Heat Carriers in Liquids: An Introduction (SpringerBriefs in Physics) 3031511085, 9783031511080

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SpringerBriefs in Physics Jaeyun Moon

Heat Carriers in Liquids: An Introduction

SpringerBriefs in Physics Series Editors Egor Babaev, Department of Physics, Royal Institute of Technology, Stockholm, Sweden Malcolm Bremer, H. H. Wills Physics Laboratory, University of Bristol, Bristol, UK Xavier Calmet, Department of Physics and Astronomy, University of Sussex, Brighton, UK Francesca Di Lodovico, Department of Physics, Queen Mary University of London, London, UK Pablo D. Esquinazi, Institute for Experimental Physics II, University of Leipzig, Leipzig, Germany Maarten Hoogerland, University of Auckland, Auckland, New Zealand Eric Le Ru, School of Chemical and Physical Sciences, Victoria University of Wellington, Kelburn, Wellington, New Zealand Dario Narducci, University of Milano-Bicocca, Milan, Italy James Overduin, Towson University, Towson, MD, USA Vesselin Petkov, Montreal, QC, Canada Stefan Theisen, Max-Planck-Institut für Gravitationsphysik, Golm, Germany Charles H. T. Wang, Department of Physics, University of Aberdeen, Aberdeen, UK James D. Wells, Department of Physics, University of Michigan, Ann Arbor, MI, USA Andrew Whitaker, Department of Physics and Astronomy, Queen’s University Belfast, Belfast, UK

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Jaeyun Moon

Heat Carriers in Liquids: An Introduction

Jaeyun Moon Sibley School of Mechanical and Aerospace Engineering Cornell University New York, NY, USA

ISSN 2191-5423 ISSN 2191-5431 (electronic) SpringerBriefs in Physics ISBN 978-3-031-51108-0 ISBN 978-3-031-51109-7 (eBook) https://doi.org/10.1007/978-3-031-51109-7 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Foreword

Theories of matter have a long and interesting history. By the eighteenth century it was widely accepted that simple materials had three “phases”—solid, liquid, and gas— accessible by changing pressure and temperature. During the nineteenth century hope improved that microscopic theories could be found to understand these phases of matter. Three profound advances then came. The first clarified the dynamics of gases; the second did the same for solids; the third has done a lot to improve microscopic understanding of liquids, but improvements are needed. This is the content of Heat Carriers in Liquids: An Introduction. Its deepest focus is on new methods to improve understanding of liquids, the most complex, and in some ways (e.g., in biology) most important phase of matter. The first of the three major advances, giving a new theory of the gas phase, was the work of Boltzmann. It was based on the emerging notion that gases are made of particles (now known as atoms or molecules) whose mass gave the weight and whose concentration gave the density. With help from earlier work of many, most directly Maxwell, he developed the “Boltzmann equation”. It gave a theory of the average time evolution of particle positions and velocities, caused by interactions between particles. Boltzmann suffered prominent (and incorrect) criticisms of his theory. In the late nineteenth and early twentieth century, experiments showed that atoms and molecules are real, contrary to critics of Boltzmann’s theory of gases. Experiment also explained crystalline solids to be atoms closely arranged in spatially periodic positions. In 1929, after astonishing experimental and theoretical developments, Peierls finished the second major advance, clarifying the dynamics of crystals. The forces keeping atoms close to their average periodic positions can be treated in “harmonic approximation”. That is, the forces can be approximated as linearly proportional to the atoms’ displacements from their average positions. The simplification given by periodicity allowed the equations of motion to be solved: the displacements are traveling-wave oscillations. Quantum theory showed that they were also “particles”. Peierls then took Boltzmann’s theory of gas particles, and applied it to crystalline vibrational particles (called “phonons”). This gave us the modern theory of dynamical properties such as heat conduction and thermalization rates. Like Boltzmann, Peierls suffered from the criticism of his mentor, Pauli. v

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The final major advance was powerful computers. Liquids are much harder to analyze microscopically than either gases or solids. Starting in the 1970s, and improving steadily ever since, trajectories of molecules of a liquid can be computed if the forces are carefully modeled. The equations of motion can now be integrated for long times and for millions of atoms, enabling computation of dynamics. However, it is frustrating that the results defy both simple visualization and detailed experimental observation. This provides the main motive of this book—clarifying liquid dynamics by adopting and unifying methods from the theory of solids and gases. There is still much work to be done. Students and other readers will be introduced to an open subject to stimulate their curiosity. Hopefully this excellent book will provoke further advances in the important and challenging problem: a deeper understanding of liquids, and greater unification of the understanding of matter. November 2023 Philip B. Allen Department of Physics and Astronomy Stony Brook University Stony Brook, NY, USA

Preface

Flow of liquids is taught in fluid mechanics/dynamics classes in several disciplines including but not limited to mechanical engineering, aerospace engineering, biomedical engineering, and applied physics. Here, some materials properties of liquids such as density, viscosity, and heat capacity are used in governing equations of fluid dynamics (continuity, momentum, and energy equations) to describe the nature of liquid flow, typically in the continuum limit. However, the microscopic picture of these fundamental materials properties of liquids is rarely discussed in classes or in textbooks, showing a sharp contrast to numerous dedicated solid-state physics classes that describe these properties in depth. This lack of comprehensive and widespread discussions on liquid materials properties motivated me to write this short introductory book with a focus on thermal properties of liquids. As shall be clear in this introductory text, this book is not meant to be a fulllength book with extensive discussions. Rather, the aim of this text is to discuss and summarize recent, exciting theoretical developments with some historical background and to hopefully lower the barrier to conducting liquid state physics research for those new to the subject. It is important to emphasize that the literature reference list provided in each chapter is not meant to be complete. The list is intended to provide a helping hand in finding relevant literature, a stepping stone to the additional literature dedicated to the field. This book is written to be readable by senior undergraduate students or first-year graduate students. The only prior background knowledge required is elementary calculus and basic working knowledge of thermodynamics and statistical mechanics. Finally, I thank Profs. Philip B. Allen and Simon Thébaud and Dr. Xun Li for their critical feedback on this Book. I am grateful to Dr. Lucas Lindsay for our nearly everyday, exciting scientific discussions regarding the liquid state physics. I

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am thankful to Prof. Takeshi Egami for his guidance during my postdoctoral training and for his support. October 2023 Jaeyun Moon Sibley School of Mechanical and Aerospace Engineering Cornell University Ithaca, NY, USA

Contents

1 What Is a Liquid? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Liquids from Gas Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Liquids from Solid Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 How to Approach Liquid State Physics? . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 4 5 7

2 Normal Mode Analysis of Atomic Motion in Solids . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 What Are Normal Modes? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Example: Monatomic Linear Chain Crystal . . . . . . . . . . . . . . . 2.2.2 Formal Description of Normal Modes in Solids . . . . . . . . . . . . 2.2.3 Types of Normal Modes in Solids: Phonons, Propagons, Diffusons, and Locons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Thermal Properties of Solids from Normal Modes . . . . . . . . . . . . . . . . 2.3.1 Energy and Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 12 14

3 Time Correlations and Their Descriptions of Materials Properties . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Time Evolution in Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Time Correlation Functions and Their Properties . . . . . . . . . . . . . . . . . 3.4 Time Correlation Functions and Macroscopic Materials Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Velocity Autocorrelation Spectra as Phonon Density of States . . . . . . 3.6 Velocity Autocorrelation Spectra Describing Atomic Diffusion . . . . . 3.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 36 37

16 19 19 24 32 32

38 38 41 46 49 51

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3.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Weiner Khinchin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Green–Kubo Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 52 57

4 Continuity of the Solid, Liquid, and Gas Phases of Matter . . . . . . . . . . . 4.1 Instantaneous Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Self-diffusion Coefficient Described by Instantaneous Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Heat Capacity of Liquids Described by Instantaneous Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Comments on Instantaneous Normal Modes of Liquids . . . . . 4.1.4 Extending Instantaneous Normal Mode Analysis to Simple Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Velocity Autocorrelation Decomposition of Atomic Degrees of Freedom in Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Bridging the Gap Between INM(ω) and VACF(ω) . . . . . . . . . 4.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 62 63 65 67 67 82 83 85 87 87

Chapter 1

What Is a Liquid?

Thermal properties of liquids are important considerations in various applications including thermal storage and power plants where specific heat and thermal conductivity of liquids determine the amount of heat to be stored and transferred [1, 2]. However, our microscopic understanding of thermal properties of liquids lags far behind that of solids and gases [3–5]. In solids (especially crystals), atoms mostly vibrate around their respective mean positions known as equilibrium positions. Therefore, perturbation theory approaches based on these equilibrium positions have been very successful in describing atomic motion in solids and hence, their materials properties. For instance, phonon quasi-particles (quanta of vibrations in crystals) are used to describe heat capacity, thermal expansion, and thermal conductivity in dielectric solids. These can now be predicted with first-principles accuracy. On the other end of the spectrum of matter, atomic interactions are weak in dilute gases and real atomic collisions are used as the basis for theory developments. Some examples include kinetic theory of thermal conductivity and viscosity and the ideal-gas law. However, liquids have neither of these ‘small’ parameters that alleviate challenges in developing rigorous theories. Liquids have dynamically disordered structures that lack spatial periodicity, yet their atomic densities are similar to solids and have strong atomic attractions, first demonstrated by their capillary action recorded by Leonardo da Vinci in the late 15th century [6] in narrow siphons. Gases are similarly dynamically disordered but atoms are nearly free. Regarding these difficulties, Lev D. Landau, Nobel Laureate in physics, together with Evgeny Lifshitz once wrote, “Liquids do not allow a calculation in a general form of the thermodynamic quantities or even of their dependence on temperature.” and suggested that it is nearly impossible to obtain rigorous, microscopic characterizations of liquid materials properties [7]. To this day, this statement is not far off in describing our state-of-art understanding of various liquid properties.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. Moon, Heat Carriers in Liquids: An Introduction, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-51109-7_1

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Despite these challenges, we are not without hope. Thanks partially to the rise of high performance computers and exciting new experimental setups, I believe the time is now more ripe than ever to revisit liquid-state physics with collective efforts and develop rigorous theories that enable accurate materials property predictions in liquids. We can now perform large-scale molecular dynamics simulations with hundreds of millions of atoms and/or with tens of millisecond simulation times which enable direct comparisons between simulation results and experimental results from conventional university lab settings [8–10]. From the experimental side, there are several multi-billion-dollar user facilities available to the public such as Spallation Neutron Source at Oak Ridge National Laboratory and Linac Coherent Light Source at SLAC National Accelerator Laboratory that enable probing materials properties with Angstrom and femtosecond resolutions. If one has an idea or hypothesis, these tools are readily available to help test it, and could eventually help its crystallization into a rigorous theory without a lot of defects. In this chapter, we provide some succinct historical background into how people have viewed liquids and their nature.

1.1 Liquids from Gas Perspectives We begin our discussions with Cagniard de la Tour’s discovery of critical phenomena. Prior to his time, experiments on steam engines in the 17th and 18th centuries ignited interests in the nature of liquids and gases at high temperatures and pressures. During the 1820s, Cagniard de la Tour had interests in acoustics and conducted an experiment involving a flint ball placed in a partially filled digester with liquids. As the device rolled, a splashing noise occurred when the solid ball crossed the interface between the liquid and vapor. Upon heating the system well beyond the liquid’s boiling point, Cagniard de la Tour observed that the splashing sound ceased beyond a specific temperature. This observation demonstrates discovery of the supercritical fluid phase. In this phase, there exists no surface tension because the distinction between the liquid and gas phases ceases to exist. In Annales de Chimie et de Physique [11], Cagniard de la Tour described how he heated ether, alcohol, and water in separate sealed glass tubes under pressure (See Fig. 1.1). He observed the disappearance of the liquid, transforming into vapor, making the tube seem vacant. With cooling, a thick, dense cloud emerged, recognized today as critical opalescence. The discovery of critical phenomena soon led to some uncertainties on how to define liquid, gas, and the ‘Cagniard de la Tour state’ among scientists including Herschel, Faraday, Whewell, Dumas, and Mendeleev [12–15]. Herschel qualitatively proposed that Cagniard de la Tour’s work was a supporting evidence for the lack of a sharp distinction between the three states of matter as he wrote, “Indeed, there can be little doubt that the solid, liquid, and aëriform states of bodies are merely stages in a progress of gradual transition from one extreme to the other; and that, however strongly marked the distinctions between them may appear, they will ultimately turn out to be separated by no sudden or violent line of demarcation, but shade into each

1.1 Liquids from Gas Perspectives

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Fig. 1.1 One of early experimental setups used by Cagniard de la Tour in the early 1820s that led to the discovery of critical phenomena [11]

other by insensible gradations. The late experiments of Baron Cagnard de la Tour may be regarded as a first step towards a full demonstration of this.” Faraday and Whewell had letter discussions on how to better describe and name the Cagniard de la Tour’s recently discovered state. Whewell who was a wordsmith credited with coining the terms such as scientist or physicist suggested to call these supercritical states vaporiscents and described liquids crossing the critical point as liquids being destroyed or disliquified. Faraday was not content with these suggestions. Mendeleev had a different take and considered that liquids turned into vapors instead. A few years later, Mendeleev published works that would lead to periodic tables that we use today [16]. Andrews’ liquefaction experiments on carbon dioxide in the 1860s resulted in some clarifications to the ‘Cagniard de la Tour state’ by demonstrating that both liquids and gases approached a common fluid state at what he called, ‘the critical point’. He published this work in his celebrated Bakerian Lecture to the Royale Society with the title, ‘On the continuity of the gaseous and liquid states of matter’ [17, 18]. With these works describing the critical point and supercritical state, we have a more comprehensive phase diagram that we use today as demonstrated in Fig. 1.2. Subsequent to Cagniard de la Tour’s groundbreaking experiments, a significant intellectual curiosity and investigation ensued. In 1873, van der Waals demonstrated in his doctoral dissertation that Andrews’ experimental findings could be elucidated by a straightforward expansion of the ideal gas law, which incorporated molecular

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Fig. 1.2 A representative phase diagram (temperature vs. pressure) of matter. Slopes of the boundaries depend on the materials

attraction and hard-core repulsion terms [19]. Intriguingly, van der Waal’s thesis was also titled, ‘On the continuity of the gaseous and liquid states’. The term ‘fluid’ started to encompass both gases and liquids, as evident by Ref. [20] and van der Waal’s equation was eventually referred to as the “equation of fluids”. van der Waal’s work has also been attributed to the development of the hard-sphere paradigm that is widely used to study soft matter, granular materials, gases, and liquids [21, 22]. It is worth mentioning in passing that the Navier–Stokes equation describing flows of viscous fluids (liquids and gases) was also developed around the time of Cagniard de la Tour’s and Andrews’ works in supercritical phases.

1.2 Liquids from Solid Perspectives Similar to approaching liquids from more well-characterized gas perspectives, liquids have been considered from solid perspectives as well. It is not clear when similarity between solid and liquid states at low temperatures became the basis for theoretical developments for liquids. As mentioned before, capillary action first recorded by da Vinci in the 15th century is a demonstration of the importance of strong intermolecular interactions in liquids, similar to atoms in solids. In addition to the critical phenomena, van der Waals also investigated microscopic mechanisms of capillary action in his thesis. More concrete treatments of solids and liquids together from theoretical perspectives began as early as 1903 when Mie considered liquids as an array of harmonic oscillators on lattice sites [23]. For nearly a century since, many have modelled liquids by atoms in mean positions as in crystals or crystal structures with interstitial defects [24–26]. One example called interstitialcy theory (IT) is shown in Fig. 1.3 where liquids are modelled as crystals with interstial defects and various thermodynamic quantities in liquids have been studied with this theory.

1.3 How to Approach Liquid State Physics?

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Fig. 1.3 An example of approaching liquid state physics from solid perspectives: Interstitialcy theory treating liquids as crystals containing a few percent of interstitial defects in thermal equilibrium, adopted from Ref. [26]

In the same time period, Yakov Frenkel instead considered liquids and solids as a continuity [27–30] and considered these phases of matter together under “condensed bodies”. Interestingly, one of Frenkel’s papers bears the title, “Continuity of the Solid and the Liquid States”, showing a contrast to Andrews’ and van der Waals’ works. In his works, Frenkel proposed that when the characteristic time of interest is less (or greater) than .τ = η/N where .η is viscosity and .N is rigidity modulus, material behaves like a solid (or liquid) [29]. One everyday example of this is splashing water in the swimming pool. When we splash water with the palm of our hands quickly, water feels as if it is rigid and solid-like. However, if we gradually submerge our hands, there is very little resistance. Frenkel’s seminal ideas have led to many recent works considering thermodynamics of liquids from phonon quasi-particles as in solids [31–42]. The phrase, “condensed matter” or “condensed bodies”, has existed for many centuries before but similar to fluids describing liquids and gases together, the use of the term, “condensed matter”, has been since popularized by Anderson and Heine to include liquid state physics in the 1960s.

1.3 How to Approach Liquid State Physics? So what is a liquid? How should we view a liquid as? From the solid perspectives as a condensed form of matter? Or from the gas perspectives as a fluid? Unfortunately, theories used in solids and gases are typically incompatible. For instance, vibrations (phonon theories in solids [43, 44]) cannot be used to describe atomic collisions in gases and vice versa (See Fig. 1.4). Rather than restricting ourselves and starting a

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Fig. 1.4 Three classical phases of matter. In solids, atomic motion is described in terms of vibrations called phonons and various thermal properties of solids are explained in terms of these vibrations. In the other end of the spectrum, real atomic motions such as collisions and translations are the basis for theory developments

theory from one specific phase, I believe we should consider heat carriers in all three classical phases of matter simultaneously in a unified manner with each phase having an equal footing that can reduce down to the phonon quasi-particle picture in solids or real atomic dynamics in gases, similar to Herschel’s continuous views of matter. This Book is written in a way that reflects this unified approach [45–48] with the focus on atomic motions in liquids and their relations to thermal physics. Electronic and magnetic properties are not considered here. The rest of the book is organized in the following manner. In Chap. 2, we first go through the phonon (or normal mode) theory of solids that has been successful in describing various thermal properties microscopically. Phonons and normal modes are used interchangeably in the literature and textbooks [49]. Different categorizations of normal modes are demonstrated depending on the structural symmetry and mode interaction mechanisms. Using the normal mode picture, we describe various thermal properties such as heat capacity and thermal conductivity in dielectric solids. Historically significant models including Debye and Einstein models are introduced. In Chap. 3, time-correlation functions and their properties are discussed. General descriptions of macroscopic viscosity and thermal conductivity for all phases by time-correlation functions are demonstrated. Velocity autocorrelation functions crucial for studying microscopic mechanisms of these macroscopic properties are examined. In the final chapter of the book, Chap. 4, a survey of recent efforts to extend the normal mode formalism to the liquid phase for describing their thermal properties is provided. Connections between the classical normal mode formalism and velocity autocorrelation functions are discussed. Finally, we discuss how one could use both the normal mode formalism and velocity autocorrelation functions to describe heat carriers in all three phases in a unified manner.

References

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23. Mie, G.: Zur kinetischen Theorie der einatomigen Körper. Annalen der Physik 316(8), 657–697 (1903). https://doi.org/10.1002/andp.19033160802. Accessed 18 July 2022 24. Mott, N.F.: The resistance of liquid metals. Proc. R. Soc. Lond. Ser. A (Containing Papers of a Mathematical and Physical Character) 146(857), 465–472 (1934). https://doi.org/10.1098/ rspa.1934.0166. Accessed 27 July 2022 25. Lennard-Jones, J.E., Devonshire, A.F.: Critical phenomena in gases - I. Proc. R. Soc. Lond. Ser. A - Math. Phys. Sci. 163(912), 53–70 (1937). https://doi.org/10.1098/rspa.1937.0210. Accessed 18 July 2022 26. Granato, A.V.: The specific heat of simple liquids. J. Non-Crystalline Solids 307–310, 376–386 (2002). https://doi.org/10.1016/S0022-3093(02)01498-9. Accessed 19 July 2022 27. Frenkel, J.: Über die Wärmebewegung in festen und flüssigen Körpern. Zeitschrift für Physik 35(8), 652–669 (1926). https://doi.org/10.1007/BF01379812. Accessed 18 July 2022 28. Frenkel, J.: Continuity of the solid and the liquid states. Nature 136(3431), 167–168 (1935). https://doi.org/10.1038/136167a0. Accessed 18 July 2022 29. Frenkel, J.: On the liquid state and the theory of fusion. Trans. Faraday Soc. 33, 58 (1937). https://doi.org/10.1039/tf9373300058. Accessed 18 July 2022 30. Frenkel, J.: Kinetic Theory of Liquids. Oxford University Press (1947) 31. Trachenko, K., Brazhkin, V.V.: Heat capacity at the glass transition. Phys. Rev. B 83(1), 014201 (2011). https://doi.org/10.1103/PhysRevB.83.014201. Accessed 06 Aug 2023 32. Bolmatov, D., Brazhkin, V.V., Trachenko, K.: The phonon theory of liquid thermodynamics. Sci. Rep. 2, 421 (2012). https://doi.org/10.1038/srep00421. Accessed 21 April 2018 33. Andritsos, E.I., Zarkadoula, E., Phillips, A.E., Dove, M.T., Walker, C.J., Brazhkin, V.V., Trachenko, K.: The heat capacity of matter beyond the Dulong-Petit value. J. Phys.: Condens. Matter 25(23), 235401 (2013). https://doi.org/10.1088/0953-8984/25/23/235401. Accessed 22 June 2020 34. Trachenko, K., Brazhkin, V.V.: Collective modes and thermodynamics of the liquid state. Rep. Prog. Phys. 79(1), 016502 (2016). https://doi.org/10.1088/0034-4885/79/1/016502. Accessed 03 June 2020 35. Wang, L., Yang, C., Dove, M.T., Fomin, Y.D., Brazhkin, V.V., Trachenko, K.: Direct links between dynamical, thermodynamic, and structural properties of liquids: modeling results. Phys. Rev. E 95(3), 032116 (2017). https://doi.org/10.1103/PhysRevE.95.032116. Accessed 13 Feb 2020 36. Yang, C., Dove, M., Brazhkin, V., Trachenko, K.: Emergence and evolution of the $k$ gap in spectra of liquid and supercritical states. Phys. Rev. Lett. 118(21), 215502 (2017). https://doi. org/10.1103/PhysRevLett.118.215502. Accessed 08 Dec 2020 37. Brazhkin, V.V., Fomin, Y.D., Ryzhov, V.N., Tsiok, E.N., Trachenko, K.: Liquid-like and gas-like features of a simple fluid: an insight from theory and simulation. Physica A: Stat. Mech. Appl. 509, 690–702 (2018). https://doi.org/10.1016/j.physa.2018.06.084. Accessed 18 Feb 2022 38. Tomiyoshi, Y., Ueda, D.: Heat capacity of simple liquids in light of hydrodynamics as U(1) gauge theory. Phys. Rev. E 100(1), 012103 (2019). https://doi.org/10.1103/PhysRevE.100. 012103. Accessed 15 Aug 2019 39. Khusnutdinoff, R.M., Cockrell, C., Dicks, O.A., Jensen, A.C.S., Le, M.D., Wang, L., Dove, M.T., Mokshin, A.V., Brazhkin, V.V., Trachenko, K.: Collective modes and gapped momentum states in liquid Ga: experiment, theory, and simulation. Phys. Rev. B 101(21), 214312 (2020). https://doi.org/10.1103/PhysRevB.101.214312. Publisher: American Physical Society. Accessed 08 June 2020 40. Kryuchkov, N.P., Mistryukova, L.A., Sapelkin, A.V., Brazhkin, V.V., Yurchenko, S.O.: Universal effect of excitation dispersion on the heat capacity and gapped states in fluids. Phys. Rev. Lett. 125(12) (2020). https://doi.org/10.1103/PhysRevLett.125.125501. Accessed 08 Dec 2020 41. Zaccone, A., Baggioli, M.: Universal law for the vibrational density of states of liquids. Proc. Natl Acad. Sci. 118(5), 2022303118 (2021). https://doi.org/10.1073/pnas.2022303118. Accessed 14 April 2021

