Carbon Nanothreads Materials (Materials Horizons: From Nature to Nanomaterials) 9811909113, 9789811909115

Table of contents :
Preface
Contents
1 Introduction
1.1 Introduction
1.2 Structural Morphology of Carbon Nanothreads
1.3 General Development and Current Situation of Carbon Nanothreads in Nanoscience and Nanotechnology
1.4 Fundamental Properties and General Behaviors of Carbon Nanothreads
References
2 Experimental Aspect
2.1 Introduction
2.2 Fabrication Methods
2.2.1 Pressure-Induced Polymerization of Benzene
2.2.2 Mechanochemical Synthesis
2.2.3 Pressure-Induced Polymerization of Non-Aromatic Molecules
2.3 Testing Technologies
2.3.1 X-ray Diffraction
2.3.2 Nuclear Magnetic Resonance
2.3.3 Raman Spectroscopy
2.3.4 Other Characterization Techniques
2.4 Potential Application and Researching Significance
2.4.1 Biotechnology
2.4.2 Nanoelectronics
2.4.3 Nanocomposites
References
3 Topological Structure
3.1 Introduction
3.2 Enumeration Rules
3.2.1 Degree-Two Nanothreads
3.2.2 Degree-Four Nanothreads
3.2.3 Degree-Six Nanothreads
3.3 Reaction Pathways
3.4 Stone Wales Defects
References
4 Mechanical Properties of Carbon Nanothreads
4.1 Introduction
4.2 Computational Methods
4.2.1 Density Functional Theory
4.2.2 Molecular Dynamics
4.3 Mechanical Properties
4.3.1 Fundamental Mechanical Properties
4.3.2 Morphology and Temperature Dependent Tensile Properties
4.3.3 Defect Dependent Tensile Properties
4.3.4 Structure Dependent Ductility
4.3.5 Functionalized Carbon Nanothreads
References
5 Electronic Properties of Carbon Nanothreads
5.1 Introduction
5.2 Computational Methodology
5.3 Electronic Band Structure
5.4 Comparison of Band Structure
5.4.1 Carbon Nanotube
5.4.2 Boron Nitride Nanotube
5.4.3 Carbon Nitride Nanotube
5.4.4 Silicon Carbide Nanotube
5.4.5 Graphyne Nanotube
References
6 Thermal Properties of Carbon Nanothreads
6.1 Introduction
6.2 Computational Methodology
6.2.1 Direct Method for Thermal Conductivity
6.2.2 Green–Kubo Method for Thermal Conductivity
6.2.3 Model for Interfacial Thermal Resistance
6.2.4 Effect of Force Field on Thermal Conductivity
6.2.5 Generalized Debye-Peierls/Allen-Feldman Model for Thermal Conductivity
6.3 Thermal Conductivity of Carbon Nanothreads
6.3.1 Phonon and Localization
6.3.2 Thermal Conductivity
6.3.3 Interfacial Thermal Resistance
6.4 Comparison of Thermal Conductivity
6.4.1 Carbon Nanotube
6.4.2 Boron Nitride Nanotube
6.4.3 Silicon Carbide Nanotubes
6.4.4 Graphyne Nanotubes
References
7 Carbon Nanothreads-Reinforced Polymer Nanocomposites
7.1 Introduction
7.2 Computational Methodology
7.2.1 Coarse-Grained Method
7.2.2 Dissipative Particle Dynamics
7.3 Mechanical Properties of Polymer Nanocomposites
7.3.1 Tensile Properties of PE Composites
7.3.2 Tensile Properties of PMMA Composites
7.3.3 Glass Transition Temperature of PMMA Composites
7.3.4 Tensile Properties of Epoxy Composites
7.3.5 Tensile Properties of PE Composites with Curved Carbon Nanothreads
7.4 Thermal Properties of Polymer Nanocomposites
7.5 Coarse-Grained Study of Polymer Nanocomposites
References
8 Arrangements of Carbon Nanothreads
8.1 Introduction
8.2 Carbon Nanothread Rods
8.3 Carbon Nanothread Forests
8.4 Carbon Nanothread Nanomeshes and Nanoforms
8.5 Carbon Nanothread Cubanes
8.6 Carbon Nanotube Superstructure
8.6.1 Carbon Nanotube Yarn
8.6.2 Carbon Nanotube Sponge
8.6.3 Carbon Nanothread Pillared-Graphene
References
9 Technologically Relevant Applications
9.1 Methodology for Nanofiltration
9.1.1 Dual Control Volume Grand Canonical Molecular Dynamics
9.1.2 Force-Based Non-equilibrium Molecular Dynamics
9.1.3 Mixed Non-equilibrium Molecular Dynamics
9.1.4 Concentration Gradient-Driven Non-equilibrium Molecular Dynamics
9.2 Nanoresonator
9.3 Gas Membrane
9.4 Water Membrane
9.5 Energy Storage
References

Citation preview

Materials Horizons: From Nature to Nanomaterials

Kim Meow Liew Wei-Ming Ji Lu-Wen Zhang

Carbon Nanothreads Materials

Materials Horizons: From Nature to Nanomaterials Series Editor Vijay Kumar Thakur, School of Aerospace, Transport and Manufacturing, Cranfield University, Cranfield, UK

Materials are an indispensable part of human civilization since the inception of life on earth. With the passage of time, innumerable new materials have been explored as well as developed and the search for new innovative materials continues briskly. Keeping in mind the immense perspectives of various classes of materials, this series aims at providing a comprehensive collection of works across the breadth of materials research at cutting-edge interface of materials science with physics, chemistry, biology and engineering. This series covers a galaxy of materials ranging from natural materials to nanomaterials. Some of the topics include but not limited to: biological materials, biomimetic materials, ceramics, composites, coatings, functional materials, glasses, inorganic materials, inorganic-organic hybrids, metals, membranes, magnetic materials, manufacturing of materials, nanomaterials, organic materials and pigments to name a few. The series provides most timely and comprehensive information on advanced synthesis, processing, characterization, manufacturing and applications in a broad range of interdisciplinary fields in science, engineering and technology. This series accepts both authored and edited works, including textbooks, monographs, reference works, and professional books. The books in this series will provide a deep insight into the state-of-art of Materials Horizons and serve students, academic, government and industrial scientists involved in all aspects of materials research.

More information about this series at https://link.springer.com/bookseries/16122

Kim Meow Liew · Wei-Ming Ji · Lu-Wen Zhang

Carbon Nanothreads Materials

Kim Meow Liew Department of Architecture and Civil Engineering City University of Hong Kong Hong Kong, China

Wei-Ming Ji Department of Architecture and Civil Engineering City University of Hong Kong Hong Kong, China

Lu-Wen Zhang Department of Engineering Mechanics School of Naval Architecture Ocean and Civil Engineering Shanghai Jiao Tong University Shanghai, China

ISSN 2524-5384 ISSN 2524-5392 (electronic) Materials Horizons: From Nature to Nanomaterials ISBN 978-981-19-0911-5 ISBN 978-981-19-0912-2 (eBook) https://doi.org/10.1007/978-981-19-0912-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The great discovery of carbon nanothreads has sparked an innovation in material science because of their superior physical, chemical, electronic, and mechanical properties. A good understanding of these properties is a key factor for the utilization of carbon nanothreads in scientific research and practical engineering. Traditional experimental research can provide brief information about the structural morphology but incomplete atomistic details of carbon nanothreads. Furthermore, experimental research inevitably encounters challenges in property characterization when entering into the nanoscale regime. Therefore, theoretical study plays an important role in this kind of nanomaterial. This book is the first book for carbon nanothreads with complete and comprehensive knowledge covering theories, numerical methods, and properties comparisons with other carbon-based nanomaterials. For one thing, the main theoretical aspects in this book include First-Principle Calculation, Density Functional Theory, Classical Molecular Dynamics Simulation, Non-equilibrium Molecular Dynamics Simulation, and Coarse-grained Simulation. For another thing, the main research contents include Fundamental Mechanical Properties, Fracture Characteristics; Electronic and Magnetic Properties; Thermal Properties; Reinforcement in Polymer Composites, and other promising applications in engineering. The target of this book is to provide to our peers the available theoretical and numerical methods, and useful computational results of carbon nanothreads for reference. This book can be used as a comprehensive source for scientists, academics, researchers, and engineers in various areas of engineering, physical sciences, and computational modeling. In order to achieve this target, the book introduces the microstructure information of carbon nanothreads and the modeling details at full length. The tunable mechanisms of physical properties of carbon nanothreads are discussed in detail, which enable integration of these nanoscale components into high order structures for “bottom-up design” purpose. The revealed reinforced mechanisms of carbon nanothreads in polymer composites can provide theoretical guidance for engineering design of advanced polymer composites. Finally, based on the aforementioned study, we investigate the most promising applications such as nano-

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resonator, gas membrane, and energy storage. Also, a special effort has been made to have the chapters as independent from one another as possible so that the interested chapters can be individually selected for self-study. This book covers 9 chapters and is organized in the following manner. Chapter 1, Introduction, introduces the background knowledge of carbon nanothreads, including the history of discovery and commonly synthesized techniques. The prospective applications of carbon nanothreads in nanoscience and nanotechnology are also presented. Chapter 2, Experimental Aspect, comprehensively introduces experimental tools in preparing and testing carbon nanothreads as well as the challenges and approaches of synthesizing single-crystalline nanothreads with both translational and orientational orders. Chapter 3, Topological Structure, introduces Euler’s rules to systematically identify all kinds of topologically distinct carbon nanothreads and presents the formation process and controlled phase diagram of carbon nanothreads from atomistic insights. Chapter 4, Fundamental Mechanical Properties, predicts the outstanding mechanical properties of carbon nanothreads through first-principle calculation and molecular dynamics simulation, including elastic constants, bending stiffness, and torsional stiffness. Chapter 5, Electronic and Magnetic Properties, introduces the specific electronic and magnetic properties of carbon nanothreads and analyzes the influence of topological structure on the conducting/semiconducting properties of carbon nanothreads. Chapter 6, Thermal Properties, introduces the fundamental thermal properties of various carbon nanothreads and provides manipulation strategies to tune the thermal conductivity. Chapter 7, Polymer Composites, explores the promising reinforcing efficiency of carbon nanothreads in improving the thermal-mechanical properties of polymer composites from multiscale prospective and elucidates the cavitationcrazing transition mechanism for microstructure design. Chapter 8, Arrangements of Carbon Nanothreads, show other kinds of carbon nanothread-based nanostructures, such as carbon nanothreads rods, carbon nanothreads forests, carbon nanothreads nanomeshes, and Carbon nanothreads cubanes. Chapter 9, Technologically Relevant Applications, introduces technologically relevant applications of carbon nanothreads. With the increasing interests and development of nanotechnology, we believe that the carbon nanothreads will be a promising alternative to traditional carbon-based nanomaterials. The predicted properties and revealed mechanisms from computational approach will provide significant guidance for experimental design in future, and further facilitate the application of carbon nanothreads. Hong Kong, China Hong Kong, China Shanghai, China

Kim Meow Liew Wei-Ming Ji Lu-Wen Zhang

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Structural Morphology of Carbon Nanothreads . . . . . . . . . . . . . . . . . 1.3 General Development and Current Situation of Carbon Nanothreads in Nanoscience and Nanotechnology . . . . . . . . . . . . . . . 1.4 Fundamental Properties and General Behaviors of Carbon Nanothreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 6 7 8

2 Experimental Aspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fabrication Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Pressure-Induced Polymerization of Benzene . . . . . . . . . . . . 2.2.2 Mechanochemical Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Pressure-Induced Polymerization of Non-Aromatic Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Testing Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 X-ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Raman Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Other Characterization Techniques . . . . . . . . . . . . . . . . . . . . . 2.4 Potential Application and Researching Significance . . . . . . . . . . . . . 2.4.1 Biotechnology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Nanoelectronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Nanocomposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 12 17

3 Topological Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Enumeration Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Degree-Two Nanothreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Degree-Four Nanothreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 46 47 49

21 32 32 34 36 38 39 40 41 41 41

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3.2.3 Degree-Six Nanothreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Reaction Pathways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Stone Wales Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 62 64 68

4 Mechanical Properties of Carbon Nanothreads . . . . . . . . . . . . . . . . . . . . 69 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2.1 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2.2 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3 Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3.1 Fundamental Mechanical Properties . . . . . . . . . . . . . . . . . . . . 81 4.3.2 Morphology and Temperature Dependent Tensile Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3.3 Defect Dependent Tensile Properties . . . . . . . . . . . . . . . . . . . . 92 4.3.4 Structure Dependent Ductility . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.3.5 Functionalized Carbon Nanothreads . . . . . . . . . . . . . . . . . . . . 104 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5 Electronic Properties of Carbon Nanothreads . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Computational Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Electronic Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Comparison of Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Carbon Nanotube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Boron Nitride Nanotube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Carbon Nitride Nanotube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Silicon Carbide Nanotube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Graphyne Nanotube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 120 122 144 144 146 148 150 151 155

6 Thermal Properties of Carbon Nanothreads . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Computational Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Direct Method for Thermal Conductivity . . . . . . . . . . . . . . . . 6.2.2 Green–Kubo Method for Thermal Conductivity . . . . . . . . . . 6.2.3 Model for Interfacial Thermal Resistance . . . . . . . . . . . . . . . . 6.2.4 Effect of Force Field on Thermal Conductivity . . . . . . . . . . . 6.2.5 Generalized Debye-Peierls/Allen-Feldman Model for Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Thermal Conductivity of Carbon Nanothreads . . . . . . . . . . . . . . . . . . 6.3.1 Phonon and Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Interfacial Thermal Resistance . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Comparison of Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Carbon Nanotube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 157 157 157 162 165 165 166 169 169 173 181 186 186

Contents

6.4.2 6.4.3 6.4.4 References

ix

Boron Nitride Nanotube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Silicon Carbide Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphyne Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .....................................................

187 188 189 191

7 Carbon Nanothreads-Reinforced Polymer Nanocomposites . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Computational Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Coarse-Grained Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Dissipative Particle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Mechanical Properties of Polymer Nanocomposites . . . . . . . . . . . . . 7.3.1 Tensile Properties of PE Composites . . . . . . . . . . . . . . . . . . . . 7.3.2 Tensile Properties of PMMA Composites . . . . . . . . . . . . . . . . 7.3.3 Glass Transition Temperature of PMMA Composites . . . . . . 7.3.4 Tensile Properties of Epoxy Composites . . . . . . . . . . . . . . . . . 7.3.5 Tensile Properties of PE Composites with Curved Carbon Nanothreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Thermal Properties of Polymer Nanocomposites . . . . . . . . . . . . . . . . 7.5 Coarse-Grained Study of Polymer Nanocomposites . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 195 196 196 207 210 210 213 222 228

8 Arrangements of Carbon Nanothreads . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Carbon Nanothread Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Carbon Nanothread Forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Carbon Nanothread Nanomeshes and Nanoforms . . . . . . . . . . . . . . . 8.5 Carbon Nanothread Cubanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Carbon Nanotube Superstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Carbon Nanotube Yarn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Carbon Nanotube Sponge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.3 Carbon Nanothread Pillared-Graphene . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251 251 252 255 260 265 266 267 269 271 273

9 Technologically Relevant Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Methodology for Nanofiltration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Dual Control Volume Grand Canonical Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Force-Based Non-equilibrium Molecular Dynamics . . . . . . . 9.1.3 Mixed Non-equilibrium Molecular Dynamics . . . . . . . . . . . . 9.1.4 Concentration Gradient-Driven Non-equilibrium Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Nanoresonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Gas Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Water Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Energy Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

275 275

231 238 242 248

275 277 279 280 281 289 292 297 305

Chapter 1

Introduction

1.1 Introduction In 2001, Stojkovic et al. [1] proposed theoretically an extremely small-diameter (0.4 nm) carbon nanowire with large band gaps and ultrahigh stiffness, which firstly uncovered the mystery of carbon nanothreads. Until 2014, the existence of carbon nanothreads was demonstrated experimentally by Fitzgibbons et al. [2], opening a fresh chapter on nanoscience and nanotechnology. Due to the extraordinarily high specific strength predicted by modeling, carbon nanothreads have been considered as ideal building blocks for lightweight cables which can truly realize the theoretical concept of “space elevator”. From the atomistic insights, carbon nanothreads are analogous to diamond while consist of one-dimensional sp3 bonded C–C bonds with their outer surface terminated in hydrogen. The thinnest carbon nanothread has a diameter of around 0.2 nm, which is the smallest carbon nanostructure among the discovered carbon allotropes. Nevertheless, the simple sp3 -bonding carbon network leads to the most diverse and richest among nanomaterials regarding structures and structure–property relations. In the past 5 years, carbon nanothreads have drawn intensive research interests owing to their outstanding mechanical, thermal, electronic, and magnetic properties as well as ultra-low density, becoming a promising candidate for a wide range of scientific and engineering applications. For instance, carbon nanothreads are predicted with a higher mass resolution of 0.58 yg with respect to carbon nanotubes (CNTs) (about 10 yg), indicating that carbon nanothreads are appealing candidates for nanomechanical resonators [3]. The atomistic structure of carbon nanothreads is considered as a topological unit cell with at least two bonds bridging each adjacent pair of benzenes. According to Euler’s rules, there are 50 topologically distinct carbon nanothreads, 15 of which are within 80 meV/carbon atom of the most stable member based on first principle calculations [4]. These lowest energy nanothreads display three kinds of chirality, including achiral, stiff chiral, and soft chiral interconnection patterns. The crosssectional area of the carbon nanothread structure is approximated by λV0 , where V0 is the atomic volume for carbon atom in bulk diamond, and λ is the linear atom © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. M. Liew et al., Carbon Nanothreads Materials, Materials Horizons: From Nature to Nanomaterials, https://doi.org/10.1007/978-981-19-0912-2_1

1

2

1 Introduction

density [4]. Besides, it was discovered that Stone–Wales transformation defects may exist in the carbon nanothreads during geometrical construction [5]. Significantly, the diverse chirality and distributed Stone–Wales defects grants carbon nanothreads with tunable physical and chemical properties, promising carbon nanothreads as potential candidates for “bottom-up design”. The covalently tetrahedral sp3 -bonding network creates an extremely strong structure with ultralight density. Arguably, carbon nanothreads are the strongest structure with a specific strength of 4.13 × 107 N·m/kg [4], exceeding all known fabricated nanomaterials. To date, mechanical properties of carbon nanothreads have been predicted by theoretical calculations including first principle calculations and molecular dynamics (MD) simulations. It is revealed that the axial Young’s modulus of carbon nanothreads is typically in the range from 13 to 193 nN [4, 6–12], while the breaking strength of carbon nanothreads is in the range from 8.7 to 26.4 nN [6– 12], assuming that the atomic volume V0 is 5.536 Å3 [1]. Interestingly, the carbon nanothreads exhibit a structure-dependent ductility, and the brittle-ductile transition can be manipulated by tuning the distributions and density of cross-links [7]. Simultaneously, different derived geometrical patterns of carbon nanothreads, such as diamond nanomeshes [13], nanofoams [13], diamond nanothread forests [14], and carbon nanothreads from polycyclic aromatic hydrocarbon [15] have been theoretically investigated. More recently, the 2D and 3D orders of carbon nanothreads nanoarchitecture were synthesized successfully through solid-state diradical polymerization of cubane [16]. The peculiar geometrical shapes also endow novel properties to carbon nanothreads, which are advantageous for their applications in the design and manufacture of devices. What’s more, the physical and chemical properties of carbon nanothreads are greatly sensitive to Stone–Wales defects, dehydration, and atomic doping, which opens new routes for designing the carbon nanothreads to achieve desired properties for multi-function devices. Hence, there are infinite possibilities in these kinds of carbon nanothreads. In the past 5 years, intensive theoretical efforts have been made to explore the promising role of carbon nanothreads as candidates for electronic devices [1, 12, 17–22]. The electronic properties are extremely sensitive to the atomic structure of carbon nanothreads which has led to rich scientific research. The bandgaps of carbon nanothreads are tunable with concentration of SW defects, atomic arrangement order and tensile strain, and the electronic behavior varies from semiconducting to insulating [12]. The carbon nanothreads can be utilized as single-molecule junctions to modulate the transport in unimolecular electronics with much slower conductance decay [17]. Due to the analogous carbon structure to CNTs, the thermal conductivity of carbon nanothreads has also attracted intensive research interests, and some theoretical explorations have been made to unveil the mystery [23–26]. Compared to the CNTs, it was predicted that the thermal conductivity of carbon nanothreads was reduced by a factor of three due to the reduction of group velocities [23]. The thermal conductivity of carbon nanothreads exhibits both temperature-dependency and length-dependency, which is similar to the thermal transport characteristics of carbon allotropes. Distinct from the CNTs, the existence of SW defects during

1.1 Introduction

3

self-assembly process endows the carbon nanothreads with a superlattice thermal transport characteristic [24]. The outstanding specific strength of carbon nanothreads enables it as excellent reinforcement for composite materials. Theoretical calculation predicted that the carbon nanothreads reinforcement possesses a comparable load transfer compacity within polymer composites with respect to the CNTs reinforcement [27]. Carbon nanothreads were also a superior reinforcement to CNTs in improving the thermal– mechanical properties of polymer composites [28–30]. The carbon nanothreads network plays a key role in nanomodulator to tune the transition between cavitation and crazing in polymer composites, which is quite important in material sciences [31]. The rapid development and great progress in the synthesis of carbon nanothreads in recent years have significantly promoted the characterizations and nanodevice exploitations. In contrast, the strong requirements of various aspects of fundamental research and practical application have been positively motivating synthetic methods in order to achieve industrial fabrication. To date, the synthesis of carbon nanothreads is consciously focusing on lowering the compressing pressure and manipulating the topological structure during polymerization process.

1.2 Structural Morphology of Carbon Nanothreads In the benzene-derived carbon nanothreads structure, each benzene ring has six sp2 hybridized carbon atoms which could transform to sp3 hybridized carbon atoms in the nanothread through adjacent interconnection. Assuming a topological unit cell of two benzene rings, three classes of topological structure can be defined according to whether the bonding pattern between adjacent rings is 1|5, 2|4 or 3|3 [4]. The entire nanothread can be assigned a class according to its configuration, given different classes should be mixed in different columns. The schematic presentation of nanothread’s bonding pattern is shown in Fig. 1.1, which shows the relative positions of carbon atoms with same bonding direction along the nanothreads axis. For classes I nanothread, there is an obvious bonding pattern: (1) (with five-prong partners not shown). Multiple four-fold rings are available in class-I due to the seriously uneven bond distribution. For class II nanothread, there are three bonding patterns: (1, 2), (1, 3), and (1, 4) (with four-prong partners not shown). For classes III nanothread, there are four bonding patterns: (1, 2, 3), (1, 2, 4), (1, 2, 5), and (1, 3, 5). Considering an up-connector pattern on ring 0 and a down-connector pattern on ring 1, the atoms in two rings can be connected in pairs. The connecting rules for class-III structure can be briefly illustrated as follows: 1. 2.

Select an up-connector pattern for benzene ring 0, such as (1, 2, 3). Select an up-connector pattern for benzene ring 1, (the next up the column of the nanothread), such as (2, 4, 6). Therefore, the benzene ring 1 only has (1, 3, 5) as a down-connector pattern.

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1 Introduction

Fig. 1.1 Connector patterns of class-I, class-II, and class-III benzene-derived nanothreads mark which atoms in a given benzene ring bond up or down the column of stacked rings

(1) Class I

(1, 2)

(1, 3)

(1, 4)

Class II

(1, 2, 3)

(1, 2, 4)

(1, 2, 5)

(1, 3, 5)

Class III

3.

Cyclically connect the up-connector pattern of benzene ring 0 and downconnector pattern of benzene ring 1: (1, 2, 3) → (1, 3, 5) (1, 2, 3) → (3, 5, 1) (1, 2, 3) → (5, 1, 3)

4.

Redo the previous step with the up-connector pattern of benzene ring 1 and the down-connector pattern of benzene ring 2, which is the same as that of benzene ring 0.

The nanothread with different connection patterns can be named by specified the up-connector and down-connector patterns of a single ring. The number in the first position indicates the bonding target for the first atom in the middle ring. If the number is underlined, the target atom is in the ring below; otherwise the target atom is in the ring above. And similar for the second through sixth integers in the sequence. The topological schematic representation of nanothread is provided in Fig. 1.2. For class-II structure, the polymer I phase of compressed benzene [32] can be expressed as (135,462). The number (1, 5) is underlined, which means atoms (1, 5) is bonded to atoms (1, 3) in the ring below. Simultaneously, atoms (6, 2, 4, 5) are bonded to atoms (2, 3, 4, 6) in the ring above. The connection patterns in the third ring follow those in the first ring. The other two examples given are the nanothread with the colloquial name “polytwistane” [33], and the lowest energy soft chiral nanothread. The polytwistane structure (CH)n , has similar molecular formula with polyacetylene, while it totally consists of chemically equivalent units C(sp3 )-H.

1.2 Structural Morphology of Carbon Nanothreads Fig. 1.2 Topological schematic and relaxed structures of three nanothreads, labeling atoms in each progenitor benzene ring to show the interconnection patterns. The polymer I phase of compressed benzene (top) is the lowest energy class-II nanothread. Polytwistane (middle) belongs to class-III and is the lowest energy nanothread overall; both rings in polytwistane have the same connector pattern. The lowest energy soft chiral nanothread (bottom) has different connector patterns between the two pairs of rings. Reprinted (adapted) with permission from [4]. Copyright {2015} American Chemical Society

5

Class II

Class II

Class III

According to the comparison between experimental and simulation work, the nanothread is also be regarded as a hydronated (3, 0) nanotube with a variation of distributed Stone–Wales (SW) transformation defects (C–C dimers rotated by 90 °C), as seen in Fig. 1.3. The existence of SW transformation defects interrupts the central hollow of the nanothread. For traditional carbon-based nanomaterials such as graphene and CNT, the SW defects play a detrimental effect on the mechanical properties under deformation, and the stress concentration around the SW defects used to cause reduction of ultimate tensile strength and brittle failure [34–36]. The occurrence of SW defects is inevitable for most of the carbon-based nanomaterials during experimental synthesis and has become a large obstacle for large-scale fabrication and even for “bottom-up” design. On the contrary, the SW defects endow carbon nanothreads with better mechanical performances during deformation [7]. The carbon nanothreads can transition from brittle to ductile behavior by increasing the proportion of SW defects while maintaining the excellent ultimate tensile strength, which make it an ideal building block for extremely strong three-dimensional nano-architectures.

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1 Introduction

Fig. 1.3 Two Stone–Wales transformations of the off-axial bonds interconvert the (3,0) tube and polymer I, showing that these two structures are members of a single structural class. A disordered admixture of Stone–Wales transformations (with periodic axial boundary conditions) yields the structure shown on the right. Reprinted (adapted) with permission from [2]

1.3 General Development and Current Situation of Carbon Nanothreads in Nanoscience and Nanotechnology With the increased understanding of carbon nanothreads, lots of potential applications have been proposed to make the best of their outstanding properties, including heat conductive nanodevices and ultrahigh-strength nanocomposites, energy storage media, ultrahigh-sensitivity nanosensors and high frequency oscillator, field emission displays, high selectivity membranes, nanometer-sized semiconductor devices, and probes, interconnects, etc. Up to now, these applications are still at the initial stage of exploring advanced devices. To date, the low production output of carbon nanothreads has been a bottleneck problem limiting their practical applications despite a variety of potential applications reported to be utilizing carbon nanothreads. It has been a challenging task to develop a more efficient synthesis approach that requires lower compressing pressure during polymerization process. To shed light on the application of carbon nanothreads, numerical modeling has become the most important tool to provide first insights into the nanostructures. The trend analysis of carbon nanothreads research and publications is provided in Fig. 1.4. Evidence from such analysis shows that carbon nanothreads research saw a stable development in recent 5 years. Noticeably, the

1.3 General Development and Current Situation of Carbon Nanothreads …

7

Fig. 1.4 Trend analysis of carbon nanothreads research and publications. The data in the circle presents the ratio of carbon nanothreads publications from year 2015 to 2020. The data in the ring presents the research field of carbon nanothreads publications from year 2015 to 2020, and all the research works are conducted through numerical modeling except for the synthesis part

numerical modeling takes a larger part in the publications. It can be seen that theoretical investigations of mechanical properties of carbon nanothreads have received the most research interests due to their superior potentials over other traditional carbon nanomaterials. Besides, the promising applications of carbon nanothreads in electronic devices and nanocomposites are also under intensive exploration. It is highly expected that with the development of fabrication technology, there will witness an upward development of carbon nanothreads research in future.

