Advances In Chromatography: Volume 43 (Advances in Chromatography)

DK1147-title 9/13/04

11:09 AM

VOLUME 43

Advances in CHROMATOGRAPHY EDITORS:

PHYLLIS R. BROWN University of Rhode Island Kingston, Rhode Island, U.S.A.

ELI GRUSHK A Hebrew University of Jerusalem Jerusalem, Israel

SUSAN LUNTE University of Kansas Lawrence, Kansas, U.S.A.

Marcel Dekker

New York

Although great care has been taken to provide accurate and current information, neither the author(s) nor the publisher, nor anyone else associated with this publication, shall be liable for any loss, damage, or liability directly or indirectly caused or alleged to be caused by this book. The material contained herein is not intended to provide specific advice or recommendations for any specific situation. Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. ISBN: 0-8247-5341-0 This book is printed on acid-free paper. Headquarters Marcel Dekker, 270 Madison Avenue, New York, NY 10016, U.S.A. tel: 212-696-9000; fax: 212-685-4540 Distribution and Customer Service Marcel Dekker, Cimarron Road, Monticello, New York 12701, U.S.A. tel: 800-228-1160; fax: 845-796-1772 World Wide Web http://www.dekker.com Copyright n 2005 by Marcel Dekker. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10

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PRINTED IN THE UNITED STATES OF AMERICA

Contents

Contributors Contents of Other Volumes 1. Gradient Elution in Liquid Column Chromatography—Prediction of Retention and Optimization of Separation Pavel Jandera

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1

I. Introduction II. Theory of Retention in Analytical Gradient-Elution Chromatography III. Reversed-Phase Chromatography with Binary Gradients IV. Normal-Phase Chromatography with Binary Gradients V. Ion-Exchange Gradient Elution Chromatography VI. Effects of the Instrumentation and of the Nonideal Retention Behavior on the Retention in Gradient Elution iii

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Contents

VII. Gradient Elution Method Development VIII. Chromatography with Ternary Gradients IX. Peculiarities of Gradient Elution Separation of High-Molecular Compounds X. Conclusion Acknowledgments Symbols References Appendix A Appendix B 2. Supercritical Fluids for Off-Line Sample Preparation in Food Analysis Prior to Chromatography Jerry W. King

109

I. II. III. IV.

Supercritical Fluids for Sample Preparation Supercritical Fluid Extraction (SFE) Integration of Cleanup Step with SFE Coupling Reaction Chemistry (Derivatization) with SFE V. Applications of Critical Fluids for Sample Preparation VI. Status of the Technique—Conclusions References Appendix A

3. Correspondence Between Chromatography, Single-Molecule Dynamics, and Equilibrium: A Stochastic Approach Francesco Dondi, Alberto Cavazzini, and Michel Martin I. II. III. IV.

Summary Introduction The Stochastic Approach of Chromatography Peak Shape Features and Experimental Errors in the Determination of the Retention Factor V. Equilibrium Conditions in Chromatography VI. Discussion

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VII. Conclusion Acknowledgments Glossary References Appendix A Appendix B Appendix C Appendix D Appendix E 4. Solid-Phase Microextraction: A New Tool in Contemporary Bioanalysis Georgios Theodoridis and Gerhardus J. de Jong I. II. III. IV. V. VI. VII.

231

Introduction Extraction Mode and Coupling Sorption Principles and Parameters Coatings Derivatization Bioanalytical Applications Possibilities and Limitations of SPME References

5. Polyelectrolytes as Stationary Phases in Liquid Chromatography Lilach Yishai-Aviram and Eli Grushka

273

I. II. III. IV.

Introduction The Principle of Dynamic Coating Column Characterization Positively Charged Polyelectrolytes as Stationary Phases V. Negatively Charged Polyelectrolytes as Stationary Phases VI. Complex Polyelectrolyte Layering References

Index

305

Contributors

Alberto Cavazzini University of Ferrara, Ferrara, Italy Francesco Dondi University of Ferrara, Ferrara, Italy Eli Grushka Department of Inorganic and Analytical Chemistry, The Hebrew University, Jerusalem, Israel Pavel Jandera University of Pardubice, Na´m. Cˇs. legiı´ , Pardubice, Czech Republic Gerhardus J. de Jong University Utrecht, Utrecht, The Netherlands Jerry W. King Los Alamos National Laboratory, Los Alamos, New Mexico, U.S.A. Michel Martin Ecole Supe´rieure de Physique et de Chimie Industrielles, Paris Cedex, France Georgios Theodoridis Aristotle University Thessaloniki, Thessaloniki, Greece Lilach Yishai-Aviram Department of Inorganic and Analytical Chemistry, The Hebrew University, Jerusalem, Israel vii

Contents of Other Volumes

Volumes 1–6

out of print

Volume 7 Theory and Mechanics of Gel Permeation Chromatography K. H. Altgelt Thin-Layer Chromatography of Nucleic Acid Bases, Nucleosides, Nucleotides, and Related Compounds Gyo¨rgy Pataki Review of Current and Future Trends in Paper Chromatography V. C. Weaver Chromatography of Inorganic Ions G. Nickless Process Control by Gas Chromatography I. G. McWilliam Pyrolysis Gas Chromatography of Involatile Substances S. G. Perry Labeling by Exchange on Chromatographic Columns Horst Elias Volume 8 Principles of Gel Chromatography Helmut Determann Thermodynamics of Liquid–Liquid Partition Chromatography David C. Locke ix

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Contents of Other Volumes

Determination of Optimum Solvent Systems for Countercurrent Distribution and Column Partition Chromatography from Paper Chromatographic Data Edward Soczewin´ski Some Procedures for the Chromatography of the Fat-Soluble Chloroplast Pigments Harold H. Strain and Walter A. Svec Comparison of the Performance of the Various Column Types Used in Gas Chromatography Georges Guiochon Pressure (Flow) Programming in Gas Chromatography Leslie S. Ettre, La´szlo´ Ma´zor, and Jo´sef Taka´cs Gas Chromatographic Analysis of Vehicular Exhaust Emissions Basil Dimitriades, C. G. Ellis, and D. E. Seizinger The Study of Reaction Kinetics by the Distortion of Chromatographic Elution Peaks Maarten van Swaay Volume 9 Reversed-Phase Extraction Chomatography in Inorganic Chemistry E. Cerrai and G. Ghersini Determination of the Optimum Conditions to Effect a Separation by Gas Chromatography R. P. W. Scott Advances in the Technology of Lightly Loaded Glass Bead Columns Charles Hista, Joseph Bomstein, and W. D. Cooke Radiochemical Separations and Analyses by Gas Chromatography Stuart P. Cram Analysis of Volatile Flavor Components of Foods Phillip Issenberg and Irwin Hornstein Volume 10

out of print

Volume 11 Quantitative Analysis by Gas Chromatography Josef Nova´k Polyamide Layer Chromatography Kung-Tsung Wang, Yau-Tang Lin, and Iris S. Y. Wang Specifically Adsorbing Silica Gels H. Bartels and P. Prijs Nondestructive Detection Methods in Paper and Thin-Layer Chromatography G. C. Barrett

Contents of Other Volumes

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Volume 12 The Use of High-Pressure Liquid Chromatography in Pharmacology and Toxicology Phyllis R. Brown Chromatographic Separation and Molecular-Weight Distributions in Cellulose and Its Derivatives Leon Segal Practical Methods of High-Speed Liquid Chromatography Gary J. Fallick Measurement of Diffusion Coefficients by Gas-Chromatography Broadening Techniques: A Review Virgil R. Maynard and Eli Grushka Gas-Chromatography Analysis of Polychlorinated Diphenyls and Other Nonpesticide Organic Pollutants Joseph Sherma High-Performance Electrometer Systems for Gas Chromatography Douglas H. Smith Steam Carrier Gas–Solid Chromatography Akira Nonaka

Volume 13

out of print

Volume 14 Nutrition: An Inviting Field to High-Pressure Liquid Chromatography Andrew J. Clifford Polyelectrolyte Effects in Gel Chromatography Bengt Stenlund Chemically Bonded Phases in Chromatography Imrich Sebestian and Istva´n Hala´sz Physicochemical Measurement Using Chromatography David C. Locke Gas–Liquid Chromatography in Drug Analysis W. J. A. VandenHeuvel and A. G. Zacchei The Investigation of Complex Association by Gas Chromatography and Related Chromatographic and Electrophoretic Methods C. L. de Ligny Gas–Liquid–Solid Chromatography Antonio De Corcia and Arnaldo Liberti Retention Indices in Gas Chromatography J. K. Haken

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Contents of Other Volumes

Volume 15 Detection of Bacterial Metabolites in Spent Culture Media and Body Fluids by Electron Capture Gas–Liquid Chromatography John B. Brooks Signal and Resolution Enhancement Techniques in Chromatography Raymond Annino The Analysis of Organic Water Pollutants by Gas Chromatography and Gas Chromatography–Mass Spectrometry Ronald A. Hites Hydrodynamic Chromatography and Flow-Induced Separations Hamish Small The Determination of Anticonvulsants in Biological Samples by Use of High-Pressure Liquid Chromatography Reginald F. Adams The Use of Microparticulate Reversed-Phase Packing in High-Pressure Liquid Chromatography of Compounds of Biological Interest John A. Montgomery, Thomas P. Johnson, H. Jeanette Thomas, James R. Piper, and Caroll Temple Jr. Gas–Chromatographic Analysis of the Soil Atmosphere K. A. Smith Kinematics of Gel Permeation Chromatography A. C. Ouano Some Clinical and Pharmacological Applications of High-Speed Liquid Chromatography J. Arly Nelson Volume 16

out of print

Volume 17 Progress in Photometric Methods of Quantitative Evaluation in TLO V. Pollak Ion-Exchange Packings for HPLC Separations: Care and Use Fredric M. Rabel Micropacked Columns in Gas Chromatography: An Evaluation C. A. Cramers and J. A. Rijks Reversed-Phase Gas Chromatography and Emulsifier Characterization J. K. Haken Template Chromatography Herbert Schott and Ernst Bayer Recent Usage of Liquid Crystal Stationary Phases in Gas Chromatography George M. Janini Current State of the Art in the Analysis of Catecholamines Ante´ M. Krstulovic

Contents of Other Volumes

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xiii

Volume 18 The Characterization of Long-Chain Fatty Acids and Their Derivatives by Chromatography Marcel S. F. Lie Ken Jie Ion-Pair Chromatography on Normal- and Reversed-Phase Systems Milton T. W. Hearn Current State of the Art in HPLC Analyses of Free Nucleotides, Nucleosides, and Bases in Biological Fluids Phyllis R. Brown, Ante´ M. Krstulovic, and Richard A. Hartwick Resolution of Racemates by Ligand-Exchange Chromatography Vadim A. Danakov The Analysis of Marijuana Cannabinoids and Their Metabolites in Biological Media by GC and/or GC-MS Techniques Benjamin J. Gudzinowicz, Michael J. Gudzinowicz, Joanne Hologgitas, and James L. Driscoll

Volume 19 Roles of High-Performance Liquid Chromatography in Nuclear Medicine Steven How-Yan Wong Calibration of Separation Systems in Gel Permeation Chromatography for Polymer Characterization Josef Janc˘a Isomer-Specific Assay of 2,4-D Herbicide Products by HPLC: Regulaboratory Methodology Timothy S. Stevens Hydrophobic Interaction Chromatography Stellan Hjerte´n Liquid Chromatography with Programmed Composition of the Mobile Phase Pavel Jandera and Jaroslav Chura´cˇek Chromatographic Separation of Aldosterone and Its Metabolites David J. Morris and Ritsuko Tsai

Volume 20 High-Performance Liquid Chromatography and Its Application to Protein Chemistry Milton T. W. Hearn Chromatography of Vitamin D3 and Metabolites K. Thomas Koshy High-Performance Liquid Chromatography: Applications in a Children’s Hospital Steven J. Soldin

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Contents of Other Volumes

The Silica Gel Surface and Its Interactions with Solvent and Solute in Liquid Chromatography R. P. W. Scott New Developments in Capillary Columns for Gas Chromatography Walter Jennings Analysis of Fundamental Obstacles to the Size Exclusion Chromatography of Polymers of Ultrahigh Molecular Weight J. Calvin Giddings

Volume 21 High-Performance Liquid Chromatography/ Mass Spectrometry (HPLC/MS) David E. Grimes High-Performance Liquid Affinity Chromatography Per-Olof Larsson, Magnus Glad, Lennart Hansson, Mats-Olle Ma˚nsson, Sten Ohlson, and Klaus Mosbach Dynamic Anion-Exchange Chromatography Roger H. A. Sorel and Abram Holshoff Capillary Columns in Liquid Chromatography Daido Ishii and Toyohide Takeuchi Droplet Counter-Current Chromatography Kurt Hostettmann Chromatographic Determination of Copolymer Composition Sadao Mori High-Performance Liquid Chromatography of K Vitamins and Their Antagonists Martin J. Shearer Problems of Quantitation in Trace Analysis by Gas Chromatograhpy Josef Nova´k

Volume 22 High-Performance Liquid Chromatography and Mass Spectrometry of Neuropeptides in Biologic Tissue Dominic M. Desiderio High-Performance Liquid Chromatography of Amino Acids: Ion-Exchange and Reversed-Phase Strategies Robert F. Pfeifer and Dennis W. Hill Resolution of Racemates by High-Performance Liquid Chromatography Vadium A. Davankov, Alexander A. Kurganov, and Alexander S. Bochkov

Contents of Other Volumes

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High-Performance Liquid Chromatography of Metal Complexes Hans Veening and Bennett R. Willeford Chromatography of Carotenoids and Retinoids Richard F. Taylor High-Performance Liquid Chromatography Zybslaw J. Petryka Small-Bore Columns in Liquid Chromatography Raymond P. W. Scott

Volume 23 Laser Spectroscopic Methods for Detection in Liquid Chromatography Edward S. Yeung Low-Temperature High-Performance Liquid Chromatography for Separation of Thermally Labile Species David E. Henderson and Daniel J. O’Connor Kinetic Analyis of Enzymatic Reactions Using High-Performance Liquid Chromatography Donald L. Sloan Heparin-Sepharose Affinity Chromatography Akhlaq A. Farooqui and Lloyd A. Horrocks New Developments in Capillary Columns for Gas Chomatography Walter Jennings

Volume 24 Some Basic Statistical Methods for Chromatographic Data Karen Kafadar and Keith R. Eberhardt Multifactor Optimization of HPLC Conditions Stanley N. Deming, Julie G. Bower, and Keith D. Bower Statistical and Graphical Methods of Isocratic Solvent Selection for Optimal Separation in Liquid Chromatography Haleem J. Issaq Electrochemical Detectors for Liquid Chromatography Ante M. Krstolovic´, Henri Colin, and Georges A. Guichon Reversed-Flow Gas Chromatography Applied to Physicochemical Measurements Nicholas A. Katsanos and George Karaiskakis Development of High-Speed Countercurrent Chromatography Yochiro Ito Determination of the Solubility of Gases in Liquids by Gas–Liquid Chromatography John F. Parcher, Monica L. Bell, and Ping L. Jin

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Contents of Other Volumes

Multiple Detection in Gas Chromatography Ira S. Krull, Michael E. Swartz, and John N. Driscoll

Volume 25 Estimation of Physicochemical Properties of Organic Solutes Using HPLC Retention Parameters Theo L. Hafkenscheid and Eric Tomlinson Mobile Phase Optimization in RPLC by an Iterative Regression Design Leo de Galan and Hugo A. H. Billiet Solvent Elimination Techniques for HPLC/FT-IR of Polycyclic Aromatic Hydrocarbons Lane C. Sander and Stephen A. Wise Liquid Chromatographic Analysis of the Oxo Acids of Phosphorus Roswitha S. Ramsey Liquid Chromatography of Carbohydrates Toshihiko Hanai HPLC Analysis of Oxypurines and Related Compounds Katsuyuki Nakano HPLC of Glycosphingolipids and Phospholipids Robert H. McCluer, M. David Ullman, and Firoze B. Jungalwala

Volume 26 RPLC Retention of Sulfur and Compounds Containing Divalent Sulfur Hermann J. Mo¨ckel The Application of Fleuric Devices to Gas Chromatographic Instrumentation Raymond Annino High Performance Hydrophobic Interaction Chromatography Yoshio Kato HPLC for Therapeutic Drug Monitoring and Determination of Toxicity Ian D. Watson Element Selective Plasma Emission Detectors for Gas Chromatography A. H. Mohamad and J. A. Caruso The Use of Retention Data from Capillary GC for Qualitative Analysis: Current Aspects Lars G. Blomberg Retention Indices in Reversed-Phase HPLC Roger M. Smith HPLC of Neurotransmitters and Their Metabolites Emilio Gelpi

Contents of Other Volumes

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xvii

Volume 27 Physicochemical and Analytical Aspects of the Adsorption Phenomena Involved in GLC Victor G. Berezkin HPLC in Endocrinology Richard L. Patience and Elizabeth S. Penny Chiral Stationary Phases for the Direct LC Separation of Enantiomers William H. Pirkle and Thomas C. Pochapsky The Use of Modified Silica Gels in TLC and HPTLC Willi Jost and Heinz E. Hauck Micellar Liquid Chromatography John G. Dorsey Derivation in Liquid Chromatography Kazuhiro Imai Analytical High-Performance Affinity Chromatography Georgio Fassina and Irwin M. Chaiken Characterization of Unsaturated Aliphatic Compounds by GC/Mass Spectrometry Lawrence R. Hogge and Jocelyn G. Millar Volume 28 Theoretical Aspects of Quantitative Affinity Chromatography: An Overview Alain Jaulmes and Claire Vidal-Madjar Column Switching in Gas Chromatography Donald E. Willis The Use and Properties of Mixed Stationary Phases in Gas Chromatography Gareth J. Price On-line Small-Bore-Chromatography for Neurochemical Analysis in the Brain William H. Church and Joseph B. Justice, Jr. The Use of Dynamically Modified Silica in HPLC as an Alternative to Chemically Bonded Materials Per Helboe, Steen Honore´ Hansen, and Mogens Thomsen Gas Chromatographic Analysis of Plasma Lipids Arnis Kuksis and John J. Myher HPLC of Penicillin Antibiotics Michel Margosis Volume 29 Capillary Electrophoresis Ross A. Willingford and Andrew G. Ewing Multidimensional Chromatography in Biotechnology Daniel F. Samain

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Contents of Other Volumes

High-Performance Immunoaffinity Chromatography Terry M. Phillips Protein Purification by Multidimensional Chromatography Stephen A. Berkowitz Fluorescence Derivitization in High-Performance Liquid Chromatography Yosuke Ohkura and Hitoshi Nohta

Volume 30 Mobile and Stationary Phases for Supercritical Fluid Chromatography Peter J. Schoenmakers and Luis G. M. Uunk Polymer-Based Packing Materials for Reversed-Phase Liquid Chromatography Nobuo Tanaka and Mikio Araki Retention Behavior of Large Polycyclic Aromatic Hydrocarbons in Reversed-Phase Liquid Chromatography Kiyokatsu Jinno Miniaturization in High-Performance Liquid Chromatography Masashi Goto, Toyohide Takeuchi, and Daido Ishii Sources of Errors in the Densitometric Evaluation of Thin-Layer Separations with Special Regard to Nonlinear Problems Victor A. Pollak Electronic Scanning for the Densitometric Analysis of Flat-Bed Separations Viktor A. Pollak

Volume 31 Fundamentals of Nonlinear Chromatography: Prediction of Experimental Profiles and Band Separation Anita M. Katti and Georges A. Guiochon Problems in Aqueous Size Exclusion Chromatography Paul L. Dubin Chromatography on Thin Layers Impregnated with Organic Stationary Phases Jiri Gasparic Countercurrent Chromatography for the Purification of Peptides Martha Knight Boronate Affinity Chromatography Ram P. Singhal and S. Shymali M. DeSilva Chromatographic Methods for Determining Carcinogenic Benz(c)-acridine Noboru Motohashi, Kunihiro Kamata, and Roger Meyer

Contents of Other Volumes

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xix

Volume 32 Porous Graphitic Carbon in Biomedical Applications Chang-Kee Lim Tryptic Mapping by Reversed Phase Liquid Chromatography Michael W. Dong Determination of Dissolved Gases in Water by Gas Chromatography Kevin Robards, Vincent R. Kelly, and Emilios Patsalides Separation of Polar Lipid Classes into Their Molecular Species Components by Planar and Column Liquid Chromatography V. P. Pchelkin and A. G. Vereshchagin The Use of Chromatography in Forensic Science Jack Hubball HPLC of Explosives Materials John B. F. Lloyd Volume 33 Planar Chips Technology of Separation Systems: A Developing Perspective in Chemical Monitoring Andreas Manz, D. Jed Harrison, Elizabeth Verpoorte, and H. Michael Widmer Molecular Biochromatography: An Approach to the Liquid Chromatographic Determination of Ligand-Biopolymer Interactions Irving W. Wainer and Terence A. G. Noctor Expert Systems in Chromatography Thierry Hamoir and D. Luc Massart Information Potential of Chromatographic Data for Pharmacological Classification and Drug Design Roman Kaliszan Fusion Reaction Chromatography: A Powerful Analytical Technique for Condensation Polymers John K. Haken The Role of Enatioselective Liquid Chromatographic Separations Using Chiral Stationary Phases in Pharmaceutical Analysis Shulamit Levin and Saleh Abu-Lafi Volume 34 High-Performance Capillary Electrophoresis of Human Serum and Plasma Proteins Oscar W. Reif, Ralf Lausch, and Ruth Freitag Analysis of Natural Products by Gas Chromatography/Matrix Isolation/Infrared Spectrometry W. M. Coleman III and Bert M. Gordon

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Contents of Other Volumes

Statistical Theories of Peak Overlap in Chromatography Joe M. Davis Capillary Electrophoresis of Carbohydrates Ziad El Rassi Environmental Applications of Supercritical Fluid Chromatography Leah J. Mulcahey, Christine L. Rankin, and Mary Ellen P. McNally HPLC of Homologous Series of Simple Organic Anions and Cations Norman E. Hoffman Uncertainty Structure, Information Theory, and Optimization of Quantitative Analysis in Separation Science Yuzuru Hayashi and Rieko Matsuda

Volume 35 Optical Detectors for Capillary Electrophoresis Edward S. Yeung Capillary Electrophoresis Coupled with Mass Spectrometry Kenneth B. Tomer, Leesa J. Deterding, and Carol E. Parker Approaches for the Optimization of Experimental Parameters in Capillary Zone Electrophoresis Haleem J. Issaq, George M. Janini, King C. Chan, and Ziad El Rassi Crawling Out of the Chiral Pool: The Evolution of Pirkle-Type Chiral Stationary Phases Christopher J. Welch Pharmaceutical Analysis by Capillary Electrophoresis Sam F. Y. Li, Choon Lan Ng, and Chye Pend Ong Chromatographic Characterization of Gasolines Richard E. Pauls Reversed-Phase Ion-Pair and Ion-Interaction Chromatography M. C. Gennaro Error Sources in the Determination of Chromatographic Peak Size Ratios Veronika R. Meyer

Volume 36 Use of Multivariate Mathematical Methods for the Evaluation of Retention Data Matrices Tibor Cserha´ti and Esther Forga´cs Separation of Fullerenes by Liquid Chromatography: Molecular Recognition Mechanism in Liquid Chromatographic Separation Kiyokatsu Jinno and Yoshihiro Saito

Contents of Other Volumes

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xxi

Emerging Technologies for Sequencing Antisense Oligonucleotides: Capillary Electrophoresis and Mass Spectrometry Aharon S. Cohen, David L. Smisek, and Bing H. Wang Capillary Electrophoretic Analysis of Glycoproteins and Glycoprotein-Derived Oligosaccharides Robert P. Oda, Benjamin J. Madden, and James P. Landers Analysis of Drugs of Abuse in Biological Fluids by Liquid Chromatography Steven R. Binder Electrochemical Detection of Biomolecules in Liquid Chromatography and Capillary Electrophoresis Jian-Ge Chen, Steven J. Woltman, and Steven G. Weber The Development and Application of Coupled HPLC-NMR Spectroscopy John C. Lindon, Jeremy K. Nicholson, and Ian D. Wilson Microdialysis Sampling for Pharmacological Studies: HPLC and CE Analysis Susan M. Lunte and Craig E. Lunte Volume 37 Assessment of Chromatographic Peak Purity Muhammad A. Sharaf Fluorescence Detectors in HPLC Maria Brak Smalley and Linda B. McGown Carbon-Based Packing Materials for Liquid Chromatography: Structure, Perfomance, and Retention Mechanisms John H. Knox and Paul Ross Carbon-Based Packing Materials for Liquid Chromatography: Applications Paul Ross and John H. Knox Directly Coupled (On-Line) SFE-GC: Instrumentation and Applications Mark D. Burford, Steven B. Hawthorne, and Keith D. Bartle Sample Preparation for Gas Chromatography with Solid-Phase Extraction and Solid-Phase Microextraction Zelda E. Penton Capillary Electrophoresis of Proteins Tim Wehr, Robert RodriguezDiaz, and Cheng-Ming Liu Chiral Micelle Polymers for Chiral Separations in Capillary Electrophoresis Crystal C. Williams, Shahab A. Shamsi, and Isiah M. Warner Analysis of Derivatized Peptides Using High-Performance Liquid Chromatography and Capillary Electrophoresis Kathryn M. De Antonis and Phyllis R. Brown

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Volume 38 Band Spreading in Chromatography: A Personal View John H. Knox The Stochastic Theory of Chromatography Francesco Dondi, Alberto Cavazzini, and Maurizio Remelli Solvating Gas Chromatography Using Packed Capillary Columns Yufeng Shen and Milton L. Lee The Linear-Solvent-Strength Model of Gradient Elution L. R. Snyder and J. W. Dolan High-Performance Liquid Chromatography-Pulsed Electrochemical Detection for the Analysis of Antibiotics William R. LaCourse and Catherine O. Dasenbrock Theory of Capillary Zone Electrophoresis H. Poppe Separation of DNA by Capillary Electrophoresis Andra´s Guttman and Kathi J. Ulfelder Volume 39 Theory of Field Flow Fractionation Michel Martin Particle Simulation Methods in Separation Science Mark R. Schure Mathematical Analysis of Multicomponent Chromatograms Attila Felinger Determination of Association Constants by Chromatography and Electrophoresis Daniel W. Armstrong Method Development and Selectivity Optimization in High-Performance Liquid Chromatography H. A. H. Billet and G. Rippel Chemical Equilibria in Ion Chromatography: Theory and Applications Pe´ter Hajo´s, Otto´ Horva´th, and Gabriella Re´ve´sz Fundamentals and Simulated Moving Bed Chromatography Under Linear Conditions Guoming Zhong and Georeges Guiochon Volume 40 Fundamental Interpretation of the Peak Profiles in Linear ReversedPhase Liquid Chromatography Kanji Miyabe and Georges Guiochon Dispersion in Micellar Electrokinetic Chromatography Joe M. Davis

Contents of Other Volumes

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xxiii

In Search of a Chromatographic Model for Biopartitioning Colin F. Poole, Salwa K. Poole, and Ajith D. Gunatilleka Advances in Physico-chemical Measurements Using Inverse Gas Chromatography Nicholas A. Katsanos and Fani Roubani-Kalantzopoulou Fundamental Aspects of Aerosol-Based Light-Scattering Detectors for Separations John A. Koropchak, Salma Sadain, Xiaohui Yang, Lars-Erik Magnusson, Mari Heybroek, and Michael P. Anisimov New Developments in Liquid-Chromatographic Stationary Phases Toshiko Hanai Non-Silica-Based Supports in Liquid Chromatography of Bioactive Compounds Esther Forga´cs and Tibor Cserha´ti Overview of the Surface Modification Techniques for the Capillary Electrophoresis of Proteins Marie-Claude Millot and Claire Vidal-Madjar Continuous Bed for Conventional Column and Capillary Column Chromatography Jia-li Liao Countercurrent Chromatography: Fundamentally a Preparative Tool Alain Berthod and Beatrice Billardello Analysis of Oligonucleotides by ESI-MS Dieter L. Deforce and Elfriede G. Van den Eeckhout Determination of Herbicides in Water Using HPLC-MS Techniques G. D’Ascenzo, F. Bruno, A. Gentili, S. Marchese, and D. Perret Effect of Adsorption Phenomena on Retention Values in Capillary Gas–Liquid Chromatography Victor G. Berezkin Volume 41 Fundamentals of Capillary Electrochromatography Ute Pyell Membrane Extraction Techniques for Sample Preparation Jan A˚ke Jo¨nsson and Lennart Mathiasson Design of Rapid Gradient Methods for the Analysis of Combinatorial Chemistry Libraries and the Preparation of Pure Compounds Uwe D. Neue, Judy L. Carmody, Yung-Fong Cheng, Ziling Lu, Charles H. Phoebe, and Thomas E. Wheat Molecularly Imprinted Extraction Materials of Highly Selective Sample Clean-Up and Analyte Enrichment Francesca Lanza and Bo¨rje Sellergren

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Contents of Other Volumes

Biomembrane Chromatography: Application to Purification and Biomolecule-Membrane Interactions Tzong-Hsien Lee and Marie-Isabel Aguilar Transformation of Analytes for Electrochemical Detection: A Review of Chemical and Physical Approaches Mark J. Rose, Susan M. Lunte, Robert G. Carlson, and John F. Stobaugh High-Performance Liquid Chromatography: Trace Metal Determination and Speciation Corrado Sarzanini Temperature-Responsive Chromatography Hideko Kanazawa, Yoshikazu Matsushima, and Teruo Okano Carrier Gas in Capillary Gas–Liquid Chromatography V. G. Berezkin Cathechins in Tea: Chemistry and Analysis Christina S. Robb and Phyllis R. Brown Volume 42 Chemometric Analysis of Comprehensive Two-Dimensional Separations Robert E. Synovec, Bryan J. Prazen, Kevin J. Johnson, Carlos G. Fraga, and Carsten A. Bruckner Column Technology for Capillary Electrochromatography Luis A. Colo´n, Todd D. Maloney, Jason Anspach, and He´ctor Colo´n Gas Chromatography with Inductively Coupled Plasma Mass Spectrometric Detection (GP-ICP MS) Brice Bouyssiere, Joanna Szpunar, Gae¨tne Lespes, and Ryszard Lobinski GC-MS Analysis of Halocarbons in the Environment Filippo Mangani, Michela Maione, and Pierangela Palma Microfluidics for Ultrasmall-Volume Biological Analysis Todd O. Windman, Barb J. Wyatt, and Mark A. Hayes Recent Trends in Proteome Analysis Pier Giorgio Righetti, Annalisa Castagna, and Mahmoud Hamdan Improving Our Understanding of Reversed-Phase Separations for the 21st Century Patrick D. McDonald Clinical Applications of High-Performance Affinity Chromatography David S. Hage

1 Gradient Elution in Liquid Column Chromatography—Prediction of Retention and Optimization of Separation ˇ legii,´ Pardubice, Pavel Jandera University of Pardubice, Na´m. Cs. Czech Republic

I. INTRODUCTION II. THEORY OF RETENTION IN ANALYTICAL GRADIENT-ELUTION CHROMATOGRAPHY A. Calculation of retention times and of retention volumes B. Bandwidths and resolution in gradient-elution LC III. REVERSED-PHASE CHROMATOGRAPHY WITH BINARY GRADIENTS IV. NORMAL-PHASE CHROMATOGRAPHY WITH BINARY GRADIENTS V. ION-EXCHANGE GRADIENT ELUTION CHROMATOGRAPHY

3 9 9 17 19 25 34

1

2 / Jandera VI. EFFECTS OF THE INSTRUMENTATION AND OF THE NONIDEAL RETENTION BEHAVIOR ON THE RETENTION IN GRADIENT ELUTION A. Effects of the dwell volume on retention in gradient elution LC. Retention data in gradient elution with an initial hold-up period. Gradient preelution and postelution B. Effects of the adsorption of strong solvents on retention VII. GRADIENT ELUTION METHOD DEVELOPMENT A. Transfer of gradient methods and effects of changing operating conditions on separation 1. Changing flow rate of the mobile phase in gradient elution chromatography 2. Changing column diameter in gradient elution chromatography 3. Changing column length in gradient elution chromatography 4. Rapid prediction of the effects of changing gradient steepness (gradient range) and initial mobile phase composition on the separation B. Optimization of gradient elution separations 1. Peak capacity and fast gradients 2. Optimization of gradients for specific separation problems VIII. CHROMATOGRAPHY WITH TERNARY GRADIENTS IX. PECULIARITIES OF GRADIENT ELUTION SEPARATION OF HIGH-MOLECULAR COMPOUNDS X. CONCLUSION ACKNOWLEDGMENTS SYMBOLS REFERENCES APPENDIX A. Correction of the retention volume in normal-phase HPLC for the column uptake of polar solvents

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37 48 55 56 58 59 61

62 69 69 71 78

81 90 90 92 96

Gradient Elution in LC Chromatography during gradient elution (solventdemixing effect) APPENDIX B. Schematics of a spreadsheet program for optimization of gradient elution

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3

104 107

I. INTRODUCTION Many complex samples contain compounds that differ widely in retention, so that HPLC in isocratic elution mode with a mobile phase of fixed composition often does not yield successful separation of the individual solutes. To keep the time of analysis within acceptable limits, the retention factors of the most strongly retained sample components, k, usually should be lower than 10. Once the appropriate chromatographic column is selected, the retention can be controlled by setting appropriate flow rate, column temperature and—most efficiently—the composition of the mobile phase. In the isocratic elution mode, the working conditions are kept constant during the separation run and in many cases satisfactory results are obtained. However, for some complex samples weakly retained compounds elute as poorly—if at all—separated bands close to the column holdup time under the conditions adjusted for adequate retention of strongly retained solutes (Fig. 1A). On the other hand, with the mobile phase adjusted to achieve desired resolution of weakly retained compounds, the elution of strongly retained sample components can be slow, their peaks are broad and their concentration in the eluate may even fall down below the detection limits (Fig. 1B). To obtain satisfactory separation of both weakly and strongly retained sample compounds, the operating conditions controlling the retention should be varied during the chromatographic run [1–5]. This can be achieved by gradually increasing the temperature, the flow rate or the elution strength of the mobile phase (as in Fig. 1C). Flow programming in HPLC is limited by maximum operation pressure, usually 30–40 MPa, and has little advantage when smallparticle packed columns are used. Although recently introduced monolithic columns are more suitable for the programmed flow rate operation because of their lower flow resistance [6], the retention factors are independent of the flow rate, hence the improvement of separation is only marginal in comparison with techniques relying on gradual decreasing of the retention factors during the analysis. The

4 / Jandera

Fig. 1 Reversed-phase separation of 1,2-naphthoylenebenzimidazole alkylsulphonamides. Column: Lichrosorb RP-18, 10 Am (300
4 mm i.d.). Mobile phase: (A) 80% methanol in water, (B) 95 methanol in water, (C) linear gradient, 70–100% methanol in 20 min, 1 mL/min. Numbers of the peaks agree with the numbers of carbon atoms in the alkyls.

resolution can be improved by using simultaneous gradient elution and flow programming [7]. The retention in HPLC usually decreases with increasing temperature, but temperature programming is rarely used in conventional HPLC, in contrast to gas chromatography. One reason is a relatively slow response of the temperature inside the conventional

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columns to a change in the temperature setting in an air-heated thermostated compartment, which might cause poor retention data reproducibility in short analyses requiring a steep temperature ramp. This limitation does not concern packed capillary HPLC columns with rapid radial heat transfer [8,9]. Further, some HPLC packing materials are not stable enough at elevated temperatures. A large rise in temperature during the run is usually needed to reduce significantly the retention of strongly retained small molecules. Hence, the temperature programming can offer results comparable to gradient elution over a relatively narrow range of the elution strength [10]. Only moderate change in the elution strength in the course of separation is usually sufficient for the separation or fractionation of large molecules such as synthetic polymer samples, where temperature programming offers promising alternative to gradient elution technique [11]. Anyway, potential merits of temperature programming in HPLC are still to be proven. On the other hand, simultaneous optimization of the temperature and of the gradient time in gradient-elution HPLC offers interesting possibilities for the separation of complex samples [12–16]. Gradient elution still remains the most widely used programming technique in liquid chromatography, since its introduction in 1952 [17–19]. Gradual increase in the elution strength of the mobile phase allows decreasing the retention factors of small molecules by two to three orders of magnitude in a single gradient run, which results in shorter separation time, increased peak capacity and more regular band spacing of compounds with large differences in affinities to the stationary phase with respect to isocratic separation. The instrumentation for gradient elution is more sophisticated and more expensive than in isocratic liquid chromatography, as two or more components of the mobile phase should be accurately mixed according to a preset time program. Binary gradients are formed by mixing two mobile phase components: the concentration of a strong solvent B with a higher elution strength in a weak solvent A with a lower elution strength increases during the gradient run. Binary gradients are used more frequently than ternary gradients prepared from three mobile phase components, whereas quaternary or more complex gradients are rarely necessary for optimum separation performance. The gradient program can be composed of a few subsequent isocratic steps, or the composition of the mobile phase can be changed continuously during the gradient run. It is also possible to employ gradients composed of several continuous steps with different slopes,

6 / Jandera if necessary combined with isocratic hold-up steps. The profile of a continuous gradient is characterized by three adjustable parameters: 1) the gradient range (i.e., the initial and the final concentrations of the solvent B); 2) the steepness (i.e., the gradient time); and 3) the shape (curvature), which all affect the elution time and the spacing of the peaks in the chromatogram and should be taken into account in the development of gradient separations. According to the shape, gradients can be classified as linear (the most common), convex, or concave. A few examples of various linear, concave, and convex gradient profiles are shown in Fig. 2. Because of a higher number of experimental variables that should be taken into account, the development of gradient elution methods is more complicated than the development of isocratic methods and the retention behavior is more difficult to describe in quantitative terms. Column dimensions and the flow rate of the mobile phase affect the retention in gradient elution in a more complex way than the isocratic retention. The effective use of gradient elution technique is easier if the theoretical principles of gradient elution are well understood.

Fig. 2 Examples of linear, concave, and convex gradients from 0% to 100% stronger eluent, B. c—concentration of B, V—volume of the eluate from the start of the gradient with various values of the gradient shape parameter j (Eq. (7)). j = 1 for linear, j > 1 for concave, and j < 1 for convex gradients.

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The actual impact of the gradient program on the separation of the sample depends on the effect of the mobile phase composition on the retention in the HPLC mode used (conveniently characterized by isocratic retention factors, k). To describe the retention in gradientelution chromatography, the dependence of the instantaneous retention factors on the gradient program should be known. For this purpose, either an equation based on more or less exact retention model or merely an empirical equation can be used, as far as it describes accurately enough the experimental data. Snyder [20,21] developed a widely used theory of linear solvent strength (LSS) binary gradients assuming linear change in the logarithms of retention factors, k, with the time, t, elapsed from the start of the gradient run: log k ¼ log ka  bs

t tm

ð1Þ

Here, ka is the value of k at the start of the gradient, t = 0, tm is the column hold-up time, and bs is a measure of the gradient steepness. The LSS theory facilitates the comparison of the retention behavior in isocratic chromatography and in gradient elution chromatography, but it is not always straightforward to preset an exact LSS gradient program in real chromatographic systems. Most easy to employ are linear concentration gradients, which correspond to LSS gradients in the chromatographic systems where the isocratic retention can be described by a simple retention equation—Eq. (2). The LSS gradients are often (but not always) reasonably well approximated in reversedphase (RP) chromatography where the gradient elution is applied most frequently [22–25]: log k ¼ a  Su

ð2Þ

Here, the constant a is the extrapolated (not necessarily the real) value of the logarithm of the retention factor in pure weak solvent A (water in RP systems) and the constant S is a measure of the solvent strength of the strong solvent B contained in concentration u in a binary mobile phase. The gradient steepness in LSS gradients is defined as: bs ¼

t0 SDu Vm SDu ¼ tG tG F m

ð3Þ

where tG is the gradient time corresponding to the change Du from the start to the end of gradient elution, Vm is the column hold-up

8 / Jandera volume, and Fm is the flow rate of the mobile phase. In other HPLC modes, i.e., in ion-exchange and in normal-phase liquid chromatography, Eq. (2) can be used only over a narrow range of mobile phase concentrations and the utility of the LSS model is limited [26]. Stout et al. [27] and other workers [28] advocated applicability of the empirically corrected LSS model for gradient-elution separations of multiply charged proteins. Recently, Snyder and Dolan published an excellent review of the LSS gradient approach [4], hence the LSS approach will not be discussed here in more detail. Gradient elution is often used in ion-exchange chromatography of ionic compounds such as charged biopolymers. Despite being described most early, the applications of gradient elution in normalphase LC (liquid–solid adsorption chromatography) have been so far less frequent than in other LC modes, but they are becoming increasingly popular, especially for the separation of noncharged industrial polymer samples. Whereas reversed-phase gradient elution with aqueous–organic mobile phases provides excellent results for the separation of peptides, proteins, and other biopolymers [29–34], gradient-elution chromatography on nonpolar chemically bonded phases or on polar adsorbents with increasing concentration of a polar organic solvent in a nonpolar one [35–37] often shows better selectivity than RP separations of synthetic nonionic oligomers and polymers containing polar monomer units such as surfactants [38], homopolymers [11,39,40], and copolymers [41–46]. As the range of HPLC phase systems in which gradient elution is applied becomes increasingly broader, accurate approaches are more urgently needed for the prediction and optimization of gradientelution separations in various HPLC modes. Earlier, we reviewed the possibilities of using a general non-LSS approach for various liquid chromatography modes with binary gradients [2,3]. The present review is focused on some more recent developments of the non-LSS gradient elution approach for binary and ternary gradients in various HPLC modes, including reversed-phase, normal-phase, and ion-exchange systems. Prediction and optimization of the retention in gradient elution are discussed together with problems arising from various sources of nonideal behavior and possible ways to suppression or compensation of their impact on the accuracy of the predicted gradient-elution data. Finally, peculiarities of the applications of gradient elution theory to the separation of high-molecular compounds are addressed.

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II. THEORY OF RETENTION IN ANALYTICAL GRADIENT-ELUTION CHROMATOGRAPHY A. Calculation of Retention Times and of Retention Volumes The theory of gradient-elution chromatography is now elaborated to the degree that it allows to predict the gradient-elution behavior of sample compounds from their isocratic retention data (or from two initial gradient experiments) and to optimize the profile of the gradient in various reversed-phase, normal-phase, and ion-exchange systems [2–4]. Two or more initial gradient runs can also be used to estimate the optimum composition of the mobile phase for isocratic separations [26,47–51]. Calculation of the retention in gradientelution chromatography is possible using adequate equations describing the dependence of the retention on the parameters characterizing the profile of the gradient. In isocratic liquid chromatography, the elution times, tR, or volumes, VR, are simply related to the retention factors, k, that remain constant during the separation run, tRtm = tRV = ktm; VRVm = VRV = kVm. This simple relationship cannot be used in gradient elution chromatography, where the retention factors decrease during the run. The band migration velocities change during the separation run and the final elution times depend on the solute and the HPLC phase system which control the relationship between the k and the actual mobile phase composition. Figure 3 shows decreasing instantaneous k of neburon during its migration along the column in 20-min gradients starting at the initial k = 50, both in reversed-phase chromatography (RPC) on a C18 column (a gradient from 57.5% to 100% methanol in water) and in normal-phase chromatography on a silica gel (a gradient from 0.84% to 17% 2-propanol in hexane). The retention data in gradient elution can be described assuming that the gradient elution is represented by the sum of consequent elementary migration steps in which the solute migrates a differential distance along the column corresponding to a differential increment of the column hold-up time d(tm) or column hold-up volume d(Vm). Unlike the isocratic LC, the relationship between the retention factor and the retention volume is defined only for a differential step by differential Eqs. (4a,b). Only during such a differential step can the change in concentration of the strong eluent B in the mobile phase,

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Fig. 3 Examples of changing instantaneous retention factors, ki, of neburon at the fraction distance X from the top of the column migrated by the sample zone during reversed-phase (full line) and normal-phase (dashed line) gradient elution. Gradient volume VG = 20 Vm. RPC: Silasorb C18, 57.5–100% methanol in water, a = 4, m = 4 in Eq. (11), NPC Silasorb silica gel, 0.84– 17% 2-propanol in hexane, k0 = 0.076, m = 1.3 in Eq. (15). l = column length, li = fractional distance from the top of the column at the retention factor ki.

u(V ), be neglected and the retention factors, k, of all sample solutes remain constant. Each elementary step contributes to the final retention time, tR, and retention volume, VR, by the increments d(t) and d(V ), respectively: dðtÞ ¼ kdðtm Þ;

dðV Þ ¼ kdðVmÞ

ð4a; bÞ

Equation (4a,b) can be integrated after introducing the actual dependence of k on the time, t (or on the volume of the eluate, V, passed through the column) from the start of the gradient until the

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elution to yield the expressions for the net gradient-elution retention times, tRV, or volumes, VRV, respectively: ð tR V 0

1 tm

dðtÞ ¼ 1; k

ðV RV 0

1 Vm

dðV Þ ¼1 k

ð5a; bÞ

The approach for the calculation of the gradient-elution retention volumes or times based on the solution of Eq. (5a,b) was first suggested almost 50 years ago [52,53] and since that time this approach has been applied to calculate the retention data in various specific gradient elution separation applications, see the survey in Refs. [2] and [3]. The solution of Eq. (5a,b) is simple for linear solvent strength gradients where log k is a linear function of V and was presented by Snyder et al. [4,26,27]. Theoretically, it should be possible to compensate for any nonlinear dependence of log k on u by designing an appropriate complementary gradient profile, but setting a suitable program for an exactly linear change in log k during the gradient elution can be impractically tedious and often is not feasible with many commercial instruments. An easier approach, which can be used for a great variety of combinations of gradient programs and chromatographic phase systems, divides the function characterizing changes in k with increasing V (or t) into two partial contributions [2,3,54] 1. The retention function describes the dependence of k on the concentration of the strong eluent B in the mobile phase, u, controlled by the thermodynamics of the distribution process of a sample solute, which differs in various reversed-phase, normal-phase, and ionexchange chromatographic systems [the retention equation k = f V(u)]. 2. The gradient function u = f(V ) controls the gradient profile (the change in u with time t or with the volume V of the eluate) and is adjusted by the operator. To avoid confusion, it is important to fix the time (place), which corresponds to the actual mobile phase composition described by the gradient function—at the time the mobile phase components are mixed in a low-pressure or high-pressure part of gradient chromatograph, or at the time the peaks leave the column and are detected. The first option is more practical as it corresponds directly to the gradient program set by the operator and is therefore consequently used in this work. On the other hand, the second possibility corresponds better to the effect of the gradient on the

12 / Jandera retention behavior of the individual compounds, and it should be kept in mind that the actual gradient composition at the detector corresponds to the composition at the outlet from the gradient mixer before the time elapsed necessary for the mobile phase to migrate to the top of the column (the gradient dwell volume) and through the column (the column hold-up volume). This gradient delay can be respected in the calculation of the retention data, as shown in Sections III–VI. Linear concentration gradients are employed most frequently because they are most simple to understand and can be generated in all modern gradient-elution instruments. However, in some cases curved gradients may yield better separation and more regular band spacing by increasing the resolution in the shallower parts of the gradient and speeding up the elution of the bands in the steeper parts of the gradient program, as illustrated by the examples of a convex and a concave gradient in Fig. 4. By more regular band spacing, nonlinear gradients can increase the peak capacity, especially for the separation of polymers and oligomers, as it is schematically shown in Fig. 5 for normal-phase gradient elution of lower oligostyrenes on a silica gel column.

Fig. 4 Effect of the gradient shape on the band spacing in the chromatograms. Convex (A) and concave (B) gradients of acetonitrile in water. Sample and other separation conditions as in Fig. 1.

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Fig. 5 Normal-phase gradient-elution separation of lower oligostyrenes on two Separon SGX, 7 Am, silica gel columns in series (150
3.3 mm i.d. each), using the optimized linear and convex gradients of dioxane in heptane. Flow rate = 1 mL/min. Normalized response relates to the original concentrations of the oligomers in the sample, c0.

14 / Jandera In the early stage of liquid column chromatography, 30–40 years ago, the most frequently used instruments for gradient elution employed the exponential dilution in a chamber of a fixed volume containing originally solvent A. Solvent B was delivered by a pump at a constant flow rate into the chamber, the contents of which was continuously mixed. Such devices, generating nonlinear (convex) gradients, are neither accurate nor flexible and hence are no more employed in contemporary practice of HPLC on conventional columns, but are still useful in microbore or packed capillary HPLC [55–57] or in electrochromatography [58], because of technical problems connected with the design of precise low-volume gradient instruments operating with flow rates in the range of microliters per minute or even lower. The linear gradients are described by the gradient function: u ¼ A þ B Vt ¼ A þ

Dut B VV DuV ¼Aþ ¼ A þ BV ¼ A þ tG Fm VG

ð6Þ

Here, A is the initial concentration u of the strong eluent B in the mobile phase at the start of the gradient, and B = Du/VG or B V = Du/tG is the steepness (slope) of the gradient, i.e., the increase in u per the time unit or per the volume unit of the eluate, respectively. VG and tG are the gradient volume and the gradient time during which the concentration u is changed from the initial value A = u0 to the concentration uG = A+Du at the end of the gradient; Du = (uGA) is the gradient concentration range. It has been found earlier [54] that a convenient curved gradient function for characterization of various convex and concave gradient profiles can be conveniently described by Eq. (7):  1 j u ¼ Að j Þ þ BV ð7Þ where A = u0 is the initial concentration of the strong eluent B at the start of the gradient, B = [uG(1/j)  A(1/j)]/VG is the steepness (slope) of the gradient and uG is the concentration of B at the end of the gradient. j is the gradient shape parameter characterizing the curvature of the gradient: j = 1 for linear gradients, whereas j < 1 for convex and j > 1 for concave nonlinear gradients (as illustrated by several examples in Fig. 2). The gradient function described by Eq. (7) is especially useful for prediction and optimization of retention in normal-phase and ion- exchange chromatography. Many gradient

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instruments do not allow direct setting of a continuous nonlinear gradient, and curved gradients should be substituted by linear segmented gradients consisting in a series of subsequent linear gradient steps with gradually increasing or decreasing slopes, B. After introducing the appropriate retention equation and gradient function, Eq. (5a,b) can be solved to enable calculations of elution volumes in various HPLC separation modes. A survey of the equations describing a variety of possible solutions can be found elsewhere [2,3]. Even in cases where the integration of Eq. (5a,b) results in an equation that does not allow the separation of the variables, the retention data can be easily calculated using numerical iteration approach; a plethora of suitable software packages are now available for such purposes. The equations for gradient times (volumes) most useful in reversed-phase, normal-phase, and ionexchange gradient modes are discussed in Sections III–V. The calculation of the elution times or of the elution volumes by integration of Eq. (5a,b) in ‘‘ideal gradient elution’’ is based on several simplified assumptions concerning both the phase equilibrium in the column and the function of the instrumentation used. 1. The interactions between various compounds in the separation column should not change the column properties and the gradient profile should not change as it moves along the column, so that its profile does not change from the start of the gradient till the elution of the last sample solute. This may not always be the case, as any part of the column is at any time in equilibrium with a multicomponent mobile phase of changing composition, from which one or more components may be preferentially adsorbed on to the surface of the stationary phase in the column. In some systems this effect can become significant, so that not only the composition of the adsorbed layer, but also the profile of the gradient may change in the course of gradient elution in dependence on time elapsed and on the distance along the column. This problem is discussed in Section VI.B. 2. The kinetics of the chromatographic process is fast enough to allow instantaneous establishment of the distribution equilibrium between the mobile and the stationary phases. This can be expected during a gradient on columns packed with fine particle materials used in modern practice of HPLC. However, the reequilibration of the column to the initial mobile phase with a lower elution strength after the end of the gradient can be rather slow if one or more mobile phase components are strongly adsorbed on the column. Generally, approx-

16 / Jandera imately 15 column hold-up volumes are necessary to reestablish the initial equilibrium after the end of the gradient, but the exact volume necessary for reequilibration depends on the chromatographic system and on the initial mobile phase at the start of the gradient. Further, the establishment of the equilibrium can be less than perfect with fast generic gradients used for high sample throughput in the laboratory. 3. It is assumed that the solute concentration is low enough for its distribution isotherm between the mobile and stationary phases to be linear, so that the retention factor is independent of the concentration of the solute. This problem is essentially the same as in isocratic-elution chromatography, and the application of Eqs. (4a,b) and (5a,b) is limited to the linear range of the sample distribution isotherms, common in analytical HPLC. On the other hand, preparative chromatography on overloaded separation columns usually employs nonlinear range of adsorption isotherms and requires different approach to the description of the retention behavior and of the band profiles. 4. The solution of Eq. (5) assumes a constant value of the column hold-up volume during the gradient elution. This is not always straightforward as the hold-up volume can change to a certain extent with changing mobile phase [59–61]. Several different methods were suggested for the determination of the Vm (see, e.g., Ref. [34]). The determination of the mobile phase volume in the column by weighing method using two solvents of different densities gives the Vm independent of the mobile phase composition [35]. However, this method may be not accurate enough and it is more practical for routine practice to define the hold-up volume in gradient elution LC by convention. A useful definition of Vm is the isocratic elution volume of a nonretained compound in pure strong eluent B as the mobile phase. 5. In gradient elution, the trailing edge of the solute zone moves along the column in the mobile phase with a higher elution strength, i.e., faster than the leading edge. This leads to additional sharpening of chromatographic bands with respect to isocratic elution [4,25,26], but usually little affects the migration of the centers of gravity of the solute zones and can be neglected in calculations of the elution volumes. The more efficient the column is, the narrower are the zones and the less significant is the band sharpening effect. 6. Correct function of the gradient instrument is essential for successful theoretical description of the experimental data. Differ-

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ences of the actual gradient profile from the preset gradient program are an inevitable consequence of any error in flow rate caused by pump failure or of possible errors in the mixing of the gradient components caused either by device malfunction or by improper construction design (for example, poor flow rate matching in the pumps delivering the components of the gradient). Such errors can occur both with high-pressure and low-pressure gradient instruments [62,63]. Other deviations from the preset gradient profile which can occur even with properly functioning devices are the rounding of the gradient at the beginning and at the end of the gradient and the gradient delay. These effects depend on the construction design of the instrument and are discussed in more detail in Section VI.A. 7. It is important to set properly the integration limits when solving Eq. (5a,b) by considering either the volume of the mobile phase that has flown through the column since the sample introduction or the volume that has passed through the solute zone maximum. Both ways of derivation give correct solutions and have been reported in the earlier literature, but they should not be confused with one another [70], as the first approach yields the equation for the uncorrected elution volume, VR = VRV+Vm, whereas the second one results in the equation for the corrected elution volume, VRV. The solution of Eq. (5a,b) considering the volume of the mobile phase which has passed through the peak maximum is more simple and therefore is used in this work. 8. Another problem can arise when calculating the retention data for large molecules which are partially excluded from the pores of the packing material by size-exclusion. This effect can be corrected by using the size-exclusion volume, VSEC, instead of the hold-up volume, Vm, in the calculations of the elution times or elution volumes in gradient elution [26]; it is assumed that size exclusion does not affect the phase ratio in the column.

B. Bandwidths and Resolution in Gradient Elution LC Once the elution volume of a solute is calculated, bandwidths wg and resolution Rs in gradient elution can be determined. Exact determination of bandwidths in gradient elution chromatography necessitates calculation of the complete profile of the elution curve using numerical methods [71]. However, this approach is not practical for routine application and a simplified procedure is often used for this

18 / Jandera purpose. To first approximation, the bandwidths in gradient elution can be set equal to the isocratic bandwidths in the mobile phase of the same composition as the instantaneous composition at the column outlet at the time of elution of the band maximum. Using this assumption, the gradient bandwidths can be predicted from Eq. (8) and the resolution from Eq. (9), introducing the value of the instantaneous retention factor kf in the mobile phase with the concentration of the strong eluent (solvent B), uf, at the elution time of the band maximum. kf can be calculated from the elution volume introducing the gradient function uf = f(VR) into the appropriate equation kf = f V(uf) describing the dependence of the retention on the concentration of B in the chromatographic system [2–4,54,72]: wg ¼

4Vm ð1 þ kf Þ pffiffiffiffiffi N

ð8Þ

Rs ¼

VRð2Þ  VRð1Þ wg

ð9Þ

VR(1), VR(2) are the elution volumes of sample compounds with adjacent peaks, N is the number of theoretical plates determined under isocratic conditions, and Vm is the column hold-up volume. It should be kept in mind that the correct plate number value cannot be determined directly from a gradient-elution chromatogram, unlike the isocratic elution where the retention factors are constant. The instantaneous retention factor kf at the peak maximum decreases as the steepness of the gradient increases and is not very different for various sample solutes eluted during a gradient run. Often, the values of kf are in between 1 and 2 [4,54]. Consequently, all sample components have approximately equal bandwidths in gradient elution, which are considerably narrower than in isocratic elution, especially for late eluting compounds [1–4,21,49,73,74], and hence the sensitivity in gradient elution is higher than under isocratic conditions [4], even at increased baseline noise usual in gradient elution. This also means that the sample structure effects on the separation efficiency are generally less important in gradient than in isocratic runs, at least for small molecules. Equation (8) neglects an additional band compression in gradient elution resulting from a faster migration of the trailing edge of the band in a mobile phase with a higher elution strength, whereas the leading edge moves along the column more slowly. In exact calcu-

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lations, the bandwidths calculated using Eq. (8) can be corrected by a band compression factor, G [73,75,76]. The effect of the additional band compression in most cases results in approximately 10% reduction of the experimental wg [4,23,77]. However, other—yet not well understood—effects often contribute to additional band broadening in gradient elution and largely compensate for the gradient band compression [23,26,77–79], so that the errors caused by neglecting these effects usually are not very significant, except for very steep gradients, where the experimental bandwidths can be broader than the calculated values by as much as 20–50% [78].

III. REVERSED-PHASE CHROMATOGRAPHY WITH BINARY GRADIENTS Reversed-phase chromatography is nowadays by far the most widely used liquid chromatography mode, because it is likely to result in satisfactory separation of a great variety of samples, containing nonpolar, polar, and even ionic compounds. Gradient elution RPC is the technique of choice for separations of complex mixtures according to the hydrophobicity and (or) size of sample compounds [80,81]. Figure 6 shows an example of reversed-phase gradient elution separa-

Fig. 6 Separation of a polyethylene glycol sample PEG 1000 with 4–27 oxyethylene monomer units on an Alltima C18, 5 Am, column, 250
4.6 mm i.d., by linear gradient elution, 30–50% methanol in water in 40 min at 0.75 mL/min and 40jC. Evaporative-light scattering detector SEDEX 75 (Sedere, France), 60jC nebulizer temperature, nitrogen pressure 3.4 bar.

20 / Jandera tion of a polyethylene glycol sample (PEG 1000) according to the number of ethylene oxide units. The stationary phase in RPC— usually a nonpolar hydrocarbon chemically bonded on an inorganic support—is less polar than the mobile phase, normally an aqueous solution of one or more organic solvents. The most useful solvents for RPC are—in order of decreasing polarities—acetonitrile, methanol, dioxane, tetrahydrofuran, and propanol. The sample retention increases as its polarity decreases and as the polarity of the mobile phase increases. For successful separation of ionic, acidic, or basic substances, it is necessary to use additives to the mobile phasebuffers, neutral salts, weak acids, or compounds forming molecular associates with ionized sample solutes. By appropriate choice of the type of the organic solvent, selective polar dipole–dipole, proton– donor, or proton–acceptor interactions with analytes can be either enhanced or suppressed and the selectivity of separation adjusted. The retention is most efficiently controlled by setting the concentration(s) of the organic solvent(s) in the mobile phase. Despite widespread applications, the exact mechanism of retention in RPC is still controversial. Various theoretical models of retention for RPC were suggested such as the model using the Hildebrand solubility parameter theory [22,24,82,83], or the model supported by the concept of molecular connectivity [84], models based on the solvophobic theory [85,86] or on the molecular statistical theory [87]. Unfortunately, sophisticated theoretical RPC retention models introduce a number of physicochemical constants which are often not known or are difficult and time-consuming to determine, so that such models are not very suitable for rapid prediction of retention data. To first approximation, the interactions in the nonpolar stationary phase are less significant than the polar interactions in the mobile phase, which are the main factor controlling the retention. Hence the transition of a nonpolar or of a moderately polar solute molecule from the bulk mobile phase to the surface of the stationary phase is attributed primarily to a decrease in the contact area of the solute molecules with the strongly polar mobile phase, which results in a decrease of the energy in the chromatographic system. This solvophobic effect is the principal driving force of the retention in the absence of strong (polar) interactions of the solute with the stationary phase. In the real world, attractive interactions with the stationary phase contribute more or less to the retention. The nonpolar bonded stationary phases have properties similar to liquid alkanes, but the

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alkyl chains bonded to a solid support cannot move freely like alkane molecules in a bulk liquid phase. Further, specific polar interactions of residual silanol groups in silica-based bonded phases can contribute to the retention of polar, especially basic, solutes. Finally, organic solvents used as the mobile phase components in reversed-phase systems can be preferentially sorbed by the nonpolar bonded stationary phase and modify its properties [88,89]. The elution times in RPC are controlled by the concentration(s) of the organic solvent(s) in the mobile phase. If a relatively small entropic contribution to the retention and secondary interaction effects are neglected, various retention models such as semi-empirical model of interaction indices [90], the regular solution theory [20,24,82,83,91], or the molecular statistical theory [87] yield, with some simplification, a quadratic equation describing the effect of the concentration of organic solvent, u, in a binary aqueous–organic mobile phase on the logarithm of the retention factor of a solute, log k: log k ¼ a  mu þ du2

ð10Þ

The constants a, m, d depend on the type of the organic solvent in the mobile phase and of the solute. The quadratic term du2 in Eq. (10) explains the occasionally observed nonlinearity of log k vs. u plots, which increases with decreasing polarity of the organic solvent and with increasing size of the solute molecules, as illustrated in the experimental plots of k of alkyl-3,5-dinitrobenzoates in methanol– water and in tetrahydrofuran–water mobile phases in Figs. (7) and (8) [92]. The quadratic term in Eq. (10) usually is not very significant over a limited concentration range of methanol–water and acetonitrile– water mobile phases, where Eq. (10) reduces to the well-known and widely used semi-empirical Eq. (11), formally identical to Eq. (2) [1–5,22–25,49]: log k ¼ a  mu

ð11Þ

Equation (11) was first introduced in thin-layer chromatography by Soczewin´ski and Wachtmeister [93] to describe the dependence between RM (equivalent of log k) on the concentration of water in mixed aqueous–organic solvents for thin-layer chromatography. The constant a in Eqs. (11) and (12) increases as the polarity of the solute decreases and as its size increases and theoretically should be equal to the logarithm of the solute retention factor in pure water, kw.

22 / Jandera

Fig. 7 Dependence of the retention factors, k, of homologous n-alkyl-3,5dinitrobenzoates on the concentration, u (vol% 102), of methanol in water on a Silasorb SPH C8 (7.5 Am) column (300
4.0 mm i.d.). Sample compounds: methyl-(1)-n-hexyl (6) esters. Points—experimental data, lines— best fit linear regression plots of Eq. (11).

Fig. 8 Dependence of the retention factors, k, of homologous n-alkyl-3,5dinitrobenzoates on the concentration, u (vol% 102), of tetrahydrofuran in water on a Silasorb SPH C8 (7.5 Am) column (300
4.0 mm i.d.). Sample compounds: methyl-(1)-n-hexyl (6) esters. Points—experimental data, lines— best fit non-linear regression plots of Eq. (10).

