« uuuu ~ 'gHE 939? 3 12 This is to certify that the dissertation entitled Fundamental Studies of Capillary Columns in Liquid Chromatography presented by Christine Esther Evans has been accepted towards fulfillment of the requirements for DQQIQraI degreein (Qhemjsjmy rumm I’flcjktuwvv Major professor Date January 3, 1991 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 4‘70 ll’Vl"! M‘ F LméitARY 1 University Michigan State A PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. " DATE DUE DATE DUE DATE DUE MSU Is An Affirmative Action/Equal Opportunity Institution c:\c|rc\datedm.pm3-o.‘ FUN! FUNDAMENTAL STUDIES OF CAPILLARY COLUMNS IN LIQUID CHROMATOGRAPHY By Christine Esther Evans A Dissertation Submitted to Michigan State University in partial fullfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1990 import hindei during chror ABSTRACT FUNDAMENTAL STUDIES OF CAPILLARY COLUMNS IN LIQUID CHROMATOGRAPHY By Christine Esther Evans The correlation between theory and experiment is of fundamental importance in modern separation science. Such correlations are frequently hindered by the inability to measure solute zone profiles directly and accurately during the separation process. With the advent of optically transparent columns and laser-based detection methods, direct examination of high-efficiency chromatographic columns is now feasible. In this dissertation, an on-column detection scheme is developed which makes possible the accurate measurement of solute retention and dispersion directly on the column. This detection scheme employs two or more identical, laser-induced fluorescence detectors to monitor the solute zone profile as it traverses the chromatographic column. By measuring the difference in the zone characteristics between detectors, local retention and dispersion processes can be examined. Solute retention or capacity factor is determined directly from the difference in the first statistical moment (retention time), which is measured as the solute zone passes sequentially through each detector block. Likewise, the solute dispersion or plate height between detectors is calculated from the difference in the second statistical moment (variance). In this design, the mobile detectors allow any specific region to be isolated from the remainder of the column as well as from any extra-column effects. delecic reverse nearly dispee veeyim eslebl eonsie sesle eelue dispe ZONE After initial fabrication and verification of system performance, this detection scheme is applied to the direct examination of separation processes in reversed-phase liquid chromatography. Initial measurements performed under nearly ideal chromatographic conditions indicate a gradient in both retention and dispersion with distance along the column. Further studies accomplished under varying inlet pressure conditions, while maintaining a constant pressure gradient, establish a clear dependence of retention on the local pressure which is consistent with solubility parameter theory. Finally, zone dispersion is systematically evaluated in the inlet and exit regions of the chromatographic column. Although often assumed to be unimportant, substantial changes in zone dispersion arise from the abrupt transition in retention encountered by the solute zone. Thus, the detection scheme described herein offers the unique opportunity to probe hydrodynamic and physicochemical interactions directly on the chromatographic column. Although investigations in this dissertation are limited to liquid chromatography, this experimental approach should also be compatible with gas and supercritical fluid chromatography, as well as high-voltage capillary electrophoresis. Copyright by Christine Esther Evans 1990 Dedicated to the memory of Jimmie Anders, for challenging everything and for believing in me. ACKNOWLEDGEMENTS It is not often when the opportunity presents itself to acknowledge publicly those people who have been instrumental in your learning. People who, in a variety of capacities, have made possible the impossible, allowing you to see farther than you ever dreamed. People who create a sense of the wonder which becomes part of you. People who will share with you the moments which arise after much hard work when "ah ha" is the only word to describe the feeling of discovery. People who do not consider giving up on you, even when the going gets quite rough. Everyone has their own group of people who have offered them nothing short of the best, and I would like to take this opportunity to thank my people and introduce them to you. I would like to begin with my mother and father, who instilled me the fundamental understanding that I could do anything and, just as important, the desire to give it the very best I've got. I wish to thank the rest of my family for their support as well, especially Carol and Lark, who continually amaze me with their understanding and capacity for life. I would also like to introduce you to my extended family, who are as various and wonderful as anyone could dream of: Adrianne Allen, who routinely checks up on me and my reality; Marie deAngelis, with whom I can be crazy and sane at the same time; Lindy Smith, who’s irrepressible self keeps me honest; and Jarl Nischan, who joins me for coffee and wades through life's many confusions. I also wish to express my heartfelt appreciation to Pat Hughes. Without her clear sight and patience, I would still be travelling in circles. Finally, my very special thanks to Clint Jones, who's encouragement and support throughout the years are more important than I can express. vi peelee¢ eoninl tonsil assist diseu lil he marl wort can el r Crc In addition to my personal friends, my gratitude is also extended to my professional colleagues. I would like to thank the McGuffin group for their continued support, especially the original members, John Judge and Jon Wahl. My special thanks to Jong Shin Yoo for his technical expertise in designing and constructing the temperature controller (Chapter 8), and to Shu Hui Chen for her assistance in the successful completion of the pressure studies (Chapter 6). Many thanks to Dr. George Yefchak for his support and many enlightening discussions, to Dr. Kevin Hart for his sanity maintenance skills, and to Kate Noon for her careful reading of this dissertation. I also wish to acknowledge the MSU machine shop (Russ Geyer, Deak Waters, and Dick Menke) for their excellent workmanship in fabricating the on-column detection system. In addition, the importance of review and discussion in scientific research cannot be overemphasized. For this reason, I would like to thank the members of my committee for their advice and guidance. A special thanks to Dr. Stanley Crouch for his clear and honest discussions. My gratitude is also extended to those members of the scientific community at large for their excellent research manuscripts, many of which are cited herein. Finally, I cannot begin to express my appreciation to Dr. Vicki McGuffin for her support throughout my stay at Michigan State. The insistence on clarity of thought, and the high standards of quality which are inherent in her approach, have improved my research and communication abilities immeasureably. Last, but not in any way least, I would like to take this opportunity to dedicate this dissertation to the memory of Jimmie Anders. Mr. Anders was an extraordinary junior high school teacher, as honest and straightforward as anyone I've ever met. He made possible a level of learning which was always your best, and would not allow you to trick yourself into any less. I will treasure his gift and carry it with me always. vii lislol lislo lisle TABLE OF CONTENTS of Tables of Figures of Symbols apter 1: Historical Background: Theoretical and Experimental Evaluation of Chromatographic Separations 1.1 Introduction 1.2 Column Characteristics 1.3 Solute Retention Theoretical Prediction Experimental Evaluation 1.4 Solute Dispersion Theoretical Prediction Open-Tubular Columns Packed Columns Experimental Evaluation Calculational Methods Extra-Column Variance 1.5 Direct On-Column Detection Scheme 1.6 Literature Cited in Chapter 1 rendix 1: Additivity of Variances tpter 2: Fabrication of On-Column Detection System 2.1 Introduction 2.2 Instrumentation Chromatographic System Microcolumn Preparation On-Column Laser-Induced Fluorescence Detector Calculations 3 Summary of Detector Fabrication 4 2. 2. Literature Cited in Chapter 2 viii xi xiii xix AN—* 10 41 46 52 61 62 67 68 rapter 3: Verification of On-Column Fluorescence Detection System Performance 3.1 Introduction 3.2 Experimental Methods Reagents Chromatographic System Detection and Calculations 3.3 Results and Discussion Plate Height versus Linear Velocity Injection Variance Temporal Variance Small Variance 3.4 Summary of Performance Verification 3.5 Literature Cited in Chapter 3 rapter 4: Retention and Dispersion along the Chromatographic Column 4.1 Introduction 4.2 Experimental Methods Analytical Methodology Chromatographic System Detection System 4.3 Results and Discussion Solute Retention Solute Dispersion ' 4.4 Conclusions 4.5 Literature Cited in Chapter 4 apter 5: The Influence of Pressure on Mobile-Phase Velocity 5.1 Introduction 5.2 Theoretical Considerations 5.3 Application to Experimental Conditions 5.4 Experimental Evaluation 5.5 Conclusions 5.6 Literature Cited in Chapter 5 apter 6: - The Influence of Local Pressure on Solute Retention 6.1 Introduction 6.2 Theoretical Considerations 6.3 Experimental Methods Analytical Methodology Chromatographic System Detection System 6.4 Results and Discussion 6.5 Conclusions 6.6 Literature Cited in Chapter 6 ix 69 70 71 85 86 87 95 98 111 112 113 116 122 133 134 135 137 141 145 165 166 apter 7: The Influence of Solvent Composition on Solute Retention 7.1 Introduction 7.2 Theoretical Considerations 7.3 Experimental Methods Analytical Methodology Chromatographic System Detection System 7.4 Results and Discussion 7.5 Conclusions 7.6 Literature Cited in Chapter 7 apter 8: The Influence of Transitions at Column Inlet and Exit on Retention and Dispersion 8.1 Introduction 8.2 Theoretical Considerations: Inlet Region 8.3 Experimental Methods: Inlet Region Analytical Methodology Chromatographic System Detection System 8.4 Results and Discussion: Inlet Region Retention of Solute Zones Methanol Injection Solvent Variation in Injection Solvent Dispersion of Solute Zones Predicted Extra-Column Effects Off- to On-Column Transition Plate Height Measurements Injection in Pure Solvents Solute Zone Retention Solute Zone Dispersion . 8.5 Theoretical Considerations: ExrtRegron 8.6 Experimental Methods: Exrt Region Analytical Methodology Chromatographic System Detection System Temperature Controller . . 8.7 Results and Discussion: Exrt Region On/Off-Column Detection . Variation in Solvent Composition and Temperature 8.8 Conclusions 8.9 Literature Cited in Chapter 8 apter 9: Future Studies Literature Cited in Chapter 9 168 169 172 173 181 182 183 186 187 189 225 229 232 259 261 263 267 LIST OF TABLES Typical characteristics of liquid chromatographic columns. Maximum variance, volume, and time constant for injection and detection systems. Fractional increase in column variance (82) expected from extra- column sources. Calculated fractional increase in variance (82) arising from extra- column sources. Effect of pressure on single- and dual-mode measurements of solute capacity factor. Effect of pressure on single- and dual-mode measurements of solute selectivity. fEstimates of parameters for the prediction of the solute capacity actor. Comparison of theoretically estimated and experimentally measured constants for Equation [6.4]. Difference in the solubility parameters of adjacent solutes (8,- calculated from Equation [6.5]. Slope (c4) and intercept (ch-i determined by linear regression analysis of dual-mode measurements. Single-mode measurements of capacity factor for 90.0% methanol/water and pure methanol mobile phases weth distance. xi 40 72 97 149 150 157 158 164 176 Constants determined by nonlinear regression analysis of Equation [7.3]. 180 Capacity factors (ka) for derivatized fatty acid standards as a function of solvent composition. 192 Effect of injection solvent composition on the measured length variance ratio and plate height for the n-Cm, derivative. 207 Effect of pure injection solvents on the measured length variance ratio and plate height forthe n-Cm, derivative. 224 Retention and variance of fatty acid derivatives at 25 °C by simultaneous on- and off-column detection. 239 xii l-l Sche mass it Sche on 2 ll Exa l-l Era Gar 1-5 Scl 1-6 lllu sol l-l Ell lhe LIST OF FIGURES Schematic illustration of longitudinal diffusion (B) and resistance to mass transfer (C) processes on zone length variance. Schematic illustration of multiple pathways through a packed bed (A) on zone length variance. Example variance measurements on a Gaussian zone profile. Example variancemeasurements on an exponentially modified Gaussian zone profile. Schematic diagram of statistical moments on a zone profile. Illustration of the on—column measurement of variance for a single solute zone. Effect of change in linear velocity (dx/dt) on solute zone variance in the time, length, and volume domains. Effect of change in volumetric flowrate (dV/dt) on solute zone variance in the time, length, and volume domains. Schematic diagram of microcolumn liquid chromatography system with dual on-column laser fluorescence detection system: I, injection valve; T, splitting tee; R, restricting capillary; L, lens; F, filter; FO, fiber optics; FOP, fiber-optic positioner; M, monochromator; PMT, photomultiplier tube; AMP, amplifier; REC, chart recorder or computer. Plate height versus linear velocity: single mode (:1), dual mode (A , Golay theory (—); column, Rcor = 20 pm, L = 397 cm (single mode , L = 337 cm (dual mode); chromatographic condrtrons as described in text. 22 27 31 34 44 55 58 64 75 Effect of injection volume on plate height: single mode (Ci), dual mode (A), Golay theory (-); u = 0.12 cm/s. Conditions same as given in Figure 3-1. Effect of detector time constant on plate height: single mode (:1), dual mode (A), Golay theory (—); u = 0.12 cm/s. Conditions same as given in Figure 3-1. Measured versus theoretical volumetric variance for small variance values: single mode (D), dual mode (A), Golay theory (—-); u = 0.12 cm/s. Conditions same as given in Figure 3-1. On-column chromatogram of derivatized fatty acid standards detected at 50 cm along the column. Solutes: n-szo, n-Cmo, n- 014:0, n-C16ZO, n-C,8;o, n-Cmo fatty acids derivatized with 4- bromomethyI-7-methoxycoumarin (100 nA full scale). Chromatographic and detection conditions described in the text. Capacity factor versus carbon number for all fatty acid derivatives, as measured directly on-column at a distance 50 cm from the inlet. Chromatographic and detection conditions described in the text. Fluorescence excitation and emission spectra of 4-bromomethyI-7- methoxycoumarin in methanol (-—) and n-decane (— —). Retention time versus distance along the microcolumn for nonretained SOIUte (9): n'C1O:0 (0). ”"0120 (0)» ”'Cfazo (A), ”'Ciszo (0): n-C1820 (v), and n-Cgo,0 (+) derivatized fatty acids. Chromatographic and detection conditions described in the text. Capacity factor evaluated using single (A) and dual (8) detection as a function of distance travelled along microcolumn. Solutes are as shown in Figure 4-4. Plate height evaluated using single (A) and dual (B) detection as a function of distance travelled along microcolumn. Solutes are as shown in Figure 4-4. Calculated reduced density as a function of pressure for methanol (Equation [5.7]). . Local density (px) versus fractional distance along the column (x/L) calculated from Equation [5.10] ( D ) compared with linear interpolation (-—). 78 81 84 9O 92 94 100 104 107 119 121 Theoretically predicted ratio of the local mobile-phase linear velocity (u,) to the linear velocity at atmospheric pressure (uo) versus the fractional column distance (x/L) (—-), together with the cumulative velocity ratio (,,/uo) (- —). Comparison of theoretically predicted local (—) and cumulative (- -) velocity ratio to experimental measurement of the cumulative velocity ratio (,,/uo) using single-mode detection (0). Chromatographic conditions: Methanol F = 0.72 pL/min, Pi = 4000 psi (270 bar). Experimental measurement of the mobile-phase linear velocity ratio as a function of the pressure difference (Px - Po) on the column at 30 cm (0 ), 52 cm (:1), and as the dual (A) measurement (u = 0.014 to 0.11 cm/s). Theoretical prediction ( ) calculated based on Equations [5.7] and [5.9]. Apparent increase in the total porosity with pressure difference (Px - Po) resulting from the compression of the mobile phase shown in Figure 5-5. Experimental conditions as described in Figure 5-5. Schematic diagram of detection system allowing simultaneous measurement at six points along the chromatographic column with only two monochromator/photomultiplier systems. I: injection valve; T: splitting tee; R: restricting capillary; MONO: monochromator; PMT: photomultiplier; AMP: current-to-voltage converter/amplifier. Single- (top) and dual- (bottom) mode measurements of solute capacity factor (L = 26.2 and 23.5 cm, respectively) for n-Cmo as a function of the average pressure encountered by the solute. Experimental conditions as given in the text. Single- (top) and dual- (bottom) mode measurements of solute capacity factor as a function of distance along the column (LTOT .—. 43.9 cm) under high and low inlet pressure conditions. Solutes: n- 010:0 (,0); ”“0120 (F1)? ”“014:o,(A)If7'Ciezo (0 )i ”‘Ciap (V); ”sz0 (+)- Experrmental conditions as grven rn the text. Single- (top) and dual- (bottom) mode measurements of logarithmic dependence of capacity factor on carbon number under varying inlet pressure conditions. Experimental conditions as given in the text. 56 Measurements of logarithmic dependence of capacity factor on the mobile-phase reduced density for all solutes, together with nonlinear regression analysis of Equation [6.4] (-~). Experimental conditions as given in the text. 125 127 130 132 144 147 153 160 Measurements of logarithmic dependence of selectivity on the mobile-phase reduced density for adjacent solute pairs, together with regression analysis of Equation [6.5] (—). Experimental conditions as given in the text. Solute retention with carbon number measured in the single mode at 30 cm (top) and 90 cm (middle) along the column, together with the dual-mode measurement (bottom). Mobile-phase composition: 90.0% v/v (v ), 92.5% v/v (Q ), 95.0% v/v (A), 97.5% v/v (II ), 100 % WV (0) methanol/water. Experimental measurement of the logarithmic dependence of the capacity factor (k) on the volume fraction of methanol (¢METHANOL) evaluated at 30 cm (top) and 90 cm (middle) in the single mode, and as the dual mode (bottom). Average pressure determined from the experimental inlet pressure is listed for each mobile-phase composition at the top of each plot. Solutes: n-Cw.0 (o); n-C“;o (r3); ”'C12:O(A)i”'C13:o(<>)§”'C14:0(V);n'C15:o(+)- Capacity factor versus distance along the column measured in single- (top) and dual- (bottom) mode for derivatized fatty acid standards ”‘Cfoc (0 )J n‘szzo (0 )i "‘Ciazo (A )3 ”'Cfec (0 )i ”'Ciszo ( v ); n-Czon (+ ). Chromatographic and detection conditions as described in Experimental Methods. Effect of the injection solvent composition on the single-mode measurements of capacity factor at L = 26.2 cm. Injection solvent: 90% v/v methanol/acetone ( v ), 95% v/v methanol/acetone (<> ), Ene)thanol (A ), 95% v/v methanol/water (r3 ), 90% v/v methanol/water o . Fractional increase (82) in the column variance caused by extra- column sources under (A) ideal conditions and for (B) methanol injection solvent. Chromatographic and detection conditions as described in Experimental Methods and solute as shown in Figure 8- Effect of the injection solvent composition on the fractional increase (62) in the column variance caused by extra-column sources. Injection solvents: (A) 90% v/v methanol/acetone, (B) 95% v/v methanol/acetone, (C) methanol, (D) 95% v/v methanol/water, (D) 90% v/v methanol/water. Chromatographic and detection conditions as described in Experimental Methods and solute as shown in Figure 163 175 179 191 195 199 202 Measured ratio of the length variance on-column (0L2)ON, to that immediately prior to the packed bed, (0L2)OFF for methanol injection solvent: Theoretical prediction (-—), Experimental measurement (0). Chromatographic and detection conditions as described in Experimental Methods. Plate height versus distance along the column for single- (top) and dual— (bottom) mode determinations. Injection solvent: methanol. Chromatographic and detection conditions as described in Experimental Methods and solutes as shown in Figure 81. Effect of injection solvent on single-mode measurements of plate height. Injection solvents: (A) 90% v/v methanol/acetone, (B) 95% v/v methanol/acetone, (C) methanol, (D) 95% v/v methanol/water, (D) 90% v/v methanol/water. Chromatographic and detection conditions as described in Experimental Methods and solute as shown in Figure 8-1. Effect of injection solvent on dual-mode measurements of plate height. Injection solvents: (A) 90% v/v methanol/acetone, (B) 95% v/v methanol/acetone, (C) methanol, (D) 95% v/v methanol/water, (D) 90% v/v methanol/water. Chromatographic and detection conditions as described in Experimental Methods and solute as shown in Figure 8-1. Development of chromatogram of derivatized fatty acid standard, n- Cm, injected in pure organic solvents. Detector positions at 0.4 cm before the column (L = -0.4 cm) as well as 10.4 cm and 20.9 cm along the column. Other chromatographic and detection conditions as described in Experimental Methods. I Measured capacity factor versus distance along the column for injection of n-C 0:0 in pure organic solvents: 2-propanol (0 ); acetone (:1); methanol (0 ); acetonitrile (A ). Chromatographic and detection conditions as described in Experimental Methods. Illustration of the predicted influence of elution nonequilibrium on solute length variance (6L2) and concentration (C) as a function of capacity factor (k). {Schematic diagram of on-column detection in exit region. I :- injection valve, T = splitting tee, R = restricting capillary, FOP .—. fiber optic positioner, L = lens, F = filter, M = monochromator, PMT = photomultiplier tube, AMP = current-to-voltage converter and amplifier, REC = recorder/computer. Schematic diagram of temperature controller at column exit. 205 21-0 213 215 219 222 228 231 234 iii 8-16 Chromatograms of n-Cmo to n-Cgofl fatty acid derivatives on and off the chromatographic column at 25 °C. Conditions as given in Experimental Methods. Ratio of the length variance measured on- and off-column versus (1+k)2 in the exit region (25 °C). Theoretical prediction (——). Normalized ratio of maximum concentration measured on- and off- column versus (1+k) in the exit region (25 °C). Theoretical prediction (—). Effect of solvent composition on the off- to on-column length variance ratio. Mobile-phase composition, methanol/water: 90.0% v/v (ale), 92.5% WV (0 ), 95.0% v/v (A), 97.5% v/v (I), 100% WV (0 ), Theory (-—)- Effect of temperature on the retention behavior of fatty acid derivatives: logarithm of capacity factor versus carbon number. Conditions given in Experimental Methods. 9Effect of temperature on the retention behavior of fatty acid derivatives: logarithm of capacity factor versus 1/T. Conditions grven in Experimental Methods. 3 Temperature variation with distance in the column exit region. 1 Effect of temperature on the ratio of the length variance measured on- and off-column versus (1+k)2. Theoretical prediction (-—-) based on Equation [8.9]; T = 40 °C (0). 50 °C (13). 