EDITORIAL BOARD BRUCE, J BERNE, Department of Chemistry, Columbia University, New York, New York, U.S.A KURT BINDER, Institut fUr Physik, Johannes Gutenberg-Universitiit Mainz, Mainz, Germany A WELFORDCASTLEMAN,JR., Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania, U.S.A DAVID CHANDLER,Department of Chemistry, University of California, Berkeley, California, U.S.A M S CHILD, Department of Theoretical Chemistry, University of Oxford, Oxford, U.K WILLIAMT COFFEY,Department of Microelectronics and Electrical Engineering, Trinity College, University of Dublin, Dublin, Ireland F FLEMINGCRIM, Department of Chemistry, University of Wisconsin, Madison, Wisconsin, U.S.A ERNESTR DAVIDSON,Department of Chemistry, Indiana University, Bloomington, Indiana, U.S.A GRAHAMR FLEMING,Department of Chemistry, University of California, Berkeley, California, U.S.A KARLF FREED,The James Franck Institute, The University of Chicago, Chicago, Illinois, U.S.A PIERRE GASPARD,Center for Nonlinear Phenomena and Complex Systems, Brussels, Belgium ERICJ HELLER, Institute for Theoretical Atomic and Molecular Physics, HarvardSmithsonian Center for Astrophysics, Cambridge, Massachusetts, U.S.A ROBINM HOCHSTRASSER, Department of Chemistry, The University of Pennsylvania, Philadelphia, Pennsylvania, U.S.A R KOSLOFF,The Fritz Haber Research Center for Molecular Dynamics and Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel RUDOLPHA MARCUS,Department of Chemistry, California Institute of Technology, Pasadena, California, U.S.A G NICOLlS,Center for Nonlinear Phenomena and Complex Systems, Universite Libre de Bruxelles, Brussels, Belgium THOMASP RUSSELL,Department of Polymer Science, University of Massachusetts, Amherst, Massachusetts DONALDG TRUHLAR,Department of Chemistry, University of Minnesota, Minneapolis, Minnesota, U.S.A JOHND WEEKS,Institute for Physical Science and Technology and Department of Chemistry, University of Maryland, College Park, Maryland, U.S.A PETERG WOLYNES,Department of Chemistry, University of California, San Diego, California, U.S.A This book is printed on acid-free paper @ Copyright © 2001 by John Wiley & Sons, Inc All rights reserved Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744 Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ@WILEY.COM For ordering and customer service, call 1-800-CALL-WILEY Library of Congress Catalog Number: 58-9935 ISBN 0-471-40541-8 Printed in the United States of America 10 I CONTRIBUTORS TO VOLUME 116 BIMAN BAGCHI, Solid State and Structural Chemistry of Science, Bangalore, India Unit, Indian Institute SARIKA BHATTACHARYYA,Solid State and Structural Indian Institute of Science, Bangalore, India Chemistry Unit, MICHAEL E CATES, Department of Physics and Astronomy, Edinburgh, Edinburgh, United Kingdom University J W HALLEY, School of Physics and Astronomy, Minneapolis, MN of Minnesota, University of JOSEPH KLAFTER, School of Chemistry, Tel Aviv University, Tel Aviv, Israel RALF METZLER, School of Chemistry, Tel Aviv University, Tel Aviv, Israel and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA D L PRICE, Department of Physics, University of Memphis, WOLFFRAMSCHROER, Institut fiir Anorganische Universitat Bremen, Bremen, Germany PETER SOLLICH, Department of Mathematics, London, London, United Kingdom Memphis, TN und Physikalische Chemie, King's College, University of S W ALBRAN, Forschungszentrum Jiilich GmbH, Institut fuer Werkstoffe und Verfahren der Energietechnik (IVW-3), JUlich, Germany PATRICK B WARREN, Unilever Research United Kingdom HERMANN WEINGARTNER, Physikalische Bochum, Bochum, Germany Port Sunlight, Bebington, Chemie Wirral, II, Ruhr-Universitat v INTRODUCTION Few of us can any longer keep up with the flood of scientific literature, even in specialized subfields Any attempt to more and be broadly educated with respect to a large domain of science has the appearance of tilting at windmills Yet the synthesis of ideas drawn from different subjects into new, powerful, general concepts is as valuable as ever, and the desire to remain educated persists in all scientists This series, Advances in Chemical Physics, is devoted to helping the reader obtain general information about a wide variety of topics in chemical physics, a field that we interpret very broadly Our intent is to have experts present comprehensive analyses of subjects of interest and to encourage the expression of individual points of view We hope that this approach to the presentation of an overview of a subject will both stimulate new research and serve as a personalized learning text for beginners in a field I PRIGOGINE STUART A RICE VB CONTENTS CRITICALITY MODE COUPLING LIQUID OF IONIC FLUIDS By Hermann Weingartner THEORY and Wolffram Schroer ApPROACH TO 67 STATE DYNAMICS By Biman Bagchi and Sarika Bhattacharyya ANOMALOUS STOCHASTIC DYNAMICS FRAMEWORK: DISPERSIVE TRANSPORT, PROCESSES IN THE FRACTIONAL FOKKER-PLANCK EQUATION, AND NON-ExPONENTIAL 223 RELAXA TION By Ralf Metzler and Joseph Klafter MOMENT FREE ENERGIES FOR POL YDISPERSE SYSTEMS 265 By Peter Sollich, Patrick B Warren, and Michael E Cates CHEMICAL PHYSICS OF THE ELECTRODE-ELECTROLYTE 337 INTERFACE By J WHalley, S Walbran and D L Price AUTHOR INDEX 389 SUBJECT INDEX 403 ix ADVANCES IN CHEMICAL PHYSICS VOLUME 116 CRITICALITY OF IONIC FLUIDS HERMANN WEINGARTNER Physikalische Chemie /I, Ruhr-Universitiit Bochum, Bochum, Germany WOLFFRAM SCHROER lnstitut fur Anorganische und Physikalische Chemie, Universitiit Bremen, Bremen, Germany CONTENTS Introduction II Background III Survey of Experimental Results A Liquid-Vapor Transitions in One-Component Ionic Fluids B Liquid-Liquid Demixings in Binary Electrolyte Solutions I Forces Driving the Phase Separation Coexistence Curves Scattering and Turbidity Viscosity Electrical Conductance Interfacial Properties The Ion Distribution Near Criticality C Liquid-Vapor Transitions in Aqueous Electrolyte Solutions D Liquid-Liquid Demixings in Multicomponent Systems E Summary IV Theoretical Methods at the Mean-Field Level A Models for Ionic Fluids B Monte Carlo Simulations C Analytical Theories of the Restricted Primitive Model l General Issues Pairing Theories D Lattice Theories E Beyond the Primitive Models Advances in Chemical Physics, Volume 116, edited by Prigogine and Stuart A Rice ISBN 0-471-40541-8 © 2001 John Wiley & Sons, Inc HERMANN WEINGARTNER AND WOLFFRAM SCHROER F Mean-Field Theories of Inhomogeneous Fluids and Fluctuations G Summary V Results from Mean-Field Theories A The Restricted Primitive Model I Critical Point and Coexistence Curve The Ion