) and contour (B). m s of the CRS response surface as a function e t e sovent zone length (cm) for 70% acetonitrile and 80% methanol. Predicted chromatogram for the isomeric PAH with a 75 cm octadecylsilica column using the optimal solvent modulation sequence of 70% acetonitrile and 80% methanol in zones of 203 and 634 cm length, respectively. Solutes as given in Figure 4.4. Experimental chromatogram obtained under the predicted optimum conditions. Column: 200 um ID. x 75 cm fused silica capillary, packed with 5 pm octadecylsilica. Mobile phase: solvent modulation se uence of 70% acetonitrile and 80% methanol in zones 0 203 and 634 cm length, respectively, 0.95 uL/min. Detectors: (A) UV-visible absorbance at 254 nm to indicate solvent modulation seqUence, (B) Laser-induced fluorescence with excitation at 325 nm and emission at 420 nm to indicate solute retention. Salutes: (0 injection solvent, (1) pyrene, 2 triphen lene, (3) benzo c]phenanthrene, (4; chrysene, 5 it 7 perylene, 8 benzo[a]pyrene, (9) dibenz[a,c]anthracene, (1 0 dibenz[a,h]anthracene. Schematic illustration of the parametric modulation concept for the simultaneous optimization of mobile phase and temperature. Structure of polynuclear aromatic hydrocarbons. Effect of temperature on the selectivity of polynuclear aromatic hydrocarbons using 100% methanol as mobile phase. Experimental conditions as given in Figure 5.3. Salutes: 1) Benzoic]phenanthrene, (2) pyrene, 3; phenanthrop enanthrene, (4) benz[a]anthracene, 5 tetrabenzonaphthacene, (6) chrysene, (7) benzo[e]pyrene, (8) perylene, (9) benzo[a]pyrene. Effect of temperature on logarithm of capacity factor for polynuclear aromatic hydrocarbons using 100% methanol as mobile phase at 1.2 uL/min. Column: 200 mm ID. x 75 cm fused silica capillary, acked with 5.5 mm polymeric octadecylsilica. Detector: er-induced fluorescence with excitation at 325 nm and emission at 420 nm. Salutes: ( A ) benzo[e]phenanthrene, ( O ) pyrene, ( Cb phenanthrophenanthrene, ( A ) benz[a]anthracene, ( tetrabenzonaphthacene, ( I ) ch sene, ( O benzo[e]pyrene,( O )perylene,( V benzo[a]pyrene. Effect of temperature on logarithm of capacity factor for polynuclear aromatic hydrocarbons using 100% acetonitrile XV 127 132 134 140 148 151 155 Figure 5.6 Figure 5.7 Figure 5.8 Figure 6.1 Figure 6.2 Figure 6.3 Figure 6.4 Figure 6.5 as mobile phase at 1.2 uL/min. Experimetal conditions and solutes as given in Figure 5.3. , Topographic map (A) and contour map (B) of the CRS response surface as a function of the column length (cm) at temperatures 23° C (Temp. 1) and 40° C (Temp. 2). Predicted chromatogram for the - polynuclear aromatic hydrocarbons on poymeric octadecylSllica column using the optimal conditions of 5 cm of column maintained at 23° C and 85 cm maintained at 40° C; using 100% methanol as the mobile phase. Solutes: (1) benzo[c]phenanthrene, (2) pyrene, (a; phenanthrophenanthrene, (4) benz[a anthracene, 5 tetrabenzonaphthacene, (6) chrysene, ( benzo[e]pyrene, (8) perylene, (9) benzo[a]pyrene. . Experimental chromatogram obtained under the predicted optimum conditions. Column: 200 mm ID. x 75 cm fused Silica capillary, packed with 5.5 mm polymeric octadecylsilica Mobile phase: 100% methanol at 1.2 uL/min. Detector: Laser-induced fluorescence with excitation at 325 nm and emission at 420 nm. Solutes as given in Figure 5.7. Structures of corticosteroids. Chromatograms of corticosteroids analyzed individually and in mixtures. Column: 47 x 0.46 cm i.d., packed with Slim octadecylsilica material. Mobile phase: 35% aqueous acetonitrile; 0.5 mUmin. Detector: UV-VlS absorbance detector (240 nm, 0.005 AUFS). Solutes: (A) cortisone (10-6 M), (B tetrahydrocortisol (10-3 M), (C) mixture of cortisone an tetrahydrocortisol. Chromatograms of corticosteroids analyzed individuall and in mixtures. Solute: (A) methylprednisolone (10-6 M , (B) tetrahydrocortisone 104 M), (C) mixture of methylprednisolone an tetrahydrocortisone. Experimental conditions as given in Figure 2. ‘ The optimum configuration for interaction between A) two cortisone molecules, (B) two tetrahydroco isol mo ecules and (C) cortisone and tetrahydrocortisol. ( )carbon, (0) hydrogen, C ) oxygen, (- - -) nonbonding atoms that meet the define energy and distance constraints for hydrogen bonds according to Equation [6.3]. ' The optimum configuration for interaction between (A) two methylprednisolone molecules, (B) two tetrahydrocortisone molecules, and C) methylarednisolone and tetrahydrocortisone mo ecules. ( ) carbon, ( 0 ) hydrogen, ( O ) oxygen, (- - -) nonbonding atoms that meet xvi 157 162 166 168 177 182 188 193 the defined energy and distance constraints for hydrogen bonds according to Equation [6.3]. 202 Figure A1.1 A flow chart of the computer program used to o timize chromatographic parameters by the parametric mo ulation method. 222 . xvii CHAPTER 1 . . FUNDAMENTAL CONSIDERATION OF LIQUID CHROMATOGRAPHIC SEPARATIONS AND THEIR OPTIMIZATION 1.1 Introduction Chromatography is a separation technique in which molecules or ions are distributed between two immiscible phases, one of which is stationary and the other mobile. Because the molecules or ions are distributed between the two phases to different extents, they migrate at different velocities, thus leading to their separation. The goal of any chromatographic technique is to achieve maximum separation of all the components in their mixture. The degree of separation between any two solutes, known as the resolution, is governed by two independent processes, namely solute retention and band broadening. The functional relationship between resolution and these two processes is given by (1) 2 (tin " ti) (WH1 + WI) Ri,l+1 = [1-11 where t represents the retention time, and w represents the band widths of adjacent solutes i, M. The retention time is controlled mainly by thermodynamic processes, whereas the band width is controlled by kinetic processes such as the rate of diffusion and mass transfer processes. The above resolution equation 2 can be expressed in terms of fundamental parameters of the chromatographic process. It can be shown (1) that for symmetrical Gaussian peaks, Equation [1 .1] can be represented in a useful form as shown below NV2 «91 k' Rs - T . [1.2] a 1+k where N is the number of theoretical plates, k' is the average capacity factor, and a is the selectivity factor. The capacity factor is a thermodynamic measure of retention and is given by ti - to k = — [1.3] to where t. and to are the elution time of a retained and unretained solute, respectively. The selectivity factor is the ratio of capacity factors of two adjacent solutes expressed as km / k; such that a 2 1. Optimization of resolution can generally be achieved by optimizing each term in the resolution equation separately. For a given stationary phase, the selectivity and capacity factor are generally optimized by controlling the type and composition of organic modifier in aqueous solution, respectively. The plate number is be optimized by changing column length, column diameter, and particle size. Of the three terms, selectivity has the greatest influence on resolution (2). Thus, chromatographic optimization is readily achieved by Optimizing selectivity. In order to influence solute selectivity, an understanding of the different modes of liquid chromatography and the model of solute retention are necessary. In this introductory chapter, the various modes of liquid chromatography are presented, followed by a discussion of the various theories 3 which have been proposed to describe the partition model of solute retention. Also, a review of current methods of univariate and multivariate optimization is presented. Then, a novel strategy for univariate and multivariate optimization called parametric modulation is introduced and the advantages over current optimization methods are discussed. 1.2 Modes of Liquid Chromatography Retention in liquid chromatography is thought to occur by different mechanisms. These different mechanisms give rise to the different types of liquid chromatographic methods. Adsorption. Separations by an adsorption mechanism result largely from the attraction of polar functional groups on the solute to discrete adsorption sites on the surface of the stationary phase. The strength of these polar interactions determine solute retention and selectivity. The type of chromatography in which adsorption is the predominant retention mode is known as adsorption chromatography or liquid-solid chromatography. It involves the use of polar stationary phases with relatively nonpolar mobile phases. Silica gel is the most widely used adsorbent (1 ), although alumina, clay, diatomaceous earth, charcoal and other solids have also been used. The adsorption characteristics of silica gel and alumina are due predominantly to surface hydroxyl groups which provide their polar surface. Additionally, these surfaces are known to have one or more layers of adsorbed water molecules. Thus, the activity of the surface depends . not only on the number and orientation of the hydroxyl groups but also on the amount of adsorbed water (3). Partition. Partition is a separation mechanism in which solute molecules are distributed between the two immiscible liquid phases, similar to liquid-liquid extraction. Thus, unlike adsorption, the partition process involves interaction of the solute with the bulk mobile and stationary phases (1,4). The type of. chromatography where partition is the predominant mode at retention is called liquid-liquid chromatography. This is divided into two categories: normal-phase ”chromatography where the stationary phase is more polar than the mobile phase, and reversed-phase chromatography where the mobile phase is more polar than the stationary phase. For normal phase separations, typical stationary phases are xwavortporrwitmlixor orv5 auworrpornyxorhxa, whereas for reversed phase stationary phases separations, typical include octylsilica (08) and octadecylsilica (C18). Retention by a partition mode is highly suited for the separation of homologues and oligomers. Various theories have been proposed to explain the partition mode .of solute retention and these are discussed in Section (1.3). Ion Exchange. The ion-exchange mechanism involves the dynamic interaction of charged solutes with the stationary phase. The stationary phase is characterized by the presence of a weak or strong acid or base bearing exchangeable counter ions. In this mode of chromatography, sample ions in the mobile phase compete with the counter ions for sites (S) on the stationary phase A’ + S+ 3' (-) B+ + S+ A' (Anion exchange) C+ + S' B+ (-) B"’ + S'C" (Cation exchange) Retention in ion exchange is governed by the extent to which the ions compete with B for the charged sites. Thus, selectivity is obtained for ionic species such 5 as organic acids and bases with different ionization constants. An important parameter used to control selectivity is pH of the mobile phase (5,6). In addition to the ion-exchange mechanism, it is argued that retention in ion-exchange chromatography also occurs by partitioning (3) and ion-pairing (7,8) mechanisms. In ion pairing, solutes form an idn-pair with a counter ion and partition into the stationary phase which is often an organic polymeric support. The formation of the ion-pair increases the lipophilic character of the solute and increases its affinity for the stationary phase. The true mechanism is controversial and it is suggested that it may be a convolution of the two mechanisms discussed (8). ‘ Size Exclusion. In size exclusion, the column packing is a porous material with uniform size distribution. Molecules that are too large are excluded from all the pores, where small molecules can penetrate most of the pores. Consequently, small molecules are retained longer than large molecules. Separation by this mode is governed mainly by molecular size (9). Additionally, size exclusion may occur in restricted cavities, such a cyclodextrins and crown ethers. Retention in these cavities is thought to occur by inclusion complexation; thus solute retention is driven mainly by entrapic rather than enthalpic interaction with the restricted cavity. 1.3 Partition Theory of Solute Retention Since solute retention in reversed-phase liquid chromatography is thought ’ to occur mainly by a partition mechanism (1), the various theories advanced for this mode of chromatography are discussed. These theories provide an insight 6 into the important parameters which must be controlled in order to Optimize chromatographic retention and selectivity. The theoretical models can be divided into two categories, classical thermodynamic treatments and statistical thermodynamic treatments. The classical thermodynamic models include the Hildebrand solubility parameter (10,11) and the solvophobic theory of Horvath and co-workers (12,13). The statistical thermodynamic models include the lattice model by Martire, Boehm and Dill (14-16) Solubility Parameter Model. The concept of solubility parameter arises from the regular solution theory of Hildebrand and Scott (10). This theory defines the square root of the cohesive energy density of a pure liquid as the solubility parameter (8), which is characteristic of its polarity. 5 = HEM)"2 [1.4] where E is the cohesive energy required to transfer one mole of the substance from the ideal gas state into its liquid state and vi is the molar volume of the liquid. While the model in its original form considered only dispersive interactions (8d), it has been extended to include other types of molecular interactions such as permanent dipole orientation (5°), dipole induction (5m), and hydrogen bonding interactions of acidic (53) and basic nature (50). By assuming that these interactions are independent of one another, the overall solubility parameter can be expressed as 52 - 5,,2 + 5p2 + 5,,2 . [1.5] 7 where the polar (8p) and the hydrogen bonding (8h) components are defined as 8P2 as 502 +25de [1.6] 8h2 ' 28aab - [17] Thus, the magnitude of the solubility parameter is a measure of the overall polarity or strength of the solvent, with non-polar solvents having low 5 values (around 7 cal"2 cm”) and polar solvents having high 8 values (around 10 cal"2 cm”). The solubility parameter can be extended to mixed solvent systems and is expressed as the arithmetic sum of the individual solubility parameters weighted according to their volume fraction (e) in the mixture. 5.2qq he] According to the regular solution theory, the activity coefficient (1) for solute (i) dissolved in solvent (j) is related to the individual solubility parameters by the following equation RT In 7; . vi (5, - 5,)2 [1.9] where v. is the molar volume of the solute and is presumed identical to the solvent at infinite dilution, R is the gas constant, and T is the absolute temperature. The capacity factor of the solute (k) is related to the distribution constant (K), which in turn is mathematically related to the ratio of solute activity coefficient in the mobile and stationary phases (11). Inki - In [fig] = "Fifi (bl—5MP ‘(51'58)2] - Inn [1.10]. Thus, solute retention is a function of the solubility parameters of the solute (6), the mobile phase (8”), and the stationary phase (85) as well as the volume ratio of the mobile and stationary phases ([3). According to Equation [1.10], separation is most favorable when the difference in solubility parameter of the mobile and stationary phases is greatest. * If the solubility parameter model accounts for all solute-solvent and solvent-solvent interactions, then a quadratic relationship should exist between the logarithm of the capacity factor and the volume fraction of one component in the mobile phase. This quadratic relationship is obtained by combining Equations [1.9] and [1.10] for a simple binary mobile-phase mixture of component 1 and 2, and expressing the result as a function of the volume fraction (¢1) "”9 = A0 + Al‘l’1 + A11¢12 [1-111 where the coefficients are equivalent to V. 2 ] 3 —— — _ 8- 2 _ I Ac RT [ (5l 52) (i 5s) “l3 A - 39—[ (5i-5lz-(8-5zlz - (5 we] 1 - RT 1 I 1 Vi A11 = -—(51"52)2 RT Generally, the predictive value of the model is only qualitative or semi- quantitative at best. This limitation arises from the assumptions inherent in the model (11). First, Equation [1.8] assumes that solvent-solvent interactions are negligible, and thus each solute is assumed to interact independently with each solvent component in a mixture. Secondly, the solute and mobile phase solvent are randomly mixed, thus neglecting clustering which may be significant in polar solvents. More importantly, the model assumes that the stationary phase is isotropic and independent of the mobile phase composition. In view of these assumptions, the solubility parameter model may not always adequately represent molecular interactions in mixed solvents. Solvophobic Theory. The solvophobic theory developed by Horvath and co- workers (12,13) is derived from the solvophobic theory of Sinanoglu (17). Unlike the solubility parameter model, the solvophobic theory emphasizes the importance of only the mobile phase in the retention of solutes. According to this theory, the driving force for solute retention is the repulsion of solute from the more polar mobile phase, rather than attraction between solute and stationary phase. Thus, this theory propounds that solute retention is a passive process rather than an active process. Horvath et al. (12,13) have carried out a detailed theoretical analysis of the factors responsible for solute retention in liquid chromatography using the ~solvophobic theory. They characterize the solvophobic strength of the eluent by its physical properties, among which the surface tension plays the dominant role. This model is successful in predicting trends in retention upon changes in mobile phase composition and upon altering certain molecular properties of the solute, principally its size. However, the 1Ir9 10 model is incomplete because it does not provide a detailed explanation of the dependence of retention on stationary phase variables (18). Cole and Dorsey (18) have observed that stationary phase bonding density plays a significant role in the retention mechanism of reversed-phase chromatography. Consequently, they concluded that the stationary phase must be considered in order to accurately describe the retention process. Lattice Model. ' The lattice model is a statistical thermodynamic model developed by Martire and Boehm todescribe retention and selectivity in liquid chromatography (14-16). Unlike the solvophobic theory, this model emphasizes the role of the mobile phase as well as the stationary phase in solute retention and selectivity. It is based on the competitive equilibrium at the molecular level ' among solute and solvent molecules distributed between non-ideal mobile and stationary phases. In the general form of the model, the stationary phase is considered as a heterogeneous planar surface containing many types of chemically or energetically different adsorption sites. Solute retention is considered in terms of both enthalpic and entropic contributions in the mobile and stationary phases. The partition coefficient (K) may be computed from the chemical potentials of the solute S in the mobile phase mixture of components A and B and in the stationary phase consisting of a bonded alkyl chain C (16). If A and B with relative concentrations ¢A and tie, respectively, are randomly mixed, then the partition coefficient is given as Wt l= [ (xSA-XSB) + ¢eOcse - XSA - m) + cs? he] - I" 13 [1-12] 11 where x is the binary interaction parameter. Thus, both the solubility parameter model and the lattice model predict a quadratic dependence of the logarithm of solute capacity factor on the mobile phase composition. It has been suggested (16) that this quadratic dependence can be expressed in a more convenient form for the purpose of graphing the dependence of retention on mobile phase composition 1 k ' [— ] In [—] = [(158 " XSA " erl + ¢eXAel '[1-131 lite Re where k0 is the value of k when its a 0. By combining Equations [1.10] and [1.12], it can be shown that the solubility parameter model is related to the lattice ’ model through the dependence of the interaction parameters and the solubility parameters (19). .m = constant (5A - 53)2 [1.14] 1.4 Optimization Procedures In chromatography, optimization refers to the process of identifying the experimental conditions required to achieve the most desirable separation in the most time- and cost-effective manner. Because of the numerous experimental parameters that directly and indirectly influence chromatographic retention and band broadening, optimization is a nontrivial process and has been the subject of numerous theoretical and practical studies (1 ,2,20,21). In this section, a review of the present strategies for univariate optimization and their logical extension to 12 multivariate optimization are presented. The existing methods are described, together with recent applications, and their advantages and limitations are evaluated. ‘ in order to determine unequivocally the most appropriate experimental conditions, a systematic approach must be adopted in the optimization process. This process consists of three distinct steps: selection of the quality criterion, selection of the experimental parameters, and selection of the method to identify the most optimal value or level of each parameter. These steps are discussed sequentially below. The Quality Criteria. The first step involves the formulation of a criterion which is a quantitative measure of the quality of the separation. Because this criterion is used to guide the decision-making process, it must inherently define and rank the goals of the optimization. One of the most important goals in ' chromatography is to maximize the resolution between adjacent solutes (Rim), which may be calculated as shown in Equation [1 .2]. Several quality criteria have been developed based on resolution alone, ranging from simple functions such as the limiting resolution of the least-resolved solute pair (22-24) to more complex functions such as the sum or product of the resolution for all adjacent solute pairs, without or with weighting factors (25,26). In addition to resolution, other goals have been incorporated or applied in either a simultaneous or sequential manner (20). These more complex quality criteria include the chromatographic response function (CRF) (27) and the chromatographic optimization function (COF) (28), both of which optimize resolution and analysis time. The CEF function developed by Vajda and Leisztner (29) optimizes the resolution and the number of solutes, whereas the chromatographic resolution statistic (CRS) of Schlabach and Excoffier (30) attempts to optimize resolution of 13 all solutes, provide uniform spacing, and minimize analysis time simultaneously. Each of these quality criteria has inherent advantages and limitations which should be considered in view of the specific application and its goals (2,20,31- 34). The Parameter Space. Next, the experimental parameters that have the greatest influence on the quality criterion are identified. These parameters may have a limited number of discrete or finite values, or they may be continuously variable within a specified range. Among the examples of chromatographic parameters that are discrete in nature are the type of mobile and stationary phases. In addition, physical dimensions such as the particle size, column diameter, and column length may be discrete if restricted to columns that are commercially available. Parameters that are continuously variable in nature include the composition of the mobile and stationary phases, as well as the temperature, pressure, and flow rate. The proper selection of these parameters is crucial, since the success of any optimization strategy is dependent exclusively upon them. It is, therefore, assumed that all experimemal factors not explicitly chosen as parameters will be held constant during the optimization. Depending upon the number and nature of the parameters, the relationship with the quality criterion can be represented by individual point(s), by line(s) in two-dimensional space, by plane(s) in three-dimensional space, or by higher-order surlace(s) in N-dimensional space. For simplicity in the subsequent discussion, the relationship between the parameter(s) and the quality criterion will be referred to as the response surface, regardless of its continuity and dimensionality. 