MOLECULAR MODELING THEORY AND APPLICATIONS IN THE GEOSCIENCES

— MOLECULAR MODELING THEORY — APPLICATIONS IN THE GEOSCIENCES

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Reviews in Mineralogy and Geochemistry

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FOREWORD The review chapters in this volume were the basis for a short course on molecular modeling theory jointly sponsored by the Geochemical Society (GS) and the Mineralogical Society of America (MSA) May 18-20, 2001 in Roanoke, Virginia which was held prior to the 2001 Goldschmidt Conference in nearby Hot Springs, Virginia. As a new series editor for Reviews in Mineralogy and Geochemistry, I thank Randy Cygan and Jim Kubicki for a wonderful job of coercing manuscripts from authors (all of them on time!) and excellent technical editing. They made my “d ebut performance” an enjoyable experience. Paul Ribbe also deserves credit for his many hours in training me to do this job. Thank you for always answering my never-ending barrage of e-mails! Also, thanks to Mike Hochella for making this all possible. Finally, I mention my infinitely patient and understanding family, Kevin and Ethan. Without them, I couldn’t have taken on this new responsibility or done the job required of me.

Jodi J. Rosso, Series Editor West Richland, Washington March 19, 2001   

DEDICATION Dr. William C. Luth has had a long and distinguished career in research, education and in the government. He was a leader in experimental petrology and in training graduate students at Stanford University. His efforts at Sandia National Laboratory and at the Department of Energy's headquarters resulted in the initiation and long-term support of many of the cutting edge research projects whose results form the foundations of these short courses. Bill's broad interest in understanding fundamental geochemical processes and their applications to national problems is a continuous thread through both his university and government career. He retired in 1996, but his efforts to foster excellent basic research, and to promote the development of advanced analytical capabilities gave a unique focus to the basic research portfolio in Geosciences at the Department of Energy. He has been, and continues to be, a friend and mentor to many of us. It is appropriate to celebrate his career in education and government service with this series of courses in cutting-edge geochemistry that have particular focus on Department of Energy-related science, at a time when he can still enjoy the recognition of his contributions.  

PREFACE AND ACKNOWLEDGMENTS Molecular modeling methods have become important tools in many areas of geochemical and mineralogical research. Theoretical methods describing atomistic and molecular-based processes are now commonplace in the geosciences literature and have helped in the interpretation of numerous experimental, spectroscopic, and field observations. Dramatic increases in computer power—involving personal computers, workstations, and massively parallel supercomputers—have helped to increase our knowledge of the fundamental processes in geochemistry and mineralogy. All researchers can now have access to the basic computer hardware and molecular modeling codes needed to evaluate these processes. The purpose of this volume of Reviews in Mineralogy and Geochemistry is to provide the student and professional with a general introduction to molecular modeling methods and a review of various applications of the theory to problems in the geosciences. Molecular mechanics methods that are reviewed include energy minimization, lattice dynamics, Monte Carlo methods, and molecular dynamics. Important concepts of quantum mechanics and electronic structure calculations, including both molecular orbital and density functional theories, are also presented. Applications cover a broad range of mineralogy and geochemistry topics—from at mospheric reactions to fluid-rock interactions to properties of mantle and core phases. Emphasis is placed on the comparison of molecular simulations with experimental data and the synergy that can be generated by using both approaches in tandem. We hope the content of this review volume will help the interested reader to quickly develop an appreciation for the fundamental theories behind the molecular modeling tools and to become aware of the limits in applying these state-of-the-art methods to solve geosciences problems. As with previous volumes in the Reviews in Mineralogy and Geochemistry series, we appreciate the efforts of the series editors, Jodi Rosso and Paul Ribbe. The diligent hard work and editorial skills of Jodi Rosso were critical in combining a diverse set of author styles and word processing formats to create a coherent and readable volume. Paul Ribbe provided significant guidance during the early stages of the book production. Virginia Sisson and Scott Wood were helpful in getting approval for the short course and review volume from the Mineralogical Society of America and the Geochemical Society, respectively. The society business directors, Alex Speer of MSA and Seth Davis of GS, provided sound advice and support during hectic times. Also, we appreciate the organizational efforts and guidance of Michael Hochella in helping to coordinate the short course with the 2001 Goldschmidt Conference. We thank all of the contributing authors for their willingness to participate in the short course and authorship of this volume. Their time and dedication in producing this book under strict deadlines—often with persistent and seemingly never-ending e-mail reminders—are greatly appreciated. We are also grateful for the critical comments and suggestions provided by the group of competent individuals who reviewed the original manuscripts. We are extremely thankful for the financial support provided by Molecular Simulations Inc. and the Office of Basic Energy Sciences of the U.S. Department of Energy (Grant No. DE-FG02-01ER151127 – Amendment No. A000). MSI and their talented scientific and programming staff have pioneered the development of commercial molecular modeling software. We appreciate their support. We are grateful for the efforts of Nick Woodward of the Geosciences Research Program at the Office of Basic Energy Sciences of DOE in funding a significant part of the short course and review volume. This book is the first in a series of short course review volumes on cutting-edge geochemistry and mineralogy that are in tribute to William C. Luth and his leadership while at the Office of Basic Energy Sciences. Dr. Luth’s broad interest in understanding fundamental geochemical processes and their applications to national problems has been

a continuous thread throughout both his university and government careers. Randall T. Cygan Albuquerque, New Mexico James D. Kubicki University Park, Pennsylvania March 9, 2001  

RiMG Volume 42 MOLECULAR MODELING THEORY: Applications in the Geosciences Table of Contents

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Molecular Modeling in Mineralogy and Geochemistry Randall T. Cygan

INTRODUCTION ........................................................................................................................... 1 Historical perspective ......................................................................................................... 2 Molecular modeling tools ................................................................................................... 3 POTENTIAL ENERGY .................................................................................................................. 6 Energy terms ...................................................................................................................... 7 Atomic charges ................................................................................................................. 10 Practical concerns ............................................................................................................. 11 MOLECULAR MODELING TECHNIQUES .............................................................................. 11 Conformational analysis ................................................................................................... 11 Energy minimization ........................................................................................................ 13 Energy minimization and classical-based equilibrium structures .................................... 14 Quantum chemistry methods ............................................................................................ 15 Energy minimization and quantum-based equilibrium structures .................................... 18 Monte Carlo methods ....................................................................................................... 20 Molecular dynamics methods ........................................................................................... 23 Quantum dynamics ........................................................................................................... 25 FORSTERITE: THE VERY MODEL OF A MODERN MAJOR MINERAL ........................... 26 Static calculations and energy minimization studies ........................................................ 27 Lattice dynamics studies .................................................................................................. 27 Quantum studies ............................................................................................................... 27 THE FUTURE ............................................................................................................................... 28 ACKNOWLEDGMENTS ............................................................................................................. 28 GLOSSARY OF TERMS ............................................................................................................. 29 REFERENCES .............................................................................................................................. 30

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Simulating the Crystal Structures and Properties of Ionic Materials From Interatomic Potentials Julian D. Gale

INTRODUCTION ......................................................................................................................... 37 INTERATOMIC POTENTIAL MODELS FOR IONIC MATERIALS....................................... 37 Long-range interactions .................................................................................................... 39 Short-range interactions ................................................................................................... 40 Energy minimization ........................................................................................................ 41 CRYSTAL PROPERTIES FROM STATIC CALCULATION .................................................... 44 Elastic constants ............................................................................................................... 44 Dielectric constants .......................................................................................................... 44

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Piezoelectric constants ..................................................................................................... 45 Phonons ............................................................................................................................ 45 DERIVATION OF POTENTIAL PARAMETERS ...................................................................... 47 Simultaneous fitting ......................................................................................................... 47 Relaxed fitting .................................................................................................................. 49 SIMULATING THE EFFECT OF TEMPERATURE AND PRESSURE ON CRYSTAL STRUCTURES ....................................................................................................................... 50 FUTURE DIRECTIONS IN INTERATOMIC POTENTIAL MODELLING OF IONIC MATERIALS.......................................................................................................................... 56 Structure solution and prediction...................................................................................... 58 ACKNOWLEDGMENTS ............................................................................................................. 59 REFERENCES .............................................................................................................................. 59

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Application of Lattice Dynamics and Molecular Dynamics Techniques to Minerals and Their Surfaces Steve C. Parker, Nora H. de Leeuw, Ekatarina Bourova, David J. Cooke

INTRODUCTION ......................................................................................................................... 63 METHODOLOGY ........................................................................................................................ 63 LATTICE DYNAMICS ................................................................................................................ 64 MOLECULAR DYNAMICS ........................................................................................................ 67 SIMULATION OF MINERAL-WATER INTERFACES ............................................................ 74 CONCLUSIONS ........................................................................................................................... 80 REFERENCES .............................................................................................................................. 81

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Molecular Simulations of Liquid and Supercritical Water: Thermodynamics, Structure, and Hydrogen Bonding Andrey G. Kalinichev

INTRODUCTION ......................................................................................................................... 83 CLASSICAL METHODS OF MOLECULAR SIMULATIONS ................................................. 86 Molecular dynamics ......................................................................................................... 86 Monte Carlo methods ....................................................................................................... 87 Boundary conditions, long-range corrections, and statistical errors................................. 89 Interaction potentials for aqueous simulations ................................................................. 90 THERMODYNAMICS OF SUPERCRITICAL AQUEOUS SYSTEMS .................................... 95 Macroscopic thermodynamic properties of simulated supercritical water ....................... 96 Micro-thermodynamic properties ..................................................................................... 97 STRUCTURE OF SUPERCRITICAL WATER......................................................................... 101 HYDROGEN BONDING IN LIQUID AND SUPERCRITICAL WATER ............................... 104 MOLECULAR CLUSTERIZATION IN SUPERCRITICAL WATER ..................................... 109 DYNAMICS OF MOLECULAR TRANSLATIONS, LIBRATIONS, AND VIBRATIONS IN SUPERCRITICAL WATER .................................................................. 113 CONCLUSIONS AND OUTLOOK ........................................................................................... 120 ACKNOWLEDGMENTS ........................................................................................................... 121 REFERENCES ............................................................................................................................ 121

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Molecular Dynamics Simulations of Silicate Glasses and Glass Surfaces Stephen H. Garofalini

INTRODUCTION ....................................................................................................................... 131 MOLECULAR DYNAMICS COMPUTER SIMULATION TECHNIQUE.............................. 131 Interatomic potentials ..................................................................................................... 135 Periodic boundary conditions ......................................................................................... 137 MD SIMULATIONS OF OXIDE GLASSES ............................................................................. 140 Bulk glasses .................................................................................................................... 140 Bulk SiO2 ........................................................................................................................ 141 Multicomponent silicate glasses ..................................................................................... 145 MD SIMULATIONS OF OXIDE GLASS SURFACES ............................................................ 147 SiO2 ................................................................................................................................ 147 Multicomponent silicate surfaces ................................................................................... 162 SUMMARY ................................................................................................................................ 162 ACKNOWLEDGMENTS ........................................................................................................... 164 REFERENCES ............................................................................................................................ 164

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Molecular Models of Surface Relaxation, Hydroxylation, and Surface Charging at Oxide-Water Interfaces James R. Rustad

INTRODUCTION ....................................................................................................................... 169 SCOPE ........................................................................................................................................ 170 THE STILLINGER-DAVID WATER MODEL ......................................................................... 172 IRON-WATER AND SILICON-WATER POTENTIALS AND THE BEHAVIOR OF FE3+ AND SI4+ IN THE GAS PHASE AND IN AQUEOUS SOLUTION .................... 174 CRYSTAL STRUCTURES ........................................................................................................ 177 VACUUM-TERMINATED SURFACES ................................................................................... 179 HYDRATED AND HYDROXYLATED SURFACES .............................................................. 183 Neutral surfaces .............................................................................................................. 183 Surface charging ............................................................................................................. 188 SOLVATED INTERFACES ....................................................................................................... 191 REMARKS .................................................................................................................................. 193 ACKNOWLEDGMENTS ........................................................................................................... 193 REFERENCES ............................................................................................................................ 194

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Structure and Reactivity of Semiconducting Mineral Surfaces: Convergence of Molecular Modeling and Experiment Kevin M. Rosso

INTRODUCTION ....................................................................................................................... 199 BACKGROUND CONCEPTS ................................................................................................... 200 Experimental approaches ............................................................................................... 200 Semiconductors and their surfaces ................................................................................. 201

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THEORETICAL METHODS ..................................................................................................... 212 Theory–Hartree-Fock versus density functional theory ................................................. 213 Basis sets–Gaussian orbital versus plane waves ............................................................ 216 Surface model–Cluster versus periodic .......................................................................... 221 Codes–Crystal vs. CASTEP ........................................................................................... 223 APPLICATIONS......................................................................................................................... 226 Sulfides ........................................................................................................................... 226 Oxides............................................................................................................................. 248 CONCLUDING REMARKS AND OUTLOOK ........................................................................ 260 ACKNOWLEDGMENTS ........................................................................................................... 262 REFERENCES ............................................................................................................................ 262

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Quantum Chemistry and Classical Simulations of Metal Complexes in Aqueous Solutions David M. Sherman

INTRODUCTION ....................................................................................................................... 273 Experimental methods .................................................................................................... 273 Continuum models ......................................................................................................... 274 Atomistic computational methods .................................................................................. 274 QUANTUM CHEMISTRY OF METAL COMPLEXES: THEORETICAL BACKGROUND AND METHODOLOGY ...................................................................................................... 275 Quantum mechanics of many-electron systems ............................................................. 275 Bonding in molecules and complexes ............................................................................ 280 Calculating thermodynamic quantities from first principles .......................................... 283 Simulations of solvent effects ........................................................................................ 284 APPLICATIONS OF QUANTUM CHEMISTRY TO METAL COMPLEXES IN AQUEOUS SOLUTIONS ........................................................................................................................ 285 Group IIB cations Zn, Cd and Hg .................................................................................. 285 Group 1B cations Cu, Ag, and Au.................................................................................. 292 Iron and manganese ........................................................................................................ 296 Alkali earth and alkali metal cations .............................................................................. 299 Post-transition metals ..................................................................................................... 299 CLASSICAL ATOMISTIC SIMULATIONS OF METAL COMPLEXES IN AQUEOUS SOLUTIONS ........................................................................................................................ 301 Background .................................................................................................................... 301 Interatomic potentials ..................................................................................................... 302 Molecular dynamics ....................................................................................................... 304 Metropolis Monte Carlo simulations .............................................................................. 305 Applications ................................................................................................................... 305 THE NEXT ERA: AB INITIO MOLECULAR DYNAMICS .................................................... 310 Application to copper(I) chloride solutions. ................................................................... 311 SUMMARY AND FUTURE DIRECTIONS ............................................................................. 311 ACKNOWLEDGMENTS ........................................................................................................... 312 REFERENCES ............................................................................................................................ 312

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First Principles Theory of Mantle and Core Phases Lars Stixrude

INTRODUCTION ....................................................................................................................... 319 THEORY ..................................................................................................................................... 321 Overview ........................................................................................................................ 321 Total energy, forces, and stresses ................................................................................... 324 Statistical mechanics ...................................................................................................... 326 SELECTED APPLICATIONS .................................................................................................... 332 Overview ........................................................................................................................ 332 Phase transformations in silicates................................................................................... 332 High temperature properties of transition metals ........................................................... 336 CONCLUSIONS AND OUTLOOK ........................................................................................... 339 Scale ............................................................................................................................... 339 Duration .......................................................................................................................... 339 Materials ......................................................................................................................... 340 ACKNOWLEDGMENTS ........................................................................................................... 340 REFERENCES ............................................................................................................................ 340

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A Computational Quantum Chemical Study of the Bonded Interactions in Earth Materials and Structurally and Chemically Related Molecules G. V. Gibbs, Monte B. Boisen, Jr., Lesa L. Beverly, Kevin M. Rosso

INTRODUCTION ....................................................................................................................... 345 BOND LENGTH AND BOND STRENGTH CONNECTIONS FOR OXIDE, FLUORIDE, NITRIDE, AND SULFIDE MOLECULAR AND CRYSTALLINE MATERIALS .......... 345 Bond lengths and crystal radii ........................................................................................ 345 Bonded interactions ........................................................................................................ 346 Pauling bond strength and bond length variations.......................................................... 347 Brown and Shannon bond strength and bond length variations ..................................... 348 Bond strength p and bond length variations ................................................................... 348 Bond number and bond length variations ....................................................................... 350 Nitride, fluoride and sulfide bond strength and bond length variations ......................... 351 Bond strength and crystal radii ....................................................................................... 352 FORCE CONSTANTS, COMPRESSIBILITIES OF COORDINATED POLYHEDRA, AND POTENTIAL ENERGY MODELS ............................................................................ 353 Force constants and bond length variations.................................................................... 353 Force constants and polyhedral compressibilities .......................................................... 354 Force fields and bond length and angle variations ......................................................... 355 Generation of new and viable structure types for silica ................................................. 357 CALCULATED ELECTRON DENSITY DISTRIBUTIONS FOR EARTH MATERIALS AND RELATED MOLECULES .......................................................................................... 358 Bond critical point properties and electron density distributions ................................... 358 Bond critical point properties calculated for molecules ................................................. 359 Bond critical point properties calculated for earth materials .......................................... 361 Variable radius of the oxide anion.................................................................................. 362

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BOND STRENGTH, ELECTRON DENSITY, AND BOND TYPE CONNECTIONS ............ 365 SITES OF POTENTIAL ELECTROPHILIC ATTACK IN EARTH MATERIALS ................. 367 Bonded and nonbonded electron pairs ........................................................................... 367 Bonded and nonbonded electron lone pairs for a silicate molecule ............................... 369 Localization of the electron density for the silica polymorphs ...................................... 370 Nonbonded lone pair electrons for low albite ................................................................ 372 CONCLUDING REMARKS ...................................................................................................... 373 ACKNOWLEDGMENTS ........................................................................................................... 375 REFERENCES ............................................................................................................................ 376

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Modeling the Kinetics and Mechanisms of Petroleum and Natural Gas Generation: A First Principles Approach Yitian Xiao

INTRODUCTION ....................................................................................................................... 383 AB INITIO METHOD ................................................................................................................. 385 KEROGEN DECOMPOSITION AND OIL AND GAS GENERATION .................................. 390 Introduction .................................................................................................................... 390 The kinetics and mechanisms of hydrocarbon thermal cracking ................................... 394 Computational methods .................................................................................................. 396 Initiation reaction (homolytic scission) .......................................................................... 397 Hydrogen transfer reaction ............................................................................................. 400 Radical decomposition (β scission) ................................................................................ 403 Elementary reactions versus overall hydrocarbon cracking ........................................... 406 Summary ........................................................................................................................ 407 ISOTOPIC FRACTIONATION AND NATURAL GAS GENERATION ................................ 408 Introduction .................................................................................................................... 408 Transition state theory and gas isotopic fractionation .................................................... 409 Natural gas plot .............................................................................................................. 410 Carbon kinetic isotope effect: homolytic scission verses β scission .............................. 411 Biogenic gas versus thermogenic gas ............................................................................. 415 Summary ........................................................................................................................ 416 POSSIBLES ROLES OF MINERALS AND TRANSITION METALS IN OIL AND GAS GENERATION ..................................................................................................................... 416 Introduction .................................................................................................................... 416 Acid catalyzed isomerization of C7 alkanes and light HC origin .................................. 417 Transition metal catalysis and natural gas generation .................................................... 420 WATER-ORGANIC INTERACTIONS AND THEIR IMPLICATIONS ON PETROLEUM FORMATION ....................................................................................................................... 423 Introduction .................................................................................................................... 423 Why don’t oil and water mix? ........................................................................................ 424 The kinetics and mechanisms of water-organic (kerogen) interaction ........................... 425 Hydrolysis of ether linkages ........................................................................................... 425 Hydrolysis of ester linkages ........................................................................................... 427 Water-hydrocarbon radical interactions ......................................................................... 428 Hydrolytic disproportionation and kerogen oxidation.................................................... 430 CONCLUSIONS ......................................................................................................................... 431 ACKNOWLEDGMENTS ........................................................................................................... 431 REFERENCES ............................................................................................................................ 431

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Calculating the NMR Properties of Minerals, Glasses, and Aqueous Species John D. Tossell

INTRODUCTION ....................................................................................................................... 437 BASIC THEORY OF NMR SHIELDING.................................................................................. 437 A BRIEF HISTORY OF NMR CALCULATIONS ON MOLECULES .................................... 439 PRESENT STATUS OF NMR CALCULATIONS ON MOLECULES .................................... 439 CALCULATION OF SI NMR SHIELDINGS IN ALUMINOSILICATES .............................. 443 CALCULATIONS OF SHIELDINGS FOR OTHER ELECTROPOSITIVE ELEMENTS: B, P, SE, NA AND RB .................................................................................. 446 CALCULATION OF ELECTRIC FIELD GRADIENTS AT O IN ALUMINOSILICATES ........................................................................................................ 448 CALCULATION OF NMR SHIELDING OF O IN OXIDES ................................................... 449 CALCULATION OF NMR SHIELDINGS FOR TRANSITION METAL COMPOUNDS AND HEAVY MAIN-GROUP METAL COMPOUNDS.......................... 450 CALCULATIONS OF C NMR SHIELDINGS IN ORGANIC GEOCHEMISTRY ................. 450 APPLICATIONS OF NMR SHIELDING CALCULATIONS IN GEOCHEMISTRY AND MINERALOGY .......................................................................... 451 A FINAL WORD ON INTERPRETATION OF CALCULATED NMR SHIELDINGS .......... 453 CONCLUSION ........................................................................................................................... 454 ACKNOWLEDGMENTS ........................................................................................................... 454 REFERENCES ............................................................................................................................ 454

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Interpretation of Vibrational Spectra Using Molecular Orbital Theory Calculations James D. Kubicki

INTRODUCTION ....................................................................................................................... 459 ENERGY MINIMIZATIONS ..................................................................................................... 460 CALCULATION OF SPECTRA ................................................................................................ 461 CALCULATION OF FREQUENCIES ...................................................................................... 462 CALCULATION OF IR AND RAMAN INTENSITIES ........................................................... 463 Infrared intensities .......................................................................................................... 463 Raman intensities ........................................................................................................... 465 VIBRATIONAL BANDWIDTHS .............................................................................................. 466 EXAMPLES AND COMPARISON TO EXPERIMENT........................................................... 467 Gas-phase ....................................................................................................................... 467 Aqueous-phase ............................................................................................................... 469 Mineral surfaces ............................................................................................................. 473 Minerals .......................................................................................................................... 475 Glasses ............................................................................................................................ 475 CONCLUSIONS AND FUTURE DIRECTIONS ...................................................................... 478 ACKNOWLEDGMENTS ........................................................................................................... 478 REFERENCES ............................................................................................................................ 479

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Molecular Orbital Modeling and Transition State Theory in Geochemistry Mihali A. Felipe, Yitian Xiao, James D. Kubicki

INTRODUCTION ....................................................................................................................... 485 TRANSITION STATE THEORY .............................................................................................. 486 Conventional transition state theory ............................................................................... 486 Potential energy surfaces and MO calculations.............................................................. 490 Other rate theories .......................................................................................................... 494 DETERMINATION OF ELEMENTARY STEPS AND REACTION MECHANISMS ........... 496 Stationary-point searching schemes ............................................................................... 496 Transition state initial guesses ........................................................................................ 498 Optimization to stationary points ................................................................................... 501 MO-TST STUDIES IN THE GEOSCIENCES ........................................................................... 504 Introduction and definitions ........................................................................................... 504 Reaction pathways of mineral-water interaction ............................................................ 505 Atmospheric reactions of global significance ................................................................ 511 ACCURACY ISSUES ................................................................................................................ 517 Basis sets ........................................................................................................................ 517 Basis set superposition error........................................................................................... 518 Methods .......................................................................................................................... 518 Long-range interactions .................................................................................................. 519 Activation energies and zero point energies ................................................................... 519 Quantum tunneling ......................................................................................................... 520 CONCLUSIONS AND FUTURE DIRECTIONS ...................................................................... 521 ACKNOWLEDGMENTS ........................................................................................................... 522 LIST OF SYMBOLS................................................................................................................... 522 REFERENCES ............................................................................................................................ 524

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Molecular Modeling in Mineralogy and Geochemistry Randall T. Cygan Geochemistry Department Sandia National Laboratories Albuquerque, New Mexico, 87185-0750, U.S.A. “A theory is something nobody believes, except the person who made it. An experiment is something everybody believes, except the person who made it.” Attributed to Albert Einstein “A theory has only the alternative of being right or wrong. A model has a third possibility: it may be right, but irrelevant.” Manfred Eigen

INTRODUCTION At what underlying fundamental level of understanding does geosciences research need to attain in order to evaluate the complex processes that control the weathering rate of silicate minerals? To investigate the formation of ore deposits and oil reservoirs, or the leaching of mine tailings into watersheds and the eventual contamination of groundwater? To predict the crustal deformation of long-term underground waste storage sites, or the stability of lower mantle phases and their effect on seismic signals? Or, for that matter, to examine tectonic uplift and cooling rates associated with orogenies? These and numerous other examples from mineralogy and geochemistry often require an understanding of atomic-level processes to identify the fundamental properties and mechanisms that control the thermodynamics and kinetics of Earth materials. Molecular models are often invoked to supplement field observations, experimental measurements, and spectroscopy. Theoretical methods provide a powerful complement for the experimentalist, especially with recent trends in which atomic-scale measurements are being made at synchrotron and other high-energy source facilities throughout the world. Such analytical methods and facilities have matured to such an extent that mineralogists and geochemists routinely probe Earth materials to evaluate bulk, surface, defect, intergranular, compositional, isotopic, long-range, local, order-disorder, electronic, and magnetic structures. Molecular modeling theory provides a means to help interpret the field and experimental observation, and to discriminate among various competing models to explain the macroscopic observation. And ultimately, molecular modeling provides the basis for prediction to further test the validity of the scientific hypothesis. This is especially significant in the geosciences where the conditions in the interior of the Earth, and other planets, preclude observation or are not achievable through experiment. The explosion of computer technology and the development of faster processors and efficient algorithms have led to the development of specialized molecular modeling tools for computational chemistry. Combined with user-friendly interfaces and the porting of molecular modeling codes to personal computer platforms, these tools are increasingly being used by non-specialists to help interpret experimental and field observations. These tools are no longer limited to a specialized few who can understand the complex logic of thousands or millions of lines of software code, or those having access to government or university supercomputers. Commercial molecular modeling software is available to most researchers and is being used to examine an ever-increasing number mineralogical and geochemical problems. But what level of theory is required to best examine and 1529-6466/01/0042-0001$05.00

DOI:10.2138/rmg.2001.42.1

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solve a particular problem? Can the problem even be solved on a personal computer, a Unix workstation, or does the researcher need a massively-parallel supercomputer? What is the theory, what are the limits of the various modeling methods, and how does one apply these modeling tools to the complex nature of Earth materials? These are the critical concerns addressed by this book. The quote noted above and attributed to Albert Einstein describes the natural skepticism that might exist in linking experimental (or field) observations to molecular models. Experimentalists and theoreticians as members of their own research specialty will have a natural tendency to be misjudged by others. The inherent heterogeneous nature and complexity of the geosciences makes the connection between observation and theory even more complicated, yet numerous successes in other scientific disciplines, such as pharmaceuticals and materials science, have made molecular simulation an accepted approach. The critical success of molecular modeling and computer simulation in solving mineralogical and geochemical problems will ultimately be judged by the entire geosciences community. Historical perspective Modern molecular modeling technology combines the most sophisticated and efficient, graphical-based software with a variety of computer platforms ranging from personal computers (and even hand-held devices) to massively-parallel supercomputers. The last decade has seen the most dramatic improvement in our ability to visualize structural models of molecules and periodic systems. Interestingly, it was not more than ten years ago that almost every introductory chemistry and mineralogy class required students to manipulate physical ball-and-stick models of molecules and crystals to help visualize and understand the structure and arrangement of atoms. In fact, for almost two hundred years this was de rigueur for most chemists. John Dalton, the founder of atomic theory, first introduced the concept of a molecular model in 1810 with his use of wooden balls connected by sticks to describe molecules (Rouvray 1995). Previously in 1808, the English chemist William Wollaston used hand-drawn sketches of atoms to visualize the tetrahedral coordination about a central atom (Rouvray 1997). The Dutch chemist Jacobus van’t Hoff built upon these early models by developing the first set of structural models for organic compounds based on the tetrahedral arrangement of hydrogens and other chemical groups about a central carbon atom. This work helped to explain the nature of organic isomers and optical activity that had confused chemists at that time (van't Hoff 1874). Further advances in the development of molecular modeling were led by the by the series of scientific breakthroughs in the late nineteenth and early twentieth century. These include the discovery of the electron in 1897 by the English physicist J. J. Thompson, and the development by Neils Bohr and Ernest Rutherford in 1911-1912 of an atomic model comprised of quantized electrons orbiting around a dense nucleus. In 1924, the French physicist Louis de Broglie recognized the wave-particle duality of matter that ultimately led to the 1926 publication of the famous wavefunction equation (Hψ=Eψ) by the physicist Erwin Schrödinger. The quantum description of many-electron chemical systems was developed in the 1930’s by the efforts of Douglas Hartree and Vladimir Fock using an exact Hamiltonian and approximate wavefunctions. Refinements on the use of electronic structure calculations were later introduced by Kohn and Sham (1965) and by Hehre et al. (1969). Ultimately, these pioneering efforts in quantum chemistry methods led to the awarding of the Nobel Prize for chemistry in 1998 to Walter Kohn for developing density functional methods and John Pople for developing molecular orbital theory. The structural analysis of molecular systems, especially proteins and other

 

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macromolecules, was of significant interest starting in the mid-twentieth century primarily due to the advances in crystallographic and spectroscopic methods. Physical molecular models needed to visualize large biochemical molecules were introduced by Robert Corey, Linus Pauling, Walter Koltun, and Andre Dreiding in the 1950’s and 1960’s. Kendrew et al. (1958) published the first three dimensional model of a protein (myoglobin) based on X-ray analysis and a wire-mesh representation of the structure. Advances in computer technology in the 1960’s brought computer visualization to the forefront of biochemistry and aided in the analysis of protein structure and protein folding (Levinthal 1966). The trend increased through the 1970’s and 1980’s as the drug industry recognized the usefulness of computer visualization methods to help design new pharmaceuticals and organic molecules. The modern era of molecular modeling probably began with the introduction of empirical-based energy forcefields, such as the one developed by Lifson and Warshel (1968), to assist with the conformational and configuration analysis of simple organic compounds. Computationally-fast energy calculations (as opposed to costly quantum methods) could now be performed on a large number of molecular configurations allowing one to determine the lowest energy structures (i.e., the most stable). Combining these molecular mechanics approaches with the interactive visualization provided by fast graphical computer displays allowed molecular modeling to quickly expand in the 1990’s. Calculations involving inorganic compounds, including a good number of mineral phases, were not performed using molecular mechanics methods until the 1970’s and 1980’s. William Busing, Richard Catlow, and Leslie Woodcock (e.g., Busing 1970; Catlow et al. 1976; Woodcock et al. 1976; Catlow et al. 1982) pioneered much of the early work associated with the simulation of oxides and silicate minerals. The use of quantum methods in mineralogy was being done at the same time, with much credit going to the pioneering studies of Gerald Gibbs and John Tossell (e.g., Gibbs et al. 1972; Tossell and Gibbs 1977, 1978; Gibbs 1982). Molecular modeling tools In general, computer simulation techniques cover a broad range of spatial and temporal variation. This is best demonstrated in the schematic diagram presented in Figure 1. Modeling geologic-scale processes pushes the distance and time scales to even larger values. Traditional continuum and finite element methods of simulation often reach to kilometer (field scale) or greater length scales and times involving millions of years (geological times). In contrast, molecular modeling methods fall at the opposite extreme where distances are typically on the order of Ångstroms (level of atomic separations) and times are on the order of femtoseconds (time scale of molecular vibrations). The transition between these two modeling extremes includes the analysis of electrons for quantum chemistry, atoms for molecular mechanics models, molecular fragments for mesoscale models, and macroscopic units for the larger-scale field models. Although the boundaries in this representation are in practice quite diffuse and significant overlap of the techniques occurs, each method provides the necessary detail for the respective scale of the modeling. Obviously, there is a greater span of scales needed to link molecular models to the large scale geological applications in the upper right of the diagram. Mesoscale modeling methods are not discussed in this book, but several recent reviews and examples of the various techniques are available (e.g., Stockman et al. 1997; Coles et al. 1998; Flekkoy and Coveney 1999). There are several excellent handbooks and texts that provide comprehensive reviews of molecular modeling methods. Noteworthy among these are Clark (1985) and Allen and Tildesley (1987), and the more recent volumes by Frenkel and Smit (1996) and Leach (1996). The recent publication by Schleyer (1998) presents an outstanding and

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log Time (s)

Figure 1. Schematic representation of the various computer simulation methods as a function of spatial and temporal variables. Boundaries between methods are approximate and diffuse to represent overlap of the techniques.

thorough review of computational chemistry including numerous, and almost exhaustive, discussions of theory, methods, forcefields, and software. However, the significant size (five volumes and over three thousand pages) and associative cost may prevent any practical access to the information. Molecular modeling tools concentrate, in general, on calculating the total energy of the molecular or periodic system under investigation. Two fundamental approaches are typically used in this effort: molecular mechanics and quantum mechanics. Figure 2 provides a schematic representation and flow chart of how these methods are related and used to examine the structure and energy of either a molecule or periodic system. The molecule can be treated as an isolated entity (gas phase molecule) or solvated (by using an advance modeling approach) ion or molecule. Periodic systems include crystalline structures, glasses, and other amorphous materials. Glasses and explicitly solvated molecules often rely on the use of large periodic simulation cells to realistically represent the long-range disorder of solution molecules or glass components while avoiding edge and surface effects. Molecular mechanics methods rely on the use of analytical expressions that have been parameterized, through either experimental observation or quantum calculations, to evaluate the interaction energies for the given structure or configuration. Various modeling schemes are then used to evaluate the potential energy and forces on the atoms to obtain optimized or equilibrated configurations for the molecule or periodic system. Energy minimization, conformational analysis, molecular dynamics, and stochastic methods are important tools in molecular mechanics. Molecular dynamics simulations directly involve the calculation of forces based on Newtonian physics (F=ma) and provide a deterministic basis for evaluating the time evolution of a system on the time scale of pico- and nanoseconds. In contrast, quantum mechanics uses first principles methods without the need of empirical parameters, for most instances, to evaluate the

5

Modeling in Mineralogy & Geochemistry Extenmrip Ab initio

+

Focr efield

Stcur erut

Stcur erut

Molecular Model Molecular Mechanics

Quantum Mechanics

F = ma

Hψ= Eψ Clterus

Pedicro

Har-ert Fock and DFT

Eng re y Miinzm atoin Cofn a mro ita no l Anasiyl Latict e Dya n icm s Mocel luar Dya n cim s Moetn Carlo

Stucr eurt Phcyis al eistrpo Thna yodmer mics Kiicetn s Specocsrt yop

. . .

Vala id i t n o

EM, CA, MD

Figure 2. Flow diagram for molecular mechanics and quantum mechanics methods showing input requirements, various approaches, and output possibilities. Molecular model can be comprised of an isolated molecular cluster or a periodic cell.

energy of the system. The Schrödinger wave equation—or more exactly, an approximation to the Schrödinger equation—is so lved by a variety of methods to obtain the total energy of the molecule or periodic system. As with molecular mechanics, minimization and dynamics methods can be implemented, however, these advanced quantum techniques can lead to extreme computational costs especially for large-atom systems. Ultimately, either approach leads to the prediction of structure and physical properties, and the determination of thermodynamic, kinetic, and spectroscopic properties. A successful molecular simulation will provide validation with experiment and lead to further refinement of the model to support its relevance to the physical world. This chapter provides an overview of the theory, methods, and philosophy of molecular modeling and simulation. Although meant to address specific applications associated with mineralogical and geochemical problems, numerous examples of simple molecular and crystalline models, some involving organic compounds, are presented. The level of the content is geared towards the novice and assumes no previous experience with molecular simulation. More detailed reviews are offered in the following chapters, or in the numerous references cited in this and other chapters of the book. Due to the scope and complexity of the subject matter, the reader will be subjected to presentations in this volume that involve various measurement units, especially those for energy. Rather than conform to one single unit system throughout the book, the chapters rely on the conventional units associated with the modeling method, and which have typically

6 

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evolved with the literature for that particular discipline. It is obvious that chemists and physicists may never come to an agreement on the use of a consistent unit system. Table 1 provides a helpful set of conversion units to sort through these various unit schemes. Several values for the universal constants are also included. A glossary presented at the end of the chapter may also be useful in sorting through the terms and methods used throughout this volume. An important reminder on the use of molecular modeling is provided by the second of the quotes presented at the beginning of this chapter. Manfred Eigen, a Noble-winning electrochemist, succinctly identified the number one failing common to those using molecular modeling methods. No matter how rigorous or uncompromising the theory is behind the model used to examine a chemical process, the model may completely miss the mark and be totally irrelevant. Tread carefully, and maintain a strong sense of validation with experimental and field observations! POTENTIAL ENERGY The most important requirement of any molecular mechanics simulation is the forcefield used to describe the potential energy of the system. An accurate energy forcefield is the key element of any successful energy minimization, Monte Carlo approach, or molecular dynamics simulation. The forcefield includes interatomic potentials that collectively describe the energy of interaction for an assemblage of atoms in either a molecular or crystalline configuration. Analytical expressions of the forcefield are typically obtained through the parameterization of experimental and spectroscopic data, or in some cases, by the use quantum mechanical calculations. The potential energy can then be presented as a function of distance, angle, or other geometry measurement. The analytical functions typically are quite simple and describe two- three- or four-body interactions. It is then possible to describe the potential energy of a complex multi-body

Table 1. Physical constants and conversion factors. Avogadro constant Boltzmann constant Gas constant Elementary charge Faraday constant Planck constant

NA k R = kNA e F = eNA h

= = h/ 2π Bohr radius Mass of electron Velocity of light Permittivity of vacuum 1 kJ/mol 1 erg 1 eV 1 rydberg 1 hartree 1 cm-1

6.022045 × 10 23 /mol 1.38066 × 10 -23 J/K 8.31441 J/K mol 1.602177 × 10 -19 C 9.6485 × 10 4 C/mol 6.62618 × 10 -34 J s 1.05459 × 10 -34 J s

0.5292 Å 9.10939 × 10 -31 kg 2.99792458 × 10 8 m/s 8.85419 × 10 -12 C2/J m

ao me c

εo = = = = = =

0.2390 kcal/mol 1.4393 × 10 13 kcal/mol 23.0609 kcal/mol 318.751 kcal/mol 627.51 kcal/mol 2.8591 × 10 -3 kcal/mol

Modeling in Mineralogy & Geochemistry

7

systems by the summation of all energy interactions over all atoms of the system. In principle, an accurate description of the potential energy surface of a system can be obtained by the forcefield as a function of the geometric variables. Energy terms The total potential energy of a system can be represented by the addition of the following energy components:

ETotal = ECoul + EVDW + E Bond Stretch + E Angle Bend + ETorsion

(1)

where ECoul, the Coulombic energy, and EVDW, the van der Waals energy, represent the so-called nonbonded energy components, and the final three terms represent the explicit bonded energy components associated with bond stretching, angle bending, and torsion dihedral, respectively. The Coulombic energy, or electrostatics energy, is based on the classical description of charged particle interactions and varies inversely with the distance rij:

E Coul =

e2 4πε o

∑ i≠ j

qi q j rij

(2)

Here, qi and qj represents the charge of the two interacting atoms (ions), e is the electron charge, and εo is the permittivity (dielectric constant) of a vacuum. The summation represents the need to examine all possible atom-atom interactions while avoiding duplication. Equation (2) will yield a negative and attractive energy when the atomic charges are of opposite sign, and a positive energy, for repulsive behavior, when the charges are of like sign. In the simple case, the Coulombic energy treats the atoms as single point charges, which in practice is equivalent to spherically-symmetric rigid bodies. Simulations involving crystalline materials or other periodic systems require the use of special mathematical methods to ensure proper convergence of the long-range nature of Equation (2); the 1/r term is nonconvergent except for the most simple and highly symmetric crystalline systems. In practice, it is therefore necessary to employ the Ewald method (Ewald 1921) or other alternative method (e.g., Greengard and Rokhlin 1987; Caillol and Levesque 1991) to obtain proper convergence and an accurate calculation of the Coulombic energy. The Ewald approach replaces the inverse distance by its Laplace transform that is decomposed into two rapidly convergent series, one in real space and one in reciprocal space (Tosi 1964; de Leeuw et al. 1980; Gale, this volume). The Coulombic energy in ionic solids typically dominates the total potential energy and, therefore, controls the structure and properties of the material. Purely ionic compounds such as the metal halide salts (e.g., NaF and KCl) are examples where the formal charge is used to accurately represent the electrostatics. In molecular systems where covalent bonding is more common, the Coulombic energy is effectively reduced by the use of partial or effective charges for the atoms. The Coulombic energy for non-periodic systems can be evaluated by direct summation without resorting to Ewald or related periodic methods. The van der Waals energy represents the short-range energy component associated with atomic interactions. Electronic overlap as two atoms approach each other leads to repulsion (positive energy) and is often expressed as a 1/r12 function. An attractive force (negative energy) occurs with the fluctuations in electron density on adjacent atoms. This second contribution is referred to as the London dispersion interaction and is proportional to 1/r6. The most common function for the combined interactions is provided by the Lennard-Jones expression:

8 

Cygan  EVDW

 

6 ⎡⎛ R ⎞12 ⎛ Ro ⎞ ⎤ o = ∑ Do ⎢⎜ ⎟ − 2⎜ ⎟ ⎥ ⎜r ⎟ ⎥ ⎢⎜⎝ rij ⎟⎠ i≠ j ⎝ ij ⎠ ⎦ ⎣

(3)

where Do and Ro represent empirical parameters. Although various forms of the 12-6 potential are used in the literature, the form presented above provides a convenient expression that equates Do to the depth of the potential energy well and Ro to the equilibrium atomic separation. This association would only apply for the interaction of uncharged atoms (e.g., inert gases), however, the functionality is used in practice for partial and full charge systems. Alternatively, a 9-6 function or a combined exponential1/r6 (Buckingham potential with three fitting parameters), among other functions, can be used to express the short-range interactions. In contrast to the long-range nature of the Coulombic energy, the van der Waals energy is non-negligible at only short distances (typically less than 5 to 10 Å), and, therefor e in practice, a cutoff distance is used to reduce the computational effort in the evaluation of this energy. Some energy forcefields are based on the simple ionic Born model such that only the first two terms of Equation (1) are used. If properly parameterized, the inclusion of just the Coulombic and van der Waals (short-range) terms for the total potential energy is more than satisfactory for successfully modeling the structure and physical properties of numerous oxides and silicates phases (e.g., Lewis and Catlow 1986). Often the shell model of Dick and Overhauser (1958) is used as a refinement of the ionic model by incorporating electronic polarization of the ions. The shell model uses two point charges joined by a harmonic spring (based on a 1/2 kx2 potential) to represent the polarization of an ion; the negatively-charged electron shell is associated with a positive nucleus-like core. The modification provides a necessary extension of the ionic model for modeling point defects in solids and surface structures where large asymmetric electrostatic potential fields will induce significant polarization among the ions, especially polarizable anions like oxygen. Elastic, dielectric, diffusion, and other materials properties can be accurately derived using the refinement provided by the shell model. Alternative polarization models (e.g., Agnon and Bukowinski 1990; Zhang and Bukowinski 1991) have also proven to be reliable in simulating oxide systems. The shell model is an attempt to treat a form of covalency in an ionic solid. However, the total-energy treatment of bonded systems requires the addition of several so-called bonded terms. The first of the bonded terms of Equation (2), the bond stretch term can be represented as a simple quadratic (harmonic) expression: E Bond Stretch = k1 (r − ro ) 2

(4)

where r is the separation distance for the bonded atoms, ro is the equilibrium bond distance, and k1 is an empirical force constant. This relation ensures that the two atoms will interact through a potential that allows vibration about an equilibrium bond distance. In fact, the force constant k1 can be obtained directly from analysis of the vibrational spectrum. Alternatively, a Morse potential can be used to provide a more realistic description of the energy of a covalent bond: E Morse = Do [1 − exp{1 − α (r − ro )}]

2

(5)

Here, Do represents the equilibrium dissociation energy and α is a parameter related to the vibrational force constant. Figure 3 provides a comparison of the two potential functions used to describe the carbon-hydrogen bond stretch based on the DauberOsguthorpe et al. (1988) forcefield parameters. Although both represent the equilibrium

9

Modeling in Mineralogy & Geochemistry 300

C-H bond stretch Harmonic

Poneta i l Eng re y (kcal/)lom

200

Morse

100

Do 0

0.0

ro 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Distance (Å) Figure 3. Comparison of harmonic and Morse potentials to represent the bond stretch energy of the carbon-hydrogen bond. The Morse potential is more appro-priate for modeling significant deviations from the equilibrium atom separation distance ro; Do is the bond dissociation energy.

bond distance of 1.105 Å, the anharmonic nature of the Morse potential provides a more satisfying description of the C-H dissociation that would be expected at large bond distances. The harmonic potential is only suitable at near-equilibrium configurations where only small distortions of the bond occur. Nonetheless, unless a structure is perturbed to extreme C-H distances (beyond 0.2 Å), the harmonic potential represents the potential energy for the bond quite well. Non-bonded interactions, as discussed above, are usually ignored once a bond has been defined between two atoms. A harmonic potential is typically used to describe the angle bend component for a bonded system. Equation (6) provides this energy expression in terms of an angle bend force constant k2 and the equilibrium bond angle θo: E Angle Bend = k 2 (θ − θ o ) 2

(6)

This expression necessarily requires a triad of sequentially bonded atoms, such as H-O-H in water or H-C-H in methane, where θ is the measured bond angle for the configuration. As with the harmonic potential for bond stretch, deviations from an equilibrium value will increase the energy and destabilize the configuration. The final bonded term of Equation (1) is that for the four-body torsion dihedral interactions. The dihedral angle ϕ is defined as the angle formed by the terminal bonds of a quartet of sequentially bonded atoms as viewed along the axis of the intermediate bond. An example of the analytical expression for the torsion energy is provided by: ETorsion = k 3 (1 + cos 3ϕ )

(7)

where k3 is an empirical force constant. The use of the trigonometric function ensures that a periodicity is followed for the dihedral angle variation, which is related to the atomic orbital hybridization of the intermediate atoms (e.g., 120° period for sp3 hybridization). The geometry measurements for the bond angle and torsion terms are represented in Figure 4 for the case of methane and dichloroethane.

10 

Cygan 

  Dica hteor l en

Mea ht en C-C ra dehi l axis

θ ro

ϕ

Figure 4. Geometry parameters for bond stretch and angle bend as noted on the energy-minimized structure of methane (left), and for bond torsion for dichloroethane (right). The axis used for defining the torsion angle is indicated along C-C bond in the energy-minimized structure of dichloroethane (upper right) with ϕ = 180° for the Cl-C-C-Cl torsion. A less stable configuration based on a smaller dihedral angle is presented in a conformer structure viewed looking down the C-C bond (bottom right).

Additional terms can be added to the total potential energy expression of Equation (2), such as an out-of-plane stretch term for systems that have a planar equilibrium structure (e.g., CO32- groups). More sophisticated energy forcefields, usually involving well-characterized organic systems, often incorporate cross terms among each of the bonded energy terms in order to accurately model the experimental vibrational frequencies of molecules. Unfortunately, details on these complex modes of interaction for most geological materials are unknown—th eir contributions are quite small—and therefore the cross terms are ignored in the parameterization. Finally, external perturbations to the molecular system can be included in the total potential energy expression. These include energy terms for the addition of a hydrostatic pressure or for directional stresses and electric fields. Atomic charges

Atomic charges are an integral part of any energy forcefield and are not to be assigned arbitrarily. The non-bonded Buckingham potential typically incorporates a full ionic charge to represent the charge on the atom. The inclusion of a shell model in the Buckingham potential requires that the ionic charge be proportioned between the core and shell components to collectively produce the full ionic charge. Molecular models relying on a bonded potential will always be represented by reduced partial charges. A bonded potential assumes that the Coulombic energy associated with an atom is reduced by the transfer of the valence electrons to the bond. The bond stretch energy is introduced to represent this contribution, thereby requiring that the charges on the atoms be reduced. There are various schemes available to assign these partial charges, one of which is the charge equilibration scheme of Rappé and Goddard (1991) based on the geometry, ionization potentials, electron affinities, and radii of the component atoms. There are other simpler empirical schemes that use the coordination, connectivity, and bond order to assign partial charges. Experimental approaches, usually based on deformation electron densities derived from high-resolution X-ray diffraction analysis, often provide

Modeling in Mineralogy & Geochemistry

11

accurate charge values (Coppens 1992; Spasojevicde-Bire and Kiat 1997). However, the most helpful and convenient approach for charge assignment relies on high-level quantum mechanical calculations. Typically, these calculations are performed on clusters or simple periodic systems that best represent the chemical environment. The electrostatic potential (ESP), derived from the electron densities, are then used in a leastsquares fit to obtain the optimum atomic charges that reproduce the electrostatic potential. Programs such as CHELPG (Chirlian and Francl 1987; Breneman and Wiberg 1990) are helpful in obtaining these ESP-based atomic charges. Similarly, Mulliken electron analysis (Mulliken 1955) can be used to derive atomic charges based on the populations of the molecular orbitals and contributing atomic orbitals, however, this method is less sophisticated and often leads to ambiguous charge assignments. Practical concerns

The exact nature of the analytical functions used to express any of the potential energy components is not the critical point of this discussion. It is important that the parameterization be as accurate as possible toward reproducing the observed data (experimental or quantum-based) to ensure that the molecular simulation reproduces the correct energies (and approximate shape of the energy surface) for the molecular model. A greater number of parameters for an energy function may ensure a more accurate representation, however, the computational cost may become prohibitive as the more complex functions are evaluated at each stage of a simulation, often over a million times. Methods to reduce the computational effort, especially for large molecular systems and simulation cells, are required. Additionally, symmetry, cell constraints, or fixed atomic positions can be incorporated in the molecular mechanics simulation. In theory, quantum chemical methods could be used to calculate the potential energy surface of a system and therefore forego with the parameterization of a forcefield. Essentially, the Schrödinger equation is solved to obtain a set of molecular orbitals that represent the lowest energy state for the molecule or periodic system. However, in practical terms, the computational cost becomes prohibitive, especially for large systems (typically greater than 20 atoms), as numerous geometries and configurations require calculation of their electronic structure and potential energy. Even for the case of approximate or semi-empirical quantum methods, or those using a limited atomic basis set, energy calculations would be impractical for most molecular modeling needs. Nonetheless, some progress has been made in this research area, specifically in quantum dynamics simulations using massively-parallel computers (see below). MOLECULAR MODELING TECHNIQUES Conformational analysis

One of the more valuable uses of molecular mechanics is the ability to test the energetics and relative stabilities of various molecular configurations. Conformational analysis provides a means of monitoring the relative stabilities of various conformers for a molecular system. Conformers, or conformational isomers, represent the various arrangements of atoms that can be converted into one another by rotation about a single bond. Figure 5 provides an example of the relative stabilities of various conformations of the carcinogen dichloroethane based on the torsional rotation about the carbon-carbon bond. The 1,2-dichloroethane isomer, also known as dichloroethylene, has several stable configurations represented by the three minima in the total energy plot (lower part of Fig. 5). The lowest energy conformer is the anti configuration where the chlorine atoms are furthest apart. The other two minima are associated with conformers in the stable gauche configuration where the one chlorine atom is staggered between the other chlorine and a

12 

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8

Van der Waals

6

4

Poa itne l Eng re y (kcal/)lom

Torosni 2

0

Coulombic

-2 15

Toa t l Eng re y (kcal/)lom

10

5

0

-5

0

60

120

Tosrnoi Angel (

180

240

300

360

ϕ)

Figure 5. Component (upper) and total (lower) potential energy for dichloroethane as a function of the torsion angle defined by Cl-C-C-Cl. Structural models corresponding to the three stable conformers (two local minima and one global minimum) and the least stable transition structure are provided in the total energy plot.

hydrogen. The least stable conformer is the transition configuration having the chlorine atoms fully eclipsed. The upper part of Figure 5 presents the components of the total potential energy as a function of the torsion angle. The component energies were obtained using the forcefield parameters of Dauber-Osguthorpe et al. (1988) in which seven bonds, twelve angles, and nine torsion terms, in addition to the nonbonded Coulombic and van der Waals energies, were evaluated for each molecular configuration. Bond distances and bond angles were kept fixed while evaluating the energy changes associated with the carbon-carbon torsion. The coincidence of the component energy minima, especially with the strong influence of the Coulombic energy, helps to stabilize the anti configuration. The shortrange repulsive component of the van der Waals energy controls the destabilization of the eclipsed configuration. The relatively small energy barriers associated with the gauche to anti transitions (approximately 2 kcal/mol) suggest that at room temperature all three of the most stable conformers would exist. This assumes the forcefield is accurately

Modeling in Mineralogy & Geochemistry

13

representing the enthalpy of these interactions. The anti to gauche transition has an energy barrier of 4.5 kcal/mol and would also occur at room temperature. In contrast, the large energy barrier associated with the eclipsed conformation is substantial and one would expect significant inhibition toward this transition. Although, at first, this example might be considered chemically intuitive, the use of molecular mechanics provides a strong theoretical basis to evaluate and identify the contributing components that control the stabilization of the molecule. Furthermore, larger and more complex molecules and periodic systems that have significantly greater configuration possibilities are only amenable to conformational analysis through computational methods. Sampling of optimal configurational space for large systems becomes more of a fine art than a simple matter of brute force energy calculations. Techniques such as Monte Carlo analysis and thermal annealing assist in this sampling effort, and are discussed later in the chapter. Energy minimization

Energy minimization, also referred to as geometry optimization, is a convenient method in molecular mechanics (and quantum mechanics) for obtaining a stable configuration for a molecule or periodic system. The procedure involves the repeated sampling of the potential energy surface until the potential energy minimum is obtained corresponding to a configuration where the forces on all atoms are zero. The energy of an initial configuration is first determined then the atoms (and cell parameters for a periodic system) are adjusted using the potential energy derivatives to obtain a lower energy configuration. This procedure is repeated until defined tolerances for the energy difference and derivatives between successive steps are achieved. Careful attention is needed for complex systems where structures associated with local energy minima may be obtained rather than the most stable configuration at the true global energy minimum. Multiple initial configurations or more advanced modeling techniques are required to ensure the attainment of the global energy minimum structure. Several algorithms are typically used in energy minimization procedures. Line searches and steepest gradient methods, and the more complex conjugate gradient and Newton-Raphson methods are often used in this effort. They can be used independently or collectively to obtain the lowest energy configuration. The Newton-Raphson approach evaluates both first and second derivatives of the energy to identify an efficient search path for locating the energy minimum configuration. Leach (1996) provides an excellent description of the various energy minimization techniques. Special conditions or constraints on the chemical system can be imposed during the energy optimization or other molecular simulation. Molecular and crystallographic symmetry can be constrained during the optimization or the atomic positions can be fixed. Periodic systems can have all cell parameters vary to simulate constant pressure conditions so that no net force occurs on the simulation cell boundaries. Fixing the cell parameters corresponds to a constant volume optimization, but this may result in the significant buildup of forces on the cell faces, especially if the cell parameters are far from their equilibrium values. A successful energy optimization is often performed without constraints of any kind. For a periodic system, this corresponds to a simulation cell having P1 symmetry where there is no symmetry imposed on the atomic positions (other than translational symmetry) and all six cell parameters are allowed to vary. Lattice dynamics simulations provide a powerful extension of energy minimization methods by evaluating the dynamical matrix that relates forces and atomic displacements for a crystal. Originally developed by Born and Huang (1954), this method incorporates a statistical mechanics approach to determine the vibrational modes and thermodynamic

14 

Cygan 

 

properties of a material. Examples of lattice dynamics calculations are noted in a later section of this chapter, and Parker et al. (this volume) presents a detail discussion of the technique for use in examining various minerals and mineral surfaces. Energy minimization and classical-based equilibrium structures

An example of how two charged atoms interact to form an equilibrium configuration is provided in Figure 6 for the case of an isolated magnesium and oxygen. A Buckingham potential is used to describe the interatomic potential based on the rigid ion parameters of Lewis and Catlow (1986) and Jackson and Catlow (1988) and which use full formal charges for the atoms. The total potential energy expression is given by: E MgO = k

q Mg qO rMgO

+ A exp(−rMgO / ρ ) −

C

(8)

6 rMgO

where k is a unit conversion factor, and ρ and C are empirical parameters. The short-range contribution to the potential energy is positive and rapidly increases at short distances. The Coulombic energy associated with the oppositely charged ions is negative and leads to stabilization as the two ions approach each other. The summation of the two terms provides the total energy characterized by an energy minimum that corresponds to the equilibrium separation distance for the atoms. Alternatively, partial charges less than the formal charge can be used to describe the same Mg-O interaction. Figure 7 presents a comparison of the full charge Mg-O and O-O Buckingham potentials (Lewis and Catlow 1986; Jackson and Catlow 1988) with those derived from quantum methods and using partial charges (Teter 2000). The latter potential uses reduced charges of qMg = 1.2 and qO = -1.2. Both sets of OO potentials are included to show the destabilization of similarly-charged ions with decreasing distance, where the total energy does not exhibit an energy minimum. In contrast, the Mg-O total energy curves exhibit minima denoting equilibrium distances of 1.48 Å and 1.75 Å, respectively, for the full charge and partial charge potential models. Note that these distances are significantly shorter than the Mg-O bond distance (2.10 Å) in 1000

Buckingham Potential Shtro rag n e

500

Poa itne l Enregy (kcal/)lom

0

Total Energy -500

Coulombic -1000

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Distance (Å) Figure 6. Potential energy as a function of separation described by a Buckingham potential for Mg-O ionic interactions. The total energy curve is characterized by a minimum corresponding to the equilibrium separation distance.

15

Modeling in Mineralogy & Geochemistry 1000 O-O Ful chag r e

Total Energy

500

O-O Partial chag r e

Poitna e l Eng re y (kcal/om)l

0

Mg-O Partial chag r e -500 Mg-O Ful chag r e

-1000

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Distance (Å) Figure 7. Comparison of partial charge and full ionic charge Buckingham models for the potential energy of Mg-O interactions as a function of separation distance.

crystalline periclase (MgO). The full charge potentials for Mg-O and O-O are characterized by larger contributions of the Coulombic energy leading to greater destabilization for the OO interaction and a deeper potential well for the Mg-O interaction. Yet, given these differences in interatomic potentials, both sets of forcefield potentials provide excellent results for the simulating the crystalline structure of periclase. The results for periclase simulations are presented in Figure 8 where the total potential energy is plotted as a function of periclase cell parameter. Periclase has the rock salt structure and is characterized by perfect regular octahedral coordination. Because of the high symmetry limiting structural variation to only the cell parameter, calculation of the total potential energy—which for crys talline materials is known as the lattice energy—is a straightforward matter. Both sets of forcefield parameters provide similar energy-minimized structures with nearly identical cell parameters that are in excellent agreement with the observed value of 4.211 Å (Hazen 1976). Also, both potential sets provide similar shapes for the energy-distance curves, of which the curvature represents the vibrational characteristics of the material. Two additional sets of potentials that incorporate a shell model to describe the oxygen polarization provided comparable results. The significant displacement of the full charge potential to lower energy is related to the greater significance of the Coulombic term in the full charge potential. In contrast to molecular systems where bonded forcefields are typically used, it is difficult to relate the forcefield parameters used in non-bonded potentials to the results of calculations where the long-range Coulombic forces are strong and occur across all anion-anion and cation-cation interactions, and not just cation-anion pairs. This is the reason for the large differences in distance values at the energy minima for the two-atom examples in Figure 7 and those for the full crystal periodic simulations of Figure 8. Quantum chemistry methods

The application of quantum mechanics to topics of mineralogical and geochemical interest is perhaps the most intriguing and challenging task for computational chemists. In implementing these electronic structure calculations, the modeler is no longer restricted to

16 

Cygan 

 

-185

ao = 4.195 Å

Partial chag r e

-190

Ponea ti l Eng re y (kcal/)lom

-195 -460

-465

-470 3.6

ao

ao = 4.211 Å

3.8

4.0

4.2

Ful chag r e

4.4

4.6

4.8

Cel Parametr (Å) Figure 8. Comparisons of potential energy for the crystal structure of MgO as a function of cell parameter ao based on partial charge and full charge Buckingham potentials.

the classical description of using the balls and springs of molecular mechanics methods to describe the complex interactions of atoms and molecules. Now, by solving the Schrödinger equation for larger and more complex systems, albeit through approximate methods, the quantum chemist can obtain energies, molecular and crystalline structures and properties, electrostatic potentials, an analysis of spectroscopic data, thermodynamic properties, a detailed description of reaction mechanisms, and non-equilibrium structures. A quantum chemistry approach brings the electrons to the forefront of the molecular model by allowing the modeler to probe the distribution of electrons among the mathematical wavefunctions that describe the molecular orbitals for the system. The time-independent Schrödinger equati on is given by the following eigenfunction relation: HΨ = EΨ

(9)

where H is the Hamiltonian differential operator, Ψ is the wavefunction, and E is the total energy of the system. The Hamiltonian is comprised of kinetic and potential energy components just as in a classical mechanics. Equation (9) can therefore be restated as: ⎛ h2 ⎜− ⎜ 8π 2 ⎝

1

∑m ∇ +∑ i

i

2

i≠ j

ei e j ⎞ ⎟ Ψ = EΨ rij ⎟⎠

(10)

where h is Planck’s constant, m is the mass, ∇2 is the Laplacian operator, and e is the charge of the particles (either electrons or nuclei) at separation distance rij. The second term of this expression represents the potential energy associated with the Coulombic interactions of all nuclei and electrons of the system. There are several restrictions on the nature of the wavefunction in order to satisfy the Schrödinger equation for electronic structure calculations (e.g., symmetry, Pauli exclusion, and choice of eigenstates). Additionally, the wavefunction provides the critical role in determining the probability

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distribution function for electrons in configurational space (i.e., orbital geometry), and for obtaining the energy of the system as the expectation value of the Hamiltonian. Unfortunately, Equation (10) has an exact analytical solution for only the one electron system, and therefore approximations must be made to apply quantum mechanics to the many-electron systems of molecules and materials of interest. The Born-Oppenheimer approximation that effectively decouples nuclear and electronic motions, and the combining of one-electron orbitals to describe the total wavefunction contribute to this effort. Excellent discussions of the various quantum methods that are commonly used today to solve the Schrödinger equation are provide d in several review articles and textbooks. Among those that are noteworthy, especially with regard to their readability and application to inorganic and crystalline materials, are Hehre et al. (1986), Labanowski and Andzelm (1991), Springborg (1997), and especially the recent book of Cook (1998). The comprehensive volume by Tossell and Vaughan (1992) is very helpful in providing numerous geochemical examples involving quantum methods. Of course, several of the following chapters in this book provide a state-of-the-art perspective on quantum methods and applications to the geosciences. Also of special note are the reviews of Gillan et al. (1998) and Billing (2000) in which they discuss the role of quantum chemistry in modeling surfaces and molecule-surface interactions. Lasaga (1992) presents a similar review but with particular application to mineral surface reactions. Quantum chemistry methods can be divided into four distinct classes: ab initio Hartree-Fock methods, ab initio correlated methods, density functional methods, and semi-empirical methods (Hehre 1995). Ab initio refers to “from the beginning”, and consequently these first principles methods do not use any empirically or experimentallyderived quantities. Hartree-Fock methods use an antisymmetric determinant of oneelectron orbitals to define the total wavefunction. Electrons are treated individually assuming the distribution of other electrons is frozen and treating their average distribution as part of the potential. The wavefunction orbitals and their coefficients are refined through an iterative process until the system reaches a steady result, or selfconsistent field. Correlated methods extend the Hartree-Fock approach by introducing a term in the Hamiltonian that corrects for local distortion of an orbital in the vicinity of another electron. The Hartree-Fock approach assumes the entire orbital is affected in an averaged sense. Standard Hartree-Fock methods still perform quite favorably in predicting equilibrium geometries compared to correlated or density functional methods, however the lack of electron correlation typically leads to inaccurate force constants and vibrational frequencies. Perturbation calculations associated with the correlated methods can often become quite costly for the sake of improving calculations to this level of accuracy. Gibbs (1982), and Lasaga (1992) provide insightful reviews of applications of ab initio methods to mineralogy and geochemistry. The third class of quantum methods includes those based on density functional theory (DFT) that incorporate exchange and correlation functionals of the electron density based on a homogeneous electron gas, and evaluated for the local density of the system. The density of the electrons rather than the wavefunction is used in DFT to describe the energy of the system. The theory was developed in the early 1960’s (Hohenberg and Kohn 1964; Kohn and Sham 1965), and led to the awarding of 1998 Nobel Prize in chemistry to the Walter Kohn. A general review of DFT methods and applications is provided by Jones and Gunnarsson (1989). The local density approximation (LDA) provides quite accurate results for a wide range of molecules and crystalline systems (Kohn and Sham 1965). A more sophisticated refinement of DFT is the generalized gradient approximation (GGA) in which the gradient of the charge density is utilized (Perdew et al. 1996). DFT methods have become the method of choice

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in recent years among computational chemists primarily due to the economy in efficiently scaling with the number of electrons in the system—at N 3 compared to N4 or greater for standard Hartree-Fock methods. Plane-wave pseudopotential methods, originally developed by the solid-state physics community, provide a computationally efficient DFT approach for periodic systems in which only the valence electrons of the atoms are explicitly treated, and represented by a plane-wave expansion. Teter et al. (1989), Payne et al. (1992), and Milman et al. (2000) provide excellent reviews of the theory and applications of plane-wave pseudopotential and DFT methods to large-atom periodic systems. Additionally, a hybrid quantum approach that combines the electron densities derived from standard Hartree-Fock theory with the DFT functionals has also been widely used (Gill et al. 1992; Oliphant and Bartlett 1994). The semi-empirical methods involve some empirical input into obtaining approximate solutions of the Schrödinger equa tion. Typically, this class of approximate methods avoids the computational cost of evaluating the numerous electron repulsion integrals that make ab initio methods so computationally expensive. A general description of the various semi-empirical methods is provided by Pople and Beveridge (1970). Because of the success of DFT methods and access to faster and more powerful computers, in addition to the inaccuracies and limitations of the approach, semi-empirical methods are no longer as common in the chemistry literature as they were twenty years ago. Energy minimization and quantum-based equilibrium structures

As with classical molecular mechanics, quantum methods provide a means for obtaining equilibrium configurations based on an analysis of the total energy using a minimization procedure. Figure 9 presents the energy-minimized structure obtained from the gas phase analysis of methane (isolated molecule), for comparison with that derived from classical methods (cf. Fig. 4). A DFT approach involving GGA functionals and a

1.5

1.098 Å

1.0

H

0.47 e/Å

3

0.5

109.47°

C

0

-0.5

0.1 e/Å

3

H

-1.0

0.09 /e Å

3

-1.5 -1.0

-0.5

0

0.5

1.0

Figure 9. Energy-optimized structure of methane (left) derived from a high-level DFT calculation and showing the extent of the 0.1 e/Å 3 electron density contour superimposed onto the ball-and-stick representation of the molecule. A slice of the electron density taken through one of the C-H-H planes (right) shows the covalent nature of the C-H bond with the buildup of charge along the C-H axes.

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double numeric basis set including polarization functions was used to calculate the equilibrium structure (Delley 1990). Geometry optimization was based on an efficient gradient scheme that involves the internal coordinates of the molecule (Baker 1993). The calculated C-H bond distances and H-C-H bond angles are in excellent agreement with experimental values. As expected, the C-H bond distance is similar to the equilibrium value associated with the Morse potential as described earlier for use in a classical forcefield. Analysis of the final wavefunctions for the optimized methane structure provides the electron density, or charge density, for determining the distribution of the ten electrons in the molecule. The extent of the 0.1 e/Å 3 isosurface is superimposed on the usual ball-and-stick model of methane in Figure 9. Also shown in Figure 9 is a slice of the electron density taken through one of the H-C-H planes. Both diagrams indicate the diffuse and asymmetric distribution of electrons in the molecule with electron buildup along each of the C-H bond axes, representing the strongly covalent bonds of methane, and at the atomic nuclei, representing the less significant role of the inner electrons in the molecular bonding. Evaluation of the wavefunctions and electron densities provide theoretical dipole moments, optical polarizabilities, electrostatic potentials, atomic charges, and spatial distributions of the molecular orbitals. Frontier orbital theory based on the analysis of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) provides insights into molecular reactivity as these are the orbitals most commonly involved in chemical reactions (Hehre 1995). Further analysis of molecular bonding using the Laplacian of the electron density to determine bond critical point properties and valence shell electron pair repulsions has been useful in evaluating the structure and reactivity of molecules and crystalline materials (Bader 1990; Gibbs et al., this volume). More sophisticated electronic structure calculations requiring gradient calculations (energy with respect to atomic displacement) for an energy-minimized structure are useful in obtaining vibrational frequencies (Kubicki, this volume) and NMR chemical shifts (Tossell, this volume). Additionally, transition states for reactive molecular configurations can be determined by identifying transition state maxima in the potential energy surface associated with the nearby stable minima (see Felipe et al., this volume). The results of a plane-wave pseudopotential DFT calculation for periclase are presented in Figure 10. The periodic structure was optimized using the GGA method with

Mg

4

O 0.13 /e Å

Mg 3

3

O

2

O

Mg

1 0.47 /e Å

Mg

0 0

3

O 1

2

Mg 3

4

Figure 10. Slice of the electron density of MgO obtained from an optimization of the periodic structure using a nonlocal DFT approach with planewave pseudopotentials. The development of charge density and associated critical points between Mg and O atoms indicates the existence of covalent character in this material.

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ultrasoft potentials and a kinetic energy cutoff of 380 eV for the plane-wave expansion (Payne et al. 1992; Teter et al. 1995). Energy minimization was performed using the BFGS scheme described by Fischer and Almlöf ( 1992). The optimized structure for MgO has a cell parameter of 4.270 Å that is slightly la rger than the observed value of 4.211 Å (Hazen 1976) but is still respectable for the plane-wave pseudopotential technique. Part of this discrepancy is related to the use of the pseudopotential to describe the core electrons. Although the calculation is computationally faster than all-electron methods, there is a slight loss of accuracy in obtaining correct geometries. A slice of the electron density that passes through the atomic centers (Fig. 10) indicates that the charge is, as expected, lowest between Mg-Mg and O-O pairs. Most noteworthy in the electron density is the slight buildup of charge between neighboring Mg-O pairs suggesting the existence of some covalent bonding due to the overlap of orbitals. A Mulliken population analysis of the electron density and orbitals suggests approximately 30% covalent character for this material, although this may be high compared to experimental evidence (Souda et al. 1994). A purely ionic compound would exhibit spherically-symmetric charge density contours about the nuclei without any directional structure of the contours between atoms. As with molecular systems, similar analysis of the wavefunctions and electron densities for periodic systems can help in evaluating various physical properties of the solid. These include electrostatic potential, spatial distribution of molecular orbitals including HOMO and LUMO, transition states, and vibrational frequencies. Additionally, equations of state and bulk moduli for the material can be derived from energy-volume curves (Cohen 1991; Stixrude et al. 1998; Stixrude, this volume). Geometry optimizations can be performed with fixed cell parameters (constant volume conditions), or the six lattice parameters can be allowed to vary (constant pressure conditions). Crystallographic symmetry can be imposed to constrain the atomic positions to symmetry sites during the energy minimization. Because of the high computational costs of obtaining fully optimized periodic structures with quantum chemistry codes, the use of space group symmetry and other constraints is extremely important. Monte Carlo methods The stochastic analysis of the energetics of a chemical system is best represented by a Monte Carlo scheme in which a random sampling of the potential energy surface is performed in order to obtain a selection of possible equilibrium configurations. Monte Carlo-based molecular simulations predate molecular dynamics methods having been first introduced by Metropolis et al. (1953) for deriving the equation of state for a system comprised of two-dimensional rigid spheres. This approach obviates the need to calculate an entire regular array of configurations within a canonical ensemble; only a random sampling is required. The basics of the so-called Metropolis Monte Carlo method is described by the flow diagram presented in Figure 11. After an initial configuration for a system is defined and the total potential energy is determined, the model is randomly displaced (positioned) to a new configuration and a new energy is calculated. If the new configuration is more stable than the original, then the configuration is accepted and the spatial displacement operation is continued again. However, if the new configuration energy is greater (less stable) than that for the original configuration, then the energy difference as part of a Boltzmann distribution is compared to a random number. If the value is less than the random number, then the configuration is accepted and used as the basis for a new displacement. If, however, the value is greater than the random number, then the new configuration is rejected and the previous configuration is used for the next displacement. The option to selectively accept initially less stable configurations ensures that the potential energy surface is fully sampled within the given stochastic constraints.

An excellent example of a Monte Carlo approach in molecular modeling is provided

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Inia t l Configuration Displacement Accept Yes

Enew < Eold Reject

No

exp ⎜⎜⎜ − ΔE ⎟⎟⎟ < rand (0,1) kT ⎠ ⎝ ⎛

Yes

⎞

No

Figure 11. Flow diagram for the generalized Monte Carlo method in which a very large number of molecular configurations are compared to derive an optimal set of energetically-favored configurations.

by Newsam et al. (1996) in their determination of the cation positions in zeolite materials. Knowledge of the positions of alkali metal cations in the aluminosilicate framework zeolites is vital to the design and control of the sorptive and catalytic properties of these industrially-important materials. X-ray diffraction determination of the optimal sites is often tedious and difficult due to the lack of quality single crystals, so the molecular simulation approach provides a convenient alternative. The simulation cell of the synthetic zeolite A (NaSiAlO4) is comprised of an equal amount of Al and Si atoms forming the framework structure having the characteristic zeolite rings and channels, and with twelve sodium ions counterbalancing the negative framework charge. The Monte Carlo packing simulation (Freeman et al. 1991) starts with a fixed framework and a potential energy surface defined by a set of Coulombic interactions and short-range interaction terms. Twelve Na ions are successively introduced into the framework structure ensuring that each new configuration leads to an acceptable energy via the scheme presented in Figure 11. Several thousand configuration attempts can be used to ensure that a statistically-sound sampling of the framework potential surface has been probed while avoiding any bias in identifying the Na ion sites. Further refinement of the thirty most favorable configurations was then performed using standard energy minimization techniques to arrive at eleven favorable configurations that agree with the experimental structure (Pluth and Smith 1980) having 8 Na+ on the six-membered rings, 3 Na+ on the eight-membered rings, and 1 Na+ adjacent to one of the four-membered rings (see Fig. 12). Similar Monte Carlo approaches have been successfully used to characterize the sorptive properties of zeolites for alkanes (Smit and Siepmann 1994; Smit 1995; Nascimento 1999; Suzuki et al. 2000), for aromatic organic compounds (Bremard et al. 1997; Klemm et al. 1998), for water (Channon et al. 1998), and for the sorption and

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6 6

6 6 4 8

8

6 6

 

Figure 12. Perspective view of zeolite A showing one of the low energy configurations for the distribution of twelve Na ions within the structure as determined using a Monte Carlo sampling approach. The label on each Na ion represents the size of the Si-Al ring structure that the cation is associated with.