References

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42. Baggioli, M., Zaccone, A.: Explaining the specific heat of liquids based on instantaneous normal modes. Phys. Rev. E 104(1), 014103 (2021). https://doi.org/10.1103/PhysRevE.104. 014103. Publisher: American Physical Society. Accessed 20 April 2022 43. Debye, P.: Zur theorie der spezifischen Wärmen. Annalen der Physik 344(14), 789–839 (1912) 44. Einstein, A.: Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme. Annalen der Physik 327(1), 180–190 (1907). https://doi.org/10.1002/andp.19063270110 45. Croxton, C.A.: Introduction to Liquid State Physics. Wiley, London (1975) 46. Moon, J.: Examining normal modes as fundamental heat carriers in amorphous solids: the case of amorphous silicon. J. Appl. Phys. 130(5), 055101 (2021). https://doi.org/10.1063/5. 0043597. Publisher: American Institute of Physics 47. Moon, J., Thébaud, S., Lindsay, L., Egami, T.: Normal mode description of phases of matter: application to heat capacity. Phys. Rev. Res. 6(1), 013206 (2024). https://doi.org/10.1103/ PhysRevResearch.6.013206. Publisher: American Physical Society 48. Moon, J., Lindsay, L., Egami, T.: Atomic dynamics in fluids: normal mode analysis revisited. Phys. Rev. E 108(1), 014601 (2023). https://doi.org/10.1103/PhysRevE.108.014601. Publisher: American Physical Society. Accessed 10 July 2023 49. Ashcroft, N.W., Mermin, N.D.: Solid State Physics. Saunders College Publishing, New York (1976)

Chapter 2

Normal Mode Analysis of Atomic Motion in Solids

Abstract Decomposition of atomic motion into individual normal modes has led to remarkable success in microscopically understanding thermal properties and thermodynamics in simple solids. We start this chapter with an example of decomposing atomic motion of a simple monatomic linear chain crystal into normal modes followed by a more general, classical normal mode formalism. Different classifications of normal modes such as phonons, propagons, diffusons, and locons are introduced. Finally, heat capacity and thermal conductivity predictions from the normal mode formalism are demonstrated. Keywords Normal modes · Phonon · Propagon · Diffuson · Locon · Heat capacity · Thermal conductivity · Perturbation theory · Molecular dynamics · Lattice dynamics

2.1 Introduction Characterization of atomic degrees of freedom is crucial in describing and understanding various materials properties microscopically. In simple, perfect solids, atoms vibrate around their equilibrium positions. Normal mode (sinusoidal planewave) decomposition of atomic motion in solids is, therefore, a natural way of understanding the atomic degrees of freedom. In this chapter, we describe succinctly what normal modes are and how they can be used to describe some materials properties such as heat capacity and thermal conductivity in solids. Some historically significant and simple models such as Debye [1] and Einstein [2] models are discussed. We will primarily focus on normal modes in crystals in this chapter but how the nature of normal modes changes going from crystals to amorphous solids is briefly discussed.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. Moon, Heat Carriers in Liquids: An Introduction, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-51109-7_2

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2 Normal Mode Analysis of Atomic Motion in Solids

2.2 What Are Normal Modes? 2.2.1 Example: Monatomic Linear Chain Crystal To get a physical intuition for what normal modes are, we first begin by a simple textbook example of describing the equation of motion of a monatomic linear chain crystal in terms of normal modes as shown in Fig. 2.1. We assume that interatomic interaction is described by a simple spring motion, ignoring higher order terms in the potential. This treatment of the potential is also known as the harmonic approximation. Anharmonic terms can be included in the potential for more accurate descriptions of interatomic interactions, often by perturbation methods (described in Sect. 2.3.2.2). Harmonic approximation is useful in that it allows us to obtain many physical characteristics of the system with a relatively small effort. In this linear chain crystal, each atom has the same mass, .m. For simplicity, each atom only interacts with the nearest neighbor atoms that are apart by a lattice constant, .a, by identical springs with a spring constant, .κ. As atoms vibrate, we define displacement of an atom, . j, from its equilibrium position by u j = x j − x 0j

.

(2.1)

where .u j is the displacement, .x j is the instantaneous position, and .x 0j is the equilibrium position of atom . j. Net force on atom . j is then given by .

Fj = m

d 2u j = κ(u j+1 − u j ) − κ(u j − u j−1 ). dt 2

(2.2)

As atoms vibrate around their equilibrium positions, it is reasonable to expect that atomic displacements will be a superposition of sinusoidal ∑ waves or modes. We then look for a plane-wave solution form for .u j as .u j = k Ak e−i(ωk t−kx) where .ωk is the angular frequency and .k is the wavevector. Rather than a continuous coordinate, . x is restricted to the values of . x = ja due to the discrete and periodic nature of this atomic system. This also means that we have limits on the values that .k can take in = 2πm , where .m is an integer, . N is the number this linear chain crystal: .k = 2πm Na L of atoms, and . L is the length of the chain. Discreteness of wavevectors, .k, merits more discussions. However, in other phases without a periodic structure which are

Fig. 2.1 Schematic of a linear atomic chain crystal. .κ is the harmonic force constant or the spring constant and interatomic distance is given by .a

2.2 What Are Normal Modes?

13

Fig. 2.2 Dispersion relations (solid black curves) of a monatomic linear chain crystal as given by Eq. 2.4 in (.k ∗ = k/(π/a) reduced units√ and .ω ∗ = ω/ κ/m). At low wavevectors and frequencies, we observe linear dispersions, which are the basis for the Debye model

the foci of the book, the meaning of wavevectors becomes more obscure. Hence, we will not delve into this subject. Plugging in the plane-wave solution and simplifying, we obtain 2 ika . − mωk = κ(e + e−ika − 2). (2.3) With some algebra, we obtain the following dispersion relations (.ω = f (k)) (Fig. 2.2) / ωk =

.

| ( ka )|| κ || |. sin m| 2 |

(2.4)

The example given above provides us wealth of information regarding the atomic motion in a crystal. First, dispersion relations shown above demonstrate that atomic motion can be decomposed into individual normal modes with frequencies and wavevectors described by Eq. 2.4 under the harmonic approximation simply from the structural features (lattice constant, number of atoms, and atomic mass) and k ) and phase velocities (.v p = ωkk ) force constants. Second, group velocities (.vg = dω dk relevant to transport properties and elastic moduli can be found from the dispersion relations. One can, therefore, naturally think that transport properties such as thermal conductivity can be engineered by optimizing some features in the dispersion relations. Third, recognizing that the Taylor expansion of .sin(x) is .

sin(x) = x −

x3 x5 x7 + − + · · ·, 3! 5! 7!

(2.5)

the long wavelength limit of the dispersion is ωk (k → 0) ∼ ck.

.

(2.6)

where .c is a constant. This linear dispersion at low wavevectors and low frequencies is observed in various real bulk systems ranging from crystals and glasses to liquids and is the basis for the Debye model [1]. The Debye model together with Einstein model [2] will be discussed in detail in Sect. 2.3.1.1. In this linear dispersion, group

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2 Normal Mode Analysis of Atomic Motion in Solids

velocity and phase velocity are identical to each other at .c, also known as the sound velocity. Normal modes in crystals are used interchangeably with phonons in crystals. In one of the widely used textbooks for solid state physics, Ashcroft and Mermin introduce normal modes as equivalent to phonons as they write, “Although the language of phonons is more convenient than that of normal modes, the two nomenclatures are completely equivalent.” [3] However, as we will find out in Chap. 4, normal modes can also describe other types of atomic motions besides the atomic vibrations and are a more general concept than phonons.

2.2.2 Formal Description of Normal Modes in Solids In the above example, we examined a simple linear chain crystal with one atom per unit cell and its single dispersion curve. More generally, we consider here an arbitrary number of atoms per unit cell with different atomic masses and in three dimensions, which lead to several dispersion curves or branches for a given wavevector. In three dimensions with .n atoms per unit cell, there are .3n branches, of which three are acoustic branches (longitudinal and two transverse) and .3n − 3 are optical branches. Acoustic branches are related to sound waves in solids as demonstrated in the linear chain example and .ω goes to zero as . k goes to zero. On the other hand, .ω is non-zero as. k goes to zero for optical modes. The name optical comes about as optical phonons are easily excited by light. The equation of motion for atom. j in the unit cell. p under harmonic approximation is given by ∑ ¨ j, p (t) = − .m j u ϕ j, p; j ' , p' · u j ' , p' (t) (2.7) j ' , p'

where matrix elements of the force constants, .ϕ j, p; j ' , p' , is given by αβ

ϕ j, p; j ' , p' =

.

∂ 2U β

∂u αj, p ∂u j ' , p'

(2.8)

with .α and .β being the Cartesian directions and .U being the potential energy. Equation 2.7 is essentially a more general case of . F = −κx. As in the linear chain example, we decompose the atomic motion into plane-wave modes as ∑ . u j, p (t) = A( j, k, ν)ei[k·r j, p −ω(k,ν)t] (2.9) k,ν

where . r j, p is the equilibrium position of the atom . j in unit cell . p and . A( j, k, ν) is the amplitude vector for atom . j for mode with wavevector . k and branch .ν and is independent of . p as the differences in motions between unit cells are described

2.2 What Are Normal Modes?

15

in the exponential phase factor. Plugging in Eq. 2.9 into Eq. 2.7 and simplifying, we obtain ∑ 2 .m j ω (k, ν) A( j, k, ν) = ϕ j, p; j ' , p' · A( j ' , k, ν)ei k·[r j ' , p' −r j, p ] . (2.10) j ' , p'

In a more compact form, above equation can be re-written as .

ω 2 (k, ν)e(k, ν) = D(k) · e(k, ν)

(2.11)

with the eigenvector .e(k, ν) having .3n by 1 matrix elements. .e(k, ν) is related to A( j, k, ν) by ⎤ ⎡√ m 1 A x (1, k, ν) √ ⎢ m 1 A y (1, k, ν)⎥ ⎥ ⎢√ ⎢ m 1 A z (1, k, ν) ⎥ ⎥ ⎢√ ⎢ m 2 A x (2, k, ν)⎥ ⎥ ⎢√ ⎢ m 2 A y (2, k, ν)⎥ ⎥ ⎢ (2.12) . e(k, ν) = ⎢ √m 2 A z (2, k, ν) ⎥ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎦ ⎣ . √ m n A z (n, k, ν)

.

.

D(k) is known as the dynamical matrix and is defined as .

αβ

D j j ' (k) = √

∑ αβ 1 ϕ j, p; j ' , p' ei k·[r j ' p' −r j p ] . m j m j ' p'

(2.13)

Dynamical matrices can be interpreted as the mass-reduced Fourier transform of αβ the force constant matrices .ϕ j, p; j ' , p' . For each . k as shown in Eq. 2.13, the size of the dynamical matrix is .3n by .3n. The eigenvalue matrix .Ω(k, ν) = ω 2 (k, ν) has the dimensions of .3n by .3n with only diagonal elements: .ω 2 (k, 1), ω 2 (k, 1), ... , ω 2 (k, 3n). Some important characteristics of Eq. 2.11 are worth discussing here. Dynamical matrices are Hermitian such that .

D(k) = ( D∗ (k))T

(2.14)

and subsequently, eigenvalues, .Ω(k, ν), are always real. This means eigenfrequencies, .ω(k, ν), can have imaginary or real values. Further, corresponding eigenvectors which are generally complex, are orthonormal as .

e(k, ν)T · e(k, ν ' )∗ = e(k, ν)T · e(−k, ν ' ) = δνν '

(2.15)

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Nowadays, once we know the structural information and force constants, we can routinely do lattice dynamics calculations and solve Eq. 2.11 via widely available softwares such as GULP [4], LAMMPS [5] and Phonopy [6]. It is worth mentioning here that primitive unit cells (smallest unit cell to describe the lattice) are conventionally used in Eq. 2.11 for crystalline materials, but use of other unit cells is equally valid. Different unit cells lead to different dispersion relations from those of primitive unit cells but the spectral distribution function of eigenfrequencies is identical regardless of the choice of the unit cell. For systems with no periodicity (e.g. glasses), the entire domain simulated in computers is considered as a unit cell which means only zero wavevector (. k = 0) or .[-point dynamical matrices are used [7, 8]. When only . k = 0 is considered, we no longer have acoustic or optical branches in a traditional sense as we do not have any other wavevectors. However, algebra becomes more intuitive as dynamical matrices and eigenvectors become real. One can then plot the eigenvector of all atoms for each mode and intuitively understand what individual mode motion looks like. For instance, if a mode is suspected to be spatially localized through some analysis tools, which typically means it contributes negligibly to thermal transport, one can simply visualize the mode and check if only a few atoms are indeed participating in the mode. As normal modes describe harmonic atomic motion, we expect that normal modes of amorphous solids which lack long range structural order look or behave differently from those of simple crystalline solids. One example is how far these normal modes can travel before scattering, relevant to thermal conductivity that we will discuss. Imagine waves in the nice calm ocean with some breeze. We expect waves to decay slowly and progressively: these waves can travel long distances similar to phonons in simple crystals. On the other hand, if we have many large and small rocks where the waves are traveling through, the waves will be able to travel only a small distance before scattering. Therefore, we expect normal modes in glasses which are full of defects (like rocks) to have much shorter mean free paths (distance before scattering). To discern different types of normal mode characters, prior works have tried to categorize normal modes in solids as phonons, propagons, diffusons, and locons depending on the scattering mechanisms and degrees of structural order [7–16]. This taxonomy is widely used in studying thermal properties of solids.

2.2.3 Types of Normal Modes in Solids: Phonons, Propagons, Diffusons, and Locons Phonons, propagons, diffusons, and locons are all vibrational in nature and follow Bose–Einstein statistics. Phonons are quanta of vibrations with well-defined frequencies and wavevectors. As phonon wavevectors are typically associated with spatial periodicity and structural order, phonon terminology is primarily used in crystals. Propagons are similar to phonons in that they are propagating delocalized normal modes that typically possess long wavelengths compared to the interatomic spacing.

2.2 What Are Normal Modes?

17

They too have relatively well-defined wavevectors (sounds also travel to your ears when you place your ears on top of window glasses and tap on them) but not in a traditional sense from periodicity of the structure. One important distinction between phonons and propagons is the uniqueness vs non-uniqueness of a mode. For example, say we did lattice dynamics calculations and obtained normal mode frequencies for a crystal and a glass of the same size that are made of the same compositions. Then, we choose a phonon and propagon mode from each structure. If we enlarge the structure by, say, a factor of two and re-do the lattice dynamics calculations from the new structures, the identity or uniqueness of the phonon mode will be conserved while the propagon mode frequency will be different. Therefore, propagons should be considered in the averaged sense. Diffusons are modes that scatter over a distance less than interatomic distance and thus transport heat as a random-walk. Locons are non-propagating and localized modes that are very inefficient to transport heat. Propagon, diffuson, and locon terminologies are used primarily when studying disordered systems including glasses [9, 10, 12, 14, 15, 17–19]. Representative eigenmode motions for phonons, propagons, diffusons, and locons for a silicon crystal and glass using Stillinger-Weber potential [20] are shown in Fig. 2.3 for a visual aid. Clear changes in the mode behavior are demonstrated. There are several proposed ways in categorizing normal modes into propagons, diffusons, and locons obtained from diagonalizing dynamical matrix at . k = 0 by treating the entire domain as a unit cell. There are largely two ways to distinguish propagons and diffusons: transport properties and periodicity of eigenmotion. Regarding the first criterion, prior works have determined the crossover frequency (Ioffe-Regel frequency) when (i) thermal diffusivity of normal modes assuming purely propagon picture or diffuson picture overlaps [16] or (ii) the mean free paths of normal modes become comparable to interatomic distance or their vibrational periods [9, 11, 21]. Another way to determine the Ioffe-Regel frequency is to use the equilibrium atomic positions and eigenvectors of individual normal modes then look for planewave-like periodicity in the eigenvectors [22]. The eigenvector periodicity (EP) of a mode is further normalized by a fictitious normal mode that has pure sinusoidal modulation, which provides information about the degree of propagation of that particular mode. Either method typically yields the crossover frequency to be on the order of 1 THz in various amorphous solids. For distinguishing diffusons and locons, inverse participation ratio (IPR) has been a widely used metric to find the diffuson to locon crossover frequency (mobility edge) and is also based on the eigenvector characters. IPRs (. pn−1 ) are found by [23] .

pn−1 =

∑(∑ i

∗ eiα,n eiα,n

)2 (2.16)

α

where .eiα,n is the eigenvector component for atom .i in direction .α for the mode .n. IPR gives a measure of how many atoms participate appreciably in the motion of a particular mode; therefore, IPR has been used to qualitatively provide information about the degree of localization of modes for various complex systems [18, 19]

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2 Normal Mode Analysis of Atomic Motion in Solids

(a)

(b)

(c)

(d)

Fig. 2.3 2D view (along. y direction) of eigenvector of an individual mode (brown arrows) projected on atomic coordinates for (a) acoustic phonon at 1.52 THz for crystalline silicon and (b–d) propagon (1.15 THz), diffuson (12.39 THz), and locon (18.15) for amorphous silicon, respectively. Black circles represent atomic positions. In the normal mode calculations, Stillinger-Weber potential [20] was utilized and both the crystal and glass have the same density. Eigenvector magnitudes are scaled for better visualization. All normal mode calculations were done at . k = 0

and even scattering behaviors of acoustic and optic phonons in binary compound crystals [24]. The IPR is conventionally interpreted such that .1/N describes all atoms participating in the mode (delocalized) and 1 if the mode motion is completely localized to a single atom. An example of a locon is shown in Fig. 2.3d. On the other hand, if the normal mode is delocalized, there will be many atoms participating in that mode as demonstrated in Fig. 2.3a–c. Historically, normal modes in amorphous silicon has been studied extensively as a representative amorphous solid due to its simple single-element composition and wide usage in applications such as solar cells [25] and gravitational wave detectors [26]. A representative normal mode density of states of amorphous solids partitioned into propagons, diffusons, and locons is depicted for amorphous silicon in Fig. 2.4. Most normal modes found in amorphous solids are considered to be diffusons similar

2.3 Thermal Properties of Solids from Normal Modes 0.15

Density of states (THz-1 )

Fig. 2.4 Vibrational density of states of amorphous silicon using Stillinger-Weber potential [20] and 4096-atom size, adapted from [28]. Each shaded area represents propagon, diffuson, and locon regions from lighter to darker grey according to Allen and Feldman proposed taxotomy [9]

19

Propagon Diffuson Locon

0.1

0.05

0

0

5

10

15

20

Frequency (THz)

to Fig. 2.4. Locons are usually located in the high frequency region of the density of states while some exceptions have also been shown [27]. With formal treatment of normal mode decomposition of atomic motion in both ordered and disordered solids, we next discuss understanding thermal properties of solids from normal mode analysis with emphasis on crystals.

2.3 Thermal Properties of Solids from Normal Modes 2.3.1 Energy and Heat Capacity One important quantity to characterize is the system energy, from which we can obtain important materials properties such as heat capacity and thermal conductivity. For harmonic oscillators, mode energy is given by .

( 1) E(k, ν) = hω(k, ν) n(k, ν) + 2

(2.17)

[ ]]−1 where .n(k, ν) = n(ω, T ) = exp(hω(k, ν)/k B T ) − 1 is known as the phonon occupation number or Bose–Einstein distribution. System energy under the harmonic approximation is then given by the summation of all the normal mode energy | | . contributions. Individual mode specific heat is defined by .C V (k, ν) = ∂ E(k,ν) ∂T V As mentioned before in the linear chain example, the meaning of .k is not welldefined in systems with no periodicity and it becomes challenging to obtain dispersion relations. It is, therefore, beneficial to develop a formalism that relies only on the frequency distribution. We define here a quantity called the density of states, .g(ω), which describes the population distribution of available normal modes such that { it obeys . g(ω)dω = 3N with .3N being the total translational atomic degrees of freedom. A simple analogical description of density of states is shown in Fig. 2.5. Imagine a building where normal modes reside. Floors represent different frequency

20

2 Normal Mode Analysis of Atomic Motion in Solids

Fig. 2.5 Analogical description of density of states in terms of building occupations. Each floor/row of the building represents frequency levels in normal modes. Note that in the actual density of states, frequency levels are not evenly distributed like the building floors shown

bins. Building registry is written in the table with some floors with more residents than others. The population distribution of these registered residents (available normal modes) per floor (frequency) is the density of states (.g(ω)). However, the actual population of the building can be different from the registry itself as residents may go on vacation, go work during the day, etc. The actual population distribution is a quantum phenomenon and can be described by .g(ω)n(ω, T ). The total energy will be then the energy per floor (.hωg(ω)n(ω, T )) integrated by all the floors. More formally, the total system energy in terms of normal modes in frequency is given by { .

E=

( 1) dωhω n(ω, T ) + g(ω) 2

(2.18)

with addition of . 21 arising from the zero point motion. Constant volume heat capacity can then be written as | { ( 1) ∂ ∂ E || dωhω n(ω, T ) + = (2.19) .C V = g(ω). | ∂T V ∂T 2 The only unknown in Eqs. 2.18 and 2.19 is the density of states. As mentioned briefly when discussing dispersion relations, density of states can now be routinely found from doing lattice dynamics calculations in readily available softwares. A representative example of phonon density of states of a crystal (germanium) by density functional theory (DFT) [29] alongside with Debye [1] and Einstein [2] densities of states proposed more than a century ago is shown in Fig. 2.6. Despite the inaccuracy of Debye and Einstein densities of states in depicting the phonon density of states, they are still used today in complex systems [7, 16, 30–32] for their simplicity, their ability to predict the specific heat of various solids well,

2.3 Thermal Properties of Solids from Normal Modes

21

Fig. 2.6 Phonon density of states of germanium by density functional theory (solid curves) adopted from Ref. [29]. Debye (blue dashed curve) and Einstein (red dotted line) densities of states are also shown for comparisons. In this figure, densities of states are normalized such that the integral over frequency is unity

and their simple analytical form of density of states. For completeness, we go over these models in detail next.