14 The Optimization Method. Once the quality criterion and the experimental parameters are chosen, a logical procedure must be used to locate the optimum value of the criterion on the response surface. Because the response surface may be highly complex, with many local minima and maxima, this procedure must be sufficiently robust and reliable to locate the primary or global optimum rather than secondary or local optima. The procedures that have been adopted to locate the optimum can be classified into four categories, as described below. Simultaneous Method Simultaneous methods are those in which all of the parameters to be varied, as well as their levels, are chosen a priori. Experimental data: are acquired at each level for each of the parameters chosen. The optimum is then identified by the experimental conditions that yield the best value of the quality criterion. The simultaneous method is also referred to as a grid search or mapping method, since the parameter space can be likened to a grid or framework of experimental conditions, generally spaced in uniform increments. The number of experiments to be performed depends on the number of the parameters and their levels (2); for n parameters at q levels, the number of experiments is q". In order to describe the response surface completely and accurately, each parameter must be examined over a wide range in small increments of the level. However, this may require a prohibitively large number of experiments. In order to reduce the number of experiments, constrained parameters may be chosen to reduce the degrees of freedom of the parameter space. A constrained parameter is one in which the sum of the levels is constrained to a constant value; for example, the sum of the volume fractions of the mobile phase solvents is unity. Another approach to reduce the number of experiments is through experimental design, common examples of which are the factorial and) star design methods (35,36). 15 Because of the inherent nature of simultaneous methods, they are most suitable for parameters that are discrete and independent. An advantage of these methods is that they do not require or rely on a theoretical or mathematical model to locate the optimum. These methods can be used to elucidate the complete response surface if the parameter levels are varied over a wide range in small increments. Thus, it is possible in principle to identify the global optimum in the presence of other local optima. The most significant disadvantage is that the number of experiments increases rapidly with the number of parameters and their levels. Sequential Method In the sequential method, a few initial experiments are performed at selected levels of the parameter(s) and the results are used to ‘ direct and guide subsequent experiments. The success of this method is highly dependent upon the type of quality criterion and the type of search routine used to locate the optimum. A variety of search routines have been developed (36,37), the most common of which is the simplex routine (38-43). A simplex is a geometric figure that consist of N+1 vertices in Ndimensional space; for example, a two-dimensional simplex is a triangle. To illustrate the search procedure, consider the response surface shown as a contour diagram in Figure 1.1 The vertices 1, 2, and 3 define the initial simplex and represent the initial levels of the parameters at which an experiment is performed. The results of these experiments are then compared by means of the quality criterion; in this example, vertex 2 has the worst response. Consequently, the simplex routine replaces this vertex with one obtained by reflection through the midpoint of the line connecting vertices 1 and 3. The simplex is now described by vertices 1, 3, and 4. An experiment is performed at the new vertex 4, and the quality criterion is compared to that obtained previously at vertices 1 and 3. Vertex 1 now has 16 Figure 1.1 The sequential method of aptimization for two parameters using the simplex approach. he initial simplex is comprised of vertices 1, 2, and 3. The vertex with worst response is identified, then replaced b projection through the midpoint of the remaining vertices. T e simplex eventually converges at the optimum, which is located in the vicinity of vertex 6 in this example. PARAMETER 2 17 Figure 1.1 PARAMETER 1 18 the worst response, and reflection in the opposite direction leads to vertex 5. The Objective of this iterative procedure is to direct successive experiments toward more favorable regions of the response surface. This procedure is continued within the defined boundary conditions of the parameters until the optimum is obtained in the vicinity of vertex 6. In the sequential simplex method originally proposed by Spendley et al. (38), the vertices are equally spaced to create a simplex that is an equilateral polyhedron. Such equilateral simplexes may not readily converge, in some cases circling around the Optimum or becoming stranded on a surface ridge. These limitations have been largely overcome by the modified procedure of Nelder and Mead (39), which allows for expansion of the simplex in a favorable direction and contraction in an unfavorable direction during the search procedure. The simplex and related sequential methods are most suitable for parameters that are continuous, since an experiment must be performed at each new vertex that is calculated. An important advantage is that these methods are able to Optimize several parameters simultaneously with no prior knowledge about their interaction. Additionally, these methods do not rely on any theoretical or mathematical model to identify the Optimum conditions. However, the sequential procedures have some disadvantages, foremost among which is that an indeterminate number of experiments are required to locate the optimum. Moreover, for response surfaces consisting of many local maxima and minima, a local optimum may be erroneously identified as the global optimum. This problem may be circumvented by repeating the Optimization procedure with the initial simplex of various sizes and in various locations on the response surface. A convergence Of the simplex to the same Optimum indicates that a global Optimum has been found. Additionally, because the quality criterion is only evaluated at selected points, a limited insight is provided about the response 19 surface. Despite these potential drawbacks, sequential optimization procedures such as the simplex method are versatile with wide applicability in chromatography (40-43). Regression [Method RegressiOn methods, othenrvise known as interpretive methods, rely on the use of a semi-empirical mathematical model to describe the quality criterion as a function Of the experimental parameters. In these techniques, a chromatographic property, such as capacity factor, plate number, etc., that is related to the quality criterion is measured at certain levels Of the parameters. The resulting data are analyzed by the least-squares method in order to determine the best value for the coefficient(s) in the model equation. The number of experiments depends on the mathematical model; a linear model requires a minimum of two experiments, a quadratic model requires three experiments, etc. However, it is generally desirable to acquire data at additional levels Of the parameter in excess of the minimum requirements in order to allow for greater statistical confidence in the results of the regression. Once the regression coefficients have been determined, the equation may be used to calculate the chromatographic property and, hence, the quality criterion at any intermediate level of the parameter. Numerical or graphical methods may then be used to represent the response surface and to determine the most optimal value of the parameter. A well-known method of representation for a single parameter is the window diagram approach, originally developed by Laub and Purnell (44-46). As shown in Figure 1.2, the quality criterion (usually the resolution for each solute pair) is graphed as a function (usually linear) Of the parameter. The regions of the graph beneath the limiting resolution for the least- resolved solute pair are shaded to create “windows”. The Optimum value of the parameter may then be identified from the highest point in the highest window, Figure 1.2 20 The regression method of Optimization for one parameter using the window diagram approach. The resolution or other-quality criterion is calculated by regression for each solute pair. The shaded window regions re resent the limiting resolution of the least-resolved solute pair. he optimum is located at the highest point in the highest window, which is approximately 0.8 for parameter 1 in this example. RESOLUTION q 21 Figure 1.2 l l l l d and ‘ uni-I 05> m3 ' . K . \~':~‘- \ \V": ‘ - teas. .. ‘9) 1“- {iv ’\\\\\1¢\‘\ \\ .\\\‘(\ ‘( \\‘ \23‘1?.~\\\\\'\ v) \‘\"\ \> \- .‘ . é ' A" Sfifi‘? .3‘ .\ -. ‘0’ flux ?" “(>34 \. "’.\ I Ave. .~.\\ 3'. . AVA Mum\\\\:fiv.\%¢\ AA\.\\\ PARAMETER 1 22 which represents the maximum value of the quality criterion. This approach may be extended to two parameters by using the overlapping resolution mapping (ORM) technique described by Glajch and Kirkland-(47). In this case, the resolution of each solute pair is represented by a surface, rather than a line, and the regions of intersection must be determined in order to create the three- dimensional “windows“. A simpler graphical approach, illustrated in Figure 1.3, is tO shade those regions with resolution less than a selected threshold value (e.g., Rim .. 1.5) for each! solute pair and then overlap these maps to identify the optimum value Of the two parameters. The extension of these graphical techniques to three or more parameters remains a challenging problem (48). When compared with the simultaneous and sequential methods, the regression techniques require fewer experiments to describe the complete response surface. However, the successful application of this method is greatly reliant on the accuracy of the mathematical model. For this reason, regression methods are most appropriate for continuous parameters with a well-defined and verifiable relationship to the chromatographic property or quality criterion, such as mobile phase composition and temperature. Theoretical Method Theoretical methods, like regression methods, use a mathematical equation to describe the effect of the parameter on a chromatographic property such as capacity factor, plate number, etc. The important distinction is that the equation is theoretically derived and requires no preliminary experimental data for implementation. This approach is most appropriate for discrete or continuous parameters that have an invariant and theoretically predictable effect, such as flow rate, column diameter, and column length. The optimal value of the parameter may be determined by any Of the numerical or graphical methods discussed previously. 23 Figure 1.3 The regression method of optimization for two parameters using the overlapping resolution mapping approach. The resolution or Other uality criterion is calculated by regression for each solute pair. he regions with resolution less than a threshold value (9. 9., RM = 1.5) are shaded. The optimum is located by . overlapping the maps for all solute pairs, which is approximately 0. 7 for parameter 1 and 0. 6 for parameter 2 in this example. PARAMETER 2 24 Figure 1.3 \\\i\ .\-\~.\ \‘R PARAMETER 1 25 1.5 Univariate Optimization Univariate Optimization involves the optimization of a single parameter or Of multiple parameters in a wholly independent manner. This is achieved by varying a given parameter to investigate its influence on the quality criterion, while all other parameters are held constant. When an optimum value for that parameter has been established, its value is held constant while the next parameter is varied to determine its own optimum value. This approach is relatively simple and is adequate for parameters that do not interact with one another, but may be problematic for mutually dependent parameters. Univariate optimization can be accomplished by any Of the procedures discussed previously. In this section, the univariate Optimization of important chromatographic parameters will be reviewed. Mobile Phase Composition. The mobile phase composition is, by far, the most common parameter used for Optimization in liquid chromatography (1). As such, it has already been the subject of several excellent books (1,2,20) and comprehensive review papers (26,4951). Mobile phase optimization will generally include the type of solvent or other modifier, [which influences the selectivity, as well as its concentration, which influences the strength. Because of the discrete nature of solvent or modifier type, Optimization is most Often performed by the simultaneous method. This approach has been utilize to select from among several organic solvents, buffers, ion pair agents, etc. (50.52.53). However, the simultaneous approach is time-consuming and ineffective to optimize solvent or modifier concentration, which is most Often performed by regression methods (1 ). 25 1 .5 Univariate Optimization Univariate Optimization involves the optimization of a single parameter or Of multiple parameters in a wholly independent manner. This is achieved by varying a given parameter to investigate its influence on the quality criterion, while all other parameters are held constant. When an optimum value for that parameter has been established, its value is held constant while the next parameter is varied to determine its own optimum value. This approach is relatively simple and is adequate for parameters that do not interact with one another, but may be problematic for mutually dependent parameters. Univariate optimization can be accomplished by any of the procedures discussed previously. In this section, the univariate Optimization of important chromatographic parameters will be reviewed. Mobile Phase Composition. The mobile phase composition is, by far, the most common parameter used for Optimization in liquid chromatography (1 ). As such, it has already been the subject of several excellent books (1 ,2,20) and comprehensive review papers (26,49-51). Mobile phase Optimization will generally include the type of solvent or other modifier, (which influences the selectivity, as well as its concentration, which influences the strength. Because of the discrete nature of solvent or modifier type, optimization is most often performed by the simultaneous method. This approach has been utilize to select from among several organic solvents, buffers, ion pair agents, etc. (50,52,53). However, the simultaneous approach is time-consuming and ineffective to optimize solvent or modifier concentration, which is most Often performed by regression methods (1 ). 26 For nonnal- and reversed-phase chromatography, regression models have been developed to describe solute retention as a function Of mobile phase composition. By extension of classical thermodynamics to non-interacting binary solvent mixtures, the capacity factor can be described as a quadratic function of the volume fraction 4», of each component "I" " AO 4' A191 + All $12 'I' Az‘I’z + A224’:2 1' A12 ¢1¢2 “-151 where A, are the regression coefficients. If the volume fraction is a constrained parameter (e.g., o, z 1 - in), this expression can be simplified as follows (54-56) Ink 8 A0 ‘1' A1 $1 + A11¢12 [1.16] In order to reduce the number of experiments required for-optimization, this quadratic model is frequently approximated by a linear equation (50,57-63) Ink = A0 + A,¢, ' ' [1.17] With this simplification, retention measurements at two solvent compositions are sufficient to describe the complete response surface. The linear approximation is accurate over a more limited range of solvent compositions than the quadratic model, and should not be extrapolated beyond the range of available experimental data In addition, deviations from both linear and quadratic behavior may arise due to nonideal solute-solvent and solvent-solvent interactions, especially in polar solvents (64,65). By extension of classical thermodynamics to non-interacting ternary solvent mixtures, the capacity factor can be described as a quadratic function of the volume fraction of each component 27 "I" 'Ao +Ai¢l 'I'AlrlIr2 +A2¢2 +A22¢22 +Aa¢3 +Aea¢32 4' A12 ¢1¢2 4’ A13¢1¢3 1‘ A23¢2¢3 [1-18] It the volume fraction is a constrained parameter (e.g., O3 = 1 - e, - o2), this expression can be simplified as follows "Ik =80 1' A1¢i 1' A11¢12 'I' A21’: 'I' A221k2 4' [A12 ¢l¢2 [L19] Alternatively, the semi-empirical expression proposed by Wayland er al. (66) can be Obtained by neglect Of the second-order terms in Equation [1.18] and inclusion Of a term for first-order ternary solvent interactions INK ' A0 1’ Al¢t 1’ A24’2 1' A3¢3 + A12¢i¢2 + Ara¢i¢a 1' A23¢2¢3 1' Aizalir ¢2¢a [1'20] This latter polynomial expression requires retention measurements at seven solvent compositions to describe the complete response surface. The equations cited above have been widely and successfully applied for univariate Optimization of mobile phase composition in normal- and reversed- phase chromatography, as well as in ion exchange, ion pair, and other chromatographic methods (1 ,2,20,26,49-51). Although there have been many noteworthy accomplishments, the most exemplary are those by Snyder, Dolan, ef al. (50,59-63) for binary mobile phases and by Glajch, Kirkland, et al. (28,47) for ternary and quaternary mobile phases. In the latter work, a mixture-design statistical method is applied which combines the simultaneous method with regression techniques to optimize both selectivity and strength of the mobile phase. The mobile phases are chosen such that they exhibit the greatest possible difference in chemical interactions, including proton donor (methanol), ' proton acceptor (tetrahydrofuran), and dipole (acetonitrile) interactions (1). A minimum Of seven preliminary experiments are performed, and the quality 28 criterion is evaluated by using the overlapping resolution mapping technique in order to locate the Optimum ternary or quaternary mobile phase. In addition to isocratic optimization, gradient mobile phases have also been optimized by a variety of methods (51 ,60-63,67-75). These methods involve optimization of the solvent selectivity, as discussed above, the strength of the initial and final solvent concentration, as well as the shape and duration Of the gradient. The sequential simplex approach has been used by Watson and Carr (27), whereas regression techniques have been favored by Jandera, Churacek, et al. (69-73) and by Snyder, Dolan, et al. (51.60-63.68). Kirkland and Glajch (74) extended the mixture-design statistical method described above to gradient elution. Experimental data were acquired using a series of seven gradients with isO-eluotropic mobile phases of aqueous methanol, tetrahydrofuran, and acetonitrile. From these results, the quality criterion was evaluated by using the overlapping resolution mapping technique. Stationary Phase Composition. In many cases, a single stationary phase is sufficient to achieve the desired chromatographic separation. Because it is a discrete parameter, the type Of stationary phase is most Often optimized by the simultaneous method (1.2.20). This approach is clearly illustrated in the work of Glajch et al. (52). who examined octyl—, cyanO-, and phenethyl-modified silica stationary phases for the separation of phenylthiohydantoin derivatives of twenty common amino acids. They applied the mixture-design statistical method to optimize the mobile phase composition separately on each column. By comparison of the three overlapping resolution maps, the cyano-modified silica was identified as the optimal stationary phase. When a single stationary phase is not sufficient. mixed or multimodal stationary phases have been used to combine dissimilar but compatible retention 29 mechanisms. Mixed phases that have been successfully used include reversed- phase with size exclusion (76-79), cation or anion exchange with size exclusion (79), reversed-phase with cation or anion exchange (80-84), and cation with anion exchange (83-85). These stationary phases may be combined in several ways: by incorporating or chemically bonding the stationary phases on the same particles (chemically mixed support), by incorporating the phases on different particles that are mixed within a single column (physically mixed support), or by incorporating the phases on different particles that are contained in separate columns connected in series (84). . Isaaq et al. (78) have investigated solute retention