50 °C (A), 90 °C (V)- 2 Effect of temperature on the normalized ratio of the concentration measured on- and off-column versus 1/(1+k). Theoretical prediction (——) based on Equation [8.12]; T = 40 °C (0), 50 °C (1:1), 60 °C (A), 90 °C (v). xviii 237 241 244 247 249 251 254 256 258 LIST OF SYMBOLS multiple path contribution to dispersion specific permeability longitudinal diffusion contribution to dispersion in the mobile phase longitudinal diffusion contribution to dispersion in the stationary phase solute concentration detected off-column solute concentration detected on-column resistance to mass transfer contribution to dispersion in the mobile phase resistance to mass transfer contribution to dispersion in the stationary phase stationary phase film thickness diffusion coefficient in the mobile phase particle diameter diffusion coefficient in the stationary phase interaction energy arising from dispersion or London forces interaction energy arising from induction forces interaction energy arising from orientation forces volumetric flowrate on-column height equivalent to a theoretical plate measured height equivalent to a theoretical plate ionization energy solute capacity factor in the injection solvent solute capacity factor in the mobile phase distribution coefficient distance along the chromatographic column viewed length of the flowcell total column length xix zeroth statistical moment; area of zone profile first statistical moment; retention time second statistical moment; zone variance third statistical moment; zone asymmetry number of theoretical plates pressure at the column inlet pressure at the column outlet pressure as a function of distance along the column radial position in cylindrical open tube mean distance between molecules 1 and 2 gas constant (1.987 cal/K mol) column radius connecting tube radius flowcell radius skewness of zone profile retention time of nonretained solute retention time of retained solute temperature off-column linear velocity of mobile phase on-column linear velocity of mobile phase linear velocity of mobile phase as a function of distance spatial average linear velocity of the mobile phase linear velocity of solute zone detection volume injection volume molar volume of solute i volume occupied by the mobile phase total volume of unpacked column volume between packed particles polarizability of molecule 1 chromatographic selectivity between molecules i andj phase ratio of the column Hildebrand solubility parameter of solute i Hildebrand solubility parameter of the mobile phase Hildebrand solubility parameter of the stationary phase XX NJ 10 l'OTAL intra-particle porosity total porosity inter-particle porosity permitivity of free space tortuosity factor in the mobile phase tortuosity factor in the stationary phase packing structure constant dipole moment of molecule 1 density at critical temperature and pressure density at the column inlet density at the column outlet reduced density (= p/pC) density as a function of distance along the column spatial average of density zone variance in length units zone variance in time units zone variance in volume units variance arising from detector variance arising from injector variance arising from detector time constant total variance of zone profile detector time constant viscosity flow resistance parameter volume fraction of solvent n fractional increase in the column variance from extra-column sources xxi CHAPTER 1 HISTORICAL BACKGROUND: THEORETICAL AND EXPERIMENTAL EVALUATION OF CHROMATOGRAPHIC SEPARATIONS 1.1 Introduction The separation of complex mixtures is an essential process in a wide range of research and industrial applications. By taking advantage of the different extent to which solute molecules interact with a stationary medium, Martin and Synge (1) discovered the powerful separation technique known as chromatography. In this technique, a flowing mobile phase is continuously introduced onto a chromatographic column, and selective interaction of solutes with the stationary phase causes their differing migration rates through the column. The resulting solute zones are then monitored sequentially as they pass a detector. This method allows samples of environmental and biochemical significance to be analyzed in a remarkably short period of time. Since this discovery in 1941, researchers in fields as diverse as oceanography and medicine have benefited from the ability of this technique to separate the constituents present in very complex sample matrices. 1 2 The effectiveness of a separation in chromatography is determined by the differential migration rate of solutes as well as by the broadening or variance of the solute zones. In the analysis of complex mixtures where the sample components are distributed statistically (2), altering the migration rate merely transposes the order of solute elution while not improving the separation to any significant extent. However, minimizing the spreading of solute zones on the chromatographic column allows a greater number of species to be separated simultaneously. Thus, understanding the influence of the various separation parameters on solute zone dispersion is central to the advancement of the field of separation science. For this reason, the primary emphasis in this chapter will be on the theoretical prediction and experimental measurement of zone dispersion, with solute retention discussed in less detail. 1.2 Column Characteristics It may be helpful to begin by describing the types of chromatographic columns presently utilized in high-performance liquid chromatography (HPLC). Column design has advanced considerably in the past ten years, and several types of HPLC columns are commercially available or under active development. The most common of these and their characteristics are shown in Table 1.1 (3). The predominant trend in developing new columns has been to decrease inner diameters (id) and increase column lengths (4,5, and references cited therein). Although most theories predict chromatographic efficiency of packed columns to be independent of column diameter (Section 1.4), clear advantages are realized in miniaturization. As shown in Table 1.1, the low flowrates required with decreased diameter allows an increase column length, leading to an increase in labl Table 1.1: Typical characteristics of liquid chromatographic columns. Ii COLUMN TYPE I.D. LENGTH FLOWRATE N o’ ; conventional 4.6 mm 25 cm 1 mL/min 10,000 500-1000 packed capillary 50-500 urn 1-10m 0.5-2 uL/min 100,000 500-1000 open-tubular capillary 1-10um 1-10m <1 uL/min 1,000,000 32 4 the chromatographic performance. Diminished flowrates also yield the practical advantage of decreased solvent consumption and related disposal problems. In addition, a variety of new detection options are possible with these smaller columns (5), including mass spectrometric detection (6). Because many of the smallest columns are fabricated from fused-silica capillaries, detection directly on the chromatographic column is also feasible (7). In the research presented here, this technological advancement is used to advantage by probing the hydrodynamic and kinetic behavior of packed-capillary columns directly. 1.3 Solute Retention The selective interaction of solutes with the stationary and mobile phases forms the basis for separations in liquid chromatography. Differences in interaction energies, often less than 100 cal/moi between similar solute molecules, cause differential migration rates along the chromatographic column. Although the subject of considerable investigation, the exact nature of this phenomenon is not yet fully understood. In the reversed-phase separations of interest in this study, the stationary phase is composed of a porous silica substrate which has been chemically modified with straight-chain alkyl moieties. The mobile phase is a polar organic solvent (i.e., methanol, acetonitrile, tetrahydrofuran), and is often utilized in aqueous mixtures. A variety of mechanisms have been proposed to describe the retention of solutes under these conditions (8-17), although conclusive proof- favoring any specific mechanism is presently lacking (17). A number of factors contribute to the difficulty in assessing the precise mechanism(s) involved in a separation. First, the chemical and physical nature ol the 5 these re individu hewlo a Gill-ll mobile [ll-20 incom slloxa solute also r sepa and Iraqi vise 5 of the stationary phase is quite complex. The environment at the surface of these modified particles is not accurately modeled by bulk solution chemistry, as individual alkyl chains are anchored to the silica substrate (18). It is also unclear how to model the presence of surface-adsorbed mobile phase, which may act as a chemically distinct stationary phase. Furthermore, the composition of the mobile phase is known to alter the stationary phase environment and structure (18-20). Finally, derivatization of the silica substrate inevitably leads to incomplete surface coverage, often leaving an unknown number of silanol and siloxane sites available on the surface for adsorption interactions with the solutes. If this were not enough, the chemical composition of the mobile phase is also quite complex. The organic solvents commonly utilized in reversed-phase separations are quite polar, exhibiting an appreciable extent of self interaction and hydrogen bonding. Moreover, aqueous mixtures of these organic solvents, frequently utilized to optimize separation conditions, often behave anomalously (21-24), displaying minima and maxima in physical properties (e.g., density, viscosity, surface tension, etc.) as a function of composition. Theoretical Prediction. The prediction of solute retention is central to the control and optimization of separations, and, thus, has been studied extensively (8-17). Among these theories, the solvophobic theory of Horvath (9,10) and the solubility parameter approach (11-13) are the most commonly employed. However, recent advances in the develooment of a unified theoretical approach to separations by Martire (15,16) shows much promise. In this statistical thermodynamic approach, a lattice model is utilized to describe both the absorption (partition) phenomenon of interest here (15,16) and the adsorption mechanisms (25) as well. In addition, numerous empirical correlations are routinely employed (26), including the recent use of solvatochromic indicators as e measurr environme To hegenerr Dispersior are lhe n lrleraclio Elli = ' .‘ I where a he area may be Ellll = ' where are in based lndepr 6 a measure of the polarity of the mobile (27,28) and stationary (29) phase environments. To begin the discussion of solute retention, it is important to understand the general types of chemical interactions which are possible between molecules. Dispersion or London forces arising from induced dipole-induced dipole forces are the most common and are present to some extent in all interactions. The interaction energy (End) between two identical molecules may be described by 2 End = ' EL “-11 rate where a, is the polarizability of molecule 1, I1 is the ionization energy, and r11 is the mean distance between molecules. Interactions between different molecules may be described in an analogous manner, E11d=- 3a1a2 l1l2 [1.2] 2 "12‘s ('1 + '2) where the polarizability for each molecule and the reduced ionization potential are incorporated. Dispersion forces between molecules are often estimated based on group polarizabilities. If group interactions are considered independent, dispersion forces in molecules of a homologous series are expected to increase directly with carbon number. It is clear from Equation [1 .2] that these forces are quite weak and decrease rapidly with distance. Nonetheless, these nonspecific interactions are the primary forces acting between solute molecules and the nonpolar stationary phase in reversed-phase separations. In addition, a permanent dipole in one molecule may induce a temporary dipole in a second molecule. The energy of dipole-induced dipole or induction interactions (E12,) is given by whee mole one inle 1 28392 H [La Era =- 411280 r125 where So is the permitivity of free space and p, is the dipole moment of the molecule. Alternatively, if both molecules contain permanent dipole moments, orientation or electrostatic interactions are possible. The energy of orientation interactions (E120) can be expressed as I 2 [1121.122 C05 8 [14] 4Tt80 3 k T r126 where k is the Boltzmann constant, T is the absolute temperature, and 8 is the angle formed by the vectors of the two dipoles. Based on Equation [1.4], the interaction energy is the greatest when the two molecules are aligned. In addition to these weak forces, molecules may also form hydrogen bonds. This specific interaction is slightly more energetic and is often described as a combination of donor-acceptor and electrostatic interactions. Interactions , between ions and molecules, not considered here, may also be important in separation schemes other than reversed phase. Although solute molecules may encounter any or all of these interactions, because of the alkyl-based stationary phase, dispersion forces are by far the most prevalent in reversed-phase separations. On traversing the chromatographic column, solute molecules will interact with the mobile and stationary phases, residing for varying amounts of time in each phase. This retention of solutes is given by the capacity factor (k) k = ‘8'“ [1.5] to 8 where tR is the time required for the solute to traverse the column and to is the column void time. In the simplest case, the distribution of the solute between these phases can be described by the equilibrium constant, K. For an isothermal process, the distribution constant is given by k = [3K = [3 exp (-AG°/RT) = p exp [(-ETOT/RT) + (AS/R)] [1.6] where B is the volume ratio of the mobile to stationary phase, and AG0 is the change in Gibbs' free energy associated with the transfer between the mobile and stationary phases resulting from a change in interaction energy of ETOT. Theoretical approaches to the retention process often diverge at this point. The solubility parameter approach is chosen for discussion here because the fundamental basis provides a reasonably good physical understanding of the separation process. Based on regular solution theory (30), the interaction energies between molecules are directly additive and assumed to be independent. In addition, this model assumes that the molecular volume is fixed and molecules are randomly distributed, with no preferred orientation. Although these assumptions seem to exclude most conditions present in chromatographic separations, the model is quite good in describing the dispersion interactions which predominate in reversed-phase separations (12-14). In the solubility parameter approach, the sum of the interaction energies (E), which were described above, per molar volume (V) are incorporated in a cohesive energy density or solubility parameter (6 = -EN). This quantity, often a direct indication of the solvent polarity, can be related to the capacity factor of solute i (k,) via Equation [1.7] (14)- In is 1);} [(SI-SMP-(fa-5s)2]+|n(1/l3l [1.71 9 In this expression, the retention of solutes is described by the difference between the solute (8,) and mobile phase (8M) parameters and the solute (8.) and stationary phase (53) parameters. For reversed-phase separations, the mobile phase is always more polar than the stationary phase (8M > 85), with the solute parameter usually at some intermediate value. If the solute parameter (8) is exactly centered between the mobile- and stationary-phase parameters, the solute will spend half of the time in each phase (k = 1). Selectivity between solutes i and] (9'11) may, likewise, be written in terms of solubility parameters (14), 2vi (5r - 5,) (5M - 53) [1-8] RT In co,- = In ki = In k, assuming equal molar volumes (Vi). Based on this expression, the separation of similar compounds is dependent not only on the difference in their solubility parameters (8 - 8,), but that of the mobile and stationary phases (6,, - 83) as well. Thus, the successful separation of solutes which are quite similar (8, ~ 8) is facilitated by a large difference in the mobile and stationary phase environments (8M >> 63). Solubility parameters have been applied to liquid chromatographic separations under a wide variety of conditions, including variations in temperature (31), mobile-phase composition (32), and pressure (33). Although the assumptions necessary to derive these solubility parameter expressions are quite limiting, the use of this approach in predicting trends in retention has been surprisingly successful. Solubility parameters are presently tabulated for a number of compounds of interest in reversed-phase liquid chromatographic separations (34), including recent evaluations of a variety of commercially available stationary phases (19.20)- Experlme measurlnl nonretain determine ewaluatin this dete proposlli (35.36). marker, material condillc lorexar 1.4 Sr 10 Experimental Evaluation. Solute retention is determined experimentally by measuring the time required for a solute to elute from the column (tn) relative to a nonretained species (to). Although the peak maximum is often utilized for this determination, the first statistical moment is the most accurate means of evaluating the true solute retention time (vide infra). Based on Equation [1.5], this determination of the solute capacity factor (k) appears to be a simple proposition. However, the selection of a truly nonretained solute is quite difficult (35,36). It has been proposed that each solute requires a different nonretained marker, as the accessibility of solutes to the pore structure of the packing material may differ (36). In addition, the precise control of experimental conditions (temperature, flowrate, mobile-phase composition, etc.) is necessary for exact measurements of solute retention (37-39). 1.4 SoluteDIsperslon In addition to interactions which act to separate solutes, a variety of forces act to increase the entropy of the system, often resulting in broadening or dispersion of the solute zone as it traverses the chromatographic column (40). These dispersion processes are detrimental to the separation performance, causing the overlap of neighboring solute zones. Thus, the column peak capacity is reduced and the maximum number of solutes that can be successfully separated is decreased (2.41.42). Although the subject of considerable study, the fundamental processes contributing to the total dispersion in chromatographic separations are not yet fully understood. Nonetheless, as the samples of interest become more complex, the need for truly optimal separation conditions requires a clearer knowledge of dispersion processes. In this section, the present status at It dlspe lheor 11 of theoretical predictions and experimental measurements of solute zone dispersion is addressed. The assumption of additivity of variance, implicit in most theoretical and experimental approaches, is discussed in Appendix 1. Theoretical Predictions. In the isolation of individual components from a mixed sample, both separative and dissipative forces act on the solute zone. Separative forces, by the various interactions discussed above, selectively displace solutes to different spatial/temporal locations. Simultaneously, however, dissipative forces are acting to increase disorder, diluting and remixing solute zones. Thus, the success of any separation is based on the ability to design a system which favors separative'forces (40). The understanding of solute zone dispersion is central to this success, especially in the separation of mixtures containing a large number of components. In this section, an overview of the prevalent theories regarding the chromatographic dispersion of solute zones will be presented. The discussion will begin with open-tubular columns, which are well characterized and well understood, and continue with the more complex, packed columns. Open-tubular Columns. The dispersion arising in straight, smooth open tubes is well understood for the laminar flow regimes of interest in chromatographic separations. The pressure-driven flow of fluid in this simple system can be modeled as sheaths or layers. Momentum is transfered between adjacent layers resulting in a radial velocity profile, u,, across a tube of radius R, which may be expressed in the following parabolic form (43). u, = 2u [1 - (r/R)2] [1.9] 12 where u represents the average linear velocity of the fluid. Under these laminar flow conditions, the maximum linear velocity (2u) occurs at the tube center (r = 0). The dispersion arising under these flow conditions was first derived by Taylor and Arts (44,45) for open-tubular columns with no stationary phase. In this equation, the dispersion of a solute zone is expressed as the plate height (H) or length variance (0.?) per unit length (L) along the column. H: 0r? =ZDM+ RZU [1.10] L U 24 DM This equation has been shown to predict accurately the dispersion arising when a solute with diffusion coefficient DM is injected into a fluid with an average linear velocity u (46). However, Equation [1.10] is only valid if the time spent on the column is long compared with R2/DM (47-49). Although not directly applicable to chromatographic columns, the Taylor-Aris equation is valid for predicting dispersion in most connecting tubes utilized in chromatographic systems. The dispersion arising from open-tubular columns which contain a thin coating of stationary phase was first derived in the classic manuscript by Golay (50). In this well-known equation, the dispersion of a solute zone of capacity factor k is expressed as the plate height (H) for a stationary phase thickness d, and solute diffusion coefficient in the stationary phase D5. H: 68:20,,+ (1+6k+11k2)R2u +2 k d,2u [1.11] L u 24(1+k)2 0,, 3 (1+k)2 os = BM/U + CM U ‘1" CS U In Golay's original development, three factors contribute to the dispersion of solute zones in open-tubular columns: longitudinal diffusion in the mobile phase 13 (BM), and resistance to mass transfer in the mobile (CM) and stationary (Cs) phases. All of these factors are presumed to be independent in this theoretical development and, therefore, the length variances are considered to be additive (Appendix 1). Longitudinal diffusion in the stationary phase (BS) was considered negligible in Golay's original development, but will be included here for completeness. A physical description of each of these individual processes is illustrated in Figure 1-1. The B term in the Golay equation represents longitudinal diffusion resulting from the Brownian motion of molecules. The resulting flux, given by the second law of thermodynamics, arises from entropic driving forces. This diffusional process is given by Fick's laws (51), where the first law describes the solute flux (J) or mass flow per unit area. J=-oacox 0121 Assuming an initial concentration C at time t = 0, this broadening process is driven by the solute concentration gradient (BC/8x) along the column axis (x). The resulting change in concentration with time (BC/8t) is given by Fick's second law, acrar = o aczraxz [1.13] where the diffusion coefficient (D) is considered constant. For a narrow injection profile, the concentration as a function of time calculated from Equation [1.13] yields a Gaussian form. 0(1) = 1 exp (-x2/4Dt) [1,14] 2(1: poi/2 Figure 1-1: Schematic illustration of longitudinal diffusion (B) and resistance to mass transfer (C) processes on zone length variance. an s... V///////././////////. 