1.4 Fundamental Properties and General Behaviors of Carbon Nanothreads Lots of theoretical works have been carried out to explore the physical and chemical properties of carbon nanothreads. Such exploration is necessary and significant, which can provide instinctive knowledge of underlying properties and basic behaviors of carbon nanothreads, which can further offer theoretical guidance for experiment. Chen et al. [37] performed theoretical examination of the linearly polymerized benzene arrays and generated a large number of degree-four polymers. They enumerated many pathways which may possibly lead transform the benzene stacks to carbon nanothreads with different saturation degrees. Gao et al. [38] conducted ab initio calculations to reveal the roles SW defects play in the structure stability and investigate how to manipulate its molecular structure to achieved desired mechanical properties. They found that SW can significantly affect the stability of carbon

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1 Introduction

nanothreads and the SW transition barrier increased with the applied strain. A strain– temperature–stretching rate phase diagram was also proposed as a guideline to modify the structural configuration in order to obtain excellent mechanical properties of carbon nanothreads. Wang et al. [39] investigated the torsional properties of carbon nanothreads bundles through large-scale MD simulations. Their findings indicate that the torsional behavior of carbon nanothreads bundles is strongly dependent on the loading direction. The mechanical properties of the bundle structure are tunable by changing the enantiomer ratios, which indicates that carbon nanothreads are promising candidates for the next-generation high-performance carbon nanofiber. Zhan et al. [40] found that carbon nanothread bundles have similar mechanical energy storage capacity with (10,10) CNTs bundles. However, the carbon nanothread bundles have unique advantage. The diverse structure of the carbon nanothreads enables full potential of mechanical energy storage subject to tensile deformation, with a gravimetric energy density of up to 1.76 MJ kg−1 , which is 4 to 5 orders higher than that of a steel spring (~0.14 kJ kg−1 ) [41] and up to three times compared to Li-ion batteries (~0.43–0.79 MJ kg−1 ) [42]. Silveira and Muniz [43] investigated the selectivity of carbon nanothreads membranes for gas separation through MD simulations. They demonstrated that carbon nanothreads membranes not only have better mechanical properties than conventional membranes, but also have excellent strain-dependent selectivity for pairs of gases. Zhan et al. [44] explored the application of carbon nanothreads as a reinforcement for polymer nanocomposites. They found that interfacial shear strength of the carbon nanothread/polyethylene interface is comparable with that of the CNT/ polyethylene interface, which offers a first understanding of the load transfer between carbon nanothread and polymer. Zheng et al. [45] explored the effects of nitrogen on the mechanical properties of nitrogendoped carbon nanothreads. Their results have shown that the doping of nitrogen atoms into carbon nanothreads introduced extra flexibility to the structure which delayed the failure of the structure, leading to a larger failure strain. Fu et al. [46] further studied the effects of morphology and temperature on the tensile characteristics of nitrogen-doped carbon nanothreads. They revealed that the structure exhibited extreme ductility due to the formation of a linear polymer via 4-step dissociation and reformation of bonds at extremely low temperatures, which offers a feasible way to design a kind of lightweight material that can be used in harsh deep space environment. Silveira and Muniz [13] proposed the synthesis of analogous two- and three-dimensional porous nanostructures, and the resulting structures exhibited a good combination of high mechanical strength, high flexibility, light density, high porosity as well as high specific surface area.

References 1. Stojkovic D, Zhang P, Crespi VH (2001) Smallest nanotube: breaking the symmetry of sp3 bonds in tubular geometries. Phys Rev Lett 87(12):125502 2. Fitzgibbons TC, Guthrie M, Xu ES, Crespi VH, Davidowski SK, Cody GD, Alem N, Badding

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JV (2015) Benzene-derived carbon nanothreads. Nat Mater 14(1):43–47 3. Duan K, Li Y, Li L, Hu Y, Wang X (2018) Diamond nanothread based resonators: ultrahigh sensitivity and low dissipation. Nanoscale 10(17):8058–8065 4. Xu ES, Lammert PE, Crespi VH (2015) Systematic enumeration of sp3 nanothreads. Nano Lett 15(8):5124–5130 5. Roman RE, Kwan K, Cranford SW (2015) Mechanical properties and defect sensitivity of diamond nanothreads. Nano Lett 15(3):1585–1590 6. Silveira JF, Muniz AR (2017) First-principles calculation of the mechanical properties of diamond nanothreads. Carbon 113:260–265 7. Zhan H, Zhang G, Tan VB, Cheng Y, Bell JM, Zhang YW, Gu Y (2016) From brittle to ductile: a structure dependent ductility of diamond nanothread. Nanoscale 8(21):11177–11184 8. Zhan H, Zhang G, Bell JM, Gu Y (2016) The morphology and temperature dependent tensile properties of diamond nanothreads. Carbon 107:304–309 9. Feng C, Xu J, Zhang Z, Wu J (2017) Morphology-and dehydrogenation-controlled mechanical properties in diamond nanothreads. Carbon 124:9–22 10. Xiao J, Chen MM, Liu WJ, He J, Pan CN, Long MQ (2019) Perfect mechanical and robust electronic properties of new carbon nanothreads: a first principles study. Phys E 111:37–43 11. Silveira JF, Muniz AR (2017) Functionalized diamond nanothreads from benzene derivatives. Phys Chem Chem Phys 19(10):7132–7137 12. Demingos PG, Muniz AR (2019) Electronic and mechanical properties of partially saturated carbon and carbon nitride nanothreads. The Journal of Physical Chemistry C 123(6):3886–3891 13. Silveira JF, Muniz AR (2018) Diamond nanothread-based 2D and 3D materials: diamond nanomeshes and nanofoams. Carbon 139:789–800 14. Xue Y, Chen Y, Li Z, Jiang JW, Zhang Y, Wei N (2018) Strain engineering for thermal conductivity of diamond nanothread forests. J Phys D Appl Phys 52(8):085301 15. Demingos PG, Muniz AR (2018) Carbon nanothreads from polycyclic aromatic hydrocarbon molecules. Carbon 140:644–652 16. Huang HT, Zhu L, Ward MD, Wang T, Chen B, Chaloux BL, Wang Q, Biswas A, Gray JL, Kuei B, Cody GD, Epshteyn A, Crespi VH, Badding JV, Strobel TA (2020) Nanoarchitecture through Strained molecules: cubane-derived scaffolds and the smallest carbon nanothreads. J Am Chem Soc 142(42):17944–17955 17. Gryn’ova G, Corminboeuf C (2019) Topology-driven single-molecule conductance of carbon nanothreads. J Phys Chem Lett 10(4):825–830 18. Miao Z, Cao C, Zhang B, Duan H, Long M (2020) First-principles study on the effects of doping and adsorption on the electronic and magnetic properties of diamond nanothreads. Phys E Low-Dimens Syst Nanostructures 118:113949 19. Wu W, Tai B, Guan S, Yang SA, Zhang G (2018) Hybrid structures and strain-tunable electronic properties of carbon nanothreads. J Phys Chem C 122(5):3101–3106 20. Miao Z, Cao C, Zhang B, Duan H, Long M (2020) Effects of 3d-transition metal doping on the electronic and magnetic properties of one-dimensional diamond nanothread. Chin Phys B 21. Podlivaev AI, Openov LA (2017) Effect of hydrogen desorption on the mechanical properties and electron structure of diamond-like carbon nanothreads. Semiconductors 51(5):636–639 22. Chen MM, Xiao J, Cao C, Zhang D, Cui LL, Xu XM, Long MQ (2018) Theoretical prediction electronic properties of Group-IV diamond nanothreads. AIP Adv 8(7):075107 23. Zhu T, Ertekin E (2016) Generalized Debye-Peierls/Allen-Feldman model for the lattice thermal conductivity of low-dimensional and disordered materials. Phys Rev B 93(15):155414 24. Zhan H, Zhang G, Zhang Y, Tan VBC, Bell JM, Gu Y (2016) Thermal conductivity of a new carbon nanotube analog: the diamond nanothread. Carbon 98:232–237 25. Zhu T, Ertekin E (2016) Phonons, localization, and thermal conductivity of diamond nanothreads and amorphous graphene. Nano Lett 16(8):4763–4772 26. Zhan H, Zhang G, Zhuang X, Timon R, Gu Y (2020) Low interfacial thermal resistance between crossed ultra-thin carbon nanothreads. Carbon 165:216–224 27. Zhan H, Zhang G, Tan VBC et al (2017) The best features of diamond nanothread for nanofibre applications. Nat Commun 8(1):1–8

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28. Zhang LW, Ji WM, Liew KM (2018) Mechanical properties of diamond nanothread reinforced polymer composites. Carbon 132:232–240 29. Ji WM, Zhang LW (2019) Diamond nanothread reinforced polymer composites: ultra-high glass transition temperature and low density. Compos Sci Technol 183:107789 30. Duan K, Zhang J, Li L, Hu Y, Zhu W, Wang X (2019) Diamond nanothreads as novel nanofillers for cross-linked epoxy nanocomposites. Compos Sci Technol 174:84–93 31. Zhang LW, Ji WM, Hu Y, Liew KM (2020) Atomistic insights into the tunable transition from cavitation to crazing in diamond nanothread-reinforced polymer composites. Research 2020:7815462 32. Wen XD, Hoffmann R, Ashcroft NW (2011) Benzene under high pressure: a story of molecular crystals transforming to saturated networks, with a possible intermediate metallic phase. J Am Chem Soc 133(23):9023–9035 33. Barua SR, Quanz H, Olbrich M, Schreiner PR, Trauner D, Allen WD (2014) Polytwistane. Chem—A Eur J 20(6):1638–1645 34. Lu Q, Bhattacharya B (2005) Effect of randomly occurring Stone-Wales defects on mechanical properties of carbon nanotubes using atomistic simulation. Nanotechnology 16(4):555 35. Liew KM, He XQ, Wong CH (2004) On the study of elastic and plastic properties of multiwalled carbon nanotubes under axial tension using molecular dynamics simulation. Acta Mater 52(9):2521–2527 36. He L, Guo S, Lei J, Sha Z, Liu Z (2014) The effect of Stone–Thrower–Wales defects on mechanical properties of graphene sheets–A molecular dynamics study. Carbon 75:124–132 37. Chen B, Hoffmann R, Ashcroft NW, Badding J, Xu E, Crespi V (2015) Linearly polymerized benzene arrays as intermediates, tracing pathways to carbon nanothreads. J Am Chem Soc 137(45):14373–14386 38. Gao J, Zhang G, Yakobson BI, Zhang YW (2018) Kinetic theory for the formation of diamond nanothreads with desired configurations: a strain–temperature controlled phase diagram. Nanoscale 10(20):9664–9672 39. Wang P, Zhan H, Gu Y (2020) Molecular dynamics simulation of chiral carbon nanothread bundles for nanofiber applications. ACS Appl Nano Mater 3(10):10218–10225 40. Zhan H, Zhang G, Bell JM, Tan VB, Gu Y (2020) High density mechanical energy storage with carbon nanothread bundle. Nat Commun 11(1):1–11 41. Madou MJ (2002) Fundamentals of microfabrication: the science of miniaturization. CRC Press 42. Janek J, Zeier WG (2016) A solid future for battery development. Nat Energy 1(9):1–4 43. Silveira JF, Muniz AR (2019) Flexible carbon nanothread-based membranes with straindependent gas transport properties. J Membr Sci 585:184–190 44. Zhan H, Zhang G, Tan VB, Cheng Y, Bell JM, Zhang YW, Gu Y (2016) Diamond nanothread as a new reinforcement for nanocomposites. Adv Func Mater 26(29):5279–5283 45. Zheng Z, Zhan H, Nie Y, Xu X, Gu Y (2019) Role of nitrogen on the mechanical properties of the novel carbon nitride nanothreads. J Phys Chem C 123(47):28977–28984 46. Fu Y, Xu K, Wu J, Zhang Z, He J (2020) The effects of morphology and temperature on the tensile characteristics of carbon nitride nanothreads. Nanoscale 12(23):12462–12475

Chapter 2

Experimental Aspect

2.1 Introduction Carbon nanothreads were first reported in 1979 by Arthur C. Clarke in his scientific fiction novel The Fountains of Paradise set in the twenty-second century. In the fiction, Clarke imaged a very thin but tough “hyperfilament” (built from “continuous pseudo-one-dimensional diamond crystal”) that makes the elevator possible. Until 2001, the mystery of carbon nanothreads was first revealed by Stojkovic [1] through density functional theory (DFT) calculation. They proposed that sp2 carbon structures can be replaced by sp3 carbon structures to produce highly stable carbon nanowires with very small diameter of ~0.4 nm. The resulting carbon structures broke the tetrahedral symmetry of a sp3 -hybridized carbon structure by connecting to one through relatively weak bonding. Despite lots of efforts have been paid, the fantasy of carbon nanothreads was not realized through experimental synthesis. Even though polymerization of aromatic molecules by compression was a feasible approach to synthesize carbon nanomaterials [2, 3], this approach was thought to produce hydrogenated amorphous carbon only after a century of study [4–8]. Until 2014, the researchers in Pennsylvania State University first reported the discovery of larger scale sp3 hybridized carbon-based materials synthesized by solid-state reaction of benzene at high pressure [9]. This new carbon nanomaterial has an intriguing structure. The existence of several chemical bonds in the cross-sectional area distinguishes them from the class of conventional polymers, and the arrangement of carbon atoms differentiates them from all reported nanowires. The central compactness and surface irregularity disentitled the class “nanotube”. The phrase “nanothread” is the best description of these unique structural features. In the early findings, carbon nanothreads with tens of nanometers in size could be synthesized by high-pressure compression of liquid benzene (about 20 GPa), followed by slowly release of pressure to ambient pressure [9]. The researchers in Pennsylvania State University applied a low compression of samples to room temperature assisted by uniaxial stress, and they found a conversion of polycrystalline to single-crystalline packings of carbon nanothreads via mechanochemical synthesis © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. M. Liew et al., Carbon Nanothreads Materials, Materials Horizons: From Nature to Nanomaterials, https://doi.org/10.1007/978-981-19-0912-2_2

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2 Experimental Aspect

[10]. However, the synthesis technology for carbon nanothreads is still not applicable for large-scale production unless the synthesis pressure can be reduced to 5–6 GPa. Enlarging the reacting molecules to non-aromatic, strained molecules provides novel ways to investigate large-scale fabrication of carbon nanothreads. Recent advances have been made successfully to reduce the synthesis pressure to ~12 GPa by using strained cage-like molecules such as cubane as a precursor. It is foreseeable that the goal of large-scale industrial fabrication of carbon nanothreads will be achieved with dedicated efforts and the space elevator made of DNT is not a fantasy anymore.

2.2 Fabrication Methods Several methods have been proposed to fabricate the carbon nanothreads. Generally, there are three techniques are employed: pressure-induced polymerization of benzene [9, 11–16], mechanochemical synthesis [10, 17], and pressure-induced polymerization of non-aromatic molecules [18–22]. The distinction of these techniques can be identified from the precursor and the polymerization environment. The comparisons of different fabrication methods are provided in Table 2.1.

2.2.1 Pressure-Induced Polymerization of Benzene In this section, the technique of pressure-induced polymerization of benzene for carbon nanothreads synthesis is discussed in detail. Solid-state high-pressure reaction has been employed to synthesize advanced nanomaterials at high pressure such as polymers and amorphous compounds [23]. Generally, single-crystalline materials can be formed by topochemical reaction of single crystal reactants, and the periodicity of crystals remains the same after the reaction [24]. Substantial changes in the lattice size usually disorder the crystal phase and often result in amorphous products. Such change is not expected in the benzene molecule due to the fact that the small changes in geometry are restrained by the stable covalent C−C bonds instead of the non-covalent interactions between molecules. Prior to the discovery, benzene was believed to give rise to amorphous compounds with hydrogenated carbon (a-C:H) under compression at extremely high pressure [5, 25–28]. Compared to the products synthesized by chemical vapor deposition techniques, the higher hydrogen content arouses great interest in a high-pressure synthesis of the benzene molecules [5], indicating the existence of a larger amount of sp3 C–C bonds. The external pressure applied to the unsaturated hydrocarbons can facilitate the reaction, for instance, the transformation of sp2 -hybridized bonds to sp3 hybridized bonds generates additional spaces for chain reactions. The pressure and temperature are critical factors for controlling the reaction paths of benzene products. The phase diagram and chemical-stability boundary of benzene are provided in Fig. 2.1. The crystal structure of phase I could be obtained at room pressure by

2.2 Fabrication Methods

13

Table. 2.1 Comparisons of different fabrication methods for carbon nanothread Method

Pressure-induced polymerization of benzene

Mechanochemical synthesis

Pressure-induced polymerization of non-aromatic molecules

Polymerization environment

Solid-state high-pressure reaction, and slowly release to ambient pressure

Solid-state high-pressure Solid-state high-pressure reaction combining with reaction, and slowly uniaxial stress, and release to ambient pressure slowly release to ambient pressure

Precursor

Benzene

Benzene, cubane

Aniline, thiophene, pyridine, phenol-pentafluorophenol co-crystal

Uniaxial stress

0 GPa

30 GPa

0 GPa

Operating pressure

20 ~ 23 GPa

12 ~ 23 GPa

12 ~ 35 GPa

Operating temperature

Room temperature

Room temperature ~ 250 °C

Room temperature ~ 250 °C

Fig. 2.1 Phase diagram and chemical-stability boundary of benzene. The data of reaction threshold for annealed samples are shown in filled circles, and the data of reaction threshold for non-annealed samples are shown in open squares, and data of previously reported phase boundaries are shown with dotted lines. The phase boundary of I–II phases and the melting line are shown in full lines. Reprinted (adapted) with permission from [25]

freezing or at room temperature by compressing (up to 700 bar). The transition to phase II happens at room temperature and pressure of 1.4 GPa, and the structure of the phase II was stable up to 4 GPa. Further increment of pressure to 11 GPa leads to a formation to phase III, and it is found that the P21 /c structure corresponds well with the measured characteristics above 4 GPa in the presumed phase III. The existence of presumed phase IV has not been demonstrated via experiment yet. The

14

2 Experimental Aspect

crystal hydrogenated carbon is difficult to be obtained through purely high-pressure compression. To transform the hydrogenated amorphous carbon to a single crystal product through solid-state reaction, the control of reaction kinetics plays a significant role during synthesis. Under slow compression/decompression, the carbon nanothreads tend to show up in a style of single crystal packings through self-assembly. By increasing the restraint of topochemical reaction, new carbon nanothreads with substitution of the exterior hydrogens or heteroatoms in their backbone could be formed [10]. The carbon nanothreads were first synthesized by high-pressure compression (~20 GPa) of liquid benzene at room temperature. After maintaining the pressure for one hour, the pressure was gradually decreased to ambient pressure at an average rate of 2 GPa/h [9]. It was shown through bright-field transmission electron microscopy (TEM) imaging in Fig. 2.2a that these products had parallel stripes with long length, suggesting the feature of tubes or threads. After sonication, these parallel stripes were disrupted into narrow and curved one-dimensional structures, as seen in Fig. 2.2b. By further changing the decompression conditions, high-resolution TEM images in Fig. 2.3 show sign of axial order with a periodicity in the samples. The solid-state nuclear magnetic resonance (NMR) also identifies primarily (~80%) sp3 carbon

(a)

(b)

Fig. 2.2 Bright field TEM micrographs of carbon nanothreads. (a) TEM image of the carbon nanothreads with tens of nanometers in length are shown in striations with an interthread distance of 6.4 Å. The grid spacing of the line profile along the marked path is 2 nm. (b) TEM image of corresponding sample showing disorder lattice after sonication in pentane. Reprinted (adapted) with permission from [9]

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15

Fig. 2.3 High resolution TEM image of carbon nanothreads (left) and contour plot of the image autocorrelation function (right). The interthread spacing of the nanothreads is measured to be ~ 6.2 Å in the transverse direction, which is close to that observed in Fig. 2.2. Interthread spacing of 3.5–3.8 Å is observed along the longitudinal direction of the threads, indicating structural order in this direction. Local maxima in the axial direction are marked by + and the local minima in the transverse direction are indicated by *. Reprinted (adapted) with permission from [9]

content of the lattice structure, as seen in Fig. 2.4, with the remainder being sp2 carbon content. It was revealed that the sp2 -hybridized carbon content is related to degree-4 nanothreads together with small amounts of benzene linkers [16, 29]. The degree of saturation varies from 0 to 6 is used to classify the structures from the two end points, by counting how many sp3 carbon atoms occupy in each (CH)6 ring. For instance, the degree of saturation for benzene molecules stacking together is 0, the degree of saturation for para polymer is 2, and the [4 + 2] polymer is 4. For carbon nanothreads which are completely saturated, such as polytwistance, the degree of saturation is 6. Some reaction intermediates which would be polyradicals exist with Fig. 2.4 13 C solid-state NMR of sp3 carbon nanothreads. NMR analyses of two different samples of carbon nanothreads (13 CD red and 13 CH black curves) reveal a high sp3 carbon content, 80– 84%. The sp2 and sp3 carbon spectral regions are colored red and blue, respectively. Reprinted (adapted) with permission from [9].

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odd degrees of saturation, and their bonds are likely to form between radical centers to achieve a higher degree of saturation. The reaction mechanism that forms a sp3 nanothread from benzene molecules can be briefly illustrated as follows. First, benzene molecules are oriented in a slipped stack along both the a and b axes of the benzene II crystal structure. Then a series of [4 + 2] cycloaddition reactions occur to form benzene polymer, as shown in Fig. 2.5. Aligned olefin functions are then well oriented for a zipper cascade to give a fully sp3 hybridized nanothread. The cycloaddition and zipper reactions both have negative activation volumes and would thus be promoted under high pressure. The zipper reaction is very exothermic; the slow decompression employed in the experiments may aid in controlling this reaction. Tight binding relaxation of the [4 + 2] cycloaddition reaction product spontaneously forms the fully sp3 thread in view of its considerable thermodynamic stability. The resulting structures consist of 5-, 6and 7-aromatic rings, and they are attainable via SW transformations from the (3,0) and polymer I nanothreads.

Fig. 2.5 Proposed mechanism for the formation of nanothreads through [4 + 2] cycloaddition reaction. Reprinted (adapted) with permission from [29]. Copyright {2015} American Chemical Society

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2.2.2 Mechanochemical Synthesis Mechanochemical synthesis is a processing approach where mechanical–chemical phenomena occur on a molecular scale. It is quite significant since it is able to promote solid-phase reactions by merely utilizing nominal amounts of reactants [31–33]. The primitive idea of employing external force to induce chemical reactions has been found thousands of years ago. In ancient time, some well-known mechanical treatments such as grinding, impacting, shaking, rubbing, and rolling were utilized to generate fire, to machining materials, and to produce building materials. The available literature for mechanochemical synthesis dates back to the fourth century BC, when Greek philosopher Theophrastus made quicksilver by grinding the cinnabar (mercury sulfide) with vinegar in a copper pestle and mortar. The finding of explosion by impact also belongs to the case of mechanochemical treatment in the “pre-publication era”, which is now widely utilized for military applications. In the nineteenth and the beginning of the twentieth century, the scientific research concerning mechanochemical synthesis was first reported. Nowadays, there are various mechanical treatments using different apparatus, as is shown in Fig. 2.6. The mechanochemical synthesis has been used for synthesis of single crystal PbS nanosheets [33]. The mechanochemical treatment was first used in 2017 for large-scale synthesis of single crystal nanothreads with length of hundreds of microns [10]. It was reported

Fig. 2.6 Various mechanical treatments using different apparatus: I - shear + impact (a - ball mill, b - attritor, c - vibration mill, d - planetary mill); II - impact (a - pin mill, b - jet-mill); III - shear (a - rollers). Reprinted (adapted) with permission from [32]

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2 Experimental Aspect

that the formation of nanothread crystals is governed by uniaxial stress which induces the crystals to align to the tensile direction. In this work, the benzene mixtures of phases I and II were compressed to extreme pressure (~23 GPa) over 8 h with varying compression rates. The products were kept at target pressure for 1 h, followed by the same releasing rates with the compression. The compression rate of crystalline benzene monomer at ambient pressure is slower than previously reported [9]. The schematic representation of the Paris-Edinburgh press is provided in Fig. 2.7. The microscopic appearance of the nanothreads structure is provided in Fig. 2.8. Results showed that the single crystal structures preferred an orientation to a nearhexagonal axis which is the tensile direction. They had macroscopic stripes which are similar to one-dimensional structure (Fig. 2.8 top). These well-aligned long threads interacted via Van der Waals forces, which is different from conventional hydrocarbon polymers and other polycrystalline materials. Given the van der Waals forces are weaker, the single crystal nanothreads can be exfoliated into fibers under mechanical treatment. The easy exfoliation of single crystal nanothreads into fibers indicates that they are promising building blocks with tunable properties for bottom-up strategy. Although the crystallographic orientation and original packing geometry can affect the orientation of nanothread [34], uniaxial stress can be applied to control the

Fig. 2.7 Unaxial Stress in the Paris-Edinburgh Press. An additional stress component arises largely in the vertical direction of the applied load (blue arrows). This stress is typically uniaxial. The magnitude of this stress component depends on the strength of the material under compression but is often at least a few percent at ~ 20 GPa pressures. Benzene molecules in columns within the threedimensional benzene phase II crystal structure that are aligned parallel to this stress will be closer together than those in columns that are not parallel at that pressure. It is thus natural to anticipate that reaction may occur first in the direction parallel to the applied stress, helping to explain why nanothreads are consistently oriented parallel to it. Reprinted (adapted) with permission from [10]. Copyright {2017} American Chemical Society

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Fig. 2.8 Morphology of the single crystal nanothreads after mechanochemical treatment. (Top) A cluster of nanothreads synthesized from polycrystalline benzene mixtures of phases I and II between cross polars. (Bottom) Carbon nanothreads bundle. After rotating the fiber by 45°, there is strong birefringence as seen from the increased degree of transmission. However, rotation of the fiber by 90°restores extinction. Reprinted (adapted) with permission from [10]. Copyright {2017} American Chemical Society

reaction direction within the benzene phase II crystal. The applied tensile stress has high propensity to lead the reaction in a consistent direction. The resulting nanothread structures originate from properly aligned benzene phase II crystal, as well as from disorder, polycrystalline benzene mixtures of phases I and II. The benzene phase I is not needed in the nanothread synthesis due to the lack of suitable benzene contacts. The “slipped” molecular stacks of benzene phase II are the most likely candidates for intra-stack reaction [29]. There are three distinctive molecular stacks in the benzene phase II, and the radical and cycloaddition mechanism to form threads may be found in these stacks [29]. If the crystallites are aligned properly, the molecules in these stacks will be brought together by the applied stress. At the onset of reaction, the reaction pathway prefers to go along the uniaxial tension direction, assisted by the constraints from the pressure system. The reaction is not topochemical, as the benzene stacks are required to contract obviously in the direction of thread axis. The reaction to form well-oriented crystal nanothreads can be further promoted by an uncatalyzed reaction involving C–C bonds at room temperature. The in situ diffraction of carbon nanothreads at high pressure is provided in Fig. 2.9. After the pressure is raised above ~ 1 GPa, the sample is a polycrystalline

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(a)

(b)

(c) Fig. 2.9 In-situ diffraction of carbon nanothreads at high pressure. (a) Monoclinic benzene phase II crystal viewed down the [011] axis. (b) Monoclinic nanothread crystal viewed down the (100) direction. (c) Benzene phase II (100) and nanothread (100) inter-planar spacings vs. pressure in increasing and decreasing pressure directions in a diamond anvil cell. Reprinted (adapted) with permission from [10]. Copyright {2017} American Chemical Society

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benzene mixture of phase I and phase II. In the increasing pressure direction benzene phase II (100) diffraction arcs (black open circles) exhibit a small discontinuity at 18 GPa, decreasing in inter-planar spacing. New diffraction arcs at a slightly larger interplanar spacing than the benzene (100) ones also appear at 10 GPa in the increasing pressure direction (red open circles). At 18 GPa there is a sudden expansion of these new arcs, which appears to be correlated with the reaction to form nanothreads. For instance, at this pressure the sample also becomes photoluminescent. These arcs have an inter-planar spacings characteristic of nanothreads. After this expansion at 18 GPa upon increasing pressure, the diffraction peaks can be traced up to 23 GPa and back to ambient pressure (solid red triangles) in the decreasing pressure direction. Upon release of pressure and melting of the solid benzene, a sixfold symmetric diffraction pattern characteristic of nanothreads remains. The benzene phase II (100) diffraction arcs in the decreasing pressure direction (solid black triangles) diverge at about 14 GPa from those in the increasing pressure direction such that there is hysteresis. It is possible after benzene has collapsed into nanothreads, there is less stress in the plane perpendicular to the X-ray beam, which causes this hysteresis.

2.2.3 Pressure-Induced Polymerization of Non-Aromatic Molecules 2.2.3.1

Aniline

Aniline crystal phase-II has been proved to be chemically stable in a wide range of pressure and temperature as a result of the covalent interactions in the aniline molecule, as well as intermolecular non-covalent interactions in the length direction [35]. To assess the pressure–temperature conditions, a theoretical model taking the thermal displacements into account was employed [18]. The critical distance between the closest C–C contacts between aniline molecules was chosen as 2.6 Å in the model. The thresholds for reactivity of aniline phase-II are evaluated in Fig. 2.10. The aniline exhibits high stability in a wide pressure–temperature range, which surpasses the pressure–temperature stability of benzene [25]. Once the chemical instability boundary for aniline was understood, the aniline crystals are subject to isothermal compression at a temperature of 550 K which corresponds to a reaction condition between 30 and 35 GPa. Once the reaction condition is reached, the reaction was kept over 24 h. The temperature was then decreased to ambient temperature, accompanied by slow release of pressure. The high-resolution bright field TEM (BF-TEM) images of nanothread structure with parallel striations are shown in Fig. 2.11a, which suggests the formation of onedimensional thread or tube structure. The line profile in the left panel of Fig. 2.11a is further shown in the BF-TEM image in the right panel. Results show that the oriented one-dimensional stripes are 4.0–5.1 Å apart with lengths of tens of nanometers. The distances between stripes are comparable to the one-dimensional benzene-derived

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Fig. 2.10 Stability of pressure–temperature range of aniline. The range of stability of aniline measured in experiments is marked in blue area [35]. The estimated thresholds for reactivity of aniline phase II are depicted in blue circles, and the uncertainty is given in gray area. The induced reactivity of aniline phase II is marked in red diamond. The disclosed liquid–solid transition boundary from [36] is marked in black squares. Reprinted (adapted) with permission from [18]

(a)

(b)

Fig. 2.11 Morphology of the aniline-derived structure and associated atomistic configurations. (a) BF-TEM images of the nanothreads structure showing striations with interthread spacing of ~ 4 to 5.1 Å. The nanothread structures have a length of tens of nanometers. (b) Views of the atomistic configuration of the nanothreads derived from aniline in different directions. Reprinted (adapted) with permission from [18]

carbon nanothreads which were spaced at a distance of ~ 6.4 Å [9]. The well-aligned nanothread structure is attributed to the existence of H-bonds between NH2 groups linking adjacent molecules in the crystal aniline. The strong H-bond interaction can effectively inhibit the involvement of the NH2 groups during the reaction. Besides, the constraints imposed by the H-bonds can promote the anisotropic compression

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23

and guide the reaction direction of C–C bonds, leading to a transformation from sp2 hybridized bonds to sp3 -hybridized bonds along the growth of crystal nanothreads. Schematic views of the atomistic configuration of the aniline-derived sp3 -hybridized nanothreads are shown in Fig. 2.11b. The resulting structures have double nanothreads with a diameter of 12.8 Å formed along the growth direction, and the double nanothreads interact through H-bonds, leading to an expanded 2D triangular lattice.

