Materials Kinetics: Transport and Rate Phenomena

By John C. Mauro

Materials Kinetics: Transport and Rate Phenomena provides readers with a clear understanding of how physical-chemical principles are applied to fundamental kinetic processes. The book integrates advanced concepts with foundational knowledge and cutting-edge computational approaches, demonstrating how diffusion, morphological evolution, viscosity, relaxation and other kinetic phenomena can be applied to practical materials design problems across all classes of materials. The book starts with an overview of thermodynamics, discussing equilibrium, entropy, and irreversible processes. Subsequent chapters focus on analytical and numerical solutions of the diffusion equation, covering Fick’s laws, multicomponent diffusion, numerical solutions, atomic models, and diffusion in crystals, polymers, glasses, and polycrystalline materials.

Dislocation and interfacial motion, kinetics of phase separation, viscosity, and advanced nucleation theories are examined next, followed by detailed analyses of glass transition and relaxation behavior. The book concludes with a series of chapters covering molecular dynamics, energy landscapes, broken ergodicity, chemical reaction kinetics, thermal and electrical conductivities, Monte Carlo simulation techniques, and master equations.

• Covers the full breadth of materials kinetics, including organic and inorganic materials, solids and liquids, theory and experiments, macroscopic and microscopic interpretations, and analytical and computational approaches
• Demonstrates how diffusion, viscosity microstructural evolution, relaxation, and other kinetic phenomena can be leveraged in the practical design of new materials
• Provides a seamless connection between thermodynamics and kinetics
• Includes practical exercises that reinforce key concepts at the end of each chapter

• Language: English
• Publisher: Elsevier Science
• Release date: Nov 22, 2020
• ISBN: 9780128242162

John C. Mauro

Dr. John C. Mauro is Professor and Associate Head for Graduate Education in the Department of Materials Science and Engineering at The Pennsylvania State University. John earned a BS in Glass Engineering Science (2001), BA in Computer Science (2001), and PhD in Glass Science (2006), all from Alfred University. He joined Corning Incorporated in 1999 and served in multiple roles there, including Senior Research Manager of the Glass Research department. John holds more than 50 granted US patents and is the inventor or co-inventor of several new glasses for Corning, including Corning Gorilla® Glass products. John joined the faculty at Penn State in 2017 and is currently a world-recognized leader in fundamental and applied glass science, materials kinetics, computational and condensed matter physics, thermodynamics, statistical mechanics, and the topology of disordered networks. He is the author of over 280 peer-reviewed publications, Editor of the Journal of the American Ceramic Society, winner of numerous international awards, and a Fellow of the American Ceramic Society and the Society of Glass Technology. John is also co-author of Fundamentals of Inorganic Glasses, 3rd ed., Elsevier (2019).

