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COMPUTATIONAL MATERIALS SCIENCE

1. INTRODUCTION: WHY ATOMIC LEVEL MODELING IS ESSENTIAL FOR MODERN MATERIALS SCIENCE ­ A BRIEF REVIEW OF EXAMPLES

Structures ­ basis of understanding of material properties Complex crystal structures Non-crystalline solids and liquids Atomic clusters and nano-structures Lattice defects, surfaces, interfaces, nano-crystals Examples: Model of a nano-crystal Model of a free surface Model of a grain boundary Linking structural studies with experiments High resolution electron microscopy Scanning tunneling microscopy and atomic force microscopy Ion and electron scattering Field ion microscopy and atom probe Lattice vibrations ­ phonons Examples: Nb-saphire interface Atomic force microscope Computer experiments Phase transformations Bulk, surface and interfacial diffusion Deformation and fracture Catalysis Examples: Dislocation generation during nanoindentation Deformation and fracture of a nanotube Radiation damage Material deposition to surfaces

2.

CALCULATION OF THE ENERGY OF A SYSTEM OF ATOMS ­ PREREQUISITE OF ATOMIC LEVEL MODELING (A BRIEF OVERVIEW)

Pair potentials Pair potentials in molecules vs pair potentials in condensed matter Lennard-Jones, Born-Mayer, Morse etc. Deficiencies of pair-potentials

Many-body central force potentials for metallic materials Embedded atom method Potentials for covalent solids - semiconductors Potentials dependent on bond angles Tersoff and Stilinger-Weber potentials for Si, C etc. Basics of the density functional theory 3. METHODS OF COMPUTER MODELING AND INTERPRETATION

General aspects of atomistic computer modeling Boundary conditions: finite blocks, periodic and semi-periodic boundary conditions Energy of a system of particles, equations of motion and equilibrium conditions Concept of stress and its evaluation in systems of particles Some methods of interpretation of results: Graphical representation of structures Radial distribution function Voronoi polyhedra Atomic level stresses Molecular statics ­ minimization of the energy Steepest descent Conjugate gradient Molecular dynamics (MD) Verlet algorithm Predictor-corrector algorithm Constant volume and constant pressure simulations 'Thermostats' and their use in the control of temperature in MD calculations Some basic concepts of statistical physics Physical interpretation of MD using statistical physics: Fluctuations, correlations, autocorrelations, Monte Carlo Brief overview of necessary statistical mechanics and equilibrium thermodynamics Metropolis method Canonical ensemble: constant volume, temperature, number of particles Grand canonical ensemble: constant volume, temperature, variable number of particles Isothermal-isobaric canonical ensemble: constant pressure, temperature, number of particles Modified grand canonical ensemble: study of segregation and order/disorder Kinetic Monte Carlo Diffusion in crystalline materials

Lattice Dynamics Lattice vibration-phonons Link with thermodynamical properties of materials

4.

ATOMIC INTERACTIONS IN SPECIFIC MATERIALS (Included only if time is available. This part is not included in assignments and in any testing.)

Density functional theory ­ method applicable to all materials Local density functional theory (LDA) and principals of ab-initio electronic structure calculations Metallic materials Pair potentials Embedded atom method and other many-body central force potentials Covalently bonded solids - semiconductors Potentials with bond bending terms: Tersoff, Stilinger-Weber and Brenner potentials Notion of the bond order Tight-binding methods for metallic and covalent materials General introduction to tight binding and parameterized tight-binding method Tight-binding in bond formulation Ionic crystals Rigid ion model Shell model Ewald summations of Coulomb interactions

ASSIGNEMENTS Development of a simple code for molecular statics, molecular dynamics and Monte Carlo studies of a simple two-dimensional system. Investigation of the structure and properties of an assembly of particles interacting via a Lennard-Jones pair potential using the developed codes. PREREQUISITES Ability to write simple computer code using a programming language (Fortran, C, C++, Matlab) However, this can be learned during the course as basics of programming will be explained

Basic condensed matter physics Thermodynamics Structure of materials TEXT FOR STUDY: Handouts in the form of pdf files

ADDITIONAL LITERATURE (not used directly as a text for the course) General texts Materials Research by Means of Multiscale Computer Simulation, MRS Bulletin 26, p. 169, 2001 Atomistic Theory and Simulation of Fracture, MRS Bulletin 25, p. 11, 2000. Computer Simulations from Thermodynamic data: Materials Production and Development, MRS Bulletin 24, p. 18, 1999. Theory and Simulation of Polymers at interfaces, MRS Bulletin 22, p. 13, 1997. Interatomic Potentials for Atomistic Simulations, MRS Bulletin 21, p. 17, 1996. Molecular dynamics Rapaport: The Art of Molecular Dynamics Simulation, Cambridge University Press, 1995 Daan Frenkel and Berend Smit: Understanding Molecular Simulations, Academic Press, 1996 J.-P. Hansen and I. R. McDonald: Theory of simple liquids, Academic Press, 1986 M. P. Allen and D. J. Tildesley: Computer Simulation of Liquids, Clarendon Press, 1987 Monte Carlo M. E. J. Newman and G. T. Barkema: Monte Carlo Methods in Statistical Physics, Clarendon Press, 1999 M. P. Allen and D. J. Tildesley: Computer Simulation of Liquids, Clarendon Press, 1987 Density functional theory and total energy calculations R. G. Parr and W. Yang: Density functional theory of atoms and molecules, Oxford University Press, 1989 A. P. Sutton: Electronic Structure of Materials, Clarendon Press, 1994 D. G. Pettifor: Bonding and Structure of Molecules and Solids, Clarendon Press, 1996

Pair potentials in metallic materials I. M. Torrens: Interatomic Potentials, Academic Press, 1972 W. A. Harrison: Electronic Structure and Properties of Solids, Freeman: San Francisco, 1980 Calrlsson, A. E.: Beyond Pair Potentials in Elemental Transition Metals and Semiconductors, Solid State physics Vol. 43, p. 1, Academic Press, 1990 Interatomic Potentials for Atomistic Simulations, MRS Bulletin 21, p. 17, 1996. Embedded atom method and many-body central force potentials M. S. Daw, S. M. Foiles and M. I. Baskes: The Embedded-Atom Method - A Review of Theory and Applications, Materials Science Reports 9, 251, 1993 Interatomic Potentials for Atomistic Simulations, MRS Bulletin 21, p. 17, 1996. Potentials for Ionic solids C. R. A. Catlow, M. Dixon, M. and W. C. Mackrodt: Interionic potentials in ionic solids, Computer Simulation of Solids, Springer, 1982 J. H. Harding: Computer simulation of defects in ionic solids, Rep. Prog. Phys. 53, 1403, 1990. Potentials for covalent solids ­ semiconductors Interatomic Potentials for Atomistic Simulations, MRS Bulletin 21, p. 17, 1996. Tight-binding method C. Kittel: Introduction to Solid State Physics, John Wiley &Sons, 1986 W. A. Harrison: Electronic Structure and Properties of Solids, Freeman: San Francisco, 1980 W. A. Harrison: Tight-Binding Methods, Surface Science 300, 298, 1994 D. G. Pettifor: Bonding and Structure of Molecules and Solids, Clarendon Press, 1996

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