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However, the values of log k extrapolated to u = 0 from various experimental plots do not describe accurately the real solute retention in water [94,95], probably because of the preferential adsorption of the organic solvent on the surface of the nonpolar stationary phase [96]. The constant m increases with decreasing polarity of the organic solvent B and is a measure of its elution strength (corresponding to the parameter S in the Snyder model of linear solvent strength gradients [4]). m also increases with increasing size of the molecule of the analyte [26,97]. Other models based on the combination of adsorption and partition mechanism in RPC result in more complex equations for the retention factors [98–100], which are, however, less suitable for prediction of retention and optimization of separation. In RPC systems described by Eq. (11), the approach outlined in Section II was employed for the derivation of the equations for elution volumes VR and bandwidths wg using linear gradients of organic solvents in water [2–4,23,24,54,77]. The equations were published in various forms, which can all be formally rearranged to Eqs. (12) and (13): h i 1 log 2:31 mBVm 10ðamAÞ þ 1 þ Vm mB   4Vm 1 wg ¼ pffiffiffiffiffi 1 þ 2:31 mBVm þ 10ðmAaÞ N

VR ¼

ð12Þ ð13Þ

with the constants a and m of Eq. (11). A is the initial concentration and B is the steepness of the gradient, N is the column plate number (under isocratic conditions) and Vm is the column hold-up volume. Equation (12) describes adequately the retention in a variety of reversed-phase gradient-elution separations (see, e.g., Refs. [3–5,19– 27,38–41,46–57]). However, using Eq. (12) with the parameters a and m determined experimentally in a range of binary mobile phase composition narrower than the actually used gradient concentration range may cause significant errors in the calculated elution volumes of late eluted compounds. This should be taken into account when planning the experiments for the determination of the constants of Eq. (11). Equation (11) can often describe the effect of the concentration of organic solvents on the retention in micellar LC [101] and in ion-pair or salting-out RPC occasionally employed for the separation of

24 / Jandera organic acids or bases [102–104] such as isomeric naphthalene monoto tetrasulphonic acids (Fig. 9). Hence Eqs. (12) and (13) can also be used principally for the calculation of the retention data in the elution with organic solvent gradients in these techniques. Changing concentration of the organic solvent during such a gradient affects the distribution equilibrium of the ion-pairing reagent between the stationary and the mobile phase [105], which impairs the accuracy of the calculated elution data in gradient-elution ion-pair chromatography. However, for carefully designed gradients, short column reequilibration times in between gradient runs and, consequently, predictable and reproducible retention data can be obtained [106]. Reversedphase behavior is also observed in HPLC on silica gel dynamically modified by adsorption of a long-alkyl surfactant, where the increasing concentration of the organic solvent during gradient elution speeds up the elution not only by increasing the elution strength of

Fig. 9 Separation of 12 naphthalene sulfonic acids by gradient-elution RPC on a Separon SGX RPS column, 7 Am (250
4 mm i.d.). Solvent program: 5 min isocratic, 0.4 mol/L Na2SO4 at 0.5 mL/min, followed by linear gradient from 0.4 mol/L Na2SO4 to 40% (v/v) methanol in water in 15 min at 1 mL/ min. Detection: UV, 230 nm; column temperature 40jC. naphthalene sulfonic acids: 1,3,5,7-tetra (1), 1,3,6-tri (2), 1,3,5-tri (3), 1,3,7-tri (4), 1,5-di (5), 2,6-di (6), 1,6-di (7), 2,7-di (8), 1,3-di (9), 1,7-di (10), 1-(11), 2-(12), unidentified less polar impurities (X).

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the mobile phase, but also by simultaneously decreasing the amount of the adsorbed surfactant stationary phase [107]. From Eqs. (12) and (13) it follows that a lower parameter B (a less steep gradient) is required to compensate for a higher parameter m to obtain comparable retention volumes. This is important especially for compounds with higher molecular weights, as m usually increases with increasing size of the molecules [26,97,108] and has the following practical consequences: 1) Shallow gradients are frequently required for separations of oligomers or polymers, so that appropriate selection of a suitable combination of the gradient parameters A and B is more critical here than for the separation of small molecules; 2) for the separation of samples with a broad range of molecular masses a flatter gradient at the end of the chromatogram than at its start provides more regular band spacing than linear gradients and a convex gradient can be more useful than a linear gradient [109], see, e.g., the example in Fig. 4A. For very large molecules m can be so great that a very small change in the concentration of the organic solvent, u, may increase the retention even by several orders of magnitude, causing an abrupt change from ‘‘full retention’’ to ‘‘complete elution’’ [34]. Hence, isocratic fractionation of large molecules is more difficult than their separation using gradient elution (if possible at all). This is why gradient elution with acetonitrile in aqueous buffers at a low pH is normally required for separating peptide and protein samples in RPC [1]. For reversed-phase systems where the retention is controlled by the quadratic Eq. (10), the equation for the elution volume (time) is not obtained in the analytical form with separated variables and should be solved by numerical methods to calculate the retention data [2,3,83]. However, linear approximation of the experimental retention data using Eq. (11) usually does not affect significantly the agreement between the calculated and the experimental gradientelution retention data [110].

IV. NORMAL-PHASE CHROMATOGRAPHY WITH BINARY GRADIENTS Normal-phase chromatography (NPC) is the oldest liquid chromatographic mode. The column packings are either inorganic adsorbents (silica or, less often, alumina) or moderately polar bonded phases,

26 / Jandera most often cyanopropyl –(CH2)3–CN, diol –(CH2)3–O–CH2–CHOH– CH2–OH, or aminopropyl –(CH2)3–NH2, chemically bonded on a silica gel support. Many new chemically bonded polar phases, which can be used either in RP or in NP systems, have been introduced recently [111]. As the retention on inorganic adsorbents originates in the interactions of the polar adsorption centers on the solid surface with polar functional groups of the analytes, this mode was previously called also as adsorption or liquid–solid chromatography (LSC). The mobile phase is usually a mixture of two or more organic solvents of different polarities, such as hexane and propanol or hexane and dichloromethane. The first model of retention in adsorption chromatography developed by Snyder [112,113] is based on the assumption offlat adsorption in a monomolecular layer on a homogeneous adsorption surface. The retention in NPC results from the competition between the molecules of the solute and of the solvent for the adsorption sites on the adsorbent surface. The interactions in the mobile phase are less significant and can be neglected to first approximation. Later, corrections were introduced for preferential adsorption on localized adsorption centers [114–116]. Soczewinski [117,118] developed a similar displacement model of retention assuming the formation of association complexes of the sample solutes and of the solvents on the adsorption centers. This model was further elaborated by Jaroniec et al. [119]. Another adsorption model considering the retention as the result of probability and strength of interactions between the solutes and the adsorbent was suggested by Scott and Kucera [120,121]. The displacement and the interaction adsorption models were compared by Snyder and Poppe [122]. Martire and Boehm [123] introduced another adsorption model based on molecular statistical– mechanical theory of adsorption chromatography. Regardless of the exact retention mechanism, the stationary phase in normal-phase chromatography is more polar than the mobile phase. The sample retention is enhanced as the polarity of the stationary phase increases and as the polarity of the mobile phase decreases, opposite to the behavior observed in RPC. The retention also increases with increasing polarity and number of adsorption sites in the column. This means that the retention is stronger on the adsorbents with a larger specific surface area and that the strength of interactions with analytes generally increases in the order: cyanopropyl < diol < aminopropyl b silica gel c alumina stationary

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phases. However, strong selective interactions may change this order. Basic solutes are strongly retained by the acidic silanol groups of silica gel whereas acidic compounds show increased affinities to chemically bonded aminopropyl stationary phases. Aminopropyl and diol bonded phases prefer compounds with proton–acceptor or proton–donor functional groups (alcohols, esters, ethers, ketones, etc.), whereas other polar compounds are usually more strongly retained on cyanopropyl silica than on aminopropyl silica. Alumina favors interactions with k electrons and often yields better selectivity for the separation of compounds with different numbers or spacing of unsaturated (double) bonds than silica. The polarity and the elution strength, i.e., the ability to enhance the elution, generally increase in the following order of the most common NPC solvents: hexanecheptanecoctane<methylene chloride<methyl-t-butyl ether<ethyl acetate
28 / Jandera 3. Some samples are more soluble or less likely to decompose in organic mobile phases. 4. Normal-phase chromatography is often useful for the separation of hydrophobic samples very strongly retained in RPC. 5. The adsorption sites usually occupy fixed positions on the surface of a polar adsorbent. If the localization of the adsorption sites fits to a specific configuration of functional groups in a solute molecule with multiple functional groups, simultaneous interactions of two or more functional groups with the adsorbent are possible, which are weaker or absent for molecules with other positions of functional groups. Differences in the retention of molecules of similar polarities, but different shapes (rigid planar, rod-like or of a flexible chain structure) are often observed and can be utilized in NPC like in RPC. Hence NPC on silica gel is suitable for the separation of various positional isomers or stereoisomers of moderately polar compounds. 6. If sample pretreatment procedures involve the extraction into a nonpolar solvent, direct injection on to a RPC column may cause problems, in contrast to NPC. 7. Reversed-phase chromatography generally offers better selectivity for the separation of molecules with different sizes of their hydrocarbon part. On the other hand, gradient elution NPC is often better suited for the separation of oligomeric samples with polar repeat monomer units such as oligoethylene glycol alkylphenyl ether surfactants (Fig. 10) [124]. Normal-phase chromatography gradient elution can be applied also for the separation of samples of polymers or oligomers with bimodal distribution of polar and nonpolar repeat units. For example, using a column with bonded aminopropyl phase and a gradient of acetonitrile in dichloromethane, the separation of oligoethylene alkyl ethers with different numbers of polar oxyethylene units is enhanced whereas the separation according to the alkyl distribution (C12–C18) is suppressed (Fig. 11) [125]. Chromatography on polar adsorbents suffers from a specific inconvenience—significant preferential adsorption of more polar solvents, especially water, which may be connected with long equilibration times if the separation conditions are changed [126]. Hence the control of retention in NPC by adjusting the mobile phase composition can be less reproducible and less predictable than in RPC. These effects may become especially important in gradient elution where the composition of the mobile phase changes during the separation and are the reason for the strong bias against using

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Fig. 10 Normal-phase gradient elution separation of an oligoethylene glycol nonylphenyl ether sample (Serdox NNP4) with 1–13 oxyethylene units. Column: Separon SGX Amine, 5 Am, (200
4 mm i.d.). Linear gradient 0–45% 2-propanol in heptane in 30 min, 1 mL/min, UV detection, 230 nm.

gradient elution in NPC. However, the column reequilibration times after the end of the gradient can be short if dry mobile phases and nonlocalizing polar solvents B are used, such as dichloromethane, dioxane, or tert-butyl methyl ether [69,127]. The adsorption of even trace amounts of water—much more polar and hence more strongly retained than any nonionic organic solvent—can considerably decrease the adsorbent activity, which must be kept constant to obtain reproducible results. To this aim, approaches were recommended relying either on mobile phases prepared from ‘‘isohydric’’ organic solvents with equilibrium water concentrations [128] or on a ‘‘constant moisture system’’ with a fixed volume of solvent circulating in a closed loop through the separation column, the detector, and a large regenerating column [129]. Unfortunately, these procedures are not feasible in connection with gradient elution. The reproducibility of retention in NPC over a long period of column use can be significantly improved by using dehydrated solvents kept dry over activated molecular sieves and filtered just before the use and by accurate temperature control to F0.1jC during the separation [68,130].

30 / Jandera

Fig. 11 Normal-phase gradient elution separation of oligoethylene glycol dodecyl-, tetradecyl-, hexadecyl-, and octadecylethers with 2–14 oxyethylene units. Column: Separon SGX Amine, 7 Am, 150
3.3 mm i.d. Linear gradient from 40% to 90% acetonitrile in dichloromethane in 10 min, followed by isocratic step with 90% acetonitrile for 5 min, 1 mL/min, 40jC, APCI+ mass spectrometric detection, RIC chromatograms of [M+H+] ions. Time in minutes.

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The separation selectivity and retention in normal-phase systems is most often adjusted by selecting an appropriate composition of a two- or a multi-component mobile phase. The dependence of the retention on the composition of the mobile phase can be described using theoretical models of adsorption. With some simplification, both the Snyder–Soczewinski and the Martire models lead to identical equation describing the retention factor, k, as a function of the mole fraction of the stronger (more polar) solvent, x, in binary mobile phases composed of two solvents of different polarities [2,3,22,99,113, 116–119]: k ¼ k0 xmV

ð14Þ

k0 and mV in Eq. (14) are experimental constants, k0 being the retention factor in pure strong (more polar) solvent. The parameter mV is the stoichiometric coefficient in the displacement model of adsorption, i.e., the number of molecules of the strong solvent B necessary to displace one adsorbed molecule of the analyte. It was shown that the mole fraction x in Eq. (14) can be substituted by the volume fraction, u, of the polar solvent, to obtain empirical Eq. (15), without significant loss of the fit of the experimental data [22,131– 137]. k ¼ k0 um

ð15Þ

The constants k0 and m have similar meaning as the constants in Eq. (14). Equation (15) can be applied in NPC systems where the solute retention is very high in the pure nonpolar solvent. If this is not the case, another retention equation can be used which was derived from the original Snyder model with less simplification than adopted for derivation of Eqs. (14) and (15) [2,133]: k ¼ ða þ buÞm

ð16Þ

Here, a, b, and m are experimental constants depending on the solute and on the chromatographic system [a=1/(kA)m, where kA is the retention factor in pure nonpolar (weak) solvent A]. The suitability of Eqs. (15) and (16) to describe experimental NPC data was compared experimentally [68,130]. Equation (16) usually only slightly improves the description of the experimental data with respect to Eq. (15). Equation (16) with the exponent m = 1 can be derived from the

32 / Jandera competitive Langmuir isotherm [138] and from the Scott–Kucera model of adsorption [120,121]. Figure 12 shows the results of fitting Eqs. (15) and (16) to the experimental k of four pesticides on a bonded nitrile phase column in mobile phases containing various concentrations, u, of 2-propanol in heptane. The agreement with the experimental data is slightly improved when the three-parameter Eq. (16) is used, but in many cases the errors in the predicted retention data caused by using the two-parameter Eq. (15) are not very significant. In the course of gradient-elution chromatography in a normalphase system the concentration of one or more polar solvents in a nonpolar solvent is increased. To resolve mixtures of analytes with large polarity range (such as samples containing fatty acid esters, sterols, and sugars), ‘‘incremental gradient elution’’ or ‘‘relay gradient elution’’ was introduced by Scott and Kucera [121] and more recently advocated by Treiber [139]. This technique employs a series of consecutive linear binary gradient steps using the solvent B from the previous step as the solvent A in the next step, e.g., a three-

Fig. 12 Dependence of retention factors, k, of phenylurea herbicides on the concentration, u (vol% 102), of 2-propanol in heptane on a Separon SGX Nitrile, 7 Am, column (150
3.3 mm i.d.) at 40jC. Solutes: 1) phenuron, 2) bis-N,N V-(3-chloro-4-methyl) phenylurea, 3) neburon, 4) metobromuron. Points—experimental data, full lines—best fit plots of three-parameter Eq. (16), dashed lines—best fit plots of two-parameter Eq. (15).

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segment gradient elution starting from hexane to ethyl acetate, continuing from ethyl acetate to acetonitrile and finishing from acetonitrile to water. In this case, preconditioning of the column for the next run should be performed by sequential washing with several solvents in the order of decreasing polarities, which is not very practical. Hence a single-step gradient elution technique with two miscible solvents such as hexane and 2-propanol should be preferred, if allowed by the polarity range of sample compounds. However, relay gradient elution can provide superior resolution for some complex mixtures. Chromatography of polar compounds on silica gel and on other polar stationary phases in mobile phases containing water [140–142] can yield improved separation for some strongly polar samples, but here the retention mechanism is based rather on liquid–liquid partition between the bulk mobile phase and the adsorbed liquid layer and is known as ‘‘hydrophilic interaction chromatography,’’ HILIC [143]. With the chromatographic systems described suitably by Eq. (15), the approach described in Section II yields Eq. (17) for the elution volumes VR in normal-phase chromatography with linear gradients [2,54]: i 1 1h A ðmþ1Þ ðm þ 1ÞBk0 Vm þ Aðmþ1Þ VR ¼  þ Vm ð17Þ B B In normal-phase systems where the three-parameter Eq. (16) should be used to describe the isocratic retention, the elution volumes can be calculated using Eq. (18) [2,68,69]: o 1 1 n a þ Ab ðmþ1Þ ðmþ1ÞbBVm þ ½a þ Ab ðmþ1Þ þ Vm ð18Þ VR ¼  bB bB If the elution volumes in gradient elution are described by Eq. (17), the solution of Eq. (8) yields Eq. (19) for the solute bandwidths in normal phase gradient elution chromatography [2,54,126,144]:  m h i mþ1 4Vm wg ¼ pffiffiffiffiffi 1 þ k0 ðm þ 1ÞBk0 Vm þ Aðmþ1Þ ð19Þ N For normal-phase systems where the three-parameter Eq. (16) applies, the bandwidths should be calculated using Eq. (20) [68,69]:  m h i mþ1 4Vm ðmþ1Þ wg ¼ pffiffiffiffiffi 1 þ ðm þ 1ÞBbVm þ ða þ AbÞ ð20Þ N

34 / Jandera Equations (17) and (19) or Eqs. (18) and (20) can be used for the calculation of resolution and for the optimization of normal-phase gradient elution, as shown elsewhere [5,143]. As illustrated in Fig. 5, nonlinear gradients can often provide better band spacing and higher peak capacity than linear concentration gradients in normal-phase chromatography. For nonlinear normal-phase gradients controlled by Eq. (7) and the retention factors described by Eq. (15), the following equations for elution volumes and bandwidths were derived [2,54,145]: 1 i 1 ðj mþ1Þ ðj mþ1Þ 1h Að j Þ j VR ¼ þ Vm ðj m þ 1ÞBk0 Vm þ A  B B  h i jm ðj mþ1Þ  j mþ1 4Vm j wg ¼ pffiffiffiffiffi 1 þ k0 ðj m þ 1ÞBk0 Vm þ A N

ð21Þ ð22Þ

Reproducibility of gradient-elution retention data in normal phase systems depends on a number of experimental factors and can be significantly improved by working at a constant temperature and keeping a constant adsorbent activity by controlling the water content in the mobile phase, preferably by using dehydrated solvents kept dry over activated molecular sieves and filtered just before the use [68,130]. With these precautions, the differences between the elution volumes measured in repeated experiments on a silica gel column were lower than 0.2 mL or 2% for over 10 months of the column use. The differences between the experimental elution volumes and the data calculated from Eq. (18) based on the threeparameter retention Eq. (16) were lower than 0.25 mL for the gradients starting at nonzero concentrations of the polar solvent B [68]. Decreased column efficiency and peak asymmetry occasionally observed in dry mobile phases [140] are less significant with sol–geltype silica gel materials [127]. The reproducibility and the accuracy of the prediction of retention in normal-phase gradient-elution chromatography and the ways to avoid or to compensate possible sources of errors are discussed in Section VI.

V. ION-EXCHANGE GRADIENT ELUTION CHROMATOGRAPHY Ion exchange chromatography (IEC) is presently used more often for separations of small inorganic ions or of ionic biopolymers such as

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oligo-nucleotides, nucleic acids, peptides, and proteins than in the analysis of ionic small-molecule compounds, for which RPC or ionpair chromatography usually offers higher efficiency and better control of selectivity and resolution. Columns used in IEC are packed with fine particles of ion exchangers, which contain charged functional ion-exchange groups covalently attached to a solid matrix. The solid matrix can be either organic, such as, e.g., cross-linked styrene–divinylbenzene or ethyleneglycol–methacrylate copolymers, or inorganic—most frequently silica gel support to which a functional group is chemically bonded via a spacer—propyl or phenylpropyl moiety. The functional groups carry either a positive charge (anion exchangers) or a negative charge (cation exchangers) and retain ions with opposite charges by strong electrostatic interactions. The separation based on ion exchange requires mobile phases containing counterions (salts, buffers, ionized acids, or bases) with charges opposite to the ion-exchange functional groups. The retention in IEC increases with increasing ion-exchange capacity of the column and with increasing affinity of the ionic solutes to the matrix of the ion exchanger, whereas it decreases with increasing concentration of the counterion in the mobile phase. The decrease in retention with increasing ionic strength is enhanced for ionic solutes with a higher charge and for less charged counterions [1–3]. The retention in ion-exchange chromatography is based on the competition between the sample ions and the counterions for the ionexchange groups. This competition can be described using the stoichiometric displacement retention model of ion-exchange chromatography, which is formally similar to the competition/displacement model of normal-phase chromatography. Hence Eq. (15)—although reported in various forms—can be used in many ion-exchange systems to describe the effect of the molar concentration of the electrolyte, u, which is similar to the effect of the concentration of the polar solvent B in NPC [2,3], as it has been verified by numerous workers (see, e.g., Refs. [133,146–151] and the literature cited herein). In this case, the parameter m is the stoichiometric coefficient of the ionexchange reaction between the sample ions and the ions of the electrolyte in the mobile phase. The parameter k0 is the retention factor in the mobile phase with 1 M electrolyte. This means that the elution volumes in ion-exchange chromatography with linear gradients of the concentration of a salt or of a buffer can be calculated

36 / Jandera using Eq. (17), like in NPC systems [2,22,149], and Eq. (19) can be used for the calculation of bandwidths. Of course, this approach can be applied only to compounds whose degree of ionization does not change during gradient elution. For the separation of weak acids and bases with pH gradients, different retention equations apply. For gradients controlled by the nonlinear gradient function described by Eq. (7), the elution volumes can be calculated using Eq. (21) and bandwidths using Eq. (22). The validity of these equations was verified for ion-exchange separations of polyphosphates [152,153], oligonucleotides [154,155], peptides, and other compounds [149]. Equation (21) has been successfully applied to predict isocratic retention from two gradient-elution runs in ion-exchange chromatography [156]. Recently, so-called ‘‘slab’’ model of IEC was claimed to provide better description of the retention of proteins than the stoichiometric model. The model is based on the assumption of the control of the retention by the electrostatic forces between two planar, charged surfaces with evenly distributed surface charges of opposite sign (the surfaces of protein molecule and of the ion exchanger), in an electrolyte solution. The model predicts linear dependence of the sample retention factors and the reciprocal square root of the ionic strength of the mobile phase [157,158]. Using these assumptions, equations were derived for the prediction of the retention of proteins in gradient elution with linear ionic strength gradients [159].

VI. EFFECTS OF THE INSTRUMENTATION AND OF THE NON-IDEAL RETENTION BEHAVIOR ON THE RETENTION IN GRADIENT ELUTION In practice, nonideal behavior often affects the transport of the mobile phase gradient in the column and can result in poor reproducibility and more or less significant deviations between the experimental and calculated gradient-elution data, which can cause problems in method development, transfer, and optimization. One possible source of nonideal behavior is the instrumentation; others can be attributed to various secondary effects that complicate the distribution of sample compounds in the phase system during gradient elution. The gradient profile can change during the transport along the column because of preferential adsorption of one or more components on the column

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[69,160]. Increasing concentration of organic solvents in buffered mobile phases in reversed-phase systems can cause shifts in pH and, consequently, the degree of dissociation of weak acids or bases may change during gradient elution. Because the dissociated forms are generally less retained than the corresponding nonionized species, the change in pH induced by the gradient affects the retention times. Empirical correction factors were introduced to account for this effect, which increases for compounds with longer retention times [161].

A. Effect of the Dwell Volume on the Retention in Gradient Elution LC. Retention Data in Gradient Elution with an Initial Hold-Up Period. Gradient Preelution and Postelution The instrumental difficulties in gradient elution can originate from improper mixing ratios of the gradient components caused by failure or poor design of the pumps or of the gradient mixer. Even properly designed and correctly operating instruments have connecting tubing, mixers, and other parts whose void volumes cause rounding of the gradient profile in the initial and final parts [66] and delay the gradient with respect to the sample injection. This effect is characterized by the gradient dwell volume, VD [26,64–66]. The rounding of the gradient can reduce the retention times of the bands eluting near the start of the gradient and increase the retention times of bands eluting near the end of the gradient [64,65]. It is usually more significant in the instruments with larger volumes between the gradient mixer and the column inlet (gradient dwell volume). The most critical instrumental parts in low-pressure gradient devices are the high-pressure pump delivering the gradient mixed at its inlet to the column; in high-pressure gradient devices the gradient mixer is often inserted between the pump and the sample injector to ensure good mixing of the gradient components. Both the quality of solvent mixing and the rounding of the gradient depend on the nature of the solvents used as the gradient components and on the flow rate of the mobile phase [3,4,66]. Imperfect mixing of the mobile phase components can affect very significantly the chromatograms of polymer samples, where a small change in the mobile phase composition results in a large increase or decrease in the retention. Even small concentration pulses of the solvent B can result in alternating periods of strong and weak

38 / Jandera retention of the sample, so that pulsing sample zones move along the column until a chromatogram is obtained with a series of artefact peaks which, however, all have the same composition. Such chromatograms simulate a false separation and can be easily misinterpreted, as the artefact peak resolution can be much larger than expected on the basis of the baseline fluctuation. This is illustrated by an experimental chromatogram (Fig. 13) of a polystyrene standard in NPC gradient elution obtained in a gradient chromatograph without a gradient mixer between the pump and the sample injector, which eliminates this effect (compare with Fig. 14). A mixer also suppresses baseline fluctuations caused by imperfect mixing of the mobile phase components and improves thus the detection limits. On the other hand, it impairs gradient rounding and increases the gradient dwell volume. The gradient delay, i.e., the time necessary for the transport of the gradient from the mixing point to the column inlet is caused by the gradient dwell volume, VD. The dwell volumes in different types of instruments can differ significantly from one another, which can give rise to undesirable changes in the separation when a gradient method is transferred from one chromatograph to another [1,3,4,67]. The gradient delay can be eliminated by delaying the injection with respect to the start of the gradient, but the delayed injection tech-

Fig. 13 Effect of imperfect mixing of mobile phase components on artefact peaks in gradient elution chromatography of polymers. Sample: polystyrene standard (Mr = 470,000). Two Nova-Pak silica columns in series, 4 Am (3.9
150 mm i.d. each), gradient 47–50% dioxane in hexane in 15 min, 1 mL/min, 40jC, UV detection at 254 nm. Low-pressure gradient pump without a mixer.

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Fig. 14 Normal-phase gradient elution separation of polystyrene standard samples with Mr = 35,000, 110,000, and 470,000, respectively. Two NovaPak silica gel columns in series, 4 Am (3.9
150 mm i.d. each), gradient 47– 50% dioxane in hexane in 15 min, 1 mL/min, 40jC, UV detection at 254 nm. A column (100
4.6 mm i.d.) packed with stainless-steel beads was used as a mixer at the outlet from a low-pressure gradient pump.

nique is not compatible with automated sequence operation of some instruments. The gradient dwell volume can be determined experimentally by running ‘‘blank’’ gradients using an unretained UV-absorbing compound dissolved in solvent A as the solvent B, with disconnected and with connected separation column. In these test experiments, the extent of gradient rounding and possible deviations from the preset gradient profile is also revealed, and hence the blank gradient tests are recommended to check the performance of the gradient liquid chromatograph [1,3,62,66]. Equations (12), (13), and (17)–(22) can be used if the gradient dwell volume, VD, is low enough and can be neglected, or if the injection is delayed with respect to the start of the gradient. Unfortunately, this is often not the case and with some instruments the gradient dwell volume can be quite significant, even a few milliliters. At the start of the gradient, the dwell volume in the instrument is filled with the mobile phase of the composition corresponding to the initial gradient conditions, which should first flow through the column

40 / Jandera before the front of the gradient profile arrives at the top of the column. This behavior should be taken into account when predicting the gradient elution volumes. If the retention factors in the initial mobile phase, k1, are high enough, the sample zone practically does not move during the gradient delay time and the retention volumes can be simply corrected by the addition of the instrumental gradient dwell volume [3,4]. The correction of the retention times should consider the actual flow rate of the mobile phase. This simple correction yields accurate gradient elution volumes if the retention factors in the dwell volume mobile phase, k1, are 1000 or more, i.e., for gradients starting at low concentration A of the strong solvent B. However, if low starting concentration A is used, both the separation time and the reequilibration time after the end of the gradient may significantly increase, hence optimized gradients should start at as high an initial concentration A as allowed by the desired resolution of weakly retained compounds. As A increases and k1 decreases, simple addition of the gradient dwell volume to the calculated gradient elution volumes may cause unacceptable errors, especially with short and narrow columns. The reason for these discrepancies is that weakly retained solutes can move a significant distance along the column before the start of the gradient at a low k1 and their elution is delayed less than it would correspond to the full dwell volume. This retention behavior can be understood as two-step elution with the first, unintended isocratic step corresponding to the dwell volume, followed by the second, gradient, step. To avoid overestimated calculated gradient elution data, appropriate correction should be adopted by using the same calculation approach as for the two-step elution with the programmed initial isocratic hold-up period followed by the second, gradient, step [2,3,54,64,68,69,144]. The correction of the calculated gradient retention data for the migration of sample compounds along the column prior to the gradient in the gradient dwell volume is based on the following consideration: At the time when a sample compound is taken over by the front of the gradient, it had already migrated a part of the column hold-up volume, Vm1, at the initial isocratic conditions, so that only a part of the column hold-up volume, Vm2, remains available for its migration during the actual gradient elution: Vm2 = Vm  Vm1. Vm1 is related to Vm in the same proportion as the gradient dwell volume VD to the (hypothetical) net elution volume of the solute under initial

Gradient Elution in LC Chromatography

/

41

isocratic conditions where the retention factor of the solute is k1. Hence the gradient part of the hold-up volume available for each sample compound, Vm2, is: Vm1 ¼

VD ; k1

Vm2 ¼ Vm 

VD k1

ð23Þ

The gradient volume can be calculated in the same way as in two-step gradient elution with an initial hold-up period, i.e., the final gradient elution volume is composed of the contributions of the gradient step to the net retention volume, VR2V, which can be calculated from Eq. (12), (17) or (18) using Vm2 instead of Vm, and of the initial isocratic step contribution of the gradient dwell volume, VR1V=VD: VR ¼ VR1 V þ VR2 V þ Vm ¼ VD þ VR2 V þ Vm

ð24Þ

Consideration of the initial isocratic step results in slight modification of Eq. (12) to Eq. (25) for reversed-phase gradient elution: VR ¼

n h i o 1 log 2:31 mB Vm 10ðamAÞ  VD þ 1 þ VD þ Vm ð25Þ mB

and of Eq. (17) to Eq. (26) or of Eq. (18) to Eq. (27) for normal-phase or ion-exchange gradient elution: i 1 1h ðmþ1Þ ðm þ 1ÞBðk0 Vm  VD Am Þ þ Aðmþ1Þ B A  þ Vm B 1 n VR ¼ VD þ ðm þ 1ÞbB½Vm  VD ða þ bAÞm bB 1 o ðmþ1 a þ Ab Þ þ Vm þ ða þ AbÞðmþ1Þ  bB VR ¼ VD þ

ð26Þ

ð27Þ

The effect of the delayed migration of the gradient along the column on the retention times or volumes is more significant for larger instrumental gradient dwell volumes, lower retention factors k1 and lower column hold-up volumes, Vm, i.e., it increases with decreasing length and diameter of the HPLC column. The example in Table 1 shows that for compounds with k1 = 100 the retention volumes corrected by simple addition of VD are approximately at 4% VD greater than the accurate VR values. This corre-

42 / Jandera Table 1 Effect of the Gradient Dwell Volume on the Differences Between the Retention Volumes, VR,COR, Calculated from Eq. (25) Respecting the Band Migration in the Gradient Dwell Volume and Uncorrected Retention Volumes, VR,UNCOR, Calculated from Eq. (12)Columns C18, porosities 75%, test solute neburon, constants a = 4, m = 4 in Eq. (11). k1—retention factor in the dwell-volume step with the concentration A of methanol. Gradients from A to 100% in 20 min. % MeOH (v.), A
100 k1 Column (mm)

VG (mL)

VD = 0.1 mL 200
4 100
4 50
4, 200
2 100
2 50
2, 200
1 100
1 50
1

40 20 10 5 2.5 1.25 0.625

VD = 0.5 mL 200
4 100
4 50
4, 200
2 100
2 50
2, 200
1 100
1 50
1

40 20 10 5 2.5 1.25 0.625

VD = 1.0 mL 200
4 100
4 50
4, 200
2 100
2 50
2, 200
1 100
1 50
1 VD = 5.0 mL 200
4 100
4 50
4, 200
2 100
2 50
2, 200
1 100
1 a

50 100

57.5 50

67.5 20

75 10

82.5 5

92.5 2

Fm (mL/min)

Vm (mL)

2 1 0.5 0.25 0.125 0.062 0.031

1.88 0.94 0.47 0.235 0.118 0.059 0.029

0.096 0.096 0.096 0.096 0.096 0.096 0.095

0.090 0.090 0.090 0.090 0.090 0.090 0.090

0.074 0.074 0.074 0.074 0.073 0.073 0.072

0.052 0.052 0.052 0.051 0.051 0.050 0.047

0.027 0.027 0.027 0.026 0.026 0.024

0.006 0.006 0.006 0.005 0.005 0.003

a

a

2 1 0.5 0.25 0.125 0.062 0.031

1.88 0.94 0.47 0.235 0.118 0.059 0.029

0.478 0.478 0.478 0.478 0.478 0.477 0.476

0.451 0.451 0.450 0.450 0.449 0.447 0.441

0.368 0.367 0.366 0.363 0.358 0.343 0.290

0.258 0.256 0.253 0.245 0.229 0.183

0.134 0.131 0.125 0.113 0.086

0.027 0.025 0.021

a

a

a

a

a

40 20 10 5 2.5 1.25 0.625

2 1 0.5 0.25 0.125 0.062 0.031

1.88 0.94 0.47 0.235 0.118 0.059 0.029

0.956 0.956 0.955 0.955 0.954 0.952 0.946

0.901 0.901 0.900 0.898 0.894 0.883 0.845

0.735 0.732 0.727 0.715 0.686 0.589

0.512 0.505 0.490 0.456 0.367

0.262 0.251 0.227 0.172

0.050 0.042

a

a

a

a

a

a

a

a

a

40 20 10 5 2.5 1.25

2 1 0.5 0.25 0.125 0.062

1.88 0.94 0.47 0.235 0.118 0.059

4.776 4.773 4.767 4.753 4.718 4.549

4.497 4.484 4.455 4.381 4.075

3.619 3.539 3.333

2.411 2.184

1.070

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

(VR,CORVR,UNCOR), mL

—Pre-elution in the isocratic step before the start of the gradient.

a a

a a

Gradient Elution in LC Chromatography

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43

sponds to a negligible error of predicted elution volumes of 0.05 mL or less for the instruments with the dwell volume VD < 1 mL and slightly higher than 0.2 mL for the large-volume mixers with the dwell volume of 5 mL. However, the errors increase significantly for compounds with low initial retention factors, k1, especially when columns with small hold-up volumes are used. The contributions of the dwell volume to the gradient elution volumes decrease at low k1—see the differences between uncorrected gradient elution volumes calculated from Eq. (12) and the data corrected using Eq. (25) in Table 1. For compounds with low k1 and a large gradient dwell volume with respect to the column hold-up volume, Eq. (25) can yield even lower elution volumes than the uncorrected VR calculated from Eq. (12). Such negative errors mean that some weakly retained solutes may elute under isocratic conditions in the dwell volume mobile phase, before the front of the gradient migrates along the full length of the column. This behavior–gradient preelution is marked by the asterisks in the Table 1. As illustrated in Figs. 15 and 16 comparing the uncorrected elution volumes, the elution volumes corrected by simple addition of VD, and the elution volumes calculated using Eq. (25), the effect of the gradient dwell volume is more important for separations on short columns and especially on narrow-bore columns with i.d.V2 mm. This precludes using standard equipment with relatively large gradient dwell volumes for gradient elution in connection with capillary HPLC columns, unless a large splitting ratio of the mobile phase flow to the column is used. The gradient preelution is illustrated by an example in Fig. 17, showing a separation of a simple mixture of triazine herbicides. Increasing ratio of the gradient dwell volume to the column hold-up volume causes preelution of one compound in the isocratic dwellvolume step from a short monolith Chromolith column (Vm = 0.7 mL), but of four compounds from a superficially porous Poroshell column (Vm = 0.26 mL), whereas no compound is eluted in the dwell-volume step from a conventional column (Vm = 0.9 mL). The same approach as in RPC should be used in other HPLC gradient elution modes. The correction of the elution volumes for band migration corresponding to the gradient dwell volume calculated from Eq. (27) improved the average error of predicted retention volumes in normal-phase gradient elution starting at a nonzero concentration of propanol to 1.3% for the silica gel column and to 2.4% for the bonded nitrile column, in contrast to 12% error of

44 / Jandera

Fig. 15 Effect of the gradient dwell volume, VD, on the elution volume, VR, in reversed-phase chromatography on a conventional analytical C18 column with the hold-up volume Vm = 1 mL. Solute: neburon, retention equation Eq. (11) with parameters a = 4, m = 4. Linear gradients, 2.215% methanol/min at 1 mL/min: a) from 57.5% methanol in water in 20 min (k1 = 50); b) from 75% to 100% methanol in water in 11.75 min (k1 = 10). VR uncorrected— calculated from Eq. (12), VR + VD  VD added to VR uncorrected, VR corrected—calculated from Eq. (25).

prediction for uncorrected data under common NP gradient-elution conditions [69]. The separation of samples containing weakly retained compounds can be often improved if an intentional isocratic hold-up period is used before the start of the gradient. The composition of the mobile phase in the hold-up period determines the starting gradient concentration, A, in the second, gradient step and depends on the number and on the relative retention of weakly retained sample compounds, like the optimum duration of this period. The retention data of the compounds eluted in the gradient step can be calculated from Eqs. (25)–(27) using the sum of the intended hold-up

Gradient Elution in LC Chromatography

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45

Fig. 16 Effect of the gradient dwell volume, VD, on the elution volume, VR in reversed-phase chromatography on a microbore analytical C18 column with the hold-up volume Vm = 0.1 mL. Solute, symbols, and conditions as in Fig. 15, except for the flow rate 0.1 mL/min.

period volume and the dwell volume instead of VD. The retention volumes of sample compounds eluted in the dwell volume or in the intentional initial isocratic step can be calculated as it is common in isocratic HPLC, from the well-known equation VR = Vm(k + 1) using the appropriate retention equation for the chromatographic phase system, e.g., Eq. (10), (11), (15), or (16). Strongly retained compounds sometimes do not elute from the column before the end of the gradient. This problem can be often solved by increasing the final concentration of the solvent B at the end of the gradient, uG, or by using a less steep gradient with a longer gradient time, tG. If for any reason this simple solution is not possible, combined gradients can be used with increasing concentration of solvent B in solvent A in the first step, followed by a second step with increasing concentration of solvent C in solvent B. Figure 18 shows an example of reversed-phase separation of methyl esters, mono-, di-,

46 / Jandera

Fig. 17 Effect of the column hold-up volume on the pre-elution in the dwellvolume step (0.5 mL). Columns: A—Lichrospher 60RP-select B, 5 Am, 125 mm
4 mm i.d., Vm = 0.95 mL, gradient 50–70% acetonitrile in 2.6 min, 3 mL/min, no pre-elution; B—CHROMOLITH, 50 mm
4.6 mm i.d., Vm = 0.70 mL, gradient 50–70% acetonitrile in 5 min, 3 mL/min, compound 1 pre-eluted in the dwell volume; C—POROSHELL, 5 Am, 75 mm
2.1 mm i.d., Vm = 0.26 mL, gradient 20–40% acetonitrile in 7.1 min, 0.3 mL/min, compounds 1–4 pre-eluted in the dwell-volume; sample compounds: triazine herbicides, simazine (1), atrazine (2), methoprotryne (3), terbutylazine (4), promethryne (5), and terbutryne (6). 40jC. Arrows indicate the start of the elution in the gradient period.

Gradient Elution in LC Chromatography

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47

Fig. 18 Two-step gradient elution reversed phase separation of a sample of partially transesterified rapeseed oil. Two Nova-Pak C18 columns in series, 4 Am (3.9
150 mm i.d. each). Linear gradient from 70% acetonitrile to 100% acetonitrile in water in 20 min, followed by an isocratic hold-up step with 100% acetonitrile until 36 min and linear gradient from 100% acetonitrile to 60% 2-propanol in acetonitrile until 132 min. Flow rate 1 mL/ min, detection UV, 205 nm. Peak notation—fatty acids in methyl esters (Me), mono-, di-, and triacylglycerols: Ln—linolenic, L—linoleic, O—oleic, P—palmitic, S—stearic, G—gadoleic.

and triacyl glycerols in a sample of partially transesterified rapeseed oil using aqueous–organic gradient of acetonitrile in water followed by nonaqueous gradient of 2-propanol and hexane in acetonitrile [162]. More often, an isocratic hold-up period with final gradient concentration of the solvent B, uG, is used after the end of the gradient to allow the elution of strongly retained sample compounds, but postgradient elution can significantly increase the analysis time. The retention volumes of the compounds eluting in the postgradient step can be calculated taking into account the contributions of the gradient

48 / Jandera volume, VG, of the gradient dwell volume, VD (or of the intentional initial isocratic step), of the column hold-up volume, Vm, and of the postgradient isocratic step with the retention factor kG, considering the proportional part of the column hold-up volume in this step, Vm  VmG: VR ¼ Vm þ VD þ VG þ ðVm  VmG ÞkG

ð28Þ

The proportional part of the column hold-up volume in the gradient step, VmG, can be calculated from Eq. (12), (17), or (18), as appropriate, by setting VRV = VG and Vm = VmG. The migration of a strongly retained compound during the gradient delay period with a large retention factor k1 is usually unimportant and can be neglected. For example, this approach yields Eq. (29) for the postgradient elution volume in reversed phase chromatography, assuming the validity of Eq. (12):

 kG VR ¼ VG þ VD 1  þ Vm ½1 þ 10amuG k1 ð29Þ h i 1 mðAuG Þ 1  10  2:31 mB Similar approach can be used also for the calculation of the elution volumes in gradient elution employing several subsequent isocratic [163] or gradient segments [2,23].

B. Effect of the Adsorption of Strong Solvents on Retention From a mixed mobile phase, the component(s) with higher affinity to the stationary phase can be preferentially adsorbed. This behavior is well known in thin-layer chromatography, where it is called solvent demixing [112]. Solvent demixing does not influence isocratic separations on the column preequilibrated with the mobile phase, but may be important in gradient elution [164–169], where the stationary phase comes into contact with mobile phase whose composition changes with time. Possible uptake of strong solvent(s) on the column can significantly change the properties of the stationary phase and the actual gradient profile. The amount of the solvent uptake and its effect on the separation depends on the difference in the strength of the interactions of the components of the mobile phase with the stationary phase characterized by the adsorption isotherm of the strong solvent, which controls the saturation capacity and the degree

Gradient Elution in LC Chromatography

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49

of saturation of the stationary phase at the start of the gradient. The adsorbed solvent becomes ineffective as the eluent and the experimental elution volumes are more or less higher than the values predicted by calculation. Further, the concentration of the strong solvent in the mobile phase can suddenly increase at the breakthrough time (Fig. 19A, calculated breakthrough curve) and can displace (sweep out from the column) weakly retained sample compounds or impurities from the initial mobile phase, giving rise to a ‘‘ghost’’ peak. An example is shown in Fig. 19B as a record of a blank gradient of 2-propanol in hexane on a silica gel column. Preferential adsorption of the organic solvent on a nonpolar chemically bonded stationary phase [88] can affect the elution times in reversed-phase gradient elution [164–166]. Quarry et al. [160] calculated the positive errors caused by preferential adsorption of acetonitrile on various C18 columns from the experimental adsorption isotherms, which increased for column materials with a larger specific surface and were greater for more strongly retained compounds, but they generally did not exceed 0.05 mL [171]. A salt can be preferentially adsorbed from the mobile phases containing mixed buffers in ion-exchange chromatography with ionic strength gradients [168]. The effects of preferential adsorption are more significant in preparative than in analytical gradient-elution RPC and IEC. Because of strong affinity of polar solvents to silica gel and to other polar adsorbents in normal-phase gradient-elution chromatography, the column uptake of the polar solvent can affect the retention in normal-phase gradient elution more significantly than in RPC or in IEC systems [20,69,132]. The errors in calculated retention volumes caused by preferential adsorption are less important with the gradients that start at a nonzero initial concentration, A, of the polar solvent B [68]. If for some reason a gradient should start at A = 0, a simple, but not quite accurate correction consists in adding the experimentally determined breakthrough volume of the strong solvent to the calculated VRV [169]. However, this approach is justified only for strongly retained sample compounds which do not migrate significantly along the column prior to the breakthrough of the polar solvent B. Further, the experimental determination of the breakthrough volumes is necessary for each gradient program used, which is neither convenient nor accurate with solvents that do not absorb light in the UV region. A more rigorous approach was suggested recently that employs the experimentally determined adsorption isotherm describing the distribution of the

50 / Jandera

Fig. 19 A: Calculated breakthrough curves in normal-phase gradientelution HPLC. Simulated calculation using the experimental isotherm data and assuming N = 5000. Gradient dwell volume = 0.50 mL. B: Record of the blank gradient detector trace showing the breakthrough of propan-2-ol at 6 min and a ‘‘ghost peak’’ of impurities displaced at the breakthrough volume. Column: silica gel Separon SGX (7.5 Am), 150
3.3 mm i.d., 1 mL/min, 40jC. Gradient: 0—50% 2-propanol in 30 min (u—concentration of propan2-ol in the eluate, V—volume of the eluate from the start of the gradient).

Gradient Elution in LC Chromatography

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51

polar solvent between the binary mobile phase and the column used [69,160]. The isotherm depends on the type of the column packing material and on the mobile phase components. The distribution equilibrium of a binary solvent mixture can be often described by a simple Everett’s equation [170], which is equivalent to the two-parameter Langmuir isotherm [171] if one solvent is strongly adsorbed: q¼

qs b1 u a1 u ¼ ð1 þ b1 uÞ ð1 þ b1 uÞ

ð30Þ

Here q is the concentration of the sample compound in the stationary and u that in the mobile phases; a1, b1 are the coefficients of the isotherm; and qs is the column saturation capacity. The Langmuir model describes satisfactorily the distribution of many binary solvent mixtures in reversed-phase and normal-phase systems. However, the distribution equilibrium of the solvents with moderate differences in polarities between some combinations of binary mobile phases (such as dioxane–hexane, dichloromethane–hexane, or propanol–dichloromethane) and polar adsorbents is often better characterized by a twolayer adsorption described by the isotherm Eq. (31) [172,173]: q¼

q1s b1 u A1 u

ð1 þ b 2 u Þ ¼ þ A2 u 1 þ B1 u ð1 þ b1 uÞ

ð31Þ

where q1s is the adsorbent saturation capacity for the adsorption in the first layer and b1, b2, A1, A2, B1, are other isotherm parameters. The adsorbed volume of the polar solvent B during gradient elution is strongly affected by the type of the adsorption isotherm. We found that silica gel columns can be almost completely saturated with the polar solvent in mobile phases containing more than approximately 1% 2-propanol or 2% dioxane in heptane. Lower breakthrough volumes of dioxane with respect to 2-propanol can be attributed to stronger adsorption of the latter, more polar, solvent. The bonded nitrile column was almost saturated by adsorption of the amount of 2-propanol corresponding to 6% of Vm from 2-propanol– hexane mobile phases [69]. The amount of the polar solvent, Vads, adsorbed in the course of gradient elution starting with a pure less polar solvent (A = 0) corresponds to the saturation volume capacity of the column, Vsat, and does not depend on the steepness of the gradient. In our experiments the

52 / Jandera polar solvent uptake on a silica gel column steeply decreased with the gradients starting at A > 0 and dropped to less than 1% of Vm with gradients starting at 3–9% 2-propanol or dioxane. The amount of the polar solvent adsorbed during gradient elution represented 26%, 15%, and 10%, respectively, of the full column saturation capacity for the gradients starting at 3%, 6%, and 9% 2-propanol in hexane. With gradients of 2-propanol in dichloromethane on a silica gel column and of dioxane in hexane on a bonded nitrile column, the silica gel column did not get fully saturated during the gradient elution (curves 3 and 5 in Fig. 20) [69]. The experimentally determined equation of the adsorption isotherm can be used to calculate the solvent breakthrough curves in gradient elution chromatography by numerical simulation. Examples in Figs. 21 and 22 show the actual gradient profiles of several polar solvents on a silica gel column corrected by calculation for the column uptake. In Fig. 21 a sudden steep increase in 2-propanol concentration at the breakthrough volume occurs with gradients starting at 0%

Fig. 20 Volume of polar solvent adsorbed on a chromatographic column, Vads, in equilibrium with the concentration c of the polar solvent in the mobile phase. Column: Silica gel, Separon SGX, 7.5 Am, 150 mm
3.3 mm i.d. (Vm = 0.905 mL, phase ratio 0.418)—plots 1—3; bonded nitrile, Separon SGX Nitrile, 7.5 Am, 150 mm
3.3 mm .i.d. (Vm = 0.966 mL, phase ratio 0.328)—plots 4 and 5. Binary mobile phases: 2-propanol–heptane (1), dioxane–heptane (2), 2-propanol–dichloromethane (3), 2-propanol–hexane (4), dioxane–hexane (5).

Gradient Elution in LC Chromatography

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53

Fig. 21 Calculated breakthrough curves of 2-propanol in heptane on a Separon SGX silica gel column in normal-phase gradient-elution HPLC, simulated by numerical calculations using the experimental isotherm data and assuming N = 5000. Gradient dwell volume = 0.50 mL. Gradients: 0— 50% 2-propanol in 30 min (1), 0–25% 2-propanol in 30 min (2), 0–16.7% 2propanol in 30 min (3), 3–50% 2-propanol in 30 min (4), c—concentration of 2propanol in the eluate, V—volume of the eluate from the start of the gradient.

Fig. 22 Calculated breakthrough curves of 2-propanol in dichloromethane on a Separon SGX silica gel column in normal-phase gradient elution HPLC, simulated by numerical calculations using the experimental isotherm data and assuming N = 5000. Gradient dwell volume = 0.50 mL. Gradients: 1–50% 2-propanol in 30 min (1), 1–25% 2-propanol in 30 min (2), 1–16.7% 2propanol in 30 min (3). c—concentration of 2-propanol in the eluate, V— volume of the eluate from the start of the gradient.

54 / Jandera 2-propanol (plots 1–3), in agreement with the experimental blank gradient profile shown in Fig. 19A. The propanol uptake followed by an abrupt concentration increase can be avoided using gradients starting at a nonzero concentration of the polar solvent B, as shown by plot 4 in Fig. 21 for the gradient starting at 3% 2-propanol, where a gradient delay is apparent, but except for the shift the gradient profile is identical with the desired composition program [68]. The profiles of the breakthrough curves of 2-propanol from dichloromethane on a silica gel column determined in this way (Fig. 22) differ significantly from the gradients set by the operator and from the profiles obtained with heptane as the weak solvent in Fig. 21. Gradient curvature is observed even with the linear gradients starting at 1% of 2-propanol [69]. The Langmuir isotherm does not describe well this system, where probably multilayer adsorption of 2-propanol from dichloromethane occurs, which is satisfactorily described by Eq. (31) [173]. Hence the correction of the gradient-elution retention volumes for the uptake of 2-propanol on the column requires using Eq. (32) or (33) with the volume of the adsorbed polar solvent, Vads, calculated using Eq. (A12) in Appendix A. The retention volumes in gradient-elution chromatography can be corrected for the uptake of the polar solvent on the column taking into account that the volume of the pure polar solvent B which is necessary to elute sample compounds, Vsolv, should be increased to include the volume of B adsorbed on the column from the start of the gradient till the elution of the peak maximum, Vads. Vads can be calculated from the appropriate adsorption isotherm, e.g., Eq. (30) or Eq. (31). Using this approach, Eq. (17) is modified as follows to account for the column uptake of polar solvents in normal-phase systems [69]: VRV ¼

i 1 A mþ1 mþ1 1h ðm þ 1ÞBk0 Vm þ ðA2 þ 2 BVads Þ 2  B B

ð32Þ

and Eq. (18) to: 1 VRV ¼ bB

"

(



2

ðm þ 1ÞbBVm þ a þ b A þ 2 BVads

a þ Ab  bB

 12

1 #ðmþ1Þ ) mþ1

ð33Þ

Gradient Elution in LC Chromatography

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55

The adsorbed volume is introduced into Eq. (32) or (33) from one of the Eqs. (A6), (A11), or (A12) in Appendix A, whichever is more appropriate with respect to the isotherm controlling the distribution of the polar solvent in the chromatographic system used. Using Eq. (33) for the calculation of the elution volumes corrected for the preferential polar solvent uptake in normal-phase chromatography resulted in improved average prediction error from 1.6% to 0.8 % for gradients of propanol in heptane on a silica gel column. With the gradients of dioxane in heptane, the improvement of the corrected calculated data was only marginal; because of a lower uptake of dioxane the accuracy of the uncorrected data (average error 0.7%) was satisfactory enough. The correction did not improve either the accuracy of the elution volumes for gradients starting at 3–9% of 2propanol, where the silica gel column is already almost completely saturated with the polar solvent and the average error of prediction was 1.6%. The differences between the calculated corrected elution volumes and the experimental values for the gradients starting at 1% 2-propanol in dichloromethane were lower than 0.2 mL, with average error of prediction 1.7% in contrast to 12% error of prediction for uncorrected data. The latter system is probably controlled by a twolayer adsorption mechanism described by an associative isotherm (Eq. 31) and full column saturation with 2-propanol is not accomplished until the end of the gradient elution [69].

VII. GRADIENT ELUTION METHOD DEVELOPMENT For many samples, gradient elution provides better separation than isocratic elution [1–5]. Gradient technique is necessary for the analysis of complex samples whose isocratic separation would necessitate mobile phases providing differences in retention factors exceeding the range between k from 0.5 to 10 and especially for samples of synthetic polymers and biopolymers. Gradient elution can accelerate the elution of strongly retained sample impurities. It also allows injecting relatively large sample volumes in a weak solvent, resulting in direct on-column sample preconcentration and increased detection sensitivity. It is a valuable tool for rapid screening analysis, or as a scouting method for optimization of isocratic separations [47,174]. Prediction of isocratic retention from the gradient experiments can be used also for rapid screening of hydrophobicity of analytes including drugs and drug candidates [175].

56 / Jandera In this part, various factors affecting the development of gradient methods, method transfer between various instrumental systems, effects of the operating conditions on gradient separations, and optimization of gradient profiles are discussed. The aim of the separation should always be kept in mind when developing gradient elution methods. The transfer of gradient methods between various instrumental systems is less straightforward than in isocratic HPLC, because a change in one operation parameter usually necessitates judicious adjustment of the gradient program to result in a desired effect on the separation. These features of gradient elution are the reason for the bias against the acceptance of the gradient technique in some laboratories. However, with adequate knowledge of the principles of gradient elution, most problems can be avoided so that the transfer of gradient methods should not be difficult. One very important and often neglected factor complicating the method transfer should be always kept in mind—namely the differences between the instrumental gradient dwell volumes, VD, in various commercial instruments. The consequence of these differences is that if we use two different instruments to run a gradient method on the same column and at the same flow rate of the mobile phase, we can nevertheless observe more or less significant differences in the retention times and even unexpected changes in the band spacing and sample separation. If we measure the dwell volume, VD, of the instruments, we can calculate the changes in the elution times using one of the Eqs. (25)–(27), as appropriate—see discussion in Section VI. Dolan and Snyder [176] suggested to overcome the problem of different dwell volumes either by late injection with respect to the start of the gradient for the systems with a higher dwell volume, or by adjusting the dwell volume by introducing an isocratic hold period at the beginning of the gradient so as to increase the actual dwell period of the equipment with a lower instrumental VD. However, either approach increases the time of analysis. The instruments employing small volume microbore or capillary columns should be designed so as to minimize the dwell volume, or precolumn flow rate splitting should be used.

A. Transfer of Gradient Methods and Effects of Changing Operating Conditions on the Separation When transferring gradient methods between various instruments and columns with different dimensions and plate numbers, it should

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be kept in mind that changing column dimensions and flow rate affects the gradient volume (steepness) and, consequently, not only the column efficiency (plate number), but also the selectivity of separation may change [1–5]. The effects of changing flow rate, Fm, column length, l, and diameter, dc, on the elution time, tR, elution volume, VR, column plate number, N, resolution, Rs, and operating pressure, Dp, in isocratic and gradient HPLC are compared in Table 2. If the dwell volume does not change, such as when a gradient method is used with another column on the same instrument, the gradient program should be adjusted to fit new column dimensions and (or) flow rate. The gradient adjustment is easy and can be applied generally for various chromatographic systems and gradient profiles: In all equations for gradient-elution times or volumes [e.g., Eqs. (12), (17), (18)], the product of the net elution volume and of the gradient steepness parameter, VRV B, is independent of the flow rate and column dimensions as long as the product Vm B is kept constant [3–5,176,177]. In other words, the number of the column hold-up

Table 2 Effects of the Separation Conditions—Mobile Phase Flow Rate, Fm, Column Length, l, Diameter, dc, and Gradient Time tG—on the Separation in Isocratic and Gradient Column Liquid Chromatography: Elution Time, tR, Elution Volume, VR, Column Plate Number, N, Resolution, Rs, and Column Operating Pressure, Dp Change in operation condition by a factor f Fm
f, isocratic Fm
f, gradient Fm
f + tG/f, gradient l
f, isocratic l
f, gradient l
f + Fm
f, gradient l
f, tG
f, gradient dc
f isocratic dc
f gradient dc
f + Fm
f 2, gradient dc
f + tG
f 2, gradient tG
f, gradient tG
f + Fm/f, gradient

Change in the characteristic of separation tR

VR

N

Rs

Dp

tR/f >tR/f tR/f tR
f
VR >VR VR VR
f
N >N N >N N >N

Rs Rs >Rs Rs >>Rs >Rs >Rs

Dp
f Dp
f Dp
f Dp
f Dp
f Dp
f 2 Dp
f Dp/f 2 Dp/f 2 Dp Dp/f 2 Dp Dp/f

58 / Jandera volumes necessary to elute a sample compound is constant if the gradient volume expressed in the hold-up volume units does not change, at a constant gradient concentration range. This is similar to the description of isocratic elution by the retention factor, k, which is equal to the number of the column hold-up volumes per the net elution volume and is constant at a constant mobile phase composition and temperature. Hence any change in column length, l, or diameter, dc, at a constant gradient range (i.e., constant concentrations of the strong solvent B at the start, A, and at the end, uG, of the gradient) should be compensated by appropriate change in the gradient time, tG, or flow rate of the mobile phase, Fm, to keep the ratio Vm/VG constant: Vm Vm d2 l ¼ ¼ c ¼ const VG tG Fm tG Fm

ð34Þ

This condition has several important practical consequences (see Table 2). Changing Mobile-Phase Flow-Rate in Gradient Elution Chromatography If the flow rate of the mobile phase increases from F1 to F2 by a factor f = F2 /F1 > 1 and the gradient time tG is kept constant, the gradient steepness parameter, B = Du /(tG Fm), decreases by the factor f, the gradient volume VG increases by the same factor and the retention volumes increase, too, so that the retention times do not decrease proportionally to the increased flow rate. This can be illustrated by isocratic (Fig. 23) and gradient (Fig. 24) reversed-phase separations of alkylbenzenes at different flow rates. Whereas the time of separation is reduced to one third at a three-times-faster flow rate under isocratic conditions (Fig. 23), the gradient elution times decrease only by approximately one half at a flow rate increasing from 1 to 3 mL/min in gradient elution (Fig. 24). The reason is that the effect of a higher mobile phase velocity on the time of separation is partially compensated by gradient volume increasing three times in the same gradient time and concentration range. A less steep concentration gradient per unit volume of the eluate increases the retention times with respect to the values expected only on the basis of increased flow rate. To compensate for this effect and to decrease the elution times by a factor f, the gradient steepness parameter B should be kept con-

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Fig. 23 Isocratic reversed-phase separation of alkylbenzenes on a Purospher Star RP-18e, 3 Am, column (30
4 mm i.d.) in 60% acetonitrile at 1 mL/min and at 3 mL/min. 40jC, detection UV, 254 nm. B—Benzene, MB—toluene, EB—ethyl benzene, PB—propyl benzene, BB—butyl benzene, AB—amyl benzene, HB—hexyl benzene.

stant by decreasing the gradient time at the factor f for an f-time increase in the flow rate (Table 2). For the example of alkylbenzene separation in Fig. 24, the increase of the flow rate from 1 to 3 mL/min should be compensated by decreasing the gradient time from 3 to 1 min to keep the gradient volume constant (VG = 3 mL) and to obtain three-times lower elution times at constant elution volumes (Fig. 25) [177]. Changing Column Diameter in Gradient Elution Chromatography If the column inner diameter is increased or decreased from dc1 to dc2 by a factor f = dc2 /dc1 at a constant column length (such as when upgrading an analytical method to a semi-preparative or preparative scale or when transferring a method from a conventional analytical to a microbore column), the column hold-up volume correspondingly

60 / Jandera

Fig. 24 Gradient-elution reversed-phase separation of alkylbenzenes on a Purospher Star RP-18e, 3 Am, column (30
4 mm i.d.). Linear gradient 50–100% acetonitrile in 3 min at 1 mL/min and at 3 mL/min. 40jC, detection UV, 254 nm. B—Benzene, MB—toluene, EB—ethyl benzene, PB—propyl benzene, BB—butyl benzene, AB—amyl benzene, HB—hexyl benzene.

increases or decreases by the factor f 2. However, the retention volumes of the sample compounds do not change proportionally to Vm at a constant flow rate, so that the band spacing in the chromatogram and the selectivity of separation can be affected. To avoid such changes, a constant product Vm B should be maintained by adjusting the gradient steepness. The effects of increased or decreased column hold-up volume on the retention can be compensated for by increasing or decreasing the gradient time by f 2 to keep the ratio Vm / VG constant at a constant flow rate—see Eq. (34), which changes both the elution times and the elution volumes by the same factor f 2. This approach is not practical as it is connected either with long separation time, or with very high operating pressure. Hence any change in the column diameter by a factor f should be compensated rather by appropriate correction in the flow rate of the mobile phase,

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Fig. 25 Gradient-elution reversed-phase separation of alkylbenzenes on a Purospher Star RP-18e, 3 Am, column (30
4 mm i.d.). Linear gradient 50– 100% acetonitrile in 3 min at 1 mL/min and 50—100% acetonitrile in 1 min at 3 mL/min. 40jC, detection UV, 254 nm. B—Benzene, MB—toluene, EB— ethyl benzene, PB—propyl benzene, BB—butyl benzene, AB—amyl benzene, HB—hexyl benzene.

Fm, by the factor f 2. In such a case, the elution volumes change by the factor f 2, but the elution times and the operating pressure do not change (Table 2) [3,5]. Changing Column Length in Gradient Elution Chromatography To increase the separation efficiency, the length of the column can be increased from l1 to l2 by a factor f = l2 /l1. At a constant column inner diameter, flow rate, and gradient time, this change results in increasing the column hold-up volume, the number of theoretical plates, but also the operating pressure all by the same factor f, but the retention times, the retention volumes, and the resolution all increase by less than f (Table 2). The elution times and the elution volumes increase by the factor f when the gradient steepness parameter B is decreased

62 / Jandera by increasing the gradient time by f. The same effect is achieved by increasing the flow rate of the mobile phase at a constant gradient time, but at the cost of increased pressure drop across the column, so that this approach is feasible with monolithic rather than with packed columns. When a shorter column is used, which is the most frequent case when rapid separation methods are developed, the column hold-up volume decreases, hence the gradient time at a constant flow rate should be decreased proportionally. The elution times and volumes do not change at a constant flow rate and gradient time, if the column length increases (or decreases) by a factor f and the column inner diameter simultaneously decreases (or increases) by Mf. An increase or a decrease in the column length is connected with corresponding change in the column plate number and resolution, which should be accounted for when interpreting the results of the gradient method transfer (Table 2) [3,5]. Rapid Prediction of the Effects of Changing Gradient Steepness (Gradient Range) and Initial Mobile Phase Composition on the Separation For the development of new gradient methods and for the transfer of established gradient methods, which often require some fine tuning of operation conditions, it is useful to have a tool making possible a rapid estimation of the effect of changing gradient program on the separation. The profile of the gradient affects the retention in the same way as the composition of the mobile phase under isocratic conditions. This is illustrated in Fig. 26 on the example of RPC separation of 10 homologous derivatives of n-alkylamines with various gradient programs. The chromatograms A–C show the effect of the gradient time (gradient steepness) on the separation with linear gradients at a constant gradient range, Du, 70–100% methanol and flow rate, 1 mL/ min. As the gradient time increases from 10 to 40 min, the steepness of the gradient decreases, i.e., the increase in the gradient time has the same effect as decreasing the isocratic concentration of methanol in water—the resolution improves, but the elution times and the time of separation increase [3]. The chromatograms D–F in Fig. 26 illustrate the effect of the gradient range on the separation at a constant gradient volume. The appropriate setting of the initial concentration of the solvent B, A, is more important than the setting of the final concentration, uG, which can be adjusted so as to terminate the gradient immediately after the

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Fig. 26 Reversed-phase gradient elution separation of 1,2-naphthoylenebenzimidazole alkylsulphonamides. Column: Lichrosorb RP-18, 10 Am (300
4 mm i.d.). Linear gradients of methanol in water with a constant gradient range, but different gradient volumes (A–C) and with a constant gradient steepness (1.67% methanol/minute), but different initial concentrations of methanol (D–F). Flow rate 1 mL/min. Numbers of the peaks agree with the numbers of carbon atoms in the alkyls.