as a function of stationary phase composition for columns containing B-cyclodextrin- and octadecyl-modifled silica. They observed that retention on chemically or physically mixed supports may not be predictable as a linear function Of the I stationary phase composition. This nonlinear additivity of solute retention may arise as a consequence of interactions between the stationary phases. On the other hand. retention on serially coupled columns is generally additive and predictable because the stationary phases are physically separated. Similar results were reported by Glajch et al. (52) for columns containing cyano- and phenethyl-modified silica. These investigations suggest that the regression technique may be used to optimize stationary phase composition for serially coupled columns. Temperature. In comparison with mobile and stationary phase composition, temperature is less frequently used as an optimization parameter in liquid chromatography. The practical temperature range for most common solvents is somewhat limited, being constrained by the boiling point, freezing point. and viscosity. Nevertheless. temperature changes within this range have been 30 reported to have a significant influence on solute retention and selectivity as well as plate number (86-90). Temperature Optimization is commonly achieved by the simultaneous method. By comparing the quality criterion at several different temperatures, the Optimum temperature may be readily identified (91-92). A more systematic approach has been adopted by using a regression method based on the classical thermodynamic relationship Of van't Hoff (93.94). If the molar enthalpy and molar entropy are invariant with the absolute temperature (T ). then . A, T Ink = A0 + [1.21] By measuring retention at two or more temperatures. linear regression can be _ used to predict the capacity factor and, hence, the quality criterion at any other temperature by means of Equation [1.21]. The optimum temperature may be identified by the window diagram approach (95) or other graphical or numerical method. However, marked deviations from linearity have been observed for "graphs of In k versu‘s 1/T for many solutes (18.96.97), thus reducing the accuracy of these optimizati-On methods. ‘ In addition to isothermal optimization, temperature programming has also been implemented in liquid chromatography. This is Often accomplished by using microbore or capillary columns in order to minimize the effect of radial temperature gradients (98). Many workers have demonstrated that, under well- controlled experimental conditions, temperature programming may be comparable or advantageous to solvent programming (99-101). Although several applications of temperature programming in liquid chromatography have been reported, theoretical models that enable the prediction and optimization Of solute retention are rare (101). The optimization is particularly complicated because .._ a u. '* 31 temperature influences both solute retention and band broadening during the separation, Often in a manner that is not accurately predictable. Other Parameters. In addition to the parameters discussed above, optimization techniques have also been developed for such physical parameters as particle size, column diameter, column length, flow rate, etc. Snyder. Dolan, et al. (50.51.59—63) have applied regression methods for particle size, and theoretical methods for column dimensions and flow rate. These methods are especially successful when combined with univariate Optimization of the mobile phase composition in the DryLab l (isocratic) and DryLab G (gradient) programs (102). 1.6 Multivariate Optimization If the Optimum value of one parameter depends on and varies with the value Of another. those parameters are mutually interacting. The optimization of interacting parameters is best achieved by concurrent, rather than independent. variation. All of the optimization procedures described previously, individually or in combination, may be used to implement multivariate optimization. However, the most widely used and successful methods for univariate Optimization, simultaneous and regression methods, can become cumbersome and ineffective because the number of characterizing experiments increases rapidly with the number of parameters. In this section, the extension of these methods 10 multivariate Optimization of important chromatographic parameters will be reviewed. 32 Multiple Mobile Phase Parameters. Two or more mobile phase parameters that are not constrained may be optimized by multivariate methods. A combination Of the simultaneous and regression methods has been applied successfully for this purpose by Deming at al. (103,104). A factorial design with two parameters at three levels (32) was used to Optimize the concentrations of an organic solvent (¢,) and an ion-pairing agent (9) in reversed-phase liquid chromatography. The resulting retention data were fitted by nonlinear least- squares to a semi-empirical relationship based on the combination Of Equation [1.17] with a Freundlich isotherm k ' A0 '1’ A4 9xP(’ A1 M) (1 1’ Azil’zm'k’) I122] The selectivity factor Of the least-resolved solute pair was used as the quality criterion to construct the response surface, from which the Optimum mobile phase composition was determined. Under the optimum conditions, the experimental retention times for five substituted anilines were in close agreement with theoretically predicted values, having an average relative error of 12.3%. ‘ Otto and Wegscheider (105) also combined the simultaneous and regression techniques to optimize buffer type, pH, ionic strength, and organic solvent composition for the separation of simple organic acids, amino acids, and dipeptides. In preliminary experiments, the affects Of buffer type and pH were examined simultaneously to select from among phosphate, citrate, acetate, tartrate, and glycine buffers (47 total experiments). A semi-empirical regression equation was used to relate the capacity factor to the hydrogen ion concentration and the dissociation constants of the buffer. Using the selectivity factor and the peak asymmetry as quality criteria, the citrate buffer was determined to be the most promising alternative. Next, a fractional factorial design (32 - 21-1) was used 33 with a more complex regression equation to model retention as a function Of the hydrogen ion concentration, the dissociation constants Of the buffer, the ionic strength. and the solvent composition (18 total experiments). By using the selectivity Of the least-resolved pair as the quality criterion, a response surface was generated to identify the Optimum composition of the citrate buffer. The experimentally observed separation was in good agreement with the theoretical prediction, having an average relative error in capacity factor Of :l:3.8%. It is also noteworthy that the optimum composition, as well as the overall quality of the separation, predicted for the citrate buffer differed significantly from that for the phosphate buffer. Mobile and Stationery Phases. Mobile and stationary phase compositions have been Optimized concurrently by Glajch et al. (52) for the separation Of phenylthiohydantoin derivatives Of twenty common amino acids. In an extension of the univariate study described previously, the solute capacity factors were measured on three different columns containing octyl-, cyanO-, and phenethyl- modified silica stationary phases using seven different ternary solvent compositions Of methanol, acetonitrile, and tetrahydrofuran (21 total experiments). By using the regression model, a response surface was generated as a function of solvent composition for each solute on each column. In order to establish the Optimum composition Of the mobile and stationary phases, the solute retention on a mixed bed was assumed to be a linear combination of the retention on each individual stationary phase at the same mobile phase composition. Thus, the quality criterion was calculated for all possible combinations, from which the optimum stationary phase composition was predicted to be 28% cyano and 72% phenethyl and the optimum mobile phase composition was 9.2% methanol, 10.6% acetonitrile, -13% tetrahydrofuran, and 34 67.2% water with phosphoric acid at pH 2.1. This prediction was validated by using a single column containing a physically mixed bed as well as two serially coupled columns. A good separation Of the amino acids was achieved that was similar. but not identical. in the two systems. Unfortunately. it was not possible to assess the predictive accuracy of these methods from the results presented by Glajch at al. (52). Mobile Phase Composition and Temperature. The optimization of mobile phase composition and temperature has been accomplished by using a combination of the simultaneous and regression methods (101 ,106-108). Diasio and Wilbum (106) measured solute retention as a function of mobile phase composition and analyzed the results by linear regression. The optimum mobile phase composition was then used to measure solute retention at various temperatures, and the temperature that yielded the best quality criterion was identified as the optimum. JII‘II‘IO, and Kuwajima (108) have used a similar approach to optimize mobile phase composition and temperature separately. Their method is distinctive because it attempts to generate regression equations that are based on physicochemical properties of the solutes and, hence, can potentially be applied to predict retention for solutes in the absence of experimental data By using polynuclear aromatic hydrocarbons (PAH) as model solutes. the relevant size and shape parameters were calculated in various mobile phases and were incorporated as physicochemical constants into their model equation. The regression equations used linear relationships Of the logarithm Of capacity factor with the mobile phase composition (Equation [1.17]) and with the inverse temperature (Equation 1.21). By using this approach, the average relative error 34 67.2% water with phosphoric acid at pH 2.1. This prediction was validated by using a single column containing a physically mixed bed as well as two serially coupled columns. A good separation Of the amino acids‘was achieved that was similar, but not identical, in the two systems. Unfortunately, it was not possible to assess the predictive accuracy Of these methods from the results presented by Glajch et al. (52). Mobile Phase Composition and Temperature. The optimization of mobile phase composition and temperature has been accomplished by using a combination Of the simultaneous and regression methods (101.106-108). Diasio and Wilbum (106) measured solute retention as a function of mobile phase composition and analyzed the results by linear regression. The optimum mobile phase composition was then used to measure solute retention at various ‘ temperatures, and the temperature that yielded the best quality criterion was identified as the optimum. ‘ Jinno, and Kuwajima (108) have used a similar approach to optimize mobile phase composition and temperature separately. Their method is distinctive because it attempts to generate regression equations that are based on physicochemical properties Of the solutes and, hence, can potentially be applied to predict retention for solutes in the absence of experimental data By using polynuclear aromatic hydrocarbons (PAH) as model solutes, the relevant size and shape parameters were calculated in various mobile phases and were incorporated as physicochemical constants into their model equation. The regression equations used linear relationships of the logarithm Of capacity factor with the mobile phase composition (Equation [1.17]) and with the inverse temperature (Equation 1.21). By using this approach. the average relative error 35 in the predicted capacity factor was :l:4.4% for sixteen polynuclear aromatic hydrocarbons (PAH) in a standard reference material. In the factorial design of YOO et al. (101), solute retention was measured as a function of mobile phase composition at four different temperatures (20 total experiments), and then analyzed by linear regression. By using the resolution Of the least-resolved solute pair as the quality criterion, the optimum mobile phase composition was identified by the window diagram approach at each temperature. These results were then used to Optimize the separation Of fourteen saturated and unsaturated fatty acids under temperature programming conditions. The Optimum separation conditions enabled baseline resolution (RM 2 1.5) of the least-resolved solute pair, with reproducibility of retention time better than 11.1%. Because these simple approaches do not account for interaction between the parameters. others have adopted the use Of regression methods to describe retention as a combined function of mobile phase composition and temperature. Melander and Horvath (109) have proposed the following analytical expression ' * A.‘ TO ' Ink .in. + T“ +A2¢[1- T J [1.23] where Tc is the compensation temperature, which was examined as a constant (625 °K) and an adjustable coefficient. This expression was later extended by Melander at al. (110) in order to improve the accuracy and precision of regression results V A T. Ink = A0 + T‘ + Azo (1 - T )4. Age [1.24] 36 Inherent in these equations is the assumption of linear relationships between the logarithm of capacity factor and mobile phase composition, and between the logarithm of capacity factor and inverse temperature. As a consequence of these assumptions, the average relative error in the predicted capacity factor was $7.8% and :l:6% for Equations [1.23] and [1.24], respectively, for 53 aromatic solutes. Temperature and Flow Rate. The optimization of temperature and flow rate, as an alternative to mobile phase composition in adsorption chromatography, has been accomplished by Scott and Lawrence (111) using the simultaneous method. Methyl palmitate and dinonylphthalate were used as as models from among 11 aliphatic and aromatic solutes, and the resolution was examined as a function of linear velocity at six temperatures (26 total experiments). The optimum conditions were selected from among those studied, and the results were compared under constant (isothermal-isorheic) conditions as well as with gradient programming in both temperature and flow rate. Moore and Synovec (112) used the regression method to optimize temperature and flow rate in a microbore LC column. In their approach, the mobile phase was heated before entrance to the column, and the flow rate was controlled to create an axial thermal gradient along the column. The average temperature within each column segment was estimated by using a two- dimensional distributed parameter model for cylindrical packed beds (113). The average capacity factor within each segment was determined as a function of the average temperature by the linear model given in Equation [1.21]. The overall solute capacity factor on the column was then calculated by summation of the capacity factor in each segment of the column. This approach was used to predict retention under conditions of both constant and gradient flow rate with 37 good results for the separation of dinitrophenylhydrazine derivatives of four aldehydes. Although the effect of temperature and flow rate on the plate height was also. examined in this work, these effects were not included directly in the model equations in order to optimize resolution or other quality criterion. Jinno and Yamagami (114) used the regression method to optimize temperature and flow rate concurrently under gradient programming conditions. In preliminary studies, a linear equation was used to relate the capacity factor of phenylthiohydantoin derivatives of five amino acids with a constant parameter that is characteristic of the physicochemical properties, shape, and structure. This equation enabled the estimation of capacity factor for other amino acids in the absence of experimental data, with average relative error of :l:2.5%. Since capacity factor is independent of flow rate, this equation also allowed the prediction of retention time or volume under conditions of constant and gradient —-flow rate. In order to consider the effects cf both flow rate and temperature programming simultaneously, a quadratic expression was included to relate the logarithm of capacity factor with temperature A1 A2 T +73" Ink = A, + [1.25] This approach was validated by optimizing the separation of the phenylthiohydantoin derivatives of twenty amino acids. The results show good agreement between the predicted and observed chromatograms with an average relative error in predicted retention time of 11.7%. 38 1.7 Conclusion A variety of methods has been used to optimize various parameters in liquid chromatography. Among these, regression methods have proven to be the most widely used and successful approach for univariate and multivariate optimization. The regression approach is attractive because, by using a simple linear or quadratic model, only a few initial experiments are necessary in order to characterize the response surface. While this approach'can lead rapidly. to the identification of the most promising experimental conditions, it requires that the model used to express the relationship between the parameter and the chromatographic property such as capacity factor, plate number, etc. be well defined. Unfortunately, deviations from ideal behavior frequently lead to ' nonlinear relationships for the parameters of interest in liquid chromatography, including mobile and stationary phase composition, temperature, and flow rate. Such deviations cannot be predicted a priori and, hence, serve to limit the accuracy with which the optimum conditions can be identified. This problem, which is detrimental in univariate optimization, becomes prohibitive when two or more parameters are to be optimized simultaneously. - ln order to overcome this problem, a new approach has been developed for univariate and multivariate optimization called parametric modulation (115- 119). The fundamental strategy of this approach is that chromatographic properties may be accurately predicted if the solute is constrained to undergo interactions independently within each mobile phase, stationary phase, and temperature environment. This strategy is implemented by maintaining each of these variables in discrete and separate zones along the chromatographic column, as illustrated schematically in Figure 1.4. The'theoretical foundation and experimental verification of the parametric modulation technique is the subject of 39 Figure 1.4: Schematic illustration of the parametric modulation concept for n solvent zones in q serially coupled columns. 40 Figure 1.4 SOLVENT RESERVOIRS 1 1 z 2 o e n n COLUMN 1 COLUMN 2 O O O COLUMN q \\ 313i n1 41 this dissertation. In Chapter 2, the concept of optimization by parametric modulation will be introduced and the theoretical basis of solute retention and dispersion established. Experimental verification of parametric modulation for univariate optimization of mobile phase and a comparison of the method with conventional premixed mobile phase is presented in Chapter 3. In Chapter 4, the multivariate optimization of both mobile and stationary phases is presented, and in Chapter 5 multivariate optimization of mobile phase and temperature is discussed. In the optimization of the separation of solutes, it is often assumed there is no significant interaction between solutes in a mixture. This assumption is investigated in Chapter 6, where a theoretical and experimental study of solute-solute interactions in liquid chromatography is presented. Finally, the work discussed in this dissertation is summarized and future work is proposed in Chapter 7. 1.8 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 42 References L. R. Snyder and J. J. lGrkland, Introduction to Modern Liquid Chromatography, 2nd Edition (John Wiley-lnterscience, New York, 1979). P. J. Schoenmakers, Optimization of Chromatogrflic Selectivity, Journal . of Chromatography Library, Volume 35 (Elsevier, sterdam, 1986). S. Ahuja, Selectivity and Detectability Optimization in HPLC, John-Wiley and Sons, New York, Ch. 4 ’ C. L. Deligny, M. C. Spanjer, J. C. Vanhouwelingen and H. M. Weesie, J. Chromatogr. 301, 311 (1984) . s. Elchuk and R. M. Cassidy, Anal. Chem. 51, 1434 (1979) H. Miyagi, J. Miura, Y. Takata, S. Kamitake, S. Ganno and Y. Yamagata, J. Chromatogr..239, 733 (1982) P. T. Kissinger, Anal. Chem. 49, 883 (1979) J. H. Knox and R. A Hartwick, J. Chromatogr. 204, 3 (1981) W. W. Yau, J. J. Kirkland and D. D. BI , Modern Size Exclusion Chromatography, Wiley-lnterscience, ew York, 1979 J. H. Hildebrand, R. L. Scott, The Solubility of None/ectrolytes; Reinhold: New York, NY 1950; Chapter 23 A. F. M.- Barton, Chem. Rev. 75, 731 (1975) W. Melander and C. Horvath, In High performance Liquid Chromatogra/olrr": Advances and Perspectives; C. 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Chromatogr. 11, 2851 (1988). 8.93.1;Pietrzyk, S. M. Senne, and D. M. Brown, J. Chromatogr. 546, 101 L. R. Snyder, J. Chromatogr. 179, 167 (1979). H. Colin, J. C. Diez-Masa, G. Guiochon, T. Czajkowska, and I. Miedziak, J. Chromatogr. 167, 41 (1978). R. B. Diasio and M. E. Wilbum, J. Chromatogr. Sci. 17, 245 (1979). J. Chmielowiec and H. Sawatzky, J. Chromatogr.17,245 (1979). K. B. Sentell and A. N. Henderson, Anal. Chim. Acta 246, 139 (1991). S. U. Shaikh and J. C. Touchstone, Anal. Chem. 509, 1472 (1987). L. C. Sander and S. A. Wise, Anal. Chem. 61, 1749 (1989). ‘ 3.9%.)Melander, D. E. Campbell, and C. Horvath, J. Chromatogr. 158, 215 l(N.9;ig.)Melander, D. E. Campbell, and C. Horvath, J. Chromatogr. 185, 99 1 . R. J. Laub and J. H. Pumell, J. Chromatogr. 161, 49. (1978). - A. Nahum and C. Horvath, J. Chromatogr. 203, 53 (1981 ). W. E. Hammers and P. B. A. Verschoor, J. Chromatogr. 282, 41 (1983). H. M. McNair and J. Bowermaster, J. High Resoluf. Chromatogr. Chromatogr. Commun. 10, 27 (1987). . K. Jinno, J. B. Phillips, and D. P. Carne, Anal. Chem. 57, 574 (1985). G. Liu, N. M. Djordjevic, and F. Emi, J. Chromatogr. 592, 239 (1992.). J. Yoo, J. T. Watson, and V. L. McGuffin, J. Microcol. Sep. 4, 349 (1992). L. R. Snyder and J. W. Dolan, DryLab l User's Manual and DgyLab 6 User's Manual (LC Resources Inc., Lafayette, California, 198 ). B. Sachok, J. J. Stranahan, and S. N. Deming, Anal. Chem. 53, 70 (1981). (B1.gsagr;hok, R. C. Konk, and S. N. Deming, J. Chromatogr. 199, 317 M. Otto and W. Wegscheider, J. Liq. Chromatogr. 6, 685 (1983). R. B. Diasio and M. E. Wilbum, J. Chromatogr. Sci. 17, 565 (1979). 