Distribution of the RPM Near Criticality B Hard Spheres of Different Size and Charge C Beyond Primitive Models D Inhomogeneous Fluids and Fluctuations I Charge and Density Fluctuations The Range of Validity of Mean-Field Theories Interfacial Properties E Summary VI Theories of Critical Behavior A Critical Phenomena and Range of Interactions B Lattice Models C Monte Carlo Simulations of Fluid Models D Analytical Theories of Fluids I The Restricted Primitive Model The Unrestricted Primitive Model The Role of r-4 Dependent Interactions E Crossover Theories and Tricriticality VII Conclusions References I INTRODUCTION This chapter deals with critical phenomena in simple ionic fluids Prototypical ionic fluids, in the sense considered here, are molten salts and electrolyte solutions Ionic states occur, however, in many other systems as well; we quote, for example, metallic fluids or solutions of complex particles such as charged macromolecules, colloids, or micelles Although for simple atomic and molecular fluids thermodynamic anomalies near critical points have been extensively studied for a century now [1], for a long time the work on ionic fluids remained scarce [2, 3] Reviewing the rudimentary information available in 1990, Pitzer [4] noted fundamental differences in critical behavior between ionic and nonionic fluids In preparing the present account, we have been impressed by how much the field has changed since Pitzer's review and a similar review published by us in 1995 [5] Therefore, a comprehensive account of the present status of the field seems timely Thus, we presume that the reader is aware of the fundamentals of critical phenomena, as described in many reviews [6-8] and monographs [9-12] Earlier reviews of one or another aspect of ionic criticality by Pitzer [4,13], Levelt Sengers and Given [14], Fisher [15,16], Stell [17,18], and ourselves [5] are notable 410 SUBJECT INDEX Glass transition, mode coupling theory (MCT): isomerization in viscous liquids, 183-184 liquid-state dynamics, 130-135 Global stability, moment free energy, 296-299 Gouy-Chapman theory, electrode-electrolyte interface: continuum models, 383-384 static structure, 343-362 time and length scales, 339-342 Green-Kubo relations: mode coupling theory (MCT), background principles, 71 mode coupling theory (MCT) in liquid-state dynamics, density/current mode derivation, 120-125 Green's function, mode coupling theory in liquid-state dynamics: friction calculations, 98-105 Gaussian approximation, 105 -106 memory kernel, structure factor derivation, 129-130 Grote-Hynes theory, isomerization in viscous liquids via MCT: background, 183-184 barrier crossing rate, 186-189 diffusion-based rate determination, 189-192 principles of, 184-185 Half-width at half-maximum (HWHM), superquadratic quantum number dependence, vibrational dephasing in molecular liquids, 170-173 Hard spheres, ionic fluid criticality, 34 mean-field theories, 42-43 Hartree-Fock methods, electrode-electrolyte interface: kinetic theory, 365-383 static structure, 349-362 time and length scales, 341-342 Helmholtz free energy, fixed pressure, moment free energy for, 327-331 HMSA schematic, mode coupling theory in liquid-state dynamics: friction calculations, 113-114 small solute anomalous diffusion, 157-161 Homopolymers, moment free energy, length polydispersity, 304-312 Hydrodynamics, mode coupling theory (MCT), 70-71 general relaxation equation (GRE) and, 94 liquid-state dynamics: intermediate length scale hydrodynamics, 78 - 81 size-dependent diffusion, 156-166 variables, 72-76 Stokes relation, friction/viscosity dependence, 139-140 Hypernetted chain approximation (HNCA), ionic fluid criticality: hard sphere models, 34 inhomogeneous fluids, 45-46 inhomogeneous fluids, mean-field theories, 47 restricted primitive model theories, 29-31 Immiscibilities, ionic fluid criticality: binary solutions, liquid-liquid demixings, 9-10 liquid-vapor transition, aqueous electrolyte solutions, 23 - 24 Importance sampling, electrode-electrolyte interface, kinetic theory, 376-383 Inhomogeneous fluids and fluctuations: ionic fluid criticality, mean-field theories: charge and density fluctuations, 44-46 interfacial properties, 46-47 restricted fluid models, 34-35 validity range, 46 mode coupling theory in liquid-state dynamics, diffusion/friction decoupling from viscosity, 145-148 Instability criteria, moment free energy properties, spinodal and critical point criteria, 289-290 Interaction ranges, ionic fluid criticality, 48 Interfacial properties, ionic fluid criticality: inhomogeneous fluids, mean-field theories, 46-47 liquid-liquid demixings, binary solutions, 21 Intermediate wavenumbers, mode coupling theory (MCT), extended hydrodynamics, 78-81 Ion density, ionic fluid criticality, mean-field theories, 27-28 Ionic fluid criticality: binary system properties, 26- 27 chemical properties, 2-3 critical behavior theories: fluid analysis, 51-53 r-4-dependent interactions, 52-53 restricted primitive model, 51-52 SUBJECT unrestricted primitive mode], 52 411 INDEX scattering and turbidity, 17-18 lattice models, 48-50 Monte Carlo simulations of fluid models, 50- 51 viscosity, 19 liquid-vapor transition: aqueous electrolyte solutions, 24 range of interactions, 48 tricritica]ity, crossover theories and, 53-55 one-component ionic fluids, 7-8 restricted primitive mode], mean-field future research issues, 55-56 liquid-liquid demixings: binary solutions, 8-22 coexistence curves, 10-17 electrical conductance, 19-21 phase separation driving forces, 8- 10 scattering and turbidity, 17-19 viscosity, 19 theories, 38-40 Isomerization dynamics, mode coupling theory (MCT) in liquid-state dynamics, viscosity effects, 181-192 barrier crossing rate, 186-189 diffusion and rate determination, ] 89-] 92 frequency-dependent friction, 185-]86 Grote-Hynes theory, 184-185 interfacial properties, 2] ion distribution, 21-22 mu]ticomponent systems, 25-26 ]iquid- vapor transition: Joint probability distribution, mode coupling theory in liquid-state dynamics, Gaussian approximation, 105-1 06 aqueous electrolyte solutions, 22-24 one-component ionic fluids, 6-8 mean-field theories: hard sphere size and charge differences, 42-43 inhomogeneous fluids and fluctuations, Kadanoff/Swift theory, mode coupling theory (MCT) in liquid-state dynamics, viscosity derivation, ]25 - ]26 Kinetic theory See also Strange kinetics electrode-electrolyte interface, models of, 34-35 charge and density fluctuations, 44-46 interfacial properties, 46-47 validity range, 46 lattice theories, 33 mode] ionic fluids, 27-28 molecular models, 