64 / Jandera elution of the last sample compound. In the example shown in Fig. 26, the gradient time was set to fit to different gradient ranges at a constant steepness of the gradient (1% methanol/0.6 min). With initial concentration increasing from 50% to 80% methanol, the resolution decreases and the retention times increase, like when the concentration of organic solvent increases in isocratic HPLC. It should be noted that increasing initial concentration of methanol causes even more significant decrease in the elution times of the early eluting compounds than does increasing the gradient steepness at a constant initial concentration of methanol in the chromatograms A–C. This means that it is equally important to adjust the initial concentration of the solvent B as to set an appropriate gradient range when developing a gradient HPLC method, to keep the time of the analysis as short as possible while maintaining the desired resolution of sample compounds. Under isocratic conditions, the bandwidths regularly increase as the elution times increase on a column with an approximately constant theoretical plate number for all sample compounds. Unlike this behavior, the bandwidths in gradient-elution chromatography characterized by Eq. (8) are approximately constant both for the early and for the late eluting sample compounds. This is caused by increasing migration velocities of the bands along the column during gradient elution, so that all sample compounds eventually elute with similar instantaneous retention factors, kf, at the time they leave the column. The kf is approximately equal to a half of the mean retention factor, k*, during the band migration along the column and depends to some extent on the gradient profile, so that the bands are narrower with steeper gradients (compare the chromatograms A and C in Fig. 26). Because kf is usually significantly lower than the retention factors in isocratic separations, especially for the late eluting compounds, the peaks in gradient-elution chromatography are generally narrower and higher than the corresponding isocratic bands, which improves the detector response and the sensitivity of determination. However, the beneficial effect of gradient elution on increasing sensitivity sometimes may be counterbalanced by increased baseline drift and noise in gradient HPLC. This means that the highest purity solvents are necessary for high sensitivity in gradient-elution chromatography with UV or fluorescence detection [1,3]. The effects of changing gradient profile on the retention in various gradient-elution liquid chromatography modes can be determined

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exactly by calculation using Eqs. (12), (13), (17)–(22), (25)–(29), (32) or (33), as appropriate. However, it is possible to use a simpler approach to estimate the effects of the gradient slope, B = Du/(tG Fm), and of the initial concentration of the strong solvent, A, on retention [177]. If we neglect the effects of the gradient dwell volume and of the preferential solvent uptake on the column, a change in the net V to V R,2 V caused by changing the gradient retention volume from V R,1 parameters from B1 to B2 and from A1 to A2 in RPC can be estimated for low-molecular mass sample solutes such as simple benzene derivatives from Eq. (35) obtained from Eq. (12) by setting m = 3: V i VR;2

   1 B2 3ðA1 A2 Þ  3B1 VR;1 V log 10 10 1 þ1 3B2 B1

ð35Þ

The errors in the retention volumes introduced by using Eq. (35) are illustrated in Fig. 27 for three compounds with different parameters m (2.6 for ethylbenzene, EB, 3.1 for butylbenzene, BB, and 3.6 for hexylbenzene, HB) separated on a C18 column using gradients from

Fig. 27 The effect of changing gradient steepness parameter, B = Du/VG, on the net retention volumes, VRV, in reversed-phase chromatography with gradients from 50% to 90% acetonitrile in water. Dashed lines—calculated from Eq. (12), full lines—calculated from the retention volumes in the 20-min gradient (1 mL/min) using Eq. (34). Solutes: ethylbenzene (EB, m = 2.60), butylbenzene (BB, m = 3.09), and hexylbenzene (HB, m = 3.59). Column: Lichrospher 60RP-select B, 5 Am (125
4 mm i.d., Vm = 0.90 mL).

66 / Jandera 50% to 90% acetonitrile with different slopes, B. Dashed lines correspond to the data calculated using nonsimplified Eq. (12) and full lines to the data calculated using Eq. (35) with m = 3 set for all compounds, V measured in the gradient from the reference elution volumes V R,1 with B = 0.02 (i.e., with gradient volume VG = 20 mL). The simplified calculation yields almost identical VR as the accurate calculation for ethylbenzene and butylbenzene and small errors <0.2 mL over major part of the gradient steepness range [177]. Of course, the errors are greater for compounds with larger molecules with higher parameters m. The advantage of the simplified calculation approach using Eq. (35) is that it does not require the experimental determination of the parameters a and m of Eq. (11) and enables to estimate a change in the retention caused by changing gradient profile from the retention data obtained in a single gradient run. In normal-phase (adsorption) chromatography, one adsorbed molecule of many low-molecular mass analytes can be displaced by approximately one molecule of the polar solvent B, and hence the value of the ‘‘stoichiometric’’ parameter m in Eq. (15) is very close to 1, although numerous exceptions from this rule have been observed. According to the displacement model of adsorption chromatography, the volume of the polar solvent B that should pass through the column for the elution of the analyte with displacement stoichiometry 1:1, Vsolv = k0 Vm, is theoretically constant and independent of either the concentration of the polar solvent B in a binary mobile phase used for isocratic elution or the gradient program. (k0 is the retention factor of the solute in pure solvent B.) Consequently, a change in the net retention volume caused by a change in the gradient program can be very simply estimated by setting m = 1 in Eq. (17) and solving for k0 Vm to yield Eq. (36) [69]: V Þ2 Vsolv ¼ ðVR;1

B1 B V A1 ¼ ðVR;2 V Þ2 2 þ VR;2 V A2 þ VR;1 2 2

ð36Þ

This simplified approach enables rapid estimation of the change in retention volumes caused by increasing or decreasing the steepness of the gradient from B1 to B2 and (or) the initial concentration of the polar solvent B from A1 to A2. Equation (36) may not be valid for compounds whose parameters m of Eq. (15) differ significantly from 1 and cannot be used for reversed-phase gradient elution. The volumes Vsolv of 2-propanol and dioxane calculated using Eq. (36) from the experimental net retention volumes of phenylurea

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compounds in gradient elution with various gradient programs on a silica gel and on a bonded nitrile column with heptane, hexane, and dichloromethane as weak solvents are compared in Table 3. Volumes Vsolv are larger on a silica gel than on a bonded nitrile column, with dioxane than with 2-propanol as the polar solvent and with heptane or hexane than with dichloromethane as the nonpolar solvent. This is in agreement with the differences in polarities of the column packing materials and of the mobile phase components—stronger adsorption is expected on more polar silica gel adsorbent and with a less polar solvent A in a binary mobile phase. To verify the validity of Eq. (36), the experimental elution volumes measured with the most steep gradients (0–50% 2-propanol in 30 min or 0–100% dioxane in 30 min, respectively) were used to predict VR for other gradients—less steep or starting at a nonzero concentration of the polar solvent B—by calculation using Eq. (36). In most cases, the simple calculation yields more or less underestimated elution volumes for gradients starting at 0% polar solvent and overestimated data for gradients starting at a nonzero concentration of 2-propanol or dioxane, probably due to the preferential adsorption of polar solvents during gradient elution and to other effects that are not accounted for in the calculation. However, the average error of prediction of the retention times reported in Table 3 is approximately 7%, which is acceptable for a rapid rough estimate of the effect of changing gradient profile on the retention [69]. The main advantage of simple Eq. (36), like Eq. (35), is that it does not necessitate the determination of the parameters of the retention equations (of the dependencies of k on u) and can be used for rapid prediction of retention in gradient-elution NPC from the retention data measured experimentally in another gradient-elution program. However, it should be kept in mind that neither Eq. (36) nor Eq. (35) is exact enough to be used for fine tuning of separation conditions, where the individual sample compounds have different parameters m in Eq. (11) or (15) and do not take into account the dependence of the separation selectivity, i.e., of the relative retention (separation factor, a) on the composition of the mobile phase. Of course, as the composition of the mobile phase changes during a gradient run, a may change, too, which results in changing peak spacing in chromatograms and occasionally even in changing elution order when the gradient profile is changed. Appropriate approaches for fine tuning of gradient methods are discussed in Section VII.B.

68 / Jandera Table 3 Retention Volumes in Normal-Phase Gradient Elution HPLC Solute

0–50% P 0–25% P 0–16.7% P 3–50% P 6–50% P Vsolv, mean*

A: Silica gel, Separon SGX heptane DPU Vsolv 0.79 VR (E ) 11.31 VR (C ) – CMU Vsolv 0.67 VR (E ) 10.55 VR (C ) – IPU Vsolv 0.76 VR (E ) 11.18 VR (C ) – DCU Vsolv 1.50 VR (E ) 14.86 VR (C ) – Mean error, % –

column (150
3.3 mm i.d.), gradients of 2-propanol in 0.94 16.68 15.41 0.80 15.55 14.34 0.84 16.01 15.23 1.75 22.02 20.43 6.9

1.04 21.66 18.56 0.89 16.69 17.24 0.89 19.50 18.34 1.90 27.78 27.71 5.9

0.78 9.53 9.88 0.66 8.82 9.11 0.75 9.47 9.75 1.51 13.27 13.50 +2.9

0.74 8.01 8.64 0.62 7.25 7.89 0.73 8.00 8.67 1.48 11.90 12.23 +5.9

0.75 F 0.03

0.63 F 0.03

0.73 F 0.03

1.48 F 0.03

B: Bonded cyanopropyl, Separon SGX Nitrile column (150
3.3 mm i.d.), gradient of 2-propanol in hexane FMU Vsolv 0.33 0.35 0.35 0.33 0.32 0.33 F 0.02 VR (E ) 8.55 11.49 13.74 6.67 5.21 VR (C ) – 11.51 13.78 7.10 5.96 CTU Vsolv 0.39 0.41 0.42 0.39 0.38 0.39 F 0.02 VR (E ) 9.10 12.13 14.70 7.21 5.58 VR (C ) – 12.29 14.73 7.65 6.48 PHU Vsolv 0.52 0.53 0.53 0.53 0.52 0.52 F 0.01 VR (E ) 10.16 14.02 16.71 8.46 7.04 VR (C ) – 13.79 16.57 8.72 7.50 CMU Vsolv 0.13 0.14 0.14 0.13 0.13 0.13 F 0.07 VR (E ) 6.34 8.26 9.84 4.29 3.03 VR (C ) – 8.39 9.95 4.93 3.98 Mean error, % – +1.4 +0.6 +7.6 +17.0 Solutes: Phenylurea herbicides and related compounds; DPU—desphenuron, CMU— bis-N,N V-(3-choloro-4-methylphenyl) urea; IPU—isoproturon; DCU—deschlorometoxuron; FMU—fluorometuron; CTU—chlorotoluron; PHU—phenuron. Gradient time 30 min, 1 mL/min, 40jC. Vsolv = Vm k0 (mL)—volume of pure propanol necessary for the elution of a solute; VR (E ) (mL), experimental retention volumes; VR (C) (mL), retention volumes calculated from Eq. (36) using VR (E ) values measured in the gradients from 0% to 50% P. Mean value for Vsolv F SD for gradients ending at 50% P.

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B. Optimization of Gradient Elution Separations Peak Capacity and Fast Gradients In the development of generally suitable gradient HPLC methods, two aspects of the separation are of primary importance. First, for very complex samples containing compounds with large differences in the retention, the obvious objective is to obtain as large a number of separated peaks as possible for maximum information on the sample composition. Second, the speed of analysis is becoming increasingly important, especially in the development of generic methods for characterization of the products of automated synthesis in the pharmaceutical industry. A convenient measure of this separation aspect is the peak capacity P, which is defined as the number of peaks with a width wg that can be separated in a given time of separation (or in a given volume of the eluate) [178]. In gradient-elution chromatography, the time range of separation can be understood either as the difference between the elution times of the last and of the first peaks in the chromatogram, tR,Z and tR,1, respectively, [Eq. (37A)] [179] or as the whole gradient time, tG, or the whole gradient volume, VG, [Eq. (37B)] [180]: pffiffiffiffiffi pffiffiffiffiffi  VG tG Fm N tR;Z N P¼ 1 or P ¼ 1 þ ¼1þ 4 4 Vm ð1 þ kf Þ tR;l wg ð37A; BÞ For reversed-phase gradient elution, Eq. (37B) can be adapted by inserting the average bandwidth, wg (which is approximately constant, as discussed above), from Eq. (13): rffiffiffiffiffi N VG 1 P ¼1þ ð38Þ 4 Vm 1 þ ½10ðmAaÞ þ 2:31 mBVm 1 With shallow gradients (large ratios VG/Vm) and compounds that are strongly retained at the start of the gradient, Eq. (38) can be simplified to approximate Eq. (39) [180]: " rffiffiffiffiffi

 # 12 l aD cD DpK0 dp 2 mDui1 þ 0:58 mDu Pi1 þ 0:58 þ g H l l ð39Þ

70 / Jandera Here l is the column length, dp is the mean packing particle diameter, Dp is the operating pressure, g is the mobile phase viscosity, K0 is the column specific permeability (Darcy’s law), H is the height equivalent to a theoretical plate, Du is the gradient concentration range, and aD,cD are the constants of van Deemter equation simplified for higher linear mobile phase velocities, u, range by neglecting the contribution of longitudinal diffusion to band broadening: H ¼ a D þ cD u

ð40Þ

Equation (39) shows that the gradient peak capacity on a given column at a constant flow rate is approximately independent of the elution times of the sample compounds, as far as the product m Du is constant. This has an important practical consequence for gradientelution separations of high-molecular weight compounds, whose parameters m are significantly (even by an order of magnitude) higher than the parameters m of the compounds with small molecules (Section IX). Hence, to obtain comparable peak capacity and separation times, a significantly narrower gradient concentration range should be used for the separation of macromolecular compounds than it is common with simple organic solutes. Equation (39) further indicates that the peak capacity for a specific sample increases with a wider gradient concentration range, increasing column length, and decreasing packing particle diameter. However, long columns packed with fine particles give rise to high operating pressures. The maximum allowed operation pressure (Dp = 30 – 40 MPa with most commercial HPLC chromatographs) sets practical limits to the column length and (or) to using small particle diameter packings. This obstacle is partially overcome with monolithic silica gel-based columns, which have three to five times higher permeability than comparable packed columns [181] and allow either using long columns or high flow rates and fast gradients [182]. Another way to obtain rapid gradient separation—however, at a cost of decreased peak capacity—is by using short packed columns and simultaneously decreasing the particle size of the packing material to keep the ratio l/dp constant, e.g., with a 3-cm column packed with a 3 Am material, or even a 1-cm, 1-Am column for very fast separation instead of a conventional 5-cm, 5-Am or 10-cm, 10-Am column. For example, the change from a 10-cm, 10-Am column to a 1-cm, 1-Am column provides a 4.5 increase in peak capacity or resolution for

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a 100-sec gradient, as discussed in detail by Neue et al. [180], but this increase is due mainly to increasing ratio VG/Vm. At constant VG/Vm and l/dp ratios, the peak capacity and resolution slightly decrease when decreasing the column length. Very fast gradients with short columns require minimization of extra-column instrumental volumes by using low-volume injectors and detector cells, a short detector time constant, high signal sampling rates, and flow splitting to decrease the effects of the gradient dwell volume (which becomes very important with short gradient runs). Hence conventional analytical instrumentation should be significantly modified for fast gradient operation. Like in isocratic HPLC, gradient separations can be accelerated at an elevated temperature by reducing the viscosity of the mobile phase and hence the column operating pressure and by increasing the diffusion coefficients of the analytes speeding the mass transfer between the stationary and the mobile phase [183]. To maintain approximately constant both the time of analysis and the peak capacity for comparable chromatographic separation when using columns with different hold-up volumes, the gradient volume, VG, and the final concentration of the solvent B, uG, should be kept constant. In this case, the gradient concentration range should be adapted to changing column hold-up volume, Vm, by adapting the initial concentration of the solvent B, A, to keep a constant product B Vm = (uG  A)Vm /VG (see the discussion of the method transfer in Section VII.A). Hence a shorter column with a lower Vm generally requires a lower initial concentration of the solvent B to maintain the resolution and the gradient column peak capacity [Eq. (38)] comparable with the separation on a longer column. Optimization of Gradients for Specific Separation Problems Tailor-made optimum gradient profile for a specific separation problem can be designed using computer-assisted strategies. For appropriate characterization of the quality of separation, various ‘‘elemental’’ or ‘‘sum’’ criteria were introduced, which can be used either in isocratic or in gradient chromatography. The chromatographic optimization function (COF) can suitably characterize the quality of the separation over the whole chromatogram time range as a ‘‘sum’’ criterion. Various definitions of COF have been suggested to avoid misleading effects of possible compensation of poor resolution of

72 / Jandera some peaks by undesirable overresolution in another part of the chromatogram. The most useful COF criteria rely on the product of the RS for all adjacent bands in the chromatogram normalized with respect to the average resolution in the chromatogram, with additional terms accounting for the number of peaks in the chromatogram and for the analysis time via various weighting factors. Excellent discussion of this topic can be found elsewhere [184]. Sequential optimization methods with a COF ‘‘sum’’ optimization criterion, such as the simplex method, are often used for multiparameter optimization [185]. Their main disadvantages consist in a large number of experiments required to find optimal working conditions, a loss of the detailed information on the separation of the individual sample components and a possibility that the search method will ‘‘slide’’ into a region with a local maximum of the optimization criterion. In simultaneous single-parameter or multiparameter optimization methods, several judiciously selected initial experiments are performed to determine the constants of the equations describing the retention in dependence on one or more optimized parameters, such as the concentration of various solvents or of ionic additives in the mobile phase, pH, temperature, the gradient time, the gradient concentration range, etc. Then, appropriate ‘‘elemental criteria’’ (such as the separation factor, the resolution, or the peak separation function) describing the quality of separation for each pair of adjacent peaks in the chromatogram are calculated and are plotted as a function of the optimized gradient parameter (gradient steepness, B, initial concentration of the solvent B, A, or—if necessary—a gradient shape parameter) as a ‘‘window diagram’’ or an ‘‘overlapping resolution map.’’ In such plots, the areas are searched in which the ‘‘elemental criteria’’ for all adjacent bands in the chromatogram are equal to or larger than the desired value (e.g., RS z 1.5). Here the optimal gradient parameters are selected that either yield the maximum resolution for the ‘‘critical pair’’ of adjacent peaks most difficult to separate, or provide the desired resolution for all adjacent peaks in the chromatogram in the shortest run time [1,184]. The advantage of simultaneous predictive optimization is that it offers a detailed information about the separation of the individual sample compounds. The optimal values of the gradient parameters are determined and simulated optimum chromatograms constructed, showing the optimized separation for all sample compounds. Several

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commercial optimization software packages are available both for isocratic and gradient liquid chromatography. Probably the best known is the Dry-Lab program [1]. The software can be used to optimize subsequently more parameters such as the composition of a binary or a ternary mobile phase, pH, and temperature, but only one variable is optimized at a time and its optimal value is used in the next step for the optimization of another operation parameter. The general strategy and various attributes of this program have been described by Snyder et al. [186] and Dolan et al. [187]. Most recent advances in the software development and various applications were reviewed by Molnar [188]. In reversed-phase gradient elution chromatography, the DryLab G version of the software for the optimization of the gradient profile is based on the Snyder linear solvent strength theory [4]. The retention data from two initial gradient runs are used to adjust subsequently the steepness and the range of the gradient and—if necessary—some other working parameters, such as pH and the separation temperature [189]. This approach can be adopted also to optimize multisegment gradients [190]. However, some gradient parameters such as the gradient range and the gradient time show synergistic effects on separation. Hence simultaneous optimization of two or more parameters at a time can provide better results than their subsequent optimization. Simultaneous multiparameter optimization approach introduced originally by Glajch et al. [191] has been used since most frequently for the optimization of reversed-phase HPLC with ternary or quaternary mobile phases containing methanol, acetonitrile, and (or) tetrahydrofuran in water or in a buffer on the basis of seven or more initial experiments. The optimization programs of this type available commercially can often be directly incorporated into a chromatographic workstation. These programs can be used also for the optimization of gradient elution, especially of ternary gradients at a constant concentration ratio of the organic solvents during gradient elution (socalled ‘‘iso-selective gradients’’) [192,193]. Commercially available structure-based predictive software [194,195] (such as CHROMDREAM [196], CHROMSWORD or ELUEX) for optimization of isocratic or gradient RPC incorporates some features of the ‘‘expert system,’’ as it predicts the retention on the basis of the retention contributions by molecular structures for various sample compounds (however, not taking into account stereochemical and intramolecular interaction effects) and a data base

74 / Jandera

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summarizing some earlier knowledge on the retention behavior of model compounds on various HPLC columns. The software requires a large number of initial scouting isocratic or gradient experiments which can be run unattended overnight. The results are only approximate and predicted separation conditions usually require some subsequent fine tuning. It is not necessary to use expensive commercial software packages, if we do not require all the comfort they offer. Programs for the optimization of one or two gradient parameters at a time using a spreadsheet software for standalone PCs are easy to write in either Windows or Macintosh format. Simultaneous optimization of gradient time (steepness), initial concentration, and—if necessary— gradient shape employs predictive calculations of the retention and of the resolution of the individual pairs of sample compounds from isocratic or gradient data acquired in a few initial isocratic or gradient initial experiments [3,5,72,145,197–199], using, e.g., Eqs. (12), (13), (25), (29) for reversed-phase gradient HPLC or Eqs. (17), (22), (27), (32), (33) for normal-phase gradient HPLC. The window diagrams or resolution maps can be drawn, like with commercial software, or optimal values of the operation parameters are directly predicted by calculation. For simple linear gradients, appropriate selection of the concentration of the strong solvent B in the mobile phase at the start of the gradient, A, is equally important as the optimization of the gradient steepness, B, because each parameter influences very significantly the resolution and the time of analysis. Further, appropriately adjusting the initial concentration of the polar solvent, A, can sup-

Fig. 28 A: The resolution window diagram for NP-gradient-elution separation of phenylurea herbicides on a Separon SGX, 7.5 Am, silica gel column (150
3.3 mm i.d.) in dependence on the initial concentration of 2-propanol in n-heptane at the start of the gradient, A, with optimum gradient volume VG = 10 mL. Column plate number N = 5000, compounds: neburon (1), chlorobromuron (2), 3-chloro-4-methylphenylurea (3), desphenuron (4), isoproturon (5), diuron (6), metoxuron (7), deschlorometoxuron (8). B,C: HPLC separation with optimized gradient-elution conditions for maximum resolution in Fig. 28. (A) Using gradients from 12% to 38.6% 2-propanol in n-heptane in 7 min (B) and from 25% to 37.5% 2-propanol in nheptane in 5 min (C). Flow rate 1 mL/min, T = 40jC.

76 / Jandera press undesirable effects of the preferential adsorption of the solvent B on the retention behavior in normal-phase gradient elution, as discussed in Section VI [68]. The gradient parameters A and B can be optimized simultaneously using the following strategy [5,145]. With a preset final concentration of the strong solvent, uG, that should be attained at the end of the gradient where V = VG, the steepness parameter B of the gradient depends on the initial concentration A: B¼

ðuG  AÞ VG

ð41Þ

The setting of VG is not critical for the results of optimization, if it is large enough [198]. The elution volume VR can be calculated in dependence on a single parameter, A, introducing Eq. (41) into one of the appropriate Eqs. (12), (25), (29) for reversed-phase gradient HPLC or one of the Eqs. (17), (22), (27), (32), (33) in normal-phase gradient HPLC. The differences between the retention volumes of the compounds with adjacent peaks or the resolution, Rs, can be plotted versus A as a ‘‘window diagram’’ to select the optimum A for highest resolution of the ‘‘critical pair’’ of compounds that are most difficult to separate. The selection of the highest value of A at which the desired resolution (e.g., Rs = 1.5) is achieved for all compounds in the sample mixture in most cases automatically minimizes the time of the analysis, as the retention volumes and the run time decrease with increasing A. With optimized A, the corresponding gradient steepness parameter B can be calculated from Eq. (41) for a preset gradient volume VG and final concentration uG. This approach can be repeated for various VG to find really optimal combination of the gradient steepness B and initial concentration of the strong solvent, A. More detailed information on the optimization approach is given in Appendix B. An example of the ‘‘window diagram’’ for the optimization of normal-phase gradient-elution chromatography of eight phenylurea herbicides on a silica gel column is shown in Fig. 28A. Here two initial concentrations of 2-propanol (12% and 25%) are predicted to yield desired resolution of all sample compounds. The gradient separations with the two optimized initial concentrations of 2-propanol are shown in Fig. 28B and C. The resolution in the two chromatograms is comparable, but the gradient starting at 25% 2-propanol provides better band spacing in the chromatogram and shorter time of analysis

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Fig. 29 Top: The resolution window diagram for RP-gradient-elution separation of phenylurea herbicides on a Separon SGX C18, 7.5 Am, column (150
3.3 mm i.d.) in dependence on the initial concentration of methanol in water at the start of the gradient, A, with optimum gradient volume VG = 73 mL. Column plate number N = 5000; sample compounds: hydroxymetoxuron (1), desphenuron (2), phenuron (3), metoxuron (4), monuron (5), monolinuron (6), chlorotoluron (7), metobromuron (8), diuron (9), linuron (10), chlorobromuron (11), neburon (12). Bottom: The separation with optimized binary gradient from 24% to 100% methanol in water in 73 min. Flow rate 1 mL/min, T = 40jC.

78 / Jandera than the gradient starting at 12% 2- propanol [145]. The same approach applied to the optimization of reversed-phase separation of 12 phenylurea pesticides using linear gradient of methanol in water is illustrated in Fig. 29A and B [199]. In addition to the gradient volume and to the column plate number, the gradient shape can be adjusted [109].

VIII. CHROMATOGRAPHY WITH TERNARY GRADIENTS Like isocratic elution with ternary mobile phases, which often provides better separation selectivity for some complex samples than binary isocratic HPLC, ternary concentration gradients can improve the resolution of samples whose separation selectivity in binary gradient elution is too low. Here the concentrations of two stronger solvents i and j, ui and uj, in a ternary mobile phase are changed simultaneously during the elution, in most simple case in a linear manner: ui ¼ Ai þ Bi V ;

uj ¼ Aj þ Bj V

ð42Þ

In practice, two useful types of ternary gradients are most easy to describe and are used most frequently [197]: 1. The ‘‘elution strength’’ ternary gradients, where the concentration ratio of the two strong solvents, r = ui /uj, is preadjusted and kept constant during the gradient and the sum of the concentrations, uT = ui + uj, changes in a linear manner during the elution: uT ¼ AT þ BT V

ð43Þ

These gradients are often called ‘‘iso-selective multisolvent gradients’’ [191,192]. 2. The ‘‘selectivity’’ ternary gradients, where the sum of the concentrations of the two strong solvents, uT, is constant during the elution, but their concentration ratio in the mixed mobile phase changes in a linear manner: ui u ¼ X ¼ 0i þ BV ¼ X0 þ BV uT uT

ð44Þ

X0 is the ratio X at the start of the gradient and B = dui/dV = duj/dV is the steepness of the ‘‘selectivity’’ gradient. A ‘‘selectivity’’ gradient may improve the resolution of isomers or other structurally related compounds. For example, ternary gra-

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dient elution improved the separation of substituted phenols [200], for which neither a binary gradient of methanol in water nor a gradient of acetonitrile in water provided satisfactory separation. As the separation selectivity for early-eluting compounds was better with a gradient of acetonitrile in water, the separation selectivity for lateeluting compounds was better with a gradient of methanol in water, a ternary gradient with increasing concentration of methanol and simultaneously decreasing concentration of acetonitrile improved the resolution of the sample [200]. The ‘‘elution strength’’ (iso-selective) ternary gradients have similar effect on the solute retention as do binary gradients, hence Eq. (11), (15) or (16) with uT instead of u can be used to describe the retention. Both in reversed-phase [197–200] and in normal-phase [130,145,169] systems with ternary mobile phases, the parameters aT, bT, k0T, and mT of the appropriate retention- ternary mobile phase composition Eq. (11), (15), or (16) can be determined from the experimental data for various uT at a constant concentration ratio r, to be used instead of a, b, k0, and m in the appropriate equation for gradient elution volume [such as one of Eqs. (12), (17), (18), (21), (25)– (27), (29), (32) or (33)] for calculations of the elution volumes in ternary ‘‘elution strength’’ gradient elution using the same optimization approach as with binary solvent gradients. The optimization of a ternary ‘‘elution strength’’ gradient is illustrated in Fig. 30A,B for reversed-phase separation of the mixture of phenylurea herbicides whose optimized separation using binary gradient elution is shown in Fig. 29A,B. The optimized ‘‘elution strength’’ ternary gradient provides the separation in approximately half the time necessary for the separation with optimized binary gradient of methanol in water [199]. Another approach should be used for the prediction of retention and optimization of separation with ‘‘selectivity’’ ternary gradients, both in reversed phase and in normal phase systems. In reversed phase systems the following simple retention equations often apply in binary mobile phases composed of water and organic solvent i and of water and organic solvent j, respectively [197,198]: log ki ¼ ai  mi ui ;

log kj ¼ aj  mj uj

ð45Þ

In this instance, the elution volumes in the elution with linear ‘‘selectivity’’ ternary gradients can be calculated—to first approxi-

80 / Jandera

Fig. 30 Top: The resolution window diagram for the ‘‘elution strength’’ ternary gradient-elution separation of a mixture of 12 phenylurea herbicides in dependence on the initial sum of concentrations of methanol and acetonitrile in water at the start of the gradient, AT, with the concentration ratio of acetonitrile, X = u(acetonitrile) / (u(acetonitrile) + u(methanol)) = 0.4 optimized for isocratic ternary mobile phases and optimum gradient volume VG = 31 mL. Column and sample compounds as in Fig. 29. Bottom: The separation of the 12 phenylurea herbicides with optimized ternary gradient from 18.6% methanol + 12.4% acetonitrile in water to 60% methanol + 40% acetonitrile in water in 73 min. Flow rate 1 mL/min, T = 40jC.

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mation—from Eq. (12) with the constants A = uT /(1 + Ai /Aj), a = ai  miuT, m = (aj  ai) /uT + mi  mj Ai and Aj are the initial concentrations of the polar solvents i and j, respectively, at the start of the gradient. In normal phase systems, the retention in ternary mobile phases is controlled by the equation: 1 / k = aV + bVX + c VX 2 at a constant sum of concentrations of the two polar solvents, i and j, uT = ui + uj [X = ui/ uT; Eq. (44)]. The net retention volumes, VRV = VR  Vm, in ‘‘selectivity’’ ternary gradients controlled by Eq. (44) can be calculated from Eq. (46) [130,169]: 2

ðVRVÞ ðVRVÞ3 cVB2 þ ðbV þ 2c VX0 ÞB þ VRVðaV þ bVX0 þ cVX02 Þ ¼ Vm 3 2 ð46Þ The constants aV, bV, cV in Eq. (46) depend on the solute, on the chromatographic system, and on the preset sum of concentrations of the two polar solvents, i and j, uT, and can be determined in at least three initial isocratic experiments with ternary mobile phases.

IX. PECULIARITIES OF GRADIENT ELUTION SEPARATION OF HIGH-MOLECULAR COMPOUNDS Large molecules can be partially or completely excluded from the pores of the packing material as their size more or less limits the pore volume accessibility. This phenomenon has been used for many years for the determination of the mass distribution of polymers by sizeexclusion chromatography (SEC) under conditions where the macromolecules are not retained on the adsorption sites of the column packing material. On the other hand, the interactions with the adsorbent surface can also be utilized for separations of large molecules in the ‘‘interactive chromatography’’ (IC) of polymers. Interactive chromatography of polymers is becoming increasingly popular because it not only often provides better separation selectivity than SEC for the individual species differing by the molecular mass distribution, but principally also enables the separation of homopolymers according to the functionality (endgroups) distribution and of block copolymers according to the chemical composition and sequence

82 / Jandera distribution of the repeat monomer units in the individual blocks. To take full advantage of IC separation possibilities, as high surface area of the column packing material as possible should be accessible to the macromolecules. Therefore wide-pore materials are preferred for the separation of both biopolymers and synthetic polymers. The ‘‘polydispersity’’ of real samples of synthetic polymers may cause complex chromatograms in IC, whose interpretation can be facilitated by adequate knowledge of the retention mechanism. Many polymer samples cannot be successfully separated by isocratic IC, because a very small change in the mobile phase composition has much stronger effect on the retention of large molecules than on the retention of low-molecular compounds, so that gradient elution is generally required for successful separation (see the discussion below). The mechanism of gradient-elution polymer liquid chromatography has been for a long time—and still is—a matter of dispute. Glo¨ckner et al. [41–43] strongly advocated a precipitation–elution mechanism. On the other hand, many aspects of the retention behavior of polymers can be explained by a mechanism very similar to that of low-molecular compounds, taking into account the effect of the size of molecules on the constants of the retention–mobile phase composition equations [Eq. (10), (11), (15) or (16)], as suggested by the groups of Snyder [4,39,201], Jandera [40,81,177,202–205], Lochmu¨ler [206, 207], and Schoenmakers [208], both for reversed-phase and normalphase gradient-elution polymer chromatography. This model was found to describe adequately the retention of oligomers and lower homopolymers and copolymers up to the molecular weight 1000–3000 [38,108,125,202–205]. The differences between the retention behavior of small and large molecules can be explained as follows: Generally, various structural elements in the molecule contribute additively to the free energy of the solute distribution between the stationary and the mobile phases and hence to the logarithm of the retention factors (Martin rule) [209], so that the retention of polymers and oligomers increases with increasing number of repeat monomer units, n, according to a second-order polynomial Eq. (47) [202]: log k ¼ log b þ n log a þ n2 log c

ð47Þ

The term a characterizes the repeat unit separation selectivity, the term b the contribution of the end groups to the retention, and the quadratic term c is a measure of occasionally observed deviations

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caused by changing conformation of adsorbed large molecules and other nonideal effects. The constant c is often small enough and the quadratic term in Eq. (47) can be neglected, at least over a limited range of repeat monomer units. In this case, the constants m, a (log k0) in Eqs. (11) and (15) are directly proportional to n [202,203]: m ¼ mV0 þ mV1 n;

a ¼ log k0 ¼ a0V þ a1V n

ð48A; BÞ

These relationships can be introduced into the appropriate equation [Eq. (3), (5), or (7)] to describe the dependence of the gradient elution volumes on the number of monomer units, n. The retention behavior in agreement with Eq. (48A,B) was found in practice for many oligomer and lower polymer samples, at least over a limited monomer unit range [203]. Table 4 shows a few examples of the experimental constants of the Eq. (48A,B) for various oligomers in reversed-phase and in normal-phase systems. The negative value of m1V and null value of a1V for oligoethylene glycol alkyl ethers in acetonitrile–water mobile phases on a C18 column means that the retention decreases in the order of increasing number of monomer oxyethylene units [204,205]. Such behavior, described by some authors as ‘‘liquid exclusion-adsorption chromatography’’ [210– 212], can be explained by negative adsorption energy (greater affinity to the mobile than to the stationary phase) of an oxyethylene unit rather than by entropic effects caused by steric exclusion of these units [203], as will be discussed later. For reversed-phase LC of peptides and proteins, Stadalius et al. [213] suggested the following approximate dependence of the parameter m of Eq. (11) [the constant S in the original Snyder notation according to Eq. (2)] on the molecular weight, Mr: m ¼ 0:48 Mr0:44

ð49Þ

The necessity for using gradient elution in IC separations of polymers follows directly from the Eq. (48A,B), showing that the constants a and m of Eq. (3), (5) or (7) regularly increase with increasing number of repeat monomer units, as long as each monomer unit provides a constant contribution to the energy of retention and consequently to log k . Hence a and m may be very large for higher polymers. For example, the constants m in reversed-phase systems are in between 2 and 4.5 for alkylbenzenes (toluene to decylbenzene) with

84 / Jandera Table 4 Constants of Eq. (48A,B) for oligomer samples Oligomers PS OEG OEG OEP OEA OEA

Monomer unit

S

a0V

a1V

m0V

m1V

u range

Retention

C6H5UCHUCH2– UCH2UCH2UO– UCH2UCH2UO– UCH2UCH2UO– UCH2UCH2UO– UCH2UCH2UO–

D M P P M A

2.49 1.1 0.9 2.69 7.37 4.1

0.77 0.36 0.34 0 0 0

3.12 0.61 1.4 3.89 7.49 3.9

0.83 0.6 3.26 0 0 0.1

<93% <61% <10% <100% <95% <78%

z z z # z #

Reversed-phase systems, Separon SGX C18 column, binary mobile phases containing organic solvents (S): dioxane—D, methanol—M, 2-propanol—P, or acetonitrile—A.

Oligomers PS PS OEP OEP OEP OEP OEP OEA OEA

Monomer unit

Column, S

a0V

C6H5—CH—CH2U C6H5UCHUCH2U UCH2UCH2UOU UCH2UCH2UOU UCH2UCH2UOU UCH2UCH2UOU UCH2UCH2UOU UCH2UCH2UOU UCH2—CH2UOU

1, T 1, D 1, P 1 2, P 3, P 4, P 4, P 4, ADW

1.35 0.92 1.47 2.46 1.81 1.6 1.78 1.07 1.01

A1V

m0V

m1V

u range

Retention

0 0 0.4 0.3 0 0.2 0.2 0.2 0.1

0.5 0.3 1 3.3 0.9 0.6 1.3 0.2 1.4

0 0.11 0.1 0.3 0.14 0 0 0.1 0

<100% <100% <100% >8% <100% <100% <100% <100% >1%

z z z z z z z z z

Normal-phase systems, columns (C): (1) Separon SGX (silica), (2) Separon SGX Nitrile, (3) Silasorb Diol and (4) Separon SGX Amine, mobile phases n-hexane-polar solvents (S): tetrahydrofuran-T, dioxane-D, 2-propanol-P, ethanol-E; ADW-acetonitrile; water 99:1 in CH2Cl2; The retention either increases (z) or decreases (#) in the order of increasing number of repeat monomer units. PS—polystyrenes, OEG—oligoethylene glycols, OEP—oligoethyleneglycol nonylphenyl ethers, OEA—oligoethyleneglycol hexadecyl ethers; in concentration ranges u of strong solvents (S).

Mr = 92 – 218 [202] (Fig. 31), but in the range 25–70 for polybutylacrylates and polystyrenes with Mr in between 10,000 and 20,000 (Fig. 32) [214]. Because log k is directly proportional to the product m u [in reversed-phase systems, Eq. (11)], or to the product m log u [in normal-phase systems, Eq. (15)], a small change in the concentration of the strong solvent B, u, causes a much more significant change in the retention (k) of large molecules than in the retention of small

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Fig. 31 Dependencies of the constants a and m of Eq. (48A,B) on the number of carbon atoms, n, in the alkyl chains of alkylbenzenes. Column: Lichrospher 60RP-select B, 5 Am, 125
4 mm i.d., mobile phases acetonitrile–water, 40jC.

Fig. 32 Dependencies of the constants m of Eq. (48A) on the molecular weight, Mr, and on the number of repeat monomer units in polybutylacrylate (nBA, 1) and polystyrene (nS, 2) homopolymers on a Symmetry C18, 5 Am (100 ˚ 150
3.9 mm i.d., column in tetrahydrofuran–water mobile phases, 20jC. A),

86 / Jandera molecules and often increasing the concentration of B by even a few tenths of percent may cause transition from ‘‘full retention’’ to ‘‘no retentionufull elution’’ behavior. Consequently, only a narrow composition range of the mobile phase is available for the elution of large molecules. For example, from the data in Table 4 it can be predicted that a polystyrene sample with molecular weight 10,000 (with approximately 100 repeat styrene units) has k = 2 on a C18 column in 86.9% dioxane in water (best elution conditions), but k = 300 in 85% dioxane (very strong retention) and k = 0.3 (very low retention) in 88% dioxane. This means that such macromolecular samples are either fully retained or nonretained at all almost over the whole composition range of the mobile phases, except for a very narrow composition interval, which is often difficult to employ reproducibly for isocratic polymer separations. The elution range is even more limited for samples with molecular weights higher than 10,000. Consequently, the application of gradient elution is a prerequisite for utilization of the narrow mobile phase ‘‘composition window’’ available for the elution of the individual species in the polymer samples [203]. The width of the mobile phase ‘‘composition window’’ depends on the repeat monomer unit range in the sample and on the constant m1V of Eq. (11) or Eq. (15). m1V is a measure of the effect of the solvent B on the repeat unit selectivity and depends on the type of the solvent B, whose correct selection is essential for successful separation. Relatively steep gradients with a wide concentration range can be used for separations of low-molecular oligomers, such as linear gradient from 0% to 90% propan-2-ol in heptane for the separation of oligoethyleneglycol nonylphenyl ethers with 0–25 oxyethylene units (Mr = 220 – 1320) shown in Fig. 10 [215]. Because of the strong effects of even minor changes in mobile phase composition on the retention of samples with high molecular weights (hence with high values of m1V ), shallow gradients are necessary for normal-phase separations of polymers, such as for polystyrenes with Mr = 35,000 – 470,000 on two silica gel columns in series using a gradient from 47% to 50% dioxane in hexane in 15 min shown in Fig. 14. Here the elution volumes of narrow-distribution polystyrene standards with molecular weights 110,000 and 470,000 differ by only 7 mL, which precludes the separation of the individual high-molecular polymer species according to the number of monomer units. The polystyrenes with Mr < 20,000 are not retained under these conditions and elute from the column at the size-exclusion volume.

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The considerations based on the validity of Eq. (48A,B) lead—in agreement with numerous experimental results—to the conclusion that there is a certain upper molecular weight, depending on the polymer or oligomer type and on the chromatographic system used— which sets practical limits to the separation of the individual highmolecular polymer species by interactive chromatography reversedphase or in normal-phase systems. This can be illustrated by the example of reversed-phase separation of narrow mass distribution polystyrene standards in dioxane–water mobile phase using the data in Table 4: The polystyrene standard with 1000 units (with molar mass approximately 100,000) has k = 1.890 in 87.45% dioxane, but the next polymer with 1001 units has k = 1.892, so that the repeat unit selectivity characterized by the separation factor a is only 1.001 and a column with 36 million theoretical plates would be required to separate completely these two species differing by one styrene unit [203]. The constants a1V = 0 and m1V = 0 [Eq. (11)] in Table 4 indicate that oligoethyleneglycol nonylphenylethers coelute in propanol–water mobile phases and oligoethyleneglycol alkylethers coelute in methanol–water mobile phases over a wide range of mobile phase compositions in reversed-phase systems. The coelution of homopolymers or oligomers with different molecular weights is often described as ‘‘liquid chromatography under critical conditions’’ and is usually attributed to the compensation of the size-exclusion (entropic) and interactive (enthalpic, e.g., adsorption) effects [212]. However, real compensation of size-exclusion and interactive effects means that homopolymers are neither excluded nor retained and hence should elute close to the column hold-up volume. For some lower polymers, ‘‘critical conditions’’ providing the coelution of the species with different molecular weights are observed even at an elution volume significantly larger than the column holdup volume. In such a case the SEC has only a minor role—if any—and the coelution can be explained by the compensation of the energy of interactions of the repeat monomer units with the stationary phase on one hand and with the mobile phase on the other [203]. Such conditions can be used for the separation of homopolymers with different end groups [45]. Chromatography under ‘‘critical conditions’’ can facilitate also the analysis and characterization of block cooligomers and copolymers. Here ‘‘critical conditions’’ are used to intentionally suppress

88 / Jandera the separation according to the distribution of the repeat monomer units in one block and to enhance the separation selectivity according to the number of the repeat units in the other block. Assuming that the deviations from the Martin rule are not very significant [i.e., cc1 in Eq. (47)] and combining Eqs. (47), (48A,B), and (11) or (15), we can predict the ‘‘critical’’ concentration ucrit of the strong solvent B from Eq. (50A) in reversed-phase (RP) systems and from Eq. (50B) in normal-phase (NP) systems, as shown by Jandera et al. [81,203, 216] and later by Schoenmakers et al. [208]:   a1V a1V mV ðRPÞ; ucrit ¼ 10 1 ðNPÞ ð50A; BÞ ucrit ¼ mV1 The ‘‘critical concentration’’ ucrit in interactive LC of copolymers corresponds to the retention behavior that occurs when the interactions of the repeat monomer units in one block with the mobile phase more or less compensate their interactions with the stationary phase, so that the contribution of one type of the repeat units to the retention of the individual species in one block is (almost) null and the separation follows only the distribution of the monomer units in the other block. In practice, adjusting the conditions for coelution of one block in pure interactive chromatography is usually possible only if there are significant differences in the polarities between the monomer units in the individual blocks. In such a case, it may be possible to adjust reversed-phase conditions for the separation according to the monomer distribution in the less polar block under ‘‘critical’’ conditions for the more polar block and normal-phase conditions for the separation according to the monomer distribution in the more polar block under ‘‘critical’’ conditions for the less polar block. Equation (50A,B) fails if the values of the two constants a1, m1 of Eq. (11) or (15) are close to 0. In such an instance, the species with different numbers of repeat monomer units in this block coelute over a broad composition range of the mobile phases, hence gradient elution separation of copolymers under ‘‘critical conditions’’ is possible. This is illustrated by the examples of separation of ethylene oxide–propylene oxide (EO–PO) block cooligomers in Figs. 33 and 34. Figure 33 shows reversed-phase gradient elution separation of a three-block PO–EO–PO cooligomer (Novanik) on a C18 column using gradient of acetonitrile in water under the conditions where the separation occurs only according to the distribution of less polar propylene oxide

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Fig. 33 Gradient-elution reversed-phase separation of an ethylene oxidepropylene oxide co-oligomer Novanik 600/20 sample. Column: Separon SGX C18, 5 Am (150
3.3 mm i.d.). Linear gradient, 15–100% acetonitrile in water in 25 min, 0.5 mL/min. Evaporative light-scattering detection. Peak numbers correspond to the numbers of PO units at the conditions of coelution for the species with different numbers of oxyethylene units.

Fig. 34 Gradient-elution normal-phase separation of an ethylene oxidepropylene oxide co-oligomer Slovanik 320 sample on a Separon SGX Amine column. Linear gradient, 10–30% 2-propanol in n-heptane in 30 min. Flow rate 0.5 mL/min, light scattering detection. Peak numbers correspond to oligomers with 0–4 oxythylene units at the conditions of co-elution for the species with different numbers of oxypropylene units.

90 / Jandera units while the separation according to the distribution of more polar ethylene oxide units in the EO block is suppressed (‘‘critical conditions’’ for the EO block). On the other hand, the normal-phase gradient-elution separation of another three-block EO–PO–EO cooligomer (Slovanik) on a bonded aminopropyl column with a gradient of 2-propanol in hexane shown in Fig. 34 occurs under ‘‘critical conditions’’ for the less polar PO block and follows only the mass distribution of more polar ethylene oxide units in the EO block [203]. If a significant part of the pore volumes in the column packing material is not accessible to large molecules, the calculated gradient retention volumes can be corrected for size exclusion by using the size-exclusion volume, VSEC, determined in a strong mobile phase where the adsorption is completely suppressed, instead of the column hold-up volume, Vm, in the appropriate equation [such as Eq. (12), (17), (18), (21), (25)–(27), (29), (32), or (33)] [26], assuming that size exclusion does not affect significantly the phase ratio in the column. Linear gradients often provide satisfactory separation of oligomers and lower polymers, but nonlinear convex gradients can improve the peak capacity and band spacing in the chromatograms of macromolecular samples and generally decrease the analysis time under both reversed-phase and normal-phase conditions (see the example in Fig. 5) [109].

X. CONCLUSION Although gradient-elution HPLC is mainly used for the separation of low-molecular compounds and of biopolymers in reversed-phase systems, normal-phase gradient elution offers important advantages for the separation of certain types of samples, such as synthetic polymers or enantiomers. Normal-phase gradient elution facilitates the treatment of various samples extracted into organic solvents and often provides better separation selectivity for positional isomers with respect to reversed-phase gradient separations. The theory of gradient elution allows predicting the retention and optimizing the resolution both in reversed-phase and in normalphase systems, using the parameters of the experimentally determined dependencies of sample retention on the mobile phase composition. The effect of changing gradient time and range can be rapidly estimated using simple calculation rules. However, for exact predic-

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tion of the retention it is possible to use sophisticated calculations, taking into account nonideal behavior. The most important sources of deviations between the expected and the experimental retention data are the migration of sample compounds in the gradient dwell volume of the instrument and preferential adsorption of the strong solvent from the mobile phase in the column that can occur during the gradient run. The gradient dwell volume effects can be eliminated in some instruments by delayed sample injection after the start of the gradient, or the effects of different instrumental gradient dwell volumes can be predicted and corrected for by relatively simple calculation when transferring gradient methods between different instruments. To suppress the effect of the adsorption of polar solvents and water from the mobile phase in normal-phase chromatography, dried mobile phases are recommended and the gradients should be preferably started at a nonzero concentration of the polar solvent. If this effect cannot be avoided, it can be respected in exact calculations of the gradient retention data. A window-diagram-type approach can be used for optimization of gradient time and gradient concentration range for binary and ternary gradient elution, both in reversed-phase and in normal-phase systems. The transfer of gradient methods between various instruments and columns is less straightforward than in isocratic chromatography, as various operational parameters affect simultaneously the separation, but can be facilitated by using several simple rules based on the theory of gradient elution. The gradient time shall always be adapted when the mobile-phase flow rate, length, or diameter of the column changes, to get predictable sample separation. Rapid gradients can be accomplished on short columns packed with small diameter particles, but the speed of separation should be traded for the peak capacity. Good separation of many synthetic oligomers or of lower homopolymers according to the molar mass distribution can often be achieved either in normal-phase or in reversed-phase systems using shallow gradients over a narrow mobile-phase composition range, as an alternative to size-exclusion chromatographic separations. The suitability of a chromatographic system for the separation of homopolymers according to the end-group distribution, or of block cooligomers or copolymers under ‘‘critical’’ conditions for one of the blocks of monomers, depends on the polarities of the repeat monomer units and of the end groups.

92 / Jandera

ACKNOWLEDGMENTS This research was partly supported by the Ministry of Education of the Czech Republic under project No. 253100002 and by the Grant Agency of the Czech Republic, project No. 203/04/0917.

SYMBOLS A = u0 Ai, Aj

AT = Ai + Aj A1 A2 B BV Bi, Bj BT B1 Fm H K0 Mr N P S V VG VB

Concentration of the strong solvent B in the mobile phase at the start of the gradient Concentrations of the strong solvents i and j in the mobile phase at the start of ternary gradient elution [Eq. (42)] Total concentration of strong solvents i and j in ternary gradient elution Constant of the two-layer associative isotherm [Eq. (31)] Constant of the two-layer associative isotherm [Eq. (31)] Gradient steepness parameter, per volume unit of the eluate [Eq. (6)] Gradient steepness parameter, per time unit from the start of elution [Eq. (6)] Steepness of the concentration change of solvents i and j in ternary gradient elution Steepness of the total concentration change of solvents i and j, ui + uj, in ternary gradient elution Constant of the two-layer associative isotherm [Eq. (31)] Flow rate of the mobile phase Height equivalent to theoretical plate Specific permeability of the column [Eq. (39)] Molecular weight of the analyte Number of theoretical plates of the column Column peak capacity in gradient elution [Eq. (37A,B)] Solvent strength parameter [Eq. (2)] Volume of eluate from the start of elution Gradient volume Breakthrough volume of the polar solvent in the mobile phase [Eq. (A2)]

Gradient Elution in LC Chromatography VBV VD VR VRV VRV (C) VRV (U) V VR1

V VR2

VS VSEC Vads

Vm VmG Vm1

Vm2

Vsat Vsolv X X0

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93

Net breakthrough volume of the polar solvent in the mobile phase, VBV = VB  Vm [Eq. (A1)] Gradient dwell volume of the instrument Elution volume of a sample compound Net elution volume of a sample compound, VRV = VR  Vm Net elution volume corrected for the polar solvent uptake on the column [Eq. (A12)] Uncorrected net elution volume [Eq. (A12)] Part of the net elution volume of a sample compound contributed by the isocratic step in two-step elution or by the gradient dwell volume [Eqs. (23)–(29)] Part of the net elution volume of a sample compound contributed by the second, gradient step in two-step elution [Eq. (23)] Volume of the stationary phase in the column Size-exclusion volume for nonretained highmolecular compounds Volume of the pure polar solvent adsorbed on the column at the time of elution of a sample compound Volume of the mobile phase in the column = column hold-up volume Part of the column hold-up volume migrated by a sample compound in the postgradient isocratic step Part of the column hold-up volume migrated by a sample compound in the isocratic step of the two-step elution or in the period corresponding to the gradient dwell volume Part of the column hold-up volume migrated by a sample compound in the second, gradient step of the two-step elution Volume of the pure polar solvent necessary for full saturation of the column Volume of the pure polar solvent necessary to accomplish the elution of a sample compound Volume fraction of the solvent i in the mixture of the solvents i and j in a ternary mobile phase X at the start of a ‘‘selectivity’’ ternary gradient [Eq. (44)]

94 / Jandera a aV aD ai, aj a0V a1V a1 b bV bs b1 b2 cV cD d dc dp k kA kG ka kf ki, kj k0 k1

k*

Constants in Eqs. (10), (11), and (16) Constant in Eq. (46) Constant of van Deemter Eq. (40) a of the strong solvents i, j in binary mobile phases [Eq. (45)] Contribution to a of the end group in a polymer [Eq. (48B)] Contribution to a of a repeat monomer group in a polymer [Eq. (48B)] Constant of the Langmuir [Eq. (30)] and two-layer associative [Eq. (31)] isotherms Constant in Eq. (16) Constant in Eq. (46) Steepness of a linear solvent strength gradient [Eq. (1)] Constant of the Langmuir [Eq. (30)] and two-layer associative [Eq. (31)] isotherms Constant of the two-layer associative isotherm [Eq. (31)] Constant in Eq. (46) Constant of van Deemter Eq. (40) Constant in Eq. (10) Column diameter Mean packing particle diameter Retention factor of a sample solute, k = (VR  Vm)/ Vm k in pure nonpolar solvent A in normal-phase LC k in the isocratic postgradient step k at the start of gradient elution [Eq. (1)] k at the column outlet at time of elution of a band maximum k in binary mobile phases containing solvents i, j in concentrations ui, uj [Eq. (45)] k in pure polar solvent B in normal-phase LC, constants in Eqs. (14) and (15) k in the isocratic step prior to the gradient step in two-step elution or in the dwell volume mobile phase Mean k during the band migration along the column in gradient elution

Gradient Elution in LC Chromatography l m mi, mj m0V m1V n q qS q1S

t tm tG tR tR,1 tR,Z tRV u wg x Du Dp U a b c g

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95

Column length Constants in Eqs. (10), (11), (15), and (16) m of the strong solvents i, j in binary mobile phases [Eq. (45)] Contribution to m of the end group in a polymer [Eq. (48A)] Contribution to m of a repeat monomer group in a polymer [Eq. (48A)] Number of repeat monomer units in a polymer [Eq. (48A,B)] Concentration of the adsorbed solvent in the stationary phase Saturation capacity concentration of the adsorbed solvent in the stationary phase qS in the first adsorbed layer of the solvent in the stationary phase for a two-layer associative isotherm [Eq. (31)] Time from the start of gradient elution Column hold-up time Time of the gradient Elution time of band maximum Elution time of the first peak [Eq. (37A)] Elution time of the last peak [Eq. (37A)] Net elution time Mean linear velocity of the mobile phase in the column Bandwidth of a solute in gradient-elution HPLC [Eq. (8)] Molar fraction of polar solvent B Gradient concentration range = change in u from the start to the end of gradient elution Column operating pressure Phase ratio in the column, U = VS /Vm Repeat unit separation selectivity of a polymer [Eq. (47)] End group contribution to log k of a polymer [Eq. (47)] Contribution of nonideal behavior to log k of a polymer [Eq. (47)] Mobile phase viscosity

96 / Jandera j u uB uf ui, uj uG uT = ui + uj u0i

Gradient shape parameter [Eq. (7)] Concentration of the polar solvent in the mobile phase (or in the eluate) Breakthrough concentration of the polar solvent in the eluate Concentration of the polar solvent in the eluate at the time of elution of a sample compound Concentrations of the two strong solvents i and j in ternary mobile phases [Eq. (42)] Concentration of the polar solvent in the eluate at the end of the gradient (in time tG) Total concentration of polar solvents i and j in an elution strength ternary gradient [Eq. (42)] ui at the start of a ‘‘selectivity’’ ternary gradient [Eq. (43)]

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Snyder, L.R. Anal. Chem. 1974, 46, 1384. Snyder, L.R.; Glajch, J.L. J. Chromatogr. 1981, 214, 1. Glajch, J.L.; Snyder, L.R. J. Chromatogr. 1981, 214, 21. Snyder, L.R. J. Chromatogr. 1983, 255, 3. Soczewin´ski, E. Anal. Chem. 1969, 41, 179. Soczewin´ski, E.; Golkiewicz, W. Chromatographia 1971, 4, 501. Jaroniec, M.; Rozylo, J.K.; Oscik-Mendyk, B. J. Chromatogr. 1979, 179, 237. Scott, R.P.W.; Kucera, P. Anal. Chem. 1973, 45, 749. Scott, R.P.W.; Kucera, P. J. Chromatogr. 1975, 112, 425. Snyder, L.R.; Poppe, H. J. Chromatogr. 1980, 184, 363. Martire, D.E.; Boehm, R.E. J. Liq. Chromatogr. 1980, 3, 753. Jandera, P. Chromatographia 1988, 26, 417. Jandera, P.; Holcapek, ˇ M.; Theodoridis, G. J. Chromatogr. A. 1998, 813, 299. Snyder, L.R.; Glajch, J.L.; Kirkland, J.J. J. Chromatogr. 1987, 385, 125. Meyer, V.R. J. Chromatogr. A 1997, 768, 315. Thomas, J.-P.; Brun, A.P.; Bounine, J.P. J. Chromatogr. 1979, 172, 107. Engelhardt, H.; Bo¨hme, W. J. Chromatogr. 1977, 133, 380. ´ , M.; Holi´kova´, J. J. Chromatogr. A 1997, Jandera, P.; Kucerova ˇ 762, 15. Jandera, P.; Chura´cek, ˇ J. J. Chromatogr. 1974, 93, 17. Jandera, P.; Janderova´, M.; Chura´cek, ˇ J. J. Chromatogr. 1975, 115, 9. Jandera, P.; Janderova´, M.; Chura´cek, ˇ J. J. Chromatogr. 1978, 148, 79. Soczewin´ski, E. J. Chromatogr. 1977, 130, 23. Golkiewicz, W.; Soczewin´ski, E. Chromatographia 1978, 11, 454. Hara, S. J. Chromatogr. 1977, 137, 41. Snyder, L.R. Anal. Chem. 1974, 46, 1384. Antia, F.D.; Horva´th, Cs. J. Chromatogr. 1991, 550, 411. Treiber, L.R. J. Chromatogr. A 1995, 696, 193. Kirkland, J.J.; Dilke, C.H., Jr.; DeStefano, J.J. J. Chromatogr. 1993, 635, 19. Churms, S.A. J. Chromatogr. 1990, 500, 555. Olsen, B.A. J. Chromatogr. 2000, 913, 113. Alpert, A.J. J. Chromatogr. 1990, 499, 177. Jandera, P.; Chura´ ˇcek, J. J. Chromatogr. 1975, 104, 9.

102 / Jandera 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172.

Jandera, P. J. Chromatogr. A 1998, 797, 11. Baba, Y.; Yoza, N.; Ohashi, S. J. Chromatogr. 1985, 348, 27. Baba, Y. J. Chromatogr. 1989, 485, 143. Kopaciewicz, W.; Rounds, M.A.; Fausnaugh, J.; Regnier, F.E. J. Chromatogr. 1983, 266, 3. Parente, E.S.; Wetlaufer, D.B. J. Chromatogr. 1986, 355, 29. Souza, S.R.; Tavares, M.F.M.; deCarvalho, L.R.F. J. Chromatogr. A 1998, 796, 335. Madden, J.E.; Avdalovic, N.; Jackson, P.E.; Haddad, P.R. J. Chromatogr. A 1999, 837, 65. Baba, Y.; Yoza, N.; Ogashi, S. J. Chromatogr. 1985, 350, 119. Baba, Y.; Yoza, N.; Ogashi, S. J. Chromatogr. 1985, 350, 461. Baba, Y.; Fukuda, M.; Yoza, N. J. Chromatogr. 1988, 458, 385. Baba, Y.; Ito, M.K. J. Chromatogr. 1989, 485, 647. Sasagawa, T.; Sakamoto, Y.; Hirose, T.; Yoshida, T.; Kobayashi, Y.; Sato, Y. J. Chromatogr. 1989, 485, 533. Stahlberg, J.; Jo¨nsson, B.; Horva´th, Cs. Anal. Chem. 1991, 63, 1867. Stahlberg, J.; Jo¨nsson, B. Anal. Chem. 1996, 68, 1536. Hallgren, E. J. Chromatogr. A 1999, 852, 351. Quarry, M.A.; Grob, R.L.; Snyder, L.R. J. Chromatogr. 1984, 285, 19. Canals, I.; Valko´, K.; Bosch, E.; Hill, A.P.; Rose´s, M. Anal. Chem. 2001, 73, 4937. Holcapek, ˇ M.; Jandera, P.; Fischer, J.; Prokes, ˇ B. J. Chromatogr. A 1999, 858, 13. Jandera, P.; Chura´ ˇcek, J. J. Chromatogr. 1979, 170, 1. Lisi, D.D.; Stuart, J.D.; Snyder, L.R. J. Chromatogr. 1991, 555, 1. Velayudhan, A.; Ladish, M.R. Anal. Chem. 1991, 63, 2028. Velayudhan, A.; Ladish, M.R. Chem. Eng. Sci. 1992, 47, 233. Le Ha, N.; Ungva´ral, J.; sz. Kova´ts, E. Anal. Chem. 1982, 52, 2410. Velayudhan, A.; Ladish, M.R. Ind. Eng. Chem. Res. 1995, 34, 2805. ´ , M. J. Chromatogr. A 1997, Jandera, P.; Petra´nek, L.; Kucerova ˇ 791, 1. Everett, D.H. Trans. Faraday Soc. 1964, 60, 1803. Langmuir, I. J. Am. Chem. Soc. 1916, 38, 2221. Jandera, P.; Guiochon, G. J. Chromatogr. 1992, 605, 1.

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ˇ Jandera, P.; Skavrada, M.; Andel, ˇ L.; Komers, D.; Guiochon, G. J. Chromatogr. A 2001, 908, 3. Stadalius, M.A.; Ghrist, B.F.D.; Snyder, L.R. J. Chromatogr. 1987, 387, 21. Kaliszan, R.; Haber, P.; Baczek, T.; Siluk, D.; Valko, K. J. Chromatogr. A 2002, 965, 117. Dolan, J.W.; Snyder, L.R. J. Chromatogr. A 1998, 799, 21. Jandera, P. J. Liq. Chromatogr. 2002, 25, 2899. Giddings, J.C. Anal. Chem. 1967, 39, 1027. Horva´th, Cs.; Lipsky, S.R. Anal. Chem. 1967, 39, 1993. Neue, U.D.; Carmody, J.L.; Cheng, Y.-F.; Lu, Z.; Phoebe, C.H.; Wheat, T.E. Adv. Chromatogr. 2001, 41, 93. Bidlingmaier, B.; Unger, K.K.; von Doehren, N. J. Chromatogr. A 1999, 832, 11. Plumb, R.; Dear, G.; Mallet, D.; Ayrton, J. Rapid Commun. Mass Spectrom. 2001, 15, 986. Neue, U.D.; Mazzeo, J.R. J. Sep. Sci. 2001, 24, 921. Schoenmakers, P.J. Optimisation of Chromatographic Selectivity; Elsevier: Amsterdam, 1986. Berridge, J.C. J. Chromatogr. 1982, 244, 1. Snyder, L.R.; Dolan, J.W.; Lommen, D.C. J. Chromatogr. 1989, 485, 65. Dolan, J.W.; Lommen, D.C.; Snyder, L.R. J. Chromatogr. 1989, 485, 91. Molnar, I. J. Chromatogr. A 2002, 965, 175. Dolan, J.W.; Snyder, L.R.; Saunders, D.L.; Van Heukelem, L. J. Chromatogr. A 1998, 803, 33. Jupille, T.H.; Dolan, J.W.; Snyder, L.R. Am. Lab. 1988, 20 (12), 20. Glajch, J.L.; Kirkland, J.J.; Squire, K.M.; Minor, J.M. J. Chromatogr. 1980, 199, 57. Kirkland, J.J.; Glajch, J.L. J. Chromatogr. 1983, 255, 27. Li, S.F.Y.; Khan, M.R.; Lee, H.K.; Ong, C.P. J. Liq. Chromatogr. 1991, 14, 3153. Dolan, J.W.; Snyder, L.R. J. Chromatogr. Sci. 1990, 28, 379. Drouen, A.; Dolan, J.W.; Snyder, L.R.; Poile, A.; Schoenmakers, P. LC GC 1991, 9, 714. Galushko, S.V.; Kamenchuk, A.A. LC GC Int. 1995, 8, 581. Jandera, P. J. Chromatogr. 1989, 485, 113. Jandera, P. J. Liq. Chromatogr. 1989, 12, 117.

104 / Jandera 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216.

Jandera, P.; Proke ˇs, B. J. Liq. Chromatogr. 1991, 14, 3125. Jandera, P.; Chura´cek, ˇ J.; Colin, H. J. Chromatogr. 1981, 214, 35. Stadalius, M.A.; Gold, H.S.; Snyder, L.R. J. Chromatogr. 1985, 327, 27. Jandera, P. J. Chromatogr. 1984, 314, 13. Jandera, P.; Hol ˇcapek, M.; Kola´rˇ ova´, L. Int. J. Polym. Anal. Charact. 2001, 6, 261. Jandera, P. J. Chromatogr. 1988, 449, 361. Jandera, P. Chromatographia 1988, 26, 417. Lochmu¨ler, C.; McGranaghan, M.B. Anal. Chem. 1989, 61, 2449. Lochmu¨ler, C.; Jiang, C.; Elomaa, M. J. Chromatogr. Sci. 1995, 33, 561. Schoenmakers, P.; Fitzpatrick, F.; Grothey, R. J. Chromatogr. A 2002, 965, 93. Martin, A.J.P. Biochem. Soc. Symp. 1949, 3, 4. Trathnigg, B.; Gorbunov, A.A. J. Chromatogr. A 2001, 910, 207. Trathnigg, B.; Rappel, C. J. Chromatogr. A 2002, 952, 149. Skvortsov, A.M.; Gorbunov, A.A. J. Chromatogr. 1990, 507, 487. Stadalius, M.A.; Gold, H.S.; Snyder, L.R. J. Chromatogr. 1984, 296, 31. Kola´ ˇrova´, L.; Jandera, P.; Claessens, H.; Vonk, E.C. Chromatographia. in press. ´cek,J.J.Chromatogr. Jandera,P.;Urba´nek,J.;Proke s,B.;Chura ˇ ˇ 1990, 504, 297. Jandera, P. In Retention and Selectivity in Liquid Chromatography; Smith, R.M., Ed.; Elsevier: Amsterdam, 1995.