2.3.1.1

Debye and Einstein Model

In the Debye model, we assume a linear dispersion: .ω = ck where .c is the averaged sound velocity. In the linear chain example, each wavevector was .2π/L apart. In the 3D bulk case, each .k point has the volume of .(2π/L)3 . Now, consider a sphere of .k points. The number of .k points with wavevector between .k and .k + dk is equal to .

g(k)dk =

V 4πk 2 dk 8π 3

(2.20)

where .g(k) has the same meaning as .g(ω) but in the wavevector space and .V = L 3 is the system volume. We can convert this expression from wavevector to frequency space .g(ω) using the linear dispersion and .dω = cdk. We obtain .

g(ω)dω =

3V ω 2 dω 2π 2 c3

(2.21)

Additional factor of three arises from three polarizations that we need to consider (one longitudinal and two transverse) for 3D bulk systems. Quadratic frequency dependence in the Debye density of states is demonstrated as a parabola in Fig. 2.6. In the linear chain example, we discussed that a linear dispersion at low wavevector and frequency is typically found in real systems. This leads to the quadratic frequency dependence in the real phonon densities of states of various materials as demonstrated by the DFT calculated density of states of germanium at low frequencies. Since the

22

2 Normal Mode Analysis of Atomic Motion in Solids

total number of modes is .3N , there is a restriction of frequency ranges in the Debye density of states from .0 to .ω D . This upper limit is known as the Debye frequency and is given by ( 6π 2 c3 N ) 13 .ω D = . (2.22) V Since .V and .c are weakly dependent on temperature for crystals, .ω D is typically assumed as a constant. Using the Debye density of states, we obtain { .

ωD

E= 0

(

3V hω 3 2π 2 c3

)(

1 hω

e kB T − 1

+

) 1 dω. 2

(2.23)

For simplified expressions, we make the following substitutions .

x=

hω kB T

(2.24)

and rewrite the energy as (

( )3 { x D x3 x3 TD .E = T d x = 9N k d x, B ex − 1 T ex − 1 0 0 (2.25) where .TD = hω D /k B is the Debye temperature. Above energy expression can be solved numerically. At low temperatures, .x D can be taken as infinity such that we can make use of { ∞ x3 π4 . (2.26) dx = , x e −1 15 0 )(

3V h 2π 2 c3

kB T h

)4 {

xD

which leads to the quartic temperature dependence in energy as . E = temperatures. Specific heat for the Debye model is then CV =

.

∂ ∂T

[(

3V h 2π 2 c3

)(

kB T h

)4 { 0

xD

] x3 d x , ex − 1

V π 2 k 4B 4 T 10c3 h3

at low

(2.27)

which can be solved numerically. Nonetheless, at two limits of temperatures (.T → 0 and .T → ∞), analytical forms of the specific heat can be obtained. At low temperatures where we have quartic temperature dependence in energy, we obtain the famous 3 x .C V (T → 0) ∼ T relations. In the other high temperature limit,.e ∼ 1 + x such that .C V (T → ∞) = 3N k B , also known as the Dulong-Petit law of specific heat. In contrast to the Debye model where a linear dispersion was assumed, Einstein postulated that atomic vibrations of a solid were harmonic oscillators, all vibrating at the same frequency, now known as Einstein frequency, .ω E [2]. This means the Einstein model for the phonon density of states is given by

2.3 Thermal Properties of Solids from Normal Modes

23

Fig. 2.7 Specific heat of germanium. Circles and cross experimental data are from Refs. [33, 34]. DFT, Debye, and Einstein model predictions of specific heat of germanium are based on the densities of states shown in Fig. 2.6 and are depicted as solid black, dashed blue, and dotted red curves, respectively

.

g(ω) = 3N δ(ω − ω E )

(2.28)

as demonstrated as a delta function for germanium in Fig. 2.6. The total energy of harmonic oscillators is then { ∞ ( 1) δ(ω − ω E )dω . E = 3N hω n(ω, T ) + (2.29) 2 0 Taking the temperature derivative, the Einstein model for heat capacity is ( C V = 3N k B

.

hω E kB T

)2

hω E

e kB T

)2 ( hω E e kB T − 1

(2.30)

Einstein frequency, .ω E , can be obtained by fitting Eq. 2.30 to available specific heat −

hω E

data. At low temperatures, Eq. 2.30 scales as .C V ∼ e k B T . At high temperatures, we can also assume here .e x ∼ 1 + x such that .C V (T → ∞) = 3N k B . Specific heat predictions of germanium from DFT and Debye and Einstein models are plotted against experimental measurements in Fig. 2.7. Despite the simplicity of Debye and Einstein models and their inaccuracy in describing the actual density of states, remarkable agreements are demonstrated throughout all temperatures. At high temperatures, all models reach the Dulong-Petit limit as mentioned previously. At low temperatures, a better agreement between the Debye model and experiments in comparison to the Einstein model is observed. When plotted in the log-log scale (not shown), measurements follow .C V ∼ T 3 at low temperatures (excluding the electronic contribution) as predicted by the Debye model. The theories outlined above for harmonic crystals is applicable under constant volume and harmonic models and cannot describe thermal expansion. However, nearly all experiments are performed under constant pressure rather than constant volume and a simple conversion to the heat capacity at constant volume, .C P , can be carried out by

24

2 Normal Mode Analysis of Atomic Motion in Solids

C P = CV +

.

T V β2 KT

(2.31)

where .β is the thermal expansion coefficient, .β = V1 ( ∂V ) and . K T is the isothermal ∂T P ) . . C and . C are typically nearly the same at low compressibility, . K T = − V1 ( ∂V V P ∂T T temperatures but can differ noticeably at high temperatures near melting.

2.3.1.2

Some Comments About .C V = 3N k B at High Temperatures

High temperature limit of .C V = 3N k B found both in the Debye and Einstein models is a general result of . N classical harmonic oscillators. Each classical harmonic oscillator has the energy of . E i = E i,K E + E i,P E = 21 m i v i2 + 21 κx i2 and has three degrees of freedom. From statistical mechanics, each degree of freedom of one harmonic oscillator contributes .k B T to energy (. 21 k B T each from kinetic and potential component of energy), leading to the total energy of .3N k B T . Kinetic and potential component of energy equally contributing to energy is called equipartition theorem and derives from having the same quadratic form in energy (. E K E ∼ v 2 and 2 . E P E ∼ x ). Constant volume specific heat is subsequently .3N k B . It is important to use caution when using the result of .C V = 3N k B for harmonic oscillators in the high temperature limit. In the high temperature limit calculations (e.g. thermal conductivity) that utilize the heat capacity of solids, it is often assumed that the constant volume heat capacity is equal to .3N k B in the literature. However, at very high temperatures, atomic vibration amplitudes can be large enough that atoms can experience higher order potential surfaces even in the constant volume considerations, leading to deviations away from .3N k B in which case vibrations are not purely harmonic. Assumption of .3N k B instead of the actual heat capacity leads to difficulties in elucidating mechanisms behind certain features in materials properties convolving heat capacities such as thermal conductivity. Therefore, care must be taken to verify that constant volume heat capacity indeed follows the harmonic oscillator model and .3N k B at high temperatures.

2.3.2 Thermal Conductivity Another thermal property of crystals that can be described in terms of normal modes is thermal conductivity. Generally, there are several types of heat carriers to consider including electrons and magnons. For dielectric crystals, thermal conductivity is typically dominated by phonons whereas electrons are the major contributors to the total thermal conductivity in metallic systems. Here, we consider phononic contribution to thermal conductivity only. Using simple kinetic theory, we first derive below Fourier’s law relating thermal energy current . J and thermal conductivity .κ.

2.3 Thermal Properties of Solids from Normal Modes

25

Fig. 2.8 Simple derivation of Fourier’s law using kinetic theory arguments in one dimension. Heat carriers are represented by grey circles

2.3.2.1

Fourier’s Law

For simplicity, we only consider one dimension without loss of generality as shown in Fig. 2.8 where a small overall heat current, . Jx , flows from hot to cold. If we take an imaginary surface perpendicular to the overall heat flow direction at .x, the net heat current across this surface is the difference between the energy flow associated with all the heat carriers flowing in the upstream and downstream. Heat carriers within a distance .vx τ can go across the imaginary surface before being scattered. Here .vx is the .x component of the random velocity of the random velocity of the heat carriers and .τ is the relaxation time. .vx τ is then the distance a heat carrier travels before it is scattered and changes its direction called mean free path, denoted by .λ. The net heat current carried by heat carriers across the surface is then J =

. x

| | 1 1 (n Evx )|x−vx τ − (n Evx )|x+vx τ 2 2

(2.32)

where .n is the number of carriers per unit volume and . E is the energy carried. The factor of .1/2 implies that only half of the heat carriers move in the positive (downstream) direction while the other half move in the negative (upstream) direction in the presence of small overall heat flow. Using Taylor expansion about .x up to the first order, we can write the above relation as | | 1 1 (dn Evx ) 1 1 (dn Evx ) (n Evx )|x + (−vx τ ) − (n Evx )|x − (vx τ ) 2 2 dx 2 2 dx (2.33) (n Evx ) = −vx τ dx

Jx = .

Assuming .vx is independent of .x and isotropic such that .vx2 = (1/3)v 2 , where .v is the average random velocity, the above equation becomes 1 dU dT . J = − v2 τ 3 dT d x

. x

(2.34)

26

2 Normal Mode Analysis of Atomic Motion in Solids

Here .U = n E is the local energy density per unit volume. By introducing the heat capacity concept, we obtain the familiar Fourier’s law. dT 1 dT = −κ J = − (Cv 2 τ ) 3 dx dx

. x

(2.35)

Generally, thermal conductivity can vary spatially and temporally in a material and is a tensor quantity. In a more microscopic view, each normal mode (phonon) with wavevector . k and on branch .ν carries a portion of the heat current, and overall thermal conductivity can be considered as a summation of all contributions from each normal mode as κ =

∑

. α

k,ν

κk,ν,α =

1 ∑ C V (k, ν)vα (k, ν)2 τα (k, ν) V

(2.36)

k,ν

where .V is the crystal volume, .C V (k, ν) is the modal heat capacity as described in Sect. 2.3.1, .vα (k, ν) = dωk,ν /dkα is the group velocity in .α direction and .τα (k, ν) is the lifetime. As discussed previously, the modal specific heat and group velocity can be found from harmonic lattice dynamics calculations. The real challenge in calculating thermal conductivity is accurately determining the phonon lifetimes. In perfectly harmonic solids, there is no loss such that lifetimes are considered to be infinite. In real solids at finite temperatures, atomic motion is not purely harmonic: Bottom of the potential can no longer be approximated as quadratic in spatial coordinates as thermal energy enables atoms to explore more of the potential energy surface. Anharmonicity leads to phonons interacting and having finite lifetimes. All the ingredients for thermal conductivity calculations at the mode level then require information about the interatomic interactions whether it is harmonic or anharmonic. For more accurate calculations of the interatomic potential and force constants, density functional theory coupled with perturbation theory is typically utilized at the expense of high computational costs. On the other hand, for materials with low symmetry and large unit cells where density functional theory becomes difficult to utilize, empirical methods such as molecular dynamics can complement density functional theory in qualitatively characterizing the modal properties needed for thermal conductivity predictions. We next discuss the formalisms of obtaining lifetimes from perturbation theory and molecular dynamics. Then, we make comparisons between lifetimes obtained from both formalisms for germanium as an example utilizing the same empirical interatomic potential (Tersoff [35]). Finally, we consider temperature dependent thermal conductivity predictions utilizing these lifetimes.

2.3 Thermal Properties of Solids from Normal Modes

2.3.2.2

27

Phonon Lifetime from Perturbation Theory

We focus our phonon lifetime using perturbation theory discussions on the lowest order anharmonic term (.V3 , the 3rd order term in the Taylor expansion of the potential) as it is sufficiently accurate around room temperature for many dielectric materials and is given by .

V3 =

1 ∑ ∑ ∑ ∑ αβγ β γ ϕ j, p; j ' , p' ; j '' , p'' u αj, p u j ' , p' u j '' , p'' . 3! j, p j ' , p' j '' , p''

(2.37)

αβγ

αβγ

β

γ

where .ϕ j, p; j ' , p' ; j '' , p'' = ∂ 3 U j, p; j ' , p' ; j '' , p'' /∂u αj, p ∂u j ' , p' ∂u j '' , p'' are third derivatives of the interatomic potential at equilibrium. This third order potential interaction gives rise to three-phonon interactions. Recently, efforts have been made to consider up to 4th order anharmonic interactions (4-phonon processes) [36, 37] and it was found that higher order terms beyond the third order are needed for more accurate lifetime calculations, especially at high temperatures. Differences in thermal conductivity predictions using up to 3rd order or 4th order anharmonic interactions will be briefly discussed in Sect. 2.3.2.4. The intrinsic phonon lifetimes are given as the sum of all individual scattering probabilities .W = (2π/h)|⟨ f |V3 |i⟩|2 δ(E f − E i ) determined from Fermi’s golden rule with the lowest-order anharmonic potential as the perturbation as .

∑∑( ) 1 1 decay coalesce = Wk,ν;k' ,ν ' ;k'' ,ν '' Wk,ν;k ' ' '' '' + ,ν ;k ,ν τ (k, ν) 2 ' ' '' ''

(2.38)

k ,ν k ,ν

.

[ ' ' ]] |ψ−k,ν;−k' ,ν ' ;k'' ,ν '' |2 πh n(k , ν ) − n(k'' , ν '' ) 4N ω(k, ν)ω(k' , ν ' )ω(k'' , ν '' ) (2.39) ( [ ' '' ' '' ' '' δ ω(k, ν) + ω(k , ν ) − ω(k , ν )]Δ k + k − (k + G))

coalesce Wk,ν;k ' ' '' '' = ,ν ;k ,ν

decay

.

ψk,ν;k' ,ν ' ;k'' ,ν '' .

[ ' ' ]] |ψ−k,ν;k' ,ν ' ;k'' ,ν '' |2 πh n(k , ν ) + n(k'' , ν '' ) + 1 ' '' ' '' 4N ω(k, ν)ω(k , ν )ω(k , ν ) ( [ δ ω(k, ν) − ω(k' , ν ' ) − ω(k'' , ν '' )]Δ k − k' − (k'' + G)) (2.40) ∑ ∑ ∑ ∑ ϕαβγ ' ' '' '' j, p; j , p ; j , p β γ = ∈(k, ν)αj ∈(k, ν) j ' ∈(k, ν) j '' √ ' m j '' m m j j (2.41) j j ' , p' j '' , p'' αβγ

Wk,ν;k' ,ν ' ;k'' ,ν '' =

'

ei k ·r j ' , p' ei k

''

·r j '' , p''

for three phonons that conserve energy and momentum within a reciprocal lattice vector . G: ' '' ' '' .ω(k, ν) ± ω(k , ν ) = ω(k , ν ) (2.42) .

k ± k' = k'' + G

(2.43)

28

2 Normal Mode Analysis of Atomic Motion in Solids

(a)

(b)

Fig. 2.9 3-phonon anharmonic interactions leading to finite lifetimes

where .± corresponds to coalescence and decay processes, respectively, as shown in Fig. 2.9. Extrinsic factors such as defects and electrons can also scatter phonons and give finite phonon lifetimes and should be included in the calculation of a mode’s lifetime. One widely used way to combine different scattering mechanisms is to assume that all scattering mechanisms, both intrinsic and extrinsic, are independent processes. Overall phonon lifetime is then given by the Matthiessen rule as .

∑ 1 1 = τ (k, ν) τ (k, ν) i i

(2.44)

where subscript .i refers to a scattering mechanism. Characterization of phonon lifetimes due to extrinsic factors is discussed in detail in Ref. [38]. Anharmonic terms (3rd order and higher orders) in the potential can be often accurately described using the density functional theory but can also be determined from empirical potentials.

2.3.2.3

Phonon Lifetime from Molecular Dynamics

One major difference between density functional theory and molecular dynamics is in describing interatomic interactions. Molecular dynamics is a classical technique as it uses the Newtonian equations of motion to describe the atomic dynamics. Normal modes in an MD simulation, therefore, follow Boltzmann statistics, the high-temperature limit of Bose–Einstein statistics. Quantum corrections are often applied to thermal conductivity predictions from molecular dynamics when comparing directly with measurements at low temperatures where consideration of Bose– Einstein statistics is important. In contrast to anharmonic lattice dynamics calculations mentioned in Sect. 2.3.2.2, MD simulations intrinsically include the full anharmonicity in the interatomic potential and all orders of phonon-phonon interactions are included in phonon lifetime calculations. The Hamiltonian of a system of harmonic oscillators can be expressed equivalently in both real space and normal mode space as follows

2.3 Thermal Properties of Solids from Normal Modes

.

H=

1∑ 1 ∑ T m j |u˙ j p (t)|2 + u j p (t) · ϕ j p, j ' p' · u j ' p' (t) 2 jp 2 ' '

29

(2.45)

j p; j p

and .

H=

1∑ ˙ 1∑ 2 ˙ ω (k, ν)Q(k, ν, t)Q(−k, ν, t) (2.46) Q(k, ν, t) Q(−k, ν, t) + 2 2 k,ν

k,ν

where a dot over a variable represents a time derivative, superscript .T represent a transpose. . Q(k, ν, t) often referred to as a normal mode coordinate is the Fourier transform of the .u j p (t) and is given by .

Q(k, ν, t) =

∑

1

m j2 e−i k·r j p e∗ ( j, k, ν) · u j p (t)

(2.47)

jp

The first term and the second term in the above Hamiltonian expressions are kinetic energy and potential energy, respectively. The instantaneous, total energy of each mode of a classical system is then .

Hk,ν =

1 ˙ 1 ˙ Q(k, ν, t) Q(−k, ν, t) + ω 2 (k, ν)Q(k, ν, t)Q(−k, ν, t) 2 2

(2.48)

To account for the mode interactions, displacements from molecular dynamics (MD) at a desired finite temperature which account for all degrees of anharmonicities are used instead and projected to the normal mode coordinates. The temporal decay of the autocorrelation of . Hk,ν is then related to the relaxation time of each mode by [39, 40] Hk,ν (t)Hk,ν (0) − t = e τk,ν . (2.49) Hk,ν (0)Hk,ν (0) A typical normalized total energy autocorrelation for a mode is shown in Fig. 2.10 [8, 40]. As expected, we see an exponential decay in the normalized energy autocorrelation and the lifetime can be extracted as discussed. Normalized potential energy autocorrelation is also plotted in the same figure. The vibration frequency of the mode is half of the oscillation frequency in the potential energy autocorrelation. The normal mode lifetime calculations can also be done in frequency space in which we see a Lorentzian function that determines the mode frequency (peak location) and mode lifetime (inverse of full width at half maximum).

2.3.2.4

Phonon Lifetime and Thermal Conductivity Prediction Example: Germanium

Phonon lifetime comparisons between the perturbation theory and molecular dynamics for certain wavevector directions (.[ to X) for Germanium at 800 K is shown in

30

2 Normal Mode Analysis of Atomic Motion in Solids

Fig. 2.10 A typical normalized energy autocorrelation for the relaxation time calculations as adapted from [40]. Blue and orange solid curves correspond to normalized total energy and potential energy autocorrelations of a mode, respectively. We see an exponential decay in the total energy autocorrelation as expected. The vibration frequency of the mode is one half of the oscillation frequency observed in the potential energy autocorrelations

Fig. 2.11 [36]. For all lifetime calculations, same Tersoff potential [35] was utilized. Up to 4-phonon processes were considered. For longitudinal and transverse branches (LA and TA), we observe good agreements between lifetime predictions using both methods as shown in Fig. 2.11a. There are some small differences in the lifetimes from these two methods for the optical branches (LO and TO) in Fig. 2.11b and consideration of even higher order terms such as the 5th order may be needed. However, group velocities for these optical branches are usually much smaller than acoustic branches such that small differences in optical phonon lifetime predictions do not lead to large errors in overall thermal conductivity predictions. To calculate thermal conductivity, Eq. 2.36 is used with the lifetimes from the perturbation theory and molecular dynamics, known as the relaxation time approximation (RTA). The main simplifications made by the relaxation time approximation are that all scatterings destroy phonon momentum and relax the phonon mode into equilibrium Bose–Einstein distribution, and the process is independent of the distribution of other modes. More precisely, the deviation from equilibrium from the temperature gradient of the mode at . k and .ν does not depend on the distributions of the other phonon modes that it interacts with. Therefore, equilibrium Bose–Einstein distributions are given for phonons with . k' and .ν ' and . k'' and .ν '' rather than the actual non-equilibrium distributions that arise from a temperature gradient. For many materials including germanium, this approximation is valid; however, for some materials including high thermal conductivity materials, this approximation leads to inaccu-

2.3 Thermal Properties of Solids from Normal Modes

(a)

31

(b)

Fig. 2.11 Phonon lifetime comparisons between perturbation theory (solid curves) and molecular dynamics (circles) formalisms described in this Chapter for Germanium at 800 K (adapted from Ref. [36]). Perturbation theory here considers up to 4th order term in the potential. Tersoff interatomic potential [35] is used. Acoustic and optical branch lifetimes along . k = [ to . X are plotted in (a) and (b), respectively Fig. 2.12 Thermal conductivity predictions of germanium from 3-phonon (blue dashed curve), 3 and 4-phonon (black solid line), and normal mode lifetimes (purple crosses) from molecular dynamics (NM-MD) compared against Green–Kubo (yellow circles) thermal conductivity values (GK-MD) (adapted from Ref. [36])

rate thermal conductivity. The failure of RTA is primarily due to the existence of significant momentum-conserving phonon scattering that causes the drift motion of phonons [41–43]. This phenomenon, called phonon hydrodynamics, is similar to liquid flow and has been predicted and observed in ultrahigh thermal conductivity materials including graphitic materials [44–46]. Resulting thermal conductivity predictions from the lifetimes obtained from both perturbation theory and molecular dynamics (NM-MD) are compared against thermal conductivity from Green–Kubo formalism (GK-MD) which does not require phonon description of atomic motion and are depicted in Fig. 2.12. Good agreement between perturbation theory (up to 4th order), NM-MD, and GK-MD is demonstrated. Differences between 3-phonon and 3,4-phonon thermal conductivity highlight the importance of considering higher order terms for more accurate thermal conductivity predictions. However, at lower temperatures the differences are typically smaller and at room temperature, difference is less than .∼5% [36].