7/////////// 4% t" A... re. Mm“ he" . u" lws W v" zine: r/////%////// VA//% 2% 2% Com] waia 16 Comparison with the normalized Gaussian distribution results in the length variance for one-dimensional mass transport derived by Einstein (52). OL2=ZDl [1.15] This mathematical description of the variance, equivalent to the 8 term variance in Equation [1.11], yields the diffusional contribution to the zone dispersion. Although this contribution is shown in Equation [1.11] to be present only in the mobile phase (BM), longitudinal diffusion may occur in the stationary phase (83) as well. Because the Golay equation was originally derived for gas chromatographic applications, stationary phase effects are considered negligible as mobile-phase diffusion coefficients (DM) are typically a factor of 104 greater than those in the stationary phase (03). The stationary phase contribution cannot be neglected in liquid chromatographic applications, however, where diffusion coefficients in the mobile and stationary phases are approximately 105 and 106 cm2/s, respectively. In this case, the stationary-phase longitudinal diffusion is given by e33= 208" [1.16] u and must be included in Equation [1.11]. The resistance to mass transfer contribution to the zone dispersion is also illustrated in Figure 1-1, for both the mobile (CM) and stationary (Cs) phases. These sources of dispersion result from the finite time required for solute molecules to transfer through the mobile or stationary phase. In the mobile phase, the limited solute movement between flow streams results in different rates of movement across the chromatographic column. Thus, solute molecules within the same zone are displaced to differing extents along the column by the parabolic flow profile. In the stationary phase, the resistance to mass transfer cont raw zen lom 17 contribution arises because solute molecules reside in the stationary phase for varying amounts of time. This distribution of interaction times within the same zone yields a range of spatial displacement along the column. In contrast to the longitudinal diffusion term, however, the plate height contribution from CM and Cs actually decreases with diffusion coefficient. In this case, diffusion increases mass transfer of solutes, providing more rapid movement of molecules between the mobile and stationary phases. As seen in Equation [1.11], the plate height dependence on capacity factor is quite complex. It mass transfer in the mobile phase is predominant, plate height is predicted to increase with retention, whereas a decreasing trend is expected for stationary-phase processes. The rigorous mathematical derivation of these terms is not presented here, and the reader is referred to the excellent development in the original manuscript (50). Although dispersion in open-tubular columns has been quite well characterized (50), recent theoretical developments provide an "exact proof" of the Golay equation (53). The few terms that must be assumed to be negligible in the original derivation are unnecessary with this new approach. In addition, this recent derivation incorporates the finite equilibration time between the mobile and stationary phases. Thus, an additional term is included in the Golay equation for the contribution of equilibration rate to the zone dispersion. Packed Columns. Similar to open-tubular columns, the dispersion in packed columns arises from a combination of diffusion and hydrodynamic considerations. Unfortunately, fluid flow within a packed bed is much more complex and, as a result, much more difficult to derive from fundamental principles. Although the effect of each force within the system may be rigorously evaluated utilizing the Navier-Stokes equation (54), application of this expression to the complex 18 geometry in packed beds is intractable. Thus, the empirical approach proposed by Darcy (55) and based on resistance to flow is most often utilized. u=-_Bg_Po'F’i [1.17] an L This description of fluid flow is directly analogous to the flow of electrical current given by Ohm's law, where the average linear velocity (u a current) results from an applied pressure drop (Po - P, a voltage) per unit length (L) along the column. The resistance to fluid flow is, therefore, given by the fluid viscosity (n) and specific permeability (K0) of the packed bed. Derivation of the specific permeability, accomplished by Kozeny (56) and Carmen (57), is given below. 3, - dPZEua - W [1.18] -180 \j12(1-eu)2 _ 4" The column permeability is dependent on the particle diameter (dp) as well as the total porosity of the packed bed (ST). and may be described in terms of the flow resistance parameter (o'). In this expression, the total porosity is the fraction of the column volume which is accessible to the solute and comprises both Interparticle (eu) and intraparticle (8,) contributions (58). ET=€U+€i [1.19] The effect of the packing structure is given by the empirical constant 1412, which is 1.0 for spherical nonporous beads and 1.7 for irregular porous particles. Although the dispersion arising under these flow conditions is difficult to model, appreciable theoretical advances have been made in describing the chromatographic technique. The first theory to describe zone dispersion is the well-known plate theory proposed in the original work of Martin and Synge (1). In this approach, the chromatographic column is divided into small vessels or 19 plates. Within each plate, complete mixing and equilibrium are assumed to occur. Although effectively applied to distillation processes (59), the plate theory is inadequate in describing zone dispersion in chromatographic separations (60). The shortcomings in this theoretical approach arise from the discrete nature of the model and the assumption of equilibrium within each discrete section or plate. These assumptions are in direct conflict with the fact that chromatographic separations are dynamic and continuous in nature, comprising many processes that are driven by nonequilibrium. Although these limitations are cited in the original manuscript by Martin and Synge, this theory has been applied without regard for these assumptions for many years. Although the numerous failures of this theory have been detailed elsewhere (60), a few are included here for completeness. The stipulation of discrete, individual plates results in the inability of the plate model to account for the effect of longitudinal diffusion on zone dispersion. Moreover, this model does not yield any information regarding the effect of many primary variables present in any separation (e.g., particle size, diffusion coefficient, temperature, etc.). Thus, although the height equivalent to a theoretical plate (H) remains in common usage, as seen earlier in this section, the meaning is quite different from the original term cited by Martin and Synge. Since the plate model, many other approaches to the theoretical description of dispersion have been proposed. These are quite varied in approach and detail, and include the random walk model (62,63), the mass balance method (63-67), general nonequilibrium theory (68,69), and the rate theory (70). A clear discussion and comparison of these theories has been presented by Weber and Carr (71 ). The most widely utilized of these theories, is the rate approach proposed by van Deemter, Zuiderweg, and Klinkenberg (70). In this theory, rate constants for the various dispersion processes are evaluated individually and the total zone dispersion is calculated as the sum of the length variance contributions. lntl mul dim 20 H = A + BM/u + Bs/u + CMU + Csu [1.20] = {(6L2)A + (0L2)BM + (GL2)BS 1' (GL2)CM + (0L2)CS}/L In this familiar expression, the A term represents the dispersion arising from the multiple paths possible through the packed bed. This contribution does not affect dispersion in open tubes which contain only one path, and, thus, the A term is absent from Equation [1.11]. As illustrated in Figure 1-2, it may be necessary for solute molecules within the same zone to travel different distances to traverse the column length. The dispersion in length variance resulting from this process ((08),) may be expressed as (0L2)A = 2 A. dp L [121] where I» is a constant dependent on the homogeneity of the packed bed, typically 0.1 to 0.5 (72). In this model, the A term is assumed to be independent of the mobile-phase velocity. This presumption, later disputed by Giddings (73,74), does not account for the possibility of molecules changing flow streams. This movement, whether by diffusion or by turbulence, would allow different possible paths through the packed bed and result in the coupling of the A and C terms. In this case, the processes are not independent and, therefore, the variances are not strictly additive. Instead, the resulting coupled length variance ((cL2)Ac) has the following form: 1 1/A+1/Cu (GinAC = [122] At high linear velocities, this relationship predicts a lower plate height than expected from the van Deemter form. 21 Figure 1-2: Schematic illustration of multiple pathways through a packed bed (A) on zone length variance. Tl column directly I to the It With ‘ MES Unlike longltu particle tortuo: statior In no descr earlw belw disp (GL2 Thi sqr lnw 23 The B terms in Equation [1.20] result from molecular diffusion along the column length, as described earlier. These longitudinal diffusion terms are directly proportional to the solute diffusion coefficient and inversely proportional to the time spent in the phase of interest. (GL2)8M = 2 TM DM 1 = 2 TM DM U“ [123] (GL2)BS=ZYS Dskt=2YS DskL/U [1.24] Unlike the analogous expressions for open tubes (Equations [1.11] and [1.16]), longitudinal diffusion in packed beds may be hindered by the presence of particles or by the uniformity of the stationary phase film. The obstruction-or tortuosity factors represent the regularity of packing (7M) and the continuity of the stationary phase film (73). Typical values of 7,, for packed bed systems are 0.73 for nonporous, spherical particles and 0.63 for porous, irregular particles (75,76). Finally, the C terms in the van Deemter expression (Equation [1.20]) describe the resistance to mass transfer of solute molecules. As discussed earlier, inertial forces resist the movement of molecules between flow streams or between the mobile and stationary phases and, thus, cause an increase in zone dispersion. The mobile-phase contribution is given by 2 (0L2)0M ' k2 dp U L .. [1.25] 100 (1 + k)2 oM This relationship is analogous to that in an open tube (Equation [1.11]) with a squared dependence on particle diameter (instead of column diameter) and an inverse dependence on diffusion coefficient. However, the capacity factor dependence is quite different from that derived for an open tube. The packing structure may act to force mass transfer of the solute, thus decreasing the k dependence in a packed bed system. In contrast, the dispersion arising from resi are lit 24 resistance to mass transfer in the stationary phase in packed columns is directly analogous to that in open tubes, as shown below. defZUL [1.26] (1 + k)2 03 (0L2)CS = The form of the expressions vary only in the value of the constant c, which is equal to 2/3 for an open tube and 8/1r2 for a packed tube with uniform stationary phase film. Derivations by Giddings (77) indicate that the magnitude of c is dependent on the variation in film thickness, as well as the shape and distribution of the stationary phase in the column. In addition, the shape and depth of the pore structure is predicted to have a significant effect. Thus, determination of the expected value for this constant is difficult, if not impossible, under practical conditions. Although the van Deemter equation provides a good qualitative description of zone dispersion, the limited number of processes and the assumption of independent contributions is quite simplistic. However, the number of unknown parameters is less than for other more sophisticated theories. This overall problem has been circumvented in a number of investigations by utilizing an empirical approach (78-80). Although often helpful for diagnostic purposes (81), these curve fitting methods are not appropriate for the evaluation of fundamental dispersion processes. Thus, although many theoretical advances have been made, the accurate prediction of zone dispersion from known experimental parameters remains elusive. Experimental Evaluation. The advancement of the theoretical basis for zone dispersion has been limited, in part, by the lack of accurate, systematically 25 acquired experimental measurements of zone variance. Verification and development of the theoretical models discussed above require the accurate and precise determination of dispersion arising solely from separation processes. A variety of factors contribute to difficulties in this experimental measurement, including calculational methods and extra-column effects. Calculational Methods. Although often presumed unimportant in dispersion determinations, the accuracy and precision of the measured variance are directly affected by the method of calculation. The variety of calculational methods employed for the evaluation of variance from a recorded profile have been critically reviewed and compared in several excellent manuscripts (82-85). In the simplest case, the measured profile is modeled as Gaussian distribution (86). Ch) = _1_ exp [ , (t - 1:02 ] [1.27] (51- (21r)1/2 2 012 As illustrated in Figure 1-3, the variance of the zone is commonly calculated from the inflection points at 60.7% of the peak height (20), from the curve at 50.0% peak height (2.350), or from the tangent lines (40). The plate height may then be determined by substitution of the measured time variance (0T2) into the following expression, H = L (012/192) [1.28] where the retention time (tR) is determined at the peak maximum for a column of length, L. Although these methods are commonly utilized because of the ease of calculation, the assumption of a Gaussian profile is not always valid for actual experimental measurements. In fact, many measured profiles are asymmetric and, therefore, arise from sources which are not entirely Gaussian in nature. 26 Figure 1-3: Example variance measurements on a Gaussian zone profile. . 27 Figure 1-3 It 6e W bmmN 6N 0.0 n To mmd 1nd Isa Foo red two $5 10.0 < O; 28 Other models have been proposed for the determination of the zone variance as well, among which the exponentially modified Gaussian (EMG) is the most common (87-94). In this model, the chromatographic zone is assumed to result from the convolution of Gaussian (with variance, 062) and exponential (with variance, 12) contributions yielding the final measured variance (0?). o2 = 062 + r? [1.29] The EMG profile is modeled based on the following expression for the concentration as a function of time, C(t). C(t)=_1_exp[ “2 + (HR) )[erfE 1,, + ‘5 )+erf[ (HR) - ° fined] 21 212 ‘5 21/20 21/21 21/20’ 21/21: The literature includes many errors in the presentation of this model which have been reviewed and corrected by Hanggi and Carr (94). Using this model, the Gaussian and exponential components (0c and 1, respectively) may be determined iteratively. Because this determination is calculationally intensive, several approaches have been proposed to simplify parameter evaluation. Barber and Carr (90,94) have proposed a series of calibration curves for the determination of Ge and r graphically. This technique yields an accuracy of i 2.4% with a corresponding precision of i 5.0% in the determination of plate height on a synthetically generated peak. This corresponds to an accuracy of i 1.5% and a precision of i 3.1% in the evaluation of the second moment. An alternate approach based on least squares fitting has been examined by Foley and Dorsey (91 ). This method yields the following expression: H: L(A/B+1.25) [1.31] 41.7 (tR/WOJ) 29 where N8 is the commonly measured asymmetry and Wm is the peak width at 0.1 of peak height, as shown in Figure 1-4. This technique offers the advantage of improved accuracy (i 1.5%) and precision (i 2.5%) in the determination of plate height (with an accuracy and precision in the second moment of i 1.5% and i 2.4%, respectively), and also utilizes easily measureable parameters. However, as with any fitting method, the EMG model restricts the determination of fundamental parameters to those implicit in the chosen model. Thus, although widely accepted, use of the EMG approach assumes the zone profile includes only one symmetric and one asymmetric parameter. Moreover, it may not be reasonable to presume that both these parameters arise exclusively from column processes. Many other fitting approaches have been investigated as well (95-99). The most common of these are the Chesler-Cram equation (97,98) and the Edgeworth-Cramer series (99). Although relatively successful at fitting chromatographic profiles, both techniques require a number of adjustable parameters and provide little physical insight into the separation process. The final, commonly utilized calculational method is that of statistical moments. Unlike the preceding techniques, the evaluation of statistical moments requires no a priori assumptions regarding the shape of the chromatographic peak (92,100-105). Mathematically, statistical moments are defined as: M0 = l l(t) dt zeroth moment M1 = I t 1(1) dt / Mo first moment [1.32] M2 = l (t - 101,)2 I(t) dt/ Mo second moment Mn =1 (t - M1)n 1(1) dt/ Mo higher moments n = 3,4,5” Figure 1-4: 30 Example variance measurements on an exponentially modified Gaussian zone profile. II V 31 Figure 1-4 0.0 1.0 1N0 de 1.80 10.0 10.0 150 rwd 10.0 0; where as a math prolllr 32 where the zone profile is described as a distribution of the detector response, l(t), as a function of time, t. Although deceptively simple, this method allows mathematical treatment which gives a complete and exact characterization of the profile. Statistical moments are related to the physical and chemical behavior of the zone profile (Figure 1-5). The zeroth moment expresses the area under the chromatographic peak. The retention time or centroid of the peak can be obtained directly from the first moment. The second moment is the variance or dispersion of the zone profile. Information about the peak asymmetry is derived from the third moment, where the skewness (S) is defined as S = M3/M23/2 [1.33] It is interesting to note that the even statistical moments describe symmetric aspects of the profile, whereas the odd moments characterize asymmetric aspects. The integration variable need not be restricted to time (t) as shown in Equation [1.32], but might alternately represent a distance or volumetric displacement. Although the method of statistical moments is rigorously correct, calculations are not without difficulties (84,92,103-105). In practice, mathematical calculation is most often performed by finite summation of the detector response across the peak profile. The primary errors in this evaluation arise from the determination of the beginning/ending of the peak and the A/D conversion rate (92,103,104). In addition, as seen from the statistical moments expression (Equation [1.32]), intensity measurements that are farther from the mean are weighted to a greater extent. This increases the imprecision in statistical moment determinations by increasing the influence of data with an inherently lower signal-to-noise ratio. Nonetheless, statistical moments are the 33 Figure 1-5: Schematic diagram of statistical moments on a zone profile. 34 Figure 1-5 . Aqmm 400 nm) is utilized to reduce the scattered and second-order radiation, and the fluorescent emission is focussed on the entrance slit of a monochromator (Instruments SA, Model H1061). The resultant emission is transmitted to a photomultiplier tube (Hamamatsu, R1463), subsequently amplified and converted to the voltage domain with matching current-to-voltage converter circuits. Analog-to-digital oonversi accomp‘ lrom ea mlcroco data at sollwar Calcul delerrr discus been metho most lnlegr SUlTlll sla‘ Eqr 66 conversion (Data Translation, Model 3405/5716) of the resultant voltage is accomplished with a resolution of 16 bits over a 100 nA range. Chromatograms from each detector block may be acquired simultaneously, and stored on a microcomputer (IBM PC-XT) as well as displayed in real time. Algorithms for data acquisition and calculations are written using the Forth-based Asyst software (Macmillan). Calculations. Statistical moments are chosen as the most accurate means to determine the retention time and dispersion of each solute zone (15). As discussed in Chapter 1, the use of statistical moments in chromatography has been well established (15-20). In contrast with many other techniques, this method requires no a priori assumptions about the peak profile, thus allowing the most accurate characterization possible (15). Although rigorously defined as integrals (Equation 1.19), statistical moments are often calculated based on finite summation of the fluorescence intensity as a function of time, l(t). Mo = Z l(t) At zeroth moment M1 = 2t l(t) At/Mo first moment [2.1] M2 = Z (t-M1)2 l(t) At/Mo second moment Mn = z (t-M1)n l(t) At/Mo nTH moment The time interval, At, is chosen to be sufficiently small, so the use of the summation may be justified. To this end, all calculations use a minimum of 40 data points that are uniformly distributed across the zone profile. For single-mode detection, the solute retention time is equal to the first statistical moment (M1) calculated at each detector block. Thus, as described in Equation [1.5], the capacity factor (k) may be evaluated directly from the first moment of the solute zone and that of a nonretained zone, corresponding with the elution per unit ler second mo H= 3L where L i: mode deli as the dif Likewise, moment region m capacity column s ol the co 2.3 Sur l allows 1 Column matche betwee Chlom; leouire this d6 67 the elution of acetone . Likewise, the plate height (H), or length variance (0L2) per unit length (L), may be determined directly from the time variance (03) or second moment (M2) and the zone velocity (U), H: 6L2 = (5'1-2U2 = M2(L/M1)2 [22] L L L where L is the distance between injection and the point of detection. In dual- mode detection, the retention time in the region between detectors is calculated as the difference in the first statistical moments (M1) evaluated at each detector. Likewise, the time variance is determined based on the difference in the second moment (M2) between detectors. Therefore, the plate height in this isolated region may be determined for each solute zone using Equation [2.2]. Both capacity factor and plate height may be evaluated either as a composite of column and extra-column effects using a single detector, or in an isolated region of the column between the two detectors. 2.3 Summary of Detector Fabrication A dual on-column detection system is designed and assembled, which allows the measurement of solute zones as they traverse the chromatographic column. Although all components in the two detection systems are carefully matched, the validity of the assumption of equal dispersion characteristics between detectors must be affirmed. Furthermore, the ability of the chromatographic system to achieve near theoretical separation efficiencies requires detailed evaluation. The following chapter will discuss the verification of this detection system under a variety of experimental conditions. 2.4 Lite 68 2.4 Literature Cited in Chapter 2 9’99.“ N 10. 11. 12. 13. 14. 15. 16. 17. Eggr, J.E.; Kristensen, E.W.; Wightman, R.M. Anal. Chem. 1988, 60, 4. gigglen, K.L.; Duell, K.A.; Avery, J.P.; Birks, J.W. Anal. Chem. 1989, 61, 4. Yang, F. J. High Resolut. Chromatogr. Chromatogr. Commun. 1983, 4, 83. Karlsson, K. ; Novotny, M. Anal. Chem. 1988, 60, 1662. Guthrie, E.J.; Jorgenson, J.W. Anal. Chem. 1984, 56, 483. 233thr7ie, E.J.; Jorgenson, J.W.; Dluzneski, P.R. J. Chromatogr. Sci. 1984, , 1 . Gluckman, J.; Shelly, D.; Novotny, M. J. Chromatogr. 1984, 317, 443. McGuffin, V.L.; Zare, R.N. Appl. Spectrosc. 1985, 39, 847. Diebold, G.J.; Zare, R.N. Science 1977, 196, 1439. Hershberger, L.W.; Callis, J.B.; Christian, G.D. Anal. Chem. 1979, 51, 1444. . Fglestad, S.; Johnson, L.; Josefsson. B.; Galle, B. Anal. Chem. 1979, 54,’ 9 5. Dovichi, N.J.; Martin, J.C.; Jett, J.H.; Trkula, M.; Keller, R.A. Anal. Chem. 1984, 56, 348. Shelly, D.C.; Gluckman, J.C.; Novotny, M.V. Anal. Chem. 1984, 56, 2990. Gluckman, J.C.; Hirose, A.; McGuffin, V.L.; Novotny, M.V. Chromatographia 1983, 17, 303. Bidlingmeyer, B.A.; Warren, F.V. Anal. Chem. 1984, 56, 1583A. Grubner, 0. Adv. Chromatogr. 1968, 6, 173. Grushka, E.; Myers, M.N.; Schettler, P.D.; Giddings, J.C. Anal. Chem. 1969, 41, 889. Chesler, S.N.; Cram, S.P. Anal. Chem. 1971, 43, 1922. Chesler, S.N.; Cram, S.P. Anal. Chem. 1972, 44, 2240. Kucera, E. J. Chromatogr. 1965, 19, 273. 