33-34 Monte Carlo simulations, 28-29 restricted primitive mode], 29-33 362-383 fractional dynamics, K]ein-Kramers equation, multip]e trapping mode], 252 mode coupling theory (MCT) in liquid-state dynamics: genesis of, 81-82 power Jaw mass dependence, ]49-]55 time-dependent diffusion in twodimensional Lennard-Jones liquids, critical point and coexistence curve, 37-40 ion distribution near criticality, 40-42 solvent models, 43-44 physics of, 3-6 ]94-] 95 Kirkwood superposition approximation: diffusion and friction theory, mode coupling in liquid-state dynamics, 98-105 mode coupling theory (MCT) in liquid-state Ionic mobility, mode coupling theory (MCT) in liquid-state dynamics, concentrationdependence and viscosity in electrolyte solutions, 2] dynamics: binary derivation, ]08 binary viscosity derivation, 120 dielectric and orientationa] relaxation, Ising criticality: fluid models, restricted primitive mode] (RPM),51-52 211 friction/viscosity relations, time-dependent calcu]ations,137-138 ionic fluids, 4-5 fluid models, 50-5] lattice gas models, 50 liquid-liquid demixings, binary solutions, ]2-] Klein-Kramers equation: Brownian motion and, 225-228 fractional dynamics: multip]e trapping model, 250-252 related equations, 252-254 412 SUBJECT INDEX Klein-Kramers equation: (Continued) general relaxation equation (GRE) and, fractional Fokker-Planck equation (FFPE), 239 Kleinman-Bylander sesparable form, electrodeelectrolyte interface, static structure, 91-94 isomerization in viscous liquids, 181-184 Laplace frequency: fractional dynamics, waiting time equations, 355-362 Kolmog equation, classical Fokker-Planck orov equation, 237 231-234 fractional Fokker-Planck equation (FFPE), nonexponential mode relaxation, Kosterlitz-Thouless (KT) theory, ionic fluid criticality: 241-242 mode coupling theory (MCT) in liquid-state crossover theories and tricriticality, 55 lattice models, 49-50 Kramers' theory: Brownian motion and, 226-228 electrode-electrolyte 381-383 interface, kinetic theory, fractional dynamics: classical formulation, 244-247 protein dynamics model, 249-250 survival probability, Mittag-Leffler decay, 247-249 isomerization in viscous liquids via MCT, 181-184 Grote-Hynes theory, 185 Kubo formula, mode coupling theory (MCT), transport processes, 86-89 Kubo-Oxtoby theory, superquadratic quantum number dependence, vibrational dephasing in molecular liquids, 168-169 vibrational line shape theory, 169-173 Lagrange multipliers, moment free energy: application of, 302-304 critical point condition, determinant form, 333 homopolymers, length polydispersity, 308-312 phase coexistence beyond onset, 296 Landau-Ginzburg theory, ionic fluid criticality: crossover theories and tricriticality, 54-55 inhomogeneous fluids, 35-36 charge and density fluctuations, 45-46 Landau-Placzek formula, mode coupling theory (MCT), liquid-state dynamic variables, 74- 76 Langevin equation: fractional dynamics: Klein-Kramers equation, 250-252 Kramers escape problem, 246-247 mode coupling theory (MCT): dynamics: extended hydrodynamics, 79-81 friction calculations, 98-105 isomerization in viscous liquids via MCT, 185-186 memory kernel, structure factor derivation, 128-130 self-consistent calculations, 115 variables in, 73-76 Laplace transform: fractional Fokker-Planck equation (FFPE), Mittag-Leffler pattern, 243 Kramers escape problem, fractional dynamics, Mittag-Leffler decay, 248-249 Levy distributions and, 258 Mittag-Leffler function, 259 mode coupling theory (MCT) in liquid-state dynamics: friction calculations, Jl3- Jl4 glass transition theory, 132- 135 time-dependent diffusion in twodimensional Lennard-Jones liquids, 196-198 Large deviation theory (LDT), moment free energy, 331-332 Large solutes, mode coupling theory in liquidstate dynamics, friction/diffusion expressions, 161-164 binary contribution, 162 density mode contribution, 163 transverse mode contribution, 163-164 Lattice gas models: ionic fluid criticality: critical behavior theories, 48-50 restricted primitive model, 33 ionic fluid criticality, liquid-vapor transition, one-component ionic fluids, 7-8 Law of the rectilinear diameter, ionic fluid criticality, liquid-vapor transition, onecomponent ionic fluids, SUBJECT Legendre transform, moment free energy: combinatorial derivation, 280-283 comparison of derivation techniques, 283-285 copolymer chemical polydispersity, 321 fixed pressure, 328-331 projection/combinatorial comparisons, 283-285 properties of, 286-288 Length polydispersity, moment free energy, homopolymers, 304-312 Length scale, electrode-electrolyte interface, 339-342 Lennard-Jones potential: mode coupling theory (MCT) in liquid-state dynamics: friction calculations, 112-114 one-dimensional rods, diffusion in, 203 - 210 long-time VACF behavior, 207-208 VACF analysis, 204-206 power law mass dependence, 151-153 size-dependent diffusion, 156-166 supercooled liquids, friction/viscosity relations, 143 time-dependent diffusion, two-dimensional liquids, 192-203 kinetic theory, 194-195 mean square displacement principles, 198-199 system size dependence, 201-202 time-dependent calculations, 200- 20 I VACF calculations, 199- 200 VACF principles, 195-198 superquadratic quantum number dependence, vibrational dephasing in molecular liquids, time-dependent friction, 176-177 Lever rule, phase separation and, 266, 268-271 Levy distributions, fractional dynamics: basic principles, 255-258 long-tailed waiting time equations, 229-234 research background, 227-228 Levy-Gnedenko central limit theorem, fractional dynamics, long-tailed waiting time equations, 229-234 Liebniz theory, fractional dynamics, waiting time equations, 233-234 Liouville operator, mode coupling theory (MCT): INDEX 413 conservation equations, 82-83 general relaxation equation (GRE) and, 91-94 liquid-state dynamics, density/current mode derivation, 121-125 time-dependent diffusion in two-dimensional Lennard-Jones liquids, 194-198 Liquid-glass transition, mode coupling theory (MCT): liquid-state dynamics, 95-96 structure factor derivation, 127-128 Liquid-liquid demixings, ionic fluid criticality: binary solutions, 8-22 coexistence curves, 10-17 electrical conductance, 19-21 phase separation driving forces, 8-10 scattering and turbidity, 17-19 viscosity, 19 interfacial properties, 21 ion distribution, 21-22 multicomponent systems, 25-26 Liquid-state dynamics, mode coupling theory (MCT): basic principles, 69- 71 conservation equations, 82-83 de Gennes narrowing, 76-78 dielectric/orientational relaxation, dipolar liquids, 211-212 diffusion and friction theory, 97 -115 binary term derivation, 