32

2 Normal Mode Analysis of Atomic Motion in Solids

2.4 Concluding Remarks As demonstrated by heat capacity and thermal conductivity discussions, normal mode description of atomic motion in solids has been very successful in describing thermodynamics and various thermal properties of solids. Encouraged by the success, many efforts have been made in recent years to extend the normal mode analysis to liquid systems. Phenomenology of normal mode behaviors and its applications in liquids will be described in Chap. 4.

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14. Agne, M.T., Hanus, R., Snyder, G.J.: Minimum thermal conductivity in the context of diffuson -mediated thermal transport. Energy & Environ. Sci. 11(3), 609–616 (2018). https://doi.org/ 10.1039/C7EE03256K. http://xlink.rsc.org/?DOI=C7EE03256K 15. Zhou, Y., Morshedifard, A., Lee, J., Abdolhosseini Qomi, M.J.: The contribution of propagons and diffusons in heat transport through calcium-silicate-hydrates. Appl. Phys. Lett. 110(4), 043,104 (2017). https://doi.org/10.1063/1.4975159. http://aip.scitation.org/doi/ 10.1063/1.4975159 16. Larkin, J.M., McGaughey, A.J.H.: Thermal conductivity accumulation in amorphous silica and amorphous silicon. Phys. Rev. B 89(14), 144,303 (2014). https://doi.org/10.1103/PhysRevB. 89.144303. http://link.aps.org/doi/10.1103/PhysRevB.89.144303 17. Fabian, J., Allen, P.B.: Anharmonic decay of vibrational states in amorphous silicon. Phys. Rev. Lett. 77(18), 3839 (1996). http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.77. 3839 18. Moon, J., Minnich, A.J.: Sub-amorphous thermal conductivity in amorphous heterogeneous nanocomposites. RSC Adv. 6(107), 105,154–105,160 (2016). https://doi.org/10.1039/ C6RA24053D. https://doi.org/10.1039/C6RA24053D 19. DeAngelis, F., Muraleedharan, M.G., Moon, J., Seyf, H.R., Minnich, A.J., McGaughey, A.J.H., Henry, A.: Thermal transport in disordered materials. Nanoscale Microscale Thermophys. Eng. (2018). https://doi.org/10.1080/15567265.2018.1519004 20. Stillinger, F.H., Weber, T.A.: Computer simulation of local order in condensed phases of silicon. Phys. Rev. B 31(8), 5262 (1985). https://doi.org/10.1103/PhysRevB.31.5262 21. Zhu, T., Ertekin, E.: Generalized Debye-Peierls/Allen-Feldman model for the lattice thermal conductivity of low-dimensional and disordered materials. Phys. Rev. B 93(15) (2016). https://doi.org/10.1103/PhysRevB.93.155414. https://link.aps.org/doi/10.1103/PhysRevB.93. 155414 22. Seyf, H.R., Henry, A.: A method for distinguishing between propagons, diffusions, and locons. J. Appl. Phys. 120(2), 025101 (2016). https://doi.org/10.1063/1.4955420. http://scitation.aip. org/content/aip/journal/jap/120/2/10.1063/1.4955420 23. Bell, R.J., Dean, P.: Atomic vibrations in vitreous silica. Discuss. Faraday Soc. 50, 55–61 (1970). http://pubs.rsc.org/en/content/articlehtml/1970/df/df9705000055 24. Lindsay, L., Broido, D.A., Reinecke, T.L.: Phonon-isotope scattering and thermal conductivity in materials with a large isotope effect: A first-principles study. Phys. Rev. B 88(14), 144,306 (2013). https://doi.org/10.1103/PhysRevB.88.144306. https://link.aps.org/ doi/10.1103/PhysRevB.88.144306. Publisher: American Physical Society 25. Carlson, D.E., Wronski, C.R.: Amorphous silicon solar cell. Appl. Phys. Lett. 28(11), 671–673 (1976). https://doi.org/10.1063/1.88617. https://aip.scitation.org/doi/abs/10.1063/1.88617 26. Birney, R., Steinlechner, J., Tornasi, Z., MacFoy, S., Vine, D., Bell, A., Gibson, D., Hough, J., Rowan, S., Sortais, P., Sproules, S., Tait, S., Martin, I., Reid, S.: Amorphous silicon with extremely low absorption: beating thermal noise in gravitational astronomy. Phys. Rev. Lett. 121(19), 191,101 (2018). https://doi.org/10.1103/PhysRevLett.121.191101. https://link.aps. org/doi/10.1103/PhysRevLett.121.191101 27. Lv, W., Henry, A.: Non-negligible contributions to thermal conductivity from localized modes in amorphous silicon dioxide. Sci. Rep. 6, 35,720 (2016). https://doi.org/10.1038/srep35720. https://www.nature.com/articles/srep35720 28. Moon, J.: Thermal conduction in amorphous materials and the role of collective excitations. Ph.D. California Institute of Technology (2020). https://doi.org/10.7907/Z23D-Z566. https:// resolver.caltech.edu/CaltechTHESIS:01162020-015608435 29. Wei, S., Chou, M.Y.: Phonon dispersions of silicon and germanium from first-principles calculations. Phys. Rev. B 50(4) (1994) 30. Kim, T., Moon, J., Minnich, A.J.: Origin of micrometer-scale propagation lengths of heat-carrying acoustic excitations in amorphous silicon. Phys. Rev. Mat. 5(6), 065,602 (2021). https://doi.org/10.1103/PhysRevMaterials.5.065602. https://link.aps.org/doi/10.1103/ PhysRevMaterials.5.065602. Publisher: American Physical Society

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31. Li, X., Lee, S.: Crossover of ballistic, hydrodynamic, and diffusive phonon transport in suspended graphene. Phys. Rev. B 99(8), 085,202 (2019). https://doi.org/10.1103/PhysRevB.99. 085202. https://link.aps.org/doi/10.1103/PhysRevB.99.085202. Publisher: American Physical Society 32. Pocs, C.A., Leahy, I.A., Zheng, H., Cao, G., Choi, E.S., Do, S.H., Choi, K.Y., Normand, B., Lee, M.: Giant thermal magnetoconductivity in CrCl.3 and a general model for spin-phonon scattering. Phys. Rev. Res. 2(1), 013,059 (2020). https://doi.org/10.1103/PhysRevResearch. 2.013059. https://link.aps.org/doi/10.1103/PhysRevResearch.2.013059. Publisher: American Physical Society 33. Berger, L.I.: Semiconductor Materials. CRC Press (1997) 34. Lide, D.R.: CRC Handbook of Chemistry and Physics, vol. 85. CRC Press (2004) 35. Tersoff, J.: Modeling solid-state chemistry: interatomic potentials for multicomponent systems. Phys. Rev. B 39(8), 5566–5568 (1989). https://doi.org/10.1103/PhysRevB.39.5566. https:// link.aps.org/doi/10.1103/PhysRevB.39.5566 36. Feng, T., Ruan, X.: Quantum mechanical prediction of four-phonon scattering rates and reduced thermal conductivity of solids. Phys. Rev. B 93(4), 045,202 (2016). https://doi.org/ 10.1103/PhysRevB.93.045202. https://link.aps.org/doi/10.1103/PhysRevB.93.045202. Publisher: American Physical Society 37. Ravichandran, N.K., Broido, D.: Phonon-phonon interactions in strongly bonded solids: selection rules and higher-order processes. Phys. Rev. X 10(2) (2020). https://doi.org/10.1103/ PhysRevX.10.021063. https://link.aps.org/doi/10.1103/PhysRevX.10.021063 38. Hanus, R., Gurunathan, R., Lindsay, L., Agne, M.T., Shi, J., Graham, S., Snyder, G.J.: Thermal transport in defective and disordered materials. Appl. Phys. Rev. 8(3), 031,311 (2021). https:// doi.org/10.1063/5.0055593 39. Ladd, A.J.C., Moran, B., Hoover, W.G.: Lattice thermal conductivity: a comparison of molecular dynamics and anharmonic lattice dynamics. Phys. Rev. B 34(8), 5058–5064 (1986). https:// doi.org/10.1103/PhysRevB.34.5058. https://link.aps.org/doi/10.1103/PhysRevB.34.5058 40. McGaughey, A., Kaviany, M.: Advances in Heat Transfer, vol. 39, pp. 169–255. Elsevier (2006) 41. Lee, S., Broido, D., Esfarjani, K., Chen, G.: Hydrodynamic phonon transport in suspended graphene. Nat. Commun. 6(1), 6290 (2015). https://doi.org/10.1038/ncomms7290. https:// www.nature.com/articles/ncomms7290. Number: 1 Publisher: Nature Publishing Group 42. Cepellotti, A., Fugallo, G., Paulatto, L., Lazzeri, M., Mauri, F., Marzari, N.: Phonon hydrodynamics in two-dimensional materials. Nat. Commun. 6(1), 6400 (2015). https://doi.org/ 10.1038/ncomms7400. https://www.nature.com/articles/ncomms7400. Number: 1 Publisher: Nature Publishing Group 43. Lee, S., Li, X.: Nanoscale Energy Transport: Emerging Phenomena, Methods and Applications. IOP Publishing (2020). https://doi.org/10.1088/978-0-7503-1738-2ch1. https://iopscience. iop.org/book/edit/978-0-7503-1738-2/chapter/bk978-0-7503-1738-2ch1 44. Huberman, S., Duncan, R.A., Chen, K., Song, B., Chiloyan, V., Ding, Z., Maznev, A.A., Chen, G., Nelson, K.A.: Observation of second sound in graphite at temperatures above 100 K. Science 364(6438), 375–379 (2019). https://doi.org/10.1126/science.aav3548. https:// www.science.org/doi/full/10.1126/science.aav3548. Publisher: American Association for the Advancement of Science 45. Jeong, J., Li, X., Lee, S., Shi, L., Wang, Y.: Transient Hydrodynamic Lattice Cooling by Picosecond Laser Irradiation of Graphite. Phys. Rev. Lett. 127(8), 085,901 (2021). https://doi.org/10.1103/PhysRevLett.127.085901. https://link.aps.org/doi/10.1103/ PhysRevLett.127.085901. Publisher: American Physical Society 46. Machida, Y., Matsumoto, N., Isono, T., Behnia, K.: Phonon hydrodynamics and ultrahigh-roomtemperature thermal conductivity in thin graphite. Science 367(6475), 309–312 (2020). https:// doi.org/10.1126/science.aaz8043. https://www.science.org/doi/full/10.1126/science.aaz8043. Publisher: American Association for the Advancement of Science

Chapter 3

Time Correlations and Their Descriptions of Materials Properties

Abstract Time correlation functions of dynamical variables have been used ubiquitously to characterize various materials properties. In this chapter, we begin by discussing some general properties of time correlation functions, followed by characterizing thermal conductivity and viscosity using heat current and stress correlations, respectively, as some examples. We then move onto velocity autocorrelation functions that characterize atomic degrees of freedom, similar to normal modes presented in Chap. 2. Keywords Time correlation functions · Thermal conductivity · Viscosity · Velocity autocorrelation functions · Molecular dynamics

3.1 Introduction In the previous chapter, characterization of atomic degrees of freedom in solids via normal modes was discussed. There exists another theoretical tool that leads to .3N degrees of freedom when integrated over frequency called velocity autocorrelation spectra, VACF(.ω). Unlike normal modes, however, VACF(.ω) are used ubiquitously in both solids and non-solids to characterize various aspects of atomic motion in different phases of matter such as atomic diffusion and vibration. In this chapter, we first go over some general properties of correlation functions. Correlation functions are useful in describing various physical properties of materials as how strong a dynamical variable can stay correlated represents an intrinsic property of materials. Examples of thermal conductivity and viscosity expressions in terms of correlation functions are provided. Finally, the main topic of this chapter, velocity autocorrelation function spectra, is discussed.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. Moon, Heat Carriers in Liquids: An Introduction, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-51109-7_3

35

36

3 Time Correlations and Their Descriptions of Materials Properties

3.2 Time Evolution in Classical Mechanics Consider an isolated system consisting of . N identical, spherical particles with mass m inside a container with volume.V . In classical mechanics, state of the system at any instant is specified by coordinates . r N ≡ r 1 , ..., r N and momenta . p N ≡ p1 , ..., p N . Each (. r N , p N ) is referred to as a phase point. Hamiltonian of the entire system without the influence of external fields can be then written as

.

.

H (r N , p N ) = K E( p N ) + P E(r N )

(3.1)

where the first and second term on the right hand side are kinetic energy and potential energy, respectively. The motion of the phase point along its phase trajectory is determined by . r˙ i = ∂∂ Hp and . p˙ i = − ∂∂ rHi and can be solved from initial conditions i of . r N and . p N . This means that the trajectory of a phase point is determined by the values of . r N and . p N at any given time and that two different trajectories cannot pass through the same phase point. The distribution of phase points of systems is described by a phase-space probability density . f N (r N , p N ; t); therefore, . f N (r N , p N ; t)d r N d p N describes the probability that the system is in a microscopic state within an infinitesimal phase space element .d r N d p N . This notion implies the normalization: { { .

f N (r N , p N ; t)d r N d p N = 1.

(3.2)

The time evolution of . f N (r N , p N ; t) is governed by the Liouville equation, describing that phase points are neither created or annihilated with time. The Liouville equation can be written as .

∂fN = {H, f N } ∂t

(3.3)

where {A, B} represents the Poisson bracket N ( ∑ ∂A

{A, B} ≡

.

i

∂B ∂A ∂B · − · ∂ r i ∂ pi ∂ pi ∂ r i

) (3.4)

The Liouville equation can be re-written in terms of Liouville operator, .L ≡ i{H, } as ∂fN = −iL f N . . (3.5) ∂t The solution of the Liouville equation in terms of .L is then .

f N (t) = e−iLt f N (0)

(3.6)

3.3 Time Correlation Functions and Their Properties

37

Time dependence of any function of the phase-space variables, . A(r N , p N ) can be similarly represented in terms of the Liouville operator. The time derivative of . A is . ddtA = iLA with the solution of . A(t) = eiLt A(0). Note the sign reversal compared to Eq. 3.6. With this background information, we next discuss time dependent correlation functions.

3.3 Time Correlation Functions and Their Properties . A and . B. Their time-correlation function, Consider two ⟨ complex variables, ⟩ C A,B (t ' , t '' ) = A(t ' )B ∗ (t '' ) , can be written as

.

⟨ .

⟩

1 A(t )B (t ) = lim τ →∞ τ '

∗

''

{

τ

A(t + t ' )B ∗ (t + t '' )dt

(3.7)

0

with the convention that .t ' ≥ t '' . In equilibrium, the ensemble average ⟨ .⟨...⟩ is inde⟩ pendent of the choice of time origin and the correlation function, . A(t ' )B ∗ (t '' ) is invariant under time translation. Therefore, the time correlation function can be sim⟨ ⟩ ply denoted as .C A,B = A(t)B ∗ (0) . Time correlation functions physically describe how variables co-vary with one another on average in time. These functions are particularly useful in studying materials properties in equilibrium as how thermal fluctuations dissipate in space and time (i.e, how dynamical variables lose correlations) is an intrinsic response of materials. For example, transport coefficients such as thermal conductivity and self-diffusion coefficients can be expressed in terms of time correlation functions. Due to time translation invariance, we have the following useful relations: ⟨ .

⟩ ⟨ ⟩ ∗ ˙ A(t)B (0) = − A(t) B˙ ∗ (0) ⟨

⟩ ˙ ∗ (0) = 0 A(t)A

(3.9)

⟩ ⟩ ⟨ d2 ⟨ ˙ B˙ ∗ . A(t)B ∗ (0) = − A(t) 2 dt

(3.10)

.

.

(3.8)

Further, in the limit of .t → ∞, the dynamic variables, . A(t) and . B(t), become uncorrelated such that ∗ . lim C A,B (t) = ⟨A⟩⟨B ⟩. (3.11) t→∞

It is often more convenient to only study the time correlation of dynamic variables fluctuations such that ⟨ ⟩ C A,B (t) = [A(t) − ⟨A⟩][B ∗ (0) − ⟨B ∗ ⟩] .

.

In this convention, .limt→∞ C A,B (t) = 0.

(3.12)

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3 Time Correlations and Their Descriptions of Materials Properties

3.4 Time Correlation Functions and Macroscopic Materials Properties 3.4.1 Thermal Conductivity Thermal conductivity of materials in any phases under equilibrium conditions can be determined from heat current autocorrelation functions (HCACF) by k

. αβ

=

1 V kB T 2

{

∞

⟨Jα (t)Jβ (t)⟩dt

(3.13)

0

where Greek subscripts denote Cartesian directions,.V is system volume,.k B is Boltzmann constant, .T is temperature, and . J is heat current. Heat current is related to total energy as d ∑ . J(t) = E j (t)r j (t) (3.14) dt j where subscript . j refers to an individual atom index and . r is the atomic position. Derivations of Eq. 3.13 are discussed in the Appendix of this chapter. This description of thermal conductivity is referred to as Green–Kubo formalism [1]. To calculate the time dependent energies and positions of atoms, classical or ab-initio molecular dynamics is typically utilized. This means that all degrees of anharmonicity is naturally included in Eq. 3.13. Physically, if heat currents correlate for a long time (long-lived), we expect to see a large thermal conductivity and vice versa. For simple crystalline solids, Eq. 3.13 here and Eq. 2.35 from Chap. 2 yield the same thermal conductivity values. The main difference between the two thermal conductivity relations lies in the underlying interpretation of effective heat carriers in crystals. In Eqs. 3.13–3.14, it is shown that thermal conductivity is directly expressed through fluctuations of atomic energies and positions; therefore, thermal conductivity can be calculated for all phases using these equations. However, in Eq. 2.35 of Chap. 2, atomic motion was assumed to be vibrational in nature and thermal conductivity was explained microscopically through vibrational modes. An example of HCACF ensemble average and corresponding averaged thermal conductivity accumulation for crystalline silicon at 300 K using Stillinger–Weber (SW) potential [2] is demonstrated in Fig. 3.1. Resulting thermal conductivity is 285 .± 50 Wm–1 K–1 in agreement with prior works utilizing the same interatomic potential [3]. There are three main considerations in using Eq. 3.13 for thermal conductivity predictions: 1. choice of interatomic potentials, 2. quantum effects, and 3. size effects. Details of the interatomic potential are crucial in describing time dependent atomic energies and positions; therefore, thermal conductivity can also vary significantly depending on the interatomic potential used. Utilizing Tersoff potential [4], thermal conductivity of crystalline silicon at 300 K has been shown to be 120 .± 35 Wm–1 K–1 [5], significantly different from values described above using the SW potential.

3.4 Time Correlation Functions and Macroscopic Materials Properties

(a)

39

(b)

Fig. 3.1 a Heat current autocorrelation function (integrand of Eq. 3.13) and b cumulative thermal conductivity accumulation function of crystalline silicon at 300 K using Stillinger–Weber potential [2]

Therefore, it is important to discern which interatomic potential is more applicable to the materials properties of interest as the interatomic potential is obtained by fitting to some specific properties. Despite the differences in thermal conductivity predictions, one can still gain both qualitative and quantitative insights into physics of thermal transport. Compared to crystalline silicon, amorphous silicon exhibits two orders of magnitude lower thermal conductivity at .∼1.5 and 2.3 Wm–1 K–1 at room temperature for SW and Tersoff potentials, respectively, signifying the role of structural disorder in thermal conductivity [6, 7]. For more accurate description of interatomic interactions, recent works have utilized machine-learned interatomic potentials, taking into account many physical properties of materials [8, 9]. As discussed in Chap. 2, molecular dynamics simulations assume Newtonian classical mechanics, leading to Maxwell statistics. This means that specific heat is .k B for a harmonic mode independent of temperature. Consequently, only high temperature (above Debye temperature) thermal conductivity comparisons between molecular dynamics predictions using Eq. 3.13 and experiments are meaningful if no quantum correction is made. By decomposition of heat currents into individual normal mode contributions, prior works have utilized a quantum correction to modal specific heat [5] as x 2 ex C V,Q (ω) = x . (3.15) C V,C (ω) e −1 where subscripts, . Q and .C, represent quantum and classical approximations, respec. Representative quantum correction to the specific heat is demontively, and.x = khω BT strated in Fig. 3.2 for amorphous silicon. Considering that Debye temperature of amorphous silicon is around 600 K, we see a significant heat capacity correction effect in the 300 K thermal conductivity accumulation function as shown in Fig. 3.2a. While caution is still needed when comparing against experimental data with a large spread for amorphous silicon, quantum correction to specific heat leads to a consistent temperature dependence in thermal conductivity as experiments.

40

3 Time Correlations and Their Descriptions of Materials Properties

(a)

(b)

Fig. 3.2 a Thermal conductivity accumulation function of amorphous silicon at 300 K (black) and 800 K (red) using Tersoff potential [4]. Larger effect of quantum correction is shown for 300 K, well below the Debye temperature of .∼600 K. Solid and dashed curves are with or without quantum correction to the specific heat, respectively. b Comparison between molecular dynamics thermal conductivity using Eq. 3.13 with (blue curve) or without the quantum correction (red curve) to the specific heat [5] and measurements [10–12]. Black circles are from Ref. [10], purple circles are from Ref. [11], and orange circles are from Ref. [12]. Molecular dynamics simulation results are adopted from Ref. [5]

Classical molecular dynamics simulations typically have on the order of . O(1000) to . O(10, 000) of atoms while ab-initio molecular dynamics simulations have hundreds of atoms, far less than typical bulk sizes of .∼1023 atoms. For studying bulk materials properties, periodic boundary conditions are applied in these equilibrium simulations. The idea is to tile the simulation domain periodically in space in all directions, allowing atoms at the edge of the original domain interact with atoms on the opposite side of the boundary across the domain boundaries. In addition, atoms moving outside the domain boundary can re-enter the original domain through the opposite domain boundary under periodic boundary conditions. These additional atomic interactions through periodic boundary conditions alleviate the size effects in thermal conductivity. For successful implementation of periodic boundary conditions, the potential cutoff must be no larger than one half of the simulation domain side length.