2.2.3.2

Thiophene

Thiophene is a promising precursor for forming a thread. There are four carbon atoms in each ring, indicating that the formation of carbon nanothreads would end after only a single step on a 4 + 2 (degree 4) pathway. The simplicity of reaction pathway is believed to generate more consistent nanothread structures, which can provide more details about the structural configuration and formation mechanism. Generally, there are five stable and three metastable structural configurations at low temperatures [37–41]. Previous works tested the thiophene at high pressure (~30 GPa) using infrared (IR) spectroscopy and observed phase transformation phenomenon (solid III transforms to solid IV) around 4–8 GPa (see Fig. 2.12). However, the reaction products recover to a white polymeric solid when the pressure is lower than 16 GPa due to the incomplete chemical reaction [42]. Although the reaction can be activated at high pressure, there Fig. 2.12 Effects of temperature and pressure on the phase structure of thiophene. It can be seen from the melting curve that at temperature of 298 K and pressure of 440 MPa, the solid I, solid III, and liquid thiophene coexist. Reprinted (adapted) with permission from [42]

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Fig. 2.13 Morphology of the thiophene-derived structure and associated atomistic configurations. (a) Optical images of samples showing change of color as a variation of pressure. (b) Le Bail refinements of thiophene using two different models. (c) Pressure-normalized volume relationship of the solid thiophene. The insets show the thiophene structures in Phases III and V. Reprinted (adapted) with permission from [19]. Copyright {2019} American Chemical Society

are still lack of structural information and evidence of orientational order in the thiophene-derived carbon nanothreads. The thiophene-derived crystalline nanothreads were first reported in 2019. During the synthesis, a slow compression rate was applied to the samples to 35 GPa over 10–12 h at room temperature. The thiophene first froze at ∼0.4 GPa. When the pressure reached ~ 7 GPa, a yellow-orange hue occurred to the samples. The sample further transformed to brown-amber at a pressure of ~ 35 GPa (Fig. 2.13a). This transformation was activated only if the distance between molecules is less than ∼3 Å [43]. Besides, it belongs to a chemical reaction as the change of color was irrecoverable during decompression. The single crystal X-ray diffraction (SCXRD) patterns in Fig. 2.13b at 0.7 GPa showed that the structure belongs to space group Pnma, Phase III thiophene. With the increase of pressure to 7.3 GPa, a monoclinic Phase V was identified in the X-ray pattern. There is an undetermined zone when the pressure is in the range of 5.2 ~ 7.3 GPa, which may be due to the phase mixtures or incommensurate phase IV. The unit cell volumes of Phase V in Fig. 2.13c were fit using a third-order Birch–Murnaghan equation [44]. The characteristic X-ray diffraction patterns and atomistic configurations are provided in Fig. 2.14. It was found that the recovered sample showed purely sixfold symmetry with d-spacings at ∼5–6 Å. The deviation from hexagonal packing is the result of the low-symmetry packing of crystals, and the cross section of the resulting product-nanothread is required to exhibit low symmetry. It was shown in Fig. 2.14b

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Fig. 2.14 Morphology of the nanothreads structure and associated atomistic configurations. a Xray diffraction pattern of the thiophene-derived nanothreads after recovery showing purely sixfold symmetry. b Dependence of the atomic configuration of thiophene-derived nanothreads on the positions of sulfur atoms. The nanothreads are oriented with periodicity. c Schematic representation of one of the reaction pathways to form the anti-thiophene-derived nanothread structure. Reprinted (adapted) with permission from [19]. Copyright {2019} American Chemical Society

that the positions of sulfur atoms will significantly affect the atomic configuration of thiophene-derived nanothread by changing the reaction pathway (see Fig. 2.14c).

2.2.3.3

Pyridine

Early studies suggested that compressing the pyridine led to an amorphous product [45]. Recent studies showed that the resulting products are strongly affected by the rates of compression and decompression. Slow rates of compression and decompression can promote the synthesis of pyridine-derived carbon nitride nanothreads, with empirical formula C5 NH5 . The slower compression/depression rates were helpful to reduce the bifurcated reactions and facilitate the formation of crystalline structure [22]. In the beginning, the pyridine was put in a double-stage membrane diamond anvil cell, and then the pyridine was slowly compressed over 8–10 h. The solid pyridine first froze at 1–2 GPa and then reacted when the pressure reached ~18 GPa. The samples were hold at 23 GPa for 1 h, followed by gradual release of pressure to ambient over 8–10 h. A yellow/orange hue occurred to the recovered sample. The measured nanothread (100) inter-planar spacing with variation of pressure during synthesis is provided in Fig. 2.15. The in situ X-ray diffraction discerns new diffraction arcs that form at ~14 GPa with d = 4.53 Å that do not belong to crystalline pyridine. Upon increasing pressure to 18 GPa, there is an unusual expansion of the inter-planar spacing, similar to that observed in benzene nanothread formation; this expansion appears to correlate with nanothread formation. These arcs can be tracked

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Fig. 2.15 Nanothread (100) inter-planar spacing vs pressure during synthesis. Reprinted (adapted) with permission from [20]. Copyright {2018} American Chemical Society

to the maximum pressure achieved (23 GPa) and then down to near-ambient pressure (0.32 GPa). They ultimately assume the characteristic ambient pressure inter-planar (100) spacing of nanothreads (5.6 Å average spacing at ambient pressure, plotted in blue with the three different observed spacings for reference). Similar to the benzene-derived nanothreads structure, the carbon nitride nanothreads samples are found to compounds of degree-4 and degree-6 structures, as seen in Fig. 2.16a. For degree-4 unsaturated structures, there are C-N double bonds in each pyridine formula (CH)5 N. The synchrotron X-ray diffraction patterns of the resulting product after recovery is shown in Fig. 2.16b, displaying an obvious sixfold symmetry. Results also show pseudohexagonal diffraction patterns for the pyridine-derived nanothreads, which deviates slightly from a perfect hexagonal symmetry. The observed diffraction patterns correspond well with the predicted sixfold monocrystalline diffraction pattern of the degree-6 carbon nitride nanothreads structure. According to the 13 C NMR spectra analysis, the concentration of degree4 nanothreads is estimated about 40%, after considering the amount of degree-4 nanothreads (sp3 -hybridized carbons). The pressure–temperature conditions were explored to understand the formation of pyridine-derived nanothreads [21]. The parameter selected to characterize the reaction kinetics is the reacted pyridine percentage R(t) based on the Avrami model:    R(t) = R End 1 − exp −k(t − t0 )n

(2.1)

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27

View down the hexagonal axis

Degree-4

Degree-6

(a)

(a)

(c)

Fig. 2.16 Morphology of the pyridine-derived nanothreads and associated atomistic configurations. a Carbon nitride nanothread structures. b The synchrotron X-ray diffraction patterns of the resulting product after recovery. c Predicted diffraction pattern of the carbon nitride tube. Reprinted (adapted) with permission from [20]. Copyright {2018} American Chemical Society

where R End is the ratio between the final reacted pyridine and total pyridine, t is the reacting time with respect to the starting time t0 , k is the rate constant, and n is the dimensionality of the nanothread structures. Results indicated that increasing temperature is necessary for activating the reaction because the increased kinetic energy leads to thermal vibration and weakens the H-bond interactions. The reactivity of pyridine was observed when the temperature >400 K, and the amount of reacted pyridine is two times larger than that at room temperature and pressure between 14 and 18 GPa. Besides, it is found that decreasing pressure or increasing

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Fig. 2.17 The reaction ratio of pyridine with increasing time as functions of pressure and temperature in experiments. Reprinted (adapted) with permission from [20]. Copyright {2018} American Chemical Society

temperature lead to reduced degree of saturation in carbon structure and promote nitrogen protonation [21].

2.2.3.4

Cubane

Cubane is a pure sp3 hybridized carbon structure with structure unit of C8 H8 . Specifically, the C − C − C bond angle in the cubane structure is 90°. Because of the high energy barrier for polymerization reaction, the cubane will not react to pressure at room temperature [46]. Although a small work was created in the processes of bond cleavage and reformation, such energy is still not enough to overcome the energy barrier for cubane polymerization, leading to a high stability of cubane under compression at room temperature. The energy barriers can be overcome by additional thermal energy by increasing the temperature, thus promoting the cubane polymerization. Besides, the intermolecular interaction will be improved by increasing the pressure, which is advantageous for the reaction toward novel cubane-derived scaffolds. Early work has been carried out to synthesize cubane-derived porous frameworks [47]. Recent studies demonstrated that a sp3 hybridized, low-dimensional carbon structures can be obtained by solid-state diradical polymerization of cubane at high pressure [17]. It was observed in experiments that the resulting crystalline products were three-dimensional and maintained most the fundamental structural configuration of cubane molecules, with better hardness than the fused quartz.

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During the synthesis, variable pressure with range of 2–30 GPa was applied to compress the cubane molecular precursor in the DAC. In the beginning, the samples were kept at the target pressure and heated to 250 °C for 24 h. The temperatures and pressures were then decreased to ambient conditions. When the pressure reached the range 4–30 GPa, a translucent yellow hue occurs to the colorless cubane samples. Such occurrence belongs to chemical reaction, which is irreversible in color change. The atomic pair distribution function, g(r), for a resulting product after recovery synthesized at 12 GPa is depicted in Fig. 2.18a. The first peak at 1.55 Å is the firstneighbor C–C bonds, which corresponds well with the 1.54 Å of sp3 C–C bonds. Besides, it can be observed that the location of third-nearest neighbor is 2.57 Å, which indicates the existence of a tetrahedral carbon structures with chemical formula CH [48, 49]. These findings demonstrate the occurrence of tetrahedral network during reaction. A peak at 2.2 Å which is indicative of pristine square rings of cubane is not found in the profile. However, the second-nearest neighbor C–C bonds with a location of 2.0 Å is found, which corresponds with the crushed four-membered rings and/or sp3 C-H bonds. The density of the peak at 2.0 Å is low owing to the low scattering power of H atoms. Schematic representation of atomic configuration of cubane superstructure is proposed and compared with cubane structure in Fig. 2.18b. The cubane superstructure could be regarded as a string connected by cyclohexane rings in a sequential way or crashed square rings in an alternating vertex way. Based on the atomic pair distribution function, the proposed cubane superstructure has a first-nearest neighbor shell at 2.06 Å and the fifth-nearest neighbor shell at 3.41 Å. The peak at 3.41 Å represents the possible relaxed C8 cages which contains the backbone. Further down shift of the second-neighbor shell is possible if the ring strain can be released substantially. BF-TEM imaging of the sample carried out at cryogenic conditions in Fig. 2.19a shows long-range parallel striations, and the spacing between these striations is found to be ~5 Å. The TEM image together with corresponding selected area electron diffraction (SAED) pattern from the polycubane sample with a size of ∼1 μm is shown in Fig. 2.19b. It is shown that the SAED pattern displays good crystallinity with a hexagonal lattice along . In Fig. 2.19c, the zone of polycubane sample has angles of 100° and 80° between the sets of {101} lattice planes. As the is perpendicular to , more information about the straight packing of the stripes seen in Fig. 2.19a can be provided in the zone11. The fourfold diffraction suggests near-square packing of the constituents. Increasing temperature is a necessary approach to overcome the energy barriers for cubane polymerization. Importantly, variable pressure can be applied so that the reaction pathways can be controlled. At high pressure, the well-aligned cubane unit can be kept within Van der Waals interaction distance, which is beneficial for a diradical chain reaction. However, at low pressure, disorder structure would be created due to the production of mixing cubane isomers and branched oligomers. During the polymerization process, the unpaired spin radicals can form either neutral localized “solitons” on the backbone or a covalent bond with the neighboring cubyl radicals. The possible polymerization pathways of cubane are provided in Fig. 2.20. It can be seen that a cubane molecule has three kinds of orthogonal square rings.

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Fig. 2.18 Atomic pair distribution function (PDF) analysis and polycubane structural unit. a PDF graph shows the interatomic distance between pair of atoms in a resulting product after recovery synthesized at 12 GPa. The inset shows the proposed structural configuration of the one-dimensional cubane superstructure. b Schematic representation of the evolution of structural unit. After reaction, the angle of C–C-C bond increases from 90 to 111.6°, which is near the tetrahedral angle of 109.5°. The square rings buckle during the release of strain, which leads to a decrease of diagonal distance from 2.2 to 2.06 Å. The measure C–C distances correspond to the color-coded regions between 3 and 4 Å. Reprinted (adapted) with permission from [17]. Copyright {2020} American Chemical Society

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Fig. 2.19 Morphology of the cubane-derived structure and associated atomistic configurations. a TEM image with the inset showing Fast Fourier Transformation (FFT) pattern shows the structural morphology of the samples in striations with in an interthread distance of 0.5 nm. b SAED pattern of the cubane-derived structure observed along the [010] zone axis. The inset shows the diffraction image of single crystal. c SAED pattern of the cubane-derived structure observed along the [111] zone axis. The schematic representation of the atomic configuration in each SAED pattern is provided in the bottom of b and c, which indicates the orientation of the cubane-derived structure. Reprinted (adapted) with permission from [17]. Copyright {2020} American Chemical Society

Fig. 2.20 Proposed new scaffolds for constructing multidimensional cubane superstructures. a Possible polymerization pathways of cubane to form one-dimensional (1D) chain, two-dimensional (2D) layer, and three-dimensional (3D) network. The carbon connectivity is presented as red line. b The calculated unit volume of different dimensional cubane superstructure. The measured volume of polycubane in experiment is provided as a reference. The deviation of unit volume for different models is shown in error bar. Reprinted (adapted) with permission from [17]. Copyright {2020} American Chemical Society

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Breaking one C–C bond per cube will lead to two new radicals which are able to extend the broaden the polymerization along one-dimensional chains, as seen in Fig. 2.20a. Cleaving two orthogonal bonds on a square ring would further increase the connectivity and lead to two-dimensional structures. Similarly, cleaving three orthogonal bonds on a square ring lead to three-dimensional structures. Figure 2.20b describes how these hypothetical polymers can be generated from the original R_3 phase. Due to the transformation from Van der Walls interaction to covalent-bond interaction, the unit volume of the cubane-derived scaffolds increases with dimensionality. Therefore, the unit volume of the resulting products can be used to determine the dimensionality of the polycubane samples. For the one-dimensional chains which has only one C–C bond between square rings, it has a unit volume of ∼14.5 Å3 /CH, which is larger than two-dimensional layer (∼12.25 Å3 /CH) and three-dimensional structure (∼10.7 Å3 /CH). The proposed polycubane structure has a higher density with respect to the COFs [48] and π-conjugated macrocycles [49]. The unit volume was measured as 13.6 Å3 /CH in the experiment, which falls within the range between 12.25 and 14.5 Å3 /CH, suggesting an incomplete polymerization and existence of other low-dimensional structures which have varied C–C bonds in the polycubane structure. A variety of one-dimensional structures can be denoted by calculating the bonding location between various positions of square rings. It is found that those one-dimensional structures with two C–C bonds between square rings have a good agreement of unit volume with the experimental measurements.

2.3 Testing Technologies So far, lots of techniques have been employed to investigate the structural configuration and properties of synthesized carbon nanothreads. Besides, some techniques can be also used for selection in a reaction/separation process. Spectroscopic approaches are able to generate population statistics of the studied bulk structures. The characterization of individual carbon nanothreads is strongly related to the device application in spite of the cost and tediousness. The tube-by-tube techniques could offer deep insights into the structural configuration and origin of the reaction pathways with simplicity, efficiency, undamaged, which is easily performed at ambient conditions.

2.3.1 X-ray Diffraction X-ray diffraction has been used to provide insights into the structural details of crystalline material. During testing, the beam of X-ray will be diffracted by the crystal into multiple specific directions. The density of electrons in the crystal could be obtained by measuring the angles and corresponding intensities of the diffracted beams.

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In situ X-ray diffraction experiments have been conducted to identify the crystalline order and packing spacing of crystal nanothreads. Li et al. [10] utilized Xray diffraction to characterize a nanothread crystal and observed a sixfold pattern under illumination with a 300 μm-diameter laboratory X-ray beam, which indicates a macroscopically crystalline order. They also identified novel diffraction spots, which is indicative of the formation of nanothread crystal from a phase II single crystal when increase the pressure from 3.3 to 23 GPa. Further evidence was provided from the evolution of diffraction intensity with pressure (see Fig. 2.21) which clearly showed that nanothread crystal emerged at the compression of 23 GPa. These findings indicate that the phase I is not necessarily required during the synthesis of nanothread. The structure factor is a critical tool for the interpretation of scattering patterns obtained in synchrotron X-ray diffraction experiment. The diffuse scattering accompanying the Bragg peaks in the structural factor gives the information of pairwise correlations of C–C, C–H and H–H. Fitzgibbons et al. [9] decomposed the structure factor by Fourier transformation and represented the distribution of interatomic distance in the PDF g(r) curve. In Fig. 2.22, the identified first-nearest-neighbor Fig. 2.21 Evolution of benzene and nanothread diffraction patterns with pressure. One-dimensional diffraction patterns vs. pressure for the nanothread sample synthesized from pure single crystal phase II benzene as a function of pressure. Down-arrows indicate decompression. Reprinted (adapted) with permission from [10]. Copyright {2017} American Chemical Society

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Fig. 2.22 Pair distribution function (PDF) analysis of the carbon nanothreads obtained in synchrotron X-ray diffraction (solid line) and comparison with simulated results (dash line). The simulated results are obtained from a model of disordered sp3 -(3, 0) carbon nanothread with random axial shifts and Stone Wales defects. The partials PDFs for the first few neighbor shells are shown in gray curves, and the intensities were reduced by a factor of two for better understanding. The simulated results agree well with the experiments in the first three neighbor shells. Reprinted (adapted) with permission from [9]

C–C distance is 1.52 Å, which is indicative of predominant sp3 hybridized carbon atoms in nanothread crystal. By introducing small amounts of Stone Wales defects into an ideal (3, 0) sp3 -bonded carbon nanotube covered by hydrogen, the simulated radial distribution function g(r) of the model match well with the experimental data. However, a small shoulder shows above 2 Å in the simulated g(r) curve, which corresponds to the second-nearest neighbor shell around a fourfold ring. This new shoulder is not found in the experimental g(r) curve. The missing peaks beyond 4 ~ 5 Å also suggested existence of long-range heterogeneities which were expected to remove the superimposed long-ranged features in the g(r) curve.

2.3.2 Nuclear Magnetic Resonance Nuclear Magnetic Resonance (NMR) spectroscopy is an analytical spectroscopic technique, and it has been widely utilized to identify the purity and molecular structure of organic molecules in solution, as well as investigate the physical mechanisms, crystalline and non-crystalline materials. Solid-state NMR experiments have been used to offer details about bonding information and molecular structure of carbon nanothreads. Wang et al. [11] used NMR spectra to assist with the understanding of structural details from atomic insights. They fitted the one-dimensional NMR spectrum measured in experiments with the

2.3 Testing Technologies

35

calculated spectrum for multiple axially arranged degree-4 and degree-6 nanothreads, and they found out a few subgroups of the most-probable candidates for carbon nanothreads. Duan et al. [12] applied two-dimensional 13 C-13 C NMR to 13 C-enriched benzenederived nanothreads to determine the chemical composition of CH, CH2 , and CH3 groups, along with the nonprotonated C (without covalent bond with H). The spectrum with CH group can be attained by means of dipolar DEPT [50] and the spectrum with CH2 group can be attained by means of three-spin coherence selection [51]. The spectrum of nonprotonated C could be measured by dipolar dephasing [52]. The band close to 40 ppm in the solid-state 13 C NMR spectrum is characterized as the sp3 -hybridized carbon, and the band close to 130 ppm is characterized as the sp2 -hybridized carbon. An example of 13 C NMR spectrum of 13 C-enriched benzene-derived carbon nanothreads is provided in Fig. 2.23. After integrating the quantitative spectrum in Fig. 2.23a, it could be found that there is 28% sp2 -hybridized carbon. The spectrum of alkenes, alkyl-linked aromatic rings, and anisotropically mobile benzene could be obtained by deconvolution according to the two well-resolved peaks near 145 and 129 ppm after dipolar dephasing and cross peaks in 13 C-13 C correlation NMR spectrum. The sp2 -carbon signals could be further complied and weighted to fit the NMR spectrum. The sharp peak near 129 ppm is attributed to the free benzene which only accounts for 5% in the sample [9]. The free benzene would be cleaned up through

Fig. 2.23 NMR spectra of 13 C-enriched benzene-derived carbon nanothreads. a Quantitative multi cross-polarization (CP) spectrum of the sp2 - and sp3 -hybridized C; b Quantitative spectrum of the chemical group CH. As a reference the dashed line shows the spectrum of chemical group CH in the backbone of amorphous polystyrene; c Quantitative spectrum of the chemical group CH2 ; d Quantitative multiCP spectrum of the nonprotonated and mobile C; e Selective spectrum of the nonprotonated and mobile C, There is a peak at 40 ppm, which is residual artifact from the intense CH peak; f Selective spectrum of alkene and nearby alkyl CH; g Selective spectrum of aromatic carbons and nearby alkyl CH. h Quantification of the chemical composition based on the experimental measures in (e–g). Reprinted (adapted) with permission from [12]. Copyright {2018} American Chemical Society

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2 Experimental Aspect

solvent extraction. The proportion of sp2 -hybridized carbon is 23% after the removal of free benzene. Given the nonprotonated carbon fraction is 4% and there are two aromatic C − H per nonprotonated aromatic carbon, the content of aromatic carbon can be calculated as 3 × 4% = 12%. After excluding the nonprotonated carbon, the proportion of alkene carbon is 11%. The ratio of sp3 and sp2 carbon in degree-4 nanothreads is 4:2. The proportion of degree-4 nanothread is then obtained as (4 + 2)/2 × 11% = 33% (±9%). The content of CH2 is 3%, therefore, the fraction of CH2 containing units is calculated as 3 × 3% = 9%. Finally, the degree-6 nanothreads have the remaining all carbon which is about 32%. Li et al. [20] conducted NMR experiment to calculate the C-N bond fraction in pyridine-derived carbon nitride nanothread. In the 15 N and 13 C solid-state NMR spectra, they found chemical shifts between 50 and 95 ppm for nearly 45% of sp3 hybridized carbons, which suggests that the sp3 hybridized carbons are connected to nitrogen atoms.

2.3.3 Raman Spectroscopy Raman spectroscopy is a chemical analysis technique which has been employed to find out the vibrational modes of molecule structures. The structural information and conformation for coupled C–C bonds can be characterized by the intensity, polarization, and frequency of Raman spectrum. Generally, the multiple peaks in the Raman spectrum enable semiquantitative assessment of the structures and properties of carbon nanothreads. Fitzgibbons [9] utilized Raman spectroscopy to determine the structure of benzene-derived nanothreads. They calculated the spectra for five one-dimensional sp3 carbon structures: the (3,0) sp3 “nanotube”, the polymer I structure, and other three models constructed by introducing Stone– Wales defects into the (3,0) sp3 nanotube. All five structures have Raman-active radial breathing modes in the region from 690 to 820 cm−1 , which agrees well with the experimental measurement of a mode at 805 cm−1 . The frequency of the radial breathing mode of the sp3 carbon structures is higher than that of the sp2 carbon nanotubes [53], due to their smaller diameter. They found that the (3,0) “nanotube” and its two Stone–Wales variants with the lowest bond rotation density have Raman-active modes in the region from 930 to 990 cm−1 that are associated with a flexure of the nanothread. These modes are close to the experimentally observed mode at 1005 cm−1 . In contrast to the flexural modes of sp2 nanotubes which do not belong to Raman-active modes [54], these modes are Raman-active with optical character. Table 2.2 provides the characteristic Raman frequencies of hydrocarbon vibrations in the regions from 750 to 1100 cm−1 based on previous works [55, 56]. The existence of molecules originating from the reaction of benzene at high pressure and corresponding vibration modes can be excluded by matching the characteristic Raman frequencies, which provides confirmed proofs for the observed breathing and flexure modes at 805 and 1005 cm−1 in sp3 carbon nanothreads, respectively. Hexane dissolves small alkanes, adamantane, and diamondoids and so would extract these

2.3 Testing Technologies Table. 2.2 Characteristic Raman modes of hydrocarbon vibrations in the regions from 750 to 1100 cm−1 [56]

37 Position of peak (cm−1 )

Vibrational mode

837–905

C–C Skeletal stretch in n-Alkanes

800–900

Longitudal acoustic mode in n-Alkanes

800–810

C–C(CH)

802

Ring breathing of cyclohexane

1001

Ring breathing of cyclobutene

992

Ring breathing of benzene

1040–1100

Antisymmetric stretch of C–C-C in n-Alkanes

800

Stretch of C–C in adamantane

758

Stretch (Cage breathing) of C–C in adamantane

m ≥ 1, b  b

18

All Table 3.7 Number of two-ring cell enantiomer pairs. Reprinted (adapted) with permission from [6]. Copyright {2015} American Chemical Society

36

Two-ring cell enantiomer pairs Patterns

Count

Y0 0Y1 1

Y0 0Y1 2

Y0 0Y2 b

Y0 0Y3 (-b )

3

Y1 0Y1 1

Y1 0Y1 2

1

Y1 bY2 b

Y1 (-b)Y3 (-b )

6

Y2 0Y2 1

Y3 0Y3 2

1

Y2 0Y2 2

Y3 0Y3 1

1

Y2 1Y2 2

Y3 1Y3 2

1

Y2 0Y3 1

Y2 0Y3 2

1

Y2 1Y3 1

Y2 2Y3 2

1

All

1

16

3.2 Enumeration Rules Table 3.8 Reverse translation from IXY notation to 1–6 notation. Reprinted (adapted) with permission from [6]. Copyright {2015} American Chemical Society

59 Y1 1Y3 2

Original name

3 [4] 6 | [1] 2 5 [1] 2 6 | 3 [4] 5

Raw tagged rings

6 3 [4] | [1] 2 5 [1] 2 6 | 3 [4] 5

Incorporate bond variables

634125 126345

Drop tags, add underlines

631254 126345

Permute rows together

142365

Drop bottom row and shift by 1

To summarize, these 6 + 20 + 24 = 50 connectivities for degree-6 thread can be considered as 50 reaction pathways. The IXY notation mentioned above can also be translated to the 1–6 notation in Fig. 3.1. There is a translation key between them, as seen in Table 3.8. The first row of the table means the following. With the sites labeled 1 to 6 moving clockwise (seen from above), tagged ring I0 has two up-connectors, on sites 1 and 4, and four down-connectors, on sites 2, 3, 5, and 6. Sites 1 and 5 are tagged. Swapping the parts on either side of the vertical stroke gives the X0 pattern. Also, it may add (mod 6) any number to the site labels. Thus, I0 can be denoted not only by “[1] 4 | 2 3 [5] 6” but also by ‘[2] 5 | 3 4 [6] 1’ and ‘5 [2] | 4 [6] 1 3’. The conversion process can be simply illustrated as follows. Copy down the raw tagged rings from the table. Cyclically permute indices of the top row, separately on either side of the stroke in order to match the bond variables (1 and 2 in this case). The tags have done their job and can now be dropped. Underline the top-row indices before the stroke and erase it. Permute columns to put the bottom row in order. The superfluous bottom row is now dropped, but the top row should begin with “1”. This can be ensured by shifting all indices by the same amount mod 6, or (sometimes) by making a cyclic permutation of the list. DFT calculations show that 15 of 50 distinct carbon nanothreads have energy smaller than 80 meV/carbon atom. The atomic configurations and energies per (CH)6 formula unit relative to an individual graphane sheet of the 15 most stable carbon nanothreads are provided in Fig. 3.11. Based on the chirality, these carbon nanothreads can be categorized as achiral and chiral. There are three kinds of carbon nanothreads, which has been mentioned in previous studies, including the (3,0) sp3 nanotube [2], polymer I [3], and polytwistane [12] structure. The first two structures have the lowest energy among the achiral configurations, and the polytwistane has the lowest energy among the chiral nanothreads. Different nanothread structures possess different number of carbon atoms in the crystallographic unit cells. For instance, chiral threads 143,652 have two carbon atoms while 135,462 have six carbon atoms in the crystallographic unit cells. The chiral carbon nanothreads have the lowest energy, due to the possible longwavelength structural degree of freedom intrinsic to the helical characteristics, which

60 Fig. 3.11 Fifteen distinct carbon nanothreads optimized using DFT have the lowest energy relative to graphane per unit of (CH)6 . The marked letters denote the representative structures a sp3 (3,0) tube, b polymer I, and c polytwistane. Reprinted (adapted) with permission from [6]. Copyright {2015} American Chemical Society

3 Topological Structure

3.2 Enumeration Rules

61

enable them to adjust the coordinates through relaxation. In conclusion, the biding energies of carbon nanothreads are close to those of well-known Cx Hx hydrocarbons. The reference state of the additional hydrogen needs to be known if comparison is made to other sp2 -hybridized carbon nanomaterials. If the molecular hydrogen is considered for the reference state, the energy of the most stable nanothread (polytwistane) is 0.68 and 0.90 eV per six carbons lower than that of graphene and a (10, 10) carbon nanotube [13], respectively.

3.2.3.3

Substituted Nanothread

The isomeric possibilities for nanothreads with substituent can be determined by chemical formula ((CH)5 X)n (X = heteroatom or −CR substituent) [8]. The chemical formula of carbon nitride nanothreads is expressed as ((CH)5 N)n . The enumeration methods of pure carbon nanothread can be adopted, and an integer string can be added to denote the locations of N atoms in the six-membered rings. An example polytwistane polytwistane_153 is given here, which indicates that the carbon atoms in the positions 1, 5, and 3 together with the bonded hydrogen atoms are replaced with nitrogen atoms, as seen in Fig. 3.12.