106-108 frequency-dependent friction derivation, 110-112 friction calculation techniques, 112-114 Gaussian approximation, 105-106 ring collision terms, 108-110 self-consistent field calculations, 114-115 diffusion size dependence, 155-166 density mode contribution, 163 friction/diffusion expressions, 164 hydrodynamic crossover, solute-solvent interaction, 164-166 large solute expressions, 161-164 small solute anomalous diffusion, 157-161 transverse mode contribution, 163-164 dynamical critical phenomena, 81-89 dynamic structure factor, 126-130 memory kernel derivation, 128-130 eigenfunction construction, 83-84 friction/viscosity relations, 135-140 414 SUBJECT Liquid-state dynamics, mode coupling theory (MCT): (Continued) frequency dependence YS Maxwell relation, 138-139 Stokes relation, 139-140 time dependence, 137-138 generalized relaxation equation (GRE), 91-94 glass transition theory, 130-135 hydrodynamic techniques, 72-76 intermediate length scale, 78-81 ionic mobility, electrolyte solutions, 212 mixtures, 212-213 one-dimensional Lennard-Jones rods, diffusion in, 203-210 long-time VACF behavior, 207-208 VACF analysis, 204-206 polyelectrolyte solutions, 213 power law mass dependent-diffusion, 149-155 renormalized kinetics, 89-91 short-terrn!1ong-term dynamics, 210-211 supercooled liquids, friction/viscosity relations, 140-149 dynamic structure factor, 142-145 inhomogeneity and decoupling effect, 145-149 supercriticalliquid diffusion, 214 superquadratic quantum number dependence, vibrational dephasing, 166-181 attraction/repulsion forces, 173-174 dephasing rates, 178-181 vibrational line shape theory, 169-173 vibration-rotation coupling, 174-178 transport processes, 84-89 two-dimensional Lennard-Jones fluids, timedependent diffusion, 192-203 kinetic theory, 194-195 mean square displacement principles, 198-199 system size dependence, 201- 202 time-dependent calculations, 200-201 VACF calculations, 199-200 VACF principles, 195-198 variable structures, 94-96 viscoelastic models, 96-97 viscosity theory, 115-126 binary viscosity derivation, 117-120 density/current contribution derivation, 120-125 INDEX Kadanoff/Swift YS Geszti, 125-126 viscous liquid isomerization dynamics, 181-192 barrier crossing rate, 186-189 diffusion and rate determination, 189-192 frequency-dependent friction, 185-186 Grote-Hynes theory, 184-185 wavevector-dependent transport, 214 Liquid-vapor transition, ionic fluid criticality: aqueous electrolyte solutions, 22-24 one-component ionic fluids, 6-8 Local density approximation, ionic fluid criticality, inhomogeneous fluids, 36 Local stability, moment free energy, 296-299 Log errors, moment free energy, copolymer chemical polydispersity, 317-321 Long-time dynamics: fractional dynamics, waiting time equations, 229-234 mode coupling theory (MCT) in liquid-state dynamics: one-dimensional Lennard-Jones rods, diffusion in, 208-210 short-term dynamics YS., 210-212 Long-wavelength fluctuations, mode coupling theory (MCT) in liquid-state dynamics, time-dependent diffusion in twodimensional Lennard-Jones liquids, 202-203 Lorentzian line shape: Levy distributions, 257-258 superquadratic quantum number dependence, vibrational dephasing in molecular liquids, 170-173 Low critical solution temperature (LCST), ionic fluid criticality, binary solutions, liquidliquid demixings, 9-10 Lyotropic series, ionic fluid criticality, liquidliquid demixings, binary solutions, 10 Maclaurin series expansion, superquadratic quantum number dependence, vibrational dephasing in molecular liquids, 171-173 Marcus theory, electrode-electrolyte interface, kinetic theory, 375-383 Markovian master equation, fractional dynamics, 228-229 boundary values, diffusion equation, 234-236 Klein-Kramers equation and, 254 SUBJECT long-tailed waiting times, 229-234 Mass action law, ionic fluid criticality, pairing theories, 31-33 Matrix elements, electrode-electrolyte interface, kinetic theory, 379-383 Maxwell relation, mode coupling theory (MCT) in liquid-state dynamics: friction calculations, 102-105 friction/viscosity relations, 136-140 time-dependent diffusion in two-dimensional Lennard-Jones liquids, 194-195 Mayer functions, ionic fluid criticality: pairing theories, 31-33 restricted primitive model, 31 Mean-field theories, ionic fluid criticality: hard sphere size and charge differences, 42-43 inhomogeneous fluids and fluctuations, 34- 35 charge and density fluctuations, 44-46 interfacial properties, 46-47 validity range, 46 lattice theories, 33 model ionic fluids, 27-28 molecular models, 33-34 Monte Carlo simulations, 28-29 restricted primitive model, 29-33 critical point and coexistence curve, 37-40 ion distribution near criticality, 40-42 solvent models, 43-44 Mean spherical approximation (MSA) See also "Generalized MSA" (GMSA); "Pairing MSA" ionic fluid criticality: hard sphere models, 34 hard sphere parameters, mean-field theory, 43 Monte Carlo simulations, 47-48 pairing theories, 32-33 restricted primitive model theories, 29-31 mean-field theory, 39-40 solvent models, 43-44 Mean square displacement (MSD): fractional Fokker-Planck equation (FFPE), Ornstein-Uhlenbeck process, 244, 246 mode coupling theory (MCT) in liquid-state dynamics: one-dimensional Lennard-Jones rods, diffusion in, 203-210 power law mass dependence, 152-153 self-consistent calculations, 114-115 INDEX 415 time-dependent diffusion in twodimensional Lennard-Jones liquids, 198-203 Memory, mode coupling theory (MCT) in liquid-state dynamics, friction calculations, 99-105 Memory function equation: diffusion and friction theory, velocity autocorrelation function (VACF), 97-105 mode coupling theory (MCT): general relaxation equation (GRE) and, 93-94 liquid-state dynamics, glass transition theory, 132-135 Memory kernel, mode coupling theory (MCT): general relaxation equation (GRE) and, 94 liquid-state dynamics, structure factor derivation, 128-130 Metal-aqueous systems, electrode-electrolyte interface, static structure, 342-362 Metallic fluids, criticality, liquid-vapor transition, one-component ionic fluids, 7-8 Microscopic analysis, mode coupling theory (MCT), renormalized kinetics and, 89-91 Mittag-Leffler pattern: basic principles of, 258-259 fractional dynamics, 227-228 boundary value problems, diffusion equation, 234-236 theoretical background, 227-228 waiting time equations, 234 fractional Fokker-Planck equation (FFPE): basic properties, 239 nonexponential mode relaxation, 240-242 Ornstein-Uhlenbeck process, 244-245 relaxation modeling, 242-243 Kramers escape problem: fractional dynamics, 246-249 protein dynamics model, 