3.1 Intro lr column Becausl well 001 column an ODE [1.11]), dual 0r evalua Capllla than a illtlivlc 01 the CHAPTER 3 VERIFICATION OF ON-COLUMN FLUORESCENCE DETECTION SYSTEM PERFORMANCE 3.1 Introduction Initial characterization and verification of the performance of the dual on- column detection system is accomplished utilizing open-tubular capillaries. Because the hydrodynamic behavior of these columns is well understood and well documented (1), they are ideal in determining the accuracy of the dual on- column technique. As shown in Chapter 1, the dispersion of the solute zone in an open tube can be readily determined from known parameters (Equation [1.11]). The goal of these preliminary studies is to determine the accuracy of the dual on-column detection system for the measurement of zone variance. This evaluation is accomplished by injecting a fluorescent dye onto an open-tubular capillary column and measuring the zone profile first near the column inlet and then again near the exit. Determination of the zone variance at each detector individually and as the difference between detectors allows the direct comparison of the measured variance values with those predicted based on the Golay 69 aquatic lheoret ol the I while t Wilke- emplo capac ill at Golay this 11 Oil-CC velor optir cont the disc 70 equation (Equation [1.11]). To make this direct comparison possible, all theoretical predictions utilize known experimental parameters. The linear velocity of the mobile phase (u), column radius (R) and length (L) are measured directly, while the diffusion coefficient in the mobile phase (DM) is determined utilizing the Wilke-Chang equation (2). For preliminary investigations, no stationary phase is employed to decrease the uncertainty in the known parameters; therefore, the capacity factor (k), stationary phase diffusion coefficient (D5), and film thickness (d,) are all zero. Thus, for an open-tubular capillary with no stationary phase, the Golay equation reduces to Equation [1 .10], the form developed by Taylor (3). In this way, the plate height measured for the open-tubular capillary using the dual on-column system can be directly compared to that predicted by Equation [1 .10]. In preliminary measurements of plate height as a function of linear velocity, extra-column sources of variance have been carefully minimized for optimal chromatographic performance. In subsequent studies, extra-column contributions to variance are systematically increased to examine the ability of the dual on-column technique to eliminate these detrimental sources of dispersion. 3.2 Experimental Methods Reagents. The fluorescent dye 7-(diethylamino)-4-methylcoumarin is obtained from Aldrich Chemical Co. and used without further purification. High-purity, distilled-in-glass grade methanol is obtained from Baxter Healthcare Products (Burdick and Jackson). The coumarin solute is dissolved in the pure methanol at a concentration of 3 x 10*1 M. Chrom illustral 400-cn 15 nL columr surfaci Delec emiss Corn; respo calcu dislril mom evalt Char vari: Tab is I “III All 71 Chromatographic System. A diagram of the chromatographic system is illustrated in Figure 2-1. In this study, the coumarin solute is introduced onto a 400-cm length of 40—um i.d. open-tubular capillary at volumes ranging from 1 to 15 nL (split ratios of 121000 to 1:65, respectively). This fused-silica capillary column is obtained from Polymicro Technologies and utilized without further surface modification. Detection and Calculations. When this column is coupled with the 500 um emission fiber, the viewed volume at each detector is no greater than 1.1 nL. Computer data acquisition is not employed for these studies, and the detector response is displayed directly onto a chart recorder. Statistical moments are calculated manually for each detector by finite summation of 30-50 points equally distributed across a peak. The arithmetic difference in the first and second moments are then determined and the linear velocity and length variance are evaluated as described in Chapter 2. Extra-column contributions to the variance are calculated as described in Chapter 1 (Equations [1.37], [1.38], and [1.391). The fractional increase in the variance (62) that is expected for these studies (Equation [135]) is shown in Table 3.1. 3.3 Results and Discussion Verification of the accuracy of the dual on-column measurement technique is accomplished by systematically varying the parameters of interest. Initially, the linear velocity of the mobile phase is varied under optimal detection conditions. All known sources of extra-column variance are kept well below 1% for this study (Table 3.1), and the dependence of the measured variance on the mobile-phase Table 5 study HVS.I More Wine 72 Table 3.1: Fractional increase in column variance (62) expected from extra- column sources. SIUdY UICm/SI VINJInL) 13(3) (92)INJ (921% H VS. U 0.010 1.0 0.10 0.080% 0.00015‘% 0.050 1.0 0.10 0.080% 0.0038°/o 0.10 1.0 0.10 0.080% 0.015% 0.20 1.0 0.10 0.080% 0.080% (<52)INJ 0.12 1.0 0.10 0.080% 0.022% 0.12 5.0 0.10 2.0°/o 0.022% 0.12 10 0.10 8.0% 0.022°/o 0.12 15 0.10 18°/o 0.02270 (62)Rc 0.12 1.0 0.01 0.080% 0.0022°/o 0.12 1.0 0.10 0.080% 0.022% 0.12 1.0 1.0 0.080°/o 2.2°/o 0.12 1.0 10 0.080% 220% H = 16.5 um; L = 400 cm; Floor. = 0.0020 cm; V051 = 1.1 nL, (evzloer = 0.096%. linear subser detern detest variar varlet syste dispe an e optin succ bea Plat met zon ace the dot are let hi hi] 73 linear velocity is determined for single- as well as dual-mode detection. In subsequent studies, extra-column sources of variance are reintroduced to determine if the column variance can be measured accurately utilizing the dual- detection method. For this purpose, two different sources of extra-column variance are chosen to determine the utility of dual-mode detection under a variety of chromatographic conditions. First, the injection volume is systematically increased (Table 3.1), contributing a symmetrical source of dispersion. Second, the detector time constant is varied (Table 3.1), contributing an exponential and, thus, asymmetrical source of dispersion. Finally, under optimal detection conditions, the distance between the detector blocks is successively decreased to determine the minimum volumetric variance that may be accurately determined using the dual on-column detection system. Plate Height versus Linear Velocity. By varying the linear velocity (u) and measuring the length variance (0L2) or corresponding plate height (H) of a solute zone, agreement of experimental results with Golay theory can be examined. To accomplish this goal, a single solute is injected onto the open-tubular column and the variance is measured at a single detector and as the difference of two detectors. The results of varying linear velocity in the region near the optimum are shown in Figure 3—1 for the 40 um i.d. capillary. Although great care was taken to minimize extra-column effects in the chromatographic system design, the single-detector measurements show substantial divergence from theory at higher zone velocities. In contrast, the dual-detector mode exhibits excellent agreement with Golay theory over the entire linear velocity region examined. Thus, the dual on-column detection scheme successfully measures the variance due only to the column proper as predicted by the Golay equation. 74 Figure 3-1: Plate height versus linear velocity: single mode (:1), dual mode (A). Golay theory (-—); column, RCOL = 20 um, L = 397 cm (srngle mode), L = 337 cm (dual mode); chromatographic conditions as descnbed in text. 75 Figure 3-1 Am\EoV Eood> mfiz: 0N0 0 ’0 N70 00.0 F r . _ L _ r 00.0 0— 0N" rd: 70F rm— Tum rNN r¢N rmN tmm 10». 1N». rfin 0m. (um) .Lr-iorar-r some inlluen extra-r varian deterr inlluer at hig detec trans accu empl woul colur pron disp acc Bot exe UIII 76 Figure 3-1 is also a good illustration of the difficulty in evaluating the influence of experimental variables on the column variance in the presence of extra-column sources of variance. Because both the column and extra-column variance are functions of the linear velocity, the linear velocity dependence determined from a single~detector measurement is a composite of both these influences. Therefore, evaluation of the mass transfer contribution from the slope at high linear velocity results in a significantly greater C term for the single- detector measurementthan is accurate for this capillary. In contrast, the mass transfer contribution measured utilizing the dual-mode approach is quite accurate. It is important to note that most commonly a single detector is employed. In this case, erroneous estimation of the mass transfer component would arise without our knowledge, even though all known sources of extra- column variance have been minimized. Thus, the dual-detection mode appears promising in the accurate determination of the various flow contributions to dispersion in chromatographic separations. Further validation of the dual on-column detection technique is accomplished by systematically varying extra-column sources of dispersion. Both volumetric and temporal sources of extra-column variance are adjusted to examine the ability of the dual-detector mode to eliminate these variances unrelated to the actual column variance. Injection Variance. Volumetric extra-column variance is systematically increased by varying the injection volume (Table 3.1). Sample volumes from 1 to 15 nL are injected onto the open-tubular column utilizing the split injection technique. Values for the plate height measured at a single detector and as the difference between detectors are illustrated in Figure 3-2. Substantial deviation of measured from theoretical variance values can be seen for the single-detector Figure 3-2: 77 Effect of injection volume on plate height: single mode (El). dual mode (A), Golay theory (—-); u = 0.12 cm/s. Conditions same as given in Figure 3-1. 78 Figure 3-2 35 m230> 20:09.2 9 .1 m, 9 m m 0F T:00 10h 100 100 Too— IO—p ONF (um) J.HDIBH HIV-Id node. T injected 1 however, lractional detection allowing Temper: an actiw circuit, 1 to to s. detector in the p‘ other e hour a] increas and th: Howev elperir OI ms the d] colum Small Illate detec per L 79 mode. The single-detector response shows a notable dependence on volume injected for volumes greater than approximately 4 nL. Dual-mode detection, however, corresponds very closely to theory, even up to 15 nL injected volume, a fractional increase (82) over the column variance of 18%. Thus, dual-mode detection successfully eliminates this volumetric form of extra-column dispersion, allowing accurate measurement of column variance. Temporal Variance. Temporal extra—column variance is adjusted with the aid of an active lowpass filter. By varying the resistance and the capacitance of this circuit, the time constant (c) of the amplifier for both detectors ranged from 0.01 to 10 5. These data, shown in Figure 33, clearly illustrate the inability of a single detector to accurately measure the column variance as predicted by Golay theory In the presence of an exponential time constant. Even for very low time constant, other extra-column sources of dispersion prevent the single-detector response from approaching the theoretically predicted value. As the time constant is increased, the disparity between the variance measured by the single detector and that predicted by theory increases drastically as expected from Table 3.1. However, when the variance is measured in the dual-detection mode, experimental variance accurately reflects theoretical predictions over three orders of magnitude in time constant. These results clearly demonstrate the ability of the dual on-column detection system to eliminate temporal sources of extra- column variance. Small Variance. Thus far, dispersion has been discussed only in terms of the plate height or length variance per unit length. Yet, one of the advantages of this detection scheme lies in the ability to measure accurately very small variances per unit volume. Preliminary studies have been conducted to determine the 80 Figure 3-3: Effect of detector time constant on plate height: single mode ([1). dual mode (A). Golay theory (—); u = 0.12 cm/s. Condrtrons same as given in Figure 3-1. single mode ([1). Conditions same 81 Figure 3-3 50 45— 40— I I I!) O P’) V) (W1) .LHSIEIH BLV'Id 25— [II 20 1) 15 10.0 1.0 0.10 0.01 TIME CONSTANT (s) smalles on-colu I, who predict changi paramr measu theory to ine statist mean scher l4-7I. deter metl the r Iron as me van eve air the the 82 smallest volumetric variance it is possible to measure accurately using the dual on-column fluorescence detection system. This feature is illustrated in Figure 3- 4, where the direct correlation of measured volumetric variance with that predicted by Golay theory can be seen. The column variance is adjusted by changing the column length between the two detectors, with all other system parameters kept constant. Substantial discrepancy is seen for single-detector measurements while the dual-detection mode shows excellent agreement with theory for variances as low as 10 nL2. Variation in the data is thought to be due to inequality of detector characteristics as well as to difficulty in calculating statistical moments by manual methods. Thus, the values reported here by no means represent the minimum variance accurately measurable by this detection scheme. Although small variance measurements have been previously reported (4-7), none, to our knowledge, are this accurate. Although this measurement scheme has many advantages for the in situ determination of separation processes, the limitations inherent in any difference method still prevail. Thus, the precision of dual-mode determinations is limited by the reproducibility of single-mode measurements. These fluctuations may arise from variations in the mobile-phase flowrate, temperature, and pressure, as well as errors in the evaluation of statistical moments. In addition, difference measurements performed in the presence of a large extra-column or column variance will inevitably lead to a decrease in precision due to the necessity for evaluating a small difference in large values. Therefore, although this technique allows the accurate determination of local retention and dispersion processes, the resulting precision in such measurements is limited by the reproducibility of the single-mode measurements. Figure 3-4: 83 Measured versus theoretical volumetric variance for small variance values: single mode (:1), dual mode (A), Golay theory (—); u = 0.12 cm/s. Conditions same as given in Figure 3-1. ’2 MEASURED VOLUMETRIC VARIANCE (ril__ ) MEASURED VOLUMETRIC VARIANCE (m?) 84 Figure 3-4 90 80 A 70 . 60 - so - 4o 7 so 1 20 - 10--4 DO U D fi I T T T fi 20 40 50 THEORETICAL VOLUMETRIC VARIANCE (m3) 1 80 3.4 Sum AI is capab chromat sources for dell hydrodj column optimiz were I possib 85 3.4 Summary of Performance Verification As demonstrated here, the dual on-column fluorescence detection system is capable of accurately determining the variance of solute zones directly on the chromatographic column. Even in the presence of symmetrical and asymmetrical sources of extra-column variance, dual-mode detection provides a reliable means for determining the true column variance. Thus, fundamental studies of hydrodynamic as well as chemical processes in both open-tubular and packed columns are feasible utilizing this technique. In particular, the examination and optimization of solute retention, band broadening, and nonequilibrium, which were hindered in previous investigations by extra-column dispersion, are now possible. 3.5 Literatu 1. Golay New 2. Wilkr 3. Tale 4. Sept 198i 5. van 6. Fole l. Mcl Chr 86 3.5 Literature Cited in Chapter 3 1. Golay, M.J.E. Gas Chromatography 1958; Desty, D.H., Ed.; Academic: New York, 1958; PP. 36-55. 2. Wilke, C.R.; Chang, P. AIChE J. 1955, 1, 264. 3. Taylor, G. Proc. Roy. Soc. (London) 1953, 219A, 186. 4. Sepaniak, M.J.; Vargo, J.D.; Kettler, C.N.; Maskarinec, MP. Anal. Chem. 1984. 56. 1252. 5. van Vliet, H.P.M.; Poppe, H. J. Chromatogr. 1985, 346, 149. 6. Folestad, S.; Galle, B.; Josefsson, B. J. Chromatogr. Sci. 1985, 23, 273. 7. McGuffin, V.L.; Higgins, J.W. International Symposium on Column Liquid Chromatography, dinburgh, Scotland, 1985. 4.1 Int tubular packer elucid In par lheorr nearly DhasI Simpl chroI com] mea CHAPTER 4 RETENTION AND DISPERSION ALONG THE CHROMATOGRAPHIC COLUMN 4.1 Introduction Although verification of the dual on-column detection system utilized Open- tubular columns, most applications in liquid chromatography rely on the use of packed columns. The remainder of the studies, therefore, focus on the elucidation and characterization of the fundamental factors affecting separations in packed beds. In an effort to correlate experimental measurements with theoretical expectations, these preliminary studies address separations under nearly ideal conditions. Only reversed-phase separations, where the mobile phase is more polar than the stationary phase, are utilized because of the relative simplicity of their theoretical treatment. In the first Of these studies, a single chromatographic column is examined in detail under constant mobile-phase composition and velocity conditions. Solute zone retention and dispersion are measured as a function of distance along the high-efficiency packed column. 87 l the chr well bl homolv metho‘ and m showr packir capac methr peak numl Final lluor who to b mec thes rem cha chr sui chr 88 Preliminary investigations Of these fundamental parameters necessitates the choice of solutes which are chromatographically as well as spectroscopically well behaved. TO facilitate direct comparison with theoretical predictions, a homologous series Of fatty acids, labeled with 4-bromomethyl-7- methoxycoumarin, have been chosen as suitable model solutes. The separation and concomitant detection of these solutes along the packed microcolumn are shown in Figure 4-1. 'When separated on a reversed-phase octadecylsilica packing material, the straight-chain, saturated fatty acids exhibit a wide range of capacity factors under isocratic conditions (approximately 0.5 to 5 in pure methanol). In addition, these solutes behave ideally, displaying symmetrical peak shapes and the logarithmic dependence of capacity factor on carbon number expected from Equation [1.6] for a homologous series (Figure 4-2). Finally, the fluorescence characteristics of these derivatives indicate favorable fluorescence in the methanol mobile phase, while no fluorescence is detected when n-decane is used to mimic the stationary phase (Figure 4-3). This appears to be due to lack of solubility Of the polar coumarin molecule in the nonpolar media. It is hypothesized that on a chromatographic column, the alkyl portion of these molecules resides in the stationary phase, whereas the coumarin moiety remains in the more polar mobile phase. This condition results in fluorescence characteristics that are not a function Of solute retention. Thus, both chromatographically and spectroscopically, these model solutes seem ideally suited as systematic probes of retention and dispersion along the chromatographic column. Figure 4-1: 89 On-column chromatogram Of derivatized fatty acid standards detected at 50 cm along the column. Solutes: n-Cmfl, n-Cmo, n- C14:o, ”C1620, n'C1830, n'CZOD fatty aCids derivatized With 4' bromomethyl-7-methoxycoumarin (100 nA full scale). Chromatographic and detection conditions described in the text. 90 Figure 4-1 / \ 1r / \ __u - ALEJLC \ L \ 91 Figure 4-2: Capacity factor versus carbon number for fatty acid derivative n- . 20:0, as measured dIrectly on-column at a distance 50 cm from the inlet. ChromatographIc and detection conditions described in the ex . 0m we m: E mmmE/SZ ZOmE<0 .: E N? . 92 Figure 4-2 m0 ION IOW Ice -90 0.0 (>I) HOIOW AllOVdVO 93 Figure 4-3: Fluorescence excitation and emission spectra of 4-bromomethyI-7- methoxycoumarin in methanol (—) and n-decane (--). 94 Figure 4-3 E5 Eozmd>§> com one oov 0mm com 0mm IIIIIWIIIIIH IIIII .IIIII..- IIIII E: mmm w 3a E: omv w 6}“ 20.3.5 h zofi .l 93 A. IO m 6 C < 6 00 R. .8082 E Ar 00 8 fl SVVI. C 7 e N 0 r A 1w I +0680 IO T. H 4 % co 8 on o oV++ # % IO 2 A B A _ _ _ _ _ 0 5 O 5 O 5 O 5 O 11 1| 4| 4| A83 30E: ESQ single I the her consta increa: evalua pressr be a t to the dittus‘ along extre initial exhil pres: inllu we sing vari dec inc are an to pit 108 single detector, again reinforcing extra-column variance as the probable cause Of the trends seen in Figure 4-6A. Although the local plate height values are more constant with distance than those evaluated at a single detector, the slight increase shown in Figure 4-6B appears to be statistically significant based on evaluation Of the slope. Because many Of the factors affecting dispersion are pressure dependent (Equations [1.23] to [126]), this increase in plate height may be a direct outcome Of the pressure gradient along the column (16). In addition to the changes in local linear velocity and capacity factor described earlier, the diffusion coefficient (11,17) also varies with pressure and, thus, with position along the column. Because the influence Of pressure is expected to reach an extreme in the region near the injector, studies are being extended to include this initial portion of the column. Recent results reported by Novotny eta]. (18) do not exhibit this change in plate height with position, perhaps due to the low inlet pressure and nearly nonretained solute utilized in their study. Although the influence of the local pressure on plate height has been discussed theoretically (17), it has been previously inaccessible to direct experimental measurement. I In addition to the differing trends of plate height with column length for the single- and dual-mode detection, it is interesting to note a difference in the variation in plate height with capacity factor. In the single mode, the plate height decreases with increasing capacity factor. In contrast, the dual mode exhibits an increase in plate height with increasing capacity factor. Since the measurements are performed at linear velocities slightly greater than the optimum, the mobile- and stationary-phase mass transfer terms (CM and Cs, respectively) are expected to have the predominant influence on plate height. If a van Deemter form of the plate height dependence on capacity factor is assumed (Equations [1.25] and [1.261), the effect Of the mobile-phase term would be an increase in plate height with capacity factor {f(k) = k2/(1+k)2}, while except for k < 1, the opposite trend is expect: measu the dUI is the j columr contrlt could (Equa ideal Altho adso simp chen varie over cont 8ch detr sev che 109 expected for the stationary phase {f(k) = k/(1+k)2). Thus, the single-mode measurements would indicate that the stationary phase effects predominate, but the dual-detector measurements suggest that mass transfer in the mobile phase is the primary contribution. Again, this discrepancy may be caused by the extra- column influence on the single-mode measurements. That is, extra-column contributions are expected to have a larger effect on less retained solutes and could reverse the actual on-column trend of plate height with capacity factor (Equations [1.35] and [1.361). It is apparent from these studies that the retention and dispersion Of even ideal solute zones along a chromatographic column is not yet clearly understood. Although it will be necessary in the future to extend investigations to include adsorption mechanisms and nonideal solutes, it is necessary to understand simple systems before proceeding to separations which are physically and chemically more complex. In addition, the influence Of extra-column sources of variance on the accurate characterization of on-column dispersion cannot be . overemphasized. The variation in the fraction of extra-column variance contributed at each detector position and to each solute profoundly affects the accurate measurement of dispersion using the single-mode technique. The dual- detection mode, which can eliminate these detrimental effects, clearly Offers several distinct advantages for the more accurate measurement of solute zone characteristics. 4.4 Conclusions Although presumed in many theoretical developments, it appears that retention and dispersion may not be truly constant along the column length. This anomaly I process a physical c phase ve variation will be m 110 anomaly has clear implications for Optimization and evaluation of the separation process and merits further investigation to determined whether the origin is physical or chemical in nature. In the following chapters, the gradient in mobile- phase velocity will be examined in more detail. In addition, the systematic variation in retention behavior with distance and, apparently, with local pressure will be measured directly. 111 4.5 Literature Cited in Chapter 4 I" NP’P‘PSD 15. 16. 17. 18. McGuffin, V.L.; Zare, R.N. Appl. Spectrosc. 1985, 39, 847. Gluckman, J.C.; Hirose, A.; McGuffin, V.L.; NOvotny, M. Chromatographia 1983, 17,303. Martin, M.; Eon, C.; Guiochon, G. J. Chromatogr. 1975, 108, 229. Huber, J.F.K.; Rizzi, A. J. Chromatogr. 1987, 384, 337. Martin, M.; Blu, G.; Guiochon, G. J. Chromatogr. Sci. 1973, 11, 641. Martire, D.E. J. Chromatogr. 1989, 461, 165. Foley, J.P.; Crow, J.A.; Thomas, B.A.; Zamora, M. J. Chromatogr. 1989, 478, 287. Jacob, L; Guiochon, G. J. Chromatogr. Sci. 1975, 13, 18. Giddings, J.C. Sep. Sci. 1966, 1, 73. Bidlingmeyer, B.A.; Rogers, L.B. Sep. Sci. 1972, 7, 131. Katz, E.; Ogan, K.; Scott, R.P.W. J. Chromatogr. 1983, 260, 277. Tanaka, N.; Yoshimura, T.; Araki, M. J. Chromatogr. 