1 O7. 108. 1 09. 1 10. 111. 112. 113. 114. 115. 118. 117. 118. 119. 47 H. J. Issaq, S. D. Fox, K. Lindsey, J. H. McConnell, and D. E. Weiss, J. Liq. Chromatogr. 10, 49 (1987). K. Jinno and M. Kuwajima, Chromatographia 22, 13 (1986). w.’ R. Melander and c. Horvath, Chromatographia 18, 353 (1984). {1925.)Melander, B. K. Chen, and C. Horvath, J. Chromatogr. 318, 1 1 . R. P. W. Scott and J. G. Lawrence, J. Chromatogr. Sci. 7, 65 (1969). L. K. Moore and R. E. Synovec, Anal. Chem. 65, 2663 (1993). B. A. Flnlayson, Chem. Eng. Sci. 26, 1081 (1971). ,K. Jinno and M. Yamagami, Chromatographia 27, 417 (1988). J. H. Wahl, C. G. Enke, and V. L. McGuffin, Anal. Chem. 62, 1416 (1990). J. H. Wahl, C. G. Enke, and V. L. McGuffin, Anal. Chem. 63, 1118 (1991). J. H. Wahl and V. L. McGuffin, J. Chromatogr. 485, 541 (1989). P. H. Lukulay and V. L. McGuffin, J. Chromatogr. 691, 171 (1995). P. H. Lukulay and V. L. McGuffin, Anal. Chem, manuscript in preparation. CHAPTER 2 THEORY OF PARAMETRIC MODULATION In this chapter, the theoretical basis of solute retention and dispersion under the conditions of parametric modulation is introduced. The advantages of this method over existing popular methods are also presented. Although this discussion will be focused on liquid chromatography, the general concepts described herein are applicable to gas and supercritical fluid chromatography as well. 2.1 Theoretical Development Theory of Solute Retention. In order to optimize separations under the conditions of parametric modulation, the theoretical basis of retention must be established. The inherent assumption underlying this theory is that solute retention is controlled independently within each solvent, stationary and temperature zone. In addition, it is assumed that the solute is able to achieve steady-state conditions rapidly after each change of environment. This latter requirement may restrict the selection of both mobile and stationary phases. The validity of these assumptions has been demonstrated previously for a single column (1-3). . The capacity factor (Kip), which is the thermodynamic measure of solute retention, may be expressed as follows: 49 .JJL-1 2.1 kip top I] where ti“, and to“, are the elution time of a retained and a nonretained solute, respectively, in solvent j and on column p. It may be readily shown (1,4) that the residence time of a solute in a solvent zone of length xi is given by: 1 .. 1,], s {1;— H32) 1221 where up is the mobile-phase linear velocity on column p. The total retention time (ti) of the solute is the summation of residence times in each solvent and on each column q n n r, = 221,. = f. 2 —xl— [flue] ’ [2.3] p-1 j-O ’9 p-1 i=0 up "in - Because the volumetric flow rate (F) of the mobilephase remains constant within the chromatographic system, the linear velocity on each column is related by F = 1r rp2 8p up - [2-4] where rp is the radius and EP is the total porosity of column p. By rearrangement of Equation [2.4] and substitution into Equation [2.3], the total retention time may be expressed in terms of the known column parameters i2": 1”,? spxj (“kip 25 in... F 17,: "‘ 50 The limit of the summation index (n), which represents the total number of solvent zones required to elute the solute from column p, is determined by evaluating the expression 2 _"L . (p [2.6] If the limit has a noninteger value for the first column (p=1), the summation in Equation [2.5] is performed in the normal manner for the integer solvent zones 0 to n, and the fraction is treated as a multiplier for the last solvent zone. This solvent zone subsequeme enters the second column, where the remainder is used as a multiplier for the first solvent zone (n=0) on the second column (p=2). The computation is performed in an analogous manner for all subsequent columns. Theory of Solute Band Broadening. In order to optimize the separation of solutes under conditions of parametric modulation, the variance of the solute peak must be determined at the exit of the last environment. If the broadening arises primarily from the packed bed and is relatively independent of solute and solvent, the variance in length units (85“,).2 can be expressed as: (Gib)? 8 hp dp 1p [2.7] where hp is the reduced plate height, tip is the particle diameter, and lp is the length of column p. Since variances are only independent and additive in the time domain (4,5), the length variance within each solvent zone on each column must be converted to the corresponding temporal variance (aim)? 51 2 . M (if 2.8 (obit ”pg top i I where the elution time of a retained solute is given by Equation [2.2] and the elution time of a nonretained solute (top) is given by top , JE— . [2.9] . up By substitution of these expresSions into Equation [2.8], the total temporal variance can be expressed as a summation of the contributions within all solvent zones and all columns 9 n q . h d at2 5px 1+ki 2 .2 : cc 2 = p p p i —p (6') E “E (0'9) _p.21 I, [jg F [ kill: D [2'10] Finally, the effective number of theoretical plates (N) for the coupled column system under the conditions of solvent modulation is given by I? N = — (0012 [2.11] where the total retention time and temporal variance are evaluated from Equations [2.5] and [2.10], respectively. Theory of Solvent Zone Broadening. In order to ensure that solute retention is controlled independently within each solvent zone, it is necessary that the 52 zones exhibit minimal mixing at the boundaries. Such mixing will adversely influence the accuracy of predicting solute retention from the theoretical models. Moreover, distortion and even splitting of solute peaks may occur if they elute at or near the solvent zone boundary. The zone purity can be expressed as the ratio of the lengths of the unmixed fraction of the final solvent zone to that of the original zone 21%]. = 1 .. g [2.12] When defined in this manner, the zone purity approaches unity when no interrnixing occurs and approaches zero when the solvent zone is completely mixed by column broadening processes. Determination of the zone purity by means of Equation [2.12] requires an estimation of the mixed fraction (5) of the solvent zone. This mixed fraction may be estimated by assuming that the initial solvent zone has an ideal rectangular profile of width xi and that column broadening processes are adequately described by a Gaussian profile with variance given by Equation [2.7], as illustrated schematically in Figure 2.1. The mixed fraction is then calculated, with a statistical accuracy of 95%, by presuming that mixing extends 2(00). at each boundary of the solvent zone, so that the total mixed region is 4(co).. The mixed fraction is, thus, given by g ,3 5%); [2.13] where the total variance for a nonretained solvent zone (on).2 in length units is 53 Figure 2.1 Schematic illustration of solvent zone broadening (1 - é :- 0.95). . 54 Figure 2.1 55 q q E )2 (Zn d I r4 2) . (so); . (9" 1,, p" p 'p p p 6p [2.14] (25. mi The effect of solvent zone length on the calculated zone purity is shown in Figure 2.2. It is apparent that the reduction of particle diameter and column length is important to maintain the integrity of the solvent zone. If the minimum acceptable value for the zone purity is arbitrarily defined to be 95%, as illustrated in Figure 2.1, the requisite solvent zone length may be determined from Equation [2.13] as follows x,- = 80(00). [2.15] For example, a solvent zone length greater than 12.7 cm would be required to maintain 95% purity for a column with 5 pm particle diameter and 25 cm length. The limiting zone length given by Equation [2.15] is the minimum value that may be used with confidence under the conditions of solvent modulation in order to ensure adherence to the assumptions of the theoretical models. 2.2 Optimization Strategy The strategy for optimizing separations requires preliminary measurement of the solute capacity factors in each solvent system on each column to be employed. From these measurements, the retention time and variance can then be predicted by using Equations [2.5] and [2.10], respectively, for any given 56 Figure 2.2A Effect of solvent zone length on zone purity according to Equations 2.12 — 2.14. Constant particle diameter (5 pm) with varying column lengths of 10 cm (—), 25 cm H, and 100 cm (-). 57 Figure 2.2A E3 :5sz mzoN Emiom .9. 9 m o ._..___.__...._._.._....md _. ; 2.. . a .. ; .90 < 2 . - .. ; x \ inc ; , t \ 1. \\\ \\ mo \\\ \ 1 \\\\\\\ \\ \ lmd 1111111111111 1 Il\\ \ O; (3 — l) Allancl BNOZ“ 58 Figure2.2B Effect of solvent zone length on zone rity according to Equations 2.12 - 2.14. Constant column ength (25 cm) )gwith varying particle diameters of 3 pm (—), 5 pm H, and 10 um (- ). 59 Figure 2.28 AEov :5sz mzoN azmiom _ mm o . on _ m n p h n — n h P h » - b b F b h h z; lmo m .2‘ - s a i x .. q . \\\\ . lmd \ \ - l ...... M H. \11.\ \ 04 (3 — l) Amend 3Noz 60 solvent zone length and column length in any sequence. After the retention time and variance are calculated for each solute, the extent of separation between adjacent solutes is given by the resolution (Rl,i+1) (. _ . . Ru” . t't‘ t') [2.16] 2[(°l+1)r+ (0011 The quality of the overall separation is then assessed using a modified form of the multivariate function developed by Schlabach and Excoffier (6), known as the Chromatographic Resolution Statistic (CRS) m—1 m—1 ,. my; (“1902 ] _‘l_ CRS [g (RUM-Rm Run + 5-21 ("I-1) Reva: m [2'17] where m is the total number of solutes, t, is the elution time of the final solute, Rap. is the optimum or desired resolution, Rm," is the minimum acceptable resolution, and Ravg is the average resolution which is given by m 1 RM .. fi- 2:. am, [2.18] The first term of the CRS function is a measure of the extent of separation between each pair of adjacent solutes in the chromatogram. This term approaches zero when the individual resolution elements approach the optimum value, and approaches infinity when the individual resolution elements approach the minimum value. The second term of the CRS function reflects the uniformity of spacing between solutes, and approaches a minimum value of unity when the individual resolution elements are equal to the average value. The final term of 61 the CRS function is intended to minimize the analysis time, and may be neglected if this is not a primary goal of the optimization. In the original expression defined by Schlabach and Excoffier (6), (Ru-+1 — Rom? passes through a minimum and 1/(RLM - Rum)2 passes through a maximum as Rim increases. As a result, whenever any individual value of Ri,l+1 approaches Rm]... the CRS value becomes extremely high and may no longer be representative of the overall separation quality. More importantly, however, values of RIM that are equidistant from Rm", whether higher or lower in magnitude, are ranked equally. The detrimental consequence is that deceptively low CRS values may be assigned when an individual resolution element approaches zero (7). This problem may be addressed by altering the form of the CRS function such that, if Rl.i+1 is less than or equal to Rm." + 0.01, the first term in Equation [2.17] is replaced by the constant term R —R - 2 1 °“ """ — [2.19] 0.01 lam," In order to optimize the separation, the experimental conditions that yield the minimum value of the CRS function must be determined. The optimum conditions may be determined in two ways: (1) by varying the sequence and length of the solvent zones and columns in a systematic manner to construct a complete CRS response surface, from which the optimum is identified by visual inspection, or (2) by using an iterative search routine such as the simplex method (8,9). ' 62 2.3 Conclusion The theoretical basis of the parametric modulation method has been developed for univariate and multivariate optimization. This approach has several advantages over traditional methods of chromatographic optimization. First, the predicted retention is inhereme more accurate because it is calculated by summation of the known and measured behavior in discrete environments rather than by numerical regression in unknown or poorly characterized mixed environments. Second, the number of preliminary experiments necessary _to fully characterize the chromatographic system is dramatically reduced in comparison with traditional methods. For a system composed of n mobile phases, q stationary phases, and t temperature zones, only n ~ q ~ t retention measurements are necessary for each solute. From these preliminary measurements, the solute retention time and variance for any combination, sequence, and length of the zones may be calculated. Other physical parameters of interest given in Equations [2.5] and [2.10], such as particle size, column dimensions, and flow rate, may be optimized simultaneously with no addition experimental measurements. Thus, a complete multivariate response surface may be systematically generated and the optimum may be identified by visual inspection or by computer-assisted search methods. The potential limitations of this approach arise from the assumptions inherent in development of the theoretical equations. First, it is assumed that solute retention is controlled Independently within each individual environment. This requires that the zones for each mobile phase, stationary phase, and temperature exhibit minimal mixing at the boundaries. A statistical treatment may be used to estimate the minimum zone length required to maintain an acceptable level of zone purity (1 ,2,8,10). This approach also requires that the solute be 63 able to achieve steady-state conditions rapidly after each change in environment. This may limit the selection of mobile and stationary phases to those exhibiting rapid kinetics and linear isotherms for the solutes of interest. The validity of these assumptions has been examined for both univariate and multivariate optimization (1 ,2,10). 64 2.4 References 1. J.H. Wahl, C.G. Enke, and V.L. MoGuffin, Anal. Chem. 62, 1416 (1990). J.H. Wahl, C.G. Enke, and V.L. MoGuffin, Anal. Chem. 63, 1118 (1991). J.H. Wahl, Ph.D. Dissertation, Michigan State University, 1991. 99".” J.906.SGiddings, Dynamics of Chromatography, Marcel Dekker, New York, 1 . S” J.C. Stemberg, Adv. Chromatogr. 2, 205 (1966). TD. Schlabach and J.L. Excoffier, J. Chromatogr. 439, 173 (1988). M.F.M. Tavares and V.L. MoGuffin, Anal. Chem, accepted for publication. J.H. Wahl and V.L. MoGuffin, J. Chromatogr. 485, 541 (1989). SN. Deming and S.L. Morgan, Anal. Chem. 45, 278A (1973). 10. P. H. Lukulay and V. L. MCGUffin, J. Chromatogr. 591, 171 (1995). SOPNS” CHAPTER 3 UNIVARIATE OPTIMIZATION OF MOBILE PHASE BY PARAMETRIC MODULATION 3.1 Introduction In this chapter, the specific case of mobile phase optimization by parametric modulation (solvent modulation) is discussed. Experimental verification of this technique is provided and compared with premixed mobile phases for the separation of corticosteroids. The separation is achieved on an octadecylsilica column using aqueous methanol and acetonitrile mobile phases. Based on these results the optimization methods are compared with respect to accuracy, total analysis time, critical resolution, and the overall quality of the separation. 3.2 Theoretical Concepts and Optimization Strategies Premixed Solvents. In reversed-phase liquid chromatography, a mixture of aqueous and organic solvents is often employed to affect the separation of solutes. Various approaches have been utilized to optimize the composition of these mixed solvents (1 -3), among which the commercially available optimization program known as DryLab IT“ is one of the most popular and successful (4-9). By using the regression approach, this program combines semi-empirical models of chromatographic retention and dispersion with a few initial experiments for the 65 CHAPTER 3 UNIVARIATE OPTIMIZATION OF MOBILE PHASE BY PARAMETRIC MODULATION 3.1 Introduction In this chapter, the specific case of mobile phase optimization by parametric modulation (solvent modulation) is discussed. Experimental verification of this technique is provided and compared with premixed mobile phases for the separation of corticosteroids. The separation is achieved on an octadecylsilica column using aqueous methanol and acetonitrile mobile phases. Based on these results the optimization methods are compared with respect to accuracy, total analysis time, critical resolution, and the overall quality of the separation. 3.2 Theoretical Concepts and Optimization Strategies Premixed Solvents. ln reversed-phase liquid chromatography, a mixture of aqueous and organic solvents is often employed to affect the separation of solutes. Various approaches have been utilized to optimize the composition of these mixed solvents (1-3), among which the commercially available optimization program known as DryLab IT" is one of the most popular and successful (4-9). By using the regression approach, this program combines semi-empirical models of chromatographic retention and dispersion with a few initial experiments for the 65 66 solutes of interest in order to optimize their separation. In this approach, a linear model is employed to relate the solute capacity factor (k) and the mobile phase composition (o) Iogk = IogliW - St) [3.1] where the slope (s) is a constant that is characteristic of each solute within the chromatographic system, and the intercept (log lg”) is the logarithm of the solute capacity factor using pure water as mobile phase. The solute capacity factor is calculated from the experimental data as k (tr "' to) [3.2] =— to where tr and to are the elution times of a retained and nonretained solute, respectively. Practical application of this technique requires that solute retention be measured by using at least two solvent compositions. This enables the estimation of the coefficients 9 and IrW for each solute by linear regression. Based on the values of these coefficients, the solute capacity factor can be predicted at other mobile phase compositions by means of Equation [3.1]. From the predicted capacity factors, the resolution between adjacent solutes can be calculated as follows NV2 ot-1 k' a, . 4 [. MW) 1.... where N is the number of theoretical plates, or is the selectivity factor, and k' is the average capacity factor. The quality of the separation is evaluated by means of the resolution between the least-resolved solute pair, called the critical resolution (Ran). In order to optimize the separation, the critical resolufion is 67 mapped as a function of the solvent composition (8, 10,11). From this resolution map, the mobile phase composition that yields the highest value of the critical resolution can be determined by visual inspection. Solvent Modulation. The general concept and theory of solvent modulation have been discussed previously in Chapter 2. In this technique, the overall retention of solute i is given by (1,2) ixt—ii kt =j-o kfl L — 1 [34} where kij is the capacity factor of solute i in solvent j, xi is the solvent zone length, and the limit of summation n represents the number of solvent zones required to elute the solute from a column of length L. Thus, in solvent modulation, the overall retention is varied by means of the type, sequence, and length of the solvent zones applied to the column. The strategy for optimization by this technique requires preliminary measurement of solute capacity factor in each solvent of interest. Based on these measurements, the overall capacity factor under the conditions of solvent modulation can be estimated by means of Equation [3.4]. Next, the resolution between adjacent solutes i and i+1 is calculated using the form of the resolution equation below N"2 khf‘ki - R = —. 3.5 "M 2 [2+8 +4.1] [ ] 68 The quality of the separation is then assessed by using a modified form of the multivariate function known as the Chromatographic Resolution Statistic (CRS) developed by Schlabach and Excoffier (12) [m—I R R 2 1 m-f (R )2 .. .. .. I CRS = 2[ m1 om) _ + 2 n+1 ) _i_ 3.6 H Ri,l+1"Rmin Ri,i+1 I-1 ("I-1) Ravgz m I l where m is the total number of solutes, t, is the elution time of the final solute, Rom is the optimum or desired resolution, Rmin is the minimum acceptable resolution, and Ravg is the average resolution calculated according to Equation [2.18]. In order to optimize the separation, the solvent zone lengths which yield the minimum value of the CRS function must be determined. This minimum CRS value may be determined in two ways: 1) by varying the length of each solvent zone systematically to produce the complete response surface from which the optimum is determined by visual inspection, or 2) by using an sequential search routine such as the simplex method (13,14). The former method is time consuming, but provides a detailed view of the complete response surface. The latter method is more efficient but, because the surface may contain many local maxima and minima, care must be taken to ensure that the global optimum is identified. Consequently, a combination of these approaches is desirable. 69 3.3 Experimental Methods Materials and Methods. The following corticosteroids are utilized in this investigation: cortisone (1781,21-dihydroxy-pregn-4-ene-3,1 1 ,20-trione), hydrocortisone (11[3,1781,21-trihydroxy-pregn-4-ene-3,20-di-one), tetrahydro- ' cortisone (3c,1781,21-tri-hydroxy-5j3-pregnane—11,20—dione), tetrahydrocortisol (381,1 1 [3,1 781,21-tetrahydroxy-5j3—pregnane-20-one), prednisone (1781,21 - dihydroxy-preg na-1 ,4-diene-3,11 ,20-trione), prednisolone (1 1 [1,1 7a,21-trihydroxy- pregna-1,4~diene-3,20-dione), methylprednisolone (11B,17c,21-trihydroxy-6c- methyl-pregna-1 ,4-diene-3,20-dlone), and dehydrocorticosterone (21-hydroxy- pregn-4-ene-3,11,20-trione). These corticosteroids shown in Figure 3.1 are obtained from the Sigma Chemical Company (St. Louis, MO, USA) and are used without further purification. Standard solutions are prepared in methanol at 10'3 M concentration for tetrahydrocortisone and tetrahydrocortisol, and at 10'6 M concentration for all other corticosteroids. Organic solvents are high-purity, distilled-in-glass grade (Baxter Healthcare, Burdick & Jackson Division, Muskegon, MI, USA); water is deionized and double distilled in glass (Model MP- 3A, Corning Glass Works, Corning, NY, USA). Experimental System. A chromatographic pump equipped with two 40-mL syringes (Model 140, Applied Biosystems, Foster City, CA, USA) is used to deliver the mobile phase at 0.5 mL/min. Sample introduction is achieved by using a 10-uL injection valve (Model EQ 60, Valco Instruments 00., Houston, TX, USA). The chromatographic column (47 cm x 0.46 cm i.d.) is packed with octadecylsilica material (Spherl-S RP-18, 5 pm, Applied Biosystems) to have a total plate number (N) of approximately 10,000 for the solutes of interest. Solute detection is accomplished by using a variable-wavelength UV-visible absorbance Figure 3.1 Structures of corticosteroids. 