3.4 Time Correlation Functions and Macroscopic Materials Properties

41

Fig. 3.3 Size effect studies in thermal conductivity of amorphous silicon using Tersoff potential [4] adopted from Ref. [15]. A clear size effect is shown. In all systems, periodic boundary conditions were applied. Above 4096 atoms, thermal conductivity appears to show small size dependence

Even with the periodic boundary conditions, molecular dynamics domain size effects in materials properties should be carefully examined. For instance, the largest phonon wavelength that can exist in the molecular dynamics is on the order of the domain size; therefore, if low frequency, large wavelength phonon properties are of importance, a large domain size is necessary [13, 14]. Size effects in thermal conductivity calculated using Eq. 3.13 for amorphous silicon using Tersoff potential [4] are demonstrated in Fig. 3.3 [15]. In these calculations, periodic boundary conditions were applied. Initial large dependence on the system size followed by weak dependence in thermal conductivity is shown. Ideally, the larger the domain more accurate the thermal conductivity prediction is. However, for practical reasons such as computational costs, many prior works have studied thermal transport in amorphous silicon with a few thousands of atoms.

3.4.2 Viscosity Viscosity is a measure of the materials resistance to flow. Similar to thermal conductivity discussed above, viscosity (.η) of materials can also be characterized by correlation functions as { ∞ V .η = ⟨σαβ (t)σαβ (0)⟩dt (3.16) kB T 0 where .σαβ is the .αβ component of the stress tensor, .T is the temperature, .V is the volume of the system, and Greek subscripts denote Cartesian directions. Stress tensor is given by σ (t) =

. αβ

1 V

(∑ N j

m j v jα (t)v jβ (t) +

N ∑ j

) r jα (t)F jβ (t)

(3.17)

42

3 Time Correlations and Their Descriptions of Materials Properties

where.r jα (t) and.v jα (t) denote time dependent position and velocity along.α direction for an atom . j, respectively and .m and . F represent mass and force, respectively. Derivations of Eq. 3.16 is given elsewhere [16]. These time dependent quantities are typically calculated from ab-initio and classical molecular dynamics. To mimic bulk materials, periodic boundary conditions are utilized. Viscosity is related to the corresponding relaxation time often referred to as Maxwell relaxation time (.τ M ) as τ

η G∞

(3.18)

V ⟨(σαβ (0)2 ⟩ kB T

(3.19)

. M

where .

G∞ =

=

is the high frequency shear modulus. If the time scale of interest is shorter than .τ M , the system is considered to be rigid and behave like a solid, whereas if it is longer, it behaves in a fluidic/viscous manner [17]. With these as background information, we next discuss some interesting findings in viscosity reported in literature.

3.4.2.1

Arrhenius to Super-Arrhenius Crossover in Viscosity

At high temperatures, atoms in liquids move relatively independently without collective reorganization of their respective local environment, leading to Arrhenius EA

behavior in viscosity (.η ∼ e k B T where . E A is activation energy) [18]. However, as liquids cool down, atomic dynamics becomes more collective with increasing dynamic heterogeneity. Viscosity becomes non-Arrhenius with decrease in temperature. Temperature at which viscosity transitions from Arrhenius to non-Arrhenius dependence in temperature is often referred to as .T A . Some examples of this crossover in atomic dynamics in liquids are demonstrated by viscosity measurements for metallic liquids in Fig. 3.4. Similar crossover behavior has also been shown in many molecular dynamics simulations and viscosity measurements of various types of liquids [19– 22]. The Arrhenius to non-Arrhenius crossover has attracted much attention recently and some findings are highlighted below. • Experimental studies have shown that .T A is related to glass transition temperature, . Tg , by . T A ∼ 2Tg for various metallic systems [24, 25]. A different relation is found for non-metallic glasses [25]. These relations signify that underlying glass properties are also embedded in high temperature liquid characteristics. • Microscopic, unified picture of viscosity remains unclear across a wide range of temperatures. However, prior works have suggested that above .T A , dynamics of atomic connectivity determines the macroscopic viscosity of liquids: the average time for an atom to lose or gain one nearest neighbor (.τ LC ) is approximately equal to Maxwell relaxation time [19] as demonstrated in Fig. 3.5a. There have been recent experimental efforts to measure .τ LC from Van Hove function measurements using inelastic X-ray and neutron scattering [26–28].

3.4 Time Correlation Functions and Macroscopic Materials Properties

(a)

43

(b)

Fig. 3.4 Temperature dependent viscosity measurements for a Zr64 Ni36 and b Cu50 Zr50 by oscillating drop technique on electrostatically levitated samples, adopted from Ref. [23]. Black lines are Arrhenius fittings. Crossover from Arrhenius to super-Arrhenius behavior is evident near T A for both systems

• Electrical resistivity commonly used for liquid metals is related to the static structure factor, . S(q) (Fourier transform of pair distribution function, .g(r )) by 3πVat 4he2 v 2F k 4F

ρ(T ) =

.

{

2k F

S(q, T )|u(q)|2 q 3 dq

(3.20)

0

where .Vat is the atomic volume, .e is the electronic charge, .v F is the Fermi velocity, and .k F is the Fermi momentum, .q is the wavevector, and .u(q) is the pseudopotential. Because the integral is heavily weighted for large wavevectors, resistivity can be approximated as ρ(T ) ∼ S(2k F , T )|u(2k F )|2 .

.

(3.21)

As Eq. 3.21 suggests, resistivity in metallic liquids is largely dependent on the liquid structure and temperature dependent resistivity could give structural evidence for the onset of atomic cooperativity in liquids at .T A . Recent electrical resistivity measurements for liquid Zr64 Ni36 at the International Space Station demonstrate qualitative changes in the temperature dependence of electrical resistivity at .T A as shown in Fig. 3.5b, providing supporting evidence for structural and dynamical changes across the crossover temperature [23]. As the use of containers can affect the intrinsic liquid state dynamics, levitation or micro-gravity environments are preferred for accurate characterizations of liquid properties [29–31]. • Stokes–Einstein relations relate the self-diffusion coefficient, shear viscosity, and temperature as . D = cT /η where .c is a constant related to the atomic diameter. Stokes–Einstein relations work well for high temperature liquids [32]. As liquids cool down, Stokes–Einstein relations have been shown to break down [33–35]. While different theories exist to explain the violation of the Stokes–Einstein relations, various works have reported the on-set of atomic cooperativity and resulting

44

3 Time Correlations and Their Descriptions of Materials Properties

Fig. 3.5 a Maxwell relaxation time (.τ M ) normalized by bond-cutting time (.τ LC ) for various liquids in molecular dynamics adopted from Ref. [19]. Divergence of this ratio at .T = T A is demonstrated. EAM and AIMD represent embedded atom model empirical potential for classical molecular dynamics and ab-initio molecular dynamics, respectively. b Temperature dependent electrical resistivity measurements of levitated Zr64 Ni36 liquids under the microgravity conditions aboard the International Space Station, adopted from Ref. [23]. Saturation of the resistivity is notable above . T A . c Temperature dependent Stokes–Einstein ratio for Zr atoms in Cu64 Zr36 from molecular dynamics, adopted from Ref. [22]. The Stokes–Einstein relationship becomes violated below .T A . Solid black line is a guide to the eye where the ratio is constant

(a)

(b)

(c)

dynamic heterogeneity appearing at .T A responsible for this violation [21, 22]. An example of the breakdown of the Stokes–Einstein relations is demonstrated in Fig. 3.5c for Cu64 Zr36 from molecular dynamics [22]. These examples demonstrate that viscosity in liquids is strongly coupled to other materials properties such as structure and transport properties. When studying liquids at the atomic scale, various aspects of atomic structure and dynamics need to be considered simultaneously for a more complete picture.

3.4 Time Correlation Functions and Macroscopic Materials Properties

3.4.2.2

45

Quantum of Viscosity

Viscosity of fluids is generally system-dependent and varies by many orders of magnitude across temperatures and pressures. However, minimum viscosity values of various fluids are similar to each other within about an order of magnitude [36]. A recent work surveyed temperature and pressure dependent viscosities of noble, molecular, and network fluids and proposed the universal minimum kinematic viscosity (.νm = ηm /ρ) in terms of fundamental physical constants as ν =

. m

h 1 √ 4π m e m

(3.22)

where .m e is the electron mass, .m is the mass of the molecule set by the nucleon mass, .h is the Planck’s constant [37]. Equation 3.22 was derived from considering the viscosity from two limits: low temperature limit of gas-like viscosity and hightemperature limit of liquid-like viscosity [37]. As the presence of .h suggests, the proposed minimum kinematic viscosity is quantum mechanical in nature. Comparison between predicted minimum kinematic viscosity and measurements is shown in Table. 3.1. Predicted minimum viscosity values are in the same order of magnitude as the measurements. Despite the success of on-the-order estimation of minimum viscosity using quantum arguments, it is imperative to consider whether the minimum viscosity can be explained from classical arguments or not. If the minimum viscosity is indeed quantum mechanical in nature (and not classical) as suggested by Eq. 3.22, we expect failure of predicting minimum viscosity from classical calculations. Therefore, we carried out classical molecular dynamics simulations of argon fluids at 20 MPa and computed their temperature dependent viscosities using Eq. 3.16 and their corresponding kinematic viscosities from 100 to 1000 K. Results are compared against NIST suggested values [38] as shown in Fig. 3.6. Generally, a good agreement between cal-

Table 3.1 Calculated and experimental minimum kinetic viscosity, adopted from Ref. [37]. Kinematic viscosity units are in .×108 m2 s–1 .νm,calc .νm,ex p Ar (20 MPa) Ne (50 MPa) He (20 MPa) N2 (10 MPa) H2 (50 MPa) O2 (30 MPa) H2 O (100 MPa) CO2 (30 MPa) CH4 (5.4 MPa)

3.4 4.8 10.7 4.1 15.2 3.8 5.1 3.2 5.4

5.9 4.6 5.2 6.5 16.3 7.4 12.1 8.0 11.0

46 Fig. 3.6 Temperature dependent a viscosity and b kinematic viscosity of argon at 20 MPa. Black curve is from NIST [38] and blue circles are from this work utilizing Lennard-Jones potential [40]

3 Time Correlations and Their Descriptions of Materials Properties

(a)

(b)

culations and measurements is demonstrated within errorbars. Clear minima in both viscosity and kinematic viscosity are depicted from the classical molecular dynamics calculations, suggesting that minimum viscosity may be a classical phenomenon. A recent work has also demonstrated the minimum viscosity in hard-sphere systems in classical molecular dynamics, in support of this view [39]. Role of quantum versus classical mechanics in viscosity of fluids merit further investigations. We have covered various aspects of time correlation functions in this chapter from their basic properties to describing fundamental materials properties through the fluctuation-dissipation theorem (Green–Kubo). Next, we discuss, for the rest of this chapter, another time-correlation function, velocity autocorrelation function, that can provide important insights into the nature of heat carriers.

3.5 Velocity Autocorrelation Spectra as Phonon Density of States Here, it is demonstrated that velocity autocorrelation spectra can be used to describe phonon density of states in solids as

3.5 Velocity Autocorrelation Spectra as Phonon Density of States

g(ω) =

3N ∑

.

δ(ω − ωn ) =

n=1

1 kB T

{

N ∞∑ 0

⟨ ⟩ m i v j (t) · v j (0) eiωt dt.

47

(3.23)

j=1

In Chap. 2, we introduced the normal mode coordinate, . Q(k, ν, t) related to the atomic displacements and velocities by 1 ∑ u j, p (t) = √ e( j, k, ν)ei k·r j, p Q(k, ν, t) mj

(3.24)

1 ∑ ˙ u˙ j, p (t) = √ e( j, k, ν)ei k·r j, p Q(k, ν, t), mj

(3.25)

.

k,ν

.

k,ν

respectively. As. r j, p is the time independent equilibrium position of atoms, only time dependence is in the normal mode coordinate and we have the following relations: ˙ . Q(k, ν, t) = −iω(k, ν)Q(k, ν, t) . For generalization purposes (for example, for glasses without unit cells), we drop the unit cell notation, . p, and we use .n index to indicate normal modes from 1 to .3N . The autocorrelation function of velocities in terms of normal mode coordinates is then /∑ \ N 3N ∑ 3N ∑ ⟨ ⟩ . m j v j (t) · v j (0) = ωn ωl Q(n, t)Q(l, 0) (3.26) n=1 l=1

j=1

Using orthonormal properties, we obtain the following expressions /∑ 3N ∑ 3N .

\ ∑ 3N ⟨ ⟩ ωn ωl Q(n, t)Q(l, 0) = ωn2 |Q(n)|2 e−iωn t

n=1 l=1

(3.27)

n=1

In the classical limit, we obtain 3N ∑ .

⟨ ⟩ ωn2 |Q(n)|2 e−iωn t = k B T e−iωn t

(3.28)

n=1

The velocity autocorrelation function then becomes N ∑ .

⟨ ⟩ m j v j (t) · v j (0) = k B T e−iωn t

j=1

We take the Fourier transform on each side as

(3.29)

48

3 Time Correlations and Their Descriptions of Materials Properties

Fig. 3.7 Vibrational density of states of crystalline silicon at 1 K using Stillinger–Weber potential [2]. Density of states is normalized such that the areas under the curve for both NM(.ω) (shaded) and VACF(.ω) (blue circles) are equal to unity

{

N ∞∑ 0

3N { ∑ ⟨ ⟩ m j v j (t) · v j (0) eiωt dt = k B T n=1

j=1

.

= kB T

3N ∑

∞

ei(ω−ωn )t

0

(3.30) δ(ω − ωn )

n=1

Rearranging the above relations, we obtain the final expression of 3N ∑ .

n=1

δ(ω − ωn ) =

1 kB T

{

N ∞∑ 0

⟨ ⟩ m j v j (t) · v j (0) eiωt dt

(3.31)

j=1

Phonon density of states of crystalline silicon using Stillinger–Weber potential [2] is shown in Fig. 3.7. Both normal mode spectra (NM(.ω)) described in Chap. 2 and velocity autocorrelation spectra (VACF(.ω)) discussed here are depicted [41]. Consistent agreement between the two methods is demonstrated. As with normal mode spectra, when integrated over frequency, we recover .3N degrees of freedom in VACF(.ω). We used the harmonic approximation to derive that VACF(.ω) represents phonon density of states. Intrinsically, higher order potential interactions (anharmonicity) is included in atomic velocities from molecular dynamics; therefore, finite temperature anharmonic effects on phonon frequencies are included in VACF(.ω). Temperature dependent phonon densities of states for crystalline silicon approximated by VACF(.ω) are demonstrated in Fig. 3.8. Weak dependence of phonon frequencies in temperature is observed from 100 to 1500 K. These anharmonic effects are not included in harmonic lattice dynamics discussed in Chap. 2 and anharmonic renormalization of phonons must be done for more accurate description of phonon frequencies at finite temperatures. As discussed before, velocity autocorrelation function can also be used to describe atomic diffusion processes in addition to vibrations. We next describe atomic diffusion using velocity autocorrelation function.

3.6 Velocity Autocorrelation Spectra Describing Atomic Diffusion

49

Fig. 3.8 Vibrational density of states (VACF(.ω)) of crystalline silicon at 100 (blue circles) and 1500 K (red circles) using Stillinger–Weber potential [2]. Each density of states is normalized such that the area under the curve is equal to unity. A general softening of phonon frequencies is demonstrated

3.6 Velocity Autocorrelation Spectra Describing Atomic Diffusion The displacement in a time interval t of a particle, .i, is {

t

r (t) − r i (0) =

. i

v i (t ' )dt '

(3.32)

0

When squared and averaged over initial conditions, above equation becomes \ { t / \ /{ t 2 ' ' '' '' |r i (t) − r i (0)| = v i (t )dt · v i (t )dt 0

{

t

=2

.

{

dt '

0 t

=6

dt '

0

{ {

0

t'

dt '' ⟨v i (t ' ) · v i (t '' )⟩

(3.33)

0 t' 0

1 dt '' ⟨v i (t ' − t '' ) · v i (0)⟩ 3

Using a change of variable from .t '' to .s = t ' − t '' followed by an integration by parts with respect to .t ' leads to /

|r i (t) − r i (0)|

.

2

\

{

t

{

t'

1 ds ⟨v i (s) · v i (0)⟩ 3 0 0 { t( s )1 1− = 6t ⟨v i (s) · v i (0)⟩ds t 3 0

=6

dt

'

(3.34)

Recalling that self-diffusion coefficient is defined by .

⟨|r i (t) − r i (0)|2 ⟩ , t→∞ 6t

D = lim

(3.35)

50

3 Time Correlations and Their Descriptions of Materials Properties

Fig. 3.9 a Time dependent mean square displacement (MSD(t)) of liquid silicon atoms at 5000 K using Stillinger–Weber potential [2]. Clear linear dependence in time on the mean square displacement is shown at large time. b Velocity autocorrelation function (VACF(t)) of the same system. Both methods lead to the same self-diffuson coefficient of silicon atoms

(a)

(b)

we obtain the following relation between the self-diffusion coefficient and velocity autocorrelation function (VACF) as { .

D= 0

∞

1 ⟨v i (t) · v i (0)⟩dt 3

(3.36)

The numerator in Eq. 3.35 is known as the mean square displacement (MSD) of atoms. The MSD and VACF of liquid silicon at 5000 K are plotted in Fig. 3.9 as an example. Initially, atoms freely move within cages that leads to MSD .α t 2 . In solids, these cages are relatively rigid such that atomic diffusion occurs through hopping processes whereas cages can dynamically change in other phases. A clear linear time dependence on the MSD is shown for large times and diffusion coefficient can be accordingly extracted from a linear fit to the MSD in the large time limit. In the VACF, a sharp decay followed by an oscillatory behavior within 0.1 ps is shown. This oscillation is reminiscent of atomic vibrations. Oscillations in the VACF after 0.3 ps can be considered as numerical noise arising from limited time of integration in the integral limits. In perfect solids without atomic diffusion, oscillations in the VACF lead to . D ∼ 0 when integrating VACF over time. Both calculations of 5000 K silicon lead to diffusion coefficients of 6.14 .± 0.05 and 6.25 .± 0.12 Å2 ps–1 , in agreement with each other within an errorbar.

3.8 Appendix

51

3.7 Concluding Remarks Time correlation functions are useful in that how strongly dynamic variables stay correlated is a measure of an intrinsic materials property. Here, we briefly discussed some fundamental properties of time correlation functions and demonstrated some examples of these functions to describe macroscopic properties such as thermal conductivity and viscosity. We then discussed the versatility of velocity autocorrelation functions in describing different types of atomic motions: atomic vibrations in solids and atomic diffusions in non-solids. Due to the intrinsically anharmonic nature of atomic velocities from molecular dynamics simulations, anharmonicity in vibrations is also included in VACF(.ω). With these ingredients prepared along with our discussions of normal modes in Chap. 2, we next consider recent efforts to characterize heat carriers in liquids.

3.8 Appendix 3.8.1 Weiner Khinchin’s Theorem The autocorrelation, . R(t), of a function, . f (x) is defined as { .

R(t) ≡

∞ −∞

f¯(τ ) f (t + τ )dτ

Recall that the inverse Fourier transform (IFT) of . F(ν) is defined by { ∞ . f (τ ) = F(ν)ei2πντ dν

(3.37)

(3.38)

−∞

with a complex conjugate of . f¯(τ ) =

{

∞

−i2πντ ¯ dν F(ν)e

(3.39)

−∞

¯ ) and . E(t + τ ) into the autocorrelation gives Plugging in . E(τ R(t) = = .

= =

{ ∞ [{ ∞ −∞

−∞

−i2πντ dν ¯ F(ν)e

{ ∞ { ∞ { ∞ −∞ −∞ −∞

{ ∞ { ∞

−∞ −∞ { ∞ −∞

][ { ∞ −∞

] ' F(ν ' )ei2πν (t+τ ) dν ' dτ

' )e−i2πτ (ν−ν ' ) ei2πν ' t dτ dνdν ' ¯ F(ν)F(ν

' F(ν)F(ν ' )δ(ν ' − ν)ei2πν t dνdν '

|F(ν)|2 ei2πνt dν

] [ = IFT F(ν)2

(3.40)

52

3 Time Correlations and Their Descriptions of Materials Properties

Weiner Khinchin’s theorem can be conveniently used to calculate the Green– Kubo thermal conductivity, viscosity, and velocity autocorrelation function as they are based on autocorrelation functions.

3.8.2 Green–Kubo Thermal Conductivity The Green–Kubo formalism is based on statistical thermodynamics. There are several derivations that result in the same formula but in this chapter, I will mainly adapt the derivations by Helfand and Kaviany [42]. The energy equation can be written as ncv

.

∂ E o (r, t) = k∇ 2 E o (r, t) ∂t

(3.41)

where . E o (r, t) = E(r, t) − ⟨E(r, t)⟩, the time and position dependent energy deviation. . E(r, t) is the actual energy and .⟨E(r, t)⟩ denotes the expectation energy. .n and .cv correspond to density and specific heat. For discrete particle systems, ∑ . E o (r, t) = j E o, j (r, t)δ[r − r j (t)] where index . j denotes the atom identity. ˜ Let us define the Fourier transform of . E o (r, t) as . E(r, t) { ∑ ˜ . E(r, E o, j (t)eiκ·r j (t) (3.42) t) = E o (r, t)eiκ·r d r = j

˜ The energy equation can then be written in terms of . E(r, t) as

.

˜ κ2 k ˜ ∂ E(r, t) =− E(r, t) ∂t ncv

(3.43)

by using the Fourier transform identity that . F T ( dd xf ) = −iκF(κ) where . F T is the Fourier transform operation and . F(κ) is the Fourier transform of . f (x). The solution ˜ with the initial condition, . E(r, 0), is then κ2 kt

.

˜ ˜ E(r, t) = E(r, 0)e− ncv

(3.44)

Multiplying the complex conjugate of the initial condition to both sides, we have .

˜ E(r, t) E˜ ∗ (r, 0) =

∑ j

and we can further write

E o, j (t)eiκ·r j (t)

∑ l

E o,l (0)eiκ·r l (0)

(3.45)

3.8 Appendix

∑ .

53

∑

E o, j (t)eiκ·r j (t)

j

E o,l (0)eiκ·r l (0) =

[∑

l

E o, j (0)eiκ·r j (0)

j

∑

] κ2 k E o,l (0)eiκ·r l (0) e− ncv

l

(3.46) Applying ensemble average on both sides and rearranging, /∑∑ \ . E o, j (t)E o,l (0)eiκ·(r j (t)−r l (0)) j

.