6 6 8

transport rates of inorganic gases (Douguet et al. 1996; Shen et al. 1999). Grand canonical methods are often implemented in these Monte Carlo studies to ensure a constant chemical potential μ during the simulation. Use of the μVT ensemble allows for a computationally fast approach for attaining an equilibrium configuration, especially for a model that includes multiple phases such as the simulation of a gas or a fluid interacting with a solid. The temperature and chemical potential are externally imposed and the number of atoms or molecules is allowed to vary during the simulation. Details of grand canonical methods for use in Monte Carlo and molecular dynamics simulations are provided in Allen and Tildesley (1987) and Frenkel and Smit (1996). The extensive literature on the molecular simulation of zeolites attests to the vast number of industrial applications requiring unique catalysts, nanoporous materials, and molecular sieves. Monte Carlo simulations have been similarly used to analyze the structure of species in the interlayer of clays. The structure and dynamics of interlayer water molecules and solvated cations are difficult to ascertain through conventional experimental and spectroscopic methods. In part, these difficulties are related to 1) their extremely fine grain size (typically less than 1 μm) of clay minerals; 2) their low crystallographic symmetry; 3) their complex chemistry with multiple components, cation disorder, and vacancies; and 4) the occurrence of stacking disorder that precludes long range ordering. Therefore, simulation methods, and, in particular, Monte Carlo techniques, are often used to develop a model for the detailed atomistic structure of the clay. Simulations of the swelling behavior of smectite clays have become quite commonplace in the mineralogical literature (e.g., Delville 1991; Skipper et al. 1991; Delville 1992; Beek et al. 1995; Chang et al. 1995; Delville 1995; Skipper et al. 1995a; Skipper et al. 1995b; Karaborni et al. 1996; Chang et al. 1997; Greathouse and Sposito 1998; Sposito et al. 1999). Recently, Spositio et al. (1999) determined the optimum positions of water molecules and cations in the expanded two layer hydrate of Na- and K-montmorillonite. The simulations involve several stages of generating acceptable Monte Carlo configurations based on the movement of water molecules, interlayer cations, and clay layers. The K-montmorillonite simulations required more than 1,700,000 steps to attain a data set suitable for evaluating the optimized configuration of interlayer water and cations. Radial distribution functions for interlayer water derived from their simulation results suggest a strong influence of the smectite tetrahedral sheets in modifying the tetrahedral coordination that exists in bulk

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water. This effect was more pronounced for the K-montmorillonite where the weak solvation of K+ is more readily influenced by the clay layers. Molecular dynamics methods

Molecular dynamics simulation is a deterministic technique to model the equilibrium and transport properties of a chemical system based on a set of interatomic potentials or forcefield terms. A large assemblage of atoms can be examined as either a cluster or periodic system whereby Newton’s equations of motion involving forces and velocities are iteratively solved to provide a classical description for a many-body system, here comprised of atoms. A molecular dynamics simulation first requires the input of an initial configuration for the system with an assignment of a velocity for each of the atoms. Usually a Boltzmann distribution of velocities is initially imparted onto all or a subset of the atoms contained in the simulation cell. The velocities are then scaled to provide the appropriate mean kinetic energy for the system to meet the desired temperature. Forces are derived based on the given forcefield, and then the equations of motion are integrated over the selected time interval. Time increments, usually on the order of a femtosecond or less, are then chosen so that all atomic motions are resolvable for the time step (i.e., the time increment is significantly less than the period of any vibrational mode associated with the model). Typically, the Verlet algorithm (1967) or similar method is used to calculate the new atomic positions and velocities that are then used to loop through the integration for the next time step. The procedure is repeated for a large number of iterations, typically on the order of several hundred thousand times, allowing the system to evolve to an equilibrium configuration (tens to hundreds of picoseconds of simulation time). Values for the temperature, and potential and kinetic energies can be evaluated throughout the molecular dynamics simulation via instantaneous or running averages. An NPT canonical ensemble (isobaric and isothermal with a constant number of atoms) can be used for the simulation of an unconstrained periodic system, allowing for the examination of the pressure and density of the simulation cell as a function of time. Allen and Tildesley (1987), Frenkel and Smit (Frenkel and Smit 1996), and Haile (1997) provide excellent descriptions of the procedures associated with a molecular dynamics simulation. Molecular dynamics simulations overcome some of the limitations associated with energy minimization schemes by allowing the kinetic energy of the system to assist atoms in better sampling of the potential energy surface. In this respect, molecular dynamics comes closest to describing the many aspects of a real experiment. Although the goal of optimizing a molecular configuration through the static energy minimization approach is to attain the most stable configuration associated with the global energy minimum, the method does not allow one to monitor the evolution of the chemical system. Temperature is explicitly incorporated in a molecular dynamics simulation and the kinetic energy assists molecular and atomic motions to overcome potential energy barriers. Thermal annealing methods allow a wide range of potential molecular configurations that would be inaccessible through the standard energy minimization technique. Impulse dynamics methods are often used to direct the transport of atoms or molecules toward a reactive site or through a diffusion pathway. Additionally, thermodynamic integration and analysis of various ensemble averages at state points can be used to derive thermodynamic properties (Allen and Tildesley 1987). This is of particular significance for grand canonical ensembles, where estimates of Gibbs and Helmholtz free energies and entropy values can be derived. Applications of molecular dynamics in mineralogy and geochemistry are often associated with the simulation of the structure and transport properties of fluids and melts due to the relatively rapid dynamics of the species in these systems. Melt and glass

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simulations (e.g., Kubicki and Lasaga 1988; Belonoshko and Dubrovinsky 1995; Stein and Spera 1995; Chaplot et al. 1998; Nevins and Spera 1998) have often provided atomistic details of mineral/melt systems at extreme conditions that are not necessarily observable under laboratory conditions. Fluid behavior at ambient, hydrothermal, and supercritical conditions have been successfully modeled using molecular dynamics simulations (e.g., Brodholt and Wood 1990, 1993; Duan et al. 1995; Kalinichev and Heinzinger 1995; Driesner et al. 1998; Driesner and Seward 2000). Although less amenable to the modeling technique due to the larger time scale associated with solids, molecular dynamics simulations of mantle phases have also helped to constrain phase transitions and their associated geophysical discontinuities (e.g., Matsui 1988; Miyamoto 1988; Matsui and Price 1992; Winkler and Dove 1992). Similarly, the molecular dynamics method has been successfully used to evaluate the structure and dynamics of water, interlayer cations, and environmental contaminants in clays (e.g., Teppen et al. 1997; Hartzell et al. 1998; Smith 1998; Teppen et al. 1998; Kawamura et al. 1999), and water and interlayer anions in layered double hydroxides such as hydrotalcite and other related phases (Aicken et al. 1997; Kalinichev et al. 2000; Wang et al. 2001). An example of the use of molecular dynamics to examine the behavior of interlayer waters in clay minerals is provided in the recent study of Cygan et al. (2001). A smectite clay corresponding to a Na-montmorillonite composition was simulated using a fully flexible forcefield, developed for clays and hydrous minerals, in which all atoms of the simulation cell were free to translate during the simulation. An NPT canonical ensemble provided complete freedom of the clay layers to expand with the sequential addition of water molecules to the simulation cell. The anhydrous system was first equilibrated for 40 picoseconds using 1 femtosecond time steps, then a single water molecule was added to the interlayer region of the smectite. Molecular dynamics was continued for 40 additional picoseconds before the addition of another water molecule and further equilibration, and so on until the smectite clay was expanded to more than 21 Å at a water content of 0.45 g H 2O/g clay, corresponding to the addition of 73 water molecules to the clay interlayer. Figure 13 presents the results of the mean basal d-spacing based on the last 20 picoseconds of simulation time for each smectite structure as a function of water content. The experimental water adsorption data for smectite (Fu et al. 1990; Berend et al. 1995) is also included to show the general agreement between the molecular dynamics simulation results and the experimental values. The fine detail of the expansion of the smectite layers is reproduced by the model as the first hydrate layer is introduced into the interlayer. The clay expands to approximately 12 Å with the initia l introduction of water and stays approximately at that value as water molecules fill in the interlayer voids and fully solvate the interlayer Na cations. A critical water amount is met at approximately 0.14 g H2O/g clay (23 water molecules in simulation cell) where the smectite expands to approximately 15 Å with formation of the st able two-layer hydrate. Each expansion of the clay represents the critical point where the energy of the clay layer expansion overrides the energy gain in forming a hydrogen bonded water network in the interlayer. The molecular dynamics simulations provide a basis for the continued expansion of the smectite clay with the addition of more water molecules. However, further expansion of the Na-montmorillonite beyond the 15 Å two-la yer hydrate is not observed in nature. Smith (1998) uses a molecular dynamics approach and the various representations of the hydration energy to demonstrate the relative stabilities of each of the stable hydration states for a Cs-montmorillonite. Grand canonical molecular dynamics and an analysis of the free energy of swelling were later used to confirm the stable clay hydration states (Shroll and Smith 1999).

25

Modeling in Mineralogy & Geochemistry 26 24

Sia lum itno

(001) d-Spacing (Å)

22 20 18 16 14

Exa tnemir p l

12 10 8 0.0

Na3(Si 31Al)(Al 0.1

0.2

0.3

14Mg2)O80(OH)16• nH 2O

0.4

0.5

MH O / Mclay 2

Figure 13. Swelling behavior for a smectite clay derived from molecular dynamics simulations of montmorillonite. The equilibrium d-spacing is presented as a function of water content of the clay. The plateaus in the experimental and simulation results at 12 Å and 15 Å represe nt the stabilization of, respectively, the one-layer (insert structure) and two-layer hydrates. No further expansion of the smectite is observed in nature beyond the two-layer hydrate. The simulations suggest that further swelling of the clay is possible although not thermodynamically favored.

Quantum dynamics

Perhaps the ultimate molecular modeling method available to date is that of quantum dynamics, or ab initio molecular dynamics, in which molecular dynamics and quantum mechanics methods are combined. Simple classical-based forcefields and interaction parameters are replaced by the more complex quantum methods of Hartree-Fock and DFT to determine the energy and forces of interaction for the system. Rather than rely on simple interatomic potentials to describe the complex many-body interactions, quantum dynamics solves the Schrödinger equation fo r each dynamics time step to explicitly obtain the electronic structure for the entire system. This approach dispenses with the inherent limitations of the empirical method for deriving interaction parameters and the uncertainty associated with knowing the range of validity. Furthermore, quantum dynamics allows a closer match to reality where the electronic properties and atomic dynamics are dependent. This is especially critical for reactive systems where dissociation and bond formation occurs on the time scale of the simulation. The quantum dynamics approach was first pioneered by Car and Parinello (1985; 1987) by combining accurate DFT methods with dynamics to examine the equilibrium structure of melts and amorphous semiconductors. As expected, due to the high computational cost of performing these simulations, most quantum dynamics studies are limited to short simulation times (on the order of one to two picoseconds) and relatively small simulation cells. An example of the technique as applied to silicon surfaces is presented in Terakura et al. (1997) while Radeke and Carter (1997) provide a review of molecule-surface interactions. A recent comprehensive review of quantum dynamics methods is provided by Tuckerman (2000).

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Although less common in the geosciences literature, quantum dynamics methods have been successfully used to examine the stabilities of potential phases of the lower mantle. The stability limits of MgSiO3 perovskite were derived by an optimization scheme using quantum dynamics with the local density approximation by Wentzcovitch (1993). Their simulations suggested the stability of the orthorhombic perovskite relative to the cubic phase increased with pressure (up to 150 GPa). The modeling approach was later used to examine the stability of the MgSiO3 ilmenite phase (Karki et al. 2000). The simulations suggested that the ilmenite phase would transform to the perovskite phase at 30 GPa. Haiber et al.(1997) examined the various phases of Mg2SiO4 (olivine and spinel polymorphs) and analyzed the dynamics of a sorbed proton at elevated temperatures (400 to 1600 K). Recently, in an application related to catalysis and environmental concerns, Hass et al. (1998) examined the dissociation of water on hydrated alumina surfaces. The quantum dynamics studies examined a relatively large simulation cell comprised of 135 atom alumina substrate that was subsequently hydrated at two different water coverages. The simulations indicated, within the one picosecond simulation time, water dissociation and proton transfer reactions between the adsorbed molecular water and the hydroxide surface. Similarly, Lubin et al. (2000) successfully used quantum dynamics to examine the solvation of hydrolyzed aluminum ions in water clusters and determine the mechanisms of proton transfer. FORSTERITE: THE VERY MODEL OF A MODERN MAJOR MINERAL

The crystal structure and physical properties of forsterite (Mg2SiO4) have been determined by a variety of molecular modeling methods and therefore are represented by a fair number of papers in the mineralogical literature. Forsterite, as the magnesium endmember of the orthosilicate olivine series, is the most abundant phase of the upper mantle of the Earth. The elastic properties of forsterite are expected to control the rheology of this region (Evans and Dresen 1991; Duffy and Ahrens 1992) and will influence plate tectonic processes of the crust, while the electrical conductivity of forsterite is critical to field investigations involving geomagnetic and magnetotelluric surveys (e.g., Jones 1999; Neal et al. 2000). The crystal structure of forsterite is depicted in Figure 14. This energy-optimized structure was obtained using a Buckingham potential with the partial charges and interaction parameters of Teter (2000) while maintaining

M2

Figure 14. Energy-optimized structure of the orthorhombic unit cell of forsterite (Mg2SiO4) obtained with an ionic model. Magnesium sites M1 and M2 and the silicon tetrahedron with O1, O2, and two O3 oxygens comprise the asymmetric unit.

O2

M1 O1 O3

O3

b c

a

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orthorhombic Pbnm symmetry during the optimization. Cell parameters and Mg-O and Si-O bond lengths are in excellent agreement with the experimental structure (Fujino et al. 1981). Static calculations and energy minimization studies

Early models of forsterite relied on classical molecular modeling methods to describe the interaction of ions using formal charges for the Coulombic interactions and empirically-derived parameters for the short-range interactions. Static energy calculations and energy minimization techniques were used to evaluate and optimize Mg-O and Si-O bond lengths and cell lengths of the orthorhombic unit cell. Lasaga (1980), Post and Burnham (1986), and Catti (1989) developed models that successfully mimicked the observed crystallographic structure of forsterite. Similarly, Matsui and Busing (1984) accurately modeled the forsterite structure but used a set of potentials based on Mg ions and rigid SiO44- groups (Matsui and Matsumoto 1982). The Catti model and Matsui and Busing models both provided reasonable values for the elastic properties of forsterite derived from the second derivatives of the energy matrix for the optimized structure. The Lasaga approach evaluated the point defect structure of forsterite and successfully predicted the anisotropic behavior of Mg diffusion (Lasaga 1980). An evaluation of cation site preference energies (M1 versus M2 octahedral site) for various endmember compositions of olivine was completed by Bish and Burnham (1984) using a combined approach of distance-least-squares method of structural analysis and lattice energy calculations. More recently, a molecular mechanics method was used to evaluate the surface structure and energies of forsterite (Watson et al. 1997). Surface energies obtained in the analysis of the various relaxed surfaces provided an accurate model of the crystal morphology. Lattice dynamics studies

Lattice dynamics modeling of forsterite has provided significant new insights into the dynamical nature of a complex silicate structure and the link between atomistic structure and macro-scopic thermodynamics. Lattice dynamics methods examine the interaction of lattice vibrations as weakly interacting phonons; a phonon being a particle representation of low frequency sound waves. Basically, a lattice dynamical model for a crystal is represented by a tensor that combines the coupling between forces and atomic displacements. Born and Huang (1954) and Wallace (1972) provide excellent comprehensive discussions of the basic theory of lattice dynamics. Combined with inelastic neutron scattering experiments, lattice dynamics provides a powerful tool for evaluating phonon dispersion, vibrational energies, and thermodynamic properties such as heat capacity and entropy. Early attempts by Iishi (1978) and Kieffer (1980) were successful in predicting the temperature dependence of the heat capacity and the vibrational spectrum for forsterite. Besides providing a strong theoretical basis for the experimental calorimetric studies on minerals in the 1970’s (e.g., Robie et al. 1978), this early theoretical work pioneered the way for more accurate lattice dynamics simulations of forsterite and related phases (e.g., Price et al. 1987; Rao et al. 1988; Patel et al. 1991; Kubicki and Lasaga 1992). Quantum studies

Quantum methods were first applied in the theoretical analysis of forsterite in the 1990’s due to the advances in computer processors and development of efficient quantum software programs for periodic systems. Computer technology had matured so that it was finally possible to routinely calculate the electronic structure of complex minerals using sophisticated quantum chemistry tools. A Hartree-Fock pseudopotential method was used by Silvi et al. (1993) to evaluate the relative energies of the Mg2SiO4 polymorphs and the

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local bonding environments. Brodholt et al. (1996) used a DFT approach and local density approximation to optimize the forsterite structure and to ascertain the compressional behavior of the phase up to 70 GPa. Wentzcovitch and Stixrude (1997) determined that the Mg octahedra and Si tetrahedra in forsterite compress nearly isotropically for pressures up to 25 GPa. They used the local density approximation and DFT method, in combination with a modified molecular dynamics approach, to obtain the optimized structure of forsterite at each pressure. The modeling results were in agreement with those of Brodholt et al. (1996) and confirmed that forsterite did not experience any changes in compression mechanism with pressure. This suggests the possibility that compression changes in the pressure medium, rather than in the forsterite crystal, were being measured in the experimental study (Kudoh and Takeuchi 1985). More sophisticated calculations of forsterite using a DFT approach with the generalized gradient approximation were recently reported by Brodholt (1997) and Winkler et al. (1996). The former study examined the energetics of various Mg and O defects in forsterite and determined that Mg diffusion was dominated by a diffusion pathway involving jumps between M1 sites. The results are in agreement with the classical approach used by Lasaga (1980) as discussed previously. The Winkler et al. (1996) study used the non-local DFT approach to obtain the electric field gradient tensors associated with NMR active nuclei in forsterite (25Mg and 17O). THE FUTURE

Molecular modeling has come a long way since John Dalton first used wooden balls in the early nineteenth century to represent molecular structures. Rapid changes in computer technologies and hardware, the introduction of the personal computer, the development of massively-parallel supercomputers, the use of new and efficient algorithms and visually-based programming, the intelligence of neural networks, and the ability of the internet to distribute complex computational problems across thousands, if not potentially millions of networked computers, have all influenced the rapid growth of computational chemistry over the last two decades. How further technological developments will affect how we do molecular modeling in the geosciences on larger and more complex chemical systems is uncertain. However, it is certain that molecular modeling theory and computational methods will play a more significant role in how mineralogists and geochemists examine the complex phases and processes of the Earth. ACKNOWLEDGMENTS

The content of this chapter benefited from discussions and reviews provided by James Kubicki, David Teter, and Henry Westrich. The author is appreciative of funding provided by the U.S. Department of Energy, Office of Basic Energy Sciences, Geosciences Research and the U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin company, for the United States Department of Energy under contract DE-AC04-94AL85000. Additionally, the author is extremely grateful to the many students, post-docs, colleagues, and collaborators who have contributed to the research efforts in using molecular simulations to understand the complex nature of minerals and geochemical systems.

 

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GLOSSARY OF TERMS Ab initio—First principles quantum mechanical a pproach for obtaining the electronic properties of a molecule based on the approximate solutions to the many-electron Schrödinge r equation, using only fundamental constants, and the mass and charge of the nuclear particles; literally “from the beginning”. Basis function—Functions describing the atomic orbitals that when linearly combined make up the set of molecular orbitals in a quantum mechanics calculation; Gaussian basis sets and Slater type orbitals are examples of basis functions. Born-Oppenheimer approximation—A method for separating electronic motion from that of the nuclei in quantum mechanics; the nuclei having greater mass are assumed stationary while the electrons are moving around them. Buckingham potential—Function used for describing the ene rgy of the Coulombic and short-range interactions of ionic or partially-ionic compounds; incorporates a two-parameter exponential and oneparameter dispersion term. Correlation energy—The difference between the experimental energy and the Hartree-Fock energy in quantum mechanics; related to the neglect of local distortion in the distribution of electrons in the calculation. Coulombic energy—The energy associated with the elect rostatic force between two charged bodies (atoms or ions) that is inversely proportional to the distance separating the two charges; like sign charges repulse each other (positive potential energy) while opposite sign charges attract (negative potential energy). Density functional theory—Class of quantum methods in which the total energy is expressed as a function of the electron density, and which the exchange and correlation contributions are based on the solution of the Schrödinger e quation for an electron gas. Electron density—Function that provides the number of electrons per volume of space. Electrostatic potential—Function describing the energy of in teraction for a positive point charge interacting with the nuclei and, in quantum mechanics, the electrons of a molecular system. Energy minimization—Computational procedure for altering the configuration of a molecular model until the minimum energy arrangement has been attained. Approach is used in molecular dynamics, Monte Carlo simulation, and quantum mechanics methods. Forcefield—A set of parameterized analytical expressions used in molecular mechanics for evaluating the contributions to the total potential energy of a molecular system; forcefields typically, but not always, include contributions for bond stretching, angle bending, dihedral torsion, van der Waals, and Coulombic interactions. Frontier orbital—Concept of molecular reactivity in quantum mechanics involving the location of the largest electron density associated with the HOMO and LUMO of the molecule. Hamiltonian—Operator function that describes the tota l energy of a molecule; operates on the wavefunction, and is part of the Schrödinger equation. Hartree-Fock method—Quantum mechanics approach that computes the energy of a molecular system with a single determinant wavefunction; a trial wavefunction is iteratively improved until self consistency is attained. Hessian—A matrix of second derivatives of the energy (force constants) with respect to the atomic coordinates of the molecular system; Hessian can be derived from various molecular mechanics and quantum mechanics approaches. HOMO—Highest occupied molecular orbital in a quantum mechanics calculation. Kohn-Sham equations—Quantum mechanics approach used for expressing the energy of a multi-electron system as a function of electron density; basis of density functional theory. Lattice dynamics—Statistical mechanics approach for eval uating the vibrational frequencies (phonons) of a material based on classical mechanics and assuming harmonic vibrational modes; useful for the derivation of phonon dispersion curves and thermodynamic properties.

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LCAO—Linear combination of atomic orbitals; method used in Hartree-Fock methods to describe multielectron molecular wavefunctions. LUMO—Lowest unoccupied molecular orbital in a quantum mechanics calculation. Molecular mechanics—Molecular modeling method based on the em pirical parameterization of analytical expressions to describe the energy of a molecular system in terms of various energy components (e.g., Coulombic, van der Waals, bond stretch, angle bend, etc.). Molecular dynamics—Deterministic molecular modeling tool that evaluates the forces on individual atoms using an energy forcefield, then uses Newton’s classical equation of motion to compute new atomic positions after a short time interval (on the order of a femtosecond); successive evaluation for a large number of time steps provides a time-dependent trajectory of all atomic motions. Molecular orbital—Quantum mechanics function, compri sed of atomic-based basis functions, for describing the delocalized nature of electrons in a molecule. Monte Carlo simulation—A stochastic modeling method for obtai ning optimized molecular structures and configurations based on the analysis of a large number of randomly-generated trial configurations. Quantum mechanics—Molecular modeling method that examines the electronic structure and energy of molecular systems based on various schemes for solving the Schrödinger e quation; based on the quantized nature of electronic configurations in atomic and molecular orbitals. Self-consistent field—Iterative method used in quantum mechan ics to obtained refinements to various approximations for solving the Schrödinger equation; a SCF calculation is complete when the molecular orbitals and energy are identical to those obtained in the preceding step. Semi-empirical—Methods used in quantum mechanics to obt ain approximate solutions to the Schrödinger equation by incorporating empirical parameters. Van der Waals energy—Energy associated with the short-range interactions between closed-shell molecules; includes attractive forces involving interactions between the partial electric charges, and repulsive forces from the Pauli exclusion principle and the exclusion of electrons in overlapping orbitals. Wavefunction—Eigenvector result from the Schrödinger wave properties of a molecular system.

equation that describes the dynamical

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Teppen BJ, Rasmussen K, Bertsch PM, Miller DM, Schafer L (1997) Molecular dynamics modeling of clay minerals. 1. Gibbsite, kaolinite, pyrophyllite, and beidellite. J Phys Chem B101:1579-1587 Teppen BJ, Yu C, Miller DM, Schafer L (1998) Molecular dynamics simulations of sorption of organic compounds at the clay mineral / aqueous solution interface. J Comput Chem 19:144-153 Terakura K, Yamasaki T, Uda T, Stich I (1997) Atomic and molecular processes on Si(001) and Si(111) surfaces. Surf Sci 386:207-215 Teter DM (2000) Accurate and transferable ionic potentials from density functional theory. Phys Rev Lett:submitted Teter DM, Gibbs GV, Boisen MB, Allan DC, Teter MP (1995) First-principles study of several hypothetical silica framework structures. Phys Rev B: Condens Matter 52:8064-8073 Teter MP, Payne MC, Allan DC (1989) Solution of Schrödi nger equation for large systems. Phys Rev B: Condens Matter 40:12255-12263 Tosi MP (1964) Cohesion of ionic solids in the Born model. Solid State Phys 131:533-545 Tossell JA, Gibbs GV (1977) Molecular orbital studies of geometries and spectra of minerals and inorganic compounds. Phys Chem Miner 2:21-57 Tossell JA, Gibbs GV (1978) The use of molecular-orbital calculations on model systems for the prediction of bridging-bond-angle variations in siloxanes, silicates, silicon nitrides and silicon sulfides. Acta Crystallogr, Sect A: Found Crystallogr 34:463-472 Tossell JA, Vaughan DJ (1992) Theoretical Geochemistry: Applications of Quantum Mechanics in the Earth and Mineral Sciences. Oxford University Press, New York Tuckerman ME, Martyna GJ (2000) Understanding modern molecular dynamics: Techniques and applications. J Phys Chem B 104:159-178 van't Hoff JH (1874) A suggestion looking to the extension into space of the structural formulas at present used in chemistry, and a note upon the relation between the optical activity and the chemical constitution of organic compounds. Arch Neerland Sci Exact Natur 9:445-454 Verlet L (1967) Computer 'experiments' on classical fluids: I. Therodynamical properties of Lennard-Jones molecules. Phys Rev 159:98-103 Wallace DC (1972) Thermodynamics of Crystals. Dover Publications, Mineloa, New York Wang J, Kalinichev AG, Kirkpatrick RJ, Hou X (2001) Molecular modeling of the structure and energetics of hydrotalcite hydration. Chem Mater 13:145-150 Watson GW, Oliver PM, Parker SC (1997) Computer simulation of the structure and stability of forsterite surfaces. Phys Chem Miner 25:70-78 Wentzcovitch RM, Martins JL, Price GD (1993) Ab initio molecular dynamics with variable cell shape: Application to MgSiO3. Phys Rev Lett 70:3947-3950 Wentzcovitch RM, Stixrude L (1997) Crystal chemistry of forsterite: A first-principles study. Am Mineral 82:663-671 Winkler B, Blaha P, Schwarz K (1996) Ab initio calculation of electric-field-gradient tensors of forsterite. Am Mineral 81:545-549 Winkler B, Dove MT (1992) Thermodynamic properties of MgSiO3 perovskite derived from large-scale molecular dynamics simulations. Phys Chem Miner 18:407-415 Woodcock LV, Angell CA, Cheeseman P (1976) Molecular dynamics studies of the vitreous state: Simple ionic systems and silica. J Chem Phys 65:1565-1577 Zhang HY, Bukowinski MST (1991) Modified potential-induced breathing model of potentials between closed-shell ions. Phys Rev B: Condens Matter 44:2495-2503

36 

Cygan 

 

2

Simulating the Crystal Structures and Properties of Ionic Materials From Interatomic Potentials Julian D. Gale Department of Chemistry Imperial College of Science, Technology and Medicine South Kensington, London, SW7 2AY, U.K. INTRODUCTION

Over the past decade computer simulation techniques have become an increasingly valuable tool in science as an aid to the interpretation of experimental data and as a means of yielding an atomic level model (Catlow et al. 1994; Wright et al. 1992). The scope of such methods has advanced alongside the developments in computational hardware, as has their accuracy, to the point where predictions can now be made ahead of experiment (Couves et al. 1993). The development of the methodology for the simulation of inorganic and organic materials has largely evolved independently to date. For organic materials, interatomic potential calculations have utilized the natural connectivity of covalent systems to develop the molecular mechanics approach (Allinger 1977). The pioneering programs in this field, such as WMIN of Busing (1981) and PCK6 of Williams (1984) were able to simplify the problem by working with rigid molecules and therefore only intermolecular potentials had to be considered. However, varying degrees of intramolecular flexibility could also be introduced by defining molecules as a series of coupled rigid fragments. In contrast inorganic materials, particular oxides and halides, have tended to be simulated starting from the concept of formally charged ions without covalent bonding. For many cases this leads to close-packed materials with relatively regular, high symmetry, structures. Deviations from such environments can be explained by inclusion of polarization of the anion, and occasionally the cation (Wilson et al. 1996a). The aim of this chapter is to highlight some of the methods being used based on interatomic potentials in the simulation of mineral structures under various conditions, but with the emphasis on static approaches, as opposed to dynamical techniques. INTERATOMIC POTENTIAL MODELS FOR IONIC MATERIALS The basis on which interatomic potential methods are built is that the energy of a system can be expressed as a sum over many-body interaction terms, where the number of bodies runs from 1 through to infinity: N

N −1

i =1

i =1

E = ∑ Ei + ∑

N

N −2

N −1

i =1

j =i +1 k = j +1

N

∑E + ∑ ∑ ∑E

j =i +1

ij

ijk

+ .....

(1)

This decomposition is only useful if the terms become progressively smaller, thus enabling the truncation of the series at a suitable point. Fortunately this is usually the case, particularly for systems that are electronically insulating. Furthermore, much is known about the typical functional forms suitable to describe each of the energy terms in many situations based on an understanding of the physical interactions that occur. In the simulation of ionic materials a convenient starting point is to assume that the 1529-6466/01/0042-0002$05.00

DOI:10.2138/rmg.2001.42.2

38

Gale

solid is composed of formally charged ions and thus the electrostatic interactions are the dominant term. This was recognized a long time ago in the simple lattice energy expressions of Born-Landé and Born-Meyer, not to mention the empirical formula of Kapustinskii (1956). All though there is no absolute requirement to use formal valence charges, and indeed there have been many partial charge models as well (van Beest et al. 1990), this is the most versatile approach as it maximizes transferability between different materials and allows defect calculations to be performed in a straight forward manner. In addition to the electrostatics we have to include other terms with a physical basis. Most importantly there must be a short-range repulsive term, such as an exponential or powerlaw form, which represents the Pauli repulsion due to overlap of electron densities. The key feature that allows the ionic model to be successful in modelling many materials is the inclusion of ion polarizability. According to how the electron density is partitioned, it is possible to view many features of semi-ionic materials equal as well as covalency effects or ion polarization. Hence, providing the necessary polarizability terms are included, it is possible to get good results with formal charges despite the fact that a solid may generally be viewed as appreciably covalent. An example of such a case is the family of silicate minerals (Sanders et al. 1984). The inclusion of polarization is also the mechanism by which low symmetry phases become stable as opposed to regular close packed structures. Polarization of ions can be included in one of two ways. The natural approach is to use point ion polarizabilities, which has been successfully explored by Wilson and Madden (1996). An alternative, which has been used for many decades, is the so-called shell model (Dick and Overhauser 1958) as illustrated schematically in Figure 1. This is a simple mechanical model, in which an ion is represented by two particles—a core and a shell—where the core can be regarded as the representing the nucleus and inner electrons, while the shell represents the valence electrons. As such, all the mass is assigned to the core, while the total ion charge (qt = qc + qs) is split between both of the species. The core and shell interact by a harmonic spring constant, Kcs, but are Coulombically screened from each other. The polarizability is then given by:

α=

q s2 (K cs + Fs )

(2)

where Fs is the force constant acting on the shell due to the local environment. The reason why the shell model has been used in preference to point ion polarizabilities is

Figure 1. Schematic representation of the dipolar/breathing shell model for polarizability.

Calculating the Structure & Properties of Ionic Materials

39

that it naturally couples the polarizability to the environment of the ion and avoids the socalled “polarization catastrophe,” that can befall the alternative model. This occurs if the polarizability or dispersion interaction is left undamped as the interionic distance tends to zero. Hence, for the purposes of this work we will be concerned with the shell model for ionic materials. There is a further refinement of the shell model that is occasionally used, known as the “breathing” shell model (Schröder 1966). He re the shell is given a finite variable radius on which the short-range repulsive potential acts. In addition a harmonic restoring force is included about the equilibrium radius. The coupling of forces via variable radii creates a many body force that allows for the change in ionic environments between different materials. Having defined the basic nature of the model, the practical calculation of the energetics of a three-dimensional system theoretically involves the evaluation of interactions between all species, be they cores, shells or united atom units, within the unit cell and their periodic replications to infinity. As this is clearly unfeasible, some finite cut-off must be placed on computation of the interactions. We can decompose the components of the lattice energy into two classes—long- and short-range potentials. These categories can then be treated differently. The summation of the short-range forces can normally be readily converged directly in real space until the terms become negligible within the desired accuracy. However, other terms may decay slowly with distance, particularly since the number of interactions increases as 4πr2Nρ, where Nρ is the particle number density. In particular, the electrostatic energy is conditionally convergent since the number of interactions increases more rapidly with distance than the potential (which is proportional to 1/r) decays. Hence, the two classes of energy components will be considered separately. Long-range interactions The electrostatic energy is the dominant term for many inorganic materials, particularly oxides, and therefore it is important to evaluate it accurately. For small- to moderate-sized systems this is most efficiently achieved through the Ewald summation (Ewald 1921) in which the inverse distance is rewritten as its Laplace transform and then split into two rapidly convergent series, one in reciprocal-space and one in real-space. The distribution of the summation between real- and reciprocal-space is controlled by a parameter η. The resulting expression for the energy is;

⎛ G2 exp⎜⎜ − π 1 4 ⎛ ⎞ ⎝ 4η Erecip = ⎜ ⎟ ∑ G2 ⎝2⎠ V G

E real

⎞ ⎟ ⎟ ⎠

∑∑ qi q j exp(− iG ⋅ rij ) i

(3)

j

1 qi q j erfc⎛⎜η 2 rij ⎞⎟ 1 ⎠ ⎝ = ∑∑ 2 i j rij

(4)

where the sums for i and j are over pairs of ions within the unit cell and the factors of a half are to allow for double counting of individual pairs. In real space the sums are also over translational images out to a cut-off radius. Likewise in reciprocal space the sum over reciprocal lattice vectors extends out to a maximum cut-off. The Ewald sum has a scaling with system size of N3/2. This is achieved when the

40

Gale

optimal value of η is chosen (Perram et al. 1988). Selection of this value can be made based on the criterion of minimizing the total number of terms to be evaluated in realand reciprocal-space, within the respective cut-offs, weighted by the relative computational expense for the operations involved, w: 1

ηopt

⎛ nwπ 3 ⎞ 3 ⎟ = ⎜⎜ 2 ⎟ ⎝ V ⎠

(5)

where n is the number of species in the unit cell, including shells and V is the unit cell volume. The above formula is as per the form derived in the literature (Jackson and Catlow 1988), except that the value of w is not implicitly assumed to be unity. It generally is found that the parameter, w, which reflects the ratio of the computational expense in reciprocal- and real-space, is not a constant but is rather a function of system size due to implementational factors. Recently there has been increasing interest in many techniques which achieve linear or NlogN scaling for the evaluation of the electrostatic contributions, such as the fast multipole method (Petersen et al. 1994) and particle mesh approaches (Essmann et al. 1995). These methods are clearly beneficial for very large systems, but have a larger prefactor and there is some debate as to where the crossover point with the Ewald sum occurs. The best estimates indicate that this happens at close to 10,000 ions. Since we are currently largely concerned with crystalline materials, most systems to be studied will be considerably smaller than this and so the Ewald technique represents the most efficient solution. However, in large-scale molecular dynamics other approaches will often be the method of choice. Short-range interactions

For many ionic materials the predominant short-range potential description used is the Buckingham potential, which consists of a repulsive exponential and an attractive dispersion term between pairs of species. For more general systems, such as molecular organics, semiconductors, metals and inert gases, a wider range of functional forms is required. An alternative approach, commonly used in computationally intensive simulations, is to represent each interaction by a tabulation of energy versus distance and then to use a spline to interpolate between points. This is also advantageous when an energy surface can be determined by quantum mechanical means as it can potentially remove the need to approximate the underlying distance dependence. In the most commonly-used interatomic potentials, the so called “short-range” cutoff is controlled by the dispersion term as represented by -C/r-6, as the exponential repulsion and terms dependant on higher powers of the distance decay more rapidly. Unfortunately, these dispersion terms can often be significant even when summed out to twice the distance needed to converge the repulsive terms. Such truncation of the dispersion terms generally leads to small, but noticeable, discontinuities in the energy surface which can lead to termination of an optimization before the gradient norm falls below the required tolerance. As pointed out by Williams (1989), it is straightforward to accelerate the convergence of the dispersion energy by the same procedure as for the electrostatic energy. When transformed partially into reciprocal space the resulting expressions for the dispersion energy are:

Calculating the Structure & Properties of Ionic Materials C6 Erecip

⎛ 3 ⎜ π2 1 = ∑∑ − C ⎜ ij ⎜ 12V 2 i j ⎝

⎡ 1 ⎞ ⎛ ⎟ ⎜ G 3⎢ 2 ⎟∑ exp(iG ⋅ r )G ⎢π erfc⎜ 1 ⎟G ⎜ 2 ⎢ ⎝ 2η ⎠ ⎣

C6 E self =

E

C6 real

1 ⎞ ⎤ ⎞ ⎛ 3 ⎟ ⎜ 4η 2 2η 2 ⎟ ⎛ G 2 ⎞⎥ ⎜ ⎟ ⎟ exp⎜ − ⎟+⎜ 3 − ⎟⎥ G ⎟ ⎟ ⎜ G ⎝ 4η ⎠⎥ ⎠ ⎝ ⎠ ⎦

3 C ij ⎡ C iiη 3 1 ⎤ 2 ( ) − + πη ∑∑ 3 ⎢⎣ ⎥⎦ ∑ 6 2 i j i

C ij ⎛ 1 η 2r 4 2 = ∑∑∑ − 6 ⎜⎜1 + ηr + 2 i j cells r ⎝ 2

⎞ ⎟⎟ exp − ηr 2 ⎠

(

41 (6)

(7)

)

(8)

The additional computational overhead to perform this summation is small and, when combined with the reduction in the real-space cut-off, the CPU time taken to achieve a particular target accuracy should be greatly diminished. Beyond the simple Buckingham potential there are many alternative two-body functional forms though, such as the Tang-Toennes potential which allows for damping of the dispersion interaction at short range. In particular, it is common to employ different forms when describing molecular or partial covalent entities within minerals, such as hydroxyl groups and the carbonate anion. Here the interaction is most often described by a Morse or harmonic potential, while also excluding the Coulomb term. Energy minimization

The most fundamental task to the simulation of any crystal structure is energy minimization since in the low temperature limit any system will be within a local minimum. In all systems there is the complication that there will be more than one local minimum—for example MgO could adopt the Na Cl, CsCl, or a whole host of other MX structures, each one of which may be locally stable. Depending on the system we may want either a metastable minimum or a global one. In the case of microporous silicates we would always want the local minimum rather than to end up at the α-quartz structure every time. In general, the location of global minima is very difficult and there can rarely be any guarantee of success. A brief mention of how this problem can be approached will be given later, but for now we shall consider the simplest method only, which is to minimize each candidate structure to its local minimum and to compare energies. Efficient minimization of the energy is an essential part of the simulation of solids as it is a pre-requisite for any subsequent evaluation of physical properties and normally represents the computationally most demanding stage. The most efficient minimizers are those which are based on the Newton-Raphson method, in which the Hessian or some approximation to it is used. The minimization search direction, x , is then given by; x = − H −1 g

(9)

where H is the Hessian matrix and g is the corresponding gradient vector. The best compromise, between the cost of evaluating the Hessian and increasing the rate of convergence, is to use the exact second derivative matrix, calculated analytically, to initialize the Hessian for the minimization variables. It can then be subsequently updated from one cycle to the next using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm (Press et al. 1992). This is done so as to avoid the recalculation of second derivatives and matrix inversion at every point, these being the major bottlenecks of calculations for large systems. The Hessian is only explicitly recalculated when either the energy drops by more than a certain criterion in one step (which usually only happens at

42

Gale

the start of a minimization, when the system is in a non-quadratic region) or the angle between the gradient and search vectors becomes unacceptably large. The above approach generally leads to rapid convergence within a few cycles for most systems, except where there are particularly soft modes in the Hessian. Difficulties of this nature can be overcome by use of more sophisticated techniques, such as the Rational Function Optimizer (RFO) (Banerjee et al. 1985), which attempts to remove imaginary modes from the Hessian by diagonalization and application of a level shift. The use of RFO can lead to rapid convergence in cases where the standard Newton-Raphson approach has difficulty, though the downside is that it is much more expensive per cycle. A useful feature of the RFO approach is that it can be made to search for stationary points with any number of imaginary modes and thus provides a mechanism for locating transition states (see Chapter 13 by Kubicki). Two families of materials where energy minimization has been used extensively as a complement to experimental methods, especially crystallography, are zeolites and aluminophosphates. Both of these categories comprise many different metastable polymorphs of SiO2 and AlPO4, respectively, with microporous environments of importance in catalysis and molecular sieving. Starting from shell model potentials derived based on the high density end members, α-quartz and berlinite, Henson et al. (1994, 1996) have made systematic studies of both families comparing structures and the correlation of heats of formation with experiment. For the silicates, the worst disagreement in cell parameters is less than 2% and most agree to better than 1%. Similar levels of agreement are found for the aluminophosphates. Where the simulations are most valuable is when there is an ambiguity concerning space groups. For example, VPI-5 (Fig. 2) has been reported to have both the space groups P63cm and P63, either from hydrated samples or with averaging of the T sites during refinement (Rudolph and Crowder 1990; McCusker et al. 1991). Simulations demonstrate that the space group P63cm leads to imaginary modes and that the pure material is best described in P63. In another case, the crystallographic symmetry of AlPO4-5 has been examined using

Figure 2. Structure of the microporous aluminophosphate VPI-5 as viewed along the z-axis. Tetrahedra represent the alternating aluminium and phosphorous cations, cross-linked by corner sharing at oxygen anions.

Calculating the Structure & Properties of Ionic Materials

43

potential models by several groups of workers as the experimental space group of P6cc forces a number of Al-O-P bond angles to be linear. The conclusion of all of this work suggests that, provided a model that incorporates polarizability is used, then the space group should be P6, allowing the bond angles to relax away from 180o. Although the energy difference between the constrained and unconstrained structures is small, the true situation is probably a disordered arrangement of oxygen about the Al-P vector. Similarly Njo et al. (1997) have recently proposed that the synthetic zeolite MCM-22 (Fig. 3) should have a space group of P6/m instead of P6/mmm or Cmmm as currently thought based on theoretical results. A whole host of other structural aspects of these materials have been examined using shell model minimization, including extra-framework cation locations (Jackson and Catlow 1988, Grey et al. 1999), proton binding sites (Schröder 1992) and the nature of silicon islands (Sastre et al. 1996). One of the most promising applications of these methods has been its use in helping to refine previously unsolved structures, such as DAF-1 (Wright et al. 1995) and MAPO-36 (Wright et al. 1992). Furthermore, where the structure is known in the presence of a templating agent the crystallographic data for the calcined material may be predicted (Girard et al. 2000). Beyond basic energy minimization for the localization of minima there is often the need to determine more dynamic information, such as the rates of diffusion of ions within ionic materials. While some fast ion conductors are amenable to molecular dynamics, the time scales involved are usually too long for the direct determination of diffusion coefficients and related properties. Hence, the natural approach is to utilize transition state theory by determining the activation energy required for diffusion. This has been done for a number of materials (Islam 1993; Islam and Ilett 1994) and in many studies this was achieved by mapping out the energy surface by constrained two-dimensional energy minimization. A more efficient route to the accurate location of transition states, as already mentioned, is to use the eigenvector following method within the RFO technique to find the point at which the forces are zero under the constraint of one imaginary mode of vibration. Because the evaluation of second derivatives is relatively inexpensive for interatomic potential models this latter approach turns out to be far more efficient and benefits from the absence of a need to make assumptions about the pathway that the ion takes.

Figure 3. Structure of the synthetic zeolite MCM-22.

44

Gale

An example of how this procedure can be useful comes from the study of immobilizing radioactive species within mineral hosts (Meis and Gale 1998). Here the defect sites of both uranium(IV) and plutonium(IV) cations were located within the zircon structure, as well as the lowest energy pathway for diffusion of the ions. Given the activation energy for diffusion that was determined, it was then possible to estimate the diffusion co-efficients for both ions as a function of temperature using either the Langmuir-Dushman (Langmuir and Dushman 1922) or Bradley-Wheeler (Bradley 1937) approximations to the prefactor. The results obtained verified that the rate at which these cations will leach from zircon should be negligibly small, thus making the material a suitable host. CRYSTAL PROPERTIES FROM STATIC CALCULATION

Once a structure has been optimized, there is a wide range of properties that can be calculated in the solid state for comparison with experiment. Conversely, these properties can also be used in the empirical derivation of interatomic potentials as will be discussed later. The properties that are readily available can be divided into the categories of mechanical, electrical and phonon properties. All of them utilize the ability to readily determine higher order derivatives (usually second) to which the observables are related. Elastic constants

The elastic constant tensor is a 6 × 6 matrix that contains the second derivatives of the energy density with respect to external strain:

E=

[

1 W ss − W scW cc−1W cs V

]

(10)

where W ss is the strain-strain second derivative matrix, W cc is the Cartesian-space coordinate second derivative matrix, W cs is the mixed Cartesian-strain second derivative matrix, and V is the volume of the unit cell. It is important to note that the elastic constant matrix, in general, depends on the orientation of the unit cell relative to the Cartesian axes. From the elastic constant matrix, or its inverse the compliance matrix, it is possible to calculate the bulk modulus, shear modulus, Poisson’s ratio and a number of other related mechanical quantities. Generally speaking, the ability of shell model potentials to reproduce the elastic properties of ionic materials is much more limited, as compared to structures, with errors typically being an order of magnitude larger. This is a consequence of the fact that the perturbation of a structure about its equilibrium form is much more sensitive to higher order polarizabilities than the minimum itself, where any errors can be readily subsumed into the parameterization. A classic example is the failure of the dipolar shell model to reproduce the Cauchy violation in the elastic constants of simple cubic oxides, such as MgO (Catlow et al. 1976). Dielectric constants

The dielectric constants can be readily calculated both in the high frequency and low frequency, or static, limits where the deviation of the high frequency values from unity is a reflection of the shell model polarizability within the material. The elements of the 3 × 3 matrices are given by:

ε αβ = δ αβ +

4π T −1 q W cc q V

(11)

Calculating the Structure & Properties of Ionic Materials

45

where q is a vector containing the charges of each species, and α and β are the Cartesian directions. For the static dielectric constant matrix the matrix operations run across all species, including cores and shells, whereas for the high frequency case only the shells are considered. Closely related to the dielectric constant tensor are the refractive indices. These can be determined by diagonalizing the former quantity, to place it into a unique axis system and then taking the square root of the eigenvalues. If, as is usual, the corespring constant is fitted then the shell model is usually capable of reproducing either the high or low frequency limits of the dielectric constant matrix, but for complex materials can rarely reproduce both simultaneously with complete accuracy. Piezoelectric constants

There are two variants of piezoelectric constant matrices, piezoelectric stress and piezoelectric strain. The second of these can be obtained from the former by multiplying by the inverse elastic constant matrix. For many materials the piezoelectric constants are zero by symmetry if there is a centre of inversion. The piezoelectric stress constants are derived from the second derivative matrices according to the relationship: Pαi = −

4π T −1 q W cc W cs V

[

]

αi

(12)

Phonons

One of the main properties that can be calculated from the Cartesian second derivative matrix is the set of vibrational frequencies. These are obtained by diagonalizing the so-called dynamic matrix that consists of the mass-weighted Cartesian second derivatives for an isolated cluster or for a solid at the gamma point: −

1

D = m 2 W cc m

−

1 2

(13)

The vibrational frequencies are the square root of the eigenvalues of the dynamical matrix. Hence, if there are any negative eigenvalues the corresponding vibrational frequencies will be imaginary, thus implying that the system is unstable with respect to a distortion given by the eigenvector of the imaginary mode. In particular, at the gamma point the first three vibrational frequencies should be equal to zero as they correspond to the translation of the lattice. The above equation for the dynamical matrix is modified in the case where a shell model is being used as these particles have no mass, yet they must be involved in the second derivatives:

D=m

−

1 2

[W

core − core

]

−1 − W core− shellW shell − shell W shell − core m

−

1 2

(14)

In the case of a periodic solid the vibrational modes become phonons and the dynamical matrix becomes a function of a reciprocal lattice vector k chosen from the Brillouin zone. This means that in constructing D(k) all interactions are multiplied by the phase factor exp(ikrji), where rji is the interatomic vector. A more detailed discussion of the theory of phonons can be found elsewhere (Dove 1993; Chapter 13 by Kubicki). If we calculate how the frequencies vary between different points in the Brillouin zone the results are a series of phonon dispersion curves. More generally, the distribution of frequencies in reciprocal space may be sampled by inelastic neutron scattering as the scattering function, S(Q,ω), which may also be calculated via interatomic potential methods.

46

Gale

In general, we are most often concerned with the phonon density of states for a solid, since the integral of this quantity multiplied by some other property that is a function of vibrational frequency leads to the average value that would be observable. This is employed in deriving thermodynamic quantities via statistical mechanics, as will be discussed later. While full analytical integration across the Brillouin zone is not readily carried out, this integral can be approximated by a numerical integration. We can imagine calculating the phonons at a grid of points across the Brillouin zone and summing the values at each point multiplied by the appropriate weight (which for a simple regular grid is just the inverse of the number of grid points). As the grid spacing goes to zero the result of this summation tends to towards the true result. The standard scheme for choosing a regular mesh of reciprocal space points was developed by Monkhorst and Pack (Monkhorst and Pack 1976). This is based around three so-called shrinking factors one for each reciprocal lattice vector. These specify the number of uniformly spaced grid points along each direction. The only remaining choice is the offset of the grid relative to the origin. This is chosen so as to maximize the distance of the grid from any special points, such as the gamma point since this gives more rapid convergence. In many cases it is not necessary to utilize large numbers of points to achieve reasonable accuracy in the integration of properties, such as phonons, across the Brillouin zone. For high symmetry systems several schemes have been devised to reduce the number of points to a minimum by utilizing special points in k space (Chadi and Cohen 1973). Often it is not necessary to integrate across the full Brillouin zone either due to the presence of symmetry. By using the Patterson group (the space group of the reciprocal lattice) the integration region may be reduced to that of the asymmetric wedge which could only be 1/48 of the size of the full volume (Rameriz and Böhm 1988). In order to make comparison between theoretical phonon spectra and experiment it is important to know something about the intensity of the vibrational modes. Of course the intensity depends on the technique being used to determine the frequency spectrum as different methods have different selection rules. Approximate values for the intensity of peaks in the infra-red spectra can be determined according to the following simple formula (Dowty 1987): I IR

⎞ ⎛ ⎟ ⎜ ∝ ⎜ ∑ qd ⎟ ⎜ all species ⎟ ⎠ ⎝

2

(15)

where q is the charge on each species and d is the Cartesian displacement associated with the normalized eigenvector. This is clearly very approximate since it depends on how realistic the charges assigned to the atomic centers are and neglects the coupling of charge with displacement. Furthermore, the influence of polarizability on the change in dipole moment is ignored. Estimation of the Raman phonon intensities is even more complex, though a model has been proposed for this quantity that is suitable for potential based methods (Kleinman and Spitzer 1962). The electric susceptibility tensor is given by:

χ = ∑∑ (rij d i )(rij rij ) i

j

and the intensity is then related to this quantity and a frequency factor:

(16)

Calculating the Structure & Properties of Ionic Materials

I Raman

⎛ ⎞ ⎜ ⎟ ⎜ ⎟ 2 1 = ⎜1 + ⎟χ ⎞ ⎛ υ h ⎜ exp⎜ ⎟ ⎜ k T ⎟⎟ ⎟ ⎜ ⎝ B ⎠⎠ ⎝

47

(17)

Note that the intensities calculated in this way are very approximate and assume that all bonds are the same in the material. Hence this approach has found application primarily for silica polymorphs and zeolites. As well as being important in their own right for comparison with experiment and predictions, the above properties are crucial in the empirical determination of potential parameters, as will be discussed in the next section. DERIVATION OF POTENTIAL PARAMETERS

Two general classes of method for potential derivation exist, empirical and theoretical. In the former approach a training set of experimental data is constructed which the forcefield is then required to reproduce. This always includes structural data for one or more configurations, supplemented by observables that contain information concerning the curvature of the energy surface, such as elastic constants or phonon frequencies. The alternative approach of theoretical derivation can encompass anything from combination rules based on atomic data through to quantum mechanical energy hypersurface fitting (Harrison and Leslie 1992; Gale et al. 1992). Clearly, the more widely varying the information included, the more transferable and robust the forcefield will be, particularly if the functional form used mirrors the underlying physical interactions that are of importance. The derivation of potential parameters is a vast and important topic, which cannot possibly be covered comprehensively here. Hence, the focus will be on two topics concerning the particular approaches used for empirical shell model potential derivation for ionic materials. However, it is noted that derivation of parameters from ab initio energy surfaces will increasingly become the method of choice for more complex materials due to the lack of suitable experimental data. Simultaneous fitting

In conventional fitting, as has been widely used within this community in the past, the gradients and properties have been calculated at the experimental crystal structure and the potential parameters have been varied so as to minimize the error in these calculated quantities. This approach takes the experimental gradients to be zero at the observed atomic co-ordinates. A problem arises when using any form of shell model, be it dipolar or breathing shell. Formally we can equate the core of an ion with the nucleus since it is assigned the atomic mass in dynamical calculations. Hence we know from a crystal structure the desired core positions to which we wish to fit, formally speaking, provided the diffraction data was obtained using neutrons. However, in the case of the shells we have no a priori information about where to place them, except in the rare case where the electron density has been determined precisely by crystallography and we can obtain information concerning the ion dipoles directly. In many cases the shells have been assumed to be coincident with the cores for early empirical potential fits, which is often true for high symmetry crystallographic sites. For low symmetry sites this is clearly an erroneous assumption which, as it will be demonstrated later, leads to a poor quality of fit.

48

Gale

There are two approaches to handling the general case in shell model fitting. Firstly an optical (shell only) energy minimization could be performed at each point in the fitting procedure and the residual sum of squares calculated as before. Alternatively, the symmetry reduced shell model co-ordinates, and radial parameters if appropriate, could be included as variables in the fit so that they are adjusted to obtain the lowest possible sum of squares. The inclusion of the shell co-ordinates as fitted parameters is countered by adding an equal number of conditions that the corresponding gradients must be zero. Hence the inclusion of the shell model leads to no change in the difference between the number of observables and fitted parameters. These above two methods yield slightly different results, if properties other than the crystal structure are included in the fit, since in the first technique the shells are purely minimized with respect to the energy, whereas in the second case the shells are optimized with respect to the sum of the squares of the residuals. Experience in applying these two approaches suggests that the latter method is more readily convergent and computationally efficient. The ability to relax shells during potential derivation has been automated in the program GULP and has been given the name “simultaneous fitting” (Gale 1996). One example of where simultaneous fitting has proved to be crucial is in determining interatomic potentials for aluminophosphates. These materials also raise other questions, such as can we really expect the ionic model to handle unphysically large formal charge states as +5? Work on deriving potential parameters for berlinite (Gale and Henson 1994) suggests that a formally charged model is indeed feasible and performs as well as other more physical partially charged models. Table 1 shows a comparison of calculated and experimental structure and properties for berlinite. Although the quality of reproduction of properties is not exceptional for everything, it should be remembered Table 1. Comparison of experimental and that only two parameters were calculated structure and properties for αberlinite based on a shell model potential for actually fitted to this particular system aluminophosphates (Gale and Henson 1994). and the rest were transferred unmodified from alumino-silicates. Observable Experiment Calculated The shell model can allow the simulation of a significantly covalent a/Å 4.9423 4.9109 material using an ionic model because c/Å 10.9446 10.9564 of the similarity between polarization Al x 0.4665 0.4670 and covalency – both are just shifts in Px 0.4669 0.4698 the electron density distribution, but 63.4 81.8 C11 (GPa) with different partitioning. In this 2.3 15.9 C12 (GPa) case, the dipolar shell model can 5.8 22.2 C13 (GPa) subsume covalent effects because of -12.1 -10.9 C14 (GPa) the low symmetry at oxygen. Modeling of silicates within the ionic model employing formal charges is now well established (Catlow and Cormack 1987), however, earlier attempts to extend the scope of such calculations to their aluminophosphate analogues had proved unsuccessful because the cores

C33 (GPa) C44 (GPa) C66 (GPa) εo11 εo33 ε∞11 ε∞33 P11/1012 CN-1 P14/1012 CN-1

55.8 43.2 30.6 5.47 5.37 4.60 4.48 -3.30 1.62

106.7 44.0 32.9 5.25 5.42 2.08 2.11 -2.30 1.09

Calculating the Structure & Properties of Ionic Materials

49

and shells were concentric during fitting. In the case of berlinite, conventional fitting, in which the cores and shells are assumed to be coincident, gives a final sum of squares of 884977.0 whereas simultaneous fitting yields 22.0, indicating that several orders of magnitude improvement may be achieved in extreme cases. This demonstrates that for the shell model to be effective in subsuming errors in charge states it is necessary to allow the core and shell to separate during fitting. Relaxed fitting

In the previous section it has been demonstrated that the problem of the shell positions can be dealt with, but now we turn to address the question of how to fundamentally improve the fitting process. Practical experience has shown that in conventional fitting lowering the sum of squares is actually no guarantee of better results when the potentials are actually applied to energy minimization. The main criterion used for deciding the accuracy of a potential model is normally not the forces at the equilibrium geometry, but instead the displacements of the optimized structure away from the experimental configuration. If the gradient vector is g and the Hessian matrix is H , then the displacements that would occur on optimization Δ , assuming the local energy surface is quadratic, will be given by; Δ = − H −1 g

(18)

Hence we could minimize the displacement vector with respect to the fitted parameters in place of the gradients. However, in many cases the quadratic approximation is not sufficient and in some cases the Hessian may not even be positive definite so we would have to include further tests to ensure that the fit is valid. There is also a second flaw in the conventional approach to fitting in that the curvature related properties are only strictly calculable directly from the second derivative matrix when the gradients are zero. Unless the fit to the structure is already perfect then trying to reproduce elastic and dielectric constants at the experimental structure is far from ideal. Both of the above difficulties can be resolved by performing a full optimization of the structure with a subsequent property calculation for each point during the fitting procedure. This method, which has become known as “relaxed” fitting, thus yields the exact displacements and genuine physical properties (Gale 1996). An illustration of the use of relaxed fitting comes from the work of Fisler et al. (2000), who used this approach to derive a set of potentials capable of describing the polymorphs of calcium carbonate, calcite and aragonite, as well as a range of other metal carbonates that are iso-structural with calcite. This work was distinguished from previous ones (Pavese et al. 1992; Dove et al. 1992) by the use of a shell model within the carbonate anion, but while retaining a molecular mechanics description of the intramolecular forcefield. All cell parameters were reproduced to better than 1% for the pure phases and even when transferred to the mixed carbonate, dolomite (MgCa(CO3)2) the error only just exceeded this. Many of the physical properties of calcite and aragonite were also examined (Table 2) and the quality of reproduction was generally very good, though, as is typically found, the errors were significantly greater than for the structural data. It should be remembered that the amount of data included in the fitting procedure is much greater than the number of parameters and that the uncertainties in experimental measurements of quantities such as elastic constants are also greater than for crystallographic information.

50

Gale Table 2. Comparison of experimental and calculated properties for calcite and aragonite (Fisler et al. 2000). Property

Calcite Experimental Calculated

C11(GPa) C12(GPa) C13(GPa) C14(GPa) C22(GPa) C23(GPa) C33(GPa) C44(GPa) C55(GPa) C66(GPa) Bulk Modulus (GPa) εo11 εo33 ε∞11 ε∞33 Asymmetric C-O stretch (cm-1) Symmetric C-O stretch (cm-1) Out of plane (CO3) (cm-1) Bend (CO3) (cm-1)

145.7 55.9 53.5 -20.5 145.7 53.5 85.3 33.4 33.4

140.9 63.7 62.6 -19.5 140.9 62.6 85.8 33.4 33.4

73.0 8.5 8.0 2.75 2.21 1463

77.0 9.28 8.30 2.69 3.02 1465

1088

Aragonite Experimental Calculated 85.0 15.9 36.6

89.9 48.0 55.9

159.6 2.0 87.0 42.7 41.3 25.6 48.0

2.86 2.34 1473

155.3 54.7 104.2 23.3 36.7 12.4 73.0 7.84 8.26 3.05 2.50 1500

1082

1086

1124

881

878

873

781

714

612

705

627

A particular feature of the carbonate potential derivation is the presence of the molecular anion. Within this grouping it is necessary to use a more complex potential model than for other ionic materials with a combination of shell model and molecular mechanics terms being necessary. This aspect of the model was validated by comparison of the intramolecular vibrational frequencies. While the positions of the modes are relatively precise for calcite, it proved difficult to correctly obtain the shifts in the frequencies in aragonite. One final aspect of the above carbonate model that is worth noting is that the transition pressure for conversion of calcite through to aragonite is accurately predicted to be 2.4 kbar, as compared to experimental estimates of 2.5 kbar (Crawford and Hoersch 1972). This transition pressure is very sensitive to the relative energies of the two polymorphs and requires a good description of the polarization contribution for the two materials. SIMULATING THE EFFECT OF TEMPERATURE AND PRESSURE ON CRYSTAL STRUCTURES

When discussing energy minimization no explicit mention of temperature was made. The majority of such studies are simulated at absolute zero or at an effective room temperature, depending on how the interatomic potentials were derived. In many cases this is sufficient to reproduce a crystal structure within the limits of the accuracy of the

Calculating the Structure & Properties of Ionic Materials

51

method. However, increasingly we would like to be able to simulate trends in structure as a function of temperature and pressure, and also to access phases that are not stable under ambient conditions. This is of particular importance in mineralogy where many materials are formed only under extremes of temperature and pressure. Inclusion of a uniform external pressure into an energy minimization is relatively trivial since this only requires the addition of the term pV to the internal energy, which is normally calculated, to make the objective quantity the enthalpy. However, the problem becomes more difficult when considering the cases of uni- or bi-axial stress. Following on from the earlier energy minimization studies of aluminophosphates, it is an even more demanding test to examine whether potential models can reproduce the pressure dependence of the structure, as well as under ambient conditions. In the case of α-berlinite, the aluminophosphate analogue of α-quartz, there are both experimental measurements (Sowa et al. 1990) and first principles calculations (Christie and Chelikowsky 1998) performed using the total energy planewave pseudopotential method within the local density approximation. In Figure 4, the ratio of the volume at a given pressure to the unstressed volume is plotted for both experiment and calculation based on shell model potentials (Gale and Henson 1994). It can be seen that the agreement between the two sets of data is excellent, demonstrating that the potential model is capable of reproducing the trend. There is a systematic error, as the initial volume at zero pressure is under estimated by 1.1%, though this is smaller than the 2-3% error found in the first principles case. Furthermore, the reproduction of the volume decrease with pressure is equally as good, if not better, despite the fact that the potential overestimates the bulk modulus (42 GPa) as compared to experiment and LDA (36 GPa). A particular weakness highlighted in the first principles study was in the description of the change of the phosphorous x fractional co-ordinate, as shown in Figure 5. While the potential model is less accurate in matching the experimental value at atmospheric pressure, the trend with decreasing volume is better reproduced. There have been many other examples of the introduction of pressure into static lattice energy minimization calculations, particularly for silicates. For example, there has been a detailed study of the effect of pressure on α-quartz using a range of different models (de Boer et al. 1996). Beyond the consideration of structural trends, this work also evaluated the pressure dependence of some physical properties as well. In particular

Figure 4. Variation in the volume relative to that at 0 GPa of αberlinite with pressure. The solid line represents the results of a shell model calculation, while the open squares represent experimental measurements.

52

Gale

Figure 5. Variation in the x fractional coordinate of phosphorous in α-berlinite (AlPO4) with pressure. The solid line represents the results of a shell model calculation, the open squares the experimental measurements and the circles are the results of density functional calculations.

the changes in the elastic constants and six lowest Raman frequencies were computed. These results were found to be especially sensitive to the particular model and parameterization, with some potentials even yielding the wrong sign for the variation of selected elastic constants. This again demonstrates the fact that potential models, in general, are better for reproducing the changes in structure than properties that relate to the second or even third derivatives. Introducing temperature into a simulation is more complex and there are several approaches that can be utilized. Two standard techniques for modelling systems at finite temperature are molecular dynamics and Monte Carlo methods. Both represent numerical integrations of the system properties to determine the ensemble average, the former having the additional advantage that information in the time domain is also yielded, though typically only for small amounts of real time. These methods also have the benefit that information about the distribution of atoms can be obtained to compare with thermal ellipsoids derived from diffraction experiments. While both methods are very useful for many problems they have two disadvantages. Firstly, they are only strictly valid for solids at elevated temperatures as they neglect the effect of vibrational quantum effects, such as the zero point energy. For many minerals the heat capacity only truly obeys the classical Dulong-Petit result in excess of 1000 K (Dove 1993), which is sometimes higher than the conditions often used for experimental studies. Secondly, the statistical uncertainty in the ensemble averages only decreases as the inverse square root of the simulation size, by the run length or number of atoms. Hence, numerical integration also represents a relatively expensive route to simulating the effect of temperature when the ions in a system are principally just vibrating about their lattice sites. The free energy of a solid can readily be calculated using statistical mechanics via the vibrational partition function, which is obtained as an integral over the Brillouin zone as described previously. Hence this offers an attractive route to simulating the properties of materials as a function of temperature by minimizing the free energy instead of the internal energy. This approach removes the statistical uncertainty associated with the numerical integration and is therefore considerably faster. The main restriction is that it relies on the validity of the quasi-harmonic approximation. This typically restricts the temperature range that can be studied to about half the melting point unless further corrections are included for anharmonicity. Nonetheless, for ionic materials with high melting points this covers many of the conditions of interest except for phase transitions.

53

Calculating the Structure & Properties of Ionic Materials

Historically the difficulty with minimizing the free energy has been to obtain the derivatives of the free energy with respect to the structural parameters. Hence the majority of the free energy minimization studies to date have relied on some degree of approximation. A number of schemes have been proposed recently for practical calculations. Sutton (1992) has developed the idea of using the moments of the dynamical matrix with an approximate functional form for the phonon density of states, which has the correct asymptotic limits to produce an analytic expression for the free energy. While the inspiration for this originally came from tight binding theory, the use of the moments of the dynamical matrix had been previously demonstrated by Montroll (1942). This avoids the need for matrix diagonalization and allows straightforward differentiation to be performed. LeSar et al. (1991) have introduced a variational approach which integrates the potential function over a Gaussian distribution which depends on the temperature. Both of the above methods have been used primarily for the study of metals and alloys so far. Within the silicate field, Parker and co-workers (Parker and Price 1989; Tschaufeser and Parker 1995) have used free energy minimization with success for modelling thermal expansion. Their approach is based on the assumption that the dominant effect of temperature is on the unit cell dimensions, rather than the internal fractional co-ordinates. If this is the case then it becomes feasible to numerically determine the strain derivatives of the free energy by finite differences as there are at most six components to evaluate and for many materials, with symmetry taken into account, there may be considerably less than this. The theory required for the determination of analytical free energy derivatives was recently developed by Kantorovich and applied to alkali halide crystals (Kantorovich 1995). Subsequently the method has been refined by Taylor et al. (1997) who have discussed many of the details of its implementation. However, as the approach is relatively new, a summary of the main features will be given here. The Helmholtz free energy can be written as the sum of the static internal energy, Ustatic, the quantity that would be calculated in a conventional energy minimization, the vibrational energy, Uvib, and the term arising from the vibrational entropy, Svib: A = U static + U vib − TS vib

(19)

This assumes that there is no contribution from configurational disorder, which must be corrected for separately, if relevant. For convenience, the sum of the vibrational energy and entropy term can expressed together, due to the cancellation of a common term, as: ⎧⎪ 1 ⎡ ⎛ hω m (k ) ⎞⎤ ⎫⎪ ⎟⎟⎥ ⎬ U vib − TS vib = ∑∑ ⎨ hω m (k ) + k B T ln ⎢1 − exp⎜⎜ − 2 k T k m ⎪ B ⎠⎦ ⎪⎭ ⎝ ⎣ ⎩

(20)

where the sum over k points is used to approximate the integral over the Brillouin zone of the phonon density of states. The derivatives of the free energy with respect to structural parameters can be related to the derivatives of the eigenvalues or frequencies squared: ⎧ h ⎛1 ⎞⎛ ∂ω 2 1 ⎛ ∂A ⎞ ⎛ ∂U static ⎞ ⎟⎟⎜⎜ ⎜⎜ + ⎟ + ∑∑ ⎨ ⎜ ⎟=⎜ ⎝ ∂ε ⎠ ⎝ ∂ε ⎠ k m ⎩ 2ω m (k ) ⎝ 2 exp(hω m (k ) / k B T ) − 1 ⎠⎝ ∂ε

⎞⎫ ⎟⎟⎬ ⎠⎭

(21)

Hence the key is to obtain the derivatives of the eigenvalues. Through the application of

54

Gale

perturbation theory these derivatives can be related to derivatives of the elements of the dynamical matrix projected onto the eigenvectors of each phonon mode: ⎛ ∂ω 2 ⎜⎜ ⎝ ∂ε

⎞ ⎛ ∂D ( k ) ⎞ ⎟⎟ = em (k )⎜ ⎟e m ( k ) ⎝ ∂ε ⎠ ⎠

(22)

The first derivatives of the dynamical matrix elements are just the third derivatives with respect to either three Cartesian co-ordinates, for internal degrees of freedom, or two Cartesian co-ordinates and the external strain in the case of the unit cell derivatives. Both must also be multiplied by the appropriate phase factor for the point in the Brillouin zone. The above scheme generates both internal and external derivatives with respect to the free energy. However, for comparison we would also like to be able to perform calculations within the zero static internal stress approximation (ZSISA) (Allan et al. 1996), as used previously in the numerical formulation. In this case the internal variables must be minimized with respect to the internal energy while only the strain variables are minimized with respect to the free energy. To achieve this we must first neglect the thermal contribution to the internal forces. However, there will also be a correction term arising for the strain derivatives associated with the fact that the internal energy must remain at its minimum point as the cell is strained. This is analogous to the internal second derivative contribution to the elastic constant tensor. The formal result for the strain correction is as follows: dA d 2 A ⎛ d 2 A ⎛ dA ⎞ ⎜⎜ − ⎜ ⎟ = ⎝ dε ⎠ qh dε dεdα ⎝ dαdβ

−1

⎞ dA ⎟⎟ ⎠ dβ

(23)

As we wish to avoid calculating the second derivatives with respect to the free energy due to the complexity and computational cost we can approximate the two second derivative matrices by the static-only components. Because one matrix is multiplied by the inverse of the other there will be a significant cancellation of errors and this turns out to be a good approximation in practice. Free energy minimization, both with and without inclusion of the internal derivatives with respect to this quantity, was first applied to simple high-density ionic materials. For instance, in the case of MgF2 (Barrera et al. 1997) very little difference was found in the results according to how the internal derivatives were approximated. When analytical FEM was first applied to a more complex and open material (Gale 1998), namely that of quartz, a significant observation was made concerning the difference between the two approaches. As illustrated in Figure 6 and Table 3, the predicted thermal expansion of quartz between 4 K and 298 K is appreciably larger when the free derivatives of the internal degrees of freedom are included. In general, the agreement with experiment is also improved, though in one case it is overestimated and in the other it is underestimated. However, the major observation is that complete free energy minimization fails in the region of 300 K, a finding that is true for all zeolites so far tested, as well as quartz. This is because imaginary modes begin to appear, incorrectly suggesting that the symmetry should be lowered. This failure can be understood since the internal co-ordinates are directly coupled to the vibrational frequencies, but not in the uniform scaling way that unit cell parameters are. Consequently, the way to lower the free energy as rapidly as possible is to generate soft modes where the free energy tends to negative infinity as the frequency tends to zero. The solution to this problem is that anharmonicity must be accounted for in the calculation of the phonons, which

Calculating the Structure & Properties of Ionic Materials

55

Figure 6. Temperature dependence of the unit cell dimensions of αquartz calculated according to full free energy minimization and within ZSISA.

Table 3. Change in the structural parameters of α-quartz between 13 and 298 K as determined according to diffraction (Lager et al. 1982) and free energy minimization (Gale 1998). Change in parameters Δa (Å) Δc (Å) Si Δx (frac) O Δx (frac) O Δy (frac) O Δz (frac)

Experiment

Full FEM

ZSISA

+0.0120 +0.0073 +0.0020 +0.0007 -0.0035 -0.0026

+0.0102 +0.0079 +0.0030 +0.0013 -0.0049 -0.0041

+0.0064 +0.0048 +0.0008 +0.0003 -0.0012 -0.0009

significantly complicates the methodology. For now, it is best to regard the temperature range of applicability of full free energy minimization as limited, and as a result most practical calculations have been performed in the ZSISA approximation. A particularly important phenomenon in the current materials literature is that of negative thermal expansion. It has been known for quite a while that some solids contract along some lattice directions as they are heated. However, for use in the construction of zero thermal expansion ceramics a material must ideally show this property uniformly along all axes. Hence much interest was aroused when it was reported that the cubic material ZrW2O8 demonstrated negative thermal expansion over a wide temperature range (Mary et al. 1996), even on passing through a phase transition. Pryde et al. (1996) were able to rationalize the behavior of this system by demonstrating that there exist Rigid Unit Modes (RUMs) within the Brillouin zone which allow the polyhedra to rotate at very low energies, thus leading to contraction. Free energy minimization, based at the time on numerical methods, was also able to reproduce this effect in a more quantitative fashion. Prior to the above work it had been predicted from free energy minimization techniques that some zeolites and aluminophosphates would also show negative thermal expansion, a fact that was subsequently verified by experiment (Couves et al. 1993). In

56

Gale

the quest for further cubic materials that would contract on heating, the search returned to the arena of microporous materials where the naturally open structures of corner-sharing tetrahedra make ideal candidates for RUMs. Experimentally, faujasite, a microporous form of SiO2 with 12-ring channels, was found to demonstrate strong negative thermal expansion (Attfield and Sleight 1998). Based on this, free energy minimization was used to compare the properties of several structures based on sodalite units, including faujasite, zeolite-A and sodalite (Gale 1999). Both faujasite and zeolite-A where calculated to shrink on heating, while sodalite showed regular positive thermal expansion. This can be understood simply from the connectivity of the sodalite units. In sodalite itself these structural motifs are directly fused via four rings which removes the flexibility for rotation of the units. Hence the dominant effect is just the lengthening of the Si-O bonds as the temperature rises. The applications performed to date have demonstrated that free energy minimization is a useful complement to other finite temperature methods in the low temperature regime and that with the advent of analytical derivatives its application can be considered more routine. However, the temperature dependence of structure must be regarded as a severe test of a potential model and a challenge that most are only equal to to a qualitative degree. FUTURE DIRECTIONS IN INTERATOMIC POTENTIAL MODELLING OF IONIC MATERIALS Improved potential models The quality of the results of interatomic potential modelling will always depend on the particular choice of functional form chosen and how well it mimics the underlying physical interactions. Hence there is always a need to strive towards improved, more complex forcefields, though it is important that they remain substantially faster to evaluate than a full quantum mechanical calculation, otherwise there is no advantage except for systems where current solid state quantum theories fail. The shell model approach used in the work described above performs remarkably considering its simplicity. However, there are many cases that fail because there are aspects of the underlying physics that are missing. One well-documented failure is the fact that dipolar models predict that corundum is not the most stable polymorph of alumina under ambient conditions. It has been shown that the factor that stabilizes this particular structure relative to others is actually the quadrupolar polarizability (Wilson et al. 1996b). While the shell model can be extended to the elliptical breathing form that allows for higher order polarizability effects, the more appealing route is to employ point ions with induced moments, provided the dispersion series is damped at short range.

A further limitation of many of the models currently used is that they fail to show the correct behaviour in the dissociation limit. This is a consequence of the use of fixed charges regardless of environment. Although for small perturbations about equilibrium this is reasonable, if we want forcefields to be transferable to gas phase molecular clusters and surfaces then this is clearly more of a harsh approximation. In addition, for many cases there is an important many body contribution to the binding energy which arises from the increasing ionic character in the condensed phase. This is demonstrated clearly for water, where the binding energy per hydrogen bond is greater in ice than it is in the water dimer. One solution to the above difficulties that has been applied to ionic materials is to use a variable charge potential model. Here the charges are usually determined according to an electronegativity equalization scheme, wherein the energy of an ion is expanded as

Calculating the Structure & Properties of Ionic Materials

57

a quadratic function of its charge about the neutral state, involving the parameters of the electronegativity and hardness:

Ei = Ei0 + χ i0 qi +

1 0 2 μ i q i + ∑ qi q j J ij 2 j

(24)

Here the final term is the interaction of the charge with the potential due to other ions. The term Jij can be taken to be just the inverse distance between the ions (unscreened Coulomb potential) or more realistically it can be calculated as a two-centre integral of some form leading to damping at short range. The former approach is typified by the method of Mortier and co-workers (van Genechten et al. 1987), while the later method is utilized in the QEq scheme of Rappe and Goddard (1991). The chemical potential, which must be the same for all ions, is then given by the first derivative of the above expression. A matrix can be formulated and solved for the ion charges that satisfy this criterion, subject to the condition that the system remains charge neutral. This process can be repeated at any given geometry to yield the required charge distribution for the energy calculation. As the charges so found usually are the ones that minimize the electrostatic energy, the calculation of forces is no more complex than for a conventional forcefield due to the Hellmann-Feynman theorem. The variable charge approach was applied by vos Burchart et al. (1992) to silicates and aluminophosphates. However, they employed a three-body term with a harmonic form, which implies that the forcefield still was unable to handle the dissociation limit. More recently, Demiralp et al. (1999) have proposed the MS-Q forcefield model in which the charges are calculated according to the QEq scheme and the short range interactions are described by a Morse potential. This form indeed leads to an energy of zero when the material is dissociated into isolated atoms. Again this forcefield has been applied to microporous materials, including silica and aluminophosphate polymorphs, and shown to give reasonable results. The extension to MgO has also now been published (Strachan et al. 1999). A system that has attracted particular interest for variable charge models is that of rutile (TiO2) where two such forcefields have been designed and applied (Streitz and Mintmire 1995; Ogata et al. 1999). However, the range of use was still relatively narrow and the full benefits of a variable charge scheme not exploited as the transferability to a range of environments was not explored. Recently the MS-Q model has been extended to the study of various titanium oxides (Swamy and Gale 2000), but unlike previous variable charge models the parameters were fitted to reproduce the structure and properties of a range of polymorphs, including some which contain titanium in oxidation states lower than Ti(IV). Consequently the full potential of the variable charge scheme is realized by a forcefield that can describe multiple oxidation states with the same parameters. A comparison of cell parameters and bulk moduli against experiment for some of the phases is given in Table 4. As can be seen, the overall quality of reproduction is quite reasonable, despite the diversity of data included in the fit, with most structures reproduced to within a few percent. Even hongquiite (TiO) is only in error by 6% despite the fact that no information concerning Ti(II) phases was included in the training set. There have been previous shell model studies of multiple oxidation states of the titanium oxides (le Roux and Glasser 1997). However, in this case a distinct Ti-O potential is needed for each valence state of titanium leading to a model with more parameters than the nine fitted for the MS-Q model. Furthermore, in the mixed valence phases there is no need to make any assumptions concerning the assignment of titanium oxidation states to particular crystallographic sites.