Book preview

Materials Kinetics

Transport and Rate Phenomena

John C. Mauro

The Pennsylvania State University, University Park, Pennsylvania

Table of Contents

Cover image

Title page

Copyright

Dedication

Foreword

Preface

Acknowledgments

Chapter 1. Thermodynamics vs. Kinetics

1.1. What is Equilibrium?

1.2. Thermodynamics vs. Kinetics

1.3. Spontaneous and Non-Spontaneous Processes

1.4. Microscopic Basis of Entropy

1.5. First Law of Thermodynamics

1.6. Second Law of Thermodynamics

1.7. Third Law of Thermodynamics

1.8. Zeroth Law of Thermodynamics

1.9. Summary

Exercises

Chapter 2. Irreversible Thermodynamics

2.1. Reversible and Irreversible Processes

2.2. Affinity

2.3. Fluxes

2.4. Entropy Production

2.5. Purely Resistive Systems

2.6. Linear Systems

2.7. Onsager Reciprosity Theorem

2.8. Thermophoresis

2.9. Thermoelectric Materials

2.10. Electromigration

2.11. Piezoelectric Materials

2.12. Summary

Exercises

Chapter 3. Fick's Laws of Diffusion

3.1. Fick's First Law

3.2. Fick's Second Law

3.3. Driving Forces for Diffusion

3.4. Temperature Dependence of Diffusion

3.5. Interdiffusion

3.6. Measuring Concentration Profiles

3.7. Tracer Diffusion

3.8. Summary

Exercises

Chapter 4. Analytical Solutions of the Diffusion Equation

4.1. Fick's Second Law with Constant Diffusivity

4.2. Plane Source in One Dimension

4.3. Method of Reflection and Superposition

4.4. Solution for an Extended Source

4.5. Bounded Initial Distribution

4.6. Method of Separation of Variables

4.7. Method of Laplace Transforms

4.8. Anisotropic Diffusion

4.9. Concentration-Dependence Diffusivity

4.10. Time-Dependent Diffusivity

4.11. Diffusion in Other Coordinate Systems

4.12. Summary

Exercises

Chapter 5. Multicomponent Diffusion

5.1. Introduction

5.2. Matrix Formulation of Diffusion in a Ternary System

5.3. Solution by Matrix Diagonalization

5.4. Uphill Diffusion

5.5. Summary

Exercises

Chapter 6. Numerical Solutions of the Diffusion Equation

6.1. Introduction

6.2. Dimensionless Variables

6.3. Physical Interpretation of the Finite Difference Method

6.4. Finite Difference Solutions

6.5. Considerations for Numerical Solutions

6.6. Summary

Exercises

Chapter 7. Atomic Models for Diffusion

7.1. Introduction

7.2. Thermally Activated Atomic Jumping

7.3. Square Well Potential

7.4. Parabolic Well Potential

7.5. Particle Escape Probability

7.6. Mean Squared Displacement of Particles

7.7. Einstein Diffusion Equation

7.8. Moments of a Function

7.9. Diffusion and Random Walks

7.10. Summary

Exercises

Chapter 8. Diffusion in Crystals

8.1. Atomic Mechanisms for Diffusion

8.2. Diffusion in Metals

8.3. Correlated Walks

8.4. Defects in Ionic Crystals

8.5. Schottky and Frenkel Defects

8.6. Equilibrium Constants for Defect Reactions

8.7. Diffusion in Ionic Crystals

8.8. Summary

Exercises

Chapter 9. Diffusion in Polycrystalline Materials

9.1. Defects in Polycrystalline Materials

9.2. Diffusion Mechanisms in Polycrystalline Materials

9.3. Regimes of Grain Boundary Diffusion

9.4. Diffusion Along Stationary vs. Moving Grain Boundaries

9.5. Atomic Mechanisms of Fast Grain Boundary Diffusion

9.6. Diffusion Along Dislocations

9.7. Diffusion Along Free Surfaces

9.8. Summary

Exercises

Chapter 10. Motion of Dislocations and Interfaces

10.1. Driving Forces for Dislocation Motion

10.2. Dislocation Glide and Climb

10.3. Driving Forces for Interfacial Motion

10.4. Motion of Crystal-Vapor Interfaces

10.5. Crystalline Interface Motion

10.6. Summary

Exercises

Chapter 11. Morphological Evolution in Polycrystalline Materials

11.1. Driving Forces for Surface Morphological Evolution

11.2. Morphological Evolution of Isotropic Surfaces

11.3. Evolution of Anisotropic Surfaces

11.4. Particle Coarsening

11.5. Grain Growth

11.6. Diffusional Creep

11.7. Sintering

11.8. Summary

Exercises

Chapter 12. Diffusion in Polymers and Glasses

12.1. Introduction

12.2. Stokes-Einstein Relation

12.3. Freely Jointed Chain Model of Polymers

12.4. Reptation

12.5. Chemically Strengthened Glass by Ion Exchange

12.6. Ion-Exchanged Glass Waveguides

12.7. Anti-Microbial Glass

12.8. Proton Conducting Glasses

12.9. Summary

Exercises

Chapter 13. Kinetics of Phase Separation

13.1. Thermodynamics of Mixing

13.2. Immiscibility and Spinodal Domes

13.3. Phase Separation Kinetics

13.4. Cahn-Hilliard Equation

13.5. Phase-Field Modeling

13.6. Summary

Exercises

Chapter 14. Nucleation and Crystallization

14.1. Kinetics of Crystallization

14.2. Classical Nucleation Theory

14.3. Homogeneous Nucleation

14.4. Heterogeneous Nucleation

14.5. Nucleation Rate

14.6. Crystal Growth Rate

14.7. Johnson-Mehl-Avrami Equation

14.8. Time-Temperature-Transformation Diagram

14.9. Glass-Ceramics

14.10. Summary

Exercises

Chapter 15. Advanced Nucleation Theories

15.1. Limitations of Classical Nucleation Theory

15.2. Statistical Mechanics of Nucleation

15.3. Diffuse Interface Theory

15.4. Density Functional Theory

15.5. Implicit Glass Model

15.6. Summary

Exercises

Chapter 16. Viscosity of Liquids

16.1. Introduction

16.2. Viscosity Reference Points

16.3. Viscosity Measurement Techniques

16.4. Liquid Fragility

16.5. Vogel-Fulcher-Tammann (VFT) Equation for Viscosity

16.6. Avramov-Milchev (AM) Equation for Viscosity

16.7. Adam-Gibbs Entropy Model

16.8. Mauro-Yue-Ellison-Gupta-Allan (MYEGA) Equation for Viscosity

16.9. Infinite Temperature Limit of Viscosity

16.10. Kauzmann Paradox

16.11. Fragile-to-Strong Transition

16.12. Non-Newtonian Viscosity

16.13. Volume Viscosity

16.14. Summary

Exercises

Chapter 17. Nonequilibrium Viscosity and the Glass Transition

17.1. Introduction

17.2. The Glass Transition

17.3. Thermal History Dependence of Viscosity

17.4. Modeling of Nonequilibrium Viscosity

17.5. Nonequilibrium Viscosity and Fragility

17.6. Composition Dependence of Viscosity

17.7. Viscosity of Medieval Cathedral Glass

17.8. Summary

Exercises

Chapter 18. Energy Landscapes

18.1. Potential Energy Landscapes

18.2. Enthalpy Landscapes