=

l

/∑∑ j

\ κ2 k E o, j (0)E o,l (0)eiκ·(r j (0)−r l (0)) e− ncv

(3.47)

l

Next, we expand both sides as a Taylor series about .κ = 0 and assuming small perturbations, we keep up to the second order. For simplicity, we will only consider the .x direction without losing generality. The left hand side is then /∑∑ \ /∑∑ [ ]\ . E o, j (t)E o,l (0) + iκ E o, j (t)E o,l (0) x j (t) − xl (0) j

l

j

.

l

[ ]2 \ κ/ ∑ ∑ E o, j (t)E o,l (0) x j (t) − xl (0) 2 j l

−

(3.48)

We now examine each term. In equilibrium, the expectation value at time .t of a dynamic property that depends on the particle positions and velocities is same as that at .t = 0. /∑∑ \ /∑∑ \ . E o, j (t)E o,l (0) = (E j − ⟨E j ⟩)(El − ⟨El ⟩) j

l

=

j

/∑∑ /

.

j

l

(E j El − E j ⟨El ⟩ − El ⟨E j ⟩ + ⟨E j ⟩⟨El ⟩)

l

= E t2 − 2E t ⟨E t ⟩ + ⟨E t ⟩2 / \ = (E t − ⟨E t ⟩)2

\

\ (3.49)

= N cv k B T 2 ∑ where . E t = n E n is the total energy of the system and N is the total number of atoms. In the last step, I have used the fact that for canonical ensemble systems, / .

\ ∂ 2 ln Z (E t − ⟨E t ⟩)2 = ∂β 2

where . Z is the partition function and .β = k B T . The first order expansion term can be written as

(3.50)

54

3 Time Correlations and Their Descriptions of Materials Properties

iκ

/∑∑

.

j

= iκ

[/ ∑ ∑ j

.

= iκ

[ ]\ E o, j (t)E o,l (0) x j (t) − xl (0)

l

\ /∑∑ \] E o, j (t)E o,l (0)x j (t) − E o, j (t)E o,l (0)x j (0)

l

[/ ∑ ∑ j

j

\

E o, j (0)E o,l (0)x j (0) −

l

l

/∑∑ j

E o, j (0)E o,l (0)x j (0)

\]

(3.51)

l

=0 The second order term can be written as .

.

=

[ ]2 \ −κ2 / ∑ ∑ E o, j (t)E o,l (0) x j (t) − xl (0) 2 j l

[ \ /∑∑ \ −κ2 / ∑ ∑ E o, j (t)E o,l (0)x j (t)2 − 2 E o, j (t)E o,l (0)x j (t)xl (0) 2 j l j l .

+

/∑∑ j

E o, j (t)E o,l (0)xl (0)

2

\]

l

[ \ /∑∑ \ −κ2 / ∑ ∑ . = E o, j (t)E o,l (t)x j (t)2 − 2 E o, j (t)E o,l (0)x j (t)xl (0) 2 j l j l .

+

/∑∑ j

E o, j (t)E o,l (0)xl (0)

2

\] (3.52)

l

If we assume that the particles are densely populated, we can write /∑∑ .

j

\ /∑ \ ∑ E o, j (t)E o,l (t)x j (t)2 = E o, j (t)x j (t) E o,l (t)xl (t)

l

j

(3.53)

l

Therefore, the second order correction term becomes .

/ )2 \ −κ2 ( ∑ x j (t)E o, j (t) − x j (0)E o, j (0) 2 j

(3.54)

The left hand side of the Eq. 3.47 is then .

N k B T 2 cv −

/ )2 \ κ2 ( ∑ x j (t)E o, j (t) − x j (0)E o, j (0) 2 j

(3.55)

3.8 Appendix

55

Now, we consider the right hand side of Eq. 3.47 and expand up to the second order about .κ = 0. With the same argument as the zeroth term on the left hand side, the zeroth term on the right hand side, /∑∑ .

j

\ E o, j (0)E o,l = N k B T 2 cv

For .κ = 0, the first order term can be written as /∑∑ \ .iκ E o, j (0)E o,l [x j (0) − xl (0)] j

(3.56)

l

(3.57)

l

With the same argument as the first term on the left hand side, the first order term on the right hand side is equal to 0. If we define \ /∑∑ a= E o, j (0)E o,l (0)eiκ·(r j (0)−r l (0)) j l . (3.58) b=e

−κ2 k ncv

For the second order term, we need to evaluate the derivative .

∂ 2 (ab) ∂2b ∂a ∂b ∂2b =a 2 +2 + 2 ∂κ ∂κ ∂κ ∂κ ∂κ2

(3.59)

The derivative of the exponential is zero at .κ = 0. Carrying out the algebra (not shown here), the second order term is equal to .−V k B T 2 k x κ2 . Therefore, the right hand side of Eq. 3.47 is 2 2 2 . N k B T cv − V k B T k x κ (3.60) Equating the left hand side and the right hand side of Eq. 3.47 and solving for .kx, k =

. x

/( ∑ )2 \ 1 [E (t)x (t) − E (0)x (0)] j,o j j,o j kB T 2V j

(3.61)

In order to obtain the Green–Kubo formula, we transform the summation to the integral form. We use the fact that /( ∑ )2 \ { = [E j,o (t)x j (t) − E j,o (0)x j (0)]

t

.

j

and we define . S(t) in .x-direction as

0

d ∑ E j,o (t ' )x j (t ' )dt ' dt j

(3.62)

56

3 Time Correlations and Their Descriptions of Materials Properties

S (t) =

. x

d ∑ E j,o (t)x j (t) dt j

(3.63)

Here . S(t) is often referred to as the heat current or energy current. However, this is misleading as . S(t) has the units of .W m. The Green–Kubo formula I have used in previous chapters use actual heat currents with the units of .W m −2 . Hence, the volume, .V , is located in the numerator instead. Using the heat current as defined above, the ensemble average can be written as /( ∑ \ { t )2 \ / { t = [E j,o (t)x j (t) − E j,o (0)x j (0)] Sx (t ' )dt ' Sx (t '' )dt '' 0

j

= .

=

{ t{ t/ 0

0

0

0

{ t{ t/

0

\ Sx (t ' )Sx (t '' ) dt ' dt '' \ Sx (t '' − t ' )Sx (0) dt ' dt ''

{ t( \ τ )/ Sx (τ )Sx (0) dτ 1− = 2t t 0 (3.64) In the last step, I have the used the identity that { t{ .

0

t

''

'

'

{

''

t

f (t − t )dt dt = 2t

0

(1 −

0

τ ) f (τ )dτ t

(3.65)

Therefore, the thermal conductivity in .x-direction can be written as .t → ∞, k =

. x

1 V kB T 2

{

∞

/

\ Sx (t)Sx (0) dt

(3.66)

0

For three dimensional systems, the thermal conductivity tensor can be written as 1 .ki j = V kB T 2

{

∞

/

\ Si (t)S j (0) dt

(3.67)

0

where the indices .i, j refer to the cartesian directions and heat current vector is .

S(t) =

d ∑ E j,o (t)r j (t) dt j

(3.68)

Once the heat current vectors are calculated (e.g. using LAMMPS [43]), thermal conductivity can be calculated using the Green–Kubo formalism outlined above.

References

57

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3 Time Correlations and Their Descriptions of Materials Properties

21. Jaiswal, A., Egami, T., Zhang, Y.: Atomic-scale dynamics of a model glass-forming metallic liquid: Dynamical crossover, dynamical decoupling, and dynamical clustering. Phys. Rev. B 91(13) (2015). https://doi.org/10.1103/PhysRevB.91.134204. Accessed 14 Feb 2020 22. Soklaski, R., Tran, V., Nussinov, Z., Kelton, K.F., Yang, L.: A locally preferred structure characterises all dynamical regimes of a supercooled liquid. Philos. Mag. 96(12), 1212–1227 (2016). https://doi.org/10.1080/14786435.2016.1158427. Accessed 25 March 2020 23. Van Hoesen, D., Gangopadhyay, A., Lohöfer, G., Sellers, M., Pueblo, C., Koch, S., Galenko, P., Kelton, K.: Resistivity saturation in metallic liquids above a dynamical crossover temperature observed in measurements aboard the international space station. Phys. Rev. Lett. 123(22), 226601 (2019). https://doi.org/10.1103/PhysRevLett.123.226601. Accessed 17 Feb 2020 24. Blodgett, M.E., Egami, T., Nussinov, Z., Kelton, K.F.: Proposal for universality in the viscosity of metallic liquids. Sci. Rep. 5(1) (2015). https://doi.org/10.1038/srep13837. Accessed 17 Feb 2020 25. Jaiswal, A., Egami, T., Kelton, K., Schweizer, K.S., Zhang, Y.: Correlation between fragility and the arrhenius crossover phenomenon in metallic, molecular, and network liquids. Phys. Rev. Lett. 117(20) (2016). https://doi.org/10.1103/PhysRevLett.117.205701. Accessed 20 March 2020 26. Iwashita, T., Wu, B., Chen, W.-R., Tsutsui, S., Baron, A.Q.R., Egami, T.: Seeing real-space dynamics of liquid water through inelastic x-ray scattering. Sci. Adv. 3(12), 1603079 (2017). https://doi.org/10.1126/sciadv.1603079. Accessed 13 April 2018 27. Shinohara, Y., Dmowski, W., Iwashita, T., Wu, B., Ishikawa, D., Baron, A.Q.R., Egami, T.: Viscosity and real-space molecular motion of water: Observation with inelastic x-ray scattering. Phys. Rev. E 98(2), 022604 (2018). https://doi.org/10.1103/PhysRevE.98.022604. Accessed 13 May 2019 28. Ashcraft, R., Wang, Z., Abernathy, D.L., Quirinale, D.G., Egami, T., Kelton, K.F.: Experimental determination of the temperature-dependent Van Hove function in a Zr80Pt20 liquid. J. Chem. Phys. 152(7), 074506 (2020). https://doi.org/10.1063/1.5144256. Publisher: American Institute of Physics. Accessed 09 Feb 2021 29. Brillo, J., Pommrich, A.I., Meyer, A.: Relation between self-diffusion and viscosity in dense liquids: new experimental results from electrostatic levitation. Phys. Rev. Lett. 107(16) (2011). https://doi.org/10.1103/PhysRevLett.107.165902. Accessed 25 May 2019 30. Mohr, M., Wunderlich, R.K., Zweiacker, K., Prades-Rödel, S., Sauget, R., Blatter, A., Logé, R., Dommann, A., Neels, A., Johnson, W.L., Fecht, H.-J.: Surface tension and viscosity of liquid Pd43Cu27Ni10P20 measured in a levitation device under microgravity. npj Microgravity 5(1), 1–8 (2019). https://doi.org/10.1038/s41526-019-0065-4. Number: 1 Publisher: Nature Publishing Group. Accessed 14 Nov 2023 31. Bendert, J.C., Kelton, K.F.: Containerless measurements of density and viscosity for a Cu$$_48$$Zr$$_52$$Liquid. Int. J. Thermophys. 35(9), 1677–1686 (2014). https://doi.org/ 10.1007/s10765-014-1664-7. Accessed 14 Nov 2023 32. Wei, S., Evenson, Z., Stolpe, M., Lucas, P., Angell, C.A.: Breakdown of the Stokes-Einstein relation above the melting temperature in a liquid phase-change material. Sci. Adv. 4(11), 8632 (2018). https://doi.org/10.1126/sciadv.aat8632. Accessed 12 Feb 2020 33. Costigliola, L., Heyes, D.M., Schrøder, T.B.: Revisiting the Stokes-Einstein relation without a hydrodynamic diameter. Chem. Phys. 7 (2019) 34. Cao, Q.-L., Wang, P.-P., Huang, D.-H.: Revisiting the Stokes-Einstein relation for glassforming melts. Phys. Chem. Chem. Phys. 22(4), 2557–2565 (2020). https://doi.org/10.1039/ C9CP04984C. Publisher: Royal Society of Chemistry. Accessed 08 Aug 2023 35. Sengupta, S., Karmakar, S., Dasgupta, C., Sastry, S.: Breakdown of the Stokes-Einstein relation in two, three, and four dimensions. J. Chem. Phys. 138(12), 12–548 (2013). https://doi.org/10. 1063/1.4792356. Accessed 11 March 2021 36. Purcell, E.M.: Life at low Reynolds number. Amer. J. Phys. 45(1) (1977) 37. Trachenko, K., Brazhkin, V.V.: Minimal quantum viscosity from fundamental physical constants. Sci. Adv. 6(17), 3747 (2020). https://doi.org/10.1126/sciadv.aba3747. Publisher: American Association for the Advancement of Science Section: Research Article. Accessed 16 June 2020

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38. NIST: Thermophysical properties of fluid systems. https://webbook.nist.gov/chemistry/fluid/ Accessed 01 Aug 2023 39. Khrapak, S.A., Khrapak, A.G.: Minima of shear viscosity and thermal conductivity coefficients of classical fluids. Phys Fluids 7 (2022) 40. Lennard-Jones, J.E.: Cohesion. Proc. Phys. Soc. 43(5), 461–482 (1931). https://doi.org/10. 1088/0959-5309/43/5/301. Publisher: IOP Publishing. Accessed 28 July 2022 41. Moon, J., Thébaud, S., Lindsay, L., Egami, T.: Normal mode description of phases of matter: application to heat capacity. Phys. Rev. Res. 6(1), 013206 (2024). https://doi.org/10.1103/ PhysRevResearch.6.013206. Publisher: American Physical Society 42. Kaviany, M.: Heat Transfer Physics. Cambridge University Press, New York, NY (2014) 43. Plimpton, S.: Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117(1), 1–19 (1995)

Chapter 4

Continuity of the Solid, Liquid, and Gas Phases of Matter

Abstract In previous chapters, we discussed the success of normal mode decomposition of atomic motion in solids in describing various thermal properties microscopically and considered how time correlation functions such as velocity autocorrelation function are useful to describe macroscopic materials properties. Here, we survey recent literature on characterizations of heat carries in liquids through these analysis tools and discuss how one could use both the normal mode formalism and velocity autocorrelation functions to describe heat carriers in all three phases in a unified manner. Keywords Instantaneous normal modes · Phonons · Propagons · Diffusons · Locons · Collisons · Translatons · Velocity autocorrelation functions · Molecular dynamics · Liquids · Heat capacity · Diffusion coefficient In Chap. 2, we discussed the normal mode formalism in solids. Using equilibrium lattice sites, atomic motion is decomposed into orthonormal oscillatory planewaves known as normal modes. Various thermal properties including heat capacity and thermal conductivity can be characterized accurately and microscopically using the normal mode formalism. In solids, all .3N normal modes are considered vibrational in nature where . N is the number of atoms. These normal modes have been characterized as phonons, propagons, diffusons, and locons depending on the presence of translational symmetry and mode interaction mechanisms. So what happens to the normal mode formalism to describe atomic motion and macroscopic properties when we no longer have equilibrium positions? Ultimately, the normal mode formalism is simply a mathematical tool to understand the atomic motion using planewave decompositions. We can certainly apply the formalism to any types of phases. Recently, motivated by the success of normal mode formalism in solids, there has been a lot of interest to extend the normal mode formalism to liquid systems where instantaneous snapshot structures are used instead of the equilibrium lattice [1–9]. Normal mode decomposition of atomic motion in liquids has the potential to be useful as the number of normal modes in the system is still equal to the number of atomic degrees of freedom. However, physical interpretations of normal modes in liquids are no longer straightforward and would not be in general the same as those with atoms vibrating around equilibrium positions. Therefore, we make © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. Moon, Heat Carriers in Liquids: An Introduction, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-51109-7_4

61

62

4 Continuity of the Solid, Liquid, and Gas Phases of Matter

Fig. 4.1 Representative instantaneous normal mode and velocity autocorrelation spectra of liquid LJ argon at 1000 K and 2.6 GPa. Vertical black dash line is the y-axis at .ω .= 0 as the guide to the eye. Both INM(.ω) and VACF(.ω) are normalized such that the area under the curve is equal to unity

a distinction here. Normal modes obtained from an instantaneous snapshot structure are defined as instantaneous normal modes and those obtained from equilibrium positions are referred to as equilibrium/relaxed normal modes in this chapter. In Chap. 3, it was demonstrated that equilibrium normal mode and velocity autocorrelation spectra for harmonic solids are equivalent. Beyond vibrations in solids, velocity autocorrelation function is also useful to describe atomic motion in nonsolid systems as it characterizes diffusion coefficients. Then, how do velocity autocorrelation spectra (VACF(.ω)) compare against instantaneous normal mode spectra (INM(.ω)) in liquids? Representative INM(.ω) and VACF(.ω) are shown for liquid argon using Lennard-Jones potential [10, 11] at 1000 K in Fig. 4.1. While both spectra recover.3N when integrated over frequency, we observe stark differences between the two spectra. VACF(.ω) includes time dependent atomic dynamics directly and the effect of anharmonicity is naturally included. However, atomic-level qualitative interpretation of VACF(.ω) is lacking and we are unable to resolve each atomic degree of freedom in VACF(.ω). INM(.ω) provides the snapshot information of the potential energy landscape in a mode resolved manner but only includes harmonicity and need some simpler descriptors to understand INM(.ω). It is the goal of this chapter to describe these differences and how one may make the connections between these two apparently different spectra. This chapter is organized as follows. We will first discuss instantaneous normal modes and what they mean physically in the context of the equilibrium normal modes. We then discuss features in the velocity autocorrelation spectra in liquids. Finally, we make connections between them and describe how these spectra can be used to characterize thermal properties in liquids.

4.1 Instantaneous Normal Modes Recall that dynamical matrices are Hermitian such that eigenvalues, .Ω = ω 2 , are real. Therefore, eigenfrequencies, .ω, can be either imaginary or real. When discussing solids and equilibrium normal modes, we have only observed modes with

4.1 Instantaneous Normal Modes

63

real eigenfrequencies (.ω ≥ 0). The most distinctive difference between instantaneous normal modes for non-solids and equilibrium normal modes for solids is perhaps the appearance of modes with imaginary frequencies as demonstrated in the instantaneous normal mode spectra in Fig. 4.1. Negative frequencies denote modes with imaginary frequencies. As such, a lot of efforts have focused on the nature of imaginary modes and what they can tell us about the system while real modes are typically considered similarly with equilibrium normal modes in solids as harmonic oscillators. Prior works have shown that as temperature is increased, the portion of the imaginary modes becomes larger as measured by the area under the curve in INM(.ω), commonly observed in many systems [1]. In the limit of minute vibrations in solids at low temperatures, instantaneous normal modes approach equality with the equilibrium normal modes. Instantaneous and equilibrium normal modes physically reflect the curvature of the potential energy landscape as they are both associated with eigenvalues of second derivatives of the potential; therefore, negative eigenvalues (imaginary modes) represent unstable configurations such as saddle points. For this reason, imaginary modes have been used to describe various non-equilibrium processes including atomic diffusion [1, 4, 12, 13], glass transition [14–16], melting [17], and plasticity [18]. Despite these efforts, the physical interpretation of imaginary modes is not yet clear. In the following subsections, we will go over some examples of how imaginary modes have been used to microscopically characterize atomic motion and materials properties of liquids such as diffusion coefficients and heat capacities and what they tell us about the nature of imaginary modes.

4.1.1 Self-diffusion Coefficient Described by Instantaneous Normal Modes There have been a few different models to describe self-diffusion coefficients using instantaneous normal modes motivated by Zwanzig’s model [19]. In his work, Zwanzig made efforts to derive the Stokes–Einstein model from the velocity autocorrelation function expression of self-diffusion coefficients (. D) that was discussed in Chap. 3. In this model, it is assumed that the atomic motion in liquids is described by liquid’s configurations oscillating in basins of the potential energy landscape until it finds a saddle point or a bottleneck and moves to another basin. The time dependence in the velocity autocorrelation function is then replaced by normal mode contributions with each contribution proportional to .cos ωt until a move to another basin interrupts the motion. Each normal mode is interrupted at different times. This t is accounted for by introducing a factor .e− τω , which characterizes the waiting time distribution in the basins. The resulting expression for the self-diffusion coefficient using normal modes is then given by kB T .D = 3m N

{

∞

dt 0

∑ ω

cos (ωt)e− τω = t

τω kB T ∑ 3m N ω 1 + ω 2 τω2

(4.1)

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4 Continuity of the Solid, Liquid, and Gas Phases of Matter

(a)

(b)

Fig. 4.2 Isobaric temperature dependent diffusion coefficients for Lennard-Jones argon at a 0.5 GPa and b 1.0 GPa, adopted from Ref. [4]. Solid lines are from molecular dynamics calculations and red dashed lines are from Eq. 4.2

where the sum is over all .3N modes. To obtain the self-diffusion coefficient using Eq. 4.1, one needs the .τω ’s and .ω’s (normal mode spectra). In Zwanzig’s paper, the sum is replaced by a modified version of the Debye model and .τω ’s are estimated from the longitudinal and shear viscosities. However, validity of using the Debye model in liquids is questionable as it deviates significantly from the actual normal mode spectra as highlighted in Fig. 4.1. Keyes and others have proposed a further development of Zwanzig’s diffusion model to connect the hopping times between basins and the imaginary part of INM(.ω) [4, 20]. Particularly, it was suggested that diffusion coefficient has the form of .

D = c⟨ω⟩ I f I

(4.2)

where .c is a constant, .⟨ω⟩ I is the averaged imaginary mode frequency, . f I is the fraction of the imaginary modes out of all modes. Comparisons between the model (Eq. 4.2) and diffusion coefficients calculated in molecular dynamics are shown for Lennard-Jones argon in Fig. 4.2. A decent agreement is shown. However, it was pointed out that at high temperatures in solids with no diffusion, atoms are displaced from their respective equilibrium sites significantly, leading to presence of imaginary modes and discrepancies with Eq. 4.2 [6, 14, 21]. An alternate proposal is that a localized mode can only contribute to the local rearrangement in liquids, contributing negligibly to the overall diffusion, whereas delocalized modes can bring about large scale structural rearrangements that lead to diffusion [21]. Therefore, only the delocalized fraction of imaginary modes (. f I,DL = f I − f I,L where . f I,L is the localized imaginary mode fraction) is considered in characterizing diffusion as A

.

D ∼ e T [aln( f I,DL )+b]

(4.3)

65

D (Arb. Unit)

Fig. 4.3 Diffusion coefficient comparisons between Eq. 4.3 (solid line) and molecular dynamics simulations (circles) of KA liquids, adopted from Ref. [21]. After fitting, constants, .a and .b, were found to be 0.5 and 2.0, respectively. Inset shows temperature dependent MD diffusion coefficients. Dashed line represents Arrhenius law

10

D (Arb. Unit)

4.1 Instantaneous Normal Modes

-2

10-2 10-4 0.5

1

1.5

2

2.5

1/T (Arb. Unit)

10-4

0

1

2

3

4

[T(aln(f I,DL) + b)] -1

where . A is a constant activation energy, ln is the natural log, .T is temperature, and a and .b are constants. Comparisons between Eq. 4.3 and diffusion coefficients from molecular dynamics are demonstrated for binary Kob-Anderson (KA) liquids [22] in Fig. 4.3 [21]. Here, delocalized imaginary modes are identified by inverse participation ratios that were discussed in Chap. 2. Temperature dependent diffusion coefficients are shown in the inset. Arrhenius to non-Arrhenius transition occurs at .1/T ∼ 1.7. It appears that Eq. 4.3 is able to describe the diffusion coefficients in a wide range of temperatures across the crossover. While these results show a promising potential for the usefulness of instantaneous normal modes to describe atomic diffusion, more rigorous theory developments are still needed. The proposal that diffusion is governed by delocalized imaginary modes does not address the fact that glass-forming liquids are dynamically heterogeneous, suggesting that localized imaginary modes could play an important role in structural relaxation and diffusion at low temperatures below .T A , the Arrhenius to non-Arrhenius crossover temperature. In addition, these works assume that only imaginary modes are responsible for diffusion, which has not been verified.