58

Gale

Table 4. Comparison of calculated (Calc) and experimental (Exp) unit cell parameters and bulk moduli for titanium oxides according to the MS-Q model (Swamy and Gale 2000). Phase TiO2 Rutile TiO2 Anatase TiO2 Brookite Ti2O3 Ti3O5(L) TiO TiO2-II

a (Å) Exp Calc

b (Å) Exp Calc

β (°)

c (Å) Exp

Calc

Exp

Calc

K (GPa) Exp Calc

4.594

4.587

2.959

2.958

210

229

3.785

3.850

9.512

9.063

59/360

176

9.174

9.115

5.158 9.748 4.293 4.532

4.928 9.433 4.034 4.506

5.449

5.451

5.138

5.167

3.801

3.825

13.611 9.441

13.406 9.567

5.502

5.502

4.906

4.965

211

91.53

90.26 98/253/ 260

284 131 333 218

The use of variable charge models clearly has great potential for future use as a compromise between simpler force-field treatments and full-blown quantum mechanics. A combination of the electronegativity equalization with point ion polarizability, where the polarizability is coupled to the charge-state, may prove even more accurate and powerful. Structure solution and prediction

An important aim of simulation methods is to be able to predict crystal structures in advance. This is perhaps a bit ambitious for the time being as it requires both interatomic potentials which are reliable over a wide range of distances and methods which can sample vast regions of conformation space. However, an aim which can, and is, being achieved is the solution of structures given a unit cell and composition, both of which can normally be readily obtained even when a structure proves difficult to solve completely by conventional crystallographic means. There are several possible approaches to try to locate a global minimum for this type of problem. Simulated annealing is widely used, in which the temperature in a Monte Carlo calculation is gradually quenched. However, an alternative is to use genetic algorithms in which a large number of starting configurations evolve according to similar principles to the natural world. In essence, the configurations “mate” as pairs with the characteristics of the best parent tending to predominate in the next generation, but with there being the chance of mutations and other modifications of the breading process. Bush et al. (1995) have solved the structure of Li3RuO4 by combining an initial genetic algorithm run, using a cost function based on target co-ordination numbers and excluding unrealistic interatomic distances, followed by energy minimization of the best of the final configurations. This process yields a number of possible structures of similar energy that can then be used for calculation of a trial diffraction pattern for comparison with experiment. More recently this work has been refined and demonstrated to be successful for a wide range of binary and ternary oxide materials, including perovskite, spinel and pyrochlore structures (Woodley et al. 1999). As the number of different cations increases, it is found that genetic algorithms rapidly evolve to the correct structure, except for the cation ordering. Hence, it becomes necessary to introduce cation exchange as a possible pathway for structure evolution. Genetic algorithms have also

Calculating the Structure & Properties of Ionic Materials

59

been used as a successful aid to structure solution for molecular crystalline systems (Kariuki et al. 1997), where the basic structure of the individual units is known, but the packing arrangement within the cell has to be found. Despite the rapid advances being made in solid state quantum mechanics, interatomic potentials will still remain useful tools for the modelling of minerals for many years to come. Although routine athermal optimization of bulk structures will soon no longer be necessary with potential models, there are many other aspects were the rapidity is necessary for now, such as mapping out extensive phase diagrams and as an aid to structure solution. More sophisticated and transferable potential models will be needed in the future to extend the utility of such methods, improving their reliability in comparison to higher level techniques by learning from the detailed physical insights the latter have to offer. ACKNOWLEDGMENTS

The author would like to thank the Royal Society for a University Research Fellowship and funding, as well as EPSRC for provision of computing facilities. REFERENCES Allan NL, Barron THK, Bruno JAO (1996) The zero static internal stress approximation in lattice dynamics, and the calculation of isotope effects on molar volumes. J Chem Phys 105:8300-8303 Allinger N (1977) Conformational analysis. 130. MM2. A hydrocarbon forcefield utilizing V1 and V2 torsional terms. J Am Chem Soc 99:8127-8134 Attfield MP, Sleight AW (1998) Strong negative thermal expansion in siliceous faujasite. Chem Commun 601-602 Banerjee A, Adams N, Simons J, Shepard R (1985) Search for stationary points on surfaces. J Phys Chem 89:52-57 Barrera GD, Taylor MB, Allan NL, Barron THK, Kantorovich LN, Mackrodt WC (1997) Ionic solids at elevated temperatures and high pressures: MgF2. J Chem Phys 107:4337-4344 Beest BWH van, Kramer GJ, van Santen RA (1990) Force-fields for silicas and aluminophosphates based on ab initio calculations. Phys Rev Lett 64:1955-1958 Boer K de, Jansen APJ, van Santen RA, Watson GW, Parker SC (1996) Free-energy calculations of thermodynamic, vibrational, elastic, and structural properties of α-quartz at variable pressures and temperatures. Phys Rev B 54:826-835 Bradley RS (1937) The rate of unimolecular and bimolecular reactions in solution as deduced from a kinetic theory of liquids. Trans Faraday Soc 33:1185-1197 Bush TS, Catlow CRA, Battle PD (1995) Evolutionary programming techniques for predicting inorganic crystal structures. J Mater Chem 5:1269-1272 Busing WR (1981) WMIN, A Computer Program to Model Molecules and Crystals in Terms of Potential Energy Functions; ORNL-5747; Oak Ridge National Laboratory; Oak Ridge Catlow CRA, Bell RG, and Gale JD (1994) Computer modelling as a technique in materials chemistry. J Mater Chem 4:781-792 Catlow CRA, Cormack AN (1987) Computer modelling of silicates. Int Rev in Phys Chem 6:227-250 Catlow CRA, Faux ID, Norgett MJ (1976) Shell and breathing shell model calculations for defect formation energies and volumes in magnesium oxide. J. Phys. C: Solid State Phys. 9:419-429. Chadi DJ, Cohen ML (1973) Special points in the Brillouin zone. Phys Rev B 8:5747 Christie DM, Chelikowsky JR (1998) Structural properties of α-berlinite (AlPO4). Phys Chem Miner 25:222-226 Crawford WC, Hoersch AL (1972) Calcite-aragonite equilibrium from 50o to 150oC. Am Miner 57:995998 Couves JW, Jones RH, Parker SC, Tschaufeser P, Catlow CRA (1993) Experimental-verification of a predicted negative thermal expansivity of crystalline zeolites. J Phys-Condensed Matter 5:L329-L332 Demiralp E, Cagin T, Goddard WA (1999) Morse stretch potential charge equilibrium force field for ceramics: Application to the quartz-stishovite phase transition and to silica glass. Phys Rev Lett 82:1708-1711

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Dick BG, Overhauser AW (1958) Theory of the dielectric constants of alkali halide crystals. Phys Rev 112:90-103 Dove MT (1993) Introduction to lattice dynamics. Cambridge Topics in Mineral Physics and Chemistry 4, Cambridge University Press, Cambridge Dove MT, Winkler B, Leslie M, Harris MJ, Salje EKH (1992) A new interatomic potential model for calcite: Applications to lattice-dynamics studies, phase-transition, and isotope fractionation. Am Miner 77:244-250 Dowty E (1987) Fully automated microcomputer calculation of vibrational spectra. Phys Chem Miner 14:67-79 Essmann U, Perera L, Berkowitz ML, Darden T, Lee H, Pedersen LG (1995) A smooth particle mesh Ewald method. J Chem Phys 103:8577-8593 Ewald PP (1921) Die berechnung optischer und elektrostatischer gitterpotentiale. Annalen der Physik 64:253-287 Fisler DK, Gale JD, Cygan RT (2000) A shell model for the simulation of rhombohedral carbonate minerals and their point defects. Am Miner 85:217-224 Gale JD (1996) Empirical potential derivation for ionic materials. Phil Mag B 73:3-19 Gale JD (1997) GULP - A computer program for the symmetry adapted simulation of solids. J Chem Soc Faraday Trans 93:629-637 Gale JD (1998) Analytical free energy minimisation of silica polymorphs. J Phys Chem B 102:5423-5431 Gale JD (1999) Modelling the thermal expansion of zeolites, in Neutrons and Numerical Methods – N2M. Johnson MR, Kearley GJ, Büttner HG (eds), Th e American Institute of Physics, p 28-36 Gale JD, Catlow CRA, Mackrodt WC (1992) Periodic ab initio determination of interatomic potentials for alumina. Model and Simul in Mater Sci and Eng 1:73-81 Gale JD, Henson NJ (1994) Derivation of interatomic potentials for microporous aluminophosphates from the structure and properties of berlinite. J Chem Soc Faraday Trans 90:3175-3179 Genechten KA van, Mortier WJ, Geerlings P. (1987) Intrinsic framework electronegativity: A novel concept in solid state chemistry. J Chem Phys 86:5063-5071 Girard S, Mellot-Draznieks C, Gale JD, Ferey G. (2000) A predictive computational study of AlPO 4-14 : crystal structure and framework stability of the template-free AlPO4-14 from its as-synthesized templated form. Chem Commun, in press Grey TJ, Gale JD, Nicholson DN, Peterson BK (1999) A computational study of calcium cation locations and diffusion in chabazite. Mesoporous and Microporous Solids 31:45-59 Harrison NM, Leslie M (1992) The derivation of shell model potentials for MgCl2 from ab initio theory. Mol Simul 9:171-174 Henson NJ, Cheetham AK, Gale JD (1994) Theoretical calculations on silica frameworks and their correlation with experiment, Chem Mater 6:1647-1650 Henson NJ, Cheetham AK, Gale JD (1996) Computational studies of aluminium phosphate polymorphs. Chem Mater 8:664-670 Islam MS (1993) Simulation studies of lithium intercalation in transition metal oxides. Phil Mag A 68:667675 Islam MS, Ilett DJ (1994) Defect structure and oxygen migration in the La2O3 catalyst. Solid State Ionics 72:54-58 Jackson RA, Catlow CRA (1988) Computer simulation studies of zeolite structure. Mol Simul 1:207-224 Kantorovich LN (1995) Thermoelastic properties of perfect crystals with non-primitive lattices. I. General theory. Phys Rev B 51:3520-3534 Kapustinskii AF (1956) Lattice energy of ionic crystals. Quart Rev Chem Soc 10:283-294 Kariuki BM, Serrano-Gonzalez H, Johnston RL, Harris KDM (1997) The application of a genetic algorithm for solving crystal structures from powder diffraction data. Chem Phys Lett 280:189-195 Kleinman DA, Spitzer WG (1962) Phys Rev 125:16 Lager GA, Jorgensen JD, Rotella FJ (1982) Crystal-structure and thermal-expansion of α-quartz SiO2 at low-temperatures. J Appl Phys 53:6751-6756 Langmuir I, Dushman S (1922) Phys Rev 20:113 Le Roux H, Glasser L (1997) Transferable potentials for the Ti-O system. J Mater Chem 7:843-851 LeSar R, Najafabadi R, Srolovitz DJ (1991) Thermodynamics of solid and liquid embedded-atom-method metals. A variational study. J Chem Phys 94:5090-5097 Mary TA, Evans JSO, Vogt T, Sleight AW (1996) Negative thermal expansion from 0.3 to 1050 Kelvin in ZrW2O8. Science 272:90-92 McCusker LB, Baerlocher C, Jahn E, Bülow M (1991) The triple helix inside the large-pore aluminophosphate molecular-sieve VPI-5. Zeolites 11:308-313 Meis C, Gale JD (1998) Computational study of tetravalent uranium and plutonium diffusion in zircon. Mater Sci Eng B 57:52-61

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Monkhorst HJ, Pack JD (1976) Special points for Brillouin zone integration. Phys Rev B 13:5188-5192 Montroll EW (1942) Frequency spectrum of crystalline solids. J Chem Phys 10:218-228 Njo SL, Koningsveld H van, Graaf B van de (1997) A computational study on zeolite MCM-22. Chem Commun 1243-1244 Ogata S, Iyetomi H, Tsuruta K, Shimojo F, Kalia RK, Nakano A, Vashishta P (1999) J Appl Phys 86:30363041 Parker SC, Price GD (1989) Advances in Solid State Chemistry 1:295 Pavese A, Catti M, Price GD, Jackson RA (1992) Interatomic potentials for the CaCO3 polymorphs (calcite and aragonite) fitted to elastic and vibrational data. Phys Chem Miner 19:80-87 Perram JW, Petersen HG, Leeuw SW de (1988) An algorithm for the simulation of condensed matter which grows as the 3/2 power of the number of particles. Mol Phys 65:875-893 Petersen HG, Soelvason D, Perram JW, Smith ER (1994) The very fast multipole method. J Chem Phys 101:8870-8876 PressWH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in FORTRAN. Second edition. Cambridge University Press, Cambridge Pryde AKA, Hammonds KD, Dove MT, Heine V, Gale JD, Warren MC (1996) Rigid unit modes and the negative thermal expansion in ZrW2O8. J Phys Condensed Matter 8:10973-10982 Ramirez R, Böhm MC (1988) The use of symmetry in r eciprocal space integrations - asymmetric units and weighting factors for numerical-integration procedures in any crystal symmetry. Int J Quantum Chem 34:571 Rappe AK, Goddard III WA (1991) Charge equilibration for molecular dynamics simulations. J Phys Chem 95:3358-3363 Rudolph PR, Crowder CE (1990) Structure refinement and water location in the very large-pore molecular sieve VPI-5 by X-ray Rietveld techniques. Zeolites 10:163-168 Sanders MJ, Leslie M, Catlow CRA (1984) Interatomic potentials for SiO2. J Chem Soc Chemical Commun 1271-1273 Sastre G, Lewis DW, Catlow CRA (1996) Structure and stability of silica species in SAPO molecular sieves. J Phys Chem 100:6722-6730 Schröder K-P, Sauer J, Leslie M, Catlow CRA, T homas JM (1992) Bridging hydroxyl-groups in zeolitic catalysts – a computer simulation of their structure, vibrational properties and acidity in protonated faujasites (H-Y zeolites). Chem Phys Lett 188:320-325 Schröder U (1966) A new model for lattice dynamics (“breathing shell model”). Solid State Commun 4:347-349 Sowa H, Macavei J, Schulz H. (1990) The crystal structure of berlinite AlPO4 at high pressure. Zeitschrift für Kristallographie 192:119-136 Strachan A, Cagin T, Goddard WA (1999) Phase diagram of MgO from density functional theory and molecular-dynamics simulations. Phys Rev B 60:15084-15093 Streitz FH, Mintmire JW (1994) Charge-transfer and bonding in metallic oxides. J Adhesion Sci Tech 8:853-864 Sutton AP (1992) Direct free energy minimisation methods: application to grain boundaries. Phil Trans Roy Soc London A 341:233-245 Swamy V, Gale JD (2000) A transferable variable charge interatomic potential for atomistic simulation of titanium oxides. Phys Rev B, in press Taylor MB, Barrera GD, Allan NL, Barron THK (1997) Free-energy derivatives and structure optimization within quasiharmonic lattice dynamics. Phys Rev B 56:14380-14390 Tschaufeser P, Parker SC (1995) Thermal-expansion behaviour of zeolites and ALPO(4)s. J Phys Chem 99:10609-10615 Vos Burchart E de, van Bekkum H, van de Graaf B, Vogt ETC (1992) A consistent molecular mechanics force field for aluminophosphates. J Chem Soc Faraday Trans 88:2761-2769 Williams DE (1984) QCPE Bulletin 4:82 Williams DE (1989) Accelerated convergence treatment of R-n lattice sums. Crystallography Rev 2:3-25 and 163-166 Wilson M, Madden PA (1996) ‘Covalent’ effects in ‘ionic’ systems. Chem Soc Rev 339-351 Wilson M, Madden PA, Peebles SA, Fowler PW (1996) Cation polarization and the crystal structure of SnO. Mol Phys 88:1143-1153 Wilson M, Exner M, Huang Y-M, Finnis MW (1996) Transferable model for the atomistic simulation of Al2O3. Phys Rev B 54:15683-15689 Woodley SM, Battle PD, Gale JD, Catlow CRA (1999) The prediction of inorganic crystal structures using a genetic algorithm and energy minimisation. Phys Chem Chem Phys 1:2535-2542

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Wright PA, Natarajan S, Thomas JM, Bell RG, Gai-Boyes PL, Jones RH, Chen J (1992) Solving the structure of a metal-substituted aluminium phosphate catalyst by electron microscopy, computer simulation and X-ray powder diffraction. Angewante Chemie. Int Ed Eng 31:1472-1475 Wright PA, Sayag C, Rey F, Lewis DW, Gale JD, Natarajan S, Thomas JM (1995) Synthesis, characterisation and catalytic performance of the solid acid DAF-1. J Chem Soc Faraday Trans 91:3537-3547

3

Application of Lattice Dynamics and Molecular Dynamics Techniques to Minerals and Their Surfaces Steve C. Parker1, Nora H. de Leeuw2, Ekatarina Bourova1, and David J. Cooke1 1

Department of Chemistry University of Bath Bath, BA2 7AY, U.K. 2 Department of Chemistry University of Reading Reading, RG6 6AD, U.K. INTRODUCTION A central challenge for atomic level simulations of minerals is to be able to model the crystal structure, thermodynamics and atom transport. Clearly, if the same technique is employed then the underlying relationships between these properties can be examined. There are two atomistic simulation techniques that have been used to model these three properties for minerals, lattice dynamics (LD) and molecular dynamics (MD). The aim of this chapter is to describe these techniques and show, via a series of examples, how these methods can be applied. Both techniques involve solving Newton’s Laws of Motion but differ in the approximations made. Lattice dynamics involves analytical solution of the equations of motion followed by the use of a statistical mechanical treatment to obtain the thermodynamic properties, such as free-energy and heat capacity. The central assumption of LD is that the vibrational modes in the material are harmonic. Hence this approach can not be used reliably for liquids but has been used extensively for modeling the thermal properties of solids (e.g., reviews by Born and Huang 1954; Cochran 1973; and Barron et al. 1980). In contrast, molecular dynamics uses a numerical solution to the equations of motion where the atom positions and velocities are updated regularly and hence is not constrained to solids and can be readily applied to fluids (e.g., Allen and Tildesley 1989 and references therein). The consequence of using a numerical solution is that it requires much more computer CPU time, particularly when evaluating thermodynamic data. However, by not making assumptions about the potential energy surface, MD has the potential to be more accurate. The expansion in the application of both of these techniques to minerals in the last few years has resulted in many publications. The full range of applications is considerable and beyond the scope of this chapter. However, in this chapter we introduce the two techniques and then illustrate the scope of the techniques by giving examples where they have been used to model structure, thermodynamics and atom transport in oxides and minerals. Finally, we discuss the modeling of the mineral-fluid interface, which is one of the most challenging areas of active study. METHODOLOGY Simulation of minerals using both LD and MD requires the calculation of the total interaction energy, Ulatt, and the force on each atom, Fi. The dynamical contribution is evaluated via the equations of motion: 1529-6466/01/0042-0003$05.00

DOI:10.2138/rmg.2001.42.3

64

Parker, de Leeuw, Bourova & Cooke  Fi = mi

∂ 2 ui ∂t 2

(1)

where u and m are the displacement and the mass of atom i. The interaction energy and the forces on the atoms can be calculated in two ways, first using interatomic potentials where the interactions are defined using simple parameterized analytical functions and second, using electronic structure simulations where the energy and forces are calculated directly. The ideal solution would be to use full electronic structure simulations where the calculated forces are virtually guaranteed to be appropriate to all of the geometries adopted by the atoms. However, the computational resources required by such methods are still beyond routine use; and hence, these methods have only been used for a few systems with more than a few tens of atoms. As a consequence, most of the applications continue to use interatomic potentials to calculate the forces, and each simulation cell routinely contains hundreds of atoms. In addition, there are a wide range of data sets containing reliable transferable potentials (e.g., Lewis and Catlow (1985) and Gale, this volume), that have been exploited to model a wide range of minerals. One general result, which has long been known for polar solids (Cochran 1977), is that when reliable treatment of the dynamics is required, there needs to be some treatment of the electronic polarizability. One of the most successful has been the Dick and Overhauser shell model (1958) where a mass-less shell is attached to the core representing the nucleus and core electrons by a spring. The polarizability is related to the charge on the shell and the magnitude of the spring constant. There are also more sophisticated models available such as Rustad et al. (1995; this volume), Matsui et al. (2000), Harrison and Leslie (1992) and Madden and Wilson (1996). In the following sections we will give a brief description of the LD and MD methodologies and illustrate their use with some recent applications. LATTICE DYNAMICS The advantage of using lattice dynamics in the treatment of solids is that it allows the direct calculation of the vibrational frequencies (phonons). The assumption is that the normal modes are harmonic (i.e., the displacement of the atoms along the normal modes are directly proportional to displacement). Thus, for the atoms the equation of motion becomes: dU latt ∂ 2u = mi 2 i du i ∂t

(2)

where Ulatt is the lattice energy following a displacement u, and

U latt = U o + Wu 2

(3)

where U0 is the minimum lattice energy and W is the second derivative of lattice energy with respect to displacement. This is sometimes referred to as the quasi-harmonic approximation because the second derivatives, W, are assumed to be harmonic with respect to displacement but will vary with cell volume. Solution of these equations gives a simple eigenvector equation from which the vibrational frequencies can be extracted. The only subtlety is that the periodic nature of the solid must be taken into account by including the dependence of the displacements, and second derivatives, on wave-vector, k, which gives the frequencies at all possible wavelengths.

65

Lattice & Molecular Dynamics Applied to Minerals & Surfaces  u = u 0 exp(2πi (k • r − νt ))

(4)

W = W0 exp(2πi (k • r − νt ))

(5)

and The frequencies can be plotted as a function of wave-vector, k, called the phonon dispersion curve. The value of examining the dispersion curve is that it can give insight into the dynamical stability of the mineral and give such information as the onset of a displacive phase transition. This is illustrated by work by Watson and Parker (1995a,b) who investigated the amorphization of quartz at pressure. The presence of a soft mode indicates that the structure is unstable and will undergo a phase transition. When this softening occurs away from the zone center, k=(0,0,0), the new structure forms a supercell. They found that on applying high pressure to quartz that one of the vibrational modes softened at (0.3333, 0.3333, 0), (i.e., the frequency approached zero at a given value of k (see Fig. 1)), which in this case corresponds to the formation of a supercell that was three times bigger in the a and b directions. Further pressure on the supercell caused the crystal to undergo a catastrophic relaxation where the material went amorphous. Another useful way of displaying the vibrational frequencies is to integrate over all values of k, which generates a phonon density of states, g(v). The resulting function gives the number of phonons as a function of frequency, and can be used to identify frequencies where there is either a large number of vibrational modes or even gaps. The density of states can also be used to evaluate thermodynamic properties such as the internal energy, E, the Helmholtz free energy, A, and the Gibbs free energy, G.

E0 = U latt +

1 hνg (ν )dν 2∫

Figure 1. The phonon dispersion curve for quartz at high pressure, showing the vibrational frequencies in the (110) direction and the softening of the acoustic mode at k (0.3333, 0.3333, 0).

(6)

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Parker, de Leeuw, Bourova & Cooke 

E = E0 +

A = E0 +

∫

∫

⎡ ⎤ ⎢ ⎥ ⎢ hνg (ν ) ⎥ dν ⎢ ⎛ hν ⎞ ⎥ ⎟⎟ − 1⎥ ⎢ exp⎜⎜ k T ⎝ B ⎠ ⎦ ⎣

⎡ ⎛ hν g (ν )k B T ln ⎢1 − exp⎜⎜ − ⎝ k BT ⎣

S=

(7)

⎞⎤ ⎟⎟⎥dν ⎠⎦

(8)

(E − A) T

G = A + PV

(9) (10)

A further advantage of this approach for calculating the thermodynamic properties is that it incorporates quantum effects, for example E0, which represents the zero point energy. As noted above, LD is essentially a static tool for the solid state. However, with care it can also be used to model the transport of isolated atoms or vacancies. For example, the self-diffusion coefficient, Dself, of a given vacancy will depend on the number of such vacancies and their diffusivity and can be written as

Dself = N V DV

(11)

where Nv is the number of vacancies and DV is the diffusion coefficient of the vacancy. The number of vacancies at low temperatures, called the extrinsic regime, will depend on the number of aliovalent impurities and at high temperatures, the intrinsic regime, the vacancies will be thermally degenerated and hence depend on the free-energy of formation of the vacancies, ΔGf

N V = exp( − ΔG f / k B T )

(12)

where the free energy of formation of a vacancy, ΔGf, is the difference in free-energy of a simulation cell containing a vacancy compared to the free-energy of the pure material. The second term in the expression for Dself, the self diffusion coefficient, namely DV, depends on structural information and is associated with the local geometry around the vacancy and the free-energy of migration, ΔGm. The free-energy for migration is the difference between the free-energy of a simulation cell containing a vacancy at the lattice site and a simulation cell with an atom at a saddle-point midway between two vacancies. At the saddle point there will be one imaginary mode, but as demonstrated by Vineyard (1957) this can be ignored. Vocadlo et al. (1995) applied this approach to model oxygen and magnesium vacancy migration in MgO and found the expression for the coefficient for vacancy diffusion is given by: 2

Z⎛ a ⎞ DV = ⎜ ⎟ exp( − ΔGm / k B T ) 6⎝ 2⎠

(13)

where Z is the coordination environment of the diffusing species and a/√2 is the jump distance where the cell parameter is a. Alternatively the expression can be written as:

67

Lattice & Molecular Dynamics Applied to Minerals & Surfaces  2

Z⎛ a ⎞ DV = ⎜ ⎟ exp(−ΔS m / k B ) exp(−ΔH m / k BT ) 6⎝ 2⎠

(14)

and found, for example, that the self diffusion coefficient for oxygen is Dself (O ) = N V 2.84 x10 −6 exp( −1.99 / k B T )

(15)

where the activation or migration energy is 1.99 eV. The only difficulty of this approach is that the saddle point must be identified, which for a complex mineral or at an interface may not be straightforward. In summary, LD is an efficient tool for modeling the structure and thermodynamics of minerals and their surfaces and it can be used to investigate dynamical processes such as the onset of phase transitions and atom migration. Its major limitation is that the underlying harmonic approximation does not easily allow for the treatment of anharmonic effects. There are notable examples (Allan et al. 1989) where anharmonic effects are incorporated. However, when anharmonic effects dominate MD is the most viable technique as it incorporates anharmonicity explicitly. MOLECULAR DYNAMICS

Molecular dynamics differs from lattice dynamics because the particles are effectively involved in time-dependent motion. Its major appeal is that it is an intuitive way of modeling time-dependent phenomena, such as diffusion but as noted above the drawback is that it is CPU time-consuming and can be computationally expensive. To a large extent, this has been offset with the development of more efficient simulation packages and the advancement of computer technology. This makes it possible to undertake MD simulations on a desktop PC. In its simplest form MD, considers a box of N particles and monitors their relative positions, velocities and accelerations by solving Newton’s laws of motion at regular finite time intervals. Initially the particles are assigned pseudo-random velocities. These are often determined from a Maxwell-Boltzmann distribution and are required to meet certain conditions. These are that the kinetic energy of the system is such that the simulation temperature is fixed and that there is no net translational momentum. The forces acting on each particle, together with their velocities and positions are calculated for all subsequent time steps by considering Newton’s Laws of Motion. If the time step is infinitely small then the acceleration, a, of an atom can be calculated from the force. a=

F m

(16)

Similarly the velocity, v, and the new atom position, r can be calculated: v(t + δt ) = v(t ) + a(t )δt

(17)

r (t + δt ) = r (t ) + v(t )δt

(18)

In practice molecular dynamics is run with finite time steps. Using the equations above would therefore lead to the introduction of inaccuracies (Biesiadecki and Skeel 1993). A number of algorithms have been developed to overcome this difficulty. One of the most widely used is the Verlet Leapfrog Algorithm (VLA), modified from Verlet’s original algorithm (Verlet 1967) which uses the velocity at the mid-step v(t+½ δt).

68

Parker, de Leeuw, Bourova & Cooke  r (t + δt ) = r (t ) + v(t + v(t + δt ) = v(t −

δt 2

δt

)δt

(19)

) + a(t )δt

(20)

2

Since the simulation cells are small, compared to real crystals, it is unlikely therefore, that initially it will be at thermodynamic equilibrium. This causes the simulation temperature to fluctuate at the beginning of a simulation. In order to take consideration of this, the velocities are re-scaled at regular intervals throughout the initial run. In doing so, it enables the kinetic energy of the system to converge to a point where it corresponds to the chosen temperature (Jacobs and Rycerz 1997). This process is termed “equilibration.” Any data recorded during this initial period is not considered when calculating the properties of the system. Once the system has achieved the required Maxwell-Boltzmann distribution of velocities the simulation begins. A timestep, δt, must be chosen such that it is shorter than the period of any lattice vibrations. However, a shorter timestep leads to an increase in the number of iterations over which the simulation must run in order for the total sampling time to be of the desired length. This adds significantly to computational time. As a compromise the timestep is set so as to use the maximum amount of computer time reasonably available. A usual compromise is to set the timestep to 1 fs. The exception is when using a shell model, so as to include a representation of electronic polarisability as described above. Here the shell is given a small finite mass, between 0.1 to 0.5 a.u. so that the dynamics can be performed but care must be taken to ensure that there is no exchange of energy between core-shell vibrations and the usual vibration modes. However, the small shell mass requires a smaller timestep, typically, 0.1-0.2 fs (de Leeuw and Parker 1998). A further consideration is the conditions under which the simulation is run. We have described the approach where the energy and volume are kept constant but there are wellestablished techniques that maintain constant temperature (Nose 1984, 1990) and constant pressure (Parrinello and Rahman 1981). One problem that is particularly well suited for examination by MD is investigating the crystal structure at high temperatures. One such example where MD has been used to examine the study high temperature behavior of mineral structures is that of Bourova et al. (2000) on cristobalite. The low-pressure polymorphs of SiO2, such as cristobalite and quartz, show similar thermal behavior, where each has two configurations, one observed at low-temperature, α, and one observed at high temperature, β. The transition between α−β is a first-order displacive transition. Experimental work has shown that these high temperature phases are disordered but can not explain the nature of the high temperature structure. Another point requiring clarification is the volume decrease of β-phases with temperature prior to fusion. The comparison of the MD predictions with the experimental data, from X-ray diffraction, for the volume dilatation of cristobalite shows that MD can reproduce the behavior and give reliable results (Fig. 2). A clearer picture of the nature of the α−β transition and the disorder of β-phases at high temperature can be drawn from the radial distribution function (RDF) (Fig. 3) and the distribution of Si-O-Si inter-tetrahedral angle (Fig. 4) which were obtained by analysis of the atom positions. The RDF represents the average distance between the different order neighboring atoms and can often be compared with NMR experimental data (Dove et al. 1997). The angle distribution gives the statistical distribution of the Si-O-Si angle in space and time. The RDFs for Si-O and O-O distances show that

69

Lattice & Molecular Dynamics Applied to Minerals & Surfaces  1.12

T = 548 K tr

1.10 1.08

Figure 2. Molecular dynamics calculation of the cristobalite volume on heating (solid circles) and cooling (open circles) as compared with the experimental data where V0 is the room temperature volume.

1.06 1.04 - Berger et al. (1966)

1.02

- Schmahl et al. (1992) - Bouro va and Richet (1998)

1.00 500

1000 T (K)

1500

2000

(a)

Coesite (T=1800 K)

2000 K 1900 K 1800 K 1700 K 1600 K 1500 K 1400 K 1300 K

Figure 3. Radial distribution functions of cristobalite against temperature as compared with RDFs of other polymorphs.

1100 K 900 K 800 K 600 K 400 K 300 K

2.2

3.3 r Si-O (Å)

4.4

5.5

(b) Coesite (T=1800K)

Coesite (T=1800 K)

(c)

Quartz (T=2000K)

2000 K 1900 K

2.7

3.6 r

4.5 (Å)

O-O

5.4

2000 K 1900 K 1800 K 1700 K 1600 K 1500 K 1400 K 1300 K

1800 K 1700 K 1600 K 1500 K

1100 K

1000 K

900 K 800 K 600 K 400 K 300 K

900 K

6.3

1400 K 1300 K 1200 K 1100 K

800 K 600 K 400 K 300 K

3

4

5

6

r

Si-Si

(Å)

7

8

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coesite (T=1800 K)

Frequency

cristobalite α -phase

cristobalite β -phase

Figure 4. Molecular dynamics distribution of Si-O-Si angles in cristobalite, coesite and in liquid silica.

liquid (3400 K)

80

100

120

140

160

180

Angle Si-O-Si (°)

the α−β transition affects the arrangement of second order neighboring. The first peak is pronounced and does not widen significantly. This agrees with the accepted idea of the α−β transition, which is associated with the rotation of rigid SiO4 tetrahedron without any internal distortion. At high temperature the observed decrease in the cell volume corresponds to the decrease of the first Si-Si distance. The large thermal motions of the oxygen atoms, observed by tracing of each O atom trajectory, and the strong Si-O bond cause the neighboring tetrahedra to approach each other. The α−β transition is then accompanied by the increase of disorder, which can be seen from the RDFs, particularly from the longer length scales, i.e., beyond the level of SiO4 tetrahedron. The liquid-like Si-O-Si angle distribution at high temperature also clearly shows that dynamical disorder in bond angle is present (Fig. 4). In the α−phase, the Si-O-Si angle distribution is narrow with a peak at around 144o and varies slightly with temperature. At high temperature, the distribution widens markedly without any localized peak which suggests the phase becomes totally disordered and is not simply different domains of stable α-phase. The MD simulation of coesite also illustrates that disorder can be produced in the crystal structure but without requiring a transition. The RDFs and angle distribution for high temperature coesite are similar to those obtained for quartz and cristobalite at similar temperatures (Fig. 3). In addition, the simulated structure of high temperature coesite is different from the structure calculated at ambient conditions. The reason is that at high temperatures there are large thermal displacements causing the atoms to move away from the local well potential well. However, the structure does not have enough energy to break Si-O bonds and there are no other potential energy wells nearby, thus the structure becomes increasingly disordered but without undergoing a transition. The disorder transitions occur only when the potential energy surface has multiple potential wells, such as in quartz and cristobalite. The data from MD that we have used to study atomic transport are the Mean Square Displacements (MSD). The MSD represent the average displacement of an ion type from its initial coordinates. This is calculated periodically; and, if no increase in MSD is observed with time, the atoms are merely vibrating about their mean lattice sites. If the MSD of one ion type increases with time, then diffusion of that ion type is indicated. The diffusion coefficient, D, is the gradient of the graph of the MSD with time;

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Lattice & Molecular Dynamics Applied to Minerals & Surfaces  r 2 = 6 Dt + B

(21)

where B is the Debye-Waller factor for the atom considered which corresponds to the average displacement of atom from its lattice site due to thermal motion. Watson et al. (1992) studied atom transport in the perovskite-structures KCaF3, Figure 5, and is a structural analogue to the lower mantle-forming phase of MgSiO3. KCaF3, however, shows an increase in the MSD of fluorine with time from temperatures 150 K below its MD melting point. The MSD shown in Figure 6, illustrates that that the fluorine atoms are diffusing by a steady increase in the fluorine MSDs. The diffusion coefficient is the gradient of the slope. In contrast, K and Ca are simply vibrating about their lattice positions. The fluoride ions trajectories were animated to study their diffusion mechanism. The fluorine vacancies move through the system by hopping of the fluoride ions, often in correlated motion, involving between 1 and 5 ions. An example of a two ion correlated hop is shown in Figure 7, which took 0.35 ps to occur. The mechanism thus postulated is that of a vacancy mechanism with partially correlated hopping of the fluoride ions across the edges of the fluoride octahedra. The limitation of this approach is that when the activation energy for atom transport is in excess of kT there is little probability of an atom or vacancy migrating during a simulation run, and the diffusion constant will appear to be zero. Thus, constraints need to be introduced to force the atom to move. One approach for identifying the diffusion

Figure 5. Perovskite-structures KCaF3, showing the Ca atom (large medium grey sphere) in an octahedral site, surrounded by F atoms (light grey), and K atoms (black) occupy the corners of the cube.

18 16 14

Figure 6. MSD at 2300K illustrating that F is diffusing whilst the K and Ca are stationary.

MSD

12

F

10 8 6

K

Ca

4 2 0 0

100

200

300

time

400

500

600

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Figure 7. Showing diffusion mechanism of fluoride ions in KCaF3.

pathways and activation energies directly is described by Harris et al. (1997). A small force is added to an atom in the direction of a vacant site adjacent to the atom. By ensuring that the net force on the moving atom always contains a small component in the direction of the vacancy and is not constrained to move along a particular trajectory. Thus the moving atom can take an indirect path to the vacant site. This is essential because both Duffy and Tasker (1986) and Vocadlo et al. (1995) for NiO and MgO respectively showed that the migration path was not always linear between the two lattice sites. The simulation temperature is kept low so the rest of the crystal can relax as the atom moves from one site to another. Simulations using the modified MD code were performed for the bulk structure with a single vacancy. The activation energies for the migration of magnesium and oxygen vacancies were calculated by Harris et al. (1997) to be 1.94 ± 0.1 eV and 2.12 ± 0.1 eV respectively. This compares well with the values calculated by Vocadlo et al. (1995) using LD of 1.99 eV and 2.00 eV, respectively. The ion followed a linear pathway between the starting and finishing sites and the energy plot for magnesium transport is given in Figure 8.

Figure 8. Variation of energy with distance as a magnesium atom hops to an adjacent vacant site in MgO.

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In addition to studying diffusion pathways in the bulk material it is also possible to consider vacancy migration along a grain boundary. One of the key results was the activation energy for vacancy migration was found to be lower going down the dislocation pipes than across them (Fig. 9). For example, considering the {410}/[001] boundary in MgO the diffusion path down a single dislocation pipe is achieved by crossing the boundary to the opposite face, (e.g., AL to AR) The route AL to AR was the preferred route with an activation energy 1.05 ± 0.1 eV compared to 1.94 ± 0.1 eV for bulk. The moving ion did not significantly enter the dislocation core for this diffusion pathway. This implies that diffusion along the grain boundaries is enhanced by a lower activation energy. However, it is energetically easier to form a vacancy at the boundary than in the bulk so that there will also be an increase in diffusivity over the bulk because of the increased number of vacancies. In addition, Harris at al. (1997) also found that the presence of this increased number of vacancies has an impact on the vacancy migration. For example, an oxygen vacancy was introduced into the pipe and the magnesium vacancy migration from AL to AR was recalculated. The effect was to increase the activation energy for this move from 1.05 to 1.87 eV compared to the bulk value of 1.94 eV. Thus, in the highly defective boundaries that may be expected in rock matrices, it is conceivable that the activation energies for grain boundary migration may not be lowered and that enhanced diffusivity is simply due to the increased number of charged carriers. Unlike the lattice dynamics technique, it is often more difficult to obtain reliable thermodynamic data from molecular dynamics. This is partly due to the large number of configurations that need to be sampled. However, such calculations have been undertaken widely within the biochemistry community (see review by Kollman 1993; Osguthorpe and Dauber-Osguthorpe 1992). Except for a few notable exceptions (Harding 1989; Matsui 1989) there are surprisingly few applications to solids. We are considering a number of approaches for modeling thermodynamic data from MD. Two of which are showing promise, the first is to use the velocity information to generate the density of states and then follow the same procedure as outlined above for the lattice dynamics treatment. The second approach is simply to use the molecular dynamics to sample configurational space. The approach we adopted for obtaining the density of states was to use the analysis code FOCUS (Osguthorpe and Duaber-Osguthorpe 1992) and we have used it to calculate the surface free-energies. One of the program’s features is that it extracts density of states spectra from molecular dynamics trajectories using digital signal

Figure 9. Activation energies for migration at the {410}/[001] grain boundary in MgO.

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processing techniques. The approach involves taking the velocity (or coordinate) trajectory of each atom and Fourier transforming to the frequency domain, using a discrete Fourier transform: n

F `(ν k ) = ∑ Q (ν a ) a =1

2

` a

(22)

where Qa` = 1

− jν k t i ⎞ va (t i ) exp⎛⎜ N ⎟⎠ ⎝ i =1 N

N∑

(23)

and v is the velocity. After appropriately weighting and converting the frequencies to the phonon density of states (DOS) can be obtained: g (ν k ) =

1 F ` (ν k ) N kbT 2

(24)

which is closely related to the energy spectrum: K (ν k ) =

1 F ` (ν k ) 2 2N

(25)

We modeled three crystal structures using this approach, namely MgO, TiO2, and Fe2O3. The simulations were run at 300 K and density of states spectra were generated (see Fig. 10 for Fe2O3). We used the code DL_POLY (Forester and Smith 1995) and chose a mass of 0.5 a.u., which is small compared to the mass of the other atoms in the system. The time step was 0.1 fs, in order to ensure the stability of the system, since a shell model was being used and the total molecular dynamics run represented a period of approximately 20 ps. Thermodynamic data was then generated from the phonon density of states using the statistical thermodynamics expressions, described above, and the results are given in Table 1. For comparison, the results of lattice dynamics simulations on the same systems are given. The good agreement between the two techniques was especially noticeable for the calculated vibrational entropy of TiO2 and Fe2O3 where the differences were less than 1 kJmol-1. Similarly, comparison of the density of states between LD and MD (Fig. 10 for bulk Fe2O3) shows good agreement. One of the key differences will be that the MD density of states will contain anharmonicity effects, which are absent from the LD approach. However, we note that the MD takes typically a factor of 50 in CPU time greater than the LD method. In the final section, we describe work on the simulation of mineral–water interfaces which can only be modeled with molecular dynamics. SIMULATION OF MINERAL-WATER INTERFACES

The first step in modeling the mineral-water interface is to develop a reliable and consistent model for the interaction of water with solid surfaces. There is a wealth of different water potentials available (e.g., Duan et al. 1995; Jorgensen et al. 1983; Brodholt et al. 1995a,b). However, we require a potential that simulates polarizability and is compatible with our potential models for solid phases. Thus, we included polarizability by using the shell model (Dick and Overhauser 1958) for the oxygen atom of the water molecule.

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Lattice & Molecular Dynamics Applied to Minerals & Surfaces  50

4000

(a) 40

g(ν)

30 2000 20

Integral of g(ν)

3000

1000

10

0 0

200

400

600

800

0 1000

Figure 10. Density of states diagrams as calculated by (a) MD and (b) LD for bulk Fe2O3.

wavenumber / cm -1 1600

(b)

30

20

g(ν)

800

10

Integral of g(ν)

1200

400

0 0

200

400

600

Wavenumber / cm-1

800

0 1000

Table 1. Comparison of bulk vibrational energies per cation calculated using lattice dynamics and molecular dynamics. Lattice Dynamics

Molecular Dynamics

15.07 20.06 26.07 12.24

16.70 22.06 29.28 13.27

25.36 32.32 37.28 21.14

28.73 35.68 37.36 24.47

16.73 23.85 38.33 12.35

21.17 28.18 38.90 16.51

MgO Zero Point Energy / kJmol-1 Vibrational Enthalpy / kJmol-1 Vibrational Entropy / Jmol-1K-1 Vibrational Free Energy / kJmol-1

TiO2 Zero Point Energy / kJmol-1 Vibrational Enthalpy / kJmol-1 Vibrational Entropy / Jmol-1K-1 Vibrational Free Energy / kJmol-1

Fe2O3 Zero Point Energy / kJmol-1 Vibrational Enthalpy / kJmol-1 Vibrational Entropy / Jmol-1K-1 Vibrational Free Energy / kJmol-1

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The potential parameters for the water molecule were empirically fitted to reproduce the experimental dipole moment, O-H bond length and H-O-H angle of the water monomer and the structure of the water dimer and infra-red data. Molecular dynamics simulations were then used to calculate the self-diffusion coefficient, radial distribution functions (RDFs) and energy of evaporation of liquid water. The computer code DL_POLY 2.6 code (Forester and Smith 1995) was employed. We simulated a box containing 256 water molecules at a temperature of 300 K where the conditions were initially set at the experimental density of ρ = 1.0 g/cm3 and run with an NPT ensemble. We chose a mass for the oxygen shell of 0.2 a.u., which is small compared to the mass of the hydrogen atom of 1.0 a.u. However, due to the small shell mass we needed to run the MD simulation with the small timestep of 0.2 fs in order to keep the system stable. With this timestep we obtained data at constant pressure and temperature for a period of 100 picoseconds. The properties calculated from the MD simulation were radial distribution functions, average energy, density, specific heat capacity, compressibility and MSDs from which the self-diffusion can be evaluated. The self-diffusion coefficient was calculated to be 1.15×10-9 m2s-1 (exp. 2.3×10-9 m2s-1 at 298 K). This value is low compared to the experimental value at 298 K, but agrees with an experimental value of 1.17×10-9 m2s-1 for a water temperature of 275 K (Krynicki et al. 1978). Although the calculated diffusion coefficient is too low for the simulation temperature of 300 K, it still falls within the range for liquid water. As we were interested in obtaining hydration energies for the adsorption of water molecules onto solid surfaces, a good test of our potential model would be to obtain an energy of vaporization from our MD simulations. We calculated this vaporization energy from the interaction energies between the water molecules in the system. The energy of vaporization hence calculated is 43.0 kJmol-1 which is in excellent agreement with the standard experimental value of 43.4 kJmol-1 at 310 K. Other results from the MD simulation that can be checked against experimental data are the radial distribution functions (RDF) of the various ions in the system. Figure 11 shows the RDFs for the O-O, O-H and H-H pairs where the peaks due to intramolecular interactions have been omitted. The RDF between oxygen atoms shows a very clear peak at 2.97 Å and a broader area between 5 and 6 Å. The first peak is in good agreement with experimental findings (2.88 Å) (Soper and Phillips 1986), although the experimental value for the second peak at 4.6 Å is somewhat smaller than the calculated value, although this is in line with other water potential models (c.f. 5.4 Å for a flexible TIPS model) (Dang and Pettitt 1987). The heights of the peaks, 3.8 and 1.3, also compare well to experimental values of 3.1 and 1.1 (Soper and Phillips 1986) indicating that our model shows ordering of the water molecules which agrees adequately with experimental findings. The first peak of the O-H RDF at 2.12 Å. is again at a somewhat larger distance than that found by Soper and Phillips (1986) (1.9 Å) although the second maximum at 3.13 Å agrees well with experimentally observed RDFs (3.2 Å). The heights of the peaks of 0.9 and 1.3 compare favorably with experimental values of 1.0 and 1.3 (Soper and Phillips 1986). Finally, the H-H RDF shows a peak at 2.6 Å of height 1.3, a shoulder at about 3.5 Å (height ≅ 1.0) and another peak at 5.7 Å of height 1.1. This compares with experimental peaks at 2.3, 3.7 and 4.9 Å, heights 1.3, 1.2 and 1.0 respectively, which again is in good agreement. Overall, the simulated and experimental systems show similar ordering of the water molecules. Once a reliable and consistent model is available for water the mineral-water interface can be considered. The work of Rustad (this volume) provides further examples. However, we will describe two systems MgO and CaCO3. The mineral considered initially was MgO. It has a relatively simple structure (i.e., face-centered cubic with six coordinate oxygens and cations) and its importance both as a support for metal catalysts

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1.6

(a)

3.5

(b)

3

1.2

RDF

RDF

2.5 2

0.8

1.5 1

0.4

0.5 0

0

1

2

3

4

5

6

7

1

2

3

r(O-O) (A)

4

5

6

7

r(O-H) (A)

3.5

(c)

3

RDF

2.5 2

1.5 1 0.5 0 1

2

3

4

5

6

7

r(H-H) (A)

Figure 11. (a) O-O, (b) O-H and (c) H-H radial distribution functions, omitting intramolecular OH and HH interactions.

and as a catalyst in its own right, make it an attractive model system and appropriate to test the applicability of the water potential. The MgO {100} surface was simulated as a repeating slab and void, the slab consisting of a 4×4×4 supercell of 256 MgO units and this system consisting of the pure surface in vacuo was run under NVT conditions. The void was then filled with NPT equilibrated bulk water and the entire system of MgO slab and surrounding liquid water was simulated under NPT conditions. The gap between the surfaces of the repeated cell was 30 Å containing 275 water molecules, the whole system consisting of 1868 species including shells. The average surface energy of the unhydrated {100} surface obtained from the NVT simulations in vacuo was calculated to be 1.31 Jm-2 at 300 K, comparable to that obtained from previous static calculations (1.25 Jm-2) (de Leeuw et al. 1995). After running the MgO slabs with the water molecules under NPT conditions the average surface energy was calculated to be 2.89 Jm-2 indicating that the {100} surface in liquid water is not very stable. This is further confirmed by the average hydration energy of +28.5 kJmol-1 which shows that hydration of the {100} under liquid water conditions is an endothermic process. The RDFs between magnesium ions and the oxygen atoms of the water molecules and between surface lattice oxygen ions and hydrogen atoms are shown in Figure 12. The first peaks at 2.0 and 1.8 Å respectively are in accord with the experimentally found Mg-Owater distances in hydrated magnesium salts and hydrogen-

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1.2

1.2

(b) 1

0.8

0.8

RDF

RDF

(a) 1

0.6

0.6

0.4

0.4

0.2

0.2

0

0 1

2

3

4

5

6

7

1

2

r(Mg-O) (A)

3

4

5

6

7

r(O-H) (A)

Figure 12. (a) Mg-Owater and (b) Olattice-H radial distribution functions of the equilibrated NPT simulation of the MgO {100} surface in water at 300K.

bonding. The self-diffusion coefficient of the water molecules between the slabs of MgO was calculated to be 4.7×10-9 m2s-1, a large increase from the value of 1.15×10-9 m2s-1 for the system of pure water. This is probably due to the fact that the density of the water molecules between the slabs has decreased from the pure water value of 1.3 gcm-3 to 1.00 gcm-3 between the MgO surfaces. As such the water molecules have scope to move more freely. The decrease in density may imply that the water is repelled by the MgO surfaces or at least that the MgO surface disrupts the hydrogen bonding in the water. However, when we look at a histogram of the number of water molecules as a function of distance from the MgO slab (Fig. 13) it is clear that the water density is greatest near the MgO surface and that there is a clear preferred orientation on the surface. This disrupts the bonding with the next layer of water and hence the density decreases in the next few layers towards a fairly level density midway between the two slabs. Together with the lower density, the implication is that the adsorption pattern on the surface forces the water molecules in subsequent layers to form an intermolecular configuration which is more open than in the system of pure water. Although rather more speculative, the oscillatory behavior in the density (Fig. 13) with two low density areas at 9-10 Å from

number of molecules

2.5

2

Figure 13. Histogram of the water molecules between the slabs of MgO {100} showing the average number of water molecules as a function of the position coordinate normal to the surface, where the two {100} surfaces are at coordinate positions 0 and 31 Å.

1.5

1

0.5

0 0

5

10

15

height (A)

20

25

30

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Lattice & Molecular Dynamics Applied to Minerals & Surfaces 

the slab surfaces, may indicate an even longer range disruption of the bulk water structure than just the monolayer adsorbed on the surfaces. Of course, this effect may have been exacerbated by the relatively small number of water molecules in the system. It would therefore be interesting to model a larger system containing more water molecules but at present, due to the use of shell model potentials, the system modeled here is stretching computational resources to the limit. Having successfully studied the structure and energies of various MgO-water interfaces where we were interested in the mineral-water interface itself, we have since begun to study surface processes, such as crystal growth and dissolution, that takes place in an aqueous environment and where inclusion of a water layer is necessary to accurately model the various processes taking place at the surface. One example is our study of calcite crystal dissolution. Calcite is one of the most abundant minerals in the environment and of fundamental importance in many fields, both inorganic and biological. We have used molecular dynamics simulations to investigate the energetics of key stages in calcite dissolution, which is achieved by modeling the dissolution of CaCO3 units from two different monatomic steps on the main (104) cleavage plane, in the presence of water. The two different steps were an acute step, where the carbonate group on the edge of the step overhangs the plane below the step (Fig. 14a) and the angle between step wall and plane is 80° on the relaxed surface (cf. exp. 78°, Park et al. 1996) and an obtuse step, where the carbonate groups on the step edge lean back with respect to the plane below (Fig. 14b) with an angle between step wall and plane of 105° on the relaxed surface (exp. 102°). These two types of step are found experimentally to form the dissolving edges of etch pits (Park et al. 1996; Liang et al. 1996) and the obtuse step is found to be the fastest moving of the two. We did a series of calculations, whereby successive CaCO3 units were removed from the step edges and the dissolution energies calculated as follows: [CaCO3 ] n ( s ) → [CaCO3 ] n −1( s ) + Ca (2aq+ ) + CO32(−aq )

(26)

Figure 15 shows a schematic representation of dissolution from the two steps and gives the energies expended or released upon removing a consecutive calcium carbonate unit from the dissolving step. Removal of the first calcium carbonate unit from the acute step, introducing two opposing kink sites on the edge (Jordan and Rammensee 1998) (Fig. 15a), is energetically the most expensive at +103.7 kJmol-1. Removing a second unit from the site adjacent to the first, which does not alter the number of kink sites costs much less energy (+36.2 kJmol-1). If the energy of removing a portion of the step was constant we would have expected removal of the third unit to cost about another 36 kJmol-1. However, it is energetically favorable (-24.1 kJmol-1). Alternatively, removal of the second unit from the next nearest neighbor position from the first site introducing yet another double kink site separated by a small gap is, not surprisingly, energetically more

(a)

(b)

Figure 14. Schematic representation of (a) acute and (b) obtuse steps on the {10 1 4} surface.

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Parker, de Leeuw, Bourova & Cooke 

(a)

(b)

Figure 15. Schematic representation of the energetics of step-by-step dissolution of calcium carbonate units from (a) the acute and (b) the obtuse step edges.

expensive than removal from the site next to the first unit (+72.4 kJmol-1). This energy is not as large as the formation of an isolated double kink site (+103.7 kJmol-1) indicating that there is an energy of attraction between the double kinks. When finally the fourth calcium carbonate unit is added, annihilating all kink sites and completing the growing edge, a large amount of energy is released, at -235.4 kJmol-1 far larger than the energy expended by the removal of the first unit and introduction of the first kink sites. The process is similar at the obtuse step (Fig. 15b). The initial removal of the first calcium carbonate unit from the step at +45.8 kJmol-1 is not as energetically expensive as from the acute step. When a second unit, adjacent to the first is removed, the energy at -33.8 kJmol-1 is exothermic rather than endothermic on the acute surface (+36.2 kJmol-1). Removing the second unit from the next nearest neighbor position and increasing the number of kink sites is energetically still slightly exothermic (-2.4 kJmol-1). Finally, when the fourth calcium carbonate unit is removed energy is again released (-82.0 kJmol-1) although less than on the acute step. Thus we expect dissolution from the obtuse step to occur preferentially, in agreement with experiment (Liang 1996). On both steps, however, dissolution of the final crystal unit from the dissolving step, and hence creating a complete edge, releases about twice the energy from what is needed to dissolve the first unit from the complete edge (-235.4 vs. +103.7 kJmol-1 on the acute edge and -82.0 vs. +45.8 kJmol-1 on the obtuse edge). Therefore, the energy released on dissolution of the final calcium carbonate unit from the edge would be enough to instigate the dissolution of two crystal units from the next step edge. CONCLUSIONS

This chapter has, we hope, illustrated the scope of lattice dynamics and molecular dynamics to model the structure, thermodynamics and diffusion in oxides and minerals. Although the techniques are well-established there are many applications to minerals that still need to be addressed. One area that we have touched on is the study of the mineral-

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fluid interface, which is an area of active study. Finally, these techniques will continue to be used widely particularly with the development of electronic structure codes that will allow not only the structure and thermodynamics to be investigated but reactivity. REFERENCES Allan NL, Kenway P, Mackrodt WC, Parker SC (1989) Calculated surface-properties of La2CuO4. Implications for high-Tc behavior. J Phy-Cond Mat 1:SB119-SB122 Allen MP, Tildesley DJ (1989) Computer Simulation of Liquids. Clarendon Press, Oxford Barron THK, Collins JG, White GK (1980) Thermal expansion of solids at low temperatures. Adv Phys 29:609-724 Berger C, Eyraud E, Richard M, Riviere R (1966) Etude radiocristallographique de variation de volume pour quelques materiauw subissant des transformations de phase solide-solide. Bull Soc Chim Fr 32:628-633 Biesiadecki JJ, Skeel RD (1993) Dangers of multiple time-step methods. J Comp Phys 109 318-328 Born M, Huang K (1954) Dynamical Theory of Crystal Lattices. Oxford University Press Bourova E, Parker SC, Richet P (2000) Atomistic simulation of cristobalite at high temperature. Phys Rev B 62:12052-12061 Bourova E, Richet P (1998), Quartz and cristobalite: high-temperature cell parameters and volumes of fusion. Geophys Res Let 25:2333-2336 Brodtholt H, Sampoli M, Vallauri R (1995a) Parameterizing a polarizable intermolecular potential for water with the ice 1H phase. Mol Phys 85:81-90 Brodtholt H, Sampoli M, Vallauri R (1995b) Parameterizing a polarizable intermolecular potential for water. Mol Phys 86:149-158 Cochran W (1973) The Dynamic of Atoms in Crystals. Edward Arnold, London Dang LX, Pettitt BM (1987) Simple intramolecular model potentials for water. J Phys Chem 91:3349-3354 de Leeuw NH, Parker SC (1998) Molecular-dynamics simulation of MgO surfaces in liquid water using a shell-model potential for water. Phys Rev B-Cond Mat 58:13901-13908 de Leeuw NH, Watson GW, Parker SC (1995) Atomistic simulation of the effect of dissociative adsorption of water on the surface structure and stability of calcium and magnesium oxide. J Phys Chem 99:17219-17225 Dick BJ, Overhauser AW (1959) Theory of dielectric constants of alkali halide crystals. Phys Rev 112:90103 Dove MT, Keen DA, Hannon AC, Swainson IP (1997) Direct measurement of Si-O bond length and of orientational disorder in the high-temperature phase of cristobalite. Phys Chem Min 24:311-317 Duan Z, Moller N, Weare JH (1995) Measurement of the PVT properties of water to 25 kBars and 1600°C from synthetic fluid inclusions in corundum – Comment. Geochim Cosmochim Acta 59:2639-2639 Duffy DM and Tasker PW (1986) Theoretical studies of diffusion-processes down coincident tilt boundaries in NiO. Phil Mag 54:759-771 Forester TR, Smith W (1995) DL_POLY user manual. CCLRC, Daresbury Laboratory, Daresbury, Warrington, UK Harding JH (1989) Calculation of the entropy of defect processes in ionic solids. J Chem Soc-Far Trans 85:351-365 Harris DJ, Watson GW, Parker SC (1997) Vacancy migration at the {410}/[001] symmetric tilt grain boundary of MgO: An atomistic simulation study. Phys Rev B-Cond Mat 56:11477-11484 Harrison NM, Leslie M (1992) The derivation of shell-model potentials for MgCl2 from ab-initio theory Mol Sim 9:171-174 Jacobs PWM, Ryzcerz ZA (1997) Computer Modeling in Organic Crystallography. Academic Press, London Jordan G, Rammensee W (1998) Dissolution rates of calcite {10 1 4} obtained by scanning force microscopy: Microtopography-based dissolution kinetics on surfaces with anisotropic step velocities. Geochim Cosmochim Acta 62:941-947 Jorgensen WL, Chandrasekhar J, Madura JD, Impey RW, Klein ML (1983) Comparison of simple potential functions for simulating liquid water. J Chem Phys 79:926-935 Kollman P (1993) Free-energy calculations. Applications to chemical and biochemical phenomena. Chem Rev 93:2395-2417 Krynicki K, Green CD, Sawyer DW (1978) Pressure and temperature dependence of self diffusion in water. Faraday Discuss Chem Soc 66:199-208 Lewis GV, Catlow CRA (1985) Interatomic potential - Derivation of parameters for binary oxides and their use in ternary oxides. J Phys C 18:1149-1161

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Liang Y, Baer DR, McCoy JM, Amonette JE, LaFemina JP (1996) Interplay between step velocity and morphology during the dissolution of CaCO3 surface. Geochim Cosmochim Acta 60:4883-4887 Madden RA, Wilson M. (1996) ‘Covalent’ effects in ‘ionic’ systems. Chem Soc Rev 25:339-350 Matsui M (1989) J Chem Phys 91:489-494 Matsui M, Parker SC, Leslie M (2000) The MD simulation of the equation of state of MgO: Application as a pressure calibration standard at high temperature and high pressure. Am Min 85:312-316 Nose S (1990) Constant temperature molecular dynamics. J Phys C 2:SA115 Nose S (1994) J. Chem Phys 81 511 Osguthorpe DJ, Dauberosguthorpe (1992) P Focus. A program for analyzing molecular-dynamics simulations, featuring digital signal-processing techniques. J Mol Graph 10:178-184 Park NS, Kim MW, Langford SC, Dickinson JT (1980) Atomic layer wear of single-crystal calcite in aqueous solution scanning force microscopy. J Appl Phys 80:2680-2686 Parrinello M, Rahman A (1981) Polymorphic transitions in single crystals a new molecular dynamics method. J Appl Phys 52:7182-7190 Rustad JR, Hay BP, Halley JW (1995) Molecular dynamics simulation of iron(III) and its hydrolysis products in aqueous solution. J Chem Phys 102:427-431 Schmahl WW, Swainson IP, Dove MT, Graeme-Barber A (1992) Landau free energy and order parameter behavior of the α/β phase transition in cristobalite. Z Kristallogr 201:125-145 Soper AK, Phillips MG (1986) A new determination of the structure of water at 25°C. Chem Phys 107:4760 Verlet L (1967) Computer experiments on classical fluids, thermodynamical properties of Lennard-Jones molecules. Phys Rev A 159:98-103 Vineyard GH (1957) Frequency factors and isotope effects in solid state processes. J Phys Chem Solids 3:121-127 Vocadlo L, Wall A, Parker SC, Price GD (1995) Absolute ionic-diffusion in MgO – computer calculations via lattice-dynamics. Phys Earth Planet Int 88:193-210 Watson GW, Parker SC (1995) Dynamical instabilities in α-Quartz and α-Berlinite; A mechanism for amorphization. Phys Rev B-Cond Mat 52:13306-13309 Watson GW, Parker SC (1995) β-Quartz amorphization – a dynamical instability. Phil Mag Let 71:59-64. Watson GW, Parker SC, Wall A (1992) Molecular-dynamics simulation of fluoride-perovskites. J PhysCond Mat 4:2097-2108

4

Molecular Simulations of Liquid and Supercritical Water: Thermodynamics, Structure, and Hydrogen Bonding Andrey G. Kalinichev Department of Geology University of Illinois at Urbana-Champaign 1301 W. Green St., Urbana, Illinois, 61801, U.S.A. and Institute of Experimental Mineralogy Russian Academy of Sciences Chernogolovka, Moscow Region, 142432, Russia INTRODUCTION

Water is a truly unique substance in many respects. It is the only chemical compound that naturally occurs in all three physical states (solid, liquid and vapor) under the thermodynamic conditions typical to the Earth’s surface. It plays the principal role in virtually any significant geological and biological processes on our planet. Its outstanding properties as a solvent and its general abundance almost everywhere on the Earth’s surface has made it also an integral part of many technological processes since the very beginning of the human civilization. Aqueous fluids are crucial for the transport and enrichment of ore-forming constituents (Barnes 1997; Planetary Fluids 1990). Quantitative analysis of hydrothermal and metamorphic processes requires information on the physical-chemical, thermodynamic and transport properties of the fluid phases involved (Helgeson 1979, 1981; Sverjensky 1987; Eugster and Baumgartner 1987; Seward and Barnes 1997). These processes encompass a broad range of pressure and temperature conditions and, therefore, detailed understanding of the pressure and temperature dependencies of density, heat capacity, viscosity, diffusivities, and other related properties is necessary in order to develop realistic models of fluid behavior or fluid-mineral interactions. Aqueous fluids under high-pressure, high-temperature conditions near and above the critical point of water (P = 22.1 MPa and T = 647 K) are especially important in a variety of geochemical processes. Due to the large compressibility of supercritical fluid, small changes in pressure can produce very substantial changes in density, which, in turn, affect diffusivity, viscosity, dielectric, and solvation properties, thus dramatically influencing the kinetics and mechanisms of chemical reactions in water. Models of hydrothermal convection suggest that the near-critical conditions provide an optimal convective behavior due to unique combination of thermodynamic and transport properties in this region of the phase diagram of water (Norton 1984; Jupp and Schultz 2000). Directly measured temperatures of seafloor hydrothermal vents reach near-critical values of 630680 K, which greatly affects the speciation in these complex chemical systems (Tivey et al. 1990; Von Damm 1990). From an engineering viewpoint, supercritical water has also attracted growing attention in recent years as a promising chemical medium with a wide range of different environmentally friendly technological applications (Levelt-Sengers 1990; Shaw et al. 1991; Tester et al. 1993). From either geochemical or technological perspective, a fundamental understanding of the complex properties of supercritical aqueous systems 1529-6466/01/0042-0004$05.00

DOI:10.2138/rmg.2001.42.4

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and the ability to reliably predict them using physically meaningful models is of primary importance. It is a common knowledge that many anomalous properties of water as a solvent arise as a consequence of specific hydrogen bonding interactions of its molecules. Under ambient conditions these anomalous properties of liquid water arise from the competition between nearly ice-like tetrahedrally coordinated local patterns characterized by strong hydrogen bonds and more compact arrangements characterized by more strained and broken bonds (e.g., Stillinger 1980; Okhulkov et al. 1994; Kalinichev et al. 1999). The question of the ranges of temperature and density (or pressure) where these specific interactions can significantly influence the observable properties of water has long been considered very important for the construction of realistic structural models for this fluid (Eisenberg, Kauzmann 1969). The answer to this question varied over time, but as more experimental evidence was gained, the temperature limit for H-bonding in water predicted to be higher and higher. At first, it was thought that hydrogen bonds would disappear above ~420 K. Then, Marchi and Eyring (1964) suggested to shift this limit up to ~523 K, assuming that above this temperature water consists of freely rotating monomers. At the same time, Luck (1965), experimentally studying the IR absorption in liquid water, extended the limit for H-bonding at least up to the critical temperature, 647 K. A subsequent series of high-temperature spectroscopic experiments (Franck and Roth 1967; Bondarenko and Gorbaty 1973, 1991) demonstrated that the upper limit for hydrogen bonds in water had not been reached even at temperatures as high as 823 K. Moreover, x-ray diffraction studies of liquid and supercritical water (Gorbaty and Demianets 1983) gave indications of a non-negligible probability even for tetrahedral configurations of the H-bonded molecules to exist under supercritical conditions of 773 K and 100 MPa. Direct experimental investigations of the water structure at high temperatures and pressures represent a very challenging undertaking, and any new set of structural or spectroscopic information obtained under such conditions is extremely valuable. Recent introduction into this field of the powerful technique known as neutron diffraction with isotope substitution (NDIS) (Postorino et al. 1993; Bruni et al. 1996; Soper et al. 1997), signified a very important step forward, since this method allows one to experimentally probe all three atom-atom structural correlations in water (OO, OH, and HH) simultaneously. However, it was quite surprising when the very first results of such neutron diffraction measurements were interpreted as the direct evidence of the complete absence of H-bonds in water at near-critical temperatures (Postorino et al. 1993). Despite obvious contradiction with previous experimental data and the results of several molecular computer simulations (Kalinichev 1985, 1986, 1991; Mountain 1989; Cummings et al. 1991), this unexpected conclusion has already made its way into the geochemical literature (Seward and Barnes 1997). At the same time, amplified by the increasing demand for the detailed molecular understanding of the structure and properties of high-temperature aqueous fluids from the geochemical and engineering communities, this controversy over the degree of hydrogen bonding in supercritical water fuelled a virtual explosion of new experimental and theoretical studies in this field by means of neutron scattering (Soper 1996; BellisentFunel et al. 1997; De Jong and Neilson 1997; Botti et al. 1998; Tassaing et al. 1998, 2000; Uffindell et al. 2000), X-ray diffraction (Yamanaka et al. 1994; Gorbaty and Kalinichev 1995), optical spectroscopy (Bennett and Johnston 1994; Bondarenko and Gorbaty 1997; Gorbaty and Gupta 1998; Gorbaty et al. 1999; Hu et al. 2000), NMR spectroscopy (Hoffmann and Conradi 1997; Matubayasi et al. 1997a,b), microwave spectroscopy (Yao and Okada 1998), and computer simulations (Chialvo and Cummings

 

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1994, 1996, 1999; Fois et al. 1994; Kalinichev and Bass 1994, 1995, 1997; Löffler et al. 1994; Mizan et al. 1994, 1996; Cui and Harris 1994, 1995; Duan et al 1995; Mountain 1995, 1999; Kalinichev and Heinzinger 1995; Balbuena et al. 1996a,b; Chialvo et al. 1998, 2000; Driesner et al. 1998; Famulari et al. 1998; Jedlovszky et al. 1998, 1999; Kalinichev and Gorbaty 1998; Liew et al. 1998; Kalinichev and Churakov 1999; Matubayasi et al. 1999; Reagan et al. 1999; Churakov and Kalinichev 2000). By the early 1990s, classical Monte Carlo (MC) and Molecular Dynamics (MD) computer simulations had already become powerful tools in the studies of the properties of complex molecular liquids, including aqueous solutions (e.g., Heinzinger 1986, 1990). Being neither experiment nor theory, computer “experiments” can, to some extent, take over the task of both in these investigations. The greatest advantage of simulation techniques over conventional theoretical approaches is in the limited number of approximations used. Provided one has a reliable way to calculate inter- and intramolecular potentials, the simulations can lead to information on a wide variety of properties (thermodynamic, structural, transport, spectroscopic, etc.) of the systems under study. In the case of simple fluids, like liquid noble gases, the results of computer simulations have long been used as an “experimental” check against analytical theories (see e.g., Hansen and McDonald 1986). In the case of complex molecular fluids, like aqueous systems over a wide range of temperatures and densities, which still cannot be adequately treated on a molecular level analytically, the computer simulations can play the role of the theory. They can predict thermodynamic, structural, transport, and spectroscopic properties of fluids that can be directly compared with corresponding experimental data. Even more important, however, is the ability of computer simulations to generate and analyze in detail complex spatial and energetic arrangements of every individual water molecule in the system, thus providing extremely useful microthermodynamic and micro-structural information not available from any real physical measurement. This gives us a unique tool for better understanding of many crucial correlations between thermodynamic, structural, spectroscopic and transport properties of complex molecular systems on a fundamental atomistic level. Since the first MC (Barker and Watts 1969) and MD (Rahman and Stillinger 1971) simulations of pure liquid water, great progress has been made in the simulation studies of aqueous systems. One of the earliest significant results was the ruling out of “iceberg” formation in liquid water. Computer simulations—in spite of quite different interatomic potentials employed—have unequivocally shown that liquid water consists of a macroscopically connected, random network of hydrogen bonds continuously undergoing topological reformations (Stillinger 1980). The effects of temperature and pressure on the structure and properties of water and aqueous solutions were also the subject of early computer simulations. However, in most studies either high pressures (Stillinger and Rahman 1974b; Impey et al. 1981; Jancsó et al. 1984; Pálinkás et al. 1984; Madura et al. 1988) or high temperatures (Stillinger and Rahman 1972, 1974a; Jorgensen and Madura 1985; De Pablo and Prausnitz 1989) were applied to the system, and the range of temperatures was usually well below the critical temperature of water. Surprisingly, the first molecular computer simulation of supercritical steam (Beshinske and Lietzke 1969) was published almost simultaneously with the first ever MC simulation of liquid water (Barker and Watts 1969). However, until the last decade, molecular simulations of supercritical aqueous fluids remained relatively scarce (O’Shea and Tremaine 1980; Kalinichev 1985, 1986, 1991; Kataoka 1987, 1989; Evans et al. 1988; Mountain 1989; De Pablo et al. 1989, 1990; Cummings et al. 1991). Several reviews have already been published which summarize the state of this field of research by the early 1990s (Heinzinger 1990; Belonoshko and Saxena 1992; Fraser and Refson

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1992; Kalinichev and Heinzinger 1992). The aim of this chapter is to provide an overview of the most recent results obtained by the application of computer simulation techniques to the studies of various microscopic and macroscopic properties of supercritical water over a range of densities relevant to geochemical applications and varying about two orders of magnitude from relatively dilute vapor-like to highly compressed liquid-like fluids. The general simulation methodology will be briefly described first, followed by the discussion of interaction potentials most frequently used in high-temperature and high-pressure aqueous simulations. The thermodynamics and structure of supercritical water are further discussed in relation to a detailed analysis of hydrogen bonding statistics in supercritical water based on the proposed hybrid geometric and energetic criterion of H-bonding and intermolecular distance-energy distribution functions (Kalinichev and Bass 1994). We show that after the initial interpretation of the first supercritical neutron diffraction results (Postorino et al. 1993) was eventually corrected (Soper et al. 1997), very good consistency now exists between several independent sources of experimental data and numerous computer simulation results, which all indicate that a significant degree of hydrogen bonding still persists in water under supercritical conditions. The dynamics of translational, librational, and intramolecular vibrational motions of individual molecules in supercritical water will be discussed in the last section. A more detailed discussion of the controversy associated with the contradictions between the initial NDIS measurements and molecular-based modeling of the structure and thermodynamics of supercritical aqueous solutions, in many ways complementary to the present chapter, the reader can find in the excellent recent review by Chialvo and Cummings (1999). CLASSICAL METHODS OF MOLECULAR SIMULATIONS Two sets of methods for computer simulations of molecular fluids have been developed: Monte Carlo (MC) and Molecular Dynamics (MD). In both cases the simulations are performed on a relatively small number of particles (atoms, ions, and/or molecules) of the order of 100 < N < 10,000 confined in a periodic box, or simulation supercell. The interparticle interactions are represented by pair potentials, and it is generally assumed that the total potential energy of the system can be described as a sum of these pair interactions. Very large numbers of particle configurations are generated on a computer in both methods, and, with the help of statistical mechanics, many useful thermodynamic and structural properties of the fluid (pressure, temperature, internal energy, heat capacity, radial distribution functions, etc.) can then be directly calculated from this microscopic information about instantaneous atomic positions and velocities. Many good textbooks and monographs introducing and discussing theoretical fundamentals of statistical physics and molecular computer simulations of fluid systems are available in the literature (e.g., McQuarrie 1976; Hansen and McDonald 1986; Allen and Tildesley 1987; Frenkel and Smit 1996; Robinson et al. 1996; Balbuena and Seminario 1999). Therefore, we only briefly mention here for completeness the most basic concepts and relationships. Molecular dynamics In MD simulations, the classical Newtonian equations of motion are numerically integrated for all particles in the simulation box. The size of the time step for integration depends on a number of factors, including temperature and density, masses of the particles and the nature of the interparticle potential, and the general numeric stability of the integration algorithm. In the MD simulations of aqueous systems, the time step is typically of the order of femtoseconds (10–15 s), and the dynamic trajectories of the

 

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molecules are usually followed (after a thermodynamic pre-equilibration) for 104 to 106 steps, depending on the properties of interest. The resulting knowledge of the trajectories for each of the particles (i.e., particle positions, velocities, as well as orientations and angular velocities if molecules are involved) means a complete description of the system in a classical mechanical sense. The thermodynamic properties of the system can then be calculated from the corresponding time averages. For example, the temperature is related to the average value of the kinetic energy of all molecules in the system: T=

2 3 Nk B

mi v i2 ∑ 2 i =1 N

(1)

where mi and vi are the masses and the velocities of the molecules in the system, respectively. Pressure can be calculated from the virial theorem: P=

Nk BT ⎛ 1 ⎞ −⎜ ⎟ V ⎝ 3V ⎠

N

r ⋅F ∑ i =1 i

(2)

i

where V is the volume of the simulation box and (ri·Fi) means the dot product of the position and the force vectors of particle i. The heat capacity of the system can be calculated from temperature fluctuations: ⎛2 T2 − T CV = R ⎜⎜ − N 2 T ⎝3

