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Vibrational Thermodynamics of Materials



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Vibrational Thermodynamics of Materials Brent Fultz California Institute of Technology W M Keck Laboratory Pasadena CA 91125 USA July 6 2009 Abstract The literature on vibrational thermodynamics of materials is reviewed The emphasis is on metals and alloys especially on the progress over the last decade in understanding differences in the vibrational entropy of different alloy phases and phase transformations Some results on carbides nitrides oxides hydrides and lithium storage materials are also covered Principles of harmonic phonons in alloys are organized into thermodynamic models for unmixing and ordering transformations on an Ising lattice and extended for non harmonic potentials Owing to the high accuracy required for the phonon frequencies quantitative predictions of vibrational entropy with analytical models prove elusive Accurate tools for such calculations or measurements were challenging for many years but are more accessible today Ab initio methods for calculating phonons in solids are summarized The experimental techniques of calorimetry inelastic neutron scattering and inelastic x ray scattering are explained with enough detail to show the issues of using these methods for investigations of vibrational thermodynamics The explanations extend to methods of data analysis that affect the accuracy of thermodynamic information It is sometimes possible to identify the structural and chemical origins of the differences in vibrational entropy of materials and the number of these assessments is growing There has been considerable progress in our understanding of the vibrational entropy of mixing in solid solutions compound formation from pure elements chemical unmixing of alloys order disorder transformations and martensitic transformations Systematic trends are available for some of these phase transformations although more examples are needed and many results are less reliable at high temperatures Nanostructures in materials can alter sufficiently the vibrational dynamics to affect thermodynamic stability Internal stresses in polycrystals of anisotropic materials also contribute to the heat capacity Lanthanides and actinides show a complex interplay of vibrational electronic and magnetic entropy even at low temperatures A quasiharmonic model is often used to extend the systematics of harmonic phonons to high temperatures by accounting for the effects of thermal expansion against a bulk modulus Non harmonic effects beyond the quasiharmonic approximation originate from the interactions of thermally excited phonons with other phonons or with the interactions of phonons with electronic excitations In the classical high temperature limit the adiabatic electron phonon coupling can have a surprisingly large effect in metals when temperature causes significant changes in the electron density near the Fermi level There are useful similarities in how temperature pressure and composition alter the conduction electron screening and the interatomic force constants Phononphonon anharmonic interactions arise from those non harmonic parts of the interatomic potential that cannot be accounted for by the quasiharmonic model Anharmonic shifts in phonon frequency with temperature can be substantial but trends are not well understood Anharmonic phonon damping does show systematic trends however at least for fcc metals Trends of vibrational entropy are often justified with atomic properties such as atomic size electronegativity electron to atom ratio and mass Since vibrational entropy originates at the level of electrons in solids such rules of thumb prove no better than similar rules devised for trends in bonding and structure and tend to be worse Fortunately the required tools for accurate experimental investigations of vibrational entropy have improved dramatically over the past few years and the required ab initio methods have become more accessible Steady progress is expected for understanding the phenomena reviewed here as investigations are performed with the new tools of experiment and theory sometimes in integrated ways II B Fultz Table of Contents I Principles and Methods 1 2 2 1 2 2 2 3 2 4 2 5 3 3 1 3 2 3 3 4 4 1 4 2 4 3 5 5 1 5 2 5 3 6 6 1 6 2 6 3 6 4 7 8 8 1 8 2 8 3 8 4 8 5 9 9 1 9 2 9 3 Overview Harmonic Lattice Dynamics Partition Function Hamiltonian for Lattice Dynamics Equations of Motion The Eigenvalue Problem for the Polarization Vector Calculation of the Phonon Density of States Predictions with the Harmonic Model Mass Local Modes and Resonance Modes Long Wavelength Limit and the Debye Model Disordered Systems Bond Proportion Model Bond Proportion Model and Unmixing on the Ising Lattice Bond Proportion Model and Ordering on the Ising Lattice Monte Carlo Results Bond Stiffness versus Bond Length Model Phonon Frequencies and Bond Lengths Extending the Bond Proportion Model Chemical Effects on Bond Stiffness Heat Capacity Harmonic Heat Capacity Quasiharmonic Thermodynamics Anharmonic Heat Capacity Thermodynamic Entropy Calorimeters Neutron Scattering Elastic and Inelastic Scattering One Phonon and Multiphonon Scattering Neutron Weighting Direct Geometry Fermi Chopper Spectrometer Triple Axis Spectrometer Ab Initio Methods Density Functional Theory First Principles Phonon Calculations Molecular Dynamics 1 2 3 5 6 7 8 9 9 11 13 13 14 18 20 21 21 24 25 25 25 27 29 30 31 32 32 35 37 38 38 39 40 40 41 II Vibrational Entropy of Materials at Lower Temperatures 10 Earlier Investigations 10 1 Polymorphism 10 2 Solubilities and Defect Concentrations 10 3 Carbides Nitrides and Oxides 11 First Principles Studies of Solutions and Studies of Ordered Compounds 43 43 44 47 48 Vibrational Thermodynamics of Materials 11 1 11 2 11 3 12 12 1 12 2 12 3 13 13 1 13 2 13 3 14 14 1 14 2 14 3 15 16 16 1 16 2 17 17 1 17 2 Solubility in Al Alloys Aluminum Compounds Low Symmetry Phases at High Temperatures Chemical Order Disorder Transformations Transition Metal Aluminides B2 Ordering in CuZn and FeCo L12 Ordering Unmixing Computational Studies of Unmixing Cluster Expansion Method for Solid Solutions Fe Cr Solid Solutions Interstitial Alloys Interstitial Oxygen and Nitrogen Hydrogen and Hydrides Lithium Intercalation and Insertion Martensite and Twinning Microstructural Contributions Elastic Energy Nanostructures f Electron Metals Cerium Uranium and Plutonium III 48 49 50 52 52 56 57 62 62 64 66 68 68 69 70 73 76 77 79 84 84 87 III Vibrational Entropy of Materials at Higher Temperatures 18 Interactions


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