1987, 406, 247. Evans, C.E.; McGuffin, V.L. Anal. Chem. submitted. Giddings, J.C. Dynamics of Chromatography; Marcel Dekker,lnc.: New York, 1965; pp.79-84. Evans, C.E.; McGuffin, V.L. J. Microcol. Sep. submitted. Poe, D.P.; Martire, D.E. J. Chromatogr. 1990, 517, 3. Martin, M.; Guiochon, G. Anal. Chem. 1983, 55, 2302. Karlsson, K.; Novotny, M. Anal. Chem. 1988, 60, 1662. 5.1 | drivl in g: phar exp. the rice line visv chr chr CHAPTER 5 THE INFLUENCE OF PRESSURE ON MOBILE-PHASE VELOCITY 5.1 Introduction A spatial pressure gradient is inherent in any system where pressure is the driving force for fluid flow. This gradient has long been recognized as important ‘in gas chromatographic applications due to the high compressibility Of the mobile phase (1-3). As the pressure decreases along the column length, the gas expands to occupy an increased volume, resulting in a concomitant increase in the volumetric flowrate. Under constant mass flow conditions, this decompression of the mobile phase also yields an increase in the mobile-phase linear velocity with distance. Fortunately, while gases are quite compressible, the viscosity Of gases is low and, thus, the pressure drop necessary for chromatographic flowrates is minimal (1-2 bar). Nonetheless, for typical gas chromatographic conditions, mobile-phase compression may result in increases Of 50% in linear velocity over the length Of a packed column (3). By contrast, liquids are generally considered incompressible and the variation in linear velocity along the column is consistently ignored in liquid chromatographic applications. 112 Howev viscous necess psi). l velocit marke introdr GIIOI'S exam veloc be di velor exar phys inllu proc 113 However, the mobile-phase liquids commonly utilized are significantly more viscous than their gaseous counterparts and, thus, the applied pressures necessary for fluid flow are considerably higher, typically 50 to 350 bar (700-5000 psi). Because liquids are slightly compressible, the resulting increase in linear velocity along the column under typical conditions may be 1—5% (4). Although markedly less than for gas chromatographic applications, the systematic error introduced by ignoring this pressure gradient effect may give rise to significant errors in fundamental studies of the separation process. The on-column detection technique provides a unique opportunity to examine this effect experimentally. By measuring the mobile-phase linear velocity as a function Of distance along the column, the in situ measurement may be directly compared with theoretical predictions. In addition, the effect of this velocity gradient on the accurate determination of column porosity may be examined. While the investigations discussed in this chapter focus on the physical effects of the pressure gradient, the next chapter will explore the ‘influences of the local pressure on the chemical aspects of the separation ‘ process as well. 5.2 Theoretical Considerations Because pressure influences so many physical and chemical processes, evaluation Of pressure effects in chromatographic separations requires careful attention to detail. Even the seemingly obvious choice of pressure as the variable Of interest has been recently questioned (5). While pressure may be applicable as a variable when the fluid is an ideal gas, density is a more directly useful state variable when nonideal fluids are Of interest. Thus, because reversed-phase chromatographic conditions are investigated here, the general lormalism employed. the densit‘ function 0 are given (5) for the Th packed c This gel regime separati assume may va terms c dX=-_ Where consta (p) 114 formalism based on density that was recently introduced by Martire (5) is employed. This derivation is quite general and may be applied to any fluid where the density as a function Of pressure (i.e., compressibility), and the viscosity as a function of density are known. Although portions of the theoretical development are given here for clarity, the reader is referred to the excellent paper by Martire (5) forthe full derivation and validation. Theoretical evaluation of the local mobile-phase density (px) along a packed column is accomplished based on Darcy‘s law (6). Bo 5T TI u = - (dP/dx) [5.1] This general expression, which is valid when fluid flow is in the laminar flow regime and at constant temperature, is directly applicable to chromatographic separations. The column permeability (Bo) and total porosity (8r) are commonly assumed to be constant, while the viscosity (1]) and pressure gradient (dP/dx) may vary with distance along the column. Equation [5.1] may be rearranged in j terms of the density (p) to yield, t B jdx = -_°_ (cIP/dp)T dp [5-2] 81- n u where (dP/dp)T is equal to the reciprocal of the compressibility coefficient (8“) at constant temperature divided by the density p. The spatial average density (

x) in the region extending from distance x, to x,- may be described as the first statistical moment with distance. x-I

x = ;_ [5.3] This e assun length ilowre linear densi U1 pX Com dens dete

. th ant 115 This expression conveniently allows the determination Of the local density by assuming a small interval from x, to X], or the average density over the column length when the integration interval is 0 to L. If it is assumed that the mass flowrate is constant throughout the system, the product of the mobile-phase linear velocity and density remains constant within the column regardless of local density/pressure. u, Px = uo p0 = constant [5.4] Combining Equations [5.2], [5.3], and [5.4], and changing integration variables to density, the necessary expression for the spatial average density (

,,) may be determined. p. I ’92 n" (dP/dph dp I

x = 9 [5'5] PI‘ p.l p n-1 th/dP)T do I where p, and pi are the local density values corresponding to distance positions x,- and xi, respectively. Evaluation Of Equation [5.5] requires knowledge Of both the density and the viscosity as a function Of pressure for the fluid Of interest. For liquids, the density behavior with pressure may be evaluated using a modified form Of the Tait equation of state (4,7). Po+b = p _ 5. tm ) expts: 1) ‘6] In this empirical expression, a constant mass is assumed to be present at pressure P and atmospheric pressure P0. In the pressure region Of interest in liquid r the firs Po‘t' P—fi where press the av intere lunct mobi the l 116 liquid chromatography (30 to 350 bars), Equation [5.6] may be approximated by the first term in a series expansion, [35: )°= [1) [5.71 P + b [30 where b and c are constants for a given fluid at constant temperature, and the pressure is expressed in bars. Although this expression is valid for many liquids, the accuracy of this approximation must be determined for each mobile phase of interest. In addition to the variation of density with pressure, viscosity is also a function of the local pressure on the column. The influence of pressure on the mobile-phase viscosity (n) is Often evaluated assuming a linear dependence of the form, n = T10 ( 1 + (11°) [58} where on and no are constant for a given isothermal fluid (4). As for the density ‘ expression, the validity Of Equation [5.8] must also be assessed for each specific t set Of experimental conditions. Evaluation of Equations [5.7] and [5.8] for the mobile-phase fluid Of interest, combined with Equation [5.5] yields the average density in specific regions of the column. From this expression, the variation in linear velocity along the column may be predicted based on Equation [5.4]. 5.3 Application to Experimental Conditions For the pure methanol mobile phase utilized in this study, the behavior of the density with pressure Is accurately described by Equation [5.7], resulting in errors [5.7], t and c As sl Equal nearly chror likew only 0.53 resu the I

Attl alo 117 errors Of less than 0.4 ppt for pressures up to 350 bar. In evaluating Equation [5.7], the density Of methanol at atmospheric pressure (p0) is 0.787 g/mL, while b and c are equal to 1210 and 0.148 when the pressure is expressed in bars (4). As shown in Figure 5-1, the reduced density (PR = p/pc), determined from Equation [5.7] and the critical density for methanol (p0 = 0.321 g/cm3), exhibits a nearly linear dependence on pressure in the range of interest in liquid chromatography. Experimental measurements Of methanol viscosity (8,9) can likewise be utilized tO verify the accuracy Of Equation [5.8], resulting in an error of only 0.3 ppt in the same pressure range (with a = 4.70 x 10'4 bar '1 and Tie = 0.531 OP at 25 °C). Substitution of Equations [5.7] and [5.8] for methanol into Equation [5.5] results in the following expression for the average mobile-phase density (

x) in the specific region extending from local densities p, to Pr- p. I 19776 dP / (0.4313 + 2.8728 p575)

x = p, [5.9] p. pl Jp676 'dp / (0.4313 + 2.8728 p676) Alternatively, this expression may be written in terms of the fractional distance along the column (x/L) upon integration Of Equation [5.2]. " p t d" 1 x96-76 dp / (0.4313 + 2.8728 p676) m = 0 = pi [5.10] L p t d" t 095-76 do / (0.4313 + 2.8728 p676) ° Pr Unfortunately, this expression is cannot be solved analytically and the local density (px) used as an integration variable must be calculated as a function of the fractional distance along the column by successive approximation. As shown in Figure 5-2, the relationship between the local density 1th and fractional Figure 5-1: 118 Calculated reduced density as a function of pressure for methanol (Equation [5.71). n’) 119 Figure 5-1 O -O 1‘0 O -O N O F0 l r l fir fit 0 N CD ‘1' O O 0‘. 0‘. 0? W (\l N N ALISNEIO 0300038 PRESSURE (bar) 120 Figure 5-2: Local density (th versus fractional distance along the column-(x/L) calculated from Equation [5.101 ( [:1 ) compared wrth lrnear interpolation (—). 0.9 0A0 0A0 #gu NAU 0. 0 2 121 Figure 5-2 .mde 7.0.83 888 owed momma Tamed mam; wwomd woowd mo 5.0 we 5.0 w $5.0 . Nde (101/o) "d ‘xtISNzrcr TVOOT column I pressure 170 and h=h' Thus. tl points z evaluat (

,) t gradier 5.4 E) IS per and L methe yieldir from monit cm a linea ttOWl s, l 122 column distance (x/L) is linear for methanol within 0.5 ppt error for the Inlet pressures commonly encountered in liquid chromatography (2500 and 5000 psi; 170 and 340 bar). I). = p. - (pr - po) (XM [511] Thus, the spatial average density

,, can be determined between any two points along the column as a function of fractional distance. If the density is evaluated between two points in close proximity, the spatial average density (

x) and the local density (p,) become equal. Finally, the effect of this density gradient on the linear velocity gradient may be predicted based on Equation [5.4]. 5.4 Experimental Evaluation Experimental measurement of the linear velocity as a function of distance is performed on the identical chromatographic column described in Chapter 4 and under similar chromatographic and spectroscopic conditions. The pure methanol mobile phase is Operated in the constant flow mode (F = 0.72 uL/min), yielding an inlet pressure (P,) of 4000 psi (270 bar). A decomposition product from the coumarin label is, again, utilized as the void marker. The void time is monitored as a function of distance using two detectors positioned at 30 and 50 cm and, subsequently, at 70 and 90 cm along the column ( = 100 cm). The linear velocity at atmospheric pressure (uo) is calculated from the volumetric flowrate (F) measured at the column exit, F = 71: RCOLZ U0 8T [5.12] where II the colu F [5.41 an linear v (170 an column column compre evalua local v may a expen injectir single alway densi equiv along 5-4. pred How appr the 20m lor 123 where the column radius (RCOL = 0.0100 cm) and the total porosity (2T = 0.43) Of the column have been determined previously. Prediction Of the linear velocity behavior is accomplished utilizing Equation [5.4] and the linear approximation Of density with distance verified above. Local linear velocity ratios (ux/uo) expected for inlet pressures of 2500 and 5000 psi (170 and 340 bar, respectively) are shown in Figure 5-3 as a function of fractional column distance (x/L). Although the expected change in linear velocity along the column is less than for gas and supercritical fluid phases, which are highly compressible, the variation in the local velocity of 2 to 4% may be important in evaluating local retention and dispersion processes. In addition to calculating the local velocity ratio (uxluo), the ratio predicted as an average from the column inlet may also be evaluated. This cumulative velocity ratio (,,/uo) is that measured experimentally utilizing a single detector to determine the column void time from injection. It is clear from Figure 5-3 that the linear velocity measured using a single detector is an average Of the velocity up to the point Of detection and, thus, always underestimates the local linear velocity at that position. Because the local density is linear with distance, the spatial and temporal average velocities are equivalent and may be used interchangeably (5). Experimental measurements Of the cumulative velocity ratio with distance along the column are illustrated together with the theoretical predictions in Figure 54. Although measurements are in general agreement with theoretical predictions, the precision in replicate determinations appears to be rather poor. However, the relative standard deviation in replicate measurements Of approximately 2% is quite good, considering that both the deviation in measuring the volumetric flowrate and in calculating the first moment Of the nonretained zone contribute to the determination. The difficulty appears to lie in the necessity for measuring changes in the zone velocity Of only a few percent. Further 124 Figure 5-3: Theoretically predicted ratio Of the local mobile-phase linear velocity (u,,) to the linear velocity at atmospheric pressure (uo) versus the fractional column distance (x/L) (—), together with the cumulative velocity ratio (,,/u0) (--). 125 Figure 5-3 0.0 . _ h N0 0.0 owed _maooom. aaoonw |I ‘II ‘- II‘ I II‘ 1 I\ ‘II °n/n Figure 5-4: 126 Comparison of theoretically predicted local (— ) and cumulative (--) velocity ratio to experimental measurement of the cumulative velocity ratio (/u0) using single- mode detection ( O Chromatographic conditions. Methanol F: 0. 72 uL/min, P = 4000 psi (270 bar). .02 00; 00.0 00.0 004.0 0N0 00.0 00.0 127 Figure 5-4 examini inlet pr theoreti P,) is s the me good develo incomj the co meaSI pores deterr lluid avere the 1 mobi Althr appl chic calc 128 examination of this mobile~phase compression is accomplished by varying the inlet pressure and, thus, the linear velocity gradient along the column. The theoretical prediction of this effect as a function of the pressure difference (P,, - P0) is shown in Figure 55 together with the measured velocity ratios. Although the measured data show considerable variation, the trend with pressure is in good agreement with that expected based on the preceding theoretical development. Thus, even though methanol is usually considered an incompressible fluid, a systematic increase in linear velocity along the length of the column is theoretically predicted and is experimentally Observed. This density/velocity gradient has important implications in the measurement of fundamental column parameters. For example, the total porosity of the column, indicative Of column packing structure (10-14), is determined experimentally based on Equation [5.121 (11). However, when the fluid is compressible, the linear velocity determined from the void time is the average velocity ,,, not the velocity at atmospheric pressure uo. In this case, the 87 value determined using Equation [5.121 assuming an incompressible mobile phase is in error by the factor u0/,, as shown in Equation [5.13]. F = 7: RCOL2 x 8T (U0/x) [5.13] Although this error has been addressed in the literature for gas chromatographic applications (13), it has been assumed to be unimportant in studies Of liquid chromatographic columns. As shown in Figure 5-6, total porosity values calculated based on Equation [5.121 exhibit a clear increase with pressure drop. Moreover, this variation is in good agreement with the expectation that this error arises from the change in uolx with pressure. Thus, the Often unstated assumption Of incompressibility leads to a systematic overestimation Of the total column porosity. From Figure 5-6, the total porosity may be determined by linear Figure 5-5: 129 Experimental measurement Of the mobile-phase linear velocity who as a function of the pressure difference (P, - Po) on the column at 30 cm (0), 52 cm ([1), and as the dual (A) measurement (u = 0.014 tO 0.11 cm/s). Theoretical prediction (—) calculated based on Equations [5.7] and [5.91. 10 130 Figure 5-5 080 E I .5 com oo_ o _ . _ t 05.0 >m0mIH 1250 E0 mm H I_ E0 00 h I_ I00.— Q D O 07F Figure 5-6: 131 Apparent increase in the total porosity with pressure difference (P,- - Po) resulting from the compression Of the mobrle phase shown In Figure 5-5. Experimental conditions as described FIgure 5-5. 0.47f 080 E I .E 0am owm 0A.: 0 28 o 2.0 :5 mm n a a E0 on H I_ < o - r240 e 5 am - 1m. F. I940 5&0 (13) xusoaoa ‘IVLOI extrap meass syster measr over syster as to cclun expel COIIII 5.5 bee Thu OCC inc rep chr 133 extrapolation to zero pressure, resulting in an 8T value of 0.427. Off-column measurements based on Equation [5.121 yield a total porosity Of 0.446, a systematic error Of +45% for an inlet pressure of 270 bar. Because measurements performed after the column exit yield an average of the velocity over the length of the column, total porosity determinations will always be systematically greater than the true value. Although this error is not as extreme as for gas or supercritical fluid conditions, the determination of fundamental column parameters is significantly and systematically affected. This deviation Is expected to be even more pronounced in normal-phase separations, where commonly utilized mobile-phase solvents exhibit greater compressibility. 5.5 Conclusions The pressure/density gradient along the chromatographic column has been shown to induce a linear velocity gradient under reversed-phase conditions. Thus, the linear velocity measured at the column exit is an average of that occurring over the length of the column. Under commonly encountered experimental conditions, the compressibility Of even polar solvents results in an increase in mobile-phase velocity of ~3%. This result is consistent with recent reports by Foley er al. of errors in delivering accurate flowrates to chromatographic columns (15). This velocity gradient also has interesting implications for the dispersion of solute zones along the column (Equations [1.231 to [1.261). In addition, the total column porosity measured at the exit will appear to be a function Of the pressure drop and may be systematically overestimated by more than 4%. Thus, contrary to common misconceptions, the experimental conditions commonly encountered in liquid chromatographic separations are sufficient to induce a significant velocity gradient along the column. 5.6 L 134 5.6 Literature Cited in Chapter 5 93.01.45.031“ 10. 12. 13. 14. 15. James, A.T.; Martin, A.J.P. Biochem. J. 1952, 50, 679. Giddings, J.C. Anal. Chem. 1963, 35, 353. Guiochon, G. Chrom. Rev. 1966, 8, 1. Martin, M.; Blu, G.; Guiochon, G. J. Chromatogr. Sci. 1973 11, 641. Martire, D.E. J. Chromatogr. 1989, 461, 165. Darcy, H.P.G. Les Fontaines Publiques de la Ville de Dijon, Victor Dalmont, 1856, Paris. Tait, P.G. Scientific Papers Vol. 2, University Press: London; 1900. pp. 343-348. Bridgman, P.W. Proc. Acad. Arts and Sci. 1926, 61, 59. lsakova, N.P.; Oshueva, L.A. Russ J. Phys. Chem. 1966 40, 607. Unger, K.K. Porous Silica Elsevier: Amsterdam, 1979. Bristow, P.A.; Knox, J.H. Chromatographia1977, 10, 279. Ohmacht, R.; Halasz, J. Chromatographia1981, 14, 155. gamers, C.A.; Rijks, J.A.; Schutjes, C.P.M. Chromatographia 1981, 14, Gluckman, J.C.; Hirose, A.; McGuffin, V.L.; Novotny, M. Chromatographia 1983, 17, 303. Foley, J.P.; Crow, J.A.; Thomas, B.A.; Zamora, M. J. Chromatogr. 1989, 478, 287. 6.1 | Mar unde disti loca grar lune add alsr der ettl lur Dll CHAPTER 6 THE INFLUENCE OF LOCAL PRESSURE ON SOLUTE RETENTION 6.1 Introduction In this chapter, studies of pressure effects on separation processes are extended to include the chromatographic retention of solute zones. To understand the effect Of pressure in chromatographic systems, it is important to distinguish the effect Of the pressure/density gradient from the influence of the local pressure/density. As discussed in the previous chapter, the pressure gradient along the column causes decompression Of the mobile phase as a function Of distance, resulting in an expansion of the fluid along the column. In addition to the pressure gradient, the absolute local pressure on the column may also affect separation processes. The influence Of pressure on fundamental physical properties such as density, viscosity, diffusion coefficient, etc. is well documented (1-6). Pressure effects on equilibrium processes have been investigated as well (7,8). This fundamental understanding has made possible a number Of theoretical predictions Of the influence on mobile-phase velocity (9), solute retention (10), 135 and indl Gid chrI 136 and zone dispersion (11). The possible control and exploitation of pressure- induced equilbrium shifts for chromatographic separations was first introduced by Giddings (12,13), leading to the development of supercritical fluid chromatography. Although variations in equilibrium constants in the liquid phase may not be expected to be large when compared to those in supercritical fluids, the greater pressures commonly applied in liquid chromatographic separations may be sufficient to induce significant changes. Under high-speed conditions or with recycle systems, the increase in retention with pressure may be significant enough to limit applicability (14). Unfortunately, only a limited number of experimental investigations have been reported in the literature. Under extreme pressure conditions (20,000 psi), Rogers etal. (1517) measure up to a three-fold Increase in capacity factor and a significant change in selectivity with separations utilizing an adsorption/exchange mechanism. Likewise, Tanaka et al. (18) measure a 12% variation in capacity factor for reversed-phase separations performed under ionization control with much more modest pressure variations (3000 psi). Measurements by Katz et al. (19) under normal-phase conditions indicate a decrease in retention, which the authors attribute to an increase in temperature in the column interior. In all cases, variations in the absolute pressure, not the pressure gradient, lead to significant variations in solute retention. These findings have interesting implications for the variation in retention with distance along a chromatographic column. It is clear that if a retention gradient is present, the capacity factor measured at the column exit is only an indication of the average behavior of the solute. The determination of fundamental retention parameters becomes even more complicated if the selectivity is also a function Of the local pressure/distance. The on-column detection scheme may be used to advantage in evaluating the i meal syste pres relel add vari Fin: oli Apt 137 the influence of these pressure-dependent retention processes. The direct measurement of local solute retention on the chromatographic column allows the systematic evaluation of the influence of local pressure. By varying the inlet pressure while maintaining constant pressure gradient conditions, the local retention may be directly correlated to the local pressure on the column. In addition, the use of small diameter, packed-capillary columns minimizes variations in temperature within the column that may arise from viscous flow. Finally, the model solutes chosen for these studies probe the most universal type of interactions, induced-dipole induced-dipole. Thus, systematic measurements of the dependence of local pressure on solute retention should find general applicability for liquid chromatographic separations. 