2

⎞ ⎟⎟ ⎠

−1

(3)

where R is the gas constant. In the Equations (1)–(3), kB is the Boltzmann constant, and angular brackets denote the time-averaging along the dynamic trajectory of the system. Molecular dynamics simulations may be performed under a variety of conditions and constraints, corresponding to different ensembles in statistical mechanics. Most commonly the microcanonical (NVE) ensemble is used, i.e., the number of particles, the volume, and the total energy of the system remain constant during the simulation. The relationships in Equations (1)–(3) are valid for this case. There are several modifications of the MD algorithm, allowing one to carry out the simulations in the canonical (NVT) or isothermal-isobaric (NPT) ensembles. Relationships similar to Equations (1)–(3) and many others can be systematically derived for these ensembles, as well (Allen and Tildesley 1987; Frenkel and Smit 1996). Monte Carlo methods In MC simulations, a large number of thermodynamically equilibrium particle configurations are created on a computer using a random number generator by the following scheme. Starting from a given (almost arbitrary) configuration, a trial move of a randomly (or cyclically) chosen particle to a new position—as well as to a new orientation if rigid molecules are involved—is attempted. The potential energy difference, ΔU, associated with this move is then calculated, and if ΔU ≤ 0, the new configuration is unconditionally accepted. However, if ΔU > 0, the new configuration is not rejected outright, but the Boltzmann factor exp(-ΔU/kBT) is first calculated and compared with a randomly chosen number between 0 and 1. The move is accepted if the Boltzmann factor is larger than this number, and rejected otherwise. In other words, the

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trial configuration is accepted with the following probability: ΔU ≤ 0 ΔU > 0

⎧1, p=⎨ ⎩exp(- ΔU k BT )

(4)

Reiteration of such a procedure gives a Markov chain of molecular configurations distributed in the phase space of the system, with the probability density proportional to the Boltzmann weight factor corresponding to the canonical NVT statistical ensemble. Typically, about 106 configurations are generated after some pre-equilibration stage of about the same length. The thermodynamic properties of the system can then be calculated as the averages over the ensemble of configurations. The equivalence of ensemble- and time-averages, the so-called ergodic hypothesis, constitutes the basis of statistical mechanics (e.g., McQuarrie 1976). The ranges of maximum molecular displacement and rotation are usually adjusted during the pre-equilibration stage for each run to yield an acceptance ratio of about 0.5. If these ranges are too small or too large, the acceptance ratio becomes closer to 1 or 0, respectively, and the phase space of the system is explored less efficiently. The advantage of the MC method is that it can be more readily adapted to the calculation of averages in any statistical ensemble (Allen and Tildesley 1987; Frenkel and Smit 1996). For example, to perform simulations in the NPT ensemble, one can introduce volume-changing trial moves. All intermolecular distances are then scaled to a new box size. The acceptance criterion is then also changed accordingly. Instead of the energy difference ΔU in Equation (4), one should now use the enthalpy difference ΔH = ΔU + PΔV – kBT ln(1 + ΔV/V)N

(5)

where P is the pressure (which is kept constant in this case) and V is the volume of the system. In this ensemble, besides the trivial averages for configurational (i.e., due to the intermolecular interactions) enthalpy: Hconf = 〈U〉 + P〈V〉

(6)

Vm =〈V〉 NA/N

(7)

and molar volume:

such useful thermodynamic properties as isobaric heat capacity CP, isothermal compressibility κ, and thermal expansivity α can be easily calculated from the corresponding fluctuation relationships (e.g., Landau and Lifshitz 1980): ⎛ H2 − H C P = ⎜⎜ 2 ⎝ Nk B T

2

⎞ ⎟⎟ ⎠

⎛ V2 − V 1 ⎛∂ V ⎞ κ≡− ⎜ ⎟ =⎜ V ⎝ ∂ P ⎠T ⎝ Nk BT V

(8) 2

⎞ ⎟ ⎠

(9)

 

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⎛ H V − H ⎞ 1 ⎛∂ V ⎞ conf conf V ⎟ ⎜ ⎟ =⎜ V ⎝ ∂ T ⎠P ⎝ Nk BT 2 V ⎠

89 

(10)

The grand canonical (μVT) statistical ensemble, in which the chemical potential of the particles is fixed and the number of particles may fluctuate, is very attractive for simulations of geochemical fluids. So far, however, it has only been barely tested even for pure liquid water simulations (Shelley and Patey 1995; Lynch and Pettitt 1997; Shroll and Smith 1999a,b). At the same time, the technique of Gibbs ensemble Monte Carlo simulation (Panagiotopoulos 1987), which permits direct calculations of the phase coexistence properties of pure components and mixtures from a single simulation, was introduced and successfully used for calculations of the vapor-liquid coexistence properties of water (De Pablo and Prausnitz 1989; De Pablo et al. 1990; Kiyohara et al. 1998; Errington and Panagiotopoulos 1998; Panagiotopoulos 2000). Molecular dynamics simulations have generally a great advantage of allowing the study of time-dependent phenomena. However, if thermodynamic and structural properties alone are of interest, Monte Carlo methods might be more useful. On the other hand, with the availability of ready-to-use computer simulation packages (e.g., Molecular Simulations Inc. 1999), the implementation of particular statistical ensembles in molecular dynamics simulations becomes nowadays much less problematic even for an end user without deep knowledge of statistical mechanics. Boundary conditions, long-range corrections, and statistical errors

One of the most obvious difficulties arises in both simulation methods from the relatively small system size, always much smaller than the Avogadro number, NA, characteristic for a macroscopic system. Therefore, so-called periodic boundary conditions are usually applied to the simulated system in order to minimize surface effects and to simulate more closely its bulk macroscopic properties. This means that the basic simulation box is assumed to be surrounded by identical boxes in all three dimensions infinitely. Thus, if a particle leaves the box through one side, its image enters simultaneously through the opposite side, because of the identity of the boxes. In this way, the problem of surfaces is circumvented at the expense of the introduction of periodicity. Whether the properties of a small infinitely periodic system and the macroscopic system, which the model is designed to represent, are the same, depends on the range of the intermolecular potential and the property under investigation. For short-range interactions, either spherical or minimum image cutoff criteria are commonly used (Allen and Tildesley 1987; Frenkel and Smit 1996). The latter means that each molecule interacts only with the closest image of every other molecule in the basic simulation box or in its periodic replica. However, any realistic potential for water (not to mention electrolyte solutions) contains long-range Coulomb interactions, which should be properly taken into account. Several methods to treat these long-range interactions are commonly used (see, e.g., Allen and Tildesley 1987), of which the Ewald summation is usually considered as the most satisfactory one. (See the discussion Gale, this volume). As any experimental method, computer simulations may also be subject to statistical errors. Since all simulation averages are taken over MD or MC runs of finite length, it is essential to estimate the statistical significance of the results. The statistical uncertainties of simulated properties are usually estimated by the method of block averages (Allen and Tildesley 1987). The MD trajectory or the MC chain of molecular configurations is subdivided into several non-overlapping blocks of equal length, and the averages of every

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property are computed for each block. If 〈A〉i is the mean value of the property A computed over the block i, then the statistical error δA of the mean value 〈A〉 over the whole chain of configurations can be estimated as

(∂A)2 =

[

M 1 A2 ∑ M (M − 1) i =1

i

− A

2 i

]

(11)

where M is the number of blocks. Strictly speaking, Equation (11) is only valid if all 〈A〉i are statistically independent and show a normal Gaussian distribution. Thus, in computer simulations of insufficient length, these error bound estimates should be taken with caution, especially for the properties calculated from fluctuations, such as Equations (3), (8)–(10). The analysis of convergence profiles of the running averages for the simulated properties is very useful in this case. One can roughly estimate the limits of statistical errors as maximum variations of the running averages during the final equilibrium stage of the simulation. Interaction potentials for aqueous simulations

Interactions between water molecules are far more complicated than those between particles of simple liquids. This complexity displays itself in the ability of H2O molecules to form hydrogen bonds, making water an associated liquid. An additional difficulty in the description of water-water interactions is the existence of substantial non-additive three- and higher-body terms, studied in detail by several authors (Gellatly et al. 1983; Clementi 1985; Gil-Adalid 1991; Famulari et al. 1998), which may raise doubts on the applicability of the pair-additivity approximation ordinarily used in computer simulations. On the other hand, the analysis of experimental shockwave data for water has shown (Ree 1982) that at the limit of high temperatures and pressures intermolecular interactions of water become simpler. In this case, it becomes even possible to use a sphericallysymmetric model potential for the calculations of water properties either from computer simulations (Belonoshko and Saxena 1991, 1992) or from thermodynamic perturbation theory in a way similar to simple liquids (Hansen and McDonald 1986). However, such simplifications exclude the possibility of understanding many important and complex phenomena in aqueous fluids on a true molecular level, which is, actually, the strongest advantage and the main objective of molecular computer simulations. The pair potential functions for the description of the intermolecular interactions used in molecular simulations of aqueous systems can be grouped into two broad classes as far as their origin is concerned: empirical and quantum mechanical potentials. In the first case, all parameters of a model are adjusted to fit experimental data for water from different sources, and thus necessarily incorporate effects of many-body interactions in some implicit average way. The second class of potentials, obtained from ab initio quantum mechanical calculations, represent purely the pair energy of the water dimer and they do not take into account any many-body effects. However, such potentials can be regarded as the first term in a systematic many-body expansion of the total quantum mechanical potential (Clementi 1985; Famulari et al. 1998; Stern et al. 1999). In the last two decades both types of potentials have been extensively used in computer simulations of aqueous systems. Several studies comparing the abilities of different potentials for reproducing a wide range of gas-phase, liquid, and solid state properties of water are currently available (Reimers et al. 1982; Morse and Rice 1982; Jorgensen et al. 1983; Clementi 1985; Robinson et al. 1996; Jorgensen and Jenson 1998; Kiyohara et al. 1998; Van der Spoel et al. 1998; Balbuena et al. 1999; Floris and Tani

 

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1999; Jedlovszky and Richardi 1999; Wallqvist and Mountain 1999; Panagiotopoulos 2000). These comparisons have shown that none of the models is able to give a satisfactory account of all three phases of water simultaneously. On the other hand, they demonstrated that many properties of aqueous systems can be qualitatively and even quantitatively reproduced in computer simulations irrespective of the interaction potential used, thus verifying the reliability of the models. Typical structures of empirical water models are schematically shown in Figure 1. Historically, the very first MD simulations of water at high pressure were performed with the empirical ST2 model (Stillinger and Rahman 1974b). It is a 5-point rigid model with four charges arranged tetrahedrally around the oxygen atom (Fig. 1c). The positive charges are located at the hydrogen atoms at a distance of 1 Å from the oxygen atom, nearly the real distance in the water molecule. The negative charges are located at the other two vertices of the tetrahedron (sites t1 and t2 in Fig. 1) but at a distance of only 0.8 Å from the oxygen. The charges were chosen to be 0.23e leading to roughly the correct dipole moment of the water molecule. The tetrahedrally arranged point charges render possible the formation of hydrogen bonds in the right directions. The ST2 model is completed by adding a (12-6) Lennard-Jones (LJ) potential, the center of which is located at the oxygen atom, with σ = 3.10 Å and ε = 0.317 kJ/mol. The total interaction energy for a pair of molecules i and j consists of the Coulomb interactions between all the charged sites and the Lennard-Jones interaction between the oxygen atoms: U ij ( r ) = ∑ α ,β

qα q β rαβ

⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ +4ε ⎢⎜ ⎟ ⎥ ⎟ −⎜ ⎝ rOO ⎠ ⎥⎦ ⎢⎣⎝ rOO ⎠

(12)

where α and β are indices of the charged sites. A special switching function was added to the Coulomb term of this water pair potential in order to reduce unrealistic Coulomb forces between very close water molecules. This ST2 model was employed in the earlier series of MD simulations of aqueous alkali halide solutions (Heinzinger and Vogel 1976). Evans (1986) later proposed a modification of the ST2 potential which included atom-atom LJ terms centered both on the oxygen and hydrogen atoms, thus eliminating the need to use the switching function. This model has been employed in MD simulations of water at temperatures up to 1273 K and at constant densities of 1.0 and 0.47 g/cm3 (Evans et al. 1988) and has shown, within the statistical uncertainty, a satisfactory reproducibility of the experimental pressure in this range and at the critical point of water. Another empirical water model often used in simulations at supercritical conditions

Figure 1. Schematic diagrams of (a) 3-point, (b) 4-point, and (c) 5-point models of a water molecule.

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Kalinichev 

is the TIP4P model (Jorgensen et al. 1983). It differs from the ST2 model in several aspects. The rigid geometry employed is that of the gas phase monomer with an OH distance of 0.9572 Å and an HOH angle of 104.52o. The two negative charges are reduced to a single one at a point M positioned on the bisector of the HOH angle at a distance of 0.15 Å in direction of the H atoms (Fig. 1b), which bear a charge of +0.52e. This simplification of the charge distribution also improves the performance of the model, since it is known that the negative charges in the tetrahedral vertices of the ST2 model exaggerate the directionality of the lone pair orbitals of the water molecule and the degree of hydrogen bonding exhibited by this model. On the other hand, Mahoney and Jorgensen (2000) have recently introduced a 5-point TIP5P model, specifically designed to accurately reproduce the density anomaly of water near 4oC. So far, this model has only been tested at temperatures below 100oC, and its behavior at supercritical temperatures is not yet known. In the TIP4P model there is a (12-6) Lennard-Jones term centered at the oxygen atom with the parameters σ = 3.1536 Å and ε = 0.649 kJ/mol. This larger value for ε compared with the ST2 and TIP5P models compensates for the reduction in Coulomb energy because of the fact that the opposite charges cannot approach as near as in a 5point model. The TIP4P water model has already proved its reliability in numerous molecular simulations of various water properties over wide ranges of temperatures and pressures (densities). The TIP4P model was widely used in the investigations of thermodynamics, structure and hydrogen bonding in supercritical water (Mountain 1989; and Kalinichev 1991, 1992; Kalinichev and Bass 1994, 1995, 1997; Churakov and Kalinichev 2000) and aqueous solutions (Brodholt and Wood 1993b; Gao 1994; Destrigneville et al. 1996). Thermodynamic and structural properties of TIP4P water at normal temperature and pressures up to 1 GPa (Madura et al. 1988; Kalinichev et al. 1999) as well as at normal density and temperatures up to 2300 K (Brodholt and Wood 1990) have also been studied. Dielectric properties for this water model have been simulated by Neumann (1986) and Alper and Levy (1989). Motakabbir and Berkowitz (1991) and Karim and Haymet (1988) have simulated vapor/liquid and ice/liquid interfaces, respectively. De Pablo and Prausnitz (1989) and Vlot et al. (1999) have studied vapor-liquid equilibrium properties of the TIP4P model, and have shown that it overestimates the vapor pressure and underestimates the critical temperature of water. The empirical simple point-charge (SPC) model (Berendsen et al. 1981) and its SPC/E modification (Berendsen et al. 1987) have been most extensively used in molecular modeling of aqueous systems over the last two decades. This is a 3-site model (Fig. 1a) with partial charges located directly on the oxygen and hydrogen atoms. The SPC and SPC/E models have a rigid geometry and LJ parameters quite similar to those of the TIP4P model. Flexible versions of the SPC model have also been introduced (Toukan and Rahman 1985; Dang and Pettitt 1987; Teleman et al. 1987). Guissani et al. (1988) made the first attempt to calculate the pH value of water from MD simulations and, after all polarization effects included, achieved a rather good agreement with experiment up to 593 K. The calculated static dielectric constant of the SPC/E water model is in good quantitative agreement with experiment over a very wide range of temperatures and densities (Wasserman et al. 1995), which is important for realistic simulations of the properties of supercritical aqueous solutions of electrolytes (Balbuena et al. 1996a,b; Cui and Harris 1994, 1995; Re and Laria 1997; Brodholt 1998; Driesner et al. 1998; Reagan et al. 1999) and non-electrolytes (Lin and Wood 1996).

 

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93 

The SPC model was successfully used in the simulations of the liquid-vapor coexistence curve (De Pablo et al. 1990; Guissani and Guillot 1993; Errington and Panagiotopoulos 1998; Kiyohara et al. 1998). It is able to correctly reproduce vapor pressure, but, like the TIP4P model, underestimates the critical temperature of water. On the other hand, the SPC/E model accurately predicts the critical temperature, but underestimates the vapor pressure by more than a factor of two. The recently proposed Exp-6 water model uses a more realistic exponential functional form for the repulsive interaction in Equation (12), and was specifically parameterized to reproduce the vapor-liquid phase coexistence properties (Errington and Panagiotopoulos 1998). However, it does not do as well as the TIP4P, SPC, and SPC/E models for the structure of liquid water, especially in terms of the oxygen-oxygen pair correlation function (Panagiotopoulos 2000). Thus, none of the available fixed point charge models can quantitatively reproduce thermodynamic and structural properties of water over a broad range of temperatures and pressures. It is clear that for strongly interacting molecules, such as H2O, a simple twobody effective potential is not sufficient, and inclusion of additional interaction terms is necessary. The most important addition is likely to be an explicit incorporation of molecular polarizability. Several polarizable models for water are available in the literature (see, e.g., Robinson et al. 1996; Wallqvist and Mountain 1999 for a review). These models seem to be slightly superior over the fixed point charge models in the description of water structure, but none of them improves the description of the vaporliquid coexistence properties and critical parameters (Kiyohara et al. 1998; Chen et al. 1999; Chialvo et al. 2000; Jedlovszky et al. 2000). It is important to keep in mind that even with recent methodological developments (Martin et al. 1998; Chen et al. 2000) the explicit incorporation of polarizability in Monte Carlo calculations comes with a penalty of a factor of ten in CPU time relative to calculations with non-polarizable models (Panagiotopoulos 2000). There are also developed a number of empirical water-water potentials with fixed charges, but incorporating intramolecular flexibility (e.g., Bopp et al. 1983; Toukan and Rahman 1985; Teleman et al. 1987; Barrat and McDonald 1990; Wallqvist and Teleman 1991; Zhu et al. 1991; Corongiu 1992; Smith and Haymet 1992; Halley et al. 1993) since Stillinger and Rahman (1978) first introduced their central force (CF) model. Although incorporation of the molecular flexibility has apparently only minor effect on the thermodynamic and structural properties of simulated water, flexible models have the great advantage of permitting the investigation of the effects of temperature, pressure, and local molecular or ionic environment on the intramolecular properties of water, like molecular geometry, dipole moments, and modes of vibration. Thus, the application of such models in molecular simulations of high-temperature aqueous systems could be particularly helpful in interpretation of some geochemical data, where vibrational spectroscopic techniques are often used as in situ probes of the chemical composition, structural speciation, etc. (e.g., Frantz et al. 1993; Bondarenko and Gorbaty 1997; Gorbaty and Gupta 1998; Gorbaty et al. 1999; Hu et al. 2000). The original CF flexible model of Stillinger and Rahman (1978) consisted of only oxygen and hydrogen atomic sites, bearing partial charges. The correct geometry of a water molecule was solely preserved by an appropriate set of oxygen-hydrogen and hydrogen-hydrogen pair potentials having a rather elaborate functional form. In order to improve the description of the gas-liquid vibrational frequency shifts by the CF model, its modification, known as the BJH water model, was later introduced by Bopp et al. (1983). The total potential is now separated into an intermolecular and an intramolecular part. The intermolecular pair potential remained only slightly modified version of the CF

94

Kalinichev 

model, and is given by: U OO ( r ) =

U OH ( r ) = −

{ [

604.6 111889 2 2 . exp −4 ( r − 3.4) − exp −15 . ( r − 4.5) + 8.86 − 1045 r r

]

[

]}

⎫ ⎧ ⎫ . 302.2 26.07 ⎧ 4179 16.74 ⎬− ⎨ ⎬ + 9.2 − ⎨ r r . ) ] ⎭ ⎩ 1 + exp[ 5.493( r − 2.2) ] ⎭ ⎩ 1 + exp[ 40 ( r − 105 U HH ( r ) =

⎫ 1511 . ⎧ 418.33 ⎬ +⎨ r . )] ⎭ ⎩ 1 + exp[ 29.9 ( r − 1968

(13)

(14)

(15)

where energies are in kJ/mol and distances in Å. The first terms in these equations are due to the Coulomb interactions of the partial charges on O and H atoms. The intramolecular part of the BJH model is based on the formulation of Carney et al. (1976)

U intra = ∑ Lij ρi ρ j + ∑ Lijk ρi

(16)

with ρ 1 = (r1 – re)/r1, ρ 2 = (r2 – re)/r2, ρ 3 = α – αe = Δα, where r1, r2 and α are the instantaneous OH bond lengths and HOH angle; the quantities re=0.9572Å and αe=104.52° are their corresponding equilibrium values (Eisenberg and Kauzmann 1969). The intramolecular parameters of the BJH potential are given in Table 1. This model is quite successful in correctly reproducing vibrational spectra of supercritical water (Kalinichev and Heinzinger 1992, 1995) and in the description and interpretation of the temperature and density dependence of ionic hydration in aqueous SrCl2 solutions obtained by EXAFS measurements (Seward et al. 1999; Driesner and Cummings 1999). This model has also performed well in reproducing the dielectric properties of water at ambient and elevated temperatures (Ruff and Diestler 1990; Trokhymchuk et al. 1993). The spectroscopic properties of isotopically substituted BJH water have also been studied (Lu et al. 1996). From the family of quantum mechanical water potentials, the MCY model (Matsuoka et al. 1976) should be mentioned in the context of high-temperature simulations. This model has the 4-point geometry (Fig. 1b), but a much more complicated functional form with parameters derived initio quantum chemical from ab calculations. The flexible version for this model (MCYL) has also been developed (Lie and Clementi 1986). The MCY model was used by Impey et al. (1981) in their MD studies of the structure of water at elevated temperatures and high density, and by O’Shea and Tremaine (1980) in the MC simulations of thermodynamic properties of supercritical water. It is well known, however, that this potential reproduces poorly the pressure at a given density (or the density at a given pressure). Even the

Table 1. Potential constants used for the intramolecular part of the BJH water model in units of kJ/mol (Bopp et al. 1983). The notations are according to Equation (16). ρ1ρ2(ρ1+ρ2)

-55.7272

(ρ1 + ρ2 )Δα

237.696

(ρ1 + ρ2 )

5383.67

2

2

4

4

2

ρ1ρ2(ρ1 + ρ2 )

-55.7272

(ρ1 + ρ2 )Δα

349.151

(ρ1 + ρ2 )

2332.27

ρ1ρ2

-55.7272

(ρ1 + ρ2)Δα

126.242

(Δα)

209.860

2

3

3

2

2

2

(ρ1 + ρ2 ) 3

3

-4522.52

 

Simulations of Liquid & Supercritical Water 

95 

addition of quantum mechanical three- and four-body terms to the potential, though extremely demanding in terms of computer time, did not improve the situation significantly (Clementi 1985). A similar ab initio CC potential (Carravetta and Clementi 1984) has been used by Kataoka (1987 1989) in extensive MD simulations of thermodynamic and transport properties of fluid water over a wide range of thermodynamic conditions, including supercritical. A qualitative reproduction of anomalous behavior of these properties has been achieved. This approach has been continued by Famulari et al. (1998). A different approach to the parameterization of the “fluctuating-charge” polarizable models from ab initio quantum chemical calculations has been recently proposed by Stern et al. (1999). However, despite of the great importance of quantum mechanical potentials from the purely theoretical point of view, simple effective two-body potential functions for water seem at present to be preferable for the extensive simulations of complex aqueous systems of geochemical interest. A very promising and powerful method of CarParrinello ab initio molecular dynamics, which completely eliminates the need for a potential interaction model in MD simulations (e.g., Fois et al. 1994; Tukerman et al. 1995, 1997) still remains computationally extremely demanding and limited to relatively small systems (N < 100 and a total simulation time of a few picoseconds), which also presently limits its application for complex geochemical fluids. On the other hand, it may soon become a method of choice, if the current exponential growth of supercomputing power will continue in the near future. THERMODYNAMICS OF SUPERCRITICAL AQUEOUS SYSTEMS

The results of isothermal-isobaric MC simulations discussed in this and the following sections were obtained for a system of N=216 H2O molecules interacting via the TIP4P potential (Jorgensen et al. 1983) in a cubic cell with periodic boundary conditions. The technical details of the NPT-ensemble algorithm are described in detail elsewhere (Kalinichev 1991, 1992). More than 40 thermodynamic states were simulated covering temperatures between 273 and 1273 K over a pressure range from 0.1 to 10000 MPa, thus sampling a very wide density range between 0.02 and 1.67 g/cm3. For each thermodynamic state point the properties were averaged over 107 equilibrium MC configurations with another 5×106 configurations generated and rejected on the preequilibration stage. The convergence of all the properties was carefully monitored during each simulation run and the statistical uncertainties were calculated by averaging over 50 smaller parts of the total chain of configurations. The MD simulations discussed in the following sections were performed using a conventional molecular dynamics algorithm for the canonical (NVE) ensemble and the flexible BJH water model (Bopp et al. 1983). The systems studied consisted of 200 H2O molecules in a cubic box with the side length adjusted to give the required density. The densities between 0.17 and 1.28 g/cm3 were chosen to correspond to the pressure range of 25
96

Kalinichev 

Macroscopic thermodynamic properties of simulated supercritical water

In Figure 2 the thermodynamic results of our MC simulations (symbols) are compared with available experimental data (lines). The calculated values of density and configurational enthalpy, Hconfig, isobaric heat capacity, CP, isothermal compressibility, κ, and thermal expansivity, α, are in excellent agreement with isotherms calculated using the equation of state of Saul and Wagner (1989), which accurately reproduces all available experimental data in this range of thermodynamic parameters. For comparison with experimental data, the non-configurational contributions to the internal energy,

Figure 2. Thermodynamic properties of supercritical water simulated using the TIP4P potential. Symbols are MC results, lines are calculated from the standard equation of state for water (Saul and Wagner 1989).

 

Simulations of Liquid & Supercritical Water 

97 

enthalpy, and heat capacity of supercritical water were assumed to be identical with those for the real gas in its standard state (Woolley 1980). The experimental configurational properties were estimated from the equation of state by

U conf = U (T ,V ) - U i.g. (T )

(17)

H conf = H (T , P ) - H i.g. (T )

(18)

and with the ideal gas reference state at the given temperature. No other corrections were added to any property reported. However, since the critical point for the TIP4P water model is lower than observed experimentally (De Pablo et al. 1990; Kalinichev 1991; Vlot et al. 1999), the isotherms in Fig. 2 were scaled in order to represent the same reduced temperature, T* = T/Tc, which is especially important in the near-critical region. The corresponding values of T* are indicated in Figure 2a for each isotherm. With this correction, a very realistic representation of many thermodynamic properties of water over extremely wide ranges of temperature and density is observed in molecular computer simulations. This result gives us sufficient confidence in the following quantitative analysis and interpretation of the details of local geometric and energetic environments experienced by individual water molecules in terms of hydrogen bonding under supercritical conditions. Micro-thermodynamic properties

Atomistic computer simulations are unique in providing a vast amount of detailed thermodynamic information on the properties, which are not readily measured in any experiment. One such property is the bonding energy: N

E b, i = ∑U i j

(19)

j ≠i

which reflects the energetic environment experienced by a single water molecule. Bonding energy distributions. Bonding energy distributions are shown in Figure 3a– d for several typical near- and supercritical thermodynamic states of water. Despite the difference in the potential models used in the MC and MD simulations, they resulted in virtually the same bonding energy distributions when the thermodynamic conditions of the simulations are the same (Kalinichev and Heinzinger 1992). The effect of pressure (or density) changes along the isotherms 573, 673, 773, and 1273 K is clearly seen in Figure 3 for these distributions. As in the case of normal temperature (Madura et al. 1988; Mahoney and Jorgensen 2000), the maxima of the distributions shift to more negative energies and their widths increase with increasing pressure.

It is interesting to note that under supercritical conditions a certain number of molecules even have positive bonding energies, thus having a net repulsive interaction balance with the environment. Obviously, the fraction of such molecules cannot increase significantly with decreasing pressure due to the thermodynamic stability constraints. Thus, an asymmetry of the distribution results at lower densities. At the lowest simulated pressure (dashed line in Figs. 3a-d) the distribution becomes bimodal, indicating that water molecules can be found in two energetically different environments with bonding energies distributed roughly around -20 kJ/mol and around 0 kJ/mol. While the latter part of the distribution is obviously due to non-bonded, free water molecules, the former one might be considered as an indication of the presence of significant amounts of hydrogenbonded molecules even under these high-temperature, low-density conditions.

98

Kalinichev  7 6

7

a

5

6 5

T=573 K

4

T=673 K

4

lom %

lom %

b

3

3

2

2

1

1

0

0

-120

-80

-40

0

-120

-80

EB / kJ/mol 7 6 5

6 5

T=773 K

d T=1273 K

4

lmo %

lmo %

0

7

c

4 3

3

2

2

1

1

0

0

-120

-40

EB / Jk /mlo

-80

-40

0

EB / kJ/mol

-120

-80

-40

0

EB / Jk /mlo 10 P M a 100 P M a

1000 P M a mabenit taw re

Figure 3. Normalized distributions of total intermolecular bonding energies for a water molecule in super-critical water. Units for the ordinate are mol % per kJ/mol.

The bonding energy distribution for the water molecules in liquid water at ambient conditions is also shown in Figure 3 as a thin solid line. At 773 K and 1000 MPa the density of supercritical water is virtually the same (≈1 g/cm3) as that of liquid water at 298 K and 0.1 MPa. Therefore, the difference between the two distributions (thin solid and dash-dotted curves in Fig. 3c) can be considered as a pure effect of temperature. At the supercritical temperature the maximum is shifted by about 40 kJ/mol to higher (less negative) energies and the distribution becomes much wider. The comparison of both distributions clearly demonstrates that an individual water molecule experiences completely different energetic environments in these two thermodynamic states despite of the fact that densities (and, hence, average intermolecular distances) are virtually the same in both cases. Pair energy distributions. Pair energy distributions represent another source of useful micro-thermodynamic information easily obtainable from computer simulations, but hardly measurable in real experiments. These functions, p(Eij), represent the probability density of finding a pair of water molecules that have some particular interaction energy under given thermodynamic conditions. Figure 4 shows such functions for several typical thermodynamic states of supercritical water. Similar distribution for normal liquid water under ambient conditions is also shown in Figure 4 for comparison.

 

99 

Simulations of Liquid & Supercritical Water  0.6 a

3

ρ∼1 g/cm

lom %

0.4

298 K 773 K

0.2

1273 K

0.0 0.6 b

ρ∼0.4 g/cm

0.4 lom %

Figure 4. Normalized distributions of pair interaction energies (dimerization energies) in liquid and supercritical water. (a) At a constant density of 1 g/cm3. (b) At a constant density of ≈0.4 g/cm3. (c) At a constant temperature of 773 K.

3

673 K 773 K 1273 K

0.2 0.0 0.6

lom %

0.4

1 - 10000 P M a 2 - 1000 P M a 3 - 100 P M a 4 - 10 P M a

2 3

0.2 0.0 -30

T=773 K

1

c

4 4

-20

-10 Eij

0 / Jk /mlo

3

10

2

1

20

30

Independent of the thermo-dynamic conditions, most interactions in a bulk system involve pairs of rather distant molecules, resulting in the main peak of the distribution located around 0 kJ/mol. However, under ambient conditions (thick dotted line in Fig. 4a) there also exists a low-energy peak, which is commonly associated with the interactions of hydrogen-bonded neighbors. With decreasing density the main peak of the distribution becomes narrower with a higher maximum (beyond the scale in Fig. 4), because a relatively larger fraction of molecular pairs become separated by large inter-molecular distances with a near-zero interaction energy. The height of the low-energy peak decreases, but a distinct shoulder is clearly seen at the same range of energies, again qualitatively indicating the noticeable persistence of H-bonded molecular pairs even at the highest temperature and the lowest density studied. The comparison of the distributions at 773 K and 1000 MPa and at 298 K and 0.1 MPa in Figure 4a presents once again the opportunity to see the pure effect of a significant temperature increase on the shape of the distribution along an isochore (densities at both thermodynamic states are ≈1 g/cm3. The width and the height of the main maximum remain unchanged. However, in the attractive branch of the distribution (negative energies) a significant amount of molecular pairs are redistributed from the “hydrogen-bonding” range of energies (approximately between –25 and –15 kJ/mol) to the “non-bonding” range (approximately above –10 kJ/mol). In the repulsive (positive) branch, the probability for a molecular pair to have rather high interaction energy between +10 and +20 kJ/mol noticeably increases, confirming that repulsive interactions become a more important contribution to the thermodynamics of supercritical water. Average potential energy. Average potential energy as a function of intermolecular distance, 〈Uij(r)〉, is yet another source of helpful micro-thermodynamic information characterizing the energetic and geometric environment of an individual water molecule

100

Kalinichev 

under various thermodynamic conditions (Heinzinger and Vogel 1976; Kalinichev and Heinzinger 1992). Such functions for the BJH and TIP4P water models at near-ambient (≈330 K) and supercritical (≈680 K) temperatures and a liquid-like density of 1 g/cm3 are shown in Figure 5a along with the contributions arising from Coulomb and non-Coulomb parts of the corresponding intermolecular potentials. In contrast to the complex character of the non-Coulomb interaction in the BJH model, arising from the combination of OO, OH, and HH contributions (Eqns. 13–15), the only non-Coulomb contribution to the TIP4P model is the oxygen-oxygen Lennard-Jones term (Eqn. 12) and, hence, its contribution to 〈Uij(r)〉 is both temperature- and density-independent (dotted line in Fig. 5b). Thus, in the case of the TIP4P potential only the change of the Coulomb part determines the differences between the average potential energies for different thermodynamic states, while in the case of the BJH potential the changes in Coulomb and non-Coulomb parts both contribute to the changes in 〈Uij(r)〉 with temperature and/or density. Figure 5 demonstrates that the BJH water model, in contrast to the TIP4P model, shows somewhat more preference for tetrahedrally ordered attractive molecular configurations (shallow minimum at OO distances ≈4.5 Å) even at supercritical temperature. However, the general features of the 〈Uij(r)〉 functions are qualitatively and quantitatively very similar for both models, with the main minimum becoming less negative at a higher temperature. Together with bonding energy distributions (Fig. 3), the average potential energy is a good tool for better understanding of the average energetic environment experienced by an individual water molecule under various thermodynamic conditions. The solid and the dashed lines in Figure 6a represent, respectively, the average potential energy for supercritical water at 773 K and 10 kbar, and for water under normal conditions with the TIP4P model employed in both cases. It is clear from the comparison, that the average energetic environment of a water molecule changes drastically as the temperature increases, despite of the fact that the average intermolecular distances remain unchanged

Figure 5. Average potential energy between two water molecules at liquid-like densities as a function of the oxygen-oxygen distance at two temperatures according to the BJH (a) and TIP4P (b) water models. (a) Total (solid line), Coulombic (dashed line) and non-Coulombic (dotted line) components at T=336 K and ρ=0.97 g/cm3 (Jancsó et al., 1984). Dash-dotted line is the total energy at T=680 K and the same density. (b) Total (solid line), Coulombic (dashed line) and non-Coulombic (dotted line) components at T=323 K and ρ=1.03 g/cm3. Dash-dotted line is the total energy at T=673 K and the same density.

 

101 

Simulations of Liquid & Supercritical Water  5

a

0

0

-5

-5

〈Uij 〉 / kJ/mol

〈Uji 〉 / kJ/mlo

5

-10 -15 -20

2

3

4 R / Å

5

6

7

b

-10 -15 -20

2

3

4

5

6

7

R / Å

Figure 6. Distance dependence of the average potential energy between two water molecules in supercritical water. (a) Full line – 773 K and 1000 MPa; dashed line – 298 K and 0.1 MPa. The water density is virtually the same in both thermodynamic states. Dotted line – non-coulombic contribution to 〈Uij(r)〉. (b) Pressure dependence along a supercritical isotherm of 773 K, from the bottom up: and 30, 100, 300, 1000, and 3000 MPa.

because of the nearly constant density. This is the consequence of a change in the average mutual molecular orientation due to the increased thermal motion (see the discussion of the structural properties in the next section). The shallow minimum around 4.5 Å in the Coulomb part of 〈Uij(r)〉 which results from tetrahedral ordering of the molecules in liquid water completely disappears under supercritical conditions. The position of the potential minimum remains virtually unchanged while its depth decreases significantly with increasing temperature indicating that, on average, water molecules are bonded to each other almost half as strongly under supercritical conditions compared with ambient liquid water. Figure 6b shows the effect of pressure (or density) increase on the average potential energy of between water molecules along a supercritical isotherm. It is clear from Figures 6a and 6b that both temperature and pressure act to decrease depth of the potential energy minimum – an indication of the weakening of hydrogen bonding (Kalinichev et al. 1999). The analysis of the water-water pair energy distributions and average water-water potential energies obtained from the simulations of aqueous electrolyte solutions under ambient conditions (Heinzinger and Vogel 1976; Heinzinger 1990) demonstrated that the effect of the dissolved ions on these functions is very similar to the effects of temperature and pressure. Thus, even more profound changes in these properties can be expected in the case of supercritical electrolyte solutions. The same conclusion can be drawn from structural properties of water and solutions, which are discussed in the next section. STRUCTURE OF SUPERCRITICAL WATER

The most important information about the structure of a molecular liquid is calculated in molecular computer simulations in the form of atom-atom radial distribution functions, gij(r). These functions give the probability of finding a pair of atoms i and j a distance r apart, relative to the probability expected for a completely random distribution of atoms at the same density. The atom-atom radial distribution functions, gOO(r), gOH(r), and gHH(r) for two

102

Kalinichev 

supercritical states (773 K and 1273 K) at the density of normal liquid water, 1 g/cm3, are shown in Figure 7 together with the results of simulations at 298 K and 0.1 MPa (dotted curves). The structure of water changes dramatically as the temperature rises to supercritical values, even at a liquid-like density. The characteristic second maximum of gOO(r) at 4.5 Å, reflecting the tetrahedral ordering of water molecules due to hydrogen bonding, is completely absent under these conditions, and a pronounced minimum of the distribution appears in its place. The comparison of the gOO(r) functions at ambient and supercritical temperatures shows a significant redistribution of nearest neighbors from the “hydrogen-bonding” regions of approximately 2.7-2.9 Å and 3.8-5.0 Å to the “nonbonding” region of 3.1-3.8 Å. It has been previously shown (Tanaka and Ohmine 1987) that precisely the pairs of molecules at intermediate distances are primarily responsible for repulsive interactions in liquid water, and these repulsions obviously increase under supercritical conditions. At the same time, the number of closest neighbors at distances up to ≈3.3 Å (the first minimum of gOO(r) in normal liquid water) remains unchanged.

g HH

g OH

g OO

The sharp first peak and following deep minimum of gOH(r) for normal liquid water (at 1.8 Å and 2.4 Å, respectively) become much less pronounced under supercritical conditions (Fig. 7b). These two characteristic features of gOH(r) observed in computer simulations, as well as experimentally (Soper et. al. 1997), are the basis of a simple geometric definition of a hydrogen bond, whereby the bond is assumed to exist between any pair of H2O molecules whose respective O and H atoms are separated by less than RHB = 2.4 Å. Integration under gOH(r) up to the chosen threshold distance provides a convenient way to quantitatively estimate the average number of H3 bonds in which an individual ρ,g/cm 3 ,T K a molecule participates under various 298 1.0 773 1.0 thermodynamic conditions (e.g., 2 1273 1.0 Kalinichev 1986; Mountain 1989; 1273 0.17 Guissani and Guillot 1993). At 1 ambient conditions, this geometric criterion gives the average number of 0 1 2 3 4 5 6 H-bonds per a water molecule 〈nHB〉G=3.2. A detailed comparison of 1.5 b simulated structural functions for near-critical water with available 1.0 neutron and X-ray diffraction data 0.5 are shown in Figures 8 and 9. For the subcritical conditions of Figure 8, the 0.0 agreement of the MC-simulated 1 2 3 4 5 6 gOO(r), gOH(r), and gHH(r) functions for the TIP4P water model with the 1.5 X-ray (Gorbaty and Demianets 1983) c and neutron (Soper et al. 1997) 1.0 diffraction data is very good. In fact, it is almost within experimental 0.5 errors. The geometric estimates of 0.0 the degree of hydrogen bonding give 1 2 3 4 5 6 〈nHB〉G=2.4 for both simulated and R / Å experimental gOH(r). Figure 7. Atom-atom radial distribution functions for The disagreement of simulated and experimental radial distribution

liquid and supercritical water at 1.0 g/cm3 and 0.17 g/cm3 from MC simulations with the TIP4P potential.

 

103 

Simulations of Liquid & Supercritical Water  T=573 K ρ=0.72 g/cm

T=673 K ρ=0.66 g/cm3

3

1 0

1

2

3

4

5

NDIS -X ray C M IT P4P D M BJH

2 g OO

1

0

g OH

NDIS -X yra C M IT P4P

6

1

g OH

g OO

2

1

2

3

4

5

6

1

ab itn io D M

1

2

3

4

5

1

0

0

6

g HH

g HH

0

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2

3

4 R / Å

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

0

R / Å

Figure 8 (left). Atom-atom radial distribution functions for subcritical water (T=573 K, ρ=0.72 g/cm3) from MC simulations with the TIP4P potential (solid lines). Thick dotted line – X-ray diffraction (Gorbaty and Demianets 1983); thin dotted lines with error bars – neutron diffraction with isotope substitution, NDIS (Soper et al. 1997). [Reproduced with permission from J Phys Chem A 1997, 101, 9720-9727. © 1997 American Chemical Society.] Figure 9 (right). Atom-atom radial distribution functions for supercritical water (T=673 K, ρ=0.66 g/cm3). Solid lines – MC simulations with the TIP4P potential; dashed lines – MD simulations with the BJH potential; thick dotted line – X-ray diffraction (Gorbaty and Demianets 1983); thin dotted lines with error bars – neutron diffraction (Soper et al., 1997); thin broken line – ab initio MD simulations (Fois et al. 1994). [Reproduced with permission from J Phys Chem A 1997, 101, 97209727. © 1997 American Chemical Society]

functions is somewhat larger at the supercritical temperature of 673 K. In Figure 9 the results for the TIP4P model are presented together with two other computer simulations (Fois et al. 1994; Kalinichev and Heinzinger 1995) under very similar thermodynamic conditions. The structural functions simulated by Chialvo and Cummings (1994, 1996, 1999) for the SPC/E water model virtually coincide with results for the TIP4P model and are not shown in Figures 8 and 9. The simulations of Mountain (1995) for the ST2 and RPOL intermolecular potentials are also very close to the results for other water models. The most striking feature of the neutron diffraction data at the supercritical temperature of 673 K (Fig. 9) is the disappearance of the first maximum of gOH(r) at ≈1.8 Å. However, a significant shoulder still remains at its place. This feature is best reproduced by the ab initio MD simulations of Fois et al.(1994) where no assumptions on

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the interaction potential was a priori made. Molecular simulations for the TIP4P and SPC water models also come close to the neutron data, while the MD simulations with the flexible BJH water model show somewhat more structure in the gOH(r) function, compared to experimental data. The geometcic estimates of 〈nHB〉G again give very close values of 2.1 for both experimental and TIP4P simulated distribution functions, while the BJH simulation gives the value of 2.3. In all simulations, the gOO(r) and gOH(r) functions for several different water models are quite close to each other. However, the gHH(r) functions are significantly different in the case of the BJH potential, and demonstrate a much steeper rise of the distribution at ≈2 Å in disagreement with neutron diffraction data. In fact, the BJH potential is unable to correctly reproduce the behavior of this function because it explicitly forbids two hydrogens of different molecules to come closer than 2 Å to each other (Eqn. 15). This very strong H-H repulsion at the shortest intermolecular separations is also the main reason for the overestimation of pressure by the BJH potential (Kalinichev and Heinzinger 1992, 1995). Although the general trends of the experimental radial distribution functions (such as the shift to larger distances and the broadening of the first gOH(r) peak with increasing temperature and decreasing density (Bruni et al. 1996; Soper et al. 1997) are qualitatively reproduced in the present simulations, these effects are less pronounced than observed in neutron diffraction measurements. Nonetheless, they are clearly seen in Figure 7 where the structural results of two extremely high-temperature (1273 K) MC simulations are shown for liquid-like and gas-like densities of supercritical water. With increasing temperature and decreasing density, the first peak of gOH(r) disappears due to the shifting and broadening, thus gradually filling the gap in the distribution between approximately 2 and 3 Å. With this specific hydrogen bonding peak becoming much less distinct at supercritical temperatures, the simple geometric criterion of H-bonding based exclusively on the analysis of gOH(r) distribution functions becomes very unreliable and additional orientational or energy constraints need to be taken into account (e.g., Kalinichev and Bass 1994; Chialvo and Cummings 1994; Soper et al. 1997). HYDROGEN BONDING IN LIQUID AND SUPERCRITICAL WATER

In computer simulations, specific configurations of molecules, which can be considered as hydrogen-bonded, arise as a consequence of the charge distribution on individual water molecules. For a quantitative analysis of the H-bonding, various geometric constraints are often used in addition to the analysis of radial distribution functions. They are most frequently based on the requirement that one or several internal coordinates of a pair of water molecules (such as oxygen-oxygen distance, angles between some characteristic bond directions) fall within a certain specified range of values (Mezei and Beveridge 1981; Belch et al. 1981; Pálinkás et al. 1984; Chialvo and Cummings 1994; Mountain 1995; Soper et al. 1997). However, in their early MD simulations of liquid water, Stillinger and Rahman (1972) have also introduced a simple energetic criterion of a hydrogen bond, which considers two molecules to be H-bonded if the interaction energy between them is lower than a given negative threshold. This criterion has been successfully applied to the analysis of temperature and density effects on the H-bond distributions in liquid and supercritical water (Jorgensen et al. 1983; Kalinichev and Bass 1994, 1995, 1997). Ideally, one should expect that the application of both—purely geometric or purely energetic—criteria should result in the same physical picture of hydrogen bonding. In practice however, both descriptions are not always consistent, especially under supercritical conditions.

 

Simulations of Liquid & Supercritical Water 

105 

The selection of the specific threshold energy value in the energetic criterion of hydrogen bonding is based on the analysis of pair energy distributions discussed earlier (Figs. 4a–c). The minimum of the distribution at about –10 kJ/mol is usually taken as the energetic cutoff threshold (EHB) for hydrogen bonding under ambient conditions (Jorgensen et al. 1983). The integration under this low-energy peak up to EHB gives another quantitative estimate of the number of H-bonds per a water molecule in the system, which is completely independent of any geometric considerations. It can be easily shown that for liquid water under ambient conditions this estimate virtually coincides with the simplest geometric estimate based on the analysis of the gOH(r) distribution function (Kalinichev and Bass 1994). The variation of the energetic threshold value EHB within reasonable limits does not qualitatively affect the picture of hydrogen bonding in liquid water which remains remarkably similar for a number of water-water interaction potentials used in computer simulations (Stillinger and Rahman 1972; Jorgensen et al. 1983; Rapaport 1983; Kalinichev and Heinzinger 1992; Kalinichev and Bass 1994; Mizan et al. 1996; Mahoney and Jorgensen 2000). Thus, adopting the same universal value of EHB= –10 kJ/mol for all thermodynamic states provides a uniform and simple criterion for quantifying the picture of H-bonding at any temperature and density. It is also worth noting that this value of EHB is quite consistent with spectroscopic and thermodynamic estimates of the hydrogenbonding energy (Bondarenko and Gorbaty 1991; Walrafen and Chu 1995). The picture of hydrogen bonding can be very efficiently visualized and analyzed by plotting intermolecular distance-energy distributions (Kalinichev and Bass 1994, 1997; Kalinichev et al. 1999), as shown in Figure 10. These distributions combine both gOH(r) and p(Eij) functions into one 3-dimensional surface, where each point represents the relative probability of finding the hydrogen atom of a molecule i at a distance RO...H from the oxygen atom of a molecule j, with a specified interaction energy between the

Figure 10. Normalized intermolecular O···H distance-energy distribution functions for liquid (a) and supercritical (b) water. Arrows show the cutoff values of the H-bond definition. [Reproduced with permission from J Phys Chem A 1997, 101, 9720-9727. © 1997 American Chemical Society.]

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Kalinichev 

molecules of Eij. From Figure 10a it is obvious that for liquid water under ambient conditions either geometric or energetic criteria both quite successfully and consistently distinguish H-bonded molecular pairs from the non-bonded ones, since the former are represented by the distinct low-energy short-distance peak on the distance-energy distribution surface around –20 kJ/mol and 1.8 Å. A weaker feature at the same energy range, but at distances around 3.3 Å obviously represents the most likely positions of the second hydrogen of the H-bonded molecule and corresponds to the second maximum of gOH(r) in Figures 7–9. However, under supercritical conditions (Fig. 10b), the picture becomes much more complicated. The non-overlapping “tails” of the distance-energy distributions extend far beyond the chosen threshold values of HB energy and distance. Some of the molecular pairs considered as bonded in geometric terms often even have positive (repulsive) interaction energy, thus being obviously non-bonded in any reasonable physical sense. On the other hand, molecular pairs considered as H-bonded in energetic terms sometimes have an O⋅⋅⋅H separation as large as 3.0 Å, which also seems to be unreasonable. The simplest geometric criterion of H-bonding based on the gOH(r) distribution function can, obviously, be made more selective by introducing additional spatial constrains on other interatomic separations and relative orientations of the interacting molecular pairs (Mezei and Beveridge,1981; Belch et al. 1981; Pálinkás et al. 1984; Chialvo and Cummings 1994; Mountain 1995; Soper et al. 1997). However, this inevitably jeopardizes the simplicity and universality of the criteria by increasing the number of more or less arbitrary chosen threshold parameters in the analysis. On the other hand, the simultaneous application of just two intermolecular cutoff criteria, one— distance-based (RHB), and one—energy-based (EHB) can be considered as the best compromise between simplicity and unambiguity of the constrains imposed on any molecular pair in order to distinguish between H-bonded and non-bonded configurations. Generally speaking, a chemical bond is most naturally described in terms of its length and strength, which is directly reflected in the hybrid distance-energy H-bonding criterion. Lifetime of a bond represents one more characteristic of H-bonding (e.g., Mountain 1995, Mizan et al. 1996), and will be briefly discussed in the next section. Density and temperature dependencies of the average number of H-bonds, 〈nHB〉, as well as those of average energy 〈Eij〉HB, distance 〈RO···H〉HB, and angle 〈θ〉 = 〈∠O–H···O〉 of hydrogen bonds obtained from the present MC simulations by the application of thehybrid energetic and geometric criterion are presented in Figure 11. The average total energy of H-bonding is almost invariant over the entire density range at any given temperature. Although some smooth variation of the average HB angle 〈θ〉 is clearly observable, it, too, remains almost constant over a very wide density range. Surprisingly enough, the angular HB distributions of the angle θ are also virtually density- (pressure-) independent along a supercritical isotherm (Fig. 12). Thus, at any particular temperature, the geometry of H-bonded molecular arrangements remains essentially the same over a very wide range of densities from dilute gas-like (~0.03 g/cm3) to highly compressed liquid-like (~1.5 g/cm3) fluid states. This fact, obviously, indicates that the growing orientational disorder, which accompanies the rise in temperature, is almost the sole mechanism of H-bond breaking under supercritical conditions. Thus, we may conclude that the increase of temperature from ambient to supercritical affects the characteristics of H-bonding in water much more dramatically than the changes in density from 0.02 to 1.67 g/cm3 along any supercritical isotherm. Compared to hydrogen bonds in ambient liquid water, H-bonds at 773 K are almost 2 kJ/mol weaker, have by 0.1 Å longer O··H bond distances, and are about 10° less linear.

 

107 

Simulations of Liquid & Supercritical Water 

Figure 11. Average parameters of hydrogen bonds in liquid and supercritical water.

4

lm o %

3

2

1

0 100

mo-r tetaremp ure taw re T=773 K 3000 P M a 1000 P M a 100 P M a 50 MPa 30 MPa 10 MPa

H O

φ

θ

H

H

O H

120

θ / e r gd

140

160

180

Figure 12. Distributions of H-bonding angles θ in liquid and supercritical water. Definitions of Hbonding angles θ and φ are given on the inset.

108

Kalinichev 

Figure 13 shows all available experimental estimates of 〈nHB〉 from various sources (Gorbaty and Kalinichev 1995; Soper et al. 1997; Hoffmann and Conradi 1997; Matubayasi et al. 1997, 1999) overlapped with the results of Monte Carlo simulations for TIP4P water model under similar thermodynamic conditions. There is excellent agreement between experimental data and computer simulations (Kalinichev and Bass 1997; Chialvo and Cummings 1999). From computer simulations, the general temperature dependence of 〈nHB〉 is observed as a broad band between high-density (dash-dotted) and low-density (dotted) curves, asymptotically approaching zero at higher temperatures and lower densities, as it was previously predicted (Gorbaty and Kalinichev 1995). Since these numbers of 〈nHB〉 are well below the percolation threshold np ≈ 1.6 (Blumberg et al. 1984), the percolating (i.e., continuous and infinite) tetrahedral network of hydrogen bonds is already broken at T = 573 K and ρ<0.63 g/cm3 (Fig. 13). For temperatures above 873 K, the continuous H-bonding network is broken even for liquidlike densities of ρ ≤ 1.0 g/cm3. However, even at the highest temperature and the lowest density there persist some noticeable degree of hydrogen bonding, represented, most probably, by small clusters like dimers and trimers. Temperature and density dependencies of fractions of water molecules having a given number of H-bonds for a typical supercritical isobar and isotherm are plotted in Figure 14 based on the MC simulations for the TIP4P water model. For the thermodynamic conditions of neutron diffraction experiments (Soper et al. 1997), at 673 K and 0.66 g/cm3, approximately 12% of water molecules are estimated to be involved in 3 H-bonds, about 70% of molecules involved in 1 or 2 H-bonds (dimers and trimers), and only 17% represent non-bonded monomers. This picture is in good qualitative agreement with semi-empirical calculations using equations of state for water based on the hydrogen bonding lattice fluid model (LFHB) (Gupta et al. 1992) and associated perturbed anisotropic chain theory (APACT) (Smits et al. 1994).

4.0 IR absorptoni (Bokorenda & Gorbaty, 1973,1991) IR b a sotpr noi (Frcna k & Rot,h 1967) -X yar , 100 P M a (Gorbta y & Deteniam s, 1983) NMR shtif , 10-40 P M a (Honamf & Co,idarn 1997) Neutrno carfid tino (Srepo te .la , 1997)



3.0

C M simulatonsi , ~1 /g cm C M simulta snoi , ~0.2 /g cm C M siu m tla sion , 100 P M a

2.0

3 3

1.0 crep talo noi thser dloh

0.0 200

400

600

800

1000

1200

T/K Figure 13. Temperature dependence of the average number of H-bonds per molecule, 〈nHB〉, derived from computer simulations and experimental data. [Reproduced with permission from J Phys Chem A 1997, 101, 9720-9727. © 1997 American Chemical Society.]

 

109 

Simulations of Liquid & Supercritical Water  60

P=100 MPa

50

3

mol %

40

2

0

a

1

4

30 20 10 0 200

5 400

600

800

1000

1200

1400

T /K

60 50

0

ml o %

40

b

T=673 K

1 2

30

3

20

4

10 0 0.0

0.2

0.4

0.6

0.8

1.0

ρ / g/cm

3

5 1.2

1.4

1.6

Figure 14. Distributions of water molecules involved in a given number of H-bonds (numbers at the curves) in supercritical water: (a) along an isobar of 100 MPa; (b) along an isotherm of 673 K.

MOLECULAR CLUSTERIZATION IN SUPERCRITICAL WATER

The formation of small molecular clusters in high temperature water vapor, as well as in supercritical fluid has recently attracted significant attention (Driesner 1997; Mountain 1999; Kalinichev and Churakov 1999, 2000). The nucleation of clusters results in local structural and density heterogeneity in the fluid. In different clusters water molecules possess slightly different properties such as the dipole moment and lengths of interatomic bonds (e.g., Ugdale 2000). So, topologically different water complexes may be even considered as distinct chemical species. Quantitative understanding of the size and structure of such clusters is important for many fundamental and practical reasons. For example, recent estimates suggest that the large effects of pressure on the isotope fractionation between high temperature aqueous solutions and minerals can be explained by the varying degree of clusterization in water vapor (Driesner 1997; Horita et al. 1999). The application of the H-bonding criterion to every pair of molecules in a computergenerated configuration results in a connectivity matrix for this configuration. From it, the groups of molecules interconnected by a network of H-bonds can be easily identified to obtain cluster size distributions. By indexing the interconnected molecules, geometric and energetic characteristics of each isolated cluster can also be obtained. We consider here a molecule as belonging to a hydrogen-bonded cluster if it has at least one H-bond with some other molecule in the cluster. Following this definition, the maximum cluster size in computer simulations is limited by the total number of molecules in the simulation box. However, water molecules can be considered to form an infinite percolating Hbonded network, if a periodic image of at least one of them belonged to the same cluster as its origin.

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Kalinichev 

It has been recently shown (Mountain 1999; Kalinichev and Churakov 1999) that the size of the H-bonded network under supercritical conditions is mainly controlled by water density. Simulations over a wide range of densities 0.03 < ρ < 1.5 g/cm3 allow us to trace gradual changes in the structure of supercritical water, from dilute vapor to dense liquidlike fluid. In Figure 15 cluster size distributions (normalized probabilities for a randomly selected molecule to participate in a cluster of a given size) are shown for several supercritical temperatures and densities (T* = T/Tc, ρ* = ρ/ρc), as well as for ambient liquid water. Dilute vapor consists mostly of individual monomers. However, even at ρ*=0.12, up to 18 % of molecules form multi-molecular clusters. The largest water cluster observed under these conditions consisted of nine molecules, contributing 0.0026 % to the cluster size distribution. It is quite remarkable, that even at very low densities and high temperatures such large clusters may sometimes occur. As density increases, water molecules form even larger clusters. At a certain density, when the concentration of H-bonds reaches the percolation threshold, an infinite cluster of Hbonded molecules is formed. At a density above the percolation threshold molecules have

60

crep lota noi thser dloh

% celom ulse

40 20 0

% clome ulse

10

0

50

150

200

T*=1.29 ρ*=3.04

0 10

% lomce ulse

100

crep talo nio thser dloh

5

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% lomce ulse

T*=0.50 ρ*=3.02 (amb. cond.)

50

100

crep lota noi thser dloh

5

150

200

T*=1.12 ρ*=2.47

0 20 15 10 5 0

0

% lmceo ulse % mcelo ulse

80 60 40 20 0

100

crep talo ino thser lohd

0 40 30 20 10 0

50

150

200

T*=1.12 ρ*=1.80

50

100

150

200

T*=1.12 ρ*=0.58

0

50

100

150

200

T*=1.12 ρ*=0.12

0

50

100 Cluster size

150

200 Fi.g 15

Figure 15. Participation of water molecules (%) in H-bonded clusters of different size under ambient and supercritical conditions.

 

111 

Simulations of Liquid & Supercritical Water 

a higher probability to join the percolating cluster. That decreases the fraction of molecules forming smaller isolated complexes, and, considering finite system size N, one can observe bimodal cluster size distributions. We calculated the probability for the largest observed cluster to percolate depending on the hydrogen bond concentration, 〈nHB〉, under various thermodynamic conditions (Churakov and Kalinichev 2000). From these data, the size of a percolating cluster at T* = 1.12 is estimated to be ~65 molecules. This value divides cluster size distributions into two parts. One of them represents infinite percolating clusters, while another part represents smaller isolated clusters (Fig. 15). The topology of water clusters observed under supercritical conditions can also be easily analyzed. In this analysis, we do not make any distinction between donor and acceptor types of hydrogen bonds originating at a particular water molecule. Thus, every molecule is depicted as a structureless dot in Figure 16, where all observable topological types of 3-, 4-, and 5-mers are shown along with their relative abundances (normalized as a fraction of molecules participating in clusters of certain type) under several supercritical thermodynamic conditions listed below in Tables 4 and 5. Figure 16 demonstrates that, despite the obvious domination of the simplest linear chain-like clusters, more complex structures are also present. Since only instantaneous configurations were analyzed, some of the observed clusters might in reality have a rather short lifetime, and thus represent only transient structures. Nevertheless, all bonds in the observed clusters satisfied both geometric and energetic criteria for H-bonding, and none of the observed structures can be considered energetically unfavorable within the given definitions. H-bonded molecular clusters containing open chain-like or tree-like structures are found to predominate over clusters containing cyclic ring-like elements. Average H-bonding angles, 〈∠O··O··O〉 and 〈∠O··H–O〉, distances, 〈RO··O〉 and 〈RO··H〉, and 101

101

a

b

100

10-1

% moleculse

% moleculse

100 10-2

10-1 10-2 10-3

3a

3b

4a

4b

4c

4d

101

c

% mlcoe ulse

100 10-1 10-2 10-3

5a

5b

5c

5e

5f

A

B C D E

5d

Figure 16. Topological types of H-bonded 3-mers, 4-mers, and 5-mers, and their relative abundance in supercritical BJH water. (See Table 4 for the thermodynamic conditions of runs A-E).

112

Kalinichev 

energies 〈UHB〉, of H-bonds forming the most abundant molecular 3-mers, 4-mers and 5mers in supercritical water at T* = 1.04 and ρ* = 1.13 are reported in Table 2. It is remarkable, that the most abundant clusters, with topologies 3a, 4a, 4b, 5a, and 5b have almost identical geometric and energetic characteristics. So, H-bond lengths were found for all clusters to be ~2.04 Å, while H-bonding angles 〈∠O··O··O〉 have values ~109°, which corresponds to the geometry of an ideal tetrahedrally ordered lattice. Cluster geometry was also studied at other thermodynamic conditions. With increasing temperature and/or decreasing density H-bond distances 〈RO··O〉 and 〈RO··H〉 increase, which is in agreement with previous observations (Kalinichev and Bass 1994, 1997; Mizan et al. 1994). However, average angles 〈∠O··O··O〉 for the most abundant clusters remain almost constant at ~109° over the whole range of supercritical conditions studied. This result is in excellent agreement with earlier X-ray diffraction data for supercritical water (Gorbaty and Demianets 1983; Gorbaty and Kalinichev 1995) which show a noticeable peak of molecular distribution function at ~4.5 Å corresponding to tetrahedrally ordered second nearest neighbors. There are two ways to compare cluster energies—energy per H-bond, or energy per molecule. These quantities are different, since clusters of the same size may have different number of internal bonds (Fig. 16). Because our cluster definition is based on the strength of its bonds, we use 〈UHB〉 to characterize the energy of the clusters. Generally, one may expect the cluster abundance to be proportional to exp(–βU), where β = (kBT)–1, and U – potential energy of intermolecular interactions. At a given temperature, the most abundant clusters generally should have lower energy. This assumption, however, is not always supported by the data in Table 2. So, 〈UHB3a〉 < 〈UHB3b〉, 〈UHB4a〉 < 〈UHB4b〉 < 〈UHB4c〉. On the other hand, 〈UHB4d〉 < 〈UHB4a〉, in contrast with the low abundance of 4-member rings. Similarly, 〈UHB5e〉 ≈ 〈UHB5b〉, while the latter topology is much more abundant. We may conclude, that along with purely energetic considerations, entropic effects also play significant role in determining the abundance of particular cluster types. For instance, despite low potential energy of the configuration 5e, such pentamers can form only when very specific conditions for mutual distances and angles are simultaneously satisfied for all 5 molecules contributing to such a structure. The dynamic nature of hydrogen bonding and the effects of temperature and density on the persistence of H-bonds in supercritical water were recently studied in MD simulations by Mountain (1995) using the ST2 and RPOL intermolecular potentials, and by Mizan et al. (1996) using the SPC potential and its flexible version. These authors use two different approaches to the estimation of the hydrogen bonding lifetime, but both of Table 2. Average geometric and energetic parameters of small H-bonded * * clusters under supercritical conditions of Τ =1.04, ρ =1.13. Cluster topology

3a

3b

4a

4b

4c

4d

5a

5b

5c

5d

5e

〈∠O··O··O1〉 / °

109

60

110

107

114

87

110

108

112

107

105

86

60

〈∠O··O··O2〉 / °

60

〈∠O··O··O3〉 / °

119

〈RO··H〉 / Å

2.04

2.07

2.04

2.04

2.06

2.03

2.04

2.04

2.05

2.06

2.05

〈RO··O〉 / Å

2.90

2.89

2.91

2.91

2.90

2.90

2.91

2.91

2.91

2.90

2.92

〈∠O··H–O〉 / °

149

143

150

150

146

150

150

150

149

146

150

〈UHB〉 / kJ/mol

-16.9 -16.4 -16.9 -16.7 -16.3 -17.0 -16.9 -16.8 -16.7 -16.3 -16.8

 

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Simulations of Liquid & Supercritical Water 

them are based on the analysis of survival (with a certain temporal resolution Δτ) of Hbonds in a series of MD-generated molecular configurations. While the choice of the geometric and energetic parameters in any hydrogen bond definition is more or less determined by the shape of the energetic surface for water dimer, the choice of the time resolution interval Δτ is much less obvious. It is clear, that at Δτ →0 the resulting picture of H-bonding should be very close to the instantaneous one. On the other hand, Δτ must be sufficiently small compared to the average H-bond lifetime (Rapaport 1983). The latter, is, actually, the principal goal of the analysis, thus it cannot be known in advance. In a simplified approach, we can consider a hydrogen bond to exist continuously if it satisfies our chosen H-bonding criterion at the end of each consecutive time interval Δτ along the MD trajectory. Otherwise, the bond considered broken during this interval. An example of how temporal resolution Δτ can affect the statistics of hydrogen bonding under typical supercritical thermodynamic conditions for BJH water is presented in Table 3. It is clear that the time-averaged picture of H-bonding resulting from spectroscopic and diffraction measurements, as well as MC simulations, can only be considered an upper boundary estimate. Depending on the stringency of the lifetime Hbonding criterion used, the average number of H-bonds in the system can change by a factor of two, and even more. The average lifetime of H-bonds in supercritical water is estimated to be about 0.2-0.5 ps (e.g., Mountain 1995; Mizan et al. 1996), which is about an order of magnitude lower then typical lifetimes of hydrogen bonds in liquid water under ambient conditions (Rapaport 1983; Bopp 1987). DYNAMICS OF MOLECULAR TRANSLATIONS, LIBRATIONS, AND VIBRATIONS IN SUPERCRITICAL WATER

In MD simulations, the dynamical behavior of a molecular fluid can be monitored in terms of velocity autocorrelation functions (VACF), which are calculated as v (0) ⋅ v (t ) =

1

Nτ N

Nτ N

∑ ∑ v j (t i ) ⋅ v j (t i + t )

(20)

i =1 j =1

where N denotes the number of particles, Nτ – the number of time averages, and vj(t) – the velocity of particle j at time t. Molecular center-of-mass velocity autocorrelation functions for several supercritical states of water are shown in Figure 17. Obviously, the VACFs decay faster at the higher density. The density dependence of these functions is very similar to that for water at normal temperatures (Jancsó et al. 1984), in agreement with the similarity in the pressureinduced changes of the structural properties discussed above.

Table 3. Average number of H-bonds per water molecule and concentration of H-bonded species in supercritical BJH water at 630 K and 0.692 g/cm3 as a function of a lifetime criterion for H-bonds.

Δτ / ps

〈nHB〉

% of monomers (0 bonds)

% of dimers (1 bond)

% of trimers (2 bonds)

% of tetramers (4 bonds)

0.01

1.41

16.6

39.4

31.9

10.7

0.1

0.83

40.2

40.2

16.3

3.0

0.2

0.57

55.0

34.2

9.5

1.2

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Figure 17. Normalized center of mass velocity autocorrelation functions for the water molecules under supercritical conditions.

Self-diffusion coefficients in supercritical water can be determined from MD simulations through the molecular mean-square displacement analysis, or through the velocity autocorrelation functions (Eqn. 20) with the help of the Green-Kubo relation (Allen and Tildesley 1987): t

1 ∫ v (0) ⋅ v (t ′) dt ′ t →∞ 3 0

D = lim

(21)

The self-diffusion coefficients calculated for several supercritical thermodynamic states of BJH water are compared with available experimental data and the results of other computer simulations in Table 4. The statistical uncertainty of the calculated values is about 10%, i.e., comparable with the accuracy of experimental data. Thus, the simulated values of D agree very well with experiments. It is also quite surprising that

Table 4. MD-simulated self-diffusion coefficients of supercritical water. MD-Run

A

B

C

D

E

ρ / g cm–3

0.1666

0.5282

0.6934

0.9718

1.2840

673

772

630

680

771

T/K –5

2 –1

D / 10 cm s

196

76

37

23

11

193(a)

68(a)

44(a)

–

–

(b)

(b)

>170

(b)

68

65(c)

45

22

(b)

24(d) 23(e)

(a)

NMR spin-echo measurements of Lamb et al. (1981). MD simulations for the TIP4P water model (Brodholt and Wood 1990, 1993a). (c) MD simulations for the SPC water model. Interpolated from (Mizan et al. 1994). (d) MD results for the BNS rigid water model at 641 K (Stillinger and Rahman 1972). (e) MD results for the Carravetta-Clementi potential (Kataoka 1989). (b)

15(b)

 

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three other high-temperature MD simulations (Stillinger and Rahman 1972; Kataoka 1989; Brodholt and Wood 1990, 1993) at a density of ≈1 g/cm3, with three different intermolecular potentials, resulted in self-diffusion coefficients virtually identical to that of the run D for the BJH model. This seems to indicate that all the intermolecular potentials are able to reproduce correctly the temperature dependence of the selfdiffusion coefficients up to supercritical temperatures at least at liquid-like densities. It also indicates that calculations of self-diffusion coefficients from MD simulations are not very sensitive to the details of the particular intermolecular potential used. The pronounced effects of temperature and density are also reflected in the spectral densities of the hindered translational motions of water molecules, which can be calculated by Fourier transformation of the velocity autocorrelation functions (e.g., Allen and Tildesley 1987). Such spectra for two high-density supercritical MD simulations (Kalinichev and Heinzinger 1992; Kalinichev 1993) are shown in Figure 18 as dotted and dash-dotted lines, while similar spectra for near-ambient liquid water are shown as solid and dashed lines for the BJH and SPC models, respectively. In normal liquid water, the peaks at ~60 cm–1 and ~190 cm–1 are usually assigned to the hydrogen bond O··O··O bending motion and O··O stretching motions, respectively (e.g., Eisenberg and Kautzmann 1969). Both peaks completely disappear at supercritical temperature, indicating a significant break down of the H-bonding water structure. It is also important to note that the decrease of the spectral intensities in the low-frequency range of the Hbond bending and stretching motions is quite similar to the effect of dissolved ions (e.g., Szász and Heinzinger 1983a,b). For a flexible water model, one can calculate the changes in the average geometry of the molecules brought about by the changes of the thermodynamic conditions. The simulated average intramolecular geometric parameters of molecules in supercritical BJH water are given in Table 5. The corresponding values for an isolated BJH water molecule and for the molecules in ambient liquid water are 0.9572 and 0.9755 Å for the intramolecular OH distance, 104.52° and 100.78° for the intramolecular HOH angle, and 1.86 and 1.97 Debye for the dipole moment, respectively (Jancsó et al. 1984). Comparing these values with the data in Table 5, one can see that the increase of density, and that of temperature (at the normal liquid-like density of the run D), both lead to an elongation of the average intramolecular OH distance and a decrease of the average HOH angle. Due to

Figure 18. Spectral densities (in arbitrary units) of low-frequency translational motions for supercritical water at liquid-like densities. The spectra of liquid water under ambient conditions for the BJH (Jancsó et al. 1984) and SPC water models are given for comparison.

116

Kalinichev  Table 5. Intramolecular geometry and vibrational frequencies of supercritical BJH water. MD-Run –3

ρ / g cm T/K 〈ROH〉 / Å 〈∠HOH〉 / ° 〈μ〉 / Debye ν1max / cm–1

FWHM1 / cm–1 ν2max / cm–1 FWHM2 / cm–1 ν3max / cm–1 FWHM3 / cm–1

A

B

C

D

E

0.1666 673 0.9705 102.00 1.99 3640 3625(a) 3630(b) 3600(c) 230 1660 110 3760 180

0.5282 772 0.9750 100.82 2.02 3580 3585(a) 3595(b) 3580(c) 300 1670 130 3690 280

0.6934 630 0.9755 100.28 2.03 3500 3567(a) 3567(b) 3569(c) 340 1690 130 3675 290

0.9718 680 0.9781 99.51 2.05 3530 – 3550(b) – 360 1700 190 3630 310

1.2840 771 0.9811 98.26 2.07 3415 – – 3527(d) 470 1730 400 3570 450

(a)

Raman spectroscopic data of Frantz et al. (1993). Raman spectroscopic data of Lindner (1970). (c) IR spectroscopic data of Gorbaty (1979). (d) Raman spectroscopic data of Walrafen et al. (1988) at 443 K and 3300 MPa. (b)

the presence of partial charges on O and H atoms, these both factors increase the average dipole moment of the water molecules with increasing temperature and density. The same temperature dependence of the average intramolecular geometry can be deduced from the comparison of the run E in Table 5 and the high-density simulation of Jancsó et al., (1984) at 350 K and 1.346 g/cm3, which resulted in 〈ROH〉 = 0.9767 Å, 〈∠HOH〉 = 99.82°, and 〈μ〉 = 1.992 Debye. The rotational relaxation of water molecules is often discussed in terms of angular momentum autocorrelation functions (e.g., Stillinger and Rahman 1972; Yoshii et al. 1998). For a flexible water model, a slightly different approach can also be used. In order to separate the various modes of molecular librations (hindered rotations) and intramolecular vibrations, the scheme proposed by Bopp (1986) and Spohr et al. (1988) can be employed. The instantaneous velocities of the two hydrogen atoms of every water molecule in the molecular center-of-mass system are projected onto the instantaneous unit vectors: i) in the direction of the corresponding OH bond (u1 and u2); ii) perpendicular to the OH bonds in the plane of the molecule (w1 and w2); and iii) perpendicular to the plane of the molecule (v1, and v2). Then the following six quantities can be defined using capital letters to denote the projections of the hydrogen velocities onto the corresponding unit vectors:

RX = W1 – W2

(22)

RY = V1 + V2

(23)

RZ = V1 – V2

(24)

Q1 = U1 + U2

(25)

Q2 = W1 + W2

(26)

Q3 = U1 – U2

(27)

 

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117 

The quantities RX, RY, and RZ approximate instantaneous rotational motions around the three orthogonal principal axes of the water molecule, and Q1, Q2, and Q3 approximately describe the three normal modes of molecular vibrations usually referred to as symmetric stretch, bend, and asymmetric stretch, respectively (schematically illustrated on the inserts in Fig. 19). The dynamics of different modes of molecular librations (hindered rotations) and intramolecular vibrations in supercritical water can now be analyzed in terms of velocity autocorrelation functions for the corresponding projections (Eqns. 22–27) (Kalinichev and Heinzinger 1992, 1995; Kalinichev 1993). The velocity autocorrelation functions calculated for the quantities Qi (Eqns. 25–27) are shown in Figure 19 for two extreme cases of high-density and low-density supercritical water. The Fourier transforms of these functions result in the spectral densities of the corresponding vibrational modes. They are shown in Figure 20 for the supercritical thermodynamic states listed in Table 5. The frequencies of the vibrational peak maxima and the widths at half maximum, FWHM, are also given in Table 5. The error bounds for the calculated frequencies are estimated to be about ±30cm–1 (Kalinichev and Heinzinger 1992, 1995). There is an empirical relationship (La Placa et al. 1973) for the rate of decrease of the OH stretching frequency in response to the decreasing intramolecular OH distance (≈20,000 cm–1/Å), which seems to hold well for simulations of aqueous electrolyte solutions at room temperature and liquid-like densities (Heinzinger 1990). From this relationship and changes of the average intramolecular geometry discussed above, a red shift of about 55 cm–1 relative to liquid water at normal temperature is expected for the vibrational

Figure 19. Velocity auto-correlation functions for the three intramolecular vibrations of water molecules at supercritical temperatures: (a) 771 K, 1.284 g/cm3; (b) 673 K, 0.166 g/cm3. Q1, Q2, and Q3 denote the symmetric stretching, bending, and asymmetric stretching modes, as illustrated on the inserts.