18.3. Landscape Kinetics

18.4. Disconnectivity Graphs

18.5. Locating Inherent Structures and Transition Points

18.6. ExplorerPy

18.7. Summary

Exercises

Chapter 19. Broken Ergodicity

19.1. What is Ergodicity?

19.2. Deborah Number

19.3. Broken Ergodicity

19.4. Continuously Broken Ergodicity

19.5. Hierarchical Master Equation Approach

19.6. Thermodynamic Implications of Broken Ergodicity

19.7. Summary

Exercises

Chapter 20. Master Equations

20.1. Transition State Theory

20.2. Master Equations

20.3. Degenerate Microstates

20.4. Metabasin Approach

20.5. Partitioning of the Landscape

20.6. Accessing Long Time Scales

20.7. KineticPy

20.8. Summary

Exercises

Chapter 21. Relaxation of Glasses and Polymers

21.1. Introduction

21.2. Fictive Temperature

21.3. Tool's Equation

21.4. Ritland Crossover Effect

21.5. Fictive Temperature Distributions

21.6. Property Dependence of Fictive Temperature

21.7. Kinetic Interpretation of Fictive Temperature

21.8. Stretched Exponential Relaxation

21.9. Prony Series Description

21.10. Relaxation Kinetics

21.11. RelaxPy

21.12. Stress vs. Structural Relaxation

21.13. Maxwell Relation

21.14. Secondary Relaxation

21.15. Summary

Exercises

Chapter 22. Molecular Dynamics

22.1. Multiscale Materials Modeling

22.2. Quantum Mechanical Techniques

22.3. Principles of Molecular Dynamics

22.4. Interatomic Potentials

22.5. Ensembles

22.6. Integrating the Equations of Motion

22.7. Performing Molecular Dynamics Simulations

22.8. Thermostats

22.9. Barostats

22.10. Reactive Force Fields

22.11. Tools of the Trade

22.12. Summary

Exercises

Chapter 23. Monte Carlo Techniques

23.1. Introduction

23.2. Monte Carlo Integration

23.3. Monte Carlo in Statistical Mechanics

23.4. Markov Processes

23.5. The Metropolis Method

23.6. Molecular Dynamics vs. Monte Carlo

23.7. Sampling in Different Ensembles

23.8. Kinetic Monte Carlo

23.9. Inherent Structure Density of States

23.10. Random Number Generators

23.11. Summary

Exercises

Chapter 24. Fluctuations in Condensed Matter

24.1. What are Fluctuations?

24.2. Statistical Mechanics of Fluctuations

24.3. Fluctuations in Broken Ergodic Systems

24.4. Time Correlation Functions

24.5. Dynamical Heterogeneities

24.6. Nonmonotonic Relaxation of Fluctuations

24.7. Industrial Example: Fluctuations in High Performance Display Glass

24.8. Summary

Exercises

Chapter 25. Chemical Reaction Kinetics

25.1. Rate of Reactions

25.2. Order of Reactions

25.3. Equilibrium Constants

25.4. First-Order Reactions

25.5. Higher Order Reactions

25.6. Reactions in Series

25.7. Temperature Dependence of Reaction Rates

25.8. Heterogeneous Reactions

25.9. Solid State Transformation Kinetics

25.10. Summary

Exercises

Chapter 26. Thermal and Electrical Conductivities

26.1. Transport Equations

26.2. Thermal Conductivity

26.3. Electrical Conductivity

26.4. Varistors and Thermistors

26.5. Summary

Exercises

Index

Copyright

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About the Cover Art: Moments in Love: Mean, Variance, Skew, Kurtosis

Jane Cook, 2018 Corning, New York

Acrylic and tissue paper on canvas, 61 cm x 45 cm

The title of this painted collage is a word-play on the title of the song Moments in Love by the 1980s new wave band Art of Noise. The canvas is painted red and overlaid with crinkled red tissue, over which four roughly rectangular forms are positioned essentially squared with each other and the canvas. The piece is the artist's nerdy commentary on the utility and futility of analysis in matters of the heart. One might find joy amongst the scattered, undulating hills and valleys of the underlying function of love without knowing more details of that function; or, one can deploy a knowledge of statistics to extract the moments of the distribution of highs and lows. Neither method of experience is superior – they each provide unique insights.

Publisher: Matthew Deans

Acquisitions Editor: Dennis McGonagle

Editorial Project Manager: Chiara Giglio

Production Project Manager: Nirmala Arumugam

Cover Designer: Miles Hitchen

Typeset by TNQ Technologies

Dedication

Dedicated to my loving wife and daughter,

Yihong and Sofia Mauro

Foreword

Tis true without lying, certain & most true.

That which is below is like that which is above & that which

is above is like that which is below to do the miracles of one

only thing

And as all things have been & arose from one by the

mediation of one: so all things have their birth from this

one thing by adaptation.

Sir Isaac Newton, excerpt from The Emerald Tablet

The original author of The Emerald Tablet, a seminal work of alchemical thinking, was a now-unknown 6th-8th Century Arabic mystic. A millennium later, Newton made his translation from version in Latin, with the first key lines given above.

Although the phrase As above, so below is now popularized in contemporary metaphysical lore, I think it likely that Newton could sense, if not know, that a deeper truth of Nature was at the core of this concept.

But it would be 300 years before Albert Einstein, another genius who could see beyond the common dimensional and mathematical restrictions of his time, would unite the above and below in his diffusion equation that connected the microscopic motion of individual atoms to the macroscopic flow of large systems.

And that one thing, from which this scale-bridging understanding arises, is what floats now before your eyes: Materials Kinetics.

In the song Heaven by the Talking Heads, David Byrne croons:

Heaven

Heaven is a place

A place where nothing ever happens.

This view of a heavenly state is a state of thermodynamic equilibrium. There are no reservoirs of potential energy to dissipate. No chemical reactions to go forward; no elevated weights on pulleys waiting to do work; no electrons waiting to discharge from capacitive plates. It would seem an engineering student's Heaven.

But reality is the here-and-now. And stuff is happening! And if it's not happening, it's probably trying to. Things ARE kinetics!