6.2 Theoretical Considerations As discussed in Chapter 1, solubility parameter theory can be utilized to predict retention behavior in liquid chromatography. Thus, insight into the possible relationship between pressure and solute retention can be gained from this theoretical approach. The cohesive energy density of interaction or solubility parameter (5), which is utilized to estimate the species polarity, is given by (20) 82 = -EN = (aE/aV)T [64} where E is the interaction energy and V is the molar volume. More generally, the solubility parameter is described as the partial derivative of energy with molar volume under constant temperature conditions (21). If the interaction energy is approximated from van der Waal's theory (21), 5 can be expressed in terms Of the van der Waal's coefficient a and the species molecular weight m. Der allc relr dev thir int 138 6 = (aVZN) = (am/m) (3 [62] Density (p), not pressure, is chosen as the state variable in this expression to allow more general applicability, as discussed in Chapter 5. Implicit in this relationship is the assumption that increased density/pressure acts solely to decrease the volume, without altering the nature or energy Of interaction. While this assumption may be questionable for species with multiple modes Of interaction, it appears to be reasonable if only dispersion interactions are present. The weak variation in the interaction energy with distance for dispersion forces (the) makes the presumption of little change with pressure a viable one (Equation [1.21). For subsequent comparison with other theories, it is convenient to write Equation [6.2] as a function of the reduced density (pH = p/pc). 5i = (PC allZ/m) PR [6'3] In this expression, the density is expressed as a fraction of the critical density (pc). Based on Equation [6.3], the solubility parameter and, therefore, the species polarity is expected to increase linearly with the reduced density. This relationship has interesting implications for the expected behavior of solute retention with pressure/density. As discussed in Chapter 1 (Equation [1.71), the capacity factor Of solute i (k,) may be described by the following expression (22): "1kg: V i [(51 ' 5M)2 ' (8r ' 5312] ' In I3 RT Unfortunately, theoretical advances are presently insufficient to allow the prediction Of the influence of pressure/density on all of these parameters. However, the low concentration of solute species i allows the approximation that the pres pha met our our exy pre 139 the solute molar volume (V,) and solubility parameter (8) are independent of the pressure/density. In addition, the stationary phase solubility parameter (53) and phase ratio (13) are assumed to be less affected by the pressure/density than the mobile phase. Although not rigorously correct, this supposition is reasonable considering the stationary phase is attached to a solid support, limiting the compressibility compared to a bulk phase (10). Thus, consistent with the few experimental measurements in the literature (15-19), the primary effect Of pressure/density is assumed to arise from the mobile phase. Based on this assumption, Equation [6.3] for the mobile phase may be combined with Equation [1 .71 to give the solute capacity factor as a function of the mobile-phase reduced denSity (pRM)' 1” k1 = CI pRM2 1’ C2 PRM 1' Ca [6-41 where 01 = Vi POM 8M"2 2 RT mM c2 = ' 2V1 POM 3M”2 5' RT mM c.= [_V__) {5.2-(5i-3sPHnl3 RT In this expression, O1 represents mobile-phase/mobile-phase interactions, where pm is the reduced density of the mobile phase, aM is the van der Waal's interaction coefficient, and mM is the molecular weight. As expected, interaction Of mobile-phase molecules increases with reduced density/pressure, leading to an increase in solute capacity factor. In the second constant (Cg), the solute/mobile-phase interaction is expressed in terms of the mobile-phase F It. r 140 interaction coefficient (aM) and the solute solubility parameter (8). As the reduced density of the mobile phase increases this parameter becomes more negative, resulting in an overall decrease in retention. Finally, solute/stationary- phase interactions are described in the third constant (c_,). As assumed in the derivation, this term is independent of the mobile-phase reduced density. An increase in these interactions results in the intuitively predicted increase in retention. This expression (Equation [6.41) is similar to predictions based on statisitical thermodynamics developed by Martire (10). However, Martire's lattice model approach is more rigorous in describing interactions between species assumed to be independent in solubility parameter theory. Unfortunately, the estimation of these interaction parameters is quite difficult in the liquid phase. The expected variation in the selectivity (01) or differential movement of solutes can likewise be predicted as a function of pressure/density. Combining the definition of selectivity with Equation [1.7] results in an expression for 01 as a function of solubility parameter values (22): rnoe.=rni=y Vi )ea-eirsu-asr [6.51 k, RT As derived above, substitution of Equation [6.3] results in an expression for the solute selectivity as a function of the mobile-phase reduced density. C4 = [ 2V1 J [ POM 3M1”) (5' _ 5]) RT mM Cs= [___2Vt )(5i' 5)) 55 RT From this expression, a simple linear variation in In 01,-,- with the mobile-phase 141 reduced density is predicted. The slope (O4) in Equation [6.6] corresponds to interactions between the mobile phase and the solutes, while the intercept is indicative of solute/stationary-phase interactions. From the above expressions, both solute retention and selectivity are predicted to be affected by the reduced density of the mobile phase and, thus, the local pressure on the column. Unfortunately, the absolute magnitude of this influence is difficult to predict a priori. The requirement for accurately estimating five unknown variables (8M, 85, [3, 8,, and Vi), combined with the quadratic dependence on a logarithmic term (Equation [64]) lead to large errors in the prediction of solute retention. However, direct comparison of theoretically predicted trends with experimental measurements should provide a good qualitative understanding of the parameters affecting retention. 6.3 Experimental Methods Analytical Methodology. Saturated fatty acid standards from n-Cmo to n-Cmo are derivatized with 4-bromomethyI-7-methoxycoumarin as described In Chapter 4. Standards are dissolved in methanol and injected individually at a concentration of 5 x 104 M. Chromatographic System. The chromatographic system is similar to that described in Chapter 2 and illustrated in Figure 2-1. In this study, however, a 25.7 cm length Of Open-tubular capillary (0.0050 cm i.d.) connects the injector to the column. This arrangement makes possible the placement of detectors in the high-pressure region near the column inlet. The microcolumn utilized for these pressure studies is fabricated as 142 described in Chapter 4. Before packing, the polyimide coating is removed from a 43.9 cm length of fused-silica tubing at 5 cm intervals to facilitate on-Oolumn detection. The resulting column has a plate height Of 9.5 pm, a total porosity of 0.43, and a flow resistance parameter Of 550 under standard test conditions (23). For all measurements, a pure methanol mobile phase is delivered to the column with a single-piston, reciprocating pump (Beckman, Model 114M) under constant pressure conditions. The inlet pressure is systematically varied from 100 to 350 bar (1500 - 5000 psi) using a 20 pm i.d. restrictor at the column exit. The length of the restrictor and splitter are altered in the course of this study to allow the volumetric flowrate (F = 0.70 uL/min) and injection volume (VjNJ = 11 nL) to remain constant under all pressure conditions. Detection System. The Optical detection conditions are identical to those described in Chapter 2. As shown in Figure 6-1, however, six matched detector blocks are positioned along the column with optical fibers from alternate blocks connected to each detection system. Since all detectors are active continuously, solutes are injected individually, thus allowing multiple detection points for a single injection with only two monochromator/photomultiplier detection systems. The first detection block is positioned on the open tube 0.4 cm before the packed bed, while the remaining five detectors are placed on the packed bed at 4.9, 10.4, 15.5, 20.9, and 26.2 cm from the column head. Themaximum viewed volumes under these conditions are 1.8 nL off-column and 12 nL on-column. Data acquisition is accomplished under computer control at 5 Hz with a 0.06 s time constant. Figure 6-1: 143 Schematic diagram of detection system allowing Simultaneous measurement at six points along the chromatographic column wrth only two monochromator/photomultiplier systems. I: Injectron valve, T: splitting tee;'R: restricting capillary; MONO: monochromator. PMT: photomultiplier; AMP: current-to-voltage converter/amplrfrer. KMFDQEOO 144 Figure 6-1 0202 ZEDJOU 0401 _ @231 00¢ ..... mmm<4 UoOI 145 6.4 Results and Discusslon Experimental evaluation Of the pressure/density dependence of solute capacity factor is accomplished by systematically increasing the inlet pressure from 102 to 337 bar (pmm = 0.796 to 0.816 g/cm3, respectively, for methanol), while maintaining a constant pressure differential of 102 bar along the column. By measuring the capacity factor using single- and dual-mode detection, the average between the injector and the point of detection as well as the local capacity factor between detectors may be determined directly on the column. In addition, measurements are performed simultaneously at five positions in the high-pressure region of the column, allowing the evaluation of the local capacity factor at several positions for a single injection. Experimental measurements performed in this manner allow the direct, in situ determination of solute retention as a function of local pressure under nearly ideal chromatographic conditions. Experimental measurements of capacity factor for both single- and dual- mode detection (L = 26.2 and 23.5 cm, respectively) are shown in Figure 6-2. A significant increase in the retention of n-Cm, is Observed as a function of pressure for both single and dual modes, with n-Cwo exhibiting an increase greater than 16% with a precision in duplicate determinations of 0.5%. This increase is not limited to n-szo, and a notable increase in k is measured for all solutes under practical inlet pressure conditions. The magnitude of this increase is somewhat surprising, however, based on the common belief that reversed- phase solvents are quite incompressible. It is even more unexpected for these model solutes, where only dispersion interactions are controlling retention. However, an increase in k is theoretically predicted based on the variation in the mobile—phase solubility parameter or polarity with pressure or reduced density (Equation [6.31). As shown in Chapter 5, the reduced density of the methanol 146 Figure 6-2: Single- (top) and dual- (bottom) mode measurements Of solute capacity factor (L = 26.2 and 23.5 cm, respectively) for n-Cmo as a function of the average pressure encountered by the solute. Experimental conditions as given in the text. CAPACITY FACTOR 147 Figure 6-2 2.2 — 9 2.0 4 n 9 1 as: 3E 1.8 I I l I l l E 2.2 -— _ ‘- e 2.0 — :1: _ E 1.8 1 I # I l I 0 100 200 300 AVERAGE PRESSURE (bor) 148 mobile phase is approximately linearly dependent on the applied pressure (Figure 5-1). If only mobile-phase effects are considered, an increase in reduced density is predicted to yield a concomitant increase in solute retention (Equation [6.41). As will be discussed later, determining the theoretically expected magnitude of this increase is nontrivial. Although all solutes exhibit this increase in retention with pressure (Table 6.1), the experimentally measured percent increase (Ak/k) is systematically greater for longer chain solutes. This apparent dependence on chain length is theoretically predicted (Equation [6.41) based on the direct dependence Of In k,- on solute molar volume (V,) and, consequently, on carbon number. Thus, while the energy Of interaction between the solute and mobile phase is assumed to remain constant, it is expected that the increase in the energy density with pressure will be greater when the solute occupies a larger volume. For this reason, the absolute pressure is expected to affect not only retention, but selectivity as well. Experimental evaluation of the solute selectivity with pressure clearly demonstrates this enhancement (Table 6.2). All adjacent solute pairs show a systematic rise in selectivity with average pressure, regardless of detection mode. Although the solutes chosen for this study are clearly separated, it is not difficult tO imagine the analysis of a complex mixture where a 2% increase in selectivity would determine the success of the separation. Thus far, solute retention and selectivity have been discussed only in terms of the predicted and Observed direction of change, not the magnitude. However, as illustrated for n-szo (Figure 6-2), this increase is not a simple linear function of the average pressure for all solutes. This observed nonlinear increase is consistent with theoretical predictions based on the variation in capacity factor with reduced density. Although the reduced density varies nearly linearly with applied pressure (Figure 5-1), the quadratic dependence of In k on 149 Table 6.1: Effect of pressure on single- and dual-mode measurements of solute capacity factor. CAPACITY FACTOR (k) SINGLE MODEa DUAL LOOEb SOLUTE PAVG=72 bar 235 bar Ak/k 48 bar 158 bar Ak/k n-Cm, 0.501 0.547 +92% 0.520 0.571 +9.8% no,” 0.784 0.874 +11.5% 0.818 0.921 +12.6% no,” 1.208 1.383 +14.5% 1.261 1.441 +14.3% n-C,,,,o 1.829 2.141 +17.1% 1.923 2.245 +16.8% no,” 2.712 3.284 +21.1% 2.827 3.419 +20.9% n-sz 3.997 4.970 +24.3% 4.197 5.205 +24.1% a L: 26.2 cm b L = 23.5 cm 150 Table 6.2: Effect of pressure on single- and dual-mode measurements of solute selectivity. SELECTIVITY (a) SINGLE MODEa DUAL M_C)DEb SOLUTE PAVG=72 bar 236 bar Aor/oc 48 bar 158 bar Act/01 ”OW”, 1.565 1.598 +21% 1.574 1.613 +25% ”CW“.o 1.541 1.583 +27% 1.541 1.564 +15% ”Cum” 1.514 1.548 +22% 1.525 1.558 +22% mam“, 1.483 1 .534 +34% 1.470 1 .523 +36% homo, 1.474 1.513 +27% 1.485 1.523 +26% a L=26.2 cm b L=23.5 cm 151 the reduced density (Equation [6.31) leads to a predicted nonlinear increase in capacity factor with pressure. From Figure 62, this rise appears to be independent of detection mode, indicating that the pressure/density variation between the injector and detector induces only a small change in k. In this case, the average pressure provides a reasonable indication of the pressure experienced by the solute, regardless Of the interval. Based on this hypothesis, however, the absolute magnitude of the capacity factor is expected to be greater for single-mode measurements than for the lower pressure dual-mode determinations, and the Opposite trend Is Observed (Figure 6-2; Table 6.1). These results indicate the possibility of another source of variation in k along the column. Based solely on the increase in solute retention and selectivity with pressure/density, it is expected that these fundamental separation parameters will not be constant along the column length, as is often assumed. In fact, capacity factor is expected to vary nonlinearly from the column inlet to exit, becoming successively less retained with distance. Thus, the capacity factor determined at the column exit is expected to be a measure of the average retention behavior of the solute along the column, and is not equal to the local retention as is Often assumed. This deduction is consistent with previous measurements along the column shown in Chapter 4. As seen in Figure 6-3, however, single-mode measurements of capacity factor on this column show a clear increase with distance under different local pressure conditions. In contrast, dual-mode measurements, which are expected based on the average pressure to exhibit a capacity factor decrease of ~5%, are relatively constant with distance. These results are further evidence that the pressure/density encountered by the solute is not the only factor determining retention with distance in this system. In addition, because the average pressure encountered Figure 6-3: 152 Single- (top) and dual- (bottom) mode measurements of solute capacity factor as a function Of distance along the column (L101 = 43.9 cm) under high and low inlet pressure conditions. Solutes: n- C10:0 (0); "'01220 (El); ”014:9 1A); ”0160 (,0); "'018zo (V); and ’7' 020,0 (+). Experimental condrtrons as grven In the text. €50 8253 E50 8253 0m. ON 0— 0 on ON 0— 0 . _ . _ . 0 . _ . _ t 0 o o O o o o O o . o a o O — O O o a r— ‘ Q 4 d d d d < T . o o o o I IN 0 o o o fN b b > > fl rm. In > b b > v e 3 T. . . . . 14 O . V 3...... . .A w "mm o o v 4 fim m m .0 x F t _ . C i 0 . _ . _ t 0 I: V O o c o o . o o O O o r 3 o o o o o a o o u u m 4 r— < 4 4 4 4 r_. HO Q Q G G o o o o o o o o o 0 TN IN C p D D P D D u D I” In > > p C . v 0 TV 0 o 4 IV 0 .9 0 v . . In La. .mm 000— “memmme #042. .mm 000? ”mm—Dmmwmn‘ #042. 154 by the solute is always higher for single-mode determinations, it is expected that capacity factor measurements at a single detector will be greater than those evaluated in the dual mode. As seen in Table 6.1, experimental measurements show the opposite trend for all solutes. These anomalies in retention are not presently understood and may be due to nonuniformity in the stationary phase along the column caused by long-term usage, and further investigations Of the column inlet region should elucidate this more clearly (Chapter 8). As shown in Figure 6-4, however, solutes continue to behave nearly ideally, exhibiting the theoretically expected logarithmic dependence Of capacity factor on carbon number for both single- and dual-mode determinations. Finally, quantitative prediction of retention is attempted for all solutes based on Equation [6.4]. As stated previously, the theoretical evaluation of capacity factor requires the accurate estimation of a number Of unknown variables. Solubility parameters cited in the literature for octadecylsilica stationary phases (25,26) and methanol (20) are shown in Table 6.3. Mobile- phase solubility parameters are chosen for both bulk alkane estimates (22) as well as recent thermodynamic measurements on actual packing materials (25). Data on the coumarin-labeled fatty acids are lacking in the literature, however, requiring the approximation Of molar volume based on group contributions to the van der Waal's volume (27) and the arbitrary placement of the solute solubility parameters between those of the mobile and stationary phases. Finally, the phase ratio for this column is estimated based on a 12% carbon loading on spherical particles. The resulting numerical prediction Of the coefficients in Equation [6.4] is shown in Table 6.4. Comparison with experimental coefficients is accomplished by regression analysis Of all dual-mode measurements. As shown in Figure 6-5, the retention behavior of all solutes is adequately described by the nonlinear regression of Equation [6.4]. White the theoretically predicted 9 ; 331E513 155 Figure 6-4: Single- (top) and dual- (bottom) mode measurements Of logarithmic dependence Of capacity factor on carbon number under varying inlet pressure conditions. Experimental conditions as grven In the text CAPACHW’ FACTOR 156 Figure 6-4 G) 404 1 I s 20H g 1.0 —_y 0.8 -1 9 0.6 — l 0.4 f r ‘1 I T fl 1 ‘1 I 4.0 — ii 2.0 — e e 1'0 J a v 1500 05' " I 0'8 7:1 o 2400 psi 0.6 A 3350 psi ii El 4300 psi 0.4 r y f t r y 0I 49150 DST IO 12 i4 16 18 20 CARBON NUMBER 157 Table 6.3: Estimates of parameters for the prediction Of solute capacity factor. SOLUTE PARAMETERS (FATTY ACID DERIVATIVES) 81 Vi a (cal 1’2/cm3’2) (cm3/mol) n-Cm, 14.2 207 n-Cm, 13.9 228 ”014:0 13.6 248 ”016:0 13.3 268 ”018:0 13.0 289 n-Cmo 12.7 31 0 MOBILE-PHASE PARAMETERS (METHANOL) 6M = 14.5 call/Z/Om3/2 b pa = 0.271 g/Om3 c p = 0.787 g/Om3 at atmospheric pressure 0 STATIONARY-PHASE PARAMETERS (OCTADECYL SILICA) 83: 7 d and 12.5 call/i’lcm3/2 9 0:20 SYSTEM PARAMETERS T = 301 K R = 1.987 cal/K mOI *1 Bondi, A. J. Phys. Chem. 1964, 68, 441. b Barton, A.F.M. Chem. Rev. 1975, 75,731. . ° Reid, R.G.; Prausnitz, J.M.; Shenvvood, T.K. The Properties of Gases and Liquids; McGraw-Hill: New York, 1977. . d Schoenmakers, P.J.; Billet, H.A.H.; de Galan, L. Chromatographra1982, 15. 205. ° Yamamoto, F.M.; Rokushika, S. J. Chromatogr. 1990, 515, 3. 158 Table 6.4: Comparison of theoretically estimated and experimentally measured constants for Equation [6.4]. c1 c2 03 SOLUTE EST. MEAS. EST. MEAS. EST. MEAS. n-Cm, 8.63 10.6 -49.1 -61.9 49.0a(65.8)b 89.7 ac,” 9.50 10.6 -52.9 -61.7 53.1 (69.9) 89.4 n-C1m 10.3 14.2 -56.3 -82.5 56.3 (73.2) 120 n-C,.,,o 11.2 14.3 -59.5 -82.8 59.2 (76.0) 121 no,” 12.0 14.2 -62.7 -81.7 61.7(78.5) 118 0020.0 12.9 14.2 -65.7 1-81.9 63.8 (80.6) 119 8 Calculated based on 53 = 7 call/«°-/cm3/2 b Calculated based on as = 12.5 call/Zlom3/2 159 Figure 6-5: Measurements of logarithmic dependence of capacity factor on the mobile-phase reduced density for all solutes, together wrth nonlinear regression analysis of Equation [6.4] (—). Expenmental conditions as given in the text. LNk 160 Figure 6-5 0.40 1.70 14:0 4' 20:0 A A '1 ++ A 1.60 —' A + 0.30 -—I A — 1' AA A + AA AA 1.50 — ++ 1' A A A _ y+ 0.20 A y 1 f y 140 *;y y 1 r 2.92 2.96 3.00 2.92 2.96 3.00 REDUCED DENSITY REDUCED DENSITY 161 coefficients shown in Table 6.4 are systematically lower than those determined from experimental measurement, the overall agreement appears to be quite reasonable. The choice of stationary-phase solubility parameter value appears to influence 03 markedly, with the more commonly used bulk alkane value of 7 call/Zlcm3/2 resulting in a greater underestimation based on measured values. In both cases, however, actual predictions of the solute capacity factor are approximately a factor Of 1000 lower than experimental measurements. This disheartening result arises from the inability to estimate the variables in Table 6.4 with sufficient accuracy, coupled with the small range of reduced density values. These difficulties have been noted in the literature and are even more pronounced for reversed-phase separations (27). Nonetheless, experimental measurements are well correlated with qualitative predictions based on Equation [6.4], providing corroborating evidence for this theoretical approach. In like manner, the relationship between selectivity and reduced density can be explored in more detail. Unfortunately, a priori prediction is even more difficult for selectivity, because very accurate knowledge of the differences in solute solubility parameters is required (Equation [6.51). However, experimental measurements can be utilized to predict differences in solubility parameters for adjacent solutes. As illustrated in Figure 6-6, although the measured change in selectivity is not large, all solutes exhibit the predicted linear dependence of In or on the mobile-phase reduced density. Coefficients C4 and c5 determined from linear regression analysis are listed in Table 6.5, together with the correlation coefficient (r). Due to difficulty in measuring such small changes in 01, the error in the coefficients is approximately 10%. Nonetheless, differences in the solubility parameters for these solutes are quite small, as expected for dispersion interactions within a homologous series. In addition, calculated difference in solute solubility parameters (8, - 8,) exhibit a small, but systematic decrease Figure 6-6: 162 Measurements of logarithmic dependence Of selectivity on the mobile-phase reduced density for adjacent solute pairs, together with regression analysis of Equation [6.5] (—). Experimental conditions as given in the text. . 163 FIgure 6-6 O 10:0/12:O D 0.46 —- LNoe 12:0/14:O 1:1 A LNoe 14:0/1 6:0 2. 92 F 3.00 REDUCED DENSITY 2.96 o 16:0/18:0 0.43 -J 6’ v 189/20:0 0.43 — 0.41 e 0.39 —t I T I 2.96 3.00 REDUCED DENSITY 164 Table 6.5: Difference in the solubility parameters of adjacent solutes (8-65 calculated from Equation [6.5]. Slope (C4) and intercept (c5 determined by linear regression analysis of dual-mode measurements. SOLUTE c4 (Si-8]) c5 (23,-6,) r PAIR 1020/1210 0.188 0.052 -0.0997 0.011 0.73 12:0/14:0 0.388 0.098 -0.708 0.071 0.86 14:0/16:0 0.387 0.089 -0.706 0.066 0.92 1620/1820 0.311 0.067 -0.514 0.044 0.75 18:0/20:0 0.338 0.068 -0.602 0.048 0.84 165 in magnitude with carbon number in all except the least retained pair. This result is consistent with that expected from Equation [6.1] for a homologous series. As the solute chain lengthens, the increase in volume arising from additional methylene groups becomes greater than the interaction energy added. In this case, solutes are separated based on change in molar volume more than any small change in interaction energy. As seen here, however, this is sufficient to produce an appreciable dependence of selectivity on the pressure/density. 