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Kalinichev 

Figure 20. Spectral densities (in arbitrary units) of the intramolecular vibrations of supercritical water from MD simulations. ν1, ν2, and ν3 denote the symmetric stretching, bending, and asymmetric stretching modes, respectively.

frequencies of the run D (at a supercritical temperature of 680 K, but at a density close to that of normal liquid water, ≈1 g/cm3). However, based on the symmetric stretching frequency for liquid BJH water under ambient conditions, ν1=3475 cm–1 (Bopp 1986), a blue shift of approximately the same magnitude is estimated in reasonable agreement with Raman spectroscopic data (Lindner 1970; Kohl et al. 1991) where a blue shift of about 120 cm–1 was observed. Clearly, the simple empirical relationship is not applicable any more when both temperature and density (pressure) can simultaneously affect the average intramolecular geometry. Raman and infrared spectroscopic data for supercritical water (Lindner 1970; Gorbaty 1979; Frantz et al. 1993), interpolated where necessary, are given in Table 5 for comparison with MD simulations. It is a common practice in high-pressure spectroscopic experiments to describe results in terms of spectral shifts per unit of pressure, cm–1/MPa

 

119 

Simulations of Liquid & Supercritical Water 

(Walrafen 1973; Klug and Whalley 1984; Farber et al. 1990). However, density would be a much more appropriate variable, as the one more directly related to the changes of interatomic distances taking place with increasing pressure. Frantz et al. (1993) have shown that while Raman spectra of water demonstrate significant pressure dependence along an isotherm, as well as temperature dependence along an isobar, they are virtually independent of temperature at any particular density over a very wide range from vaporlike to liquid-like densities at temperatures above 520 K. The density dependence of the frequency of spectral maxima ν1max obtained in the MD simulations of BJH water is compared in Figure 21 with all available experimental data. One point from high-pressure, high-temperature Raman measurements of Walrafen et al. (1988) is also plotted in Figure 21 for comparison, although it corresponds to a somewhat lower temperature and higher pressure (443 K and 3300 MPa, respectively) than those used in the simulations. Except for one point, there is a remarkable agreement between simulated and measured values. The large discrepancy for the run C at 0.6934 g/cm3 (see Table 5) could possibly be related to the fact that it was the only MD-simulation in this series actually performed at subcritical temperature, 630 K, and thus could correspond to a metastable thermodynamic state. An indirect indication of this might also be seen in the discrepancy of pressure calculated for the same point (Kalinichev and Heinzinger 1992; 1995), which was also far beyond the qualitatively reasonable representation of the equation of state observable for all other simulation points. However, more extensive simulations for the BJH water over wider ranges of temperature and density need to be performed before the phase diagram for this particular water model could be better understood. The bending frequency ν2 of liquid BJH water under ambient conditions is 1705±5 cm–1 (Bopp et al. 1983; Heinzinger 1990) compared to 1644–1 in real liquid water (Falk 1990). This frequency does not change much with temperature in MD simulations (cf. run D in Table 5) in perfect agreement with Raman measurements of Ratcliffe and Irish (1982) who found that bending frequency shows no temperature dependence within 3700

3600

3500

ν

xam 1

/ cm -1

isolated (non-bonde) molecules

3400

3300 0.0

D M simulations Raman data (Linder, 1970) IR data (Gorbaty, 1979) Raman data (Frantz et al., 1993) Raman data (Warenlaf et .al , 1988)

0.5 Densti y / g/cm

1.0 3

1.5

Figure 21. Spectral density maximum of the symmetric stretching vibration, ν1max, as a function of water density under supercritical conditions. Big open squares – MD simulations for the BJH model. Other symbols – experimental data from various sources.

120

Kalinichev 

experimental error up to 573 K. In MD simulations of high-density BJH water Jancsó et al. (1984) obtained ν2 = 1729±5 cm–1 at 350 K and a density of 1.346 g/cm3, quite close to that of the run E in Table 5. Comparing these values, we see that the bending frequency remains unchanged over a wide temperature range at high densities, as well. On the other hand, the blue shift of ν2 with increasing density at 673 K (runs A and D) and 773 K (runs B and E) is also in good agreement with Raman spectroscopic data (Farber et al. 1990; Kohl et al. 1991). With the gas-phase harmonic vibrational frequencies for the BJH water model being ν1 = 3832 cm–1, ν2 = 1649 cm–1, and ν3 =3943 cm–1 (Bopp et al. 1983), it is also interesting to note that at the lowest density studied (run A in Table 5) the bending frequency is already close enough to its gas-phase value, while the two stretching frequencies are still about ≈200 cm–1 red-shifted with respect to their corresponding gasphase values. This clearly indicates that the stretching vibrational modes in supercritical water are much more sensitive to the changes in local molecular environment, and they are much more strongly influenced by the effects of hydrogen bonding and molecular clusterization, than the bending vibrational mode, even at relatively low vapor-like densities. High-temperature experimental data on the asymmetric stretching vibrations, ν3, are available only over a limited range of relatively low pressures, P<50 MPa (Bondarenko and Gorbaty 1973). However, the general temperature and density dependence of our simulated spectra seem to be in qualitative agreement with these data. The shapes of the vibrational spectra obtained from the MD simulations using the procedure described above, which employs only simple classical-mechanical considerations, can not be directly compared with the shapes of experimental Raman or infrared spectra. However, it is nevertheless instructive to analyze how the widths of the simulated spectra change with density. The width of the ν1 stretching band increases almost linearly with the rate of ≈200 cm–1/(g⋅cm–3) over the entire density range (Table 5 and Fig. 20). This density broadening agrees surprisingly well with experimental results of Lindner (1970), from which a rate of about ≈175 cm–1/(g⋅cm–3) can be derived at a supercritical temperature of 673 K over approximately the same density range. From the Raman data of Frantz et al. (1993) a rate from 100 to 160 cm–1/(g⋅cm–3) can be estimated at temperatures between 630 and 780 K and over a somewhat narrower density range. Thus, we again see quite satisfactory agreement with experimental data. CONCLUSIONS AND OUTLOOK

The results of numerous computer simulations in recent years have demonstrated that hydrogen bonding is still noticeable in supercritical water even at temperatures as high as 800 K and very low densities. After some controversy, these results now seem to agree well with all available experimental data from several independent sources. Obviously, more experimental diffraction and spectroscopic measurements under supercritical conditions are necessary before these structural data can be used for reliable reparameterization of the available and the development of new intermolecular potential functions used in computer simulations for specifically supercritical conditions. Some of this work is already under way (Errington and Panagiotopoulos 1998; Stern et al. 1999; Chen et al. 2000; Guillot and Guissani 2000). However, even in their present form the effective semi-empirical potentials like TIP4P, SPC, or BJH are able to qualitatively, and often even quantitatively, reproduce the structure and properties of supercritical water. The predictive value of classical computer simulations for aqueous systems is still questioned by some researchers (Brodsky 1996). Yet, the case of the interpretation of the picture of hydrogen bonding in supercritical water clearly proves that such simulations

 

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can yield extremely useful information, when performed and analyzed carefully. The detailed picture of hydrogen bonding in liquid and supercritical water seems to be quite adequately described by classical simulations using relatively simple empirical potentials where the only reason for intermolecular H-bonding arises due to electrostatic interactions between partially charged atoms. On the other hand, recent ab initio MD simulations (Tukerman et al. 1997) and experiments (Isaacs et al. 1999; Martin and Derevenda 1999) have shown that up to 10% of H-bonding interactions is due to covalent forces, at least in liquid water and ice. Of course, the purely classical-mechanical approach to the high-frequency vibrational motions accepted here is limited by the exclusion of any quantum effects. However, these effects are expected to cancel out, if we are mainly interested in the changes of the vibrational frequencies, and not so much in their absolute values (Bopp et al. 1983). This is especially true for the supercritical conditions discussed in this chapter. Computer simulations are sometimes used in the geochemical literature with the sole objective to predict thermodynamic PVT properties of molecular fluids at high temperatures and pressures (e.g., Belonoshko and Saxena 1991, 1992; Duan et al. 1992; Fraser and Refson 1992). However, the ability to improve our physical understanding of the complex chemical behavior of geochemical fluids and to unravel fundamental molecular-scale correlations between the structural, transport, spectroscopic, and thermodynamic properties of supercritical aqueous fluids, seems to be a much more important feature of these techniques. Being an invaluable source of information intermediate between theory and experiment, the methods of molecular computer simulations will definitely continue to greatly stimulate the development of both theoretical and experimental studies of geochemical fluids in the near future. With the continuing progress in computer technology, which makes the use of powerful supercomputers less expensive and easily accessible for large-scale simulation studies by almost any geochemical laboratory, the prospects of molecular computer simulations of aqueous geochemical fluids look now more promising than ever. ACKNOWLEDGMENTS

The author is extremely grateful to YE Gorbaty, K Heinzinger, JD Bass, SV Churakov, and RJ Kirkpatrick for many years of fruitful collaboration some results of which are presented in this chapter. The financial support of the Russian Basic Research Foundation (grants 95-05-14748 and 97-03-32587), INTAS (grants UA-95-0096 and 961989), CRDF (grant RC1-170 to YE Gorbaty), NSF (grant EAR-9305071 to JD Bass, and grant EAR 97-05746 to RJ Kirkpatrick), and U.S. Department of Energy (grant DEF0200ER1528 to RJ Kirkpatrick), is most gratefully acknowledged. The computations were partly supported by the National Computational Science Alliance (grant EAR 990003N) and utilized NCSA SGI/CRAY Origin 2000 computers. REFERENCES Allen MP, Tildesley DJ (1987) Computer Simulation of Liquids. Oxford University Press, New York Alper HE, Levy RM (1989) Computer simulations of the dielectric properties of water: Studies of the simple point charge and transferable intermolecular potential models. J Chem Phys 91:1242-1251 Balbuena PB, Johnson KP, Rossky PJ (1996a) Molecular dynamics simulation of electrolyte solutions in ambient and supercritical water. 1. Ion solvation. J Phys Chem 100:2706-2715 Balbuena PB, Johnson KP, Rossky PJ (1996b) Molecular dynamics simulation of electrolyte solutions in ambient and supercritical water. 2. Relative acidity of HCl. J Phys Chem 100:2716-2722

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5

Molecular Dynamics Simulations of Silicate Glasses and Glass Surfaces Stephen H. Garofalini Department of Ceramics and Materials Engineering Rutgers University Piscataway, New Jersey, 08854, U.S.A. INTRODUCTION

A variety of classical (Newtonian) computational approaches have been developed to address the atomistic structure and behavior of materials, such as molecular dynamics (MD), monte carlo (MC), and molecular mechanics (MM) computer simulations. For the purposes of the current review, the discussion will be limited to the MD simulation technique. A major advantage of the classical MD technique is the ability to study large numbers of atoms, O(103-107), for relatively long times (from the standpoint of certain molecular events), O(ns-μs). By gathering data over these large numbers and times, the simulated time and number averages can be compared to the experimentally obtained time and number averages that are inherent in most experiments. If the simulations reproduce the experimental data (with some expected level of accuracy), the simulations offer the advantage of enabling an evaluation of the discrete events that caused the time and number averages. That is, because all of the atoms are labeled in the simulations, the exact atomistic behavior that cause the averaged data can be followed in order to determine the molecular mechanisms. Although algorithms have been developed that employ some level of electronic structure calculations with MD or MC schemes, system sizes are still fairly limited because of computational complexity. This chapter will discuss some aspects of the MD technique used in our laboratory to study glasses, glass surfaces, and interfaces, and some specific results. MOLECULAR DYNAMICS COMPUTER SIMULATION TECHNIQUE The data presented in this review were generated using the classical molecular dynamics (MD) computer simulation technique. Classical MD simulations involve solving Newton’s equations of motion for a system of interacting particles (atoms, ions, or hard spheres). The earliest MD simulations involved hard spheres (Alder and Wainwright 1957, 1959), but have been followed by decades of research into new algorithms and new interatomic potentials of varying complexity, as documented in several excellent books discussing either MD simulations in particular or molecular level computational techniques in general (Allen and Tildesley 1987; Haile 1992; Robinson et al. 1996). Throughout the following review, the terms “atoms” or “ions” will be used interchangeably without regard to whether the particle was neutral (atom) or charged (ion), although much of the data presented below will be generated from studies of ions. The molecular dynamics computer simulation technique involves moving atoms according to a force derived from an assumed interatomic potential via a predictorcorrector algorithm. In MD simulations, atoms are given x, y, and z coordinates within a given volume or box. Each atom is also given an initial velocity within a Maxwellian distribution such that the average velocity is consistent with the average initial temperature based on the virial relation. While various schemes exist, two prevalent ones 1529-6466/01/0042-0005$05.00

DOI:10.2138/rmg.2001.42.5

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include the Verlet algorithm (Verlet 1967) and the Nordsieck-Gear predictor-corrector algorithm (Nordsieck 1962; Gear 1966). In the predictor-corrector scheme, the positions and time derivative(s) at time t + δ t are predicted from the current values at time t. After all atom positions and time derivatives are updated to new values at t + δ t , they must be corrected to account for the stepwise nature of the prediction based on the value of δ t , the timestep. Ideally, δ t would approach zero, making for a continuous variation in position, not a discrete stepwise variation. Extremely small values would make the atom trajectories more continuous, but would be computationally excessive and unnecessary; too large a value and the predicted positions would be too inaccurate and solutions to the equations would be unstable. Generally, δ t is a small fraction of the vibrational period. In order to correct for the fact that the stepwise movement of atoms did not take into account the continuously varying positions of all the atoms during the motion, forces are calculated from the predicted positions at t + δ t and used in an algorithm to correct the positions and time derivatives. The forces are obtained from G Fi = −∇ rGi Φ i

(1) G where Fi is the force on atom i and ΦGi is the interatomic potential between atom i and all other atoms as a function of position ri . In the simulations performed in our lab and discussed in this chapter, the fifth-order Nordsieck-Gear algorithm was used. In this algorithm, Taylor series expansions of the existing positions and time derivatives of each atom are used to predict the next positions and time derivatives for that atom. Using a particular value for the timestep ( δ t ) in the expansion, each atom is moved some predicted distance and the predicted fifth-order time derivatives (velocity, acceleration, etc.) are also updated. The force on each atom is calculated from the predicted positions and the difference between the predicted G acceleration and that obtained in the force calculation (from F i = mi ai ) is used to correct the position and derivatives. As a result of this prediction and correction, the system at t + δ t is displaced from the previous system at t by a small increment in time, δ t . Reiteration of the calculations and movement of atoms creates a time evolution of the system from which time averaged properties can be obtained. In summary, positions and time derivatives are predicted, forces are calculated from the predicted values, and positions and time derivatives are corrected based on these forces. Various structural, thermodynamic, and kinetic data can therefore be obtained in the simulations. Examples of structural data relevant to the oxides presented later in this chapter include X-ray diffraction, the static structure function, the radial distribution function (RDF), the pair distribution function (PDF), coordination numbers, bond lengths and bond angles, ring structures, Qn species (from NMR nomenclature). The radial distribution function, g(r), is given as g (r ) =

1 N N 〈 ∑∑ δ (r − rij )〉 N ρ i j ≠i

(2)

where ρ is the number density, N is the number of atoms, δ is the Dirac function, and the angle brackets indicate a time average. While g(r) is summed over all central atoms i and all neighbors j ≠ i, a variation called the pair distribution function (PDF) would involve a similar summation, but over particular types of atoms i and particular neighbors j. An example would include the PDF for Si-O interactions, in which the first summation would only be over central Si atoms and the second summation over only O atoms. Of

 

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course, the second summation could also be over other atoms, not just O, depending on the type of data one wishes to obtain. The PDF enables delineation of the specific types of structural correlations and, in the case of glasses, can be compared to EXAFS or NMR data. Coordination numbers can also be obtained from the radial and pair distribution functions by integrating over the first peak in these distribution functions. For some species in a glass, such as Si or Al, the first peak in the RDF goes to zero after the first maximum, making the integration over the first peak fairly simple (see Fig. 1a); other species, such as alkali or alkaline earth ions, may have a less ordered local structure in the glass. In the latter cases, the first minimum after the first maximum in the R-O PDF (R=alkali or alkaline earth) does not cleanly go to zero but only shows a broad minimum, thus making the integration over the first peak and the resultant coordination number somewhat dependent on the outer distance limit of the integration (see Fig. 1b). Thermodynamic properties such as the internal energy, U, pressure, p, average temperature, T, and constant volume heat capacity, Cv, to name a few, can also be readily obtained in the simulations from N 3 U = NkT + ∑ Φ i 2 i

p=

(3)

NkT 1 N G G + 〈 ∑ ri ⋅ Fi 〉 3V i V

(4)

1 N 〈 ∑ mi vi2 〉 3Nk i

(5)

T=

3 〈Φ 2 〉 − 〈Φ〉 2 Cv = Nk + kT 2 2

(6)

where k = Boltzmann's constant, v = velocity, V = volume, and other terms are given above. Kinetic properties such as diffusion coefficients and vibrational spectra can also be calculated. The diffusion coefficient, D, can be obtained from D=

1 1 N 〈 ∑ [ri (t0 ) − ri (t0 + τ )]2 〉 2ατ N i

(7)

2.0

0.4

(b) INTENSITY

INTENSITY

(a) 0.3 0.2 0.1

1.5 1.0 0.5 0.0

0 1

1.5

2

DISTANCE (Å)

2.5

2.0

2.3

2.5

2.8

3.0

3.3

3.5

DISTANCE (Å)

Figure 1. (a) First peak in the Si-O pair distribution function in silica. (b) First peak in the Na-O pair distribution function.

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Garofalini 

where α is the dimensionality of diffusion (1, 2, or 3), t0 is the initial time, τ is the time increment over which diffusion is measured, and ri is the position of an atom at a time t0 or t0 + τ. The summation is over all atoms N, although N could also be some restricted set of atoms of particular interest, such as only cations, or all Al ions, or only those ions near the surface, etc. In order to remove any memory or correlation with an initial set of positions, various initial times, t0, are used. Values of τ from short time to the length of a simulation run are used. Various autocorrelation functions can be determined, such as the velocity autocorrelation function, C(t), which is given as N

C (t ) =

〈 ∑ vi (t0 ) ⋅ vi (t0 + t )〉 i

N

〈 ∑ vi (t0 ) ⋅ vi (t0 )〉

(8)

i

where vi(t0) is the velocity of atom i at an initial time t0 and vi(t0+t) is that atom’s velocity at some later time t. The angle brackets again indicate a time average. Such a selfcorrelation function measures the correlation between atom i's velocity at some initial time t0 and its velocity at time t0 + t, summed over all atoms and normalized to the initial velocities. The frequency spectrum G(ω) (actually the power spectrum) is obtain from the Fourier transform of the velocity autocorrelation function G (ω ) ∝

∫

∞

0

C (t ) cos ω t dt

(9)

Other vibrational spectroscopies such as neutron, Raman, or IR spectroscopy can also be calculated from the simulations. Although the various analyses of the simulations shown above can be performed during a simulation (that is, during the calculated time evolution of the system), the low cost of storage devices means that relevant data, such as positions and velocities (from which most of the structural and dynamic properties shown above can be obtained) can be frequently stored in the simulation run and evaluated after the run is complete. Although every configuration can be saved for subsequent analysis, configurations can often be saved less frequently. For instance, given a timestep, δ t , that is much smaller than a vibrational period, positions and velocities of all atoms can be saved every tenth move, or configuration, usually without loss of important data relevant to some average. The importance of such storage is that unexpected atomistic behavior that occurs in the simulation can be subsequently evaluated in more detail in the stored data. Without this, the more costly whole simulation run would have to be redone, with the previously unexpected specific behavior now analyzed in the run itself. This is clearly inefficient since any analysis is much less costly than the full simulation run itself. As such, one of the major advantages of an MD simulation is the ability to evaluate properties over the whole system or over specifically interesting parts of the system of atoms. Therefore, if the overall time and number averages over the whole simulated system correlate well with available experimental data, the simulations offer the advantage of enabling evaluation of specific mechanisms by following the behavior of specific atoms or groups of atoms. That is, in the equations above, data can be gathered over specific atoms (N in the equations above could be a subset of the whole system). The trajectories of the atoms can be followed by plotting the x, y, and z coordinates of each atom as a function of configuration (time). Such and used to evaluate molecular

 

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mechanisms, diffusion paths, reactions, etc. By putting the saved positions of the atoms into a program that draws a sphere centered on each position, graphics and movies of the system of interacting atoms can be made. Such visualization often helps in determining the type of analyses that would provide the most relevant information with respect to mechanisms. Interatomic potentials

Since forces are derived from the interatomic potential, the potentials are vitally important to the success of the simulations. Initially, pair, or two-body, potentials were used. While hard sphere pair potentials were initially used in the earliest simulations, soft sphere pair potentials were soon introduced (Rahman 1964) and are the common form of the two-body terms used in current simulations. An excellent fundamental discussion of intermolecular potential functions is available (Hirschfelder et al. 1963). One of the simplest soft sphere pair potentials in widespread use is the Lennard-Jones 12-6 potential, discussed by Cygan (this volume). The LJ12-6 potential, shown in Figure 2, offers a simple model description of the general form of most pair potentials; that is, a short range repulsive term and a longer range attractive term. All variations of pair potentials are conceptually similar, although the precise forms can become more complicated. The potential function currently used for the MD simulations presented below include both two-body and three-body terms

Φ i = ∑Vij(2) + j ≠i

∑V

k ≠ j ≠i

(3) jik

(10)

The two-body term is given in Equation (11) ⎛ r ⎞ qq ⎛ r ⎞ Vij(2) = Aij exp ⎜ − ij ⎟ + i j ξ ⎜ ij ⎟ + VijCSF (11) ⎜ ρ ⎟ r ⎜β ⎟ ij ⎠ ij ⎝ ⎝ ij ⎠ G G where rij = ri − r j , qi = the formal ionic charge on atom i and Aij, ρij, and βij are adjustable pair parameters. ξ is the complimentary error function used to reduce the formal charges (qiqj) as a function of separation distance, thus creating a screened coulomb potential. In the form of the potential used here, the βij is dependent on the i-j pair, which offers significant flexibility in altering the product of the charges (Garofalini and Melman 1986). Of course, the coulomb term is a slowly decaying function in r. In order to reduce the computational cost, schemes such as the Ewald method (Ewald 1921), the cell multipole method (Ding et al. 1992), or the fast multipole method (Shimada et al.

10

POTENTIAL

8 6

Figure 2. Lennard-Jones 12-6 interatomic potential. Distance in units of the atom diameter.

4 2 0 -2

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

-4

DISTANCE

1.7

1.8

1.9

2

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Garofalini 

1994) are often used. VijCSF is used to add additional structure to a potential where the smooth variation shown in a simple pair potential (e.g., Fig. 2) is insufficient and is given as x ⎛ ⎞ aij( ) = ∑⎜ ⎟ ( x) ( x) ⎟ ⎜ x =1 1 + exp(bij ( rij − cij )) ⎝ ⎠ 6

CSF ij

V

(12)

VijCSF is zero for most pair interactions presented below, but is useful in simulations involving H interactions (Feuston and Garofalini 1990). It was also useful in simulations of the crystalline forms of layered oxides, V2O5 and γ-LiV2O5 (Garcia et al. 1998; Garcia and Garofalini 1999).

The three-body potential can be used to account for any bond directionality in the bonding between atoms in a system and is given in Equation (13) as:

V jik(3) = (λ jik exp[

γ ij rij − r

o ij

+

γ ik rik − riko

]Ω jik )

(13)

when rij < rijo and rik < riko . Otherwise, V jik(3) = 0 . Ωjik, the angular term, is a function of θjik, the angle formed by atoms j, i, and k with atom i at the vertex. The three-body term is a penalty function that increases the potential (and repulsive forces) whenever a three atom triplet deviates from an ideal angle. Ωjik is given as o 2 Ωtetra jik = (cosθ jik − cosθ jik )

(14a)

where, for tetrahedrally coordinated ions i, such as Si, θ ojik is the tetrahedral angle, 109.47º (Feuston and Garofalini 1988), while for trigonally coordinated ions in an oxide glass, such as B, θ ojik is 120°. For purely octahedrally coordinated species, Ωjik can be 2 Ω octa jik = (sin θ jik cosθ jik )

(14b)

For species that readily display both tetrahedral and octahedral coordination in common minerals (such as Al ions in γ-Al2O3), the angular term given below can be used (Blonski and Garofalini 1993) − octa Ωtetra = {(cosθ jik − cosθ ojik )sin θ jik cosθ jik }2 jik

(14c)

where θ ojik is again the tetrahedral angle. In this case, minima in the 3-body force occur at 90°, 109°, and 180°. The strength of the 3-body term depends on λ, while the shape of the 3-body repulsion depends on γ, and the farthest distance at which the 3-body term acts between an i-j pair of the triplet is rijo (similarly for the i-k pair in the triplet). An important part of the success of the use of a 3-body term is related to its stiffness, or how quickly the 3-body term rises from zero below the cut-off distance, rijo . If this is too stiff (3-body repulsion rises too quickly), then non-equilibrium structures would be inhibited and possible reaction states would be precluded from forming in the simulations. An example of this is the formation of the 5-coordinated Si that is believed to form during the reaction of a water molecule with silica or the polymerization of silicic acid (H4SiO4) molecules. Pentacoordinated Si has been observed in experimental studies (Stebbins and McMillan 1993) and in previous simulations of glassy SiO2 (Angell et al. 1982a,b) and its

 

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137 

ionic analog, glassy BeF2 (Brawer 1981, 1985), quenched from the liquid state using pair potentials alone. However, the concentration of these defect species in these early simulations were excessively high. That was due to the lack of bond directionality in the pair potential. The addition of the tetrahedral 3-body term on the Si reduces the concentration of these overcoordinated defects in the glass (Feuston and Garofalini 1988), but nonetheless allows for their formation, as will be discussed in more detail below. An example of the pentacoordinated Si is shown in Figure 3. This figure shows the condensation reaction between two silicic acid molecules in the formation of the siloxane (Si-O-Si) bond and a water molecule. The formation of the pentacoordinated Si in a trigonal bipyramidal structure as a reaction intermediate is shown in the central portion of the figure. Ab initio calculations show the pentacoordinated Si to be a stable intermediate (Damrauer et al. 1988; Lasaga and Gibbs 1990; Kubicki and Sykes 1993), not a transition state complex. MD simulations using the multibody potential shown above reproduce the mechanism shown in Figure 3 (Feuston and Garofalini 1990; Garofalini and Martin 1994; Martin and Garofalini 1994). The activation energy for formation of the Q3 species was found in the simulations to be 12 kcal/mol (Garofalini and Martin 1994); the Q3 allows for cross-linking, and hence gelation. The activation energy for gelation of silica in neutral pH was found experimentally to be 12 kcal/mol (Rabinovich et al. 1990). The simulations also showed the preferential growth chains prior to rings (Garofalini and Martin 1994). The dominant ring structures that grew changed with time. Detailed MD simulations also showed the role of H ions in the proximity of the O leaving the pentacoordinated Si in the reaction shown in Figure 3 (Martin and Garofalini 1994). Detailed reviews of the important ab initio calculations related to water interactions with silica have been presented (Lasaga 1990, 1995). The important feature in the simulations using the multibody potential is that the overcoordinated species are allowed to form, but are at a higher energy state and are therefore less stable than the tetrahedrally coordinated Si. Thus, within the timeframe of a simulation, the pentacoordinated Si relaxes back to the 4-coordinated state. Periodic boundary conditions

Currently, most MD simulations involve systems containing O(103-105) atoms, although simulations using O(106-107) atoms have been performed (Omeltchenko et al. 2000) and will become more prevalent as both computer power increases and the need to study such large systems becomes more important. In the more common, smaller simulations, the relatively small system size means that a significant proportion of the

(a)

(b)

(c)

Figure 3. Polymerization reaction between silicic acid (H4SiO4) molecules showing formation of a 5-coordinated Si in a trigonal bipyramidal symmetry in (b).

138

Garofalini 

atoms are at the edges or surface of the system. In trying to study bulk properties, these surfaces would have a detrimental effect and must be negated in some manner. A common approach is the use of periodic boundary conditions (PBC). A two-dimensional (2D) representation of the role of PBC in simulations is presented in Figure 4. Atoms are given x, and y (and z in 3D) coordinates that put them in the box with dimensions xl and yl (and zl in 3D) that is in the center of the figure and is highlighted by the bold lines. Four atoms, labeled 1-4, are shown in the central box (I) in bold. While the central box ends at the bold lines drawn in the figure, thus creating surfaces, the use of PBC creates “images” that removes these surfaces. The “image” atoms do not really exist in the simulation. Rather, they appear at appropriate moments in the simulations as needed to remove edge or surface effects. Specifically, atom 1 calculates the interactions between itself and all neighbors out to some cut-off distance, say rc. In many circumstances the cut-off radius is 0.5L, where L is the box length. While atom 3 is within this cut-off

(a)

(b)

(c)

Figure 4. (a) Periodic boundary conditions (PBC). Heavy solid lines indicate location of PBC that surround Box I, which is the central cell where atoms are located. Assume the third dimension, Z, is similarly drawn. (b) Periodic boundary conditions (see text). (c) Periodic boundary conditions in two dimensions (assuming the PBC in the third dimension is unchanged).

 

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139 

radius in the central box, atom 2 is beyond it. However, by using PBC, the “image” of atom 2 is within the cut-off radius in box VIII. Because the maximum cut-off radius is 0.5L, only one “image” of atom 2 is ever seen by atom 1. This 1-2 interaction means that atom 1 does not see a surface on its left side because of the presence of the image atom 2 in box VIII; similarly, atom 2 does not see a surface on its right side because the 1-2 x x = − F1-2 ). interaction is equivalent to a 2-1 interaction ( F2-1 In order to describe the previous paragraph mathematically, a brief fortran algorithm showing the PBC features for closest neighbor distance is given below in one dimension xij = x(1)-x(2) if (xij.gt.0.5L) then xij=xij-L if (xij.lt.-0.5L) then xij=xij+L

where x(1) is the x coordinate of atom 1 (x(2) for atom 2) and L is the box length (L) in the x direction. gt (lt) stands for greater than (less than). Thus, if xij > 0.5L, then subtracting L from this distance makes xij < 0.5L and acts like putting atom 2 into box VIII, although atom 2 is never actually moved. A faster approach uses integer truncation to produce PBC. In using such PBC, the light dotted line in the outer boxes (II-IX) indicates the maximum distance into each adjoining box that an atom in the central box (I) could possibly “see” an “image” atom (one-half the distance into each adjoining box). Any larger distance would mean that the neighbor in the central box is < 0.5L from the central atom. That is, assume atom 1 was at the edge of the central box (see Fig. 4b, where atom 1 is moved leftward from its position in Fig. 4a), and let atom 2 also be moved leftward so that it is slightly greater than 0.5L from atom 1 in the central box in the x direction (horizontal in the Fig. 4b). Then the algorithm above puts image atom 2 just short of the dotted line in box VIII and atom 2's “image” in box VIII is closest to atom 1. However, if atom 2 had been moved leftward slightly more, then atom 2 in the central box (I) would be < 0.5L from atom 1 in the central box and its “image” in box VIII is beyond the dotted line in VIII. In addition, going back to Figure 4a, the resultant forces on atom 2 may be such that it moves horizontally to the right, as shown by the dashed arrow. Atom 2 in the central box would move out beyond the bold line on the right, but because of PBC, atom 2's “image” would move from the left, as shown by the arrow on image atom 2. In reality, this means that atom 2's position suddenly changes from a value just above L (L+δx) to δx (from L+δx -L). With respect to the movement of atoms across the boundary, a mathematical description shown as a short fortran algorithm could be xx=x(2)*xli where xli is preset to 1/L and x(2) is atom 2's x coordinate m1=xx m2=xx-1.0 where m1 and m2 are integers x(2) = (xx - m1 -m2)*L

In order to create surfaces, the PBC are removed in the appropriate dimensions. Thus, for a free cluster, no PBC would be employed. In the simulations presented below for surfaces, the PBC are removed in one dimension and kept in the other two dimensions. Figure 4c shows a schematic of the system with a surface (in the Y dimension), where the bold lines again indicate PBC. In a three dimensional system, X and Y could contain PBC and Z would contain free surfaces, thus creating a thin slab that is “infinite” in X and Y. Since surface relaxation or reconstruction can occur, the system dimension in the Z direction must be sufficiently thick so that the surface phase (the volume influenced by the formation of the surface) on one side does not extend to the

140

Garofalini 

surface phase on the other side. Surface relaxation is a term indicating some symmetry conserving change in spacing between atoms at or near the surface, often as movement perpendicular to the free surface. Surface relaxation has been of significant interest in the last several decades, especially with the application of surface-specific experimental techniques that provide information with regard to surface structure, such as surface EXAFS (Citrin 1986), LEED, and medium ion scattering spectroscopy (MEIS) (Walls and Smith 1994). A example of surface relaxation in oxides is the reduction of the Al-O planar spacing in the (0001) single Al terminated surface of α-Al2O3 as seen in ab initio calculations (Batirev et al. 1999) and MD simulations (Blonski and Garofalini 1993). Surface reconstruction indicates a change in the surface structure that changes the symmetry from that of the bulk such that the surface is considerably different than a simple truncation of the bulk-like structure. A prime example of surface reconstruction is the (7×7) structure of the (111) surface of Si, in which the (1×1) 2D unit cell structure parallel to the surface is altered to a (7×7) unit cell (Feenstra 1994). MD SIMULATIONS OF OXIDE GLASSES Bulk glasses

Bulk glasses (three dimensional PBC) are usually made via a simulated melt-quench technique, in which a system of atoms is either given a crystalline structure or a random distribution within the central cell with three dimensional periodic boundary conditions. This system is brought to a high temperature in order to generate a simulated melt. The melt temperature should be sufficiently high and run for a sufficiently long time such that the melt has no memory of the initial configuration. With the multibody potential shown above, the superheated, short time (10 ps) liquidus temperature of non-defect silica is near 7500 K. Of course, this is not an accurate thermodynamic melting point in comparison to experimental data, but is rather a homogeneous mechanical melting process. Superheating in simulated surfaceless systems is well known; Clancy showed the benefits of obtaining melt temperature using a solid/liquid system with a shared interface (Chokappa and Clancy 1988). A lively discussion regarding problems that occur in simulations of melting has been recently presented (Belonoshko 2001; Chaplot and Choudhury 2001). Experimentally, melting is initiated at surfaces, grain boundaries, and point defects (Cahn 1986) and is a heterogeneous process (Wolf and Yip 1992). It is probably more appropriate to study melting using free surfaces (Wolf and Yip 1992) or interfaces (Belonoshko and Dubrovinsky 1996). Thus, the high temperature defect free simulations used for our melt/quench procedure are not designed to study melting, but rather are only used to generate a three dimensional bulk liquid from which a glassy state can be made. In our simulations, melting silica or silicates at 8000 K to 10000 K for 20 ps–50 ps is more than sufficient to meet the condition of loss of memory of the starting positions of the atoms, regardless of the composition. Of course, the volume is expanded from the room temperature value to the elevated temperature. Upon cooling, the volume is similarly rescaled using the thermal expansion coefficient and room temperature density (Mazurin et al. 1983). Cooling can occur via several protocols based on lowering temperature, T, from the melt temperature to room temperature via intermediate temperatures. The temperature can be instantaneously dropped from the higher T, Th, to the next lower T, Tl, by rescaling velocities to the new temperature, or it can be dropped as a fraction of the ratio of the desired temperature and the actual temperature (based on velocities). In either protocol, the velocities are rescaled to the desired temperature for some appropriate number of timesteps. In simulations using the microcanonical ensemble, NVE (constant number of atoms, N, volume, V, and energy, E), the atoms have their velocities rescaled for some initial number of timesteps, after

 

141 

MD Simulations of Silicate Glasses & Glass Surfaces 

which no rescaling occurs and the temperature should remain stable. Often, this initial time period requires no more than a few thousand timesteps for 20,000-50,000 timestep runs, depending on the temperature and the size of the timestep. In constant pressure NPT simulations, the temperature is rescaled throughout the run (Berendsen et al. 1984). Analysis of the glass at room temperature occurs via the equations shown above. Bulk SiO2

Figure 1a above showed the first peak in the pair distribution function for Si-O pairs in silica. The full RDF is shown in Figure 5, while the simulated static structure function is compared to the experimental data in Figure 6 (Feuston and Garofalini 1988). The bond lengths observed in the simulations are: 1.62 Å Si-O, 2.62 Å O-O, and 3.12 Å Si-Si. The O-Si-O bond angle is 109°, with a full width at half-max of 14°, consistent with calculations based on electron spin resonance, ESR, results (Feuston and Garofalini 1988). 5-membered and 6-membered rings dominate the structure, although smaller 3-membered rings are observed (Feuston and Garofalini 1988). An n-membered ring contains n tetrahedra in a closed loop. 3-membered rings are believed to be the cause of the 606 cm-1 peak in the Raman spectrum of silica (Galeener and Mikkelsen 1981; Galeener 1982, 1983). No 2-membered rings, or edge-sharing tetrahedra, are observed in the bulk simulations using the multibody potential (although, as presented below, they are found in the simulated surfaces (Levine and Garofalini 1987; Feuston and Garofalini 0.3

RDF

0.2

Figure 5. Radial distribution function of silica. 0.1

0 0.0

1.5

2.5

3.5

4.5

5.5

DISTANCE (Å)

2.0

SIMULATION Figure 6. Static structure function of silica glass; solid line from molecular dynamics simulations, asterisks from experimental data (see text).

S(q)

EXPERIMENT

1.0

0.0 0

2

4

6

q(Å-1)

8

10

12

14

142

Garofalini 

1989; Garofalini 1990), consistent with experimental data (Bunker et al. 1989a,b) Self-diffusion of ions in silica has been measured both experimentally and computationally. Experimentally, Mikkelsen (Mikkelsen Jr. 1984) used SIMS (secondary ion mass spectroscopy) to measure the concentration profiles of labeled O deposited on silica to obtain the diffusion coefficient. Brebec et al (Brebec et al. 1980) used labeled Si and SIMS analysis for determining Si diffusion. Hetherington (Hetherington et al. 1964) used viscosity measurements of commercial silica to determine diffusion constants. Their experimental data are shown in Table 1, along with results from several computational studies. While Table 1 shows both the bulk and surface diffusion constants, only the bulk data will be discussed here, with the surface data discussed below. Figure 7 shows the mean square displacement (Eqn. 7) of Si and O in simulated bulk vitreous silica at 6000K. Note that significant diffusion occurs at this temperature. Simulations of diffusion require sufficiently long times at temperature to go beyond large vibrational amplitudes that may misleadingly appear like diffusive behavior if the simulations are not run long enough. In the simulations discussed here (Litton and Garofalini 1997), runs of 100 ps at T ≥ 4800 K were used. Using simulations of bulk vitreous silica at T ranging from 4800 K to 7200 K, a series of diffusion constants were obtained from Equation (7). These are shown in the Arrhenius plot in Figure 8. An Arrhenius plot of the experimental data is shown in Figure 9. While Table 1 shows a similarity among simulation results, it also shows significant differences among the experimental data as well as between the experimental data and the simulation results. A possible reason for the differences among the experimental data may be the different types of silica used in the experiments and the effect of impurities in each glass on either D or D0 in D = D0 exp( and

Q ) kT

(15)

⎛ ΔS ⎞ Do = γ ao2ν exp ⎜ ⎟ ⎝ R ⎠

(16)

Table 1. Activation energy (Q) and pre-exponential factor (D0) for diffusion in silica from simulations and experiments. Q (kcal/mol)

a

Si 115 113

O 114 113

120 110 138 122-170

110 120 108 -

D0 (cm2/s) Si 0.21 3D 0.26 2D 0.39 0.009 0.18 328 -

O 0.26 3D 0.33 2D 0.50 0.01 0.45 2.6 -

T range (K)

Reference

4800-7200 4800-7200

(Litton and Garofalini 1997)a (Litton and Garofalini 1997)b

8000-10000 6000-8000 1473-1673 1383-1683 1173-1673

(Soules 1982)c (Kubicki and Lasaga 1988)c (Mikkelsen 1984) (Brebec et al. 1980) (Hetherington et al. 1964)

bulk diffusion surface diffusion using α = 3 in Equation 7 (3D) or α = 2 (2D) c calculated from the Arrhenius plots found in each reference b

 

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143 

Figure 7. Mean square displacement for Si and O ions in bulk silica glass at 6000 K.

Figure 8. Arrhenius plot of the natural logarithm of D versus 1/T for Si and O diffusion in bulk silica.

where γ is a geometrical term, ao is the jump distance, ν is the vibrational frequency, and ΔS is the activation entropy. Heatherington et al. (1964) found that viscous flow is affected by both hydroxyl content and impurity metal content. The former affects Q, the activation energy in Equation (15), while the latter affects D0. Consideration of such effects helps explain the differences between the experimental data by Mikkelsen (1984) and the data by Brebec et al. (1980), as discussed previously in more detail (Litton and Garofalini 1997). The activation energies observed in the simulations shown in Table 1 are consistent with the rupture of an Si-O bond which has a bond energy near 110 kcal/mol. Figure 10 shows one diffusive mechanism of an O ion in bulk silica at 6000 K. The diffusing O (O(2) in Fig. 10) is attached to a 5-coordinated Si (Si(1)) in (a). Si(1) is in a

144

Garofalini 

Figure 9. Natural logarithm of D versus 1/T for experimental data (see text).

Figure 10. Short distance diffusion of an O (O(2)) from (a) to (d) (see text).

trigonal bipyramidal structure, as discussed above. In (b), O(2) breaks the O(2)-Si(1) bond and in (c) Si(1) relaxes towards O(6), enabling its other three O to relax to the tetrahedral configuration from the planar one in (a) and (b). In doing so, these three O screen Si(1) from O(2), which moves towards Si(3). In (c), Si(3) forms a 5-coordinated Si in the trigonal bipyramidal structure. The Si(3)-O(4) bond ruptures, letting Si(3) go back to 4-coordination. Larger scale diffusive motion is shown in Figure 11, in which configuration (b) is

 

MD Simulations of Silicate Glasses & Glass Surfaces 

taken 12 ps after configuration (a). The labeled Si has diffused a large distance, with significant exchange of O neighbors. Also note the diffusion of the O from (a) to (b). An important aspect in the diffusion of Si or O in silica is the formation of the pentacoordinated Si intermediate, as discussed here and above. Such formation is not as important in diffusion on silica surfaces, as discussed below.

145 

Diffusing Si DIFFUSING Si

a. a.)

Multicomponent silicate glasses

Introduction of additional species into silica creates silicate glasses with properties quite different from those of the parent pure silica system. Structurally, the addition of alkali metals to silica alters the structure, creates non-bridging oxygen (NBO), lowers the liquidus, and decreases durability in aqueous solution (Weyl and Marboe 1964; Doremus 1979; De Jong and Brown 1980; De Jong et al. 1981; Schramm et al. 1984; Murdoch et al. 1985; Schneider et al. 1987; Stebbins 1987; Zhang et al. 1996). Additions of alkaline earths to alkali silicates reduces corrosion behavior and small amounts of Al enhance durability even more.

b.

DIFFUSING Si Diffusing Si

b.)

Figure 11. Large scale diffusion of a Si ion in bulk silica. All O that were bonded to this diffusing Si sometime during this run are drawn as spheres, with the largest spheres being those O initially attached to this Si.

Simulations of alkali silicates using pair and multibody potentials show the formation of NBO associated with the alkali ions (Soules 1979; Soules and Busbey 1981; Huang and Cormack 1990, 1991; Melman and Garofalini 1991). Channel formation is observed in the simulations (Huang and Cormack 1990, 1991; Melman and Garofalini 1991), consistent with the Modified Random Network Model (Greaves 1985). The Na-O PDF (Melman and Garofalini 1991) observed in MD simulations of sodium silicates using a multibody potential is similar to the experimental X-ray and neutron data (Waseda and Suito 1977) and EXAFS data (Greaves et al. 1981). There have been many studies of alkali silicate glasses and a brief review is available (Cormack and Cao 1997). Addition of Al to a sodium silicate glass alters properties such as the viscosity and the activation energy for Na diffusion. Various models have been proposed to account for the maximum in viscosity and minimum in activation energy for Na diffusion as the Al/Na ratio goes through ~1.0. Early on, the traditional view of a change in coordination of the Al from 4-fold to 6-fold has been proposed as an explanation for the extrema in these properties (Isard 1959; Day and Rindone 1962a,b,c; Graham and Rindone 1964; Taylor and Rindone 1970). However, another model involving formation of oxygen triclusters (three cations around an O ion) was proposed (Lacy 1963). Direct structural information from NMR and EXAFS studies showed no 6-coordinated Al above the Al/Na equivalence point in these glasses at normal pressures (McKeown et al. 1984, 1985; Ohtani et al. 1985; McKeown 1987). Later studies indicate the presence of oxygen triclusters (Stebbins and Xu 1997; Toplis et al. 1997).

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One of the major advantages of MD simulations is the ability to design a simulation to test specific ideas that would be impossible to perform experimentally. In order to test the early interpretation that the experimentally observed extrema in viscosity and activation energy for sodium diffusion implied by a change in Al coordination from 4- to 6-coordinated, MD simulations were performed that biased the Al to remain 4 coordinated by placing the tetrahedral form of the 3-body potential (Eqn. 14a) on the Al ions (Zirl and Garofalini 1990). (Normally, one would use Eqn. 14c as the 3-body term to describe Al, since Eqn. 14c allows for both octahedral and tetrahedra coordination, as used in simulations of α- and γ-alumina (Blonski and Garofalini 1993).) Evaluation of the simulated glasses showed that the Al ions remained 4-coordinated, yet the glasses nonetheless showed the extrema in properties at the Al/Na ratio ~1.0 (Zirl and Garofalini 1990). Such results indicate that invoking the Al-coordination change is not necessary for generating the change in macroscopic properties. Other structural mechanisms were observed in the simulations that would affect the macroscopic property changes. These included the formation of O triclusters, closing of alkali-rich channels by Al, loss of NBO, and the formation of 3-membered ring structures. Combined, all of these could be used to explain the change in viscosity and activation energy for Na diffusion (Zirl and Garofalini 1990). Additional simulations of transport in alkali silicate glasses, usually at high pressures, have been presented (Stein and Spera 1995, 1996; Bryce et al. 1997, 1999). Vibrational studies (Kamitsos et al. 1994) support the presence of a large number of Si-O-Al bonds and some Al-O-Al bonds that were observed in the simulations. The formation of small rings in the simulations is interesting in that Al ions were found to preferentially form in the small 3-membered rings (three tetrahedra in a closed ring structure). Although Si ions still predominate in the 3-membered rings, more Al are found in the 3-membered rings than their concentration would warrant. Since 3-membered rings have smaller bond angles than the larger rings, this preference for Al to be in the 3-membered rings is consistent with ab initio calculations of equilibrium Al-O-Si bond angles and lengths (Geisinger et al. 1985). That is, while the average Si-O-Si bond angle in silica glass is near the upper 140°s, the Al-O-Si bond angle is in the upper 130°s. Since the T-O-T (T = tetrahedral cation) angle is ~135° in the 3-membered ring, the incorporation of Al into the ring costs less energy than having all Si in the ring. Molecular orbital calculations also support the preferential formation of 3-membered rings in aluminosilicate glasses (Kubicki and Sykes 1993). Simulations of more complex silicate glasses were performed. The composition of these E-type glasses are shown in Table 2. The Si-O, Al-O, and B-O PDF's are shown in Figure 12a-d. As a function of composition, the Si-O PDF shows the least variability while the B-O PDF shows the most change. The B-O PDF shows the presence of both 3and 4-coordinated B ions in the glasses, the concentration of which changes with composition. Figure 12d highlights the difference between a Table 2. Composition (mole %) of glass with nearly equivalent multicomponent glasses. amounts of Al and B (EG2) versus one with significantly Label SiO2 Al2O3 B2O3 CaO Na2O excess Al (EG4). With excess Al, EG 1 72 3 9 11 5 there are fewer 4-coordinated B, consistent with the idea that the EG 2 73 6 5 11 5 Al draws Na ions away from the EG 3 63 3 18 11 5 B, thus decreasing the concenEG 4 65 13 5 12 5 tration of 4-coordinated B.

 

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Figure 12a,b. (top; a) Si-O pair distribution functions for four multicomponent silicate glasses (see Table 2 for compositions). First Si-O peak unaffected by composition, but second Si-O peak changes slightly, with EG3 slightly shorter distance than other compositions. (bottom; b) Al-O pair distribution functions for four multicomponent silicate glasses (see Table 2 for compositions). Slight differences in structure with composition, especially in the second peak in EG3.

MD SIMULATIONS OF OXIDE GLASS SURFACES SiO2

Glass surfaces can be made from the bulk simulated glasses by removing periodic boundary conditions (PBC) in one dimension, as presented above (Fig. 4c). In the simulations performed in our lab, the PBC in the Z dimension is removed at 300 K and the system is run for a few ps in order to stabilize the truncated surfaces. The system is then heated to 1000-1500 K for several ps in order to enable relaxation of the surface atoms, followed by a cool down through intermediate temperatures to 300 K. Data are

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Figure 12c,d. (top; c) B-O pair distribution functions for four multicomponent silicate glasses (see Table 2 for compositions). First B-O peak shows contributions from both 3- and 4- coordinated B, with greater split in EG2 and EG3 than the other two glasses. (bottom; d) B-O pair distribution functions for two multicomponent silicate glasses (see Table 2 for compositions), showing effect of Al/B ratio on B coordination, with more 4-coordinated B in EG2 vs EG4.

collected at 300 K. Depending on the type of run and the questions to be answered, either one free surface or both free surfaces can be studied (see Fig. 13). In the former, the atoms on one side are frozen, thus creating one free surface of moving atoms (Fig. 13a). The upper surface in the figure would be analyzed and adsorbates or interface formation with other phases (gas, liquid, or solid) could be studied. The figure also shows a region of atoms called the “thermal sink'. In cases where gas phase atoms are allowed to adsorb onto the free surface, the heat of adsorption is pumped into the substrate atoms. While in a real system this energy would be dissipated throughout the rest of the substrate and sample holder, etc., in the simulations the energy must be specially removed. By rescaling the velocities of the atoms in the “thermal sink” to room temperature, the heat pulse that passes from the surface upon adsorption towards the bottom of the substrate is

 

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(b)

Figure 13. (a) Schematic drawing of simulated glass surface system with bottom 5-6 Å frozen, a thermal sink that has temperature of atoms in this volume rescaled to the desired T of the simulation, and gas phase adsorbate atoms above the free surface. PBC in the other two dimensions of a 3-D system. (b) Schematic drawing of simulated glass surface system with two free surfaces and gas phase adsorbate atoms on both sides the free surface. Although PBC are only shown in two dimensions, a 3D PBC could be used as long as the glass surfaces do not interact. This also allows the adsorbate atoms to exchange across the top and bottom of the figure.

removed. This enables an appropriate increase in thermal energy near the surface caused by adsorbate-substrate interactions, but it also allows for dissipation of the heat. In some cases, both surfaces (Fig. 13b) are used, such as in calculations of surface energies. Results of simulations of silica surfaces (Feuston and Garofalini 1989; Feuston and Garofalini 1990), with and without exposure to water molecules, are shown in Table 3, along with experimental data. While the data for “dry” surfaces is precise in the simulations in that there are no water molecules or hydroxyls present, in the experimental cases, the term “dry” means that the surfaces were dried to sufficiently high temperatures to remove water molecules and condense most silanol sites, although some isolated silanols may still be present. (Even though the “dry” ESR data were collected on samples fractured in UHV conditions, the presence of H2O in the silica is possible.) Thus, the simulated “dry” silica is not meant to model any real surface, since any such surface would contain hydroxyls (even high-fired silica glass surfaces that initially appear hydrophobic eventually hydroxylate). While 5- and 6-membered rings (5-6 tetrahedra per ring) dominate the simulated structure, small 2- and 3-membered rings and undercoordinated ions (NBO and 3-coordinated Si) are present in the dry simulated surfaces, consistent with the experimental data shown in Table III. No 2-membered rings exist in the simulated bulk silica (or seen experimentally, so their presence is induced by the formation of the dry surface. Exposure to water molecules in the simulations removes the most reactive sites,

150

Garofalini  Table 3. Concentrations of surface structures observed in simulations and experimental studies of silica after drying or exposed to moisture. Simulation

Experimental Technique

Dry Surfaces 2 member rings (0.13 /nm2) 3 member rings (4.2 Si3 memb /nm2) Si E's increase at surface

0.13 /nm2 FTIRa Ramanb consistent with ESRc

Wet Surfaces Removal of 2 member rings 3 member rings (2.1 Si3 memb /nm2) Removal of E's Geminal sites (~18%) Silanol concentration (3.4 /nm2)

0.04 /nm2 FTIRa 2.2-4.5 Si3 memb /nm2 Ramanb consistent with ESRc 15 - 20% NMRd 2 – 6 /nm2 IRe

a Bunker

et al. 1989 et al. 1990 c Antonini and Hochstrasser 1972; Hochstrasser and Antonini 1972 d Maciel and Sindorf 1980 e Iler 1979; Zhuravlev 1987 b Brinker

the strained small rings and saturates the undercoordinated ions, forming silanols (SiOH's). The higher reactivity of the small rings is consistent with ab initio calculations and experimental data regarding bond energy (and bond length) versus bond angle in silica (Gibbs et al. 1972). Smaller rings have smaller siloxane bond angles and are expected to be more reactive with moisture, as both experimental studies (Bunker et al. 1989a, 1990) and the simulations shown here indicate. The simulations show that the defect structure induced by the surface is localized to the top 7-8 Å, below which bulklike density and structure exist (Feuston and Garofalini 1989). Calculations of the surface energy, Es, of pure silica obtained in simulations in our lab gave a value ~1.2 J/m2. This energy was calculated from: Es = (Ef - Ei)/2A

(17)

where Ef is the total energy of the system with two free surfaces (Figs. 4c and 13b), Ei is the total energy of the bulk system (Fig. 4a), and A is the area of one surface. Figure 14 shows a snapshot of the surface topography of silica, in which the size of the O ions drawn in the figure is exaggerated in order to see the height differences between the O. (Si ions present in the system are hidden behind the O.) The atomistic surface roughness is apparent from the image, with openings into the subsurface caused by the ring structure of silica. An example of the ring structure is shown in Figure 15, where again atom sizes are not realistic in the image. Bonds are drawn between Si (small dots) and O (large dots) that are within 2.0 Å, consistent with the first peak in the PDF shown in Figure 1. Different size rings are apparent in the figure. The presence of the smaller ring structures in the glass surface alters the Si-O-Si bond angle distribution in comparison to the bulk distribution, as will be mentioned below. Adsorption of water molecules in the simulations showed reaction with strained siloxane bonds and undercoordinated defects, forming silanols (SiOH) via dissociative chemisorption (Feuston and Garofalini 1990; Garofalini 1990), as mentioned above.

 

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Figure 14. Top view of amorphous silica surface, with O ions drawn with exaggerated size for the image to highlight height differences between the oxygen. Si ions are hidden by the large O size. Atomistic roughness is easily discernable in the image.

Figure 15. Network connectivity of silica surface. Small light spots are Si, larger dark spheres are O. Sizes for image only. Bonds drawn between Si and O within 2 Å of each other. Note different ring sizes (4, 5, 6, and 7 membered rings are easily discernable).

Additional water molecules above the hydroxylated glass surface then physisorb onto these surface silanols via hydrogen bonding, but do not uniformly cover the surface, as shown in Figure 16. Rather, after the first few water molecules physisorb onto the silanols, additional water molecules hydrogen bond to these physisorbed molecules, forming water clusters (Fig. 16). Such behavior was previously inferred from experimental data (Anderson and Wickersheim 1964; Zettlemoyer et al. 1975). The behavior of these water clusters is important as two amorphous silica surfaces come into contact, either for technological reasons (such as wafer bonding technology (Stengl et al. 1989; Maszara 1992; Tong and Gosele 1994)), or for natural geological reasons. The inherent surface roughness will allow for a change in the distribution of water molecules between the surfaces as well as potential pathways into the subsurface. MD simulations (Litton and Garofalini 2001) have shown the penetration of water molecules into the

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Garofalini 

OS

HS

Hw Ow

Figure 16. Top view of hydroxylated silica surface with excess water molecules physisorbed on the surface. Sizes of ions are exaggerated in the image. OS are O on silica; HS are H at silanol sites; HW are H in water; OW are O in water. Note clustering of water molecules (see text).

subsurface as surfaces are brought together in the presence of water, as discussed below. Such water migration would have a potentially deleterious effect on fracture and dissolution behavior because of the effect of water on siloxane bond rupture. The normal ring structure of silica creates openings formed by larger rings that can create preferential sites (pathways) in the surface for migration of adsorbates into the subsurface. Several MD simulations of adsorbate/silica interactions addressing this have been reported (Levine and Garofalini 1986, 1988; Zirl and Garofalini 1989; Athanasopoulos and Garofalini 1992; Kohler and Garofalini 1994). Absorption into the subsurface was found to depend on the strength of the adsorbate/silica interaction parameters in the potentials. For relatively strong adsorbate/substrate interactions, little penetration occurred until after a contiguous thin film layer formed on the glass (Levine and Garofalini 1988). For less strongly interacting systems, such as inert gas adsorption onto silica, penetration was compositionally dependent (Kohler and Garofalini 1994). An example of migration into the subsurface is shown in the density profiles shown in Figure 17 (Kohler and Garofalini 1994). The figure shows the density profiles of Ne atoms that were placed as a gas phase above (a) a silica surface, (b) a sodium disilicate (N2S) glass surface, and (c) a sodium alumino-silicate (NAS) glass surface and allowed to adsorb at 77 K. The glass substrate density profile is not shown, but the surface is located at 0 in the figure, with the interior towards the left (negative direction). The Ne atoms and their interactions with the substrates were described by the LJ12-6 potential discussed above. After a run of over 300,000 timesteps, the final 100,000 timesteps were used to generate each density profile shown in the figure. While there is a build-up of Ne at the surface, located at 0 in the figure, Ne atoms are seen to penetrate nearly 10 Å into the silica substrate. This penetration required no siloxane bond rupture, but rather was dependent on the open structure inherent in amorphous silica. Since the addition of an alkali oxide to the glass opens the network structure by disrupting siloxane bonds, creating NBOs, it might be expected that penetration of Ne into the sodium disilicate glass would be greater than that in the pure silica. However, that is not the case, as shown in Figure 17b. Similar migration was observed in the sodium disilicate glass as for the

 

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153 

(a) SiO2

(b) N2S

(c) NAS -10

-5

0

5

10

DISTANCE FROM SURFACE (at 0) (Å) Figure 17. Density profiles of Ne atoms that were placed as a gas phase above (a) a silica surface, (b) a sodium disilicate (N2S) glass surface, and (c) a sodium alumino-silicate (NAS) glass surface and allowed to adsorb at 77 K. Glass surface located at 0, bulk to left (-10 direction), gas phase to right.

pure silica (Fig. 17a). This could be attributed to the role of the Na ions. While their presence in the glass allows for less network bonding and more NBOs, the Na ions themselves block the resultant open structure, preventing enhanced Ne penetration. Thus, there is a compensating effect of the alkali opening the structure, but also blocking it. This idea was tested using additional simulations of Ne penetration into a sodium trisilicate glass under two conditions: one in which the normal glass composition is used and the second in which the Na ions of the same glass were removed from the glass just prior to Ne adsorption. Since the simulations were run at 77 K, no relaxation of the glass network structure occurred to compensate for the removal of the Na ions. The resultant density profiles are shown in Figure 18. The curve labeled “original” shows the Ne profile on the stoichiometric sodium trisilicate glass while the curve labeled “Na depleted” shows the Ne profile on the same glass with Na ions removed. Clearly, Ne penetrates significantly deeper when the blocking Na ions are removed but the NBO remain. The sodium alumino-silicate glass shown in Figure 17c alters glass structure and reduces Ne penetration. The Al ions close the open structure by removing the NBO caused by the Na ions. The Na ions associate with the BO near the Al ions. Thus, the open structure caused by the Na ions (and NBO) is removed by the Al and yet the blocking effect of the Na ions remains. These two combine to inhibit Ne penetration (Kohler and Garofalini 1994). These open channels in pure silica caused by the normal ring structure may also allow for penetration of water molecules into silica (Litton and Garofalini 2001). Simulations of two hydroxylated silica surfaces coming into contact in the presence of water between the surfaces (shown schematically in Fig. 19) showed significant reactions between the water and the silica surfaces, increasing the concentration of silanols on the surfaces. This increased reactivity was a result of the increased pressure as the two

154

DENSITY

Garofalini 

ORIGINAL N3S

DISTANCE FROM SURFACE (0) (Å) Figure 18. Heavy line: density profile of Ne atoms that were placed as a gas phase above a sodium trisilicate glass surface (ORIGINAL) and allowed to adsorb at 77 K. Light line: same glass, but Na ions removed from the glass surface (Na depleted), showing greater penetration when Na ions removed. See text for details. Glass surface located at 0, bulk to left (−10 direction), gas phase to right.

SURFACE

SURFACE

SUBSTRATE PERIODIC BOUNDARY CONDITIONS Figure 19. Schematic of set-up of simulations of two silica surfaces being brought into contact with water molecules between them. The two surfaces are slowly brought together and reactions and bonding behavior are evaluated as a function of separation distance.

surfaces closed the distance between them without removal of the excess water molecules between the surfaces. This latter condition is relevant in processes where the water becomes trapped between surfaces as they come into contact, such as when the contacting surfaces are large and the water is not able to diffuse out of the edges. Technologically, wafer bonding of 10 cm wafers is such a case (Tong and Gosele 1998), but geologically relevant conditions could be envisioned. However, as the two surfaces get within 2 Å of each other, siloxane bonding between the surfaces occurs (which is the goal in wafer

 

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bonding), with an O ion attached to a Si ion from each surface. Figure 20 shows schematic drawings of the forces between the two surfaces as they are brought together as well as the number of siloxane bonds across the interface as a function of separation distance. At the farther distance, attractive forces between the surfaces occur due to the presence of moisture. At closer distances, repulsive forces dominant due to the compression of the water between the surfaces. The more water that is initially present between the surfaces causes an increase in the repulsive slope. The repulsive forces reach a plateau in all cases, concurrent with the formation of siloxane bonds across the interface. Water molecules play a complex role of both enhancing and limiting siloxane bond formation across the interface (Litton and Garofalini 2001). The simulations showed that water molecules react with the silica surfaces, increasing the concentration of Q2 Si species. These Q2 Si species are more labile than Q3 or Q4 sites because of the fewer bonds that the Q2 Si has to the rest of the silica network. The simulations showed, for instance, that a Q2 Si tetrahedron can rotate more freely around the axis of the two BO, enabling the NBOs at the Q2 site to sample a larger area than would a Q3 and thus increase the probability of finding an appropriate site to which to bond across the interface. Lower QN species enhance siloxane bond formation between the two surfaces. Thus, water molecules have the role of breaking siloxane bonds on a surface, but this also affords formation of other siloxane bonds across the interface. This role of water on siloxane bond rupture and bond formation is therefore more complex than might otherwise be anticipated.

Si-O-Si Bonds OR Force

In addition, water molecules were observed to migrate into the subsurface along an open channel, as shown in Figure 21. The figure was taken 2 ps after the first siloxane bond across the interface was formed. In this figure, Si and O in silica are not drawn, although the bonds between these ions are drawn. The large spheres that are drawn

FORCE

MORE MOISTURE

BONDS

REPULSIVE

ATTRACTIVE

Separation Distance (nm)

Figure 20. Schematic drawing taken from simulation data of five paired silica surface systems with different amounts of water between the surfaces. Drawing shows the force between the two surfaces and the Si-O-Si bond formation between the surfaces as a function of separation distance of the surfaces. Surfaces show initial attraction at longer distances (right) followed by repulsive forces at closer distances. More water between the surfaces causes the repulsion to increase more rapidly. Plateau in repulsive forces occurs at the same distance at which siloxane bonding across the interface occurs.

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Garofalini 

Figure 21. Snapshot of two silica surfaces in contact with water between them. Atoms in glass not drawn, although bonds between Si and O are drawn. The large spheres that are drawn represent the O in water, Ow, the small dark spheres are the H in water, and the small light spheres are the H that are at the silanol sites. Arrows denote interface where most water remains, but water diffuses in upper right into silica. (see text)

represent the O in water, Ow, the small dark spheres are the H in water, and the small light spheres are the H that are at the silanol sites. The arrows in the figure indicate the interface between the two surfaces where most of the water and hydroxyls are concentrated. However, there are clearly Ow and H that have diffused into the silica, especially along the upper right side. Evaluation of this behavior showed that the water had diffused into an open channel in the glass surface caused by the normal network ring structure of silica. Such migration of water into subsurface regions, albeit only a nanometer or so, is important in technologies such as wafer bonding (Stengl et al. 1989; Maszara 1992; Tong and Gosele 1994) and chemical-mechanical polishing. Another important feature observed in the simulations of adsorption onto silica surfaces was the effect of the adsorbate on the structure of the glass surface. Simulations of adsorption of model Lennard-Jones Pt atoms onto silica showed that the siloxane bond angle distribution of the silica surface shifted to smaller bond angles (Athanasopoulos and Garofalini 1992). Figure 22 shows the siloxane bond angle distributions averaged over the top 5 Å of five systems before and after adsorption of Pt atoms. There is an

AVERAGED OVER TOP 5Å OF SURFACE

Intensity

Before After

8

10

12

14

16

18

Si-O-Si Bond Angle Figure 22. Siloxane bond angle distribution before and after deposition of a multilayer amount of Pt atoms onto silica surfaces, averaged over the top 5 Å of five (5) surfaces. Shift to smaller siloxane bond angles caused by overlayer formation.

 

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157 

increase in the concentrations of smaller siloxane bond angles and a decrease in the concentration of the larger bond angles. Similar behavior was observed in simulations of adsorbate atoms described by the embedded atom method (EAM) rather than by the LJ12-6 potential (Webb and Garofalini 1994). The shift to smaller siloxane bond angles would imply an interface containing more reactive siloxane bonds, as discussed above. In order to test this, additional simulations were performed using a Lennard-Jones crystal that was brought into contact with the silica surface, slightly compressed, and subsequently removed, as shown in Figure 23 (Webb and Garofalini 1994, 1998). The results of the contact and slight compression (Fig. 24) were similar to those observed in the thin film formation from the gas phase adsorption (Athanasopoulos and Garofalini 1992). That is, the simulations showed that contact with the crystalline phase led to a shift to smaller siloxane bond angles (Fig. 24b). Exposure of the original glass surface to moisture in the simulations resulted in the preferential loss of smaller siloxane bond angles, as shown in Figure 25. This figure shows the bond angle distribution before and after exposure to water, as well as the difference curve (Fig. 25b). The simulations are consistent with the results of IR studies of silica surfaces (Parada et al. 1996). Shifts were observed in the IR spectrum for silica exposed to moisture that corresponded to a decrease in the bond angle distribution near 130° and an increase in the region near 150°, similar to the simulation results. No such change in bond angle distribution was observed in silica not exposed to moisture. The shift in the bond angle distribution observed in Figure 24b with the crystal in contact with the glass remained after the crystal was removed (Fig. 24c). After contact and removal of the crystal from the glass surface, the resultant silica surface was brought into contact with water molecules and, as anticipated, the siloxane bonds at the smaller bond angles showed preferential reaction with the water and rupture of these siloxane bonds (Webb and Garofalini 1998), as shown in Figure 26. The difference curve here (Fig. 26b) shows a much greater reduction of siloxane bond in the 130° region than seen in Figure 25b. Thus, contact with the crystal created a surface that was much more reactive with moisture. Microhardness tests of silica showed that both the crack initiation load and the Knoop hardness were constant with time in a dry non-aqueous environment, but

APPROACH

WITHDRAWAL

Z Y X Figure 23. Snapshots of LJ crystal brought into contact with a silica glass and then removed. PBC in two dimensions (X, Y), free surfaces in the third dimension (Z).

158

Garofalini  (c)

Figure 24. Siloxane bond angle distribution in the top 5 Å of a glass surface at three stages of interaction with an overlayer crystal, as shown in Figure 23: (a) before interaction, pristine glass surface, showing non-bulk-like bond angle distribution caused by small ring structures in the surface; (b) dotted curve, at contact with the crystal, showing the bond angle shift to smaller angles and loss of larger angles; (c) thickest curve, after removal of the crystal, showing a remnant effect and a resultant increase of smaller bond angles in comparison to the original surface.

(b)

(a)

SILOXANE BOND ANGLES

Intensity

b.)

Delta Curve Before Water After Water

a.)

100

110

120

130

140

150

160

170

180

Si-O-Si Bond Angle (degrees) Figure 25. Effect of exposing the original glass surface to moisture. (a) bond angle distributions before and after exposure to water. (b) difference between intensity of bond angle distribution before and after exposure (Iθafter − Iθbefore). Note loss of intensity at bond angles less than 140°.

decreased with time when exposed to ambient or water conditions (Hirao and Tomozawa 1987). The authors showed that this behavior was dependent on the presence of water. The simulations presented above are consistent with the effect of a Knoop hardness indentor in contact compression with a silica surface exposed to moisture. The results of the above-mentioned simulations have interesting implications with respect to the role of contact between silica and other solid phases. The change in the

 

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159 

Intensity

b.)

Delta Curve Before Water After Water

a.)

100

110

120

130

140

150

160

170

180

Si-O-Si Bond Angle (degrees) Figure 26. Effect of exposing the glass surface to moisture after contact with and removal of crystal (rightmost image in Fig. 23). (a) bond angle distributions before and after exposure to water. (b) difference between intensity of bond angle distribution before and after exposure (Iθafter − Iθbefore). Note significant loss of intensity at bond angles less than 140°.

structure of the silica surface alters its reactivity with moisture (and potentially other polar molecules), thus altering the overall properties of the glass phase. In addition, the equilibrium siloxane bond angle is ~132° when a H+ ion is adsorbed onto the bridging oxygen (Gibbs 1982; Geisinger et al. 1985; Edwards and Germann 1988). The bond energy is ~30 kcal/mol (Geisinger et al. 1985; Edwards et al. 1994). The increase in the concentration of siloxane bond angles in the 130°s below an adsorbate layer or after contact with a crystalline phase, as discussed above, potentially means an increase in sites that will adsorb H+ ions at strengths much less than the normal O-H bond (~100 kcal/mol), but much stronger than a hydrogen bond (~5-10 kcal/mol). This makes for interesting speculation regarding the acidity of such sites. Diffusion of Si and O on silica surfaces was studied and compared to such diffusion in the bulk (Litton and Garofalini 1997). Results are shown in Table 1, where simulation results for diffusion in the bulk and on the surface are shown, as well as experimental data. The surface diffusion coefficients were calculated using either 2- or 3-dimensional diffusion (α = 2 or 3, respectively in Eqn. 7). Figure 27 shows the density profiles of the Si and O ions at several temperatures in the glasses used in the simulations, with the surfaces near 20-25 Å. Mean square displacements of atoms that continuously have their z coordinates above a chosen z cutoff (see Fig. 27) were averaged over the amount of time above that coordinate. That is, it is possible for an atom to start above a chosen z cutoff, say 20 Å, but shortly move below that value and get trapped in a bulk site, or spend time deeper in the glass and move upwards above the z cutoff for some short time in the simulation. If the mean square displacement of these atoms over the whole run were included in the data regarding surface diffusion, they might skew the data inappropriately. Thus, by including only atoms that were above some chosen value of z for the time they are at these higher z (surface) values, we guarantee data associated with surface behavior. Mean square displacements of oxygen above three selected z cutoffs will be presented shortly.

160

NUMBER DENSITY (ATOMS/Å3)

Garofalini 

Z-COORDINATE (Å) Figure 27. Density profile of atoms in silica glass as a function of distance (Z) perpendicular to the free surface (which is nominally located near 20-25 Å) at three temperatures.

Figure 28 shows the Arrhenius plot for Si and O ions that had z coordinates above 20 Å (see Fig. 27) using 3 for α in Equation (7) (3D case). The diffusion coefficients are a factor of 2 to 3 greater than that in the bulk (see Fig. 28 vs. Fig. 8). Table 1 shows the resultant activation energies for these results as well as when α equals 2. The activation energies are slightly lower than the bulk data and the pre-exponentials (from Eqn. 15) are slightly higher. The main difference between bulk versus surface diffusion was the importance of overcoordinated species in bulk diffusion that was not significant in surface processes. Since rupture of the Si-O bond still dominated the process, the activation energies for bulk and surface diffusion were similar. However, the lower surface density (see Fig. 27), presence of the vacuum phase, and the greater concentration of undercoordinated O in the surface enabled larger jump distances in the surface; this increased Do in Equation (16), affecting D in Equation (15). The NBO in the surface would create a tetrahedron with only three bridging oxygen (BO) (Q3 Si) or two BO (Q2 Si) connecting it to the network, thus enabling greater motion than if it had four BO connecting it to the network (Litton and Garofalini 1997). Figure 29 shows the mean square displacements, MSD, for O ions that were above three different z cutoffs during the simulations. (Species that evaporated from the surface into the vacuum were not included in any of the data presented here.) For those O in the thickest volume (>18 Å), the MSD is fairly uniform for 50-60 ps. However, with thinner surface volumes, the MSD curves are quite different. For O above 20 Å and 21 Å, a large increase in the MSD occurs in each. These increases are due to the O that show significantly greater displacement than O farther below the surface (again, evaporating species are not included). While such O are present in the volume >18 Å, their concentration is sufficiently low that the MSD behavior of all O in this volume is dominated by the other O. However, in the thinner volumes above the 20 Å or 21 Å z cutoff, the longer time MSD is dominated by these O that show large MSDs. (Note that the short time MSDs are very similar in all three cases and the MSD curve used to generate Fig. 28 for O diffusion above 20 Å used the portion of the MSD curve for times

 

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Figure 28. Arrhenius plot of diffusion of Si and O ions in the simulated surface of silica. Results are averaged over Si and O that remain above a 20 Å cutoff in the Z dimension (see Fig. 26) for the last 90 ps of the 100 ps run.

18 > 18Å > 20Å > 21Å

16

MSD (cm2/s)(x10-15)

14 12 10 8 6 4 2 0 0

10

20

30

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50

60

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ELAPSED TIME (ps) Figure 29. Mean square displacement for O ions on the surface at 6000 K. Data calculated from O ions that remain above the cut-off in Z shown in the inset for the last 90 ps of the100 ps run over which the data were collected. The cut-offs pertain to the density profile at 6000 K shown in Figure 26. (see text)

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less than 30 ps in Fig. 29.) Thus, while short time displacements were similar, there are clearly large displacement mechanisms at work for O migration that are enhanced for those O that are at the outer surface, where density is lowest. If just these outer species were used in generating the diffusion coefficients, much larger values of D would be observed than those given in Figure 28 for surface diffusion. However, atoms only a few Å below the outer surface appear to have displacements fairly similar to the bulk. A more detailed discussion of these simulation results are available (Litton and Garofalini 1997). Multicomponent silicate surfaces Addition of alkali ions into silica results in preferential segregation of K and Na ions to the glass surface, with no such segregation by Li (Garofalini 1984, 1985; Garofalini and Levine 1985), consistent with experimental data (Kelso et al. 1983). Simulations of more complex silicates showed that cations with a higher cation field strength (ion charge/ion size) showed less surface segregation. For instance, while the larger alkali ions segregated to the surface, Li, alkaline earths (Ca and Mg), and the +3 and +4 cations (Al, B, Si) do not segregate to the surface. Figure 30a shows the density profile of a bulk multicomponent silicate glass containing ~60 mole % SiO2, ~20% Na2O, with the rest being boria, calcia, alumina, and magnesia. Figure 30b shows this system after formation of the free surface (Fig. 13a). Similar to earlier simulations and experimental studies mentioned above, O and Na ions relax outward slightly. At the outer surface (~22 Å), the Na/O ratio is much higher than in the bulk. As a comparison, the surface of a borosilicate glass shown in Figure 30c showed O and Na again moving outward, but no such relaxation for B, Si, or other higher cation field strength ions. In a study of leaching of glass surfaces, the high soda glass (Fig. 30b) was exposed to water at elevated temperatures (to assist in migration and reactions). Figure 30d shows the result on the distribution of O, Na, and H from the water at the surface. Also shown is the original density profile of the Na prior to exposure to water (Na start). Note the migration of Na and O into the water (delineated by the peak in the H profile at the glass surface), indicating leaching of these species with exposure. H ions migrate into the interior of the glass, apparently replacing the Na that have leached out. Additional simulations of this system will be continued for further analysis. Reviews of weathering of silicates has been presented (White and Brantley 1995). SUMMARY Results of molecular dynamics computer simulations of silica and silicate glasses and glass surfaces have been presented, along with a basic methodology for performing such simulations. The simulations have been shown to reproduce many of the important features of these systems. A multibody potential that includes both 2- and 3-body terms was presented and discussed in terms of its applicability for providing reasonable results for conditions ranging from molecules, bulk glasses, and glass surfaces, with varying compositions. Many of the simulation results build upon the work of earlier ab initio calculations of small molecular systems and have been corroborated by additional experimental and computational studies. Pentacoordinated Si intermediates are observed in reactions of water with silica, or between silicic acid molecules, or during diffusion. The ring structure of silica enables penetration of adsorbates into the glass under specific conditions, with composition and pressure affecting penetration. The glass surface structure also changes upon adsorption of overlayer species or interface formation, such that the surface becomes more reactive with moisture. Such results have been used to

 

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(a)

O

(b)

O

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Vacuum

Si

Si

Na

Na

Al, Ca, Mg...