3.2.3.4

Cubane-Derived Nanothread

Cubane (C8 H8 ) is a pure sp3 hybridized hydrocarbon molecule with cubic shape and octahedral symmetry. Specifically, the C − C − C bond angle in the cubane structure is 90°. Breaking one bond on the four-membered rings will create two radicals, which could grow and promote the formation of covalent bonds along length direction. A family of one-dimensional nanothread structures can be built by creating bonds between the different cube locations (1, 2, 3, and 4) of adjacent faces within a molecular stack. The reason for building connectivity between four-member rings is introduced in Fig. 3.13. The 1–3 connection leads to structure Cub-1. Cub-1 is a flexible polymer chain with cross-linked cubyl cage backbone, which possesses higher relative energy of 0.86 eV/CH. In addition, the 2–2 connection could lead to a rigid rod-shape molecule with minimized rotational degrees of freedom, which is a more energetically stable structure. Among all these linear structures, Cub-13, which can also be denoted as 2,2-metapolycubane, is the most energetically favored structure with relative energy of 0.7 eV/CH. Cub-12 and Cub-14 are alternative structures, which can be constructed based on 2–2 connection on the ortho sites, which maintain the square ring. They, hence, have a higher relative formation energy of 0.82 eV/(CH) and 0.79 eV/(CH), respectively. Cub-14 is also known as ladderane because of its ladder-like geometry. It should be noted that the precursor molecular cubane has a very high relative energy of 1.21 eV/CH compared to all these structures. All the enumerated structures are more stable thermodynamically than the starting cubane.

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Fig. 3.12 Ring numbering for four types of benzene nanothreads. The orange-numbered ring denotes the first ring; the blue-numbered ring below is the second ring and so on. The nomenclature tube (3, 0)_1245 means this pyridine nanothread is based on the structure of the tube (3, 0) benzene nanothread, substituting N atoms for the C-H bonds at position 1 in 1st ring, position 2 of 2nd ring, position 4 of the 3rd ring and position 5 of the 4th ring; the repeat unit contains four rings. Reprinted (adapted) with permission from [8]. Copyright {2018} American Chemical Society

3.3 Reaction Pathways Based on the prementioned enumeration method for the structures of degree-two, four, and six carbon nanothreads, some possible pathways that are able to transform benzenes to carbon nanothreads with larger degrees of saturation could be proposed [7]. Starting from benzenes to degree-two nanothreads, one double bond in a sixmembered ring turns to sp3 C–C bond, and two bonds are created in opposite direction, one connect to the upward ring and the other connect to the downward ring. There are totally three kinds of cycloadditions, which could transform benzenes to degree-two nanothreads, including [2a + 2a ] (generating −o+ − and −o+ − o−− ),

3.3 Reaction Pathways

63

Fig. 3.13 Rationale of building linear sp3 structures from four-member rings. Four candidate structures are enumerated by two types of connectivity, 1–3 connection and 2–2 connection. Reprinted (adapted) with permission from [14]. Copyright {2020} American Chemical Society

[4a + 2a ] (−p− o+ − ), and [4a + 4a ] (−p − ). In going from degree-two nanothreads to degree-four nanothreads, not only the [4a + 2a ], [4a + 4a ], and [4 s + 4 s ] cycloadditions but also the double-bond zipping can turn a degree-two nanothread to a degreefour nanothread. Generally, there are four connection methods for the double-bond zipping, including two “zig-zag” (zz1 and zz2) and two “armchair” (ac1 and ac2) types, as seen in Fig. 3.14. The zz1 type could also be the zz2 in certain bonding sequences, so as the ac1 to ac2. All the degree-four nanothreads considered here are syn isomers. Note that only the syn isomers can further turn to degree-six nanothreads, rather than the anti–isomers, which have dead ends. The transformation diagram is provided in Fig. 3.15. Fig. 3.14 Transformation from benzene to degree-two nanothread. a Bonding position and direction; b Four connection methods for the double-bond zipping. Reprinted (adapted) with permission from [7]. Copyright {2015} American Chemical Society

(a)

(b)

64

3 Topological Structure

Fig. 3.15 Possible reaction ways to transform benzene to nanothreads with various degrees of saturation. Reprinted (adapted) with permission from [7]. Copyright {2015} American Chemical Society

3.4 Stone Wales Defects According to the comparison between experiment and simulation work, the carbon nanothread is also be regarded as a hydronated (3,0) nanotube with various distributions of Stone Wales (SW) defects (rotating C–C bond angle by 90 °C), as seen in Fig. 3.16a. The radial distribution function (RDF) g(r), of a thread structure with a

3.4 Stone Wales Defects

65

Fig. 3.16 Proposed atomic structure of carbon nanothreads. a Snapshot of carbon nanothread with two SW defects located ~2 nm from each end. The inset shows the details of SW defects. b RDF curves of carbon nanothreads with different amounts of SW defects, which shows ignorable difference. Reprinted (adapted) with permission from [15]. Copyright {2015} American Chemical Society

length of ~8 nm at a temperature of 300 K, was compared with the previous work. It is found that the simulated RDF curves in Fig. 3.16b agree with previous findings. The first-neighbor shell that is indicative of sp3 C–C bonds is observed at ~1.52 Å. The domain peak at ~1.1 Å corresponds well with the proposed 1:1 C/H stoichiometry. Consecutive peaks are observed because of the second nearest C−C and C−H, H−H

66

3 Topological Structure

Fig. 3.17 Effect of SW content on the stability and energy of carbon nanothreads. a The connection way between components H-tube (3,0) and SW defect; b The energy of carbon nanothread with variation of SW content χ. Reprinted (adapted) with permission from [9]

correlations. Of note, it is necessary to introduce certain amounts of SW defects on the thread in order to represent the real structure in experiment. Gao et al. [9] utilized DFT to explore the structural characteristics of carbon nanothreads with variation of SW content. The computational methodology is illustrated as follows. Generalized gradient approximation (GGA) with the Perdew– Burke–Ernzerhof (PBE) functional [16] was employed to express the exchange– correlation interaction. The projector-augmented wave (PAW) method [17] was utilized to represent the core electrons. A planewave basis kinetic energy cutoff of 500 eV and an energy convergence criterion of 10–5 eV were considered throughout the calculations. A conjugate-gradient algorithm was applied to optimize the ions with a force convergence criterion of 0.01 eV Å−1 . To calculate the SW transition energy, the climbing-image nudged elastic band (cNEB) method [18] was employed to find minimum energy paths with a force convergence criterion of 0.02 eV Å−1 . The structural block of SW defect is depicted in the red region in Fig. 3.17a, there is a kinked connection between the components H-tube (3,0) and SW defect. It is plotted in Fig. 3.17b the effects of SW content (χ) on the energy of carbon nanothreads. Results show that the energy first reduces with increasing SW content χ from 0 to 0.5, and then increases when the χ exceeds 0.5. The most stable carbon nanothread with minimum energy is obtained when the ratio between H-tube (3, 0) and SW composition is 1:1, and such energy is even ~0.43 eV per unit cell smaller than a sp3 H-tube (3,0) without SW defects. The introduction of a SW defect into a perfect sp3 H-tube (3, 0) can be realized by breaking C–C bonds and rotations. Figures 3.18 shows the atomic structures of initial state (Fig. 3.18a), transition state (Fig. 3.18b), and ending state (Fig. 3.18c) during the formation of a SW defect, and their corresponding transformation energy. The transformation process starts by breaking two r1, r2 bonds while forming two new r3, r4 bonds. Once a SW defect is formed, about 1% strain is released, resulting in longer equilibrium length of the new carbon nanothreads (signed as DNT1/8 ). The energy barriers depend strongly on the applied tensile strain. The energy barrier of

3.4 Stone Wales Defects

67

Fig. 3.18 Transformation process of SW defect and energy evolution during deformation. a Sp3 H-tube (3,0); b Structure in transition state; c Structure after SW transformation; The energy barrier during SW transformation at tensile strains of d 1%; e 6%; f 11%; g 13%. Reprinted (adapted) with permission from [9]

SW transformation under 1% tensile strain is plotted in Fig. 3.18d. It can be seen that the DNT1/8 structure is more stable than the sp3 H-tube (3, 0). The first energy barrier corresponding to the rotation of first C–C bond is 5.04 eV, and the second energy barrier corresponding to the rotation of the second C–C bond is 4.74 eV. These energy barriers are close to the SW transformation in carbon nanotubes [19]. The energy barrier could be further decreased by increasing the tensile strain, as seen in Figs. 3.18d–g. The maximum strain applied here does not reach the fracture strain of carbon nanothreads. It can be seen that the maximum energy barrier used to be the first barrier, indicating that the first barrier governs the SW transformation in nanothread.

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References 1. Fitzgibbons TC, Guthrie M, Xu ES, Crespi VH, Davidowski SK, Cody GD, Alem N, Badding JV (2015) Benzene-derived carbon nanothreads. Nat Mater 14(1):43–47 2. Stojkovic D, Zhang P, Crespi VH (2001) Smallest nanotube: breaking the symmetry of sp3 bonds in tubular geometries. Phys Rev Lett 87(12):125502 3. Wen XD, Hoffmann R, Ashcroft NW (2011) Benzene under high pressure: a story of molecular crystals transforming to saturated networks, with a possible intermediate metallic phase. J Am Chem Soc 133(23):9023–9035 4. Barua SR, Quanz H, Olbrich M, Schreiner PR, Trauner D, Allen WD (2014) Polytwistane. Chemistry—A Eur J 20(6):1638–1645 5. Olbrich M, Mayer P, Trauner D (2014) A step toward polytwistane: synthesis and characterization of C2-symmetric tritwistane. Org Biomol Chem 12(1):108–112 6. Xu ES, Lammert PE, Crespi VH (2015) Systematic enumeration of sp3 nanothreads. Nano Lett 15(8):5124–5130 7. Chen B, Hoffmann R, Ashcroft NW, Badding J, Xu E, Crespi V (2015) Linearly polymerized benzene arrays as intermediates, tracing pathways to carbon nanothreads. J Am Chem Soc 137(45):14373–14386 8. Li X, Wang T, Duan P, Baldini M, Huang HT, Chen B, Juhl SJ, Koeplinger D, Crespi VH, Schmidt-Rohr K, Hoffmann R, Alem N, Guthrie M, Zhang X, Badding JV (2018) Carbon nitride nanothread crystals derived from pyridine. J Am Chem Soc 140(15):4969–4972 9. Gao J, Zhang G, Yakobson BI, Zhang YW (2018) Kinetic theory for the formation of diamond nanothreads with desired configurations: a strain–temperature controlled phase diagram. Nanoscale 10(20):9664–9672 10. Houk KN, Lin YT, Brown FK (1986) Evidence for the concerted mechanism of the Diels-Alder reaction of butadiene with ethylene. J Am Chem Soc 108(3):554–556 11. Dewar MJ, Olivella S, Stewart JJ (1986) Mechanism of the Diels-Alder reaction: reactions of butadiene with ethylene and cyanoethylenes. J Am Chem Soc 108(19):5771–5779 12. Enyashin AN, Ivanovskii AL (2011) Graphene allotropes. Phys Status Solidi (B) 248(8):1879– 1883 13. Chen B, Wang T, Crespi VH, Li X, Badding J, Hoffmann R (2018) All the ways to have substituted nanothreads. J Chem Theory Comput 14(2):1131–1140 14. Huang HT, Zhu L, Ward MD, Wang T, Chen B, Chaloux BL, Wang Q, Biswas A, Gray JL, Kuei B, Cody GD, Epshteyn A, Crepsi VH, Badding JV, Strobel TA (2020) Nanoarchitecture through strained molecules: cubane-derived scaffolds and the smallest carbon nanothreads. J Am Chem Soc 142(42):17944–17955 15. Roman RE, Kwan K, Cranford SW (2015) Mechanical properties and defect sensitivity of diamond nanothreads. Nano Lett 15(3):1585–1590 16. Perdew JP, Burke K, Ernzerhof M (1996) Generalized gradient approximation made simple. Phys Rev Lett 77(18):3865 17. Blöchl PE (1994) Projector augmented-wave method. Phys Rev B 50(24):17953 18. Henkelman G, Uberuaga BP, Jónsson H (2000) A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J Chem Phys 113(22):9901–9904 19. Nardelli MB, Yakobson BI, Bernholc J (1998) Mechanism of strain release in carbon nanotubes. Phys Rev B 57(8):R4277

Chapter 4

Mechanical Properties of Carbon Nanothreads

4.1 Introduction Ever since the successful synthesis of carbon nanothreads through solid-state reaction of benzene at high pressure [1], carbon nanothreads have been considered as a promising candidate for carbon-based nanomaterials for scientific and engineering applications, becoming one of the most intensive research focuses. However, due to the harsh environments for mechanical properties characterization, there is no existing experimental results yet for the carbon nanothreads. To explore the promising role of carbon nanothreads and provide theoretical guidance for experimental characterization, computational simulations including Density-functional theory (DFT) calculation and molecular dynamics (MD) have been widely employed and considered as powerful and convincing tools for studying the mechanical properties. Because the structural configuration will significantly affect the mechanical performance of carbon nanothreads, lots of efforts have been exerted to study the structural characteristics and corresponding applications [2–6]. It has been demonstrated recently that the carbon nanothreads have an extremely high Young’s modulus and large failure strain [7, 8]. DFT calculation is a modelling technique based on quantum mechanics, and it has been utilized in physics, chemistry and computational materials science to study the electronic characteristics of multibody systems. According to the DFT, the fundamental properties of a multibody system are predictable by empolying specific functionals. The functionals in the DFT are related to the electron density in space. Referring to the computational materials science, the DFT calculation is able to predict the material performance based on quantum mechanics, while allowing the intrinsic material properties to be unknown. Although the DFT has theoretical backgrounds from the Thomas–Fermi model, it was first aroused by Walter Kohn and Pierre Hohenberg in their theoretical frameworks, which is the well-known Hohenberg–Kohn theorem (HK) [9]. Walter Kohn and Lu Jeu Sham further developed the HK theorem and proposed the Kohn–Sham DFT [10]. In the context of this framework, it simplified the complicated multibody problem of interacting electrons in a © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. M. Liew et al., Carbon Nanothreads Materials, Materials Horizons: From Nature to Nanomaterials, https://doi.org/10.1007/978-981-19-0912-2_4

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4 Mechanical Properties of Carbon Nanothreads

static external potential and introduced a concept of noninteracting electrons moving in an effective potential. The introduced potential contains the external potential and the influences of the electron–electron Coulomb interactions, e.g., the exchange and correlation interactions. However, the vital issue for DFT calculation is the precise functional to describe the exchange and correlation interactions in multibody system, excluding the homogeneous electron gas. The local-density approximation (LDA) is one of the most popular functionals to approximate the exchange and correlation interactions. It is assumed in the LDA functional that the electron density is solely dependent on the coordinate of electron. Due to the assumption, the exchange energy of system used to be underestimated and the correlation energy is overestimated [11]. The errors brought by the portions of exchange energy and correlation energy are likely to recompense each other and reduce to a certain content. To improve the accuracy and represent the heterogeneity of the electron density, the generalized gradient approximations (GGA) is proposed to consider both the density and gradient of density at each point [12]. The GGA functional have been demonstrated with high accuracy in calculations of molecular geometry and ground-state energy. More accurate functional can be referred to the meta-GGA functionals which were proposed on the basis of the GGA [13]. The meta-GGA functionals consider the second derivative of the electron density into the exchange–correlation potential, while the original GGA functional only consider the first derivative of the electron density. There are multiple types of meta-GGA functionals, such as the TPSS (named after the authors’ initials) [14] and the Minnesota Functionals [15]. These functionals extend the GGA function with additional terms, including electron density, gradient of the density, and non-interacting kinetic energy density. However, the meta-GGA functionals are confronted with difficulties in accurate description of the exchange energy. This issue could be further solved by calculating the exchange energy by Hartree–Fock theory, whose functionals are called as hybrid functionals [16]. Despite the development of DFT, the size of the many-body system is very small due to the extremely high computational cost. The classical MD simulation can overcome the timescale and spatial limitations of DFT, with ability to investigate large-scale system with billions of atoms. The theory of MD method was first put forward by Alder and Wainwright [17] to study a condensed phase fluid system in 1957. Before their research works, the problem whether a solid–fluid phase transition occured to a hard spheres system was very difficult to solve by existing static simulations including MM method. Therefore, they attempted to numerically solve Newton’s equations of motion and performed the earliest MD simulation. In the simulations, the atoms and molecules could interact with each other, and their motions were tractable as a function of time, which provided possibility to study physical problems. Their results demonstrated that MD simulation is capable to investigate atoms and molecules behaviors with time evolution and provide dynamic information for molecular systems, which shed light to the application of MD in statistical mechanics. With the fast development of computer technology, Rahman [18] firstly applied MD simulation to a realistic fluid model concerning both the Newton’s equations of motion and Lennard–Jones potential. This

4.1 Introduction

71

classic paper demonstrated that MD simulation could precisely obtain dynamic quantities including pair-correlation function and the constant of self-diffusion, which set up the framework for subsequent MD simulation. As Verlet [19] proposed the famous Verlet algorithm for integrating Newton’s equation of motion, the immense advances of MD simulation occurred from 1970 to 1990s. In this period, ensembles were introduced to MD simulation for thermodynamic system. In 1980, Andersen [20] began to introduce an isothermal–isobaric ensemble with a constant number of particles, pressure and temperature (N, P, T) into MD simulation, which was further extended by Parrinello and Rahman [21] to study the crystal structures. Subsequently, Nosé and Hoover [22, 23] introduced the famous Nosé-Hoover thermostat into MD simulation to realize the realistic canonical ensemble with constant molecules, volume and temperature (N, V, T). In addition, a variety of force fields which represent the functional forms and associated parameters for energy calculation, were developed to study more complex structures. Some typical and popular force fields could be listed as CHARMM [24], AMBER [25], GROMOS [26], OPLS [27] and COMPASS [28]. For example, the first three force fields were widely used for biomolecule systems, while the latter two were more suitable for condensed phase systems. With the development of computer simulation, MD simulation has been widely employed in the field of materials science, biophysics and nano-engineering. The accuracy of MD simulation results is strongly dependent on the potentials used to describe the interatomic interactions. To date, various potentials including Morse potential, Tersoff potential, Brenner potential and reactive force field (Reaxff) are proposed to study the properties of carbon-based nanomaterials. To represent the bond formation and breakage during reaction, Abell [29] proposed the bond order formalism based on quantum–mechanical concept. and developed a simple model of bonding energetics. He demonstrated this model could well represent the bonding behaviors in metallic and molecular system. Tersoff [30, 31] propose a new concept of bond order to describe the bonding behavior, and the so-called Tersoff potential were found to predict the properties of covalent materials well. Nordlund et al. [32] also made modification to the Tersoff potential to consider the interlayer interaction of covalent materials. Brenner [33] further improved the Tersoff potential by adding additional terms into the bond order function. The so-called Brenner potential is shown with better reliability in solving conjugacy problem, while providing accurate descriptions of the bond order, bond formation and bond breakage. Based on the Brenner potential, another potential called the adaptive intermolecular reactive bond order (AIREBO) potential was developed to consider the bonding behavior, noncovalent atomic interactions, and single-bond torsion torsional interactions [34]. The potentials mentioned above use empirical energy functions to calculate the system energy, and the interatomic distances are needed only during the calcualtion. More accurate empirical potential could be referred to the Reaxff potential which could characterize the bond order during the reaction process through bond order formalism, and the bond order is determined based on the interatomic distances. In the Reaxff potential, the chemical bonding derived by the electronic interactions are treated implicitly, which allows it to study the chemical reaction without referring to the quantum mechanics. In order to use one single atom type for each element to cover

72

4 Mechanical Properties of Carbon Nanothreads

the information of bond breaking and formation, the ReaxFF potential incorporates lots of parameters into the functionals. To derive these parameters, it is required to perform a large amount of training tests, including the bonding strength, angle stretch, energies of activation and reaction, surface energy, and so on. Normally, these parameters are obtained by using electronic structure methods. For example, DFT has been used as a practical way to provide the training data for the ReaxFF. Among the techniques for parameterization of the ReaxFF, global optimization is the most suitable one to describes the training data [35]. So far, the ReaxFF has been used to study the reaction process at the interface in multiple-phase system, and the simulation results provide important insights for experimental testing.

4.2 Computational Methods 4.2.1 Density Functional Theory According to the Born–Oppenheimer approximation, the nuclei of the studied atoms or molecules are not allowed to move, and only the electrons outside the nuclei can move freely, which is a common approach used in many-body electronic system. This will generate a static external potential V. The total energy of the electronic system is expressed by a wavefunction (r1 , …, rN ), which is time independent and satisfies the Schrödinger equation: ⎡  ⎤ N N N   2    

 H = T + V + U  = ⎣ ∇2 + V (ri ) + U ri , r j ⎦ = E. − 2m i i 







i=1

i=1

i< j

(4.1) ˆ is the Hamiltonian, E is the energy of the Here, N is the number of electron, H ˆ is the external potential brought by the positive system, Tˆ is the kinetic energy, V charge of the nuclei, and Û is the interaction energy between electrons. The operators Tˆ and Û refer to universal operators, and they will be consistent for any system with ˆ is dependent on the system. Due to the N electrons. In comparison, the operator V interaction energy term Û, this complex equation involing many particles cannot be further separated into simpler single-particle equations. Numerous experienced ways have been proposed to solve the Schrödinger equation according to the expansion of the wavefunction in Slater determinants. Among these approaches, the Hartree–Fock method is the easiest. Other experienced approaches built on the basis of Hartree–Fock method are termed as post-Hartree– Fock methods. However, these approaches have one common issue-ultrahigh computational cost when dealing with large-scale system, which limits their applications. To improve the applicability, the DFT simplifies the many-body problem with term Û to a single-body problem without term Û. The important parameter in DFT is the

4.2 Computational Methods

73

electron density n(r), which is expressed by a normalized wavefunction : n(r) = N ∫ d3 r2 · · · ∫ d3 r N  ∗ (r, r2 , . . . , r N )(r, r2 , . . . , r N ).

(4.2)

Note that there is a reversable correlation in the above equation. Once the groundstate density n0 (r) is known, thoeretically, it is plausible to calculate the corresponding ground-state wavefunction  0 (r1 , …, rN ). That is to say,  is only dependent on n0 : 0 = [n 0 ],

(4.3)

and as a result, the ground-state expectation value of an observable Ô is dependent on n0 : ˆ O[n 0 ] = [n 0 ]|O|[n 0 ]

(4.4)

Particularly, the ground-state energy can be expressed as a functional of n0 :    ˆ |[n 0 ] , E 0 = E[n 0 ] = [n 0 ]O

(4.5)

      where the external potential term  Vˆ  could be given as a function of density n: V [n] = ∫ V (r)n(r)d3 r.

(4.6)

Similar to the operators mentioned above, the functionals T[n] and U[n] are universal functionals which are dependent on the system, while the functional V[n] is dependent on the system. Considering a system with specified V[n], the energy functional needs to be minimized with respect to the n(r): E[n] = T [n] + U[n] + ∫ V (r)n(r)d3 r,

(4.7)

on condition that the functionals T[n] and U[n] are accurate. The gound-state density n0 could be obtained successfully after minimization of the energy functional, and other ground-state valuables could be also calculated. To solve the variational problem in the energy functional during minimization, Lagrangian method of undetermined multipliers could be applied. It can be started by an energy functional without explict expression of the interaction energy between electrons,      ˆ s s [n] , E s [n] = s [n]Tˆ + V

(4.8)

74

4 Mechanical Properties of Carbon Nanothreads

ˆ s is an effective potential where the here, Tˆ is the kinetic-energy operator, and V particles are moving. On the basis of the E s functional, the Kohn–Sham equations of the system can be expressed as:   2 2 − ∇ + Vs (r) ϕ(r) = εi ϕi (r). 2m

(4.9)

The orbitals ϕ i which characterize the density n(r) can be obtained accordingly: n(r) =

N 

|ϕi (r)|2 ,

(4.10)

i=1

Finally, the effective potential for single particle is given as: Vs (r) = V (r) + ∫

n(r  ) 3  d r + VXC [n(r)], |r − r  |

(4.11)

where V (r) is the external potential, the second term describes the Coulomb repulsion between electrons, and the VXC in the third term is the exchange–correlation potential which contains all the interactions in manu-body electronic system. As seen from the Eq. (4.11), the second and third terms is a function of n(r), which is determined by the orbitals ϕ i . The orbitals ϕ i is in turn determined by the Vs . Therefore, solving the Kohn–Sham equation is practible using a self-consistent approach, such as iteration. Starting with an initial guess of n(r), the corresponding Vs can be obtained, and the Kohn–Sham equations can be solved to obtain the corresponding oribitals ϕ i . Once a new n(r) is calculated, this process is repeated again until convergence. In addition to the iterative method, a non-iterative approximate formulation called Harris functional DFT provide another way to solve the Kohn–Sham equation.

4.2.2 Molecular Dynamics In MD system, the total energy of the system is expressed in terms of atomic coordinates. Every atom in the system is considered as a point carrying mass. The interactions between atoms are determined by a known potential energy E(r1 , r2 ,...,rN ), where ri is the vector position of the ith atom. The potential energy of the system is updated at each time step, and the position and velocity of each atom are obtained by solving the Hamilton’s classical equation of motion from Newton’s second law: Fi = m i ×

d 2 ri = −∇i E(r1 , r2 , . . . r N ), dt 2

(4.12)

4.2 Computational Methods

75

Here, m i and ri are the mass and coordinates of the ith atom, respectively, E is the potential energy of the system which is strongly dependent on the force field parameters applied to atoms. To integrate the Newton’s equations of motion, the Verlet method [19] and other high-order methods have been used to calculate the trajectory of each atom at each time step.

4.2.2.1

AIREBO Potential

In the REBO potential, the bond energy between atoms i and j is expressed as:





 E B ri j = VR ri j − Bi j V A ri j .

(4.13)

Here, ri j is the distance between two closest atoms i and j, and the interatomic repulsion between them, such as the core-core interactions, is represented by the function VR (ri j ), and interatomic attraction originating from the valence electrons is represented by the function V A (ri j ). The functions VR (ri j ) and V A (ri j ) are expressed as:

 D (e) −√2Sβ (ri j −R (i) )  f c ri j , V R ri j = e S−1

(4.14)

 D (e) S −√2/Sβ (ri j −R (e) )  e f c ri j . V A ri j = S−1

(4.15)

f c is a smooth cutoff function that limits the reactive distance between atoms, as given by: ⎧ ⎫ 1ri j R

(4.16)

where R (1) and R (2) are the effective cutoff distance for the function. The variable Bi j in Eq. (4.13) describes multibody interaction between the bond i-j and the local environment of atom i: ⎡ Bi j = ⎣1 +



⎤−δ

 G θi jk f c (rik )⎦ ,

(4.17)

k =i, j

where θi jk is the angle between the two bonds i-j and i-k, and G is the angle function expressed as:

76

4 Mechanical Properties of Carbon Nanothreads

 

 c02 c02 G θi jk = a0 1 + 2 −

2 . d0 d02 + 1 + cos θi jk

(4.18)

where the parameters D (e) , S, β, R (e) , δ, R (1) ,R (2) , a0 , c0 , d0 are obtained by fitting the simulated properties with experimental measurements. In the context of AIREBO potential, the total energy of the system can be calculated by adding the interatomic interactions, including covalent bonding REBO interactions, Lennard–Jones (LJ) interactions as well as torsional interactions: ⎛ ⎞  1  ⎝ R E B O + E iLj J + E kitorjls ⎠, Ei j E= 2 i j =i k =i, jl =i, j,k

(4.19)

The LJ potential is used to describe the vdW interaction between atoms:

 "

#



  E iLj J = S tr ri j S tb bi∗j Ci j ViLj J ri j + 1 − S tr ri j Ci j ViLj J ri j ,

(4.20)

where S(t) is a universal switching function. It is a unity when t < 0 and zero when t > 1. For any value t between 0 and 1, the S(t) is switched smoothly using a cubic spline function: " # S(t) = (−t) + (t) (1 − t) 1 − t 2 (3 − 2t) .

(4.21)

Here, the (t) is the Heaviside step function. The interactions of LJ potential between a pair of atoms are determined by their distance, which is expressed in the S(tr (ri j )) term with the scaling function tr (ri j ):

 tr ri j =

ri j − riLJmin j riLJmax − riLJmin j j

.

(4.22)

The scaling function serves to rescale to domain of the switching function S(t). The S(tr (ri j )) term will be zero once the distance ri j is largen than the riLJmin , and j hence the LJ potential will not be affected by the pair distance. ViLj J is expressed as LJ 12–6 potential: $ ViLj J

= 4 i j

σi j ri j

%12

$ −

σi j ri j

%6  ,

(4.23)

where ri j is the distance between a pair of atoms i and j, is the depth of the potential well (also called as dispersion energy), and σ is the critical distance where the interatomic potential V goes to zero.

4.2 Computational Methods

77

The second parameter affecting the LJ potential is the bond switch term S(tb (bi∗j )) in Eq. 4.20, where tb (bi∗j ) is the scaling function given by:

 tb bi j =

b − bimin j bimax − bimin j j

.

(4.24)

The scaling function tb (bi∗j ) change the bond-order term bi j in REBO potential to a range which can be used in the cubic spline function of S(t). In the non-covalent part of AIREBO potential, the bond-order term bi∗j can be hypothetically regarded as a bi j term which is given as riLJmin , j bi∗j = bi j |ri j = rimin j .

(4.25)

It is presumed that the distances from atoms i and j to each of their neighbors are constant when calculating bi∗j . If the intermediate atoms connecting atoms i and j are larger than two, the LJ potential is govened by another switching function: Ci j = 1 − max{wi j (ri j ), wik (rik )wk j (rk j ), ∀kwik (rik )wkl (rkl )wl j (rl j ), ∀k, l}, (4.26) where wi j (ri j ) is a weighting function ranging from 0 and 1:



 wi j ri j = S  tc ri j .