249-250 Mixed liquids, mode coupling theory (MCT) in liquid-state dynamics, 212-213 Mode coupling theory (MCT): ionic fluid criticality, liquid-liquid demixings, binary solutions, viscosity, 19 liquid-state dynamics: basic principles, 69-71 conservation equations, 82-83 416 SUBJECT Mode coupling theory (MCT): (Continued) de Gennes narrowing, 76-78 dielectric/orientational relaxation, dipolar liquids, 211-212 diffusion and friction theory, 97-115 binary term derivation, 106-108 frequency-dependent friction derivation, II 0-112 friction calculation techniques, 112-114 Gaussian approximation, 105-106 ring collision terms, 108-11 self-consistent field calculations, 114-115 diffusion size dependence, 155-166 density mode contribution, 163 friction/diffusion expressions, 164 hydrodynamic crossover, solute-solvent interaction, 164-166 INDEX inhomogeneity and decoupling effect, 145-149 supercriticalliquid diffusion, 214 superquadratic quantum number dependence, vibrational dephasing, 166-181 attraction/repulsion forces, 173-174 dephasing rates, 178-181 vibrational line shape theory, 169-173 vibration-rotation coupling, 174-178 transport processes, 84-89 two-dimensional Lennard-Jones fluids, time-dependent diffusion, 192-203 kinetic theory, 194-195 mean square displacement principles, 198-199 system size dependence, 201- 202 time-dependent calculations, 200-20 I large solute expressions, 161-164 small solute anomalous diffusion, 157-161 transverse mode contribution, 163-164 dynamical critical phenomena, 81-89 dynamic structure factor, 126-130 memory kernel derivation, 128-130 VACF calculations, 199-200 VACF principles, 195-198 variable structures, 94-96 viscoelastic models, 96-97 viscosity theory, 115-126 binary viscosity derivation, 117-120 density/current contribution derivation, eigenfunction construction, 83-84 friction/viscosity relations, 135-140 frequency dependence vs Maxwell 120-125 Kadanoff/Swift vs Geszti, 125-126 viscous liquid isomerization dynamics, relation, 138-139 Stokes relation, 139-140 time dependence, 137-138 general relaxation equation (GRE), 91-94 glass transition theory, 130-135 hydrodynamic techniques, 72-76 intermediate length scale hydrodynamics, 78-81 181-192 barrier crossing rate, 186-189 diffusion and rate determination, 189-192 frequency-dependent friction, 185-186 Grote-Hynes theory, 184-185 wavevector-dependent transport, 214 polymer dynamics, 213-214 ionic mobility, electrolyte solutions, 212 mixtures, 212-213 one-dimensional Lennard-Jones rods, diffusion in, 203-210 long-time VACF behavior, 207-208 VACF analysis, 204-206 Molecular dynamics (MD) simulations: electrode-electrolyte interface: kinetic theory, 369-383 time and length scales, 342 ionic fluid criticality: liquid-vapor transition, one-component polyelectrolyte solutions, 213 power law mass dependent-diffusion, 149-155 ionic fluids, 6-8 mean-field theories, 28-29 Molecular liquids, superquadratic renormalized kinetics, 89-91 short-terrnllong-term dynamics, 210-211 supercooled liquids, friction/viscosity relations, 140-149 dynamic structure factor, 142-145 quantum number dependence, vibrational dephasing, 166-181 attraction/repulsion forces, 173-174 dephasing rates, 178-181 vibrational line shape theory, 169-173 SUBJECT vibration-rotation coupling, 174-178 Molten salt, ionic fluid criticality: liquid-liquid demixings, binary solutions, 14-17 liquid-vapor transition, one-component ionic fluids, -8 mean-field theories, 27-28 Moment free energy: application of, 302-304 derivation: combinatorial method, 276-283 comparision of techniques, 283-285 projection method, 271-276 polydisperse systems: copolymers, 312-324 with solvent, 321-324 without solvent, 314-321 critical point criterion, determinant form, 332-333 density distribution, 324-327 fixed pressure, 327-331 homopolymers,304-312 mixing entropy, large deviation theory, 331-332 spinodal criterion, exact free energy, 332 properties of, 285-301 cloud point and shadow, phase coexistence, 292-293 criteria for spinodals and multi -critical points, 288-290 critical points, 291- 292 density distribution space geometry, 299301 global and local stability, 296-299 phase coexistence beyond onset, 293-296 spinodals,290-291 strengths and weaknesses, 324-327 theoretical principles, 267-271 Momentum density, mode coupling theory (MCT), liquid-state dynamic variables, 72- 76 Monte Carlo simulations: electrode-electrolyte interface, static structure, 343-362 ionic fluid criticality: fluid models, 50-51 mean-field theories, 28-29 principles of, 5-6 restricted primitive models, 47-48 moment free energy, application of, 303-304 INDEX 417 Mori continued fraction, mode coupling theory (MCT): liquid-state dynamics: density/current mode derivation, 120-125 friction calculations, 113-114 viscoelastic model (VEM), 97 -98 Morse potential, superquadratic quantum number dependence, vibrational dephasing in molecular liquids, 171-173 Multicomponent systems, liquid-liquid demixings, ionic fluid criticality, 25-26 Multi-critical points, moment free energy properties, 288-290 Multiple trapping model, fractional dynamics, Klein-Kramers equation, 250-252 Navier-Stokes hydrodynamics, mode coupling theory (MCT): liquid-state dynamic variables, 76 power law mass dependence, 150-155 Near-metal region, electrode-electrolyte interface, static structure, 347-362 Newton-Raphson algorithm, moment free energy, application of, 302-304 Nonexponential mode relaxation, fractional Fokker-Planck equation (FFPE), 240-242 Non-Markovian Langevin equation (NMLE), isomerization in viscous liquids via MCT, diffusion-based determination, 191-192 Non-Markovian rate theory (NMRT), isomerization in viscous liquids via MCT, 182-184 Grote-Hynes theory and, 184-185 Nonquadratic quantum number dependence, vibrational dephasing in molecular liquids, molecular liquids, 167-169 Number density, mode coupling theory (MCT): de Gennes narrowing, 77 - 78 liquid-state dynamic variables, 72-76 One-component ionic fluids, criticality, liquidvapor transition, 6-8 One-dimensional Lennard-Jones rods, mode coupling theory (MCT) in liquid-state dynamics: diffusion behavior, 203-210 long-time VACF behavior, 207-208 VACF analysis, 204-206 418 SUBJECT One-electron equations, electrode-electrolyte interface, static structure, 347-362 Orientational relaxation, mode coupling theory (MCT) in liquid-state dynamics, dipolar liquids, 210- 211 Ornstein-Uhlenbeck process: fractional dynamics, Klein-Kramers equation, 253-254 fractional Fokker-Planck equation (FFPE), 243-245 Omstein-Zemike (OZ) equation: ionic fluid criticality, restricted