70 71 Figure 3.1 $HZOH CHZOH cHon o CHJC=0 CH O CORTISONE DEHYDROCORTICOSTE RONE ([214on ' C=O HO CHJ'uoH CH 110'" TETRAHYDROCORTISOL 011,011 CH20H I C- I c=o i CH3 ‘0 HO CH3 "OH 3 “OH HO «0H CH3 o o 0 5 . CH, f Pnsomsoue PREDNISOLONE METHYLPREDNJSQLQNE 72 detector (240 nm, 0.005 AUFS, Model 166, Beckman Instruments, San Ramon, CA, USA). ‘ Computer-Assisted Optimization Programs. The optimization program for premixed mobile phases, DryLab lTM lsocratic HPLC Simulation/Optimization Program (LC Resources Inc., Lafayette, CA, USA), is executed on an IBM- compatible computer with 80486 microprocessor. From the initial measurements of solute retention times, this program uses linear regression to calculate the capacity factor as a function of the mobile phase composition. In addition, the total analysis time, selectivity factor, and critical resolution are calculated from Equations [3.1] to [3.3], assuming a plate number of 10,000. The optimum conditions are identified from a graph of the critical resolution as a function of mobile phase composition (11). The optimization program for solvent modulation is written in the Fortran 77 language and executed on a VAX Station 3200 computer (Digital Equipment, Maynard, MA, USA) (15-17). From initial measurements of solute capacity factors, the overall capacity factor of each solute is calculated for the conditions of solvent modulation by using Equation [3.4]. The resolution of each solute pair is calculated by using Equation [3.5], and the overall quality of the separation is assessed by means of the CRS function in Equation [3.6], where the selected values for the optimum and minimum acceptable resolutions are 1.5 and 0.5, respectively. The optimum conditions are identified by two methods (18). In the topographic mapping method, these calculations are performed while systematically incrementing each solvent zone length within a prescribed range. By graphing the resulting CRS values as a function of the solvent zone length, a complete response surface is constructed. The minimum CRS value is then located by visual inspection of this response surface (17). In the sequential 73 search method, the modified simplex algorithm of Nelder and Mead (14) is employed. This algorithm permits expansion and contraction of the simplex during the search and will converge at the optimum position. In order to ensure the identification of the global optimum, both the size and the location of the initial simplex are varied systematically in 200 independent searches (17). For each initial simplex, the calculations are performed according to Equations [3.4] to [3.6] at each successive vertex, and the best conditions are continuously updated and stored in a file (18). This file contains the solvent zone lengths, the solute elution order, the predicted capacity factors, the predicted resolutions, and the corresponding CRS value for the separation. 3.4 Results and Discussion In this work, solvent modulation and premixed solvents are compared by using computer-assisted optimization programs to optimize the separation of eight corticosteroids. The practical utility of both techniques requires that solute retention be measured in solvents of different compositions, from which the optimum conditions for the separation can be predicted. The capacity factor of each corticosteroid was measured on an octadecylsilica column using 60% and 75% methanol as well as 35% and 50% acetonitrile. Table 3.1 summarizes the retention measurements for the corticosteroids. In the methanol mobile phases, all of the corticosteroids are well separated except for prednisolone and hydrocortisone. On the other hand, in the acetonitrile mobile phases, the corticosteroids are separated with the exceptions of prednisone and 74 Table 3.1 Capacity Factors for Corticosteroids on Octadecylsilica Stationary Phase using Methanol and Acetonitrile Mobile Phases CAPACITY FACTOR (k) CORTICOSTEROIDS Methanol-Water Acetonitrile—Water 60% 75% 35% 50% Prednisone 1 .57 0.41 1.66 0.50 Cortisone 1 .73 0.48 1.83 0.58 Prednisolone 2.14 0.55 1 .40 0.41 Hydroccrtisone 2.18 0.56 1 .69 0.50 Dehydrocorticosterone 2.97 0.59 4.09 1.39 Methylprednisolone 3.79 0.86 2.53 0.70 Tetrahydrocortisol 4.29 0.96 1 .94 0.50 Tetrahydrocortisone 5.17 1 .08 2.53 0.63 75 hydrocortisone as well as methylprednisolone and tetrahydrocortisone. Thus, the least-resolved solute pairs vary with the type of organic modifier. Premixed Solvents. To optimize the separation of corticosteroids using premixed solvents, the DryLab I" program is utilized. Based on the preliminary measurements of the capacity factors in Table 3.1, this program calculates the retention of the corticosteroids at other solvent compositions according to Equation [3.1]. In order to determine the optimum solvent composition, the critical resolution of the least-resolved solute pair is mapped as a function of the mobile phase composition. The resolution maps for the aqueous acetonitrile and methanol mixtures are shown in Figures 3.2 and 3.3, respectively. For the acetonitrile mixtures, the composition of 30% acetonitrile is predicted to yield the highest value of the critical resolution. The least-resolved .solutes are prednisone and hydrocortisone, with a predicted resolution 010.3. Because this region of the resolution map is irregular, slight variations in the mobile phase composition may result in a large change in the critical resolution. For the methanol mixtures, however, a more rugged optimum region is observed between 55% and 60% methanol. The least-resolved solutes are prednisolone and hydrocortisone, with a predicted resolution of 0.3. Because this region is relatively broad and flat, slight variations in the mobile phase composition will not be as detrimental. Thus, 56% methanol was chosen as the optimum mobile phase composition and was used to obtain the separation shown in Figure 3.4. From this chromatogram, it is apparent that prednisolone and hydrocortisone are completely overlapped (R5 at 0.3), whereas all other solutes are fully resolved. The experimentally measured capacity factors are in good agreement with the theoretically predicted values from Equation [3.1], as summarized in Table 3.2, with an average relative error of 13.22%. Figure 3.2 76 Critical resolution as a function of the mobile phase composition for aqueous acetonitrile mixtures. Column: 47 x 0.46 cm i.d., packed with octadecylsilica material. Solutes: (1) prednisone, 2 cortisone, (3) prednisolone, (4) hydrocortisone, 5 dehydrocorticosterone, (6) methyl-prednisolone, tetrahydrocortisol, (8) tetrahydrocortisone. 77 Figure 3.2 00. LO fig N. #- d: u- N I T i I l O O O O “’7 C‘! ' "‘. O. O O O O NOIiflTOSEB TVOIIIHO 0 L0 20 ACETONITRILE 78 Figure 3.3 Critical resolution as a function of the mobile phase composition for aqueous methanol mixtures. Experimental conditions as given in Figure 3.2 79 Figure 3.3 3,4 3,4 N 1- 0.30- 0.10— NOIUTIOSEIEI ‘lVQIiIEIO 0.00 60 70 80 % METHANOL 50 40 Figure 3.4 80 Experimental chromatogram of corticosteroids obtained under the predicted optimum conditions for premixed mobile phases. Column: 47 x 0.46 cm i.d., packed with octadecylsilica material. Mobile phase: 56% methanol, 0.5 mUmin. Detector: UV-visible absorbance detector, 240 nm, 0.005 AUFS. Solutes: 1 prednisone, (2) cortisone, (3) prednisolone, (4) hydrocortisone, 5 dehydrocorticosterone, (6) methylprednisolone, tetrahydrocortisol, (8) tetrahydrocortisone. 81 Figure 3.4 3.4 O O ~r~ (x ”‘8 to __O In In 40 TIME (min) 20 10' 82 Table 3.2 Comparison of Experimental and Theoretical Capacity Factors under the Predicted Optimum Conditions for Premixed Mobile Phases CAPACITY FACTOR (k) CORTICOSTEROIDS TheoryT Experiment Relative Error (%)* Prednisone 2.46 2.25 —8.54 Cortisone 2.60 2.54 -2.31 Prednisolone 3.27 3.1 1 -4.89 Hydrocortisone 3.32 3.1 1 -6.32 Dehydrocorticosterone 4.62 4.55 —1 .52 Methylprednisolone 5.84 5.80 —0.68 Tetrahydrocortisol 6.61 6.60 -0.15 Tetrahydrocortisone 8.08 8.1 9 1 .36 Average $3.22 1’ Calculated by using Equation [3.1]. 1 Calculated as 100 x (Experiment - Theory)/Theory. 83 Solvent Modulation. To optimize the separation of corticosteroids using solvent modulation, the four solvent systems shown in Table 3.1 are utilized. Although there are twelve possible permutations of a two-solvent modulation sequence, the computer-assisted search routines provide a rapid and effective means to identify the most promising permutation. For each permutation, the sequential simplex method is used to determine the minimum CRS value on the complete response surface. The results of this preliminary search are summarized in Table 3.3. From these results, the most promising solvent modulation sequence is identified to be 50% acetonitrile followed by 60% methanol (CRSmin = 1.9). The least-resolved solutes are cortisone and hydrocortisone, with a predicted resolution of 1.30. Although this permutation initially appears to be very promising, a more detailed inspection by the topographic mapping method reveals that the response surface is highly irregular. Variations in the solvent zone length as small as :l:1.0 cm alter the identity of the least-resolved solute pair and cause a significant change in the critical resolution. As a consequence of this limitation, the next most promising permutation of 35% acetonitrile followed by 60% methanol (CRSmin = 3.6) is selected for further study. The chromatograms corresponding to each of these solvents are shown in Figure 3.5. The least-resolved solutes in 35% acetonitrile are methylprednisolone and tetrahydrocortisone, whereas those in 60% methanol are prednisolone and hydrocortisone. Although the separation of all solutes is not achievable in either of these solvents individually, the results in Table 3.3 suggest that the modulation of these solvents may provide a more beneficial separation. In order to determine the optimum conditions for solvent modulation, the complete response surface was constructed by the mapping method. The topographic and contour maps of the CRS response surface are shown in Figure 3.6 as a function of the solvent zone length. When expressed in terms of the 84 Table 3.3 Evaluation of the Permutations for a Two-Solvent Modulation Sequence Solvent 1 Solvent 2 t, Rem CRS,“n (min) 60% Methanol 75% Methanol 29.3 0.20 95 75% Methanol 60% Methanol 28.8 0.19 96 35% Acetonitrile 50% Acetonitrile 32.3 0.15 113 50% Acetonitrile 35% Acetonitrile 33.9 0.15 104 60% Methanol 35% Acetonitrile 50.3 0.72 26 35% Acetonitrile 60% Methanol 51.9 0.99 3.6 60% Methanol 50% Acetonitrile 30.3 0.86 8.1 50% Acetonitrile 60% Methanol 31.1 1.30 1.9 75% Methanol 35% Acetonitrile 18.9 0.20 97 35% Acetonitrile 75% Methanol 19.3 0.18 98 75% Methanol 50% Acetonitrile 18.6 0.18 100 50% Acetonitrile 75% Methanol 18.6 0.18 101 85 Figure 3.5 Separation of corticosteroids in individual mobile phases for solvent modulation. Mobile phase: (A) 60% methanol, (B) 35% acetonitrile, 0.5 mUmin. All other experimental conditions as given in Figure 3.4. 86 Figure3.5 1 3.4 6 5 2 7 8 0"1'07 20 V 30140 U 50 50 6,8 1.4 B 7 2 3 5 ('10 '20 1 50 ' 40 5° 50 TIME (min) 85 Figure 3.5 Separation of corticosteroids in individual mobile phases for solvent modulation. Mobile phase: (A) 60% methanol, (B) 35% acetonitrile, 0.5 mL/min. All other experimental conditions as given in Figure 3.4. 86 Figure 3.5 1 3.4 6 5 r 2 c3 710 7 20 30 1 4'07 50 80 6,8 1.4 B l, 2 TIME (min) 87 Figure 3.6A Topographic map of the CRS response surface as a function of the fractional zone lengths for 35% acetonitrile and 60% methanol. 89 FIGURE 3.6B Contour map of the CRS response surface as a function of the fractional zone lengths for 35% acetonitrile and 60% methanol. 90 Figur83.68 C0 0. a! '9 "- °‘ 0’ tntn ‘* ‘1' 7' 7. *- s \ w W'U \\W K \cz or: 0‘32 WW 0 O O Q ~ 0 r2 1 J l l l 1 oq a: .0 1x. ‘12 ‘Q in ‘I' 7. 7r 0.4 0.5 35% ACETONITRILE 0.3 91 fractional length (xi/L), these maps may be used to determine the optimum conditions independent of the column length. From these maps, the minimum CRS is predicted for zones of 35% acetonitrile and 60% methanol in fractional lengths of 0.4 and 4.8, respectively, which correspond to absolute lengths of 19 and 226 cm, respectively, forthe 47 cm column utilized in this study. The experimental chromatogram in Figure 3.7 shows good separation of all corticosteroids with the exception of the least-resolved solute pair, prednisolone and hydrocortisone (Rem ~ 0.7). The experimentally measured capacity factors agree well with the theoretically predicted values from Equation [3.4], as summarized in Table 3.4, with an average relative error of :I:2.70%. Comparison of Solvent Modulation and Premixed Solvents. The optimization methods used in this work may be compared on the basis of the following criteria: average relative error in predicted capacity factor, total analysis time, critical resolution, and overall quality of the separation assessed by the CRS function. The average relative error is i2.70% for the solvent modulation technique, compared with :I:3.22% for premixed solvents. When optimized with solvent modulation, the corticosteroid separation is achieved experimentally in 52 minutes with a critical resolution of 0.7, and a CRS value of approximately 26. When optimized with premixed solvents by DryLab I”, the corticosteroid separation is achieved experimentally in 73 minutes with a critical resolution of 0.3, and a CRS value of approximately 146. On the basis of these criteria, the separation achieved by using solvent modulation is at least comparable to and, in some respects, significantly better than that achieved by using premixed solvents. Figure 3.7 92 Experimental chromatogram of corticosteroids obtained under the predicted optimum conditions for solvent modulation. Mobile phase: solvent modulation sequence of 35% acetonitrile and 60% methanol in fractional zone lengths of 0.4 and 4.8, respectively, 0.5 mUmin. All other experimental conditions as given in Figure 3.4. 93 Figure 3.7 9. 3:5 wzc. _ on _ F ow or Figure 3.7 92 Experimental chromatogram of corticosteroids obtained under the predicted optimum conditions for solvent modulation. Mobile phase: solvent modulation sequence 0135% acetonitrile and 60% methanol in fractional zone lengths of 0.4 and 4.8, respectively, 0.5 mUmin. All other experimental conditions as given in Figure 3.4. 93 Figure 3.7 om 0.. 35 mi; _ on _ ow o_. Table 3.4 Comparison of Ex 94 rimental and Theoretical Capacity Factors under the Pre icted Optimum Conditions for Solvent. Modulation CAPACITY FACTOR (k) CORTICOSTEROIDS TheoryT Experiment Relative Error (%)* Prednisone 1 .59 1 .58 —0.63 Cortisone 1 .76 1.78 1 .14 Prednisolone 1 .93 2.08 7.77 Hydrocortisone 2.06 2.10 1 .94 Dehydrocorticosterone 3.08 2.97 —3.57 Methylprednisolone 3.59 3.60 0.28 Tetrahydrocortisol 3.82 4.01 4.97 Tetrahydrocortisone 4.76 4.82 1 .26 Average :I:2.70 T 2 Calculated by using Equation [3.4]. Calculated as 100 x (Experiment - Theory)/Theory. 95 3.5 Conclusions Solvent modulation is a practical alternative to premixed mobile phases in liquid chromatography. Because the solvent zones are spatially and temporally separated from one another, solute retention is a simple time-weighted average of the retention in each individual solvent. Consequently, optimization of the separation is more accurate and requires fewer preliminary experiments with solvent modulation than with premixed solvents. In this study, the separation of corticosteroids was optimized by each technique and compared with respect to accuracy, total analysis time, critical resolution, and overall quality of the separation. The solvent modulation approach compares favorably in all of these respects and, hence, is demonstrated to be a very promising optimization strategy. 3.6 509°.“ng 11. 12. 13. 14. 15 16. 17. 18. 96 References H. A H. Billiet, L. de Galan, J. Chromatogr. 485, 27 (1989) J. L. Glajch, L. R. Snyder, Computer-Assisted Method Development for High Performance Liquid Chromatography, Elsevie r, Amsterdam, 1990 L. R. Snyder, J. J. Kirkland, Introduction to Modern liquid Chromatography, 2nd ed., John-Wiley, New York, 1979 M. A. Quarry, R. L. Grob, L. R. Snyder, J. Chromatogr. 285, 1 (1984) M. A. Quarry, R. L. Grob, L. R. Snyder, J. Chromatogr. 285, 19 (1984) M. A Quarry, R. L. Grob, L. R. Snyder, Anal. Chem. 58, 907 (1986) L. R. Snyder, M. A. Quarry, J. L. Glajch, Chromatographia 24, 33 (1987) L. R. Snyder, J. W. Dolan, D. C. Lommen, J. Chromatogr. 485,65 (1989) , J. W. Dolan, D. C. Lommen, L. R. Snyder, J. Chromatogr. 485, 91 (1989) R. J. Laub, J. H. Pumell, J. Chromatogr. 112,71 (1975) L. R. Snyder, J. W. Dolan, DryLab I User's Manual, LC Resources Inc., Lafayette, CA, 1987 T. D. Schlabach, J. L. Excoffier, J. Chromatogr. 439, 173 (1988) W. Spendly, G. R. Hext, F. R. Hinsworth, Technometrics 4, 44 (1962) J. A Nelder, R. Mead, Comput. J. 7, 308 (1968) J. H. Wahl, C. G. Enke, V. L. MoGuffin, Anal. Chem. 62, 1416 (1990) J. H. Wahl, C. G. Enke, V. L. MoGuffin, Anal. Chem. 63, 1118 (1991) J. H. Wahl, V. L. MoGuffin, J. Chromatogr. 485, 541 (1989) J.H. Wahl, Ph.D. Dissertation, Michigan State University, 1991. CHAPTER 4 MULTIVARIATE OPTIMIZATION OF MOBILE AND STATIONARY PHASES BY PARAMETRIC MODULATION 4.1 Introduction The subject of this chapter is the experimental verification of multivariate optimization by the parametric modulation approach. In the work described herein, the use of serially coupled columns and solvent modulation is investigated as a viable strategy for the optimization of both stationary and mobile phases. By maintaining the stationary phases in separate columns and the mobile phases as separate zones, the solute will interact independently in each environment (Figure 1.4). Thus, overall solute retention will be a simple, predictable function of the capacity factor weighted by the time that the solute is exposed to each stationary and mobile phase. This approach should allow accurate prediction of solute retention and, hence, allow optimization of multidimensional separations with a minimum number of preliminary experiments. 4.2 Theoretical Development The optimization of the separation requires that the overall retention time (ti) be calculated when the solute is exposed to each stationary phase and 97 98 mobile phase. This is accomplished by summing retention within each individual environment as shown below q n r 2 x 1 u . + .. ,.22 ,.,, {—8) 1...] pal j-0 F: kip where kfip is the capacity factor of solute i in solvent j on column p, rp and 8,, are the radius and total porosity of column p, respectively, xi is the length of solvent zone j, F is the volumetric flow rate, 11 and q are the limits of summation of number of solvent zones and columns, respectively. In a similar manner, the variance (of) of the solute zone, in temporal units is calculated by independent summation as q n q (002 = Z Z (inlz = 2 h: dp p21 j-0 p=1 p [z nrpzfixj (14919))? [4.2] "0 kip where hp is the reduced plate height, tip is the particle diameter, and ID is the length of column p. The resolution between adjacent solute zones (Rii +1) is then calculated by substitution of Equations [4.1] and [4.2] in the following expression R- . = “1+1 Tti) "'*‘ ZIIO'mlt + 10.1.] [4.3] Finally, a criterion such as the chromatographic resolution statistic (CR8) (1) shown below is used to evaluate the overall quality of the separation 99 [ ‘ R R 2 1 1 (R )2 I — I t CRS a 2 {—9—321-‘89 ) — + ,2 “+1 j) —' 4.4 where Flop, and Rmin are the optimum and minimum resolution, respectively, Ravg is the average resolution, t, is the total analysis time, and m is the number of solutes. The computer program used to accomplish these calculations is given in Appendix [A1.2]. 4.3 Experimental Methods Materials and Methods. Reagent-grade polynuclear aromatic hydrocarbons (PAH) are obtained from Sigma Chemicals (St. Louis, MO, USA). Standard solutions are prepared at 10'4 M by dissolution of the PAH in spectroscopic- grade methanol. Organic solvents are high purity distilled-in-glass grade (Baxter Healthcare, Burdick and Jackson Division, Muskegon, MI, USA); water is deionized and doubly distilled in glass (Model MP-3A, Corning Glass Works, Corning, NY, USA). Experimental System. The chromatographic system consists of a dual syringe pump (Model 140, Applied Biosystems, San Jose, CA, USA), which is programmed to deliver solvent zones for a specified time period at a nominal flow rate of 1 pL/min. Sample introduction is achieved by using a 1.0 pL injection valve (Model ECI4W1, Valco lnstnlments, Houston, TX, USA), after which the effluent stream is split (75:1) and applied to the chromatographic column. The 100 microcolumns are prepared from fused-silica capillary tubing (200 um l.D., Polymicro Technologies, Phoenix, AZ, USA), whose length can be readily adjusted, and terminated with a quartz wool frit. The microcolumns are packed with octadecylsilica and B-cyclodextrinsilica materials (5 pm diameter, Advanced Separation Technologies, Whippany, NJ, USA) according to the slurry packing procedure described previously (2). The octadecylsilica and B—cyclodextrinsilica columns, respectively, have total porosity (8p) of 0.81 and 0.80, separation impedance (dip) of 890 and 720, and reduced plate height (hp) of 3.2 and 12.9 using pyrene as a model solute under standard test conditions (2). The polynuclear aromatic hydrocarbon solutes are detected using both laser-induced fluorescence and UV-visible absorbance in a 75 um I.D. fused- silica capillary flow cell. In the fluorescence detection system [37,38], a helium- cadmium laser (325 nm, 25 mW, Model 3074-40M, Omnichrome, Chino, CA, USA) is reflected by a dielectric mirror and focussed on the capillary with a quartz lens. The fluorescence emission is collected perpendicular and coplanar to the excitation beam with another quartz lens, spectrally isolated with a bandpass interference filter (420 nm, SIO-420-F, Corion, Holliston, MA, USA), and focussed onto a photomultiplier tube (Model Centronic 042498, Bailey Instruments, Saddle Brook, NJ, USA). The photocurrent is amplified and converted to voltage by a picoammeter (Model 480, Keithley Instruments, Cleveland, OH, USA). A variable-wavelength UV-absorbance detector (Model Uvidec 100-V, Japan Spectroscopic Co., Tokyo, Japan), operated at 254 nm,is placed in series after the fluorescence detector. Data are acquired simultaneously from both detectors using an analog-to-digital converter (Model DT2805-5716, Data Translation, Marlborough, MA, USA) together with a personal computer (Model ZMF-248-40, Zenith Data Systems, St. Joseph, MI, USA). ‘ The data acquisition programs are written in the Forth-based 101 programming language ASYST (Version 2.1, Keithley Asyst, Rochester, NY, USA) Computer-Assisted Optimization Program. The computer optimization program is written in the FORTRAN 77 language to be executed on a VAXstation 3200 computer (Digital Equipment, Maynard, MA, USA). This program generates a complete CRS response surface by using a systematic mapping procedure. The lengths of each column and solvent zone are varied independently within prescribed limits; column lengths are varied from 0 to 75 cm and solvent zones are varied from the minimum length that maintains 95% purity (Equation [2.15]) to the maximum length required to elute all solutes. For each combination of column and solvent lengths, the total retention time and variance of each solute peak are calculated by means of Equations [4.1] and [4.2], respectively. The resolution between each pair of adjacent solute peaks is calculated by using Equation [4.3]. Finally, the overall quality of the separation is assessed by means of the modified CRS function (Equations [4.4] and [2.19]), using selected values for the optimum and minimum acceptable resolution of 1.5 and 0.3, respectively. By graphing the CRS value as a function of the column and solvent zone lengths, a complete multidimensional response surface can be constructed. The minimum CRS value is then Identified by visual inspection of the response surface. To assist in identifying the minimum CRS value, the five best conditions are continuously updated and stored in a file by the computer program. 