We live life in a universe in motion. We, and all around us, are verbs, as Buckminster Fuller pointed out: …integral functions of Universe. We are motivated fundamentally by the possibilities of change.

Thermodynamics tells us what can be, what is favored, what is desired. But it is Kinetics that tells us how it can happen, what it'll take, the ultimate, integrated pain vs. gain.

It is the traffic report for the freeways of materials science.

I grew up in the 1970s in the suburbs north of Los Angeles, a city famous for its Car Culture. A sprawling megalopolis, LA offered 16 year-old Me uncountable worlds to explore. But along with mastering the critical skills of safe vehicle operation, I needed to know the freeways, highways, boulevards, side streets -- and even the alleyways -- to get to places safely, timely, and with low-stress. I had a paper map in my bedroom that I would study before heading out, and another in the glovebox. But maps were never enough. I also needed the radio's traffic reports -- the real-time data -- so I'd know, for example, that the southbound I-405 over the pass was backed up to Victory Boulevard due to a vehicle fire, and so it would, surprisingly, be faster to take I-5 through downtown to I-10 to get to the beach.

Thermodynamics told me I wanted to get to the beach, that there I'd be chill and at deep local equilibrium; my heaven was a spot of sand just north of Pepperdine University in Malibu. Kinetic pathways of maps and traffic reports provided the options for roads to take. It was the complex interplay of those two, moment to moment on that journey that defined the actual lowest energy pathway that my car should take: Reality is our thermodynamic motivations playing out on the diamond lanes, stoplights, fender-benders, and alleyway shortcuts of serial/parallel kinetic mechanisms.

This textbook unites the elegant potentialities of thermodynamics with the real universe of shifting chemical activity gradients, thermal transients, alternating electromagnetic fields, and deviatoric stresses. Materials Kinetics shows how one thing by adaptation from Newton's alchemical picture manifests the Einsteinian miracle of the one thing, the astonishing myriad subtlety of the physical universe.

Jane B. Cook

The Pennsylvania State University, University Park, PA, United States of America

Notes:

http://webapp1.dlib.indiana.edu/newton/mss/norm/ALCH00017.

Heaven lyrics by David Byrne and Jerry Harrison, on Fear of Music 1979, Sire Records.

I Seem to be a Verb by Buckminster Fuller, ISBN-13 : 978-1127231539.

Preface

Kinetic processes are the agents of change in materials science and engineering. Despite the ubiquitous importance of kinetics in materials science, a comprehensive textbook on this topic has been sorely lacking. I hope the current volume will fill this gap, providing a textbook that covers the full breadth of materials kinetics, spanning all classes of materials and covering both the liquid and solid states from the macroscale down to the atomic level. In writing this book, I have attempted to strike a balance among fundamental theory, modeling and simulation techniques, experimental methods, and practical applications in materials design.

The book is written in a pedagogical fashion, aiming to provide a rigorous treatment of materials kinetics in a format that is accessible to both first-year graduate students and upper-level undergraduates in materials science and engineering. It should also be useful as a reference for professionals in the field. Emphasis has been placed on developing the fundamental concepts of materials kinetics and the importance of these concepts in the understanding and design of materials. Real-world examples are given throughout the text, and each chapter ends with a series of exercises that are meant to stimulate critical and creative thinking around relevant concepts.

This volume emerged from teaching MATSE 503, Kinetics of Materials Processes, my first-year graduate course at Penn State. The organization of the book follows exactly how I teach the course. I am happy to provide additional course content—including lecture slides—to any instructors who ask. Please feel free to email me at [e-mail] to request a copy of this material.

The book begins with an overview of important thermodynamic concepts and the difference between thermodynamics and kinetics (Chapter 1). Thermodynamics is an elegant subject, but one that is often difficult to grasp for many students. I hope the overview provided in Chapter 1 will help students develop an intuitive understanding of some key thermodynamic concepts which are vital to materials science and engineering. One of my goals with this book is to provide a seamless connection between the thermodynamics and kinetics of materials. With respect to transport and rate phenomena, thermodynamics can often be viewed as the cause while kinetics is the effect.

With this foundation in place, we introduce the subject of irreversible thermodynamics (Chapter 2), which provides the thermodynamic driving force for kinetic processes. Aside from the excellent chapter in Callen's classic thermodynamics textbook, there is almost no good overview of irreversible thermodynamics in the literature that is both physically rigorous and clearly accessible to introductory readers. Such was my goal with Chapter 2, which took me more time to write than any other chapter in this book. I hope the reader will find this to be a lucid and useful introduction to Onsager's formulation of irreversible thermodynamics, as well as its practical importance in materials science.

In Chapter 3, we introduce Fick's laws of diffusion. Chapter 4 is devoted to analytical solutions of the diffusion equation. Here, I have tried to select an assortment of solutions that I have seen most commonly used in practical problems. Next, in Chapter 5 we consider multicomponent diffusion, a problem first rigorously addressed by my advisor (Arun Varshneya) during his Ph.D. studies at Case Western Reserve University. Of course, many diffusion problems are too difficult to be solved analytically. Chapter 6 is, therefore, devoted to numerical solutions of the diffusion equation using the finite difference method.