primitive model theories, 29-31 mode coupling theory (MCT) in liquid-state dynamics: friction calculations, 113-114 one-dimensional Lennard-Jones rods, diffusion in, 206-210 time-dependent diffusion in twodimensional Lennard-Jones liquids, 197-198 Overall density distribution, moment free energy, global and local stability, 296-299 Oxygen density, electrode-electrolyte interface, kinetic theory, 367 -383 "Pairing MSA," ionic fluid criticality, 32-33 Pairing theories, ionic fluid criticality, restricted primitive model, 31-33 Parent density distribution: moment density, 268-271 moment free energy: combinatorial derivation, 278-283 copolymer chemical polydispersity, 314-324 homopolymers, length polydispersity, 305-312 phase coexistence, cloud point and shadow, 293 Percus-Yevick (PY) closure: ionic fluid criticality, restricted primitive model theories, 29-31 mode coupling theory in liquid-state dynamics, small solute anomalous diffusion, 157-161 Perturbation theory, superquadratic quantum number dependence, vibrational dephasing in molecular liquids, 172-173 INDEX Phase coexistence, moment free energy properties: beyond onset, 293-296 cloud point and shadow, 292-293 Phase equilibria, free energy surface, 266- 267 Phase separation driving forces: lever rule, 266 liquid-liquid demixings, binary solutions, ionic fluid criticality, 8-10 scattering and turbidity, 18-19 Phase space correlation function: Klein- Kramers equation, fractional dynamics: multiple trapping model, 250-252 related equations, 252-254 mode coupling theory (MCT) in liquid-state dynamics, friction calculations, 103-105 Plane-wave calculations, electrode-electrolyte interface, static structure, 347-362 Point charges, ionic fluid criticality, lattice gas models, 33 Point of zero charge (pzq, electrode-electrolyte interface: static structure, 343-362 time and length scales, 339-342 Poisson-Boltzmann (PB) equation: electrode-electrolyte interface, static structure, 352-362 ionic fluid criticality, restricted primitive model, 30- 31 Polydisperse systems: moment free energy: applications, 302-304 combinatorial derivation, 276-283 copolymers, 312-324 with solvent, 321-324 without solvent, 314-321 critical point criterion, determinant form, 332-333 density distribution, 324-327 derivation method comparisons, 283-285 fixed pressure, 327-331 homopolymers, 304-312 mixing entropy, large deviation theory, 331-332 projection derivation, 271-276 properties of, 285-301 cloud point and shadow, phase coexistence, 292-293 criteria for spinodals and multi-critical points, 288-290 SUBJECT critical points, 291- 292 density distribution space geometry, 299-301 global and local stability, 296-299 phase coexistence beyond onset, 293-296 spinodals,290-291 spinodal criterion, exact free energy, 332 thermodynamics of, 266-271 Polyelectrolyte solutions, mode coupling theory (MCT) in liquid-state dynamics, 213 Polymer dynamics, mode-coupling theory of, 212-213 Power law mass dependent-diffusion: fractional dynamics, long-tailed waiting time equations, 229-234 mode coupling theory (MCT), liquid-state dynamics, 149-155 calculation, 153-155 theoretical principles, 151-153 polydisperse systems, 267-271 Prior distribution, moment free energy, projection derivation, 273-276 Probability density function (pdf): Brownian motion, 224-228 classical Fokker-Planck equation, 237 fractional dynamics: boundary value problems, diffusion equation, 234-236 Klein-Kramers equation, 250-252 long-tailed waiting time equations, 229-234 fractional Fokker-Planck equation, OrnsteinUhlenbeck process, 244-245 Levy distributions, principles of, 257-258 Probability distribution, mode coupling theory in liquid-state dynamics, Gaussian approximation, 105-106 Projection method, moment free energy: defined, 270 derivation techniques, 271-276 Projection operator technique, mode coupling theory (MCT), 71 general relaxation equation (GRE) and, 91-94 transport processes, 87-89 Protein dynamics model, Kramers escape problem, 249-250 Pseudopotentials: INDEX 419 electrode-electrolyte interface, static structure, 348-362 moment free energy, strengths and weaknesses, 326-327 Quantum mechanics, electrode-electrolyte interface, static structure, 348-362 Quasi-species densities, moment free energy: combinatorial derivation, 278-283 projection derivation, 273-276 properties of, 286-288 Raman spectrum: mode coupling theory (MCT), liquid-state dynamic variables, 74-76 superquadratic quantum number dependence, vibrational dephasing in molecular liquids, 166-169 vibrational line shape theory, 169-173 Random walk theory See also Continuous-time random walk theory Brownian motion and, 226-228 Rayleigh-Benard convection, Brownian motion and, 226-228 Rayleigh-Brillouin spectrum, mode coupling theory (MCT), liquid-state dynamic variables, 74-76 Rayleigh equation: Brownian motion analysis, 225-228 classical Fokker-Planck equation, 237 fractional dynamics, Klein-Kramers equation, 253-254 fractional Fokker-Planck equation (FFPE), Mittag-Leffler pattern, 243 R -4-dependent interactions, fluid models, critical behavior theories, 52-53 Rectilinear behavior, ionic fluid criticality, liquid-liquid demixings, binary solutions, 16-17 Redissociation, ionic fluid criticality, liquidliquid demixings, binary solutions, 22 Reentrant phase transitions, liquid-liquid demixings, multi component systems, ionic fluid criticality, 25-26 Refined potentials, ionic fluid criticality, meanfield theories, 28 Relaxation equations: fractional Fokker-Planck equation (FFPE), Mittag-Leffler pattern, 242-243 mode coupling theory (MCT), 71 420 SUBJECT Relaxation equations: (Continued) liquid-state dynamics, structure factor derivation, 127-130 Renormalization group (RG) analysis: fluid models, restricted primitive model (RPM), 51-52 ionic fluid criticality, 3-4 liquid-liquid demixings, binary solutions, viscosity, 19 INDEX Scattering behavior: ionic fluid criticality, liquid-liquid demixings, binary solutions, 17-19 mode coupling theory (MCT) in liquid-state dynamics: diffusion/friction decoupling from viscosity, 147-148 power law mass dependence, 152-153 Schrodinger equation, electrode-electrolyte Renormalized kinetic theory, mode coupling theory (MCT): liquid-state dynamics, 89-91 principles of, 70-71 Repeated ring theory, mode coupling theory (MCT), renormalized kinetics and, 90-91 interface: static structure, 348-362 time and length scales, 341-342 Schulz distribution, moment free energy, homopolymers, length polydispersity, 305-312 Second-order moment densities, moment free Repulsive forces, superquadratic quantum