Al, Ca, Mg... 10

20 DISTANCE (Å)

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Na (start) B Al, Ca, Mg... 10

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Na

Na

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DISTANCE (Å)

Figure 30. Density profiles of species. (a) A multicomponent silicate bulk glass containing significant amount of Na. (b) The surface of the multicomponent glass shown in (a) Note migration of O and Na towards vacuum, while higher charged species (Si, Al, etc.) show no such segregation to outer surface. (c) The surface of a multicomponent glass that contains a significant amount of B. Note O moves outward towards vacuum, while Si and B do not. Of the minor components, only Na shows migration towards surface. (d) O and Na species in the surface of the multicomponent glass after exposure to water. Only H from water molecules shown. “Na start” shows the original Na distribution in this glass surface prior to water exposure (see b).

30

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explain recent experimental results of Knoop hardness studies. Preferential surface segregation of specific species, similar to experimental data, has been observed in the simulations. Such behavior affects surface structure, crack initiation, and leaching behavior. The past and current series of molecular dynamics simulations, either those predominantly discussed here or the many others using other potential forms, have provided useful insight into molecular behavior in glass and crystalline systems. However, newer and better techniques will come forward with the advance of faster computers and more accurate interatomic potentials. While ab initio techniques will also advance in kind, the use of the more simplified models that incorporate the most important features of a system of interest will enable reasonably accurate simulations of much larger systems (O(109-1012)) or longer time frames than currently available. Such large scale calculations will then fit much more closely to the experimental world and provide better links to experimental data and, more importantly, data interpretation. ACKNOWLEDGMENTS Much of the data presented here comes from the excellent work of the post-doctoral associates and graduate students in our laboratory. Support from the Department of Energy, Office of Basic Energy Sciences is greatly appreciated. Additional support from the Army Research Office, Corning, Inc., the New Jersey Commission on Science and Technology, and the Center for Ceramic Research at Rutgers University are also gratefully acknowledged. REFERENCES Alder BJ, Wainwright TE (1957) Phase transition for a hard sphere system. J Chem Phys 27:1208-1209 Alder BJ, Wainwright TE (1959) Studies in molecular dynamics I. General method. J Chem Phys 31:459466 Allen MP, Tildesley DJ (1987) Computer simulation of liquids. Oxford University Press, New York Anderson JH, Wickersheim KA (1964) Near infrared characterization of water and hydroxyl groups on silica surfaces. Surf Sci 2:252-260 Angell CA, Cheeseman PA, Tamaddon S (1982a) Pressure enhancement of ion mobilities in liquid silicates from computer simulations studies to 800 kilobars. Sci 218:885-887 Angell CA, Cheeseman PA, Tamaddon S (1982b) Computer simulation studies of migration mechanisms in ionic glasses and liquids. J de Physique C9:381-385 Antonini J, Hochstrasser G (1972) Surface states of pristine silica surfaces II. UHV Studies of the CO2 adsorption-desorption phenomena. Surf Sci 32:665-686 Athanasopoulos DC, Garofalini SH (1992) Molecular dynamics simulations of the effect of adsorption on SiO2 surfaces. J Chem Phys 97:3775-3780 Batirev IG, Alavi A, Finnis MW, Deutsch T (1999) First principles calculations of the ideal cleavage energy of bulk niobium(111)/alpha-alumina(0001) interfaces. Phys Rev Let 82:1510-1513 Belonoshko AB (2001) Molecular dynamics simulation of phase transitions and melting in MgSiO3 with the perovskite structure-comment. Am Mineral 86:193-194 Belonoshko AB, Dubrovinsky LS (1996) Molecular dynamics of NaCl(B1 and B20 and MgO(B1) melting: two-phase simulation. Am Mineral 81:303-316 Berendsen H, Postma J, van Gunsteren W, DiNola A, Haak J (1984) Molecular dynamics with coupling to an external bath. J Chem Phys 81:3684-3670 Blonski S, Garofalini SH (1993) Molecular dynamics simulations of α-alumina and γ-alumina surfaces. Surf Sci 295:263-274 Brawer S (1981) Defects and fluorine diffusion in sodium fluoroberyllate glass: A molecular dynamics study. J Chem Phys 75:3516-3521 Brawer SA (1985) Relaxation in viscous liquids and glasses. American Ceramic Society, Columbus, OH Brebec G, Seguin R, Sella C, Bevenot J, Martin JC (1980) Diffusion du silicium dans la silice amorphe. Acta Metall 28:327-333

 

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Molecular Models of Surface Relaxation, Hydroxylation, and Surface Charging at Oxide-Water Interfaces James R. Rustad W.R. Wiley Environmental Molecular Sciences Laboratory Pacific Northwest National Laboratory P.O. Box 999, MSIN K8-96 Richland, Washington, 99352, U.S.A. INTRODUCTION

Increasingly sophisticated experimental techniques are resolving detailed aspects of the surface chemistry of oxide and silicate materials. Structural characterization can now be carried out on remarkably complex systems (Brown et al. 1999). Recent examples include the distribution of iron in dioctahedral smectites (Manceau et al. 2000), the arrangements of protons on the hematite (012) surface (Henderson et al. 1998), the relaxation of iron atoms at the surface of hematite (001) (Thevuthasan et al. 1999), the arrangements of defects in the γ-Fe2O3 corrosion film formed on metallic iron (Ryan et al. 2000), the existence of two terminations of magnetite (001) (Stanka et al. 2000), and the structure of the Cr(III) passivation layer formed on magnetite as a result of magnetiteinduced reduction of aqueous Cr(VI) (Peterson et al. 1997). Mesoscale structural studies include measurement and quantification of surface morphology using scanning probe and x-ray scattering methods (Eggleston and Stumm 1993; Weidler et al. 1998a,b). Similar developments are taking place in the measurement of reaction energetics and kinetics, such as temperature programmed desorption studies of the binding energies of water molecules on oxide and sulfide surfaces (Bebie et al. 1998; Stirniman et al. 1998; Henderson et al. 1998; Peden et al. 1999), the binding of phosphate on hematite (Nooney et al. 1996), and the measurement of surface energies through high resolution calorimetric investigations (McHale et al. 1997; Laberty and Navrotsky 1997). As these developments allow investigation of surfaces on increasingly small scales, it becomes more difficult to collect these experiments together into a coherent model, to link them with experiments on macroscopic systems in the laboratory (Sposito 1999), and bring the results to bear on complex, multicomponent natural systems. For example, measurement of the amount of surface relaxation taking place on vacuum-terminated hematite (001) gives some insight into the chemical characteristics of the energies and forces associated with the Fe-O bond. This information is highly specific to the particular structural environment in which this bond is formed. It is difficult, for instance, to draw any conclusions about the arrangements of protons on hematite (012) from knowledge about the shortening of the surface Fe-O bond on cleavage of vacuum-terminated hematite (001), even though the latter observation in principle forms an important constraint on the former. Furthermore, knowledge of the proton distribution on monolayer-hydroxylated hematite (012) is obviously valuable in understanding its surface charging behavior, yet one cannot directly use the proton speciation information determined in (Henderson et al. 1998) to predict the surface charge as a function of pH. Other questions of this nature are: Is the high threshold pressure for hydroxylation of corundum relative to hematite (Liu et al. 1998) related in any way to the lower acidity of Al(H2O)63+ relative to Fe(H2O)63+ in aqueous solution? How do the observations of the structural characteristics of reduced nontronites worked out by Manceau et al. (2000) 1529-6466/01/0042-0006$05.00

DOI:10.2138/rmg.2001.42.6

170

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affect the interpretation of the XPS experiments of Ilton et al. (1997) on the oxidation of biotite by chromate? Computer simulation can play an important role in answering these types of questions, particularly in forming the important connection between high-vacuum surface science techniques and sorption in solvated systems. Computer models can be used to calculate the amount of relaxation on vacuum-terminated hematite (001) (Wasserman et al. 1997; Wang et al. 1998), the difference in acidity between Fe(H2O)63+ and Al(H2O)63+ (Rustad et al. 1999a), and potentially, the energies of the various proton and iron(III) distributions proposed for reduced nontronite. Then, if these calculations agree with experimental observations, the same models can, at least in principle, be used to simulate more complex processes, such as the distribution of protons on hematite (012), the hydroxylation pressure of hematite and corundum, and the oxidation of biotite by chromate. In practice, computational power limits (possibly severely) the length and time scales available to simulations (see Harding 1997 for an excellent review). However, there is little doubt that simulation is the only realistic path toward answering such questions, and it is playing an increasingly important part in the unification and correlation of the results of these complex experiments as well as providing a major avenue between these molecular scale experiments and thermodynamics (Blonski and Garofalini 1993; McCarthy et al. 1996; McCoy and Lafemina 1997; Gibson and LaFemina 1997; Felmy and Rustad 1998; Hartzell et al. 1998; Henderson et al. 1998; Rustad et al. 1999b; Brown et al. 1999; Halley et al. 1999; de Leeuw et al. 1996, 1998, 2000; Brown et al. 2000; Rustad and Dixon 2001). SCOPE First it should be pointed out that this review is focused primarily on examples drawn from our own research group, and does not constitute a comprehensive review of the literature. Good introductions to molecular dynamics methods abound (Frenkel and Smit 1996; Allen and Tildesley 1989). An exceptionally clear discussion can be found in Bennett (1975). Reviews of general aspects of computational modeling of surfaces and interfaces with a geochemical emphasis are provided in Gibson and LaFemina (1997) or Rustad and Dixon (1998). This review is meant to complement those articles by considering some dominant issues arising in our research on solvated/hydroxylated oxide surfaces. The discussion of these specific issues likely will be useful to those designing similar research programs focused on inorganic aqueous interfacial chemistry. Molecular simulation has a broad scope, ranging from calculation of the trajectory of a 100 atom system for a few picoseconds using direct dynamics with ab initio forces (Car and Parrinello 1985), to systems with thousands of atoms simulated for a few nanoseconds (deLeuuw et al. 1996, 1998; Wasserman et al. 1997; Jones et al. 2000), to large-scale molecular dynamics studies using millions of atoms (Vaidehi and Goddard 2000; Campbell et al. 1999) to kinetic Monte Carlo (McCoy and LaFemina 1997) and dissipative particle dynamics simulation of crystal growth over length scales of microns (Dzwinel and Yuen 2000; see Yuen and Rustad 1999 for additional references). All of these categories are concerned with simulation, meaning that they are essentially numerical experiments, subject to statistical uncertainties, careful control of experimental conditions such as pressure/stress, temperature, and composition, and concerns about reproducibility and independence of initial conditions. In this review, we focus on the thousand atom/nanosecond timescale range. For convenience, we refer to these types of simulations as "classical" MD. "Parameterized" MD is actually a better term; such simulations can treat the dynamics quantum mechanically if so desired (Strauss and Voth 1993).

 

171

Molecular Models of Oxide‐Water Interfaces 

One reason for this focus is that there have been very few simulation studies on systems of geochemical relevance outside this scale. Partly, this lack of studies results from the fact that the thousand atom nanosecond timescale studies on interfacial systems are still in infancy themselves. In some sense, this ground must be understood thoroughly before studies at other scales can be justified/directed. For example, a picosecond direct dynamics simulation of a model mineral-water surface is not sufficient to define any interesting energetic quantities, or even structural averages, that can be regarded as independent of the initial conditions of the numerical experiment. These initial conditions must therefore be chosen carefully, with some knowledge of the approximate behavior of the system. The shorter the timescale of the simulation, the greater the knowledge required to define the initial conditions. At the other extreme are kinetic Monte Carlo studies in which molecular degrees of freedom are eliminated and the simulation parameters are chosen to represent the frequencies of “fundamental events” such as the probabilities of attachment of an atom at various surface sites presumed to exist at the interface. At both ends of this spectrum, the influence of the practitioner on the simulation results is high: at the lower end, through the choice of initial conditions; on the upper end, through the choice of event frequency parameters. Both of these techniques can be very effective but both benefit from the information generated at the classical MD scale to reduce the arbitrary aspects of the practioner influence. As computers evolve, these considerations will be modified; one can imagine the Car-Parrinello methods at some point will completely encompass the parameterized MD studies at the length and time scales under consideration here, particularly as pseudopotentials are treated using more general methods (Bloechl 1994; Vanderbilt 1990). But parameterized methods have survived the decade of vector to parallel revolution without being pushed into irrelevantly large system sizes. With the increasing recognition of role nanoscale heterogeneity in controlling structure-reactivity relationships (Weidler 1998a,b; Zhang et al. 1999), parameterized methods will likely survive through the next decade as well. As a first step, the simulation of a mineral-aqueous interface requires treatment of the issue of surface hydroxylation, which is fundamentally tied to the dissociation of water and the energetics of acid-base reactions on mineral surfaces (Blesa et al. 2000). Even just setting the problem up requires some knowledge of the protonation states of the oxide ions at the surface; are they aquo, hydroxo, or oxo functional groups? If one cannot describe the processes behind Figure 1, it is not possible to go further. This description is (a)

(b)

H

H

O

M

O

M

O

M

O

M

O

M

O

M

O

M

O

(c)

H

H

O

H

O

O

H H

O

H

O

M

O

M

O

O

M

O

M

O

M

O

M

O

M

M

O

M

O

M

O

M

O

M

O

O

M

O

M

O

Figure 1. Schematic illustration of terminology for oxide surfaces used in this paper (a) vacuumterminated surface, (b) hydrated surface with no dissociation of water, (c) hydroxylated surface with adsorbed water fully dissociated.

172

Rustad 

the major bottleneck in applying molecular dynamics methods to the aqueous mineral interface. There are potentially many ways to treat this problem using analytical force fields. The simplest is to leave the surface speciation issue to empirical methods. There are empirical estimates of surface pKas even for multisite models of surface protonation (Hiemstra et al. 1996). One could use these methods to estimate a population of surface sites, then use a well-understood water model, such as SPC/E or TIP4P to solvate this surface. The advantage of this approach is that we can effectively bypass the problem of surface hydroxylation and focus on larger-scale problems, such as the structure of the electric double layer using a water model known to give good transport and dielectric properties and which already has been used in the parameterization of alkali and alkaline earth cations and several anions. The main argument against this approach is that while the empirical methods can be effective in representing or fitting macroscopic phenomena such as surface charging, they may have little basis in molecular reality. It would be dangerous to use “fitted” molecular species from a thermodynamic analysis as a literal representation of molecular level structure. Given that the primary influence on the electric double layer will come from the speciation of the proton, the most important potential-determining ion, conclusions drawn about the interface without a firm picture of the solid side of the hydroxylated interface would be of limited defensibility. Halley et al. (1993) following Stillinger and David (1980) have attempted to treat the water dissociation problem explicitly using the potential functions. In this approach, the O-H interaction is treated like any other interaction, with a finite range and a meaningful well depth. For example, a collection of O2- and H+ ions, if placed at randomly in a computational cell, should give rise to something like water. The water molecule forms as a result of the potential and is not put in explicitly a priori. The rest of this review is a discussion of the results of applying the Stillinger-David water model to problems related to hydroxylated mineral surface structures and energies. THE STILLINGER-DAVID WATER MODEL Stillinger and David (1980) developed an ionic model for water that was capable of dissociation into ions. In addition to being the only dissociating parameterized model for water, this was also one of the first polarizable water models. The explicit incorporation of oxygen polarizability allowed the investigation of charged clusters composed of H+, OH-, and water molecules. Stillinger and David (1980) used the model to study small ionic clusters of H+-OH--H2O including H5O2+, H3O2-, and the water octamer. The model uses formal charges of +1 on the proton and -2 on the oxide ion, ensuring the existence of a meaningful reference energy for the reaction H2OyH+ + OH-. Note that dissociation into ions (also called heterolytic dissociation) is certainly not favored in the gas phase in the absence of aqueous solvent. However, because the model is designed for problems in aqueous chemistry, it was built around the hypothetical heterolytic dissociation energy, which is taken to be 395 kcal/mol. This limits the model to applications involving ionic dissociation (for a parameterized model that reproduces both the homolytic and heterolytic dissociation energies, see Corrales 1999). Besides the dissociation energy, the parameters are chosen so that the equilibrium structure of the water molecule is good by construction. The ions retain their charges in molecular water, shown in Figure 2. Keeping these charges would result in a model with a very large dipole moment. If these charges are maintained with no other modifications, the dipole moment of a gas-phase water molecule with rOH = 95.84 pm and θ0 = 104.45˚ will be 5.64 D. This dipole moment, as

 

Molecular Models of Oxide‐Water Interfaces 

173

defined by the positions of the ions, is reduced by the polarization of the oxide H2Og → H+g + OH-g ; 395 kcal/mol ion. The interesting feature of the H Stillinger-David model is the fact that H +1 + 104.45° +1 the H ions polarize the oxide ion, even within the molecule. In general, the polarization of an ion is given by μ = α E , where E is the electric field μind 0.9584 Å and α is the polarizability of the ion. For O -2 convenience, Stillinger and David took the polarizability of the oxide ion to be 1.444 Å3, the experimental polarizability μ=1.85 D of the water molecule, although the exact role of the polarizability in the α=1.444 Å3 model is different from that in the experimental measurement. If the oxide Figure 2. Stillinger-David water model showing formal charges, induced point dipole on oxygen, ion were subject to the full polarizing equilibrium geometry, overall dipole moment, and power of two protons at a distance of dissociation energy. 95.84 pm, the induced dipole would be too large (the total dipole moment, pointing in the opposite direction, would be too small). The idea of the Stillinger-David model is to reduce the OH charge dipole interaction by a function S(r), which approaches 1 at long distances but cuts off the interaction at short distances. Physically, the cutoff function S(r) can be rationalized by covalent bond formation at short range; as the hydrogen ion takes on some electron density within the water molecule, its polarizing power is reduced. The value of the cutoff function S(r) at the equilibrium OH separation in the water molecule is fixed by the dipole moment of water, 1.85 D. This requires that S(re)=0.4091. The derivatives of S(r) at re were estimated by Stillinger and David from experimental information and used to fix other parameters appearing in the functional form for the OH interaction. Note that the reason water is bent in the Stillinger-David model is the reduction in energy of the OH charge dipole interaction obtained by the polarization of the oxide ion. In its original formulation, separate functions were used for the force and the energy, that is, the force was not equal to –grad V. Halley and co-workers (1993) modified the Stillinger-David model to be dynamically consistent. They also added short-ranged bond bending three body interactions to recover exactly the vibrational frequencies of the model. The major modification, however, was in the O-O interaction. In the original Stillinger-David model, the O-O interactions were fixed by what was then known about the water dimer. In the model of Halley et al. (1993), the O-O interactions were fitted to neutron scattering data for the O-O radial distribution function at room temperature (Soper and Phillips 1986). The functional form of the modified Stillinger-David potential is as follows

Φ OO =

3r r 1 AOO BOO qO qO qO ( μ ⋅ rOO ' ) 1 + 6 + + + μO ( I − OO2' OO ' ) μO ' ∑ 3 O ∑ O ' 12 2 rOO ' rOO ' rOO ' rOO ' 2 rOO '

(1)

where rij = ri − rj , I is the 3×3 unit matrix, and μO is the dipole vector on the oxygen. The functional form of the O-H interaction is

174

Rustad  Φ OH = ∑ O ∑ H aOH

e( − bOH rOH ) −e (r −r )2 + {cOH (rOH − rOOH ) 2 − dOH (rOH − r0OH )} e OH 0OH rOH

q q q (μ ⋅ r ) + H O + H O3 OH SOH (rOH ) rOH rOH

(2)

where S (r ) =

r3 r 3 + f (r )

(3)

and f OH (r − rO )e − gOH ( r − r0 ) + hOH e − pOH r f (r ) = 1 + e sOH ( r − tOH )

(4)

H-H interactions are purely coulombic: Φ HH =

1 qH qH ∑ ∑ 2 H H ' rHH '

(5)

For water molecules, there is a three-body term of the form 1 ⎧ ⎫ 2 ⎪aHOH ( rOH − r0 )(rOH ' − r0 ) + bHOH (θ − θ 0 ) ⎪ eHOH ⎡⎣⎢( rOH −r0OH )2 +( rOH ' −r0OH )2 ⎤⎦⎥ Φ HOH = ∑ O ∑ H ∑ H ' ⎨ 2 ⎬e ⎪⎩+cHOH (rOH + rOH ' − 2r0 )(θ − θ 0 ) + d HOH (θ − θ 0 ) ⎪⎭ (6) where r0OH and θ0 are the desired bond length and bond angle in the water molecule. These parameters are chosen such that the isolated water molecule has the correct vibrational frequencies. Table 1 lists the potential parameters in units of e (charge), Å (length), and e2/Å (energy). The dipole moment, dissociation energy, and equilibrium structure are used to construct the model. OH

OH

OH

The model then was used to predict the gas-phase proton affinity of water, which played no part in the model parameterization. This is a fairly rigorous test, as one is faced with computing the energy of a covalent bond based on an ionic parameterization of the structure and dissociation energy of water into ionic fragments. The proton affinity of water, involving an association between a charged species and a neutral species, is much harder to reproduce, for an ionic model, than the energy of dissociation into ions. The modified Stillinger-David model predicts 163 kcal/mol, which is about 8.6 kcal/mol lower than the ΔEelec at the coupled cluster level and at the complete basis set limit (Peterson et al. 1998). This is remarkably good considering the simplicity of the model. The H-O-H angles in H3O+ are predicted to be nearly 100˚, whereas the angles calculated using the best MO methods are 110˚. The hydroxide affinity of water, the energy for the reaction H3O2− y H2O + OH−, is 38.1 kcal/mol. This compares with 37.6 at the DFT BP/DZVP2 level. It is interesting that an essentially ionic model gives such reasonable results for minimum energy structures and energetics. IRON-WATER AND SILICON-WATER POTENTIALS AND THE BEHAVIOR OF Fe3+ AND Si4+ IN THE GAS PHASE AND IN AQUEOUS SOLUTION

As a first step in extending the Stillinger-David approach to mineral systems, an FeO and Si-O potential was introduced (Rustad et al. 1995; Rustad and Hay 1995). The basis for parameterization of the Fe-O potential was the Fe3+-H2O potential surface calculated in Curtiss et al. (1987). The surface is given in Figure 3.

 

175

Molecular Models of Oxide‐Water Interfaces  Table 1. Parameters for the molecular statics model of the magnetite surface. AOO BOO

2.02 1.35

aOH bOH cOH dOH eOH fOH gOH hOH pOH sOH tOH r0OH

10.173975 3.69939 -0.473492 0.088003 16.0 1.3856 0.01 48.1699 3.79228 3.0 5.0 0.9584

qH qO qFe qFe

1+ 23+ tet. sites 2.5+ oct. sites

aHOH bHOH cHOH dHOH eHOH θ0HOH

-0.640442 0.019524 -0.347908 -0.021625 16.0 104.45˚

AFeO BFeO CFeO DFeO EFeO FFeO

1827.1435 4.925 -2.136 -74.680 1.0 1.8

α 1.444 Å3 note that the short-ranged FeO parameters are the same for 2.5+ and 3+ sites.

Note: When used in conjunction with Equations 1–9, energies in e2/Å are generated. For reference, the water molecule at equilibrium geometry has an energy of 3.11595 e2/Å; the Fe(H2O)3+ complex has Fe-O = 1.8506 Å, an induced dipole moment μO of 0.232222 eÅ (positive side toward the Fe), and a binding energy of 0.48598 e2/Å relative to Fe3+ and H2O; the hexaaquo Fe(H2O)63+ complex has Fe-O=2.0765 Å, μO=.577682 eÅ (positive side toward the Fe), and a binding energy of 2.124873 e2/Å relative to Fe3+ and 6 H2O.

-20

Eb (kcal/mol)

-40 -60 -80 -100 Curtiss et al (1987)

-120

model

-140 -160 1.5

2

2.5

3

3.5

4

4.5

5

r(Å) 3+

Figure 3. Fe -H2O potential energy surface from Curtiss et al. (1987), and fit from parameterized Fe-O potential function. Eb is the binding energy defined as E(FeH2O3+)-E(H2O) (the energy of Fe3+ is zero in the context of the ionic model).

176

Rustad  For the Fe-O interactions, the functional form is Φ FeO = ∑ Fe ∑ O AFeO e − BFeO rFeO −

CFeO DFeO qFe qO qFe ( μO ⋅ rFeO ) S FeO (rFeO ) + 12 + + 3 rFeO r6 rFeO rFeO

(7)

where S FeO (r ) = 1 −

1 e

EFeO ( r − FFeO )

(8)

For the gas phase Fe(H2O)63+ ion, the Fe-O bond length is 207 pm, a bit longer than the range of experimental distance of 200-205 pm, with the lower part of the range being the most reliable, having been determined from x-ray diffraction on Fe3+ cesium alum (Beattie et al. 1981). Thus one is faced with the question of whether to refit the potential, including not only information from the ab initio surface, but also the Fe-O bond distance in the hexaaquo ion. Several issues govern this type of decision. For one, the Fe-O distance is measured in aqueous solution (or, in the case of the Cs alum, in a condensed phase), not in the gas phase. For another, QM calculations on Fe(H2O)63+ complexes available at the time also gave Fe-O distances that were too long at 206 pm (Akesson et al. 1994). If the Fe-O distance in the gas phase hexaaquo complex is in error, there is no way of knowing whether the problem is in the Fe-O potential or the water-water potential. Given this second issue, little guidance could therefore be obtained from the quantum mechanical studies in sorting out the right way to improve the model. Furthermore, there is a very strong motivation for keeping the O-O interactions the same for the hexaaquo ion as they are for bulk water. Sacrificing this assumption is unwise as the O-O interaction then becomes dependent on coordination environment, and there will be plenty of instances where there will be no basis at all for assigning coordinationdependent parameters, for example, on surfaces. Taken together, these factors suggest that the simplest approach be used and that one should retain the fit to the ab initio surface, leaving the Fe-O bond length in the hexaaquo complex as a prediction of the model based on the Fe3+-H2O potential surface of Curtiss et al. (1987). The Si-O potential was introduced by fitting to the structure and vibrational frequencies of H4SiO4 as calculated by Hess et al. (1988). The gas-phase acidities of both models were then assessed. The energy required to remove a proton from H4SiO4 is 355 kcal/mol, in good agreement with recent DFT calculations (Ferris 1992; Rustad et al. 2000a). The energy required to remove a proton from hexaaquo ferric iron is 40 kcal/mol. This is quite a bit higher than the range of values calculated from ab initio theory 20-30 kcal/mol (Rustad et al. 1999a; Martin et al. 1998), but very close to the value calculated for Al(H2O)63+ using a variety of ab initio methods. The correlation with size-charge ratio for these ions is known to be poor (witness the lesser acidity of the smaller Al3+ ion). Despite the fact that DFT calculations were shown to correlate very well with acidities, the electronic structural reasons for the trends observed in the acidities of the trivalent ions are unclear. Until the underlying reasons for these trends are better understood, it seems we should accept the Al3+-like value of 40 kcal/mol, realizing that we are not now really talking about “iron” but a model trivalent ion. More complex molecules may be used as surrogates for surface sites on oxides. Consider the Fe3(OH)7(H2O)62+ molecule in Figure 4. This molecule, the simplest polynuclear cluster having an Fe3OH functional group, was optimized by Rustad et al. (2000b) using density functional theory. The deprotonation energy calculated using the generalized gradient approximation was 179 kcal/mol. Calculations using the molecular

 

Molecular Models of Oxide‐Water Interfaces 

model presented above also gave 179 kcal/mol, showing that the model is capable of giving highly accurate results and is properly accounting for the influence of multiple Fe-O bonds on the acidity of the OH functional group. This provides an immediate point of assessment not available to empirical models such as the MUSIC model, which, because no information is available on the acidities of solution phase functional groups bound to more than one metal, implicitly assumes the additivity of Pauling bond strengths on the surface hydroxide functional groups. 4+

3+

177

Figure 4. Fe3(OH)7(H2O)62+ cluster used in the ab initio calculation of the gas phase acidity of the Fe3OH functional group. Large shaded atoms are irons, spotted atoms are oxygen, white atoms are hydrogen.

and Fe potentials Both the Si yielded qualitatively reasonable behavior when used in a simulation of a single aqueous ion in a solution of 216 water molecules. For the Fe3+ system, the potential gave a six-fold coordinated aquo ion Fe(H2O)63+. In contrast, the Toukan-Rahman flexible model (Toukan and Rahman 1985) gave an eightfold coordination for the Fe3+ in solution when fit to the Fe3+-O interaction fit to the Fe3+-H2O surface calculated by Curtiss et al. (1987). The improvement in the modified Stillinger-David model is presumably due to the better transferability of the polarizable model. The dipole moment of the Toukan-Rahman water model (2.3 D) is an effective dipole moment that would be changed if the water is not in an environment typical of bulk water. The charge-dipole interaction between the Fe3+ ion and the coordinating oxygen nearly nullifies the induced moment, greatly increasing the net dipole moment of the coordinating water molecules. These kinds of many-body effects are important in stabilizing the hexaaquo complex in solution. The Si4+ when placed in the 216 water system immediately hydrolyzed four water molecules to make an orthosilicic acid molecule and four hydronium ions. This is reasonable, considering that Si4+ is so acidic that its pKa is not measurable and, at a pH of about 1.7, the orthosilicic acid molecule should not be protonated. CRYSTAL STRUCTURES

Having established that reasonable results are obtained for the aqueous ions using the simple potential functions, the natural next step is to try the potential functions on crystal structures. This was carried out for goethite, lepidocrocite, akaganeite, and hematite in the Fe3+-O-H system for the structures by Rustad et al. (1996a). Just using the simple parameterization from the single Fe3+-H2O surface, quite good results were obtained with stable crystal structures for all the FeOOH polymorphs as well as hematite. As with the hexaaquo ion, the Fe-O bonds were about 5% too long. As for the hexaquo complex, it could not be determined whether the long Fe-O bond lengths were primarily a result of the Fe-O interaction being incorrect or the O-O interaction being incorrect. It might be supposed that the relative energies of the different polymorphs could be used to further test the model. Recent calorimetric studies have provided valuable information on the relative energies of the FeOOH polymorphs. These energies are compared in Figure 5 with calculated potential energies at the optimized lattice and fractional coordinates both for the model derived in Rustad et al. (1995) and also from density

178

Rustad 

a) Structures

guyanaite lepidocrocite

grimaldiite

goethite

akaganeite

b) Relative Energies

16

kcal/mol

gri

14 12 10

Laberty and Navrotsky (1998)

Ab Initio

80 60

Parameterized MD model

gri goe aka guy lep

40

lep aka goe

20

aka lep goe

0

guy

Figure 5 (a) Structures and (b) relative energies of FeOOH polymorphs calculated with parameterized model, plane-wave pseudopotential methods, and experiment (Laberty and Navrotsky, 1998).

 

Molecular Models of Oxide‐Water Interfaces 

179

functional calculations using CASTEP, taken from Rosso and Rustad (2001). The major conclusion to be drawn is that the relative energies of the FeOOH polymorphs are very close. The ordering of the DFT total energies of the different polymorphs was dependent on whether GGA or LDA was used, with LDA giving the correct ordering of diaspore < boehmite. One concludes that the problem of obtaining theoretically the energy differences among the various polymorphs is beyond current capability of electronic structure methods. It may be that an empirically derived exchange-correlation functional analogous to B3LYP, but parameterized on thermodynamic data on oxides, will be a useful approach in future work. VACUUM-TERMINATED SURFACES

The next level of complexity is to create a surface from the bulk material. First, it is necessary to choose the type of model to be used to represent a surface. There are two possible choices: (1) a semi-infinite surface (Jones et al. 2000) or (2) a slab. The semiinfinite model uses a two-region approach, whereas the slab model uses 2-D periodic Ewald methods to treat long-range forces. In either case, it is necessary to demonstrate convergence: in the former case, with respect to the partitioning of the two regions; in the latter case, with respect to the thickness of the slab. Vacuum truncation of an oxide surface produces unusual coordination environments for surface atoms. In the process of producing a molecular model of surface, be it in a vacuum or otherwise, some guess must be made about the surface termination. In principle, this could be arrived at from the simulation itself. One could imagine simulating the crystal growth process of, say, molecular beam epitaxy and allowing the surface structure to evolve from the simulation. One could also start with a bulk crystal and apply a uniaxial tension along the direction of the surface of interest, forcing the crystal to break along the surface of interest but allowing the atomistic structure of the surface to be determined from the simulation. As many vacuum surfaces are produced by sputtering and annealing, one might also attempt to simulate this process. In practice, nearly all surface terminations are arrived at through a combination of common sense and a set of rules defined by Tasker (1979), which basically states that surfaces should be constructed such that they exhibit zero dipole moment. Another way to think of this is that the surface will cleave in such a way as to give the same structure on either side of the cleavage surface. This principle is illustrated in Figure 6 for the (001) surface of hematite. Much recent experimental work has focused on the structures of vacuum terminated oxide and carbonate surfaces of geochemical interest (Chambers et al. 2000; Thevuthasan et al. 1999; Sturchio et al. 1997; Fenter et al. 2000; Guenard et al. 1998; Charlton et al. 1997). For the (001) surfaces of corundum and hematite, there is good agreement between a variety of theoretical calculations and experimental results, at least in the sense that the gross feature of the surface relaxation, the large inward relaxation of the metal terminated surface, is the same for both the experimental and theoretical models. Parameterized models, quantum mechanical calculations, and experiment all agree that the uppermost Fe layer relaxes inward by approximately 50% after cleavage. However, there are some significant differences between both the ab initio and parameterized models and experiment. For example, in Table 2, we see that the theoretical methods agree well with each other, but may underestimate or overestimate the relaxation in the layers underneath the top layer. Magnetite (001) is geochemically relevant and, because magnetite is a good conductor, is also amenable to various high vacuum techniques requiring good sample

180

Rustad 

Fe O Fe Fe O

Unrelaxed

Relaxed

Figure 6. Relaxation of corundum (001). Small atoms are metals, large atoms are oxygens. (a) Bulk structure showing cleavage plane between adjacent iron layers, (b) Relaxation of top iron layer into the surface.

Table 2. Percent change in z component upon relaxation after cleavage of hematite along (001) as shown in Figure 6. XPD

a

b

GIXS (Al2O3)

Model

c

LAPW

d

Fe -41

-51

-49

-57

+18

+15

-3

+7

-8

-29

-41

-33

+47

+18

+21

+15

O Fe Fe O a

X-ray photoelectron diffraction on hematite (001) from Thevuthasan et al. (1999) Grazing incidence x-ray scattering on Al2O3 (001) from Guenard et al. (1998) c Calculations on hematite (001) from Wasserman et al. (1997) d Linearized augmented plane wave calculations from Wang et al. (1998) b

conductivity. The spinel structure may be represented as a sequence of neutral stacking units parallel to (001) as shown in Figure 7. This stacking sequence builds up the octahedral (B layer) and tetrahedral (A layer) sites characteristic of the spinel structure. The unit cell of the stacking layer is (√2×√2)R45 relative to the bulk. In the bulk, a tetrahedral atom, coming from the overlying stacking unit, is bonded to the oxygens at the center of the unit cell. This shrinks the cell by a factor of 1/√2 and rotates the cell by 45˚. In magnetite, Fe3+ occupies the tetrahedral sites, while the octahedral sites are filled with an equal mixture of Fe2+ and Fe3+. Above the Verwey temperature (119 K), the electrons in the rows of octahedral B sites are delocalized; these sites may be thought of as being occupied by Fe2.5+ ions. Termination of the stacking sequence in Figure 7 will result in a neutral, stoichiometric, autocompensated surface with two-fold coordinated Fe3+ and five-fold coordinated Fe2.5+ sites (Kim et al. 1997). This surface is referred to as the tetrahedral or "A" layer termination. It was found in Rustad et al. (1999c) that the two-fold coordinated Fe3+ ions at the surface are unstable in their bulk positions and rotate downwards to occupy the adjacent five-fold half octahedral site in the plane of the B layer. This relaxation mechanism is illustrated in Figure 8. This was rationalized in

 

Molecular Models of Oxide‐Water Interfaces 

181

Figure 7. Neutral stacking sequence generating magnetite parallel to (001) large medium gray atoms are oxygens, small light gray atoms are octahedral irons, small dark atoms are tetrahedral irons. Bulk unit cell is shown by the solid box; the surface unit cell is shown by the dashed box.

(a)

(b)

Figure 8. Relaxation of vacuum terminated magnetite (001) according to Rustad et al. (1999). (a) unrelaxed structure assuming (√2×√2)R45 termination in Kim et al. 1997), down [110] direction as indicated in the Figure. (b) relaxed structure. Only a single row between the 2.5+ sites is shown. Tetrahedral sites are dark, small ions, octahedral 2.5+ sites are medium gray ions, and large light ions are oxygens.

terms of the driving force resulting from the overcoordination of the oxygens attached to the two-fold coordinated Fe3+ in terms of Pauling’s rules. This surface structure provides a compelling interpretation of STM images of magnetite (001), where curious dimeric structures were observed in between the Fe2.5+ rows along [110]. The observation of “dimers” would then simply result from the second layer tetrahedral Fe3+ being pushed up to the surface in response to the relaxing first layer tetrahedral Fe3+. Recent experiments, however, have been more consistent with a bulk termination without extensive surface relaxation. It is possible that the discrepancy results from full oxidation

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of the surface ions, in which case the MD model does not relax as in Figure 8. This interpretation does not provide a ready explanation of the STM images; however, it may be possible that probing the surface structure with the STM changes the oxidation states of the surface sites, allowing the relaxation mechanism in Figure 8. Two important aspects of vacuum-terminated surfaces are 1) the extent of surface relaxation upon cleavage of the crystal structure across the surface and 2) the energy required to cleave the crystal. Under vacuum conditions, only the surface relaxation is accessible to experiment. Measurements of cleavage energies, at least for the oxide crystals of interest in the work described here, are exceedingly difficult and have not produced convincing results. These energies are readily calculated theoretically, however, and this provides an important connection between quantum mechanical and parameterized methods. The surface energy is defined by

σ=

( Eslab − Ebulk ) 2A

(9)

where Eslab is the total energy of the slab and Ebulk is the energy of an equivalent number of atoms in the bulk crystal. This equation assumes that the slab is stoichiometric. Nonstoichiometric slabs are possible if changes take place in redox state. The basic approach used for nonstoichiometric slabs is described in Wang et al. (1998). To illustrate the general idea, let us define the slab free energy of an iron oxide surface at zero temperature and pressure (so both PV and TS are zero) as Ω slab = Eslab − N Fe μ Fe − N O μO

(10)

Consider specifically a magnetite slab. In addition to the “A” termination discussed above, another termination has been postulated (Voogt 1998) and observed with scanning tunneling microscopy investigations (Stanka et al. 2000). As this termination involves the removal of FeO from the “A” terminated surface, the relative stabilities of the two surfaces will depend on the partial pressure of oxygen in the system. Rustad et al. (2000c) analyzed the relative stabilities of these surfaces using the molecular model presented above. Because of the interdependence between the chemical potentials of magnetite, oxygen, and iron 1

4

μ Fe = μ Fe O − μO 3

3 4

3

(11)

Equation (10) may therefore be rewritten: 1

4

3

3

Ω slab = Eslab − N Fe μ Fe3O4 + ( N Fe − N O ) μO

(12)

At zero pressure and zero temperature, the first two terms are just Eslab-Ebulk, similar to Equation (9) except that we have not yet divided out the surface area. The last term is zero if the slab is stoichiometric magnetite. If the stoichiometry is different from magnetite, however, the free energy of the slab will depend on the partial pressure of oxygen in the system. If the surface is oxidized relative to magnetite (NO>4/3NFe), the surfaces chemical potential will decrease with pO2, if it is reduced, the chemical potential will increase with pO2. Wang and coworkers (1998) were able to calculate the chemical potential of O2 gas directly using DFT methods. Rustad et al. (2000c) used an indirect approach in which the differences in total energies of bulk FeO, Fe3O4, and Fe2O3 were used to fix the μO values at the magnetite-hematite, magnetite-wustite, and wustitehematite buffers. This allowed the establishment of an empirical relation between μO and pO2 that could be used to evaluate Equation (11) allowing the relative stabilities of slabs

 

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having different oxidation states to be calculated as a function of the oxygen pressure. Calculations on charge-ordered magnetite slabs indicate that, within the context of the ionic model presented here, the surface energy of the “A” termination of magnetite is lower than that of the “B” termination over a wide range of oxygen fugacities (Rustad et al. 2000c). Hydroxylation has a negligible effect on the relative energies of the “A” and “B” surfaces. HYDRATED AND HYDROXYLATED SURFACES Neutral surfaces

The next level of complexity involves hydrating the vacuum terminated surface with a single layer of water. Hydration and hydroxylation of mineral surfaces have received significant attention in both theoretical and experimental investigations. Many studies have focussed on the energy difference between molecular and dissociative adsorption of water (Lindan et al. 1996; Stirniman et al. 1996; Wasserman et al. 1997; Langel and Parrinello 1994; Giordano et al. 1998; Henderson et al. 1998; Shapalov and Truong 1999). This energy difference is a direct reflection of the acidities and basicities of surface functional groups. Dissociative water adsorption will be favored for acidic =MOadH2 functional groups and basic MnOlat where Oad represents the oxygen of an adsorbed water molecule and Olat represents a lattice oxygen. Other things being equal, the acidities of =MOH2 functional groups would be expected to be proportional to the size to charge ratio of the metal and the coordination number (Parks and deBruyn 1962; Parks 1965; Yoon 1979; Hiemstra et al. 1989). The process of surface hydration was indicated in Figure 1. Hydroxylation energies, defined as the difference between the vacuum-terminated surface energy (Eqn. 8) and the adsorption energy of water, are negative for minerals in the iron oxide system discussed here. This means that the crystal would prefer to break apart and hydroxylate rather than remain whole in the presence of aqueous solution. This is wrong, as experimental studies have shown that the enthalpy of dissolution of goethite decreases (becomes more negative) as total surface area increases. There may be important entropic and zero-point effects, but, taken at face value, this indicates that either the dry cleavage energy is too easy or that the binding of the water to the undercoordinated iron ions on the surface is too large. Although ab initio calculations agree well with the cleavage energies obtained by the classical model discussed here for hematite (001), we have shown that plane-wave pseudopotential calculations also give negative hydroxylation energies for Al2O3 (see Fig. 9). Several experimental techniques can be used to examine the extent of water dissociation on a mineral surface. One of the most informative is the simple temperature programmed desorption experiment (Masel 1996). In this experiment, a sample is placed in a vacuum chamber at low temperature, and a measured amount of water vapor is introduced. The sample is then heated at a definite rate and the amount of water vapor desorbing from the sample is measured as a function of temperature. In general, the desorption is not uniform with temperature, and several TPD “states” are manifested as maxima in the flux off the sample. These experiments are performed under several water doses. At very high doses, a maximum at 160 K is observed independent of the substrate. The presence of this 160 K “ice peak” indicates the formation of multiple layers of ice forming on the sample. As the exposure is decreased to one monolayer, this peak gradually disappears. Peak positions may change with coverage. This experiment in principle provides a “fingerprint” of the energies of various sites for water binding on an oxide surface. In practice, interpretation of TPD data can be

184

Rustad  -1.3 -1.4

EA (J/m2)

-1.5 -1.6

Cleavage energy (1.64 J/m

Figure 9. Adsorption energies for water on the αAl2O3 as a function of the extent of dissociation on the surface. The cleavage energy is represented by the solid line at 1.64 J/m2, implying that the crystals are unstable. Zero point energies are not considered.

2

)

-1.7 -1.8 -1.9 -2 -2.1 0

20

40

60

80

100

Percent dissociation difficult; in a typical experiment, both the coverage and the temperature are changing at the same time. For example, consider Figure 10 taken from Henderson et al. (1998) on the hematite (012). One observes one peak at 260 and another at 350 K. One possible interpretation of this data is that the low-temperature peak represents a small amount of molecularly adsorbed water, and the high-temperature peak represents the recombination of OH and H of water molecules that have dissociatively adsorbed to the surface. The problem with this interpretation is that the 350 K peak is independent of coverage. This is not typical of recombinative desorption; at lower coverages, the recombining species have less frequent encounters, and the desorption temperature should increase. This observation, along with other evidence from secondary ion mass spectrometry experiments, lead to more detailed consideration of the calculated hydration/hydroxylation energies for hematite (012) reported above. In particular, it would be interesting to know hydration energies at intermediate points between 0, 50% and 100% dissociation. Perhaps there were low configurations having 10% or 20% dissociation that were lower in energy than the 50% dissociated configuration. Rustad et al. (1999b) considered a 2×2 supercell, containing eight adsorbed water molecules. The results of these calculations are shown in Figure 11, essentially confirming the extensive dissociation predicted by the model for the 1×1 cell. The larger scale investigations predicted 75% dissociation. Even with these modest cell sizes the number of proton configurations becomes enormous. In the particular case of hematite (012), we examined 1,296 possible proton configurations. This number could have been reduced by nearly a factor of two by accounting for symmetry, but this still leaves approximately 650 possible proton configurations, each of which would ideally require a conformer searching procedure to find the lowest energy tautomers. This is well beyond what would be possible with ab initio methods unless they were “grand-challenge” types of efforts. The hydroxylation problem is essentially a rare-events problem in that the barriers to proton motion are large. Even in a full MD simulation with solvent present, one cannot reasonably expect to let the system naturally sample all protonation states by itself. Conformer searching by molecular dynamics techniques on this scale is not a reasonable proposition with Car-Parrinello methods at the present time.

 

185

Molecular Models of Oxide‐Water Interfaces  5 158

4

m/e = 18 QMS signal (cps, x 10

4

)

353

Figure 10. Temperature programmed desorption curve of water on hematite (012) from Henderson et al. (1998) at various coverages. Peak at 158 K is from ice formed at coverages in excess of 1 monolayer. Note first-order coverage-independent behavior of desorption peak at 353 K.

3

2

1

0 100

200

300

400

500

Temperature (K)

-2.0

EA(J/m2)

-2.2 -2.4 -2.6 -2.8 -3.0 -3.2

0

20

40

6

80

10

Percent Hydroxylation Figure 11. Adsorption energy as a function of percent hydroxylation calculated from energy minimization studies using the model on hematite (012). Cleavage energy is close to 2 J/m2. Each point on the curve represents the lowest energy structure at each point on the x axis, taken from 1,296 total energy minimizations.

Additional experiments using isotopically-resolved TPD and vibrational spectroscopy using HREELS have confirmed the theoretical calculations of approximately 75% water dissociation on hematite (012). In the isotopically resolved TPD experiments, O18 water was deposited on a bulk terminated hematite (012) surface made with O16 (by annealing in O16 gas). The peak at 350 K at monolayer coverages comprises approximately equal mixtures of O18-O16, almost surely requiring that the water dissociate. HREELS work showed the presence of δ Fe-OH vibrations at 960 cm-1, indicative of dissociated water. Regarding the first-order behavior of the TPD spectrum,

186

Rustad 

one straightforward explanation is that the hydroxyls are not mobile on the surface. The water arrives at the surface and dissociates, and both hydroxyls are essentially immobile until they recombine upon desorption. The mixing of the lattice and adsorbed oxygens serves as proof of water dissociation but is troubling nonetheless. One problem is that given the canonical surface structure one is faced with exchanging the Fe3OH and FeOH sites, as shown in Figure 12, which would be expected to have a very high barrier of activation energy. Another problem is that this is supposed to happen without the dissociated products being mobile enough to yield second order behavior of the 350 K TPD peak. Remember that this exchange is pervasive and is not happening only at defects. One possible hypothesis is that some mobile defect is present which, like a plow, mixes up the adsorbed and lattice oxygens as it migrates through the surface. Another possible interpretation is that the vacuum structure of the surface is wrong altogether, and, in fact, the true structure contains a singly-coordinated Fe-O group. This would give rise to a mixed desorption peak because after adsorption of water, equivalent Fe-OlatticeH and Fe-OadsorbedH groups exist on the surface. LEED patterns show that the surface structure is 1×1, but this does not mean that the surface has the simple bulk-terminated structure. To help resolve this issue, Bylaska and Rustad (unpublished) have carried out ab initio Car-Parrinello simulations of the canonical surface in the Al2O3 system. They did not observe any structural rearrangements to configurations with Al-O groups, nor did they observe any reconstructions on the hydrated surface that could explain the mixing. These authors chose to work with the analog Al2O3 in these calculations because of the complications of treating transition metals with plane-wave pseudopotential methods. The analogy may not be valid; the mixing experiments on the Al2O3 (021) surface have yet to be performed. Suffice it to say that although we have agreement between theory and experiment about the extent of dissociation of water on hematite (021), some fundamental pieces of the puzzle are still missing. Similar difficulty exists for the (001) surface of Al2O3 (Nelson et al. 1998; Hass et al. 2000), but this is less surprising given the relative complexity of the (001) relaxation. For the magnetite (001) surface, calculations were carried out on three sets of hydroxylated slabs, including both the relaxed and unrelaxed “A” terminations and the “B” termination (Rustad et al. 2000c). For the “A” termination, four waters are added per

Figure 12. Hydroxylated (012) surface of α−Al2O3 from ab initio molecular dynamics simulation (Bylaska and Rustad, unpublished). Arrows show the mixing that must somehow take place during isotopic mixing.

 

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187

unit cell to the octahedral sites and two waters per unit cell are added to the tetrahedral sites (see Fig. 13). Assuming each of the sites has at least one proton, there are 12!/(6!×6!) = 924 possible tautomers for each unit cell. An exhaustive search through these possible tautomers yielded the structure shown in Figure 13 as the lowest-energy tautomer. Because of the large number of tautomers within the unit cell, it was not possible to examine arrangements outside the k = 0 (all unit cells the same) approximation as was done for hematite (012) (Rustad et al. 1999b). Total water binding energies for both surfaces were about 2.32 J/m2, indicating that the presence of water will have little effect, at least in a thermodynamic sense, on which surface is observed. It is of interest that the unrelaxed “A” te rminated surface is lower in energy than the relaxed “A” surface upon hydroxylation; the pr esence of water in the system should “undo” the surface relaxation predicted in Figure 8. This in fact explains an apparent paradox suggested by temperature-programmed desorption studies on magnetite (001) (Peden et al. 1999). These investigators showed the existence of three peaks in the (001) TPD spectrum at 225 K, 260 K, and 325 K. Each peak contributes approximately equal amounts to the TPD spectrum. In one possible interpretation, the 225 K and 260 K peaks are contributed by octahedral Fe2+ and Fe3+ sites, while the peak at 325 K is coming from the Fe3+ tetrahedral sites. The octahedral 2.5+ site would be charge-ordered in this interpretation. An objection to this interpretation is that one would not expect the two waters on the tetrahedral sites to desorb at the same temperature. Once one of the waters has desorbed, the remaining water should be held more tightly, as the surface Fe 3+ is now only threefold coordinated. It seems reasonable to expect two peaks at high temperature because the desorbing waters are coming from the same site. This puzzling lack of two peaks at high temperatures can be rationalized by calling on the large surface relaxation energy to reduce the binding energy of the final water removed from the surface. Calculation of binding energies for each of the waters on the tetrahedral sites shows that the binding energy of the second water (43 kcal/mol) is in fact the same as that of the first (44 kcal/mol). The similarity in binding energies arises because the “A” surface relaxation mechanism is not accessible until the second water is removed from the surface. After this water is desorbed, the system gains 0.72 J/m2 of

Figure 13. Surface sites on hydroxylated magnetite (001). “a” sites are acid/donor sites. “b” sites are basic/acceptor sites. Large atoms are oxygen, small dark atoms are iron, small light atoms are protons.

188

Rustad 

surface energy in relaxation, thus decreasing the total binding energy of the second water to a value very close to that of the first water. Surface charging

One of the oldest and most fundamental experiments on solvated oxide surfaces is the measurement of the amount of charge accumulated on the surface as a function of the pH of the solution. This is closely related to the issue of the dissociation of water on the hydrated surface discussed in the previous section. There is a close relationship between surface and aqueous hydrolysis. Aqueous hydrolysis reactions are structurally much less ambiguous than surface hydrolysis reactions and therefore are useful in the interpretation of surface hydrolysis data. This concept is fundamental and goes back to the pioneering work of Parks and deBruyn (1962). As pointed out by Hiemstra et al. (1989), the main difficulty of this approach is that all aqueous hydrolysis data are based on mononuclear MO, MOH, MOH2 functional groups whereas surfaces also will have bi- and tri-nuclear surface functional groups such as M2OH and M3OH. Other obvious differences are that solvation effects would presumably be very different between surfaces and aqueous complexes, and internal solid-state relaxation effects are absent entirely. The obvious approach to using molecular modeling to address surface charge is to calculate the energies required to remove protons from the neutral surface and the energies gained by adding protons to the neutral surface. In periodic slab systems, the repeating cell must be neutral overall to define the potential energy using the Ewald summation. Therefore, the calculation actually performed is, in the case of loss of a proton, the energy of moving the proton from a localized positive charge to a positive charge dispersed uniformly throughout the 2-D periodic plane defining the unit cell of the slab (see Fig. 13). This quantity is sufficient to calculate relative values of proton affinities or gas-phase acidities within the same cell as the energy of the uniform compensating charge can be shown to be independent of the positions of the atoms in the cell. There is, however, a systematic dependence of the deprotonation/protonation energies arising from defect-defect interactions across image cells which should be taken into account (Wasserman et al. 1999 and references therein). To illustrate the gas-phase proton binding approach to calculating logK for the surface charging, the neutral surface of magnetite (001) is shown in Figure 14. To simplify the task of assigning locations for proton addition and removal, assume protons are added in such a way as to maintain the Pauling bond strength at the oxide ion in the range –1<0<1. Pauling bond strength (PBS) is defined as ΣZi/CNi where Z is the charge of the ith coordinating cation and CN is the coordination number of the ith coordination cation. Therefore, we eliminate surface species such as Fe3OH2 (PBS=+3/2) or FeO (PBS=-3/2). The various surface sites and their associated energy changes (positive for proton removal, negative for proton addition) are shown in Table 3. Also shown in Table 3 are the predictions from the revised MUSIC model (Hiemstra et al. 1996). Because of the sensitive dependence of the site acidity on bond length, the MUSIC model predicts that the triply coordinated =Feoct2FetetOH sites and the singly-coordinated FetetOH2 are much more acidic than would be indicated by the relative gas-phase acidities. The molecular model does recover the differences on bond length between the FetetOH and FeoctOH bonds, but the influences on the gas-phase acidities are not systematic. To make some prediction about an experimentally measurable quantity, like the pH of zero charge, requires that some relationship be developed between the gas-phase proton affinities and acidities of surface sites and their associated pKas in solution. To do this, we used the classic method of correlating the known pKas of aqueous species with

 

189

Molecular Models of Oxide‐Water Interfaces 

Figure 14. Protonation/deprotonation of neutral slab in periodic boundary conditions. Slab remains neutral due to addition of uniform positive (deprotonation) or negative (protonation) background charge. The energetic contribution from the uniform charge does not depend on the positions of the atoms, only on the size and shape of the unit cell.

Table 3. Sites present in Figure 13. Site

Fe-O bond lengths (Å) on neutral surface

GPA GPPA

LFER pKa

CDMUSIC

Fe2.5+OH Fe2.5+ 2Fe3+O Fe2.5+ 2Fe3+O Fe3+OH Fe3+OH Fe2.5+ 2Fe3+O

2.09 1.99, 2.06, 1.84 2.02, 1.97, 1.89 1.90 1.91 2.06, 2.01, 1.85

-340 -352 -343 -343 -340 -352

10.2 10.5 10.3 10.3 10.2 10.5

11.5 4. 5 4. 5 5.1 5.1 4. 5

Fe2.5+3OH Fe2.5+OH2 Fe2.5+ 2Fe3+OH Fe2.5+3OH Fe2.5+OH2 Fe2.5+OH2

2.13, 2.08, 2.14 2.17 2.02, 2.21, 2.14 2.19, 2.11, 2.10 2.21 2.16

307 288 288 311 288 288

9.3 8.7 8.7 9.4 8.7 8.7

10.9 11.5 4. 5 10.9 11.5 11.5

Site Type

Bulk: Fe3+O=1.94 Fe2.5+O=2.12

Acceptor Sites b1 b2 b3 b4 b5 b6 Donor Sites a1 a2 a3 a4 a5 a6

Note: Sites are listed by donors and acceptors. The list includes site type as identified by coordinating metal ions and protons, the bond lengths to the coordinating Fe ions, the gas-phase acidity/gas-phase proton affinity (GP/GPPA), the log K for the reaction using the relationship from Rustad et al. (1996), and the prediction of the revised MUSIC model described in Hiemstra et al. (1996). MUSIC model predictions are given in terms of proton loss from protonated acceptor sites, for example FeOH in the acceptor list and FeOH2 in the donor list imply the same reaction and have the same pKa. Bond lengths are those predicted by the molecular dynamics model discussed here, which are systematically too long by about 5%.

190

Rustad 

some calculated quantity and applying the same relationship to surface species. In Rustad et al. (1996b), gas-phase acidities for Fe(H2O)63+ -Fe(OH)3(H2O)3 were calculated using the model. The calculated acidities were correlated with the solution pKas, and this correlation was used to predict solution pKa for the surface species. This correlation was subsequently empirically modified by Felmy and Rustad (1998) to get a somewhat better fit to the surface charging data at multiple ionic strengths. Applying the correlation from Felmy and Rustad (1998) to magnetite gives the surface pKas listed in Table 3. The prediction using the LFER derived from molecular modeling incorrectly predicts a basic surface. The PZC of the (100) surface predicted by the MUSIC model, however, is in good agreement with experiment (see also estimates by Wesolowski et al. (2000)). I make the bold claim that the MUSIC model is getting the right answer for the wrong reason, and that the model of Rustad et al. (1996b) gets the wrong answer for the right reason. This may be difficult to accept, especially given the ease of use of the MUSIC model (requiring only paper and pencil) and its great generality of application (nearly the entire periodic table). In defense of the molecular model, it must be pointed out that it has successfully explained experiments on the simpler systems described above. Of particular relevance here is the amount of water dissociation on hematite (021), the relaxation of hematite (001), and the agreement with quantum mechanical calculations of the gas-phase acidity of the trimeric Fe3(OH)7(H2O)62+ polynuclear cluster. These calculations taken together suggest that the model can correctly deal with both structural and energetic characteristics of the triply coordinated surface oxygens. It is true that the model has been somewhat controversial on magnetite (001), but even if it fails here, it is because of the complex electronic structure of magnetite. This should also cause anomalies in the MUSIC model. The empirical MUSIC model was not designed to make any predictions about these simpler systems and cannot be tested in these less complicated proving grounds. It is worth emphasizing that the arguments leading up to the construction of the MUSIC model are gas-phase arguments based on “ionic” concepts such as Pauling bond order. This essential physics is present in both approaches. It cannot be argued that the MUSIC model is accounting for subtle quantum mechanical effects not present in ionic model because the MUSIC model rests entirely on an ionic framework. To advocate this line of reasoning, there must be a convincing explanation of the cause of the disagreement with experiment. There are two possibilities: 1) aqueous solvent effects, and 2) electronic structural effects not accounted for by the ionic model of magnetite. Relevant to this second point is the recent work of Sverjensky (1994) and Sverjensky and Sahai (1996). These authors emphasized the role of the mineral dielectric constant in determining mineral surface reactivity. According to their model, the major reason for the enhanced acidity of magnetite is its high dielectric constant, which was shown to be positively correlated with site acidity, other things (like Pauling bond orders) being equal. A possible physical explanation is that if the distribution of cation charges rearranges itself in response to surface deprotonation, the relaxation energy is large, and this makes the mineral is more acidic. The same thing is true of excess positive charge added to the surface, but, since the adsorbed positive surface charge should be “further away” from the bulk, it elicits less of a response than positive charge removed from the neutral surface, which should be closer to the bulk mineral. The problem with this argument, in the case of magnetite, is that maghemite (γ-Fe2O3), which has effectively the same structure as magnetite, but without mobile electrons giving rise to a large dielectric constant, should be less acidic than magnetite. There have not been many studies, but it appears that maghemite has about the same

 

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acidity as magnetite (Watanabe and Seto 1990). Also, if bulk relaxation effects caused the high acidity of magnetite, it might be expected that this would be evident in the TPD experiments on magnetite, perhaps giving rather high binding temperatures due to the increased “solvation” of the protons and OH groups on the surface. Recent TPD experiments on magnetite (001) by Peden et al. (1999) as discussed above give OH recombination TPD peaks at roughly the same temperatures as seen by Henderson et al. (1998) on hematite (012), showing no evidence of enhanced retention of either the molecular or dissociated products. Furthermore, spin-polarized STM studies (Wiesendanger et al. 1994) indicate that the effective Verwey temperature of the surface of magnetite may be much larger than that of the bulk. This results from surface induced band-narrowing, giving rise to a lesser degree of quantum “delocalization” energy, which tends to trap the electrons on Fe2+ sites. It is emphasized that these arguments apply to one case only (magnetite), and that the above arguments are not intended to call into question the general utility of considering the mineral dielectric constant in surface complexation theory. In the particular case of magnetite, then, the remaining issue is the effect of aqueous solvent. This brings us to the final section on solvated systems although the issue is by no means resolved here. SOLVATED INTERFACES

Calculations on solvated systems are very expensive computationally, even for parameterized molecular dynamics methods. A reasonable size is on the order of 25×25×35 Å, or about 1500 to 2000 atoms. For the Fe-oxide-water model of Rustad et al. (1995), such system sizes require about 18 sec/timestep on a single 500 MHz pentium III processor (note that this model is more computationally demanding than simpler, nondissociating water models such as PSPC or TIP4P). The methods used for million-atom types of MD simulations do not help much in this regime, as it is close to the crossover in terms of computational overhead for the O(N) methods (Kutteh and Nicholas 1995). In other words, one can go to larger scales than this at relatively small cost, but the time-savings at this particular scale is minimal. Because little is known about surface structural features above this scale, there is currently no compelling reason to extend the spatial scale. Parallelism can be conveniently exploited, however, to vastly reduce computation time. Because the memory requirements are minimal, these calculations are well suited to distrubuted memory “Beowulf” type parallel computers (Sterling 1999) with relatively inexpensive nodes. On 64 500 MHz pentium III nodes, the time to execute one timestep is about 0.29 seconds, making nanosecond simulations a reasonable proposition. Rustad et al. (unpublished) investigated the stability of the magnetite surface shown in Figure 14 in a solvated environment. The starting configuration consisted of a 20 Å layer of ice sandwiched between a 15-Å hydroxylated slab of magnetite (001). The simulation was run for about 2 nanoseconds at 300 K ion temperature on 64 500-MHz processors connected with a GIGANET network, consuming about 20 days of real time. A major effect of solvation is an extensive secondary hydroxylation of the surface, with additional protons being picked up by the octahedral FeoOH and tetrahedral FetOH surface functional groups, with the concomitant creation of hydroxide ions in the solution. This amount of charge picked up is shown in Figure 15, and a breakdown of the various populations of the surface functional groups is shown in Figure 16. The majority of the population changes take place within the first 10 picoseconds. The active surface functional groups are the singly coordinated sites. Triply coordinated surface sites have nearly constant populations throughout the simulation. This illustrates how the gas-phase calculations can be misleading. Because the gas phase proton affinities

192

Rustad  200

150

Figure 15. Accumulation of charge attendant upon solvation of the magnetite (001) surface. This charge is the result of extensive “secondary” hydroxylation of the surface.

mC/m2

100

50

0

-50

0

10

20

30

40

50

60

t (psec) 70

FeOOH2

FeO2FetOH

60

50

FeO3OH

40

FetOH

30

FeO2FetO

20

FeOOH

10

t

Fe OH2 OH-

0 0

10

OH- (Fe)

20

30

40

50

60

t (picoseconds)

Figure 16. Evolution of the populations of various functional groups in the simulation of the solvated magnetite (001) surface. Octahedral FeoOH and tetrahedral FetOH groups accumulate charge by acquiring protons from the solution.

of the triply coordinated sites are comparable to the other sites, one might expect them to be actively involved in proton exchange with the solution. Evidently the solvation effects make these groups relatively inert. The inertness is a result of the difficulty of solvating the more deeply buried triply coordinated surface sites. Because the solvent relaxation on deprotonation is ineffective, the sites are in fact less acidic than would have been predicted by the gas-phase proton binding energies. The sites also are much less acidic

 

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than would be predicted by the MUSIC model. The significant number of FetOH2 sites is consistent with the gas-phase binding energies but remains in disagreement with the MUSIC predictions that nearly all the singly coordinated tetrahedral sites should be in the FetOH state. This small simulation is meant only as an illustration of what kinds of systems are now within reach for full molecular dynamics simulations of the solvated, hydroxylated oxide surface. More work will be needed on this system to determine whether this view of the surface is viable. Numerical experiments in progress include virtual surface titrations involving the addition of acid (HClO4), base (NaOH), and variations in ionic strength [NaClO4]. These experiments will yield valuable insight into the structure of the electrical double layer and the effect of surface ion pairing. REMARKS

The approach taken in the research described here is characterized by a very simple model construction phase, followed by an extensive series of tests involving information from ab initio electronic structure theory, mineralogy, aqueous chemistry, and high vacuum surface science. As of the time of this review, classical models are the only models simple enough to perform the necessary benchmarking calculations in all these areas and also are the only models that can be extended to 10,000 atom/nanosecond timescale simulations of interfacial phenomena. Molecular modeling methods comprise the only reasonable theoretical framework for connecting this diverse range of experimental observations. In the future, ab initio methods will eclipse at least some of the range of applicability of classical simulations. At the same time, the length and time scales available to classical simulations will increase. Advances will take place in potential function formulation, allowing a greater range of phenomena to be investigated with classical simulation. There is room for employing electronic structure investigations to arrive at a better representation of the fundamental physics of the chemical bond (Parr and Pearson 1983; Gibbs et al. 1998) and to use this information to construct more accurate and flexible potential functions. A recent example is the development of potential functions capable of describing the oxidation of aluminum (Cambpell et al. 1999) and water (Burnham et al. 1999). The major limiting factor in large-scale simulation will be the creation of a body of experimental observation which allows posing meaningful molecular level problems at scales of 10 to 1000 nm. Some chemical problems at these scales are already being addressed experimentally (Weidler et al. 1998a,b; Banfield et al. 2000). If it turns out that there are important influences of nano-mesoscale structure on chemical reactivity in geochemistry, simulation will play a large role in sorting out the factors responsible for these nanoscale structure-function relationships. Such factors might involve fluid dynamics and mixing at the interface, the influence of crystal surface roughness on intrinsic reactivity and on the surface potential, and chemomechanical effects such as stress-induced dissolution (Scudiero et al. 1999). ACKNOWLEDGMENTS

This work, along with the bulk of the research described herein, was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Engineering and Geosciences Division, Contract 18328. The author is grateful to Nick Woodward in particular for his program management support from this office, which included funds for the purchase of the 96-node “pile of PCs” used for the computationally intensive parts of this research. Discussions and input from Eric Bylaska, Scott Chambers, Dave Dixon,

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Andy Felmy, Mike Henderson, Steve Joyce, Chuck Peden, Kevin Rosso, Theva Thevuthasan, and Evgeny Wasserman of Pacific Northwest National Laboratory were essential. Many of their ideas have been instrumental in forming the basis for the research reviewed here. The National Energy Research Supercomputing Center is acknowledged for a generous grant of computer time. Scott Jackson and Ryan Braby are acknowledged for their help in keeping the “pile of PCs” functioning. The author is very grateful for technical reviews by Jim Kubicki and two anonymous reviewers and to Andrea Currie and Jamie Benward for editorial assistance. Pacific Northwest National Laboratory is operated for the U.S. Department of Energy by Battelle Memorial Institute under Contract DE-AC06-76RL01830. REFERENCES Akesson R, Pettersson LGM, Sandstrom M, Wahlgren U (1994) Ligand field effects in the hydrated divalent and trivalent metal ions of the first and second transition periods. J Am Chem Soc, 116:86918704 Allen MP, Tildesley DJ (1989) Computer Simulation of Liquids. Oxford University Press Banfield JF, Welch SA, Zhang HZ, Ebert TT, Penn RL (2000) Aggregation-based crystal growth and microstructure development in natural iron oxyhydroxide biomineralization products. Science 289:751-754 Beattie JK, Best SP, Skelton BW, White AH (1981) Structural studies on the cesium alums, CsM(III)[SO4]212H2O. J Chem Soc - Dalt Tran 10:2105-2111 Bebie J, Schoonen MAA, Strongin DR, Fuhrmann M (1998) Surface charge development on transitionmetal sulphides: An electrokinetic study. Geochim Cosmochim Acta 62:633-642 Bennett CH (1975) Exact defect calculations in model substances. In: Diffusion in Solids: Recent Developments. Nowick AS, Buron JJ (eds), p 73 Blesa, MA, Weisz AD, Morando PJ, Salfity JA, Magaz GE, Regazzoni AE (2000) The interaction of metal oxide surfaces with complexing agents dissolved in water. Coord Chem Rev 196:31-63 Blonski S, Garofalini SH (1993) Molecular dynamics simulations of α-alumina and γ-alumina surfaces. Sur Sci 295:263-274 Bloechl PE (1994) Projector augmented wave method. Phys Rev B - Cond Mat 50:17953-17979 Brown GE, Henrich VE, Casey WH, Clark DL, Eggleston C, Felmy AR, Goodman DW, Gratzel M, Maciel G, McCarthy MI, Nealson KH, Sverjensky DA, Toney MF, Zachara JM (1999) Metal oxide surfaces and their interactions with aqueous solutions and microbial organisms. Chem Rev 99:77-174 Brown GE Jr., Chambers SA, Amonette JE, Rustad JR, Kendelewicz T, Liu P, Doyle CS, Grolimund D, Foster-Mills NS, Joyce SA, Thevuthasan T (2000) Interaction of aqueous chromium ions with iron oxide surfaces, ACS EMSP Symposium Proceedings, in press Burnham CJ, Li JC, Xantheas SS, Leslie M (1999) The parametrization of a Thole-type all-atom polarizable water model from first principles and its application to the study of water clusters (n=2-21) and the phonon spectrum of ice. IH J Chem Phys 110:4566-4581 Campbell T, Kalia RK, Nakano A, Vashishta P, Ogata S, Rodgers S (1999) Dynamics of oxidation of aluminum nanoclusters using variable charge molecular-dynamics simulations on parallel computers. Phys Rev Let 82:4866-4869 Car R, Parrinello M (1985) Unified approach for molecular dynamics and density functional theory. Phys Rev Let 55:2471-2474 Chambers SA, Thevuthasan S, Joyce SA (2000) Surface structure of MBE-grown Fe3O4(001) by X-ray photoelectron diffraction and scanning tunneling microscopy. Surf Sci Let 450:L273-L279 Charlton G, Howes PB, Nicklin CL, Steadman P, Taylor JSG, Muryn CA, Harte SP, Mercer J, McGrath R, Norman D, Turner TS, Thornton G (1997) Relaxation of TiO2(110)-(1×1) using surface X-ray diffraction. Phys Rev Let 78:495-498 Corrales LR (1999) Dissociative model of water clusters. J Chemical Physics 110:9071-9080 Curtiss LA, Halley JW, Hautman J, Rahman A (1987) Nonadditivity of ab-initio pair potentials for molecular dynamics of multivalent transition metal ions in water. J Chem Phys 86:2319-2327 de Leeuw NH, Parker SC, Catlow CRA, Price GD (2000) Proton-containing defects at forsterite (010) tilt grain boundaries and stepped surfaces. Am Min 85:1143-1154 de Leeuw NH, Parker SC (1998) Surface structure and morphology of calcium carbonate polymorphs calcite, aragonite, and vaterite: An atomistic approach. J Phys Chem B 102:2914-2922

 

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Structure and Reactivity of Semiconducting Mineral Surfaces: Convergence of Molecular Modeling and Experiment Kevin M. Rosso W.R. Wiley Environmental Molecular Sciences Laboratory Pacific Northwest National Laboratory P.O. Box 999, MSIN K8-96 Richland, Washington, 99352, U.S.A. INTRODUCTION

Semiconducting minerals are a unique class of materials that have a broad range of influence in near-surface geochemical environments. Many are electroactive and play an important role as natural solid phase electron sources and sinks. This property gives them the capacity to control the availability of electrons in the environment, with direct and indirect effects on many redox-based geochemical and biogeochemical processes. Examples include the natural cycling of redox-active elements (Calmano et al. 1994; Nimick and Moore 1994), dissimilatory metal reduction in microbial respiration (Lovley and Phillips 1988; Lovley et al. 1991), the oxidative degradation of organic pollutants (Schwarzenbach and Gschwend 1990; Matheson and Tratnyek 1994), and precious metal ore deposit formation (Nash et al. 1981; Bakken et al. 1989). The surfaces of semiconducting minerals are therefore, in a sense, an important threshold for the natural flux of electrons in the subsurface. The transfer of electrons across the boundary occurs in the form of redox reactions at the surface, and these are influenced by specific behavior at the mineral surface. Aside from this important characteristic, this class of minerals also presents chemically distinct surfaces to the environment that can uniquely influence non-redox based processes. For instance, sites in the structures of these materials are compatible with the incorporation of many types of heavy metals (Manceau and Combes 1988; Ford et al. 1997). Their surfaces can also catalyze organic decomposition reactions (Bell et al. 1994). The significance of semiconducting mineral surfaces in natural settings, as well as in technological applications, have generated a great deal of cross-disciplinary interest in understanding their physical and chemical properties at a fundamental level. Research with the goal of determining the structure of these surfaces and the physical basis underlying the way in which they react is a vast and currently very active field. A large component of this field falls in the surface science domain, where high resolution microscopies and spectroscopies are used to directly characterize the upper few nanometers of the surface region under well controlled conditions. This approach monopolizes the quantum physics of interacting electrons or photons with the electrons of surfaces to probe for information about the arrangement of atoms and the distribution of their electrons. At this level of observation, it is natural to suspect that theoretical models of the atomistic properties of the surface could amount to an important complimentary resource. Indeed, many important advances in mineral surface science have been facilitated by molecular modeling calculations. For semiconducting surfaces in particular, where surface properties and reactivity are often controlled by the unique electronic characteristics of the solid, calculations using first principles or ab initio methods have been invaluable. It is the principal goal of this review to demonstrate the methodological application 1529-6466/01/0042-0007$10.00

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and the utility of molecular modeling techniques for fundamental studies of semiconducting mineral surfaces. Essential background material pertaining to the physics and chemistry of this type of material and their surfaces is included. Because of the vastness of this scientific area, the scope should be defined. A focus is placed on ab initio methods because of the importance of specific electronic and magnetic characteristics of semiconductors. Of these, the so-called one-electron methods are highlighted because they are widely available and commonly in use today. The general methods and practical issues of both periodic and aperiodic treatments are presented. Ab initio dynamic simulations of processes at semiconductor surfaces are currently computationally prohibitive, therefore the methods and applications sections are slanted towards static total energy and optimized geometry calculations. Recent applications of these modeling techniques and parallel experimental results on the structure and reactivity of some mineral surfaces are reviewed. Surfaces of the sulfides galena and pyrite, and the oxides hematite and magnetite are addressed in detail because of their ubiquity in nature and the broad range of geochemical influences they bring to bear in near-surface environments. The research was compiled from a wide base of chemical surface science and geochemical literature with a goal of presenting a uniform synthesis of the two. BACKGROUND CONCEPTS Experimental approaches Molecular modeling is sometimes the only means of exploring atomistic phenomena at surfaces. But more often, modeling fills the role of either aiding in the development of reasonable hypotheses to be experimentally tested, or as an interpretative tool for experimental data. Hence, especially in the latter case, it is prudent for the modeler to be familiar with the general “in's and out's” of the relevant experiments. A plethora of microscopic and spectroscopic techniques are in use for mineral surface studies today and not even a few could adequately be reviewed here. Therefore, some familiarity with the capabilities of the common methods is assumed, but not enough to mire the interested reader. A few techniques are repeatedly mentioned, most of which are necessarily ultra-high vacuum (UHV) based techniques. Low energy electron diffraction (LEED) is based on the diffraction of an electron beam at a surface and therefore samples reciprocal space (e.g., Van Hove et al. 1986). It can provide detailed information regarding the long-range periodic arrangement of atoms at surfaces. Probes of the electronic structure commonly rely on the photoelectron effect which is an energy conserved relationship between incident photons and ejected electrons, and vice versa. These are called photoelectron and inverse photoelectron spectroscopies, which provide area-averaged information on the surface electronic structure. Valence band photoelectron spectra are often collected using low energy monochromatic radiation, either in the form of an ultra-violet source (UPS) or tunable synchrotron radiation. UPS is a very common source of electronic structure information. The general principles and experimental implementation of photoelectron spectroscopy can be found in Hochella (1988). A good introduction to UPS and inverse UPS can be found in Henrich (1985). Finally, scanning tunneling microscopy and spectroscopy (STM/STS) are scanned probe techniques based on the tunneling of electrons across an Ångstrom-scale gap between a sample surface and a sharp metal tip electrically connected in a biased circuit. It can provide both geometric and electronic structure information. In contrast to LEED, it locally probes direct space for this information with atomic resolution. It is extremely surface sensitive compared to the other mentioned techniques, which sample information

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with nanometer-scale penetration. Moreover, it may be performed in UHV, air, solution, and likewise less restrictive environments. Excellent introductions to the underlying theory, implementation, and capabilities of STM/STS are available (Bonnell 1993; Chen 1993). Ab initio modeling is very useful for the interpretation of STM/STS data and various theoretical methods of modeling electron tunneling and their successes have recently been reviewed (Briggs and Fisher 1999). Semiconductors and their surfaces The unique electronic properties of semiconducting minerals lead to surface properties that differ considerably from those of metals and insulators. One purpose of this section is to concisely introduce the important characteristics underlying this behavior. In doing so, the conventional terminology of semiconductors is also covered. Many surface studies to date have focused on the pure well characterized material and avoid the complications of impurities and defects commonly found in naturally occurring semiconductor minerals. Yet, the potential for a sensitive dependence of surface reactivity on imperfections makes this one of the most important considerations in this field. Therefore, it is prudent to be aware of any differences in the chemical properties of a pure phase and how it may be typically found in nature, which are also visited below. Finally, the topics in this section can also be viewed as present day target areas that modelers and experimentalist are attempting to achieve convergence upon. Even for pure phases, these two approaches do not always agree. In that sense, it is also hoped that this brief introduction will carry enough information to allow the reader to understand the difficulties in accurately modeling these surfaces. Excellent detailed descriptions of the chemistry and physics of semiconductor surfaces are available elsewhere (Hannay 1976; Henrich 1985; Zangwill 1988; Prutton 1994; Miller et al. 1995; Noguera 1996). The semiconductor bulk. The bulk electronic structure of semiconductors is often best understood in terms of the distribution and character of energy bands. For the purpose of illustration, consider changes in the electronic structure of a collection of infinitely separated identical atoms, N in number, as they condense to form a crystalline solid. At large separation, the electrons around individual atoms are distributed in discrete electronic levels or orbitals (states). Atomic orbitals of a particular type (s, p, d, ....) are grouped at equivalent energies. The highest energy electrons, i.e., higher potential energy, are the least tightly bound to their respective nuclei (lower binding energy) and therefore correspond to the outermost electrons (valence electrons). As the atoms are brought together, a distance of separation is reached at which the outermost electrons begin to interact with each other, causing degenerate valence electrons to split into states of slightly different energies. As the separation is further decreased, stronger interaction of the valence electrons occurs and splitting of core electron states can occur as well. At some equilibrium separation between the atoms, the degenerate energy levels of any particular atom will have been separated into new groupings of closely spaced states. This process is equivalent to combining overlapping atomic orbitals into new bonding and anti-bonding orbitals described by molecular orbital theory (Fig. 1). The more atoms that are brought together, the finer the splitting within groupings becomes. Considering the resulting elemental solid phase now, the number of possible states throughout depends on the number of levels an individual atom supplied multiplied times N. Because electrons can be either spin up or spin down, this number is further multiplied by 2 for the number of individual electron states. For solids, N is extremely large (~1022 atoms/cm3) resulting in numerous energy levels that are so closely spaced within the groupings that they form essentially continuous energy bands throughout the lattice. Bands can be considered groupings of crystalline orbitals, energetically separated by gaps where no electronic states are found. To measure the number of electronic states over an energy

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Figure 1. Energy level diagram illustrating the formation of energy bands in a solid from discrete energy levels in atoms. Atomic orbital energy levels of individual atoms combine to form new bonding and antibonding orbitals in a dimerized pair. Continuation of the process to form an aggregate of atoms likewise leads to new levels that are closely spaced together. The separation becomes smaller as more atoms are incorporated. In a macroscopic solid, where the number of atoms is enormous, the states are so closely spaced that they effectively form energy bands throughout the solid.

interval is to measure the density of states. For natural semiconducting minerals, where transition metals lead to partially filled d-bands, band energies are typically closely related to the energies of the free atom states and band widths can be relatively narrow. Thus, bands can often be spoken of in terms of their atomic orbital character (e.g., s-p band, d-band, etc.). Nevertheless, the close spacing and high degree of covalent bonding between the atoms can lead to bands that are significantly mixed in their atomic orbital character. All the available electrons in the system fill the energy bands starting from the lowest available state and proceeding upwards until there are no more electrons. The highest occupied states collectively form the valence band because they are typically composed of valence atomic orbital contributions. When mixing of atomic orbitals is significant, associated electrons may exhibit delocalized behavior within the mixed band which is smoothly varying across atom types in the crystal. This is especially true for valence electrons which can be viewed as being shared by the material on the whole. Orbital mixing is effectively what leads to bond covalency. Unoccupied states immediately higher in energy than the valence band form the conduction band, because of their important role in electrical conduction through metals and semiconductors. If a gap exists between the top of the valence band and the bottom of the conduction band, it is referred to simply as the band gap. It is the most important gap for the determination of the electrical properties of the material because conduction requires the close juxtaposition of filled and unfilled states (i.e., partial occupation of a band). Charge carriers usually arise from a fraction of electrons thermally excited or photo-excited to the conduction band, temporarily leaving a hole behind in the valence band; large band gaps are not amenable to such promotions. Metals have no band gap between the highest filled states and the lowest unfilled states and therefore have a very high availability of mobile charge carriers and conduct electricity very well. Semiconductor band gaps typically range from 0 < X < 2 eV (Table 1) and are still small enough to support a steady population of charge carriers. Above ~ 2 eV, most materials are insulators. The Fermi level, defined as the energy at which the chemical potential of electrons in the solid is zero, is also the principal reference level for the solid at which the binding energy of an electron is zero. This means that an electron at the Fermi energy is not bound to any particular atomic core and is free to move about the solid. Additional energy, called the workfunction, is needed to remove the electron from the solid and is

Semiconducting Mineral Surfaces:  Modeling & Experiment Table 1. Room temperature electrical properties of some naturally occurring semiconducting minerals. Mineral Name

Formula

Band Gap (eV)

Log Resistivitya (Ω⋅m)

Some Observed Conductivity Types

0.1b

-5

metallicb

-3 to 0

pc

~ 0e

n, pd,f

Magnetite

Fe3O4

Ilmenite

FeTiO3

Hematite

Fe2O3

1.9 to 2.3d

Wuestite

Fe1-xO

2.4g

Pyrolusite

MnO2

f

0.2

-3 to 1

Braunite

Mn2O3

2-3h

-1 to 0

Manganite

MnOOH

Cuprite Melaconite

Cu2O CuO

n, pd,f

-2 to -1 i

1

i,j

3

f

1 to 2

2.29 2.17

Rutile

TiO2

3.05

Uraninite

UO2

2.14k

Cassiterite

SnO2

2.2 to 4.3

n, pf

0 to 2 l

-4 to 4

m

nl pn

Troillite

FeS

0.04

Pyrite

FeS2

0.95o

-1 to 0

n, pp

Marcasite

FeS2

0.34q

-1 to 0

pq

Pyrrhotite

Fe7S8

0.2r

-4 to 0

Arsenopyrite

FeAsS

Loellingite

FeAs2

Enargite Chalcopyrite

Cu3AsS4 CuFeS2

-4 to 1 -4 to 0 1.28

s

-2 to -1

t

-3 to 2

0.53 u

np

Chalcocite

Cu2S

1.4

Digenite

Cu1.8S

1.55u

Covellite

CuS

1v

-5 to -1

nw

Millerite

NiS

0.10x

-7 to 0

py,o

Vaesite

NiS2

0.37z

Galena

PbS

0.4v

Alabandite

MnS

3.02aa

Molybdenite

MoS2

1.97o

Laurite

RuS2

1.15

-4 to 1

po

ab

n, pp

-1 to 0

n, po n, po

a

o

b

p

Carmichael (1989); unless otherwise noted Park et al. (1997); above the Verwey temperature c Andreozzi et al. (1996) d Gleitzer (1997) and references therein e Gharibi et al. (1990) f Goodenough (1971) and references therein g Lee and Oh (1991); stoichometric phase (x=0) h Tian et al. (1997); composite phases i Joseph and Pradeep (1994); optical gap j Kumar et al. (2000) k Killeen (1980) l Mishra et al. (1995) m Gosselin et al. (1976) n Shimada et al. (1998)

-6 to -2

Jaegermann and Tributsch (1988) Pridmore and Shuey (1976) q Jagadeesh and Seehra (1980) r Shirai et al. (1996) s Boldish and White (1998) t Berger and Pamplin (1991) u Nair et al. (1998); optical gap v Shuey (1975) w Rosso and Hochella (1999) x Barker and Remeika (1974); optical gap y Nakamura et al. (1994) z Kautz et al. (1972) aa Lokhande et al. (1998); optical gap ab Bichsel et al. (1984)

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discussed in a later section. The Fermi level is always located between the valence and conduction bands and all available states below it are spontaneously filled (Fig. 2). The position of the Fermi level in the band gap is dependent on the relative availability of charge carriers. For a pure semiconductor, also called an intrinsic semiconductor, the charge carriers are perfectly balanced and the Fermi level is in the middle of the band gap. The incorporation of certain impurities in a structure or the presence of bulk defects leads to bulk states in the band gap that can significantly alter the balance between electron and hole charge carriers by donating electrons (donors) to the conduction band or holes (acceptors) to the valence band. To maintain charge neutrality, the Fermi level responds by shifting to either higher or lower energy within the gap. If the electron constituent in the conduction band are the majority carriers, then the Fermi level shifts higher in energy toward the bottom of the conduction band and the semiconductor is ntype. If holes are the majority carrier, then the Fermi level shifts to lower energies towards the top of the valence band and the semiconductor is p-type. These impuritybased effects are characteristic of extrinsic semiconductors. Most natural semiconducting minerals are impure and defective to some degree and are therefore more prone to exhibit n- or p-type behavior, depending on the types of structurally compatible impurities and defects present. In some minerals, both donor and acceptor elements can be compatible with a structure leading to the ability to be either n- or p-type (e.g., Pridmore and Shuey 1976). As discussed below, these types of nominal variations can have a substantial impact of the electronic properties of the mineral surface and therefore its reactivity as well.