6.5 Conclusions Although not Often considered a separation variable, the local pressure/density is shown to contribute substantially to solute retention and selectivity. Even though the separation studied here is based solely on dispersion interactions and utilizes a mobile phase which is only slightly compressible, significant increases in both retention and selectivity are measured under modest pressure conditions. Because dispersion interactions are universal in nature, the variation in retention with reduced density has important Implications for all partition-based separations in liquid chromatography. These variations may become especially important for separations utilizing very small particles or high speeds, where changes In pressure and, hence, in retention and selectivity along the column length are expected to be significant. In addition, whether theoretical or experimental in approach, fundamental studies of solute retention must consider the influence of pressure under all separation conditions. Finally, with further study, these alterations may clearly be used to advantage in the design and Optimization of difficult separations. 166 6.6 Literature Cited in Chapter 6 10. 11. 13. 14. 15. 16. 17. 18. 19. 20. 21. Bridgman, P.W. Proc. Acad. Arts and Sci. 1926, 61, 59. Hirschfelder, J.O.; Curtiss, C.F.; Bird, R.B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954. Reid, R.C.; Prausnitz, J.M.; Sherwood, T.K. The Properties of Gases and Liquids; MOGraw-Hill: New York, 1977. Sgeghan, K.; Lucas, K. Viscosity of Dense Fluids; Plenum: New York, Ilgill‘elygstEJi; (715132.55; Tanaka, Y. Properties of Inorganic and Organic , . - , phere Publishing. New York,19 8. Kestin, J.; Wakeham, W.A. Transport Pro erties of Fluids: Thermal Conductivity, Viscosity, and Diffusion Coe cient; Vol. l-1; Hemisphere Publishing: New York, 1988. Hamann, S.D. J. Phys. Chem. 1962, 66, 1359. Macko, T.; Soltes, L.; Berek, D. Chromatographia 1989, 28, 189. Martin, M.; Guiochon, G. Anal. Chem. 1983, 55, 2302. Martire, D.E. J. Liq. Chromatogr. 1987, 10, 1569. Poe, .D.P.; Martire, D.E. J. Chromatogr. 1990, 517, 3. Giddings, J.C. Separ. Sci. 1966, 1,73. f(337iddings, J.C.; Myers, M.N.; McLaren, L.; Keller, R.A. Science 1968, 162, van der Wal, S. Chromatographia 1 986, 22, 81. Bidlingmeyer, B.A.; Hooker, R.P.; Lochmuller, C.H.; Rogers, L.B. Separ. Sci. 1969, 4, 439. Bidlingmeyer, B.A.; Rogers, L.B. Separ. Sci. 1972, 7, 131. Prukop, G.; Rogers, L.B. Separ. Sci. 1978, 13,59. Tanaka, N.; Yoshimura, T.; Araki, M. J. Chromatogr. 1987, 406, 247. Katz, E.; Ogan, K.; Scott, R.P.W. J. Chromatogr. 1983, 260, 277. Barton, A.F.M Chem. Rev. 1975, 75, 731. Hildebrand, J.H.; Scott, R.L. Solubility of Nonelectrolytes; Reinhold: New York, 1950, pp. 429-431. 22. 23. 24. 25. 26. 27. 167 Schoenmakers, P.J.; Billiet, H.A.H.; deGalan, L. Chromatographia 1982, 15. 205. Gluckman, J.C.; Hirose, A.; McGuffin, V.L.; Novotny, M. Chromatographia 1983, 17,303. Yamamoto, F.M.; Rokushika, S. J. Chromatogr. 1987, 408, 21. Yamamoto, F.M.; Rokushika, S. J. Chromatogr. 1990, 515, 3. Bondi, A. J. Phys. Chem. 1964, 68, 441. Schoenmakers, P.J.; Billiet, H.A.H.; Tijssen, R; deGalan, L. J. Chromatogr. 1978, 149, 519. CHAPTER 7 THE INFLUENCE OF SOLVENT COMPOSITION ON SOLUTE RETENTION 7.1 Introduction Studies up to now have been limited to the use of a pure methanol mobile phase. The availability of known physical constants made this choice the most viable for reversed-phase separations. In addition, more ideal experimental conditions were maintained by eliminating the possible selective partitioning of mobile-phase components into the stationary phase. However, the influence of the mobile-phase composition on solute retention is of central importance to the practical application of liquid chromatography. Because solvent composition is the primary variable utilized to control separations, a detailed understanding of the influence on solute retention is essential for the design of universally successful optimization schemes. Due the complexity of the chemical and physical environments present on a liquid chromatographic column, however, most theoretical descriptions of solute retention are empirical or semiempirical in nature (1-13). As discussed in Chapter 1, these include the solvophobic approach proposed by Horvath and coworkers (1,2), the statistical 168 169 thermodynamically model developed by Martire (7,8), and the solubility parameter theory adapted for liquid chromatography by Tijssen and coworkers (3-5). Of the models available and in use, these are the most rigorously correct and the least empirical in nature. Verification of these models requires the accurate measurement of solute retention under carefully controlled experimental conditions. Moreover, because more than one of these theories predicts a variation in retention along the column length for isocratic conditions (Chapter 6), detection after the column exit is inadequate in assessing the validity of these models. The on-column detection scheme, thus, provides a unique opportunity to measure solute retention directly on the chromatographic column. in this preliminary study of solvent composition effects, solute retention is measured at two points on the column under both constant and varying solvent composition conditions. By determining the capacity factor in this manner, solute retention can be characterized as a function of solvent composition and the resulting variation in local pressure arising from differences in viscosity with composition. 7.2 Theoretical Considerations Similar to Chapter 6, the theoretical development based on solubility parameter theory is chosen to predict retention behavior. In the development presented here, however, the influence of solvent composition on solute retention is explored. As discussed previously, the solute capacity factor (k) may be described in terms of the solubility parameters for the solute, mobile phase, and stationary phase (8,, 8M, and 83, respectively) (4): In N =___‘F"iT [(5r - am (a. - 53V] - MB [1.71 170 where V, and [5 represent the solute molar volume and phase ratio, respectively. For mixed solvent systems, the mobile-phase parameter (5M) can be calculated from solubility parameters for the individual components (5), weighted by their respective volume fractions (oi). 8M=2¢jsl [7.1] in this expression, the chemical interactions of the individual components are presumed to be independent and volumes must be additive. Although these conditions are rarely true for the polar solvents commonly encountered in reversed-phase separations, Equation [7.1] is often utilized as a first approximation. For a simple binary system, the solubility parameter of the mixed mobile phase is given by 5M = (DA 5A + ¢B 5B =(1 ' ¢BI 5A + ¢B 59 [72] where ¢A and 5A are the volume fraction and solubility parameter of the weak component (A), and (>3 and 63 represent the strong component (B). Substitution of Equation [7.2] into Equation [1.7] results in an expression for solute retention as a function of the volume fraction of the strong mobile-phase component (its) (4)1 ln ki = 01%") + 02 ¢e 1' 03 [7'3] where q= [ W Ira-er RT < %= [ i )ro-ax-m-arxa-am RT 171 c3: [1%.] [is-sxrz-iar-ssrr-Inrs where c1 represents the interaction of mobile-phase components, oz describes the solute/mobile-phase interaction, and ca predicts the retention of the solute in the weak mobile phase. Because the solubility parameter for the solute is always intermediate in value between that of the mobile and stationary phases (85, < a < 5M), the 02 coefficient must be negative while c, is expected to be slightly positive. Thus, as seen in practice, the solute capacity factor (k,) is predicted to exhibit a marked decrease with the volume fraction of the strong solvent (cps). In addition to the change in chemical environment with composition, mobile-phase physical properties may be altered as well. Variations in solvent composition may yield physical effects on solute retention in addition to the chemical influences described above. Prediction of this physical effect is accomplished by substitution of the relationship between the solvent solubility parameter and reduced density (Equation [63]) into Equation [7.2]. 5M = (1 ' (be) _.____(p°A aA1/2) PRA + ¢a(_—p°B 881/2) PR8 I7-4I mA m8 This expression indicates that if the interactions between individual solvents is independent, the solubility parameter of the mixed mobile phase is expected to increase with the reduced density of components A (pm) and B (pRB). The effect of the density on retention is compounded by the influence of solvent composition on the mobile-phase viscosity and, thus, the reduced density. Unfortunately, the viscosity of mixed solvent systems often cannot be theoretically predicted (14) and must be measured experimentally. The effect of this change in viscosity on the pressure and reduced density can then be calculated based on Darcy's law (Equation 5.1) and the Tait equation of state (Equation 5.7) as described in Chapter 5. Thus, the solubility parameter of the binary mobile phase is expected 172 to vary with the volume fraction of B (cps) due to changes in the pressure/density with composition as well as the change in chemical environment. 7.3 Experimental Methods Analytical Methodology. Saturated fatty acid standards from n-Cm, to n-C15,0 are derivatized with 4-bromomethyl-7-methoxycoumarin as described in Chapter 4. Mixed standards are dissolved in acetone and injected at concentrations of 5 x 10*i M. lsocratic mobile-phase solvents of 90.0%, 92.5%, 95.0%, 97.5% and 100% v/v methanol/water are prepared from stock mixtures of 90.0% methanol/water and pure methanol. Chromatographic System. The chromatographic system is identical to that described in Chapter 4 and illustrated in Figure 2-1. The column utilized for this study is also identical to that specified in Chapter 4. The mobile-phase composition is systematically varied from 90.0 to 100% methanol/water under isocratic conditions. A dual-syringe pump (Applied Biosystems, MPLC) operated in the constant-flow mode is utilized to deliver the mobile phase at slightly greater than the optimum flowrate (F = 0.70 pL/min; 0.080 cm/s). Under isocratic conditions, this flowrate resulted in an inlet pressures of 256 bar for 100%, 272 bar for 97.5%, 340 bar for 95.0%, 360 bar for 92.5%, and 375 bar for 90.0% methanol/water mobile phases. Detection System. The optical detection conditions are identical to those described in Chapter 2. Two detectors are positioned at approximately 30 and 90 cm along the 100 cm length of column. Data acquisition is accomplished under computer control at 1 Hz with a 0.1 s time constant. 173 7.4 Results and Discussion in preliminary investigations examining the influence of solvent composition on solute retention, coumarin-derivatized fatty acids from n-Cm, to n-C152° are evaluated under isocratic conditions. With detectors positioned at approximately 30 and 90 cm along a 100-cm packed-capillary column, capacity factors are measured for mobile-phase compositions ranging from 90.0% to 100% methanol/water. As shown in Figure 7-1, solutes appear to be well behaved under all mobile-phase conditions, exhibiting the expected logarithmic relationship between capacity factor and carbon number at both detector positions and as the local capacity factor between detectors. In addition, experimental measurements also show the decrease in k predicted with increasing methanol composition (Equation 7.3). However, the magnitude of solute capacity factors measured at 30 cm from the column inlet (Figure 7-1; top) appear to be greater than those measured at 90 cm (Figure 7-1; middle), with the dual detector measurements being the lowest (Figure 7-1; bottom). The slope of log k versus carbon number also exhibits a small but systematic decrease with distance along the column. This apparent decrease in k with distance is illustrated more clearly in Table 7.1 where retention in 90.0 and 100% methanol/water solvents are compared. A systematic decrease in solute capacity factor is seen from 30 to 90 cm along the column for both solvent systems. This appears to be a direct result of the difference in the average pressure/density encountered by the solutes, as described in detail in Chapter 6. As seen in Table 7.1, the difference in average pressure between detectors is substantial for 90.0% methanol/water and is appreciably less when a pure methanol mobile phase is employed. This difference in pressure between detectors results in a larger decrease in k with distance for the mobile phase with Figure 7—1: 174 Solute retention with carbon number measured in the single mode at 30 cm (top) and 90 cm (middle) along the_ column, togetherwrth the dual-mode measurement bottom . Mobile-phase composrtion. 90.0% v/v (v), 92.5% (9 ), 95.0% (A), 97.5% (I), and 100 °/o (O) methanol/water CAPACITY FACTOR (k) CAPACITY FACTOR (k) CAPACITY FACTOR (k) 175 Figure 7-1 10.6.)-4 6.0-: a N O 1 0100 ‘l—ILI‘ 0.07‘ .0 is v O U'I 10.0: A A > 1.0: 0.8- 0.6- A 6.0-: 4.0-i l 2.0-l w v v O U‘ 9 997‘ 4> @030 A __LL.L P 5‘ 9’ o o o \\ ‘ O CARBON 13 NUMBER _. Ui 176 Table 7.1: Single-mode measurements of capacity factor for 90.0% methanol/water and pure methanol mobile phases with distance. CAPACITY FACTOR (k) 90%,METHANOL/WATER METHANOL . L =30 cm 90 cm 30 cm 90cm SOLUTE PAW-5:313 bar 201 bar Ak/k 210 bar 137 bar Ak/k n-Cw.0 2.60 2.31 -11.2% 0.520 0.507 -2.5% n—C":0 3.64 3.22 -1 1 .5% 0.659 0.642 -2.6% n-Cm0 5.06 4.49 -11.3% 0.832 0.809 -2.8% n-Cmo 7.05 6.24 -11.5% 1.05 1.02 -3.1% n-CM;0 9.80 8.66 -11.6% 1.31 1.27 -3.0% no,” 13.6 12.1 -1 1.0% 1.65 1.58 -4.2% ,______f 177 higher water content. In addition to the pressure difference, however, the absolute magnitude of the pressure/density is greater for mixtures containing water, leading to a larger change in k for an equivalent pressure difference. Thus, under identical flowrate conditions, the capacity factor gradient along the column is expected to be greater for mobile-phase compositions containing water. Measurements of capacity factor with mobile-phase composition are shown in Figure 7-2. Retention for all solutes follows the general trend in In k versus volume fraction methanol (¢METHANOL) predicted using Equation 7.3, with the quadratic regression line shown for each solute. The coefficients determined as a function of position are shown in Table 7.2. Although the regression analysis appears successful based on Figure 72 standard errors in the measured coefficients range from approximately 400% for c1 to 100% for ca. In addition, negative values are determined for 01 (Table 7.2), which are not possible based on strict solubility parameter theory (Equation 7.3). This latter result indicates that the interaction of methanol with water in the mixed mobile phase is sufficient to yield erroneous predictions of retention behavior. Unfortunately, the combination of these two factors does not allow meaningful comparison of theoretical predictions to experimental measurements. Thus, although qualitative evaluation of both theory and experiment clearly indicate that changes in solvent composition produce an additional dependence of capacity factor on the variation in local pressure/density, quantititative comparison is not possible. Figure 7-2: 178 Experimental measurement of the logarithm of the capacity factor (k) versus the volume fraction of methanol ((METHANOL) evaluated at 30 cm (top) and 90 cm (middle) in the single mode, and as the dual mode (bottom). Average pressure determined from the experimental inlet pressure is listed for each mobile-phase composition at the top of each plot. Solutes: n-Cm, (O I. n'C11:0 (DI: "'C1zzo (A). ”013.0 (0), ”C1420 (VI: and "'Crsn (+)- 1 79 Figure 7-2 210 bar f x I Z 1 .01 .J r 0.0-l l . I ’ — 1 0 1’ T— T 201 194 183 I46 137 bar r I I IOI 2.0 MI ual .1 lie ' Z 1 0 159 .1 -‘ (IIII . - ' r 0.0 . _ r l ' u D — 1 .C I i l i 150 144 136 109 102 bar 2. Z I .1 —1.0 0.900 0.025 0.675 1 % .000 Table 7.2: [gosnlstants determined by nonlinear regression analysis of Equation c1 02 C3 SOLUTE L(cm) MEAS. MEAS. MEAS. I'l-Cum 30 -3.0 :1: 13.9 -10.4 :I: 26.4 12.7 i 12.6 90 2.4:1: 12.3 -19.9i23.4 16.8:1: 11.1 dual 5.5 :I: 11.5 -25.3 :1: 21.9 19.2 21:10.4 mom, 30 -3.5 1: 14.8 -1o.4 :i: 28.1 13.5 1: 13.4 90 3.1 i12.7 -22.3 :I: 24.1 18.7 :I: 11.4 dual 7.2 i 11.6 -29.7 :1: 22.1 22.0 :I: 10.5 no,” 30 -4.5 :t 16.0 -9.7 i 30.4 13.9 i 14.4 90 2.7 i135 -22.6 i 25.6 19.6 i 12.1 dual 7.1 i121 -30.4 :I: 23.1 23.1 i' 10.9 n-Cm, 30 -5.3 :t 17.3 -9.2 :l: 32.9 14.5 i- 15.6 90 1.9 :I: 15.0 -22.0 :I: 28.4 20.1 21:13.5 dual 6.1 i 13.8 -29.5 :1: 26.2 23.4 :I: 12.4 no,“ 30 -5.9 :t 18.3 -9.2 1: 34.8 15.4 i 16.5 90 -0.69 $17.6 -18.2 i" 33.4 19.1 i 15.8 dual 2.2 i 17.5 -23.1 :1: 33.3 21.1 i' 15.8 n-C15;o 30 -7.2 i 19.5 -7.9 i 37.1 15.6 i 17.6 90 -2.4 i 20.7 -16.1 i 39.4 18.9 i 18.7 dual 0.40 :1: 22.3 -20.8 i 42.5 20.8 i 20.2 181 7.5 Conclusions Experimental measurements of solute retention as a function of solvent composition are in qualitative agreement with theoretical predictions. In addition to the chemical effect of altering the mobile-phase composition, variations in the local pressure arising from differences in viscosity result in a physical influence on solute retention as well. This local pressure effect yields a capacity factor gradient along the column that is dependent on the composition of the mobile phase, even under isocratic conditions. Thus, the behavior of solute capacity factor under gradient conditions is expected to be even more complex, withTthe composition and local pressure varying simultaneously on the column. These preliminary results indicate that the variation in local solute retention along the chromatographic column may be more complicated than previously considered, and further investigations are clearly warranted. 182 7.6 Literature Cited in Chapter 7 SOPNP’F" 10. 12. 13. 14. Horvath, Cs.; Melander, W.; Molnar, I. J. Chromatogr.1976, 125, 129. Horvath, Cs.; Melander, W. J. Chromatogr. Sci. 1977, 15, 393. Tijssen, R.; Billiet, H.A.H.; Schoenmakers, P.J. J. Chromatogr. 1976, 122, 185. Schoenmakers, P.J.; Billiet, H.A.H.; de Galan, L. Chromatographia 1982, 15, 205. Dill, K.A. J. Phys. Chem. 1987, 91, 1980. Locke, D.C. J. Chromatogr. Sci. 1977, 15,393. Martire, D.E. J. Liq. Chromatogr. 1987, 10, 1569. Martire, D.E.; Boehm, HE. J. Phys. Chem. 1987, 91, 2433. Jandera, P.; Churacek, J. Gradient Elution in _ Column Li uid Cgrfingtgsg’ra/gllryé 1Theory and Practice; J. Chromatogr. LIb.; Elsevrer: ew Katz, E.D.; Ogan, K.; Scott, R.P.W. J. Chromatogr. 1986, 352,67. Kowalska, T. Chromatographia 1989, 27, 628. KowalSka, T. Chromatographia 1989, 28, 354. Cheong, W.J.; Carr, P.W. J. Chromatogr. 1990, 499, 373. Janz, G.J.; Tomkins, R.P.T. Nonaqueous Electrolytes Handbook, Academic Press: New York, Vol. 1, 1972, pp. 83-118. CHAPTER 8 THE INFLUENCE OF TRANSITIONS AT THE COLUMN INLET AND EXIT ON RETENTION AND DISPERSION 8.1 Introduction Thus far, retention and dispersion have been discussed exclusively in terms of processes occuring on the chromatographic column. In practice, however, the solute is introduced onto the column from a nonretentive injection valve and is most commonly eluted from the column before detection. These aanpt transitions may affect the measured chromatographic performance, leading to misinterpretation of the fundamental mechanisms of separation occurring on the column itself. In simplified chromatographic theory, separations are generally modeled as an equilibrium process. In reality, however, the transfer of solute molecules between the mobile and stationary phases is rarely instantaneous. Due to the finite rate of exchange, solute molecules in the mobile phase will travel some distance along the column before transfer occurs, while molecules in the stationary phase are fixed. Consequently, the solute zone profile in the mobile 183 184 phase is slightly in advance of that in the stationary phase. This discrepancy results in broadening of the solute zone, due only to nonequilibrium processes (1). Although this phenomenon occurs throughout the column, it is expected to reach an extreme at the entrance and exit of the chromatographic column, due to the abrupt changes in solute retention in these transition regions. In the inlet , region, the effective rate of solute transfer from mobile to stationary phase, which is zero prior to the column, becomes a finite value upon entering the region containing the stationary phase. Likewise, upon elution from the column, the solute zone passes from a retentive region (on-column) into an open tube where no stationary phase is present (off-column). In these transition regions. substantial changes in solute velocity occur across the solute zone profile. As the solute enters the column, the rate of movement of the front portion decreases due to solute interaction with the stationary phase, while the rear portion continues at the faster mobile-phase velocity. Thus, the solute zone passes from a region where it Is nonretained (off-column) to a retentive region (on-column) at a rate dictated not only by the mobile-phase linear velocity but also by the solute capacity factor. This injection nonequilibrium is expected to result in a decrease in the zone length variance and a concomitant increase in solute concentration. In like manner, upon elution from the column, the front portion of the solute zone increases to the mobile-phase velocity, while the rear of the zone remains at the slower mean zone velocity. This elution nonequilibrium is predicted to result in an increase in length variance and a concomitant decrease in maximum concentration of the solute zone detected off- column. The extent to which these discontinuities in the physical and chemical environment affect chromatographic performance has been much debated for 185 both gas and liquid chromatography (2-11). Although methods of theoretical treatment vary widely, it is generally agreed that the influence of these retention discontinuities is dependent on the initial peak profile and the equilibration of solutes between the mobile and stationary phases. Theoretical predictions of solute behavior in the inlet region are further complicated because the solute zone itself may affect the local environment, thus altering the local equilibrium conditions (7). This condition may arise, for example, if the increase in the solute concentration upon entering the column is sufficient to exceed the linear range of the equilibrium isotherm (12). In this case, the shape of the zone profile could be substantially altered, and the measured column efficiency adversely affected. In this chapter, on-column detection is utilized to measure retention and dispersion in the inlet and exit regions of the column. Although the theoretical treatment of these two regions is directly analogous, experimental measurements utilizing the on-column detection approach are somewhat different. At the column inlet, five detectors are positioned along the packed bed with one detector placed immediately prior to the head of the column. With this experimental design, the solute retention and dispersion may be measured» as a function of distance in the inlet region and the initial injection profile may be monitored as well (13). For the inlet region studies, a pure methanol mobile phase Is chosen to eliminate the selective partitioning of individual solvents into the stationary phase which may occur with mixed solvent systems. This design also allows the influence of the injection solvent composition to be systematically evaluated. Mixtures of methanol/acetone and methanol/water are Chosen as injection solvents for this study because of the wide range of solute retention possible. In the elution study, two detectors are utilized with one positioned before the frit and the other immediately after the column exit. By isolating this exit region, the broadening of solute zones upon elution from the column may be 186 measured directly (14). In addition, the influence of the mobile-phase linear velocity, the mobile-phase composition, and a spatial temperature gradient at the column exit are explored as well. For ease in discussion, studies of the inlet and exit regions of the column are addressed separately in this chapter. 