While Chapters 1–6 deal with macroscopic thermodynamics and kinetics, in Chapter 7 we dive into the microscopic description of diffusion in terms of atomic jumping. The connection between the microscopic and macroscopic descriptions of diffusion is made via the Einstein diffusion equation. Then in Chapter 8 we specifically deal with atomic jumping and diffusion in single crystals, starting first with perfect crystals and then moving to those having point defects. In Chapter 9 we cover diffusion in polycrystalline materials, accounting for the impact of grain boundaries and free surfaces. Chapter 10 then deals with the kinetics of dislocation and interfacial motion. Chapter 11 concludes our treatment of polycrystalline materials by studying various types of morphological evolution, including particle coarsening, grain growth, diffusional creep, and sintering. In Chapter 12, we turn our attention to diffusion in polymers and glasses, including reptation and ion exchange processes.

Next, in Chapter 13 we cover the thermodynamics and kinetics of phase separation, including droplet nucleation and spinodal decomposition. Chapters 14 and 15 are devoted to crystal nucleation and growth. Chapter 14 presents classical nucleation theory, while Chapter 15 covers several types of advanced nucleation theories.

In Chapter 16 we turn our attention to liquid viscosity, including fundamental theory, experimental measurement techniques, and models describing the viscosity-temperature relationship. This naturally leads to the topics of nonequilibrium viscosity and the glass transition in Chapter 17. The glass transition is a particularly interesting topic since it is intrinsically a kinetic transition, and one with profound thermodynamic consequences, i.e., kinetics is the cause and thermodynamics is the effect!

Chapter 18 introduces the notion of an energy landscape, which is one of the most powerful and versatile ways of describing both the thermodynamics and kinetics of materials. In Chapter 19, we cover the vitally important—but usually overlooked—topic of broken ergodicity, which is of critical importance for nonequilibrium systems displaying long relaxation times. The kinetics of broken ergodic systems can be rigorously calculated in terms of master equations, which are detailed in Chapter 20. Next, Chapter 21 applies this knowledge to the study of long-time relaxation in glasses and polymers. This chapter includes detailed coverage of fictive temperature and the stretched exponential relaxation function.

The next pair of chapters focuses on useful computer simulation techniques for modeling kinetic phenomena at the atomic level. Chapter 22 covers the fundamentals of molecular dynamics, and Chapter 23 is devoted to Monte Carlo techniques, including the kinetic Monte Carlo approach for accessing long time scales.

In Chapter 24, we discuss fluctuations in condensed matter. Atomic scale fluctuations in time and space are critically important across all of materials science and engineering. However, this topic is rarely given much attention in standard materials science curricula. With Chapter 24, I hope to bring more attention to this important topic.

Chapter 25 provides an introduction to chemical reaction kinetics, i.e., kinetics from the perspective of chemistry and chemical engineering. Finally, we conclude the book with a brief chapter on thermal and electrical conductivities (Chapter 26), topics that are already part of the standard curriculum in solid state physics courses. More in-depth treatment is left for solid state physics textbooks (see either of the canonical texts by Kittel or by Ashcroft and Mermin).

Whether you are an experimentalist or theorist, a metallurgist, ceramist, glass scientist, or polymer scientist, I hope there is something of interest here for you, and I hope that you will enjoy reading this book as much as I have enjoyed writing it!

John C. Mauro

The Pennsylvania State University

University Park, Pennsylvania

Acknowledgments

I am blessed with an amazing group of students here at Penn State, whose passion for materials research and making the world a better place is a daily inspiration to me. I would like to thank all my students for making my role as a professor such a fulfilling experience. With respect to the current volume, I would especially like to thank the following Penn State students who provided figures from their research to help elucidate various concepts throughout this book: Sun Hwi Bang, Nicholas Clark, Anthony DeCeanne, George Kotsonis, Rebecca Welch, Collin Wilkinson, and Yongjian Yang. I would also like to thank all the students in my MATSE 503 Kinetics of Materials Processes class for all their great questions and enthusiasm for learning. I would especially like to thank my teaching assistant, Karan Doss, who is a wellspring of great ideas and insights. Discussions with Karan have improved both the course itself and the content of this textbook. I would also like to thank Matthew Mancini and Daniel Cassar for their helpful suggestions on this book. I owe an ocean of thanks to Brittney Hauke, who designed 37 of the figures in this book. I really appreciate Brittney's ability to clearly capture key concepts in her illustrations, which will be greatly helpful to students learning the course content for the first time. Brittney has also been amazingly generous with her time, providing careful word-by-word proofreading of most of this book. Thank you, Brittney!

Thanks also to my colleagues at Penn State for all of their support and encouragement. In particular, Venkat Gopalan has been a huge source of encouragement throughout the writing of this book. Sorry for clogging your email with so many chapter files! I'd also like to thank Susan Sinnott, Long-Qing Chen, Jon-Paul Maria, and Clive Randall for their ongoing support and for sharing some of their exciting new research for inclusion as part of this volume. I also owe a huge thanks to Carlo Pantano, Seong Kim, and John Hellmann for their amazing support of both me personally and my research group at Penn State.

I am deeply thankful to Jane Cook for all of her encouragement, inspiration, and friendship over so many years, first at Corning Incorporated and now here at Penn State. It is difficult to express how deeply meaningful it is to me that she would write the Foreword for this book and also offer her beautiful artwork, Moments in Love, both as the cover for this book and as one of the figures in Chapter 7. Jane's work at the intersection of art and science is profoundly inspirational to me, as I know it is to our students.