(4.27)



with S (t) a switching function:

 1 S  tc ri j = (−t) + (t) (1 − t) [1 + cos(π t)], 2

(4.28)

The torsional interactions expressed in Eq. 4.29 is incorporated in the AIREBO potential for all dihedral angles in the system, which is proportional to the weighting  function wi j ri j  

 1 256 10 & ω ' cos − , V tors ωi jkl =

405 2 10 E tors =

1    wi j (ri j )w jk (r jk )wkl (rkl ) × V tors (ωi jkl ). 2 i j =i k =i, j l =i, j,k

(4.29) (4.30)

78

4 Mechanical Properties of Carbon Nanothreads

4.2.2.2

ReaxFF Potential 

In the context of ReaxFF, it is assumed that the bond order BOi j between a pair of atoms i and j is obtainable with the known interatomic distance ri j :   T pbo,4   π π pbo,6    $ % pbo,2  ri j ) ri j ri j  + exp pbo,3 · BOi j = exp pbo,1 · + exp pbo,5 · . ro ro ro

(4.31) Three exponential terms needs to be considered in the above equation, including (1) the sigma bond ( pbo,1 and pbo,2 ) which is unity if the bond length is below ∼1.5 Å but negligible if the bond length is over ∼2.5 Å; (2) the first π bond ( pbo,3 and pbo,4 ) which is unity if the bond length is below ∼1.2 Å and negligible if the bond length is over ∼1.75 Å, and (3) the second π bond ( pbo,5 and pbo,6 ) which is unity if the bond length is below ∼1.0 Å and negligible if the bond length is above ∼1.4 Å. The sigma bond will only be concerned for C-H and H–H bonds, and correspond ingly the maximum bond order BOi j is 1. In the cases of overcoordination or residual 1–3 bond orders in valence angles, the  bond orders BOi j needs to be adjusted accordingly, as shown in Eqs. 4.32a-f. Note that for the overcoordination correction (Eqs. 4.32b–4.32d), it is only applicable to C–C bonds. The residual 1–3 bond correction (Eqs. 4.32e and f) is applicable to all the bonds in a molecular system. Eventually, the bond orders in the molecule could be determined by multiplying the bond orders from Eq. 4.31 by the correction factors from Eqs. 4.32. &  ' &  ' &  '    BOi j = BOi j · f 1 i , j · f 4 i , BOi j · f 5 j , BOi j , ⎛

&

'

&

(4.32a) '

⎞     Vali + f 2 i , j Val j + f 2 i , j 1 ⎝ &  ' &  '+ &  ' &   ' ⎠, f 1 i , j = · 2 Vali + f 2 i , j + f 3 i , j Val j + f 2 i , j + f 3 i , j



&  ' & ' & '   f 2 i , j = exp −λ1 · i + exp −λ1 · j , (  & &  ' ' & ') 1 1   f 3 i , j = · exp −λ2 · i + exp −λ2 · j , · ln λ2 2 &  ' 1 



, f 4 i , BOi j =    1 + exp −λ3 · λ4 · BOi j · BOi j − i + λ5 &  ' 1  .



f 5 j , BOi j =    1 + exp −λ3 · λ4 · BOi j · BOi j − i + λ5 The bond energies can be calculated from the corrected bond order BOi j

(4.32b) (4.32c) (4.32d) (4.32e) (4.32f)

4.2 Computational Methods

79

& '  p ! E bond = −De · BOi j · exp pbe,1 1 − BOi jbe .

(4.33)

The valence angle terms can be calculated by. 





 

2  , E val = f 7 BOi j · f 7 BO jk · f 8 j · ka − ka exp −kb o − i jk ' &

 f 7 BOi j = 1 − exp −λ11 · BOiλj11 ,

(4.34a) (4.34b)

2 + exp(−λ13 · j ) · [λ14 − (λ14 − 1) 1 + exp(−λ13 · j ) + exp( pv,1 · j ) 2 + exp(λ15 · j ) (4.34c) · 1 + exp(λ15 · j ) + exp(− pv,2 · j ) *  %λ16 + neighbors $  ( j) 1 SBO = j − 2 · 1 − exp −5 ·

j BO jn,π , + 2 n=1 f 8 ( j ) =

j,2 = j if j < 0,

j,2 = 0 if j ≥ 0, SBO2 = 0 if SBO ≤ 0

(4.34d)

SBO2 = SBOλ17 if 0 < SBO < 1, SBO2 = 2 − (2 − SBO)λ17 if 1 < SBO < 2, SBO2 = 2 if SBO > 2, 0 = π − 0,0 · {1 − exp[−λ18 · (2 − SBO2)]}.

(4.34e)

In terms of torsional interaction, the Eqs. (4.35a–4.35c) can be used to calculate the energy of torsion ωijkl for BO → 0 and for BO greater than 1. The V 2 -cosine term in Eq. 4.35a is related to the the bond order of the central bond BO jk . Its magnitude will reach the largest value if the torsion has a central double bond (BO jk = 2).

 E tor s = f 10 BOi j , BO jk , BOkl · sin i jk · sin ikl  

2



1 · 1 − cos 2ωi jkl V2 · exp BO jk − 3 + f 11 j , k · 2

80

4 Mechanical Properties of Carbon Nanothreads



1 (4.35a) + V3 · 1 + cos 3ωi jkl , 2

 "

# f 10 BOi j , BO jk , BOkl = 1 − exp −λ23 · BOi j · "

# " # 1 − exp −λ23 · BO jk · 1 − exp(−λ23 · BOkl ) , (4.35b) #

"

 2 + exp −λ24 · j + k # " # . (4.35c) "



f 11 j , k = 1 + exp −λ24 · j + k +exp λ25 · j + k Not only the covalent interactions but also the repulsive and attractive interactions are dependent on the overlap. The non-covalent interactions including vdW and Coulomb forces are concerned in all interatomic interactions, which can avoid inconvenient modifications in the description of energy upon bond breakage. To add the vdW interactions into consideration, a Morse potential (Eqs. 4.36a, 4.36b) dependent on the distance is employed. The unreasonably large repulsion force between bonded atoms (1–2 interactions) and atoms sharing a valence angle (1–3 interactions) is excluded using a shielded interaction (Eq. 4.36b): *





E vdWaals = Di j · exp αi j · 1 −





 + "1 f 13 ri j # f 13 ri j −2 · exp · αi j · 1 − , rvdW 2 rvdW

(4.36a)  $ %λ28 1/λ28

 1 λ29 f 13 ri j = ri j + . λw

(4.36b)

In addition to the vdW interaction, Coulomb interactions between pairs of atoms are also considered. A shielded Coulomb potential is introduced to avoid orbital overlap between atoms at very short distances. The Electron Equilibration Method (EEM) method [36, 37] is used to calculate the atomic charges: qi · q j E Coulomb = C · 

3 1/3 . ri3j + 1/γi j

(4.37)

4.3 Mechanical Properties

81

4.3 Mechanical Properties 4.3.1 Fundamental Mechanical Properties Xu et al. conducted DFT calculations to study the mechanical properties of 15 most stable degree-6 carbon nanothreads [38]. Before mechanical characterization, geometry optimization was applied to the nanothread structures with at least eight benzene rings. The interatomic interactions were described by a MMFF94 potential [39, 40] without considering bond dissociation. The energy per (CH6 ), Young’s modulus, linear atom density λ as well as bond length of the 15 most stable carbon nanothreads are provided in Table 4.1. Due to the difficulty in calculating the cross-sectional area of carbon nanothreads, the stiffness is quantified with regard to linear atom density λ. In the context of Young’s modulus, the cross-sectional area is approximated by multiplying λ by V 0 which is the volume per carbon atom in bulk diamond. Two kinds of boundary conditions are concerned, including free-end condition which allows the rotation and pin-end condition which forbids the rotation. The results show that the (3,0) sp3 nanotube possess the largest Young’s modulus of 1.16 TPa among the 15 most stable carbon nanothreads, while the soft carbon nanothread with identifier Table 4.1 Properties of carbon nanothreads. Reprinted (adapted) with permission from [38]. Copyright (2015) American Chemical Society Energy per (CH6 ) (ev)

Young’s modulus (free, pinned) (TPa)

λ atoms/Å

Length of C–C (Å)

123456

0.73

1.16

2.79

1.54–1.57

135462

0.82

0.98

2.41

1.53–1.60

143562

0.95

0.93

2.38

1.53–1.59

135462

0.97

0.90

2.60

1.54–1.58

153624

1.01

0.59

2.60

1.53–1.59

143562

1.04

1.08

2.44

1.51–1.67

143652

0.57

(1.11, 1.14)

2.45

1.54–1.57

136254

0.62

(0.73, 0.74)

2.75

1.53–1.58

136425

0.70

(0.64, 0.64)

2.63

1.53–1.57

135462

0.81

(0.63, 0.76)

2.64

1.54–1.57

135246

0.64

(0.31, 0.37)

2.66

1.53–1.58

132546

0.66

(0.35, 0.37)

2.72

1.53–1.58

134562

0.69

(0.08, 0.10)

2.91

1.53–1.58

145263

0.75

(0.19, 0.26)

2.74

1.53–1.58

136524

0.96

(0.41, 0.45)

2.38

1.54–1.59

Identifier Achiral

Stiff, Chiral

Soft, Chiral

82

4 Mechanical Properties of Carbon Nanothreads

of 134,562 exhibit the smallest Young’s modulus of 0.08 TPa. These findings indicate that the mechanical properties of carbon nanothreads are strongly dependent on the structure morphology. The widely varied mechanical properties enable carbon nanothreads as promising building blocks for multi-functional structure through bottom-up design. Silveira and Muniz [2] carried out DFT calculations to explore the mechanical performance of selected degree-6 carbon nanothreads under tensile deformation. To obtain the stress–strain curves, the nanothread structure after equilibration was deformed at a constant engineering strain rate. After each loading step, the structure was optimized with a force tolerance of 10–5 Ry/bohr to relax the stress. Therefore, it can be regards as a quasi-static loading method. The Young modulus, ultimate stress and failure strain can be obtained from the stress–strain curves. The one-dimensional stress (force units) was considered because of the difficulty in calculating the crosssectional area of the carbon nanothreads. They also conducted MD simulations to study the mechanical performance of selected degree-6 carbon nanothreads. In the simulations, the time step was set as 0.1 fs, and strain rate was 5 × 10–4 ps−1 . Similarly, the nanothread structure was deformed at a constant engineering strain rate. After each loading step, geometry optimization was applied to the system. The deformation continued until complete fracture. To exclude the effect of thermal fluctuations on the ultimate stress of carbon nanothread, a temperture of 0.1 K was considered. The interatomic interactions were described by the AIREBO potential with the C–C cutoff distance set as 2.0 Å to produce reasonable response at larger bond strains. It should be noticed that the C–C cutoff distance ranged from 1.92–2.0 Å have been widely chosen for studying the tensile strength of carbon-based nanomaterials. For instance, Zhang et al. [8] showed some results for the tensile strength of carbon nanothreads using different values for the cutoff parameter within this range, and decided to employ 1.945 Å as the cutoff in further calculations based on the agreement with the ReaxFF predictions. However, the ReaxFF used to overpredict the strength and fracture strain, as pointed out by Silveira and Muniz [2]. They showed the predictions of AIREBO employing the C–C cutoff parameter of 2.0 and 1.95 Å for configurations DNT-B and DNT-C in Fig. 4.1. The artificial hardening and increase of strength and fracture strain are evident. Reducing the parameter would just lead to larger deviations regarding the DFT predictions. For this reason, Silveira and Muniz suggested the use of 2.0 Å for this parameter, considering there is no other easy way of fixing this AIREBO shortcoming unless the potential is modified. The atomic configuration of the four studied carbon nanothreads are provided in Fig. 4.2. They are labeled as DNT-A, DNT-B, DNT-C and DNT-D, and the first two structures are the (3,0) sp3 -tube [41] and the Polymer I phase [42], respectively. The tensile stress–strain curves are plotted in Fig. 4.2, and the ultimate stress and failure strain are obtained from the maximum point of the curves. It can be seen that there is an abrupt reduction in stress when the strain exceeds the failure strain, indicating a brittle behavior of carbon nanothread. The abrupt reduction in stress can be ascribed to the change of structural configuration during bond dissociation.

4.3 Mechanical Properties

83

Fig. 4.1 Stress–strain curves of a carbon nanothread using AIREBO potential with different C–C cutoff parameters. Reprinted (adapted) with permission from [8]

The mechanical properties including the one-dimensional Young’s modulus Ym , ultimate stress σmax and failure strain εmax are provided in Table 4.2. It is shown that the DNT-A has the largest stiffness with Ym = 168 nN and highest ultimate stress with σmax = 15.7 nN. These outstanding mechanical properties are due to the similar atomic configuration of DNT with bulk diamond. The DNT-D has the smallest stiffness with Ym = 64 nN and lowest ultimate stress with σmax = 9.7 nN, but it has the largest failure strain εmax ≈ 0.23. Although the atomic configurations of DNT-B and DNT-C are different, they exhibit similar mechanical properties. The calculated Young’s modulus is in good consistence with reported results [4]. To conclude, the carbon nanothreads have a specific strength as high as 2.6 × 107 N m/kg and specific stiffness as high as 2.8 × 108 N m/kg, which indicates the carbon nanothreads are promissing candidates with superior mechanical properties for carbon-based nanomaterials. As the defects in the structure will degrade the mechanical performance, the mechanical properties predicted here could be considered as the upper bound for a carbon nanothread. The results in Fig. 4.2 and Table 4.2 demonstrate that the MD simulation results did not agree well with the DFT calculation, but the trends in regard to the relative strength among different carbon nanothreads are consistent. Compared with the DFT calculation, the ReaxFF-CHO and ReaxFF-FC potentials are found to overestimate the ultimate stress and failure strain, with the largest deviations reach 50% and 65% respectively. While the AIREBO potential underestimates the ultimate stress with deviation over 22%, and the failure strain is overestimated with deviation over 32%. However, the AIREBO potential has good predictions about both the ultimate stress and failure strain. It has been demonstrated the AIREBO potential is suitable to predict the diamond-like nanostructures. As expected, the mechanical properties of DNT-A which has a similar structure configuration with

84

4 Mechanical Properties of Carbon Nanothreads

Fig. 4.2 Structural configurations of four carbon nanothreads denoted as a DNT-A, b DNT-B, c DNT-C, and d DNT-D viewed along the longitudinal axis and transverse axis. The stress– strain curves predicted by MD simulation and DFT are shown together. Reprinted (adapted) with permission from [2] Table 4.2 Mechanical properties of the three carbon nanothreads obtained from DFT calculations and MD simulations. Ym is the 1-D Young modulus, σMAX and εMAX are the ideal strength and fracture strain. For ReaxFF, the two entries correspond to predictions according the ReaxFF-CHO [45] and ReaxFF-FC [46] parameterizations respectively. Reprinted (adapted) with permission from [2] σMAX (nN)

Ym (nN)

εMAX

DFT AIREBO ReaxFF DFT AIREBO ReaxFF

DFT

AIREBO ReaxFF

DNT-A 168

214

136/136 15.7 15.5

24/21.5

0.175 0.169

DNT-B 130

82

138/88

12.6

9.8

16.1/17.9 0.168 0.182

0.19/0.206

DNT-C 125

55

144/87

11.3

9.2

17/14.6

0.206/0.187

DNT-D 64

55

78/83

9.7

8.7

13.4/13.7 0.23

0.143 0.178 0.303

0.248/0.29

0.297/0.327

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85

diamond can be well-predicted by the AIREBO potential. Besides, the negative radial Poisson ratio predicted in DNT-A using DFT is also observed in the simulation using AIREBO potential. For the ReaxFF-FC and AIREBO potentials, the predicted Young’s modulus is smaller than the one predicted by DFT. The ReaxFF-CHO, however, is found to predict a relatively higher Young’s modulus. For ReaxFF, the two entries correspond to predictions according the ReaxFF-CHO [43] and ReaxFF-FC [44] parameterizations respectively. The strain–stress curves in Fig. 4.2 showed significantly qualitative variation. For instance, according to the prediction of AIREBO potential, there is a small relief of stress at an intermediate strain of the DNT-C. Other potentials show that there is a plateau of stress before the failure strain for the DNT-B and DNT-D, which indicates a plastic behavior of carbon nanothreads. The stress distribution and valence electron density maps of different carbon nanothreads at different strains are shown in Fig. 4.3. It is shown from the DFT calculations that with increasing strain, the electron density of marked C–C bonds reduces dramatically, which provides

Fig. 4.3 The stress distribution and valence electron density maps at various strains for the a DNTA, b DNT-B, c DNT-C, and d DNT-D structures. Atoms are colored based on their stress and electron density. Reprinted (adapted) with permission from [2]

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4 Mechanical Properties of Carbon Nanothreads

confirm proofs for MD simulation that these bonds are the most likely ones that will undergo bond dissociation during tensile deformation. The stress is homogeneouly distributed along the backbond structure of DNT-A, as seen from the stress distributions in Fig. 4.3a. The uniforom stress distribution enables the DNT-A with the largest Young’s modulus and highest ultimate stress. However, stress concentration is found in some vulnerable regions in other carbon nanothreads, as seen in Figs. 4.3b, c. The vulnerable regions used to have the smaller number of C–C bonds at the cross section, e.g. two C–C bonds only. Figure 4.3d shows that the DNT-D structure which can be considered as a chain of interconnected U-shaped components changes to a flatter structure during tensile deformation. This structural evolution is attributed to the bond rotation as well as bond stretching. Hence, the DNT-D has the smallest Young’s modulus and lowest ultimate stress. Besides, the bond rotation could absorb additional strain energy, which enables the DNT-D to sustain a larger strain without fracture. Due to the intensive bond rotation, there are large differences of the bond angle before and after the deformation. The bond angle could even increase from ∼120° to ∼140°, which leads to severe strain localization into the involved C–C bonds and hence fracture. The evolutions of bond length and bond angle as a function of strain for four carbon nanothreads are shown in Fig. 4.4. It is shown that the structural information of DNTC and DNT-D is improperly produced by the AIREBO potential. The bond distortion, as evidenced by the decrease of bond length with increasing strain, is responsible for the deviation of predicted mechanical properties. The bond distortion is not observed in the ReaxFF before bond dissociation, indicating that the ReaxFF can well characterize the structural symmetry. Qualitatively, the MD simulation results correspond well with the DFT calculation results. The quantitative discrepancies in the bond length and its evolution with strain predicted by MD and DFT are ascribed

Fig. 4.4 Evolutions of bond length and bond angle as a function of strain for four carbon nanothreads. Reprinted (adapted) with permission from [2]

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87

to the improper computation of bond energy. The tremendous increase of bond length will induce strain localization into some regions, and hence decrease the strain energy shared in other C–C bonds. As seen in the Figs. 4.4b, c, the length of most strain C–C bond in DNT-B predicted by ReaxFF agrees with the DFT prediction to some degree, but not the length of 2nd most strain C–C bond. These results demonstrate that the improper description of bond energy in the potentials is responsible to the deviations of mechanical properties.

4.3.2 Morphology and Temperature Dependent Tensile Properties Zhang et al. [3] employed MD simulations to study the mechanical behaviors of three representative carbon nanothreads, marked as NTH-I (achiral), NTH-II (stiffchiral) and NTH-III (soft-chiral), respectively. The mechanical properties of different diamond NTHs were studied using tensile simulation. The strain–stress curves of these carbon nanothreads are presented in Fig. 4.5. It can be seen brittle fracture occurs to all the carbon nanothreads, as evidenced by an abrupt decrease of stress after reaching a critical value. The abrupt change of stress is attributed to the bond dissociation upon deformation. The critical value of stress in the stress–strain curves is considered as the ultimate stress, and the corresponding strain is taken as failure strain. Results showed that the NTH-I and NTH-III possess similar ultimate stress and failure strain, while the NTH-III has the highest ultimate stress (141 GPa) and largest failure strain (~ 22%). The influences of temperature on the mechanical properties of carbon nanothreads are studied in Fig. 4.6. The ultimate stress and failure strain of all carbon nanothreads degrade with increasing temperature, while the brittle behavior is the same at all temperatures. In addition, the temperature appears to play a negligible effect on the

Fig. 4.5 a Stress–strain curves of three carbon nanothreads at temperature of 1 K; b–d Stress distribution of atoms in carbon nanothreads long the tensile direction at the strain of 12%. Atoms are coloured based on their values of atomic stress. Reprinted (adapted) with permission from [3]

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4 Mechanical Properties of Carbon Nanothreads

Fig. 4.6 Effects of temperature on the stress–strain curves at different temperatures for: a NTH-I, b NTH-II, and c NTH-III. d Influences of temperature on the relative failure/ultimate strength of three carbon nanothreads. The standard variation of the ultimate stress for each case is shown as error bar. Reprinted (adapted) with permission from [3]

Young’s modulus. The ultimate stress and failure strain of the NTH-II at 300 K reduce by ~18% and ~45% with respect to those at 1 K. Similar findings are found in the NTH-I and NTH-III structures. The ultimate stress of NTH-I decreases from ∼90 GPa at 1 K to ∼40 Gpa at 300 K, and the failure strain decreases from ~ 16% to 7%. The ultimate stress of NTH-III decreases from ∼79 GPa at 1 K to ∼27 Gpa at 300 K, and the failure strain decreases from ~18% to 6%. Although the mechanical properties are dependent on the temperature, the trends of the relative Young’s modulus, ultimate stress, and failure strain among the three carbon nanothreads are almost consistent. The NTH-II structure has the largest Young’s modulus, highest ultimate stress and largest failure strain at any temperature than the NTH-I and NTH-III. The NTHIII structure has the smallest Young’s modulus, lowest ultimate stress and smallest failure strain at any temperature. The influences of temperature on the relative failure/ultimate stress of three carbon nanothreads are shown in Fig. 4.6d. The relative ultimate stress (σ fr ) is determined as the ratio between the ultimate stress at the temperature of n (σ fn ) and temperature of 1 K (σ f0 ). Four independent simulations are conducted for each point, and their average value together with the error bar are shown in the curves. In the four simulations, the duration for dynamic equilibration varies from 1 to 2.5 ns, in order to ensure different initial condition prior to deformation. It has been found that this methodology produces similar stress–strain curves of a given carbon nanothread.

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89

Fig. 4.7 a Schematic work showing how to induce a curvature by virtually attaching the carbon nanothread to an artificial wall; the interactions between the atoms and wall are described by a 9–3 LJ function; b Effects of curvature on the potential energy normalized by length (E r /L). Reprinted (adapted) with permission from [47]

The relative ultimate stress is found to decrease with increasing temperature. The NTH-II has the least reduction of relative ultimate stress, while the NTH-III has the largest. The flexibility of the two NTHs including NTH-I and NTU-II is investigated through bending stiffness test. A curvature was built by virtually attaching the carbon nanothread to an artificial wall (see Fig. 4.7a). The interactions between the atoms and wall are described by a 9–3 LJ function: E = ξ[

σ 3 2 σ 9 ( ) − ( ) ], 15 r r

(4.38)

where the parameters ξ and σ were set as 0.65 eV and 2 Å, respectively [46]. The degree of curvature is set in a range from 0.0083 to 0.033 Å−1 . Before the bending deformation, the structure was optimized to relax the internal stress. Then the bending energy E b at each degree of curvature was calculated by E b = E r − E 0 , where E r was the current potential energy, and E 0 is the potential energy after geometry optimization (no deformation). In the same time, based on the continuum elastic theory, the elastic bending energy is given by: Eb =

1 2 Dρ L . 2

(4.39)

where D is the bending stiffness, ρ is the degree of curvature, and L is the length of carbon nanothread.

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4 Mechanical Properties of Carbon Nanothreads

In Fig. 4.7b, the potential energy is normalized by length (E r /L), and the effects of curvature on the normalized energy are shown. The correlation between the normalized energy and curvature is found to agree well with the continuum elastic theory. Using Eq. (4.39), the bending stiffness for NTH-I and NTH-II is calcualted as ~64 eV Å (1480 kcal/mol·Å) and ~ 88 eV Å (2030 kcal/mol·Å), respectively. In addition, the persistence length L p can be also calcualted by L p = D/kB T, with kB the Boltzmann constant and T the temperature. The persistence length for NTH-I and NTH-II at 300 K is calcualted as ~250 nm and ∼340 nm, respectively. The persistent length can help describe the flexibility of carbon nanothreads. For instance, if the carbon nanothread has a length shorter than the persistence length, it will act like a flexible elastic rod. The carbon nanothreads (NTH-I and NTH-II) have larger rigidity than the reported polymers (L p ≈ 1 nm) and double-stranded DNA (L p ≈ 45–50 nm) [47], but more flexible than the reported CNTs (L p ≈ 10–100 μm). Feng et al. studied the influences of morphology and temperature on the mechanical properties of 15 most stable carbon nanothreads under tensile deformation [4]. The ReaxFF is utilized to characterize the interaction between C and H atoms [4]. The atomic structures carbon nanothreads are shown in Fig. 4.8. The tensile properties at temperature of 1 K are shown in Fig. 4.9. The diameter of both achiral and stiff-achiral nanothreads is approximated as 0.5 nm, and the diameter of soft-achiral nanothreads are measured directly. For both achiral and stiff-achiral nanothreads, the stiffnesses decrease with increasing strain. However, the stiffnesses of soft-achiral nanothreads shows the opposite trend in the early deformation due to their helical morphologies. The stiffnesses of soft-achiral nanothreads are lower than those of achiral and stiff-achiral nanothreads. The mechanical responses in Fig. 4.9 show that the atomic structure has significant influences on the mechanical properties of carbon nanothreads. For tensile stiffness, the DNT #6 containing 1:1:1 of pentagon, hexagon, and heptagon along the length has the highest value due to the fact that it has similar atomic arrangement with the hydrogenated graphene-nanoribbon. In comparison, the DNT #14 also made of 1:1:1 of pentagons, hexagons, and heptagons has the lowest stiffness due to its largest helical coiled diameter. In terms of ultimate stress, the DNT #1 has the highest value due to the perfect symemetry of hexagon units. The DNT #4 and #5 whose structures are dominated by pentagon have the lowest ultimate stress due to the least number (two) of C–C bonds at the cross section. For fracture strain, the largest flexibility is found in the DNT #14 with a failure strain of ~ 32% due to largest helical coiled diameter and similar atomic arrangement with the hydrogenated graphenenanoribbon on one surface. The smallest failure strain is found in the DNT #6 due to the pre-deformed C–C bonds and least C–C bonds in paralle to the tensile direction. For mechanical toughness (per weight), the DNT #1 has the highest toughness, while the DNT #5 exhibits the lowest toughness. The tensile properties of three kinds of carbon nanothreads (DNT #1, #8 and #14) under temperatures of 1–2000 K are shown in Fig. 4.10. Similar to previous findings [3], the ultimate stress and failure strain decrease as the temperature increases. A brittle-to-ductile transition is observed for both DNT #8 and #14 structures when

4.3 Mechanical Properties

91

Fig. 4.8 Atomic structures of six carbon nanothreads viewed along the longitudinal and transverse directions. (1)-(6) Achiral carbon nanothreads. (7)-(10) Stiff-chiral carbon nanothreads. (11)-(15) Soft-chiral carbon nanothreads. The pitch and diameter of the soft-chiral carbon nanothreads are measured and shown. Reprinted (adapted) with permission from [4]

the temperature is 2000 K, while this phenoemon is not observed in the DNT #1 at 2000 K. It is found that the atomic stress is well distributed along the DNT #1 due to symmetrical hexagons, leading to strongest strength and brittle behavior. For DNT #8, although made of hexagons, undergoes bond dissociation at 2000 K under tension. In the beginning, two C–C bonds in the DNT #8 will dissociate at larger strain, which transforms the hexagonal structure to sp2 -carbon pentagonal structure. The newly

92

4 Mechanical Properties of Carbon Nanothreads

Fig. 4.9 Tensile stress–strain curves for 15 most stable carbon nanothreads. Reprinted (adapted) with permission from [4]

Fig. 4.10 Tensile stress–strain curves for three selected carbon nanothreads at temperature ranging from 1 to 2000 K. a DNT #1. b DNT #8. c DNT #14. Reprinted (adapted) with permission from [4]

formed pentagonal structure will dissociate again until complete bond rupture at the cross section, leading to ductile behavior. For DNT #14 made of hexagons and heptagons on the surface, bond dissociation also happen with increasing strain at 2000 K. As the strain increases, continous bond rupture occurs, resulting in a transformation in specfic regions from nanothread structure to cross-linked polyethylene molecule. As a result, the bonds connecting the carbon nanothread and polyethylene structure break, leading to formation of cyclic polyethylene molecule.

4.3.3 Defect Dependent Tensile Properties Roman et al. [46] explored the influence of Stone–Wales (SW) defects on the ultimate stress, Young’s modulus, and failure strain of carbon nanothreads using steered molecular dynamics (SMD) simulations. The interatomic interactions were described by the ReaxFF potential. The SW defects were introduced to the nanothread structure, as seen in Fig. 4.11, and then the nanothreads were subjected to uniaxial tensile deformation. To obtain the tensile stress–strain curves, the nanothread structure was deformed at a constant engineering strain rate. After each loading step, geometry

4.3 Mechanical Properties

93

Fig. 4.11 Snapshot of a sp3 -(3,0) carbon nanothread with two SW defects at two ends. The atomic details of SW defects are shown in the inset. Reprinted (adapted) with permission from [46]. Copyright {2015} American Chemical Society

optimization was applid to relax the stress. Therefore, it can be regards as a quasistatic loading method. The stress–strain curves for a carbon nanothread with two SW defects at two ends are shown in Fig. 4.12a. The cross-section of the nanothread was estimated by as 0.5 nm using the vdW radii of opposite hydrogen atoms. The deformation continued until complete fracture. The ultimate stress of carbon nanothread is obtained as 134.3 GPa, and the failure strain is 14.9%. After obtaining the stress–strain curves, the axial stiffness (E) can be further obtained using the following equation: E = σ/ ε.