F

Figure 2. Sketch illustrating typical features in the density of states (DOS) of semiconducting minerals. The Fermi level (EF) marks the energy where the chemical potential of electrons is zero. Available states below EF are filled. The highest energy occupied states arise from valence orbitals of the constituent atoms, and form the valence band (VB). Above EF, available states are empty and the lowest energy unoccupied states form the conduction band (CB). The VB and CB are energetically separated by the band gap.

Surface atomic structure. The arrangement of atoms at a surface is the single most important factor affecting its physical and chemical behavior and, thus, is the clear starting point. Atoms introduced to a surface environment, either by cleavage of a solid or deposition otherwise, will spontaneously adjust their positions to minimize the free energy excess at the interface, the surface energy. A relaxation is an adjustment between layers that preserves the periodicity and lateral symmetry of the bulk material in the nearsurface region whereas a reconstruction is a wholesale rearrangement that does not. Surface energy minimization may also involve a compositional adjustment in the near surface region (e.g., creation of vacancies). All semiconductor surfaces can be expected to exhibit either relaxation or reconstruction.

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The driving force behind these structural changes can begin to be understood by treating the atoms in a solid as ions of fixed charges and using simple electrostatic arguments regarding the starting surface structure. Tasker (1979) presented a classification scheme for types of ionic or semi-ionic crystalline surfaces and discussed their stabilities in terms of their surface dipole moments. This scheme is useful and widely applicable for defining terminations of a crystal structure that will lead to the electrostatically stable surfaces (see Applications section). Surfaces can be considered as a repeating sequence of atomic planes with the sequence starting from the outermost plane and proceeding into the bulk. From any chosen starting plane, a repeat unit may be defined by constraining the repeat unit to maintain the bulk stoichiometry. If the chosen plane leads to a repeat unit that has no permanent electric dipole moment normal to the plane, then the repeat is charge neutral and the surface is electrostatically stable. Three possible surfaces arise in this context (Fig. 3). Type I surfaces consist of planes containing both cations and anions, with each plane being charge neutral. Every cut parallel to a type I surface is electrostatically stable. Type II surfaces are composed of alternating planes of anions and cations (charged planes) that are arranged in such a way that no net surface dipole moment is possible with some cuts. Type I and II surfaces are referred to as non-polar. Type III surfaces are also composed of alternating charged planes, but are arranged in such a way that there will always be a permanent dipole moment normal to the surface. A type III surface is termed polar and has an increasingly larger surface energy with increasing numbers of repeat units stacked together. Thus for crystals of any appreciable size, the electrostatic surface energy becomes huge. Type III surfaces are therefore unstable and one can begin to understand the phenomena of relaxation or reconstruction as being driven by the need to minimize the dipole moment, and therefore the excess free energy, at the surface. Type III surfaces often become nonpolar through a reconstruction or a change in the stoichiometry of the surface with respect to the bulk (see Noguera 2000). This general description is incomplete for several reasons. Electrons also redistribute at the interface, leading to an additional means of achieving decreases in the surface

Figure 3. Three types of surface structures based on electrostatics along the surface normal arising from the arrangement of cations and anions in the bulk (Tasker 1979). The surface plane is represented by the dashed line. A repeat unit can be found proceeding from the surface inward (maintaining the bulk stoichiometry) each of which has a different dipole structure. Type I surfaces consist of stacks of charge neutral atomic planes. Type II surfaces consist of alternating charged planes of cations and anions that are arranged is such a way as to have no net dipole moment. Both type I and type II surfaces are electrostatically stable. The repeat unit for type III surfaces has a non-zero dipole moment. Stacking these units causes the dipole moment at the surface to increase and diverge rapidly for crystals of macroscopic sizes. Type III surfaces are polar and unstable.

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energy. This is especially important for semiconductors because of the typical strongly covalent bonding interactions that stabilize these structures. For ionic crystals, charge distributions around atoms can be said to change very little from that for the isolated constituent ions. In contrast, although ionic bonding character may be present in certain semiconductors, more characteristic electron sharing in covalent bonds leads to delocalization of electronic density from the constituent atoms over the bonding regions of the lattice. The decreased number of bonds around atoms at a semiconductor surface therefore presents a substantially different environment for electrons. Hence, altering the charge distribution at the surface can be an important source of surface energy decreases for these materials. For example, electrons will be highly driven to avoid spending time outside the solid in orbitals dangling into empty space where a former bond may have been, so-called dangling bonds. By breaking bonds to form a surface, electrons left to occupy dangling bonds will attempt to transfer out of these high energy environments into any available lower energy orbitals. This may involve backfilling remaining bonds or electron transfer between surface atoms. Surface energy decreases arising from the redistribution of charge by these electronic means can be considerable, and may even be sufficient to stabilize polar surfaces. If not, then a large driving force remains for the creation of new bonds (i.e., resulting from a reconstruction). This electronic drive to redistribute dangling bond electrons into bonding orbitals is thought to be the most fundamental driving force behind surface relaxations and reconstructions in semiconductors (LaFemina and Duke 1991). In fact, where a variety of surface terminations are possible, surface energies may often be energetically ranked relative to one another based solely on the number of dangling bonds produced (Zangwill 1988). When combined with the electrostatic arguments of Tasker (1979), one has a framework to make intelligent decisions about surface structures, their stabilities, and the tendency to reconstruct. At this point, it should be noted that energy barriers to reconstruction are often present which can impede transformation to a lower energy surface structure. This leads to the important consideration, for both experiment and molecular modeling methods, that any particular surface structure may only be metastable with respect to the lowest energy arrangement. Also, the formation of new bonds at a surface entails changes in the lengths and angles of other bonds at the surface. As a result, surface energy compensation arising from redistribution of electrons into new bonds directly competes with energy increases associated with straining other surface bonds (Gibson and LaFemina 1996). The interplay between these electronic processes is another important consideration for semiconductor surfaces because of the highly directional nature of covalent bonds. The adsorption of charged ions at the surface can also provide electrostatic stabilization of what would otherwise be a polar, unstable surface. Surface dipole moments can be compensated by the dipoles of water molecules in the solvent layers near the surface. The specific adsorption of uncharged, relatively unreactive species at a relaxed surface can effectively restore the coordination sphere of underbonded surface atoms and thereby stabilize the surface back towards a pre-relaxed configuration (Noguera 1996). Electrons occupying dangling bonds can be stabilized in new bonds formed between surface atoms and adsorbates. Charge transfer into or from adsorbates may substantially change the strength of surface bonds and can alter the tendency to reconstruct. Complicating issues further is the fact that adsorbate-induced surface structures can be coverage dependent as well. To varying degrees, these effects may result in a structurally different surface than what is stable in the vacuum-terminated case. An implication is that the chemical behavior of a mineral surface can depend on its environment. Although true, it is not intended to leave the reader with the impression that this precludes the ability to establish relationships between the structure of a surface in

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Figure 4. A one-dimensional model for a surface state, modified from Zangwill (1988). Black dots represent the lattice of atoms in a crystal, forming the periodic potential field for constituent electrons. Electronic states are perfectly oscillating over this field for an infinite crystal as indicated by the finely dashed wave. Where the lattice is terminated at a surface, new electronic wave functions are formed which are distinctly different from those in the bulk, represented by the coarsely dashed wave. These are localized in the region of the surface, decaying both into the bulk and outward from the surface.

different environments. In fact, there are many good correspondences on structural and reactivity issues between studies considering the vacuum-terminated and adsorbatecovered (or in solution) cases (Stuve and Kizhakevariam 1993). Some of these correspondences will be examined in the Applications section. Surface electronic structure. For semiconductors, a surface imposes a significant perturbation to the electronic structure of the crystal in the surface region. The discontinuation of the periodic potential of the crystal lattice leads to new electronic states that are localized at the surface which decay both into the bulk and outward from the surface (Fig. 4). Surface states can arise from a variety of other sources including surface reconstructions, defects, adsorbates, or discrete surface phases. The lateral density of surface states is often directly related to the surface free energy (Zangwill 1988). If a surface reconstructs to lower its surface energy then the bonding for surface atoms will be different than for bulk atoms and surface states will likely be present. Surface defects such as vacancies or the intersection of bulk defects with the surface plane such as dislocations can produce surface states. Surface reaction products can form surface states. Another example of an important type of surface state for semiconductors is the dangling bond. When bonds in a crystal are broken during cleavage to form a surface, crystalline orbitals localized on surface atoms destabilize back towards their free atom orbital character and energies. Because bonding states usually lie in the valence band, bond breaking tends to energetically displace surface states into the gap (Fig. 5). For binary semiconductors (two component), there are typically two surface states created, one associated with the dangling bond on the cation and the other with the dangling bond on the anion. This leads to either a reduction of the band gap at the surface, or causes isolated states to appear within the gap. As long as the conditions are such that these dangling bonds cannot form new bonds, either through recombining with each other through a surface reconstruction or combining with adsorbate orbitals, they will be present at the surface. The reactivity of the surface with respect to heterogeneous redox reactions is often significantly altered by the presence of surface states. The highest occupied states at the

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Figure 5. Sketch illustrating how surface states are typically manifested in the density of states at a semiconducting surface. Surface states are often located in the bulk band gap, causing the gap to narrow at the surface. Their location and the degree of narrowing can vary, including essentially no band gap narrowing to narrowing completely, i.e., surface states located at EF. In the latter case, a metallic surface is created, where partially occupied surface states allow metallic conduction along the surface plane.

surface are energetically predisposed to be the source of electrons in an oxidation reaction, and the lowest unoccupied states are predisposed to accept electrons in a reduction reaction. Following an analogy with frontier molecular orbital theory, the former constitutes the HOMO and the latter constitutes the LUMO. For a redox reaction to proceed, the HOMO must be energetically higher than the LUMO and the orbital symmetries of the two must be compatible. The simple presence of surface states can alter both and impart a fundamentally new propensity for a surface to react in electron transfer reactions. For example, a surface oxidation reaction may be driven to saturate dangling bond surface states. This may lead to a new surface phase that can either passivate the surface towards continued oxidation, catalyze further reaction, or alter the oxidation mechanism. Because of the varied origins of surface states, it follows that they can be present under a variety of conditions, from UHV to solution. Therefore, they form one of the most important properties of semiconductor surfaces to be considered. The surface responds to the availability of new states by redistributing charge. This electronic energy minimization process is best thought of as the rearrangement of electron density around the near surface atoms, driven by the need to re-establish charge neutrality at the surface. It may involve charge transfer between surface atoms and/or between surface and bulk atoms. The former is a process local to the uppermost surface and is an issue of relative electron affinities of the surface atoms. In this local context, it is reasonable to assume that surface state electrons would prefer to shift from surface cations to anions for reasons of electronegativity. It is common for the anion dangling bond states to lie energetically below the Fermi level and cation states above, therefore a degree of electronic energy stabilization arises from charge transfer from cations to anions. This assumption is implicit in the electron counting principles of surface autocompensation which have found great success in predicting stable surface structures (Harrison 1980; Pashley 1989; Gibson and LaFemina 1996). The model predicts that a stable surface results when the number of available electrons in the surface layer exactly fills all the dangling-bonds below the Fermi level, i.e., localized on the anion. Such surfaces are said to be autocompensated and are, by default, uncharged with respect to the bulk. The charge redistribution may also partially involve charge shifting from surface atoms to bulk atoms. Dangling bond electrons may shift out of surface states and into existing bonds in a process known as backbonding. These established principles work quite well for understanding the electronic structure and stability of pure semiconductor surfaces.

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Another driving force is present to alter the population of surface states. Conduction band electrons or bulk donor electrons will populate any lower lying empty surface states made available to them to lower their energy. Likewise, valence band holes will “displace” electrons from higher lying filled surface states (equivalent to allowing electrons to drain from filled surface states). For extrinsic semiconductors, where an imbalance is present between the types of charge carriers, a net charge flow results between the surface and bulk where majority carriers are spontaneously driven into surface states. This process is known as charge trapping. For many extrinsic semiconductors, the carrier density at the surface (~108–1012/cm2) is significantly lower than the surface state density (~1015/cm2). Hence the driving force to fill the surface states can strip a wide near-surface region of charge carriers. The process is then accompanied by the formation of a surface potential extending from the surface into the bulk as the charge distribution at surface atoms increasingly becomes different from the near surface bulk atoms. For metals, surface charge is screened from bulk atoms by the high concentration of conduction band electrons. Therefore the potential drop occurs entirely at the surface. For semiconductors, the surface potential can penetrate several hundred nanometers into the bulk. The resulting space charge layer gradually modifies the energies of the bulk bands across that region resulting in what is known as band bending. Considering as an example a surface of an n-type sample with an electron accepting surface state below the Fermi level (Fig. 6). The flow of conduction band electrons into lower lying empty surface states strips the electron donor atoms, causing the surface to become more negative and leaving behind a positively charged depletion layer. Viewing this layer as highly resistive to carrier electrons, or equivalently viewing the surface atoms as being more negatively charged than the bulk atoms, an upward bending of the bulk bands ensues as they approach the surface. At some point, thermodynamic equilibrium is achieved between the energy gains arising from infilling the surface states and the increasing energy cost to extract electrons from the bulk. If the surface state density is high enough, electrons may

Figure 6. Energy level diagram illustrating the formation of band bending due to surface states in an ntype semiconductor. Impurities in the structure lead to electron donor states in the band gap (ED), giving rise to a small population of electrons at the bottom of the conduction band. These constitute the majority charge carriers and electron holes are the minority carriers. If empty states become available in the band gap, such as from surface states, a charge flow ensues of conduction band electrons into the available states. For semiconductors, surface state densities are typically much higher than charge carrier densities and the flow strips a wide surface region of conduction band electrons. Electrons infilling the surface states form an increasingly negative potential at the surface that can be viewed as repulsive to conduction band electrons, manifested by an upward bending of the bulk bands as they approach the surface (EBB).

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be drawn from the valence band as well, which would be represented in Figure 6 as contact between the upward bending valence band edge and the infilling surface state. If this was the case, the band bending would cause the Fermi level to be closer to the valence band edge than the conduction band edge at the surface, converting it to p-type, resulting in an inversion layer. For surfaces where band bending is severe (on the order of tenths of volts), electron binding energy shifts can be directly observed in experimental surface core-level photoelectron spectra (SCLS). This is because the energy of photoejected core electrons is modified by the potential at the surface. In the previous discussion, we have ignored the situation where a semiconductor surface is in contact with another phase such as a solution or another solid. In this section, we briefly turn our attention to the effects of another important spontaneous driving force for charge redistribution at semiconductor surfaces, establishing thermodynamic equilibrium at an interface between two phases. At equilibrium, the chemical potential of electrons in both phases must be equal. If this is not the case at contact, then charge will redistribute between the two phases with many similarities to the surface-bulk equilibration process described above. As charge flows, the electronic states of the two phases shift relative to one another until the Fermi level of the semiconductor matches the equivalent energy in the opposite phase. In semiconductors, should electrons be allowed to reside at the Fermi level, in surface states for example, then they are completely unbound within the solid surface and can be viewed as free to move about the surface. If the opposing phase at the interface is a solution, there are no free electrons in this sense. At this point, it is useful to define a reference energy external to the interface. The most useful reference is the energy of an isolated electron at rest in a vacuum, or the so-called vacuum level (EV). In solid state physics and in ab initio modeling, this energy is 0 eV. For solutions, the vacuum level is related to solution phase redox potentials by the halfcell potential of the standard hydrogen electrode (SHE) by Ev = EoSHE + 4.5 V (Gratzel 1989). In other words, the pseudo-Fermi level of the solution is zero on the standard hydrogen electrode (SHE) scale which aligns with the Fermi level of the semiconductor at equilibrium. For the semiconductor, the Fermi level may be located with respect to the vacuum level by the workfunction (φ). The workfunction is defined for any solid phase as the energy difference between an electron at the Fermi level in the solid and the vacuum level (Fig. 7). Although it is defined independent of the location of the removed electron (i.e., bulk vs. surface), when it is experimentally measured there are specific surface contributions to φ that arise from the fact that an electron at the Fermi level will interact with the particles in the solid as it is removed. This interaction continues well outside the solid and is sensitive to the surface structure and charge distribution, causing φ for any

Figure 7. Energy level diagram illustrating the definition of the workfunction (φ) and electron affinity (χ) of semiconductors. Both are referenced to the vacuum level (EV). χ is based on the energy of the bottom of the conduction band at the surface.

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single crystal to be anisotropic depending on the particular surface considered. It is also quite sensitive to adsorbates, which decrease or increase the workfunction depending on whether the negative end of adsorbate dipoles are oriented toward or away from surface plane, respectively. Changes in the value of the workfunction can therefore generally be thought of as a measure of changes in the affinity of a surface for an electron. For metals, this is exactly the case. For semiconductors, because states may or may not be available at the Fermi level, a more general definition of the electron affinity (χ) of the surface is required, namely the energy to remove an electron from highest occupied state to the vacuum level (Fig. 7). At room temperature in any semiconductor, regardless of n- or ptype behavior, this typically is the bottom of the conduction band (Sze 1981). With the Fermi levels of the two phases thermodynamically referenced to the vacuum level, the direction of charge flow at an interface and its concomitant effects can be determined. A relatively simple but important starting example is the metal-insulatorsemiconductor interface used in STM. The insulator can be a vacuum gap or another phase that can be ignored for the purposes of this discussion. Consider the case where a n-type semiconductor equilibrates with a metal probe across an electron tunneling contact (Fig. 8a). Because of the n-type characteristic, the semiconductor workfunction will usually be lower than for the metal. As the distance between the metal tip and the semiconductor surface is reduced and electron tunneling contact is established, charge carrier electrons will transfer from semiconductor conduction band to the metal conduction band until the Fermi levels match. Because this process can be thought of as equivalent to conduction band electrons infilling surface states discussed earlier, the tip can be viewed as becoming increasingly negatively charged with respect to the semiconductor. Likewise, a positive space charge layer then builds up in the semiconductor near-surface that upwardly bends the bulk bands. The upwardly bent conduction band can be viewed as a barrier for further transfer of conduction band

Figure 8. Energy level diagram illustrating tip-induced changes in the electronic structure of an n-type semiconductor surface with (a) and without (b) the presence of surface states. The separation between the metal tip and the sample decreases from the top of the figure to the bottom. At large separations, the tip and sample are not in electrical contact and out of equilibrium. Decreasing the separation to the point of tunneling contact causes conduction band electrons to flow from the sample to the tip until the two Fermi levels match. Without surface states, the build up of negative charge on the tip causes the sample bands to bend upwards, with a sensitive dependence on the separation distance. With surface states, electrons bound at the sample surface can screen the influence of the potential at the tip, leaving bands in the sample unaffected.

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electrons to the tip. The degree of band bending will depend on the final tip-sample separation. This tip-induced band bending is effectively an electrostatic coupling of the tip to the sample (Kaiser et al. 1988; Weimer et al. 1989; McEllistrem et al. 1993). The associated modification of band positions of the sample is usually an unwanted effect in STM and can be a significant source of discrepancies between STM and molecular modeling attempts to determine the surface electronic structure. If a bias voltage is applied across the interface, the band bending is free to be modified by this voltage and the surface is, in a sense, unpinned. This has deleterious effects on the collection of I(V) spectra where the tunneling current (I) is acquired over a bias voltage ramp (V) while the tip-sample distance is held constant. In this case, the applied voltage ramp continuously modifies the band bending and the facility with which current can flow across the interface. However, for semiconductors, if occupied surface states are present in the band gap, the surface bound electrons will often be numerous enough to effectively screen the semiconductor bulk states from the buildup of negative charge at the tip surface, including that induced by the bias voltage. In this case, the semiconductor bands at the interface are less likely to be affected by the presence of the tip (Fig. 8b). For interfaces with electrolyte solutions, the equilibration process is more complicated because the semiconductor surface charge induces a double layer of opposite charge (ions) in the solution called the Helmholtz layer (e.g., Miller et al. 1995). Semiconductor-solution interfaces can be viewed as a balance of two opposing double layers of charge across the interface, one in the semiconductor and one in the solution (Fig. 9). In a narrow region within a few Ångstroms at the interface, the semiconductor surface charge is matched by a layer of adsorbed species having a net opposite charge. Similar to the above example, if the n-type semiconductor Fermi level is higher than the pseudo-Fermi level of the solution, equilibration involves the transfer of electrons from the semiconductor to the solution until the Fermi levels are matched. However, in this case, the potential drop across the Helmholtz layer affects the degree of band bending. The equilibrium surface charge on the semiconductor results from the interplay between the majority carrier concentration, the surface state density, and adsorbed species in the Helmholtz layer. Because the carrier concentration is reflected in the position of the Fermi level, it is intuitive to determine if a relationship exists between the surface workfunction and adsorption. Indeed, a well known linear relationship occurs between the workfunction and the point of zero charge in solution with a slope near unity (Bockris and Reddy 1973; Morrison 1976; Parsons 1987; see also Mullins and Averbach 1988; Mullins 1989). This relationship has been observed for many types of materials from metals to insulators, but perhaps is most significant in the study of semiconductor surface reactivity where the Fermi level position is a highly variable electronic characteristic of the material. THEORETICAL METHODS Current computing resources often limit the range of properties and reactions that can be modeled on semiconducting mineral surfaces. Most molecular modeling applications are restricted to address properties and processes at a thin near-surface region of the material. Here, some flexibility in the number of atoms to be considered is available based on the particular semiconductor and the property being addressed. Although the focus of this review is on ab initio methods, it should be mentioned that molecular mechanics calculations (interactions based on parameterized potentials) have the ability to treat certain problems at semiconductor surfaces (see Rustad, this volume). For this application, the accuracy can be expected to depend on whether or not the

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Figure 9. Energy level diagram illustrating the semiconductor-electrolyte interface, modified from Miller et al. (1995). Two opposing double layers of charge are set up, one in the form of the semiconductor surface potential, and the other in the form of ordering of ions in solution. The H+/H2 redox couple is the pseudo-Fermi level of the solution. In this case, because the sample is n-type, equilibration at the interface involves electron transfer from the semiconductor to the solution, forming the upward band bending and the positive space charge layer in the solid surface. The Helmholtz layer is a layer of specifically bound ions at the surface. It causes a potential drop (EH) across the interface that modifies the band bending in the semiconductor. The relationship between the electrochemical scale (SHE) and the vacuum scale is shown graphically.

property modeled is best treated in the ionic limit. For example, good accuracy can be expected in the treatment of surface relaxation where the driving force is largely electrostatic and all other possible electronic contributions to the surface structure happen to be small by comparison. It should be pointed out that no current molecular modeling method has the capacity to capture the properties of all types of semiconductor surfaces at the same time. The space charge layer is a primary example, whose inherent dispersion over the thick surface region entails too large a section to consider explicitly using an atomistic treatment. Local effects can be treated such as the influence of impurities or defects in the structure and properties of the surface. However, because the effect of the space charge layer is largely to rigidly shift bands at the surface, ab initio modeling results for the near-surface region can usually be reconciled with experiments on samples with extrinsic modifications. For most semiconductors, electronic effects such as surface states and charge redistribution cannot be neglected. Therefore, ab initio theory is a more general approach to these systems. Theory–Hartree-Fock versus density functional theory In the current section, two ab initio methods that are currently in common use for surface applications are reviewed. A detailed presentation of the quantum mechanical foundations and mathematical implementation of modern ab initio codes is beyond the scope of this review. The interested reader is referred to Hehre et al. (1986), Sherman

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(this volume), and Xiao (this volume) for Hartree-Fock (HF) methods and Parr and Yang (1989), Sherman (this volume), and Stixrude (this volume) for Density Functional Theory (DFT). The focus of this discussion is on the similarities and differences between HF and DFT methods that are important considerations in their practical use for semiconductor surfaces. The long standing ab initio method of choice has been HF, primarily due to good computational performance and because of the ability to provide usefully accurate results in a wide range of applications. HF performs well for many types of systems ranging from atoms to molecules to solids. Hence, it is not surprising that it forms the basis for many modern ab initio methods. It utilizes the so-called one-electron method that assumes each electron can be considered separately, moving in a mean potential field. One electron basis functions (discussed later) are expanded into orbitals and the total wave function for a system. The potential field approximates the effects of all the other particles in the system on the electron. This approach models nuclear screening, electron-nuclei, and nuclei-nuclei interactions accurately but electron-electron interactions are incompletely described. Two important types of electron-electron interactions are electron exchange and electron correlation. Electron exchange is a short range interaction between electrons that comes as a consequence of the set of quantum mechanical rules that govern the possible set of quantum numbers for many-particle systems (e.g., the Pauli exclusion principle). To treat the exchange problem properly, electrons of like spin must be spatially separated in the ab initio calculation, which gives rise to a slight reduction in the total energy called the exchange energy. An exact expression for electron exchange is included in the HF approach. Electron correlation is the mutual Coulombic repulsion between electrons that has the effect of spreading out the electrons. Correlation is not included in the HF approach which leads to an overbinding of electrons to nuclei. This deficiency is the primary drawback of HF which is overcome by DFT. DFT was developed by Hohenberg and Kohn (1964) and Kohn and Sham (1965). In the former contribution, it was proved that the ground state wave function and total energy can be derived entirely from knowledge of the electron density distribution, while the latter developed the highly useful equations for the practical implementation of DFT. In pure DFT, the ground state wave function and energy are expressed as functionals of the electron density distribution, which is typically solved self-consistently using the Kohn-Sham equations. In practice for DFT, an exchange-correlation functional is generated by parameterizing a combination of electron correlation and exchange terms from the self-consistent electron density. Various functionals are available and they are generally of two types, with important differences in approximation between them. The original implementation of DFT utilized the local density approximation (LDA) to describe the exchange-correlation energy (EXC). This assumes that EXC at a point in a system is equal to EXC in a uniform electron gas of the same density. Several parameterizations of the LDA are available based on slightly different treatments of the electron gas, but each functional leads to similar results in practice (Payne et al. 1992). A comparison between HF and some LDA approaches against experiment shows that LDA performed slightly better in the description of atomic wave functions. Using energy difference calculations, LDA also outperformed HF for atomic ionization energies but HF outperformed LDA in total energy calculations for atoms (Jones and Gunnarsson 1989). LDA is well known to perform very well for structural properties and total energies of solids. Bulk moduli, bond lengths, and lattice constants for crystalline compounds are accurately modeled using LDA, but it typically overestimates cohesion energies (Jones and Gunnarsson 1989), i.e., the energy to infinitely separate the constituent atoms of the solid. In fact, LDA will generally overbind atoms in most types of model systems, leading to overestimates of

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bond energies. This occurs because LDA, while successful to describe regions of high electron density near the core, fails to properly describe the decay of electron density in the low density region, i.e., wave function tails. This leads to the overall good performance for tightly bonded systems such as solids, but poor performance for more weakly interacting particles and at surfaces. Correcting this deficiency was the principal motivation for the development of the generalized gradient approximation (GGA), the second type of functional. GGA attempts to correct LDA by incorporating a dependence on the gradient of the electron density that more accurately describes the exponential decay in the low density region. This resulted in significantly improving calculated bond energies. Furthermore, for high density regions, GGA differs very little from LDA and therefore it performs nearly as well for properties that depend mostly on the local density such as bond lengths. For magnetic structure calculations, LSDA (local spin density approximation) and GGS (generalized gradient spin) are the spin unrestricted counterparts of LDA and GGA. Also, it is fairly well established that gradient corrected methods are superior to LDA for surface applications and systems with hydrogen-bonded components. Although the correlation energy makes up a relatively small part of the total energy of a compound, differences in correlation energy can be a significant part of total energy difference calculations involving different bonding configurations. For structural changes that conserve the numbers and types of bonds, the correlation energy tends to remain constant (Parr and Yang 1989). When this is not the case, the correlation energy difference is often substantial. This general rule is probably the best first criterion when deciding between HF and DFT methods. Strongly correlated systems, such as semiconductors with highly mixed atomic orbital contributions to the valence band, are best treated using DFT. Moreover, if electron correlation is adequately accounted for in the calculation, theoretical estimates of surface workfunctions can agree with experiment to within 0.1 eV (Prutton 1994). As is the case with any type of computational method, because none is yet perfect, one at least must be able to rely on predictable shortcomings. A general failure in both the HF and DFT methods is to calculate band gaps in semiconductors. Band gaps are typically greatly overestimated by HF and underestimated by LDA by about 50-60% (Bertoni 1990). GGA can improve the latter by about 20% but it is considered to be mostly fortuitous. That these deficiencies could be the case is somewhat intuitive when it is considered that these approaches are ground state one electron methods that only approximate many particle interactions by single particle-mean field methods. HF theory includes an exact treatment of the electron exchange energy that stabilizes occupied states only, leading to band gap overestimation. Accurately estimating band gap energies often requires excited state calculations (such as configuration interaction calculations) that can treat contributions from, among other things, the energy to create an electron hole near the top of the valence band. Such methods are available but quickly become overly computationally expensive for practical use on semiconductor surfaces. This does not preclude the usefulness of the one-electron methods to model surface states however, and much success in this area has been demonstrated (Bertoni 1990; LaFemina 1992; Prutton 1994). Finally, it is also noteworthy that observed relationships between structure, reactivity, and the topological properties of electron density distributions have close ties to DFT (Parr and Yang 1989; Bader 1990). DFT expresses the total energy of a compound from the electron density. This idea was extended by Bader to calculate the energy and chemical properties of subsystems of the compound, namely atoms, based on their electron density distributions. The theory defines the boundaries of its constituent

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atoms by three dimensional surfaces of zero flux in the gradient field of the electron density, so-called zero flux surfaces. Such a surface also bounds the complete system at infinity, in a sense resulting in a framework that treats the compound and its constituent atoms on an equal footing (Bader 1990). Therefore, Bader's methods are a natural extension of DFT. The reader is referred to Gibbs et al. (this volume) for insight into the utility of this approach for geoscience applications. Basis sets–Gaussian orbital versus plane waves As described earlier, electronic states of atoms are dispersed in energy bands when condensed into a crystalline solid phase. The descriptive model chosen allows a qualitative understanding of the origins of bands, but neglects many details such as the effects of atomic periodicity and crystal symmetry. Quantitative treatment of an allowed electronic state in a solid (ψ) is based on Bloch's theorem which states that ψ(r) = eik⋅r u(r) where r is a location in the unit cell, k is the wave vector, and u(r) represents a periodic electrostatic potential (see Ashcroft and Mermin 1976). The term eik⋅r represents the oscillating part of the wave and u(r) represents the potential that modulates the wave. The expression describes the continuous wave-like motion of an electron moving through the periodic potential field created by an array of nuclei (Fig. 10). To continue in much more detail is beyond the scope of this review, except for some important related points. The wavevector k refers to lattice vectors in the reciprocal unit cell which define the size and shape of the zone over which the states are continuous. For a particular structure, a unique set of k-points exists over which the state energies are dispersed into bands. Accurate total energy calculations for solids require the calculation of the electronic states over an adequate sampling of k-points. This applies primarily to what now should be defined as a solid state or band structure type of ab initio code, where the calculation involves a periodic treatment of the system. For the implementation of Bloch's theorem in band structure codes, a mathematically convenient means to express an electronic state in a solid is required to minimize the computational overhead. In practice, this is accomplished using a finite set of analytic functions to construct (or expand) the electronic state in Bloch wave form. Two prevalent methods are used for the expansion. One method utilizes a set of linearly combined atomic orbital functions (LCAO) (local functions) and the other utilizes a set of plane waves (periodic functions). The set of functions comprising these expansions is called a basis set. It is worthwhile to visit basic concepts behind these two types of basis sets to discuss important considerations in their application to surfaces.

Figure 10. Sketch illustrating the periodic potential of ions in a crystalline structure along one dimension, modified from Ashcroft and Mermin (1976). Repeating ions are represented by the black dots (a = periodicity). The solid curves show the potential along the line of ions and the dashed curve shows the potential between the planes of ions.

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The LCAO method uses functions localized at nuclei to represent atomic orbitals (φ). Atomic orbitals are known to be well described by Slater-type functions. An example of a Slater-type orbital (STO) is a 1s orbital, which can be written as φ1s STO(ζ,r) = (ζ3/π)0.5exp(-ζr) where ζ is the exponent and r is the radial distance from the nucleus. While very accurate, these types of functions are too mathematically cumbersome for use in ab initio codes. The most common recourse is to mimic this form using gaussian functions. Atomic orbital functions expanded in this way are called gaussian-type orbitals (GTO's). By comparison with the STO above, a 1s GTO has the form φ1s GTO(α,r) = (2α/π)0.75exp(-αr2), where α is the exponent. The largest differences between the two occur at r = 0 and large r. At r = 0, the STO has a finite slope while the GTO has a zero slope. At large r, the exp(-αr2) dependence of GTO's deviates substantially from the exp(-ζr) dependence of STO's, with the GTO overestimating the orbital dropoff. This problem is solved by utilizing more gaussian functions, which are linearly combined (or contracted) to give a net resulting GTO that properly mimics the true behavior. This typically involves at least a relatively narrow gaussian function to correct the r = 0 deviation, and a broad gaussian function to correct the decay deviation at large r. The width of each gaussian function is determined by their respective exponents α, with smaller values of α resulting in wider (or more diffuse) functions. An example of a contracted GTO is the STO-3G 1s orbital for hydrogen, where the Slater behavior (ζ = 1) is approached using a combination of three gaussian functions with exponents 2.2, 0.41, and 0.11 (Fig. 11). As a rule of thumb, for any particular orbital, a satisfactorily complete description begins when exponents are incremented by no more than a factor of ~ 3-4, assuming they are scaled properly to the orbital. Diffuse functions are often required to adequately describe the correct asymptotic behavior of valence atomic orbitals where relatively loosely bound electrons reside. For large elements and transition metals, one often uses a basis set that replaces the core state functions with a simple spherical potential that mimics the electrostatic properties of the screened core. This potential function, called a pseudopotential, therefore sacrifices an explicit treatment of core states to reduce the computational burden. Because core states usually do not play a substantial role in bonding in a solid, the sacrifice is minimal and little accuracy is lost. In fact, basis sets utilizing pseudopotentials are considered more accurate for heavy elements where relativistic effects on core states must be included. For more details on GTO's, the reader is referred to Hehre et al. (1986) and Xiao (this volume). A completely different method of expanding electronic states in solids is to use a plane wave basis set. In contrast to localized gaussian functions, plane waves are continuous oscillating functions that are, in fact, Bloch functions in their generalized form. In principle, Bloch's theorem indicates that an infinite number of plane waves is required to completely describe the electronic wave functions (Payne et al. 1992). However, in practice a finite set of plane waves is sufficient. To illustrate how a finite set of continuous waves can be used to represent an electronic state, consider the following conceptual exercise. Building on the illustration of the H 1s orbital discussed in the previous paragraph, it can be shown that the Slater functional form of the H 1s orbital can be represented by a truncated Fourier series of a relatively small number of continuous waves (cosines). A function F(r) can be found that results from a summation of cosine waves each multiplied by some expansion coefficient f(n), where n is the expansion number. The differences in the behavior of F(r) due to changing the truncation of the series is shown in Figure 12. Even at small values of n, F(r) rapidly approaches the behavior of the Slater function. At n = 32, the fit is perfect by inspection. Increasing the value of n leads to diminishing returns in the improvement of fit. Thus, at some finite cutoff the number of component waves becomes sufficient to represent this orbital in a usefully accurate way.

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Figure 11. Graph of the H 1s Slater-type orbital (STO) and its approximation using a contracted gaussiantype orbital (GTO). Three gaussian functions, centered at r = 0, are allowed to vary in a fitting routine that adjusts their relative heights (leading to the contraction coefficient) and widths (exponents) until their sum best matches the STO behavior. The best fit is found with the exponents 0.11, 0.41, and 2.2 for gaussians 1-3 respectively, giving the STO-3G basis set. Although in this case the decay of the STO is well approximated by the STO-3G basis set, the cusp near r = 0 is not.

Figure 12. Graph of the expansion of the H 1s STO in terms of a truncated Fourier series of cosine waves. The truncation of the expansion (n) is the number of waves comprising each curve. The fit is improved with increasing n. The curve for n = 32 overlays the STO function perfectly by inspection. This concept is analogous to the use of a truncated series of plane waves to expand electronic states in a crystalline system.

This concept is analogous to the use of a truncated plane wave basis set to describe an electronic state in a solid. The truncation of a plane wave basis set is defined by a kinetic energy cutoff (or hereafter just the energy cutoff), which can be thought of as a frequency cutoff or the expansion number n in the above exercise. For electronic states in solids, it turns out that the expansion coefficients for lower energy plane waves (lower frequency) are more significant than those for higher energy plane waves (Payne et al. 1992). It should be noted here that higher energy plane waves are important to describe the rapid oscillation of states in the vicinity of the core but, as will be discussed below,

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this is remedied using core pseudopotentials. This effectively means that the modulation of the plane waves over the periodicity of the bonds in a structure is more important than the higher frequency modulation associated with fine structure within bonds. This fact is what leads to the success of using a truncated plane wave basis set. Thus, the above exercise is probably an exaggeration how finely modulated plane waves would be for the solid state case. Nevertheless, higher cutoff energies always amount to more completeness in the basis set and inevitably lead to more accurate wave functions and total energies. In this way, plane wave basis sets are analogous to gaussian orbital basis sets in that the final description of the electronic states in a solid improves with increasing numbers of basis functions. Also similar to gaussian orbitals, pseudopotentials can be used to mimic the electronic properties of the screened cores. But in the case of a plane wave basis set, pseudopotentials are essential and drastically reduce the number of plane waves required to accurately compute the total energy of the system. In the vicinity of an atomic core, the large potential causes core states to be tightly bound and therefore highly “cusped”, and valence electronic states are caused to be rapidly oscillating (Payne et al. 1992). Therefore, to expand the wave functions with plane waves in the vicinity of the core would require a prohibitively large cutoff energy for systems of any appreciable size. Instead, the cusps are effectively removed by replacing the large core potential with a much weaker one within some cutoff distance (rc) that acts on a set of pseudo wave functions (Fig. 13). The potential is chosen such that the pseudo wave functions are smoothly decaying in the region of the cores, but also best match the behavior of the true valence wave functions outside the cores. Plane wave pseudopotentials may be designed to be local or non-local, depending on whether or not they have a built in dependence on the angular momentum of the valence electrons (direction dependence). The former uses a direction independent spherical potential and are therefore computationally less expensive, but also generally less accurate. The non-local pseudopotentials were developed to overcome this deficiency in the core description, but require increasing the computational demand. Pseudopotentials may also be norm-conserving if designed in such a way that the electron density in the core region calculated using the pseudo wave functions equals that calculated using the true core wave functions. This guarantees that the match between the wave functions

Figure 13. Sketch illustrating the plane wave pseudopotential concept, modified from Payne et al. (1992). The all electron wave function (ψV) is rapidly oscillating at small r due to the strong ionic potential of the core (Z/r). A very large number of plane waves would be required to mimic this cusped behavior. Instead, the Z/r potential of the core is replaced by a weaker pseudopotential (Vpseudo) that acts on a set of pseudo wave functions (ψpseudo) up to the cutoff distance rc. Vpseudo is chosen such that ψpseudo is smoothly varying inside the core and best matches the behavior of ψV outside the core.

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outside the core is exact. Non-local norm-conserving pseudopotentials often come in the form described by Kleinman and Bylander (1982) which are known to perform quite well for many types of systems. This form allows accurate pseudopotentials to be generated for atoms in a wide variety of chemical environments and is therefore said to be highly transferable or soft. Another efficient and accurate non-local pseudopotential was put forth by Vanderbilt (1990). These are constructed in such a way that the smoothness of the pseudo wave function can be optimized, made possible by relaxing the normconservation constraint. This allows larger rc values and fewer plane waves to expand the wave function (Fig. 14). The resulting ultrasoft pseudopotential effectively speeds up calculations at little sacrifice in accuracy for many applications. The reduction in the number of plane waves arising from the use of pseudopotentials leads to the efficacy of plane wave methods in ab initio calculations. It also turns out that expanding electronic states using plane waves leads to a particularly simple expression of the Kohn-Sham equations (Payne et al. 1992). Plane wave pseudopotential based codes can therefore deliver the full accuracy of DFT to total energy calculations of a variety of large and complicated systems. A plane wave basis set has many important advantages over the local orbital method. Perhaps the foremost benefit is the fact that the plane wave basis set is universal and depends only on the energy cutoff, as opposed to the local orbital method where basis sets must be empirically constructed for each individual atom. Moreover, several significant advances in plane wave DFT methods were developed by Car and Parrinello (1985). Among other things, they showed that the minimization of the total energy with respect to the total wave function and atomic coordinates can be solved simultaneously, and that molecular dynamics can also be performed at only a slight increase in computational expense. But as with any modeling method, artifacts are present that must be properly managed. Because a plane wave basis set is necessarily incomplete, a small error is introduced in the computed total energy. This also leads to the possibility of an artificial non-zero contribution to the stress on the unit cell, called the Pulay stress (Payne et al. 1992). The magnitude of the total energy error is estimated by re-performing the calculations using different cutoff energies and comparing the total energies computed. Increasing the cutoff energy will usually decrease the total energy. When the decrease becomes small enough with respect to the number of atoms in the system, the calculation can be considered converged with respect to the basis set. One way to quantify the convergence criterion is to compute the value of |(dET/dlnEcutoff)|/N where dET/dlnEcutoff is the change in the total energy per unit change in the logarithm of the cutoff energy, and N is the number of atoms in the unit cell. Values of 0.01 eV/atom indicate good convergence (Cerius2 User Guide 1997).

Figure 14. Sketch illustrating the Vanderbilt ultrasoft pseudopotential approach, modified from Payne et al. (1992). The strongly peaked wave function in the core (solid curve) represents tightly bound core orbitals. To mimic this behavior with a plane wave basis set requires a large cutoff energy. If the norm-conserving constraint is lifted, then the resulting wave function (dashed curve) can be mimicked using a lower cutoff energy.

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Surface model–Cluster versus periodic At this point, we turn our attention to consider the advantages and disadvantages of different types of atomistic models for surface configurations. The distinction between the models, as they are defined here, will have some dependence on whether the modeling code used treats the system as aperiodic or periodic. Using the former means that the model is not repeated in space, and using the latter means that the code has the ability to utilize translational symmetry to effectively repeat the model infinitely in any direction. Historically, aperiodic codes are predominantly molecular orbital theory codes designed for application to molecules. The aperiodic treatment has the ability to handle non-neutral models. For most practical purposes, periodic codes restrict the model to charge neutrality, but methods are becoming increasingly available to handle charged systems (Marx et al. 1995; Jarvis et al. 1997). Given these distinctions, there are three ways to represent a solid surface, namely a cluster, finite slab, or supercell slab (Fig. 15). Aperiodic codes can only utilize the cluster model for solids. A cluster is defined by excising a discrete set of atoms from the surface, with a face of the cluster to represent the surface of interest. Depending on whether or not translational symmetry is applied to the representation, the cluster model can be used in either aperiodic or periodic codes. Slab models are by definition infinitely repeating in at least two dimensions and are generated using periodic codes. The two types of slab models differ in that the finite slab is bounded in the third dimension by infinite vacuum space, whereas the supercell slab is repeated in the third dimension along with some vacuum space. The supercell method is borne out of necessity to always describe the model as a three dimensional system, as is the case when a plane wave ab initio method is used. For each model type, the goal is to adequately mimic both the surface and bulk electronic structures simultaneously, with the latter typically requiring a large proportion of non-surface atoms to be built into the model. For semiconductors, where bonding is often strongly covalent, attaining this goal can be a difficult problem. Broken bonds at the surface are a substantial perturbation that can have long range influences on the electronic structure of the fully coordinated atoms. Charge redistribution at the top few monolayers is often significant and electrostatics subsequently may drive surface relaxation several layers into the bulk. Long-range lateral electrostatic forces arising from dipoles are also important. Semiconductors can also have delocalized behavior induced at the surface by surface states, leading to metallic behavior when the surface states are partially occupied. For metals, smaller model sizes can be very successful because the difference in the electronic structure on going from the surface to the bulk decays rapidly over a thin layer at the surface. But for semiconductors, long range forces often require larger models. Cluster models have been highly successful for bulk electronic structure calculations for semiconductors (see Tossell and Vaughan 1992 and references therein). But compared to slab models, it is clear that cluster models have more difficulty in accurately describing the electronic structure at semiconductor surfaces because of the unwanted electronic perturbations arising from the atoms at the sides and bottom of the cluster, sometimes called edge effects. The drastic undercoordination at edge and corner atoms leads to unstable high energy sites that can significantly change the overall electronic structure of the cluster. Their effects can become even more significant when open shell calculations are performed. The loss of bonding overlap at these sites can catalyze electron unpairing and lead to lower total energies for spin polarized solutions where there should be none. The cluster must be chosen that the separation between the cluster edges and the area representing the surface is maximized, but at the same time not sacrificing the number of atoms representing the underlying bulk. Increasing the number

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Figure 15. Sketch illustrating the different types of model systems and their representation within the three common types of ab initio modeling methods. A 3D system is a crystal, a 2D system is a slab, a 1D system is polymer-like, and a 0D system is a molecule. For periodic calculations, infinitely repeating directions in a model are indicated by black lines. Plane wave calculations require the system to be repeated in all three directions. In this case, slab models are created within a supercell containing vacuum space. Periodic LCAO based slabs are only repeated along the two lateral directions, and are finite in the third direction. Clusters may be approached by all three modeling methods.

of atoms is, of course, counterbalanced by the typical N3–N4 increase in computational expense. When using a cluster model then, in principle, one should repeat calculations using a variety of cluster sizes to determine if the modeled properties of interest at the surface are converged with respect to the cluster size. It is typical for edge effects to drive up the required cluster size to >100 atoms, often beyond the current limit of ab initio computational practicality. However, the inflation of the cluster size can be minimized by combating edge effects more intelligently using

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either of two methods. One is to terminate edge atoms with hydrogen bonds, which serve as surrogate hosts for electron density as it should be distributed around atoms in the bulk. In the ideal case, the geometry optimized distribution of hydrogen atoms around the sides of the cluster will mimic the symmetry of the missing bulk atoms. This strategy has been shown to work very well for cluster models of bulk insulators (Gibbs 1982; Gibbs et al. 1999). On the other hand, differences in valence between the hydrogen and the atoms being replaced can easily lead to problems with an unphysical excess of net charge on the cluster. Another method is to embed the cluster in a larger model treated at a lower level of theory. The cluster may be embedded in an array of point charges around the sides and bottom that model the Coulombic interactions for those atoms in the bulk. Embedding methods have been successful to improve cluster convergence (Siegbahn and Wahlgren 1991). Both hydrogen termination and embedding methods are useful approaches to satisfactorily minimize edge effects. Their successful implementation allows one to recover many of the advantages of utilizing a cluster approach, namely computational facility and the flexibility to treat non-neutral models (for aperiodic treatments). It should also be pointed out that edge sites in a cluster model can also be of interest to mimic the electronic structure of surface defect sites such as step edges, kinks, and corners. Slab models are the more ideal approach to model surfaces because edge effects are not present. Unlike the cluster model, this approach has the ability to capture delocalized electronic behavior as well as long-range electrostatic effects. But similar to cluster models, there are important issues concerning model size. The inner planes of atoms in the slab model represent the bulk, while the outer planes of atoms are two equivalent surfaces. For both finite and supercell slab models, convergence should be demonstrated with respect to the slab thickness. This insures that the two surfaces do not interact with each other through the inner planes of the slab, and allows for an accurate representation of the bulk. One useful test is to compare the DOS for the inner atoms of the slab with that calculated for the three dimensional bulk case, all else being equivalent. The computational expense of periodic ab initio methods often limits the thickness of slabs to 2-3 repeat cells. Fortunately, this turns out to be sufficient for most semiconductor surfaces (excluding extrinsic effects such as band bending). For supercell slabs, convergence should also be demonstrated with respect to the vacuum layer thickness to insure that no significant interaction is taking place between the slabs across the vacuum layer. Codes–Crystal vs. CASTEP Up to this point, general aspects of ab initio theory and modeling methods have been presented, with a slant towards surface applications. This background information only takes us far enough to understand the important basics. In practice, there are many codespecific features, pitfalls, and important advantages and disadvantages that a user should also be aware of for successful modeling. There are only a handful of ab initio codes currently being used routinely in the geosciences for mineral surface modeling, which facilitates a more code-specific review. Aperiodic codes (e.g., Gaussian, GAMESS, HONDO, HyperCHEM, Spartan) are very similar to each other in many respects and will not be reviewed here because of the many excellent tutorials elsewhere and in other chapters in this volume. Periodic codes can be subdivided cleanly into LCAO (e.g., Crystal, ADF) and plane wave (e.g., CASTEP, VASP, DOD plane wave). Because of popularity and author bias, Crystal (Saunders et al. 1998) and CASTEP (Payne et al. 1992) will be briefly reviewed here. Notably missing is the code NWCHEM which has dual functionality, having both an aperiodic LCAO capability and a newly added periodic plane wave DFT capability (Anchell et al. 1999). NWCHEM is freely distributed by Pacific Northwest National Laboratory.

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Crystal. Crystal is a LCAO ab initio code that can apply Hartree-Fock or KohnSham DFT methods to both aperiodic and periodic systems (see Pisani 1996). A wide variety of exchange and correlation functionals are incorporated, including the ability to build hybrid functionals such as B3LYP and combinations needed for LDA and GGA formulations. Systems can be infinitely repeated in 0–3 dimensions, with the 0D case comprised of a cluster of atoms treated aperiodically, similar to Gaussian. For periodic systems, one may model either a 1D polymer, a 2D slab (finite slab), or a 3D crystal. It is a true band structure code in the sense that the full symmetry of the system can be incorporated and the wave function is solved self-consistently in reciprocal space. A set of reciprocal space sampling points is generated using a mesh (Monkhorst net) defined from the lattice vectors and shrinking factors that control the mesh spacing. Crystal can perform spin unrestricted calculations (UHF or UDFT) and manually set initial spin configurations are allowed. Crystal allows the calculation of most ground state properties of the system from the wave function. Especially useful for crystals and surfaces are the total and projected DOS (atomic orbital weighted), band structure, electron density maps, and atomic charges. Forces on atoms are not explicitly calculated from energy derivatives so optimizations must be performed numerically. The external shell routine LoptCG (Zicovich-Wilson 1998) is supplied with the code to automate the potentially laborious process of performing total energy difference calculations and minimizations numerically as a function of structure parameters (such as atomic coordinates or lattice parameters). Basis sets are a particularly important part of performing calculations with Crystal. Similar to aperiodic LCAO codes such as Gaussian, GTO's are used exclusively as the basis set with the option to utilize core pseudopotentials. But in contrast to Gaussian, the GTO's in Crystal are used to compose Bloch functions that have the periodicity of the lattice. Also, a wide variety of GTO's have been developed for molecular calculations but, it turns out, that these are often problematic to use in Crystal for solid state calculations because they routinely use very diffuse gaussian functions (Pisani 1996). Diffuse functions are important for an accurate description of the decay of electron density around loosely packed atoms, but are overcomplete descriptions for closely packed atoms in the solid state. This is because the tails of diffuse wave functions around atoms in a crystal are found in regions where there is already large variational freedom with the basis functions on other atoms. Such diffuse functions are not required to accurately expand the electron states into Bloch functions in a semiconducting crystal. Attempting to include diffuse functions then usually leads to a rapid explosion in the computational expense, convergence problems, and the possibility of numerical instabilities. Therefore, basis sets must be carefully crafted for specific systems for use in Crystal. Many published Crystal basis sets are available as starting points. Exponents and coefficients are typically optimized for a system (e.g., using LoptCG) in an initial best guess structural configuration, before forces on the atoms or cell parameters are minimized. The diffuse function constraint can have significant drawbacks for surface applications, where the additional degrees of freedom at the surface cause the decay of electron density away from surface atoms to be less like that for their solid state counterparts, and more like isolated atoms. The tails of surface atom wave functions are poorly described by Crystal basis sets in both directions perpendicular to the surface. This can lead to accuracy problems for surface interactions with loosely bound adsorbates and difficulties in the estimation of the tunneling current in theoretical STM image calculations (see Becker et al. 1996). Fortunately, convergence tools are available to increase the chances of success when such cases call for diffuse basis functions. Level shifting techniques are useful for achieving a starting converged wave function that can be used as an initial guess for more accurate subsequent calculations. Fermi function methods are supported to improve the description of the occupancy of states around the

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Fermi level and allow faster convergence. Convergence tools such as these can make the difference between defeat by an unwieldy calculation and successful surface modeling with Crystal. CASTEP. The Cambridge Serial Total Energy Package (CASTEP), or its parallel processing equivalent CETEP, is a popular implementation of plane wave pseudopotential Car-Parrinello method. DFT is the default ab initio theory, supporting LDA, GGA, and their spin unrestricted equivalents (LSDA and GGS respectively). Pseudopotentials may either be local or non-local with Vanderbilt and norm-conserving pseudopotentials supported. Molecular dynamics is available at little extra computational expense. As was mentioned previously, the Car-Parrinello method is a very computationally efficient form of DFT. The electronic and structural optimizations are performed simultaneously, with the useful option of simultaneously optimizing the atomic coordinates and cell parameters on an equal footing, with or without the influence of a fixed external stress. Varying the cell size for a fixed cutoff energy leads to discontinuities in the basis set and Pulay stress on the unit cell, so it is important to demonstrate that the system has converged with respect to the cutoff energy (Payne et al. 1992). Symmetrization of the wave function is also available to speed up calculations for symmetric systems. Similar to Crystal, CASTEP samples k-points over a reciprocal space mesh, the spacing of which is controlled by the user. Increasing the k-point sampling leads to more accurate total energy calculations, so it is also important to demonstrate that the total energy is sufficiently converged with respect to the number of k-points chosen. The code is equally well suited for all types of materials including metals, semiconductors, and insulators, although metals must be treated somewhat differently using partial occupancy methods and more dense k-point sampling. Several different types of electronic minimizers are available with unique strengths for different types of systems, and convergence problems are rare. CASTEP makes fully relaxed geometry optimizations of a wide variety of large complicated mineral structures accessible. As a plane wave code, it is restricted to the supercell method to maintain a three dimensional periodic treatment of the model system. Therefore surfaces are treated by building in a vacuum layer. There is no size restriction on the unit cell size but, because plane waves are continuous functions, the vacuum layer is filled with basis states and therefore increasing its thickness increases computational expense. Also, because a plane wave basis set is the ideal expansion method for electronic states at surfaces, CASTEP calculations have the inherent ability to accurately describe the decay of electron density from surface atoms perpendicular to the surface plane. Although CASTEP calculations can be spin unrestricted, it lacks spin terms necessary to describe spin-spin interactions, hence CASTEP is not well suited to calculate magnetic properties. For common magnetic minerals, it has been found to be somewhat unpredictable in its ability to find the proper magnetic ground state, with successes strongly depending on the pseudopotentials, the use of LSDA or GGS, and the electronic minimizer chosen. Norm-conserving pseudopotentials outperformed ultrasoft pseudopotentials in this area for hematite (Rosso et al. in prep). The antiferromagnetic spin configuration of goethite was found to agree with experiment (Rosso and Rustad 2001). The Car-Parrinello scheme for minimization of the electronic wave functions involves an orthogonalization step, the computational expense of which increases as the square of the number of bands (Payne et al. 1992). Therefore, to increase the speed, unoccupied states are typically not included in this process. For systems such as insulators and semiconductors, their exclusion does not pose a problem because the electronic ground state is often easily found from linear combinations of lowest occupancy initial states. In the converged wave function then, unoccupied states are not

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automatically available for evaluation. Because these are of use in modeling the conduction band and for interpretation of empty state STM images, unoccupied states must be calculated a posteriori from the density arising from the occupied states. It is also noteworthy that in CASTEP, independent of whether or not LDA or GGA functionals are chosen, the pseudopotentials are generated using LDA. This introduces a systematic inconsistency for GGA calculations because the core is modeled using LDA. However, because of the similarities of LDA and GGA in the core region, this approximation turns out to be acceptable in practice (e.g., Bridgeman et al. 1996). APPLICATIONS Sulfides Galena–Bulk. The rocksalt structure of PbS is a deceptively simple structure for the application of ab initio methods (space group Fm3m) (Fig. 16). Accurate treatment of the heavy element Pb requires either the incorporation of relativistic effects for core electrons or electron core pseudopotential methods. All-electron treatment requires f-orbital basis sets as well. Although it is a small band gap semiconductor (~0.4 eV), fully occupied dand f- series lead to less convergence problems than are typical for small band gap transition metal compounds. It is an intrinsic semiconductor but is often impure and/or defective in its natural state which alters the position of the Fermi level in the band gap and gives rise to extrinsic semiconduction. It may be either n- or p-type, depending on the type of impurities and defects present (Hemstreet 1975; Pridmore and Shuey 1976). Commonly found Sb and Bi substitutions for Pb, or S vacancies, lead to n-type behavior. P-type behavior arises from Ag → Pb substitution or Pb vacancies. A variety of cluster and periodic ab initio modeling strategies have been applied to pure bulk galena (Tung and Cohen 1969; Rabii and Lasseter 1974; Hemstreet 1975; Tossell and Vaughan 1987, 1992 ; Mian et al. 1996; Gurin 1998; Gerson and Bredow 2000). Theoretical estimates of band energies, widths, and densities of states of the valence band are largely in excellent agreement with photoemission experiments (Grandke et al. 1978; Santoni et al. 1992; Ollonqvist et al. 1995). The top of the valence band consists of non-bonding S 3p states, overlying a Pb 6s–S 3p bonding band (Fig. 17). The conduction band is less thoroughly studied, but recent photoabsorption experiments (Santoni et al. 1992) and inverse photoemission experiments (Ollonqvist et al. 1995) are in good agreement with periodic LCAO calculations (Mian et al. 1996). These studies show that the conduction band is

Figure 16. The rocksalt structure of galena. Pb (black balls) and S (white balls) are both in octahedral coordination in a face-centered cubic arrangement. The bulk unit cell edge is ~ 6 Å.

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Figure 17. Calculated total and projected DOS for bulk galena using data from Becker and Rosso (2001). The calculations were performed using an geometry optimized wave function generated using an LCAO periodic treatment (Crystal) at the B3LYP level of theory. The Fermi level (zero on the energy scale) is arbitrarily located at the valence band edge. The top of the valence band is comprised of predominantly S 3p states and the bottom of the conduction band is comprised of predominantly Pb 6p states.

dominated by Pb 6p states with a minor admixture of S 3p states (Fig. 17). Calculated overlap populations, orbital-weighted densities of states, and electron density maps indicate that bonding in galena is best characterized as ionic with minor covalent character (Mian et al. 1996). The high ionicity of this mineral makes it unique among the others included in this review, which are predominantly based on covalent bonding interactions. Galena–(100) surface. The cubic surface of any mineral with the rocksalt structure should be quite stable when considered in both the ionic and covalent limits. The surface consists of stacked layers each of which is charge neutral and electrostatically stable (Fig. 18). The similar 6-fold coordination for both the cations and anions and the 1:1 ratio of dangling bonds across (100) planes means that charge transfer from surface cations to anions will completely fill the anion dangling bonds and this surface is autocompensated. Also, because of the high coordination symmetry across the surface, lateral relaxation can be expected to be insignificant. Collectively, these qualitative arguments predict that the (100) surface structure will be resistant to differ significantly from a bulk termination when viewed down the surface normal direction. However, fine adjustments via relaxation along the surface normal direction are possible coming in the form of layer-by-layer displacements, relative displacements of atomic sublattices (called rumpling), or a combination thereof. The mechanisms that drive the relaxation depends on the bonding character of the material (Noguera 1996). Covalent materials will tend to respond to the loss of coordination at the surface by significantly redistributing charge between surface cations and anions. This is the case for the other minerals discussed in this chapter. However, because of the largely ionic character of galena, charge transfer between Pb and S at the surface will be small. For the unrelaxed structure, charge transfer from cation to anion has been estimated to be only 0.13 e- (Allan 1991). In the ionic limit then, galena (100) relaxation predominantly arises from a competition between short range repulsive forces and the Madelung attractions at

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Figure 18. The unrelaxed structure of the (100) surface of galena. Both Pb and S are equally arranged in a face-centered cubic array across the surface. Perpendicular to the surface plane, the surface can be seen to be built up from stacks of charge neutral atomic planes. The surface is type I and electrostatically stable (see Fig. 3).

the surface (Noguera 1996). Because the latter is stronger, layer-by layer relaxation is typically manifested as a contraction of the uppermost atomic plane towards the bulk. This type of relaxation can be expected to be small because it involves bond compression and bond lengths are significantly stiffer than bond angles (e.g., Gibbs 1982). For the majority of rocksalt materials, the contraction is usually small but nonetheless significant, entailing inward displacements of < 5% of the bulk lattice constant (Gibson and LaFemina 1996). To-date, all ab initio calculations utilizing a wide variety of approaches suggest similar behavior for galena (100), with the magnitude of the contraction tightly clustering around 1.5–2.5% of the bulk lattice constant (Fig. 19) (Becker and Hochella 1996; Becker et al. 1997; Wright et al. 1999a,b; Becker and Rosso 2001). SCLS spectra are also consistent with a contraction of the outermost plane (Leiro et al. 1998). At opposite extremes for this surface are the predictions of a 15% contraction based on an electrostatic model (Allan 1991), and an X-ray standing wave measurement of 0% (Kendelewicz et al. 1998), with the latter study concluding that galena (100) is an ideal bulk termination. These extreme results call for additional experimental work in this area. As of yet, no quantitative LEED studies have been performed on this surface, but data on isostructural and isoelectronic PbTe (100) indicate a contraction of 7% (Lazarides et al. 1995). Surface rumpling is driven by a different mechanism than the layer contraction. For ionic materials such as galena, rumpling arises from the differential polarizability of the cations and anions (Noguera 1996) and is counterbalanced by the driving force to maintain the zero dipole moment within (100) atomic planes. The rumpling amplitudes of offset cation and anion sublattices are typically small for materials having the rocksalt structure, typically 1–2% of the bulk lattice constant (Gibson and LaFemina 1996). The small displacement makes this component of surface relaxation an experimental challenge to identify. Not surprisingly, this issue is even more controversial than the uppermost layer contraction. The rumpling of galena (100) has been experimentally

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Figure 19. The general pattern of relaxation for the (100) surface of galena as calculated by a variety of ab initio modeling methods (see text). The uppermost atomic plane is contracted towards the bulk by ~ 1.5–2.5% of the bulk lattice constant. The Pb and S sublattices of this plane rumple with the Pb sublattice outward and the S lattice inward. Rumpling amplitudes range between ~ 0.3–1.5% of the bulk lattice constant. The reverse rumpling relationship is typically observed for rocksalt structures (Noguera 1996).

indicated to involve either the S sublattice projecting further outward than the Pb sublattice (Leiro et al. 1998), or no rumpling at all (Kendelewicz et al. 1998). The relative displacements found in the former result are consistent with Lazarides et al. (1995) and the known rumpling trends for most surfaces of both ionic and covalent materials (Noguera 1996). In contrast, molecular modeling has uniformly indicated the opposite relative displacement. A wide variety of ab initio approaches has been applied to this problem, including cluster DFT and finite slab HF (Becker and Hochella 1996; Becker et al. 1997), embedded cluster DFT (Becker et al. 1997; Wright et al. 1999a), finite slab DFT (Wright et al. 1999b; Becker and Rosso 2001), and supercell plane wave GGA calculations (Wright et al. 1999b). All the calculations predict that the Pb sublattice should project further outward than the S sublattice with rumpling amplitudes of ~ 0.02– 0.09 Å (0.3–1.5%) (Fig. 19). In this review, this is perhaps the best example of where high level ab initio calculations consistently “fly in the face” of both experiment and conventional wisdom. An explanation is not apparent at this time and more work is needed to rectify this detail. The small surface relaxation of galena (100) changes the surface structure very little, but the directions of the displacements can be important pieces of information for understanding the electronic characteristics of a surface. The widespread disparity of results on these subtle issues for galena point to the difficulties of routinely capturing these characteristics using state-of-the-art methods, even using a combined theoretical and experimental approach. On the other hand, galena (100) is also a very good illustration of where the combined approach is essential. STM has been routinely performed at the atomic scale on (100) surfaces in ambient atmosphere, oil, solution, and UHV environments (Zheng et al. 1988; Hochella et al. 1989; Cotterill et al. 1990; Sharp et al. 1990; Eggleston and Hochella 1991, 1993, 1994; Liao et al. 1991; Ettema et al. 1993; Laajalehto et al. 1993; Kim et al. 1994, 1995, 1996; Higgins and Hamers 1995, 1996a,b; Becker et al. 1997; Eggleston 1997; Vaughan et al. 1997; Becker and Rosso 2001). Because of the identical structure of the Pb and S sublattices, deriving meaningful interpretations of the STM data, and its bias and tip-sample separation dependence, has

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proven to be tenuous without taking into account ab initio models of the surface electronic structure. These have come in the form of both cluster and finite slab calculations by Becker and co-workers (Becker and Hochella 1996; Becker et al. 1997; Becker and Rosso 2001). STM images at negative sample bias are predicted to show the S sublattice as higher tunneling current sites over the Pb sublattice, in accord with the conclusions drawn in most STM studies to date (Fig. 20). The reverse is predicted for positive sample biases because of the predominance of Pb 6p states at the bottom of the conduction band. In contrast, many STM workers have concluded that the S sublattice is also imaged at positive bias, although tunneling current from both sublattices has been observed (Zheng et al. 1988; Eggleston and Hochella 1990). Simultaneously collected dual-bias images collected under oil suggested that highest tunneling current sites were commensurate and therefore likely arising from the S sublattice at both positive and negative bias (Eggleston and Hochella 1990). Likewise high tunneling current sites in empty state images collected in solution have been attributed to S sites (Higgins and Hamers 1996b). There are several possible explanations for the discrepancy. One is that the electronic structure at the surface in the experiments is altered from the intrinsic semiconducting behavior by mid-gap states arising from impurities or defects (e.g., Hemstreet 1975). Reduced band gap conditions have been observed on this surface (Grandke and Cardona 1980; Eggleston and Hochella 1990). Tip-induced band bending may also alter the tunneling current magnitude by forming a bias-dependent Schottky barrier at the surface that can vary with tip location. Another arises from the poorly understood influence of the overlying non-vacuum medium for these particular experiments. The last explanation presumably could be ruled out by

Figure 20. A comparison of experimental (a) (Eggleston and Hochella 1990) and theoretical (b) STM data for the (100) surface of galena [Used by permission of Elsevier Science, from Becker and Hochella (1996), Geochim Cosmochim Acta, Vol. 60, Fig. 2, p. 2417]. The experimental STM image was collected under oil at -600 mV sample bias and 2.9 nA setpoint current. The surface unit cell is outlined (~ 6 Å). Because of the similar arrangement of the Pb and S atoms, an assignment of the high tunneling current sites cannot be deduced from the experimental STM data without independent information. The theoretical image reproduces this data and allows highest tunneling current sites to be assigned to surface S atoms. The predominance of S 3p states at the top of the valence band gives rise to this behavior.