I am also very fortunate to have a wonderful network of friends and collaborators from around the globe. I appreciate all the good times working together to address our latest research challenges. In the context of the current book, I would like to extend special thanks to Roger Loucks (Alfred University); Morten Smedskjaer (Aalborg University, Denmark); Edgar Zanotto (Federal University of São Carlos, Brazil); Doug Allan, Adam Ellison, Ozgur Gulbiten, and Matt McKenzie (Corning Incorporated); Ken Kelton (Washington University in St. Louis); and Prabhat Gupta (The Ohio State University). Prabhat, in particular, has been like a second advisor to me. We worked very closely together in developing many of the models and techniques described in this book, including the MYEGA equation for liquid viscosity, temperature-dependent constraint theory, the statistical mechanics of continuously broken ergodicity, and a general technique for solving large sets of master equations over disparate time scales.

I would also like to express my deepest gratitude to my advisor, Arun Varshneya from Alfred University, who is both my glass guru and my academic father. I owe so much to him, for everything he has taught me directly and for all the doors of opportunity he has opened for me. Arun has taught me that being an academic advisor is a lifelong commitment. He is the role model whom I am striving to follow, in his engaging teaching style, his outstanding research at the interface of science and engineering, and (most importantly) in his deep commitment to his students.

I would also like to thank my family for their unfailing love and support. From birth to the present day, my parents, Ron and Susie Mauro, have always been such wonderful role models for how to live my life. Finally, the biggest thanks of all goes to my wife, Yihong, and our daughter, Sofia, for their unbounded love and support, especially during the countless hours I've spent writing this book. I struggle to find the words to express the depth of my love and gratitude for you.

John C. Mauro

The Pennsylvania State University

University Park, Pennsylvania

Chapter 1: Thermodynamics vs. Kinetics

Abstract

Thermodynamics concerns the relative stability of the various states of a system, whereas kinetics concerns the approach to equilibrium and the intermediate states visited along the way. Thermodynamic processes are spontaneous if they result in a decrease in the Gibbs free energy of the system. The change in Gibbs free energy has both enthalpic and entropic contributions. However, the spontaneity of a process says nothing about the rate of its kinetics. As proposed by Boltzmann, entropy is a measure of the number of microstates visited by a system to produce a specific macrostate. Justification of both the second and third laws of thermodynamics can be made in terms of the underlying statistical mechanics of the system and Boltzmann's equation for entropy.

Keywords

Thermodynamics; Kinetics; Free Energy; Enthalpy; Entropy; Statistical Mechanics

1.1. What is Equilibrium?

The field of classical thermodynamics is primarily concerned with equilibrium states, i.e., the states to which systems will eventually evolve and become stable. Although the concept of equilibrium is seemingly intuitive, this simplicity can be deceptive. Perhaps the most insightful definition of equilibrium is given by Richard Feynman [1], who said that equilibrium is when all the fast things have happened but the slow things have not.

Surely you're joking, Mr. Feynman! we may be tempted to reply. While Feynman's definition may seem somewhat flippant at first, it incisively captures the importance of time scale in determining what constitutes equilibrium.

Perhaps we should rephrase our response to Mr. Feynman. If equilibrium is when all the fast things have happened but the slow things have not, this begs the questions, What is fast? and What is slow? Indeed, fast and slow are intrinsically relative terms and depend on your perspective as an observer [2].

Maybe some caffeine will help accelerate our understanding of the relative nature of equilibrium. Let us consider the simple example of a mixture of coffee and cream in a mug, depicted in Figure 1.1. Initially the coffee and the cream are two separate phases. When the cream is poured into the coffee, the mixture homogenizes on a time scale of several seconds. Hence, the coffee-cream mixture achieves an equilibrium within the mug on a fairly short time scale. This would be an appropriate time to take a sip, because if we wait longer, the hot coffee-cream mixture will cool and reach thermal equilibrium with the room temperature environment. This second equilibrium occurs on a longer time scale, e.g., on the order of tens of minutes. Now let us extend the time scale again, this time to several days. On this much longer time scale, the thermally equilibrated coffee-cream mixture will reach a vapor equilibrium with the atmosphere, and our thirsty reader will be left with only some residue of what was formerly coffee in the bottom of the mug.

Figure 1.1 Example of coffee-cream equilibrium. Different stages of equilibrium can be observed on different time scales.

Of these three equilibria depicted in Figure 1.1, which one is the true equilibrium? The answer to this question depends on you, dear reader. Specifically, what is the objective of your experiment? If you intend to study vapor equilibrium, then the longest time scale is the most relevant, as your experiment must be conducted on a long enough time scale to reach the desired vapor equilibrium. However, if you are a disciple of Fourier and wish to study the approach to thermal equilibrium, then the intermediate time scale is the appropriate choice, since the experiment can conclude when thermal equilibrium has been achieved, i.e., when there is no further change in the temperature of the coffee-cream mixture. In this case, the subsequent vapor equilibrium is irrelevant. Or perhaps the reader simply wishes to enjoy a refreshing cup of coffee, in which case the relevant equilibrium is the homogenization of the coffee-cream mixture. The reader can enjoy the outcome of the experiment within several seconds.

It should now be apparent to our caffeinated reader that the question of thermodynamic equilibrium is really a question of time scale. We may think of an experiment as having two relevant time scales: an internal time scale on which the approach to the relevant equilibrium occurs, and an external time scale on which the system is being observed or measured. The internal time scale of equilibration is essentially a time scale for the relaxation of atomic structure (structural relaxation) over which a system loses any memory of its preceding states. The external time scale (or observation time scale) defines the time over which the system is being measured. This measurement can be made either directly by a human observer or using an instrument capable of accessing time scales not available to an unaided human, i.e., on either ultra-fast or ultra-long time scales. Following Feynman's definition above, a system is considered to be in equilibrium if all of the relevant relaxation processes have taken place, while the remaining slower processes are essentially frozen on the external (observation) time scale of interest [3].