APPENDIX A. CORRECTION OF THE RETENTION VOLUME IN NPHPLC FOR THE COLUMN UPTAKE OF POLAR SOLVENTS DURING GRADIENT ELUTION (SOLVENT-DEMIXING EFFECT) When a solvent B is preferentially adsorbed from a mixed mobile phase during gradient elution, its concentration in the mobile phase is lower than expected for the preset gradient and the column effluent contains only the pure weak solvent A until the breakthrough of the

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strong solvent B occurs when the column is saturated with B. The net breakthrough volume, VBV , can be calculated as the volume of the mobile phase necessary to bring the column into equilibrium with the mobile phase by adsorbing the volume Vads of pure solvent B. A linear gradient running from the initial concentration of B, u = A, to the final concentration of B, u = uG, in the time tG at a flow rate Fm is described by Eq. (6) and VBV can be determined by integration of a simple equation: ð VRV B 2 Vads ¼ cdV ¼ AVBV þ VBV ðA1Þ 2 0 from which we obtain Eq. (A2) for the breakthrough volume, VB: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2 þ 2BVads þ Vm þ VD VB ¼ VBV þ Vm þ VD ¼ ðA2Þ B and Eq. (A3) for the corresponding breakthrough concentration of the solvent B, uB: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA3Þ uB ¼ A þ BVBV ¼ A þ A2 þ 2 BVads (Vm is the column hold-up volume and VD is the gradient dwell volume.) For a gradient starting from zero concentration of the polar solvent B, A = 0: rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Vads þ Vm þ VD ; VB ¼ uB ¼ 2 BVads ðA4; A5Þ B Vads can be determined from the experimental adsorption isotherm of the polar solvent B between the column packing material and a two-component mobile phase. If the distribution of the solvent B between the stationary and the mobile phase is controlled by the Langmuir isotherm, Eq. (30), with a steep initial slope and column saturation with the solvent B is achieved at a low concentration of B in the mobile phase (because of a high value of the isotherm parameter b1) for a gradient starting at zero concentration of B, A = 0, Vads can be calculated as the volume of B, Vsat, necessary to reach the column saturation capacity concentration, qs: Vads ¼ Vsat ¼

UVm a1 ¼ UVm qs b1

ðA6Þ

106 / Jandera where U is the phase ratio, i.e., the ratio of the volumes of the stationary, Vs, and of the mobile, Vm, phases in the column. In some cases, the distribution isotherm does not allow accomplishing the full saturation of the column with the polar solvent B at the time of elution of sample compounds during the gradient run. If so, the volume of the polar solvent adsorbed on the column, Vads, is controlled by the actual elution volume, VRV , which depends on the individual solute and gradient program. Then, Vads can be determined by integrating the product of the volume of the stationary phase in the column and of a differential increase in the adsorbed concentration of the solvent B, q, from the initial equilibrium value at the start of the gradient, q0, to the adsorbed concentration at the solute elution time, qf: ð uf  ð qf dq Vads ¼ Vs dq ¼ UVm du ðA7Þ du q0 A Here q is expressed as the concentration of B in the whole volume of the stationary phase in the column, VS, which is—for simplicity sake—set equal to the part of the volume of the column that is not occupied by the mobile phase. The volume of the mobile phase in the column, Vm, is equal to the column hold-up volume, U = VS /Vm is the column phase ratio, and (dq/du) is the first derivation of the adsorption isotherm for the solvent B on the column packing. The first derivation of the Langmuir isotherm is described by the following equation:

 dq a1 ðA8Þ ¼ du ð1 þ b1 uÞ2 and the first derivation of the associative bi-layer isotherm, Eq. (31), by:

 dq a1 ðb1  a2 Þ a1 a2 þ ðA9Þ ¼ dc b1 b1 ð1 þ b1 cÞ2 After introducing the appropriate equation for (dq/du) we can solve Eq. (A10) for Vads:   UVm a1 1 1  ðA10Þ Vads ¼ 1 þ b1 A 1 þ b1 ðA þ BVRV Þ b1

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for the chromatographic systems where the Langmuir isotherm, Eq. (30), applies, or:   UVm a1 ðb1  a2 Þ 1 1  Vads ¼

1 þ b1 A 1 þ b1 ðA þ BVRV Þ b21 ðA11Þ UVm a1 a2 BVRV þ b1 for the systems controlled by the two-layer associative isotherm, Eq. (31). The uptake of the polar solvent B on the column occurring during normal-phase gradient elution can be accounted for by assuming a constant volume of the polar solvent B necessary to accomplish the elution, Vsolv. Hence the volume of B adsorbed on the column, Vads, should be added to Vsolv to correct the elution volume, VRV (U ), calculated from Eq. (17) or (18) for the adsorption effect. In this way, we obtain the following equation for the corrected elution volume, VRV (C ): Vsolv ¼ ½VRV ðU Þ 2

B B þ VRV ðU ÞA þ Vads ¼ ½VRV ðCÞ 2 þ VRV ðCÞA 2 2 ðA12Þ

APPENDIX B. SCHEMATICS OF A SPREADSHEET PROGRAM FOR OPTIMISATION OF GRADIENT ELUTION A. Input 1. The constants of a suitable retention–mobile phase composition equation [such as a, b, d, m, k0 in Eq. (10) or (11) for reversedphase HPLC, Eq. (15) or (16) for normal-phase HPLC or for ionexchange LC] are determined for each sample solute in at least two to three isocratic or gradient experimental runs. 2. The values of the appropriate constants a, b, d, m, k0, of the column plate number, N, and the column hold-up volume, Vm, the gradient dwell volume, VD, the gradient volume, VG, or the gradient time, tG, and the flow rate, Fm (the gradient curvature parameter, if necessary), are introduced as the input values. 3. Diagrams of the isocratic retention volumes, VR, and of the resolution, RS, are constructed for all sample components in depen-

108 / Jandera dence on the concentration of the solvent B in the mobile phase, u, to check if gradient elution is necessary.

B. Output 1. Diagrams of VR and of RS are constructed for all sample components in dependence on the concentration of the solvent B at the start of the gradient, A with VG, N, Vm ( Fm, VD) as adjustable parameters. 2. Optimum A is determined from the RS – A plots for the highest resolution of the ‘‘critical pair’’ of solutes or for the desired resolution obtained in minimum time. The corresponding VR is determined for all sample compounds from the VR – A plots. The concentration of the solvent B providing the elution of the whole sample at the end of the gradient, uG, is determined from the VR of the last eluting compound. 3. With optimized conditions, the expected chromatogram can be calculated and plotted. 4. If the optimized separation is not satisfactory, VG or other parameters (N, Vm) are varied to find optimum chromatographic conditions. VG can be varied also if the calculated retention volumes of more strongly retained compounds indicate the elution after the end of the gradient (VR > VG + Vm + VD). If such variation still does not result in the elution of the last compound before the end of the gradient even with 100% B final concentration (uG = 1), an isocratic final hold-up step with uG = 1 is included into the gradient program. 5. If further refinement of the separation is required, the use of curved or segmented gradients can be attempted and the whole procedure repeated.

2 Supercritical Fluids for Off-Line Sample Preparation in Food Analysis Prior to Chromatography Jerry W. King Los Alamos National Laboratory, Los Alamos, New Mexico, U.S.A.

I. SUPERCRITICAL FLUIDS FOR SAMPLE PREPARATION II. SUPERCRITICAL FLUID EXTRACTION (SFE) A. Basic Principles of SFE B. Types of Extraction and Instrumentation C. The Sample Matrix and Its Preparation for SFE D. The Problem with Coextractives and Water E. Collection of the Extracted Analyte III. INTEGRATION OF CLEANUP STEP WITH SFE A. Fluid Density-Based Fractionation B. Use of Adsorbents with SFE C. Inverse SFE D. Variation in Extraction Fluid Type or Composition

110 112 112 117 122 125 128 131 132 135 139 140

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IV. COUPLING REACTION CHEMISTRY (DERIVATIZATION) WITH SFE A. Reactions in Supercritical Fluid Media (SFR) B. Types of Derivatizations Used in SFE C. Utilization of Catalysts with SFR V. APPLICATIONS OF CRITICAL FLUIDS FOR SAMPLE PREPARATION A. Analysis of Trace Components B. Proximate Fat Analysis C. SFE Prior to Gas Chromatography (GC) D. SFE with HPLC or SFC E. SFE Integrated with Selected Chromatographic/ Spectroscopic Techniques (IR, MS) VI. STATUS OF THE TECHNIQUE—CONCLUSIONS REFERENCES APPENDIX A

142 142 144 146 149 150 153 156 158 164 166 167 174

I. SUPERCRITICAL FLUIDS FOR SAMPLE PREPARATION Sample preparation prior to chromatography has been an integral step of analytical method development that has received increasing emphasis in recent years. The rationale for this trend is the increasing complexity of chemical analysis, which continues to place a burden on the analyst using chromatographic methods. Hence improvements in sample preparation prior to analysis via chromatographic techniques can substantially reduce the complexity of such assays as well as reduce the attrition on columns and associated instrumentation. There are a plethora of sample preparation methods available, and this review will focus the attributes of supercritical fluids and similar compressed media as agents for this process. There are good reasons to consider the use of supercritical fluids in sample preparation prior to chromatography, particularly because the fluid is easily removed from the sample matrix after extraction or sample cleanup. The most widely used supercritical fluid, supercritical carbon dioxide (SC-CO2), is relatively inexpensive, nonflammable, and environmentally benign. This last feature has been a key factor in the development of critical-fluid-based methods in the early 1990s when legislative and regulatory mandates on the use of hazardous solvents were officially promulgated [1]. Such acts as the Environmental Protection Agency’s (EPA) Pollution Prevention Act, the

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Superfund Amendments and Reorganization Act (SARA), the Resource, Conservation, and Recovery Act (RCRA), and Montreal protocols were designed to reduce or eliminate the use of carcinogenic or environmentally adverse compounds such as chlorinated solvents, aromatics (i.e., benzene), and fluorinated hydrocarbons. Thus supercritical fluid extraction (SFE) using SC-CO2 provides a viable alternative to using the above solvent media and potentially totally reducing the analyst’s dependence on organic solvents altogether. Supercritical fluid extraction also competes with an assortment of other, relatively new sample preparation techniques that have been developed toward reducing the use of copious amounts of organic solvents. Practically all of these new sample preparation methods share some of the same generic features: a substantial reduction in the amount of solvent, reduction in the sample size of the sample matrix, and high sample throughput via automated, unattended operation. When one considers that classic extraction methods, such as the Soxhlet extraction technique, have been used for over 90 years [2], it is somewhat surprising that newer methods have not evolved at a faster pace in the interim. Modern supercritical fluid technology is documented in many books and reviews, which cover both processing as well as analytical utilization of these unique fluids. Some interesting primers for the novice to the field are by Taylor [3], Clifford [4], and Luque de Castro et al. [5]. Although engineering theory and applications might appear as having questionable relevance to the analytical chemist or chromatographer, nothing could be farther from the truth. The theory and fundamental principles of SFE share intradisplinary application as will be emphasized in the review that follows. Key tomes involving the fundamentals and processing utilization of critical fluids are by Brunner [6], McHugh and Krukonis [7], and Mukhopadhyay [8]. In addition, there are approximately 45 other major references dealing with the subject of supercritical fluids, and the author has listed these in Appendix A. This review admittedly focuses on the use of off-line SFE and its variants in preparing samples prior to chromatography. There are some researchers who share the view that analytical SFE is a derivative of activity in the field of analytical supercritical fluid chromatography (SFC) [9]. Although both analytical techniques share a common physicochemical basis, analytical SFE stands on its own merits, whether expedited in either the on-line vs. off-line mode. For semantic purposes, on-line SFE will refer to the technique

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when directly linked in tandem with another analytical technique, such as gas chromatography (GC), Fourier-transform infrared spectroscopy (FTIR), high-performance liquid chromatography (HPLC), or mass spectrometry (MS). Whereas on-line SFE, particularly when coupled with SFC, preceded off-line SFE in terms of development, it is the latter technique that has seen widespread use and resulted in commercial instrumentation [10]. Unfortunately, such on-line SFE techniques require a relatively high level of operator training to facilitate their use routinely in an analytical laboratory; consequently, there is a paucity of instrumentation that can be purchased outright for conducting on-line SFE. Despite these limitations, the reader is encouraged to read the volume by Ramsey [11] to appreciate the merits of on-line SFE methodology. In this review, some of the basic principles of supercitical fluids (SFs) will be presented, including their optimization during analytical SFE. Types of extraction and instrumentation will also be discussed, as well as the preparation of the sample prior to SFE, and collection of the extracted analyte after SFE. Integration of sample cleanup during and after SFE will be covered with an emphasis on handling the problem of coextractives that can plague chromatographic separation of the extracted analytes. Coupling selective reaction chemistry (derivatization) in SFs is another viable alternative toward easing the burden of sample preparation for the analyst. Such supercritical fluid reactions (SFRs) can be directly integrated in situ with the sample preparation step, directly on-line with chromatographic instrumentation. Finally, selected applications of critical fluids for sample preparation prior to chromatography will be cited with a bias toward our own method development in trace toxicant and lipid analysis [12].

II. SUPERCRITICAL FLUID EXTRACTION A. Basic Principles of Supercritical Fluid Extraction The supercritical fluid state for any substance maybe defined as existing above a specific temperature, known as the critical temperature, Tc, and a specific pressure, Pc, the critical pressure. Its relationship to other states of matter in the case of carbon dioxide is shown in Fig. 1. Here both Tc and Pc define a critical point on a pressure– temperature diagram, and correspondingly define a critical density,

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Fig. 1 Phase diagram of carbon dioxide.

which in the case of CO2 is approximately 0.45 g/mL [13]. One of the practical implications of these defined properties is that CO2 cannot be converted to its liquid state no matter what pressure is applied, as long as it is held above Tc. As we shall see, this confers some unusual and exploitative properties on critical fluids that can be used to advantage in performing SFE. The so-called ‘‘near critical fluids’’ have also been used to advantage, by operating the extraction in a temperature range slightly below the critical temperature (usually in the range of 0.85–0.95) in terms of the reduced temperature, Tr = T/Tc. When a fluid meets the above criteria, it exhibits physical properties that are intermediate between those of a gas or liquid, and its density can be changed by varying the applied pressure on the fluid. Therefore when a fluid is in a state of high compression, it takes on a high density, approximating those associated with liquid solvents. Under such conditions, the supercritical fluid has the capability of dissolving a variety of materials, just as liquids do. Also, to a more limited extent, the selectivity of a supercritical fluid can be changed by altering its density, akin to changing liquid solvents in conventional extraction. However, at high densities, the extraction selectivity of supercritical fluids is lost and their molecular specificity approximates that found for nonpolar to moderate polar solvents (in the case of CO2).

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To illustrate the above concepts, the changes that occur for a model solute, naphthalene, in SC-CO2 are shown in Fig. 2. Here a solubility of 5.2 mol% is found for naphthalene at a pressure of 300 atm and a temperature of approximately 55jC, parameters that correspond to conditions used in the extraction stage of SFE (E1). Separation of the naphthalene from the compressed CO2 after SFE can then be affected in one of two ways as depicted in Fig. 2. The separation of the solute from the SC-CO2 can take place at a constant pressure (300 atm) while the temperature is lowered to 20jC, thereby affecting a solubility change of 4.0 mol% (S2). However, an even larger change in naphthalene’s solubility in SC-CO2 can be achieved by reducing both the pressure and temperature at the separation stage (S2), by adjusting the pressure to 90 atm and temperature to approximately 45jC. Under these conditions, only 0.1 mol% of naphthalene is left in the SC-CO2 phase. The concept of ‘‘threshold pressure’’ with respect to supercritical fluids has it origins in the early studies of ‘‘dense gas chromatography’’ [14], the historical forerunner to SFC. This is defined as the pressure (at a specific temperature) at which the analyte can first be solubilized and detected in the extraction fluid. Threshold pressures

Fig. 2 Solubility of naphthalene in SC-CO2 under conditions corresponding to those used during extraction and separation (sample collection).

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are dependent on the detection method employed for estimating the initial solubilization of the solute in the SF, and can vary over magnitudes of concentration [15], depending on whether the detection technique is an element-specific GC detector, a TLC spot test, or a gravimetric balance. Threshold pressures will also be dependent on the sample matrix to some extent for a common solute; hence this factor should also be specified when quoting threshold pressures [16]. Despite these factors, the threshold pressure is a useful concept because it allows the analyst to know the minimum pressure conditions required for extracting the analyte from a given sample. Threshold pressures tend to have a weak dependence on temperature and molecular weight, and can be estimated for SFE and SFC using the guidelines developed by King [17]. A solute’s maximum solubility in a SF may or may not be of importance to the analyst depending on the specific analysis problem being considered. For example, for trace analysis purposes, there is usually sufficient solute solubility in the supercritical fluid based just on solubility considerations. However, if one is analyzing the fat content of a food matrix, then conditions for affecting high lipid solubilities in SFs are desired to minimize extraction time. Solute solubilities can be calculated as a function of fluid density from equations of state [18,19], or estimated from solubility parameter theory [17,20]. In some cases, the use of higher extraction fluid densities is desired to remove the target analyte from a very adsorptive sample matrix. When solutes are dissolved in supercritical fluids, they exhibit higher diffusivities than they do in liquids, thereby facilitating rapid mass transfer of the solutes from the sample matrix. Corresponding mass transport properties or dimensions also take on intermediate values between those of a dilute gas and liquid, and exhibit a dependence on fluid density. The end result of these trends is that faster extraction fluxes can be achieved using SFs, corresponding to a substantial reduction in extraction time. Indeed, for effective analytical SFE to occur, the triad noted in Fig. 3—the analyte’s solubility, diffusion, and interaction with the sample matrix—must all be considered in designing optimal extraction conditions. The rate of solute removal from a sample matrix using a SF is similar to those found while using liquid extraction solvents, except the time required is usually less. This is illustrated in Fig. 4 for the extraction of fat from a ham matrix. Initially, the extraction kinetics are governed by the solubility of the lipid in SC-CO2, that is to say,

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Fig. 3 The analytical SFE triangle.

there is an approximately linear removal of lipid into the SC-CO2 from the ham matrix [21]. This then gives way to a transition region in which the removal of the fat becomes rate limiting, followed by an asymptotic approach to the final lipid content with passage of the extraction fluid (SC-CO2). Such extraction curves have been modeled by several investigators and generalized in a ‘‘hot ball’’ kinetic model by Bartle et al. [22]. Knowledge of the extraction kinetics in SFE is important because it determines the time and quantity of extraction fluid required to complete the extraction. In addition, such extraction rate curves can be diagnostic, suggesting that if the extraction takes too long, the addition initially of a static extraction sequence may be beneficial.

Fig. 4 Supercritical fluid extraction of ham sample as a function of extraction time.

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Supercritical fluid extraction can be facilitated using both the dynamic and static modes of extraction, either individually or coupled in a stepwise sequence. In the dynamic extraction mode, the fluid is conveyed continuously through the sample matrix, while in the static mode, extraction fluid is pumped into the vessel containing the sample, and held for a predetermined time, prior to initiating dynamic extraction of the sample. In certain SFEs, just a single static extraction step is sufficient to yield the desired extract [23].

B. Types of Extraction and Instrumentation Instrumentation for analytical SFE evolved from home-built equipment, frequently constructed on a modular basis using several types of fluid delivery options (pumps, compressors, etc.), and pressure reduction/collection devices crafted from commercial vales/regulators, or even silica capillary tubing. These basic units were improved upon by instrumentation companies, resulting in the commercial instrumentation of today, which feature unattended operation, multisample capability, and several analyte collection options. Table 1 lists some of the desired features for performing analytical SFE. Most modern instrumentation is capable of conducting SFE up to pressures of 680 atm (10,000 psi), temperatures in excess of 100jC, and flow fluid rate ranges to 10 L/min (CO2 at STP). Sample size, a subject that was initially quite controversial in the early development of analytical SFE, is typically 1–10 g on commercial instruments, but options do exist to easily extract up to 50 g on certain instrumentation or home-built equipment. Collection options will be discussed in Sec. E, while typical automated instrumentation will be cited later in this section. Depending on how instrumentation is configured, it is possi-

Table 1 Desired Features in Analytical SFE Pressure, temperature and flow rate ranges Sample size range Variety of collection options Size and portability of instrument Automation Cosolvent capability Ability to interface with other instruments Delivery system for carbon dioxide

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ble to process in excess of six samples in one run, and up to 24 on one particular instrument. Cosolvent addition capability, which is highly desired in many applications, is available as an add-on feature. Figure 5 illustrates a basic, home-built extractor design that has been successfully utilized in our laboratory for over 20 years [24]. The unit consists of a gas booster unit, which delivers pressurized gaseous CO2 from a cylinder (A). The booster pump can easily provide extraction pressures of 680 atm and high flow rates for processing larger samples. The carbon dioxide is delivered without heat tracing to an oven enclosure (dotted line), and can be diverted downward or upward into a vertically positioned extractor vessel using double switching valves (SV-1, SV-2). Conversion of the fluid to the supercritical state prior to extracting the sample is achieved using a generous length of coiled tubing (HC-1, HC-2). Extraction cells can be fabricated using 316 stainless-steel tubing of varying lengths. Depending on the wall thickness, these extraction vessels can hold 50–70 or 100–140 mL of material for 70- or 140-MPa extractions. The extract is conveyed out of the extractor through another dual switching valve (SV-3, SV-4) into a heated micrometering valve. This valve must be heated to counteract the effects of Joule–Thomson cooling caused by depressurization (expansion) of the extraction fluid (e.g., CO2). This can be accomplished using a heating tape, hot air gun, or cartridge heaters inside an encasing aluminum block. As shown in

Fig. 5 Generic laboratory SFE unit. A = CO2 cylinder; TP = cylinder pressure gauge; CV = check valve; F = filter; C = air-driven gas booster compressor; RV = relief valve; SV = on/off switching valve; PG = pressure gauge; HC = equilibration coil; Tc = thermocouple; MV = micrometering valve; FM = flow meter; GT = gas totalizer.

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Fig. 5, the expanded gas can be conveyed through a receiver vessel through a rotameter (FM) and into a gas totalizer device (dry test meter). The performance of this basic and relatively inexpensive SFE unit is well documented in the literature [25–27]. An expansion of this basic extractor has been expanded to yield a versatile multisample SFE unit shown in Fig. 6 [28]. The basic principle behind this multisample instrument will allow simultaneous extraction of several samples in a parallel mode, thereby emulating classical Soxhlet extraction instrumentation. The fluid delivery system is very similar to that described for the unit in Fig. 5. A series of flow control and shutoff valves, operating in series, provides manually adjustable and stable flow rates to each of six extraction vessels. Flow is primarily controlled by the addition of a micrometering valves after each extraction vessel, including a two-stem valve, inserted before the micrometering valve, to relieve pressure from the column if required, or to measure fluid flow. A novel flow restrictor, developed by Hopper

Fig. 6 Simultaneous, parallel multisample SFE unit.

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[28], is also inserted in-line right before the extract is totally decompressed, to suppress the formation of volatile aerosols that can interfere with deposition of the extract into the collection vessels. Note that both the collection and extractor vessels have their own ovens, juxtaposed to house each array of vessels in its own thermally controlled environment. This instrument module was specifically designed to process relatively large samples (20–50 g), mandated for established regulatory protocols. Commercial instrumentation for SFE has largely been developed in the United States in the 1990s and is marketed throughout the world. Isco, Inc. (Lincoln, Nebraska) was one of the first companies to provide instrumentation for off-line SFE: their Model SFX 2–10 and SFX 220. Both units deliver the extraction fluid via syringe pumps with varying capacity and pressure range, although the 5000 and 10,000 psi pump modules are normally purchased for use with SFE. Extraction cells of 0.5, 2.5, and 10 mL are offered in three different cell materials: stainless steel,aluminum, and a high-temperature compatible polymeric composition (9-mL disposable cartridges). Although the SFX 2–10 module is entirely manual in operation, the extraction cells can be sealed without the need of wrenches (hand tightened); permitting two extractions to be conducted in parallel. Depressurization of the solute-laden fluid is normally accomplished through the use of either a fixed flow rate or adjustable flow rate, heated coaxial backpressure restrictor. Control of the fluid delivery flow rate, extraction cartridge temperature, and restrictor temperature is achieved by microprocessor control. The SFX 220 is an automated version of the SFX 2–10 with automated valve operation for increase sample throughput. An advantage of the above units is their modularity, which permits the analyst to design and alter the extraction unit. Cosolvents are delivered with the aid of an additional pump through the microprocessor controller. There is considerable flexibility in collecting the extract; both neat and solvent-based collection tubes can be interfaced with the coaxial-heated restrictors. The analyst also has the advantage of designing about any type of collection system with these units, including the use of sorbent-laden cartridges for on-line collection of lipids and volatiles for further sample preparation or offline analysis. Isco, Inc. also produces the Model SFX 3560, which allows up to 24 samples to be sequentially extracted. This module can operate

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unattended overnight through an interactive 80  24 microprocessor display, allowing both high sample throughput as well as automated method development. Programmable fluid ‘‘wash’’ cycles between each SFE is an integral part of the instrument’s operation, and both static and dynamic extraction modes can be performed on individual samples up to 10,000 psi and 150jC. Extract collection is accomplished using empty or solvent-filled vials using an automated, feed back-controlled heated restrictor to prevent icing. To aid in extract collection, the 20-mL glass tubes used for collection can be cooled to as low as 20jC as well as be pressurized above ambient conditions. Applied Separations, Inc. (Allentown, PA) offers several extraction units based on prototypes developed in the U.S. Department of Agriculture (USDA) laboratories [24] that offer considerable flexibility with respect to sample size and experimental design. These units can be purchased as single modules (the Spe-ed 2 or 4) having the capability of extracting two to four samples in parallel. Supercritical fluid extractions can also be conducted at high temperatures (250jC) and up to 10,000 psi (680 bar) with these units. Extractor vessel sizes can range from several milliliters to 1 L if required. With these units, the analyst has considerable choice with respect to the type of extract collection system that can be employed with the Spe-ed units. Leco Corporation (St. Joseph, Michigan) produces a Total Fat Analyzer designated the Model TFE 2000. Although lacking the modularity of the above-described instrumentation, the unit is carefully designed (and marketed) for total fat/oil determination using SC-CO2 as the extraction agent. The unit accommodates 10 mL extraction cells and operates up to 10,000 psi and 150jC. Flow rates from 0 to 5 L/min (expanded CO2 flow rate) are regulated using heated variable restrictors. A single module will accommodate up to three extraction cells, but separate units can be ‘‘piggybacked’’ to allow extraction of up to 24 samples in parallel. With all of the above instrumentation, CO2 is the preferred extraction fluid for reasons previously cited. For total fat or oil extractions, high-purity, SFE-grade CO2 is not always required; however, the impurity and moisture levels in various industrial grades of CO2 can accumulate and adversely effect instrumental performance. This accumulation of contaminants is of particular concern when analyzing for trace components using SFE, because the extraction step will tend to concentrate these contaminants in the collection stage. Such an accumulation of contaminants can ulti-

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mately interfere with the off-line analysis techniques, such as gas or liquid chromatography. Ultra-high-purity grades of CO2 (SFE and SFC grade-CO2) are available from several vendors of laboratorygrade gases, but they are relatively expensive. Alternatively, fluid purification schemes, such as those reported by Hopper et al. [28], or CO2 purification using a microporous ceramic oxide catalyst [29], can be employed to purify even welding grades of CO2. When utilizing analytical SFE, one should avoid the use of helium headspace-padded CO2 cylinders. This technique, originally developed to avoid the use of circulating coolers with fluid pumps, introduces small quantities of helium into the CO2 phase in the pressurized cylinder. Several investigators [30,31], however, have shown that the presence of He admixed with CO2 can reduce the solubility of solutes relative to their solubility in neat CO2. For example, the presence of helium in CO2 will reduce the solubility of lipids in SC-CO2 from 33% to 50%, depending on the chosen extraction pressure and temperature. Therefore use of such CO2 sources can lead to lower analyte recoveries from sample matrices and hence inconsistent analytical results.

C. The Sample Matrix and Its Preparation for Supercritical Fluid Extraction The choice of sample size for any analytical determination or preparation is perhaps more crucial than many analysts realize, and this applies equally as well when using SFE. In recent years, there has been an increasing trend toward smaller sample sizes because of two factors: improved sample comminution methods and the desire to have smaller analytical instrumentation in the laboratory environment, i.e., smaller ‘‘footprints’’ on the benchtop. The latter factor, to some extent, guided the design of the initial SFE instrumentation offered commercially, which as noted previously, accommodated an average sample size of approximately 10 g. Sample sizes smaller than this puts a premium on assuring sample homogeneity through mixing, grinding, and similar processes. However, such sample treatments must not alter the sample matrix via mechanical or thermal means, so that even the ‘‘homogenized’’ sample is no longer representative of the original whole sample. An application of analytical SFE where sample size becomes important is in the SFE of aflatoxins from corn and similar seed/ grain matrices, an extraction that usually requires the use of a

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cosolvent to achieve suitable analyte recoveries [32,33]. Aflatoxins in this particular case are generated on the corn matrix from infestation of a fungus, Aspergillus flavus, and evolve and spread from a specific site, leading to a potential maldistribution of the target analyte on a single kernel of corn, throughout a corn ear, or resulting in ‘‘hotspots’’ within a corn elevator. Therefore obtaining a representative sample for SFE or any other extraction/sample preparation procedure is difficult, considering that the toxicity of the analyte does not make it very amenable to many standard homogenization techniques. Table 2 shows the recovery results of aflatoxin B1 from different quantities of the same corn sample for both solvent and supercritical fluid extraction. The SFE result in this case was obtained on a 3-g sample. Obviously, comparison of the SFE-extracted sample to a 50 g solventextracted sample (CB method and 50 g—Method 1) could lead to a low recovery in the SFE case. However, comparison of the SFE result to a 3 g solvent-extracted sample indicates that both extraction techniques produce similar recoveries, if a 3-g sample is representative of the true aflatoxin. There is little doubt, based on statistical sampling theory [34], that using a larger sample size in any type of extraction yields more precise results. An example of this is shown in Fig. 7 for the determination of the fat content of potato chips using SFE. Here one obviously sees the tradeoff between sample size and precision of analysis. These results have ramifications in terms of comparing SFE with traditional extraction methods, which are usually based on much larger sample sizes. Nevertheless, with proper homogeneity,

Table 2 Extraction of Aflatoxin B1 from Corn Method (N = 5)

Average recovery (ppb)

Relative standard deviation

441.4 519.6 549.2 515.1

3.2 6.2 3.5 10.5

CB methoda SFE methodb Method 1a Method 1b a b

Fifty-gram samples. Three-gram samples.

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Fig. 7 Effect of sample size on extraction time and precision.

even SFE using small sample sizes has yielded good precision indices [35], comparable with those found via established methodologies. Sample preparation prior to SFE consists of the following: comminution of the sample if needed, controlling the amount and effect of water on the sample matrix, and dispersion of the sample matrix prior to SFE. Mechanical grinding of the sample prior to SFE to decrease the particle size will usually increase the mass transfer of the target analyte, resulting in a faster SFE [36]. Likewise, the use of sorbent mulling, e.g., matrix solid phase dispersion (MSPD) [37], can effectively disrupt the sample matrix, aiding recovery or retention of the desired analyte from the sample matrix. Because samples intended for SFE are often placed in tubularconfigured extraction cells, it behooves the analyst to try and produce a particle size that will yield an optimal extraction. In this regard, chromatographic theory may be applicable for optimizing the particle diameter of the sample to column diameter ratio [38]. Although SFE does not usually occur under conditions of turbulence in a packed bed, SFE can be modeled as a chromatographic process [39], a factor that should aid the analyst in optimizing the particle diameter consistent with the dimensions of the extraction cell. This will assure mixing of the extraction fluid with the sample matrix and reduce conditions that lead to channeling in the bed, and subsequently, incomplete extraction.

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In some instances, sample grinding can be detrimental in SFE, particularly when the analytes ofinterest are located on the surface of the sample particle. This is particularly true when dealing with samples containing potential unwanted coextractives, which may interfere in the analysis and require cleanup of the supercritical fluid extracted sample. In this specialized case, SFE on the neat sample may prove more efficacious. For example, on-line SFE of the seeds of the desert smoke tree, Dalea spinosa, yielded extract compositions that depended on the comminuting of the sample [40]. Grinding the seed sample allowed the SFE of higher molecular weight components, i.e., triglycerides, which were not desired. Therefore in this case, extraction of unground sample is to be preferred vs. sample grinding. The use of pelletized celite, i.e., Hydromatrix, when mixed with a sample matrix, solves many of the sample preparation problems in SFE [41]. This patented concept uses large particle size diatomaceous earth to disperse many sample types very effectively, and is marketed by SFE instrumentation companies under various product designations. An additional benefit of using Hydromatrix is that it will also adsorb approximately twice its weight in water, and hence can be used to successfully prepare samples, with high water content, for SFE.

D. The Problem with Coextractives and Water As previously noted, some degree of fractionation can be achieved in SFE by adjustment of the fluid density, thereby allowing proximate separation of solutes that differ considerably in their molecular weight and/or volatility. Because of the propensity of SC-CO2 to dissolve fats or oils, lipid-type compounds are frequently extracted along with the desired analytes, particularly from biological sample matrices. Such samples along with environmental matrices also contain water that can also be coextracted during SFE. Solubility data on lipid solutes in SC-CO2 have been determined over a range of pressures and temperatures [42,43], and can be used to minimize carryover of lipids into the final extract when using SFE. Figure 8 shows the dependence of triglyceride solubility in SC-CO2 as a function of temperature and pressure. Note that a relatively low weight percent solubility in SC-CO2 (5%) is found for triglycerides at 40jC and 50jC, but as the temperature is increased upward from 50jC to 60jC, there is a pronounced increase in triglyceride solubility,

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Fig. 8 Solubility of soybean oil triglycerides in SC-CO2 as a function of temperature and pressure.

particularly at higher pressures. Further increases in temperature substantially enhances the triglyceride solubility and can result in solubilities that exceed 40 wt.% in SC-CO2 at 700 bar. Such solubility trends in SC-CO2 have been routinely employed to perform oil and fat extractions using SFE [44]. Based on the density dependence of fat solubility in SC-CO2, Gere [45] has defined a ‘‘fat band’’ of fluid densities for SFE, which should not be exceeded to prevent coextraction of lipid moieties. Likewise, water has a finite solubility in SC-CO2 as shown in Fig. 8 [46], a factor that can lead to problems in SFE, or its potential exploitation in assuring a more efficient extraction. As noted in Fig. 9, over the range of pressures and temperatures commonly employed in SFE, there is a monotonic solubility of water as a function of pressure in SC-CO2. Despite this low solubility level of water in SC-CO2, water can be problematic, leading to irreproducible results, contamination of the extract, and problems associated with restrictor function and/or collection devices. A convenient way of suppressing the effect is to add a desiccant to the sample matrix to adsorb the water. Alternatively, a quantity of desiccant can also be added to the extraction cell downstream of the sample matrix, but this can lead to problems in facilitating the extraction. The choice of desiccant is critical to avoid caking of the sample matrix, which could impede the extraction process. The choice

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Fig. 9 Solubility of water in SC-CO2 as a function of temperature and pressure.

of drying agent can be made by consulting the study of Burford et al. [47] in which several common desiccants were evaluated with respect to their efficiency in analytical SFE. Considerable success has been achieved using Hydromatrix, which embraces not only many of the properties of the ideal sample dispersant, but aids in the retention of modest water levels in moist samples. Control of water during the SFE is also important for minimizing the plugging of restrictors, because the attendant Joule–Thompson effect that is present during the expansion of SC-CO2 to atmospheric pressure can result in ice formation at the restrictor orifice, resulting impedance of fluid flow. The role of water in SFE can be twofold: that of a synergist in facilitating extraction or as an inhibitor in sterically blocking contact between the analyte and the extraction fluid. It has been noted in engineering-scale studies of SFE [48] that water can modify the morphology of the sample matrix, leading to improved mass transport of the extract (analyte) out of the sample. The most often cited case of this phenomenon is the extraction of caffeine from coffee beans, which can only be effectively accomplished with a moist bean matrix. The natural presence of water in such a sample matrix al so facilitates its use as an in situ ‘‘cosolvent,’’ because its presence during the extraction of polar analytes can lead to enhanced extraction recoveries [49].

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Conversely, large quantities of water in the sample matrix, which is frequently the case during the SFE of foods, natural products, and biological tissue, can inhibit extraction because of a reduction of contact between the fluid and analytes. Nowhere is this more prevalent than in the extraction of lipids from moist tissue samples [50]. Figure 10 illustrates the dramatic effect of dehydrating the sample prior to SFE with carbon dioxide on a ham sample containing over 70 wt.% water. Quantitative recovery of total fat content, which is desired for pesticide residue analysis, is clearly inhibited by the presence of such a relatively large quantity of water. Gentle dehydration of the sample in an oven prior to SFE, or freeze drying, will rapidly facilitate the removal of water, and ultimately fat (Fig. 10), reducing both the time and mass of CO2 required for the extraction.

E. Collection of the Extracted Analyte As noted by Taylor [51], optimization of the collection method for the resultant extract from SFE is often neglected, resulting in incomplete analyte recoveries that are falsely associated with an incomplete SFE. There are several techniques for collecting or trapping the extracted analytes in analytical SFE and each is effected by temperature. The most often utilized options are open vessel, liquid, and

Fig. 10 Effect of sample moisture content on the SC-CO2 extraction of a smoked ham sample.

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sorbent trapping of analytes. Some empirical and experimentalbased studies have been reported by various researchers, particularly Taylor et al. [52], who have studied the impact of various experimental parameters on analyte trapping efficiency. Collection in a empty vial or vessel has been successfully practiced by a number of investigators and is particularly appropriate for bulk extraction of fat and similar exhaustive extractions. It is also applicable, however, for the extraction of trace levels of analytes, such as pesticides, but larger collection vessels are required for capturing such trace analytes to minimize their loss. Avoidance of entrainment of analytes in the escaping fluid stream can be minimized by adding a glass or steel wool or ball packing to the empty container. The chosen material should be chemically inert, provide a high surface for condensing the analyte from the rapidly expanding fluid, but allow for the ready description of the analyte after completion of the extraction. A novel scheme for inserting an open collection vessel prior to a sorbent trap allow for trapping of both nonvolatile and volatile constituents from food matrices is shown in Fig. 11. Here the initial collector serves to capture coextracted lipid constituents, while volatile species are isolated downstream on sorbent-filled tube (B). In addition, it is also possible to design a collection scheme that

Fig. 11 Schematic of a supercritical fluid extraction device for collecting nonvolatiles (R) as well as volatiles on a Tenax trap.

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permits the concentration of volatile species in coextracted oil under pressure. King and Zhang [53] have modeled solute trapping in a open vessel in terms of the retention efficiency of the analyte being collected and shown that trapping efficiency is related to the relative vapor pressures of the solute (analyte) and the solvent (supercritical fluid). Because CO2 upon decompression has a large fugacity, it is not unusual for the ratio of solvent/solute vapor pressures to exceed 103. Despite this favorable phase separation, it is best to use a collection vessel packed with a surface area material, i.e., glass beads or wool to avoid entrainment of target analyte in the escaping fluid [54]. Taylor et al. have studied collection efficiency using both neat and modified collection solvents [55–57], for both model test solutes and fat-soluble vitamins. Similarly, Langenfeld et al. [58] measured the effect of collection solvent parameters as well as extraction cell geometry for over 65 different compounds in 5 different solvents. More recently, Vejrosta et al. [59] have reported optimizing the collection device, for low boiling compounds having a vapor pressure similar to the collection solvent, where significant analyte losses can occur. Analyte collection in a sorbent-filled collection device has been utilized in analytical SFE for many years, and has been an integral component in older instrumentation that is no longer commercially produced (i.e., Hewlett Packard Model HP 7680 and Suprex Autoprep 44). Successful application of this mode of collection requires an appreciation of the potential of analyte breakthrough off the collection sorbent as the extraction continues. Breakthrough characteristics for a number of common volatile compounds have been measured by gas– solid chromatography using CO2 as a carrier gas [60], and have been shown to be considerable less than those found with helium as a carrier gas. This result is a direct reflection of the enhanced interaction between low pressure CO2 and typical organic solutes, i.e., indicating that CO2 is a favorable medium for extracting volatile compounds at very low pressures. Taylor et al. [61,62] have conducted studies to measure the trapping efficiencies of various adsorbents with neat and modified supercritical carbon dioxide (SC-CO2), the variance in the trapping capacity for different types of solid phases, and the effect of trap temperature on analyte recovery. Chaudot et al. [63] also studied the effect of modifier (cosolvent) content on the trapping efficiencies of analytes on various adsorbents, and showed that analyte retention

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was possible even when the modifier content of carbon dioxide was quite high (e.g., 10% methanol). While faster flow rates may yield rapid extraction rates, the analyst may have to consider the tradeoff between the speed of extraction and good collection efficiencies. Experience has shown that a high fluid velocity through the extraction cell may result in too large a flow rate of expanded extraction fluid into the collection device. This can lead to lower collection efficiencies and entrainment of the analytes out of the collection device into the expanding gaseous stream.

III. INTEGRATION OF CLEANUP STEP WITH SUPERCRITICAL FLUID EXTRACTION Analytical SFE is capable of yielding crude fractionations by changing the fluid density, but it is rare to obtain a ‘‘clean’’ extract unless the sample matrix is insoluble in the supercritical fluid, or the compounds to be isolated differ substantially in their physicochemical properties (i.e., polarity, vapor pressure, molecular weight). For example, the separation of fat from a food matrix or contaminants in a soil sample can be handled quite adequately by SFE. On the other hand, the isolation of pesticides from a food sample that contains appreciable quantities of fat or water may be more problematic. In some cases, a judicious choice of extraction fluid density may provide an extract that is perfectly acceptable for analysis without the need for further cleanup of the extract. Supercritical fluid extraction can be made potentially more selective then liquid extraction, because the density or solubility parameter of the fluid can be varied with extraction pressure or temperature. However, it is common place in the practice of SFE to use a sorbent, either in the cell, or after decompression to further fractionate the extract. Sorbents used for this purpose tend to be classified chromatographically as normal phase chromatography adsorbents, because SC-CO2 is in some ways analogous to nonpolar solvents. Therefore SFC may be useful as a screening tool to chose the optimal sorbent for cleanup of the extract under SFE conditions. As shown in Table 3, there are a number of ways for simplifying a supercritical fluid-derived extract. These include of course varying the pressure, temperature, and time of extraction to yield an extract containing the target analytes of interest, thereby reducing the total

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Table 3 Options for Integrating Sample Cleanup with SFE Fluid density-based fractionation Supercritical fluid adsorption chromatography Integration of selective adsorbents Alternative fluids to carbon dioxide On-line SFE/chromatography methods Inverse SFE SF-modified size exclusion chromatography (SEC) Use of binary gas mixtures

number of coextractives (if there are any). Other options can include changing the type of extraction fluid or fractionating the extract according to individual solute (analyte) threshold pressures. Such relatively simple approaches do not always work well because the resolving power of SFE is rather limited. It is for this reason that SFE cleanup methods frequently use in situ adsorbents, e.g., adding the sorbent, usually after the sample to be extracted, to impart additional selectivity over that which can be achieved by changing the variables that control SFE. Variations in this theme include ‘‘inverse’’ SFE and coupling matrix solid phase dispersion with supercritical fluids.

A. Fluid Density-Based Fractionation The variation of fluid density as a function of pressure and temperature for compounds in their supercritical and near-critical fluid state are available, or can be computed with a fair degree of accuracy from thermodynamic equations of state [64]. The density of a supercritical fluid goes through a substantial change at its critical point making control of its density and hence solvent power difficult to regulate in this region. Beyond the critical point, further compression yields a modest increase in density and hence solvent power; however, by increasing both the extraction pressure and temperature beyond the critical temperature and pressure, an increase in a solute’s solubility in a supercritical fluid can be affected. Hence with SC-CO2, extraction of higher molecular weight or polar compounds can be accomplished under such conditions. As noted by King [65], it is the relative solvent power of the supercritical fluid, i.e., its solubility parameter to that of the target

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analyte, that to a first approximation determines the extent of solubility of the analyte in the SF. Therefore if there are significant differences in the solubility parameters of the extractable components making up the sample matrix, fractionation may be feasible. Table 4 lists some of the estimated solubility parameters of components found in sample matrices. For maximum solubility of a solute to occur in a supercritical fluid, their respective solubility parameters must be equal; however, this condition in analytical SFE is only rarely required (e.g., extraction of fat). Because the solubility parameter of the extraction fluid is directly proportional to the fluid density, it should be noted that when the solubility parameter of the solute and solvent are within 2.5 Hildebrand units (d, cal1/2/mL3/2), some degree of mutual miscibility is assured. For SFE of trace components, even this criterion may be relaxed, and a much lower fluid density or solubility parameter will suffice for extracting the target analyte. Solubility parameters between 0 and 9.0 can be attained using SC-CO2. From the values in Table 4, it is apparent that some solubilization of fats and lipids can occur in SC-CO2 depending on the extraction fluid density. The high solubility parameters associated with carbohydrate and proteins or amino acids suggest that these moieties will not be soluble in SC-CO2 to any great extent. Indeed, this is what is found experimentally [66] and only at very high pres-

Table 4 Characteristic Solubility Parameters (y) and Their Relationship For maximum solubility: ysolute = ySF For miscibility: (ysolute

ySF)f2.5

Characteristic solubility parameters (y) Compound SC-CO2 Fats/lipids Water Carbohydrates: Glucose (Cal’c) Sucrose in H2O

y

Compound

y

0.0–9.0 8.5–10.0 23.5

Proteins/amino acids: Valine Histidine Tryptophan Glycine (Cal’c) BSA Blood serum

11.0 15.3 13.1 13.0 11.7–14.7 21.7

18.9 22.0

BSA = bovine serum albumin.

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sures can any recorded solubility of these substances occur. This fact makes SFE with SC-CO2 attractive for separating many compounds from protein- and carbohydrate-sample matrices. It should be noted that the addition of cosolvents to SC-CO2 can increase the fluid’s propensity for extracting more polar analytes. For example, addition of ethanol to SC-CO2 has been shown to extract polar lipids, such as phospholipids, that hardly exhibit any solubility in neat SC-CO2. In some cases, careful selection of the extraction fluid density can provide the desired degree of sample cleanup for the final analysis method. Gere and Derrico [67] have suggested that extraction fluid densities less than 0.4 g/mL will minimize the coextraction of interfering lipids during SFE. A further increase in extraction fluid density will increase the degree of lipid solubilization, and a fluid density of 0.6 g/mL will in most cases, assure the coextraction of lipids. For example, David et al. [68] found that for the SFE of polychlorinated biphenyls from cod liver oil, an extraction fluid density of 0.50 g/mL for SC-CO2 coextracted only 2 wt.% of the interfering fats, while at an extraction fluid density of 0.75 g/mL, SC-CO2 successfully extracted all of the cod liver oil. Other examples of partial fractionation that may be useful for analysis purposes are the separation of essential oils, bitter acids, and triglycerides from hops by changing the density of SC-CO2, or the separation of antioxidant, Irganox 1076, from a high molecular weight polyethylene film matrix [69]. Despite the examples of optimizing selectivity in SFE by changing the extraction fluid density alluded to in the previous paragraph, some mention of the special problems posed by biological and natural product matrices when conducting SFE should be noted. Although it is a mute argument as to what sample matrices are the most difficult to extract specific analytes from via SFE, it is probably fair to say that the molecular complexity of many natural and biological samples poses specific problems to SFE with respect to extraction specificity. Whereas SFE of environmental matrices are simplified somewhat because of their high content of inorganic matter, the level of extractables in food and biological matrices is quite variable, despite the fact that carbohydrate and protein matter have limited solubility in SCCO2 under typical analytical SFE conditions. It is the high propensity of SC-CO2 to extract lipid matter from natural products; however, that causes much of the selectivity problem in the SFE of these materials, and has lead to the integration of cleanup techniques into the SFE schemes.

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The extraction of volatiles and semivolatiles from biological matrices by SFE offers some improvement over other techniques such as headspace or purge/trap methods because of its benign nature. When performing SFE using SC-CO2, the sample matrix in the extraction cell is in a nonoxidative environment (CO2). When this factor is coupled with a relatively low extraction temperature, there should be minimal degradation of the target analyte during the extraction process. Supercritical fluid extractions using SC-CO2 can frequently be performed under 200 bar and at temperatures slightly above the critical temperature of the extracting fluid, e.g., 35–45jC for SC-CO2. Such SFEs conducted at lower pressures also avoids the simultaneous extraction of oil or fats, which can interfere in the final analytical method, e.g., gas chromatography.

B. Use of Adsorbents with Supercritical Fluid Extraction As previously noted, integration of adsorbents into the analytical SFE process, either prior to or after SFE, can often produce a sample sufficient ‘‘clean’’ for direct analysis. Table 5 lists typical sorbents and materials that have found use in analytical SFE. The sorbents listed generally tend to be ‘‘normal’’ phase column packing materials according to a liquid chromatographic or HPLC classification system. This fact is not coincidental because the elutropic strength of SC-CO2 even at high pressures is equivalent to nonpolar to medium polarity liquid solvents. Sorbents such as aluminas, silicas, surface bonded silicas, diatomaceous earths, and Florisil have all been cited in the SFE literature. These sorbents can be directly added to the sample or into the extraction vessel as a segregated bed apart from the sample matrix. As noted by Randall [70], SC-CO2 is a weak elutropic solvent, and the analyst must be careful in selecting the proper sorbent, or tailoring its surface activity, so as to permit elution of the desired

Table 5 Sorbents Used for Fractionation of Extract Aluminas Silicas Celite Silyated silicas

Silica gel Florisil Hydromatrix Synthetic resins

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analytes. As in normal adsorption chromatography-based cleanup systems, sorbent strength can be tempered by addition of water to the sorbent before SFE to lower their surface energy. Organic-synthesized sorbents, such as adsorbent disks and synthetic resins/foams, are low-surface-energy sorbents, and are compatible for use with SFE, although they may be subject to morphological change to the effect of pressure or sorption of the SF (i.e., plasticization) [71]. Use of the above-normal, phase-adsorption technique requires that several factors be assessed and controlled for the technique to work in the supercritical fluid mode. The analyte retention characteristics must be assessed as a function of the total quantity of supercritical fluid eluent passed through the sorbent bed to successfully capture the analytes. This is illustrated in Fig. 12 where the breakthrough of three organochlorine pesticides from an alumina cleanup sorbent with SC-CO2 at 250 atm and 50jC follows a classic sigmoidal frontal breakthrough curve. This elution pattern, expressed in terms of total expanded volume of CO2 through the sorbent bed, was accomplished using 1.8 g of alumina in a 3.5-mL extraction cell. In

Fig. 12 Percent pesticide recovery through an extraction cell loaded with alumina for sample cleanup.

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this case, approximately 0.2 g of sample was initially put on top of the alumina bed. The use of minichromatographic columns or cartridges in series with the SFE extraction vessel has also been reported [72] and can involve all of the retention mechanisms well known to chromatographers; namely adsorption, size exclusion, and complexation. This post-SFE trapping can include the use of traditional solid phase extraction (SPE) materials, or fabricated traps as has been reported for the capture of more volatile compounds. This sorbent-based chemistry can be used in several modes when coupled with SFE. Figure 13 illustrates the use of Hydromatrix both as a sample dispersant and mild desiccant, as well as void volume filler in the extraction cell. Figure 13 also illustrates how alumina as a sorbent can be used in its traditional format after SFE to segregate the target analytes from fat coextractives, or within the SFE cell to retain more polar target analytes that can then be eluted off the alumina bed. In the latter case, the isolated analytes can be removed from the alumina by increasing the SFE temperature and pressure, or by incorporating an organic cosolvent with SC-CO2, or by simply emptying the cell and using a liquid to elute the target analytes off the alumina. Such methods have been extensively used by Maxwell et al. [73,74] for the analysis of antibiotics in biological tissues.

Fig. 13 Integration of sorbent collection device based on SPE cartridge, for both off- and in-line trapping.

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An excellent example of the use of sorbent technology with analytical SFE is for the isolation of pesticides from lipid matter as shown in Fig. 14 employing the principle discussed in Fig. 12. Here neutral alumina, initially thermally activated, with subsequent adjustment of its final activity level via additional of water, is used to retain interfering lipid moieties while the pesticides, endrin, dieldrin, and heptachlor epoxide, are eluted with high recovery with SC-CO2 relative to conventional cleanup techniques [75]. Similar fractionations for cleanup have been reported using SFE in the selective isolation of polynuclear aromatic hydrocarbon and PCPs from environmental matrices. The addition of small quantities of cosolvent can aid in analyte recovering using the supercritical fluid cleanup (SFCU) technique, but care should be taken that breakthrough of the undesired species (lipids) does not occur.

Fig. 14 Comparison of packed column GC/ECD chromatograms of incurred pesticide residues in SFE extract from poultry adipose tissue: (A) supercritical fluid cleanup; (B) blank CO2 collection (20 min) prior to injection; (C) fraction collected immediately following sample, 10-min collection time; (D) conventional cleanup methodology.

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C. Inverse Supercritical Fluid Extraction Another sorbent-based SFE invented by the author is ‘‘inverse’’ SFE. Here a sorbent is incorporated into the extraction cell to isolate the target analyte of interest under SFE conditions, while interfering compounds are removed by the extraction fluid. The concept is illustrated in Fig. 15 and contrasted with the normal SFCU technique. As shown in the first two sequences in Fig. 15, the addition of an adsorbent into the extraction sequence is normally utilized to yield a simplified extract containing the analytes of interest. In inverse SFE, the sorbent is added to the extraction vessel, or in-line as a separate bed, to facilitate the removal of the solutes that are unwanted or would interfere in the subsequent assay [76]. This is frequently performed by using neat SC-CO2 to remove the unwanted compounds, followed by a cosolvent/SC-CO2 mixture or organic solvent to remove the target analytes from the sorbent bed. Examples of inverse SFE include the separation of lipids from leucogentian violet, a coccidiostat found in poultry tissue, cleanup of extracts containing aflatoxin M1, isolation of polymyoxin B sulfate from a pharmaceutical cream, and reduction of the interfering lipids in extracts containing cholesterol. With respect to the last case of determining cholesterol in the presence of other lipid coextractables, such a sample cleanup is of considerable importance in determining cholesterol levels in foods and biological fluids. The determination of cholesterol can be accomplished with SC-CO2 as reported in the literature [77–79]; however, the problem from an analytical perspec-

Fig. 15 Sequence of steps for inverse SFE vs. normal SFE.

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tive are the large number of coextractives that also exhibit sufficient solubilities in SC-CO2, which are coextracted along with the desired analyte. Using ‘‘inverse’’ SFE [76], it has proven possible to retard the target analyte of interest, cholesterol, while removing the interfering coextractives first with neat SC-CO2. For example, amino-bonded silicas (from SPE cartridges) will selectively retard sterols relative to other lipid components in the presence of SC-CO2. By using a threefold excess of this sorbent to sample by weight in the extraction cell, interfering triglycerides were fractionated away from cholesterol at 500 atm and 80jC. Then, by using 6 vol.% of methanol in the SC-CO2, for the same extraction time and conditions used for the neat SC-CO2 extraction, the cholesterol could be eluted off the sorbent, relatively free of interfering lipids (based on results from the capillary SFC analysis of the collected extract fractions).

D. Variation in Extraction Fluid Type or Composition Although CO2 is by far the most commonly used extraction fluid for the reasons noted above, there are several other candidates that have utility also, or niche applications. Of particular note are the fluorocarbons such as HC-134, SF6, and fluoroform. Levy [80] has shown that SF6 under appropriate conditions can selectively extract alkanes with respect to aromatic hydrocarbons. The hydrogen-bonding propensity of fluoroform (HCF3) allows differential SFE to be accomplished on polar analytes, such as opium alkaloids. For example, Stahl et al. [81] demonstrated that the alkaloid, thebaine, is preferentially solubilized over codeine and morphine by HCF3. Fluoroform also has a lower propensity to extract lipids, which makes it attractive for extracting analytes from fat-containing matrices. This property of HCF3 has been exploited by King and Taylor [82] to selectively extract pesticides from poultry fat as shown in Fig. 16. More recently, Taylor et al. [83] have used the selective extraction properties of HCF3 to extract drugs devoid of extraneous lipid coextractives. Relative to SC-CO2, HCF3 under the right extraction conditions can be used rather then SC-CO2, resulting in an extract with 100 times less fat than that obtained with SC-CO2 under identical extraction conditions (250 atm, 50jC, 50 mL of HCF3 or CO2). The result from using HCF3, as shown in Fig. 16, is that the derived extract can then be diluted and directly injected for GC/ECD

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Fig. 16 Gas chromatography/electron capture detector chromatogram of incurred pesticides extracted from chicken fat using fluoroform.

analysis of the organochlorine pesticides. The resultant chromatogram in Fig. 16 allows detection of the three organochlorine pesticides from poultry adipose tissue at the 1–3 ppm level, relatively free of any interference. This is indicative of the superior discriminating power of the HCF3 relative to lipid coextractives. Another approach that has been found to be effective is the use of binary supercritical fluid mixtures for fractionating target analytes and coextractives. In this case, a fluid is used that has a considerably lower critical temperature relative to the principal solvating fluid (i.e., SC-CO2), but both fluids are in their supercritical state. This type of binary fluid mixture has less solvating power that that possessed by the fluid having the higher critical temperature [84], but sufficient solvating power to selectively extract trace levels of target analytes from interfering substances, such as coextracted lipids. This is one of the reasons that SFE using a 70 mol% CO2/30 mol% N2 mixture will give extracts containing less than 5 mg of fat at 8000 psi and 60jC or 80jC, while assuring complete recovery of pesticides at the ppm level [85].

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Table 6 Pesticide Recoveries and Lipid Extracted from Poultry Fat as a Function of Fluid Composition Fluid composition (mol%) Pure CO2

95% CO2/ 5% N2

75% CO2/ 25% N2

20% CO2/ 80% N2

3800

1820

110

0

Pesticide recovery (%) 100 Heptachlora Dieldrin 100 Endrin 100

100 100 100

70 70 65

6 11 9

Lipid (mg)

a

As epoxide.

An indication of the moderating effect of the fluid with the lower critical temperature, N2, is shown in Table 6 where the amount of pesticide recovered along with the quantity of lipid coextracted, as a function of fluid composition at 10,000 psi and 70jC, is noted. Both pure CO2 and a 95 mol% CO2/5 mol% N2 extracted 3.8 and 1.82 g of lipid, respectively, under the above conditions. A fluid composition of 20 mol% CO2/80 mol% N2 extracts approximately zero fat, but as noted in Table 6, the pesticide recoveries are very low. An intermediate composition of carbon dioxide with nitrogen (75 mol% CO2/25 mol% N2) reduces the coextraced lipids to 110 mg, while yielding 70% recoveries of the target analytes. Further optimization of the binary fluid composition (8000 psi, 60jC, 70 mol% CO2/30 mol% N2) permitted the recoveries of pesticides noted above with minimal coextractives.

IV. COUPLING REACTION CHEMISTRY (DERIVATIZATION) WITH SUPERCRITICAL FLUID EXTRACTION A. Reactions in Supercritical Fluid Media Analytical reactions conducted during SFE provide the analyst with another variable to improve extraction selectivity, analyte detection of extracted analytes via derivatization, an increase in analyte volatility,

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and enhanced solubility of the target analyte(s) in the extraction fluid. This can encompass quite a wide range of analyte types, ranging from lipids, pesticides, to inorganic species as shown in Table 7. Many of the cited reactions utilize well-known derivatizing agents that have been used in GC and HPLC methodology, or as a reactant designed to improve the solubility of a sparingly insoluble analyte in SC-CO2. The latter is best illustrated by the use of fluorinated ‘‘designer’’ ligands for enhancing the solubility of metal species in SC-CO2 [86]. One must be cautious when using derivatizing agents as reactants in SFE, particularly with matrices that are complex, because the resultant extract may turn out to have more extracted components than obtained through a conventional SFE approach. Reactions conducted under pressurized supercritical fluid conditions, hereafter referred to as supercritical fluid reactions (SFR), accrue many of the same benefits when applying pressure to facilitate or accelerate particular reaction chemistry. The rate constant associated with the reaction may increase (and in some cases decrease) and ultimately favor a particular reaction pathway or product. Likewise, catalysts that show minimal effect under ambient conditions may be better at facilitating a reaction under supercritical fluid conditions. Therefore by judiciously combining SFE with SFR, one gains some of the same benefits as changing the reaction solvent in the condensed liquid state, as well as the possibility of further fractionating the resulting end products for analysis.

Table 7 Reactions and Derivatization Applied in Analytical SFE Alkylating agents/BF3—acidic herbicides Ion-pairing agents/TMPA—ionic surfactants Pentafluorobenzyl esters/TEA—phenols, etc. Silylation reagents—matrix derivatization 5% HCl/methanol/XAD-4 resin—fatty acids Trimethylphenylammonium hydroxide—fatty acids Acetic anhydride/AG-1-X8 resin—phenols Esterification—tocopherols Lipase/alcohol transesterification—fatty acids Alumina/alcohol—fatty acids Ligand reactions—assorted metals/inorganics

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Exploiting an SFR for the purpose of analytical derivatization has frequently been carried out on an empirical basis with little optimization. This includes the choice of the best derivatizing agent as well as the optimal conditions for affecting derivatization for a particular class of analytes. A multivariate SFR/SFE optimization scheme has been used by Cela et al. [87] to study the acetylation of phenolic analytes from soil samples in which nine experimental variables were optimized. In a more recent study, King and Zhang [88] examined five reagents with respect to their efficacy as derivatizing agents for carbamate pesticides. Heptafluorobutyric anhydride (HFBA) was found to be the best reagent when performing the derivatization in tandem with SC-CO2 extraction of the target analytes, with identification of the resultant derivative confirmed by GC/MS. A standard HPLC postcolumn derivatization method was used to ascertain the completeness of the reaction as well as facilitate a comparison of the SFR method, with the derivatization performed in a typical organic solvent, benzene. Derivatizations run in SC-CO2 were found to be faster and more complete than that achieved in benzene as a derivatization medium. Similarly, Chatfield et al. [89] demonstrated the general advantages of resin-mediated methylation of acidic analytes in SC-CO2 vs. acetonitrile using methyl iodide.