102 4.4 Results and Discussion The goal of these preliminary studies is to demonstrate the conceptual basis and to test the theoretical models of solvent modulation using serially coupled columns. To facilitate these goals, the stationary phases were selected for their dissimilar retention mechanisms, but compatible mobile phase requirements. Octadecylsilica retains solutes predominantly by a partition mechanism, whereas fi-cyclodextrinsilica forms inclusion complexes based on the size and shape of the solute. Although both of these phases interact with solutes by van der Waals forces, monomeric octadecylsilica is dominated by enthalpic effects while B—cyclcdextrinsilica has a significant entropic contribution. This difference in retention mechanisms may be advantageous for the separation. of isomeric four- and five-ring polynuclear aromatic hydrocarbons (PAH), whose structures are shown in Figure 4.1. The capacity factors for each PAH were measured on the octadecylsilica and [i-cyclodextrinsilica stationary phases using mobile phases with compositions of 80% and 90% aqueous methanol as well as 50% and 70% aqueous acetonitrile. On the octadecylsilica phase (Table 4.1), many of the isomeric four- and five-ring compounds are well separated. However, there are several critical pairs that are difficult to resolve in all mobile phases, most notably chrysene and benz[a]anthracene, as well as benzo[e]pyrene and perylene. Although PAH molecules are reported to form inclusion complexes with [3- cyclodextrinsilica (3-7), the capacity factors are very small under all experimental conditions examined herein (Table 4.2). Only by using the- weakest mobile phases is the PAH retention measurable and the resolution partially achieved. Although methanol mobile phases show no selectivity based on either solute size or shape, the acetonitrile mobile phases appear to be able to discriminate 103 Figure 4.1 Structures of isomeric four- and five-ring polynuclear aromatic hydrocarbons. 104 Figure 4.1 8 8 @909 6’6’8 88 PYRENE TRIPHENYLENE BENZO[C]PHENANTHRENE 88% 888‘9 CHRYSENE BENZ{a]ANTHRACENE @ ‘ 0 89 .899 .19 PERYLENE BENZO[a]PYRENE BENZO[e]PYRENE oreswzlachNTHnAcsrlE DIBENZ[a,h]ANTHRACENE 105 H: Em men . 8.: mZmoEIezscafizmma 8.2 0.9. 3.4 0.9 mzmoéxezsoefizmma mam 0mm 55 of mzmmEEoszm m3 ode . Ed 2: mzmuEmn 2.8 as Ed 2: mzmmiEoszm ear. cam . 5N 8s mzmoéEzsenzmm New. mam 6a 5.8 mzmmEIo Re 8.8 a: 8.8 mzwmxez1 5.5.2052 52,252 0.2.205 as; mob/E >Eoomu£oo co 202823; oszoLm 58.8258 .8 A JV £203 £8380 fie 838... 106 :0 00.0 00.0 90 02020052393sz05 00.0 00.0 00.0 00.0 mzwoézezsoanzmma 000 0.0.0 000 00.0 mzmmEEoszm :0 00.0 . 3.0 00.0 mzmimmc 3.0 00.0 00.0 0.0.0 mzmm>$$oszm 00.0 00.0 00.0 20 0202010235me 00 0 :0 00.0 00.0 02000610 00.0 00.0 00.0 00.0 mzmmxazI 3.002052 0023415: 0.2205. 20000 005% E055 5000:2500 .mmmmga 0:208 0:502QO new .050 BE maomscm 050: 8093 >095sz mo___mc_0§mno_o>o-n co mcoemoen : oszoLw 000.20:on 02 A0 v 9200.0 3.0300 Nd 83.0... 107 between the four- and five-ring classes. The most noteworthy and useful result is that perylene may be readily separated from both benzopyrene isomers on the [3- cyclodextrinsilica phase. Based on these results, it seems reasonable to expect that resolution of the PAH can be achieved by serial coupling of the octadecylsilica and B- cyclodextrinsilica columns. However, there are 64 possible permutations for the columns and solvents examined in this study: 8 one-column/one-solvent systems, 24 one-column/two-solvent systems, 8 two-column/one-solvent systems, and 24 two-column/two-solvent systems. It would clearly be nearly impossible to examine all of these possibilities by time-consuming, trial-and-error experimental measurements. However, the computer program described previously provides a rapid and effective [means to identify the most promising permutation. The individual capacity factors (kijp) for the PAH solutes shown in Tables 4.1 and 4.2 are used to calculate the retention time and variance according to Equations [4.1] and [4.2], respectively. The resolution of each adjacent pair of solutes is calculated from Equation [4.3], and the overall quality of the separation is assessed by the CRS function in Equation [4.4]. For each possible combination and sequence of columns and solvents, the minimum CRS values are stored in a file and are used to identify the most promising permutation for further study. The results of these preliminary simulations are summarized in Table 4.3 for the permutations of a two-columnl‘two-solvent chromatographic system. If the minimum CRS value is used as the criterion for evaluation, then the best column sequence is predicted to be octadecylsilica followed by Bayolodextrinsilica, and the best solvent sequence is 80% methanol followed by 50% acetonitrile. These conditions should provide a minimum resolution 010.97 with a total analysis time of 113 minutes. 108 000.0 00.0 000. 10010 .000 20010 .000 00-0 000 . 000.0 00.0 3.0 20010 .02 100.0 .000 00-0 000 000 00.0 000 20010 .000 20010 .000 00-0 000 000 00.0 0.00. 20020 .000 10020 .000 00-0 000 00 0 00.0 .0 00 10010 .000 20010 .02 00-0 000 00. 00.0 000 20010 .02 10010 .000 00-0 000 00 00.0 00. 10000 .000 20000 .000 00-0 000 00 00.0 0: 20010 .000 10010 .000 00-0 000 000 00.0 3.0 20010 .000 20010 .000 00-0 000 000 :0 000 20010 .000 20010 .000 00-0 000 000 .00 000 10010 .000 10010 .000 00-0 000 0.0.0 .00 0.0.0 10010 .000 10010 .000 00-0 000 . .000 E8000 SE.2... 0. .0 0. 020300 0 020300 0 2.2300 . 255-.00 .0020 0000 0000.008 E20000 000 0. 05000000000 000002 Asemmov 0.00.006 00.5.0000. 0.000.020.0200 E:E_0.E 000 .0.E¢+..E. 00m2000. E:E_0.E .3. 0E: 0.00.000 000. .m.v 000 0.0 00.000. 0. 0020 00 .z 0:0. 0.0002000 00.... .IO 10. 60050.0 .0 0053:. 00000.00 00 0000200 00. .50 -9 00.._00_=0000.0>0.mv 000 awn. V 00.....03000080 000 00E0.00 00... 0000000000.... 0.00E000 000.00.00.00 .0 00.00.0000 05 .2 E0000 0... 00020000000 .00>_0m-03<0E0.00-o§ 0 .0 00000500000 05 .0 00:00.03 0.0 0.000 109 If resolution and not analysis time is considered to be the primary goal, the final term may be neglected in evaluation of the CRS function in Equation [4.4]. Under these conditions, the results for the permutations of a two-column/two- solvent chromatographic system are summarized in Table 4.4. In some cases, the optimum separation is virtually identical to that identified in Table 4.3; in other cases, a significant increase in resolution is achieved, usually at the expense of an increase in analysis time. Again, the most promising choice for column sequence is octadecylsilica followed by B-cyclodextrinsilica, and the most promising solvent sequence is 80% methanol followed by 50% acetonitrile. This permutation will be evaluated in further detail below. In order to identify the most favorable experimental conditions, the computer program is used first to determine the optimum lengths of the octadecylsilica and B—cyclodextrinsilica columns. The topographic and contour maps of the CRS response surface are shown in Figure 4.2 as a function of the column length. It is apparent that the overall quality of the separation improves dramatically with increasing length of the octadecylsilica column. Although the extent of the improvement after 25 cm is relatively small, the minimum CRS value is achieved for a column length of 75 cm. In contrast, the B—cyclodextrinsilica column has no beneficial effect upon the separation and, thus, has an optimal column length of 0 cm. The computer program is used next to determine the optimum lengths of the 80% methanol and 50% acetonitrile solvent zones. The topographic and contour maps of the CRS response surface are shown in Figure 4.3 as a function of the solvent zone length. There are several regions that yield low values for the CRS function; the overall minimum value is observed for solvent lengths of 167 cm of 80% methanol and 318 cm of 50% acetonitrile. 110 00. 00.0 000 10010 .000 200.00 .02 00-0 000 00. 00.0 :0 20000 .02 00010 .000 00-0 000 0. 2.0 000 10010 .000 20010 .000 00-0 000 0. 2.0 00.. 20010 .000 10010 .000 00-0 000. 0.0 00.0 000 00.00 .000 200.00 .02 00-0 000 0.0 00.0 000 20010 .02 100:0 .000 00-0 000 0.0 00.. 000 10010 .000 20010 .000 00-0 000 0.. 00.. 000 20000 .000 100:0 .000 00-0 000 0.0 2.0 3.0 20010 .000 20010 .02 00-0 000 0.0 .00 0.0 20000 .02 200.00 .000 00-0 000 0. .0.0 000 100.00 .000 10010 .000 00-0 000 0. .0.0 20 10010 .000 100:0 .000 00-0 000 SE000 00.2.0. 0.“... 0 020300 .020300 0 25.300 2.2300 000020.000 00.. .0 .000 0 00 0. 0.00.0000 .00 0. 0000 0.02000 0002. .0020 0.0 00000008 0020000 00.. o. 000000000000 00000:. A Emmo. 00000.0 00020000. 0.000.020.0200 0.0.0.0? .00 .50.... 0.. 00.. 0.0.0280 000 .10 .1 0 000. 00000.00: .3. 0.00 0.02000 0.... .00 000 0... 00.00.. 0. 0020 00 .200... 0. 6000.00. .0 00.0.x.E 0:00:00 0.0 0.00200 00.. .50.... 000.000.08.020 000 800. 000.0300 0.00 0.0 00.00.00 00 .- 0000.08.00... 0000.20 00.000200 .0 02.0.0000 0... .0. E9000 0.000. 0.0.0900 20200-0300820003. 0 .0 0000050000 0... .0 00000.0>m 06 030... 111 Figure 4.2A Topographic mafi of the CRS response surface as a function of the column lengt (cm) for octadecylsilica and B-cyclodextrinsilica. 112 Figure 4.2A I 113 Figure 4.28 Contour map of the CRS response surface as a function of the column length (cm) for octadecylsilica and B-cyclodextrinsillca. 75 U! D CYCLODEXTRINSILICA 114 Figure 4.28 4 ”11111111111 25 50 OCTADECYLSILICA 50 115 Figure _4.3A Topographic map of the CRS response surface as a function of the solvent zone length (cm) for 80% methanol and 50% acetonitrile. F iiiiiiiii 117 Figure 4.38 Contour map of the CRS response surface as a function of the solvent zone length (cm) for 80% methanol and 50% acetonitrile. 118 Figure 4.38 coo - BOO _ soo : son 1.. i... i... 50% ACETONITRILE N o o I J:- 200 100 ’ Q .1, 1 0 100 200 80% MEFHANOL O 119 The predicted separation of the isomeric four- and five-ring PAH with a 75 cm octadecylsilica column is shown in Figure 4.4. The optimal solvent modulation sequence of 80% methanol and 50% acetonitrile in zones of 167 and 318 cm length, respectively, provides a good overall separation (CRS = 1.9) with an analysis time of 666 min. Under these conditions, the critical solute pair is chrysene and benz[a]anthracene (Ri,i+1 = 1.39). All other solute pairs have resolution greater than the optimum value (1.5), which is desirable for accurate qualitative and quantitative analysis. . Although this optimized separation initially appears to be very promising, a more detailed examination of the CRS response surfaces is warranted. In the immediate vicinity of the optimum conditions shown in Figure 4.3, the CRS function is found to vary in a significant and abrupt manner. For example, as the solvent zone length of 80% methanol changes by :I:5 cm, the CRS value increases by approximately two orders of magnitude. Thus, small variations in the experimental conditions may have a highly detrimental effect upon the quality of the separation. The next most favorable permutation identified from Table 4.4, in which the column sequence is octadecylsilica followed by B-cyclodextn'nsilica and the solvent sequence is 70% acetonitrile followed by 80% methanol, does not suffer from this limitation. This permutation will be evaluated in further detail below. The topographic and contour maps of the CRS response surface are shown in Figure 4.5 as a function of the column length. The overall quality of the separation improves continuously with increasing length of the octadecylsilica column, up to the limit of 75 cm. As noted for the previous case, B- cyclodextrinsilica is relatively ineffectual and, thus, has an optimal length of 0 cm. The topographic and contour maps of the CRS response surface are shown in Figure 4.6 as a function of the solvent zone length. There is a broad Figure 4.4 120 Predicted chromatogram for the isomeric PAH with a 75 cm octadecylsilica column using the optimal solvent modulation sequence of 80% methanol and 50% acetonitrile in zones of 167 and 318 cm length, respectively. Solutes: (1) pyrene, (2) triphenylene, (3) benzo[c]phenanthrene, S4) chrysene, 25 benz[a]anthracene, (6) benzo[e]pyrene, (7 perylene, 8 benzo[a]pyrene, (9) dibenz[a,c]anthracene, (10 dibenz[a,h]anthracene. 121 Figure 4.4 700 NOLL‘V’HLNEIONOC) 3Ali‘v’738 o - a _ o —o J “O 01 fl - o —o J L0 00 - - . r\ J _8 ‘0 - ~<1~ 8 In J m V m 0 j] 0 N —0 j N o -o I I I I I I I I I I I O O ”0. ‘0. V“ N O. ‘— o o o o 0 TIME (min) 122 Figure 4.5A Topographic mafi of the CRS response surface as a function of the column lengt (cm) for octadecylsilica and B—cyclodextrinsilica. 123 Figure 4.5A 124 Figure 4.58 Contour map of the CRS response surface as a function of the column length (cm) for octadecylsilica and B-cyclodextrinsilica. 75 U! D CYCLODEXTRINSILICA El 125 Figure 4.58 --50 25 A. I 75 25 _ so OCTADECYLSILl CA 124 Figure 4.58 Contour map of the CRS response surface as a function of the column length (cm) for octadecylsilica and B-cyclodextrinsilica. 75 U! D CYCLODEXTRINSILICA bi 125 Figure 4.58 25 _ so OCTADECYLSILI CA 75 50 25 126 Figure 4.6A Topographic map of the CRS response surface as a function of the solvent zone length (cm) for 70% acetonitrile and 80% methano. 128 Figure 4.68 Contour map of the CRS response surface as a function of the solvent zone length (cm) for 70% acetonitrile and 80% methanol. 129 Figure 4.68 1.020.153. mom 0 000 000 00¢ 000 000 00. _ _ .é. . . _ 000.1 .I 7 r .9 0 a. . 000 I fur 00 _ _ _ .fi. _ _ . . _ 0 .1 000 000 00+ 000 000 00. HWIHilNOBOV 95OL 130 region of the surface having relatively low values for the CRS function; the minimum value is achieved for solvent lengths of 203 cm of 70% acetonitrile and 634 cm of 80% methanol. The predicted separation of the isomeric PAH with a 75 cm octadecylsilica column is shown in Frgure 4.7. The optimal solvent modulation sequence of 70% acetonitrile and 80% methanol in zones of 203 and 634 cm length, respectively, provides an excellent separation (CRS = 5.0) with an analysis time of 319 min. All solute pairs have resolution greater than the minimum acceptable value (0.3), whereas two pairs have resolution less than the optimum value (1.5). These critical solute pairs are chrysene and benz[a]anthracene (Rm +1 = 0.88), as well as benzo[e]pyrene and perylene (Rim = 0.88). Because the CRS response surfaces shown in Figures 4.5 and 4.6 are relatively flat in the vicinity of the optimum, small discrepancies in column length and solvent zone length should have little effect upon the quality of the separation. Thus, the analytical method should be relatively reproducible and rugged. In order to demonstrate the practical application of this method, the separation of the isomeric four- and five-ring PAH was performed under the predicted optimum conditions. The experimental chromatogram (Figure 4.8) shows excellent separation of all PAH standards with resolution comparable to that in the predicted chromatogram (Figure 4.7). The experimental retention time and peak width for each PAH standard agree well with the theoretically predicted values from Equations [4.1] and [4.2], respectively, as summarized in Table 4.5. The average relative error is :l:3.5% for retention time and 121% for peak width. Thus, the theoretical models developed herein can accurately predict the experimental results for solvent modulation in serially coupled columns. 131 Figure 4.7 Predicted chromatogram for the isomeric PAH with a 75 cm octadecylsilica column using the optimal solvent modulation se uence of 70% acetonitrile and 80% methanol in zones of 203 an 634cm length, respectively. Solutes as given in Figure 4.4. 132 Figure 4.7 350 O .__...2 ‘— fi 0 —-o P”) ___..——J 07 a .._..-a O 00 - ~10 of [\ .J to 77 o —o 01 O 10 __ 00 —10 V1” __——- J r- m 3 (\r - 'r— 7“ O -o __o 10 I 1 I I I I r I I I I O O 00 “O ‘1”. “f O. 0' o o o o o NOIi‘v’EliNBONOO BAILV-IBH TIME (min) Figure 4.8 133 Experimental chromatogram obtained under the predicted optimum conditions. Column: 200 um ID. x 75 cm fused silica capillary, packed with 5 pm octadecylsilica. Mobile phase: solvent modulation sequence of 70% acetonitrile and 80% methanol in zones of 203 and 634 cm length, respectively, 0.95 pL/min. Detectors: (A) UV-visible absorbance at 254 nm to indicate solvent modulation sequence, (8) Laser-induced fluorescence with excitation at 325 nm and emission at 420 nm to indicate solute retention. Solutes: (0) injection solvent, (1) pyrene, (2) triphenylene, (3) benzo[c]phenanthrene, (4) chrysene, (5 benz[a]anthracene, (6) benzocge]pyrene, (7) perylene, 8 benzo[a]pyrene, (9) ibenz[a,c]anthracene, (10 dibenz[a,h]anthracene. omm oom .000.000. 000 00. 134 Figure 4.8 or 01.11.1111 135 .00. 0 .0 0.23 2000 0 00.0.0000 00.... 02.003 0.0.. 00.0.0200 0 >0001...>0001. .. Exw 00. u 00000 .00 .. .0. 00000 0 0.0.. 00.0.0200... .00n 0.00 00<00>< 00 0.0 0.0 .0- 000 0.0 02020120002000 0.01 0.0 .0 . 0...- .00 .00 0200<01.2<.0.0.N200_0 0... 0... .... 0.0.. 000 :0 02050005200 00- 0.. 0.0 0.0 000 .00 020.000.. 00- 0... 0.0 0.0 :0 0.0 0200>0.0.0N200 0. .0 .0 .... 0... 00. 02020102300200 0. 0.0 .0 0.0 .0... .0. 0209.010 00 0.0 0.0 3. 00. 0... 02001.2<2010.0.0~200 t 0.0 00 0.0 00. 00. 020020120. 0. 0.0 0.0 5. .0. .0. .. 0200.... Emmfi 0.00001. .50 0mmM00 0.00001. .00 02000<0000>1 0.2200... 0...... 1.8.2. 0...... 0.2.. 20:20.00 «200002.300 .0... 000 0... 00.00.“. 0. 00.0000... 002.0000 0.0.0000 02200.0 00. .0000 0.22. 2000 000 08.. 00000.0. CEOmIb .0000.000. 000 memv .0.00.0..00x0 .0 000000080 0... 0.00... 136 4.5 Conclusion The complete optimization of separations in multimodal or multidimensional liquid chromatography is a nontrivial task due, in part, to the large number of variable parameters and, in part, to deviations from ideal retention behavior. The computer program developed in this work appears to present a promising approach for examining all possible permutations of parameters, such as stationary and mobile phases, that may be used in multidimensional optimization. Because the individual stationary and mobile phases are spatially separated from one another, solute retention is a simple, time-weighted average of each environment to which the solute is exposed. For a system comprised of q columns and n solvents, only q x n measurements of the solute capacity factor are necessary. From these measurements, retention may be accurately predicted for any sequence and length of the stationary and mobile phases. This approach was demonstrated for the separation of isomeric polynuclear aromatic hydrocarbons using octadecylsilica and B-cyclodextrinsilica stationary phases with aqueous methanol and acetonitrile mobile phases. Based on these results, this approach appears to be a viable strategy for the simultaneous optimization of stationary and mobile phase environments. 4.6 137 References J.H. Wahl, Ph.D. Dissertation, Michigan State University, 1991. J.C. Gluckman, A. Hirose, V.L. McGuifin, and M. Novotny, Chromatographia 17, 303 (1983). l(-l.J. Issaq, D.W. Mellini, and T.E. Beesley, J. Liq. Chromatogr. 11, 333 1988 . . D.W. Armstrong, W. DeMond, A. Alak, W.L. Hinze, T.E. Riehl, and K.H. Bui, Anal. Chem. 57, 234 (1985). D.W. Armstrong, A. Alak, W. DeMond, W.L. Hinze, and T.E. Riehl, J. Liq. Chromatogr. 8, 261 (1985). M. Olsson, L.C. Sander, and SA Wise, J. Chromatogr. 477,277 (1989). PR. Fielden and A.J. Packham, J. Chromatogr. 516, 355 (1990). CHAPTER 5 MULTIVARIATE OPTIMIZATION OF MOBILE PHASE AND TEMPERATURE BY PARAMETRIC MODULATION 5.1 Introduction The goal of this chapter is to present an experimental verification of the simultaneous optimization of mobile phase and temperature by the parametric modulation approach. By maintaining different portions of a polymeric octadecylsilica column at different temperatures and the mobile phases as separate zones, the solute will interact independently in each environment (Figure 5.1). Thus, the overall solute retention will be a simple, predictable function of the capacity factor weighted by the time that the solute is exposed to _ each temperature region and mobile phase. This approach should allow accurate prediction of solute retention and, hence, allow optimization of multidimensional separations with a minimum number of preliminary experiments. 