number dependence, vibrational dephasing in molecular liquids, 173-174 energy, properties of, 286-288 Self-consistent calculations, mode coupling theory (MCT), liquid-state dynamics, Restricted primitive model (RPM): fluid analysis, 51-52 ionic fluid criticality: mean-field theories, 27-28, 29-33 critical point and coexistence curve, 37 40 94-96 friction/diffusion states, 114-115 glass transition theory, 130-135 Shadow curves: moment free energy: homopolymers, length polydispersity, inhomogeneous fluids and fluctuations, 34-36 ion distribution near criticality, 40-42 pairing theories, 31-33 principles of, 5-6 Riemann-Liouville fractional operator, fractional dynamics, waiting time equations, 231-234 Ring collision term, mode coupling theory (MCT) in liquid-state dynamics: 307-312 phase coexistence, 292-293 polydisperse systems, 267-271 Short-term dynamics, mode coupling theory (MCT) in liquid-state dynamics, longtime dynamics VS., 210-211k Sine-Gordon theory, ionic fluid criticality, lattice gas models, 33, 50 Size-dependent diffusion, mode coupling theory (MCT) in liquid-state dynamics, derivation techniques, 108-110 friction calculations, 104-105 memory kernel, structure factor derivation, 130 time-dependent diffusion, 192-194 Scaling laws: electrode-electrolyte interface, time and length scales, 339-342 fractional Fokker-Planck equation (FFPE), nonexponential mode relaxation, 241242 ionic fluid criticality, 3-4 mode coupling theory (MCT), transport processes, 89 155-166 density mode contribution, 163 friction/diffusion expressions, 164 hydrodynamic crossover, solute-solvent interaction, 164-166 large solute expressions, 161-164 small solute anomalous diffusion, 157-161 transverse mode contribution, 163-164 Slow diffusion, fractional dynamics and, 227228 Slow kinetics: fractional dynamics, long-tailed waiting time equations, 229-234 mode coupling theory (MCT), liquid-state dynamics, large wavenumbers, 76-78 SUBJECT Small solutes, mode coupling theory (MCT) in liquid-state dynamics, anomalous diffusion, 157-161 Smoluchowski equation: Brownian motion analysis, 225-228 classical Fokker-Planck equation, 237 Smoluchowski limit of Kramers' theory, isomerization in viscous liquids via MCT, 182-184 Smoluchowski- Vlasov equation, mode coupling theory (MCT), de Gennes narrowing, 76-78 Solute-solvent interactions, mode coupling theory (MCT) in liquid-state dynamics: friction calculations, 100-105 Gaussian approximation, 106 power law mass dependence, 151-153 Solvation barriers, electrode-electrolyte interface, kinetic theory, 378-383 Solvent models: ionic fluid criticality, mean-field theory, 43-44 moment free energy, copolymer chemical polydispersity, 314-324 Solvent-solvent interactions, mode coupling theory (MCT) in liquid-state dynamics, crossover behavior, 164-166 Solvophobic demixing, ionic fluid criticality: binary solutions, 10 liquid-liquid demixings, multicomponent systems, 25-26 Spinodal criterion: moment free energy: copolymer chemical polydispersity, 321-324 exact free energy, 332 general criteria, 288-290 homopolymers, length polydispersity, 307 - 312 truncatable systems, 290- 291 polydisperse systems, 269 Static correlations, mode coupling theory (MCT), liquid-state dynamics, 95-96 Static structure, electrode-electrolyte interface, 342-362 Stem layers, electrode-electrolyte interface, static structure, 344- 362, 351-362 Stochastic analysis, fractional dynamics, theoretical background, 228 Stokes-Einstein equation: INDEX 421 ionic fluid criticality, liquid-liquid demixings, binary solutions, electrical conductance, 20-21 mode coupling theory (MCT) in liquid-state dynamics: large solute friction/diffusion expression, 164 power law mass dependence, 149-155 size-dependent diffusion, 155-157 supercooled liquids, friction/viscosity relations, 141-149 variables, 75-76 Stokes relation, mode coupling theory (MCT) in liquid-state dynamics: dynamic variables, 74-76 friction/viscosity relations, 135-140 renormalized kinetics and, 90-91 Strange kinetics, Brownian motion and, 226-228 Stretched Gaussian form, fractional dynamics, waiting time equations, 232-234 Structural relaxation, mode coupling theory (MCT) in liquid-state dynamics, Maxwell relation, friction/viscosity dependence, 139-140 Structure factor, mode coupling theory (MCT) in liquid-state dynamics: de Gennes narrowing, variables in, 77 - 78 density/current mode derivation, 123-125 derivation, 126-130 memory kernel derivation, 128-130 friction calculations, 104-105 supercooled liquids, friction/viscosity relations, 142-145 Subdiffusion, fractional dynamics and, 227 -228 Subspaces, moment free energy, projection derivation, 272-276 Supercooled liquids, mode coupling theory (MCT) in liquid-state dynamics, friction/ viscosity relations, 140-149 dynamic structure factor, 142-145 inhomogeneity and decoupling effect, 145-149 Supercritical fluid models, mode coupling in liquid-state dynamics, diffusion in, 213 Superquadratic quantum number dependence, vibrational dephasing, mode coupling theory (MCT) in liquid-state dynamics, 166-181 attraction/repulsion forces, 173-174 422 SUBJECT Superquadratic quantum number dependence, vibrational dephasing, mode coupling theory (MCT) in liquid-state dynamics (Continued) dephasing rates, 178-181 vibrational line shape theory, 169-173 vibration-rotation coupling, 174-178 Surface molecule arrangements, electrodeelectrolyte interface, static structure, 356-362 Survival probability, Kramers escape problem, fractional dynamics, Mittag-Leffler decay, 247-249 Tangent plane distance calculations, moment free energy: application of, 302-304 global and local stability, 296-299 Taylor series, superquadratic quantum number dependence, vibrational dephasing in molecular liquids, vibration-rotation coupling, 174-175 Temperature fluctuations, mode coupling theory (MCT), liquid-state dynamic variables, 73- 76 Thermal conductivity, mode coupling theory (MCT), transport processes, 88-89 "Through metal interaction," electrodeelectrolyte interface, static structure, 350-362 Through-metal repulsion, electrode-electrolyte interface, static structure, 349-362 Time-dependent diffusion, mode coupling theory (MCT) in liquid-state dynamics, two-dimensional Lennard-Jones fluids, 192-203 kinetic theory, 194-195 mean square displacement principles, 198-199 system size dependence, 201-202 time-dependent calculations, 200-201 VACF calculations, 199-200 VACF principles, 195-198 Time-dependent friction, superquadratic quantum number dependence, vibrational dephasing in molecular liquids, 176-177 Time-dependent shear