(4.40)

The maximum strain used for a linear fit is set as 4% in order to fulfill the hypothesis of linear elasticity. The axial stiffness for a carbon nanothread with two SW defects is calculated as ~ 850 GPa. An alternative calculation of stiffness is conducted by plotting the potential energy density U versus strain ε (see Fig. 4.12b). The stiffness can be expressed as: E = ∂ 2 U/∂ε2 .

(4.41)

After fitting the plotted curves with a fourth-order polynomial, the axial stiffness can be obtained as 774 GPa at ε = 2%. The carbon nanothread with two SW defects has a ultimate stress of 134.3 GPa, which is two times larger than the ultimate stress of carbyne. The carbyne is an emerging nanomaterial made of single carbon trains with outstanding mechanical properties [48]. The specific strength T of the carbon nanothread could be further calculated by T = σ ult /ρ, where σ ult is the ultimate stress and ρ is the density. Note that the specific strength is affected by the assumption in cross-section area, thus it will be more appropriate to characterize the material properties. The calculated specific strength is ~ 4.13 × 107 N·m/kg, indicating the carbon nanothread is the strongest nanomaterial ever reported. The flexibility of carbon nanothreads with SW defects is characterized by bending stiffness test. The bending stiffness is calculated by:

94

4 Mechanical Properties of Carbon Nanothreads

Fig. 4.12 Tensile deformation of carbon nanothreads. a Stress–strain curves for a carbon nanothread with two SW defects at two ends using quasi-static loading method. Brittle fracture occurs to the carbon nanothreads due to stress concentration in the SW defects. b Potential energy densitystrain curve for stiffness calculation. The curve is fitted by a fourth-order polynomial. Reprinted (adapted) with permission from [46]. Copyright {2015} American Chemical Society

4.3 Mechanical Properties

95

U=

1 Dκ 2 L , 2

(4.42)

where U is the potential energy, D is the bending stiffness, and κ is the predetermined curvature. In Fig. 4.13, the bending stiffness could be obtained as ~ 770 (kcal/mol)·Å, or 5.35 × 10–28 N ·m2 . This value is much smaller than the (5,5) CNT which has a bending stiffness on the order of 100,000 (kcal/mol)·Å, but largern than the carbine which has bending stiffness of 30 ~ 80 (kcal/mol)·Å) [48]. Therefore, the carbon nanothreads is much flexible with respect to the CNT. In addition, the introduction of SW defects will increase the persistent length of carbon nanothread. The carbon nanothread is separated to five segments to explore the influences of SW defects on the rigidity. The stress–strain curves for different segments are plotted in Fig. 4.14. It can be clearly seen thin the SW defects will significantly reduce the rigidity of carbon nanothreads. The pristine segments which have no SW defects have an effective stiffness of ~ 1089 GPa, while the SW defect regions have a much lower stiffness. The calculated stiffnesses for two SW defects are ~ 306 GPa and ~ 347 GPa, respectively. Compared to the pristine segments, there is obvious vibration in the magnitude of stress for the the SW defect regions, owing to the relatively short length and very low strain rate. The carbon nanothread with SW defects can be regarded as a system of serial springs (Fig. 4.14b). For a given number n of the SW defects, the defect density is determined by ρ d = n/L 0 with L 0 the length of the carbon nanothread. Using a simple rule of mixtures, the stiffness of carbon nanothread could be theoretically approximated as: E(n) = [

(L 0 − n L d ) −1 nLd + ] . Ed L 0 E0 L 0

(4.43)

Here, L d is the effective length of the segment with SW defect, E d and E 0 the stiffness of defect segment and pristine segment, respectively. The effective length L d is estimated as 4.8 Å by measuring the longitudinal length over the entire defect region. The stiffness of carbon nanothread with two SW defects E(2) ranges from 833 to 867 GPa, depending on the value of E d . This result is consistent with the predicted result 850 GPa using molecular mechanics method. The SMD simulations were carried out to exam the proposed model. Based on the SMD simulation results (see Fig. 4.15), the stiffness of carbon nanothreads with SW defect number n = 2, 3, and 4 is obtained as 808 GPa, 763 GPa and 665 GPa, respectively. According to the Eq. (4.43), the calculated stiffness of carbon nanothreads with SW defect number n = 3 and 4 is 746 ~ 786 GPa and 675 ~ 720 GPa, respectively, which corresponds well with the SMD simulation results. The theoretical model indicates that there is a negative correlation between the defect density and stiffness in carbon nanothreads, while a positive relationship is observed between the defect density and failure strain.

96

4 Mechanical Properties of Carbon Nanothreads

Fig. 4.13 Bending deformation of carbon nanothreads. a Schematic work showing how to induce a curvature by virtually attaching the carbon nanothread to an artificial wall; the interactions between the atoms and wall are described by a 9–3 LJ function; The degree of curvature is set in a range from 0.01 to 0.05 Å−1 . b Effects of curvature on the potential energy. The potential energy is optimized after bending. The bending stiffness can be obtained by fitting the curve with Eq. 4.42. Reprinted (adapted) with permission from [46]. Copyright {2015} American Chemical Society

4.3 Mechanical Properties

97

Fig. 4.14 Influence of defects on the rigidity of carbon nanothreads. a Stress–strain curves for different segments in carbon nanothreads. The carbon nanothread is separated to five segments. The pristine segments have no SW defects, while the defect segments have SW defects. b Serial spring model of nanothread. c Stress–strain curves for carbon nanothreads with different numbers (n = 2, 3, 4) of SW defects. Reprinted (adapted) with permission from [46]. Copyright {2015} American Chemical Society

98

4 Mechanical Properties of Carbon Nanothreads

Fig. 4.15 Stiffness predicted by SMD and theoretical model versus number of defects in the carbon nanothread. Reprinted (adapted) with permission from [46]. Copyright {2015} American Chemical Society

4.3.4 Structure Dependent Ductility Zhan et al. [8] investigated the mechanical behaviors of carbon nanothreads by performing tensile tests performed via MD simulations. The AIEREBO potential was employed to describe the interactions between carbon and hydrogen atoms. To overcome the nonphysical high tensile stress produced by the AIEREBO potential, the cut-off distance (rcmin ) of C–C bond for the DNT is comprehensively studied. The strain–stress curves at a temperature of 300 K as obtained by changing the cutoff distance of the AIEREBO potential from 1.9 Å to 2.0 Å are compared with the ReaxFF potential in Fig. 4.16a. The yield strain (together with the yield strength) receives a significant reduction when the cutoff distance increases from 1.9 to 2.0 Å. A similar changing profile has also been observed from the testing under temperatures of 200, 100 and 50 K as compared in Fig. 4.16b. Referring to the results from ReaxFF potential, the intersection value suggests that a cut-off distance between 1.94 ~ 1.95 Å would lead to a similar yield strain by using AIREBO potential, which shows marginal dependency on the temperature. The influence of strain rate on the mechanical response of carbon nanothread is investigated. Strain rate including 1 × 10–6 , 5 × 10–7 , 1 × 10–7 , 5 × 10–8 and 1 × 10–8 are examined. The stress–strain curves under different strain rates are shown in Fig. 4.17. It can be observed that there is a positive relationship between the ultimate stress/failure strain and strain rate. The strain rate has minor impacts on the mechanical properties when it is lower than 1 × 10–7 fs−1 . Despite of the varied ultimate stress and failure strain, the fundamental deformation mechanisms are consistent. During tensile deformation, stress concentration occurs to the regions with SW defects, leading to bond dissociation and fracture in these regions. The influence from the locations of SW transformation defect is studied. Four carbon nanothread models with unevenly distributed SW transformations are examined, as seen Fig. 4.18. The strain–stress curves in Fig. 4.19 show that there is an ignorable influence from the location of SW defect on the Young’s modulus.

4.3 Mechanical Properties

99

Fig. 4.16 a The stress–strain curves of DNT-8 as obtained from ReaxFF potential and AIREBO potential with modified cut-off distance ranging from 1.9 to 2.0 Å at 300 K. b Comparisons of the normalized yield strain under different cut-off distance with varying simulation temperature. The green triangles are the normalized yield strength obtained from graphene with different cut-off distances [49], which are normalized by the value at 1.90 Å and shifted 0.2 for comparison clarity. The circled ‘X’ markers denote three representative cut-off distances that are adopted to study the mechanical properties of graphene-based systems, including 1.92 for irradiated graphene with vacancy [50]; 1.95 for bilayer graphene with sp3 bonds [51]; and 2.00 for grapheme nanoribbon [52]. The cyan stars are the normalized yield strength for ultra-thin (2,2) CNT. Reprinted (adapted) with permission from [8]

Fig. 4.17 Effects of strain rate on the stress–strain curves for the carbon nanothreads unit cell with 17 poly-benzene rings between two adjacent SW defects (DNT-17) (sample size ~24 nm). Reprinted (adapted) with permission from [8]

However, the yield strength and yield strain are intensively influenced by the location change of the SW defect. Different locations of the SW defect will obviously influence the stress distribution, which introduces stronger local variances, and thus induces different failure strain and ultimate stress. The length-dependency tensile properties of three groups of carbon nanothreads (DNT-8, DNT-14 and DNT-20) are investigated. Figures 4.20a, b show the calculated yield strain (failure strain) and yield strength (ultimate stress) of different carbon

100

4 Mechanical Properties of Carbon Nanothreads

Fig. 4.18 The carbon nanothread models with four evenly and unevenly distributed Stone–Wales defects. All models have periodic boundary conditions along the length direction and an identical size of ~ 31 nm. Reprinted (adapted) with permission from [8]

Fig. 4.19 Influence of the location of SW defect on the tensile properties of carbon nanothread. a The stress–strain curves are obtained from the carbon nanothreads with evenly and unevenly distributed SW defects. b The normalized distribution of the virial atomic stress for carbon atoms along the length direction at the strain of 7.5%, showing that the location of the SW defects will influence the stress distribution during tensile deformation and thus make the carbon nanothreads exhibit different yield strength and yield strain. Reprinted (adapted) with permission from [8]

nanothreads subjected to tensile deformation. A negative relationship between the ultimate stress/failure strain and the carbon nanothreads length can be seen. Both the ultimate stress and failure strength exhibit fast reductions in the range L < 30 nm, and then saturate to a certain value when L is larger than 30 nm. In Figs. 4.20a, the carbon nanothreads containing shorter poly-benzene units are inclined to have a higher saturated value of failure strain. For instance, the failure strain of DNT-8 is ~ 9.0% when L > 40 nm, which is larger than that of ~7.8% of the DNT-14. In Figs. 4.20b, the saturated value of ultimate stress is insensitive to the length of polybenzene units. Besides, the ultimate stress tends to be independent of the length of poly-benzene units in the range 13 nm < L < 92 nm.

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101

Fig. 4.20 Poly-benzene length-dependency tensile properties for carbon nanothreads DNT-n, with n being the number of SW defects. a Yield strain versus the length of poly-benzene unit; and b Yield stress versus the length of poly-benzene unit. c Length distribution of bond length; d Atomic stress distribution in carbon nanothread along the axial direction at the strain of 4.6%; e Bond breaking configuration in the SW defect (top panel with strain 11.2%), resulting in a complete fracture when strain is increased to 13.4%. Reprinted (adapted) with permission from [8]

The atomic stress distribution in carbon nanothread along the axial direction at the strain of 4.6% is shown in Figs. 4.20d. Higher atomic stress is observed around the SW defect during tensile deformation, indicating a stress concentration. The higher atomic stress is ascribed to the bond structure of the SW defect. The C–C bonds in the pentagon of SW defects are relatively longer than other bonds, and these bonds cause strain localization in the SW defects upon stretching. Due to the higher atomic stress, the SW defects are the vulnerable regions for bond breaking and fracture. A seen in Figs. 4.20e, the C–C bonds in the pentagon of SW defects break first, followed by complete fracture at the cross section with increasing strain. This finding provides confirmed proof there is stress concentration around the SW defects. The fracture mode is also observed in other carbon nanothreads with different number of SW defects. Generally, carbon nanothread with a longer length has more SW defects, which induces more regions of stress concentration, leading to early fracture. Although the ultimate stress and failure strain are significantly affected by the length of carbon nanothread, the Young’s modulus is found to be insensitive. As seen in Fig. 4.21, the predicted Young’s modulus for DNT-8 (with eight SW defects) slightly decreases from 831.8 to 799.1 GPa when the length increases from 15.7 to 78.4 nm. Similar trends are also found in the DNT-14 (with eight fourteen defects) and DNT-20 (with twenty SW defects). The Young’s modulus of the DNT-14 and DNT-20 are obtained as 872.5 GPa and 898 GPa, respectively. It is shown that carbon nanothreads with smaller numbers of SW defects consistently have higher Young’s

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4 Mechanical Properties of Carbon Nanothreads

Fig. 4.21 Effects of sample length on the Young’s modulus of carbon nanothreads DNT-n, with n being the number of SW defects. Reprinted (adapted) with permission from [8]

modulus, which can be explained by the theoretical model in Eq. 4.43. This finding is consistent with the previous results in Fig. 4.15. The carbon nanothreads exhibit tunable mechanical behaviors by altering the number of SW defects. The mechanical properties of carbon nanothreads with different numbers of SW defects are depicted in Fig. 4.22a. All carbon nanothreads have a length of 42 nm. Results show that carbon nanothreads with smaller numbers of SW defects are relatively brittle, and the stress increases monotonically with the strain before fracture. However, carbon nanothreads with larger numbers of SW defects, e.g. the DNT-2, have monitonic increase of stress at smaller strains and strain hardening at larger strains. The strain hardening could postpone the fracture of carbon nanothread, leading to a relatively ductile behavior. The strain hardening is more evident in carbon nanothreads with larger numbers of SW defects. Taking the DNT-2 (with 32 SW defects) and DNT-48 (with 2 SW defects) as examples, the predicted failure strain of DNT-2 is about two times largen than that of DNT48. Therefore, a relatively brittle-to-ductile transition can be induced by increasing the number of SW defects normalized by the length ( or defect density) in carbon nanothreads. In Fig. 4.22b, the ultimate stress is found to fluctuate around a certain value. Such fluctuation is related to the variation of atomic stress in the SW defect regions. The C–C bonds in the SW defect regions used to have different initial lengths, and they are the initial locations for bond breakage. The ununiform bond distribution will induce different degrees of strain localization under thermal perturbations. As a result, the atomic stress in the SW defect regions vary significantly, leading to the fluctuation of ultimate stress. In Fig. 4.22c, the failure strain tends to increase with the number of SW defect, which is attributed to the two reasons, including the extended hardening process and stress relaxation in the SW defect regions. For a carbon nanothread with a determined length, introducing more SW defects will effectively alleviate the stress concentration, thereby leading to a larger failure strain. The atomic stress distribution

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103

Fig. 4.22 Tensile properties of carbon nanothreads with variation of SW defect number. a Stress– strain curves for carbon nanothreads showing a brittle-to-ductile transition with increasing SW defects; b Effects of SW defect number on the yield strength; c Effects of SW defect number on the yield strain; d Atomic stress distribution along the tensile direction. The green arrows mark the maximum stress in the SW defect regions. Reprinted (adapted) with permission from [8]

in carbon nanothreads along the tensile direction is shown in Fig. 4.22d. It can be seen that the maximum stress in the SW defect regions decreases with increasing SW defects. It has been demonstrated thin the SW defectsexhibit ductile characteristics. To prove this, the stress–strain response in a segmental region with only SW defects is tracked. The cutoff distance of C–C bond in the AIREBO potential is increased to 2.0 Å in order to avoid any artificial effect caused by the switching function in the AIREBO potential. It is shown in Fig. 4.23 that the segmental region without SW defects has brittle characteristics (curve P-20). The brittle behavior is not affected by the length of the region, as can be seen from the comparison between curves P-20 and P-180. Here a carbon nanothread with a length of N Å is denoted as P-N. However, the segmental region with SW defects has ductile characteristics (black curve in Fig. 4.23a). The strain hardening phenomenon accommodates the deformation, leading to a large failure strain of 25%. The failure strain of a segment with SW defects is more than two times larger than that of a segment without SW defects. The stress–strain curve of a segment with SW defects can be divided into two stages. When the strain increases from point A to point B in Fig. 4.23a, only the bond elongation occurs to the structure. When the strain reaches point B, two C–C bonds in the SW defect region start to break, which initiates the strain hardening.

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4 Mechanical Properties of Carbon Nanothreads

Fig. 4.23 Tensile properties of segmental regions with and without SW defects. a Stress–strain curves of different segmental regions. A carbon nanothread without SW defect and with a length of N Å is denoted as P-N. The P-20 indicates a pristine segmental region a length of 20 Å. A carbon nanothread with one SW defects and a length of N is denoted as N. The 208 indicates A carbon nanothread with one SW defects and a length of 208 Å. The atomic details of the SW defects are provided in the inset. b Effects of segment length on the tensile stiffness. The SWD and PBR means the segmental region has SW defects and poly-benzene rings (no defects), respectively. Reprinted (adapted) with permission from [8]

Further increase of strain leads to continuous bond dissociation in the SW defect region, which provides additional accommodation of deformation, leading to ductile fracture finally. The ductility of carbon nanothread is tunable by changing the density defects, and the carbon nanothread will exhibit relatively brittle behavior if the SW defects are removed. As seen in Fig. 4.23b, the tensile stiffness increases from 480 to 900 GPa with increasing length of the segmental region, indicating a ductile-to-brittle transition phenomenon. The length and structure dependent failure strain and ultimate stress predicted by MD simulations is found to agree well with the proposed serial spring model Eq. (4.43), as seen in Fig. 4.24. However, a large difference is found for carbon nanothreads with larger amounts of SW defects, which can be possibly explained by the brittle-to-ductile transition. These findings indicate that structural modification is an efficent strategy to manipulate the mechanical properties of carbon nanothreads.

4.3.5 Functionalized Carbon Nanothreads Benzene which is fundamental building block of carbon nanothreads has many aromatic derivatives. Some of these aromatic derivatives could be used as precursors for constructing functionalized carbon nanothreads with excellent mechanical performance. The mechanical properties of functionalized carbon nanothreads is studied by DFT calculations [53]. The nanothread structures with a structural unit of C6 H5 -X were

4.3 Mechanical Properties

105

Fig. 4.24 Effects of SW defect number on the tensile stiffness of carbon nanothreads with different lengths. The MD simulation results are compared with spring model in Eq. (4.43). The solid lines are fitted using the data from carbon nanothreads with constituent units longer than DNT-6. The underlined numbers represent the number of poly-benzene rings between two neiboring SW defects in the carbon nanothread structure. Reprinted (adapted) with permission from [8]

constructed by substituting one of the hydrogen atoms in the benzene rings with functional groups (-CH3 , -NH2 , -OH, -F). The nanothread structures with a structural unit of C5 H5 -X were constructed by substituting one of the carbon atoms in the benzene rings with functional groups (-N). The structural unit is depicted in Fig. 4.25. Prior to the mechanical deformation, the relative stability of these nanothread structures is analyzed by calculating the relative specific energy Er s : Er s = E D N T − E D N T ms

(4.45)

where E D N T is the energy of a carbon nanothead with a specific substituting position of functional group, and E D N T ms is the lowest energy among all the substituting positions using the same functional group. It was found in Table 4.3 that the substituting position has a limited effect on the structural stability of carbon nanothreads, suggesting that the functional groups could attach to the carbon nanothreads with random substituting position. The Young’s modulus, ultimate stress and failure strain of various functionalized carbon nanothread were calculated by performing uniaxial tensile test. In Table 4.4, the results show that the Young’s modulus and ultimate stress are not significantly affected by the presence of functional group. However, the introduction of functional group deteriorates the ductility of carbon nanothreads. The tensile properties of carbon nitride nanothreads (CN-NTHs) with a structural unit of C5 H5 -N are investigated through MD calculations [54]. The ReaxFF developed by Mattsson et al. [55] is utilized to characterize the interaction between C and N atoms. Three types of morphology were considered, including sp3 (3,0) tube, polymer I and polytwistane structures.

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4 Mechanical Properties of Carbon Nanothreads

Fig. 4.25 a Structural unit of C6 H5 -CH3 , C6 H5 -OH, C6 H5– NH2 , C6 H5 -F and C5 H5 -N (from left to right). b Arrangement of benzene molecules to form the pristine carbon nanothread. The indexes (1–6) indicate the connecting position of the neighboring ring. There are two kinds of notation for the repeating units. For index (1, p):FG, p (1 ≤ p ≤ 4) is the relative location of the substituted hydrogen in the adjacent ring with respect to the first benzene ring. For index (1, p2 , p3 , p4 , p5 , p6 ):FG, pi (2 ≤ p ≤ 6) is the relative position of the substituted hydrogen in the i-th neighboring ring with respect to the first benzene ring. For (1,4):pyr, the substituting atom is carbon. (c-f) Structural configurations of four functionalized carbon nanothreads viewed along the longitudinal and transverse directions. The fundamental repeat units are provided in the dashed boxes. The C, H, N, O and F atoms are colored in gray, white, blue, red and green, respectively. Reprinted (adapted) with permission from [53]

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107

Table 4.3 Relative energies Er s of functionalized carbon nanothreads. Reprinted (adapted) with permission from [53] Er s (meV/C6 H5 -X unit) CH3

NH2

OH

F

Pyr

(1, 2)

54

63

36

87

88

(1, 3)

43

13

36

32

72

(1, 4)

59

65

34

45

66

(1, 2, 3, 4)

34

48

28

30

36

(1, 4, 2, 5)

37

0

32

0

37

(1, 2, 3, 4, 5, 6)

0

4

0

32

0

(1, 4, 2, 5, 3, 6)

36

17

16

5

49

Table 4.4 Young’s modulus (Y m ), ultimate stress (τ max ) and failure strain (εmax ) of functionalized carbon nanothreads. Reprinted (adapted) with permission from [53] Pristine

(1, 4):CH3

(1, 4):NH2

(1, 4):OH

(1, 4):F

Y m (nN)

54

63

36

87

88

τ max (nN)

15.8

14.5

14.8

15.4

14.4

εmax

0.176

0.166

0.157

0.166

0.161

For Polymer I group, CN_NTU_a1 structure refers to Polymer I_3-3_25 in Fig. 3.10, CN_NTU_a2 refers to Polymer I_4-2_36. For polytwistane group, CNNTU_b refers to Polytwistane_153. For Tube (3,0) group, CN_NTU_c1 refers to Tube (3,0)_1245, CN_NTU_c2 refers to Tube (3,0)_123456. In Fig. 4.26 the tensile stress–strain curves indicate that the influence of nitrogen on the mechanical properties is affected by the morphology. The length of the C–C bonds will be redistributed due to the additing of nitrogen atoms, which further affects the stress distribution during deformation. The carbon nitride nanothreads obtained from polymer I and sp3 (3, 0) tube used to exhibit higher fracture strain and ultimate strength than the carbon nanotheads (C-NTHs), while the polytwistane carbon nitride nanothreads has lower strain and strength than its carbon counterpart. For all the CN-NTHs and C-NTHs,

Fig. 4.26 Tensile stress–strain curves for CN-NTHs and C-NTHs. a Polymer I morphology; b Polytwistane morphology; and c Sp3 (3,0) tube morphology. Reprinted (adapted) with permission from [53]

108

4 Mechanical Properties of Carbon Nanothreads

the fracture initiates at the C–C bonds, and the C-N bonds could sustain a higher strain during deformation. It should be mentioned that there is a discrepancy between the DFT calculation results [53] and MD simulation results [54]. The disagreement may be attributed to: 1) structure variation; (2) force field selection. In terms of the last reason, there are numerous ReaxFF potentials for C-H-N systems, which could lead to different mechanical responses of carbon nitride nanothread. Besides, it has been proved that ReaxFF potentials used to overestimate the fracture properties of carbon nanothreads [2]. Zheng et al. also investigated the bending stiffness of CN-NTHs and C-NTHs [54]. The bending stiffness was characterized by a free vibration test. A sinusoidal velocity excitation v(z) = asin(π z/L) was applied to the structure. The natural frequency f was calculated from the external energy trajectory according to the Fast Fourier Transformation (FFT). Due to the anisotropic characteristic of nanothreads, the vibration was deposed along two lateral directions with different frequencies. The bending stiffness (EI) is expressed by the Euler–Bernoulli beam theory: f =

ω , E I /ρ A 2π L 2

(4.46)

where ω is the eigenvalue, L is the sample length,ρ is the mass density and A is the cross-sectional area. It was found that the bending stiffness of CN-NTHs was higher than that of C-NTHs. The calculated bending stiffnesses for C-NTHs were 1.27 × 10−27 and 2.42 × 10−27 N m2 in the two vibration directions, respectively. The EI were obtained as 1.63 × 10−27 and 3.79 × 10−27 N m2 for CN-NTH_a1 in the two lateral directions, respectively. These findings demonstrate that addition of nitrogen is effective to improve the bending rigidity of carbon nanothreads. The large difference between two bending stiffnesses also signified a strong anisotropic mechanical property. Further comparisons showed that the bending stiffness of CNNTH was over an order of magnitude higher than that of cumulene carbyne chain, but much lower than that of (5,5) CNT. Demingos et al. conducted DFT calculations to study the tensile properties of fully and partially saturated carbon nanothreads and carbon nitride nanothreads [56]. The nanothread structures were shown in Fig. 4.27, and they were labeled based on the notation mentioned in Fig. 3.12. The syn structure indicates two consecutive C– C double bonds are placed on the same side, and the anti structure indicates two consecutive C–C double bonds on the opposite side. The syn-anti structure indicates that two successive C = C bonds are placed alternatively along the length. There are three kinds of hybridization for two N atoms in the unit cell of carbon nitride nanothreads, including both sp2 (-N22), both sp3 (-N33), and sp2 combining with sp3 (-N23). The relative stability of fully and partially saturated nanothreads is characterized by the bending energy (E b ). The binding energy is defined as the energy difference between the energy of the structure after optimization and the energy of the molecule from which it derived. Table 4.5 shows that the binding energy of the fully

4.3 Mechanical Properties

109

Fig. 4.27 Structural configurations of the fully and partially saturated carbon nanothreads and carbon nitride nanothreads. (a − e) Carbon nanothreads; (f − j) Carbon nitride nanothreads. The sp2 - and sp3 -hybridized C, N and H atoms are marded in magenta, gray, blue and white, respectively. The fundamental repeat unit is marked in dashed boxes. Orange arrows indicate the saturated counterparts. Reprinted (adapted) with permission from [54]. Copyright {2019} American Chemical Society

saturated nanothread is smaller than the partially saturated one, indicating a higher stability of the fully saturated nanothread. For all the partially saturated nanothreads, the syn isomers have relatively lower stability than the anti and syn-anti isomers. The structural instability of syn isomer can be explained by the stronger vdW interactions/repulsions between the unsaturated atoms due to the shorter interaction distance. For all the carbon nitride nanothreads, the ones with remaining C-N double bonds connected to the backbone has the highest stability.

110 Table 4.5 Binding energy E b of different nanothread structures. Reprinted (adapted) with permission from [54]. Copyright {2019} American Chemical Society

4 Mechanical Properties of Carbon Nanothreads Nanothread

E b (eV)

anti

0.006

syn

0.567

syn-sat syn − anti

−0.465 0.018

syn − anti-sat

−0.495

anti-N22

−0.177

syn-N22 syn-N22-sat syn-N33 syn-N33-sat

0.025 −0.069 1.144 −0.105

The tensile responses in Fig. 4.28 indicate that a decrease of degree of saturation leads to reduction of ductility of both carbon nanothreads and carbon nitride nanothreads. Besides, the nanothread structures with lower degree of saturation have lower Young’s modulus (82 versus 103 nN for syn nanothreads and 40 versus 55 nN for syn-anti nanothreads) and lower ultimate stress (10 versus 12.8 nN for syn nanothreads and 7.7 versus 10.7 nN for syn-anti nanothreads). The presence of N atoms plays a limited effect on the tensile properties of carbon nanothreads. For all the nanothread structures, the syn-sat nanothread has the best mechanical properties, and its Young’s modulus and ultimate stress are comparable with the Polymer-I nanothread. The anti- and syn-anti nanothreads have the lowest Young’s modulus but the largest failure strains. The outstanding flexibility is attributed to the structural configurations, including corrugated chain and alternative bonding patterns. For all the fully and partially saturated carbon nanothreads and nitrogen-doped nanothreads, the fracture used to initiate at the C–C single bonds. Fig. 4.28 Tensile stress–strain curves for carbon nanothreads and carbon nitride nanothreads. Reprinted (adapted) with permission from [54]. Copyright {2019} American Chemical Society

4.3 Mechanical Properties

111

Fu et al. also conducted MD simulation to investigate the mechanisms of morphology and temperature dependent mechanical properties of carbon nitride nanothreads [57]. Different from previous work [54], a ReaxFF potential developed by Budzien et al. [58] is used. Seven kinds of CN-NTHs including Tube (3,0)_123456, Tube (3,0)_1245, Polymer I_3-3_25, Polymer I_4-2_36, Polytwistane_153, Zipper polymer_24, and IV-7 are concerned. The atomic configurations have been introduced in Fig. 3.9. The tensile responses in Fig. 4.29a indicate that all the carbon nitride nanothreads exhibit nonlinear mechanical responses. There is a small drop of tensile stress during the elastic deformation due to the transformations of C–C bond and C–C–C angle. The transformations are recoverable after unloading. All the carbon nitride nanothreads except the Polymer I_3-3_25 fail in a brittle manner.