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observations of the pristine surface in UHV, of which there have only been four reports (Zheng et al. 1988; Sharp et al. 1990; Becker et al. 1997; Becker and Rosso 2001). In these four studies, unfortunately, dual mode images were not collected, most likely due to the commonly encountered but poorly understood difficulty of obtaining high resolution images at negative sample bias on this surface in UHV (Zheng et al. 1988; Becker and Rosso 2001). Steps and point defects are commonly observed at the atomic scale by STM on cleaved galena (100) surfaces (Eggleston and Hochella 1990, 1991, 1993; Laajalehto et al. 1993; Higgins and Hamers 1995, 1996a,b; Becker and Rosso 2001). In addition, the surface outcropping of bulk dislocations and defects arising from impurities have been inferred from STM observations (Zheng et al. 1988; Sharp et al. 1990). Understanding the structure at defects and the strained areas around them at the surface is currently at an embryonic stage. Perhaps best understood for this surface is the structure at step edges. Following from electrostatic arguments, step edges oriented along cubic surface directions are non-polar and therefore predicted to be the most stable. Relaxation at such step edges should follow similarly from the above discussion for (100) surface relaxation. All other surface directions would have a polar component that would most likely be stabilized by charge compensating vacancies. Thus, in the absence of other stabilizing influences, one expects to find step edges principally aligned along cubic axes, a prediction confirmed in UHV (Zheng et al. 1988; Becker and Rosso 2001). However, for galena (100) in solution, step edges along surface diagonal directions are found to be more stable, conceivably due to a strong interaction between step edge sites and electrolyte species in solution (Higgins and Hamers 1995; 1996a,b). It is noteworthy that for the rocksalt derivative pyrite (FeS2), both cubic and cubic diagonal step edges are found to be stable in UHV, with charge compensating vacancies along the diagonal step edges likely arising from S vacancies (Rosso et al. 1999a, 2000) (see pyrite section). UHV STM images of a cubic step edge at the atomic scale on galena (100) show that edge sites have a higher density of empty states relative to terrace sites, and these perturb the electronic structure of the upper terrace on the nanometer scale (Fig. 21) (Becker and Rosso 2001). The perturbation is seen in the form of deforming the surface symmetry of the high tunneling current sites along a direction perpendicular to the step edge. The data implies that relaxation effects associated with step edges penetrate over longer ranges compared to the relaxation effects normal to the (100) surface, not unlike effects observed previously in air on pyrite (100) (Eggleston and Hochella 1992a). What cannot be determined from the image data is whether this deformation arises from purely an electronic effect, or from changes in the actual positions of atoms at the surface. Ab initio DFT calculations were performed using a periodic treatment with Crystal and an embedded cluster treatment with Gaussian (see Theoretical Methods section) to model sections of the step edge structure (Becker and Rosso 2001). Step edge relaxation followed the trends predicted for the (100) surface by laterally contracting towards the upper terrace by ~ 0.2 Å and rumpling Pb outward and S inward with an amplitude of ~ 0.06 Å (Fig. 22). Using Crystal, calculated STM images at low positive sample bias correctly predicted that the tunneling current should be predominated by the Pb sublattice due to Pb 6p states at the bottom of the conduction band. Crystal results also suggested that the tunneling current should become substantially higher over Pb sites at the step edge, consistent with the STM data. However, the periodic treatment failed to mimic the observed deformation because of cancellation effects arising from the symmetry in the periodic step edge model. In contrast, the cluster calculations performed well in this regard. Theoretical STM images reflected deformation of the terrace electronic structure consistent with the image data. These also suggested that high tunneling current sites occur at points “off” the step edge due to a superposition of Pb 6p orbitals projecting

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Figure 21. Atomic scale UHV STM images of a step edge on galena (100) from Becker and Rosso (2001). The tunneling conditions were 0.25 V sample bias and 1 nA setpoint current. The images capture a (100) terrace terminated by a kinked step edge at the left side of the images (denoted by a dashed line). Bright spots correspond to Pb sites. The apparent height of step edge Pb atoms is increased relative to terrace Pb sites as seen in the topographic channel (a). The current channel (b) clearly shows the atomic periodicity over the terrace, and a surface unit cell is outlined (~ 6 Å). Based on changes in the symmetry of the tunneling current sites approaching the step edge, the image also suggests that the electronic structure of terrace atoms is perturbed by the step edge over nanometer distances. This is most easily seen in the FFT filtered version (c) where the apparent widening of the PbPb-Pb angle is denoted.

outward from step edge Pb sites with those projecting upwards from the Pb atoms below (Fig. 23). The cluster calculations allowed the conclusion that the deformation observed in the images was purely an electronic effect. In light of the short range relaxation effects known for the (100) surface, the long range effect of the step edge on the surface electronic structure is curious and has yet to be adequately explained. The galena (100) surface is known to oxidize in air and air-saturated solutions. XPS investigations have indicated that this process is somewhat sluggish, with the initial reactions in air predominantly producing Pb oxides, hydroxides, and carbonate as opposed to oxidized S species (e.g., Buckley and Woods 1984; Laajalehto et al. 1993). Nevertheless, at the atomic scale, the initial formation of oxidation products has been documented using STM within minutes of exposure to air (Cotterill et al. 1990; Eggleston and Hochella 1991, 1993, 1994; Liao et al. 1991; Laajalehto et al. 1993; Kim et al. 1994; Eggleston 1997). Surface oxidation can be seen to proceed at negative sample bias by the birth and outward spread of low tunneling current patches of oxidized areas, typically bounded by surface diagonal directions (e.g., Eggleston and Hochella 1991). The oxidation patches consist of areas where the density of occupied S 3p states is decreased

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Figure 22. The results of relaxed step edge calculations for the (100) surface from Becker and Rosso (2001) using Crystal. In a finite slab treatment starting from the relaxed surface structure, a row of atoms was removed from one surface of the slab (a) and the step edge atoms on one side of the “trench” were allowed to relax (marked with “r”). Similar to the relaxation behavior normal to the (100) surface, the relaxed step edge contracts towards the terrace by ~ 0.2 Å (b). The minimum energy configuration was used to calculate theoretical STM images for positive sample bias to model the experimental data shown in Figure 21. Current data is for a constant tip location of 4 Å above the uppermost atomic plane. The positions of atoms in the image is vertically commensurate with the structure models. Because of the predominance of Pb 6p states at the bottom of the conduction band, high tunneling current sites are predicted to be surface Pb sites, in agreement with previous calculations. At the step edge, the current density is predicted to increase, in agreement with the experimental observations.

as a result of electron transfer from S to the oxidant. Very little is known in regards to the stepwise reactions involved in the oxidation process and would be quite difficult to extract from the STM data alone. This is one area where molecular modeling can be enormously effective. Simple model oxidation reactions between oxygen and galena (100) were theoretically treated in detail using cluster calculations by Becker and Hochella (1996). The adsorption of molecular oxygen with the surface was shown to be energetically downhill (but not activation-less because of a spin transformation for O2 from paramagnetic to diamagnetic). The geometry optimized sorbed O2 is configured with one end of the molecule below the surface plane, and alignment of the O2 axis along surface diagonal directions with bonds to a S and two Pb atoms. Similar to the case for pyrite oxidation (see below), the O-O double bond is destabilized by the transfer of electrons from surface S2- to antibonding π* orbitals of O2, indicating facile dissociation. Dissociated oxygen atoms draw electron density from surface S atoms, depopulating the S 3p states at the top of the valence band, consistent with the STM observations at negative sample bias (Fig. 24). The agreement between experiment and theory was found to be striking in other aspects as well. Increased tunneling current has been routinely observed at sites adjacent to oxidation patches and was likewise manifested in theoretical STM images of oxidized areas of the surface. Also, the calculations predicted that oxidized surface S sites should have a higher empty state density at the bottom of the conduction band relative to unoxidized S sites. Therefore, they should appear brighter at positive sample bias, a prediction later confirmed by Eggleston (1997).

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b

a

c

Figure 23. Calculated STM image data for a step edge on the (100) surface using Crystal (a) and Gaussian (b) (modified from Becker and Rosso 2001). The 2D periodic slab treatment using Crystal reproduces the regular tunneling current pattern for galena terraces at low positive sample bias. Aspects of the observed deformation in the tunneling current near the step edge (see Fig. 22) are described by the images calculated with an embedded cluster model using Gaussian (b) (see text). The model is defined in (c) with embedding charges located at lattice sites outside the solid white box, and the calculated image area used to build the image in (b) is located within the thin dashed white lines. Some Pb positions are marked with white dots and the step edge is denoted with the thick dashed line.

The formation of oxidation patches is generally not found to be associated with surface defects, although they have been observed to form preferentially at point defects (e.g., Laajalehto et al. 1993). The combined STM image and modeling data of Becker and Rosso (2001) is suggestive of the possibility of a strong influence of step edges on (100) surface reactivity. Indeed, the electrochemical dissolution of galena has been directly observed at the atomic scale to proceed preferentially at step edges (Higgins and Hamers 1995). This important area has been recently explored theoretically using both embedded cluster and slab calculations at 0 K to model the interaction of H2O with galena (100) and step edges thereon (Wright et al. 1999a,b). The calculations showed that the molecular adsorption of H2O to (100) terraces is energetically downhill, but dissociation was found to be both strongly activated and energetically uphill. This weakly interacting, physisorbed behavior was unaffected by the presence of surface vacancies. However, at step edges the dissociation reaction was found to become exothermic and the activation energy barrier was reduced by an order of magnitude. At step edges, the precursor state is formed when the negative end of the H2O dipole interacts with a step edge Pb site (Fig. 25). The minimum energy dissociated configuration consists of a hydroxyl bound between Pb sites of the step edge and lower terrace, and an adjacent protonated step edge S site. In between, a transition state was located, which was found to

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Figure 25. Sketch showing the dissociation of H2O at a step edge on galena (100) as calculated using LCAO embedded cluster methods at the B3LYP level of theory [Used by permission of Elsevier Science, from Wright et al. (1999a), Chem Phys Let, Vol. 299, Fig. 1, p. 529]. The H2O molecule interacts with a Pb site (black balls) at the step edge (a). Dissociation is activated by ~ 20-30 kJ/mol, with a transition state involving the formation of an additional Pb-O bond and an S-H bond at the step edge (b). The dissociated structure leaves a hydroxyl bound to Pb sites and a protonated S step edge site. Although hydrolysis is energetically uphill on (100) terraces, it is downhill at step edges, indicating a defect driven catalytic behavior for this surface.

Figure 24. A comparison of experimental (a) (Eggleston and Hochella 1990) and theoretical (b) STM data for the oxidized (100) surface of galena [Used by permission of Elsevier Science, from Becker and Hochella (1996), Geochim Cosmochim Acta, Vol. 60, Fig. 6, p. 2420]. The experimental STM image was collected at -405 mV sample bias and 2.2 nA setpoint current. The surface unit cell is outlined (~ 6 Å) in (b). Oxidation due to air exposure leads to the development of “dark” patches (labeled A) of low tunneling current areas on the surface (a). Adjacent to the patches, sites of enhanced tunneling current can be found (labeled B). Finite slab calculations were performed using Crystal to locate the optimized position of an oxygen atom over a 3×1 surface supercell. Theoretical STM images based on this supercell for negative sample bias show both the loss of S 3p electron density to O and the enhancement of tunneling current at adjacent Pb sites due to oxidation, consistent with experimental data.

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entail the formation of the Pb-OH2 bond with the Pb site of the lower terrace and the formation of the S-H bond. The activation energy was estimated to be ~ 20–30 kJ/mol. These calculations demonstrate that hydrolysis should be quite facile at step edges on galena (100) at room temperature, similar to the theoretically predicted behavior for MgO (Langel and Parrinello 1994, 1995). Wright and co-workers also suggest that the nucleophilic attack of hydroxyls at Pb step edge sites is energetically favored, providing an efficient mechanism for the formation of Pb hydroxide species at the surface. This is consistent with XPS observations indicating a rapid formation of Pb-O bonds and sluggish oxidation of S in air. It is also consistent with the results of Becker and Rosso (2001) which show that the density and energy of empty Pb 6p states is significantly increased at step edge Pb sites. Because the former can be viewed as increasing the capacity to accept electrons and the latter can be viewed as increasing the driving force to take them, the experimental and theoretical STM image data predict an increased susceptibility to nucleophilic attack at step edge Pb atoms. Whether or not hydrolysis at step edge sites would play a synergistic role in the oxidation of galena surfaces by O2 is currently unknown, partially because it is unknown whether oxygen in the end product S-O species is derived primarily from dissociated O2 or H2O. However, it seems clear that this process could at least operate in tandem with oxidation reactions. The oxidation of galena (100) by Fe3+ has been recently addressed from a theoretical standpoint (Becker et al. 2001). UHF cluster calculations were designed to investigate the possibility of delocalized electron transfer from surface S to remotely located sorbed oxidants, or so-called proximity effects. A 32 atom cluster was used to model the (100) surface (4 atomic rows square, two atomic planes thick), with cluster edges left vacuum terminated so as to mimic the structure at step edges and corner sites. A dissociated H2O molecule and surface adsorption sites were geometry optimized in the form of Pb-OH and S-H species at interior surface sites on one side of the cluster. On the other side, the position of a sorbed Fe3+ ion in its high spin ground state (5 unpaired spins) was independently optimized. As the ferric ion approached the surface, a coupled exchange ensued where electron density was transferred from the cluster to the sorbed Fe ion and spin density transferred in the opposite direction (Fig. 26a). The oxidative exchange was found to involve primarily two corner S sites, leaving them partially oxidized and spin polarized. By lifting the geometric constraint on the dissociated H2O species, the hydroxyl group was found to preferentially migrate towards the closest spin polarized corner S site (Fig. 26b). Among other things, this result demonstrated that oxidation can be strongly influenced by defects and spin polarization. In this way, corner S sites were found to be predisposed to oxidation and the preferred adsorption site of a hydroxyl group was significantly modified. But, more importantly, the calculations suggest that S oxidation can occur remotely via non-local electron transfer to the oxidant through the substrate, even when the oxidant is an appreciable distance away. This proximity-based aspect of coupling redox reactions on this surface is a largely unexplored phenomenon with obvious implications on the way semiconducting surface reactivity should be viewed. Pyrite–Bulk. Pyrite is face-centered cubic with a structure that is closely related to the rocksalt structure type (space group Pa3) (Fig. 27). Each Fe atom is coordinated to six S atoms in a slightly distorted octahedron, and each S atom is coordinated to three Fe atoms and one S atom in a distorted tetrahedron. The S-S bond is oriented along body diagonals of the cubic cell. This disulfide bond is a unique characteristic of the pyrite structure type, and the S2 group is often best viewed as a distinct structural and chemical unit. Impurities in natural pyrite can lead to both n- and p-type semiconducting behavior, with the former more common and usually arising from (Co, Ni, Cu) → Fe cation substitutions, and the latter from As → S anion substitutions (Pridmore and Shuey 1976).

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Figure 26. Cluster modeling results for the oxidation of galena by Fe3+ suggesting the importance of proximity effects to surface reactivity on galena, modified from Becker et al. (2001). The adsorption of an Fe3+ ion to one side of the cluster results in a coupled exchange of electron density from corner sulfur sites to the ferric ion and spin density (indicated by vertical arrows) from the ferric ion to the corner sulfur sites (a). This causes the equilibrium position of a hydroxyl on the opposite side of the cluster to migrate towards the nearest spin polarized corner site (b). The calculations demonstrate that delocalized orbitals typical of semiconducting minerals can couple reactants spatially separated by several bond lengths in the substrate.

Figure 27. The pyrite structure type. The structure is closely related to the rocksalt structure, with Fe atoms (black balls) and the centers of S2 dianion pairs (white balls) arranged similarly in a face-centered cubic cell (~ 5.4 Å). Fe is octahedrally coordinated to S atoms and S is tetrahedrally coordinated to three Fe and one S atoms.

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The electronic structure of bulk pyrite has been thoroughly elucidated using ab initio aperiodic and periodic models (Li et al. 1974; Tossell 1977; Bullett 1982; Fujimori et al. 1996; Raybaud et al. 1997; Eyert et al. 1998; Rosso et al. 1999a; Gerson and Bredow 2000) and a variety of experiments (Bither et al. 1968; Burns and Vaughan 1970; Li et al. 1974; Ogawa et al. 1974; Schlegel and Wachter 1976; van der Heide et al. 1980; Folkerts et al. 1987; Folmer et al. 1988; Ferrer et al. 1990; Huang et al. 1993; Mosselmans et al. 1995; Bocquet et al. 1996; Charnock et al. 1996; Fujimori et al. 1996). The electronic structure is most easily explained in the language of molecular orbital theory. Because S valency is isoelectronic with that of O, the valence molecular orbitals of the S2 moiety are qualitatively analogous to those for an O2 molecule, (e.g., Luther 1987). Degenerate antibonding π* orbitals, paramagnetically filled by S valence electrons, are completely filled by the electrons donated from Fe, giving Fe2+ and S22(Fig. 28). These orbitals constitute the HOMO for the S2 moiety, whereas the LUMO is of 3p σ* character. Fe 3d states are split into antibonding eg* and nonbonding t2g states, with the former because the dz2 and dx2-y2 orbitals are oriented along Fe-S bonding directions, and the latter because the dxz, dyz, and dxy orbitals project into the spaces between Fe-S bonds. The repulsive energy to pair the 3d electrons in the t2g set is less than the energy to partially occupy the higher lying eg* set, so a low spin configuration is adopted. Thus, for Fe the HOMO is the nonbonding t2g set and the LUMO is the empty eg* set. The complete electronic structure description in terms of band theory then depends essentially on the relative energies of the individual Fe2+ and S22- HOMO's and LUMO's and the degree of interatomic orbital mixing. The wealth of ab initio and photoelectron experiments in this area is largely in agreement, including HF and both LDA and GGA DFT calculations, but unsettled issues do exist. All the studies agree that the top of the valence band, the HOMO, consists of the non-bonding Fe 3d t2g states. The calculations predict that these form a distinct narrow (~ 1 eV) band. This is nicely complementary to UPS spectra, where a characteristic peak in the density of states at the top of the valence band is easily resolved (Fig. 29). This lies slightly above a highly mixed S 3p–Fe 3d bonding band composed of the σ, π, and π* S22- 3p states and the eg Fe 3d states. The high degree of mixing found between cation and anion states is indicative of the strongly covalent bonding interactions in pyrite. Regarding the bottom of the conduction band, the LUMO, most studies have attributed it to a mixed S 3p–Fe 3d band, composed of the σ* S 3p and eg* Fe 3d orbitals. In contrast, Eyert and co-workers (1998) suggested that the bottom of the conduction band is exclusively due to S 3p states, which is attributed to the larger predicted splitting of S 3p states arising from the strong S-S σ interaction. This interpretation is based on an accurate LDA calculation using the most rigorous integration over reciprocal space for Figure 28. Energy level diagram illustrating the relative energies of molecular orbital states for Fe2+ and S22-, leading to the known band structure of pyrite. The octahedral ligand field about Fe splits d-orbitals into the fully occupied non-bonding t2g and empty antibonding eg* set. Fe 4s electrons (white) are donated to the lowest unoccupied states of the S2 group which are the antibonding π* states. This leaves t2g states as the highest occupied states in the bulk.

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Figure 29. A comparison of features in He I UPS spectra with calculated total and projected densities of states for pyrite using Crystal, from Rosso et al. (1999a). The characteristic peak at 1 eV, well reproduced by the calculations, is attributed to Fe 3d t2g states. S 3p states predominate deeper in the valence band.

pyrite to date. Also, interestingly, a weak π-bonding component between the Fe 3d t2g and S 3p orbitals was found, supporting an early idea that the t2g states are not completely non-bonding (Burns and Vaughan 1970). The well accepted experimental bulk band gap is between 0.9–0.95 eV. As previously discussed, the one-electron ab initio methods are not capable of meaningful accuracy in this regard. For example, within DFT methods, GGA calculations underestimate the band gap by 0.6 eV (Raybaud et al. 1997), whereas, in a fortuitous result, LDA calculations predict 0.9 eV (Eyert et al. 1998). Pyrite–(100) surface. Pyrite cleaves poorly along the cubic planes and such surfaces can be macroscopically characterized as conchoidal. However, air and UHV STM images of cleaved pyrite at the nanometer scale have demonstrated that true (100) terraces are present and can be quite prevalent (Eggleston and Hochella 1990, 1992a; Eggleston et al. 1996; Rosso et al. 1999a, 2000). But these terraces are not laterally extensive and a high density of steps is typical. Ab initio calculations of the surface structure are much fewer than those performed for the bulk. DFT cluster and periodic finite slab calculations demonstrate that cleaving Fe-S bonds along cubic planes in the structure results in a stable surface that shows very little relaxation (Fig. 30) (Rosso et al. 1999a). Recent GGA DFT periodic calculations (using CASTEP) of the fully relaxed surface structure, ideally the most accurate calculations to date for this surface, indicate that surface Fe atoms are retracted only 0.1 Å towards the next lower Fe plane, and S atoms are retracted an order of magnitude less (Rosso and Becker in prep). Lateral relaxation is insignificant and the face-centered symmetry is preserved. Structurally, this surface is very close to ideal, meaning a bulk terminated surface. These results are in excellent agreement with LEED images of these surfaces (Pettenkofer et al. 1991; Rosso et al. 1999a). Although the surface atomic structure varies little from the ideal, the redistribution of electrons at the surface can be expectedly significant because of the covalent nature of Fe-S bonding in pyrite. An ideal starting point is to consider the predictions of the electron counting principles of autocompensation, which uses the formal charges on the atoms. Each Fe atom contributes two valence electrons to six Fe-S bonds, or 1/3 e- per bond. Each S atom contributes six valence electrons to three Fe-S and one S-S bond. Taking one electron from each S to form the S-S bond, 10 are left for the six Fe bonds

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Figure 30. The pyrite (100) surface structure. Fe (black balls) and S2 centers (white balls) are arranged in a facecentered cubic lattice. A surface cell is outlined (~ 5.4 Å). Along the surface normal, it can be seen that the surface is built up by stacks of charge neutral layers consisting of a group of three S-Fe-S atomic planes. This is a type II surface and is electrostatically stable (see Fig. 3). Each surface Fe and S atom is missing one bond. Geometry optimizations indicate that the relaxed surface structure differs very little from the ideal bulk termination.

around an S2 group, or 5/3 e- per Fe-S bond. In this way, every bond in pyrite has two electrons and the structure is charge neutral. At the surface, there is a 1:1 ratio of dangling bonds on uppermost Fe and S atoms, arising because only Fe-S bonds are broken. Dangling bonds localized on Fe and S form surface states in the bulk band gap at energies that can approach those for the atomic orbital components of the free atoms. For reasons of relative electronegativity, the 1/3 electron from each Fe dangling bond is transferred to each S dangling bond, completely filling the anion dangling bonds (1/3 + 5/3). Hence, the (100) surface is autocompensated and considered structurally stable. This exercise then predicts that the sample will have a surface gap smaller than the bulk band gap. The Fermi level at the surface will be flanked energetically above by unoccupied Fe dangling bond states and below by occupied S dangling bond states. In the absence of any other information, this is a reasonable starting guess for the electron redistribution at the surface of a pure sample of FeS2. This constitutes a good reference for ab initio calculations that typically consider the pure case first. But, as will be seen, other factors complicate this simple picture. High resolution STM imaging of (100) terraces on cleaved natural and synthetic pyrites have collected detailed electronic structure information at the atomic scale (Eggleston and Hochella 1990, 1992a; Fan and Bard 1991; Siebert and Stocker 1992; Eggleston et al. 1996; Rosso et al. 1999a, 2000). All but the latter two UHV studies were performed in air or in oil, and most were confirmed to have been performed on n-type samples. Various interpretations of the images have been reported, attributing the high tunneling current sites to the S2 sublattice, the Fe sublattice, or both simultaneously. However, in only one case has the S2 sublattice been convincingly imaged and that occurred in close proximity to a step edge under oil, with interactions between the surface and solvent molecules not ruled out as a possible influence on the image data (Eggleston

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and Hochella 1992a). Otherwise, the majority of the STM results on the (100) surface in air and UHV clearly indicate that at both negative and positive sample bias around the Fermi level, the Fe sublattice dominates the tunneling current (Fig. 31). For various physical reasons, this effectively means the highest occupied and lowest unoccupied states at the surface are localized on Fe sites (Rosso et al. 1999a). This is in conflict with the autocompensation model which leads to a prediction that the S sublattice should be imaged at negative sample bias, and the Fe sublattice should be imaged at positive sample bias. Ab initio DFT calculations using a nine monolayer thick finite slab (using Crystal) were performed to model the (100) surface electronic structure (Rosso et al. 1999a). Densities of states localized on Fe and S atoms at the surface were compared with that on Fe and S atoms located in the middle of the slab, which are fully coordinated and can be taken to represent the bulk sites. According to the slab calculations, breaking an Fe-S bond and the loss of overlap with S 3p orbitals energetically destabilizes every Fe 3d orbital with a z-component (z defined to be normal to the surface plane) (Fig. 32). Figure 31. Atomic scale UHV STM image This effect is primarily seen in the shift of Fe (a) of pyrite (100), from Rosso et al. dz2 states to higher energy, but also in the (1999a). The tunneling conditions were slight destabilization of dxz and dyz states -0.2 V bias and 2 nA setpoint current. The face-centered cubic surface cell is outlined relative to dxy. At the same time, the (~5.4 Å). Based on supporting calculations occupation of dx2-y2 states increases (see text), high tunneling current sites are indicating that electron density is shifted into attributed to the Fe sublattice as shown in remaining Fe-S bonds. This is consistent (b). with an observed trend of increased covalent character of bonds at non-polar surfaces of covalent materials (Noguera 1996). The energetic displacements of the d-orbitals are somewhat similar to the ligand field theory predictions of Bronold et al. (1994b), but were not found to lead to spin unpairing as asserted therein. The ab initio results are also similar to the prediction of autocompensation in that charge is preferentially transferred out of dangling Fe 3dz2 orbitals into remaining Fe-S bonds, but there are differences in several respects. Although it has lost bonding overlap with the removed surface S atom, the dz2 orbital still has bonding overlap with S 3p states from the S below. Because electrons prefer bonding orbital environments, the states with dz2 components have lost some of their attraction as electron hosts and charge is driven out, effectively lifting the bonding dz2 component out of the valence band. Net movement of S 3p derived states is imperceptible, apparently from a balance of the destabilizing effect of losing bonding overlap at the surface and the stabilizing effect of backfilling the remaining Fe-S bonds. The ab initio calculations suggest that a surface gap will be formed with Fe dz2 states

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Figure 32. Calculated occupied densities of states for the bulk and the (100) surface of pyrite projected onto the Fe 3d orbitals, from Rosso et al. (1999a). The z axis is arbitrarily chosen to be parallel to the surface normal direction. dz2-like states are shifted to higher energy and partially depopulated at the surface. Other states with a z-component show similar trends but to lesser degrees. Nonbonding dxy states are changed very little because overlap with S 3p orbitals is unchanged in the lateral directions at the surface. The density of dx2-y2 states increases at the surface, likely indicating a shift of electron density from dangling bond dz2 states into remaining Fe-S bonds at the surface.

constituting the highest occupied states (Fig. 33). Recent plane wave GGA calculations of the surface electronic structure and theoretical STM images confirm this result and indicate that Fe dangling bonds form the surface gap (Rosso and Becker in prep). Collectively, the calculations are in excellent agreement with the STM observations in that the Fe sublattice is imaged at both negative and positive sample bias. Also, from the range of bias voltages that were found to lead to significant tunneling currents (−400 to −20 mV and +20 to +140 mV), it is likely that a surface gap is indeed present and is quite narrow (~ 40 meV) with respect to the bulk band gap (~ 0.9 eV). The small surface gap is suggestive that Fe dangling bonds impart a nearly metallic characteristic to the (100) surface. The agreement between the STM and ab initio results was perhaps best demonstrated using DFT aperiodic cluster calculations to simulate I(V) tunneling spectra. Atomic orbital contributions to the densities of states around the Fermi level at a constant height above the atomic corrugation at the surface. Spectra were calculated for tip positions directly over surface Fe and S2 group sites and compared with characteristic experimental I(V) spectra (Fig. 34). The spectra clearly show an occupied surface state, with the calculations making it possible to attribute it to a Fe dz2-like dangling bond. The lowest unoccupied states were found to be of mixed Fe dz2 and S 3p character, not in conflict with the STM data. The ab initio calculations were successful in describing the surface electronic structure of the uppermost few Ångstroms. But, because the samples showed n-type behavior, it is likely that the redistribution of electrons in response to the surface involved

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Figure 33. Energy level diagram illustrating the changes in the d-orbital structure at surface Fe atoms, determined from Crystal finite slab calculations and UHV STM experiments. The density of states with a particular d-orbital character is represented by gaussian peaks, with filled peaks corresponding to occupied states (see Fig. 32). In the bulk, the dz2 and dx2-y2 orbitals, i.e., the eg set, participate in Fe-S bonds. At the surface, the loss of bonding overlap with one S atom destabilizes primarily dz2-like states to higher (occupied) and lower (unoccupied) energy within the bulk band gap. These dangling bonds form the HOMO and LUMO at the surface.

Figure 34. A comparison of STS spectra and calculated densities of states for pyrite (100), from Rosso et al. (1999a). The area-averaged surface electronic structure is seen in the normalized (dI/dV)/(I/V) tunneling spectrum collected over a random distribution of points over the surface (a). Individual, atomically resolved tunneling spectra show site specific features in the LDOS at the surface (b). Calculated local densities of states (LDOS) for tip positions over surface sites (c) show striking similarities to the characteristic features shown in (b). Contributions to the LDOS over surface Fe and S2 sites originate primarily from Fe 3dz2-like dangling bond states and S 3p states, respectively.

a much thicker surface section. Impurities in most pyrite samples (natural and synthetic) can play a dominant role in the final occupation of surface states. The computational expense of ab initio calculations currently inhibits modeling the space charge layer, but some insight is still possible in this regard. Equilibration between the pyrite bulk and Fe dangling bond states, located in the bulk band gap, involves filling the surface states up to the position of the bulk Fermi level (see Fig. 6). The charge occupying the surface states

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constitutes a negative potential which repels bulk electrons from approaching the surface, causing an upward bending of the bulk bands at the surface. The band bending is an experimental observable based on SCLS techniques, shown nicely for n-type pyrite by Bronold et al. (1994b). It is noteworthy that the charge in the surface states was apparently sufficient to screen the pyrite bulk from electrostatically coupling to the STM tip, based on the symmetric exponential conductance behavior in I(V) spectra. It is likely that this averted possible complications from tip-induced band bending and its deleterious effects on electron tunneling data (see Fig 8). Recently, defects have become an important subject for (100) surfaces of pyrite, as demonstrated by the numerous experimental studies in this area (Buckley and Woods 1987; Birkholz et al. 1991; Karthe et al. 1993; Bronold et al. 1994a,b; Nesbitt and Muir 1994; Eggleston and Hochella 1992a; Eggleston et al. 1996; Gueveremont et al. 1997, 1998b; Nesbitt et al. 1998, 2000; Schaufuss et al. 1998; Rosso et al. 1999a, 2000). Step densities have been observed to be very high with STM. Outcroppings of planar bulk defects at the surface have been observed with STM in air (Eggleston et al. 1996). The presence of monosulfides across the cleaved or in-vacuum grown surface is well documented with XPS. It is reasonable to expect these S deficiencies to occur, among other places, along step edges oriented along surface diagonal directions, because the step edge would otherwise be charged. Steps along these directions and cubic surface directions have been documented with ambient and UHV STM. At the same time, Fe vacancies have been imaged at the atomic scale with UHV STM (Fig. 35). Collectively, these defects comprise a significant fraction of surface sites, but very little is known about their structure and influence on surface reactivity. Some recent modeling work has been performed to begin to address these issues. The stability and structure Fe vacancies and step edge sites has been addressed using a combination of experiment and theory. Ab initio aperiodic clusters were used to parameterize an electrostatic Fe adatom–surface interaction model to quantify barriers for Fe self-diffusion (Rosso et al. 2000). This data was used to interpret UHV STM observations of Fe vacancy lifetimes. The models were later extended to include step edges and dynamic simulations of diffusion (Becker and Rosso in prep). In both cases, modeling nicely complemented experiment by serving to provide physical explanations and quantification of processes occurring faster than STM examination could document. Oxidation of Fe states at the surface leading to the development of low tunneling current patches across the (100) surface has been observed using STM in air and O2/H2O

Figure 35. Atomic scale UHV STM image of the pyrite (100) surface, from Rosso et al. (2000). The tunneling conditions were -0.2 V sample bias and 2 nA setpoint current. A face-centered cubic surface cell is outlined (~ 5.4 Å). Half unit cell high step edges are outlined by white dashed lines, and are observed to follow cubic [10] and diagonal [11] surface directions. Fe vacancies are indicated as sites A and B.

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exposure experiments in UHV (Eggleston et al. 1996; Rosso et al. 1999b). Photoelectron spectra have long documented the development of Fe-O bonds at the surface and oxidized Fe end-products. Oxidation patches form by a birth and spread mechanism similar to that observed on galena (100), and are also bounded by surface diagonal directions (Fig. 36). High resolution imaging in UHV at the perimeter around oxidized areas show a gradual change in the surface electronic structure that extends over approximately one nearest neighbor row of atoms. The image data suggest that the oxidation patch delivers a change in the propensity of next-nearest neighbor Fe sites to react, consistent with the observations that patches continue to grow outward. More generally, this can be considered evidence that a reaction occurring at one surface site can remotely influence a reaction at another through the surface, with the degree of influence dependent on proximity. For semiconducting minerals, proximity effects such as these may play a significant role in the redox reactivity at their surfaces (Becker et al. 2001). Water has been known to play an important role as a reactant in the overall oxidation process, and was found to enhance the rate of oxidation in the UHV experiments. Complementary cluster calculations (aperiodic DFT, spin unrestricted) were performed to model the interaction of O2 and H2O with this surface with the goal of gaining some insight into the initial oxidation mechanism (Rosso et al. 1999b). Geometry optimizations of the position of an O2 molecule with respect to the cluster were performed spin unrestricted (spin multiplicity = 3) to accommodate the paramagnetic ground state of the free O2 molecule. The O2 molecule was found to dissociatively chemisorb to surface Fe sites. One end of an O2 molecule bonded with a surface Fe site, giving a tilted “end-on” configuration for the molecule with a preferred alignment along surface diagonal directions. Electron transfer from surface Fe into antibonding π* orbitals of the O2 molecule destabilized the O-O bond and indicated that dissociation would be effective. Subsequent geometry optimizations of Fe-O surface sites showed that oxidation to Fe3+-O− was facile, with the oxygen atoms tightly chemisorbed to Fe sites. This process

Figure 36. Dual-mode UHV STM images collected simultaneously on pyrite (100) exposed to 4 L oxygen, from Rosso et al. (1999b). The tunneling conditions were -20 mV sample bias and 3 nA setpoint current (a), and 20 mV sample bias and 3 nA setpoint current (b). The scale bar represents 10 Å. Dark patches due to oxidation occur in both images indicating that state density is removed from both the valence and conduction band edges. A local reduction in the tunneling current at sites neighboring a patch can be generally seen as circumscribed by white dashed lines. Oxidation patches are generally bounded by [11] directions across the surface.

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was found to be energetically enhanced by the presence of H2O. H2O also sorbed preferentially to Fe sites, but showed no tendency to dissociate in the absence of sorbed oxygen. For the coadsorption case, the adsorption of H2O to an Fe site adjacent to the Fe-O site had the effect of enhancing electron transfer to the oxygen of the Fe-O group, concomitantly weakening one O-H bond in the water molecule. This indicated an increased propensity for H2O to dissociate on the surface in the presence of sorbed oxygen, which was confirmed by lower total energies obtained for Fe-OH, Fe-H dissociated H2O species with respect to Fe-H2O. In this way, the calculations provide an explanation for the relative roles of O2 and H2O and their synergy in the initial formation of oxidized surface species (Fig. 37). In UHV, these reactions are all facilitated by the HOMO/LUMO states arising from Fe dangling bonds. Because site-to-site electron transfer is also demonstrated, the observations are consistent with the proximity effect inferred from the UHV STM observations of oxidation patch growth. Pyrite–(111) surface. Compared to the (100) surface of pyrite, substantially less work has been performed on (111). UHV based spectroscopies, principally XPS, have been used to assess the relative surface reactivities of the (100) and (111) planes (Guevremont et al. 1998a; Elsetinow et al. 2000), but little information is available on the surface structure of (111). The structure is prone to be much more complex than (100) because both Fe-S and S-S bonds should be expected to be broken and a higher surface energy for the ideal terminations may drive a reconstruction. Recent periodic plane wave

Figure 37. Illustration of the oxidation of pyrite (100) by coadsorption of O2 and H2O. In the absence of O2, H2O is weakly bound to the surface and does not dissociate (a). O2 dissociatively chemisorbs to Fe sites on the surface, drawing charge from the nearby Fe and S sites, leaving Fe-O groups (b). If H2O is present, the interaction of O2 with the surface is stronger because of the electron density donating ability of H 2O molecules. In this case, the interaction of H2O with the surface is also stronger and leads to cleaving one of the O-H bonds (c). S-OH bonds are predisposed to form adjacent to hydroxylated Fe sites because the transfer of electron density out of surface S sites increases their susceptibility to nucleophilic attack.

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DFT calculations have been performed with CASTEP to model the (111) surface (Rosso and Becker in prep). Also noteworthy are similar periodic HF and DFT calculations for the (111) surface of RuS2, which is isostructural and isoelectronic with pyrite (Frechard and Sautet 1995; Grillo et al. 1999). From an analysis of the bulk structure, there is only one possible stoichiometric non-polar (111) surface (Fig. 38). Although all possible variations in S deficiencies and related reconstructions have yet to be modeled, it is reasonable to assume that if a non-polar surface can be generated, it will have a low surface energy and will be resistant to reconstruct for electrostatic reasons. The non-polar surface is formed by a planar cut through the midpoint of S-S bonds along the (111) plane. The unit cell of the ideal termination is hexagonal with a cell edge ~ 7.6 Å. Full geometry optimizations of this surface using a 15 monolayer thick supercell slab (the first charge neutral repeat unit) suggest that this surface is quite stable. Like the (100) surface, relaxation is slight, with the largest movement in the form of a contraction of uppermost S atoms by ~ 0.2 Å. The calculated surface energy of 1.6 J/m2 is somewhat higher than that 1.1 J/m2 calculated similarly for (100) but is reasonable when the aerial density of broken bonds is compared. The broken bond density on (100) consists of 14Fe-S/nm2, whereas the (111) surface consists of 10Fe-S/nm2 and 7S-S/nm2. Analysis of the densities of states and electron density distribution at the optimized (111) surface provide a basis for the interpretation of STM images, which have yet to be reported for this surface. The calculations indicate that the highest energy occupied states are localized at uppermost S sites on this surface. This comes in contrast to the experimental and theoretical results for (100), where highest occupied states are at Fe sites. An implication is that STM images of this surface at negative sample bias should show a regular array of trigonally arranged high tunneling current sites associated with the uppermost S atomic plane (Fig. 39). This also predicts that S sites have a higher propensity to be oxidized relative to Fe sites by surface interaction with an oxidant such as O2. The mechanism of such an interaction has yet to be explored in detail, but recent experiments on in-vacuum cleaned natural pyrite growth surfaces have shown that

Figure 38. The arrangement of atomic planes along the [111] direction in pyrite. Two non-equivalent S planes are identified as SA and SB planes. Equivalent Fe planes sandwich four S planes consisting of two SA and two SB planes, arranged symmetrically in the sequence Fe-SB-SA-SA-SB-Fe. A cut can be found between SA-SA planes that leads to a type II surface and is electrostatically stable (see Fig. 3). The cut locations are denoted by black dashed lines (defining subsets of the full unit cell along this direction).

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Figure 39. Calculated occupied state STM data for the (111) surface of pyrite (Rosso and Becker in prep). Plane wave GGA calculations of the surface structure indicate that this termination is stable and relaxes very little from the ideal bulk termination. This leaves trigonal groups of SA atoms forming the uppermost plane at the surface. Analysis of the highest occupied states indicates that the top of the valence band is predominantly of S 3p character and localized on this upper atomic plane. STM images at low negative sample bias should therefore show a hexagonal array of trigonal groups of high tunneling current sites with a surface repeat ~ 7.6 Å (a). Each surface cell is composed of one SA group and one SB site slightly lower that does not appear as a high tunneling current site (b).

oxidation of the (111) surface by O2/H2O mixtures is facile, and somewhat more rapid initially than that for the (100) surface (Guevremont et al. 1998a; Elsetinow et al. 2000). The result was tentatively attributed to Fe availability at the surface, but the stable (111) surface structure has not yet been rigorously solved via experiment. Quantitative LEED and angle-resolved photoemission experiments would be of great value in this regard. Preferred adsorption sites on the (111) surface for an oxidant such as O2 are unknown, and may be different than for the (100) surface because of the issue of the compatibility of surface site and oxidant orbitals (e.g., Luther 1990). Based on the oxidation behavior found for the (100) surface, it also seems possible that site-to-site proximity effects could be effective on (111). Although many other open issues remain as well, the plane wave calculated surface structure is consistent with the experimental findings because the results suggest that different oxidation mechanisms may tend to operate on the two surfaces. Oxides Hematite–Bulk. Hematite (α-Fe2O3) crystallizes in the corundum structure (space group R 3 c ). It consists of hexagonally closest packed oxygen (001) planes with iron atoms filling 2/3 of the octahedral sites. Each Fe3+ is coordinated to six O2- anions in slightly distorted octahedra. The primitive rhombohedral cell contains two formula units. Along [001], hematite consists of an alternating sequence of Fe-O-Fe-Fe-O-Fe layers (Fig. 40). It is antiferromagnetic below 953 K (weakly ferromagnetic at room temperature), having equal but opposing magnetic moments canceling each other in the unit cell (nearly canceling at room temperature). All the iron atoms are in high spin d5 configurations. Fe-Fe bilayers are coupled spin parallel and adjacent bilayers are coupled antiparallel, leading to a reduction in the symmetry to R 3 when considering the magnetic cell (Fig. 40). The [001] sequence is then better described as Fe1-O-Fe2-Fe2-O-Fe1. Isostructural transition metal oxides are also commonly antiferromagnetic but often exhibit a different spin distribution, such as eskolaite (α-Cr2O3) which follows Cr1-O-Cr2-Cr1-O-Cr2.

The wide band gap of hematite (~ 2 eV, Mochizuki 1977; Ma et al. 1993) leads to effectively insulating characteristics by restricting electrical conductivity via typical band

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Figure 40. The primitive (left) and conventional (right) unit cells for hematite viewed perpendicular to [001]. Each Fe atom (black) is in octahedral coordination with O atoms (white). Octahedra share edges along (001) planes and share faces along the [001] direction. Bilayers of Fe-Fe atomic planes separated by O planes can be seen in the conventional unit cell. Based on the topology alone, each Fe atom is equivalent. The antiferromagnetic spin distribution reduces this symmetry. Fe planes within a bilayer are coupled spin parallel and adjacent bilayers are coupled antiparallel.

semiconductor means. Instead, conduction predominantly occurs by the hopping of conduction band electrons between cations (Goodenough 1971). Nearest neighbor iron atoms are closely spaced in adjacent octahedra across both shared edges in (001) planes or shared faces along [001] directions. Although the separation of iron atoms is nearly the same in both cases (2.90 Å vs. 2.97 Å respectively), the spin distribution in hematite leads to very high conduction anisotrophy, with very low activation energies for hopping within Fe-Fe bilayers along (001) planes, and high activation energies between cations in adjacent Fe-Fe bilayers along [001] (Goodenough 1971; Gleitzer 1997). Extrinsic semiconducting behavior can arise from impurities or defects and, like pyrite, hematite can exhibit both n- and p-type conduction (Goodenough 1971; Gharibi et al. 1990; Gleitzer et al. 1991). The subtleties behind the magnetic and electrical properties of hematite, and its technological importance have motivated many experimental and theoretical studies of the electronic structure of this material. The application of ab initio calculations to hematite began with the cluster calculations of Tossell and Vaughan (Tossell et al. 1973, 1974; Vaughan et al. 1974) and has been similarly followed since then by many (e.g., Sherman 1985, 1987; Sherman and Waite 1985; Fujimori et al. 1986, 1987; Armelao et al. 1995). Periodic calculations have been performed recently, using both HF/DFT LCAO (Catti et al. 1995; Catti and Sandrone 1997) and LDA plane wave methods (Sandratskii and Kubler 1996; Sandratskii et al. 1996). There is much consensus between the cluster and periodic modeling results compared to experiment, especially in regards to the band structure and 0 K magnetism. Deductions regarding the electronic structure from photoemission and inverse photoemission experiments indicate that the upper valence band consists of a wide, bonding O 2p–Fe 3d band, and the conduction band is a more narrow antibonding O 2p

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band (e.g., Fujimori et al. 1986; Lad and Henrich 1989; Ciccacci et al. 1991). A longstanding debate exists regarding the relative atomic orbital contributions of Fe 3d and O 2p states to these bands. Cluster calculations (configuration interaction) and photoemission experiments were nicely combined by Fujimori et al. (1986) to put forth a thorough explanation of the valence band, later confirmed by Lad and Henrich (1989). The results indicated that the valence band is strongly mixed with nearly equal contributions of O 2p and Fe 3d states. At the same time, Fujimori et al. (1986) suggested that the conduction band was predominantly of Fe 4s character. In contrast, using inverse photoemission experiments, the conduction band was found to be dominantly of Fe 3d character (Ciccacci et al. 1991). The strongly mixed valence band is indicative of socalled charge transfer materials because the band gap is formed by charge transfer between cations and anions, as opposed to arising from the correlation energy from d-d electron interactions (Zaanen et al. 1985). The p-d characteristic of the band gap has been reasonably well reproduced by ab initio periodic calculations, but differences remain. The HF finite slab LCAO calculations suggest that much less mixing should be present and that the top of the valence band is almost completely O 2p–like (Catti et al. 1995; Catti and Sandrone 1997). No significant difference was found by superimposing DFT corrections for the correlation energy on the self-consistent HF solution. However, plane wave LDA calculations seem to have better reproduced experiment by similarly showing that the valence band is strongly mixed (Sandratskii and Kubler 1996; Sandratskii et al. 1996). Calculated densities of states mimicked the experimental valence and conduction band spectra remarkably well (Fig. 41). No reason is apparent to explain the discrepancy between the calculations at this time. Both types of periodic calculations are consistent with the Fe 3d origin of the conduction band, and the p-d nature of the band gap. Hematite–(001) surface. By examination of the bulk structure, a low surface energy, non-polar bulk termination is found along (001) planes between the iron atoms in Fe-Fe bilayers (Fig. 42). The planar division creates two equivalent surfaces composed of a 1/3 monolayer of iron over a close-packed oxygen layer. Along a single [001] vector, the minimum repeat unit of 18 atomic planes (conventional cell) can be seen to have no net dipole moment and is thereby predicted to be stable from an electrostatic perspective. Similarly, using the covalent perspective, autocompensation predicts that this surface

Figure 41. Comparison of calculated density of states (solid line) with UPS (crosses) (Fujimori et al. 1986) and inverse UPS spectra (diamonds) (Ciccacci et al. 1991) [Used by permission of IOP Publishing Limited, from Sandratskii et al. (1996), J Phys: Cond Mat, Vol. 8, Fig. 3, p. 987]. The calculations were performed using LDA. The calculated data for the occupied and unoccupied DOS have been shifted independently on the energy scale to align with the experimental features. The ability of the calculated DOS to predict the features in the spectra is clearly remarkable.

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Figure 42. Illustration of two possible (001) terminations of hematite. Along [001], the bulk repeat unit is 18 atomic planes thick (~ 13.7 Å conventional cell). An electrostatically stable (non-polar) termination (labeled Fe-O3-Fe) is found by dividing the structure between Fe-Fe planes in a bilayer (a). This leaves a 1/3 monolayer of Fe over a close packed oxygen layer. A polar oxygen terminated surface is created by dividing the structure just above oxygen planes (labeled O3-Fe-Fe). Both have been experimentally and theoretically identified (see text). The uppermost planes of the Fe-O3-Fe termination (to a depth of two atomic planes below the uppermost O plane) consist of three Fe sites labeled A, B, and C (b). Using the same labeling scheme, the uppermost planes of the O3-Fe-Fe termination are shown in (c). The hexagonal unit cell is outlined (~ 5 Å).

should have little driving force to reconstruct. Fe contributes three valence electrons to six bonds, or 1/2 e- per bond, and oxygen contributes six valence electrons to four bonds, or 3/2 e- per bond. A 1:1 ratio of Fe to O dangling bonds is found at this surface. Transferring the 1/2 e- in each Fe dangling bond to the 3/2 e- in each oxygen dangling bond completely fills the anion dangling bonds and the surface is autocompensated. Thus, this Fe-terminated surface has long been predicted to be the most stable (001) termination of hematite. It turned out that early attempts to prepare or clean this surface in UHV using sputter/annealing cycles experienced difficulties arising from a sensitive dependence on

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the partial pressure of oxygen. In vacuum, Ar+ ion bombardment preferentially removes oxygen atoms and leads to facile reduction of the surface to FeO (McIntyre and Zetaruk 1977). High temperature annealing of such defective surfaces relies on the diffusion of oxygen atoms from the bulk to reestablish the 2:3 stoichiometry at the surface. In this case, UHV annealing can nearly, but not completely recover the Fe2O3 stoichiometry at the surface (Kurtz and Henrich 1983, 1987). To fully reoxidize the surface, annealing must be performed in the presence of an oxygen background. Subsequent UHV studies have followed some variation of a sputter/anneal (in PO2) method for this surface. Collectively, these investigations have unveiled the fascinating complexity of the T-PO2 surface phase diagram for the Fe-O system. Remarkable atomic scale UHV STM studies have demonstrated preparation conditions that lead to a variety of possible Fe3O4 (111) or Fe1-xO overlayers and coexisting surface phases on hematite (001) (Condon et al. 1994, 1995, 1997, 1998). In certain cases, the distribution of coexisting phases themselves becomes ordered across the surface and mesoscale superlattices are formed. These surface structures are generated at high temperatures in diminutive concentrations of oxygen (<10-6 mbar) near or beyond the thermodynamic stability limits of hematite. The STM observations are made at room temperature on the quenched samples, likely preserving the high temperature/low PO2 phases in a metastable state. The studies make it clear that many surface reconstructions are possible under these extreme conditions. These surfaces have yet to be approached with ab initio modeling, which would be challenging because of the structural complexity, but could prove to be highly useful in sorting out reconstruction mechanisms and the relative stabilities of these interesting surface phases. At lower temperatures and at higher oxygen availability (>10-6 mbar), the T-PO2 surface phase diagram is comparatively simpler. The possible (001) terminations follow more closely to the previous qualitative discussion on the predicted non-polar bulk termination. In this regime, two surface terminations are possible. One is the Fe termination discussed above (referred to as Fe-O3-Fe), and the other is oxygen terminated (referred to as O3-Fe-Fe) (Fig. 42). Both derive from non-reconstructed bulk-terminations and have been investigated using both modeling and experiment. The relaxation and electronic structure of the Fe-O3-Fe surface was first modeled by Becker et al. (1996) using Crystal periodic UHF calculations. Relaxation in vacuum conditions was predicted to be restricted primarily to the upper two atomic planes, with a contraction of the uppermost Fe layer by ~ 0.5 Å comprising most of the displacement. The underlying close-packed oxygen layer relaxes only very slightly, with a ~ 0.1 Å layer contraction and ~ 0.01 Å lateral adjustments which preserve the C3 surface symmetry. Similar relaxation for the equivalent basal terminations of isostructual minerals have been found using various other ab initio approaches (Causa et al. 1989; Guo et al. 1992; Manassidis et al. 1993; Rehbein et al. 1996, 1998; Rohr et al. 1997). The dramatic contraction of the uppermost metal layer and the shallow penetration depth of the relaxation has been explained as being driven by electrostatics to minimize the local dipole between the upper two atomic planes (Rehbein et al. 1996). The O3-Fe-Fe termination was first predicted to be stable using plane wave GGA calculations to evaluate the surface energies of a variety of (001) terminations as a function of the chemical potential of oxygen (Wang et al. 1998). Its structure can be understood by adding the next two higher atomic planes (one Fe and one O) to the Fe-O3Fe termination, restoring the uppermost Fe-Fe bilayer and capping all bilayer Fe atoms with trigonal groups of oxygen atoms. Although this termination is highly polar, it is thought to be stabilized through large relaxations (and electron redistribution) involving a layer contraction of the underlying Fe-Fe bilayer by nearly 80% of its interlayer spacing in the bulk. It is noteworthy that for the Fe-O3-Fe surface, these calculations also

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predicted a similar contraction for the uppermost Fe layer, but suggest significant adjustments continue to penetrate more than 6 atomic planes inward in an oscillatory manner. This includes a 33% contraction of the bilayer below the two uppermost atomic planes. A similar pattern of deeply penetrating layer adjustments have been experimentally determined for the Fe-O3-Fe termination using XPD (Thevuthasan et al. 1999). The penetration depth for covalently bonded materials is likely an issue of surface compactness (Noguera 1996) and, in this case, remains as unresolved for the Fe-O3-Fe surface. Also, the uppermost oxygen plane for both surfaces are predicted to undergo a small rotation of O3 groups about the surface normal axis (Wang et al. 1998). These results are quite different than previous predictions for the Fe-O3-Fe surface. Nonetheless, the Wang et al. (1998) calculations predicted that both terminations can be stable and are interrelated by PO2. In the Wang et al. (1998) study, evidence for both terminations was provided by atomic scale UHV STM images on the basal surface prepared by annealing at 1100 K in 10-3 mbar O2. Both terminations were found to coexist simultaneously based on periodicities of high tunneling current sites and measured step heights that were too small to connect equivalent atomic planes. This finding was recently confirmed and the PO2 dependence systematically explored using UHV STM and LEED (Shaikhutdinov and Weiss 1999). Using 800oC as the annealing temperature, oxygen background concentrations < 10-5 mbar led to biphase Fe-O3-Fe and FeO1-x (111) terminations indicating partial surface reduction. Surface coverage by the Fe-O3-Fe termination was complete for PO2 = 10-5 mbar. For PO2 between 10-4–10-1 mbar, domains of both Fe-O3-Fe and O3-Fe-Fe terminations were found to coexist in regularly varying proportions. For PO2 = 1 mbar, the O3-Fe-Fe termination covered the entire surface. Presumably then, in PO2 > 1 mbar, this suggests that the surface should be uniformly terminated by oxygen (e.g., atmospheric conditions). However, the relative stabilities of the two terminations were determined in a way that is influenced by kinetic barriers in their formation. Recent X-ray photoelectron diffraction and STM experiments found no tendency to form the O3-Fe-Fe termination under aggressive oxidizing conditions using oxygen plasma during growth (Chambers and Yi 1999). Only the Fe-O3-Fe termination was found in this study, suggesting that the O3-Fe-Fe termination is metastable by comparison, even in concentrated O2 backgrounds. This issue is currently unresolved and more work is needed to better understand the relative stabilities of the two terminations. The indication that the stable (001) termination in PO2 > 1 mbar should be the oxygen terminated O3-Fe-Fe structure has important implications for understanding the surface chemical behavior of hematite in oxygenated geochemical environments. The basal surface has long been viewed as presenting the Fe-O3-Fe termination to air and aqueous solution. In air at STP, the partial pressure of O2 is ~ 0.2 bar, opening the need to reevaluate the surface structure under ambient conditions. For both terminations, electronic structure calculations have greatly aided in the interpretation of atomic scale STM data. At first glance, differentiating between Fe and O atomic planes seems straightforward because any of the Fe planes should have ~ 5 Å periodicity and the O planes should have ~ 3 Å periodicity. But as with many surfaces, problems arise when trying to deduce the surface structure from visual inspection of the STM images. This approach requires underlying assumptions about the electronic structure at the surface, which in this case are shown to be invalid based on modeling results. In the Becker et al. (1996) study, theoretical STM/STS data were calculated for the relaxed Fe-O3-Fe termination using Crystal applied to (001) slabs. The overestimated dropoff of the gaussian orbital wave functions into the vacuum was managed using a grid of basis functions (ghost atoms) centered over surface atoms. The ghost atom grid provides improved variational freedom for electronic states to extend into the vacuum

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region. For negative sample biases, it was shown that either Fe 3d or O 2p states could make the largest contributions to the tunneling current. O 2p lobes were shown to converge over Fe centers indicating that oxygen-based tunneling current could mimic the 5 Å periodicity of the underlying Fe-Fe bilayer (Fig. 43). At positive sample biases, Fe 3d surface states make the primary contributions to the tunneling current. Fe 3d states from any or all of the three uppermost Fe planes can make the most significant contributions to the tunneling current, with bilayer Fe 3d states clearly able to project through the oxygen layer into the vacuum. For the O3-Fe-Fe termination, the Wang et al. (1998) calculations demonstrate a similar result. Fe 3d surface states from the underlying Fe-Fe bilayer project through the overlying oxygen atoms significantly, and possibly dominate the tunneling current on this surface (Shaikhutdinov and Weiss 1999). For both terminations, the ab initio calculations clearly demonstrate that simply assuming a 1:1 correspondence of high tunneling current sites with the highest exposed atomic plane is inappropriate, and a dangerous practice in general. Ab initio modeling has not yet provided a complete explanation for all the STM data collected to date on this surface, and certain open issues are noteworthy. Beside the UHV STM studies, hematite (001) has been successfully imaged at the atomic scale in air and oil (Eggleston and Hochella 1992b; Eggleston and Stumm 1993; Eggleston et al. 1998; Eggleston 1999). The latter study showed an STM image collected in air which captured two commensurate lattice types across a distinct surface boundary similar to that observed in (Wang et al. 1998), but without any topographic relief between the layers. The lack of structural correspondence to possible step edge types, and other observations, forced alternative explanations of the nature of the tunneling mechanism for hematite (001). A coherent scheme for vacuum and non-vacuum conditions was envisioned based on a two-step resonant tunneling mechanism where the uppermost Fe of the Fe-O3-Fe termination act as resonant centers for electron transfer between sample and tip. This view is consistent with the UHV interpretations in that high tunneling current sites for all

Figure 43. A comparison of experimental (a) (Eggleston 1999) and theoretical (b) (Becker et al. 1996) occupied state STM data for the (001) surface of hematite. The experimental STM image was collected under oil at -336 mV sample bias and 1 nA setpoint current. The hexagonal surface unit cell is outlined (~ 5 Å). At negative sample bias, the periodicity of high tunneling current sites is seen to be ~ 3 Å, corresponding to either the oxygen sublattice periodicity or the locations of the A, B, and C Fe sites at the surface (see Fig. 42). Theoretical images for the Fe-O3-Fe termination mimic this periodicity and are able to attribute the high tunneling current sites to atomic orbital contributions. Fe 3d states give rise to the high tunneling current over A sites, but the convergence of occupied O 2p lobes is largely the cause of the high tunneling current over B and C sites.

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conditions arise from any of the upper three Fe layers, but a distinctly different mechanism is invoked. Also, it is noteworthy that this treatment does not require the O3-Fe-Fe termination to explain images acquired in air. This area is ripe for verification using ab initio calculations, especially in regards to the electron transfer parameters at the surface (Eggleston 1999). Hematite–Adsorbates. Very little ab initio modeling work on the interaction of adsorbates with hematite (001) has been performed. The interaction of H2O and O2 with the Fe-O3-Fe termination has been investigated (Becker et al. 1996). Here, C3 surface symmetry was maintained by geometry optimizing the position of three H2O molecules over each surface Fe site (Fig. 44). The optimized surface configuration showed that ~ 20% of the uppermost Fe site contraction towards the bulk in the vacuum case is recovered by re-establishing the 6-fold coordination sphere about surface Fe atoms. Fe-OH2 bond energies compare well with measured values based on temperature programmed desorption experiments (Hendewerk et al. 1986). The sorbed H2O layer is laterally stabilized by intersorbate and sorbate-O3 layer hydrogen bonds. This calculated sorbate structure is clearly stable at low temperature under vacuum conditions, but nearroom temperature classical dynamic simulations including background H2O suggest that

Figure 44. Illustration of the calculated structure of a layer of H2O sorbed onto the Fe-O3-Fe termination of hematite (001) viewed perpendicular (a) and parallel (b) to [001], from Becker et al. (1996). In this calculation, three H2O molecules coordinate each uppermost surface Fe site, restoring the 6-fold coordination at these sites. The H2O layer is stabilized by intersorbate and sorbate-O3 layer hydrogen bonds.

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the coordination of surface Fe sites on average are better approximated by 4-fold structures (i.e., H2O-Fe-O3-Fe) (Wasserman et al. 1997). Recent experimental observations also suggest a facile tendency for water to dissociate and hydroxylate the surface at PH2O > 10-4 mbar (Kurtz and Henrich 1987; Liu et al. 1998). Optimizations of similarly configured trigonal groups of O2 molecules forming an sorbed oxygen layer showed that little change is induced in the relaxed vacuum-based surface configuration by sorbed molecular oxygen. A low Fe-O2 bond energy is predicted for this configuration and apparently very little electronic interaction occurs (Becker et al. 1996). The calculation suggest that the interaction of O2 with the Fe-O3-Fe termination is very weak, consistent with experimental observations (Kurtz and Henrich 1987) and with the idea that generating the O3-Fe-Fe termination is kinetically slow, requiring high temperatures and significant mass flux at the surface. Magnetiet B – ulk. Magnetite is one of the most complex oxides in nature because of its electronic and magnetic properties. It adopts the inverse spinel structure with mixed valence Fe cations occupying the tetrahedral and octahedral interstices of a cubic close packed array of oxygen anions (space group F3dm) (Fig. 45). Cations in the primitive unit cell consist of two tetrahedral Fe3+, two octahedral Fe3+, and two octahedral Fe2+ ions. Below ~ 119 K, the Verwey temperature (TV), the oxidation states of the octahedral cations are fixed and magnetite is an electrical insulator. The ordering of the fixed valence cations and the slight concomitant structural change below TV has been experimentally approached by many workers and has been somewhat controversial since

Figure 45. The structure of magnetite, based on two octants of the inverse spinel structure, viewed along two mutually orthogonal directions (top, middle) and along an oblique (bottom). Two Fe sublattices are indicated as A (tetrahedral) and B (octahedral). Tetrahedral Fe are Fe2+ and, because of rapid electron exchange at room temperature, the octahedral Fe can be considered Fe2.5+ (see text). All Fe sites are in a high spin configuration. The spins of the A sublattice are coupled antiparallel with the spins of the B sublattice, giving rise to a net magnetic moment and the ferrimagnetic behavior of magnetite.

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the mid-1920's (e.g., Hamilton 1958). Above TV, electrons hop between the octahedral cations extremely rapidly (10-12s/hop, Mizoguchi and Inoue 1966), blurring the ferricferrous distinction between the two. This leads to near-metallic electrical conduction for the bulk material. All the d-electrons are in high spin configurations and the well known magnetic properties of this material arise from the specific coupling of unpaired spin directions (i.e., α or β) between cations. Magnetite is ferrimagnetic, with the overall magnetic moment formed by the net result of competition between two different types of magnetic sublattices in the crystal (Fig. 45). Within the octahedral sublattice, the cations are ferromagnetically coupled with spins aligned parallel to each other, consistent with electron hopping between them without violating Hund's rules for spin distributions (see Ashcroft and Mermin 1976). Counting the number of spins in the octahedral sublattice (2 Fe3+ and 2 Fe2+), one arrives at 20α and 2β with the net difference at 18α. The spins in the tetrahedral sublattice are aligned antiparallel to the octahedral sublattice (Shull et al. 1951), contributing 10β spins (2 Fe3+). The net result is an overall magnetic moment arising from an 18α - 10β = 8α spin majority. Very few ab initio calculations have been performed on magnetite to date. The large number of unpaired d-electrons leads to enormous variational freedom for the ground state occupation of states near the Fermi level. When combined with the fact that the complex magnetic structure does not allow for an easy simplification of the unit cell, the difficulties become apparent. Periodic LDA calculations were successful to describe the cohesion energy and the antiparallel magnetic coupling of the octahedral and tetrahedral sublattices (Zhang and Satpathy 1991; Uhl and Siberchicot 1995). In contrast, periodic UHF calculations located the lowest energy magnetic structure to be ferromagnetic (18α + 10α = 28α), with both cation sublattices spin parallel (Ahdjoudj et al. 1999). Futhermore, both methods showed limitations in that, although ground state (0 K) methods were applied, two of the three calculations erroneously converged to the metallic state, one that is experimentally known to manifest itself only above TV. The difficulties encountered in modeling the bulk become even more pronounced when extending the methods to the application of modeling magnetite surfaces. Magnetite–(111) surface. Planes of atoms along the [111] direction can be described as alternating layers of octahedral and tetrahedral cation layers between cubic closepacked oxygen layers, with a fraction of the cation sites unoccupied. For the cation planes, fully occupied octahedral layers alternate with tetrahedral-octahedral-tetrahedral multi-Fe layer groupings. Under the constraint of maintaining the bulk Fe3O4 stoichiometry, the repeat unit along [111] will always have a net dipole moment and any of the six possible non-equivalent starting planes will result in a polar surface to varying degrees. Electrostatically, these terminations are predisposed to either reconstruct, change the surface stoichiometry, or compensate for the instability by relaxation and redistribution of electrons. The lowest energy termination has been the subject of recent LEED, STM, X-ray absorption, and theoretical modeling studies (Vaughan and Tossell 1978; Weiss et al. 1993; Barbieri et al. 1994; Condon et al. 1994, 1997, 1998; SchedelNiedrig et al. 1995; Lennie et al. 1996; Ahdjoudj et al. 1999; Ritter and Weiss 1999; Shaikhutdinov et al. 1999; Fellows et al. 2000). Experimental preparation of this surface in UHV has proved to be complicated, similar to hematite surfaces, by a sensitive dependence on the background oxygen pressure. The (111) surface has been prepared as an epitaxial overgrowth on Pt(111) (e.g., Ritter and Weiss 1999), by UHV sputter/annealing techniques applied to cut and polished natural single crystal surfaces (Lennie et al. 1996), as well as indirectly in the form of an coexisting phase prepared in UHV on hematite surfaces (e.g., Condon et al. 1998). In most cases, the surface has been imaged with atomic resolution using UHV STM. Observed lattice periodicities of high tunneling current sites (typically ~ 6 Å) are consistent with bulk terminations exposing

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iron atoms arising from one of the iron layers, but oxygen bulk terminations have been discerned as coexisting phases as well (Lennie et al. 1996). In this case, the coexistence of multiple surface terminations aided in the determination of the various possible surface structures because lateral registries and step height differences could be correlated with bulk atomic planes and spacings along [111]. But in the absence of any other experiments or theoretical modeling, the STM results alone leave room for ambiguities regarding the most stable surface phase. However, using quantitative LEED, (Weiss et al. 1993; Barbieri et al. 1994; Ritter and Weiss 1999) presented arguments suggestive of a nonreconstructed, relaxed surface structure terminated at the tetrahedral iron layer (Fig. 46). The surface consists of a 1/4 monolayer of formerly tetrahedral iron atoms over a close

Figure 46. A suggested stable (111) termination of magnetite. All possible bulk terminations lead to polar surfaces that are unstable in the electrostatic perspective. The termination of the structure at the Fetet1 plane (a) represents one of the most stable of the possible polar surfaces (see text). This surface consists of a 1/4 monolayer of formerly tetrahedral Fe atoms over a close packed oxygen layer (b). The periodicity of the uppermost Fetet1 plane is consistent with LEED and atomic scale UHV STM images of this surface (c) [Used by permission of the American Physical Society, from Shaikhutdinov et al. (1999), Phys Rev B, Vol. 60, Fig. 2b, p. 11064].

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packed oxygen layer, with the complete stacking sequence starting from the surface given by FeTet1-O1-FeOct1-O2-FeTet2-FeOct2 (Ritter and Weiss 1999). Regarding its stability, in the ionic limit it has one of the lowest dipole moments, and in the covalent limit it leads to the lowest dangling bond density of the bulk terminations (Condon et al. 1994; Ritter and Weiss 1999). In this case, because a stoichiometric polar surface is formed, compensation must be arising from surface relaxation and the redistribution of charge. The LEED studies differ on the issue of the atomic displacements involved in the relaxation, but a rigorous analysis indicates that a layer-by-layer relaxation is probably operative (Ritter and Weiss 1999). The outermost FeTet1 layer contracts towards the surface by an estimated 41%, with lesser but significant displacements penetrating at least the first four atomic planes. Treating the surface atoms as ion cores of fixed charge, this relaxation can account for a decrease in the surface dipole moment by ~ 1.6 e/Å, where e is the elementary charge, but a substantial dipole moment still remains (Ritter and Weiss 1999). The rest of the compensation was envisioned to arise from accompanied shifts in electron density leading to increased covalency of bonds in the near surface region. In parallel timing, ab initio plane wave GGA calculations were performed to model the surface atomic and electronic structure of the relaxed form of this bulk termination (Shaikhutdinov et al. 1999). Modeled surface relaxation was found to mimic the relaxation directions determined using LEED. The near-Fermi level electronic structure at the surface was found to be dominated by FeTet1 3d contributions, which flank the Fermi level at the surface. The calculated surface states provided a sound basis for the assignment of the high tunneling current sites in UHV STM images using both positive and negative biases to the FeTet1 layer at the surface and completed a consistent picture of the surface structure (Fig. 46). Nevertheless, these results do not explain the STM results of Lennie et al. (1996). At the same time, the results of periodic UHF finite slab calculations also disagree with the FeTet1 terminated structure (Ahdjoudj et al. 1999). In this study, a variety of stoichiometric and non-stoichiometric bulk terminated slabs were geometry optimized. For the former case, Fe6O8 polar slabs were relaxed and used to deduce lowest total energy terminations. Slabs terminated on one side by FeTet1 and consequently FeTet2-FeOct2 on the other were found to be lowest in energy, but differentiating the stabilities of these two further required the development of new slabs to isolate each surface type. In this case, slabs were chosen using a symmetry constraint with respect to the surface normal direction so as to cancel the dipole moment for the slab cell and similarly terminate the upper and lower slab surfaces. Using this approach, the FeTet2-FeOct2 termination of Fe7O8 composition was determined to be more stable, in opposition to the FeTet1 termination discussed above, but in near agreement with one of the surfaces imaged in Lennie et al. (1996). The relaxed surface was found to involve layer adjustments, some with substantial displacements, including a possible electrostatically driven inversion of the FeTet2 layer to below the underlying two oxygen layers. It should be pointed out that the latter result showed a dependence on the ferrimagnetic (antiferromagnetic coupling of the two cation sublattices) versus ferromagnetic spin structure, one that the UHF bulk calculations misidentified (Shull et al. 1951). The ferromagnetic case was predicted for the slabs and was found to be a prerequisite for the layer inversion (Ahdjoudj et al. 1999). A different spin distribution at the surface is entirely plausible but likely is an artifact of the UHF calculation and, in the absence of independent information, cannot otherwise be differentiated. Therefore, the possible occurrence of the layer inversion at the true surface is unsubstantiated. Overall, there are predominantly two conflicting conclusions on the most stable

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(111) termination of magnetite, each based on internally consistent sets of experimental and theoretical conclusions. The merits of addressing this complex system with combined experimental-theoretical approaches is clear but, as well demonstrated in this case, agreement between the two cases does not always lead definitively to the correct answer. In both cases, qualitative electrostatic and dangling bond density arguments are used successfully to aid in the rationalization of the conclusions. Further complications arise from the results of a recent LEED, X-ray photoelectron diffraction (XPD), and STM UHV study (Kim et al. 1998) that largely agrees with the surface structure described by (Ritter and Weiss 1999), except with the exposed 1/4 monolayer of FeTet1 atoms removed. The only consensus at this point appears to be that the most stable (111) surface structure arises from non-reconstructed, relaxed bulk terminations. In each case, the suggested relaxed terminations are still polar. A complete understanding of surface energy compensation mechanisms is still lacking, and rigorous treatment in this important area may very well help to resolve the conflicting results. Given the recent LDA molecular modeling successes for the bulk, it seems natural that similar modeling work should be purposefully extended into this open area. CONCLUDING REMARKS AND OUTLOOK

It is hoped that this review is a useful introduction to the characteristics of semiconductor surfaces and molecular modeling approaches to address their structure, stability, and reactivity. These materials are complex, and an impression of the current utility and pitfalls of modeling to capture the important processes operative at these surfaces should be evident. The combination of ab initio modeling and experiment is clearly an incredibly powerful tool in this area. The ability of the calculations and the surface microscopies and spectroscopies to elucidate a convergent description is remarkable, and the connection between the two is quite natural. Much more information stands to be gained when the two methods are applied in parallel, and certainly more concretely than could be gained from either approach alone. The calculations allow observables to be explained in useful physical terms, and nonobservables to be quantified. At the scale of atoms and molecules, this approach is indispensable. The examples cover the recent findings on just a small fraction of important semiconducting mineral surfaces. They differ significantly in their natural occurrence, structure, and electronic properties, and, in that sense, cover a wide a range of important types of surfaces in geochemical environments. Fundamental processes operative at these surfaces are currently the focus of intense theoretical and experimental research. It should be apparent in the details that our understanding is still in its early stages, and a wealth of information is yet to be determined. The development of more efficient modeling codes and faster computing resources undoubtedly paces the utility of ab initio calculations for application to surfaces. At the same time, the natural difficulties with the preparation of well characterized samples and the collection of high resolution experimental data, especially STM/STS spectra, seem to pace the important experimental breakthroughs. Generalizations are prone to be premature at this point because new developments are frequent. Nevertheless, it is prudent to examine the state of this art more critically. So far, the experimental work has largely been restricted to clean, well developed surfaces under non-solution conditions. Molecular modeling calculations have invariably been constrained to small cell or cluster sizes, which are inherently predestined to overlook any extended surface chemical behavior. Moreover, sorbates are often considered in the dilute limit, with only a few molecules considered at any one time. The effects of sorbatesorbate interactions are therefore not addressed at this time. Although much has been learned about these systems under these constraints, it is justified to ask what new

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insights have been gained so far? Even at this early stage important developments have emerged, and these are setting the stage for future modeling work. For any particular mineral, a range of possible surface structures may be thermodynamically stable depending on the relative chemical potentials of the constituent atoms at the surface. Recent advances in this area may be attributed to the higher pressure O2 and H2O exposure work in UHV chambers as discussed for hematite (Liu et al. 1998; Wang et al. 1998; Shaikhutdinov and Weiss 1999). This is clearly an important consideration for surfaces in general. Future ab initio calculations should account for equilibrium effects at the interfacial environment, and its influence on surface structure and adsorbate speciation. The plane wave LDA study of Wang et al. (1998) demonstrated one way to do this using static total energy calculations. However, rigorous treatment of the collective effects of an excess of free molecules near equilibrium with a solid surface must include the dynamic interactions. This philosophy should perhaps be viewed as more long term because of the natural extension to accessing kinetics. Recently, plane wave ab initio methods have been extended into this area (Hass et al. 1998; Stampfl et al. 1999a,b). These studies demonstrate that it is feasible to capture short snapshots of full interfacial system dynamics with good accuracy. This allows insight into dynamics occurring at the nanosecond time scale, such as surface diffusion, acid-base interactions, and proton transfer reactions. For processes occurring over longer time scales, the computational expense of these dynamic simulations can be reduced by dividing the simulation cell into a small cell treated quantum mechanically, with the remainder treated with empirical potentials (Shoemaker et al. 1999). Another approach can be based on Monte Carlo models with all the salient interactions accurately parameterized from ab initio calculations. Monte Carlo steps can be related to real time to describe the kinetics using the principles of kinetic Monte Carlo (e.g., Radeke and Carter 1997). These methods hold great promise in making the link between atomistic behavior and macroscopic observables. It should prove to be enormously constructive to utilize such approaches to investigate the dynamical behavior at environmentally important semiconductor-water interfaces. Relaxation imparts significant changes to the surface electronic structure, which clearly cannot be well approximated as equivalent to that in the bulk. Surface states arising from broken bonds, and likewise surface defects and impurities lead to electronic states that can catalyze reactions that are otherwise energetically uphill, and direct the surface chemical behavior. For theoretical investigations of semiconductor surfaces, these important electronic effects require ab initio methods. It has recently been shown that semiconductor surface states can strongly mix with valence orbitals of nearby water molecules, effectively spreading the electron density of surface atoms outward into the solution (Ursenbach and Voth 1995; Ursenbach et al. 1997). This solvent based delocalization of electrons at the interface has important implications for redox reactions at surfaces that start to blur the traditional distinctions of inner versus outer sphere mechanisms. It also has implications for understanding tunneling currents in solutionbased and humid air STM work (Schmickler 1995). This effect suggests that at least the first monolayer of water molecules should also be treated at the ab initio level to study redox reactions at semiconductor surfaces. The redistribution of charge between surface atoms alters bonding character and therefore bond strength at the surface, which may influence dissolution mechanisms and kinetics. Material dependent trends are only recently becoming apparent in this regard (Noguera 1996). More detailed investigations of the electron density distributions at surfaces would be valuable. The development of topological analyses of electron density distributions by Bader (1990) provides a useful framework for mapping and interpreting

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features in electron density distributions. It allows one to determine characteristics of any chemical bond such as strength and covalency on an equal footing (see Gibbs et al. chapter, this volume). The theory also exposes the relationship between the Laplacian of the electron density distribution and acid-base reactivity. It has recently been applied to understand the adsorption behavior of CO at metal and insulator surfaces (Aray and Bader 1996; Aray and Rodriguez 1996, 1998). This approach also offers a method of calculating charges on atoms that can be used to treat charge redistribution at surfaces in a way that is likely more precise than what can be achieved using traditional population analyses. The possibility of delocalized electronic properties at semiconductor surfaces can lead to the propensity for non-local electron transfer pathways in redox reactions at semiconducting surfaces. This