B. Types of Derivatizations Used in Supercritical Fluid Extraction A number of researchers have devised SFE/SFR methods that have been the subject of an excellent review by Field [90]. Some of the more popular methods for derivatizing analytes in the presence of supercritical fluid media are the use ion-pairing reagents [89,91], silylation [92], formation of pentafluorobenzyl esters [93], transesterification to form methyl esters [94,95], and the novel use of chelating agents for metal analysis [96,97]. Silylation, which is widely employed in GC derivatizations, has also been used in SFR/SFE. The major concern here is its application to complex matrices, which may yield unwanted derivatives that make the final analysis difficult as well as the presence of moisture in the sample matrix, which can negate the effect of the silylating agent. Hills and Hill [92] suggest that several benefits may accrue when using silylating agents in SFR/SFE: (1) direct reaction with the target analytes, and (2) reaction of the sample matrix surface with the

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analyte (3) and with the SFE-derived extract. Modification of sample matrix surface with the silylation agent, e.g., the end-capping of silanol groups, may aid in increasing the efficiency of the SFE. Acylation agents have also been used in SFE/SFR, particularly for the extraction and subsequent analysis of phenols from environmental samples. Using acetic anhydride as the acylating agent, the extraction, derivatization, and analysis of phenols in soil samples [87] has been accomplished in high yields and recoveries. Likewise, phenolic analytes have been isolated from water samples using initially an anion ion exchange resin-impregnated disk to capture the phenolic moieties via adjustment of the solution pH. Subsequently, acetic anhydride is then added to the disk, which is then rolled up and inserted into extraction vessel, followed by SC-CO2 extraction of the phenolic acetates. Similarly, Wells et al. [98] have collected acidic organic analytes on an anion ion exchange resin, and formed the methyl ester derivatives using methyl iodide as the methylation reagent. A wide variety of analytes can be assayed using this method, including chlorophenoxyacetic acids; pentachlorophenol; succinic, fumaric and citric acids; and albendazole. Esterification reactions of organic solutes in SC-CO2 have been extensively studied, not only for analytical purposes, but for process reaction potential [99]. Greibrokk et al. [100] were one of the earliest groups to demonstrate the possibilities of an enzymatic-initiated transesterification, both in the off-line and on-line modes of SFE, for the formation of the butyl esters of vegetable oils. Other investigators have used methanol/HCl on a cross-linked polymeric resin to methylate fatty acids or acidic alumina with methanol to form the fatty acid methyl esters of the free acids or from vegetable oils. For example, King et al. [101] demonstrated, both off-line and on-line, that the fatty acid methyl esters (FAMEs) of common vegetable oils could be formed by using methanol and a cosolvent (reactant) in conjunction with an alumina cleanup column, yielding FAME profiles equivalent to those found by GC analysis. Likewise, the methyl esters of natural pyrethrins were formed by Wenclawiak et al. [102] during SFE, using acidic alumina with methanol at a very high temperature, 270jC, at 40 MPa. A more environmentally benign method of performing transesterifications is to employ an enzymatic catalyst, such as a lipase, for FAME formation. This approach has been extensively exploited in the author’s laboratory based on the excellent qualitative and quan-

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titative results achieved with lipid standards [103], oils and fats [104,105], and for nutritional labeling analysis [106]. Using Novozyme 435 as the active lipase in the presence of SC-CO2, facile extractions and FAME formation can be achieved at pressures ranging from 10 to 30 MPa and temperatures of 40–70jC. Examples of using this SFE/SFR approach as an alternative to organic solventbased methods will be documented in the next section. Finally, recent developments in SFE utilizing SFR and special derivatization reagents have permitted the analysis of metals and radioactive species, such as lanthanides and actinides. Space does not permit a detailed discussion of this new aspect of SFE/SFR, but the publication of Lin et al. [86] provides a nice summary of the various chelating and derivatization reagents that have been found suitable for this purpose. A rationalization on the choice of ligands for SFE of toxic heavy metals (Cu+2, Pb+2, Cd+2, and Zn+2) from environmental matrices has been made on the basis of solubility parameter theory [107], and a model for the SFE of uranium with SC-CO2 was offered by Clifford et al. [108]. Such developments indicate that this new application of SFRs is becoming well characterized, and along with the SFE of more volatile elemental species, such as mercury [109] and sulfur [110], may allow a nearly total analysis of the periodic table.

C. Utilization of Catalysts with Supercritical Fluid Reactions Catalysts are used in the presence of SFs for many of the same reasons they are employed in high-pressure catalytically initiated reactions, i.e., to accelerate the desired reaction. Their efficacy in the presence of SFs must be evaluated because the dense fluid phase can compete for the available surface area or catalytically active sites [111]. Field [90] has provided some examples of typical catalystderivatization reagent pairings that have been used during SFE/ SFR. Inorganic-type catalysts as well as enzymes can be reused or regenerated in the presence of SF media, an option that is particularly attractive to the analyst. Even without the possibility of reuse, an expensive catalyst may be justified in terms of the overall time and expense associated with the sample preparation method. The mechanism of catalytic reactions the presence of SFs may be of secondary interest for analytical purposes, but the reader can consult the extensive reference by Jessop and Leitner [112] for

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further reading on the topic. However, catalysts can serve multiple purposes during SFE; for example, the use of tetraalkylammonium salts during SFE/SFR have been implicated for the following: (1) as a phase-transfer agent effecting the solubility in the SFE step; (2) as a catalyst for an alkylation reaction; (3) as a post-SFE derivatization reagent during analysis (e.g., in a GC injection port); and (4) a reagent yielding a volatile reaction by-product. Controversy has arisen for the exact mechanism in which the tetraalkylammonium salts facilitate derivatization in conjunction with SFE [90], but useful methylations are the end result. Similarly, the mode of catalytic action maybe adjusted when using enzymes in SFE/SFR. As shown in Fig. 17, alcoholysis or transesterifications of lipids containing a ester group can be facilitated using a lipase under relatively anhydrous conditions, while hydrolysis is favored using the same lipase at higher water concentrations. Likewise, in the presence of SC-CO2, the same conditions can be used in the extraction cell to prepare a sample for subsequent

Fig. 17 Mechanism of lipase-catalyzed hydrolysis (left side) and alcoholysis (right side) of triglycerides.

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analysis. As noted in the last section, triglycerides can be made to undergo methanolysis [104,105], for the SFE and formation of FAMEs from fats/oils; however, Turner et al. [113] have applied the same lipase for hydrolyzing fat-soluble vitamins in the presence of SC-CO2. Not all catalytic agents are equally active under the conditions of SFE. King et al. [114] demonstrated that the percent conversion for the reaction of methanol with oleic acid to form the methyl ester in a recirculating reactor for 2 hr at 70jC and 20.5 MPa varied with catalyst type. Conversion occurred in the following order: supported p-toluene sulfonic acid > Novozyme 435 > acidic alumina > cation exchange resin (H+ form) > titanium silicate. Only the first two catalysts proved practical for methyl ester formation. Screening of catalysts for SFE/SFR can be accomplished with the aid of automated SFE instrumentation as described in Sec. B. In this case, the SFE/SFR technique is ‘‘inversed’’ to permit the evaluation of enzymatic activity under supercritical conditions, and hence the efficacy of different enzymes for a specific task. This can be carried out quite conveniently and rapidly using automated analytical SFE instrumentation in a combinatorial mode [115]. Table 8 tabulates the results of surveying various lipases for their ability to facilitate methanolysis of the following lipid substrates in SC-CO2 at 17.2 MPa and 50jC over 80 min, for a triglyceride-containing shortening, cholesteryl stearate, and phosphatidylcholine. Note that Novozyme 435 assures methanolysis of all of the above lipid moieties under the stated conditions, while Lipase G, Lipozyme IM, and Chirazyme L-1 were slightly inferior and substrate dependent. It should be noted that eight other lipases in this study failed to show sufficient catalytic activity under the above conditions, and that there was no correspondence to their ability to hydrolyze the same substrates in an aqueous buffer solution. Turner et al. [116] ran preliminary tests on the hydrolytic activity of lipases in a SC-CO2 for the enzymatic hydrolysis of vitamin A, retinyl palmitate, at 25.3 MPa and 60jC over 25 min using lipases derived Candida antarctica, Pseudomonas cepacia, and Rhizomucor miehei. It was found that at a water activity level of 0.43 (aw = 0.43), the lipase derived from C. antarctica was best for the hydrolysis of Vitamin A in the presence of SC-CO2. Three additional lipases and one esterase that were also evaluated did not show sufficient hydrolytic activity to warrant further investigation. Thus activity levels

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Table 8 Lipase-Catalyzed Methanolysis for SFE/SFR Conversion of Lipids (%) Lipase a

Lipase PS30 Lipase La Lipase Aya Lipase MAP10a Lipase Ga Pseudomonas cepacia lipaseb Novozyme 435c Lipase from C. antarctica A.c Chirazyme L-1c Chirazyme E-1c Lipozyme Imc,d

Shortening

C18CE

PC

2 4 5 56 90 81 100 1 100 6 99

10 1 1 31 100 45 98 N.R. 98 2 96

1 N.R. N.R. 22 48 80 99 N.R. 90 1 60

C18CE = cholesteryl stearate; PC = phosphatidylchloine. a Immobilized on Accurel. b Sol–gel reaction products included 15% monoglycerides and 19% diglycerides. c Carrier-fixed (not specified by manufacturer). d Reaction products included 16% monoglycerides.

and reaction efficiencies may change with the solvent media, aw, and the type of reaction for a specific enzyme.

V. APPLICATIONS OF CRITICAL FLUIDS FOR SAMPLE PREPARATION There are many applications that have been documented using supercritical fluids for sample preparation. Key reference texts that enumerate many of these applications are cited in the introductory section. In this section, specific applications have been chosen to illustrate the value of utilizing SFs in terms of simplifying sample preparation, reducing the use of chemicals, as well as savings in terms of cost and labor. Most of the examples that have been selected are from the author’s and his coworkers’ research on method development for regulatory agency use in the United States. Despite this focus, the examples illustrate both the application and potential of SFE and SFs for sample preparation.

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Table 9 lists areas of application in which analytical SFE has been successfully applied. Within each generic class of compounds in Table 9, there are certain compounds or subclasses that have not been successfully extracted using SC-CO2, such as the beta-lactam drugs. The results obtained with SFE are also somewhat matrix dependent; therefore certain pesticides that are successfully extracted from foods maybe more problematic, or require a change in conditions, for removal from soil matrices. However, this is also true when using other sample preparation methods. Overall, pesticides as a compound class extract well using SC-CO2 or SC-CO2/modifier mixtures. Analytical SFE has also experienced success when applied for the analysis of drugs in both foods, biological matrices, and pharmaceutical preparations. In this field of application, it is not unusual to employ a small quantity of cosolvent dissolved in SC-CO2 to facilitate extraction of the drug from the sample matrix. Early success using SFE was recorded in the environmental analysis field, particularly in the extraction of organochlorine pesticides and dioxins, polynuclear aromatic hydrocarbons, and total petroleum hydrocarbons (TPH), resulting in the issuance of several official EPA methods. One of these, the TPH method, utilizes SC-CO2 as a solvent replacement for a fluorocarbon previously used in the method.

A. Analysis of Trace Components Off-line SFE has enjoyed considerable success when applied to the analysis of trace components in foods; especially as a replacement extraction technique for traditional methods that use large quantities of organic solvents. Applicable trace components that can be

Table 9 Applications of Analytical SFE Pesticides Petroleum products Environmental samples Fat and lipid analysis Drugs and antibiotics Polymer oligomers/additives Metal analysis Volatiles and flavors

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extracted and cleaned up using SFs include pesticides, antibiotics, natural toxins, and substances that are indicative of food adulteration. Quantitative extractions have been achieved down to the sub– parts per billion (ppb) level [117,118] and several standard methods using SC-CO2 have been developed and are now in routine use by regulatory agencies, particularly those involving pesticide residue analysis. A wide range of pesticides can be analyzed using SFE, although polar pesticides may require the use of a cosolvent. Initial SFE studies involved the removal of pesticides from both hydrophilic and fat-containing samples, in which control of the amount of coextracted water or fat was desired [117,118]. For example, Hopper and King [117] demonstrated that the addition of Hydromatrix to a sample not only allowed for adequate matrix dispersion, but the SFE of high-water containing samples (e.g., lettuce containing 95% water). By contrast, the use of Hydromatrix also allowed the SFE of viscous, high-fat samples such as peanut butter, which contained 52% fat and 2% moisture. The average recovery for 30 different types of pesticides was over 85% at incurred concentration levels ranging from 0.0005 to 2 ppm. Similar results were obtained by Snyder et al. [119] for the SFE of incurred organochlorine pesticides from various types of poultry tissues (peritoneal fat, breast, leg/thigh, liver). In this study, the SFE recovery results for the pesticides from liver tissue were found to be higher than those obtained by conventional solvent extraction. This result was ascribed to ability of the SF to better penetrate the sample matrix (liver) and to extract the target pesticides from this particular tissue matrix. It should be noted that this study stands in marked contrast to those which report the SFE of pesticides from ‘‘model’’ matrices, such as Celite [120], because it is important to verify that SFE can be successfully applied to an actual target matrix, preferably containing incurred residues whenever possible. Cleanup of the SF extract can be accomplished on-line, as noted in Sec. B; however, it is more common to decompress the pesticide-laden extract onto a sorbent-filled column/cartridge and use conventional liquid-based cleanup methodology. For example, Jones [121] extracted eight fortified pesticides in wool wax using SC-CO2 obtaining recoveries between 85% and 108% using collection in toluene. The resultant extracts were then cleaned-up using a silica column. Hopper [122], on the other hand, applied SFE and cleanup on organo-

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chlorine and phosphorus pesticides at 4000 psi and 95jC by decompressing the CO2 directly onto a C1 silica-based column, and then conventional cleanup methodology before final analysis. Pensabene et al. [123] applied SFE for the removal of triazine-type herbicides (both incurred and fortified) from egg matrices by decompressing onto an off-line mini-Florisil column. Subsequent extract cleanup required only 8 mL of solvent. This approach was also used by King et al. [124] to extract grain samples that contained a multiresidue mixture of both spiked and incurred pesticides using the home-built apparatus shown in Fig. 18. In this case, a Florisil trap was inserted between a micrometering valve (MV) and the gas totalizer (GT) to permit isolation of pesticide residues on the sorbent column. Table 10 tabulates a portion of the results from this study for the SFE of eight fortified pesticide residues at the 0.1-ppm level in wheat; extracted at 345 bar for three temperatures: 40jC, 60jC, and 80jC. By most standards, the listed recoveries on duplicate samples are certainly acceptable at all three extraction temperatures; however, the results are optimal at 60jC. Note that even an incurred residue, methyl chloropyrifos, was consistently found at the 0.04-ppm level. Supercritical fluid extraction can also be used to advantage for the trace analysis of marker compounds that are indicative of food

Fig. 18 Supercritical fluid extraction apparatus with sorbent trap option: TP = cylinder pressure; RD = rupture disk; CF = check valve and filter; PG = pressure gauge; SV = switching valve; TC = thermocouple; MV = micrometering valve; GT = gas totalizer.

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Table 10 Percent Pesticide Recoveries from Wheat at 0.1-ppm Fortification Level Conditions: 345 bar, 100-L CO2 (expanded) 40 C

Dimethoate Methyl parathion Pirimiphos methyl Chlorpyrifos Malathion Dieldrin Methoxychlor Carbofuran

60 C

80 C

10A

10B

11A

11B

12A

12B

88 89 96 97 93 95 94 89

87 89 95 97 95 91 94 97

82 92 101 105 102 104 85 97

101 103 108 113 109 104 107 98

77 91 99 99 96 93 97 92

84 93 100 101 97 91 103 95

0.038

0.042

0.043

0.044

0.041

Incurred residue results (ppm) Methyl chlorpyrifos 0.039

adulteration. Snyder et al. [125] used gas chromatography coupled with mass spectrometry to detect naphthalene and other aromatic hydrocarbons in meat matrices that had been exposed to fire and smoke in a storage cavern. Using statistical analysis, she was able to show that the appearance of the above analytes in a fire-exposed sample vs. a control meat sample in SC-CO2-derived extracts at 100 atm and 60jC were consistently higher (Table 11). This was found to be true even when extracting and analyzing meats that had been commercially smoked! Typical concentrations detected for the marker compounds were between 5 and 50 ppb in seven different types of meat. A similar approach using SFE at 175 bar and 40jC was used to detect irradiation treatment of meats by Hampson et al. [126]. Here the appearance of radiolytically produced hydrocarbons, i.e., long-chained olefinic compounds at ppm levels, produced signature compounds that were associated with the irradiation of the meat.

B. Proximate Fat Analysis One of major successes of analytical SFE is in the extraction of fat and similar lipid substances from foods. Studies in our laboratory and others have shown that extraction of extraction of fat or oil triglycer-

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Table 11 Concentration of Naphthalene (ppb)a in Meat Samples by SFE/GC/MS Type of meat Beef roast Boneless beef Corn beef Ham Smoked chicken Turkey breast Boneless turkey

Control (RSD)b 0 LOQc 1.7 (14.2) 2.5 (12.8) 11.7 (5.3) LOQc 0

Fire-exposed (RSD)b 10.7 3.5 14.6 21.3 50.8 4.3 6.2

(8.5) (4.1) (3.1) (4.8) (0.6) (4.0) (0.8)

a

Concentration determined using naphthalene-d8 as the internal standard. b RSD, relative standard deviation was determined from three extractions. c LOQ, limit of quantitation in 1 ppb.

ides can be best accomplished at pressures approaching 10,000 psi (70 MPa) and temperatures in the range of 70–90jC, where triglyceride solubility is maximized [42]. It should be noted that lower pressures and temperatures are frequently used for extracting other lipid species, such as fatty acids, cholesterol, or fat-soluble vitamins, while phospholipids require the addition of a cosolvent (ethanol) for successful SFE. However, simple gravimetrically based analytical SFE assays for fats in foodstuffs can be prone to error, particularly if one accepts the new definitions and analytical protocols mandated by the Nutritional Labeling and Education Act (NLEA) of 1990 [127]. This new protocol requires a pre-extraction hydrolysis of the lipids, followed by extraction, and than a high-resolution gas chromatographic analysis of the methyl esters of the constituent fatty acids, which comprise the fat moieties in the food matrix. Such a procedure presented a challenge to develop an alternative SFE-based method. To establish a baseline, a method was developed whereby all steps that were inclusive in the NLEA solvent-based-extraction protocol were utilized in a procedure incorporating SFE with SC-CO2 rather then the specified liquid solvent [128]. This off-line SFE method utilized a sorbent disk to entrap the resultant lipid precipitate from the meat sample after acidic hydrolysis of the meat sample

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via filtration. The disk containing the fat precipitate was subsequently placed inside an extraction cell and the fat removed by SFE using CO2. Trials on two different commercial SFE units indicated that the technique was not instrument dependent. Further, comparison of the results from the SFE procedure with those obtained via the traditional solvent-based protocol were equivalent for nine different meat matrices representing different levels of fat and types of meat. This procedure, however, was exacting and difficult to reproduce in the hands of an unskilled analyst. Utilizing the previously described lipase-based method for transesterifying lipids, King et al. [105,106] developed alternative methods for producing the FAMES required for the NLEA-based method for determining fat content. Both off-line and on-line modes of SFE/SFR (supercritical fluid reaction) were developed utilizing lipase-catalyzed transesterification that could be employed on small representative samples. Extraction/reaction conditions of 12.2 MPa and 50jC using Novozyme 435 were found to yield both reproducible and quantitative FAME distributions on different types of dehydrated meat matrices of varying fat content (15–40 wt.%). Comparison of the fat content determined by enzymatic formation of FAMEs using SFE vs. the FAMEs derived from chemical derivatization using solvent extraction were in good agreement. Caution must be exerted to not exceed the upper pressure and temperature limits tolerated by the lipase (approximately 275 bar and 60jC) during the SFE/SFR. Water can also be a mitigating factor in successful lipase-initiated derivatizations, and matrices having high water content should be freezedried or purged with SC-CO2 to remove excessive water. The integration of this particular SFR reaction sequence with the SFE step can take on quite a high degree of sophistication as shown in Fig. 19. Pictured is a totally automated on-line FAME synthesizer in which the methanol is blended with the SC-CO2 before the extraction cell, and then passed sequentially over the sample being extracted, followed by esterification of the extract over the lipase, contained in a segmented extraction cell. The derivatized sample is then captured on a sorbent contained in the analyte collection trap, rinsed into a using a small aliquot of solvent, and robotically transferred to a gas chromatographic autoinjector for analysis. The results from this method have compared favorably with the manual SFE-NLEA method as well as organic solvent extraction-NLEA determined fat levels on a variety of foodstuffs.

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Fig. 19 Automated SFE/SFR/GC analyzer for the determination of fat content in foods. (A) cylinder; (B) methanol; (C) high-pressure pump; (D) valve; (E) extraction cell—(1) sample, (2) glass wool plug, (3) supported lipase; (F) analyte trap; (G) hexane rinse solvent; (H) rinse solvent pump; (I) sample vial; (J) GC autoinjector tray; (K) gas chromatograph.

C. Supercritical Fluid Extraction Prior to Gas Chromatography Utilizing SFE prior to gas chromatography assumes that the extract contains analytes that are volatile enough for GC analysis, or are amenable to derivatization. This can cover quite a wide spectrum of analytes ranging from volatile or semivolatile species that may be difficult to trap for GC analysis to nonvolatile or more polar moieties that require the application of SFR in tandem with SFE. The use of SFE with GC and specific detectors, such as the electron capture detector (ECD) or flame photometric detector (FPD), does not always assure that cleanup of the extract has been accomplished, because ECD, FPD, etc. are insensitive to contaminants that can foul the GC column. Nevertheless, applying SFE with integrated extract cleanup may reduce the need to refurbish the injection end of capillary GC columns by reducing the nonvolatile solutes that are injected onto the column (see Sec. A).

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As noted in Sec. I, there have been a number of applications of analytical SFE for the analysis of lipid or lipid-derived volatile and semivolatile compounds. This is in part because of the relative benign extraction conditions used during SFE that minimize the formation of thermal or oxidative by-products. In addition, by applying SFE, the analyst can extract higher molecular weight, semivolatile compounds that are not readily extracted by other techniques, thereby providing additional information. For example, Snyder and King [129] contrasted the volatile/semivolatile profiles obtained from a thermal desorption technique with those obtained by desorption using SFE. They found two important differences between the two techniques: (1) using SFE for desorption yielded higher molecular compounds normally not accessible via thermal desorption that could be used to further characterize the oxidative state of a seed oil, and (2) there was an absence of low molecular weight degradation products in the SFE desorption profile. The latter observation suggests that the conventional thermal-based desorption technique produced artifacts from the technique, i.e., headspace analysis-purge and trap, that were not in the original sample. The absence or limited quantity of volatiles having a carbon number less than C6 at equivalent extraction (desorption) temperatures strongly supported this conclusion. Another advantage in using SFE for volatiles analysis is that a larger quantity of volatiles and semivolatiles can be extracted more rapidly then when using competitive techniques. Analytical SFE of lipid-derived volatiles/semivolatiles has been used to study additional problems. Morello [130] applied analytical SFE to the characterization of aroma volatiles in extruded oat cereals, and noted the increased intensity of hexanal, 2,4-decadienal, and pyrazine in the SFE extract. Seitz et al. [131] characterized the volatiles obtained from whole and ground grain samples using two methods: SFE and helium purge technique, characterizing both extracts by off-line GC-MS/IR (gas chromatography-mass spectrometry/infrared spectroscopy). The extraction of volatiles from the ground grain by SFE was optimal at extraction pressures less than 14 MPa in the temperature range from 50jC to 90jC; however, the helium purge method yielded a greater quantity of volatiles for analysis. Moreover, extraction using SFE proved optimal with respect to aldehydes, for 2,3-butanediols, and halogenated anisoles. An interesting example of the application of the SFE/SFR (supercritical fluid reaction) technique prior to gas or supercritical fluid

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chromatography analysis is the analysis of the fatty acid content of an industrial by-product called soapstock [132]. This rapid method consisted of mixing the sample with Hydromatrix, quickly freezedrying the mixture, and then extracting and derivatizing the extract simultaneously using the SFE/SFR technique. The benefits of using this technique are illustrated in Fig. 20, where the SFE/SFR technique is contrasted with the AOCS (American Oil Chemical Society) Official Method G3-53 [133]. Note that the AOCS method consists of many manual steps, takes 5–8 hr to perform depending on the analyst, and requires over half a liter of organic solvent. However, the SFE-based method takes only 3 hr and utilizes less than 2 mL of solvent! An alternative method, which is quite rapid but gives slightly lower results than either the AOCS or SFE/SFR method, uses capillary SFC for the analysis of the soapstock sample. This method takes only 45 min and uses only 8 mL of solvent. Such rapid methods find application in industry, thereby permitting the quick diagnosis of problematic shipments of soapstock.

D. Supercritical Fluid Extraction with High-Performance Liquid Chromatography or Supercritical Fluid Chromatography The use of off-line SFE in conjunction with either HPLC or SFC has been reported many times in the literature. The coupling of SFC with off-line SFE is a logical extension of the use of SC-CO2, as noted by King and Snyder [134], because extracts obtained using SFE with CO2 should be amenable to analytical chromatography using the same SF. The ability to use pressure or density programming in SFC for the resolution or removal of unwanted higher molecular solutes components extracted during the SFE step is another key advantage of using off-line SFE/SFC as cited by King [135]. High-performance liquid chromatography, like GC, offers the opportunity to use selective detectors such as UV, photodiode-array UV, or fluorescence, which can mask responses from unwanted and coeluting solutes in the final chromatographic assay. Nonspecific modes of detection, such as the FID in SFC and evaporative light scattering detector (ELSD) in both HPLC and SFC have found use in characterizing SF-derived extracts. The studies of Maxwell et al. in which off-line SFE was applied for the analysis of drugs are excellent examples of coupling SFE with

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Fig. 20 Comparison of AOCS official method (G3-53) for fatty acid content of soapstock with results from SFE/SFR and SFC methods.

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HPLC. For example, sulfonamides were isolated from chicken tissue using SC-CO2 at 10,000 psi and 40–60jC, using an in-line trap of alumina to trap the target analytes [136,137]. The analytes were then eluted off the sorbent with the HPLC eluent allowing a detection sensitivity of 100 ppb to be realized. Similarly, HPLC with photodiode array detection was used to quantify zoalene and its metabolites in chicken liver [138]. It was found that excessive dehydration of the liver tissue prior to SFE was deleterious to the recovery of zoalene; however, the addition of a small quantity of water to the liver tissue/ Na2SO4 mixture in the extraction cell permitted 90% recoveries of zoalene and one of its metabolites. Another method developed by Parks et al. [139] using both HPLC-UV or GC-MS employing the hexafluorobutyric anhydride (HFBA) derivative allowed the determination of melengesterol acetate in bovine fat tissue down to the 25-ppb level with over +99% recovery of the analyte. As shown in Table 12, the method developed for melengesterol acetate, which uses SFE, results in considerable savings in solvent use. The Food Safety and Inspection Service (FSIS), Food and Drug Administration (FDA),

Table 12 Comparison of Organic Solvent Consumption for Recoveries of Melengesterol Acetate Method

Recoveries (% F RSD)

Solvent used (L)

FSIS

96.7

>1.9

FDA

74.4 F 8.0

>2.2

AOAC

93.0 F 7.5

>1.7

SFE

98.4 F 4.5

0.012

Solvents Hexane Acetone Acetonitrile Hexane Methanol Ethanol Chloroform Diethyl ether Benzene MeCl2 Hexane Acetone Benzene Acetonitrile Methanoltable

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and Association of Official Analytical Chemists (AOAC) methods all require between 1.7 and 2.2 L of organic solvents; many of the solvents correspond to those that are carcinogenic such as chloroform, benzene, and methylene chloride (MeCl2). In addition, both the recoveries and precision of the SFE method for melengesterol acetate are superior to those obtained with the solvent-based regulatory agency methods. Addition information on these SFE methods that employ inline sorbent trap as discussed in Sec. B and Fig. 13 can be found in the reviews by Stolker et al. [140] and Maxwell and Morrison [141]. Multiple HPLC detectors and SF-based sequences can be coupled to advantage in the development of methods. Recently, researchers [142,143] at Lund University in Sweden have used an integrated enzyme-initiated reaction to hydrolyze fat-soluble vitamins in situ during the SFE of vitamins from a variety of food matrices. Using Novozyme 435 at 60jC and 25.9 MPa (a SC-CO2 density of 0.8 g/mL), and 5 vol.% of ethanol, they successfully extracted foods containing vitamins A and E. They found that vitamin A could be readily hydrolyzed under SFE conditions to retinol, which could then be determined by HPLC both using ultraviolet and/or fluorescence detection. However, Vitamin E moiety was sterically inhibited from entering the active site of the enzyme, and could not be hydrolyzed to any appreciable extent. Nevertheless, analyses for Vitamin E were performed successfully on many of the chosen food samples; milk powder, infant formula, liver paste, and minced meats, which contained a large quantity of only alpha-tocopherol. When using this SFE/SFR method, recoveries of retinol from the above matrices were found to be 79–119%, averaging 100.8%, when compared to a European Union-based collaborative study utilizing just the SFE mode for the extraction of vitamins. The enzyme bed (Novozyme 435) could be used in the SFE/SFR sequence up to four consecutive times without a noticeable loss in hydrolytic efficiency. A two-step method has been developed to separate and concentrate plant sterols using analytical SFE with sorbent media off-line using SFC for the final analysis. The method was initially developed using a refined, bleached, deodorized (RBD) soybean oil containing known concentrations (0.1–1.0 wt.%) of stigmasterol. Both hexaneextracted soybean oil and soybean oil extracted with SC-CO2 were used in developing the method. This technique was also applied to canola, corn, and cottonseed oils. Soybean oil (0.5 g) was mixed with 0.5 g Hydromatrix and added into a 7-mL extraction vessel. Glass

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wool was then inserted into the cell and 1.5 g NH2-Mega Bond Elut (Varian, Harbor City, California)-coated sorbent was added. The conditions used for the initial SFE step were 5000 psi, 80jC, and a flow rate of 2.0 mL/min for 60 min. The second extraction sequence on the same sample matrix used SC-CO2 with 5% methanol at 4000 psi, 80jC, and a flow rate of 1.0 mL/min for 20 min. Each fraction was analyzed by supercritical fluid chromatography (SFC) with a SBOctyl-50 capillary column, isothermally at 100jC, using pressure programming. Using the above two-step SFE fractionation method on a soybean oil containing 0.11% beta-sitosterol, 0.06% stigmasterol, and 0.04% campesterol indicated that the initial SFE step removed 95% of the triglycerides. Upon application of the second SFE step, the concentration of sterols increased from 0.21% in the initial extract to 25%. Similar results were also achieved on other vegetable oils as shown in Table 13. These results indicate that the two-step fractionation method can produce a substantial enrichment of sterols from seed oils for analytical detection. An extension of this method using four discrete SFE steps and methyl t-butyl ether as a cosolvent has been reported by Snyder et al. [144]. Extraction of polar analytes from biological matrices by SFE presents some of the same problems as SFE of analytes from environmental matrices. This is because of the fact that some analytes may be sparingly soluble in SC-CO2 and/or be tightly bound to the sample matrix so as to require the use of a cosolvent along with CO2. The choice of cosolvent and its quantity along with the extraction conditions can require many independent experiments to optimize the

Table 13 Concentration of Sterols in Seed Oils by Supercritical Fractionation Seed oil Corn oil Canola oil Cottonseed oil Soybean oil (hexane) Soybean oil (SFE)

Initial amount (%)

Amount after SFE (%)

0.2 0.7 0.3 0.2 0.2

21 33 28 18 25

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final SFE method. The use of automated SFE instrumentation along with a combinatorial evaluation approach can greatly accelerate the development of a final extraction method [145]. An illustrative case is the extraction of the mycotoxin, aflatoxin B1, from yellow corn, which requires the use of a binary modifier to obtain successful recoveries. Extraction with neat SC-CO2 proved unsuccessful, even at pressures up to 1034 bar and high temperatures (80jC). Static addition of small aliquots of several modifiers also proved insufficient relative to dynamic addition of the cosolvents. A 2:1 acetonitrile/methanol modifier mixture [146] was then tested using different extraction temperatures, pressures, and percent modifier as shown in Table 14. As indicated, 15% of the binary modifier at 5000 psi (345 bar) and 80jC proved sufficient to give recoveries equivalent to those obtained via solvent extraction. Highperformance liquid chromatography with fluorescence detection was used to determine the concentration of aflatoxin B1 in the extracts, which were derivatized with trifluoroacetic acid (TFA) to convert aflatoxin B1 to B2a for enhanced detection. Will such conditions suffice for the same or similar mycotoxins in different matrices and at different levels of contamination? Additional

Table 14 Screening for Optimal Conditions for the SFE of Aflatoxin B1 from Corn Sample Pressure (bar) 345 345 345 345 517 517 517 517 CB methodb No cleanupc a

Temperature (jC)

% Modifiera (vol)

Volume of CO2 (mL)

Recovery (ppb)

80 80 80 80 40 40 40 40

5 10 15 20 5 10 15 20

100 100 100 100 100 100 100 100

476 274 595 342 446 502 459 282 449 595

ACN/MeOH (2:1). Silica column cleanup of HCCl3 extract. c No silica column cleanup. b

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research has shown at much lower levels of aflatoxin in yellow corn (15 vs. 600 ppb in the above example) that aflatoxin B1 is only recovered at a 60% level using the above optimal conditions. This low recovery may reflect the difficulty in extracting the lower level of trace analyte from the yellow corn matrix. Similar results have also been recorded for the recovery of aflatoxin B1 from white corn, indicating that the method does not have universal applicability to a variety of sample matrices. Using the above approach, extractions were attempted of the more polar aflatoxin B1 metabolite, aflatoxin M1, from beef liver at a 0.3-ppb level. Liver is a notoriously difficult matrix to extract analytes from as noted previously, and the use of cosolvents frequently requires the need for extract cleanup after completion of SFE. Nevertheless, a reasonably clean SF extract can be achieved by conducting the extraction at 552 bar and 80jC, using only 3.3 vol.% of 2:1 acetonitrile/methanol modifier, yielding a 86% recovery [146]. Although such method development is often arduous, as it is even when using conventional liquid extraction, it demonstrates the experimental flexibility that makes analytical SFE an attractive technique.

E. Supercritical Fluid Extraction Integrated with Selected Chromatographic/Spectroscopic Techniques (IR, MS) Supercritical fluid extraction has been used in conjunction with an assortment of spectroscopic techniques, often in combination with a chromatographic separation method, to allow a virtual ‘‘alphabet soap’’ of possibilities as noted in Table 15 [147]. Many of the off-line SFE-based methods are similar to those used in integrated on-line

Table 15 The ‘‘Alphabet Soap’’ of Hyphenated Supercritical Fluid Techniques SFE-GC SFE-IR SFE-SFC-MS SFE-SFC-FTIR-MS SFE-IMS

SFE-HPLC SFE-FTIR SFE-GC-AED SFE-GC-IR-MS SFE-TLC

SFE-GPC SFE-GC-MS SPE-SFE-GC SFC-UV-FTIR-MS SFE-UV

GPC = gel permeation chromatography; AED = atomic emission detector; SPE = solid phase extraction; UV = ultraviolet spectroscopy; IMS = ion mobility spectroscopy; TLC = thin-layer chromatography.

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SFE systems, perhaps the most popular being SFE-GC-MS, SFESFC-MS, and SFE-IR. As commented on the introductory section, several references review the use of on-line SFE with various spectroscopic instrumentation, Ramsey’s [11] perhaps being the most current. To illustrate the potentially wide range of applications that can be investigated using SFE and spectroscopic detection, several selected studies are described below. Liescheski [148,149] has coupled infrared spectroscopy on-line with SFE to determine the iodine number of edible oils as well as the trans-fatty acid content of vegetable oils. In the former case, it was found that the symmetric CH2 stretching frequency could be linearly correlated with the iodine number. Direct transfer of the dissolved lipids to an on-line IR cell from an Isco SFX 2–10 unit was used in the reported experiments. Liescheski has also used the SFE-IR tandem technique to determine the total lipid content of milled rice flour. A particularly novel application of analytical SFE related to lipid technology is its use to detect irradiated foodstuffs. In a landmark study, Lembke et al. [150] used SFE and GC-MSD (mass selective detector) to characterize the hydrocarbon patterns and appearance of cyclic ketones that were characteristic of foods exposed to irradiation. By using a low extraction fluid density, 0. 25 g/mL, the marker hydrocarbons could be readily extracted avoiding the SFE of higher molecular weight fatty acid moieties. Among the irradiated foods extracted were pork meat, duck breast, pastachio nuts, and chicken soap. Both Tewfik et al. [151] and Stewart et al. [152] used analytical SFE to extract the 2-alkylcyclobutanone moieties from irradiated foods. Exposure of foods to irradiation yields straight chain hydrocarbons that are one carbon number less than the parent fatty acid, i.e., odd numbered fatty acids that are reliable markers for food exposure to irradiation. The 2-alkyl cyclobutanones arise from fatty acids of the same carbon number and have the alkyl group in a ring position; therefore fatty acids such as palmitic, stearic, oleic, and linoleic can yield trace levels of the alkylcyclobutanones. As shown in Table 16 [152], extraction using SFE shows an increasing concentration of alkylcyclobutanones with irradiation dosage for three commodity food items. It should be noted that the analytical method using SFE and GC/MS for cyclobutanone detection takes approximately 6 hr to perform, while the standard method takes 2 days to arrive at the same results.

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Table 16 Concentrations of 2-Dodecylcyclobutanone (2-DCB) and 2-Tetradecylcyclobutanone (2-TCB) Isolated by SFE from Irradiated Foods Foodstuff Chicken meat

Liquid whole egg

Ground beef

a

Irradiation dose

2-DCBa

2-TCBa

0.5 2.5 5.0 0.5 2.5 5.0 0.5 2.5 5.0

0.02 0.10 0.14 0.06 0.57 1.23 0.06 0.35 0.63

0.01 0.03 0.05 0.03 0.36 0.57 0.06 0.36 0.57

Concentrations in micrograms/10 g of sample.

Multiple couplings or uses of SFE can also put to advantage in analyzing ingredients in complex food samples. Huang et al. [153] identified and quantified the fat-reducing ingredient, Salatrim, in cookie, bonbons, and ice cream using SFE in combination with particle beam LC-MS and HPLC using an evaporative light scattering detector (ELSD). The fat content of the above matrices was also determined using SFE, while the particle beam LC-MS system using the ammonia chemical ionization mode was used off-line to determine the triacylglycerol that are characteristic of Salatrim. Quantification on the SFE samples was performed on the HPLC/ELSD system. This is a nice example of the compatibility of off-line SFE with different analytical methods.

VI. STATUS OF THE TECHNIQUE—CONCLUSIONS In conclusion, it would appear that overall SFE-based methods have a promising future in food analysis, particularly for sample preparation involving the analysis of fats, pesticides, specific drug moieties, trace toxicants, and food adulteration. It has been demonstrated that analytical SFE can be reproducibly used over a wide range of analyte concentrations, ranging from 1 to 50 wt.% down to ppb levels. Several collaborated or peer-verified methods have slowly evolved involving the determination of fat/oil levels in food and natural product matrices, for the determination of pesticide residues, and a recent

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European Union-sponsored study involving the determination of fatsoluble vitamins in foodstuffs. However, more collaborative and/or peer-validated methods will be needed in the future to substantiate SFE as a sample preparation tool. Currently, several food companies in the United States utilize multiple SFE units for routine fat determinations in a production plant environment. In one specific case, the SFE results are used to calibrate an on-line infrared analyzer used in food production lines. Additional future trends are nicely summarized by Luque de Castro and Jimenez-Carmona [154]. These include the use of pressurized fluids, such as subcritical water as an alternative to SC-CO2, as an environmentally and worker-friendly solvent. Recently, Curren and King [155] have also demonstrated the utility of pressurized water or water/ethanol mixtures for the extraction and sample preparation of pesticides from fortified meat tissues. Another area of application for analytical SFE is in the field of nutraceuticals, where it is a logical extension of SC-CO2-based processes that are used for the extraction and enrichment of key nutraceutical ingredients [156]. Therefore it becomes possible using some of the specific instrumentation discussed in Sec. B to combinatorially optimize analytical or process SFE, as would be required for the complex prodffucts that are used in nutraceutical product development.

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Salisbury, C.L.; Wilson, H.O.; Prizner, F.J. Environ. Test. Anal. 1992, 1 (2), 48. 2. Hawthorne, S.B. Anal. Chem. 1990, 62, 633A. 3. Taylor, L.T. Supercritical Fluid Extraction; Wiley: New York, 1999. 4. Clifford, T. Fundamentals of Supercritical Fluids; Oxford Univ. Press: Oxford, 1999. 5. Luque de Castro, M.D.; Valcarel, M.; Tena, M.T. Analytical Supercritical Fluid Extraction; Springer-Verlag: New York, 1994. 6. Brunner, G. Gas Extraction; Springer-Verlag: New York, 1994. 7. McHugh, M.; Krukonis, V.J. Supercritical Fluid Extraction; Butterworths: Boston, 1994. 8. Mukhopadhyay, M. Natural Extracts Using Supercritical Carbon Dioxide; CRC Press: Boca Raton, FL, 2000.

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17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

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Lee, M.L.; Markides, K.E., Eds. Analytical Supercritical Fluid Chromatography and Extraction; Chromatography Conferences Inc.; Provo, UT, 1990. King, J.W.; Snyder, J.M.; Taylor, S.L.; Johnson, J.H.; Rowe, L.D. J. Chromatogr. Sci. 1993, 31, 1. Ramsey, E.D., Ed. Analytical Supercritical Fluid Extraction Techniques; Kluwer Academic: Dordrecht, Germany, 1998. King, J.W. J. AOAC Int. 1998, 81, 9. Schneider, G.M.; Stahl, E.; Wilke, G., Eds. Extraction with Supercritical Gases; Verlag-Chemie: Weinheim, 1980. Giddings, J.C.; Myers, M.N.; King J.W. J. Chromatogr. Sci. 1969, 7, 276. McHugh, M.; Krukonis, V.J. Supercritical Fluid Extraction, 2nd Ed.; Butterworth-Heinemann: Boston, 1994; 368 pp. King, J.W.; France, J.E. In Analysis with Supercritical Fluids: Extraction and Chromatography. Wenclawiak, B., Ed.; Springer-Verlag: Berlin, 1992. King, J.W. J. Chromatogr. Sci. 1989, 27, 355. Kurnik, R.T.; Reed, R.C. Fluid Phase Equilib. 1982, 8, 93. Schaffer, S.T.; Zalkow, L.H.; Teja, A.S. Fluid Phase Equilib. 1988, 43, 45. King, J.W.; Friedrich, J.P. J. Chromatogr. 1990, 517, 449. King, J.W.; Johnson, J.H.; Friedrich, J.P. J. Agric. Food Chem. 1989, 37, 951. Bartle, K.D.; Clifford, A.A.; Hawthorne, S.B.; Langenfeld, J.J.; Miller, D.J.; Robinson, R. J. Supercrit. Fluids 1990, 3, 143. France, J.E.; King, J.W. J. Assoc. Off. Anal. Chem. 1991, 74, 1013. King, J.W. Trends Anal. Chem. 1995, 14, 474. Favati, F.; King, J.W.; Mazzanti, M. J. Am. Oil Chem. Soc. 1991, 68, 422. Friedrich, J.P.; List, G.R.; Heakin, A.J. J. Am. Oil Chem. Soc. 1982, 59, 288. King, J.W. In Lipid Biotechnology; Kuo, T.M., Gardner, H.W., Eds.; Marcel Dekker: New York, 2002; 663–687. Hopper, M.L.; King, J.W.; Johnson, J.H.; Serino, A.A.; Butler, R.J. J. AOAC Int. 1995, 78, 1072. Zorn, M.E.; Noll, R.J.; Anderson, M.A.; Sonzogni, W.C. Anal. Chem. 2000, 72, 631. Raynie, D.E.; Delaney, T.E. J. Chromatogr. Sci. 1994, 32, 298.

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Arancibia, V.; Segura, R.; Leiva, J.C.; Contreras, R.; Valderrama, M. J. Chromatogr. Sci. 2000, 38, 21. Barden, T.J.; Croft, M.Y.; Murby, E.J.; Wells, R.J. J. Chromatogr. A 1997, 785, 251. Nakamura, K. TIBTECH 1990, 8, 288. Berg, B.E.; Hansen, E.M.; Gjorven, S.; Greibrokk, T. J. High Resolut. Chromatogr. 1993, 16, 358. King, J.W.; France, J.E.; Snyder, J.M. Fresenius J. Anal. Chem. 1992, 344, 474. Wenclawiak, B.W.; Krappe, M.; Otterbach, A. J. Chromatogr. A 1997, 785, 263. Snyder, J.M.; King, J.W.; Jackson, M.A. J. Am. Oil. Chem. Soc. 1997, 74, 585. Eller, F.J.; King, J.W. Sem. Food Anal. 1997, 1, 145. Eller, F.J.; King, J.W. J. Agric. Food Chem. 1998, 46, 3657. Snyder, J.M.; King, J.W.; Jackson, M.A. J. Chromatogr. A 1996, 750, 201. Elshani, S.; Smart, N.G.; Lin, Y.; Wai, C.M. Sep. Sci. Technol. 2001, 36, 1197. Clifford, A.A.; Zhu, S.; Smart, N.G.; Lin, Y.; Wai, C.M.; Yoshida, Z.; Meguro, Y.; Iso, S. J. Nucl. Sci. Technol. 2001, 38, 433. Wang, S.; Elshani, S.; Wai, C.M. Anal. Chem. 1995, 67, 919. Louie, P.K.K.; Timpe, R.C.; Hawthorne, S.B.; Miller, D.J. Fuel 1993, 72, 225. King, J.W. In Supercritical Fluids—Chemical and Engineering Principles and Applications; Squires, T.G., Paulaitis, M.E., Eds.; American Chemical Society: Washington, DC, 1987: 150–171. Jessop, P.G.; Leitner, W. Chemical Synthesis Using Supercritical Fluids; Wiley-VCH: Weinheim, Germany, 1999. Turner, C.; King, J.W.; Mathiasson, L. J. Agric. Food Chem. 2001, 49, 553. King, J.W.; Holliday, R.L.; Sahle-Demessie, E.; Eller, F.J.; Taylor, S.L. Proceedings 4th Int. Symp. on Supercritical Fluids, 1997; Vol. C, 833 pp. Frykman, H.B.; Snyder, J.M.; King, J.W. J. Am. Oil Chem. Soc. 1998, 75, 517. Turner, C.; Persson, M.; Mathiasson, L.; Aldercreutz, P.; King, J.W. Enzyme Microb. Technol. 2001, 29, 111. Hopper, M.L.; King, J.W. J. Assoc. Off. Anal. Chem. 1991, 74, 661.

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King, J.W.; Johnson, J.H.; Taylor, S.L.; Orton, W.L.; Hopper, M.L. J. Supercrit. Fluids 1995, 8, 167. 119. Snyder, J.M.; King, J.W.; Rowe, L.D.; Woerner, J.A. J. AOAC Int. 1993, 76, 888. 120. Nemoto, S.; Sasaki, K.; Toyoda, M.; Saito, Y. J. Chromatogr. Sci. 1997, 35, 467. 121. Jones, F.W. J. Agric. Food Chem. 1997, 45, 2569. 122. Hopper, M.L. J. AOAC Int. 1997, 80, 639. 123. Pensabene, J.W.; Fiddler, W.; Donoghue, D.J. J. Agric. Food Chem. 2000, 48, 1668. 124. King, J.W.; Hopper, M.L.; Luchtefeld, R.G.; Taylor, S.L.; Orton, W.L. J. AOAC Int. 1993, 76, 857. 125. Snyder, J.M.; King, J.W.; Nam, K.S. J. Sci. Food Agric. 1996, 72, 25. 126. Hampson, J.W.; Jones, K.C.; Foglia, T.A.; Kohout, K.M. J. Am. Oil Chem. 1996, 73, 717. 127. DeVries, J.W.; Nelson, A.L. Food Technol. 1994, 48 (7), 73. 128. King, J.W.; Snyder, J.M.; Eller, F.J.; Johnson, J.H.; McKeith, F.K.; Stites, C. J. Agric. Food Chem. 1996, 44, 2700. 129. Snyder, J.M.; King, J.W. J. Am. Oil Chem. Soc. 1994, 71, 261. 130. Morello, M.J. In Thermally Generated Flavors: Maillard, Microwave, and Extrusion Processes; Parliament, T.H., Morello, M.J., McGorrin, R.J., Eds.; American Chemical Society: Washington, DC, 1994; 95–101. 131. Seitz, L.M.; Ram, M.S.; Rengarajan, R. J. Agric. Food Chem. 1999, 47, 1051. 132. King, J.W.; Taylor, S.L.; Snyder, J.M.; Holliday, R.L. J. Am. Oil Chem. Soc. 1998, 75, 1291. 133. Firesetone, D., Ed. Official Methods and Recommended Practices of the American Oil Chemists’ Society, 4th Ed.; AOCS Press: Champaign, IL, 1990. Method G3-53. 134. King, J.W.; Snyder, J.M. In New Techniques and Applications in Lipid Analysis; McDonald, R., Mossoba, M., Eds.; AOCS Press: Champaign, IL, 1997; 139–162. 135. King, J.W. J. Chromatogr. Sci. 1990, 28, 9. 136. Parks, O.W.; Maxwell, R.J. J. Chromatogr. Sci. 1994, 32, 290. 137. Maxwell, R.J.; Lightfield, A.R. J. Chromatogr. B 1998, 715, 431. 138. Parks, O.W.; Lightfield, A.R.; Maxwell, R.J. J. Chromatogr. Sci. 1995, 33, 654. 139. Parks, O.W.; Shadwell, R.J.; Lightfield, A.R.; Maxwell, R.J. J. Chromatogr. Sci. 1996, 34, 353.

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Stolker, A.A.M.; Sapoli Marques, M.A.; Zoontjes, P.W.; Van Ginkel, L.A.; Maxwell, R.J. Sem. Food Anal. 1996, 1, 117. 141. Maxwell, R.J.; Morrison, F. Handbook of Analytical Therapeutic Drug Monitoring and Toxicology; Wong, S.H.Y., Sunshine, I., Eds.; CRC Press: Boca Raton, FL, 1997; 77–105. 142. Turner, C.; King, J.W.; Mathiasson, L. J. Chromatogr. A 2001, 936, 215. 143. King, J.W.; Turner, C. Lipid Technol. Newsl. 2001, 7 (5), 107. 144. Snyder, J.M.; King, J.W.; Taylor, S.L.; Neese, A.L. J. Am. Oil Chem. Soc. 1999, 76, 717. 145. King, J.W. In A Century of Separation Science; Issaq, H.J., Ed.; Marcel Dekker: New York, 2002; 395–396. 146. Taylor, S.L.; King, J.W.; Greer, J.I.; Richard, J.L. J. Food Prot. 1997, 60, 698. 147. King, J.W. Abstracts of the 4th International Symposium on Supercritical Fluid Chromatography and Extraction, Supercritical Conferences, Cincinnati, OH, 1992; 237–239. 148. Liescheski, P.B. J. Agric. Food Chem. 1996, 44, 823. 149. Liescheski, P.B. Sem. Food Anal. 1996, 1, 85. 150. Lembke, P.; Bornert, J.; Engelhardt, H. J. Agric. Food Chem. 1995, 43, 38. 151. Tewfik, I.H.; Ismail, H.M.; Sumar, S. Lebensm. Wiss. Technol. 1998, 31, 366. 152. Stewart, E.M.; McRoberts, W.C.; Hamilton, J.T.G.; Graham, W.D. J. AOAC Int. 2001, 84, 976. 153. Huang, A.S.; Robinson, L.R.; Gursky, L.G.; Profita, R.; Sabidong, C.G. J. Agric. Food Chem. 1994, 42, 468. 154. Luque de Castro, M.D.; Jimenez-Carmona, M.M. Trends Anal. Chem. 2000, 19, 223. 155. Curren, M.S.S.; King, J.W. J. Agric. Food Chem. 2001, 49, 2175. 156. King, J.W.; Dunford, N.T.; Taylor, S.L. Proceedings of the 7th Meeting on Supercritical Fluids, 2000; Vol. 2, 537–547.

APPENDIX A Fundamentals of Supercritical Fluids, T. Clifford, Oxford University Press, Oxford (1999). Analytical Supercritical Fluid Extraction Techniques, E.D. Ramsey (ed.), Kluwer Academic, Dordrecht (1998).

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Analysis with Supercritical Fluids: Extraction and Chromatography, B. Wenclawiak (ed.), Springer-Verlag, Berlin (1992). Supercritical Fluids, T.G. Squires and M.E. Paulaitis (eds.), American Chemical Society, Washington, DC (1987). Supercritical Fluids, N. Noyori (ed.), Chem. Rev., 99 (2) (1999). Extraction with Supercritical Gases, G.M. Schneider, E. Stahl, and G. Wilke (eds.), Verlag-Chemie, Weinheim, (1980). Designing a Sample Preparation Method that Employs Supercritical Fluid Extraction, C.R. Knipe, W.S. Miles, F. Rowland, L.G. Randall, Hewlett Packard Company, Little Fall, DE (1993). High Pressure & Biotechnology, C. Balny, et al. (eds.), John Libby Eurotext, Montrouge, France (1992). High Pressure Chemistry and Physics of Polymers, A.L. Kovarskii (ed.), CRC Press, Boca Raton, FL (1994). Supercritical Fluid Extraction, L.T. Taylor, John Wiley, New York (1996) Analytical Supercritical Fluid Extraction, M.D. Luque de Castro, M. Valcarel, and M.T. Tena, Springer-Verlag, New York (1994). Supercritical Fluid Extraction, M. McHugh and V. Krukonis, Butterworths, Boston, MA (1994). Supercritical Fluid Chromatography, R.M. Smith (ed.), Royal Society of Chemistry, London (1988). Modern Supercritical Fluid Chromatography, C.M. White (ed.), Alfred HuthigVerlag, Heidelberg (1988). Analytical Supercritical Fluid Chromatography and Extraction, M.L. Lee and K.E. Markides (eds.), Chromatography Conferences, Inc., Provo, UT (1990). Supercritical Fluid Technology in Oil and Lipid Chemistry, J.W. King and G.R. List (eds.), AOCS Press, Champaign, IL (1996). Dense Gases for Extraction and Refining, E. Stahl, K.-W. Quirin, and D. Gerard, Springer-Verlag, Berlin (1986). Supercritical Fluid Engineering Science, E. Kiran and J.F. Brennecke (eds.), American Chemical Society, Washington, DC (1992). Extraction of Natural Products Using Near Critical Solvents, M.B. King and T.R. Bott (eds.), Blackie Academic, (1993).

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Fractionation by Packed Column SFC and SFE, M. Saito, Y. Yamauchi, and T. Okuyama (eds.), VCH Publishers, New York (1994). Supercritical Fluid Technology, T.J. Bruno and J.F. Ely (eds.), CRC Press, Boca Raton, FL (1992). Innovations in Supercritical Fluids, K.W. Hutchenson and N.R. Foster (eds.), American Chemical Society, Washington, DC (1995). Applications of Supercritical Fluids in Industrial Analysis, J.R. Dean (ed.), Blackie Academic, London (1993). Supercritical Fluid Technology, F.V. Bright and M.E.P. McNally (eds.), American Chemical Society, Washington, DC (1992). Supercritical Fluid Extraction and Its Use in Chromatographic Sample Preparation, S.A. Westwood (ed.), Blackie Academic, London (1993). Hyphenated Techniques in Supercritical Fluid Chromatography and Extraction, K. Jinno (ed.), Elsevier Science Publishers, (1992). Supercritical Fluid Processing of Food and Biomaterials, S.S.H. Rizvi (ed.), Blackie Academic, London, (1994). Packed Column SFC, T. Berger, Royal Society of Chemistry, London (1995). Supercritical Fluid Extraction and Chromatography, B.A. Charpentier and M.R. Sevenants (eds.), American Chemical Society, Washington, DC (1988). SFC with Packed Columns - Techniques and Applications, K.Anton and C. Berger, Marcel Dekker, Inc. (1998). Supercritical Fluids: Extraction and Pollution Prevention, M.A. Abraham and A.K. Sunol (eds.), American Chemical Society, Washington, DC (1997). Gas Extraction, G. Brunner, Springer-Verlag, New York (1994). Natural Extracts Using Supercritical Carbon Dioxide, M. Mukhopadhyay, CRC Press, Boca Raton, FL (2000). The Principles of Gas Extraction, P.F.M. Paul and W.S. Wise, Mills & Boon Ltd., London (1971). Supercritical Fluid Technology, J.M.L. Penninger, M. Radosz, M.A. McHugh, and V.J. Krukonis (eds.), Elsevier, Amsterdam (1985).

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Chemical Engineering at Supercritical Fluid Conditions, M.E. Paulaitis, J.M.L. Penninger, R.D. Gray, Jr., P. Davidson (eds.), Ann Arbor Science, Ann Arbor, MI (1983). Supercritical Fluid Science and Technology, K.P. Johnston and J.M.L. Penninger (eds.), American Chemical Society, Washington, DC (1989). Supercritical Fluids - Fundamentals for Applications, E. Kiran and J.M.H Levelt Sengers (eds.), Kluwer Academic Publishers, Dordrecht (1994). Supercritical Fluids - Fundamentals and Applications, E. Kiran, P.G. Debenedetti, and C.J. Peters (eds.), Kluwer Academic Publishers, Dordrecht (2000). High Pressure Chemical Engineering, P.R. von Rohr and C Trepp (eds.), Elsevier, Amsterdam (1996). Chemical Synthesis Using Supercritical Fluids, P.G. Jessop and W. Leitner (eds.), Wiley-VCH, Weinheim (1999). Organic Reactions in Aqueous Media, C.-J. Li and T.-H. Chan, John Wiley & Sons, New York (1997). Chemistry Under Extreme or Non-Classical Conditions, R. Van Eldik and C.D. Hubbard (eds.), Wiley, New York (1997). Thermophysical Properties of Carbon Dioxide, M.P. Vukalovich and V.V. Altunin, Collet s Publishers Ltd., London (1968). Supercritical Fluids in Chomatography and Extraction, R.M. Smith and S.B. Hawthorne (eds.), J. Chromatogr. A., 785 (1 + 2) (1997). Supercritical Fluid Extraction (SFE) and Chromatography (SFC), J.W. King (ed.), Seminars in Food Analysis, W.J. Hurts (ed.), 1 (2) (1996). Practical Supercritical Fluid Chromatography and Extraction, M Caude and D. Thiebaut (eds.), Harwood Academic Publishers, Amsterdam (1999). Extraction Methods in Organic Analysis, A.J. Handley (ed.), CRC Press, Boca Raton, FL (1999). Supercritical Fluid Methods and Protocols, J.R. Williams and A.A. Clifford (eds.), Humana Press, Totowa, NJ (2000). Symposium on Supercritical Fluids, A.S. Teja and C.A. Eckert (eds.), Ind. Eng. Chem. Res., 39 (12) (2000).

3 Correspondence Between Chromatography, Single-Molecule Dynamics, and Equilibrium: A Stochastic Approach Francesco Dondi and Alberto Cavazzini University of Ferrara, Ferrara, Italy Michel Martin Ecole Supe´rieure de Physique et de Chimie Industrielles, Paris Cedex, France

I. SUMMARY II. INTRODUCTION III. THE STOCHASTIC APPROACH OF CHROMATOGRAPHY A. General Aspects B. Basic Stochastic Model IV. PEAK SHAPE FEATURES AND EXPERIMENTAL ERRORS IN THE DETERMINATION OF THE RETENTION FACTOR A. Peak Splitting Effect B. Peak Tailing Effect C. Stochastic Bias Effect D. Injection Effect E. Unretained Tracer Selection Effect V. EQUILIBRIUM CONDITIONS IN CHROMATOGRAPHY

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VI. DISCUSSION A. Peak Splitting B. Peak Tailing C. Stochastic Bias D. Injection E. Number of Analyte Molecules F. Selection of the Holdup Time Marker VII. CONCLUSION ACKNOWLEDGMENTS GLOSSARY REFERENCES APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E

197 198 199 201 201 202 206 206 208 208 210 213 221 226 228 230

I. SUMMARY We report a detailed study concerning the correspondence between separations by chromatography, dynamic quantities coming from single-molecule measurements at the interfaces, and phase partition equilibrium by using the unifying approach of the stochastic description. The fundamental hypotheses allowing establishing the proper links between the three experimental techniques are discussed, and the full correspondence between the different quantities is determined from basic principles. The expressions of the errors on the retention factor which are intrinsically linked to the separation process, and which arise from peak splitting, peak tailing, stochastic bias, injection step, and number of the analyte molecules, are derived under general conditions and discussed in detail. Reference is made to the growing area of microsystems or nanosystems and chip technology, with numerical examples. How to determine the impact of single-molecule dynamics observations on the chromatographic peak shape of the experimentally observed sorption time distribution and, in general, of the behavior of the species at the stationary phase (surfaces, interfaces) is pointed out.

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II. INTRODUCTION Chromatography is recognized as a very powerful separation technique at disposal to chemist. Separation stems from the combination of differential thermodynamic potential and flow[1–3]. However, the chromatographic process is essentially a dynamic process, not an equilibrium one, and thus the thermodynamic equilibrium is the result of this dynamic process and not merely an independent property. This aspect was underlined since the beginning of the development of the stochastic theory of chromatography by Giddings and Eyring [4], who were able to derive the column sojourn time distribution function in the case of adsorption chromatography on identical adsorption sites in terms of adsorption–desorption event kinetics of individual molecules. After this basic work, significant advances were made both in the mathematical and numerical handing of the stochastic problem and in the consideration of more complex cases such as adsorption chromatography on two, or more, types of adsorption sites, size exclusion chromatography, and nonlinear chromatography [5–17]. However, the stochastic approach, even if attracting from a conceptual point of view, had the drawback of being conceived in terms of substantially inaccessible quantities, the individual molecule dynamics quantities. Consequently, other theoretical approaches of chromatography were preferred. The possibility of directly observing the real-time dynamics of single molecules undergoing adsorption and desorption at interfaces [18–22], as well as the axial displacement probability distribution of fluid molecules over discrete time domains by pulsed field gradient nuclear magnetic resonance [23], is adding new substantial tools for separation scientists. Indeed, the different physicochemical phenomena which compose separation processes (partition, mass transfer, diffusion, etc.) can now be described not only with the classical concepts of macroscopic phase equilibrium and with the phenomenological laws of transport phenomena, but they can also be finally expressed in terms of the dynamics of the single species involved in them and directly viewed at the submicron and submillisecond scales. Because of these new possibilities, a renewal of separation engineering can be expected. In fact, the individual molecular recognition processes can be observed, and the dynamics of individual mobile phase molecules can be followed in time and space, which provides a sound basis for a new approach in designing separation media and in optimizing the processes of separation.

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In this context, the stochastic description of chromatography appears now, unlike in the past, a quite suitable theory since it is able to fully treat in an abstract way the dynamics of differential migration from the point of view of a single molecule. The stochastic theory of chromatography expresses the chromatographic quantities—retention, efficiency, peak shape, and resolution—in terms of the statistical properties of the single-molecule dynamic quantities: individual sorption time, number of sorption–desorption steps— which are now becoming measurable. From a mathematical point of view, the so-called characteristic function approach was proved to be able to treat in a straightforward and complete way very complex cases. In this paper, we want to focus on the stochastically grounded relationships and equivalence between chromatographic retention, equilibrium distribution, and single-molecule dynamic quantities. We also intent to obtain quantitative estimates of the errors made on the determination of distribution coefficients from retention times when some basic conditions are relaxed at different degree. There are several reasons which may justify such an investigation. First of all, although fundamental in character, this question has not, to the best of our knowledge, been specifically handled even if, obviously, its basic concepts and intuitions are deeply rooted inside the main treatments of the theory of chromatography [1–3]. Second, any improvement of our understanding of the chromatographic process may lead in general to new tools for method development and separation optimization. Third, the most recent evolution of the chromatographic operating conditions, involving microsystems or nanosystems by means of the chip technology, is modifying so significantly the time, space, and sample amount scales of the separation that the validity of the assumption of equivalence between chromatographic retention and distribution equilibrium, which was not disputed during the past decades, needs to be again considered. The present treatment will be limited to the general aspects and to a simplified set of conditions for the chromatographic process, namely, by assuming that the analyte distribution between the mobile and stationary phases occurs under linear conditions and that the mobile phase velocity is constant. Nonetheless, the basic requirement allowing one to establish the equivalence conditions between chromatography and equilibrium distribution will be derived, putting forward in the mean time pertinent limiting conditions in the handled cases. These conditions can, however, be properly extended to more

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complex and more realistic chromatographic conditions which can be considered separately.

III. THE STOCHASTIC APPROACH OF CHROMATOGRAPHY A. General Aspects The basic concept in the stochastic description of chromatography considers the statistical behavior of a single molecule of a given sample component as it progresses along the chromatographic bed. The molecule undergoes a number of successive visits in the two phases of the bed (the mobile phase and the stationary phase) and of transfers between these phases. All these visits and transfers are controlled by the laws of chance, hence the Greek term ‘‘stochastic’’ used to describe this random walk process. One must expect that the results of the stochastic theory of chromatography will be related to those of the classical equilibrium theory of chromatography in the same way as the results of the statistical mechanics are related to those of the classical thermodynamics. The case of elution chromatography is considered here. Other chromatographic processes, such as that occurring in thin layer chromatography, can also be treated by the stochastic approach, but require a specific treatment [1,2] which is not developed here. When the molecule visits the mobile phase, it stays there during some time, sm, before getting transferred to the stationary phase. This time, called ‘‘ingress time,’’ may vary significantly for different molecules of the same component and for the same molecule in successive visits performed in the mobile phase. It is therefore a random variable. Similarly, when the molecule visits the stationary phase, it stays in this phase for a random ‘‘egress time,’’ ss, before going in the other phase. When it visits the mobile phase, the molecule advances through the column at some velocity which is usually the velocity, vm, of that phase. When it visits the stationary phase, the molecule stays at a fixed position in the column (assuming that axial diffusion in the stationary phase is negligible). The stationary phase is represented as an ensemble of discrete sites, some of them being visited by one given molecule during its sojourn in the chromatographic bed. Different molecules can have slightly different sojourn times in the column, and the distribution of these sojourn times makes the chromatographic peak [24].