.

4.1.2 Heat Capacity of Liquids Described by Instantaneous Normal Modes In addition to diffusion coefficients described above, thermodynamics (e.g., heat capacity) of liquids has been studied recently in terms of instantaneous normal modes [23]. It was assumed that atoms obey an overdamped equation of motion as .

dv i = −[v i dt

(4.4)

where .[ is related to a relaxation time .τ by .[ = 1/τ and .v i is the velocity of atom .i.

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Fig. 4.4 Heat capacity comparisons between experiments (NIST) [24] and Eq. 4.6 with fitting, adopted from Ref. [23]. Good agreements are generally demonstrated

Utilizing a generalization of the Plemelj identity to arbitrary integration pathways on the complex plane and associated Green’s function to Eq. 4.4, it was proposed that the density of states of liquids has the following form [23] − ω2 ω ωD e ω 2 + [(T )2 2

g(ω, T ) = c

.

(4.5)

where .c is a normalization factor and .ω D is Debye frequency. The exponential term is ω decays slowly with frequency. an arbitrarily introduced Gaussian cut-off as . ω2 +[(T )2 Utilizing this density of states expression, heat capacity of liquids proposed is given by { ∞( ( hω )−2 hω )2 .C V (T ) = k B sinh g(ω, T )dω. (4.6) 2k B T 2k B T 0 Equation 4.6 is equivalent to the harmonic oscillator heat capacity expression we encountered in Chap. 2 (See Eq. 18) but with .g(ω, T ) replaced by Eq. 4.5. Heat capacity comparisons between Eq. 4.6 (fitted) and experiments for selected liquids are shown in Fig. 4.4. Generally, good fits are shown. However, there are some concerns about this proposal of heat capacity relations. First, it is assumed that all heat carriers in liquids are considered to be harmonic oscillators as in solids. This assumption becomes questionable especially at high temperatures where liquid structures change quicker than supposed phonon periods. Second, Eq. 4.6 is derived by assuming that the mode frequencies and density of states do not change with temperature as Eq. 4.6 derives from { .

E= 0

∞

( 1) g(ω)dω hω n + 2

| { ∞ d E || dn .C V ≡ = hω g(ω)dω. | dT V dT 0

(4.7)

(4.8)

4.1 Instantaneous Normal Modes

67

However, there is temperature dependence on both .ω and .g(ω, T ). Therefore, even if the assumption of the harmonic oscillator model is valid for liquids at sufficiently low temperatures, there are some questions of validity using Eq. 4.6. Third, there has been a debate over the density of states expression in Eq. 4.5 for liquids [7]; therefore, unbiased and rigorous experimental measurements are necessary to further characterize the density of states of liquids.

4.1.3 Comments on Instantaneous Normal Modes of Liquids As demonstrated by the examples of utilizing instantaneous normal modes in describing various liquid properties given above, physical interpretation of instantaneous normal modes is typically an extrapolation of our understanding of equilibrium normal modes in solids. For instance, in the diffusion coefficient expressions, only imaginary modes are considered whereas real modes are considered similarly as harmonic oscillators in solids. In the heat capacity studies, all instantaneous normal modes were considered as harmonic oscillators. However, it is our view that we should first understand the nature of instantaneous normal modes of the other end of the spectrum of phases of matter (gas). Once the two limits of instantaneous normal modes are understood, interpretation of instantaneous normal modes of liquids becomes an interpolation rather than an extrapolation from the solid behavior alone. We next discuss some of our current efforts to characterize instantaneous normal modes of gas systems and revisit instantaneous normal modes of liquids from a unified perspective.

4.1.4 Extending Instantaneous Normal Mode Analysis to Simple Gases 4.1.4.1

Instantaneous Normal Mode Distributions from Solids to Gases

In our recent works, we extended instantaneous normal mode analysis to various single element systems such as argon (Lennard-Jones [10, 11]), silicon (StillingerWeber [25]), and iron (modified Johnson [26, 27]). We first look at various phases of Lennard-Jones argon systems under NPT (constant number of atoms, pressure, and temperature) ensemble at 1 bar in molecular dynamics simulations. Temperature dependent density of Lennard-Jones argon from solid to gas phases at 1 bar is shown in Fig. 4.5a. We observe discontinuities between 80 and 90 K and 110 and 120 K, marking the first-order phase transitions from crystal to liquid and liquid to gas phases, consistent with prior studies. ∑ Representative pair distribution functions, .g(r ) = 4π N1nr 2 i, j ⟨δ(r − |r i − r j |)⟩, for crystal (1 K), liquid (90 K), and gas (120 K) argon are shown in Fig. 4.5b. Here, . N is the number of atoms, .n is the number density, . r i is the atomic position of the

68

4 Continuity of the Solid, Liquid, and Gas Phases of Matter

(a)

(b)

Fig. 4.5 a Temperature dependent density of LJ argon from 1 to 150 K at 1 bar which covers all three phases: crystal (blue circles), liquid (purple circles), and gas (red circles). Error bars which denote standard deviations from ensemble averages are smaller than the symbols. b Representative pair distribution function, .g(r ), of crystal (1 K), liquid (90 K), and gas (120 K) LJ argon. Large changes in structures across different phases are evident. This figure is adopted from Ref. [29]

ith atom, and the angled bracket denotes an ensemble average. The pair distribution function describes the distribution of distances between pairs of particles contained within a given volume; therefore, it contains structural information of the system of interest. As expected for argon crystals, we see very sharp peaks up to very large distances. Broadened peaks and valleys are observed for the liquid phase and the pair distribution function eventually damps out to unity with increasing distance. This decay with distance has been attributed to the structural medium range order [28]. These peaks and valleys showing structural correlations no longer exist for the gas systems except for a weak accumulation at the contact distance. With these various phases of LJ argon, we performed lattice dynamics calculations and obtained corresponding eigenfrequencies and eigenvectors. Representative instantaneous normal mode densities of states, INM(.ω), for LJ argon in crystal (1 K), liquid (90 K), and gas (120 K) phases are shown in Fig. 4.6a. For the crystalline argon, all normal modes are real at low temperatures and the density of states has sharp kinks, depicting van Hove singularities typically observed near the Brillouin zone boundaries. For the liquid argon, a large number of imaginary modes starts to appear as noted previously for liquids due to instabilities of the instantaneous snapshots of liquid structures. There is a weak temperature dependence (not shown here) to the number of imaginary modes observed in the liquid phase; however, real modes still dominate the normal mode spectra as measured by the area under the curve. For lattice dynamics calculations of argon gases, a long potential cutoff distance on the order of half the domain size was necessary to build dynamical matrices satisfying translational invariance, which leads to three Goldstone modes at 0 THz similar to solids and liquids. Corresponding normal mode density of states for gas argon at 120 K and 1 bar is shown in Fig. 4.6a. Interestingly, we observe an equal number of real and imaginary modes (and in this case, symmetric) for gas phases irrespective of temperatures. From the INM(.ω) of gases, it is apparent that interpreting the real

.

4.1 Instantaneous Normal Modes

(a)

69

(b)

Fig. 4.6 a Representative normal mode densities of states for crystal (1 K), liquid (90 K), and gas (120 K) LJ argon at 1 bar. Normal mode frequencies are multiplied by a factor of 500 for the gas phase for better visualization. b Instability factors proposed as a descriptor of phase of matter as a function of temperature. Vertical dashed (crystal to liquid) and dotted lines (liquid to gas) represent phase transitions. Shaded areas represent error limits determined from standard deviations from ensemble averages. Negative frequencies denote modes with imaginary frequencies. This figure is adopted from Ref. [29]

frequency modes in high temperature gases as conventional phonon harmonic oscillators is highly questionable. Therefore, our results show that an automatic assumption of real modes as harmonic oscillators in liquids needs to be revisited. Our results of INM(.ω) in gas systems also have implications in diffusion coefficients. Diffusion coefficients are not a unique materials property to liquids alone and if prior diffusion coefficient expressions in terms of imaginary modes are valid, they should also be applicable to gas systems. However, we find that these diffusion coefficient expressions (Eqs. 4.2 and 4.3) are not compatible with our findings of normal mode spectra of gas systems for the following reasons: (i) the fraction of imaginary modes is constant and the averaged imaginary mode frequency decreases with increase in temperature for gas systems while diffusion coefficients increase with increase in temperature (not shown), and (ii) nearly all modes are spatially localized [9] by the convention given by their inverse participation ratios. We note that our calculations demonstrate that imaginary modes are nearly half the modes observed at high temperatures in the gas limit, while relaxed and stable solids have only real modes. As discussed previously, the square of the instantaneous normal mode frequency reflects the curvature of the local potential energy landscape (PEL) that atoms participating in the normal mode see at that instant, i.e., MD snapshot. At low temperatures the system largely surveys the valleys of the PEL with positive curvatures, so the eigenvalues are positive and instantaneous normal mode frequencies are real. In contrast, at very high temperatures, the system has large enough kinetic energies to overcome any peaks and valleys in the PEL and samples the configurational space with positive curvatures (valleys) as well as the positions with negative curvatures (hills) equally. In addition to this physical picture, from random matrix theory, the eigenvalue distribution of symmetric sparse random matrices results in Wigner’s semi-circle law

70

4 Continuity of the Solid, Liquid, and Gas Phases of Matter

with equal number of positive and negative eigenvalues [30]. As matrix elements for gas systems are more randomized than those of solids and liquids (though they still obey translational invariance), it is reasonable that the numbers of positive and negative eigenvalues are equal for dynamical matrices of high temperature gases. To describe this crossover of instantaneous normal mode spectra from solid to gas, we have proposed two phenomenological factors called instability factors (IF and IFalt ), built from the microscopic mode characteristics as a measure of how unstable the system is in the configurational space as IF =

.

IFalt =

.

Ni Nr

2Ni . N i + Nr

(4.9)

(4.10)

IF represents the ratio of the number of imaginary modes to that of real modes and IFalt is linearly related to the fraction of the PEL with negative curvatures that the system sees, similar to . f u in the diffusion coefficients literature. For a relaxed solid (i.e., not near a phase transition), both instability factors are zero, while for a gas both are unity. Liquid instability factors fall between these two limits. Therefore, the instability factors proposed in Ref. [29] are a measure of ‘gasness’ in the system. These instability factors describe phases of matter in a continuous, normalized manner and provide an emergent pathway to understand thermodynamic behaviors and thermal properties of liquid states microscopically. Temperature dependent instability factor values for our LJ argon system at 1 bar are shown in Fig. 4.6b. Instability factors increase with temperature and saturate at 1 above the liquid-gas phase transition. We have additionally done lattice dynamics calculations for LJ argon at 500 and 1000 K at 1 bar and find that the instability factors indeed stay at 1. We do observe some imaginary modes for solids near melting due to the instability of the largely perturbed crystalline structures. Perhaps, these instability factors near melting could be used as a crystal-liquid phase transition criterion similar to Lindemann’s melting criterion [31–33] where root mean square of the particle displacements from the equilibrium positions reaches .∼10% of the interatomic distance.

4.1.4.2

Heat Capacity of Liquids from Both Solid and Gas Perspectives Using Instantaneous Normal Modes

To test our interpretation of normal modes describing phases of matter, we have considered constant volume heat capacity as a testing ground for a variety of single element materials, argon (Lennard-Jones [10, 11]), silicon (Stillinger-Weber [25]), and iron (modified Johnson [26, 34]), from solid to gas. At this stage, we are not looking for a rigorous theory of heat capacity of matter (though it would certainly

4.1 Instantaneous Normal Modes

71

be nice). It will take some corrections and iterations to get there. The goal here is to test out our physical view of liquids as an intermediate phase between solids and gases as evident by their normal mode spectra from a unified perspective, similar to Debye’s and Einstein’s models of phonon density of states and their respective heat capacity model. Their models are indeed very different (some would say wrong) from the actual phonon densities of states. However, what they were right about was the physical picture of viewing atomic motion in solids as vibrations. Similarly, we aim to first test out our physical view of the unified perspective of studying heat carriers in solids, liquids, and gases with each phase having an equal footing with a heat capacity model based on instability factors. In order to obtain three different phases under constant volume (with equilibrated density .ρ0 at 1 K) in molecular dynamics, it was necessary to set the temperatures in an unphysically large range from 1 K all the way up to .108 K. Our systems at the highest temperature, .108 K, is even hotter than the center of our sun at a mere 7 .∼ 1.5 × 10 K. We encourage the readers not to take the temperatures here seriously but rather simply consider the temperatures here as a knob to turn our materials from one phase to the other under the assumptions of classical molecular dynamics. We first demonstrate strong correlations between the molecular dynamics heat capacity from energy fluctuations and instability factors for these materials in Fig. 4.7a and c. Despite different spectral shapes and features among argon, silicon, and iron systems [9], strong relations between instability factors and heat capacities with small spread are evident irrespective of the system, and specific heats approach .1.5N k B around instability factors equal to unity, consistent with what we have identified as gas systems in our Lennard-Jones argon systems under NPT at 1 bar. However, we have instability factors exceeding unity for our constant volume systems described here unlike our Lennard-Jones argon systems under NPT at 1 bar. Instability factors greater than unity may be due to not being able to account for low frequency modes adequately in our fixed volume systems. Motivated by these strong correlations, we developed a simple heat capacity model based on instability factors where we approximate the total energy as the linear combination of those in solid-like (first term) and gas-like (second term) states, ( .

E = (1 − x)(3N k B T ) + x

3 N kB T 2

) (4.11)

where .x is either IF (Eq. 4.9) or IFalt (Eq. 4.10). The corresponding constant volume heat capacity for solids, liquids, and gases under the classical and harmonic approximations is then simply | (3 ) d E || CV ≡ = (1 − x)(3N k ) + x N k B B dT |V 2 . ) dx (3 − N kB T . dT 2

(4.12)

72

4 Continuity of the Solid, Liquid, and Gas Phases of Matter

(a)

(b)

(c)

(d)

Fig. 4.7 a Heat capacity calculated from molecular dynamics plotted against instability factors for various constant volume materials. Transition to .C V = 1.5N k B (dot-dash line) is observed near instability factors equal to unity. b Predicted constant volume heat capacity from instability factor (Eq. 4.12), compared to heat capacity from molecular dynamics. .ρ0 represents equilibrated density for each system at 1 K. Dashed black lines represent one to one correspondence. Good agreement is generally shown within 5% of error. This figure is adopted from Ref. [29]

The last term in Eq. 4.12 originates from the fact that instability factors are also temperature dependent and captures temperature dependence in the density of states. Resulting one-to-one comparison between heat capacities from molecular dynamics and from our proposed model (Eq. 4.12) for liquids and gases are demonstrated in Fig. 4.7b and d. .C V,I Falt tends to underestimate .C V,M D . In other words, IFalt overestimates the ‘gasness’ in these materials. .C V,I F shows a better agreement within 5% of discrepancy against .C V,M D . In using IFalt , it is naturally assumed that imaginary modes are entirely responsible for ‘gasness’ in the system and real modes are solidlike. However, in our recent work [9], we have shown that both real and imaginary modes can represent gas-like collisional and translational motion, demonstrating that IF may be more physical. Agreement between .C V,I F and .C V,M D within 5% error for various liquid and gas systems gives supporting evidence for our interpretations of

4.1 Instantaneous Normal Modes

73

normal modes in solid, liquid, and gas phases from a unified, continuous perspective via microscopically defined instability factors. Despite the qualitative and quantitative agreement between .C V,I F and .C V,M D for various materials, further refinement and development of the theory are necessary. Particularly, the effect of anharmonicity in solids beyond the harmonic oscillator model is lacking in Eqs. 4.11 and 4.12. Further, in Ref. [29], normal mode spectra of only single-element systems were investigated from solid to gas. We believe that our finding for gas systems that half the normal mode population has real frequencies and the other half imaginary frequencies is generally applicable to atomic multi-element systems and that normal modes of liquids should be considered as an intermediate state from these two limits. However, we expect that some corrections may be necessary for molecular systems and long-range atomic interactions [35] in instability factors and that gas-limit normal mode spectra in these materials include internal vibrational degrees of freedoms. So far, we described the overall population (real modes and imaginary modes) distributions and how they could relate to different phases of matter. As discussed in Chap. 2, normal modes in solids are vibrational in nature and are categorized as phonons, propagons, diffusons, and locons depending on the translational symmetry and mode interaction mechanisms. However, the physical nature of individual modes across different phases remains elusive and is largely unexplored. Recently, we have studied individual mode characters from solid to gas phases of Lennard-Jones argon at constant volume (density .ρ0 ) through the lens of inverse participation ratios (IPRs) and phase quotients (PQs) [9]. As discussed in Chap. 2, IPRs (. pn−1 ) are found by [36] .

pn−1 =

∑(∑

∗ eiα,n eiα,n

)2 (4.13)

α

i

where .eiα,n is the eigenvector component for atom .i in direction .α for the mode n. IPR gives a measure of how many atoms participate appreciably in the motion of a mode; therefore, IPR has been used to qualitatively provide information about the degree of spatial localization of modes for various complex systems [37, 38] and even scattering behaviors of acoustic and optical phonons in simple crystals [39]. Two conventional limits of the IPR is that IPR .= .1/N describes all atoms participating in the mode (delocalized) and 1 if the mode motion is completely localized to a single atom. Phase quotients have also been useful in characterizing normal modes in various materials, especially disordered solids [40–42] and is defined as [40]

.

∑ .

∑

eiα,n e jα,n | ∑α | | | i, j∈m | α eiα,n e jα,n |

P Qn = ∑

i, j∈m

(4.14)

where the sum over .m represents counting only the nearest neighbor pairs among atoms .i and . j. PQ is a measure of the averaged phase relationship of an atom and

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4 Continuity of the Solid, Liquid, and Gas Phases of Matter

its neighbors for a given mode. A PQ value of 1 then describes clusters of atoms all moving in-phase (e.g. Goldstone translational modes), and a PQ value of .−1 means that all atoms participating in the mode are moving in the opposite direction with their nearest neighbors. For our systems, we defined nearest neighbors to be within 4 V 3 ; however, our overall results are not dependent on the .dc = 2r c where . πr c = 3 N choice of .dc . Using these two lenses, we first take a look at normal mode characters in LennardJones crystalline and amorphous argon (both at .ρ0 ) as representative solid systems.

4.1.4.3

Normal Mode Characters in Solids

Inverse participation ratio—Crystalline modes are typically expected to be spatially delocalized whereas glasses are expected to have some localized modes due to structural disorder. Inverse participation ratios for both crystalline and amorphous Lennard-Jones argon are shown in Fig. 4.8a and give a measure of the degree of localization for each mode. In the literature, a normal mode has been considered to be spatially localized when less than 10–20% of atoms are participating in the mode motion [41, 43, 44]. Here, modes with IPR values over IPRloc .= 0.007 (corresponding to 10% atom participation out of the total 1372 atoms in a mode) are considered to be localized. At 0 THz, we have the well-known Goldstone modes with IPR .= .1/N for both solids where all atoms are participating in the purely translational motion. As expected for phonons in simple crystals, all modes for crystalline argon have low IPR values, demonstrating that all modes are spatially extended. For our glass argon, most low frequency modes are also delocalized with some exceptions known

(a)

(b)

Fig. 4.8 a Inverse participation ratios for individual modes in solid argon systems. Black and yellow circles represent data for glass and crystal argon systems, respectively. IPRloc represents a value above which normal modes are conventionally considered as spatially localized. IPR .= 0.5 black dashed line serves as a guide to the eye. b Phase quotients for solid argon systems. At low frequencies, PQ approaches unity as the wavelengths are large. At high frequencies, PQ approaches nearly .−1 as there is a large phase factor between neighbors. This figure is adopted from Ref. [9]

4.1 Instantaneous Normal Modes

75

as quasi-localized or resonant modes [45–47]. Prior works have proposed that the localization at low frequencies is due to finite size effects [48]. These quasi-localized modes at low frequencies have also been observed in disordered crystals [49, 50]. At high frequencies, we observe modes with large IPR values greater than IPRloc , similar to that observed previously in another single element glass (amorphous silicon) [41]. These localized vibrational modes, often referred to as locons (See discussion on locons in Chap. 2), form a small fraction of the total number of modes, about 2 to 3%. Locons typically have low thermal diffusivities and conductivities compared to extended modes [41, 51, 52], though some recent works have suggested some locons can surprisingly transport heat efficiently [53, 54]. Maximum IPR values in our argon glass are around 0.3. Phase quotient.—Individual mode phase quotients (PQ) for our crystalline and amorphous argon that characterize relative mode motion of an atom with its neighbors are shown in Fig. 4.8b. Recall that PQ .= 1 for fully in-phase and PQ .= −1 for fully out-of-phase. For the crystal argon, some modes, including the 0 THz Goldstone modes, have PQ .∼ 1 at low frequencies demonstrating that atoms are moving nearly in phase with their neighbors, physically consistent with long wavelength acoustic excitations. With increase in frequency, PQ decreases and eventually appears to fluctuate between PQ .= 0 and PQ .= −0.5 and finally approaches PQ .∼ .−1. While crystalline argon does not have optical modes (typical out-of-phase vibrations in compound crystals) due to having one atom in the primitive unit cell, away from 0 THz, acoustic modes start to develop phase differences among neighboring atoms. For the argon glass, we observe a monotonic decrease in PQ values with increase in frequency, similar to other amorphous solids reported in the literature [41, 42]. The minimum PQ values found in the argon glass are around -0.9, close to the fully out-of-phase value of PQ .= −1. Prior works utilizing PQs in glasses have proposed that modes with PQ .> 0 and PQ .< 0 represent acoustic-like and optical-like modes, respectively [41, 42]. Using this notation, an acoustic to optical-like mode transition occurs around 1.2 THz for the argon glass. With this normal mode analysis for solids as background context, we next characterize normal modes in the same way for various liquid and gas systems at higher temperatures.