Fig. 4.29 a Tensile properties at 1 K of Tube (3,0)_123456, Tube (3,0)_1245, Polymer I_3-3_25, Polymer I_4-2_36, Polytwistane_153, Zipper polymer_24, and IV-7 carbon nitride nanothreads, and b Young’s modulus (E) and c yield strength of the corresponding carbon nitride nanothreads. Reprinted (adapted) with permission from [57]

112

4 Mechanical Properties of Carbon Nanothreads

From Fig. 4.29b, the tensile stiffnesses of carbon nitride nanothreads vary from around 300 to 920 GPa. The IV-7 group has the lowest tensile stiffness due to the homogeneous distribution of effective C–C bonds and armchair-like structure. In comparison, the Tube (3,0) group shows the highest tensile stiffness due to its diamond-like atomic structures. Besides, Tube (3,0)_1245 structure which has inhomogeneous doping of N has higher Young’s modulus than the Tube (3,0)_123456 with uniform N-doping. From Fig. 4.29c, the yield strength/ultimate stress of carbon nitride nanothreads vary from around 45 to 120 GPa. Similar to the trends in Young’s modulus, the IV-7 group has the lowest ultimate stress, and the tube (3,0)_1245 has the highest value. Figure 4.30 shows evolution of carbon bond orders of carbon nitride nanothreads during elongation process. The carbon nitride nanothreads undergo complicated deformations including the bond, angular and torsional deformations, as seen from the evolutions of the bond orders. For IV-7, there is a smooth reduction of bond orders with the increase of strain. For other carbon nitride nanothreads, several singularities are observed, and the total bond orders sudderenly increase when the strain

Fig. 4.30 Dynamics of carbon bond orders of carbon nitride nanothreads upon tensile deformation at 1 K. a–g Evolution of carbon bond orders of one carbon atom located in the bond breaking region for seven carbon nitride nanothreads. Insets show the corresponding atomic structures at different strains. Reprinted (adapted) with permission from [57]

4.3 Mechanical Properties

113

reaches the failure strain. The abrup changes of bond orders originate from deformations of bonds, angles and torsional angles in carbon nitride nanothreads. In particular, a sawtooth-like evolution of bond orders with strains is observed in the Polymer I_33_25 structure, which corresponds well with the sawtooth-like stress–strain curves in Fig. 4.29a. The developments of bond orders of Polymer I_3-3_25 in Fig. 4.29 show that the carbon and nitrogen atoms undergo evolve similarly during the elongation process. Prior to deformation, differences of bond orders between carbon and nitrogen atoms are observed. Upon deformation, the structural configurations change due to bond breakage, leading to significant reduction of bond orders around the breaking regions. The bond breakage starts from two ends of the carbon nitride nanothreads and then develops towards the central region. Final fracture occurs to the connection between the single atomic chains and the residual parts of carbon nitride nanothreads. There is an increase of bond orders after fracture, which is attributed to the recovery of stress (Fig. 4.31). There are three deformational stages (A, B, and C) for Polymer I_3-3_25 structure in the tensile stress–strain curve, as observed in Fig. 4.32a. At the stage B, there are six sawtooth-like regions corresponding to the continuous bond dissociations during tensile deformation. Figure 4.32b shows the bonding configurations of structure at the stage A. The configurational snapshots in Fig. 4.32c show that bond dissociation

Fig. 4.31 Dynamics of carbon and nitrogen bond orders of Polymer I_3-3_25 upon tensile deformation at 1 K. a The change of one carbon and nitrogen bond orders with strain; b Corresponding atomic structure of Polymer I_3-3_25 at different strains. The carbon and nitrogen atoms are colored based on their bond orders. Reprinted (adapted) with permission from [57]

114

4 Mechanical Properties of Carbon Nanothreads

Fig. 4.32 Tensile properties and structural configurations of Polymer I_3-3_25 at different strains. a Stress–strain curve with sawtooth-like characteristics. b–e Snapshots of atomic structures during elongation. The backbone carbon atoms are in different colors to the show the transformation from nanothread to single atomic chain. Reprinted (adapted) with permission from [57]

first occur to two C–C bonds in two pentagonal rings near the free ends, and one C–C bond is reformed in the same time, leading to the formation of polyethene segements. The final structure at the stage A is provided in snapshot #1 in Fig. 4.32c. In following, bond dissociation occurs to one C–C and one C-N bond in another two pentagonal rings, and one C–C bond is formed again, as seen in snapshot #2. This process continues at the stage B, as seen from the atomic structures in Fig. 4.32c, d, leading to developments of single atomic chain towards the central regions. The tensile stress–strain curves with variation of temperature in Fig. 4.33 show that the ductility of carbon nitride nanothread is temperature dependent. There is an inverse relationship between the failure strain and temperature. However, there is no clear correlation between the temperature and ultimate stress. The nonlinear deformation is unobvious with increasing temperature. When the temperature increases from 1 to 100 K, the Polymer I_3-3_25 structure undergoes a ductile-to-brittle fracture transition. The ductile characteristics at low temperatures are due to the homogeneous breakage and reconnection of C–C bonds through the length. At high temperatures, thermal fluctuation will inhibit the reformation of C–C bonds, causing disconnection of the Polymer I_3-3_25 at the fracture-occurring region. The ductile characterisitcs are also found in other carbon nitride nanothreads at low temperatures. These findings indicate that the carbon nitride nanothreads have significant applications at low-temperature environment.

4.3 Mechanical Properties

115

Fig. 4.33 Tensile properties of carbon nitride nanothreads with increasing temperatures from 1 to 1500 K. a–g Stress–strain curves of Polymer I_3-3_25, Polymer I_4-2_36, IV-7, Polytwistane_153, Tube (3,0)_1245, Tube (3,0)_123456 and Zipper polymer_24 structures, respectively. Reprinted (adapted) with permission from [57]

116

4 Mechanical Properties of Carbon Nanothreads

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48. Liu M, Artyukhov VI, Lee H, Xu F, Yakobson BI (2013) Carbyne from first principles: chain of C atoms, a nanorod or a nanorope. ACS Nano 7(11):10075–10082 49. He L, Guo S, Lei J, Sha Z, Liu Z (2014) The effect of Stone–Thrower–Wales defects on mechanical properties of graphene sheets–a molecular dynamics study. Carbon 75:124–132 50. Carpenter C, Maroudas D, Ramasubramaniam A (2013) Mechanical properties of irradiated single-layer graphene. Appl Phys Lett 103(1):013102 51. Zhang YY, Wang CM, Cheng Y, Xiang Y (2011) Mechanical properties of bilayer graphene sheets coupled by sp3 bonding. Carbon 49(13):4511–4517 52. Zhao H, Min K, Aluru NR (2009) Size and chirality dependent elastic properties of graphene nanoribbons under uniaxial tension. Nano Lett 9(8):3012–3015 53. Silveira JF, Muniz AR (2017) Functionalized diamond nanothreads from benzene derivatives. Phys Chem Chem Phys 19(10):7132–7137 54. Zheng Z, Zhan H, Nie Y, Xu X, Gu Y (2019) Role of nitrogen on the mechanical properties of the novel carbon nitride nanothreads. J Phys Chem C 123(47):28977–28984 55. Mattsson TR, Lane JMD, Cochrane KR, Desjarlais MP, Thompson AP, Pierce F, Grest GS (2010) First-principles and classical molecular dynamics simulation of shocked polymers. Phys Rev B 81(5):054103 56. Demingos PG, Muniz AR (2019) Electronic and mechanical properties of partially saturated carbon and carbon nitride nanothreads. J Phys Chem C 123(6):3886–3891 57. Fu Y, Xu K, Wu J, Zhang Z, He J (2020) The effects of morphology and temperature on the tensile characteristics of carbon nitride nanothreads. Nanoscale 12(23):12462–12475 58. Budzien J, Thompson AP, Zybin SV (2009) Reactive molecular dynamics simulations of shock through a single crystal of pentaerythritol tetranitrate. J Phys Chem B 113(40):13142–13151

Chapter 5

Electronic Properties of Carbon Nanothreads

5.1 Introduction Due to the ultra-light characteristics, carbon nanothreads are expected as promising integrators into electronic devices to improve the electronic property. In recent years, intensive theoretical efforts have been made to explore the promising role of carbon nanothreads as candidates for electronic devices [1–11]. However, due to experimental inaccessibility for electronic property characterization, there is no existing experimental result yet for the carbon nanothreads. To explore the promising role of carbon nanothreads and provide theoretical guidance for experimental characterization, computational simulations such as density-functional theory (DFT) calculation have been widely used and considered as powerful and convincing tools for studying the electronic property. The intriguing electronic property of carbon nanothreads is highly sensitive to their topological structure, and lots of efforts in recent years have been exerted to study the intrinsic properties and applications. This chapter aims to introduce the fundamental electronic property of carbon nanothreads, and provide manipulation strategies to improve the properties. The computational details of first-principle calculation are introduced. The effects of topological structure on the bandgaps of carbon nanothreads are illustrated, and the associating conducting/semiconducting properties are analyzed and compared with other carbon allotropes. Besides, the carbon nanothread’s electronic property can be tuned through physical approach (strain) and chemical approaches (functionalization), and the tunable mechanisms are revealed systematically. The electronic properties of other carbon-based nanotubes are also introduced and compared with each other.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. M. Liew et al., Carbon Nanothreads Materials, Materials Horizons: From Nature to Nanomaterials, https://doi.org/10.1007/978-981-19-0912-2_5

119

120

5 Electronic Properties of Carbon Nanothreads

5.2 Computational Methodology A bandgap is a crucial factor characterizing the electrical conductivity for solid materials including conductors, semiconductors, and insulators. The bandgap G is the difference of energy between the valence and conduction bands of electrons. For conductors, the bandgap almost disappears because of the overlap between the valence and conduction bands. For insulators, there is a large bandgap inhibiting electrons from the valence band from moving to the conduction band. In semiconductors, the bandgap is very small and in-between that of the conductors and insulators. The lowest conduction-band energy is given by [12]: c = E N +1 − E N ,

(5.1)

where E M is the ground-state energy of M particle, and E N is the insulating groundstate energy. Similarly, the highest valence energy is given by: v = E N − E N −1.

(5.2)

The fundamental bandgap is then obtained as: E g = c − v .

(5.3)

The Kohn–Sham density functional theory (DFT) has been the most common approach to calculate the ground-state energy and electron density of many particle systems [13]. It considers a fictitious system of electrons without interaction, but the calculated ground-state density is consistent with the real system with interaction. The effective Kohn–Sham potentials contain the external potential, Coulomb potential and the exchange–correlation potential. The density for an M-particle state can be determined by the wave functions of the Schrödinger equation [12]:   1 2 − ∇ + V (M)  j (M) =  j (M) j (M), 2

(5.4)

where V (M) is the Kohn–Sham potential for M particles:  V (M) = Vext +

un(M) + Vxc (M),

(5.5)

where Vext is the total external potential, the second term in the right side is the Coulomb potential through the interaction u due to the charge distribution n(M), Vxc (M) is the exchange–correlation potential:

5.2 Computational Methodology

121

Vxc (M) =

δ E xc,M , δn

(5.6)

E xc is the exchange–correlation energy. So far, there are some functionals for approximation of exchange–correlation energy E xc , including local-density approximation (LDA) [13], generalized gradient approximations (GGA) [14], meta-GGA [15], and hybrid GGA with exact exchange [16]. The LDA presumes that the exchange–correlation energy at every position in the system coincides with a uniform electron system of the same electron density [13]. The value of E xc is approximated by:  E xc =

ρ(r)xc (ρ)dr

(5.7)

where ρ(r) is the electron density, and xc (ρ) is the exchange–correlation energy for a uniform electron system with density ρ. The E xc is typically separated into two terms, including the exchange term E x and correlation term E c . The LDA is used to predict lower values of atomic ground-state energies, ionization energies, and exchange energies and overestimate the binding and correlation energies due to the homogeneity assumption [17]. In a real system, the density is spatially inhomogeneous. The GGA takes into account the non-homogeneity of the true electron density by expansion of the gradient of density [14]: E xc = E xc [ρ(r), ∇ρ(r)].

(5.8)

The LDA tends to generate more accurate total energies, atomization energies, structural energies’ difference, and energy barriers [17]. However, the GGA does not yield significant improvement on the prediction of ionization energies and electron affinities than the LDA [17]. The meta-GGA makes a further improvement on the GGA by adding the second derivative of the electron density [15], leading to more accurate properties such as atomization energies [17]. However, the meta-GGA is used to suffer numerical stability during calculation. The hybrid GGA is a mixture of GGA exchange–correlation functional and a Hartree–Fock exchange with a percentage [15]. The exact percentage of Hartree– Fock exchange is determined by d semiempirical fitting. The hybrid GGA makes a significant improvement for most properties over the GGA. However, it is less effective in solid-state physics owing to the challenges in calculating the exactexchange term inside a plane-wave basis set [17].

122

5 Electronic Properties of Carbon Nanothreads

5.3 Electronic Band Structure The electronic band structure of carbon nanothreads was first investigated by Dragan et al. through DFT calculation [1]. They proposed the sp3 -(3,0) and sp3 -(2,2) carbon nanothreads, which were hydrogenated carbon nanotube (CNT) with chirality of (3, 0) and (2,2), respectively. It is observed from the band structures in Fig. 5.1 that the carbon nanothreads have large bandgaps, indicating promising applications for semiconductors. Besides, the (3,0) carbon nanothread has weaker dispersive bands because of the longer C–C bond length of 1.62 Å. Silveira and Muniz investigated the electronic properties of functionalized carbon nanothreads through DFT calculation [3]. The GGA/Perdew-Burke-Ernzerhof (PBE) exchange–correlation functional was employed. The van der Waals interaction was described by Grimme’s D2 semi-empirical correction. The valence and core electrons were treated by Projector Augmented-Wave (PAW) pseudopotentials. The structural configurations are shown in Fig. 4.27. The electronic band structures and density of states are shown in Fig. 5.2, and the values of bandgaps are summarized in Table 5.1. Results show direct bandgaps of functionalized carbon nanothread. The bandgaps differ from 2.42 to 4.25 eV, depending on the type and spatial distribution of functional group. The variation is attributed to the difference of atom’s electronegativity as well as strain degree of C–C bond affected by the functional group. The (1,4):pyr carbon nanothreads, which is also carbon nitride nanothread, exhibit the lowest bandgap among the investigated structures due to the effects of the Nheteroatoms. As seen in the projected density of states in Fig. 5.3, additional states in vicinity of the edge of valence band edge are introduced in the (1,4):pyr carbon nanothreads with respect to the pristine one. The energy of additional states is closer to that of the conduction band, leading to a reduction in the bandgap [3].

Fig. 5.1 Electronic band structures of the sp3 -(3,0) and sp3 -(2,2) carbon nanothreads. Reprinted (adapted) with permission from [10]. Copyright (2001) American Physical Society

5.3 Electronic Band Structure

123

Fig. 5.2 a The electronic band structures of pristine carbon nanothread, carbon nanothread with functional groups –OH, and carbon nitride nanothread. b Density of states of pristine carbon nanothread, carbon nanothread with functional groups –OH, –CH3 , –NH2 , –F, and carbon nitride nanothreads. Reprinted (adapted) with permission from [3]

Wu et al. [4] investigated the electronic properties of carbon nanothreads with different concentrations of Stone–Wales (SW) defect under tensile strain. The GGA/PBE exchange–correlation functional was employed during the DFT calculation. Two types of carbon nanothreads, which are sp3 (3,0) tube and polymer-I,

124

5 Electronic Properties of Carbon Nanothreads

Table 5.1 Electronic bandgaps for the functionalized carbon nanothreads. Reprinted (adapted) with permission from [3] Bandgap (eV) CH3

NH2

OH

F

Pyr

(1, 2)

4.05

3.73

4.12

3.72

2.42

(1, 3)

3.97

3.57

4.25

3.78

2.44

(1, 4)

3.95

3.60

4.06

3.49

2.50

(1–4)

4.04

3.62

4.00

3.75

2.81

(1, 4, 2, 5)

4.00

3.45

4.09

3.75

2.81

(1–6)

4.06

3.43

3.86

3.90

2.88

(1, 4, 2, 5, 3, 6)

4.06

3.47

3.90

3.75

2.85

Pristine

3.76

Fig. 5.3 Projected density of states for a pristine carbon nanothread and b pyridine-based (1,4):pyr carbon nanothread. Reprinted (adapted) with permission from [3]

were considered. Different amounts of SW defect were introduced into the structures, dividing the chain domain into pristine segment and SW-defect segment, as seen in Fig. 5.4. The band structure and density of states are plotted in Fig. 5.5 [4]. The sp3 (3,0) possess a direct bandgap in the middle of Brillouin zone ( point) of about 3.92 eV,

5.3 Electronic Band Structure

125

Fig. 5.4 Atomic configurations of carbon nanothreads. a Structures of sp3 (3,0) (top panel) and polymer-I (bottom panel). b Carbon nanothreads with one, two, and three SW defects in a unit cell. The location of SW defect regions is marked in red box. Reprinted (adapted) with permission from [4]. Copyright (2018) American Chemical Society

Fig. 5.5 Electronic band structures and projected density of states for a sp3 (3,0) and b polymer-I. The Fermi level is set at valence band maximum. Reprinted (adapted) with permission from [4]. Copyright (2018) American Chemical Society

and the polymer-I possess an indirect bandgap of about 4.82 eV. The location of conduction band minimum of the polymer-I is at the  point, while the valence band maximum is at the Brillouin zone boundary. The flatter bands near the valence band maximum of polymer-I carbon nanothread indicate that a larger hole effective mass and a more localized character. Compared to the sp3 (3,0), the polymer-I possess a more prominent peak near the valence band maximum in the density of states. The effect of SW defects on the bandgap is shown in Table 5.2 [4]. Results show that introduction of SW defects widens the bandgap of sp3 (3,0) carbon nanothread. The bandgap rises from 3.92 eV without SW defects to 4.53 eV with full SW defects,

126 Table 5.2 Bandgap for the SW hybrid carbon nanothreads. Reprinted (adapted) with permission from [4]. Copyright (2018) American Chemical Society

5 Electronic Properties of Carbon Nanothreads Structures

Bandgap (eV)

Sp3 -(3,0)

3.92

(no SW)

Single SW

4.07

Double SW

4.22

Triple SW

4.53

Polymer-I (full SW)

4.82

indicating the surface hydrogenation of carbon nanothreads is an effective means to tune the conductivity. The band structures of hybrid carbon nanothreads are also shown in Fig. 5.6. It is found that there is a flat band above the original valence band maximum with single SW defect. The flat band is related to the states distributed in the pristine segments, as seen in the distribution of charge density in Fig. 5.6d. The valence band state is likely to be localized into the pristine segments by the SW defect. For the conduction band minimum, it is shown in Fig. 5.6e that the distribution of charge density is disrupted and confined into the pristine segments by the SW defect. Due to this confinement effect, the existence of SW defects tends to push up the energy of conduction band minimum state, as seen in Fig. 5.6b and c. The effect of tensile strain on the bandgap is shown in Fig. 5.7 [4]. It can be seen from the results the bandgap for the sp3 -(3,0) increases from 3.92 to 4.41 eV when the strain increases from 0 to 10%, indicating that tensile strain tends to increase the bandgap at small strains. At larger strain, there is a sudden drop of bandgap. It can be found that the existence of SW defect is able to stabilize the bandgap of structure.

Fig. 5.6 Electronic band structures of carbon nanothreads with a one SW defect; b two SW defects; c three SW defects. The Fermi level is set at the valence band maximum. Distribution of charge density in carbon nanothreads at the  point for d valence band maximum; e conduction band minimum. Reprinted (adapted) with permission from [4]. Copyright (2018) American Chemical Society

5.3 Electronic Band Structure

127

Fig. 5.7 a Effects of strain on bandgap for carbon nanothreads with various number of SW defects. b Evolution of bond length with strain in sp3 -(3,0) carbon nanothread. Reprinted (adapted) with permission from [4]. Copyright (2018) American Chemical Society

The effects of strain on the electronic band structures of sp3 (3,0) and polymer-I carbon nanothreads are analyzed in Fig. 5.8. Besides, it is shown in Fig. 5.8d the indirect bandgap of polymer-I could transform to direct bandgap with increasing tensile strain. Before tension, the polymer-I has conduction band minimum at the  point and valence band maximum at k = ± π/c. When tensile strain increases up to 8%, the valence band state at the  point moves up to be the new valence band maximum. The character of indirect band is important to improve the intensity of photoluminescence for semiconductor. These findings indicate that the optical properties of polymer-I carbon nanothread are tunable through strain. Chen et al. [8] studied the electronic properties of carbon nanothread doped with different elements. The GGA/PBE exchange–correlation functional was applied during the DFT calculation. The phonon band spectra of four types of carbon nanothreads are investigated. The carbon nanothreads are denoted as DNTs-M, with M being the doped element. Results show that these carbon nanothreads could be thermostable, and imaginary frequency is not observed in the vibrational band structure. The calculated bandgap of DNTs-C, DNTs-SiC, DNTs-Si, and DNTs-Ge is 3.93, 4.69, 3.26, and 2.70 eV, respectively, with the first two structures having direct bandgap and the last two structures having indirect bandgap. Compared with the bulk carbon materials, the DNTs-C has a smaller bandgap than the diamond with bandgap ranging from 5.45 to 5.56 eV [18, 19]. The bandgap of DNTs-SiC, DNTs-Si, and DNTs-Ge is larger than corresponding bulk phases of doped elements [18, 20, 21]. Chen et al. [8] also calculate the charge mobility μe(h) of carbon nanothreads by Boltzmann transport equation: μe(h)

e = kB T





− → − → − → − → τ (i, k )v 2 (i, k )exp[∓ εki (B kT ) ]d k . − →   → εi ( k ) − i∈C B(V B) exp[∓ k B T ]d k

i∈C B(V B)

(5.9)

128

5 Electronic Properties of Carbon Nanothreads

Fig. 5.8 a Electronic band structures of sp3 (3,0) carbon nanothread at various strains; b Electronic band structures of polymer-I carbon nanothread at various strains; c Distribution of charge density at the  point of stages A and B marked in (a); d Effects of strain on the bandgap showing an indirect-to-direct transition of bandgap of polymer-I carbon nanothread. Reprinted (adapted) with permission from [4]. Copyright (2018) American Chemical Society

− → − → Here, the minus (plus) sign indicates electron (hole), τ (i, k ), εi k , and − → − → v(i, k ) are charge relaxation time, band energy, and the group velocity at k state of the ith band, respectively. The group velocity of electron and hole can be obtained  = ∇εi (k)/.  by the equation v(i, k) The mobilities of different carbon nanothreads at room temperature are summarized in Table 5.3. Results show that the DNTs-C possesses the highest electron mobility, which is comparable to the graphene of 3 × 105 cm2 /Vs [22]. The hole mobility of DNTs-C is much smaller than the electron mobility, indicating DNTsC acts like a n-type semiconductor. However, the hole mobility of DNTs-Ge and DNTs-Si is much larger than the electron mobility, indicating they act like a p-type

5.3 Electronic Band Structure

129

Table 5.3 Room temperature mobility (μ) of different carbon nanothreads

μ(cm2 /Vs) Electron mobility

Hole mobility

DNTs-C

1.19 × 105

70.8

DNTs-SiC

228

120

DNTs-Si

24.3

264

DNTs-Ge

58.9

1.38 × 103

semiconductor. These findings suggest that element doping into carbon nanothreads could be a powerful way to adjust the electronic properties of carbon nanothreads. The decomposed charge density at the edge of bands is also studied by Chen et al. [8] to explain the charge mobility. They found that the valence band maximum of DNTs-C comprises 2px orbits with mixed s orbits, and py orbits, resulting in bad hole transport. The direction of px is parallel to x direction. Besides, it can be found that the conduction balance maximum states are composed of mixed s and p orbits. The pz orbits account for the majority. Therefore, the charge density conduction band minimum is distributed along the z direction, which is also the length direction, resulting in good electron transportation and high electron mobility. The large difference of pz orbits (large proportion in valance band maximum and small proportion in conduction band minimum) of DNTs-Si and DNTs-Ge leads to poor electron transport. To conclude, pristine carbon nanothreads and carbon nanothread doped with Ge are promising conductors. Xiao et al. [9] investigated the electronic properties of carbon nanothreads made of C–C triple bonds. The GGA/PBE exchange–correlation functional was applied during the DFT calculation. The atomic structures of different carbon nanothreads are shown in Fig. 5.9. The DNTs-C1 refers to the structure composed of C–C triple bonds between benzenes only. The DNTs-C1 refers to the structure composed of C–C triple bonds between two benzenes. The introduction of C–C triple bonds decreases the density of mass from 37.3 u/Å (DNTs) to 33.2 u/Å (DNTs-C1) and 33.8 u/Å (DNTsC2). The phonon band spectra of two types of carbon nanothreads are depicted in Fig. 5.10a. Results show that these carbon nanothreads could be thermostable, and imaginary frequency is not observed in the vibrational band structure. The band structure in Fig. 5.10b shows that all the carbon nanothreads are widegap semiconductors. The bandgap decreases by increasing the fraction of triple C– C bonds. The triple C–C bonds play important role in affecting the valence band maximum and conduction band minimum. The bandgaps of carbon nanothreads subjected to various strains are plotted in Fig. 5.11. The bandgap of DNTs increases when a tensile or compressive strain is applied. For DNTs-C2, the bandgap increases with decreasing compressive strain and is insensitive to the tensile strain. For DNTsC1, applying compressive strain can increase the bandgap while tensile strain can decrease the bandgap. It can be found that increasing the proportion of triple C–C bonds leads to less strain sensitivity of carbon nanothreads. The bandgap of DNTs increases from around 3.1–4.4 eV when the strain changes from −10 to 10%, while the bandgap of DNTs-1 decreases from around 3.6 to 3.2 eV only.

130

5 Electronic Properties of Carbon Nanothreads

Fig. 5.9 Band decomposed charge density of different carbon nanothreads. Reprinted (adapted) with permission from [9]

Fig. 5.10 The phonon spectrum a and energy band spectrum b of DNTs, DNTs-C1, and DNTs-C2. Reprinted (adapted) with permission from [9]

5.3 Electronic Band Structure

131

Fig. 5.11 Effects of strain on the bandgap of carbon nanothreads composed of different numbers of C–C triple bond. Reprinted (adapted) with permission from [9]

Xiao et al. [9] also calculated the charge mobility (μ∗ ) of DNTs, DNTs-C1, and DNTs-C2 according to one-dimensional deformation potential theory [23]: μ∗ =

e2 C 1

3

(2π k B T ) 2 |m ∗ | 2 E 12

.