52 Theoretical Development The theoretical basis of solute retention and dispersion under conditions of parametric modulation has been established in Chapter 2. For the case of optimizing mobile phase and temperature, the technique requires that the overall 138 CHAPTER 5 MULTIVARIATE OPTIMIZATION OF MOBILE PHASE AND TEMPERATURE BY PARAMETRIC MODULATION 5.1 Introduction The goal of this chapter is to present an experimental verification of the simultaneous optimization of mobile phase and temperature by the parametric modulation approach. By maintaining different portions of a polymeric octadecylsilica column at different temperatures and the mobile phases as separate zones, the solute will interact independently in each environment (Figure 5.1). Thus, the overall solute retention will be a simple, predictable function of the capacity factor weighted by the time that the solute is exposed to each temperature region and mobile phase. This approach should allow accurate prediction of solute retention and, hence, allow optimization of multidimensional separations with a minimum number of preliminary experiments. 52 Theoretical Development The theoretical basis of solute retention and dispersion under conditions of parametric modulation has been established in Chapter 2. For the case of optimizing mobile phase and temperature, the technique requires that the overall 138 139 Figure 5.1 Schematic illustration of the parametric modulation concept for the simultaneous optimization of mobile phase and temperature 140 Figure 5.1 SOLVENT RESERVOIRS TEMP 1 CONTROL COLUMN TEMP 2 CONTROL O \ 0 ° ll TEMP q CONTROL LL \\\\\\\\\\\\\ *\ 141 retention time (ti) be calculated when the solute is exposed to each mobile phase and temperature region. This is accomplished by summing retention within each individual environments as shown below q n . . 2 2 0., [1w] [5.1] "'1 M F kip where kip is the capacity factor of solute i in mobile phase j on column at temperature p, rp and 2p are the radius and total porosity of column at temperature p, respectively, xi is the length of solvent zone, F is the volumetric flow rate, n and q are the limits of summation of number of solvent and temperature zones, respectively. In a similar manner, the variance (0,2) of the solute zone in temporal units is calculated by independent summation as q 2 hp tip I P" p [2 “"2 E’x‘ (MED: [5.21 q n (5)2 = X Z (0» )2 ( ' pal i-O 'p i=0 F kip where hp is the reduced plate height, (1.3 is the particle diameter, and ID is the length of column at temperature p. The resolution between adjacent solute zones (RU +1) is then calculated by substitution of Equations [5.1] and [5.2] in the following expression (tin “til 3' = 5.3 "M 21M“): + (0th] I ] 142 Finally, a criterion such as the chromatographic resolution statistic (CBS) (1) shown below is used to evaluate the overall quality of the separation [m—l R R 2 1 m—1 (R )2 CBS = 2(M)—+Z ‘4'“ J—tt 5.4 i-1 RiJ-rt‘nmin Run i=1 ("i-1) Rat/92 m I l where Hop, and Rmin are the optimum and minimum resolution, respectively, RaVg is the average resolution, t, is the analysis time and m is the number of solutes. The computer program used to accomplish these calculations is given in Appendix [A1.2]. 5.3 Experimental Methods Chromatographic System. The experimental system is illustrated schematically in Figure 5.1. A dual syringe pump (Model 140, Applied Biosystems, San Jose, CA, USA) is programmed to deliver solvent zones for a specified time period. The sample is introduced by means of a 1.0 uL injection ' valve (Model ECI4W1, Valco lnstmments, Houston, TX, USA), after which the effluent stream is split (75:1) to yield a nominal injection volume of 13 nL, and a nominal flow rate of 1.2 uUmin. In order to ensure rapid thermal transport and equilibration, the chromatographic column is prepared from fused silica tubing of capillary dimension (350 um 0.0., 200 um l.D., Polymicro Technologies, Phoenix, AZ, USA) which is terminated at the desired length with a quartz wool frit. An irregular silica (5.5 pm IMPAQ 200, PO Corporation, Conshohocken, PA, USA) with a polymeric octadecylsilica phase (5.4 umonZ), is slurry packed 143 according to the procedure described previously (2). The resulting column has total porosity (5p) of 0.90, separation impedance (clip) of 450, and reduced plate height (hp) of 3.7 at 45°C, 43°C at 40°C, 4.7 at 35°C and 4.5 at 23°C using pyrene as a model solute under standard test conditions (2). For independent control of each temperature zone, the column is enclosed in water jackets contructed of nylon tubing (0.32 cm 0.0., 0.2 cm ID., Cole-Parmer, Niles, lL, USA) and fittings (Swagelok NY-200-3, Crawford Frtting Co., Solon, OH, USA). Water circulation is provided by thermostatically controlled baths (Model RTE 9B; Neslab Instruments, Portsmouth, NH, USA), with an operating range of —20 to 100°C (:r:0.1°0). The polynuclear aromatic hydrocarbon solutes are detected using both laser-induced fluorescence and UV-absorbance in a 75 um l.D. fused-silica capillary flow cell. In the fluorescence detection system (3), a helium-cadmium laser (325 nm, 25 mW, Model 3074~40M, Omnichrome, Chino, CA, USA) is reflected by a dielectric mirror and is focussed on the capillary with a quartz lens. The fluorescence emission is collected perpendicular and coplanar to the excitation beam with another quartz lens, spectrally isolated with a bandpass interference filter (420 nm, S1 0-420-F, Corion, Holliston, MA, USA), and focussed onto a photomultiplier tube (Model Centronic 042498, Bailey Instruments, Saddle Brook, NJ, USA). The photocurrent is amplified and converted to voltage by a picoammeter (Model 480, Keithley Instmments, Cleveland, OH, USA). A variable-wavelength UV-absorbance detector (Model Uvidec too-v, Japan Spectroscopic Co., Tokyo, Japan), operated at 254 nm, is placed in series after the fluorescence detector. Data from both detectors are displayed simultaneously by using a chart recorder (Model 585, Linear instruments Corp., Reno, NV, USA) 144 Materlals and Method. Reagent-grade chrysene, benz[a]anthracene, benzo[c]phenanthrene, pyrene, perylene, benzo[a]pyrene, and benzo[e]pyrene are obtained from Sigma Chemical Co. (St. Louis, MO USA). Tetrabenzonaphthalene and phenanthro[3,4—c]phenanthrene are obtained from the National Institute of Standards and Technology (Gaithersburg, MD, USA). Standard solutions of each polynuclear aromatic hydrocarbon (PAH) are prepared by dissolution in methanol at 104 M concentration. Organic solvents are high purity distilled-in-glass grade (Baxter Healthcare, Burdick and Jackson Division, Muskegon, MI, USA). Computer-Assisted Optimizatlon Program. The separation is optimized by means of the computer-assisted optimization routine discussed in Chapter 4. The program was executed on a VAXstation 3200 computer (Digital Equipment, Maynard, MA, USA). The program utilizes a systematic maping procedure in which the length of each parameter to be optimized is varied independently within prescribed limits. For the present study, temperature zone length is varied from 0 to 90 cm, and the solvent zone length is varied from 5 cm to the maximum length required to elute all solutes. For each combination of temperature and solvent zone length, the total retention time and variance of each solute are calculated by means of Equations [5.1] and [5.2], respectively. The resolution between each pair of adjacent solute peaks is calculated by using Equation [5.3]. Finally, the 1 overall quality of the separation is assessed by means of the modified CRS function (Equation [5.4] and [2.19]), using selected values for the optimum and minimum acceptable resolution of 1.5 and 1.0, respectively. By graphing the CRS value as a function of temperature and solvent zone lengths, a complete multivariate response surface may be constructed. The minimum CRS value is 145 then identified by visual inspection or by computer search of the response 50 rface. 5.4 Results and Discussion The goals of this preliminary study are to demonstrate the conceptual and theoretical basis of parametric modulation, and to apply this strategy to the multivariate optimization of mobile phase and temperature. In preparation to meet these objectives, it is worthwhile to elucidate and to clarify the condition under which a spatial gradient in temperature will be most beneficial. In a spatial gradient, unlike a temporal gradient, all solutes will encounter all temperature zones in an analogous manner (Equation (5.1)). Solutes that exhibit a linear relationship between In k and 1/T (Equation (1.21)) will spend the same fraction of their total retention time in each temperature zone. For these ideal solutes, there exists a single temperature that is exactly equivalent to the spatial temperature gradient, whether that gradient is discontinuous (as described herein) or continuous in nature (4). Consequently, a spatial temperature gradient can be utilized for all solutes but will only be superior to isothermal conditions for solutes that deviate from ideal thermodynamic behavior. On the basis of this consideration, the chromatographic system and solutes chosen for these preliminary studies should exhibit a distinct change in the retention mechanism with temperature. Although there are a variety of chromatographic systems that meet this criterion, octadecylsilica has been chosen as the stationary phase because of its widespread use and general applicability in liquid chromatography. This type of stationary phase has been shown to undergo a phase transition, where the transition temperature is 146 dependent upon the bonding density of the alkyl group, the alkyl chain length, the mobile phase composition and other variables (5-7). Whereas the exact thermodynamic nature of this phase transition remains the subject of continued study and controversy, it is nevertheless apparent that a significant change in the stmcture and morphology occurs. These changes have been examined by a number of analytical techniques, including nuclear magnetic resonance spectroscopy (840), Fourier-transfonn infrared spectroscopy (11,12), thermal analysis, (13), as well as chromatographic methods. At temperatures above the transition point, the stationary phase appears to be randomly oriented and flexible, so that solute retention is dominated by enthalpic interactions. As temperature is decreased below the transition point, the stationary phase becomes progressively more rigid and ordered, so that solute retention acquires a correspondingly greater entropic contribution. Under ”entropy dominated” conditions, there is a significant increase in the selectivity of octadecylsilica with respect to the size and shape of solute molecules (3). Throughout the phase transition, both AH and A8 are temperature dependent, leading to substantial deviations from ideal behavior for many solutes. For these studies, the model solutes consist of isomeric four-, five-, and six-ring polynuclear aromatic hydrocarbons (PAH), whose structures are shown in Figure 5.2. A polymeric octadecylsilica with a high bonding density of 5.4 pmoIe/m2 is utilized as stationary phase. From previous studies (14), the transition temperature for this material is known to be approximately 45° C. Because of the progressive nature of the phase transition, however, it is not known where the most optimal variations in selectivity will occur for the model solutes. Thus, it is desirable to examine a broad range of temperatures in the vicinity of the transition temperature. Representative chromatograms of a 147 Figure 5.2 Structures of polynuclear aromatic hydrocarbons cos? BENZ[a]ANTHRACENE O Ooo 6%? © 0% BENZO[c]PHENANTHRENE O BENZO[e]PYRENE C0) ©© PHENANTHRO[3,4-C]PHENANTHRENE © © 149 mixture of the model solutes are shown Figure 5.3 at various temperatures in the range from 23 to 45° C. The capacity factors of each PAH were measured at temperatures of 23, 35, 40, and 45° C using pure methanol (T able 5.1) and acetonitrile (Table 5.2) as mobile phases. In general, the capacity factors of the PAHs decrease with increasing temperature for both mobile phase. In the methanol mobile phase, a change in the elution order of chrysene and the tetrabenzonaphthalene occurred at about 35° C. In the acetonitrile mobile phase, a change in the elution order of phenanthro[3,4-c]phenanthrene and benz[a]anthracene occurred at around 35° C. In order to further examine the influence of temperature on the retention and selectivity of the solutes, a plot of In R vs. 1fl' was obtained (van't Hoff plot) for both mobile phases. Using methanol as the mobile phase (Figure 5.4), it is apparent that the logarithm of capacity factor increases linearly with temperature between 23° C and 35° C, but deviates from linearity at 35° C. This deviation is more clear with some solutes than with others. For example, tetrabenzonaphthalene and chrysene show greater deviations from linearity than the other solutes. This deviation from a linear trend of In k vs 1/T at 35° C and change in elution order of solutes suggests a change in the retention mechanism of the stationary phase. A similar break in the linear trend of In k versus 1/T was also observed when acetonitrile was used as the mobile phase (Figure 5.5). Based on the results shown above, it seems reasonable to expect that resolution of the isomeric PAHs can be achieved by modulation of the mobile phase and temperature. However, there are 64 possible permutations for the solvents and temperatures examined in this study: 8 one-solvent/one- temperature systems, 24 one-solvent/two-temperature systems, 8 two- solvent/one-temperature systems, and 24 two-solvnt/two-temperature systems. It would be clearly impossible to examine all of these possibilities by 150 Figure 5.3 Effect of temperature on the selectivity of polynuclear aromatic hydrocarbons using 100% methanol as mobile phase. Column: 200 mm ID. x 75 cm fused silica capillary, packed with 5.5 mm polymeric octadecylsilica. Mobile phase: 100% methanol at 1.2 uL/min. Detector: Laser-induced fluorescence with excitation at 325 nm and emission at 420 nm. Solutes: (1) Benzo[c]phenanthrene (2) pyrene (3) phenanthrophenanthrene 4 benz[a]anthracene (5) tetrabenzonaphthacene (6) chrysene (7) benzo[e]pyrene (8) perylene (9) benzo[a]pyrene. 151 Figure 5.3 2 “-14 1 7 9 55 ' a 45°C 2 “(’4 1 7 ss 9 40°C -431 " 2 “‘1 1 7 5 9 35°C a 3 ._J_ 4 2 23’c 1 7 6 a A j 1b IL [L t W I Y I T if 2'0 4'0 :0 20 100 120 TIME (min) 152 oo..o. ohm ohm o5. mzwmiEoszm o: o: 23 3o mzmdfimo om; 8; N5 ohm wzmmififimzwm aho who who ohm mzmm>mxo o S E; mo; oo; w zmox 402/5152 he oos 0.220% as ESE >Eoomnmfio otmEzoa no 209893: oszBm 52025.8 .5. A S 9203 bfimamo fim 03¢... 153 who 4 I 3: So. mzmmEEoszm Ko who 84 oo; mzwfimmo who ooo oho o3 mzmmiiomzmm who oho oho so; mzmmEIo rho Ko who oo; wzwoz 5.5205? .x. oos 0.220% of 555 $2.20 $303258 .mmmca 250E mm QEEBmom $03 9%: mmoamomaEE Emakfi “a nmcfifio $sz amcozflm mo___m_>omnm~oo otmEzoa :0 209893; oszBm 320258 .2 A S 920$ 363.8 mm 2an 154 Figure 5.4 Effect of temperature on logarithm of capacity factor for lynuclear aromatic hydrocarbons using 100% methanol as mo ile phase. Solutes: ( A ) Benzo[c]phenanthrene ( O ) pyrene ( 6] ; phenanthrophenanthrene A ) benz[a]anthracene ( tetrabenzonaphthacene ( I ) chrysene ( O ) benzo[e]pyrene ( O )perylene( V )benzo[a]pyrene. 155 figure 5.4 \\\) mmood O0 EDZKEEC; 40300 000.00 m00_0.0 Fmo_o.o omood ind lo; oh El OiOVj All O‘v’d‘v’O 907 156 Figure 5.5 Effect of temperature on logarithm of capacity factor for polynuclear aromatic hydrocarbons using 100% acetonitrile as mobile phase. Solutes: ( A ) Benzo[c] henanthrene ( O ) pyrene ( D ; phenanthrophenanthrene A ) benz[a]anthracene ( 0 tetrabenzonaphthacene ( I ) chrysene ( O ) benzo[e]pyrene ( O )perylene( V )benzo[a]pyrene. 157 figure 5.5 N mmood 00 EEEEEEE 430.0 0030.0 m00_0.0 50.0.0 omood Imd to; 0.0 901 HOLOVJ AilOVdVO 158 time-consuming, trial-and-error experimental measurements. However, the computer program described previously provides a rapid and effective means to identify the most promising of these permutations. In order to exploit the different selectivity of the stationary phase at different temperatures, the column should be maintained at more than one temperature. This was accomplished by maintaining different portions of the column at different temperatures as shown in figure 5.1. By using the computer program described earlier, the optimum length of column maintained at each temperature and the length of solvent necessary for the total separation of all the solutes can be determined. This is achieved by systematically varying the length of column maintained at each temperature as well as the length of solvent zone. For each combination of column and solvent lengths, the total retention time and variance of each solute are calculated by means of Equations 5.1 and 5.2, respectively, using the individual capacity factors in Tables 5.1 and 5.2. The resolution between each pair of adjacent solute is calculated by means of Equation 5.3. Finally, the overall quality of the separation is assessed by means of the modified CFlS function (15), using selected values of 1.5 and 1.0 for the optimum and minimum acceptable resolution. Table 5.3 shows a few of the possible permutations of column temperature and solvent that may be utilized for the separation of the PAH solutes. For the case of one solvent, only one sequence of temperatures is investigated and not the reverse, because solute retention will be independent of the column order. For each pairwise combination of column temperatures, the length of column is systematically varied and the minimum CBS is stored in a file and used to identify the optimum conditions. For the case of optimizing temperature alone, the best combination of temperatures is predicted to occur using columns at 23° C and 40° C. The minimum resolution predicted under this condition is 1.31 with a total 159 hm oh? o.oo Iooxo so? zooxo hoooo om ov hm mo; oso Iooxo hooos zooxo hooos o4 om. hm 4o; o.oo zoozo c\eoS 1033 so? om o4 hm mo; o..o.os zoozo hooos IooIo o\eoS oo om hm No; 35 2096 hooor mo mm to om; oxo zoozo nso? oo ow to om; oso zooxo coo? mo om hm om; ho: Ioozo o\ooe mo om hm so; hows :ooIo nso? oo om hm om; hoS Iooxo c\coS mm mm semmo 52...? $5 3 m Einm a 0.5300 “wopoafiww umxmx‘ww .520 mom 22:38 52530 9: 2 059.8950 5:22 38 $8 23sz 5:282”. ofanfiEoE E:E_c_E 0cm Erotic: 5:239 E:E_c_E é: mEz Em Raw 9:. .m.m 0cm fim menu... E .520 mm EEEBmom ncm _ocm£mE $03 2m 2:028 05 62.638098 020628 2 cE:_oo 9:. 632893: ozEBm 52023.8 3 cosmomamm m5 .6. E29? oEamomofiEoEo #:0209035.293E265 0cm EozomécoE.398E265 m .o mcozwsgma m5 .0 cozwamsw «cm 220... 160 analysis time of 120 min. For the case of optimizing both the sequence nd length of the column at two temperatures are varied. The optimum corresponds to column temperature of 23° C and 40° C and using acetonitrile and methanol. The minimum resolution predicted under this condition is 1.35 with a total analysis time of 87 min. Since the optimum obtained using both mobile phase and temperature is not significantly better than that using temperature alone, a two-temperature modulation sequence with one mobile phase was pursued for the optimization, because of the ease with which it can be implemented. The topographic and contour maps are shown in Figures 5.6A and 5.68, as a function of the column length at each temperature. It is observed that the overall quality of the separation improves continuously with the length of column at 23° C, and remains fairly constant for length of column at 40° C for all lengths greater than 5 cm. Thus, for the total column length of 90 cm, the predicted optimum length of column at 23° C and 40° C are 85 cm and 5 cm, respectively, using pure methanol as mobile phase. The predicted chromatogram shown in Figure 5.7 shows excellent separation of the PAHs. Because the CRS response surface shown in Figure 5.6A is relatively flat in the vicinity of the optimum, small discrepancies in column length at the two temperatures should have little effect upon the quality of the separation. Thus, the analytical method should be rugged. Experimental verification of the predicted optimum was obtained by the separation of the PAHs in Figure 5.8 using the predicted optimum conditions. The experimental chromatogram shows an excellent separation of all solutes. Table 5.4 shows a comparison of the experimental and predicted retention times and peak widths. There is an excellent agreement of the predicted retention times and peak width with average errors of :I: 0.9% and :t 6.9% respectively. Thus temperature and mobile phase can be used simultaneously to affect solute 161 Figure 5.6A Topographic map of the CRS response surface as a function of the column length (cm) at temperatures 23° C (Temp. 1) and 40° C (Temp. 2) 162 Figure 5.6A 163 Figure 5.6A Contour map of the CRS response surface as a function of the column length (cm) at temperatures 23° C (Temp. 1) and 40° C (Temp. 2) 8 8 COLUMN LENGTH AT TEMP. 2 (cm) a a 164 Figure 5.68 ,D. . O J l 20 4o . COLUMN LENGTH AT ~80 20 165 Figure 5.7 Predicted chromatogram for the polynuclear aromatic hydrocarbons on polymeric octadecylsilica column using the optimal conditions of 5 cm of column maintained at 23° C and 85 cm maintained at 40° C; using 100% methanol as the mobile phase. Solutes: (1; Benzogc1phenanthrene (2) pyrene (3) phenanthrophenanthrene 4 benz[a anthracene (5) tetrabenzonaphthacene (6) chrysene (7) benzo[e]pyrene (8) perylene (9) benzo[a]pyrene. 166 NOIch‘JiNHONOD BAILWEH figure5.7 J 0 07 —N fl _ o —-o m __.J _o J ”O r\’ fl o A -8 L0 4 ¢ a I- PO —1 Q N ______;7 : ~d‘ __o (\l 1 l 1 r l T l r 7 l r O o co to <1: (\I o - o o o o 0 TIME (min) 167 Figure 5.8 Experimental chromatogram obtained under the predicted optimum conditions. Column: 200 mm ID. x 75 cm fused silica capillary, packed with 5.5 mm polymeric octadecylsilica. Mobile phase: 100% methanol at 1.2 uUmin. Detector: Laser-induced fluorescence with excitation at 325 nm and emission at 420 nm. Solutes : (1) Benzo[ciphenanthrene (2) pyrene 3 phenanthrophenanthrene 4) benz[a anthracene tetrabenzonaphthacene (6) chrysene (7) nzo[e]pyrene 8 perylene (9) benzo[a]pyrene. 168 figure 5.8 9 r I 120 7 A 8 JL TIME (min) 2 1 . 5 6 - 3 TM 410 I 6° #4 169 .53 v 5 £23 xmma w 05Eammm .53 5:35 Ea: 86.3030 0 EBIEEBE .. Evan o8 u «.25 has . Ea 8%: u 52. 85.8.8“ , ooh hon.n m0< to hm hm no- oomr ho: wzmmEEOszm 4.3 o; o.m ho- too So mzmimmo 3.- o; h F to- h: is mzmeEOszm to o; 3 ho- too moo mzmmézo no I NS ho to how mzwox 0.220% 35 :52, 5.5 92; 20.55% $303250; hm ooo B oeoom s 859 05 80:: £22, xmma 0cm 0E: c2229 9.52.5 .mozmcomé 0cm mem A __ mcoEncoo EaEzao 022090 fiEmEtmaxm .o comtmano «.0 min». 170 selectivity and the optimization strategy developed in this work enables an accurate prediction of the experimental conditions for the optimization. 