4.1.4.4

Normal Mode Characters in Liquids and Gases

Inverse participation ratio.—Similar to solids, we characterize instantaneous normal modes of liquids and gases (all at the same density .ρ0 ) through inverse participation ratios and phase quotients. Inverse participation ratio spectra for select liquid and gas argon systems are shown in Fig. 4.9. Colors change from blue (liquid) to red (gas) as a function of temperature (.103 –.108 K). For simplicity, blue and purple data sets are referred to as low (.103 K) and high temperature (.105 K) liquids, respectively and red data are referred to as a gas phase (.108 K). For the low temperature liquid, we observe an asymmetric U shape with high real and imaginary frequency modes being strongly localized and low frequency

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4 Continuity of the Solid, Liquid, and Gas Phases of Matter

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4.9 a, c, e Inverse participation ratios from liquid to gas argon. Increase in temperature is denoted by the color change from blue to red. Asymptotic behaviors to IPR.= 0.5 at large frequency magnitude values are demonstrated. b, d, f Phase quotients from liquid to gas argon. Insets in e, f represent zoomed-in views. Blue, purple, and red data points represent liquids and gas systems at 3 5 8 .10 , .10 , and .10 K, respectively. This figure is adopted from Ref. [9]

4.1 Instantaneous Normal Modes

77

modes being more spatially delocalized. This IPR trend is reminiscent of the glass IPR features, although the liquid system has a large population of imaginary modes. Similar behaviors have been shown in other liquid systems [16]. In the limit of high frequency magnitudes, it appears that IPR approaches 0.5. With increase in temperature from liquid to gas, low frequency modes have higher IPR values, filling up the previously shown empty region in the U shapes found at lower temperatures. For our gas system, we have a cross-like shape with the vertical limits determined by the lowest IPR (.1/N ) from the pure translational modes and highest IPR approaching .∼0.9 and with the horizontal limits with high frequency modes having IPR .= 0.5. For the gas system, nearly all modes (over 98% of .3N ) are ‘localized’ in the conventional interpretation of having IPR .> IPRloc . Phase quotient.—Phase quotients for liquids and gases are shown in the right column of Fig. 4.9. For all systems, the phase quotient spectra appear nearly symmetric for real and imaginary modes with the exception of the long tail in the real frequency spectrum. A monotonic decrease in PQs is demonstrated moving away from 0 THz, similar to the glass system. One notable behavior seen for liquids and gases versus solids is the increasing presence of modes with PQ .= −1 with increase in temperature. In both characterizations (IPR and PQ), we do not see apparent and distinct differences between real modes and imaginary modes. Looking at the normal modes in the gas phase more closely, we find some modes with exact values of IPR .= 0.5 and/or PQ .= −1, absent in the solid systems. For a more clear visualization, bivariate probability distributions of the IPRs and PQs for argon at select temperatures are plotted in Fig. 4.10 and demonstrate a strong trend: going from solids to gases, modes with both IPR .= 0.5 and PQ .= −1 become the dominant population. In the limit that .T → ∞, we picture that this trend continues and this IPR .= 0.5 and PQ .= −1 character will dictate the instantaneous normal mode spectra. As our direction lies in using normal mode analysis to understand liquids in terms of both solids and gas phases, we look to fully understand these modes. IPR .= 0.5 & PQ .= −1 modes.—IPR of 0.5 typically means that two atoms are participating in the mode and PQ .= −1 describes a mode motion among neighboring atoms that are fully out-of-phase. From the molecular dynamics simulations, we have identified pairs of atoms that include modes with IPR.= 0.5 and PQ.= −1 and studied the roles of these pairs of atoms in all modes of the system. As a representative example, Fig. 4.11a shows all eigenvector components (.x-direction) for such a pair (atom 1: blue and atom 2: red) for all modes for the gas phase. This pair of atoms contributes negligibly to most modes (near zero amplitudes) as nearly all modes are localized to other atoms (as demonstrated by large IPR values). However, for modes with finite and sometimes large .x-components (see inset) for this pair, the mode motion of the two atoms are the same in-phase with each other, while three modes have identical amplitudes in the opposite direction. Thus, in the ocean of other mode motions, their relative motions are determined by three modes with opposite directions with IPR .= 0.5 and PQ .= −1. . y and .z eigenvector components have similar trends.

78

4 Continuity of the Solid, Liquid, and Gas Phases of Matter

(a)

(b)

(c)

(d)

Fig. 4.10 Bivariate probability distribution functions of inverse participation ratios and phase quotients for a solid argon (orange data for crystal and black data for glass) and b–d liquid to gas argon, adopted from Ref. [9]. Color schemes are identical to that of Fig. 4.9. A clear trend of IPR .= 0.5 and PQ .= −1 modes becoming more prominent going from solid and liquid to gas is demonstrated

Interestingly, the three modes that govern the motion of the pairs of atoms with IPR .= 0.5 and PQ .= −1 are composed of one real mode and two imaginary modes. Visualization of these three modes projected on the pair of atoms is shown in the inset of Fig. 4.11b. The real modes have eigenvectors along the ‘bond’ (green arrows), while atomic motion for the other two imaginary modes are perpendicular to the bond. In short, the repulsive force along the bond directs the atoms to a more stable position while there is no energy loss in the motion of the other two modes, similar to unstable rotational modes. In bulk dense solids, each atom is located in an effective potential well that is formed by the contributions from all neighboring atomic potential interactions. However, for the pairs of atoms dominating the mode motions for IPR .= 0.5 and PQ .= −1 modes, the behavior is simulating purely independent two-atom motion governed by a single pair potential (.Ui, j (r )) that defines the mode frequencies, while other interactions are negligible. For purely independent two atom interactions, we have three Goldstone modes (two atoms moving in-phase for three Cartesian degrees of freedom) and three out-of-phase modes with opposite eigenvector directions with one being along the bond. Mode frequency along the bond is strictly determined

4.1 Instantaneous Normal Modes Fig. 4.11 a Eigenvector components (.x-direction for one pair of atoms identified to have IPR .= 0.5 and PQ .= −1) for all modes for argon gas at .108 K. Blue and red colors represent atom 1 and atom 2 in the pair. Inset shows a zoomed-in view of . x-eigenvector components near 0 THz. All eigenvectors are in-phase between the two atoms except three which are fully out-of-phase, similar to pure, independent two-atom interactions. All pairs of atoms with these features are identified and their along-the-bond mode (green arrows in inset) frequencies and their distances are shown in b. Clear relations to a single LJ pair potential (black dashed line) is demonstrated. This figure is adopted from Ref. [9]

79

(a)

(b)

∂ 2 U (r)

by . ∂ri, 2j . Along-the-bond mode frequencies with IPR .= 0.5 and PQ .= −1 as a function of distance between the atomic pairs are shown in Fig. / 4.11b for both liquid

and gas systems. The nearly perfect power law behavior (. ∂ ∂rU2L J ∼ r −7 at small .r ) demonstrates that these mode frequencies are purely determined by the second derivative of a single pair potential, similar to truly independent two-atom interactions. We also observe that there is a temperature dependence in the bond distance and mode frequencies. There is a large distribution of bond lengths and frequencies at high temperatures while the spread is smaller for low temperatures. Dynamical matrices are built based purely on the potential of a snapshot structure. However, kinetic energies play an important role in determining the atomic locations that feed into the dynamical matrices, especially at high temperatures (liquids and gases). The larger the kinetic energies that atoms have, the closer they can approach each other, leading to higher frequencies for these modes. Collisons.—As noted in Chap. 2, normal modes are typically considered as a synonym to phonon quasi-particles or harmonic oscillators in solids [55]. Even in gases, there can be internal degrees of freedom for non-monoatomic species that lead to vibrations (e.g., H2 and O2 ). However, the results described above show that along-the-bond modes with IPR .= 0.5 and PQ .= −1 should be considered as collisional rather than vibrational. Even with absence of kinetic energy, if pairs of 2

80

4 Continuity of the Solid, Liquid, and Gas Phases of Matter

(a)

(b)

Fig. 4.12 Representative eigenvector maps of modes with high inverse participation ratio values: a b 0.1 THz mode with IPR.= 0.74, among the highest IPRs for the gas system, adopted from Ref. [9]. Eigenvector amplitudes are multiplied by a factor of 10 for easier visualization, same as Fig. 4.11b. We observe that for these modes, eigenvector magnitudes are concentrated to a single atom, reminiscent of a single atom translating in space. All the other eigenvector magnitudes appear as dots as the magnitudes are small

.−0.3 THz mode with IPR.= 0.82 and

argon atoms are initially placed at distances shown in Fig. 4.11b and their motions are allowed to evolve with time, their potential energy is positive (.U L J (r < 3.4 Å.) > 0) such that they dissociate after participating in the along-the-bond mode motion with kinetic energies determined by energy conservation. We, thereby, characterize these modes as two-body collisional modes or ‘collisons’. While not exclusively shown, we anticipate that there are also collisons representing many-body collisions. Translatons.—As evident from Fig. 4.9, the highest IPR values which are located near 0 THz also depend on temperature. Traditional interpretation of IPRs that IPR is a measure of the number of atoms participating in the mode does not strictly work here as there are many IPRs between. 21 (2 atoms participating) and 1 (1 atom participating). However, we find that these high IPR modes appear to have concentrated eigenvector amplitude on a single atom as demonstrated in Fig. 4.12. It is, therefore, expected that under different conditions there exist modes with IPR approaching 1 where mode motion is projected to a single atom translating through space with small frequency. This is physically consistent with dilute gas normal modes where nearly all modes have zero eigenfrequencies due to lack of or weak potential interactions. As we progressively reach lower densities, atomic interactions will become less frequent and more uninterrupted translational motion will appear, which would have .∼ zero eigenfrequency. At the same time, in the limit of .T → ∞ where atoms become point particle-like and overall collisions become less frequent, we would expect modes that purely describe translational motion, or what we may characterize as ‘translatons’, to appear.

4.1 Instantaneous Normal Modes

81

Fig. 4.13 Spectrum of eigenmodes of dynamical matrices from .T = 0 K to .T → ∞ under a constant volume, adopted from Ref. [9]. At .T = 0 limit, we have vibrational modes (phonons, propagons, diffusons, and locons depending on disorder and localization). At the high temperature limit, both real and imaginary modes are expected to be gas-like, which we characterize as collisons and translatons, different from vibrational modes. Normal modes for finite, intermediate temperature structures could potentially be described through these two limits

Our results on the instantaneous normal mode analysis on heat capacities [29], pair distribution functions, and mode characters by IPRs and PQs, suggest that in the limit of .T → ∞, particles will effectively behave as point-particles and collisional and translational modes, different from vibrational modes, will dominate the particle dynamics. This is in parallel with real gas collisions where two-atom collisions become statistically dominant with increase in temperature over many-body collisions [56]. In glasses, normal modes are vibrational within a local potential well and have been categorized into propagons, diffusons, and locons depending on the degree of localization and interaction mechanisms [41, 57]. Although there exist different methods to classify these modes [43, 51, 58], they often rely on the normal mode shapes, i.e., eigenvector characters. Similarly, we have classified normal modes at . T → ∞ into collisons and translatons from their eigenvector characters and their local atomic and potential environment. With these new insights into the behavior of normal modes of gas systems and existing vast literature regarding T .= 0 normal modes in solids, we propose that normal modes in liquids may be best considered as a collection of both solid-like vibrational and gas-like collisional and translational modes as summarized in Fig. 4.13. While the probability of having two-body collisional modes decreases with decreasing temperature, we still observe these modes even in the lowest temperature liquid argon considered here. Here for demonstration purposes, a simple Lennard-Jones pair potential was used. However, we expect that our high temperature limit results (gas) will not depend on the choice of interatomic potential (two-body, many-body, etc.) as all systems will become an atomic gas where details of the nature of the interactions become less important (whether it is a two body or many-body interaction). We anticipate that gaslike modes, collisons and translatons, would also appear in multi-element systems (for example, molecules colliding with other molecules).

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4 Continuity of the Solid, Liquid, and Gas Phases of Matter

So far in this chapter, we have explored and discussed some recent developments to generalize the use of normal modes to describe materials properties in various phases of matter. For the next part of the chapter, we describe the other spectra that lead to .3N atomic degrees of freedom when integrated over frequency: velocity autocorrelation function spectra and their features in the velocity autocorrelation spectra in liquids.

4.2 Velocity Autocorrelation Decomposition of Atomic Degrees of Freedom in Liquids In Chap. 3, we described some properties of velocity autocorrelation (VACF) functions. There are many interesting features in VACF(.ω) that indicate different physical phenomena. In harmonic solids, it was shown that VACF(.ω) is equivalent to equilibrium normal mode spectra. VACF is also useful in non-solids because integration in VACF(t) over time leads to diffusion coefficients. Fourier-transforming VACF(t) into frequency space, VACF(.ω = 0) represents the diffusion coefficients. Using these properties, there have been numerous efforts to understand features in VACF(.ω) of liquids from both theoretical and experimental perspectives [59–66]. In this chapter, we focus on describing the nature of heat carriers of liquids using VACF(.ω). One seminal example to understand the overall shapes of VACF(.ω) of liquids by Rahman [59] is demonstrated in Fig. 4.14 where VACF spectra from molecular dynamics simulations are compared against the Langevin diffusion model for a Lennard-Jones argon liquid. The difference between the two curves (peak at.β = 0.2) was thought to originate from solid-like behavior (vibrations) in liquids. However, the difference at high frequencies (black curve—red curve) leads to unphysically negative values. Motivated by some of these early works in comparing the MD VACF(.ω) and theoretical models such as Langevin diffusion and Gaussian spectral distribution, there have been more concrete proposals to qualitatively divide the VACF(.ω) into

Fig. 4.14 Velocity autocorrelation spectra of liquid argon (Lennard-Jones) at 94.4 K, adopted from Ref. [59]. The density is 1.374 g cm–3 . Black curve is from molecular dynamics and red curve is a Langevin-type diffusion 2function of the form, . λ2λ+β 2 where .λ is related to the diffusion coefficient, . D by .λ = h/m D

4.2 Velocity Autocorrelation Decomposition of Atomic Degrees of Freedom in Liquids

83

Fig. 4.15 Spectral density of liquids sodium at 373 K determined from neutron scattering experiments, adopted from Ref. [61]. Qualitative division of the total spectra into diffusive and vibratory components is proposed

two separate motions: atomic diffusion and vibrations [61]. One such example is shown for liquid sodium in Fig. 4.15 where low frequency VACF(.ω) is dominated by diffusive motion and med-to-high frequency VACF(.ω) is solely attributed to vibrations. However, quantitative framework of this categorization of VACF(.ω) was not provided and only qualitative arguments were given. Since then, there have been some efforts to quantitatively dissect VACF(.ω) into these two motions. With these seminal velocity autocorrelation works in mind, we next revisit instantaneous normal mode spectra and examine how we can bridge the gap between the two different spectra.

4.2.1 Bridging the Gap Between INM(.ω) and VACF(.ω) Following Zwanzig’s work relating the diffusion coefficient (integral of time domain velocity autocorrelation) to normal modes [19], some have attempted to directly relate these two spectra with one-to-one correspondence by connecting the imaginary part of INM(.ω) and hopping rates to the topology of the potential energy landscape in the configuration space as shown in Fig. 4.16. The proposed INM theory appears to describe the features of the time-domain VACF well though with some discrepancies in magnitudes. However, the relationship between INM(.ω) and VACF(.ω) is not obvious. In INM(.ω), curvatures of the potential energy landscape are described through normal modes based on snapshot structures and we have proposed to characterize phases of matter in terms of the averaged distribution of instantaneous normal modes, i.e., instability factors. On the other hand, VACF(.ω) captures time dependent atomic dynamics more directly. Further, anharmonic vibrations are intrinsically included when calculating VACF(.ω) in molecular dynamics, not captured by harmonic instantaneous normal mode calculations. Therefore, direct one-to-one comparisons between VACF and INM are not currently feasible. Motivated by prior works bisecting the VACF(.ω) into diffusional and vibrational atomic motion, we have proposed to connect VACF(.ω) and INM(.ω) via the instability

84

4 Continuity of the Solid, Liquid, and Gas Phases of Matter

Fig. 4.16 Time domain velocity autocorrelation function of supercooled Lennard-Jones argon at 80 K near glass transition, adopted from Ref. [62]. Solid curve is based on molecular dynamics simulation and dotted curve is based on instantaneous normal mode theory

factor. In monatomic systems where we do not have additional internal degrees of freedom, we assume that atomic diffusion describes gas-like motion and vibration is intrinsic to solids. In the case of a gas (IF.= 1 for our analysis), VACF(.ω) is Lorentzian with a peak given by. D and integral describing all.3N atomic degrees of freedom, with no phonon quasi-particles. For a relaxed solid (IF .= 0), VACF(.ω) strictly describes phonon density of states, with no free diffusing atoms. We partition VACF(.ω) into a gas-like portion, .VACFgas (ω), with negligible attractive potential interactions and a solid-like portion, .VACFsolid (ω), with strong potential interactions. We assume that .VACFgas (ω) has a Lorentzian lineshape with the height determined by the diffusion coefficient. Furthermore, its width is chosen such that { 1 (4.15) . VACFgas (ω)dω = IF 3N VACFsolid (.ω) is then determined by subtracting VACF(.ω) by VACFgas (.ω), .

1 3N

{ VACFsolid (ω)dω = 1 − IF

(4.16)

This way, anharmonic vibrations are still accounted for in VACFsolid (.ω). Calculated VACF(.ω) and corresponding gas-like and solid-like decompositions for all systems (argon, silicon, and iron) at select low, intermediate, and high temperatures are shown in Fig. 4.17. If the ‘gasness’ in the system is severely overpredicted, hence affecting the Lorenzian linewidth, .VACFsolid (ω) will be negative over a wide range of frequencies, which is unphysical. The results here satisfy this test, suggesting the soundness of this decomposition and the reasonableness of the choice of IF as a bridge between INM(.ω) and VACF(.ω).

4.3 Concluding Remarks

85

(a)

(b)

(d)

(e)

(f)

(h)

(i)

(g)

(c)

Fig. 4.17 Decomposition of spectral velocity autocorrelation function, VACF(.ω) at select temperatures of (a–c) argon (Lennard-Jones [10, 11] from 103 to 108 K, (d–f) silicon (Stillinger-Weber [25]) from .5 × 103 to 107 K, and (g–i) iron (modified Johnson [26, 27]) from .5 × 103 to 106 K. All systems were evaluated at .ρ0 (equilibrated density at 1 K). Black curves represent spectra of gaslike degrees of freedom, VACFgas (.ω), from instability factor and orange curves represent spectra of solid-like degrees of freedom, VACFsolid (.ω), obtained from subtraction of VACF(.ω) by the black curves. This figure is adopted from Ref. [29]

4.3 Concluding Remarks We started this Book with historical approaches to studying liquid state physics: from more well-established solid or gas perspectives. While these approaches have led to fruitful new findings about the nature of liquids, I believe that we should view liquids from more holistic approaches as a subset of a more general description of matter. With this in mind, we discussed two spectral methods across different phases of matter that both recover .3N atomic degrees of freedom: (equilibrium and instantaneous) normal mode and velocity autocorrelation spectra. Equilibrium normal modes are typically used as a synonym to phonons in crystals as planewave solution decompositions are good descriptors of oscillating motion

86

4 Continuity of the Solid, Liquid, and Gas Phases of Matter

such as atomic vibrations. Thermal properties of crystals can then be described microscopically via these normal modes. Some examples were shown for thermal conductivity and heat capacity. For disordered solids such as glasses, equilibrium normal modes which are still vibrational in nature are no longer spatially periodic and the notion of a wavevector of a mode loses its meaning. Different types of classifications of these normal modes (propagons, diffusons, and locons) determined by their mode characters in glasses were discussed. Following the success of equilibrium normal modes in solids, there have been recent interests to extend normal mode analysis to instantaneous structures of liquids known as instantaneous normal modes. In the limit of minute perturbations away from the equilibrium positions, equilibrium and instantaneous normal modes approach equality. In liquids where we no longer have equilibrium positions, extrapolation of our interpretation of equilibrium normal modes in solids to instantaneous normal modes is typically considered in literature. Rather than approaching liquids from one end of the spectrum (solids), we extend the instantaneous normal mode analysis to gas systems such that the interpretation of instantaneous normal modes in liquids now becomes an interpolation between the two limits. We find that nearly half the modes in the gas systems are real, raising questions towards an automatic assumption of considering instantaneous normal modes of liquids as harmonic oscillators. Further, we have proposed two new types of normal modes (collisons and translatons) based on their mode characters that can describe gas-like atomic collisions and translations, leading to a more comprehensive description of normal modes (hence, atomic degrees of freedom) across different phases from solid to gas. Through microscopically derived instability factors describing ‘gasness’ in instantaneous normal mode spectra, we have proposed a simple heat capacity model from solid to gas phases in a unified manner and observed decent agreements against independently computed heat capacities, providing supporting evidence of our holistic view of matter. However, our work is in infancy and some details including the effect of anharmonicity need to be ironed out. In solids, velocity autocorrelation function spectra and normal mode spectra are equivalent under harmonic approximations. At high temperatures where anharmonicity becomes increasingly important in solids, velocity autocorrelation function spectra intrinsically include anharmonic effects (e.g. frequency shifts) via molecular dynamics. Velocity autocorrelation function spectra are also useful in non-solids as they can also represent atomic diffusion processes. Motivated by prior works trying to decompose velocity autocorrelation spectra, we made efforts to bridge the gap between the normal mode and velocity autocorrelation spectra of liquids and gases, a necessary step towards further understanding and using these spectra for other thermal properties, through instability factors.

References

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4.4 Outlook In addition to conventional solid, liquid, and gas phases, we expect that considering heat carriers in a continuous manner could also be useful in studying thermodynamics and thermal properties of non-conventional materials including liquid crystals and solid ionic conductors. There have been a lot of recent research interests for thermoelectric power generators in certain solid ionic conductors dubbed ‘phononliquid, electron crystal’ where atoms at sub-lattice sites are fixed while others are diffusing [67–70]. This leads to a desirable low heat capacity and non-electronic thermal conductivity but the origin of these is not yet clear. Typically, only phonon descriptions or the lack thereof are utilized to explain these phenomena. It is possible that our viewpoint could help identify the mechanism behind these complex phenomena. Similar to liquid state physics, strongly correlated electrons are also notoriously challenging to characterize and understand. Perhaps, this holistic view could be useful in other disciplines in science. As demonstrated by this Book, the identity and nature of heat carriers describing atomic motions in liquids are still very active areas of research. With amazing new tools we have as mentioned in Chap. 1, we learn something new about atomic dynamics in the liquid phase everyday both computationally and experimentally. Our theories of liquids are still at infancy and are dynamically changing versus more mature, established theories in solids such as phonons. Perhaps, with more data accumulated over time, rigorous theories microscopically describing liquids materials properties will someday emerge, verified against numerous data. Similar to what we have observed in the breakthroughs of various applications ranging from solar cells to semiconductor devices through our well-grounded theories in phonons and electrons in solid state physics, rigorous theories in liquids will enable more control in materials properties of liquids and lead to numerous disruptive technological advances in the society that we haven’t even thought of. I cannot wait for what’s to come.

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