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Since they are random variables, the ingress time, sm , and the egress time, ss , have statistical regularities; that is, they have their own frequency functions, fm ðsm Þ and fs ðss Þ , and thus well-defined mean values (sm ; ss ), variances (r2s;m ; r2s;s ), and higher statistical moments, respectively [25–28]. One typical example of frequency function is the exponential one:
s 1 f ðsÞ ¼ exp  ð1Þ s s According to this type of frequency function, short times are more abundant than long times. For this frequency function, mean and standard deviation are both equal to s [28]. The exponential frequency function is well common in physical sciences and corresponds to a very simple condition: the probability for a given event to occur in an elementary time interval ds is proportional to ds [27]. For example, the process of physical adsorption over homogeneous surfaces obeys such a law [29] and the frequency function of the desorption time (i.e., egress time) is:   1 ss fs ðss Þ ¼ exp  ð2Þ ss ss where the mean desorption time is given by:   Es ss ¼ s0 G exp RT

ð3Þ

s0 is approximately equal to 1.61013 sec, R is the gas constant, T is the absolute temperature, Es is the molar adsorption energy, i.e., the energy which is liberated when an Avogadro number of molecules are adsorbed on the surface, and G is a statistical factor equal to the ratio between the partition functions in the adsorbed and bulk phases. Equation (3) is known as the Frenkel–de Boer equation [29]. Likewise, in the case of partition in a liquid phase, the mean egress time required to visit a film of depth df can be expressed by the Einstein equation [30]: s s ¼ qs

d2f Ds

ð4Þ

where Ds is the diffusion coefficient of the sample component in the liquid phase and qs is a geometrical factor depending on the config-

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uration of this phase. In this instance, one can assume that the exponential distribution [see Eq. (1)] is holding true for the stationary phase visit. However, both the surface adsorption and stationary phase diffusion processes can be significantly more complex than in the above discussed cases depending on the surface heterogeneity or the stationary phase geometry [31–33]. Equations (3) and (4) allow us to estimate the order of magnitude of the egress time in different cases. For example, in the case of homogeneous surface adsorption, the values of the mean adsorption time ss are equal to 4.01012, 1.01010, 6.4108, and 2.6102 sec when the adsorption energies Es are 10, 20, 40, and 80 kJ/mol, respectively, at a temperature of 100jC, assuming G = 1. Likewise, in the case of liquid partition, for a typical film thickness of a viscous liquid in a gas chromatographic capillary column (df = 0.5 Am) and a small molecule with a diffusion coefficient Ds = 1.3109 m2 sec1 (at 398 K, for n-dodecane in a polydimethylsiloxane–diphenyl–methylvinylsiloxane copolymer gum [34], a typical stationary phase presently used in gas chromatography), ss becomes equal to 1.9104 sec, assuming qs = 1. Recently, fluorescence imaging methods enabled to follow the adsorption–desorption dynamics of single molecules on a submillisecond timescale [19–22]. One must, however, remind that the individual step durations involved in chromatography may be considerably much shorter than this timescale and that the whole chromatographic process usually develops itself at timescales of the order of minutes, which results in an enormous number of phase exchange steps. The statistics of these exchange processes constitutes the core of the stochastic theory. In order to perform it, a stochastic model must be assumed.

B. Basic Stochastic Model In order to shed light on the essential features of the stochastic theory, a highly simplified model, based on four fundamental hypotheses, is considered here [35]. Still, relaxing some of the constraints of this model will not affect the derived conclusions, whereas some other hypotheses appear to be essential for insuring the correspondence between chromatographic retention and equilibrium distribution. 1. Constant velocity of the analyte molecules in the mobile phase. When they are visiting the mobile phase, all analyte mole-

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cules are assumed to move with the same constant mobile phase velocity, vm. The real behavior is, in general, much more complex. Indeed, the distribution of the velocities of the mobile fluid encountered either in a capillary or in packed beds gives rise to a more or less complex behavior for the velocities of individual analyte molecules in the mobile phase [1–3,23]. It must be underlined that these velocities might be different from that of the mobile phase itself because of various effects like diffusive, steric, or hydrodynamic effects. The constant velocity hypothesis is not essential as will be pointed out in the following, but it is assumed in order to introduce in the simplest way the somewhat complex features of the stochastic process. 2. Mutual independence of the different steps (visits and transfers) performed by a single molecule. This hypothesis means in practice that there is no memory effect; that is, the chance for the molecule to perform a step at a given stage in the chromatographic bed is independent of the previous step sequence. This assumption might appear incorrect in some situations. For example, in the case of a liquid stationary phase held in the pores of packing particles, one molecule which is desorbed from one position in one pore has a higher probability of being soon sorbed again in that pore. Only when it will be near to the pore outlet, it will have the chance of going into the mobile phase. However, in that case, the whole stay in the pore can be considered as a unique sorption step, characterized by a given time frequency function. Then, the pore ensemble can be considered as the ensemble of sorption sites randomly visited by analyte molecules which do not have any memory effect of their previous history. A complex multistep adsorption kinetics was also observed in DNA adsorption on local sorption sites on planar liquid–solid interfaces [21]. Consequently, the concept of ‘‘site’’ in stochastic theory is broad and the only constraint to put forward is that there is no memory effect regarding the visits in the stationary phase as well as the flights in the mobile phase. 3. Mutual independence among the sample molecules. The stochastic theory considers the statistics of the sojourn time of a given molecule obtained by ideally repeating a large

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number of times the migration process of that molecule along the chromatographic bed. The distribution of the sojourn times so obtained can represent an experimental chromatographic peak at the condition that the analyte molecules behaved independently of the presence of all the other molecules. In other words, this means that there is no effect of sample concentration on the migration process, i.e., that the chromatographic process is linear. 4. Identity of the ‘‘sites’’ of the stationary phase. This hypothesis means that all the ‘‘sites’’ of the stationary phase—considered, as mentioned above, in their broadest sense—have the same statistical properties; that is, they are identical. This corresponds to the assumption of a homogeneous column. Even if apparently simplified, this basic model of chromatography is general since no specific hypotheses have been put forward concerning both the ingress process to the stationary phase and the egress process to the mobile phase. Its usefulness, and interest, comes from the fact that one can consider a broad class of chromatographic methods. In probabilistic terms, the present basic model belongs to the class of the so-called stochastic compound processes, whose features are well described in literature [27]. First of all, this model allows us to identify one important random variable, the number of visits to sites of the stationary phase. This comes from the random character of the time spent in the mobile phase between two subsequent site visits. In turn, this is a consequence of the fact that the ingress time, sm, is a random variable. For example, if sm is exponentially distributed:   1 sm fm ðsm Þ ¼ exp  sm sm

ð5Þ

it can be proved that, provided the chromatographic process starts in the mobile phase, the probability distribution of the number of transfers (number of ingresses), ni, to the stationary phase sites is given by the Poisson law (see p. 11, Vol. II of Ref. 27):

Pðni Þ ¼

expðni Þðni Þni ni !

ni ¼ 0; . . . ; l

ð6Þ

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with ni ¼

tm sm

ð7Þ

where ni is the mean number of ingresses and tm is the time spent by the molecules in the mobile phase (holdup time) within the column. This assumption corresponds to the classical chromatographic practice of injecting the sample in the mobile phase. As the molecules exit the column also in the mobile phase, the number, nm, of visits in the mobile phase is equal to ni+1. Consequently, the mean number of visits in the mobile phase, nm , is: nm ¼ ni þ 1

ð8Þ

The effective average time for the mobile phase visits is defined as: seff;m ¼

tm nm

ð9Þ

From Eqs. (7)–(9), one can see that sm and s eff,m are different. This paradox derives from the fact that the last visit in the mobile phase does not have the same statistical properties as the other visits which end by an ingress to the stationary phase (see Ref. 27 for a discussion of this topic). However, provided that nm is large enough, this difference is negligible. Consequently, if the chromatographic process starts in the mobile phase (injection in the mobile phase) and ends in the mobile phase, one can identify the number of visits performed in the stationary phase, ns, as equal to the ingress number: ns ¼ ni

ð10aÞ

and: n s ¼ ni

ð10bÞ

If the ingress process is not described by the exponential function given by Eq. (5), the probability distribution of the ingress number is different from that given by the Poisson law [see Eq. (6)]. Whatever the ingress number distribution P(ni), some relationships hold true provided that the chromatographic process satisfies the second above hypothesis (mutual independence of the single steps). In particular, the average ingress number is always given by Eq. (7) [27].

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The most important quantity in chromatography is the mean time spent in the stationary phase, t s. For the basic stochastic model of chromatography considered in this section, one has: t s ¼ ns ss

ð11Þ

This equation, known as Wald equation, applies to compound processes including the Poisson one (see p. 601, Vol. II of Refs. 27 and 36) and then it is correctly applicable to the ‘‘basic stochastic model.’’ The mean time spent in the column, tR (called retention time), is given by: tR ¼ tm þ t s ¼ tm ð1 þ kVÞ

ð12Þ

where the retention factor, kV, is defined as: kV ¼

tR  tm ts ¼ tm tm

ð13Þ

It can be noted that Eqs. (12) and (13) are correct even if the times spent by the different analyte molecules in the mobile phase are not equal; that is, if the above hypothesis 1 is relaxed (see for instance, Ref. 37). In this case, tm represents the mean value of these times in Eqs. (7), (12), and (13). By combining Eqs. (7), (10b), and (11), it appears that k V can be simply expressed as: kV ¼

ss sm

ð14Þ

Equation (14) gives an important interpretation of k V in terms of the averages of the ‘‘microscopic’’ quantities sm and ss, sm and ss . It is absolutely general provided that hypotheses 2–4 hold true. Its relevance is apparent since it establishes the link between chromatography and quantities obtained in real-time single-molecule dynamics measurements. It is also the starting point for establishing the correspondence between chromatography and equilibrium. It is not superfluous to underline that sm is defined by Eq. (7) and not by Eq. (9).

IV. PEAK SHAPE FEATURES AND EXPERIMENTAL ERRORS IN THE DETERMINATION OF THE RETENTION FACTOR Equations (12) and (13) show that the determination of the retention factor, kV, in a chromatographic experiment requires the measure-

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ments of two experimental times, tm and tR. It is worthwhile to underline that both are first moments of specific distributions: the first one of the time spent by the analyte molecules in the mobile phase, and the second one of the total sojourn time in the column. However, only the second distribution is directly observable: this is the elution peak. The time tm cannot be directly measured. There are only indirect methods to estimate it. One of them is to use the mean elution time of another species that is supposed to be unretained, i.e., which is not visiting the stationary phase, and to spend in the mobile phase the same time as the analyte of interest. In the following, five different effects which influence the accuracy and the precision of the determination of tm and tR, and hence of k V, are described. We will not consider here the different sources of errors connected to the experimental setup (such as the constancy of flow rate, temperature,. . .) or arising from experimental measurements. The errors sources discussed below are, instead, fundamental in nature and are connected to the intrinsic process of chromatographic migration. The general expressions of the resulting errors are reported. Their detailed derivations are presented in Appendices A to E.

A. Peak Splitting Effect Peak splitting generally appears as a spike before the main analyte peak. This effect is generally neglected [38]. This spike is located at the same position as the unretained peak component. It contains that fraction of the sample amount which did not undergo retention. Its origin is explained by the fact that the probability for the ingress time to be greater than the total residence time in the mobile phase is not zero. For example, if the ingress time frequency function is exponential [see Eq. (5)], one can understand that the above case has finite probability because the frequency function of the ingress time sm is always positive for sm z tm. This condition is related to the probability for the molecule of performing just zero visits to the stationary phase. Neglecting the peak splitting produces a bias in evaluation of the first moment of the peak. In Appendix A, the computation of this bias is derived under the most general conditions and is specifically detailed for the Giddings–Eyring model of chromatography [4], in which both the ingress and egress processes are exponentially distributed. For such a model case, as for all the models in which tm is considered as constant, only the band broadening coming from the

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transfer process kinetics is relevant. In this case, the relative error in k V is:   ykV cexpðns Þ ð15aÞ kV 1 or, by combining Eqs. (7), (10b), (14), and (15a):     ykV tm cexp  kV kV 1 ss or still:   ykV cexpð2NVs Þ kV 1 where NVs is the effective number of theoretical plates:   kV 2 NVs ¼ Ns kV þ 1

ð15bÞ

ð15cÞ

ð16Þ

and Ns is the number of theoretical plates. The subscript s in Ns and N Vs emphasizes the fact that it refers to a hypothetical column where only the phase exchange process is acting on peak dispersion.

B. Peak Tailing Effect A second type of error arises when the retention time is determined from the peak maximum (which is unambiguously identified) instead of from the first moment of the analyte peak, which can be difficult to evaluate, especially when the peak tailing is significant. In the case of homogeneous surfaces, when the tailing is moderate, and when the distributions of ns and ss are Poissonian and exponential, respectively, this error is approximately given by (see Appendix A):   ykV 3 c ð17aÞ kV 2 4NVs If there are different types of sorption sites or if the sites do not exhibit an exponential distribution of ss [see Eq. (2)], the expression is:   ykV 3  const c ð17bÞ kV 2 4NVs

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where the const is a number equal to or greater than 1, depending on the site complexity or on the column heterogeneity [39,40]. In Appendix A, its value for the multiple-site case is detailed [Eq. (A-21)]. Note that Eqs. (17a) and (17b) only apply to conditions of linear chromatography. This type of tailing is referred to as kinetic tailing. It must be underlined that both Eqs. (17a) and (17b) hold true for very moderate tailing. General aspects of the peak shape are discussed in Appendix A.

C. Stochastic Bias Effect The stochastic theory proves that under linear conditions and for a homogeneous column, the retention factor, defined by Eq. (13), is equal to the ratio of the two mean times of the egress and ingress processes. It is thus apparent that the retention factor is strictly related to elementary phase exchange processes. However, before establishing a link with equilibrium conditions, another basic aspect, giving rise to a third error source, must be considered. The stochastic theory, in fact, only establishes an ‘‘a priori’’ equivalence between k V and the ss =sm ratio. On the other hand, a chromatographic run allows one to estimate k V through an experimental determination of the ss =sm ratio. The former ratio refers to the ‘‘population’’ (in statistical terminology) or to ‘‘a priori’’ parameters, whereas the experimental ratio contains ‘‘sample’’ or ‘‘a posteriori’’ quantities. There can be a discrepancy—the bias—between these two quantities, as usual in physicochemical sciences. It is well known that the bias depends on the number of replicas of the experiment. In the present case, the ‘‘experiment’’ is just performed by the sample molecules during their sojourn in the column. The core of the question is that k V just allows us to evaluate the ratio of the two average quantities ss and sm thanks to two opportunities: as each molecule experiences on the average ns times both ss and sm, and as the sample contains Nmol molecules of the analyte of interest, the experiment is repeated ns  Nmol times by the injected analyte. It can be easily proven (see Appendix B) that, for a constant mobile phase velocity model, but whatever the specific forms of the distributions of ns and ss, the relative bias in the estimation of kV is given by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r  1 kV ð18Þ ¼ NVs NVmol kV 3

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The relative error on kV is here expressed as rk V/k V, instead of yk V/k V in Eqs. 15a–15c and 17a, 17b, to emphasize the fact that the errors arising from the peak splitting and peak tailing effects are systematic, while that arising from the stochastic bias effect is a random error. Equation (18) is derived from the pertinent expressions for the propagation of errors starting from the definition of k V in Eq. (12) (as described in Appendix B). Equation (18) was applied to validate Monte Carlo simulation of the chromatographic process [17]. Under normal conditions, the bias is negligible because Nmol is usually very large. However, this conclusion becomes questionable in the context of the general trend toward miniaturization of the separation systems. Consequently, the dependence of Nmol on various operating parameters must be detailed. Increasing the number of injected molecules reduces the relative error on k V. However, increasing Nmol also leads to an increase of the injected sample volume, Vinj, and thus of the contribution, rt,inj, of the injection to the peak standard deviation. This results in a deterioration of the chromatographic resolution. If one can tolerate that rt,inj reaches a fraction h of the peak standard deviation rt generated by the chromatographic migration process [41], the maximum number of analyte molecules which can be introduced in the column is given by (see Appendix C): kV Nmol ¼ ahUNAv pffiffiffiffiffiffi Vm cM NV

ð19Þ

where N V is the column effective plate number, Vm is the volume of mobile phase in the column, cM is the analyte molar concentration in the sample, and NAv is the Avogadro number. In this expression, a is a proportionality factor in the relationship between rt,inj and Vinj defined as: rt;inj ¼

Vinj aF

ð20Þ

where F is the volumetric mobile phase flow rate. The value of a depends on the injection device and injection conditions [42,43]. In Eq. (19), U is the molecular detection efficiency, defined as the ratio of the number of detected analyte molecules to the number of injected analyte molecules.

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D. Injection Effect Because the concentration of the analyte in the sample is necessarily finite and since one usually injects more (and much more) than one analyte molecule in the column, the volume of injected sample is also finite. Accordingly, there is a distribution of the times at which the analyte molecules are introduced into the chromatographic column. Let t inj be the mean value of this distribution calculated by assuming as origin the instant when the first analyte molecule enters into the column. It can be shown that the mean value of the distribution of elution times of the analyte zone differs from tR by the quantity t inj . When the estimation of tm is made by injecting an unretained solute in the sample containing the analyte of interest, the mean elution time of this unretained solute is also delayed by the quantity t inj . If this effect of the finite injection time is not taken into account, this leads to an additional error on k V which only depends on the ratio t inj =tm and is given by (see Appendix D):   t inj =tm ykV ¼ ð21aÞ kV 4a 1 þ t inj =tm If, instead, tm is estimated by means of an independent experiment, supposedly without error, then the relative error on k V arising from the finite injection time of the analyte sample is:   t inj =tm ykV ð21bÞ ¼ kV 4b kV When the maximum sample volume compatible with a tolerated relative loss h2 of the efficiency value is injected, the relative errors given by Eqs. (21a) and (21b) become, respectively (see Appendix D): 

ykV kV

kV ach pffiffiffiffiffiffi NV ¼ kV 4a 1 þ ach pffiffiffiffiffiffi NV



ð22aÞ

and 

ykV kV



ach ¼ pffiffiffiffiffiffi NV 4b

ð22bÞ

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where NV is the effective plate number and c is a proportionality factor between mean injection time and injected volume, defined as: t inj ¼ c

Vinj F

ð23Þ

E. Unretained Tracer Selection Effect The time tm spent by the analyte in the mobile phase is sometimes estimated as the mean elution time of another species that is supposed to be unretained. If the latter is not truly unretained, but has itself a retention factor, kVu, the relative error on the true analyte retention factor is expressed as (see Appendix E):   ykV 1 þ kVo ¼ kVu ð24Þ kV 5 kVo where k Vo is the true retention factor of the analyte of interest.

V. EQUILIBRIUM CONDITIONS IN CHROMATOGRAPHY In order to establish the concept of equilibrium in chromatography, one must conceive a batch, static experiment in which the analyte is allowed to distribute itself between the two phases, within the column itself, or within a slice of the column having the same phase ratio: b ¼ Vm =Vs

ð25Þ

where Vm is the mobile phase volume and Vs is the stationary phase volume (or interfacial surface area when the retention mechanism involves an interface) until equilibrium is reached. Moreover, it is assumed that the stationary and mobile phases are thermodynamically homogeneous phases. The correspondence between chromatography and batch-equilibrium conditions is obtained if one assumes that there is a proportionality between the numbers of analyte molecules in the phases—Nm and Ns, for the mobile and stationary phases, respectively—and the mean times spent in these phases by the molecules: t s;b Ns ¼ Nm t m;b

ð26Þ

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where the subscript b indicates quantities in this batch, static experiment. This position is the ‘‘ergodic’’ hypothesis establishing the equivalence between the mean computed over the time and the mean computed over the space [44]. The latter is proportional thus to the number of molecules of the species in a phase. This equivalence holds true under ‘‘long-time’’ approximation; that is, the phase exchange process must be repeated many times, as discussed above. The mean times t s;b and t m;b are equal to the products of the mean durations of a single visit in a phase, ss;b and sm;b , by the mean numbers of visits in each phase, ns;b and nm;b , respectively. Since in the ‘‘long-time’’ approximation, for a distribution between two phases, ns;b is equal to nm;b , Eq. (26) becomes: ss;b Ns ¼ Nm sm;b

ð27Þ

It is reasonable to assume that the mean duration of a single visit in the stationary phase does not depend on whether the mobile phase is static, as in the batch experiment, or moving, as in a chromatographic run. Hence ss;b becomes equal to ss . Such an a priori identity is not obvious for the mean duration of a single visit in the mobile phase. Under typical chromatographic conditions, however, the instantaneous root-mean-square velocity of individual analyte molecules arising from their thermal energy is many orders of magnitude larger than the mobile phase velocity, even for molecules of large size, such as macromolecules. Obviously, molecules maintain their instantaneous velocity only during a very short time before bumping mobile phase molecules, hence wandering around and executing a random walk. As a consequence, the motion of individual analyte molecules in the mobile phase is essentially a diffusive one, with a comparatively very small drift induced by the flow of mobile phase [33]. In these conditions, the mean time spent by the analyte molecules in the mobile phase before visiting the stationary phase does not depend on the mobile phase velocity and sm can be reasonably assumed equal to sm;b . This assumption is reinforced by the fact that the experimental chromatographic practice indicates that k V is most generally independent of the mobile phase velocity. Then, comparing Eqs. (14) and (26) shows that: kV ¼

Ns Nm

ð28Þ

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Since Ns cs V s ¼ N m cm V m

ð29Þ

where cs and cm are the analyte equilibrium concentrations in the stationary and mobile phases, respectively, by combining Eqs. (25), (28), and (29), one obtains: kV ¼

K b

ð30Þ

cs cm

ð31Þ

where K¼

is the equilibrium distribution constant. Consequently, from Eqs. (13) and (30), K is obtained as: K ¼b

tR  tm tm

ð32aÞ

or, from Eqs. (14) and (30): K ¼b

ss sm

ð32bÞ

VI. DISCUSSION Equation (32a) establishes the fundamental equivalence between the equilibrium distribution constant K and chromatographic quantities, tR and tm, together with b. Equation (32b) establishes the same fundamental equivalence with the quantities ss and sm , which can be obtained from real-time single-molecule dynamics observations. In the following, the discussion will mainly deal with the aspects of the chromatographic measurement. However, some of the discussed questions also concern the real-time single-molecule dynamics observations since one must then correctly evaluate averaged values from the observed distributions of the step durations in the two phases.

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The parameter b in Eqs. (32a) and (32b) is the volumetric (or surface-to-volume) ratio of the two phases in the chromatographic column. Most generally, their geometrical configuration is such that they have a very large interfacial area. The distribution constant obtained from Eq. (32a) corresponds to this configuration. Therefore care must be exerted if this K value has to be extrapolated to other phase configurations when the interfacial region plays an important role in the analyte distribution between the two phases [45]. There are several effects connected to the measurement of tR which may lead to potential errors in the determination of k V. As discussed above, they are referred to as peak splitting effect, peak tailing effect, stochastic bias effect, injection effect, and unretained tracer selection effect. The resulting relative errors on k V are expressed by Eqs. (15a)–(15c), (17a) (17b), 18, (21a) (21b), and 24, respectively. The implications of these effects, which must be properly taken into account for insuring the correspondence between chromatographic and batch-equilibrium quantities, are discussed below.

A. Peak Splitting The importance of the peak splitting and of the resulting error on k V is essentially determined by the number of visits performed to the stationary phase by the analyte molecules, as shown by Eq. (15a). When this phenomenon is relevant, it must be included in the evaluation of the first moment of the chromatographic peak. This may be difficult if the split peak superimposes to the nonretained peak or to a signal perturbation due to the injection. There might be situations where the effective plate number, N sV, is small, which render this effect relatively significant. For instance, this is the case in size exclusion chromatography for large macromolecules which are nearly totally excluded from the internal porous structure of the packing particles [46]. Generally, N sV is large enough for the relative error computed according to Eqs. (15a)–(15c) to be negligible. Indeed, even for NsV as low as 10, this equation gives an error of the order of 109. As shown by Eq. (15b), peak splitting can be critical in ultrafast analysis since the amount of molecules in this peak varies as etm. Equation (15b) is seen to establish a link between a chromatographic effect (peak splitting) and a quantity determined from singlemolecule dynamics experiments (ss ). For example, when tm becomes of the order of ss , the fraction of molecules eluting within the split

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peak reaches 37% for an analyte with k V = 1. It must be remembered that Eqs. (15a)–(15c) are based on the assumption of a Poissonian distribution of ns given by Eq. (6), which assumes that the column is homogeneous in the cross section. If, on the contrary, there are inhomogeneities, for instance due to channeling through the porous bed, some of the analyte molecules will perform a quite smaller number of visits of the stationary phase than the main part of the molecules, which will result in significant peak splitting, as described by Eq. (A-10) in Appendix A. In this instance, the distribution of the number of visits of stationary phase by the analyte cannot be represented by a Poisson law, and the resulting error on k V cannot be any longer given by Eqs. (15a)–(15c), but it may be significantly larger than the latter. The full treatment of this problem is outside the goal of the present paper. One can suggest that a careful analysis of the peak eluted at the holdup time tm should be made in order to check whether it contains or not a significant fraction of the analyte of interest.

B. Peak Tailing The first moment of the peak must always be considered in evaluating equilibrium quantities. It can be quite different from the value of the peak maximum especially in the case of significant peak tailing. It must be observed that when the column is efficient, the maximum is close to the peak mean and thus the peak maximum is related to equilibrium conditions. Moreover, on increasing the column length, the peak shape converges to the Gaussian shape. This is the consequence of the central limit theorem which establishes that, in a process of addition of random and independent variables— and the chromatographic process is just a process of this type—the peak shape becomes Gaussian [26,27,47]. Equations (17a) and (17b) were derived on the basis of the central limit theorem. They just express in a rigorous way this aspect and state that the rate convergence of the maximum of the distribution toward the first moment is proportional to 1/NVs. The actual value of this rate depends on the complexity of the sorption kinetics or on the degree of the sorption surface heterogeneity. This is accounted for by Eq. (17b) where the numerical value of the constant, greater than 1, depends on the site complexity or on the column heterogeneity [39,40] (see Appendix A).

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In the most simple chromatographic cases, one can refer to Eq. (17a) since the exchange kinetics between mobile and stationary phase can be assumed to be simple and of homogeneous first order. According to this equation, one reaches the conclusion that the relative error on k V is rather small when NVs is of the order of 1000 or more. However, this conclusion cannot always hold true. Then, one has to refer to Eq. (17b). It is worth to point out that the numerical value of the constant involved in this equation can be directly evaluated by applying Eq. (A-21) of Appendix A to the distribution of sorption times which can be obtained in single-molecule dynamics experiments. Applying this equation to the data of Fig. 8 of Ref. 19 for adsorption of an organic dye from water on a C18-modified silica surface and to the data of Fig. 6A of Ref. 21 for adsorption of DNA from an aqueous solution on fused silica leads to value of const equal to 2.0 and 1.3, respectively. In the latter case, this evaluation together with Eq. (17b) allows to express in a quantitative way the correlation observed between the peak asymmetry and the complexity of adsorption kinetics. In the case of chiral separations, very often, the more retained enantiomer usually exhibits a greater tailing than the first one. This arises from the fact that the interaction of the first component on the chiral sites follows simple first-order kinetics, while the second one exhibits a more complex interaction kinetics. In these conditions, Eq. (17a) applies only for the first peak but not for the second one. For the latter, one must refer to Eq. (17b) where the value of const can be significantly greater than 1. Additionally, it is worthwhile to mention that the stochastic theory was proved able to quantitatively represent the peak shapes obtained in chiral separations just on the basis of the theoretical model employed to obtain Eq. (17b) [48]. It must be pointed out, however, that the discussion based on Eqs. (17a) and (17b) holds true only under conditions of moderate tailing (see Appendix A), but that any type of tailing deriving either from site complexity or column heterogeneity can be handled by the stochastic approach, as reported elsewhere [35,36,47]. As a conclusion of the present point, the stochastic theory is able to treat the problem of the difference between peak maximum and first moment in a rigorous and general way. It is able to evaluate and interpret the extent and the origin of the difference between peak maximum and first moment in terms of the kinetics of phase exchange processes and of the column structure. This obviously holds true only under conditions of linear chromatography.

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C. Stochastic Bias The number of injected analyte molecules together with the mean number of visits of the stationary phase [which is proportional to the effective plate number NsV according to Eq. (A-12) of Appendix A] must be large enough to allow unbiased evaluation of the first moment of the analyte peak [see Eq. (18)]. This equation is a simplified expression of the relative error on k V resulting from the stochastic bias effect. It has a clear statistical meaning since it shows that this relative error is inversely proportional to the square root of the number of effective experiments performed by the chromatographic system in order to measure kV. In fact, the product of Nmol by NsV is simply the total number of trials performed by the sample molecules in visiting the stationary phase for determining ss which is directly proportional to kV [see Eq. (14)]. Consequently, in the most usual chromatographic conditions, the error on k V is negligibly small owing to both the large number of analyte molecules contained in a given sample and the large number of individual sorption steps performed, for each analyte molecule, during the chromatographic process. Nevertheless, Eq. (18) can be potentially interesting for the growing area of microtechnologies which are rapidly developing in separation science. In this instance, the combination of very short lengths of the separation system and of the possibility of detecting even single molecules [21,22] can lead to a rather small number of total sorption steps. This limiting case is that of ‘‘single-molecule chromatography’’: the single analyte molecule will have a random sojourn time, distributed according to the peak profile that would be obtained for a large ensemble of analyte molecules. This can readily be seen from Eq. (18) when Nmol = 1. Then, noting that NsV = ðt s =rt Þ2, where rt is the standard deviation of that peak, Eq. (18) becomes (rk V/kV)3 = rt =t s . The relative error on k V is then equal to the relative standard deviation of the time spent by the analyte in the stationary phase.

D. Injection A necessarily finite volume of the sample gives rise to a distribution of the times at which analyte molecules enter the column. This induces an error in the determination of k V, expressed by Eqs. (21a) or (21b). It depends on the ratio t inj =t m . The knowledge of the time t inj is not trivial [43]. In practice, what is generally known from the experimenter is the injected sample volume, Vinj. The simplest situation

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is that of a plug injection of finite duration, pffiffiffiffiffiffiin which case a and c in Eqs. (20) and (23) are equal to 1/2 and 12 , respectively. Then, for NV = 1000 and k V = 2, if the maximum sample volume compatible with a tolerated loss of efficiency of 10% is injected (h = 0.32), the relative error on kV resulting from the neglect of the finite sample volume amounts to 3.3% or 1.7% depending on whether t m is determined by injecting an unretained species within the analyte sample or it is independently known. In practice, a will be smaller than for a plug injection, and c will be larger, but the overall effect depends on the actual injection profile. Anyway, the injection effect should be taken into account for a precise determination of k V and K, unless the sample size is much smaller than that corresponding to the above value of the tolerance factor h or the effective number of theoretical plates is much larger than in the above case.

E. Number of Analyte Molecules From the above discussion, it appears that the larger is the number of analyte molecules, the larger is the error resulting from the injection effect (for a given analyte concentration in the sample), when it is not correctly taken into consideration, but the smaller is the error resulting from the stochastic bias. As the latter is the most intrinsically linked to the chromatographic process and cannot be corrected for, it is interesting to discuss this error at the light of the recent evolution of the geometries of the chromatographic systems. As seen from Eqs. (19) and (20), the number of analyte molecules injected in the column depends on the analyte (through its molar concentration in the sample, cM, and its retention factor, kV), on the column geometry (volume of mobile phase, Vm), on the column efficiency (effective plate number, NV, and tolerated relative decrease, h2, of column efficiency resulting from sample injection), on a characteristic of the injection process (a), and on the molecular detection efficiency (U). To get numerical estimates of the relative errors on k V arising from the stochastic bias effect, one assumes that k V is equal to 2, the tolerance factor h equal to 0.32, the injection parameter a equal to 2, the effective plate number NV equal to 1000, and the molecular detection efficiency equal to 1. Then Vinj is equal to 0.04Vm. Under these conditions, the number of analyte molecules introduced in the chromatographic column becomes directly proportional to the volume of mobile phase in the column and to the mole fraction of the analyte

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in the sample. One considers five different liquid chromatographic systems called, respectively:      

The millisystem (a packed 10-cm-long, 4.6-mm i.d. column with a total porosity of 0.7) The microsystem (a packed 30-cm-long, 0.25-mm i.d. column with a total porosity of 0.7) The nanosystem (a 5-cm-long open channel on a chip with a 10  10 Am cross section) The pico-chip system (a 1-cm-long open channel with a 10 Am  80 nm cross section [49]) The pico-tube system (a 100-cm-long, 0.1-Am i.d. open tube [50]) The femtosystem (an 80-Am-long open channel with a 1  0.27 Am cross section [51])

The prefixes describing these systems are, of course, somewhat arbitrary, but they refer to the prefixes of the order of magnitude of the volume of mobile phase they contain, expressed in liter. The millisystems and microsystems are the commercialized classical and miniaturized liquid chromatographic columns. The nanosystem has size typical of those presently found in various systems used in separations on chip. The two picosystems and the femtosystem correspond to recent developments in the miniaturization of separation systems. Table 1 indicates, for each system and for various molar analyte concentrations, the relative error on k V expressed in % and the number of analyte molecules introduced in the channel, calculated according to Eqs. (18) and (19), respectively. In all cases, except for the lowest molar concentrations mentioned in the table for each system, the relative error on kV appears to be very small (lower, and usually much lower, than 0.1%). Accordingly, this table shows that, except in the extreme conditions of the analysis of an ultratrace compound in the picosystem and femtosystem, the chromatographic retention provides a very good estimate of the true equilibrium distribution coefficient, provided of course that the true first moment of the peak is used as the retention time. One should nevertheless mention that, in Table 1, it is assumed that the maximum sample volume, compatible with a tolerated loss of separation performance, is introduced in the system. If the injected sample volume is k times smaller than this

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Table 1 Number of Analyte Molecules, Nmol, and Relative Error on k V, (rk V/k V)3, Arising from the Stochastic Bias Effect for Different Chromatographic Systems Described in the Text and Different Molar Analyte Concentrations, cM, in the Sample cM ( M )

Nmol

(rk V/k V)3 (%)

Millisystem ( Vm = 1.2 mL, Vinj = 4.7  102 mL = 47 AL) 106 2.8  1013 6.0  107 9 10 10 2.8  10 1.9  105 1012 2.8  107 6.0  104 15 4 10 2.8  10 1.9  102 18 10 28 0.60 Microsystem ( Vm = 102 mL = 10 AL, Vinj 104 mL = 0.4 AL) 106 2.5  1011 9 10 2.5  108 1012 2.5  105 15 10 250 1017 2

= 4.1  6.3  106 2.0  104 6.3  103 0.20 2.2

Nanosystem ( Vm = 5  106 mL = 5 nL, Vinj = 2  107 mL = 0.2 nL) 106 1.2  108 2.9  104 9 5  10 1.2 10 9.1  103 1012 120 0.29 1014 1 3.2 Picosystems : pico-chip and pico-tube systems (Vm = 8  109 mL = 8 pL, Vinj = 3  1010 mL = 0.3 pL) 106 1.9  105 7.2  103 9 10 190 0.23 1010 19 0.73 Femtosystem (Vm = 22  1012 mL = 22 fL, Vinj = 0.86  1012 mL = 0.86 fL) 106 520 0.14 108 5 1.4 k V = 2; tolerance factor, h = 0.32; injection parameter, a = 2; effective plate number, N V = 1000.

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maximum volume, then the number of analyte molecules pffiffiffi is reduced by k and the relative error on k V increased by a factor k. Besides, the data in this table are based on the assumption that the effective plate number is equal to 1000. Accordingly, even if only one molecule is injected, as in single-molecule dynamics experiments, the relative error on k V arising from the stochastic bias does not exceed 3.2%. But the effective plate number depends on the geometrical dimensions of the chromatographic columns (it decreases with increasing crosssectional dimensions and increases with increasing length), as well as on other parameters such as the mobile phase velocity (since this influences ns ), which can be controlled independently of the size of the system. So the comparison between the various systems can be misleading if N V is grossly changing from one system to another. For instance, although the two picosystems have the same mobile phase volume, one might expect that the nanotube system will provide a larger effective plate number than the nonchip system because its larger length gives opportunity for a larger number of visits of the stationary phase. Generally, if, with regard to the above value of 1000, NV is increased by a factor l (which can be larger or smaller than 1), the relative error on k V is increased by a factor l1/2. In addition, data in Table 1 are computed assuming that all analyte molecules injected are detected and thus contribute to the determination of k V, i.e., assuming that U is equal to 1. This parameter can, in certain circumstances, be significantly lower than 1. Then, in Table 1, the number of molecules and the stochastic bias error on kV have to be multiplied respectively by U and U1/2. For example, if U = 0.017 as observed on a nanosystem [52], the injection of a picomolar analyte solution will lead to the detection of 2.1 molecules in average. The stochastic bias error on k V will then reach 2.2% if N V=1000. In batch distribution experiments and in real-time single-molecule observations, one can find sources of measurement errors similar to those discussed above for chromatography. For example, in batch experiments, the evaluation of the numbers of moles in the two phases can undergo a significant stochastic bias when these numbers are low. In fact, the number of molecules Nm in a free volume element inside the fluid phase is fluctuating because of the Brownian movement of the analyte molecules within it. As recently observed in attoliter volumes [53], this quantity is likely Poissonian and its pffiffiffiffiffiffiffiffi relative standard deviation is, in this instance, 1= Nm. This quantity can be significant when Nm is lower than 100, as in real-time singlemolecule observation experiments [21,22,51]. Likewise, the quanti-

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ties ss and sm , which can be obtained from real-time single-molecule dynamics observation, can be biased because of several error sources, similar to those found in chromatographic retention time measurements. For example, a bias effect, similar to the peak-splitting effect, can arise if a significant number of analyte molecules persist in the free phase and are not allowed to visit the adsorption surface during the observation time. Still, if the distribution of the durations of adsorption steps is strongly tailed, a great number of observations are required to insure a given precision. This has a close relationship with the peak first-moment evaluation in the case of strong tailing. Problems connected to observation of a limited number of molecules are also found, and discussed, in other separation methods, such as electrophoretic separations on microchip devices [52]. Each specific technique has its specific error sources. An exhaustive discussion of all these aspects lies beyond the aims of the present treatment.

F. Selection of the Holdup Time Marker The times tR and tm in Eq. (32a) are both associated to the analyte of interest. As discussed in a previous section, tm cannot be directly measured but is estimated from the elution time of a supposedly unretained tracer species or by means of other methods. This estimation can be a source of error. Its amount depends on the specific chromatographic system and retention mechanism. If the tracer species used to estimate tm is not truly unretained, but has, for instance, a retention factor of 0.05, or if the value of tm determined by an independent method differs by 5% from the true value, the relative error on the k V of the analyte of interest is at least 5% and increases with decreasing k V. This effect is significant and must be properly taken into consideration for accurate determinations of K.

VII. CONCLUSION The correspondence between chromatography, equilibrium, and single-molecule dynamics observation was fully established by the stochastic approach. The relationships between retention chromatographic quantities, average individual molecular dynamic quantities, and equilibrium constants were revisited in order to single out basic hypotheses assuring the correct correspondence between the three approaches. Expressions of intrinsic measurement errors of different types were derived on the same theoretical basis showing the com-

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pleteness of the stochastic approach. The main focus was dedicated to discuss the first moments of the distributions which are those directly involved in establishing the correspondence with the equilibrium conditions. By this way, several critical effects were handled in a general way. The two most important ones are the systematic error arising when replacing the first moment of the peak by the peak maximum and the effect of the number of molecules in various systems on the stochastic bias effect and on the injection effect. How these effects depend on experimental parameters such as the effective number of theoretical plates, the injection conditions, and the column parameters was rigorously derived and discussed by comparing various systems (millisystems, microsystems, nanosystems, picosystems, and femtosystems). Even if all the above discussed aspects were rigorously derived by assuming a given number of hypotheses defining the so-called stochastic model, or in certain cases the Giddings–Eyring model, the constraints could, in general, be removed and the conclusions appear quite general. In particular, the assumption of a constant velocity of the analyte molecules in the mobile phase was not stringent, and that of identity of the sorption sites (homogeneous column) was not limiting in handling problems of peak tailing. However, it must be pointed out that the column heterogeneity can heavily affect column efficiency, with the consequence that some of the above optimistic conclusions as for the stochastic bias in nanosystems, picosystems, and femtosystems ought to be reconsidered. It is worthwhile to mention that the stochastic approach enables to provide insights even in this case because it allows to exactly compute the effect on band broadening [40]. For example, at the femtomole level, Nmol is so large (of the order of 108) that the relative error in k V is as low as 0.014% even if the molecules are visiting only once, in average, the stationary phase during their sojourn in a homogeneous column [see Eq. (18) with NVs = 1/2). However, the bias becomes significantly larger if the surface is heterogeneous. It can be shown for instance that, if the standard deviation of the adsorption energies of the stationary phase is equal to 1, 2, or 4 times the kinetic energy RT, the error arising from the stochastic bias effect is 1.6, 7.4, or 3000, respectively, times the error one would have for a homogeneous column [40]. These findings underline the great interest on exploring the surface property by single-molecule dynamics observation because of the link between the kinetics and energy of adsorption [see Eqs. (2) and (3)] and the chromatographic behavior.

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Another hypothesis requires that the analyte molecules are not affected, first, by hydrodynamic chromatographic effects (i.e., by steric and/or van der Waals and electrostatic interactions with solid surfaces in the moving zone of the mobile phase), and second, by hydrodynamically induced conformational modifications. If this is not the case, one cannot expect that the chromatographic retention will lead to a meaningful equilibrium distribution coefficient. A last hypothesis concerns the mutual independence of the analyte molecules in the partition processes, i.e., the linearity conditions. It is well known that this can induce strong distortion in peak shape. However, under these conditions, neither the chromatographic peak first moment nor the peak maximum has a precise meaning. One can thus only estimate the error in the equilibrium parameter estimation—which ought to be ideally obtained at zero sample size—under given overloading conditions [3]. Unfortunately, the stochastic theory on nonlinear chromatography is not available in an analytical form. Nonetheless, a Monte Carlo model of nonlinear chromatography, recently set up [17], allows exploiting, on stochastic grounds, the multiple faces of the problem. Likewise, a Monte Carlo approach can be applied to investigate complex flow pattern and phase exchange chromatographic situations [16]. The point would require a separate handling.

ACKNOWLEDGMENTS This work was supported by research of the Italian MURST Cofin no. 2001033797, by the Galileo 2000 French-Italian governmental program, and by the University of Ferrara, Italy. Fruitful discussions with Maurizio Remelli are gratefully acknowledged.

GLOSSARY c cM const df D Es

Concentration Analyte molar concentration Constant in Eq. (17b) [see Eq. (A-21)] Stationary film depth Analyte diffusion coefficient Molar adsorption energy

Chromatography: A Stochastic Approach f( ) F F( ) G j kV k Va k Vb k Vo k Vu K L n ni N NV N N Vmol NAv pi P( ) qs R S t tR T V X

Frequency function Mobile phase flow rate Fractional peak area Statistical factor Summation index in Appendix B Retention factor of the analyte Apparent retention factor of the analyte [Eq. (D-2)] Apparent retention factor of the analyte [Eq. (D-3)] (True) retention factor of the analyte Retention factor of holdup time marker Distribution constant of the analyte Column length Number of steps Number of visits to the stationary phase Number of theoretical plates Effective number of theoretical plates Number of molecules number of analyte molecules Avogadro number Relative site abundance Probability Geometrical factor Molar gas constant Skewness Residence time as stochastic variable Retention time Temperature Volume Normalized random variable [see Eq. (A-15)]

Superscripts

 *

Mean quantity Convolution

Subscripts app b

/

Apparent quantity Quantity referring to a batch experiment

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Greek Symbols a b c y d( ) D h k l l3 r r Vk rt s seff;m so us U

Injection parameter [Eq. (20)] Phase ratio Injection parameter [Eq. (23)] Differential quantity Dirac function differential quantity Tolerance factor [Eq. (C-2)] Injection volume proportionality factor Effective plate number proportionality factor Third central moment of the peak Standard deviation Standard deviation of kV Peak standard deviation Duration of an elementary step Effective average time for steps in mobile phase [Eq. (9)] Constant in Eq. (3) Numerical factor given by Eq. (B-22) Molecular detection efficiency

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APPENDIX A A.1. Peak Splitting and Peak Tailing in Homogeneous Columns Under Constant Moving Phase Velocity From a probabilistic point of view, the chromatographic problem corresponds to the problem of finding the mathematical solution for a sum of a random number of random independent variables, as a function of their distributions [36]. The random number corresponds to the number of visits of the stationary phase and the random independent variable corresponds to the time spent in each visit of the stationary phase. In the case of constant mobile phase velocity, the time spent in the mobile phase, tm, is constant. The time spent in the stationary phase, ts, is a random variable, characterized by a frequency function f(ts). In the case of homogeneous column, i.e., of identical sorption sites, this frequency function can be written in a general way as [36]: X f ðts Þ ¼ Pðns Þfs ðss Þ*ns ns ¼ 0; . . . l ðA-1Þ ns

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where P(ns) is the distribution of the number of visits, ns, of the stationary phase and fs(ss) is the frequency function of the duration, ss, of a single visit of the stationary phase by a single molecule. Equations (2) and (6) are examples of fs(ss) and P(ns), respectively. However, Eq. (A-1) is absolutely general and holds true for any type of either fs(ss) or P(ns). The symbol ‘‘*ns’’ in Eq. (A-1) means an ns-fold convolution. This translates into a mathematical expression of the fact that those analyte molecules performing exactly ns sorption processes do not spend the same total time in the stationary phase since some visits of this phase are of a short duration while others may last a long time. For example, ‘‘*2’’ means: ð fs ðss Þ*2 ¼ f ðts ; n ¼ 2Þ ¼ fs ðts  xÞfs ðxÞdx ðA-2Þ and it represents the frequency function of the total time spent in the stationary phase by analyte molecules performing over two visits of this phase. The prefactor P(ns) in Eq. (A-1), representing the probability of performing a given value ns of sorption steps, is also equal to the fraction of the number of analyte molecules performing ns visits of the stationary phase, when the number of analyte molecules is large. Consequently, the quantity P(ns)f(ss)*ns represents the frequency function of the total sorption time of that fraction of number of molecules, whereas Eq. (A-1) is the frequency function of the whole number of analyte molecules. Equation (A-1) includes the possibility for molecules to undergo zero sorption step: Pðns ¼ 0Þfs ðss Þ*0

ðA-3Þ

which can be mathematically expressed as Pðns ¼ 0Þdð0Þ

ðA-4Þ

where d(0) is the Dirac function, i.e., a spike of unit area located at ts = 0. Thus the molecules of this category appear as a spike of total area equal to P(ns = 0). Their total sojourn time in the stationary phase is zero and they elute from the column exactly a time tm after injection. Such a class of molecules gives rise to the so-called ‘‘peak splitting’’ at the origin, as explained in the following.

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Figure 1 illustrates how the peak shape is built up from Eq. (A-1): the chromatographic peak is the sum of a large number of subpeaks, each subpeak corresponding to a given value of ns and representing the distribution of the total sojourn time in the stationary phase of those analyte molecules which perform exactly ns visits to this phase. One notes that a given subpeak is relatively broad, owing to the random character of the duration of each visit. Obviously, when ns increases, the positions of the maximum and of the first moment of the subpeak move to longer times. Due to the stochastic character of the process, there is a mutual superimposition of all these subpeaks. Two important features are noticed on Fig. 1: 1. Peak splitting is observed at the origin. This is due to the first term of the sum, P(0)d(0), which appears as a spike of area equal to P(0) located at tm, representing the elution of those molecules who did not perform any visit to the stationary phase. These molecules travel along the column at the mobile

Fig. 1 Frequency distribution of the sojourn time in the chromatographic column (full upper curve). This peak is the sum of all lower peaks which correspond the sojourn time distribution of molecules performing exactly one, two, three to six visits of the stationary phase (on increasing values of their peak maximum positions on the time axis). The vertical arrow at time tm represents the peak splitting, i.e., the fraction of the number of molecules which do not perform any visit to the stationary phase.

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phase velocity and appear thus at the holdup time. In certain cases, peak splitting can be experimentally observed as a peak at the origin, separated from the main peak. Obviously, it is somewhat difficult to distinguish it from the peak of an unretained component. In most cases, this represents only a small fraction of the total peak of a retained analyte, which means that probability of traveling along the column without visiting the stationary phase is small. It can be intuitively understood that the more efficient is the column, the less relevant is the peak splitting effect. 2. Peak tailing is evident. This likely comes from the features of both the P(ns) distribution and the single sorption time distribution fs(ss). In fact, the more disordered is the entry process (i.e., the more extended is the range of possible ns values), the longer is the tail of the peak. Likewise, if the range of sorption times, ss, is extended, each subpeak is broad because of the existence of visits of short duration and of visits of long duration. The origin of tailing is thus complex and depends on the specific chromatographic case. Peak splitting, if neglected, leads to an apparent value of the first moment of the peak which differs from the true value by the error quantity Dt s;1. Peak tailing leads a second type of error, Dt s;2, if the position of the peak maximum is used to approximate the true first moment of the peak. These two errors are discussed in the following. A.1.1. Peak Splitting The area under the peak splitting position located at ts = 0 is f ðts ¼ 0Þ ¼ Pðns ¼ 0Þ and the remaining area at positive time values is ðl Fðts ¼ lÞ ¼ f ðts Þdts ¼ 1  Pðns ¼ 0Þ

ðA-5Þ

ðA-6Þ

ts >0

The average time spent in the stationary phase is: ðl ts f ðts Þdts t ¼0 t s ¼ ðsl f ðts Þdts ts ¼0

ðA-7Þ

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The average time obtained neglecting the peak splitting, ts;app;1 will be: ðl ts f ðts Þdts ts t >0 ðA-8Þ ¼ t s;app;1 ¼ ðsl 1  Pðns ¼ 0Þ f ðts Þdts ts >0

Equation (A-8) was derived by combining Eqs. (A-6) and (A-7) and by considering that: ðl ðl ts f ðts Þdts ¼ ts f ðts Þdts ðA-9Þ ts ¼0

ts >0

From Eq. (A-8), one can see that the average time spent in the stationary phase obtained by neglecting the peak splitting is biased and greater than the true value. Finally, from Eq. (A-8), one can write: Dt s;1 t s;app;1  t s Pðns ¼ 0Þ cPðns ¼ 0Þ ¼ ¼ 1  Pðns ¼ 0Þ ts ts

ðA-10Þ

The last approximation holds true if the peak splitting effect is small, i.e., when P(ns = 0)b1. This equation is quite general. When ns is distributed according to a Poisson law [Eq. (6) in the main text], one can write: Pðns ¼ 0Þ ¼ expðns Þ

ðA-11Þ

When, in addition, ss is exponentially distributed, one gets the Giddings–Eyring model for which [37,40]: NVs ¼

ns 2

ðA-12Þ

where  2 ts NVs ¼ rt

ðA-13Þ

is the effective number of theoretical plates. rt is the peak standard deviation, which, for constant mobile phase velocity models, is equal to the standard deviation of ts. Then, the subscript s for NVs empha-

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sizes the fact that such models refer to a hypothetical column where only the mobile-stationary phase exchange process contributes to the peak dispersion. In this case, the absolute error is: Dt s;1 cexpð2NVs Þ ts

ðA-14Þ

It decreases exponentially on increasing column efficiency. A.1.2. Peak Tailing The second type of error arises in assuming the peak maximum as the first moment. This problem can be handled in a general way, in the case of the basic stochastic model, i.e., in general, in linear chromatography. In fact, as discussed in Ref. 35, this model has the important mathematical property of being a stochastic process with independent and stationary increments, and, because of this, the peak profile can be represented by the Edgeworth–Crame´r series expansion [25,28]. This series expansion is an asymptotic approximation of the peak profile, where the asymptotic quantity is the average number of sorption steps [35,36,47]. This series expansion is related to the central limit theorem of probability theory which establishes that the sum of an increasing number of equally distributed random variables converges to a Gaussian law [25–27]. More specifically, the Edgeworth–Crame´r series expansion describes the rate of convergence of the actual profile toward the Gaussian law and provides a better approximation of this profile than the Gaussian law when the limiting condition of a very large number of added variables is not satisfied and, more concretely, when the peak shape exhibits a significant tailing. When this number is large enough, the distribution function can be approximated by the Gaussian law, which is the first term of the Edgeworth–Crame´r expansion. Then, the maximum of the frequency function, tmax, becomes indistinguishable from the first moment, tR. Thanks to these properties and provided that tailing is moderate, one has [25]: Xmax u

tmax  tR Dt R;2 S ¼ c 2 rt rt

ðA-15Þ

where S is the peak skewness: S¼

l3 r3t

ðA-16Þ

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l3 is the third central moment of the peak. In a constant mobile phase velocity model, one has Dt R;2 ¼ Dt s;2 . Hence combining Eqs. (A-13), (A-15), and (A-16), it becomes: Dt s;2 S c  pffiffiffiffiffiffiffiffi ts 2 NVs

ðA-17Þ

If the duration of the individual steps in the mobile and in the stationary phases are both exponentially distributed, i.e., in the case of Giddings–Eyring model, one can prove that [37,40]: 3 S ¼ pffiffiffiffiffiffi 2 NVs

ðA-18Þ

By combining Eqs. (A-17) and (A-18), one has: Dt s;2 3 c 4NVs ts

ðA-19Þ

The major problem of Eq. (A-17)—which applies to the basic stochastic model—is that it contains the quantity S, which is difficult to estimate with accuracy, whereas Eq. (A-19) only contains the effective number of theoretical plates, which can more easily be measured. The hypothesis under which Eq. (A-19) is derived is, however, strict: it assumes a first-order kinetics of exchange between mobile and stationary phase. In general, for modern high-efficiency chromatographic systems, this approximation is acceptable. However, in a significant number of applications, the hypothesis of a column having identical sorption sites with a single first-order kinetics of phase exchange is only a rough approximation [19,21]. The matter is obviously general and many cases are possible. Situations in which a much complex kinetics is involved are numerous. One particular, but important, case is that of a heterogeneous chromatographic column containing sorption sites with different firstorder kinetics or having sites on which several first-order sorption processes are simultaneously involved. In this instance, it was proved that: S¼

3  const pffiffiffiffiffiffi 2 NVs

ðA-20Þ

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const is a numerical quantity greater than 1 and, depending on the relative strength and abundance of the sites [40], is equal to:  X
pi s3s;i const ¼

i

X

!3=2

ðA-21Þ

pi s2s;i

i

where pi is the abundance of sites of mean sorption time equal to ss;i . By combining Eq. (A-20) with Eq. (A-17), one has the expression for the multiple site case: Dt s;2 3  const c 4NVs ts

ðA-22Þ

It appears thus that the stochastic treatment allows us to evaluate the bias made in evaluating the peak first moment from the peak maximum in a sufficient number of cases of practical interest. A most significant error source is the column heterogeneity and the site sorption kinetics. The relative error in k V will be simply equal to the relative error over t s because of the proportionality between the two quantities. Thus one has:   ykV 3 c ðA-23Þ kV 2 4NVs for the case of homogeneous first-order kinetics of both the mobile phase and stationary phase times, while, for the multiple-site adsorption kinetics, one has:   ykV 3  const c ðA-24Þ kV 2 4NVs One must observe that, according to Eqs. (A-23) and (A-24), the difference between peak maximum and peak first moment are inversely proportional to N Vs and thus to the number of sorption–desorption steps [see Eq. (A-12)]. However, it is well known that the central limit theorem states that the convergence to the Gaussian law is inversely proportional to the square root number of added random variables [27]. This apparent contradiction is simply explained by the fact that

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the central limit theorem refers to the normalized random variable quantity X [see Eq. (A-15)], but by combining it with Eq. (A-18) or (A-20), this contradiction disappears. Note that the above developed handling only applies to conditions of linear chromatography. This type of tailing is referred to as kinetic tailing.

APPENDIX B B.1. Stochastic Bias Effect One considers, in the following, that the analyte retention factor, k V, is obtained from the measurements of the analyte retention time, tR, and of the elution time, t m;u , of an unretained solute (also called tracer) as: kV ¼

tR  t m;u tR ¼ 1 t m;u t m;u

ðB-1Þ

B.1.1. Relative Random Errors on k V and on tR/ t m,u Accordingly, the random error on k V, r(k V), is due to the random error on tR =t m;u , rðtR =t m;u Þ. Hence: r2 ðkVÞ ¼ r2 ðtR =t m;u Þ

ðB-2Þ

Hence from Eq. (B-2), the square of the relative error on k V is related to that on tR =t m;u by:   r2 ðkVÞ r2 ðtR =t m;u Þ 1 þ kV 2 ¼ kV kV2 ðtR =t m;u Þ2

ðB-3Þ

The random error on tR =t m;u arises itself on separated random errors on tR, r(tR), and on t m;u , rðt m;u Þ. Since tR and t m;u are independent variables, one gets, according to Ref. 54 (p. 33, Example 1-6-12): r2 ðtR =t m;u Þ ðtR =t m;u Þ2

" # r2 ðtR Þ r2 ðt m;u Þ r2 ðt m;u Þ ¼ 1 þ þ t2R t 2m;u t 2m;u

ðB-4Þ

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B.1.2. Relative Errors on tR and on t m,u The analyte retention time is the mean value of the residence time in the column of the injected solute molecules, i.e. mol 1 X tc; j Nmol j¼1

N

tR ¼

ðB-5Þ

where tc, j is the column residence time of the jth molecule of analyte and Nmol is the number of analyte molecules in the injected sample. Because of the stochastic character of the column residence process, the values of tc for the individual molecules are randomly distributed according to some frequency function f(tc) having a mean value, t c, and a variance, r2(tc), defined as usual as: ðl tc ¼ tc f ðtc Þdtc ðB-6Þ 0

and r2 ðtc Þ ¼

ðl

ðtc  t c Þ2 f ðtc Þdtc

ðB-7Þ

0

From Ref. 54 [Eqs. (25) and (26)], it is easy to show that as long as the individual variables tc, j in Eq. (B-5) are independent, i.e., for a linear chromatographic process (no overloading effect), one has: tR ¼ t c

ðB-8Þ

and N mol X 1 r ðtR Þ ¼ 2 r2 tc; j Nmol j¼1 2

!

mol 1 X 1 r2 ðtc; j Þ ¼ r2 ðtc Þ 2 Nmol Nmol j¼1

N

¼

ðB-9Þ from which one gets the square of the relative error on tR: r2 ðtR Þ 1 r2 ðtc Þ ¼ Nmol t 2c t2R

ðB-10Þ

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The chromatographic literature makes a frequent use of the concept of plate number to characterize the relative width a peak. Let N be the analyte plate number, defined as: Nu

t2R t 2c ¼ r2 ðtc Þ r2 ðtc Þ

ðB-11Þ

One gets from Eqs. (B-10) and (B-11): r2 ðtR Þ 1 1 ¼ Nmol N t2R

ðB-12Þ

This equation provides the relative error on the retention time of the solute of interest, arising from the fact that the sample contains a limited number of analyte molecules. Because this number is limited, the distribution of their residence time is not exactly equal to the probability density distribution of the individual molecules, f(tc). Equation (B-12) shows that the relative error on tR becomes vanishingly small as the number of analyte molecules increases. Repeating the above derivation for the unretained tracer used to measure t m;u , one gets: r2 ðt m;u Þ t 2m;u

¼

1 1 Nmol;u Nu

ðB-13Þ

where Nmol,u is the number of molecules of the unretained tracer in the injected tracer sample and Nu is the plate number for the unretained tracer, defined as: Nu u

t 2m;u r2 ðtc;u Þ

ðB-14Þ

Here r2(tc,u) is the variance of the frequency distribution of the residence time of the tracer molecules. B.1.3. Relative Error on k V Combining Eqs. (B-3), (B-4), (B-12), and (B-13), one gets the expression of the square of the relative error on k V as: 

   r2 ðkVÞ 1 1 1 1 1 1 1 þ kV 2 1 þ ¼ þ Nmol N Nmol;u Nu Nmol;u Nu kV kV2 ðB-15Þ

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The residence time of an analyte molecule in the column is the sum of the time, tm, that the molecule spent in the mobile phase and of the time, ts, spent in the stationary phase: tc ¼ tm þ ts

ðB-16Þ

These two times are distributed according to some frequency functions with mean values, t m and t s , and variances, r2(tm) and r2(ts), respectively. It was shown previously [37,46] that: tR ¼ t c ¼ t m þ t s

ðB-17Þ

r2 ðtc Þ ¼ ð1 þ kVÞ2 r2 ðtm Þ þ r2 ðts Þ

ðB-18Þ

and

Combining Eqs. (B-11) and (B-18), one can write: 1 r2 ðtm Þ r2 ðts Þ r2 ðtm Þ r2 ðts Þ ¼ ð1 þ kVÞ þ 2 ¼ þ 2 N t2R tR tR t 2m

ðB-19Þ

Let us define Nm in a way similar to Eq. (B-14) for the unretained tracer as: 1 r2 ðtm Þ u Nm t 2m

ðB-20Þ

If the analyte and the unretained tracer have similar diffusivities, Nu and Nm should also have similar values (note that the subscript m, for mobile phase contribution, does not appear in Nu because the whole residence of the unretained tracer in the column occurs in this phase). In addition, it is customary to define the effective plate number, NV, by relating the time variance to the square of the mean time spent in the stationary phase, instead of that of the retention time. Then: 1 r2 ðtc Þ 1 u 2 ¼ NV N ts



 1 þ kV 2 r2 ðtm Þ r2 ðts Þ r2 ðtm Þ 1 ¼ þ ¼ þ 2 2 2 kV NV s ts ts ts ðB-21Þ

where NsV can be regarded as the effective plate number arising from the stationary phase.