1.2. Thermodynamics vs. Kinetics

Hence, thermodynamics involves determining the relative stability of different possible states of a system under appropriate constraints related to chemical composition, temperature, pressure, etc. Thermodynamics tells us whether a given reaction is feasible and which product is in stable equilibrium. The approach to equilibrium is the domain of kinetics and the subject of this monograph.

The word kinetics is derived from the ancient Greek kinesis (κίνησις), meaning movement or motion. The goal of kinetics is to determine the rates of processes and the evolution of materials systems under some set of conditions. The kinetics of a given process fundamentally depend on two important factors:

1. The thermodynamic driving force for the process; and

2. The kinetic rate parameters.

The thermodynamic driving force depends on the degree of disequilibrium of the system. A greater degree of disequilibrium gives rise to a larger thermodynamic driving force for equilibration. The details of this driving force are the subject of irreversible thermodynamics, which will be introduced in Chapter 2. The kinetic rate parameters govern the time scale of the process. They determine how fast a reaction occurs and how much time is required to achieve equilibration. The kinetics of the system will also determine which intermediate states are visited along the path to equilibrium. Whereas some intermediate states are kinetically stable (i.e., on a short time scale), they may be thermodynamically unstable (i.e., in the limit of long time).

While the kinetics of materials processes is the raison d′être of this volume, it would be prudent to begin with a brief review of some key concepts in classical thermodynamics. Figure 1.2 shows a mnemonic table for remembering some of these key quantities. Let us begin in the upper-left quadrant with U, the internal energy of the system. The internal energy is an extensive parameter including all relevant contributions to the energy of the system. If a system is at constant pressure rather than constant volume (i.e., if the system is isobaric rather than isochoric), then the relevant thermodynamic variable is the enthalpy, H = U + PV, where P is pressure and V is volume. Enthalpy is given in the upper-right quadrant of Figure 1.2. The corresponding free energies are obtained in the second row of the figure by subtracting the product of the absolute temperature, T, and the entropy, S, from either the internal energy or the enthalpy. For an isochoric (constant volume) system, the relevant quantity is the Helmholtz free energy, F = U − TS. For an isobaric (constant pressure) system, the relevant quantity is the Gibbs free energy, G = H − TS. In either case, moving from the upper row to the lower row in Figure 1.2 involves subtracting TS, and moving from the left-hand column (for isochoric systems) to the right-hand column (for isobaric systems) involves adding PV.

Figure 1.2 Mnemonic for remembering key thermodynamics quantities.

Most practical experiments tend to be performed under isobaric rather than isochoric conditions. Hence, Gibbs free energy is the more commonly used quantity compared to Helmholtz free energy. Under such conditions, the relative stability of the various states of a system is determined by the differences in the Gibbs free energy among these states. The state having the lowest free energy is the thermodynamic equilibrium and the state to which the system will evolve in the limit of long time. Any number of metastable equilibrium states may also exist, which require overcoming an activation barrier to evolve to a more stable configuration having lower free energy. A system can remain in a state of metastable equilibrium for a very long time, enabling it to be treated as a thermodynamic equilibrium state over that finite interval of time. The condition of metastability implies that small perturbations to the system will not be sufficient for it to change its state—the relaxation to a more stable state can only occur if a large activation barrier is overcome.

The simplest possible system of any practical significance is the two-state model depicted in Figure 1.3. This figure shows a double-well free energy diagram with two different stable states of the system. The higher free energy well on the left-hand side of the figure represents a metastable equilibrium state. The lower free energy state on the right-hand side is the stable equilibrium. Suppose that a reaction occurs to transition the system from the initial left-hand state (the reactant) to the final right-hand state (the product). There are two relevant parameters for this reaction, as depicted in the figure:

1. The free energy of reaction, ΔG, which is the change in the free energy of the system due to the reaction, i.e., the difference in free energy between the initial and final states; and

, which is the activation barrier that must be overcome for the reaction to proceed.

The free energy of reaction (ΔG). For the reaction to proceed during a given observation time, there must be both a nonzero thermodynamic driving force and a reaction rate that allows the kinetics to occur within the time allotted for the experiment.

Figure 1.3is the activation free energy and ΔG is the free energy of the reaction.

1.3. Spontaneous and Non-Spontaneous Processes

During a spontaneous process, the system moves to a more thermodynamically stable state having a lower free energy [4]. In other words, a spontaneous process has a negative free energy of reaction: ΔG = G products − G reactants < 0. This occurs, for example, if the two-state system moves from the left state to the right state in Figure 1.3. Spontaneous processes occur without any external stimulus, i.e., without any application of work to the system. Examples of spontaneous processes in our everyday lives include:

• Ice melting at room temperature.

• Water flowing downhill.

• Sugar dissolving in water.

• Heat flowing from a hotter object to a colder object.

• A gas expanding to fill an empty vessel.

• Iron exposed to oxygen and water forming rust.

