Vibrational Thermodynamics of Materials

Download corrected draft of July 6, 2009 (4.5 MB)

Progress in Materials Science, in press.

Brent Fultz

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. Phonon-phonon "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.

Pressure-Induced Invar Behavior in Pd_3Fe

Press Release of June, 2009 (Also, try google: Invar pressure alchemy)

Phys. Rev. Lett. 102, 237202 (2009).

M. L. Winterrose, M. S. Lucas, A. F. Yue, I. Halevy, L. Mauger, J. A. Munoz, Jingzhu Hu, M. Lerche, and B. Fultz

Synchrotron x-ray diffraction (XRD) measurements, nuclear forward scattering (NFS) measurements, and density functional theory (DFT) calculations were performed on L12-ordered Pd_3Fe. Measurements were performed at 300 K at pressures up to 33 GPa, and at 7 GPa at temperatures up to 650 K. The NFS revealed a collapse of the 57Fe magnetic moment between 8.9 and 12.3 GPa at 300 K, coinciding with a transition in bulk modulus found by XRD. Heating the sample under a pressure of 7 GPa showed negligible thermal expansion from 300 to 523 K, demonstrating Invar behavior. Zero-temperature DFT calculations identified a ferromagnetic ground state and showed several antiferromagnetic states had comparable energies at pressures above 20 GPa.

This book explains concepts of transmission electron microscopy (TEM) and x-ray diffractometry (XRD) that are important for the characterization of materials. The third edition has been updated to cover important technical developments, including the remarkable recent advances in resolution of the TEM. This edition is not substantially longer than the second, but all chapters have been updated and revised for clarity. A new chapter on high resolution STEM methods has been added. The book explains the fundamentals of how waves and wavefunctions interact with atoms in solids, and the similarities and differences of diffraction measurements with x-rays, electrons, or neutrons. Diffraction effects of crystalline order, defects, and disorder in materials are explained in detail. Both practical and theoretical issues are covered. This textbook can be used in an introductory-level or advanced-level course, since sections are identified by difficulty. Each chapter includes a set of problems to illustrate principles, and the extensive Appendix includes laboratory exercises.
Excerpts from the book in Adobe Acrobat .pdf format can be found here.