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Furthermore, according to the stochastic theory of chromatography, one has:   1 r2 ðts Þ ns r2 ðss Þ þ s2s r2 ðns Þ 1 r2 ðss Þ r2 ðns Þ u u 2 ¼ ¼ þ ¼ s 2 2 NVs n n ns s s s ts ðns ss Þ s ðB-22Þ In this equation, the mean values and variances of the number of visits of a molecule to the stationary phase during its stay in the column, ns, and of the duration of one visit, ss, appear. us, equal to the term in brackets in Eq. (B-22), is a number, the value of which depends on the particular forms of the probability distribution functions of ns and ss. In the case of a Poisson distribution of ns and of an exponential distribution of ss, us is equal to 2 [35]. Combining Eqs. (B-15) and (B-20–B-22), one obtains the square of the relative error on k V as: " #    r2 ðkVÞ 1 1 1 þ kV 2 us 1 1 ¼ þ 1 þ Nmol Nm kV Nmol;u Nu ns kV2 ðB-23Þ  2 1 1 1 þ kV þ Nmol;u Nu kV This equation, like Eq. (B-15), is quite general. It can take a simpler form in some particular cases. For instance, when the dispersion process in the mobile phase is negligible, Nm and Nu vanish and one gets: r2 ðkVÞ 1 us ¼ ¼ 2 Nmol NVs Nmol ns kV The relative error on k V, r(kV)/kV, is then equal to: sffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 ðkVÞ r2 ðkVÞ us u ¼ 2 kV N kV mol n s or, with Eq. (B-22): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðkVÞ 1 ¼ kV Nmol NVs

ðB-24Þ

ðB-25Þ

ðB-26Þ

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Equation (B-25) reveals that Nmol and ns play a very similar role as concerns the error on k V. The larger they are, the more the analyte molecules ‘‘sample’’ the stationary phase. It is then equivalent to have a small mean number of visits to the stationary phase (short columns or fast carrier velocities) and a large number of molecules as to have a few molecules visiting, in average, a large number of times the stationary phase. In the case of the Giddings–Eyring model, as mentioned above, us is equal to 2. Then, one gets, together with Eq. (A-12): rðkVÞ ¼ kV

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Nmol ns

ðB-27Þ

APPENDIX C C.1. Expression of the Maximum Number of Analyte Molecules The number, Nmol, of molecules of a given analyte injected in a chromatographic column is given by: Nmol ¼ NAv cM Vinj

ðC-1Þ

where NAv is the Avogadro number, cM is the molar concentration of the analyte in the injected sample, and Vinj is the injected volume. The number of analyte molecules injected in the chromatographic column is obviously proportional to the injected sample volume. It can thus be increased by increasing this volume. However, the finite volume of the injection band contributes to the overall width of the peak and must be limited in order to limit the degradation of the chromatographic resolution. In fact, one has to tolerate some loss of efficiency due to the sample volume contribution to peak broadening. This can be expressed by stating that the contribution to the peak variance arising from the injection process, r2V;inj , must not exceed a fraction h2 of that, r2V;c , arising from the column migration process [41]. Then, its maximum value becomes: r2V;inj ¼ h2 r2V;c

ðC-2Þ

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where the variances are expressed in terms of square of volume units. Obviously, r2V;inj is related to the injected volume, which can be written as: r2V;inj ¼

2 Vinj

ðC-3Þ

a2

where a2 is a parameter which depends on the shape of the injection profile. For a plug injection, which corresponds to the least dispersive injection profile, a2 is equal to 12. In practice, a2 is smaller and has been found to lie between 3.5 and 7 in liquid chromatography and to have quite lower values in gas chromatography [42]. The column variance can be related to the column plate number, N, or to the effective plate number, NV, as: r2V;c

V2 V2 ¼ R ¼ R N NV



kV 1 þ kV

2 ðC-4Þ

where VR is the retention volume of the analyte, which is related to the volume Vm of mobile phase in the column through: VR ¼ Vm ð1 þ kVÞ

ðC-5Þ

By combining Eqs. (C-1–5), one gets the expression of the maximum number of analyte molecules which can be introduced in the chromatographic column under the accepted tolerance criterion: kV Nmol ¼ ahNAv pffiffiffiffiffiffi Vm cM NV

ðC-6Þ

This number of analyte molecules is seen to depend on a characteristic of the injection device (a), on the tolerance factor, and on the column effective plate number and to be proportional to the retention factor, to the volume of mobile phase in the column, and to the analyte molar concentration in the sample. Equation (C-6) gives the maximum number of analyte molecules which are injected in the column under the tolerance criterion. However, the number of molecules which are detected may be only a fraction, U, of this number. U is called the molecular detection efficiency. Obviously, only the detected molecules contribute to the

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determination of the retention factor. Then, the relevant number of molecules becomes: kV Nmol ¼ ahUNAv pffiffiffiffiffiffi Vm cM NV

ðC-7Þ

APPENDIX D D.1. Error Resulting from the Effect of the Injection Process on the First Moment of the Peak Because all analyte molecules do not enter simultaneously the chromatographic column, there is a systematic bias in the mean elution time of the analyte peak. In fact, the distribution of the elution times of the analyte results from the convolution of the distribution of the injection times by the distribution of the residence times in the column. Let t inj be the mean injection time. The mean elution time of the analyte peak, t el , is then: t el ¼ tR þ t inj

ðD-1Þ

If t el is used instead of tR in Eq. (B-1) to compute kV, an error in k V results. However, the mean column residence time of an unretained solute is frequently taken as a measure of the time spent by the analyte in the mobile phase, tm (or, more precisely, of the mean time spent by the analyte in the mobile phase, t m , when there is a distribution of tm due to mobile phase dispersion). If, because of the finite duration of the injection process, the mean elution time of the unretained solute, t el;u, differs from t m, by the amount t inj, an apparent value of k V, k aV , is obtained: kVa ¼

t el  t el;u tR  t m ¼ t el;u t m þ t inj

ðD-2Þ

If t m can be independently determined, supposedly without error, one gets then an apparent k V value, kVb, given by: t el  t m tR  t m þ t inj kVb ¼ ¼ ðD-3Þ tm tm The relative error resulting on k V becomes in the first case:   t inj =t m ykV kVa  kV ¼ u kV a kV 1 þ t inj =t m

ðD-4Þ

Chromatography: A Stochastic Approach and in the second case:   ykV kVb  kV t inj =t m ¼ u kV b kV kV

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ðD-5Þ

In the first case (determination of t m by means of an unretained solute injected present in the analyte sample), it appears that the difference between the first moments of the elution peaks of the analyte and of the unretained solute is not affected by the injection process so that the error on k V arises solely from the deviation of the first moment of the unretained solute. When t m is independently known, the error on k V generally decreases with increasing k V and is generally lower than in the first case. The value of t inj can be related to the injected volume, Vinj, as: t inj ¼ c

Vinj F

ðD-6Þ

where c is a proportionality factor, the value of which depends on the injection device. Then, the smaller is the value of Vinj, the smaller is the error on k V resulting from the injection effect. When the maximum volume compatible with a tolerated deterioration of the column efficiency, given by Eq. (C-3), is injected, t inj becomes using Eqs. (C-2–5) together with (D-6): kV t inj ¼ ach pffiffiffiffiffiffi t m NV

ðD-7Þ

Then, the relative errors on k V become, combining Eqs. (D-4), (D-5), and (D-7):



kV ach pffiffiffiffiffiffi ykV NV ¼ kV kV a 1 þ ach pffiffiffiffiffiffi NV

ðD-8Þ

 ykV ach ¼ pffiffiffiffiffiffi kV b NV

ðD-9Þ



and 

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APPENDIX E E.1. Error on k V Resulting from an Improper Selection of the Unretained Tracer The experimental determination of the retention factor, k V, of the solute of interest requires the measurement of the retention time, tR, of this solute and of the mean elution time, t m;u , of a supposedly unretained compound which is visiting only the mobile phase, according to Eq. (B-1). If the supposedly unretained tracer is not really unretained, but has itself a retention factor, kVu, such that:

kVu ¼

t m;u 1 tm

ðE-1Þ

where t m is the mean elution time of a truly unretained compound, there is a systematic error made on kV when using Eq. (B-1). If kVo is the true retention factor of the analyte, defined as:

kVo u

tR 1 tm

ðE-2Þ

the relative systematic error on k V is equal to: 

ykV kV

 ¼

kVo  kV kVu 1 þ kVo ¼ kVo 1 þ kVu kVo

ðE-3Þ

which, for small k Vu, becomes: 

 ykV 1 þ kVo ckVu kV kVo

ðE-4Þ

4 Solid-Phase Microextraction: A New Tool in Contemporary Bioanalysis Georgios Theodoridis Aristotle University Thessaloniki, Thessaloniki, Greece Gerhardus J. de Jong University Utrecht, Utrecht, The Netherlands

I. INTRODUCTION II. EXTRACTION MODE AND COUPLING A. Novel Devices III. SORPTION PRINCIPLES AND PARAMETERS IV. COATINGS A. Absorptive Coatings B. Solid Coatings C. Special Coatings V. DERIVATIZATION VI. BIOANALYTICAL APPLICATIONS VII. POSSIBILITIES AND LIMITATIONS OF SPME REFERENCES

232 234 238 242 246 247 249 250 255 258 264 268 231

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I. INTRODUCTION Solid-phase microextraction (SPME) has been introduced recently as a useful method in sample preparation. This technique integrates sampling, extraction, preconcentration, and sample introduction in a simple single-step procedure. Additionally, it facilitates automation and direct coupling to chromatographic analysis: gas chromatography (GC) and high-performance liquid chromatography (HPLC). When performed in the most known fiber format, SPME is based on the sorption of the analyte on an extraction phase coated on a small fused silica fiber. The fiber is mounted in a syringe-like protective holder (Fig. 1). During extraction, the fiber is exposed to the sample, either immersed in a liquid sample or exposed to the headspace above the sample. After equilibrium or a defined time, the fiber is withdrawn in the septum-piercing needle and introduced into the analytical instru-

Fig. 1 Schematic of the SPME device commercially available from Supelco.

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ment. There the analytes are either thermally desorbed (GC) or redissolved in a proper solvent for HPLC or capillary electrophoresis (CE). The technique has been commercialized in 1993 by Supelco. Initial work was exclusively aimed at SPME-GC combinations since SPME was originally introduced as a method aiming at the sample pretreatment of environmental samples. Furthermore, coupling to GC is straightforward and convenient because the fiber is introduced into the GC injector. In the few years of its practice, SPME has developed to a mature technique and a useful alternative to contemporary techniques in various scientific and research fields. Not surprisingly, SPME was one of the six ‘‘great ideas of the decade’’ as illustrated in a recent survey of Analytical Chemistry [1]. Solid-phase microextraction offers unique advantages: solventless extraction, low cost, simplicity, on-site sampling, high efficiency and reproducibility, and compatibility with numerous analytical techniques. Furthermore, due to its distinctive features (e.g., geometry, portability), the technique provides the ground for innovative approaches and designs. As a result, SPME has more specific advantages to cover niches in analytical sciences as will be exemplified in the following chapter. A clear example of the profound evolution is the continuous annual increase in the number of research papers published in peer-reviewed journals (Fig. 2). In

Fig. 2 Annual plot of the number of publications reporting on SPME.

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less than a decade of existence as a commercially available technique, SPME has risen above the landmark of 1000 publications. In this chapter, the practice and application of SPME are described with special emphasis to bioanalytical applications. Solidphase microextraction modes are illustrated with emphasis on novel technological approaches. Theoretical discussion is limited to fundamental terms and features, whereas technological aspects as coating materials and special devices are stressed. Finally, specific advantages and limitations of SPME are discussed.

II. EXTRACTION MODE AND COUPLING There are, as a rule, two extraction modes: direct immersion (DI) in liquid samples and headspace (HS) extraction. The major criteria for mode selection are the nature of the sample matrix, the volatility of the analyte, and the affinity of the analyte for the matrix. Medium volatile or nonvolatile analytes such as macromolecules and polar analytes are extracted by direct immersion of the fiber into the sample. The mass transfer rate is determined mainly by diffusion of the analyte in the coating, provided that the convection in the liquid sample is ideal and the sample is a single homogeneous phase. In practice, however, a thin boundary layer of static liquid sample is formed around the fiber, hindering the access of the analytes to the coating. This boundary cannot be removed without vigorous agitation methods. Analytes exhibiting low vapor pressure remain trapped on the fiber allowing field sampling and analysis by GC or HPLC in a second stage. For dirtier samples, the fiber can be protected by a membrane [2,3]. Employment of a membrane can enhance the overall efficiency due to the added membrane selectivity. However, mass transfer is reduced; thus increased temperatures or thin membranes are necessary for relatively short extraction times. HS-SPME is preferred for volatile compounds because it may provide cleaner extracts, greater selectivity, and longer fiber lifetime. In this case, there are three phases (coating, headspace, and sample matrix) involved in the extraction process. As a rule, equilibrium is faster in HS-SPME than in DI-SPME since diffusion in gaseous phases is typically much faster than in liquid phases. The timelimiting step is the transfer of the analytes from the sample to the

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headspace. Hence gentle heating or stirring of the sample will improve extraction. However, higher temperatures will result in reduction of the partition coefficients. Although often regarded as the less critical step in SPME, desorption is also important for a successful method. For the SPME-GC combination, analyte desorption from the fiber is straightforward. The septum-piercing needle of the SPME device is introduced into the GC injector, where the fiber is exposed to the heated chamber and the analytes are thermally desorbed. For faster desorption, elevated temperatures and a narrow bore insert are required. To eliminate carryover effects, split/splitless injection is used: desorption occurs in splitless mode, so that the main part of the desorbed amount of analyte is introduced in the GC column, where it can be cryofocused. During analysis, the fiber remains exposed in the injector (operating now in split mode); thus possible carryover is thermally desorbed without entering the column. Coupling to LC requires an appropriate interface, and one such is commercialized by Supelco. The fiber is placed into a low-volume desorption chamber with 3 ports in T-configuration. The chamber is mounted in a typical 6-port injection valve in the place of the injection loop. Desorption occurs either statically or dynamically. In static mode, the desorption chamber is filled with an appropriate solvent, and then, the fiber is introduced to the interface for a determined time. Static desorption depends on time and the composition of the desorption liquid. Switching the valve introduces the plug of the backextracted analytes to the analytical column. In dynamic desorption, the mobile phase flows within the desorption chamber, desorbing the analytes and driving them to the analytical column. Dynamic desorption is governed by the selection of the mobile phase and the flow rate and often suffers from substantial peak broadening [4]. Heating the interface was shown to enhance mass transfer rates and thus to affect desorption and separation, reducing peak broadening and carryover [5]. One possibility that seems to be overlooked so far is the off-line combination of fiber SPME with liquid chromatography with no interface. In this approach, desorption may occur in a small volume tube, (e.g., an autosampler insert) in static mode. Subsequently, an aliquot of the resulting solution is injected in HPLC. In general, however, SPME-LC offers no profound advantages to displace SPE-LC; thus it is not surprising that SPME-LC applications are limited to almost 10% of the total of SPME reports.

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In-tube SPME is an attractive alternative for the automation of SPME-LC. In this case, extraction takes place in the inside of a fused silica capillary, which is coated with the extractive phase. For sorption, an aliquot (some microliters) of the sample is aspirated and dispensed into the capillary. Desorption of the analytes is achieved by aspiring a proper organic solvent and dispensing the eluate into the injection loop (Fig. 3). In-tube SPME exhibits different geometry than fiber SPME. This method enhances full automation and can be performed with typical LC autosamplers after minor modifications. Moreover, in-tube desorption was reported to be quantitative, eliminating carryover effects. In general, sample introduction of relatively large sample volumes in capillary electrophoresis is a challenge. Although both SPME

Fig. 3 Various designs for the in-tube coupling of SPME to HPLC. Extraction capillary in place of transfer line (a). Extraction capillary in place of standard loop (b). Extraction capillary in place of flush injection loop (c). (From Ref. 2.)

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and CE are capillary techniques, it is rather difficult to couple them on line. So far, only experimental ‘‘house-built’’ interfaces have been developed. In one approach, an SPME-CE device was developed in house by gluing two CE capillaries in a Teflon sleeve containing a small amount of C18 material. Another alternative is to introduce the fiber via guides to the end of the capillary (Fig. 4) [6]. Recently, SPME has been coupled with confocal Raman spectroscopic analysis for airborne sampling of contaminated air. Raman spectroscopy is a powerful method for chemical fingerprinting of analyte molecules. Combination with SPME provided a novel procedure for identification and possible numeration of airborne particulate matter. Mass loading in SPME fibers was changed by altering the sampling time. The polydimethylsiloxane (PDMS) fiber was next subjected to confocal Raman microspectroscopic analysis. Raman spectroscopic analysis resulted in the identification of several characteristic bands enabling single particle analysis of less than 1 Am in diameter [7].

Fig. 4 Coupling of SPME to CE. (From Ref. 6.)

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A. Novel Devices Solid-phase microextraction is a new and unique extraction technique. Therefore researchers from diversifying fields have shown initiative to use SPME as a tool for a variety of purposes. This trend has resulted to numerous novel devices that exploit the features of SPME. Some noteworthy applications are the following: Portable Solid-Phase Microextraction for Field-Air Sampling A great field for future evolution for SPME is seen in field sampling. The technique is easy and convenient, portable, and not space demanding. Sampling may be performed independently of analysis or may also be combined with portable GC [8]. In the former case, innovative portable SMPE devices have been developed for sampling air, aroma, and volatiles from foods and living organisms (e.g., insect pheromones). Additionally, new coatings and methodologies are investigated [9]. The use of SPME fibers coated with adsorptive porous polymer solid phases for quantitative purposes is limited due to interanalyte displacement and competitive adsorption. For air analysis, these problems can be averted by employing short exposure times to air samples flowing around the fiber. In these conditions, a simple mathematical model allows quantification without the need of calibration curves. Portable dynamic air-sampling devices have been designed for application of this approach to nonequilibrium SPME sampling and determination of airborne volatile organic compounds (VOCs). These devices reduced total sampling and analysis time compared to the official methods (trapping in charcoal, extraction with CS2, and GC-FID analysis) [10]. Despite the reduced sampling time, method sensitivity is superior for volatile organic compounds. Hence detection limits as low as 700 parts per trillion (ppt) have been reached. Solid-phase microextraction technology combined with fast portable GC reduced the sampling and analysis time to less than 15 min. The configuration offered the conveniences of on-site monitoring that is not possible with conventional methods [8,11]. An interesting methodology in that direction may be the SPMEdirect FID method. In that scheme, the SPME was connected to an FID system but with no GC column. The idea was to measure rapidly the total sum of volatiles sorbed on the fiber. Using this scheme as an

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electronic nose, one can easily add a column to obtain qualitative insight of the analyzed samples [12]. Wire or Fiber In-Tube An ingenious approach is the incorporation of wire inside a capillary tube in order to minimize the internal volume of the capillary (compare Fig. 5A and B). A stainless-steel wire (0.20 o.d.
20 cm) was inserted in a piece (0.25 mm i.d.
20 cm) of a GC capillary column, diminishing its volume to 3.53 AL. The capillary was used as an intube SPME device, connected to micro-LC. Preconcentration of tricyclic antidepressants from human urine was accomplished, using minimal volume of organic solvent for analyte desorption [13]. With a similar experimental setup, the same group studied the configuration of fiber in tube. Approximately 280 Zylon fibers (11.5-Am

Fig. 5 Three types of extraction tubes used (in tube microextraction). (A) Intube, (B) wire-in-tube, and (C) fiber-in-tube configurations. (From Ref. 15.)

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diameter) were forced in a PEEK tube (0.25 mm i.d.) as can be seen in Fig. 5C. The tube was used for the enrichment of n-butylphtalates, providing a preconcentration factor of about 160. The authors claim that such schemes employ minimal amounts of organic solvent and facilitate direct coupling to capillary techniques such as CE and micro-LC [14,15]. Protection of the Fiber by a Membrane Protecting the SPME fiber by a membrane may serve as means to prolong fiber lifetime and enable the process of dirty samples. In this objective, a PDMS coated bar was enclosed in a dialysis membrane bag and the sampler was used for the solventless procedure for the preconcentration of triazines from aqueous matrices. In addition to enrichment, hollow fiber-protected microextraction also served as a technique for sample cleanup because of the selectivity of the membrane, which prevented large molecules and extraneous materials (e.g., humic acids) from being extracted [16]. In a similar manner, a bar coated with PDMS was enclosed in a dialysis membrane bag and the device was used for microextraction of hydrophobic analytes (cyclohexanes, PAHs) from aqueous matrices. The concept combined passive sampling with solventless preconcentration of organic solutes. Subsequently, the sampler was desorbed and the analytes were analyzed by capillary GC-MS [17]. Fiber Conditioner Fiber conditioning is performed thermally by its insertion in a GC injection port for a certain time. Usually, this means 1–3 hr for a new fiber or 5 min between different samples at 250–280jC. This may reduce available instrumentation and at the same time introduce unwanted interferences to the GC column. To address these concerns, Koziel et al. [18] developed a fiber conditioner device. The conditioner ` was assembled from a Hamilton syringe cleaner, where a 1000-U ceramic heater and a flow of N2 gas facilitated desorption and conditioning of the fiber. The device performed equal or better than GC injectors for removing test components (alkanes of varying boiling points). Solid-Phase Microextraction Electrochemistry The idea behind the combination was to integrate extraction and electrochemical reaction in a single conductive polymer coating. In this coating (a carbon steel wire with a 10-Am gold coating), the

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analytes (Hg in this case) are electrochemically oxidized (or reduced) and sorbed. Desorption was accomplished with a dedicated desorption system and analysis by GC-MS. Hg2+ in aqueous solution was reduced to Hg0 which is next determined by mass spectrometry. With this methodology, inorganic mercury and organomercury compounds were differentiated. Another attractive proposal is the construction of a SPME-electrodeposition device for the determination of putrescine and cadaverine [19]. The three-electrode system consisted of a Ag/AgCl reference electrode and a stainless-steel mesh counter electrode, which surrounded a pencil lead; the latter served as both the SPME device and the working electrode. The pencil lead was immersed in a pH 8 borate buffer, and 1.70 V potential vs. the reference electrode was applied, resulting in an electrochemical reduction of buffer solution protons. Subsequently, diamines present in the solution are converted into their free base form and retained on the electrode, which is used as the SPME fiber. The device was then transferred to a capillary GC equipped with a thermionic detector. Solid-Phase Microextraction-Mass/Atomic Spectrometry The combination of SPME with the high sensitivity and selectivity of mass spectrometry may reduce the need for chromatographic separation and allow for very rapid sample processing. Initial approaches have dealt with coupling to ion mobility mass spectrometry [20]. Recently, increased interest in this coupling is seen in different aspects such as bioanalysis and determination of heavy metals. Accordingly, direct coupling of nonequilibrium SPME to ion trap has been reported via static desorption in an SPME–HPLC interface. The 70-AL analyte plug was then directly introduced to an APCI (Atmospheric pressure chemical ionization) or an ESI interface. Polydimethylsiloxane fibers were tested for the extraction of spiked calf urine. Despite the absence of a real liquid phase separation mode, problems such as matrix complexity and ion suppression were overcome by the high selectivity of both SPME and MS. Method linearity (0.4–80 ng/mL), reproducibility (2.5–13.7% RSD), and sensitivity (0.4 ng/mL LOD) were adequate, and the combination has proven an efficient and rapid method for the determination of lidocaine in biological samples [21,22]. In a similar manner, PDMS and PDMS/DVB fibers were tested for the extraction of amphetamines and their

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methylenedioxy derivatives from urine and their quantitation by electrospray ionization–high-field asymmetric waveform ion mobility spectrometry–mass spectrometry. Desorption occurred dynamically (0.4 mL/min) within a PEEK tubing (15
0.5 mm) that was fixed in a stainless-steel T connector. Limits of detection in human urine were between 200 pg/mL and 7.5 ng/mL [23]. Solid-phase microextraction has also been used as an introductory technique for matrix-assisted laser desorption/ionization (MALDI) for mass spectrometry and ion mobility spectrometry. A silanized optical fiber served as the sample extraction surface, the support for the sample plus matrix, and the optical pipe to transfer the laser energy from the laser light source to the sample. Atmospheric pressure MALDI ion mobility spectrometer or quadrupole/time-of-flight mass spectrometer was used for the detection of nicotine, myoglobin, enkephalin, and substance P utilizing 2,5-dihydroxybenzonic acid and alpha-cyano-4-hydroxy cinnaminic acid as the ionization matrix [24]. Finally, SPME has recently been coupled directly with atomic spectroscopy techniques for the determination of metal hydrides (As, Se, Sn, Sb) and Ge hydride and chloride [25,26]. For similar purposes, SPME has been used as a sampling technique combined with radiofrequency glow discharge MS for the determination of tetraethyl lead at ppt concentrations [27]. Coupling of SPME with atomic spectroscopy techniques was so far achieved via GC or HPLC. Taking into account the growing importance of metal determination and speciation in biological systems, such a direct coupling may provide an interesting and advantageous alternative for sampling and determination of organometal compounds in various matrices, opening a new prospect for bioanalysis.

III. SORPTION PRINCIPLES AND PARAMETERS In SPME, the extraction can either reach equilibrium (complete extraction) or last for a defined time. In the former case, extraction is considered to be complete and to follow the rules of liquid–liquid extraction. For the established equilibrium of the analyte between the fiber and the solution equilibrium, the distribution constant Kfs is [28]: Kfs ¼

Cf Cs

ð1Þ

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From Eq. (1), with simple mathematics, one comes to Eq. (2) describing n as the number of moles extracted by the coating phase n¼

Kfs Vf Vs C0 Kfs Vf þ Vs

ð2Þ

where Vf is the fiber coating volume, Vs is the sample volume, and C0 is the initial concentration of the analyte. From this equation, it is derived that after equilibrium, the relationship of amount extracted and sample concentration is directly proportional. This relationship enables quantitative analysis. Care should be taken since the linear range of the method is affected not only by SPME, but also by the subsequent analytical method. For liquid absorptive fibers, it is very unlikely to observe saturation phenomena and it is thus assumed that the response of the fiber will be linear for most working concentration ranges. However, solid adsorption coatings provide less active surface and (see discussion in Sec. IV) analyte displacement may occur. It is of utmost importance to validate the method in terms of linearity using standard solutions before applying real samples (spiked or not). When the sample volume is very large, Eq. (1) is simplified to: n ¼ Kfs Vf C0

ð3Þ

Equation (3) signifies that in cases where Vs is very large, the amount extracted is independent of the sample volume. This indicates the value of SPME for field analysis. Extraction conditions affect extraction recovery to a great extent. The most critical parameters are sample volume, sample pH value, ionic strength (salt concentration), extraction temperature, extraction time, and finally convection or agitation. Detailed discussion on the effect of these parameters on extraction can be found in comprehensive reviews covering the topic [2,29] and the excellent books published recently on SPME [30–32]. Accuracy and precision of SPME can be easily influenced by the above parameters. Optimization of these conditions may lead to large enhancement of the extraction yield. Salt concentration and pH affect SPME as they also affect any extraction procedure. Salt addition can improve the extraction yield. Salts often employed include NaCl, (NH4)2SO4, and Na2CO3 in varying contents. Typically, an increase in the extraction yield is

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observed with increasing amount of salt due to ‘‘salting out’’ effect. This can be followed by a maximum and a decrease in yield with further saline increment. In this case, it is believed that polar analytes contributing to electrostatic interactions in saline environment lose their mobility and mass transfer toward the extracting phase [2]. Adjustment of pH may improve the extraction yield for compounds that can be protonated. In most of the cases, pH is adjusted in order to obtain the analyte in its neutral undissociated form to enhance extraction yield since only this form is extracted in absorptive fiber. Care has to be taken when direct-immersion SPME is used since extreme pH values (lower than 2 and higher than 10) can damage the coating. Sample volume selection should be based on the estimated partition constant Kfs. For compounds with high Kfs values, large sample volumes (z10 mL, if available) should be used. For headspace extraction, the gaseous phase volume should be minimized in order to increase the yield. Agitation of the sample is used in order to enhance the extraction recovery with time or to reduce the equilibrium time. The most common agitation methods are magnetic stirring and fiber vibration. An increase in temperature can increase the extraction yield in nonequilibrium situations as a result of diffusion enhancement. The latter will also result to decrease in the time required to reach equilibrium. However, in principle, increase in temperature decreases the distribution constant (and thus to the amount extracted) due to decrease in partition coefficient to the extraction phase. Extraction time varies greatly with times ranging from 1 to 60 min. Solid-phase microextraction is an equilibrium process, but very often, extraction is ended in a fixed time before reaching equilibrium. Equilibrium time is governed by mass transport between sample and coating and therefore affected by coating thickness, agitation method, temperature, and so forth. An attractive option to accomplish reduction in extraction time could be Multiple SPME under nonequilibrium conditions. Koster and de Jong [33,34] studied the theory and the application of performing multiple SPME experiments on the same sample. Theoretically, the yield of multiple extractions is higher than the yield of one extraction of the cumulative time. This was observed in the SPME-LC and SPME-GC analysis of lidocaine, amphetamine, and related drugs from human urine [33,34]. Alternatively, for a standard extraction time, the total yield obtained by

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multiple SPME was higher compared to single SPME. Theoretically, this enhancement is evident in the extraction of compounds with rather low Kfs. In contrast for analytes with high values of Kfs (Kfs > 10,000), multiple extraction is not likely to improve extraction performance. Characteristic Kfs values are Kfs = 125 for benzene, Kfs = 831 for xylene [28], Kfs = 221 for clozapine, and Kfs = 2671 for loxapine [35]. As a rule, an indication of the value of Kfs can be obtained by octanol–water partition coefficients (Kow). However, this should be regarded as estimation only and experiments should be made to confirm the fit. Special care has to be taken for the determination of partition coefficients. Very nonpolar compounds, such as polycyclic aromatic compounds (PAHs), may be adsorbed into glass

Fig. 6 Gas chromatogram of blank plasma and plasma spiked with 5 nmol/ mL of diazepam (peak 1) and prazepam (internal standard, peak 2). SPME modified with 1-octanol PA fiber. (From Ref. 37.)

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walls of laboratory devices (e.g., extraction vials) and Teflon coatings to a substantial extent. Such interactions must be taken into account when calculating partition coefficients. It was shown that failing to do this may lead to large errors in the value of the partition coefficient, particularly for very nonpolar compounds [36]. Krogh et al. [37] used another approach in order to improve extraction recovery. They proposed a solvent-modified extraction procedure that employs the modification of a PA fiber by sorption of 1-octanol before its direct immersion in blood plasma samples. The amount of diazepam extracted this way was twice higher compared to the amount extracted without the use of 1-octanol. The method was further optimized in a recent publication [38]. Parameters, which were found to influence analyte recovery, were studied in a factorial design and response surface methodology. Figure 6 depicts the application of this extraction scheme in the analysis of drugs in plasma. It should be noted, however, that the potential of the method is limited due to the incompatibility of SPME fibers with organic solvents.

IV. COATINGS The efficiency of a separation method is dependent to a great extent on the stationary phase. In a similar way, the efficiency of a SPME method is subject to the choice of fiber coating. The physical and chemical properties of the extracting phase govern extraction selectivity and yield. So far, six coatings are available commercially in 18 different configurations. The main characteristics of these coatings are depicted tabulated in Table 1. The most common PDMS and PA coatings are liquid polymeric phases, where absorption is the major mechanism (Fig. 7). In contrast, divinylbenzene phases have a more rigid crystalline lattice polymeric structure. In these coatings, extraction of the analyte occurs via its adsorption on the surface of the polymer. Additionally, to commercial phases, many other experimental phases have been developed. The selection of fiber coating is mainly based on the principle ‘‘like dissolves like’’ and is of utmost importance for a successful application. This effect is important especially due to the fact that SPME is (almost) never an exhaustive extraction method. The fundamental properties of the various coatings are described in detail in the following paragraphs. Finally, a parameter that should always be taken into account is the stability of coating (fiber) in organic solvents. Polydimethylsilox-

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Table 1 Characteristics and Major Properties of the Most Common SPME Fiber Coatings Fiber coating Polydimethylsiloxane (PDMS) Polyacrylate (PA)

Thickness Compatibility/ (Am) recommended for 7a 30b 100b 85

GC, GC, GC, GC,

HPLC HPLC HPLC HPLC

Polydimethylsiloxane/ divinylbenzene (PDMS/DVB)c

65

GC, HPLC

60

HPLC

Carboxen/PDMSc

75 85d 65

GC

50

HPLC

Carbowax/DVB (CW/DVB)c Carbowax/templated resin (CW/TPR)

GC

Target analytes Nonpolar organics (VOCs, PAH, pesticides, drugs, etc.) Polar organics (phenols, triazines) Aromatic hydrocarbons, VOCs Amines, polar compounds VOCs, hydrocarbons Polar organic compounds, alcohols Anionic surfactants

a

Bonded phase. Nonbonded phase. c Partially cross-linked phase. d On a Stableflex fiber. b

ane and PA are liquid phases that may exhibit swelling and shrinking. Caution is required when using chlorinated solvents as these may dissolve the epoxy glue that holds the fiber. Especially for PDMS/ DVB and CW/DVB fibers, extra caution is required. In extreme cases, the polymer coating may swell and drop off the fiber. Newer developments in fiber manufacturing have enhanced stability and tolerance for HPLC mobile phases.

A. Absorptive Coatings In absorption process, the analyte progresses from the bulk of the sample toward the fiber coating. This phenomenon is a combination of convection and diffusion; thus increasing either of these can enhance absorption. Agitation is the best way to increase convection. Diffusion can be increased by increasing the extraction temperature [2,3]. In

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Fig. 7 Scheme of the types of coatings for SPME. Liquid absorptive coatings such as PDMS and PA (left) vs. solid adsorptive coatings (e.g., template resin) where adsorption occurs in either large or small pores. (From Ref. 3.)

absorptive coatings, the analytes partition in to the extracting phase, where analyte molecules are solvated. Diffusion of the analytes in the extracting phase facilitates the penetration of the analyte molecules to the whole volume of the coating. The first coatings developed and commercialized were PDMS and PA; these remain still the most popular since they offer generic selectivity and thus adequate recovery for many types of nonpolar analytes. Furthermore, they are rugged fibers of (generally) long lifetime. Nonpolar analytes have relatively high affinity for the apolar PDMS phases. Polyacrylate is more polar and can be used for the extraction of more polar compounds, such as phenols. Mixed phases are mainly used for the

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extraction of volatile compounds. The extraction yield of these fibers is higher compared to PDMS, but their lifetime is limited. Coating thickness is selected according to the extraction yield required, the extraction time, and the nature of the analyte. The thinner the coating, the faster the partition equilibrium can be reached. The choice of coating thickness is also related to the molecular mass of the analyte: for small molecular weight compounds, high extraction yields can be obtained with relatively thick coatings [2].

B. Solid Coatings Coatings regarded as solid are the divinylbenzene phases used in combination with both GC and HPLC and other experimental coatings used in combination with HPLC for the in-tube approach. In these phases, penetration of the analyte molecules into the core of the polymeric phase is negligible. Such materials posses a well-defined structure of a highly dense network, which reduces the diffusion of the analyte within its structure. Partitioning generally follows a Langmuir isotherm with the assumptions that: (a) molecules adsorb into an immobile state, (b) sorption active sites are homogenous and capable of one to one interaction with analyte molecules, and (c) no interaction occurs between absorbed molecules and neighboring sites [2,39]. However, in multianalyte samples, competition occurs between the analytes for the coating binding sites. Due to its dense structure, the active volume of the polymer is much less and displacement of analytes of low affinity may be observed, especially during long extraction times. This phenomenon may either be considered as a desired selectivity enhancement or as possible source of inaccuracies. In case when the analyte has low affinity for the coating, nonlinearity is often observed. A way to overcome such problems is to utilize extraction times much shorter than the equilibration time and also lesser amounts of analytes. This displacement effect was seen using a homemade polyacrylic acid-coated fiber for the extraction of proteins (Fig. 8). Analytes with low affinity were extracted only when short extraction times were employed. In such a case, the amount of basic proteins adsorbed onto the fiber was found to be proportional to the concentration of the protein. In contrast, during longer extraction, displacement of week binders occurred. Proportionality was also obtained for longer extraction times provided that the protein content does not

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Fig. 8 Cation-exchange microchromatography of a mixture of model proteins. Samples: (a) the original sample consisting of myoglobin (M), cytochrome c (C) and lysozyme (L); (b and c) proteins adsorbed onto and then released from a home-made polyacrylic acid-coated fiber with extraction times of 5 and 240 sec, respectively. (From Ref. 40.)

exceed the binding capacity; otherwise, the extraction of strongly absorbed proteins was favored. In longer extraction, displacement of week binders occurred. Figure 8 shows chromatograms of the analysis proteins obtained with the micro-LC system with and without SPME. Because myoglobulin was almost in its neutral form at the used extraction conditions, it was not adsorbed on the cation-exchangercoated fiber. Besides the selectivity, Fig. 8 also shows that cytochrome c is displaced by lysozyme during extraction; that is, at longer extraction time (compare Fig. 8b and c), the amount of lysozyme is increased as the amount of cytochrome c is decreased [40].

C. Special Coatings Lately, innovative phases for SPME have been developed. Media commonly used in liquid chromatography have been validated as possible SPME media. As such, porous bonded silica LC coatings (C8, C18), sol gel media, carbon graphitized silica, molecularly imprinted polymers, and immunoaffinity media have been used for microextraction. This drive is considered a strong future trend since such combinations enhance the advantages of the corresponding methods whereas, at the same time, they suppress their failings.

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Production of SPME materials by sol gel has attracted immense interest during the last years. Such phases are reported to exhibit high thermal stability and tolerance to organic solvents. Developed sol gel fibers employ polyethylene glycol (PEG), hydroxydibenzo-14crown-4 (OH-DB14C4)/hydroterminated silicone oil or hydroxydiberizo-14-crown-4 (OH-DB14C4), dihydroxy-substituted saturated urushiol crown ether (DBUD14C4), and 3,5-dibutyl-unsymmetrydibenzo-14-crown-4-dihydroxy crown ether (DBUD14C4) coatings. The fibers were validated for the SPME-GC of several organic pollutants and proved to be very stable at high temperature (up to 340–350jC) and in different solvents [41–44]. Molecularly imprinted polymers (MIPs) are media of predetermined selectivity that have found extensive use in separations and analysis. Molecularly imprinted polymers are produced by copolymerization of the analyte (as a template) within a highly dense polymeric network. At the end of polymerization, the analyte is removed, leaving a specific cavity, which should be complementary to the analyte molecule in terms of shape and chemical interactions. If the analyte is extracted by the polymer in a later stage, selective binding will occur in the binding site due to molecular recognition. Molecularly imprinted polymers actually represent another strong trend for high selectivity in separations. The utilization of MIPs in solid-phase extraction was first reported in 1994 [45]. Now it is by far the best studied and most widespread application area for MIP technologies and the first that made it to the market. An expected development was the expansion to microextraction. This was done by two groups independently. The group of Pawliszyn used bulk polymerization to manufacture MIPs for propranolol. The MIPs were used in the fashion of a miniaturized SPE column. Ground polymer particles were packed in a 80-mm PEEK tube, and the MIP minicolumn was fitted in an HPLC system to be used for in-tube microextraction from spiked serum samples [46]. Researchers in Groningen (The Netherlands) followed the fiber geometry to fetch a templated polymer on the outer surface of a fused optical fiber. The plastic coatings of a silica fiber were removed by burning. The fiber was cleaned and treated subsequently with sodium hydroxide, hydrochloric acid, and silylation reagents. The fiber was next dipped in a prepolymer solution that contained the template clenbuterol, and polymerization with simultaneous coating of the fiber was initiated by UV radiation. The obtained coated fibers were washed with a mixture

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of acetic acid and methanol (10:90 v/v) to remove the template and free the binding active sites. The methacrylate MIP coatings had a film thickness of f75 Am. The MIP fibers were used to trap analog molecules from aqueous solutions. Subsequent washing with acetonitrile (the polymerization solvent) facilitated removal of impurities and selective binding of the template and related molecules on the fiber. As can be seen in Fig. 9, acetonitrile washing removes interfer-

Fig. 9 HPLC-ECD chromatograms following SPME of blank urine (a), urine spiked (100 ng/mL) with brombuterol and washing of the fiber (b), blank urine and washing of the fiber (c). Fibers were washed in 200 AL of acetonitrile. Injection of 20 AL of the desorption liquid. Brombuterol is the peak indicated by the arrow. (From Ref. 39.)

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ing peaks from urine samples (compare Fig. 9a and c). In contrast, the analyte brombuterol remains on the MIP fiber. The method provided efficient cleanup and a satisfactory yield (40%) [47]. Especially for these phases, capacity and therefore method linearity are limited. Immunoaffinity extraction (IAE) is another way to combine a molecular recognition mechanism with the high-resolution power of the separation techniques. In this method, antibodies specific for a given analyte are immobilized on an appropriate support. The obtained medium should exhibit specificity for the analyte, facilitating very selective binding from a variety of matrices. Immunoaffinity extraction is now established in environmental and biological analysis as a powerful tool for sample purification and analyte preconcentration. The combination of IAE with SPME faces some important limitations: (a) antibodies are proteins, which do not tolerate extreme ionic strength, pH values, and temperatures, (b) desorption should be limited to liquid, and (c) nonspecific binding on the core of the base material should be prevented. Yuan et al. [48] immobilized antitheophylline serum on a silica fiber that had been previously modified with

Fig. 10 Competitive binding of cold theophylline with [3H]theophylline to the antitheophylline antibodies immobilized on a fused silica fiber. The [3H]theophylline was kept at saturation value (4 ng/mL), whereas the concentration of added cold theophylline was varied. (From Ref. 48.)

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3-aminopropyltriethoxysilane and subsequently with glutaraldehyde. The immunofiber was used for the specific binding of theophylline or a radioactive tracer ([3H]theophylline) from human serum. Quantification was accomplished in a scintillation counter, and both competitive and noncompetitive assays were performed (Fig. 10) [48]. Further developments toward this route are expected. Lately, special attention is given to polypyrrole (PPY) coatings for the extraction of ionic analytes. Exploiting its natural anion exchange properties as a conducting polymer, PPY was examined for direct SPME of anionic species from aqueous solutions without derivatization. Polypyrrole is coated on fused silica capillary’s inner surface (GC precolumn) by chemical polymerization. The inherent multifunctionality of pyrrole polymer (k–k electrons, interactions by polar functional groups, and hydrophobic interactions) may enhance extraction efficiency for both

Fig. 11 Effect of coating polymer chemistry on extraction efficiency for intube SPME of a series of h-blockers. (From Ref. 2.)

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polar and nonpolar aromatics in aqueous samples. Various analytes have been already tested on such coatings: catechins, caffeine, aminecontaining drugs, beta-blockers, organoarsenic compounds, and aromatic compounds [49–53]. Preliminary studies indicate that adsorption governs the extraction mechanism on such polymers. Acid–base interactions and ionic properties of PPY are advantageous features for future applications. Figure 11 depicts an example of the effect of varying coating selectivity on extraction yield. Diversifying extraction efficiency for the four different types of coatings is clearly observed for their use for in-tube SPME of pharmaceuticals. It is evident that PPY provides an overall superior efficiency for the extraction of the drugs of interest. Other innovative approaches include the utilization of polycrystalline graphites in the form of pencil leads for the microextraction of a nonionic alkylphenol ethoxylate surfactant [54], fibers coated with polymeric furelenes for the extraction of BTEX, naphthalene congeners, and phthalic acid diesters from water samples [55], anodized aluminum wire for aliphatic alcohols, BTEX, and petroleum products from gaseous samples [56], and low-temperature glassy carbon films for aromatic hydrocarbons [57].

V. DERIVATIZATION Derivatization is a useful practice often encountered in contemporary analysis in order to enhance the analytical behavior and the signal obtained from certain analytes. The truth is that derivatization is often called ‘‘a necessary evil’’; however, more and more workers tend to study and develop derivatization schemes to improve their analytical results. Currently, analysis of polar compounds is a major challenge since their isolation and analysis is often problematic. Hence in chromatographic separations, derivatization mainly aims at the reduction of polarity. Another chief objective is the introduction of an appropriate moiety to enhance detection sensitivity (e.g., the introduction of fluorophore for HPLC detection). For SPME, the need for derivatization mainly aims at the improvement of chromatographic behavior in GC. Hence polarity reduction is the major goal of the derivatization process. This is achieved by introducing moieties such as alkylsilyl, acetyl, and chloroformates to couple polar groups of the analyte like hydroxyl, amino, and carboxyl active groups.

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Derivatization may be performed prior to extraction, combined with the extraction, or following the extraction procedure. In the first case, derivatization is performed in the sample itself. For derivatization in the sample, the derivatizing agent is added to the sample matrix or to an appropriate extract derived from the sample. Since the majority of the samples analyzed with SPME are aqueous, direct derivatization in situ necessitates the formation of stable derivatives in aqueous environment. As such derivatizing agents, alkylchloroformates seem promising. Several alkylchloroformates have been tested for the derivatization of primary amines in water. The reaction is well known in peptide chemistry as a protecting reaction for peptides and aminoacids. The resulting carbamates are stable in water and are satisfactorily extracted by SPME. The method has been applied to the derivatization of amphetamines according to the reaction equation [58,59]: RNH2 þ CICO  OR2 ! RNH  CO  OR2 þ HCl For the derivatization of organic acids, reagents like borates, chloroformates, and benzyl bromide have shown good results. The reaction of acetic acid with benzyl bromide in aqueous solution resulted in the formation of benzyl acetate. However, reaction of acetic acid with hexylchloroformate did not yield the desired derivative. In contrast, hexylchloroformate successfully derivatized benzoylecgonine [60]. Trimethyloxonium tetrafluoroborate was used as a derivatizing agent to modify 29 organic acids in urine samples via a rather cumbersome procedure [61]. Benzodiazepines have been derivatized in urine to benzophenones by acid hydrolysis. The derivatives can better be extracted by direct immersion SMPE rather than HSSPME [62] because they are not volatile. Alternatively, a fast liquid–liquid extraction may be employed to transfer the analytes of interest to organic environment, which is necessary for conventional derivatization reactions (silylation, alkylation, acylation) [63]. This strategy may seem more convenient and easy to adapt to existing analytical protocols but in fact shrinks the distinctive features of SPME as a solvent-less technique. Lately, however, several derivatization procedures in aqueous environments have been developed utilizing novel reagents. Thus the development of a direct derivatization in the presence of water is not such an obstacle as it used to be.

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Simultaneous extraction and on-fiber derivatization is a very promising approach. This is a straightforward scheme that may deliver high efficiency for both extraction and derivatization reactions. The best way to perform such a scheme is to introduce first the derivatizing reagent on the fiber either by dipping the fiber in the reagent solution or by exposing the fiber to its headspace. Next, the fiber is introduced to the sample. As the analyte molecules are sorbed on fiber, they are continuously converted to derivatized analogs. Since derivatization occurs on the fiber, extraction cannot proceed toward equilibrium. Notable examples of this approach are the simultaneous HS extraction and derivatization of fatty acids with pyrenyldiazomethane to produce pyrenylmethyl ester [64], formaldehyde with o(2,3,4,5,6,-pentafluorobenzyl)hydroxylamine hydrochloride [65], amphetamines with pentafluorobenzoylcholride [66] and acetic anhydride [63], and aldehydes with pentafluorophenylhydrazine to form hydrazones [67]. On-fiber derivatization after extraction is performed for analytes that exhibit adequate extraction efficiency but require enhancement for their GC analysis. Polar analytes such as carboxylic acids, amphetamines, steroids, and hydroxyl metabolites of PAHs can thus be derivatized on-fiber after extraction to improve peak shape and detection sensitivity. In this case, the fiber with the extracted analytes is exposed to the headspace of the derivatization reagent [68,69]. The enhancement achieved by derivatization is clearly exemplified in Fig. 12 where the chromatograms of high molecular mass carboxylic acids are depicted prior and after on-fiber derivatization. Following extraction, the SPME fiber was exposed to diazomethane. The resulting esters provide sharp peaks in GC (Fig. 12b). On-line derivatization may occur also with SPME-LC configurations. In such schemes, the fiber following extraction is desorbed statically in an organic solvent containing the derivatization reagent. Next, the interface/injection valve is switched driving the derivatized analytes toward the analytical column. The method was tested for the SPME of alcohol ethoxylates and their derivatization with 1-naphthoyl chloride [70]. Another alternative is to perform the derivatization inside the GC injection port during the desorption of the SPME fiber. In one approach, amphetamines were extracted by headspace SPME from whole blood. The analytes were desorbed in the hot GC injection port, where heptafluorobutyric anhydride had been injected. Thus desorp-

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Fig. 12 Improvement of peak shapes by derivatization of carboxylic acids. (From Ref. 2.)

tion and derivatization occurred simultaneously. Such an approach requires fast derivatization kinetics and on-column focusing to avoid peak tailing [71].

VI. BIOANALYTICAL APPLICATIONS Solid-phase microextraction was initially introduced as a new tool for the extraction of organic compounds from environmental samples. However, in the last few years, the method has gained a lot of interest in a broad field of analysis including food, biological, and pharmaceutical samples. Successful coupling of SPME with LC and CE enables the analysis of a variety of polar or macromolecular analytes of biological significance: proteins, polar alkaloids, pharmaceuticals, and so forth. Furthermore, the development of HS-SPME provided a powerful alternative for the sampling and pretreatment of various biological samples such as urine, blood plasma, and hair [72]. Urine is one of the most important samples in bioanalysis and certainly the

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most important sample for toxicological analysis. As a sample for SPME, urine may be used by either direct immersion or headspace extraction. Regulation of ionic strength and pH value by addition of salts may improve extraction yield. Blood plasma and serum are rather complex samples of great significance for clinical chemistry, toxicology, therapeutic drug monitoring, and other analytical aspects. Solid-phase microextraction is mostly used in headspace mode to trap volatiles or semivolatile analytes. Direct immersion and in-tube SPME protocols have also been used, but they may result to shortened fiber lifetime or capillary clogging. Therefore special attention and thorough protein precipitation may be necessary. Hair sampling has evolved as an attractive noninvasive method especially suited for toxicological analysis; thus hair is now considered the third fundamental biological specimen for drug testing besides blood and urine. Drug metabolites or nonmetabolized drug molecules are distributed in hair either incorporated in the hair shaft from blood or due to adsorption from other media from the environment (sweat, smoke, etc.). As the hair grows with a certain rate, hair specimens may provide a historical record of exposure of the individual and can be later found there when often they are not detectable in other tissue. The application of headspace SPME for hair analysis of organic compounds has recently been reviewed [73]. Noteworthy application areas of SPME in bioanalysis can be found in the following major directions.

A. Analysis of Pharmaceuticals Solid-phase microextraction has found extensive use in the determination of pharmaceuticals in either pharmaceutical preparations or biological samples. Antidepressant drugs, valproic acid, steroids, anorectic agents, anesthetics, and many other types of pharmaceutical agents have all been analyzed by GC or LC following SPME. Solid-phase microextraction provides a powerful alternative to existing methods for the extraction of blood plasma and serum. Hence SPME may also find use in therapeutic drug monitoring. An example is seen in Fig. 13: SPME with a 100 Am PDMS fiber was used to extract spiked plasma samples. Recently, comprehensive reviews on the use of SPME for the analysis of drugs have covered the field [2,72,74–77]. In these reviews, detailed information on applications and methods is given.

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Fig. 13 SPME-GC-NPD chromatogram of antidepressant drugs in human plasma. (1) Amitrytyline, (2) trimpramine, (3) imipramine, (4a) cis-doxepin, (4b) trans-doxepin, (5) nortriptylin, (6) mianserine, (7) desipramine, (8) maprotilline, (9) clomipramine, (10) desmethylchlomipramine, (IS) chlomipramine, 375 ng/mL of each analyte, extraction for 30 min. (From Ref. 75.)

B. Toxicological and Forensic Analysis Toxicological analysis is a field where routine and research are integrated to a great extent. Novel methods are often rapidly implemented to advance the tasks of toxicological laboratories (provided that the quality of a new method is evident and undoubtful). Solidphase microextraction offers great advantages to toxicological analysis in both research and routine analysis. Headspace SPME-GC-MS has proven a powerful tool in toxicological analysis. The preconcentration of the analytes obtained on PDMS and PA fibers offers great advantages compared to conventional headspace GC-MS. Therefore many toxicological laboratories have implemented SPME and developed such methods for the analysis of numerous analytes in a variety of matrices: alcohol in blood, VOCs in plasma and blood, poison agents (malathion, cyanide), nereistoxin, chlorophenols, organochlorines persistent in blood, PAHs, and mercury and other heavy metals species in blood and other biological specimens [26,72,77]. Drugs of abuse is probably the widest application field of analytical toxicology and the most common task of such laboratories. The

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combination of SPME with gas chromatography-mass spectrometry (GC-MS) has found extensive use for the determination of amphetamines, benzodiazepines, barbiturates, methadone, cannabinoids, alkyl nitrites, tricyclic antidepressants, and other drugs of abuse. The field has been covered by comprehensive reviews [72,75–77]. An example of the utilization of SPME in the analysis of drugs in hair is depicted in Fig. 14. GC-MS analysis combined with HS-SPME to recover the drugs from spiked hair [76]. Solid-phase microextraction has also been coupled to HPLC and CE for the analysis of barbiturates, benzodiazepines, and other drugs of abuse [Table 1 in Ref. 72].

C. Clinical Chemistry Solid-phase microextraction has also found use in clinical chemistry. Compared to existing techniques, it shows significant benefits and offers a good alternative to conventional methods [72,74–77]. Although the use of liquid-phase separations has by far outnumbered that of GC in clinical chemistry, SPME-GC has found a niche and has been used for a variety of studies: the investigation of drug metabolism in human keratinocyte cells [78], the study of metabolism and excretion of benzophenone [79], and the determination of putrescine and cadaverine [19], monocyclic aromatic amines in biological fluids in screening for trimethylaminuria (fish odor syndrome) [80], urinary

Fig. 14 GC-MS (single-ion monitoring) following HS-SPME of 10 mg of hair spiked with 16 drugs. Concentrations: 1 ng/mg. (From Ref. 76.)

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organic acids in metabolic studies [81], carnitine (an essential factor in the fatty acid metabolism of organisms) [82], and aminoacids [28]. Figure 15 illustrates the potential of SPME in clinical chemistry. Feces from an adult on a normal diet were dried, acidified (pH 1–2), and saturated with NaCl. A 75 Am PDMS fiber was exposed to the headspace for 30 min and desorbed in the GC injector (250jC, 2 min). Compounds derived from food products and end metabolism products (4-methylphenol, dimethylsulfide) were determined [76]. Determination of biomarkers in exhaled human breath attracts an increasing interest in clinical chemistry and diagnosis as an alternative noninvasive method. More than 100 VOCs have been identified in normal human breath by GC-MS. The methods currently used for sampling and preconcentration (chemical interaction, adsorptive binding, cold trapping) are tedious procedures, they require complex devices, and they suffer from particular problems (e.g., excess of water from the breath). Solid-phase microextraction offers an alternative that can overcome such limitations. The fiber can be directly exposed in the mouth of the subject. An inert tubing is added to a commercial SPME device in order to protect the fiber from the subject’s tongue (Fig. 16). The method demonstrates significant

Fig. 15 GC-MS profile of VOCs in the HS offeces of an adult with normal diet. Peak identities: (1) dimethylsulfide, (2) acetic acid, (3) propionic acid, (4) isobutyric acid, (5) n-butyric acid, (6) 2-methylbutyric and isovaleric acids, (7) n-valeric acid, (8) isocaproic acid, and (9) 4-methylphenol. (From Ref. 76.)

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Fig. 16 Device for the sampling of human breath. (From Ref. 83.)

advantages compared to existing extraction techniques, requiring only 1–3 min for sampling and providing detection limits in the low nanomolars per liter range [83].

D. Affinity Measurements The measurement of binding affinity is a worthy application field for SPME [84–87]. Solid-phase microextraction is rarely an exhaustive extraction method; thus it causes negligible depletion of the analytes. Hence the method is a very well suited method for the quantification of the free quantity of analytes participating in equilibria (e.g., protein binding). In contrast, in exhaustive extractions (employing solvents or a solid-phase bed), the equilibrium between matrix components (proteins) and the drug is disturbed. This leads to a shift of the equilibrium toward the freely dissolved fraction. Vaes et al. [84] used PA-coated fibers to measure the protein binding of four polar drugs (aniline, nitrobenzene, 4-chloro-3-methylphenol, and 4-n-pentylphenol). The determination of binding to bovine serum albumin (BSA) by (nondepletion) SPME gave comparable results to equilibrium dialysis. It was shown that increasing hydrophobicity results to an increase in affinity for BSA [84]. The group extended the concept to predict absorption profiles and kinetics using quantitative structure–activity relationships [85] and to investigate the correlation of membrane/water partition coefficients with free concentrations in in vitro systems [88]. Researchers at the same university employed nonequilibrium SPME for the determination of freely dissolved analytes in complex matrices (chyme) [89]. If protein binding occurs in one to one molar ratio, binding constants can be calculated. This concept was exploited by Yuan

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and Pawliszyn [86] in order to determine binding constants of diazepam to human serum albumin utilizing Scatchard plots. The method may work even in a four-compartment system: matrix (protein), solution, headspace, and fiber. Hence SPME was used to measure the concentrations of alkylbenzenes (volatile drug) in the headspace of a solution containing also BSA [90].

VII. POSSIBILITIES AND LIMITATIONS OF SOLID-PHASE MICROEXTRACTION Solid-phase microextraction exhibits certain advantages that have brought the technique to the forefront of contemporary analytical chemistry. Of these advantages, the most significant are: no use of solvents, ease of handling, in-line coupling to GC, no need for expensive sophisticated instrumentation, automation capabilities, and its nature as a microtechnique. Solid-phase microextraction is very useful in miniaturized systems as is demonstrated for the combination with micro-LC and CE. The fiber geometry allows an efficient and low dead-volume coupling with these techniques’ automation capabilities. The automation capabilities of SPME are a great advantage taking into consideration the continuous drive toward utilization of more controlled and automated methods. Compared with other extraction methods like LLE and SPE, SPME when coupled to GC does not need specific devices and can be easily automated. Direct-immersion SPME can be very easily automated by modification of conventional autosamplers in order to host the SPME fiber instead of the sampling needle. To automate HS-SPME, the autosampler should also allow controlled heating of the sample vial. Yet maybe the most important feature of SPME is the integration of sampling and extraction in one step and the subsequent straightforward sample introduction. The method eliminates steps in the analytical process and thus eliminates the sources of possible errors. Not only time and resources are saved, but also most important precision and accuracy of the method can be controlled to a better extent. However, no matter how attractive SPME seems to the reader, it is not always the same undemanding to the practitioner. The conditions of SPME should be precisely controlled in order to achieve accurate and reproducible measurements. As a dynamic multivariate

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method, SPME requires control of the most important parameters that affect the process: agitation, sampling time, temperature, sample volume, vial volume (for HS-SPME), sample matrix, and additives. Calibration methods used with SPME are external standard, internal standard, and standard addition. External calibration is probably the most widely used calibration method especially in cases of little variance between samples [2]. For biological samples, care has to be taken in order to maintain constant ionic strength. In cases of high concentrations, dilution may be necessary, whereas in trace analysis, a large urine volume can be used. In such cases, normalization of the ionic strength by salt addition is often performed. Standard addition is another alternative for variable samples. The sample is analyzed, and then a known amount of the analyte is added and the sample is processed again. Extraction can even be made from the same sample amount if negligible analyte depletion occurs in the first extraction. The use of internal standard can work satisfactory in SMPE under some conditions. Extraction time profiles should be determined also for the internal standard. If there are large differences in equilibrium times, large errors may occur. Very precise time programming should be applied to obtain reproducible results. Furthermore, the use of internal standard with adsorptive coatings may result in competitive binding, displacement, and therefore large errors. Finally, when a competing phase is present in the sample (e.g., proteins, humic acids), the use of internal standard faces another limitation. The affinity of the internal standard toward the competing face may be very different from the analyte affinity. Fiber SPME is especially attractive in the case of specific application fields such as the measurement of binding affinity or proteinfree drug concentrations. Coupling of fiber SPME with HPLC and other liquid separation methods is a practical alternative to the use of SPE especially if certain needs are thus covered, e.g., nondepletion extraction and field sampling. In other cases, the use of SPME can hardly compete with SPE since SPE offers much greater array of stationary phases and thus stationary phase selectivities. It is believed, however, that continuous research in that direction will result on the development of new innovative phases/coatings for SPME. Nevertheless, comparing the two extraction modes is a trivial effort since the two processes differ to their very nature. Proper judgment could be done when comparing on-line SPE with SPME. Additionally, SPE is often a multistep procedure employing quanti-

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tative (in most cases) trapping of the analytes on the bed, appropriate washing, and elution in a chosen solvent. Very often, the washing step(s) is the most critical stage. For example, in SPE, the most common approach is to trap the analytes in general (hydrophobic interactions), wash out unwanted impurities in subsequent washing steps, and finally, recover the analyte ofinterest in a final elution step. In contrast, selectivity of SPME seems more based on the diffusion process and the sorption of the analytes on the coating. As a rule, SPME does not employ washing steps, but only a direct desorption step. This simplicity, which is considered SPME’s major advantage, may also prove a drawback. In certain cases, simple SPME protocols may not reach the selectivity and cleanup obtained by three- or fourstage LLE or SPE [35]. Multistep protocols could also be used with SPME and with much easier handling and lower consumption of organic solvents. Such procedures are not yet developed probably because they would lessen the simplicity of the technique. In general, SPME provides low recoveries. However, this problem is overcome by the fact that the total extracted compound is subsequently determined. Solid-phase microextraction should be calibrated carefully (see above), but then, it provides very satisfactory sensitivity, linearity precision, and accuracy. It is unlikely that SPME will become a universal method. Scientists and practitioners should comprehend such methods as useful alternatives to existing methods. Solid-phase microextraction offers improvement in several characteristics of conventional practices. The method has already found a wide application area, and it is seen to find numerous additional utilizations. Furthermore, SPME may expand to cover applications such as the analysis of small samples, analysis of air samples, field sampling affinity measurements, and determination of free analyte concentration (e.g., drug in plasma). For example, as seen in Fig. 1, the increase on publications reporting on SPME originating from food/flavor analysis exhibits the strongest trend of all the fields. Such a trend is easily understood considering the superior advantages that SPME offers for this specific scientific area: sampling from individual organisms, sampling for specific time cycle of the life of the organism, convenient and reproducible field sampling, and no need for calibration of air pumps. An important field for SPME may be the automation of forensic and toxicological analysis. Samples found positive by automated

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immunoassays need to be confirmed by an independent method. In common practice of this application field, this means GC, employing LLE, SPE, and derivatization for many groups of analytes. Such laborious methods could be integrated in automated SPME protocols, provided that the further development and the validation of such methods will prove them reliable for such tasks. It is thus believed that SPME will become an established methodology in this specific but also in other bioanalytical fields, and that its applications will increase to an even greater extent in the near future.

ABBREVIATIONS BTEX BSA CE CX/PDMS CW/DVB CW/TPR DI EMIT ELISA ESI FID FPIA GC HPLC HS IAE LC LLE MIP MS PA PAH PDMS PDMS/DVB PEEK PEG PPY

Benzene, toluene, ethylbenzene, xylene Bovine serum albumin Capillary electrophoresis Carboxen/polydimethylsiloxane Carbowax/divinylbenzene Carbowax/templated resin Direct immersion Enzyme-modulated immunotest Enzyme-linked immunosorbent assay Electron spray ionization Flame ionization detection Fluorescence polarization immunoassay Gas chromatography High-performance liquid chromatography Headspace Immunoaffinity extraction Liquid chromatography Liquid–liquid extraction Molecularly imprinted polymers Mass spectrometry Polyacrylate Polycyclic aromatic hydrocarbon Polydimethylsiloxane Polydimethylsiloxane/divinylbenzene Polyether ether ketone Polyethylene glycol Polypyrrole

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SPME SPE VOCs

Solid-phase microextraction Solid-phase extraction Volatile organic compounds

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Mullett, W.M.; Martin, P.; Pawliszyn, J. Anal. Chem. 2001, 73, 2383. 47. Koster, E.H.M.; Crescenzi, C.; den-Hoedt, W.; Ensing, K.; de-Jong, G.J. Anal. Chem. 2001, 73, 3140. 48. Yuan, H.D.; Mullett, W.M.; Pawliszyn, J. Analyst 2001, 126, 1456. 49. Wu, J.C.; Lord, H.L.; Pawliszyn, J. J. Microcolumn Sep. 2000, 12, 255. 50. Wu, J.C.; Yu, X.M.; Lord, H.; Pawliszyn, J. Analyst 2000, 125, 391. 51. Wu, J.C.; Pawliszyn, J. Anal. Chem. 2001, 73, 55. 52. Wu, J.C.; Xie, W.; Pawliszyn, J. Analyst 2000, 125, 2216. 53. Wu, J.C.; Mester, Z.; Pawliszyn, J. Anal. Chim. Acta 2000, 424, 211. 54. Aranda, R.; Kruus, P.; Burk, R.C. J. Chromatogr. 2000, A888, 35. 55. Djozan, D.; Assadi, Y.; Haddadi, S.H. Anal. Chem. 2001, 73, 4054. 56. Xiao, C.H.; Liu, Z.L.; Wang, Z.Y.; Wu, C.Y.; Han, H.M. Chromatographia 2000, 52, 803. 57. Giardina, M.; Olesik, S.V. Anal. Chem. 2001, 73, 5841. 58. Ugland, H.G.; Krogh, M.; Rasmussen, K.E. J. Chromatogr. 1997, B701, 29. 59. Ugland, H.G.; Krogh, M.; Rasmussen, K.E. J. Pharm. Biomed. Anal. 1999, 19, 463. 60. Hall, B.J.; Parikh, A.R.; Brodbelt, J.S. J. Forensic Sci. 1999, 44, 527. 61. Liebich, H.M.; Gesele, E.; Woll, J. J. Chromatogr. 1998, B713, 427. 62. Guan, F.Y.; Seno, H.; Ishii, A.; Watanabe, K.; Kumazawa, T.; Hattori, H.; Suzuki, O. J. Anal. Toxicol. 1999, 23, 54. 63. Staerk, U.; Kulpmann, W.R. J. Chromatogr. 2000, B745, 399. 64. Mills, G.A.; Walker, V.; Mughal, H. J. Chromatogr. 1999, B730, 113. 65. Koziel, J.A.; Noah, J.; Pawliszyn, J. Environ. Sci. Technol. 2001, 35, 1481. 66. Koster, E.H.M.; Bruins, C.H.P.; Wemes, C.; de-Jong, G.J. J. Sep. Sci. 2001, 24, 116. 67. Stashenko, E.E.; Puertas, M.A.; Salgar, W.; Delgado, W.; Martinez, J.R. J. Chromatogr. 2000, A886, 175. 68. Pan, L.; Pawliszyn, J. Anal. Chem. 1997, 69, 196.

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5 Polyelectrolytes as Stationary Phases in Liquid Chromatography Lilach Yishai-Aviram and Eli Grushka Department of Inorganic and Analytical Chemistry, The Hebrew University, Jerusalem, Israel

I. II. III. IV.

INTRODUCTION THE PRINCIPLE OF DYNAMIC COATING COLUMN CHARACTERIZATION POSITIVELY CHARGED POLYELECTROLYTES AS STATIONARY PHASES A. Coating Procedures B. Properties of the Phases and Retention Mechanism C. Applications V. NEGATIVELY CHARGED POLYELECTROLYTES AS STATIONARY PHASES A. Coating Procedures B. Properties of the Phases and Retention Mechanism C. Applications VI. COMPLEX POLYELECTROLYTE LAYERING REFERENCES

274 276 279 282 282 284 289 289 289 291 297 298 298 273

274 / Yishai-Aviram and Grushka

I. INTRODUCTION Chemically modified silicas are used ubiquitously as stationary phase support in liquid chromatography (LC). In spite of alternative column packing, such as zirconia, alumina, titania, etc., silica continues to be the most commonly used packing material in modern LC. The main advantages of silica are its high level of mechanical strength, uniformity in terms of size and surface area, and purity. These qualities permit the formation of efficient packed beds that remain stable under high operating pressures for long periods of time [1]. The surface of silica consists of various kinds of silanols and siloxane bonds [2,3]. The siloxane (Si–O–Si) sites are hydrophobic and contribute little or nothing to the retention of polar solutes. However, the hydrophobic nature of the siloxane bond makes it possible to observe some retention of nonpolar solutes [4]. On the other hand, the silanol groups, which can be hydrated by adsorption of water, are polar and are considered to provide strong adsorption sites [3,5,6]. Basic analytes can interact strongly with the acidic silanols. The silanols can exist in three different forms [5]: single (isolated), geminal, or vicinal form. The degree of ionization of the silanols depends on the pKa values of the different silanol groups in the particular mobile phase used and on the pH of the mobile phase. Mendez et al. [7] found that for the two types of columns, silica and C18, there are two types of silanols with pKa values of about 3.6 and 6.3. Thus, the silica surface is negatively charged from a relatively low pH. In silica-based columns (normal phase as well as reversed phase,) it is possible to work only within a relatively narrow pH range, usually between 2 and 7.5. One of the approaches to overcome this limited pH range is to use new types of polymer-based phases [8] including polymer-clad silica gel particles, which are the subject of this review. These new polymeric phases have an added advantage of offering new and unique selectivities that cannot be attained using conventional silica gel. Most of the polymeric stationary phases in use are neutral [9–13]. However, the range of polymeric stationary phases can be expanded to include polyelectrolytes. For example, positively charged polymers [14–18] can be adsorbed on silica particles by electrostatic interactions over a wide range of pH due to the fact that the surface of the silica has partially negative charge. On these positively charged surfaces, it is easy to adsorb negatively charged polyelectrolytes to form a multilayer [19–21].