5.5 Conclusions The complete optimization of mobile phase and temperature is achieved for the separation of isomeric polynuclear aromatic hydrocarbons. By subjecting portions of polymeric octadecylsilica column to different temperatures, an optimum combination of temperature along the length of the column is found to provide adequate separation of the PAH molecules. Because the individual temperature regions are spatially separated from one another, lsolute retention is a simple, time-weighted average of each environment to which the solute is exposed. From these measurements, retention may be accurately predicted for any sequence and length of column at each temperature. The strategy developed in this work appears to be a promising approach for systematically searching all possible combinations of temperature and mobile phase parameters required to achieve the optimum separation. 5.6 N99?!” 9" 10. 11. 12. 13. 14. 15. 171 References T. D. Schlabach and J. L. Excoffier, J. Chromatogr., 439, 173 (1988) J.C. Gluckman, A. Hirose, V.L. McGuffin, and M. Novotny, Chromatographia 17, 303 (1983). M. A. Heindorf and V. L. McGuffin, J. Chromatogr., 464, 186 (1989) L. K. Moore and E. Synovec, Anal. Chem, 65, 2663 (1993) J. Chmielowiec, and H. J. Sawatzky, Chromatogr. Sci, 17,245 (1979) L. C. Sander and S. A. Wise, Anal. Chem, 61, 1749 (1989) K. Jinno, T. Nagoshi, W. Tanaka, M. Okamoto, J. Fetzer, J. R. Biggs, P. R. Griffiths, J. M. Oiinger, J. Chromatogr. 461, 209 (1989) R. K. Gilpin, Anal. Chem, 57, 1465A (1985) D. W. Sindorf, G. E. Marciel, J. Am Chem. Soc, 105, 1848 (1983) D. W. Sindorf, G. E. Marciel, J. Am Chem. Soc, 105, 3767 (1983) L. C. Sander, J. B. Callis, L. R. Field, Anal Chem, 55, 1068 (1983) D. E. Leyden, D. S. Kendal, L. W. Burgraf, F. J. Pem, M. Debello, Anal. Chem, 54,101 (1982) D. Morel, K. Tabar, J. Serpinet, P. Claudy, J. M. Letoffe, J. Chromatogr., 395, 73 (1987) S. H. Chen and V. L. McGufffin, Anal. Chem, Manuscript in preparation. P. H. Lukulay and V. L. McGuffin, J. Chromatogr., 691, 171 (1995) CHAPTER 6 EXPERIMENTAL AND COMPUTER SIMULATION STUDIES OF SOLUTE-SOLUTE INTERACTIONS IN LIQUID CHROMATOGRAPHY In the most complete and rigorous theoretical models of chromatography, discussed in Chapter 1, the interaction of all species is predicted to contribute to solute retention and dispersion. However, in the practical application of chromatographic theories, solute-mobile phase and solute-stationary phase interactions are considered predominant, whereas solute-solute interactions are invariably neglected. In this chapter, an experimental and computer simulation study of solute-solute interactions is presented, and its effect on solute retention and dispersion discussed. 6.1 Introduction Solute-solute interactions may be broadly defined as an intimate contact or short-range association between molecules that persists as the concentration of solute is decreased. Solute-solute interactions have been studied extensively in bulk solution by measurement of colligative properties such as vapor pressure, melting point, freezing point, conductance, etc. (1-3). in addition, infrared, NMR, and other spectroscopic methods have been used to examine solute-solute interactions at the molecular level in order to identify the bonding sites and to determine the aggregation number (4-6). By combining the information obtained 172 173 from these two types of experimental measurements, the equilibrium constant(s) for solute aggregation as well as the activity coefficients and excess thermodynamic functions for the solution may be calculated for comparison with theoretical models. From a theoretical perspective, solute-solute interactions represent a deviation from ideal solution behavior by violation of the assumption of random molecular distribution. A variety of theoretical models have been developed to account for these deviations. In the classical paper by McMillan and Mayer (7), the grand canonical ensemble method was applied to the generalized case for multicomponent gas or liquid systems. This statistical thermodynamic approach enabled the prediction of the radial distribution function and, henceforth, the thermodynamic properties of the solution. Stigter (8) combined the McMillan- Mayer theory together with simple models of van der Waals and hydrogen bonding forces to interpret the thermodynamics of aqueous solutions of sucrose and glucose. Kozak, Knight, and Kauzmann (1) similarly combined the McMillan- Mayer theory with several lattice models for aqueous solutions of hydrophobic solutes. Nemethy and Scheraga (9,10) as well as Pratt and Chandler (11,12) subsequently developed models of solute-solute interactions based on the hydrophobic theory (13-15). Although the thermodynamic consequences of solute-solute interactions are reasonably well understood in bulk solution, this understanding has not been widely applied to multiple phase systems such as extraction and chromatography. At the present time, very few studies have documented the effect of solute-solute interactions in chromatography. Amaya and Sasaki (16) investigated the gas chromatographic retention behavior of a binary mixture of chloroform and methyl ethyl ketone on a nonpolar stationary phase (Apiezon J). In addition, they examined binary mixtures of chloroform with carbon tetrachloride 174 and with toluene on a polar stationary phase (polyethylene glycol). In every case, they observed an increase in the retention time of both solutes in the binary mixture, which was attributed to solute-solute interactions in the stationary phase. These effects were qualitatively explained by using a theoretical model in which the mobile phase was treated as an ideal gas and the stationary phase as a regular solution (16). More recently, solute-solute interactions have been implicated in supercritical fluid and liquid chromatography (17-19). However, because of the innate complexity of condensed phases, a rigorous and comprehensive theoretical model has yet to be developed. Classical thermodynamic models based on regular solution theory (20-24) and the hydrophobic theory (25,26), as well as statistical thermodynamic models (27-29) have been developed for the prediction of solute retention. In practical application of such models, solute-mobile phase and solute-stationary phase interactions are considered predominant, whereas solute-solute interactions are invariably neglected. This neglect is often justified by an argument of statistical probability, since the concentration of solute molecules is small with respect to that of the phases. However, if the energy of interactions is sufficiently large, solute—solute interactions may become important despite the low concentration. In this study, the self-association and mixed association of corticosteroids are demonstrated to arise under routine operating conditions in reversed-phase liquid chromatography. In order to facilitate understanding of the origin and nature of these strong solute-solute interactions, molecular mechanics and dynamics calculations are employed. finally, these results are used to explain the observed deviations in chromatographic retention and dispersion behavior. 175 6.2 Experimental Section Materials and Methods. The following corticosteroids are utilized in this investigation: cortisone (17a,21-dihydroxy-pregn-4-ene-3,1 1 ,20-trione), tetrahydrocortisol (3a,1 1 8,17a,21-tetrahydroxy-5j3-pregnane-20—one), tetrahydrocortisone (3a,17a,21-trihydroxy-58-pregnane-1 1 ,20-dione), and methylprednisolone (1 1 8,17a,21-trihydroxy-6a-methyl-pregna-1 ,4-diene-3,20- dione). These corticosteroids, shown in Figure 6.1, are obtained from the Sigma Chemical Company (St. Louis, MO, USA). Standard solutions are prepared in methanol at 10-6 M concentration for cortisone and methylprednisolone and at 10-3 M concentration for tetrahydrocortisol and tetrahydrocortisone. Organic solvents are high-purity, distilled-in-glass grade (Baxter Healthcare, Burdick & Jackson Division, Muskegon, MI, USA); water is deionized and double distilled in glass (Model MP-3A, Corning Glass Works, Corning, NY, USA). Experimental System. A chromatographic pump equipped with two 40-mL syringes (Model 140, Applied Biosystems, Foster City, CA, USA) is used to deliver the mobile phase, 35% aqueous acetonitrile, at 0.5 mUmin. The solutes are introduced by means of a 10-uL injection valve (Model E0 60, Valco instrument 00., Houston, TX, USA) to the reversed-phase liquid chromatography column (47 cm x 0.46 cm i.d., 5-pm octadecylsilica, Spheri-S PIP-18, Applied Biosystems). Solute detection is accomplished by using a variable-wavelength UV-VIS absorbance detector (240 nm, 0.005 AUFS, Model 166, Beckman Instruments, San Ramon, CA, USA). The chromatographic figures of merit, such as area, capacity factor, plate number, skew, etc., are assessed by manual calculation according to the method of Foley and Dorsey (30) for exponentially modified Gaussian peak profiles. 176 Figure 6.1 Structures of corticosteroids. 177 figure 6.1 CIEHZOH ('31on CORTISONE TETRAHYDROCORTISOL CIJHZOH CHZOH METHYLPREDNISOLONE TETRAHYDROCORTISONE 178 Molecular Mechanics and Dynamics Simulations. Simulations of the interaction between corticosteroid molecules are performed using classical molecular mechanics and dynamics methods (BioGraf version 3.0, Biodesign lnc., Pasadena, CA, USA) on a Silicon Graphics Indigo computer (Model CMN8003, Mountain View, CA, USA). A generic force field, Dreiding (31), is employed to calculate the total energy of the molecule as the sum of the bonding and non-bonding interactions. The bonding interactions include contributions from stretching (E,), bending (Eb), and torsional (Em) energy between atoms that are covalently bonded. The non-bonding interactions consist of contributions from van der Waals (Em), electrostatic (EC), and hydrogen bond (Em) energy between atoms that are not covalently bonded. The van der Waals energy is expressed by a standard Lennard-Jones equation EVW = ARiiTIZ - Baa—6 [6.1] where Rij is the distance between atoms i and j, and A and B are empirically derived constants. The electrostatic energy (E0) is calculated by using Coulomb's law where O, and q are the net charge on atoms i and j, respectively. The dielectric constant is assumed to be that of a vacuum environment (2 = 1). The hydrogen bonding energy is expressed as: Ehb = Dhb [5 (Rm/RDA)12 " 6 Ith/ RoAlwl 005‘ (9mm) [53] where OM is the bond angle between the hydrogen donor (D), the hydrogen atom (H), and the hydrogen acceptor (A), while RDA is the distance between the donor and acceptor. D”, and R”, are the energy and the maximum distance used 179 to define the hydrogen bond, the magnitude of which depend on the convention used for assigning charges in the force field model (31). To simulate the solute—solute interactions between the corticosteroids, a systematic three-step approach is used. First, the bonding energies are minimized in order to determine the optimum three-dimensional structure and charge distribution for each corticosteroid. These individually optimized structures are then arranged in pairwise combinations. Next, a Monte-Carlo search is performed to examine all possible spatial orientations and to identify those with lowest energy (32). For this study, the steroid pairs are randomly varied in 200 different relative spatial positions and the non-bonding energy is minimized in 30 incremental steps at each of these positions. The final stage of the optimization involves a more refined energy minimization of the most promising orientations (typically 50) identified from the Monte-Carlo search. The annealed dynamics method simulates the exchange of thermal energy between the environment and the steroid pair, thereby allowing translational, vibrational, and rotational motion to minimize the total energy. For this study, the steroid pairs are simulated to be annealed in the temperature range from 300 to 600 K in 20 incremental steps. From among all of these conformations, the one with the lowest total energy is identified as the optimum stmcture for the steroid pair. The total interaction energy (AE) is calculated by subtracting the energy at the optimum separation distance from that at infinite distance. AE = Eop, - E... [6-4] In the same manner, the van der Waals (Ava), electrostatic (AE0). and hydrogen bonding (AE)”) components of the total energy can be calculated: 45m = 5mm - 5am [65] 180 AEQ = Em - E... [6.61 A6... = 5...... - E..... [6.71 When defined in this manner, the most stable solute-solute pair will have the greatest negative interaction energy. 6.3 Results Experimental Studies. In a previous study discussed in Chapter 3, the routine analytical separation of eight corticosteroids was performed by reversed-phase liquid chromatography using an octadecylsilica stationary phase and aqueous methanol and acetonitrile mobile phases. During the course of this study, it was observed that specific pairs of steroids exhibited different retention and dispersion behavior when they were analyzed individually and in mixtures (33). Because of the theoretical and practical significance, it was desirable to evaluate this anomalous behavior in greater depth and detail. The chromatograms of the steroids cortisone and tetrahydrocortisol are shown individually in Figures 6.2A and 6.28, respectively, and their mixture is shown in figure 6.20. The chromatographic figures of merit derived from these chromatograms are summarized in Table 6.1. Within the error of the manual measurements, the sum of the areas for the individual steroid peaks is approximately equal to the area for the composite peak, which confirms that the steroids are co-eluting. However, the capacity factor for the composite peak (1.47) is less than that for the individual peaks (1.56 and 1.48). The standard deviation of replicate measurements is $0.02 (n = 6), thus the difference in capacity factor is statistically significant at the 99% confidence level (34). 181 Figures 6.2A,B Chromatograms of individual corticosteroids Column: 47 x 0.46 cm i.d., packed with S-pm octadecylsilica material. Mobile phase: 35% aqueous acetonitrile; 0.5 mUmin. Detector: UV—visible absorbance detector (240 nm, 0.005 GLASS“ Solutes: (A) cortisone (10-6 M), (B) tetrahydrocortisol 182 Figures 6.2A,B l l 20 - 30 TIME (min) 183 Figure 6.20 Chromatograms of a mixture of cortisone and tetrahydrocortisol. Column: 47 x 0.46 cm i.d., packed with 5-pm octadecylsilica material. Mobile hase: 35% aqueous acetonitrile; 0.5 mUmin. Detector: UV—visib e absorbance detector (240 nm, 0.005 AUFS). 184 Figure 6.20 I o *u A‘ A fifiv w l 20 TIME (min) 185 00.. 0000 00.0 000.0 050.500.3503 3.0 0000 mod 2.0.0 00.. oom. more. tho 65.80.622.562 00.. 000.. 00.. <0 .0 30.500.030.03 00. r 005. to; 500.0 0.... 000 00.. 03.0 050.50 0.0.x.E 000.200. >>wxm 0.0.x.E .000.20:. «39232 w...<..n. 0.00%: .000.20:. m0...o..._o055 0.003 50 :0> .0 0.5:00E00 0:. 0:0 Am. >055 .05... 00.0 0. 4.0.0 0050.“. :. 56:0 00.05.000.50 .0 050050.500 00.5.00 0:. 5. .5. 85.0.0 5.55000 E0E..00 0:. .0 0:0 .8. 85.0.0 5.55000 0..:..:. .0 Anew. 055 0505: ad 0.00... 200 optimum conformation for the methylprednisolone-methylprednisolone pair (figure 6.5A) is a head-to-tail orientation that permits hydrogen bonding between I the carbonyl group at 0-3 and the hydroxyl group at C-11 of one molecule with the hydroxyl and carbonyl groups on the side chain of the other molecule. The tetrahydrocortisone-tetrahydrocortisone pair (figure 6.58) prefers a head-to-head orientation that allows interaction between the carbonyl and hydroxyl groups on the side chains. finally, the methylprednisolone-tetrahydrocortisone pair (Figure 6.50) prefers a head-to-tail orientation that allows interaction between the hydroxyl group at C-11 of methylprednisolone with the carbonyl group at C-11 of tetrahydrocortisone, between the carbonyl group at 0-3 of methylprednisolone with the side chain of tetrahydrocortisone, as well as between the side chain of methylprednisolone with the hydroxyl group at 0-3 of tetrahydrocortisone. The total energy for each pairwise combination of methylprednisolone with tetrahydrocortisone at infinite and at optimum separation distance is summarized in Table 6.3. As in the previous case, the van der Waals component remains relatively constant whereas the electrostatic and hydrogen bonding components decrease significantly as the molecules approach from infinite to optimum distance. The methylprednisolone-methylprednisolone and methylprednisolone- tetrahydrocortisone pairs have the most negative interaction energy and, hence, are the most energetically favorable combinations. From these molecular mechanics and dynamics simulations, we may conclude that significant non-bonding interactions exist between cortisone and tetrahydrocortisol and between methylprednisolone and tetrahydrocortisone. For both cases, the total interaction energy is of comparable magnitude (approximately -33 kcal/mole) and arises predominantly from electrostatic and hydrogen bonding interactions. 201 Figure 6.5A The optimum configuration for interaction between two methylprednisolone molecules. ( O ) carbon, ( o ) hydrogen, ( C ) oxygen, (- - -) nonbonding atoms that meet the defined energy and distance constraints for hydrogen bonds according to Equation [6.3]. 202 Figure 6.5A 203 Figure 6.58 The optimum configuration for interaction between two tetrahydrocortisone molecules. ( O ) carbon, ( o ) hydrogen, ( C ) oxygen, (- - -) nonbonding atoms that meet the defined energy and distance constraints for hydrogen bonds according to Equation [6.3]. 204 Figure 6.58 . .-.. 0.). ,0..- - ~; «- 21' ”<46: 07-. ‘fv \ .-.x 4. r- 1- 03:3; ‘. “v.1. 205 Figure 6.50 The optimum configuration for interaction between m8h lprednisolone and tetrahydrocortisone molecules. I carbon, ( 0 ) cydrogen,( (C ) oxygen, (- - -) nonbonding atoms that meet the efined ener y and distance constraints for hydrogen bonds according to Equation [6. 3]. 206 figure 6.50 207 0.001 0.001 0.21 0.0.1 0.0. .0. .0 0.5. 0.0.. >001 0.5. .00. 050.080.030.50 1 05.00.505.255. 0.0.1 .001 0:1 0.01 .0 0.0. N... .00. 0.00. 0.001 0.00. 0.00. 050.58.02.05... 1 050.080.0385... 0.0.1 0.51 :7. 0.01 0.. .0... 0.0 0.00 .00 .001 0.00. 0.00. 05.00.505.555. 1 05.00.505.25: 200 2......0 2.10 600 3:0 6:0 2:00 3360 2.....0 0o .20 -0 m_055 2.0.8.805 :3. > 5:0 0.003 .00 :0> .0 0.55058 0:. 0:0 .000 0. <0.0 00.00.“. :. 55:0 00.05.08.000 .0 0:0..0.00..:8 00.20%: 0:. ~ >055 05050 5 80>: 0. .996 .98 0.0 6.00.. 00 6.8.. 208 6.4 Discussion In order to understand the effect of these solute-solute interactions, it is helpful to discuss briefly the structure of octadecylsilica and the associated mechanism(s) of solute retention (45-49). Octadecylsilica is comprised of alkyl chains covalently bonded to the silica surface, with residual silanol and siloxane groups. Nonpolar solute molecules may interact predominantly with the alkyl chains, whereas polar functional groups may interact with the weakly acidic silanol groups or weakly basic siloxane groups. The former interactions arise from relatively weak van der Waals forces and, thus, tend to be rapidly reversible. In contrast, the latter interactions arise from stronger electrostatic and hydrogen bonding forces, which are characterized by slow mass transfer kinetics (50). These interactions often result in lower plate number and higher asymmetry or skew. For solutes such as the corticosteroids, which possess a hydrocarbon skeleton together with varying numbers of carbonyl and hydroxyl groups, a dual retention mechanism is likely to occur (51). In the aqueous acetonitrile mobile phases utilized for the present study, steroids with readily accessible carbonyl groups exhibit the lowest plate number and highest asymmetry. For example, the plate number for cortisone, which has carbonyl groups at C-3 and C-11, is significantly less than that for tetrahydrocortisol, which has hydroxyl groups at these positions (Table 6.1). However, in aqueous methanol mobile phases, which can form hydrogen bonds with the carbonyl groups, the plate number is significantly increased and the asymmetry is reduced. These observations suggest that interaction of the carbonyl group, which is weakly basic, with silanol groups or other Lewis-acid impurities in the silica is largely responsible for the observed peak profiles. 209 The retention and dispersion of the individual steroids and their mixtures can, therefore, be rationalized in terms of the number of carbonyl groups that are available to adsorb at the silica surface. For the case of cortisone with tetrahydrocortisol, the cortisone-tetrahydrocortisol pair is more energetically favorable than either of the homogeneous pairwise combinations. Because of the extensive intermolecular and intramolecular hydrogen bonding in the cortisone-tetrahydrocortisol pair, there are fewer free carbonyl groups than in the individual steroids. Similarly for the case of methylprednisolone and tetrahydrocortisone, the methylprednisolone-tetrahydrocortisone pair is energetically favorable and provides extensive hydrogen bonding to mask the carbonyl groups. In each case, the solute-solute interactions serve to reduce adsorption at the silanol groups, hence the composite peak is less retained and has higher plate number and lower skew than the individual steroids. Based on the interaction energy in Table 6.3, one would expect the homogeneous methylprednisolone-methylprednisolone pair to be at least as prevalent as the heterogeneous pairs discussed above. To test this hypothesis, the chromatographic peak profile was analyzed as the concentration of methylprednisolone was increased from 106 M to 10-3 M, comparable to tetrahydrocortisone. The plate number correspondingly increased from 1200 to 2100, and the skew decreased from 1.65 to 1.60. Thus, methylprednisolone appears to undergo self association in addition to mixed association with tetrahydrocortisone. The molecular mechanics and dynamics simulations may prove to be useful in predicting other cases of solute-solute interactions. 210 6.5 Conclusion By a combination of experimental and computer simulation techniques, corticosteroids are shown to undergo both self-association and mixed association at the 100 M concentration level. Like the structurally similar bile acids, the steroids interact primarily by electrostatic and hydrogen bonding forces. These interactions have a significant influence upon solute retention and dispersion under routine operating conditions for reversed-phase liquid chromatography. Because solute-solute interactions are invariably neglected in theoretical and semi-empirical models, they may have a detrimental effect on the prediction and optimization of chromatographic separations. 6.6 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 211 References .(l. J. l)