8.2 Theoretical Considerations: Inlet Region Some insight into the possible magnitude of these abrupt transitions can be gained by applying the nonequilibrium approach of Giddings (1 ). Although the theoretical development is discussed here for injection onto the column, it will be shown later that this approach is equally applicable to elution from the column. For a solute zone migrating from an open tube to a point an infinitesimal distance on-column, the time variance (072) off- and on-column may be assumed to be equaL (GTZIOFF = (GTZION [8111 Conversion to the length domain (01.2) is accomplished utilizing Equation [8.2], of = 6T2 U2 = 072 [u/(1 + k)]2 [8.2] where k is the solute capacity factor, and U and u are the mean zone velocity and mobile-phase velocity, respectively. The off- and on-column length variance can then be expressed by substituting this relationship into Equation [8.1]. (0L210N= (GLZIOFF (UON/UOFF)2 _—1..__ I8-3I (1 + k)2 187 Because the volumetric flowrate (F) is constant off- and on-column, the mobile- phase velocities are related by F = Ti ROFFZ UOFF = 7‘ RON2 UON ET [841 where R is the tube radius, and ET is the total porosity of the packed bed (on- column), which is unity for an open tube (off-column). Thus, the corresponding length variance on-column is given by the following expression: 1 1 8.5 8?“ + kINJI2 [ ] (GLZION = (GLZIOFF (ROFF4/RON4) According to this relationship, two distinct factors determine the Change in length variance upon solute injection. Length dispersion arising from the increase or decrease in solute zone volume is reflected in the radial and porosity terms. More importantly, a substantial decrease in length variance on—column is predicted as a function of the solute capacity factor in the injection solvent (kw) due solely to this transition in zone velocity. 8.3 Experimental Methods: Inlet Region Analytical Methodology. Saturated fatty acid standards are derivatized with 4- bromomethyl-7-methoxycoumarin as described in Chapter 4. Standards are injected individually at a concentration of 5 x 104i M in a variety of injection solvents. The diffusion coefficient, estimated based on the Wilke-Chang equation (15), is 3 x 10-5 cmZ/s for the n-Cm, fatty acid derivative in methanol. 188 Chromatographic System. The chromatographic system is described in detail in Chapter 2 and illustrated in Figure 2-1. As in Chapter 6, this study utilizes an open-tubular capillary (0.0050 cm I.d., 25.7 cm length) extending from the injector to 0.1 cm before the packing material to transfer the injected plug to the head of the column. Connection in this manner provides the minimum band broadening, while simultaneously allowing a detector to be placed on the open tube immediately prior to the packed bed. The microcolumn utilized for the inlet studies is fabricated using a fused- silica capillary (0.020 cm i.d., 43.9 cm length), from which the polyimide coating has been carefully removed to facilitate on-column detection. The resulting column, prepared as described in Chapter 2, has a plate height of 9.5 pm, a total porosity of 0.43, and a flow resistance parameter of 550 under standard test conditions (16,17). In all inlet region measurements, the methanol mobile phase is operated at slightly greater than the optimum velocity (F = 0.57 uL/min; u = 0.070 cm/s) resulting in an inlet pressure of approximately 1600 psi. The split ratio is 1:100 for this study resulting in an injection volume (V,NJ) of 9.8 nL. Under the experimental conditions utilized in this study, most of the extra- column variance is expected to arise from the injection process, with 14% from the injection volume and 62% from the connecting tube. Only 24% of the total extra-column contribution is predicted from detector sources, with 23% from the viewed volume and less than 1% from the time constant (vide infra). Detection System. The optical detection conditions are identical to those described in Chapter 2. As shown in Figure 6-1, however, six matched detector blocks are positioned along the column with optical fibers from alternate blocks connected to each detection system. If solutes are injected individually, this arrangement allows multiple detection points with only two 189 monochromator/photomultiplier detection systems. The first detector block is positioned on the open tube 0.4 cm before the packed bed, while the remaining five detectors are placed on the packed bed at 4.9, 10.4, 15.5, 20.9, and 26.2 cm from the column head. The maximum viewed volumes in this case are 1.8 nL off- column and 12 nL on-column. Data acquisition is accomplished under computer control at 5 Hz with a 0.06 s time constant. 8.4 Results and Discussion: Inlet Region Retention of Solute Zones. In most theoretical approaches to chromatographic separations, equilibration of the solute zone with the stationary phase is assumed to occur instantaneously. Under chromatographic conditions when this is true, and if solute-solvent and solute-stationary phase interactions are not affected by the local pressure, retention in the inlet region of a liquid chromatographic column is predicted to be constant with distance along the column (18). With the present experimental design, it is possible to examine this theoretical prediction by measuring the retention of solutes as they traverse the chromatographic column. Methanol Injection Solvent. Experimental measurements of capacity factor (k) with distance along the column using methanol as the injection solvent are shown in Figure 8-1 for both single- and dual-mode determinations. The solute capacity factor increases logarithmically with solute chain length, ranging from 0.54 for n—Cm, to 4.95 for n-Czo:0 (Table 8.1). Contrary to theoretical predictions, however, capacity factor values measured in the single mode (Figure 8-1, top) exhibit a small but systematic increase with distance travelled. This increase in k of approximately 3% in the region from 4.9 to 26.2 cm along the column appears to be independent of solute and is statistically significant at the Figure 8-1 : 190 Capacity factor versus distance along the column measured In single- (top) and dual- (bottom) mode for derivatized fatty acrd standards n-Cm, (O). n-C1220 (CI), n-Cmo (A), n-Qisn (0)3.”01820 (v), and n-Cm, (+). Chromatographic and detection condrtrons as described in Experimental Methods. CAPACITY FACTOR 191 Figure 8-1 + + V V 0 0 A A D D O O 4.1 31 2__I IIO DISTANCE (cm) 192 Table 8.1: Capacity factors (ijJ) for derivatized fatty acid standards as a function of solvent composition. SOLUTES SOLVENT "'C1o:o ”’C1zzo ”614:0 "”0160 ”'Crezo ”‘Czozo 90% methanol/acetone 0.44 0.68 1.06 1.62 2.48 3.66 95% methanol/acetone 0.47 0.74 1.16 1.80 2.77 4.19 methanol 0.54 0.85 1.34 2.09 3.23 4.95 95% methanol/water 1.20 2.20 3.80 6.80 12.5 21.0 90% methanol/water 2.60 5.10 9.90 19.0 37.0 72.0 193 95% confidence level for an average precision in replicate measurements of i0.5% relative standard deviation (rsd). Errors in the determination of to caused by slight retention of the void marker could lead to this positive trend. If so, the local capacity factor measured in the dual mode would also be expected to exhibit the same trend. However, as seen in Figure 8-1 (bottom), the local capacity factor remains constant with distance for all solutes within the average precision of i0.6% rsd (with n-Cm, and n-Cm, exhibiting anomalously poor precision at :4.4% and :1.5%, respectively). Based on these measurements, it is possible that initial equilibration upon injection is not instantaneous, but does occur well before the first detector (L = 4.9 cm). Because of this decreased retention at the column inlet, all solutes exhibit single-mode capacity factor values at the last detector (L = 26.2 cm) that are systematically 0.7% less than local capacity factor values. Thus, a small systematic error may result if the capacity factor measured at the column exit is assumed to represent the local capacity factor on the column. Variation in Injection Solvent. In this investigation, the composition of the injection solvent is systematically varied and the retention behavior is, again, evaluated as a function of distance along the column. The mixtures of methanol/acetone and methanol/water (90% and 95% v/v), chosen as the injection solvents for this study, yield a broad range of capacity factors (km) for the derivatized fatty acids (Table 8.1). Even though the retention behavior in the injection solvents varies markedly, no discernable variation is seen in the resulting capacity factors measured on the column. As shown in Figure 8-2, the retention measured at 26.2 cm along the column is unaffected by the composition of the injection solvent, and is in agreement with the theoretically predicted relationship between capacity factor and carbon number. Moreover, single detector measurements in all injection solvents exhibit the identical increase in Figure 8-2: 194 Effect of the injection solvent composition on the single-mode measurements of capacity factor at L = 26.2 cm. Injection solvent: 90% v/v methanol/acetone (v), 95% vlv methanol/acetone (O ). methanol ( A ), 95% v/v methanol/water ( [j ), 90% vlv methanol/water (Q). 1 95 Figure 8-2 2O 16 14 CARBON NUMBER rm lil- 6 O 5.0— I O. 4 0— .0 2 0— El OIOVJ All OVcIVO 10 0.5 196 capacity factor with distance seen for injections in methanol (Figure 81). Dual- detector measurements are also similar to injections in the methanol mobile phase, and are constant with distance along the column regardless of the injection solvent composition. Thus, the injection solvent has no significant effect, either temporary or persistent, on the retention behavior of these model solutes under ideal experimental conditions. Dispersion of Solute Zones. Systematic evaluation of the dispersion or broadening of solute zones is also essential to understanding the factors affecting chromatographic performance. Although the plate height is generally assumed to be constant along the column length, extra-column and nonequilibrium effects occurring upon injection may lead to unexpected behavior in the column inlet region. To evaluate the effect of the injection process on zone dispersion, the variance and plate height of solute zones with distance along the column are determined from data in the same set described above. Because the dispersion of solute zones in the inlet region is a complex phenomenon, it is instructive first to estimate the influence of extra-column dispersion, then to predict the effect of the transition onto the chromatographic column. Finally, single- and dual-detector measurements of the plate height with distance along the column may be evaluated. As in the study of retention, initial investigations focus on the methanol mobile phase as the injection solvent, while later studies explore the effect of the injection solvent composition. Predicted Extra-Column Effects. Plate height measurements performed at a single detector include dispersion contributidns from extra-column as well as column sources. In experimental determinations, these sources of dispersion arising outside the column may have a substantial influence on the accurate measurement of the column plate height (19). Unfortunately, plate height 197 measurements near the column inlet, where the solute zone has only travelled short distances on the column, may be dominated by extra-column sources of dispersion. Forthis reason, it is informative to predict the extra-column influence expected under the experimental conditions of this study. This detrimental influence may be estimated from the fractional increase in the column variance caused by known extra-column sources of variance (20). Under ideal conditions, the fractional increase in the variance (62) is described by Equations [1 .34] and [1.35]. The resulting fraction 02, calculated for the experimental conditions in this study, is shown in Figure 8-3A as a function of distance, for an assumed column plate height (HCOL) of 9.5 um. As expected, the extra-column influence is greatest at small distances and decreases as the solute zone is further broadened by the column. A substantial decrease in 62 is also seen as the solute capacity factor increases, effectively increasing the volumetric variance contributed by the column. Thus, for ideal conditions, extra—column dispersion is predicted to affect not only the magnitude of the measured plate height, but the dependence of H on distance and capacity factor as well. These estimates presuming best-case conditions (Equations [1.37] to [139]) are often the only indication of the expected magnitude of extra-column effects. In the present experimental design, a more realistic estimate of 62 may be determined from the variance of the injected profile measured immediately prior to the column (L = -0.4 cm). This in situ measurement allows a more accurate measure of the largest sources of extra-column variance, those contributed upon injection and flow through connecting tubing. With the methanol mobile phase as the injection solvent, this initial profile is approximately symmetric with an average time variance (0T2) of 4.30 i 0.32 32 (n = 12). As shown in Figure 8-3B, the 02 value calculated from the measured injection/connection variance indicates that the extra-column variance is Figure 8-3: Fractional increase (02) in the column variance caused by extra- column sources under (A) ideal conditions and for (B) methanol injection solvent. Chromatographic and detection conditions as described in Experimental Methods and solutes as shown in Figure 8-1. 199 Figure 8-3 2- 1.. O D A O i 8 8 6 a 62 . 1&7 I r 2— O C] 1_ O A I] O O A CI O O V 0 A D C] o + ,2 j ,1 .9 0 10 20 DISTANCE (cm) 30 200 substantially greater than first estimated based on ideal injection and hydrodynamic conditions (Figure 8-3). In fact, the extra-column contributions maybe as much as twice the column variance measured at the first detector. Variation in the injection solvent is ideally expected to have no effect on the fractional increase in the column variance (62). However, measurements immediately prior to the column indicate a systematic change in the variance contributed by injection/connection sources with injection solvent. When 90% and 95% v/v methanol/acetone injection solvents are utilized, the measured time variances are statistically equivalent (4.68 i 0.16 32 (n = 25)) and slightly greater than measured for methanol injections. In contrast, the measured time variances for 90% and 95% v/v methanol/water (2.72 i 0.22 52 (n = 9) and 3.58 i 0.37 32 (n = 12), respectively) are substantially less than for injections in methanol. The fractional increase in the column variance (62) calculated based on these unexpected results is shown in Figure 8-4. A substantial decrease in 62 is seen for the injection solvents containing water, solely due to this systematic decrease in the injection/connection extra-column contributions to dispersion. The origin of this decrease in injection/connection variance remains unclear, but may be due to changes in the physical properties of the solvent (viscosity, surface tension, diffusion coefficient, etc.) that influence hydrodynamic flow. Thus, not only is the magnitude of 62 substantially greater than expected based on ideal conditions, but the composition of the injection solvent also has a direct influence on the extra-column variance. Since neither of these trends is predicted theoretically, evaluation of extra-column dispersion based on Figure 8-3A would have greatly underestimated these detrimental effects. Off- to On-Co/umn Transition. The transition of the solute zone onto the chromatographic column is often assumed to have little effect on the measured plate height. However, in traveling from a nonretentive open tube to a retentive W Figure 8-4: 201 Effect of the injection solvent compositions on the fractional increase (62) in the column variance caused by extra-column sources. Injection solvents: (A) 90% v/v methanol/acetone, (B) 95% v/v methanol/acetone, (C) 95% v/v methanol/water, (D) 90% v/v methanol/water. Chromatographic and detection conditions as ge1scribed in Experimental Methods and solutes as shown in Figure 202 Figure 8-4 30 ODAQ. O 0502. out DISTANCE (cm) 203 packed bed, the solute zone undergoes an abrupt change in capacity factor. As described by Equation [8.3], the length variance of the zone on the column is a function of the mobile-phase linear velocity and the solute capacity factor. In this transition, the capacity factor in the injection solvent (kmj) is identical to that in the mobile phase (kMp) when samples are dissolved in the methanol mobile phase. This expression is further simplified in Equation [8.5] in terms of the column radius and porosity. Thus, based on only a few experimental parameters, the change in the zone length variance caused by this abrupt transition may be predicted. Experimental measurement of this phenomenon is accomplished by calculating the ratio of the on-column length variance, extrapolated to zero distance, to that measured off column ((oL2)0N/(oL2)OFF). As shown In Figure 8-5 for the methanol injection solvent, there is excellent agreement between the measured length variance ratios and those predicted using Equation [8.5]. This decrease in the length variance of the solute zone at the column inlet effectively decreases the resulting volume injected onto the column as a function of the solute capacity factor, thus reducing the detrimental effects of extra-column dispersion. Further studies of the decrease in extra-column variance are accomplished by altering the composition of the injection solvent. As seen in Equation [8.5] and later in Equation [8.6], the extra-column variance is expected to decrease markedly with the solute capacity factor in the injection solvent (kw). For highly retained solutes (large kMp), the influence of the injection solvent is expected to be minor. In contrast, solutes that are only slightly retained (small kMp) are more affected by extra-column variance and, thus, are predicted to exhibit a substantial decrease in these detrimental effects. Experimental measurement of the length variance ratios for the least retained solute, nszo, 204 Figure 8-5: Measured ratio of the length variance on-column, (0.3),»). to that immediately prior to the packed bed, (6L2)OFF for methanol Injection solvent: Theoretical prediction (—), Experimental measurement (0). Chromatographic and detection condrtrons as descnbed In Experimental Methods. iii nerl 205 Figure 8-5 I T T I I O 00 L0 <1- N O (\l *- O O O O O O O. C. o O. O O O. O O O O O O | 530(Z1D)/NO(Z1D) 0.5 0.4 0.3 1/(1 +102 0.1 0.2 0.0 206 as a function of injection solvent are shown in Table 8.2. As predicted, when the injection solvent is stronger than the methanol mobile phase (ka < km), the measured length variance ratio and plate height are both greater than for the pure methanol injection solvent. Under these conditions, the solute zone is expanded upon entering the column and the extra-column variance caused by the injection is actually more detrimental. Alternately, when the n-Cm.o fatty acid is injected in solvents that are weaker than methanol (ijJ > km), a substantial decrease in the length variance ratio and measured plate height is clearly seen. Contrary to common misconceptions (21), this decrease in the extra-column variance does not require a large change in injection solvent composition to realize a notable improvement in the measured plate height, even at a distance of 26.2 cm along the column. Plate Height Measurements. In most theoretical derivations of dispersion processes in liquid Chromatography, plate height is predicted to be independent of distance along the column (22-24). As noted above, however, single-mode measurements of plate height are strongly influenced by extra—column sources of variance and, therefore, are expected to exhibit some dependence on distance. In contrast, dual-mode measurements are expected to be in agreement with theoretical predictions, because extra-column contributions to the variance have been effectively eliminated. In addition to the effect of the injection process itself, the influence of the injection solvent composition on both these measurements can be systematically evaluated. Prediction of the extra-column influence on the measured plate height (HMEAS) may be accomplished utilizing Equation [1.36]. By assuming the variance contribution from injection volume of a Gaussian band immediately prior to the column ([6nROFF2(oL)OFF]2/36) instead of the more ideal plug injection (VINJ2/12), the magnitude of the extra-column influence can be more accurately 207 Table 8.2: Effect of injection solvent composition on the measured length variance ratio and plate height forthe n-Cm, derivative. INJECTION SOLVENT k,NJ (OLZION PLATE HEIGHT (um) (6L2)OFF single mode (L=26.2 cm) 90% methanol/acetone 0.44 0.010 14.6 95% methanol/acetone 0.47 0.0099 13.9 methanol 0.54 0.0086 13.5 95% methanol/water 1.20 0.0045 11.9 90% methanol/water 2.60 0.0022 9.0 208 estimated. If this injection volume resulting from injection/connection dispersion processes is the primary source of extra—column variance, combining Equation [1.36] with Equations [1.35] and [1.37] results in the following expression for the plate height measured at a single detector (HMEAS). (57iF1OFI=2(m<4fimCI - 000000 0010000 (\INP’DSI'LOLO ox<1<>r>EI 'l . I I'I'I'9*1V>3 o QOOLO St: (\I N 1—00 0 o 6.0 (>1) aorow ALIOVdvo 14 16 CARBON NUMBER 12 1O 250 Figure 8-19: Effect of temperature on the retention behavior of fattyacid derivatives: logarithm of capacity factor versus I/T. Condrtrons given in Experimental Methods. 8003 3 9m 0%. min NW Tm. 0AM mN _ _ _ _ _ N.Q O 4 o r O l o O x I¢.O O - o x - V x < V . IG We I 0.. mo UV 1.. - .. . - O ,. . ,. . w... 0 g 4 0 . H T < IO — 11 q o 8 V o D O + i 0 D O I o > + 0.8 + rod 8 D + Gum: D \/ I p + Cum: 0 1 {VI\ + < I X + O _ _ _ 252 retained prior to the column exit. It is hoped that the detrimental effect of elution nonequilibrium on zone variance and concentration will be minimized by effectively decreasing the abruptness of this transition In retention. This hypothesis is experimentally evaluated by heating the column exit region between the two detectors (Figure 8-13). The temperature behavior of this apparatus, shown in Figure 8-20, Is constant In the heated region and decreases to room temperature less than 1 cm from the ceramic temperature controller. With this system, the length variance and maximum concentration of each solute zone may be determined sequentially on- and off-column, as a function of temperature In this heated zone. The influence of temperature on the ratio of the length variance measured off-column to that measured on-column (GLZOFFJOLZON) versus (1+k)2 Is shown in Figure 8—21. Measured length variance ratios show excellent agreement with theoretical predictions described by Equation [8.7]. Unfortunately, this relationship appears to be Independent of temperature in the exit region of the column. This result is consistent with the prediction by Giddings (36) that the total change In the zone velocity determines the change in length variance, not the spatial location of such a change. The concomitant decrease in the maximum concentration predicted based on a Gaussian zone profile, described by Equation [8.10], is shown in Figure 8-22. The concentration ratio has been normalized to the least retained solute due to difficulty in matching the excitation intensity forthe two detectors. The measured ratios show good agreement with those predicted by Equation [8.10], independent of the temperature In the heated zone. Based on these measurements, a spatial temperature gradient is not successful in minimizing the detrimental effects of elution at the column exit. Apparently, the temperature gradient only acts to displace the retention discontinuity, and whenever the change In solute retention is spatially specific, Figure 8-20: Temperature variation with distance in the column exit region. 254 Figure 8-20 60 | O O 0 L0 (00 BantvaadwaL DISTANCE (cm) 255 Figure 8-21: Effect of temperature on the ratio of the length variance measured on- and off-column versus (1+k)2. Theoretical predrctron (—-) based on Equation [8.9]; T = 40 °C (0), 50 °C (:1), 60 °C (A). 90 °C (1)- . 256 Figure 8-21 1600 O I"? 00 <1 _0 (\I (\I A x ” + q \— V _o I I I I l I I I I I l I O O O O O C) C) O (\I (I)