Polyelectrolytes as Stationary Phases / 275 There are two main approaches for making silica-based polyelectrolyte stationary phases. One approach is to coat the silica gel particles in a batch mode outside the column. Once coated, the column is then packed using conventional high-performance liquid chromatography (HPLC) packing techniques (e.g., see Krokhin et al. [22] and Pirogov et al. [23]). The batch mode of preparing polyelectrolytecoated columns is relatively straightforward. There are three different methods to coat silica with charged polyelectrolytes. The first method forms covalent bonds [24] between the polymer and the support material by refluxing the two components in toluene for several hours [25,26]. The second approach utilizes electrostatic interaction between the silanol groups on the silica support and a cationic polymer, which can be irreversible. The third approach is physical adsorption, which can be accomplished by several different ways: thermal treatment [11], irradiation with microwave and gradiation [27], and self-immobilization [28]. In all these approaches, the stationary phases are prepared outside the column. Once prepared, the columns are packed using slurry packing techniques. The main advantage of preparing the phases outside the column is the wide range of polymers that can be used for coating [e.g., (x,y)-ionene bromide [29] or copolymers such as poly(ethylene-co-acrylic acid) [30]]. Other advantages of the ‘‘outside-the-column’’ approach are the ability to determine the exact amount of polymer bonded to the silica and the capability of characterizing the newly formed stationary phase without having to empty the column. The second approach to prepare silica-based polyelectrolyte stationary phase is to coat the column dynamically in situ. In this approach, the coating solution is passed through the column that has been packed previously with the silica support. Conventional chromatographic equipment is used for transporting the coating solution through the column. Of the two approaches, the batchwise method is, by far, the most prevalent. The dynamic coating technique is an in situ method used to prepare stationary phases for HPLC, capillary electrophoresis, and capillary electrochromatography. In the literature, we often find that the expression ‘‘in situ coating’’ refers to batch-mode methods in which the polymerization proceeds directly on silica gel particles prior to column packing (e.g., Carbonnier et al. [31] and Mao and Fung [32]). In this review, we use the term in situ to indicate a coating process in which a solution containing the coating material is passed

276 / Yishai-Aviram and Grushka through a column already packed with the appropriate chromatographic packing. As mentioned, dynamic coating is not used often to produce polymer-coated stationary phases for HPLC. However, due to its simplicity, the dynamic coating technique has potential benefits in terms of cost, ease of preparation, and stability. As a result, we will discuss the technique in some greater details in Section 2.

II. THE PRINCIPLE OF DYNAMIC COATING The term ‘‘dynamic coating’’ was first employed by Ghaemi and Wall [33] and later by Hansen [34] and Helboe et al. [35] (and references therein). These authors coated dynamically naked silica with a salt of an alkyl quaternary amine. The cationic quaternary amines in the mobile phase are attracted, by electrostatic forces, to the negatively charged silanol. After reaching equilibrium, three phases are present: the adsorbed quaternary ammonium salt as the stationary phase, unadsorbed amines, and micelles in the case where the concentration of the quaternary amines in the mobile phase exceeds the critical micellar concentration (CMC). The amount of adsorbed amine was found to depend on the silica surface and on the amine’s alkyl chain length [36]. In general, dynamic coating can be carried out on any column packed with conventional chromatographic medium (e.g., silica gel, reversed-phase material, etc.) by passing through the column a solution containing the new stationary phase component. The additive to the mobile phase adsorbs on the existing stationary phase, thus changing the nature of the stationary phase. Usually, but not necessarily, the coating procedure continues until the stationary phase is saturated with the adsorbed components. The adsorbing stationary phase (the old stationary phase) can be naked silica or a bonded phase, depending on the desired new stationary phase. Ion-pair chromatography, a well-established technique used to separate charged analytes, also takes advantage of dynamic coating [37–39]. Negatively or positively charged surfactants are added to the mobile phase to create the new stationary phase. Most often, the column contains a hydrophobic bonded phase (e.g., RP8 or RP18) and the surfactant additives are either negatively charged or positively charged, depending on the application at hand. Frequently, alkyl sulfates or sulfonates are used as negative surfactants, and tertiary

Polyelectrolytes as Stationary Phases / 277 or quaternary amines are used as positive ion-pair reagent. In ionpair chromatography, the additive is present in the mobile phase both during the stationary phase modification stage and the separation stage. On the other hand, in classical dynamic coating situations, the mobile phase additive is there only for the new stationary phase generation and is absent in the separation stage. Although ion-pair chromatography is an important variant of dynamically coated chromatographic systems, it will not be discussed here. Many reviews are devoted solely to ion-pair chromatography (e.g., see Refs. 40 and 41). In the context of this chapter, dynamically coated columns are columns coated with polyelectrolytes. The mobile phase in these cases does not contain the modifying polymers. We can expend the ‘‘classical’’ dynamic coating techniques by working with polyelectrolytes. Silica columns are easiest to coat due to the presence of negative charges. A positively charged polymer is added to a suitable mobile phase, which is then passed through the column using the chromatographic pump. As the polymer passes through the column, its positive charges interact with the Si–O
groups on the silica packing. Once the column is loaded, the polymer is no longer present in the mobile phase. It should be noted that during the polyelectrolyte loading stage, the pump pressure is high due to the viscosity of the polymer-containing solution. However, during the separation part, because the polyelectrolyte is no longer present in the mobile phase, the pressure is typical of HPLC systems. Due to the strong electrostatic interactions between the polyelectrolytes and the charged silanols groups on the silica, the modified column is highly stable. The dynamic coating technique is very common in capillary electrophoresis (e.g., Refs. 42–49) and in capillary electrochromatography (viz. Refs. 50 and 51). The use of a quaternary ammoniumbased polymer, which interacts strongly with the surface of the silica, has been proven successful. Figure 1 describes the adsorption behavior of poly(diallyldimethylammonium chloride) (PDADMAC), a polycationic electrolyte, on negatively charged silica [52]. The amount of adsorbed PDADMAC increases, irrespective of the molecular weight, with an increase in the pH of the coating solution. The increase in adsorption is due to rising surface charge density caused by the ionization of silica silanol groups. In addition to the pH, the ionic strength of the polyelectrolyte coating solution has a strong influence on the adsorption of the

278 / Yishai-Aviram and Grushka

Fig. 1 The pH dependence of PDADMAC adsorption on silica gel. (From Ref. 52. nElsevier.)

polymer [53,54]. Increase of the ionic strength increases the adsorption of the positively charged polymer (see Fig. 2) [52]. The salt has two effects: it changes the structure of the polyelectrolyte in solution, and it changes the adsorption of the altered polyelectrolyte to the silica [15]. Thus, it is highly important to maintain constant ionic strength. As in the ‘‘classical’’ dynamic coating technique, the mobile phase can be of any kind. Most often, it is a buffer mixed with methanol, acetonitrile, or tetrahydrofuran. Juskowiak [55] suggests that the pH of the ‘‘conditioning mobile phase’’ during the initial treatment of silica is a crucial factor in governing surface coverage. For the dynamic coating technique to be effective, it is important to understand the factors governing the adsorption of the coating material, such as the amount of modifier in the coating solution, its pH, the molecular weight of the polymer, and the charge of the polymer [56]. Polyelectrolytes should be charged over a wide pH range. The polymer should be soluble in the mobile phase and should not affect the detection of the analytes. There are no special requirements from the column being coated over and above the conventional chromatographic requirements such as being well packed. The dynamic coating technique has several advantages as follows. It is easy to perform and does not need any additional instru-

Polyelectrolytes as Stationary Phases / 279

Fig. 2 The ionic strength dependence of PDADMAC adsorption on silica. (From Ref. 52. nElsevier.)

mentation over and above a routine liquid chromatograph. The columns thus prepared can be applied for the separation of ionic and nonionic solutes. Last but not least, the columns prepared by dynamic coating are reproducible irrespective of the original brand of the column.

III. COLUMN CHARACTERIZATION Coated stationary phases can be characterized chemically, physically, and chromatographically (e.g., see Tonhi et al. [57,58]). Although the coated packing material described in these references was prepared outside the column (not dynamically in the context of this review), the methods of characterization used are universal and can be applied for in situ cases as well. Among the physical and chemical approaches to the characterization of the polymeric phases, we find the following: Carbon content: The percent carbon in the silica plus polymer phase can be obtained through elemental analysis, which is

280 / Yishai-Aviram and Grushka carried out before and after the polymer coating. In this way, we can evaluate the amount of stationary phase in the column. Thermogravimetric analysis (TGA): TGA allows us to ascertain the thermal stability of the coated polymer vs. the polymer by itself. Infrared spectroscopy (IR): IR evaluates the presence of residual silanols and thus indicates the efficiency of the coating procedure. In addition, IR can shed light on the interactions between the polymeric coating and the silica gel matrix underneath. Nuclear magnetic resonance (NMR): 13C cross-polarization magic angle spinning (CP-MAS) NMR can be used to analyze possible interactions between the polymer and the silica gel support. Scanning electron microscopy (SEM): SEM analysis provides morphological information on the coated polymeric stationary phase. Atomic force microscopy (AFM) [59,60]: Similar to SEM, AFM yields information on the morphology of the coated layer. X-rays techniques [31,60]: Various X-ray spectroscopy methods can be used to obtain information on the chemical composition of the coated phase as well as on the nature of the coated surface. Small-angle neutron scattering (SNAS) [61]: This technique provides a direct determination of the stationary phase thickness and bonding density. A different kind of characterization is obtained by examining the interaction of the coated polymeric stationary phase with different test solutes. This approach is the chromatographic characterization and it is complementary to the physical and chemical methods described above. The chromatographic interactions that can be examined include hydrophobic interactions, hydrogen bonding, ion exchange capacity, steric selectivity, silanol activity, etc. Kimata et al. [62], Galushko [63,64], Czok and Engelhardt [65], as well as Classens et al. [66] offer a list of solutes that can be used for evaluating these interactions. In general, these evaluations are empiric in nature, but they allow quick classification of the major interactions that will determine the elution times. An example on the use of some

Polyelectrolytes as Stationary Phases / 281 of the above chromatographic interactions can be obtained from the work of Tonhi et al. [58]. A more rigorous approach to chromatographic interactions and characterization is the linear free energy relationship. However, this approach fails for charged stationary phases including coated polyelectrolytes. More conventional chromatographic parameters, such as plate height and plate number, have also been used to characterize polymeric-coated columns [58]. We next detail the use of various polyelectrolytes to obtain new charged stationary phases. We will describe the preparation of the various columns, their characterization, and their use. Tables 1 and 2 give the names, abbreviations, and structures of the polymers described in this review. Also given in are the references of the papers using these polyelectrolytes.

Table 1 Positively Charged Polymers Polymer

Structure

References

Poly(dimethyldiallylammonium chloride) (PDADMAC)

22,72

Poly(N-ethyl-4-vinyl pyridinium bromide) (PEVP)

22

Poly(hexamethyleneguanidium hydrochloride) (PHMG)

22

x,y-Ionene

22,25,67–69

Poly(N-chloranil, N,N,NV,NV-tetramethylethylene diammonium dichloride) (PCED)

70

Polyethyleneimine (PEI)

H2N(CH2CH2NH)nH

71

282 / Yishai-Aviram and Grushka Table 2 Negatively Charged Polymers Polymer

Structure

References

Dextran sulfate (DS)

71,74–76

Heparin

77

Poly(styrene sulfonate) (PSS)

72

IV. POSITIVELY CHARGED POLYELECTROLYTES AS STATIONARY PHASES A. Coating Procedures Several groups have prepared positively charged stationary phases with polyelectrolytes. Krokhin et al. [22] used reversed-phase packing (Silasorb’s C8) to form several anion exchange columns using batchmode procedures. First, they mixed the packing material (C8) with a solution of dodecylbenzenesulfonic acid (DBSA). The DBSA adsorbs on the C8-bonded phase, forming a negatively charged surface layer. Then, a positively charged polymer solution was added to the freshly prepared negatively charged packing. Due to strong electrostatic interaction, the polymer adsorbs on the surfactant, yielding an anion exchange material. After the stationary phase is prepared, a chromatographic column is packed using conventional slurry techniques. In their study, they have prepared four different anion exchangers using four different charged polymers: poly-(dimethyldiallylammonium chloride) (PDADMAC), poly(N-ethyl-4-vinyl pyridinium bromide) (PEVP), poly(hexamethyleneguanidinium hydrochloride) (PHMG), and 2,5-ionene (ionene). They found different ion exchange capacities, stabilities, and selectivities for each of the new stationary phases. The columns were used to separate a variety of inorganic

Polyelectrolytes as Stationary Phases / 283 anions as well as some heavy metal ions using an ethylenediaminetetraacetic acid (EDTA)-containing mobile phase. Pirogov et al. [23] continued the study of Krokhin et al. [22]. They also used DBSA to ‘‘activate’’ reversed-phase packing material followed by coating with a positively charged polyelectrolyte. In later works, Pirogov et al. [67,68] extended the approach and used several ionenes with various functional groups to prepare a series of anion exchange packing. In their procedure, they applied the cationic polymer coating solution directly on the commercially available cation exchange materials. Because the cation exchange material has on it sulfonic groups, the DBSA step is eliminated from the procedure. Ionenes are water-soluble linear cationic polyelectrolytes consisting of dimethylammonium charge centers. Pirogov et al. compared several types of cation exchange material at two temperatures of ionene coating. They found that higher coating temperature gave higher ion exchange capacity. They used the polymeric anion exchange columns to separate various inorganic anions. They found that they can manipulate the selectivities by varying the ionene used to prepare the ion exchange material. Suzuki et al. [69] also used several ionenes to prepare positively charged polymeric stationary phases. In this case, the ionenes were covalently attached to the silica surface via bonded propyl amine. They characterized the chromatographic characteristics of the newly made column and compared it with conventional C18 and phenylbonded phases. The batch-mode approach was used for the preparation of the ionene stationary phase. Gupta and Prasad [70] bonded poly(N-chloranil N,N,NVNV-tramethylene diammonium dichloride) (PCED(Cl)2) to silica in batch mode and then packed the newly created stationary phase in a corning glass tube. The polycationic polymer was synthesized by them. They used this column for molecular recognition of h-lactam antibiotic. Millot et al. [71] prepared three differently charged polymeric phases. Two of the three were cationic and the third was anionic and will be described in a later section. One of the polycationic phases was obtained by mixing silica gel with a solution of polyethyleneimine (PEI) in methanol. The adsorption was carried out using sonication for 10 min at 0jC followed by shaking for 25 hr. The amount of PEI on silica was found to be 50 mg/g silica. The silica-coated PEI was then crosslinked by suspending it in 1,4-butanedioldiglycidylether

284 / Yishai-Aviram and Grushka (BUDGE) solution and sonicating the mixture for 2 hr at 60jC. After cooling and filtering, it was slurry-packed to the column. A second phase was obtained by mixing silica gel and hexadimethrine bromide (HB) in water. This mixture was kept under agitation for 24 hr, and then washed, dried, and slurry-packed. The amount of adsorbed HB per gram of silica was found to be 29 mg/g. All the polycationic stationary phases described above were prepared in a batch-mode outside the column. Once the support was coated, the columns were then packed. Recently, Aviram and Grushka [72] dynamically coated a silica gel column with PDADMAC. The PDADMAC was added to a coating solution that was passed through the silica column using an HPLC pump. PDADMAC adsorbs strongly on the silica particles due to electrostatic interactions between the negatively charged silanol groups and the positive centers of the polyelectrolyte. After the column is loaded with the polyelectrolyte, the mobile phase, which no longer contains PDADMAC, is introduced, and once the column reaches equilibrium, separations can be preformed. They found that the magnitude of the capacity factors is directly related to the amount of the adsorbed PDADMAC in the column. The capacity factors of the negatively charged solutes increased dramatically as the amount of PDADMAC increased. On the other hand, the capacity factor of the positive solutes decreased as the PDADMAC in the column increased.

B. Properties of the Phases and Retention Mechanism In all the works quoted in Section 4.1, the chromatographic behavior of the polyelectrolyte-modified columns was compared to that of conventional HPLC columns and, in every case, it was found that retention behaviors and selectivities changed markedly in the newly created stationary phases. The retention mechanism of the solutes is affected by the type of the solutes and the stationary phase. In cases discussed here, the stationary phase contains positively charged centers on the polymers, functional groups of the polymer, and nonbonded silica. Most of the articles cited above consider the positive polymers as anion exchangers and the retention mechanism is taken to be typical of ion exchange columns. Suzuki et al. [69] extended the chromatographic characterization of their ionene-based stationary phases by using some of the test compounds of Kimata et al. [62]. They found that the new phases have significant solute shape recognition ability. They showed that their

Polyelectrolytes as Stationary Phases / 285 ionene-based column can be used either as reversed-phase or as ion exchange column, depending upon the nature of the components of the analyte mixture. Krokhin et al. [22] characterized their positive polyelectrolyte columns by checking their ion exchange capacity. As might be expected, they found that the capacity of the anionic exchangers depends on the functional group density of the polymer chain. Thus,

Fig. 3 Isotherms of sorption of 4,6-ionene at different temperatures on Silasorb-S and observed chromatograms of inorganic anions: (1) at 70jC and (2) at 20jC. (From Ref. 68.)

286 / Yishai-Aviram and Grushka

Fig. 4 The retention time of a-Lact on (a) HB-based column; (b) triple-layer of adsorbed HB, DS, and HB; (c) crosslinked PEI column. (Reprinted with permission from Ref. 71. nFriedr. Vieweg and Sohn Verlagsgesellschaft mbH, 1999.)

poly(N-ethyl-4-vinylpyridinium bromide) (PEVP) has higher capacity (0.032 mmol/g) than ionine (0.010 mmol/g). The PEVP phase has one quaternary ammonium group per two atoms in the chain, whereas ionene has two groups of quaternary ammonium per nine atoms. Subsequent work from the same group [68], where ionenes were adsorbed directly on cation exchange material, showed that the amount of polyelectrolyte adsorbed is a function of the charge density of the adsorbing medium. Also, it was shown that higher adsorption temperature results in greater adsorption (see Fig. 3).

Polyelectrolytes as Stationary Phases / 287

Fig. 5 Separation of structural isomers of phenylphenol on (3,16)-ionene column. (1) o-Phenylphenol; (2) m-phenylphenol; and (3) p-phenylphenol. (From Ref. 25. nAmerican Chemical Society, 2001.)

Millot et al. [71] have also calculated the ion exchange capacity of their columns. The anion exchange capacity of the silica/HB phase was 0.05 mEq/g, whereas the silica/crosslinked PEI phase was 0.19 mEq/g for the same silica support. The adsorbed crosslink PEI was very effective in screening the underlying silanol groups. On the other hand, the surface coverage of HB was low and the underlying silica was exposed for interactions with the solutes, resulting in mixed retention mechanisms. This behavior is seen in Fig. 4, where the retention time of a-lact is shown on the HB-based column and on the crosslinked PEI column (plus on another column that will be dis-

Fig. 6 Separation of some inorganic anions on ionene modified columns. a) least hydrophobic ionenes; b) ionenes of intermediate hydrophobicity; c) most hydrophobic ionenes. (Reprinted from J. Chromatogr. A, 850, A.V. Pirogov, M.M. Platonov, O.A. Shpigun, Polyelectrolyte sorbents based on aliphatic ionenes for ion chromatography, pp. 53–63, 1999, with permission from Elsevier.)

Polyelectrolytes as Stationary Phases / 289 cussed shortly). Panel a in Fig. 4 is the HB-based phase and panel c is the crosslinked PEI phase. As can be seen in Fig. 4, the retention time is longest on the HB column even though its ion exchange capacity is the smallest. The severe tailing seen in the HB column is also indicative of mixed retention mechanism. Aviram and Grushka [72] found that on PDADAMAC, which was adsorbed (dynamically) on silica gel, the retention mechanism for positively charged solutes and for the neutral solutes is similar to than in reversed-phase chromatography. Negatively charged solutes (acids) exhibited a retention behavior that was a combination of reversed-phase and ion-pair chromatography.

C. Applications Suzuki et al. [25] found that bonded ionene stationary phases have the ability to separate positional and geometric isomers. They separated solutes with small structural differences. For example, Fig. 5 shows a baseline separation of o-phenylphenol, m-phenylphenol, and p-phenylphenol on (3,16)-ionene column under isocratic condition using a methanol/water (65:35 vol/vol) mobile phase. This separation was not possible on a reversed-phase column under the same conditions. Krokhin et al. [22,49] as well as Pirogov et al. [23,67,68] and Pirogov and Buchberger [50] used their positively charged polymeric stationary phases for the separation of inorganic anions and metal cations (as their EDTA complexes). For the separation of inorganic anions, indirect photometric detection was used. Figure 6 shows an example of the separation of some inorganic anions on ionene-modified columns [67]. The figure shows that the selectivity can be manipulated by changing the nature of the ionene stationary phase.

V. NEGATIVELY CHARGED POLYELECTROLYTES AS STATIONARY PHASES A. Coating Procedures Because silica surface is negatively charge, it is not practical to adsorb anionic polyelectrolytes on it. The most common way to prepare an anionic polyelectrolyte stationary phase is by adsorbing the positive polyelectrolyte on silica gel or on an anion exchanger and then coating the resultant (positively charged) packing with the anionic polyelec-

290 / Yishai-Aviram and Grushka trolyte. Huhn and Muller [73] prepared a cationic exchange column by first coating a vinyl-modified silica with polystyrene or with poly(glycidyl methacrylate) (PGMA) and then sulfonating the coated silica gels with concentrated sulfuric acid to achieve strong cationic exchange (SO3
). Sulfite solution was also used to sulfonate the coated polymers by ring opening. For example, the PGMA-coated silica gels were sulfonated with 1 M sodium sulfite solution in the presence of tetrabutylammonium bromide as catalyst. The result of the reaction with the sulfite is also strong cationic exchange column. Figure 7 shows the two forms of the new coated cation exchangers. Takeuchi et al. [74,75] and Safni et al. [76] dynamically coated an anion exchange column with dextran sulfate (DS). Dextran sulfate has sulfate groups in each D-glucopyranosyl unit, which interact strongly with the positively charged groups of the original stationary phase. They passed [75] aqueous solution of 1.0% sodium dextran sulfate through the column at 4.2 Al/min for 2 hr followed by washing with water until the baseline was stabilized. It was found that the amount of dextran sulfate retained on the anionic exchanger depends on the size of the dextran sulfate. The smaller is the molecular weight of the dextran, the more of it is retained on the column during the coating process. Thus, the pore size of the original anionic exchanger influences the ion exchange properties of the resulting anionic polymer stationary phase. Safni et al. [77] extended the method and dynamically coated a silica-based anion exchanger with heparin. Heparin, a mucopolysaccharide, possesses carboxyl, sulfate, and aminosulfonate groups as ionic moieties. Although the resulting new stationary phase is a cation exchanger, they used it to separate inorganic anions.

Fig. 7 Polymeric cation exchangers formed by (a) direct sulfonation, and (b) sulfite-induced ring opening.

Polyelectrolytes as Stationary Phases / 291 Millot et al. [71] prepared a double-layer polymer coating by adsorbing first HB on silica (as discussed in the previous chapter) and then adsorbing on the HB layer anionic DS. Similarly, they coated previously prepared crosslink PEI on silica with DS. The DS coating procedure calls for mixing the components overnight with gentle shaking. The HB-DS or PEI-DS packing was washed and packed in the column. Aviram and Grushka [72] used dynamic coating technique to adsorb on silica gel first positively charged PDADMAC (as discussed in the previous chapter) and then negatively charged poly(styrenesulfonate) (PSS) on the PDADMAC layer. The resulting doublelayered column could be used as a cation exchanger and also to separate neural species as well.

B. Properties of the Phases and Retention Mechanism The negatively charged polyelectrolyte stationary phase can be characterized using all the tools discussed previously. Huhn and Muller [73] confirmed the presence of the anionic polymers on the stationary phase by doing elemental analysis. From the results, they calculated the average polymer film thickness, which is used to characterize the polymer bonded to the silica gel surface. Huhn and Muller [73] found that, after the polymer coating, the average polymer thickness was 0.25–0.5 nm. After sulfonation, the film thickness was 0.12–0.28 nm. The PGMA layer was thicker than the polystyrene and, therefore, had higher pH stability as shown in Fig. 8. Conventional silica gel columns cannot be used at pH above 7.5. However, the PGMA-coated column can be operated at pH values up to 11. Huhn and Muller [73] checked the performance of the newly coated column packing material by generating reduced van Deemter plot (i.e., reduced plate height h vs. the reduced velocity m). Figure 9 shows two reduced van Deemter plots obtained by them. Some of the minimum h values that they measured on their negatively charged polymeric stationary phases were 4 when Na+ was used as the test solute and 5 when K+ was the test solute. These values demonstrate the good efficiencies that can be realized with polymer-coated silicas. Also, the gentle increase of h with increasing m above the minimum values indicates favorable resistance to mass transfer.

292 / Yishai-Aviram and Grushka

Fig. 8 pH stability on cationic column. (5) Polystyrene column; (o) poly(glycidyl methacrylate) column. (From Ref. 73. nElsevier.)

Huhn and Muller [73] measured the column resistance parameter /, which is indicative of the ‘‘goodness’’ of the column packing. Typical values for well-packed silica gel columns are between 500 and 800. The values found by Huhn and Muller for three different columns were 790, 890, and 850. These values indicate that the columns were well packed and had good flow properties.

Fig. 9 The van Deemter plot; reduced plate height (h) vs. reduced velocity (m) on PGMA column: (5) 6 mg/L Na+; (o) 8 mg/L K+. (From Ref. 73. nElsevier.)

Polyelectrolytes as Stationary Phases / 293 Takeuchi et al. [75] probed the nature of their dextran sulfatemodified column using nitrate ions. They monitored the retention behavior of nitrate as a function of the concentration of sodium sulfate in the mobile phase. They found that the unmodified column behaved as expected from an anion exchanger; namely, the retention of the nitrate decreased as the concentration of the sodium sulfate in the mobile phase increased. However, the dextran sulfate-modified columns, with the exception of the one modified with the largest dextran sulfate, behaved in the opposite direction; kV increased with increasing sodium sulfate in the eluent. Figure 10 shows this behavior. Takeuchi et al. [75] suggest that nitrate is repelled by free sulfate groups. The behavior of the 50,000 dextran sulfate-modified column was explained by its inability to cover completely all the pores of the underlying packing. Figure 10 also shows that the kV values for all the modified columns are smaller than those values obtained with the unmodified columns. The decrease in kV is attributed to the decrease in anion exchange sites after the modification with dextran sulfate. Takeuchi et al. [75] found that the retention behavior of inorganic anions was altered by the modification with dextran sulfate. Safni et al. [77] characterized their heparin-modified column, which is also a cation exchanger. They found that the retention time of nitrate (the

Fig. 10 log kV vs. log sodium sulfate in the mobile phase on modified column with different average Mw of dextran sulfate. 1 (o) = 50,000; 2 (5) =25,000; 3 (4) = 15,000; 4 (.) = 8000; 5 (n) = 5000; 6 (E) = without modification. (From Ref. 75. nFriedr. Vieweg and Sohn Verlagsgesellschaft mbH, 1999.)

294 / Yishai-Aviram and Grushka probe solute) behaved differently in the presence of inorganic salts (sodium sulfate and magnesium sulfate) in the mobile phase than in the presence of organic acids (citric acid, glutamic acid, oxalic acid, and tartaric acid) in the mobile phase. With inorganic salts in the mobile phase, the retention of the nitrate changed little when the concentrations of the salts were changed. However, with the organic acids in the mobile phase, it was found that as the concentration of the acids increased, the retention time of the nitrate increased as well. Figure 11 shows the behavior of nitrate retention as a function of the various salts in the mobile phase. Safni et al. [77] explain the behavior depicted in Fig. 11 by the existence of both cationic and anionic sites in the heparin-modified column. The cationic exchange sites are due to the heparin coverage, whereas the anionic exchange sites are due to the underlying quaternary amine groups in the original anion exchange column. The solutes can interact with both exchange centers. The strength of these interactions depends on the constituents of the mobile phase. The organic acids in the mobile phase are more effective in shielding the heparin groups, thus increasing the interactions of the nitrate

Fig. 11 Log retention time vs. log concentration of the salts in the eluent. (From Ref. 77. nElsevier.)

Polyelectrolytes as Stationary Phases / 295

Fig. 12 Separation of monovalent and divalent cations (a) in guinea pig serum, and (b) bovine serum. (From Ref. 76. nElsevier.)

296 / Yishai-Aviram and Grushka solute with the underlying anion exchange sites. The authors maintain that the retention behavior of cations is expected to be normal because the cations should interact with the cation exchange sites without any repulsion from the anionic exchange sites. Millot et al. [71] determined the ion exchange capacity for their negatively charged double-layer phases. They found that the stationary phase containing HB-DS is a cation exchanger, whereas the stationary phase consisting of PEI-DS is an anion exchanger. This behavior was explained by incomplete charge neutralization by the DS layer. As a result, a mix retention mechanism (anionic exchange from the PEI surface and cationic exchange from the DS polymer layer) exists. They also found that the ion exchange capacity is a function of the silica gel porosity; the higher is the porosity, the lower is the ion exchange capacity.

C. Applications Safni et al. [76] used their anion exchanger column modified with dextran sulfate to separate various alkali and alkaline-earth cations. They applied the technique to separate monovalent divalent cations present in guinea pig serum and bovine serum. Figure 12 shows the resulting chromatograms.

Fig. 13 Separation of anions on anion exchanger modified with heparin. (From Ref. 77. nElsevier.)

Polyelectrolytes as Stationary Phases / 297 In their paper on the heparin-modified column, Safni et al. [77] show the separation of inorganic anions. For the separation to take place, the mobile phase included tartaric acid (see discussion in Section 5.2). The detection was done indirectly using an ultraviolet (UV) detector. Figure 13 shows the resulting chromatogram. Millot et al. [71] used their negatively charged HB-DS composite column to separate some basic proteins using gradient elution. The mobile phase was a Tris buffer at pH 7 and the gradient was a NaCl gradient. Figure 14 shows the separation obtained.

VI. COMPLEX POLYELECTROLYTE LAYERING This review describes the use of polyelectrolytes to coat an existing stationary phase and generate a new stationary phases with different chromatographic properties, selectivities, and efficiencies. The coating can be done in a batch mode or by dynamic coating. The review discussed the use of both negatively and positively charged polymers to prepare new columns. The newly coated columns often provide a platform for complex retention mechanism as well as for unique selectivities. The use of polyelectrolytes to modify stationary phases opens the door to generate complex combinations for specific separa-

Fig. 14 Separation of positively charged proteins on negatively charged HB-DS composite column. (1) TRYP; (2) a -CHYMO; (3) MYO; (4) LYSO. (From Ref. 71. nFriedr. Vieweg and Sohn Verlagsgesellschaft mbH, 2003.)

298 / Yishai-Aviram and Grushka tions. A step in this direction is reported by Millot et al. [71], who describe the generation of a triple-layer polyelectrolyte-coated column. The three layers were HB–DS–HB and because the topmost layer is positively charged, the resulting column is an anion exchanger. The triple layer masked well the residual silanol groups on the silica gel particles, resulting in improved peak shapes. Although the polyelectrolyte columns provide charge centers for ion exchange-based separations, they can also provide sufficient hydrophobic centers to allow for the separation of uncharged solutes. Thus, the polyelectrolyte-coated column can separate mixtures of solutes with diverse chemical properties.

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Index

Absorptive coatings, 247–248 Acetic anhydride, 145 Acetonitrile-water mobile phase, 83 dependencies of constants, 85 Acetyl, 255 Actinides, 146 Acylation agents used in SFE/SFR, 145 Adsorbent activity, 29 Adsorbents SFE, 135–138 Adsorption chromatography, 26, 66 Adsorption isotherm equation, 52 Aflatoxin B1 extraction, 123, 163–164 AFM, 280 Airborne volatile organic compounds, 238

Alcoholysis, 147 Alkylbenzenes, 264 gradient elution reversed-phase separation, 60, 61 Alkylsiyl, 255 Aluminas, 135, 136 American Oil Chemical Society (AOCS) Official Method G3-53, 158 fatty acid content, 159 Amino-bonded silicas, 140 Aminopropyl, 26 Ammonia NH2-Mega Bond Elut, 162 Amphetamine, 244 Analyte collection, 130 Analyte molecules, 201 expression of maximum number, 226–228 number, 204

305

306

/

Index

Analyte peak, 190 Analyte polar group, 255 Analyte retention factor, 195 Analyte trapping efficiency, 129 Analytical derivatization SFR, 144 Analytical determination sample size, 122 Analytical gradient-elution chromatography retention theory, 9–18 Analytical SFE applications, 150 lipid or lipid derived volatile and semivolatile compounds, 157 detect irradiated foodstuffs, 165 drug analysis, 150 features, 117 instrumentation, 117 integration of adsorbents, 135 reactions and derivatization applied, 143 triangle, 116 yielding crude fractionations, 131 Analytical toxicology application, 260–261 Anion exchangers ionene, 282 PDADMAC, 282 PEVP, 282 PHMG, 282 Anion separation, 296 Antidepressant drugs SPME-GC-NPD chromatogram, 260 AOAC methods organic solvents, 161 AOCS Official Method G3-53, 158 fatty acid content, 159 APCI, 241 Applied Separations, Inc., 121 a priori parameters, 192 Aroma volatiles, 157

Aspergillus flavus, 123 Association of Official Analytical Chemists (AOAC) methods organic solvents, 161 Atmospheric pressure chemical ionization (APCI), 241 Atomic force microscopy (AFM), 280 Automated SFE/SFR/GC analyzer fat content determination, 156 Bandwidths calculation, 33 gradient elution chromatography liquid column, 17–18 Bandwidths increase isocratic conditions, 64 Batch-equilibrium conditions and chromatography, 195 Batch mode PCED, 283 Behavior-gradient preelution, 43 Benzene, 161 Binary gradients, 5 reversed-phase chromatography, 19–24 Binary solvent mixture distribution equilibrium, 51 Bioanalytical applications, 258–263 Biological matrices, 135 Biomarkers determination exhaled human breath, 262 Bonded ionene stationary phases, 289 Bonded nitrile column gradient elution weak solvent, 67–68 Breakthrough curves calculated, 53 BUDGE solution, 284 Butanedioldiglycidylether, 283–284 Caffeine extraction, 127 Candida antarctica, 148

Index Capillary electrochromatography, 275, 278 Capillary electrophoresis (CE), 233, 278 SPME coupling, 237 Carbon content silica plus polymer phase, 279 Carbon dioxide, 140 phase diagram, 113 SC-CO2 extraction, 132–140 Carbon graphitized silica, 250 Carboxylic acids peak shape improvements, 258 Cation(s) separation of monovalent and divalent, 295 Cation exchange material compared, 283 ionine, 286 Cation-exchange microchromatography, 250 Cationic column pH stability, 292 CE, 233, 278 SPME coupling, 237 Celite, 135, 151 Central limit theorem, 199 Chip technology, 182 Chiral separation, 200 Chloroform, 161 Chloroformates, 255 Cholesteryl stearate, 148 Chromatographic column frequency distribution of sojourn time, 215 polar solvent adsorbed, 52 Chromatographic interactions, 280 Chromatographic migration, 190 Chromatographic optimization function (COF), 71–72 Chromatographic process kinetics, 15

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307

Chromatographic quantities and equilibrium distribution constant K, 197 Chromatographic retention, 182 Chromatography batch-equilibrium conditions, 195 under critical conditions, 87–88 equilibrium conditions, 195–196 polar adsorbents, 28 polar compounds, 33 preparation prior, 110 stochastic approach, 183–188 stochastic description, 182 ternary gradients, 78–80 CHROMDREAM, 73, 75 CHROMSWORD, 73, 75 Classical dynamic coating technique, 277, 278 CMC, 276 Coating(s), 246–254 Coating polymer chemistry extraction efficiency, 254 Coating procedures polyelectrolytes, 282–283, 289–290 Coating solution PDADMAC, 284 Coextractives and water, 125–127 COF, 71–72 Column characterization, 279–281 Column hold-up volume, 48 effect, 46 preelution, 46 Column uptake of polar solvents correction of retention volume, 104–106 Complex polyelectrolyte layering, 298 Composition window, 86 Confocal Raman spectroscopic analysis, 237 Constant mobile phase velocity, 213 Corn extraction of aflatoxin B1, 123

308

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Index

Cosolvent, 127 addition capability, 118 Critical concentration interactive LC of copolymers, 88 Critical fluids sample preparation applications, 149–166 Critical micellar concentration (CMC), 276 Critical pair of adjacent peaks, 72 Cyanopropyl, 26 Dalea spinosa, 125 DBSA, 282, 283 2-DCB, 166 Dehydrated meat matrices FAME distributions, 155 Dense gas chromatography, 114 Derivatization, 255–257 analytical SFE, 143 on-fiber after extraction, 257 on-line SPME-LC configurations, 257 organic acids, 256 peak shape improvements, 258 postcolumn HPLC, 144 SFE, 144–145 and simultaneous extraction, 257 Desorption, 235 Dextran sulfate, 290 modified column, 293 structure, 282 DI, 234 SPME, 234–235 Diatomaceous earth, 135 Dibutyl-unsymmetry-dibenzo-14crown-4-dihydroxy crown ether, 251 Dihydroxy-substituted saturated urushiol crown ether, 251 Dioxane volumes, 66

Dirac function, 214 Direct immersion (DI), 234 SPME, 234–235 Direct sulfonation polymeric cation exchangers, 290 Distribution equilibrium binary solvent mixture, 51 Divinylbenzene phases, 246, 249 Dodecylbenzenesulfonic acid (DBSA), 282, 283 2-dodecylcyclobutanone (2-DCB), 166 Double-layer polymer coating, 291 Drug analysis analytical SFE, 150 off-line SFE, 158 Drugs of abuse, 260–261 Dry test meter, 119 Dynamic coating defined, 276 polymer-coated stationary phases, 276 principle, 276–278 Dynamic desorption, 235 ECD, 156 chromatogram pesticides extracted using fluoroform, 141 Edgeworth-Crame´r series expansion, 218 EDTA, 283 Egress time, 183–185 Einstein equation, 184 Electrochromatography, 14 Electron capture detector (ECD), 156 chromatogram pesticides extracted using fluoroform, 141 ELSD, 158, 166 Eluent salts retention time vs. concentrations, 294 ELUEX, 73, 75 Elution chromatography, 183

Index Elution peak, 190 Elution strength ternary gradients, 79 elution separation resolution window diagram, 80 Elution time of analyte peak, 228 Elution volumes calculation, 33 corrected, 55 gradient dwell volume, 44, 45 gradients 0% polar solvent and overestimated data, 67 Elutropic solvent, 135 Environmental Protection Agency (EPA) Pollution Prevention Act, 110–111 EO-PO, 88 cooligomer gradient elution reversedphase separation, 89 EPA Pollution Prevention Act, 110–111 Equilibrium conditions chromatography, 195–196 Equilibrium constants, 206 Equilibrium distribution, 182 constant K and chromatographic quantities, 197 Ergodic hypothesis, 196 Error on kV, 230 ESI interface, 241 Esterification reactions organic solutes, 145 Ethers butanedioldiglycidylether, 283–284 dibutyl-unsymmetry-dibenzo14-crown-4-dihydroxy crown, 251 dihydroxy-substituted saturated urushiol crown, 251

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309

[Ethers] octadecylethers separation, 30 oligoethylene glycol alkyl, 83 coelute, 87 oligoethylene glycol nonylphenyl coelute, 87 separation, 29 Ethylenediamine-tetraaectic acid (EDTA), 283 Ethylene oxide-propylene oxide (EO-PO), 88 cooligomer gradient elution reversed-phase separation, 89 Evaporative light scattering detector (ELSD), 158, 166 Everett’s equation, 51 Exhaled human breath biomarkers determination, 262 Extracted analyte collection, 128–130 Extraction aflatoxin B1, 123, 163–164 polar analytes, 162 Extraction cell pesticide recovery, 136 Extraction conditions vs. extraction recovery, 243 Extraction efficiency coating polymer chemistry, 254 Extraction fluid density, 131 cleanup, 134 preferred, 121–122 type or composition variation, 140–141 Extraction mode and coupling, 234–241 novel devices, 238–241 Extraction rates vs. flow rates, 131

310

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Index

Extraction recovery vs. extraction conditions, 243 Extraction tubes types, 239 Extraction yield increase temperature, 244 Extractor design, 118 False separation, 38 FAME, 145 distributions dehydrated meat matrices, 155 synthesizer automated on-line, 155 Fast gradients, 69–70 Fat band, 126 Fatty acid content AOCS Official Method G3-53, 159 Fatty acid methyl esters (FAME), 145 distributions dehydrated meat matrices, 155 synthesizer automated on-line, 155 FDA, 160 Femtosystem, 203–205 Fiber conditioner device, 240 Fiber-in-tube extraction, 239–240 Fiber SPME, 265 Flame photometric detector (FPD), 156 Florisil, 135 Florisil trap, 152 Flow programming HPLC, 3 Flow rates vs. extraction rates, 131 Fluid composition pesticide recovery and lipid extracted, 142 Fluid density-based fractionation, 132–134

Fluorescence imaging methods, 185 Fluorocarbons, 140 Fluoroform, 140 Food and Drug Administration (FDA), 160 Food Safety and Inspection Service (FSIS), 160 Fourier-transform infrared spectroscopy (FTIR), 112 FPD, 156 Frenkel-de Boer equation, 184 FSIS, 160 FTIR, 112 Gas chromatography (GC), 112, 232 dense, 114 pesticides extracted using fluoroform, 141 SFE, 156–157 Gas totalizer device, 119 Gaussian shape peak shape, 199 GC. See Gas chromatography (GC) Giddings-Eyring model of chromatography, 190, 207, 226 Gradient(s) effect of delayed migration, 41 linear, concave, and convex examples, 6 optimization, 71–77 Gradient delay, 38 Gradient dwell volume determined, 39–40 effect, 42, 44, 45 elution volume, 44, 45 increases, 38 Gradient elution, 5 IC separations of polymers, 83 ideal, 15 method development, 55–77

Index [Gradient elution] optimization spreadsheet program, 107–108 retention data, 9 reproducibility, 34 retention volumes calculation, 11 reversed-phase separation alkylbenzenes, 60, 61 ethylene oxidepropylene oxide cooligomer, 89 silica gel and bonded nitrile column weak solvent, 67–68 theory, 90–91 Gradient elution chromatography analytical retention theory, 9–18 changing column diameter, 59–60 changing column length, 61–62 changing flow rate mobile phase, 58–59 ion exchange chromatography (IEC), 34–35 liquid column bandwidths and resolution, 17–18 effects of dwell volume on, 37–47 optimization of separation, 3–93, 104–108 prediction of retention, 3–93, 104–108 symbols, 92–96 polymer liquid mechanism, 82 retention, 7 volumes, 54 Gradient elution separation elution strength ternary resolution window diagram, 80 high-molecular compounds, 81–89

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[Gradient elution separation] LSS model, 8 optimization, 69–77 Gradient function, 11 linear gradients, 14 Gradient instruments function, 16–17 Gradient methods transfer, 56–57 Gradient preelution illustrated, 43 Gradient profile changing, 64 Gradient program isocratic steps, 5–6 Gradient retention delay calculated, 40 Gradient rounding, 37–38 Gradient RPC optimization, 75 Gradient separation accelerated, 71 Gradient shape effect, 12 Gradient steepness changing, 65 prediction of effect, 62–68 Gradient volume calculated, 41 Gravimetric balance, 115 Hair spiked GC-MS following HS-SPME, 261 Ham SC-CO2 extraction moisture content, 128 Hamilton syringe cleaner, 240 HB, 284 Headspace (HS) extraction, 234 SPME, 234–235, 258 SPME-GC-MS, 260

311

312

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Index

Heparin structure, 282 Heptafluorobutyric anhydride, 144 Hewlett Packard Model HP 7680, 130 Hexadimethrine bromide (HB), 284 Hexafluorobutyric anhydride derivative, 160 High-performance liquid chromatography (HPLC), 14, 112, 232 designs in-tube coupling, 236 flow programming, 3 isocratic accelerated, 71 elution mode, 3 normal phase gradient elution, 50 correction of retention volume, 104–106 retention volumes, 68 off-line SFE, 158 postcolumn derivatization method, 144 reversed-phase optimization, 73 SFE, 158–163 Holdup time marker selection, 206 Homogeneous columns moving phase velocity, 213–221 peak splitting, 213–221 peak tailing, 213–221 Homopolymers coelution, 87 Hot ball kinetic model, 116 HPLC. See High-performance liquid chromatography (HPLC) HS. See Headspace (HS) Human breath sampling device, 263 Hydrolysis of vitamin A, 148 Hydrolytic activity of lipases, 148

Hydromatrix, 125, 135, 137, 158 soybean oil, 161 Hydroterminated silicone oil, 251 Hydroxydibenzo-14-crown-4, 251 Hyphenated supercritical fluid techniques, 164 IAE, 253 IC, 81 separations of polymers gradient elution, 83 Ideal gradient elution, 15 IEC, 34 ionic compounds, 8 retention, 35 slab model, 36 Immunoaffinity extraction (IAE), 253 Immunoaffinity media, 250 Imperfect mixing mobile phase components, 37 Infrared spectroscopy (IR), 280 on-line with SFE determine iodine number, 165 Ingress number distribution, 188 Ingress time, 183–185 Initial isocratic step, 41 Initial mobile phase retention factor, 40 retention factors, 40 Injection, 201 Injection process error resulting, 228–229 Instantaneous retention factors, 18 changing, 10 Instrumental gradient dwell volumes, 56 Integration limits, 17 Interactive chromatography (IC), 81 separations of polymers gradient elution, 83 In-tube coupling HPLC designs, 236

Index In-tube SPME, 236 Inverse SFE, 139 examples, 139 sequence of steps, 139 Iodine number determine infrared spectroscopy on-line with SFE, 165 Ionene anion exchangers, 282 isotherms of sorption, 285 modified columns separation of inorganic anions, 288 prepare positively charged polymeric stationary phases, 283 stationary phases bonded, 289 structure, 281 Ion exchange chromatography (IEC), 34 ionic compounds, 8 retention, 35 slab model, 36 Ionic strength dependence PDADMAC adsorption on silica, 279 Ionine cation exchange material, 286 Ion-pair chromatography, 276–277 IR, 280 on-line with SFE determine iodine number, 165 Isco, Inc., 120, 121 Isocratic conditions bandwidths increase, 64 Isocratic elution mode HPLC, 3 Isocratic elution retention factor, 58 Isocratic HPLC accelerated, 71 Isocratic LC, 9

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313

Isocratic retention calculation, 33 factors, 7 gradient-elution behavior prediction, 9 Isocratic RPC optimization, 75 Isocratic separations, 48 Isocratic steps gradient program, 5–6 Isohydric organic solvents, 29 Iso-selective gradients, 73 Iso-selective multisolvent gradients, 78 Iso-selective ternary gradients, 79, 80 Isotherms of sorption 4,6-ionene, 285 Joule-Thompson effect, 127 Kinetic tailings, 192 Langmuir isotherm, 32, 249 Langmuir model, 51 Lanthanides, 146 Leco Corporation, 121 Lidocaine, 244 Linear chromatography, 200 Linear concentration gradients, 12 Linear gradients gradient function, 14 Linear solvent strength (LSS) binary gradients, 7 Lipase(s) hydrolytic activity, 148 Lipase-catalyzed hydrolysis, 147 Lipase-catalyzed methanolysis for SFE/SFR, 149 Lipids conversion, 149 Lipid solutes in SC-CO2 solubility data, 125

314

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Index

Liquid absorptive coatings scheme, 248 Liquid chromatography under critical conditions, 87 exclusion-adsorption, 83 polyelectrolytes stationary phases, 273–298 Liquid–liquid extraction SPME, 242 LSS binary gradients, 7 MALDI, 242 Martin rule, 82, 88 Martire model, 31 Mass spectrometry (MS), 112 Matrix-assisted laser desorption/ ionization (MALDI), 242 Matrix solid phase dispersion (MSPD), 124 Mean desorption time, 184 Mean egress time, 184 Meat samples concentration of naphthalene, 154 Methanol with oleic acid, 148 polyethyleneimine (PEI), 283 in water, 22 Methylene chloride, 161 Microchromatography cation-exchange, 250 Microsystem, 203–205 Millisystem, 203–205 Minichromatographic columns, 137 MIP, 250–252 Mobile phase acetonitrile-water dependencies of constants, 85 components, 183 imperfect mixing, 37, 38 prediction of effect, 62–68

[Mobile phase] constant velocity of analyte molecules, 185 flow rate effects, 57 sodium sulfate, 293 time by analyte, 195 visits effective average time, 188 volume, 195 determination, 16 Molecular dynamic quantities, 206 Molecularly imprinted polymers (MIP), 250–252 Monolithic silica gel-based columns, 70 Monte Carlo simulation, 193 Moving phase velocity homogeneous columns, 213–221 MS, 112 MSPD, 124 Mycotoxin extraction, 163–164 Nanosystem, 203–205 Naphthalene concentration meat samples, 154 Naphthalene in SF-CO2, 114 solubility, 114 Naphthalene sulfonic acid separation, 24 Naphthoylene-benzimidazole aklylsulphonamides reversed-phase separation, 4 gradient elution, 63 Near critical fluids, 113 NH2-Mega Bond Elut, 162 Nitrile column gradient elution weak solvent, 67–68 NLEA, 154, 155 NMR, 280 Nonideal retention behavior instrumentation effects, 36–54

Index Normal phase chromatography (NPC), 25, 66 advantages, 27–38 with binary gradients, 25–33 solvents, 27 Normal phase gradient elution HPLC, 50 retention volumes, 68 optimization window diagram, 76 separation, 29, 30, 39 Normal phase HPLC correction of retention volume, 104–106 Normal phase systems separation selectivity and retention, 31 Novanik 600/20 sample, 89 Novozyme 435, 148, 161 NPC. See Normal phase chromatography (NPC) Nuclear magnetic resonance (NMR), 280 Nutritional Labeling and Education Act (NLEA), 154 based method determining fat content, 155 solvent-based-extraction protocol, 154 Octadecylethers separation, 30 Off-line SFE drug analysis, 158 and HPLC or SFC, 158 Off-line trapping, 137 Oligoethylene glycol alkyl ethers, 83 coelute, 87 nonylphenyl ether coelute, 87 separation, 29 separation, 30

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315

Oligomers separations, 25 Oligostyrenes normal-phase gradient-elution separation, 13 On-fiber derivatization after extraction, 257 On-line derivatizations SPME-LC configurations, 257 Optimization reversed-phase HPLC, 73 Organic acids derivatization, 256 Organic solutes esterification reactions, 145 Organic solvent preferential adsorption, 49 Organochlorine pesticides, 151 Overlapping resolution map, 72 Packed column comparison, 138 Packing particle diameter, 70 PAH, 245 Parallel multisample SFE unit simultaneous, 119 Parameters, 243 Partitioning, 249 PCED batch mode, 283 structure, 281 PDADMAC. See Poly (diallyldimethylammonium chloride) (PDADMAC) PDMS coated bar, 240 fiber, 237 Peak capacity, 69–70 Peak shape features and experimental errors retention factor determination, 189–194

316

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Index

Peak splitting, 198, 216–218 homogeneous columns, 213–221 Peak tailing, 199–200, 218–221 homogeneous columns, 213–221 PEEK tubing, 240, 242 PEG, 251 separation, 19–20 PEI methanol, 283 structure, 281 Pelletized celite, 125 Pesticide recovery extraction cell, 136 Pesticide recovery and lipid extracted fluid composition, 142 Pesticides organochlorine, 151 PEVP anion exchangers, 282 quaternary ammonium group, 286 structure, 281 PGMA-coated column, 291 pH adjustment extraction yield, 244 dependence PDADMAC adsorption on silica, 278 stability cationic column, 292 Phase-adsorption technique, 136 Phases properties, 284–288 Phenylphenol separation of structural isomers, 287 Phenylurea optimization window diagram, 76 Phenylurea herbicides, 32, 68 resolution window diagram, 77

Phenylurea herbicides NP-gradient elution separation resolution window diagram, 74–75 PHMG anion exchangers, 282 structure, 281 Phosphatidylcholine, 148 Pico-chip system, 203–205 Pico-tube system, 203–205 Plant sterols, 161 Plasma gas chromatography, 245 Poisson law, 187–188 Polar adsorbents chromatography, 28 Polar analytes, 257 extraction, 162 Polar compounds chromatography, 33 Polarity and elution strength, 27 Polarity reduction, 255 Polar solvent adsorbed chromatographic column, 52 Polar solvents column uptake correction of retention volume, 104–106 Poly(diallyldimethylammonium chloride) (PDADMAC), 277–278 adsorption on silica ionic strength dependence, 279 pH dependence, 278 anion exchangers, 282 coating solution, 284 layer, 291 structure, 281 Poly(hexamethyleneguanidium hydrochloride)(PHMG) structure, 281

Index Poly(N-chloranil, N,N,N ,V N V-tetramethylethylene diammonium dichloride) (PCED) batch mode, 283 structure, 281 Poly(N-ethyl-4-vinyl pyridinium bromide)(PEVP) anion exchangers, 282 quaternary ammonium group, 286 structure, 281 Poly(styrene sulfonate)(PSS), 282, 291 Polyacrylic acid-coated fiber, 249 Polycationic electrolyte, 277–278 Polycationic phases, 283 Polycationic polymer, 283 Polycationic stationary phases, 284 Polycyclic aromatic compounds (PAH), 245 Polydimethylsiloxane (PDMS) coated bar, 240 fiber, 237 Polydispersity of synthetic polymers, 82 Polyelectrolyte layering, 298 Polyelectrolytes applications, 297 coating procedures, 282–283, 289–290 stationary phases liquid chromatography, 273–298 negatively charged, 289–297 positively charged, 282–288 Polyethylene glycol (PEG), 251 separation, 19–20 Polyethyleneimine (PEI) methanol, 283 structure, 281 Polymer-coated stationary phases dynamic coating, 276

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317

Polymeric cation exchangers direct sulfonation, 290 Polymers separations, 25 Polypyrrole (PPY) coatings, 254 Polystyrene separation, 39 Poor retention data reproducibility, 5 Population, 192 Porous bonded silica LC coatings, 250 Portable solid-phase microextraction for field-air sampling, 238 Positively charged polymers, 281 Positively charged proteins separation, 297 Postcolumn derivatization method HPLC, 144 Post-SFE trapping, 137 Poultry fat pesticide recoveries lipid extracted, 142 PPY coatings, 254 Preelution behavior-gradient, 43 Preferential adsorption, 49 organic solvent, 49 Programmable fluid ‘‘wash’’ cycles, 121 Propanol in dichloromethane calculated breakthrough curves, 53 in heptane, 53, 68 in hexane, 68 volumes, 66 Proximate fat analysis, 153–155 Pseudomonas cepacia, 148 PSS, 282, 291 Pulsed field gradient nuclear magnetic resonance, 181 Purospher Star RP-18e, 60

318

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Index

Quaternary ammonium group PEVP, 286 Raman microspectroscopic analysis, 237 Raman spectroscopic analysis, 237 RBD soybean oil, 161 RCRA, 111 Real-tome single-molecule observations, 205 Refined, bleached, deodorized (RBD) soybean oil, 161 Relative error on kV, 223 Relative errors residence time, 222–223 Relative random errors, 221 Resolution gradient elution chromatography liquid column, 17–18 Resource, Conservation, and Recovery Act (RCRA), 111 Retention effects of adsorption of strong solvents, 48–54 gradient elution chromatography, 7 instrumentation effects, 36–54 liquid chromatography modes, 64 ternary mobile phases, 81 Retention behavior small and large molecules differences, 82 Retention chromatographic quantities, 206 Retention data calculating, 17 Retention factor, 189 dependence, 22, 32 determination peak shape features and experimental errors, 189–194

[Retention factor] initial mobile phase, 40 injection effect, 194 peak splitting effect, 190 peak tailing effect, 191 vs. solute concentration, 16 stochastic bias effect, 192–193 unretained tracer selection effect, 195 Retention function, 11 Retention mechanism and NPC phases, 26 properties, 284–288, 291–296 Retention prediction selectivity ternary gradients, 79–81 Retention theory analytical gradient-elution chromatography, 9–18 Retention time alpha-Lact, 286 calculation, 9–16 determined, 191 Retention volume calculation, 9–16 correction column uptake of polar solvents, 104–106 differences window diagram, 76 errors, 65 gradient elution chromatography, 54 normal-phase gradient elution HPLC, 68 Reversed-phase chromatography (RPC), 44, 45 adsorption and partition mechanism, 23 binary gradients, 19–24 elution times, 21 gradient optimization, 75

Index [Reversed-phase chromatography (RPC)] ion-pair or salting-out, 23 isocratic optimization, 75 retention, 209 Reversed-phase gradient elution, 69 separation resolution window diagram, 77 Rhizomucor miehei, 148 Rotameter, 119 RPC. See Reversed-phase chromatography (RPC) Salatrim characteristic, 166 Salting out effect, 244 Salts in eluent retention time vs. concentrations, 294 Sample size extraction time and precision, 124 SARA, 111 Scanning electron microscopy (SEM), 280 Scatchard plots, 264 SC-CO2, 110, 111, 125 extraction, 132–140 Scott-Kucera model of adsorption, 32 SEC, 81 Seed oils sterol concentration, 162 Selectivity ternary gradients, 78 retention prediction, 79–81 SEM, 280 Separation conditions effects, 57 Separation efficiency increase, 61

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319

Separation engineering renewal, 181 Separon SGX, 13 Nitrile, 32 resolution window diagram, 74–75, 77 RPS column, 24 silica gel column, 53 Sequential optimization methods with COF sum optimization criterion, 72 SF. See Supercritical fluid (SF) SFCU technique, 138, 139 SFE. See Supercritical fluid extraction (SFE) SFR. See Supercritical fluid reaction (SFR) Short packed columns, 70 Silanized optical fiber, 242 Silanols, 274 Silasorb SPH C8, 22 Silica, 135 carbon graphitized, 250 Silica-based polyelectrolyte stationary phases approaches for making, 275 Silica gel and bonded nitrile column gradient elution weak solvent, 67–68 Silica plus polymer phase carbon content, 279 Siloxane bonds, 274 Silyated silicas, 135 Silylation, 144–145 Simultaneous extraction and on-fiber derivatization, 257 Simultaneous predictive optimization advantage, 72–73 Single molecule mutual independence, 186–187

320

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Index

Single-molecule chromatography, 201 Single-molecule dynamic quantities, 182 Size-exclusion chromatography (SEC), 81 Slovanik 320, 89 Small-angle neutron scattering (SNAS), 280 Snyder, 26 Snyder model, 31 Snyder-Soczewinski model, 31 Soczewinski displacement model of retention, 26 Sodium sulfate mobile phase, 293 Sol gel media, 250 Solid coatings, 249 Solid phase extraction (SPE) materials, 137 Solid-phase microextraction (SPME), 231–268 accuracy and precision, 243 application area, 266 CE coupling, 237 clinical chemistry, 261 description, 232 designs in-tube coupling, 236 determination of pharmaceuticals, 259 device, 235 electrode position, 241 schematic, 232 DI, 234–235 electrochemistry, 240–241 fiber, 265 fiber coatings characteristics and properties, 247 filter protecting, 240 GC-NPD chromatogram antidepressant drugs, 260

[Solid-phase microextraction (SPME)] Headspace GC-MS, 234–235, 258, 260 in-tube, 236 LC, 235–236 LC configurations on-line derivatizations, 257 liquid-liquid extraction, 242 low recoveries, 266 mass/atomic spectrometry, 241–242 publications reporting, 233 references possibilities and limitations, 264–267 Solubility data lipid solutes in SC-CO2, 125 Solubility parameters characteristic, 133–134 Solute removal using SF, 115 Solute’s maximum solubility, 115 Solute trapping, 130 Solvent breakthrough curves calculate, 52 Solvent demixing, 48 effect, 104–106 Solvent-modified extraction procedure, 246 Sorbent-based SFE, 139 Sorbent collection device integration, 137 Sorbent-filled collection device, 130 Sorbents, 135 fractionation of extract, 135 Sorbent trap option supercritical fluid extraction apparatus, 152 Sorption principles and parameters, 242–245 Soxhlet extraction technique, 111

Index Soybean oil Hydromatrix, 161 triglycerides solubility, 126 Spectroscopic analysis confocal Raman, 237 SPE materials, 137 SPME. See Solid-phase microextraction (SPME) Static desorption, 235 Stationary phase, 183 identify sites, 187 mean time, 189 Steep gradients with wide concentration range, 86 Sterol concentration seed oils, 162 Stochastic approach of chromatography, 183–188 aspects, 183–184 Stochastic bias, 201 effect, 221 Stochastic model, 185–188 Stochastic theory of chromatography, 225 Stoichiometric parameter value, 66 Structural isomers separation phenylphenol, 287 Structure-based predictive software, 73, 75 Sulfite-induced ring opening polymeric cation exchangers, 290 Supelco, 232, 235 Supercritical-carbon dioxide (SC-CO2), 110, 111 extraction, 132–140 Supercritical fluid (SF), 151 extract cleanup, 151–152 for off-line sample preparation in food analysis, 110–167

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321

[Supercritical fluid (SF)] sample preparation, 110–111 solute removal, 115 Supercritical fluid chromatography (SFC), 111 fatty acid content, 159 off-line SFE, 158 SFE, 158–163 Supercritical fluid cleanup (SFCU) technique, 138, 139 Supercritical fluid-derived extract simplifying, 131–132 Supercritical fluid extraction (SFE), 111–130 acylation agents, 145 adsorbents, 135–138 apparatus sorbent trap option, 152 collecting nonvolatile and volatiles device schematic, 129 commercial instrumentation, 120 coupling reaction chemistry derivatization, 142–148 extraction of volatiles and semivolatiles, 135 fatty acid content, 159 gas chromatography (GC), 156–157 ham sample extraction time function, 116 with HPLC or SFC, 158–163 integrated with selected chromatographic/ spectroscopic techniques, 164–165 integration of cleanup step, 131–141 lipase-catalyzed methanolysis, 149 optimizing selectivity, 134 pesticides analyzed, 151

322

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Index

[Supercritical fluid extraction (SFE)] principles, 112–116 sample matrix and preparation, 122–124 with SFR combining, 143 sorbent-based, 139 technique inversed, 148 two-step fractionation method soybean oil, 162 types of extraction and instrumentation, 117–121 unit design, 118 utilizing SFR metals and radioactive species analysis, 146 Supercritical fluid reaction (SFR), 112, 143 acylation agents, 145 analytical derivatization, 144 catalysts, 146–148 fatty acid content, 159 inversed technique, 148 lipase-catalyzed methanolysis, 149 reactions, 142–143 with SFE combining, 143 Supercritical fractionation in seed oils, 162 Superfund Amendments and Reorganization Act (SARA), 111 Suprex Autoprep 44, 130 Synergist in facilitating extraction, 127 Synthetic resins, 135 Tailings, 192 Tandem off-line and in-line trapping, 137

2-TCB, 166 Tenax trap, 129 Ternary gradients chromatography, 78–80 Ternary mobile phases retention, 81 Tetraalkylammonium salts, 147 2-tetradecylcyclobutanone (2-TCB), 166 Tetrahydrofuran in water, 22 Tetrahydrofuran-water mobile phases dependencies of constants, 85 TFA, 163 TGA, 280 Theophylline competitive binding, 253 Thermogravimetric analysis (TGA), 280 Thin layer chromatography (TLC), 21, 48, 183 spot test, 115 Three-block PO-EO-PO cooligomer, 88 Threshold pressure, 114, 115 TLC. See Thin layer chromatography (TLC) Total Fat Analyzer, 121 Total petroleum hydrocarbons (TPH), 150 Toxicology analytical application, 260–261 TPH, 150 Trace components analysis, 150–152 Transesterification, 145 Transesterified rapeseed oil, 47 Trapping efficiency, 130 Trifluroacetic acid (TFA), 163 Triglycerides, 147 Trimethyloxonium tetrafluoroborate, 256

Index 2-dodecylcyclobutanone (2-DCB), 166 Two-parameter Langmuir isotherm, 51 Two-step gradient elution reversed phase separation, 47 Two-step SFE fractionation method soybean oil, 162 2-tetradecylcyclobutanone (2-TCB), 166 Unretained tracer, 224 improper selection, 230 Urine HPLC-ECD chromatograms following SPME, 252

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323

Van Deemter equation, 70 Van Deemter plot, 292 Volatile organic compounds (VOC) airborne, 238 GC-MS profile, 262 Volumetric mobile phase flow rate, 193 Wald equation, 189 Wheat pesticide recoveries, 153 Window diagram, 72 Wire-in-tube extraction, 239–240 X-rays spectroscopy methods, 280 X-rays techniques, 280