viscosity, mode coupling theory (MCT) in liquid-state dynamics: derivation, 116-117 INDEX friction/viscosity relations, 137-138 Time scales: electrode-electrolyte interface, 339-342 fractional dynamic theory and, 226-228 mode coupling theory (MCT), 71 Toukan-Rahman model, electrode-electrolyte interface, kinetic theory, 364-383 Transfer coefficient, electrode-electrolyte interface, kinetic theory, 380-383 Transition state theory (TST): isomerization in viscous liquids via MCT, Grote-Hynes theory, 185 isomerization in viscous liquids via modecoupling theory, 181-184 Transport coefficients, mode coupling theory (MCT): conservation equations, 83 general relaxation equation (GRE) and, 94 genesis of, 81-82 processes and modes, 84-89 wavevector-dependent transport, 213 Transverse degrees of freedom, moment free energy, projection derivation, 272-276 Transverse mode contribution, mode coupling theory (MCT) in liquid-state dynamics, large solute friction/diffusion expression, 163-164 Trapezoidal function, moment free energy, copolymer chemical polydispersity, 318-321 Triangle weight function, moment free energy, copolymer chemical polydispersity, 318-321 Triticritical behavior, ionic fluid criticality: crossover theories, 53-55 lattice models, 49-50 TroullierlMartins form, electrode-electrolyte interface, static structure, 355-362 Truncatable systems, moment free energy: critical point criteria, 291-292 fixed pressure, 327-331 multi-critical point criteria, 288-290 projection derivation, 271-276 spinodal criteria, 288 - 291 strengths and weaknesses, 326-327 Turbidity, ionic fluid criticality, liquid-liquid demixings, binary solutions, 17-19 Turing instability, fractional dynamics, boundary value problems, diffusion equation, 236 Two-dimensional structures: SUBJECT electrode-electrolyte interface, 357-362 mode coupling theory (MCT) in liquid-state dynamics: kinetic theory, 194-195 mean square displacement principles, 198-199 system size dependence, 201-202 time-dependent calculations, 200-201 time-dependent diffusion, 192-203 VACF calculations, 199-200 VACF principles, 195-198 Unrestricted primitive model (UPM): fluid analysis, 52 ionic fluid criticality, hard sphere parameters, 42-43 Upper critical end point (UCEP), ionic fluid criticality, liquid-vapor transition, aqueous electrolyte solutions, 24 Upper critical solution temperature (UCST), ionic fluid criticality: binary solutions, liquid-liquid demixings, 9-10 liquid-liquid demixings, binary solutions, 12-17 van der Waals equation: ionic fluid criticality, -4 inhomogeneous fluids, 35-36 restricted primitive model, mean-field theories, 38-40 polydisperse systems, 268-271 Velocity, mode coupling theory (MCT), liquidstate dynamic variables, 72-76 Velocity autocorrelation function (VACF), mode coupling theory (MCT) in liquid-state dynamics: diffusion and friction, 97-105 frequency-dependent friction expression, 110-112 one-dimensional Lennard-Jones rods, diffusion in, 203 - 210 long-time VACF behavior, 207-208 VACF analysis, 204-206 renormalized kinetics and, 90-91 self-consistent calculations, 114-115 time-dependent diffusion in two-dimensional Lennard-Jones liquids, 193-194, 195-198 kinetic theory, 194-195 INDEX 423 mean-square displacement, 198-199 results, 199-201 system size dependence, 201-203 Verlet algorithm, electrode-electrolyte interface, static structure, 348- 362 Vibrational dephasing, superquadratic quantum number dependence, mode coupling theory (MCT) in liquid-state dynamics, 166-181 attraction/repulsion forces, 173-174 dephasing rates, 178-181 vibrational line shape theory, 169-173 vibration-rotation coupling, 174-178 Vibrational line shape theory, superquadratic quantum number dependence in molecular liquids, 169-173 Vibration-rotation coupling, superquadratic quantum number dependence, vibrational dephasing in molecular liquids, 174-175, 177-178 Viscoelastic model (VEM): mode coupling theory in liquid-state dynamics, friction calculations, 113-114 mode coupling theory (MCT), liquid-state dynamics, 97-98 Viscosity: ionic fluid criticality, liquid-liquid demixings, binary solutions, 19 mode coupling theory (MCT) in liquid-state dynamics: concentration-dependence and ionic mobility in electrolyte solutions, 212 diffusion/friction decoupling from viscosity, 145-148 friction relations with, 135-140 frequency dependence vs Maxwell relation, 138-139 Stokes relation, 139-140 time dependence, 137-138 glass transition theory, 132-135 isomerization dynamics, 181-192 barrier crossing rate, 186-189 diffusion and rate determination, 189-192 frequency-dependent friction, 185-186 Grote-Hynes theory, 184-185 power law mass dependence, 155 supercooled liquids, friction/viscosity relations, 140-149 dynamic structure factor, 144-145 424 SUBJECT Viscosity: (Continued) mode coupling theory (MCT) transport processes, 87-89 Viscosity feedback, mode coupling theory (MCT), liquid-state dynamics, 95-96 Viscosity theory, mode coupling theory (MCT) in liquid-state dynamics, 115-126 binary viscosity derivation, 117-120 density/current contribution derivation, 120-125 KadanofflSwift vs Geszti, 125-126 Walden product, ionic fluid criticality, liquidliquid demixings, binary solutions, 21-22 Wave functions, electrode-electrolyte interface, static structure, 347 - 362 Wavenumbers, mode coupling theory (MCT): extended hydrodynamics, 79-81 INDEX liquid-state dynamics, viscosity calculations, 125-126 power law mass dependence, 153-155 slow liquid-state dynamics, large wavenumbers, 76-78 Wavevector-dependent traJl~port, mode coupling in liquid-state dynamics, 213 Weak anomalies, ionic fluids, liquid-liquid demixings, binary solutions, viscosity, 19 Weeks-Chandler-Anderson (WCA), mode coupling theory in liquid-state dynamics, small solute anomalous diffusion, 157-161 Wegner series, ionic fluid criticality: liquid-liquid demixings, binary solutions, 10-17 scattering and turbidity, 17-18 Weiss and Schroer (WS) theory, ionic fluid criticality, pairing theories, 32-33 ... and Michael E Cates CHEMICAL PHYSICS OF THE ELECTRODE-ELECTROLYTE 337 INTERFACE By J WHalley, S Walbran and D L Price AUTHOR INDEX 389 SUBJECT INDEX 403 ix ADVANCES IN CHEMICAL PHYSICS VOLUME 116... subjects into new, powerful, general concepts is as valuable as ever, and the desire to remain educated persists in all scientists This series, Advances in Chemical Physics, is devoted to helping... obtain general information about a wide variety of topics in chemical physics, a field that we interpret very broadly Our intent is to have experts present comprehensive analyses of subjects of interest