However, merely satisfying the condition of ΔG < 0 does not tell us anything about the kinetics of the spontaneous process. Consider the case of diamond and graphite depicted in Figure 1.4. Under ambient conditions, diamond is a metastable form of carbon, having a higher free energy than graphite. Hence, the transformation from diamond to graphite is a spontaneous process, lowering the free energy of the system. Should our readers be concerned that their diamond jewelry (and Vickers diamond indenters) will spontaneously transform into graphite? Fortunately, the answer is no, because the kinetics of this transformation are much too slow to observe at room temperature. Although the transformation of diamond to graphite is a thermodynamically spontaneous process, a diamond is kinetically confined in the diamond state, unable to overcome the activation barrier necessary to transition to graphite. Although a diamond is not forever, it should safely last for billions of years.

Figure 1.4 The conversion of carbon from metastable diamond to stable graphite is a spontaneous process. However, we do not observe this process under normal conditions since the kinetics of the transformation have time scales on the order of billions of years.

The opposite of a spontaneous process is a non-spontaneous process. During a non-spontaneous process, the system moves from a more thermodynamically stable state to a less thermodynamically stable state having a higher free energy, i.e., ΔG > 0. As the name implies, non-spontaneous processes cannot occur on their own: they require an external source of energy to drive the process. Examples of non-spontaneous processes in our everyday lives include:

• Filling a tire with air.

• Photosynthesis.

• Refrigeration.

• Skiing uphill.

• Water purification.

• Taking a Kinetics exam.

They are all non-spontaneous processes because they all require work!

Whether a process is spontaneous or non-spontaneous is determined entirely by the sign of ΔG, i.e., ΔG < 0 for a spontaneous process and ΔG > 0 for a non-spontaneous process. Given that ΔG = ΔH − TΔS, we must consider both the enthalpic (ΔH) and entropic (ΔS) contributions to the free energy of reaction. Four different scenarios are possible, as shown in Figure 1.5. If the reaction is both enthalpically and entropically favored, i.e., if the enthalpy decreases (ΔH < 0) and the entropy increases (ΔS > 0) as a result of the reaction, then the process is always spontaneous at all temperatures. This is indicated in the upper-left quadrant of Figure 1.5. Conversely, if a process involves both an increase in enthalpy (ΔH > 0) and a decrease in entropy (ΔS < 0), then that process is always non-spontaneous, as indicated in the lower-right quadrant of the figure.

Figure 1.5 Thermodynamic processes can be either spontaneous or non-spontaneous depending on their associated changes in entropy and enthalpy.

The more interesting cases are when there is a competition between the enthalpic and entropic terms. For example, the lower-left quadrant of the figure shows the case of a reaction that is enthalpically favorable (ΔH < 0) but entropically unfavorable (ΔS < 0). Since ΔG = ΔH − TΔS, the spontaneity of the reaction depends on the temperature of the system. At low temperatures, the entropic term (− TΔS) contributes less because of the smaller value of T. Hence, at low temperatures the Gibbs free energy of the reaction is dominated by the enthalpy of the reaction, i.e., the heat of reaction. This means that ΔG < 0 at low temperatures, since the enthalpy is dominant and ΔH < 0. However, at high temperatures the entropic term becomes dominant, leading to ΔG > 0. Hence, such an enthalpy-driven reaction would be spontaneous at low temperatures but non-spontaneous at high temperatures.

The opposite is true in the upper-right quadrant of Figure 1.5, which shows a reaction that is favored by entropy (ΔS > 0) but not by enthalpy (ΔH > 0). Again, the entropic term is dominant at high temperatures and the enthalpic term is dominant at low temperatures. Therefore, in this case of an entropy-driven reaction, the process is spontaneous at high temperatures but non-spontaneous at low temperatures.

One of the most common themes in thermodynamics is the competition between enthalpy and entropy. As we shall see throughout this volume, the competition between enthalpic and entropic effects has a profound impact on all of materials physics, including the associated kinetic processes of materials.

1.4. Microscopic Basis of Entropy

Every good story needs a hero, and the hero of our story is Ludwig Boltzmann, an Austrian physicist born in Vienna in 1844. While the concept of atoms, i.e., the indivisible fundamental units of matter, dates back to the ancient Greek philosophers, most notably Democritus (ca. 460–370 B.C.), Boltzmann was the first to develop the physics connecting atomic theory and thermodynamics. Boltzmann became the father of a new field of physics known as statistical mechanics, which explains the macroscopic properties of materials in terms of their underlying atomic structure or microstates. A microstate corresponds to a specific atomic configuration of the system. The macrostate of the system encompasses its set of measurable properties, e.g., volume, temperature, heat capacity, thermal expansion coefficient, etc. The fundamental postulate of statistical mechanics is that the macroscopic properties of a system are a direct result of suitable averaging over the various microstates adopted by the system. Since the microstates are not necessarily known, they must be described in terms of their probabilities of occurrence. The macrostate of a system can then be determined based on this set of microstates and their respective probabilities.

Boltzmann's theory of statistical mechanics faced stiff criticism and outright rejection from many prominent scientists in the late 19th century [5]. At that time, the field of statistics was considered an immoral branch of mathematics, suitable only for gambling and not for legitimate science. Moreover, the atomic theory of matter itself was highly controversial. For example, in a famous exchange at the conclusion of one of Boltzmann's lectures, the eminent physicist Ernst Mach rose from his seat and boldly declared, I do not believe that atoms exist, flatly rejecting Boltzmann's entire life's work. Indeed, Boltzmann's ideas proved to be a few decades ahead of their time. Unfortunately, he did not cope well with this rejection from his less enlightened contemporaries. Boltzmann took his own life on September 5, 1906, by hanging himself while on a vacation with his family near Trieste, in what is now northeastern Italy. As