(5.10)

Here, E 1 is the deformation potential constant, m* is the effective mass of charge, C is the stretching modulus, E(k) is the energy band, and a and a0 are the lattice constants with and without deformation, k B is the Boltzmann constant, T is the temperature. The charge mobility μ∗ in DNTs, DNTs-C1, and DNTs-C2 at room temperature is obtained as 5.81 × 104 , 2.32 × 102 , and 1.54 × 102 cm2 /Vs, respectively. Besides, the μ∗ in DNTs, DNTs-C1, and DNTs-C2 for room-temperature hole mobility is obtained as 3.66 × 101 , 9.79 × 103 , and 1.14 × 102 cm2 /Vs, respectively. Results show that there is a hole mobility in DNTs, due to the intense coupling between electron and phonon for hole carriers. With the insertion of triple C–C bonds, the electron mobility of carbon nanothreads decreases significantly, while the hole mobility increases. In DNTs-C2, there is no difference in order of magnitude for mobility of electron and hole. In DNTs-C1, the hole mobility is almost two times larger than the electron mobility. Therefore, DNTs-C1 is a promising p-type semiconductor. It is also found that the DNTs-C2 with applied compressed stress has the characteristics of n-type semiconductor. Demingos and Muniz [5] investigated the electronic properties of partially saturated carbon nanothreads and carbon nitride threads. The GGA/PBE exchange–correlation functional was applied during the DFT calculation. The atomic structures of different carbon nanothreads are shown in Fig. 4.29. The bandgaps of different nanothread structures are shown in Table 5.4, and some representative band dispersion curves and density of state plots are provided in Fig. 5.12. Results show that the syn nanothread possesses the narrowest bandgap

132 Table 5.4 Bandgap E g of partially saturated and saturated nanothread structures. Reprinted (adapted) with permission from [5]. Copyright (2019) American Chemical Society

5 Electronic Properties of Carbon Nanothreads Nanothread

E g (eV)

anti

3.96

syn

1.84

syn-sat

4.25

syn-anti

3.46

syn-anti-sat

4.45

anti-N22

3.82

syn-N22

3.08

syn-N22-sat

3.48

syn-N33

1.92

syn-N33-sat

3.09

Fig. 5.12 Electronic band structure and density of states for carbon nanothreads and carbon nitride nanothreads. a–d Band structure, projected and partial density of states for syn, anti, syn-anti, and syn-sat carbon nanothreads. e–h Band structure, projected, and partial density of states for antiN22, syn-N23, syn-N33, and syn-N22-sat carbon nitride threads. The partial density of states for sp2 -C, sp2 -N, and sp3 -N is plotted in different colors. Reprinted (adapted) with permission from [5]. Copyright (2019) American Chemical Society

(1.84 eV), while the syn-anti-sat nanothread presents the widest bandgap (4.45 eV). It can be found that partially saturated nanothreads have a smaller bandgap than the saturated counterparts as a result of the existence of C–C double bonds and C-N double bonds. The partial density of states in Fig. 5.12 indicates that the pz orbitals of C atoms in C–C double bonds induce energy states adjacent to the gap. When the orbitals approach each other, the interaction between the orbitals may decrease the bandgap. Figure 5.12 also shows that fully saturated carbon nitride nanothreads have smaller bandgaps than the fully saturated carbon nanothread, which is attributed to energy states adjacent to the gap induced by N element. The type of double bonds

5.3 Electronic Band Structure

133

Fig. 5.13 Electronic band structures of carbon nanothreads doped with different elements. Reprinted (adapted) with permission from [10]

(C–C or C–N) and corresponding distribution in the nanothread will strongly affect the bandgap of partially saturated configurations. The syn nanothreads have smaller bandgaps than the anti-configurations. The N22 configuration (composed of two C– N double bonds) has larger bandgaps than the N33 configuration (composed of two C–C double bonds), which is attributed to the higher electronegativity of N element. Miao et al. [10] investigated the electronic properties and magnetic properties of carbon nanothreads doped with elements Al and Ga. The carbon nanothreads are denoted as DNTs-M, with M being the doped element, and the H atom connected to the original C atom is removed. If the H atom is not removed, the carbon nanothreads are denoted as DNTs-M-H. The GGA/PBE exchange–correlation functional was applied during the DFT calculation. The bandgaps of DNT, DNT-Al, DNT-Ga, DNT-Al-H, DNT-Ga-H structures are calculated as 3.98, 2.62, 2.43, 3.08, and 3.14 eV, respectively. Some representative band dispersion curves and density of state plots are provided in Fig. 5.13a. Results show that carbon nanothreads doped with metal atoms have smaller bandgaps. The out layer of electrons of the metal atoms is transferred, which will reorganize the internal charge and change the overall energy level of the doped carbon nanothread. The loss of out layer of electrons will become local electrons and introduce an impurity state at the Fermi level of the doped carbon nanothread, as seen from the flat peaks in Fig. 5.13b and c. The introduced impurity state will decrease the bandgap of carbon nanothreads. For Al(Ga) doped carbon nanothreads with hydrogen adsorption, the impurity state further moves downward to the Fermi level. The hydrogen adsorption will increase the bandgap of Al(Ga) doped carbon nanothreads. The bandgap value of DNT-Al-H is 0.46 eV larger than that of DNT-Al, and the bandgap value of DNT-Ga-H is 0.71 eV larger than that of DNT-Ga. The total and projected density of states of Al(Ga) doped carbon nanothreads are plotted in Fig. 5.14. Results show that DNT-Al(Ga) has nonmagnetic ground states and all density of states near Fermi level are spin-degenerated. However, the adsorption of hydrogen leads to spin polarization and magnetization in the Al(Ga)

134

5 Electronic Properties of Carbon Nanothreads

Fig. 5.14 The total and partial density of states of carbon nanothreads doped with Al and Ga. a and b DNT-Al(Ga), c and d DNT-Al(Ga)-H. Spin charge density distribution of the e DNT-Al-H and f DNT-Ga-H. Reprinted (adapted) with permission from [10]

doped carbon nanothreads, as seen from the difference between spin-up and spindown density of states near Fermi level. Besides, the partial density of states indicates the high peak near Fermi level dominantly originates from the p orbitals of the three C atoms neighboring to the metal atom (C3-p), the p orbitals of the metal atoms (Al-p or Ga-p), and the s orbital of the hydrogen (H–s). The spin charge density distribution in

5.3 Electronic Band Structure

135

Figs. 5.13f and 5.14e shows that the magnetics of Al(Ga) doped carbon nanothreads with hydrogen adsorption are dominantly contributed by the doping metal atom and the neighboring three C atoms. The band structures of the Al-doped carbon nanothreads with adsorption of CO and NO molecules are shown in Fig. 5.15. As seen in Fig. 5.15a and b, the adsorption of CO molecules decreases the bandgap of Al-doped carbon nanothreads from 2.62 to 1.45 eV (C site) and 1.77 eV (O site). A spin-splitting event in the energy band

Fig. 5.15 The band structures of the carbon nanothreads with doping metal atoms and molecules adsorption. a and b DNT-Al-CO(C/O). c and d DNT-Al-NO(N/O). Reprinted (adapted) with permission from [10]

136

5 Electronic Properties of Carbon Nanothreads

is observed in Fig. 5.15c and d. The adsorption of NO molecules leads to spin polarization, as seen from the non-degenerate spin-up and spin-down band structures. For the spin-up state, the adsorption of NO molecules decreases the bandgap of Aldoped carbon nanothreads from 2.62 to 0.44 eV (N site) and 0.40 eV (O site). Similar trends of reduction with molecule adsorption are also observed in the spin-down state. Demingos et al. [11] proposed novel carbon nanothreads on the basis of different five-membered ring compounds such as thiophene, furan, and pyrrole and investigated the electronic properties of these novel carbon nanothreads. The GGA/PBE exchange–correlation functional was applied during the DFT calculation. The structure configuration of thiophene-derived carbon nanothreads with different configurations is shown in Fig. 5.16. The bandgaps of different nanothread structures are shown in Table 5.5 and Fig. 5.17. Results show that all the nanothread structures have a direct bang gap. The S-syn configuration possesses the smallest bandgap (0.87 eV), and the O-anti structure possesses the largest bandgap (4.47 eV). Compared to the bandgap of 3.76 eV for sp3 -(3,0) carbon nanothread [3], adding N and S heteroatoms in the nanothread can obviously reduce the bandgap by inducing new electronic states. Among these heteroatoms, the S atoms are more efficient to reduce the bandgap because of its larger diameter that improves the interatomic interaction.

Fig. 5.16 a Chemical structure of precursors for forming carbon nanothreads; b Structural configuration of thiophene-derived carbon nanothreads with different symmetries. Other carbon nanothreads can be obtained by substituting the S atoms. Reprinted (adapted) with permission from [11]

5.3 Electronic Band Structure Table 5.5 Bandgaps E g of thiophene-, furan-, and pyrrole-derived carbon nanothreads. Reprinted (adapted) with permission from [11]

137 Nanothread

E g (eV)

N-anti

3.65

N-syn-anti

3.36

N-syn

2.90

O-anti

4.47

O-syn-anti

4.16

O-syn

3.32

S-anti

3.61

S-syn-anti

2.92

S-syn

0.87

For carbon nanothread derived from same molecule, the bandgap of various configurations follows the order: anti > syn-anti > syn. The bandgap of syn configurations is more sensitive to the effect of heteroatoms than that of anti-configurations. Figure 5.17c shows the integrated local density of states for three energy ranges: two sulfur-induced occupied bands (A and B) and one unoccupied band (C). It is shown that the increase of bandgap is mainly contributed by the interactions between appended groups (b). For example, all the anti and syn-anti structures have indirect interactions between appended groups, leading to flatter bands around the Fermi level. In comparison, the syn structures have better interaction between appended group, resulting in well-dispersed equivalent bands within a wider energy range. Miao et al. [6] investigated the influences of doping transition metal on the electronic and magnetic properties of carbon nanothreads. The GGA exchange–correlation functional was applied during the DFT calculation. The transition metal-doped carbon nanothreads are constructed by replacing one edged carbon atom with one transition metal atom. The bandgaps and band structures are shown in Table 5.6 and Fig. 5.18, respectively. Results show that adding transition metal atoms introduces impurity band at the Fermi level and reduce the bandgap, which can transform the carbon nanothread from metal to semiconductor. The Fe-doped carbon nanothread exhibits the spin-splitting metallic property, carbon nanothreads doped with Sc, V, Cr, Mn, and Co atoms behave like semiconductor with bandgap ranging from 0.35 to 1.71 eV for spin-up states and 0.56–2.54 eV for spin-down states, respectively. The partial density of states of the transition metal atom together with the neighboring three C atoms (C3) is shown in Fig. 5.19. For carbon nanothreads doped with Si and Ti atoms, the peak of density of states are mainly contributed from the transition metal atom 3d states and some C3 2p states near Fermi level. For carbon nanothreads doped with Sc atoms, interaction is found between the 3d orbital (Sc) and 2p orbital (C3). For carbon nanothreads doped with V, Cr, Mn, Fe, Co, and Ni atoms, the peak of density of states is mainly contributed from the 3d orbital (metal atoms) at the Fermi level. The carbon nanothreads doped with transition metal atom (except Ti and Ni) have spin polarization caused by varying d orbitals. The magnetic moments in Cr- and Mn-doped structures, for example, are mainly originated from the dxz and dx2 orbitals.

138

5 Electronic Properties of Carbon Nanothreads

Fig. 5.17 a Electronic band structures, total and projected density of states of thiophene-, furan-, and pyrrole-derived carbon nanothreads. b Electronic band structures, total, and projected density of states of benzene-derived carbon nanothreads. c Integrated local density of states for three energy ranges in S-syn-anti structure. Reprinted (adapted) with permission from [11]

5.3 Electronic Band Structure Table 5.6 Bandgap E g of the carbon nanothreads doped with different transition metal atoms. Reprinted (adapted) with permission from [6]

139 Structure

E g (eV) Up

Down

Pure

3.93

3.93

Sc

1.17

1.28

Ti

2.45

2.45

V

0.35

2.41

Cr

1.45

2.06

Mn

1.71

2.54

Fe

–

–

Co

0.73

0.56

Ni

1.79

1.79

Cu

0.73

0.56

Zn

1.62

–

Fig. 5.18 Electronic band structures of carbon nanothreads doped with different metal atoms. a Sc-doped carbon nanothreads; b Ti-doped carbon nanothreads; c V-doped carbon nanothreads; d Crdoped carbon nanothreads; e Mn-doped carbon nanothreads; f Fe-doped carbon nanothreads; g Codoped carbon nanothreads; h Ni-doped carbon nanothreads. Reprinted (adapted) with permission from [6]

Gryn’ova and Corminboeuf [2] investigated the conductance of carbon nanothreads with different topological structures. The atomic structures of different carbon nanothreads are provided in Fig. 5.20. Junction geometries are constructed by anchoring the optimized carbon nanothreads to Au(111) electrodes with methylthio

140

5 Electronic Properties of Carbon Nanothreads

Fig. 5.19 The partial density of states of the transition metal atom together with the neighboring three C atoms (C3). a Sc-doped carbon nanothreads; b Ti-doped carbon nanothreads; c V-doped carbon nanothreads; d Cr-doped carbon nanothreads; e Mn-doped carbon nanothreads; f Fe-doped carbon nanothreads; g Co-doped carbon nanothreads; h Ni-doped carbon nanothreads. Reprinted (adapted) with permission from [6]

linkers. The conductance properties of carbon nanothread-Au junctions were characterized by a combination of non-equilibrium Green’s function formalism and DFT calculation with PBE pseudopotentials and DZP basis sets. Generally, the conductance decays exponentially with increasing length: G = e−β L ,

(5.11)

where L is the length of nanogap, β is a decay constant. The calculated zero-basis transmission probabilities at the Fermi level are obtained by T (E F ) = G(E F )/G0 , where G0 is the quantum of conductance, G(E F ) is the conductance at the Fermi level. The results are shown in Fig. 5.21. It is found that atomic configuration has a significant impact on the conductance of carbon nanothreads. For the acene-derived carbon nanothread in Fig. 5.21a, T(E F ) is almost the same with the increase of nanogap length L, as a result of the decrease of lowest unoccupied molecular orbital and better coupling with gold’s Fermi level [24]. For the alkane-derived carbon nanothread, the T (E F ) decreases with the increase of L due to the exponential decay rule (Eq. 5.11). For other carbon nanothreads, there is also a negative relationship

5.3 Electronic Band Structure

141

Fig. 5.20 The atomic configuration of carbon nanothreads: “zipper” refers to zipper polymer, “tube” refers to sp3 -(3,0), “twistane” refers to polytwistane, and “polymer” refers to polymer I. The Arabic numbers indicate the homolog number, and the wavy lines indicate the location of the— SCH3 linkers. Reprinted (adapted) with permission from [2]. Copyright (2019) American Chemical Society

between the T (E F ) and nanogap length L. Figure 5.21b shows that for a determined L, one-dimensional alkane-derived carbon nanothread has the lowest T (E F ), and the two-dimensional cycloalkane-derived carbon nanothreads has T (E F ) next to the lowest. Three-dimensional configurations used to have much higher conductance, as seen from the polytwistane configuration at larger L. It can be seen the cycloalkane configuration has the fastest decay (β = 1.49) of conductance with increasing L. The polytwistane configuration exhibits the lowest decay (β = 0.55), which could be attributed to the oligosilane-like nature: small angles of C–C–C and high contributions of p-orbitals (sp2.82 ). The oligosilane-like nature could reduce the overlap of σ-electron densities between neighboring C–C bonds and improve the conductance by higher zero-bias transmission.

142

5 Electronic Properties of Carbon Nanothreads

Fig. 5.21 Calculated zero-bias transmission probabilities at the Fermi level for carbon nanothreads derived from different molecules. The carbon nanothreads are anchored to Au(111) electrodes. The nanogap length larger than 10 Å is concerned to avoid direct interactions between the nanothread and Au. a Effects of decay constant denoted by Arabic number in brackets on the transmission probabilities. The Arabic numbers near the data denote homolog number; b Effects of nanogap length on the transmission probabilities. Reprinted (adapted) with permission from [2]. Copyright (2019) American Chemical Society

Gryn’ova and Corminboeuf [2] proposed a series of carbon nanothreads by introducing quaternary carbon chains into the polytwistane configuration with an aim to improve the conductance. The atomic structures of proposed carbon nanothreads are provided in Fig. 5.22. Due to the special linkage sites, these novel structures possess only one shortest conductance pathway but lots of longer conductance pathways. The calculated zero-bias transmissions are shown in Fig. 5.23. Results show that the mono-OT configuration exhibits similar conductance with the twistane configuration in Fig. 5.21. Besides, the di- and tri-OT configurations possess much lower decay (β < 0.2) and higher conductance, for instance, ∼10–5 G0 in di-OT and ∼10–4 G0 in tri-OT at over 20 Å nanogap. The excellent conductance of di- and tri-OT configurations can be attributed to the oligosilane-like behavior and multiple σ-conductance channels.

5.3 Electronic Band Structure

143

Fig. 5.22 Proposed super carbon nanothreads with numerous conductance channels. Arabic numbers indicate the homolog number, and the wavy lines indicate the location of the—SCH3 linkers. The mono-, di-, and tri-OT indicate the number of “conjoined” oligotwistane chains. Reprinted (adapted) with permission from [2]. Copyright (2019) American Chemical Society

Fig. 5.23 Effects of nanogap length on the computed zero-bias transmission probabilities at the Fermi level for carbon nanothreads derived from acene and alkane molecules, as well as super carbon nanothreads. The decay constants are denoted by Arabic number in brackets. Reprinted (adapted) with permission from [2]. Copyright (2019) American Chemical Society

144

5 Electronic Properties of Carbon Nanothreads

5.4 Comparison of Band Structure 5.4.1 Carbon Nanotube As mentioned above, an individual carbon nanothread is semiconductive. In comparison, an individual CNT could be either metal or semiconductor, which is dependent on its chirality integers (n, m) and diameter [25]. From Fig. 5.24, the chirality of CNT is determined by the chiral vector C h and the unit vectors a1 and a2 of the graphene, where the chiral vector Ch is expressed by the integers (n, m): C h = na1 + ma2 , (n ≥ m).

(5.12)

The chiral angle θ is calculated as: √

3m θ = arctan . 2n + m

(5.13)

The diameter d of CNT is determined by: √ a n 2 + m 2 + nm . d= π

(5.14)

Fig. 5.24 Schematic representation of the formation of CNT by wrapping a graphene sheet. The chiral vector OA and lattice vector OB specify the chirality. The basic symmetry operation is determined as R = (|τ ) with Ψ being the rotation angle and τ the transloation. Reprinted (adapted) with permission from [25]. Copyright (1992) American Physical Society

5.4 Comparison of Band Structure

145

Fig. 5.25 The chiral vector C h for CNTs is determined by the chirality integers (n, m). The CNTs with metallic characteristics are marked by the circled dots, and CNTs with semiconducting characteristics are marked by the circled dots. Reprinted (adapted) with permission from [25]. Copyright (1992) American Physical Society

The CNTs can be classified into three types based on the chirality, including zigzag, armchair, and chiral CNTs. CNTs with chiral angle 0 < |θ | < 30° are called as chiral nanotube. Other CNTs with chirality integers (n, 0) and (m, m) are termed as zigzag and armchair nanotubes, respectively. Based on the realistic tight-binding band-structure calculations [25], the CNTs can be either metal with chirality index satisfying (n = m), small-gap semiconductor ((n − m)/3 = integer), and a moderategap semiconductor otherwise, as seen in Fig. 5.25. Matsuda et al. [26] further calculated the bandgaps for various CNTs using DFT theory. The Becke-Lee-YangParr (B3LYP) exchange–correlation functional was applied during the DFT calculation. The bandgaps of metallic zigzag and chiral CNTs ((n − m)/3 = integer) are provided in Table 5.7. It can be seen that the bandgap scale as ~1/d 2 , but no bandgap is found for (6, 0) and beyond (24, 0). The effects of diameter on the bandgap are also plotted in Fig. 5.26. When d > 0.6 nm, the bandgap of zigzag CNTs (n = 3m, 0) increases with decreasing diameter. However, the bandgaps for zigzag (6, 0) and (5, 0) CNTs are zero (metallic). The small bandgaps for (3m, 0) with m = 3, 4, 5, and 6 are implied to arise from the intrinsic properties of CNTs rather than the strain effects such as distortions from the Au(111) substrate.

146 Table 5.7 Bandgaps (Eg ) of Metallic ((n − m)/3 = integer) zigzag and chiral CNTs employing B3LYP exchange–correlation functional. n and m are chiral index. Reprinted (adapted) with permission from [25]. Copyright (1992) American Chemical Society

5 Electronic Properties of Carbon Nanothreads N

m

Diameter (nm)

B3LYP Eg (eV)

6

0

0.489

0.00

9

0

0.713

0.079

12

0

0.951

0.041

15

0

1.182

0.036

18

0

1.42

0.028

21

0

1.655

0.021

24

0

1.855

0

27

0

2.217

0

30

0

2.317

0

5

5

0.557

0

10

10

1.366

0

8

2

0.725

0

11

5

1.121

0

16

4

1.446

0

15

6

1.478

0

Fig. 5.26 Effects of diameter on the bandgaps for zigzag (blue dots) and chiral (light blue dots) CNTs. Reprinted (adapted) with permission from [26]. Copyright (2010) American Physical Society

5.4.2 Boron Nitride Nanotube The boron nitride nanotubes can be constructed by wrapping a boron nitride honeycomb sheet. Similar to CNTs, the chirality of boron nitride nanotubes can be determined by the chiral vector C h and the unit vectors a1 and a2 of the boron nitride sheet. The atomic structures of different boron nitride nanotubes are shown in Fig. 5.27. Rubio et al. [28] studied the bandgap structures for various boron nitride nanotubes. The Slater-Koster tight-binding scheme that can reproduce the LDA exchange–correlation functional well was applied. It was found that the boron nitride

5.4 Comparison of Band Structure

147

Fig. 5.27 Atomic structures of boron nitride nanotubes: a chiral nanotube in top view; b chiral nanotube in side view; c zigzag nanotube in top view; d zigzag nanotube in side view; a armchair nanotube in top view; b armchair nanotube in side view. Reprinted (adapted) with permission from [27]

nanotubes have semiconductor characteristics, independent of the chirality and diameter. The boron nitride nanotubes were predicted to with a bandgap larger than eV. The bandgap increases with the diameter of boron nitride nanotubes. The bandgap structures of (10, 0) and (10, 10) boron nitride nanotubes are shown in Fig. 5.28. A

Fig. 5.28 Tight binding band structure along the −X direction. a (10,0) boron nitride nanotubes; b (10,10) boron nitride nanotubes. Reprinted (adapted) with permission from [28]

148

5 Electronic Properties of Carbon Nanothreads

difference of dispersion of the conduction and valence bands could be observed. It was demonstrated that the nanotubes with chirality (n, 0) are semiconductors with direct gap, while the armchair chiral (n, n) nanotubes are semiconductors with indirect gap.

5.4.3 Carbon Nitride Nanotube The carbon nitride nanotube can be formed by wrapping the graphitic carbon nitride sheets such as C3 N4 or CN. The graphitic C3 N4 sheet can be constructed either by heptazine units (C6 N7 ) by additional nitrogen atoms in Fig. 5.29a, or by triazine rings by additional nitrogen atoms in Fig. 5.29b. The graphitic CN sheet can be built by connecting C and N atoms from a porous honeycomb structure, as seen in Fig. 5.30. Following the constructing method of CNTs, the carbon nitride nanotubes can be classified as zigzag nanotubes and armchair nanotubes, depending on the chiral vector C h and the unit vectors a1 and a2 of the graphic carbon nitride sheet. Enyashin and Ivanovskii [31] calculated the bandgaps of triazine-based C3 N4 nanotubes based on tight-binding band method. It was found that all C3 N4 nanotubes were semiconductors. The bandgap is strongly dependent on the diameter d when d < 3 nm. The bandgap reaches a point of saturation at larger diameters, becoming independent on the diameter. For hexagonal C3 N4 nanotubes, the chirality has negligible effect on the saturated value of bandgap because the π electrons are completely delocalized. For orthorhombic g-C3 N4 nanotubes, the difference between zigzag and armchair configurations is about 0.6–0.8 eV. The electrons of orthorhombic C3 N4 nanotubes are delocalized both in the rings and between them, along the unit vectors

Fig. 5.29 Linking models g-C3 N4 based on a heptazine unit; a triazine unit. Reprinted (adapted) with permission from [29]

5.4 Comparison of Band Structure

149

Fig. 5.30 Atomistic configurations of a graphitic carbon nitride CN sheets; b side view of armchair (6, 6) carbon nitride nanotube; c side review of zigzag (9, 0) carbon nitride nanotube. Reprinted (adapted) with permission from [30]

a1 by connecting nitrogen atoms. Therefore, the bandgaps of orthorhombic C3 N4 nanotubes are sensitive to the chirality. Gracia and Kroll [32] studied the electronic properties of both heptazine- and triazine-based C3 N4 nanotubes. The GGA exchange– correlation functional was applied during the DFT calculation. The calculated bandgaps of different C3 N4 structures are provided in Table 5.8. It was shown that the studied C3 N4 nanotubes are semiconductors with bandgaps ranging from 1.5 to 2.2 eV . The zigzag C3 N4 nanotubes have smaller bandgaps than the armchair C3 N4 nanotubes. The bandgap of heptazine C3 N4 nanotube is smaller than that of the triazine one. Chai et al. [30] studied the electronic properties of CN nanotubes built from triazine units. The GGA/PBE exchange–correlation functional was applied during the DFT calculation. The calculated bandgaps and band structures of different CN nanotubes are shown in Table 5.9. Results show that both the zigzag and armchair CN nanotubes are semiconductors with direct bandgap and dependent on the chirality and diameter. For the zigzag CN nanotubes, the bandgaps are in the range of 1.675– 1.942 eV, and it decreases with an increase in diameter. For the armchair CN nanotubes, the bandgaps are in the range of 1.819–1.998 eV, and it decreases with Table 5.8 Bandgaps (Eg ) of different C3 N4 nanotubes and C3 N4 sheets. Reprinted (adapted) with permission from [32]

C3 N4 structure

E g (eV)

h-(12,0)-heptazine

1.5

h-(6,6)-heptazine

2.1

h-(6,6)-triazine

2.2

m-(12,0)-heptazine

1.5

m-(6,6)-heptazine

1.8

h-C3 N4 sheet-heptazine

1.9

m-C3 N4 sheet-heptazine

1.9

150 Table 5.9 Bandgaps (Eg ) of different CN nanotubes and CN sheet. Reprinted (adapted) with permission from [30]

5 Electronic Properties of Carbon Nanothreads CN structure

E g (eV)

(6, 0)

1.675

(9, 0)

1.926

(12, 0)

1.942

(15, 0)

1.921

(6, 6)

1.998

(9, 9)

1.920

(12, 12)

1.819

CN sheet

1.683

increasing diameter and converges to that of CN sheet (1.683 eV). The top of the valence band of CN nanotube is dominantly contributed from the 2p orbitals (N), and the bottom of conduction band originates from the C–C π* orbitals. In comparison, the bandgap of CN nanotube is smaller than that of the C3 N4 nanotube.

5.4.4 Silicon Carbide Nanotube The boron nitride nanotubes can be built by wrapping a silicon carbide honeycomb sheet. Similar to CNTs, the chirality of silicon carbide nanotubes can be determined by the chiral vector C h and the unit vectors a1 and a2 of the silicon nitride sheet. The structural configurations of different silicon carbide nanotubes are shown in Fig. 5.31.

Fig. 5.31 Axial-and radial-directions views of the a armchair, b zigzag silicon carbide nanotubes. Reprinted (adapted) with permission from [33]

5.4 Comparison of Band Structure

151

Fig. 5.32 Bandgaps of silicon carbide nanotubes with variation of diameter. Reprinted (adapted) with permission from [34]. Copyright (2005) American Physical Society

Zhao et al. [34] performed DFT calculations to predict the electronic properties of silicon carbide nanotubes. The GGA/PBE exchange–correlation functional was applied. The calculated bandgaps of different silicon carbide nanotubes with variation of diameter are plotted in Fig. 5.32. Results show that the silicon carbide nanotubes are semiconductors, which is due to the higher ionicity of silicon carbide with respect to the graphite. The bandgap increases with an increase in diameter and reaches a point of saturation, which may be explained by the σ-π hybridization induced by curvature. At small diameter, the rehybridization could shift the π* and σ* bands to lower and higher energies, respectively [28], resulting in a smaller bandgap. At large diameter, the repulsion between π and π* states leads to the increase of bandgap. For silicon carbide nanotubes with similar diameters, such as (6,6) and (11,0) nanotubes, the armchair nanotube has the largest bandgap, while the zigzag nanotube has the smallest bandgap. Therefore, the σ-π hybridization induced by curvature is more prominent in zigzag silicon carbide nanotubes, which downshifts the conduction bands and contributes to the increase of bandgap. The band structures of (8,0), (5,5), (16,0), and (9,3) silicon carbide nanotubes are provided in Fig. 5.33. It is shown that both armchair and chiral silicon carbide nanotubes have indirect bandgaps, whereas zigzag silicon carbide nanotubes have direct bandgaps.

5.4.5 Graphyne Nanotube The graphyne nanotubes can be built by wrapping a graphyne sheet. There are three typical kinds of graphyne sheets, including α-, β-, and γ -graphynes. The γ -graphyne is the most stable member among the graphyne family. Similar to CNTs, the chirality of graphyne nanotubes can be determined by the chiral vector C h and the unit vectors a1 and a2 of the graphyne sheet. The graphyne nanotubes can be also categorized

152

5 Electronic Properties of Carbon Nanothreads

Fig. 5.33 Electronic band structures of a (8,0), b (5,5), c (16,0), d (9,3) silicon carbide nanotubes. Reprinted (adapted) with permission from [34]. Copyright (2005) American Physical Society

into three types according to the chirality, including zigzag, armchair, and chiral nanotubes. The atomic structures of different graphyne nanotubes are provided in Fig. 5.34.

5.4 Comparison of Band Structure

153

Fig. 5.34 Atomic structures of a α-graphyne nanotube; b β-graphyne nanotube; c γ -graphyne nanotube. The left panels show the armchair nanotubes, and the right panels show the zigzag nanotubes. Reprinted (adapted) with permission from [35]. Copyright (2018) American Chemical Society

An individual graphyne nanotube is either metal or semiconductor, which is dependent on its chirality. Coluci et al. [36] investigated the electronic properties of various graphyne nanotubes using tight-binding method and DFT theory. The GGA/PBE exchange–correlation functional was applied during the DFT calculation. For α-graphyne nanotubes, the armchair structures have metallic characteristics, and the zigzag structures are either metallic or semiconducting, determined by the chirality index (n, m). Figure 5.35aa shows that the zigzag α-graphyne nanotubes are metallic with chirality index satisfying (n − m)/3 = integer, and the bandgaps decrease with increasing nanotube diameter. For β-graphyne nanotubes, the armchair structures are predicted to be metallic, and the zigzag structures are either metallic or semiconducting, determined by the chirality index (0, m). The zigzag βgraphyne nanotubes are metallic with chirality index satisfying (m)/2.72≈integer (see Fig. 5.35b). The γ -graphyne nanotubes are semiconducting, independent on chirality index (n, m). Figure 5.36 shows that the bandgaps of all γ -graphyne nanotubes are quite similar.

154

5 Electronic Properties of Carbon Nanothreads

Fig. 5.35 Bandgaps calculated by tight-binding method for a zigzag α-graphyne nanotube; b zigzag β-graphyne nanotube. Reprinted (adapted) with permission from [36]. Copyright (2003) American Physical Society Fig. 5.36 Bandgaps calculated by tight-binding method for γ -graphyne nanotubes with variation of diameter. Reprinted (adapted) with permission from [36]. Copyright (2003) American Physical Society

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25. Hamada N, Sawada SI, Oshiyama A (1992) New one-dimensional conductors: Graphitic microtubules. Phys Rev Lett 68(10):1579 26. Matsuda Y, Tahir-Kheli J, Goddard WA III (2010) Definitive bandgaps for single-wall carbon nanotubes. J Phys Chem Lett 1(19):2946–2950 27. Sargazi-Avval H, Yoosefian M, Ghaffari-Moghaddam M, Khajeh M, Bohlooli M (2021) Potential applications of armchair, zigzag, and chiral boron nitride nanotubes as a drug delivery system: letrozole anticancer drug encapsulation. Appl Phys A 127(4):1–7 28. Rubio A, Corkill JL, Cohen ML (1994) Theory of graphitic boron nitride nanotubes. Phys Rev B 49(7):5081 29. Wang Y, Gao B, Yue Q, Wang Z (2020) Graphitic carbon nitride (gC3 N4 )-based membranes for advanced separation. J Mater Chem A 8(37):19133–19155 30. Chai G, Lin C, Zhang M, Wang J, Cheng W (2010) First-principles study of CN carbon nitride nanotubes. Nanotechnology 21(19):195702 31. Enyashin AN, Ivanovskii AL (2004) Electronic structure of nanotubes of layered modifications of Carbon Nitride C3 N4 . In: Doklady physical chemistry, vol 398, no 1. Kluwer Academic Publishers-Plenum Publishers, pp 211–215 32. Gracia J, Kroll P (2009) First principles study of C3 N4 carbon nitride nanotubes. J Mater Chem 19(19):3020–3026 33. Moradian R, Behzad S, Chegel R (2008) Ab initio density functional theory investigation of structural and electronic properties of silicon carbide nanotube bundles. Physica B 403(19– 20):3623–3626 34. Zhao M, Xia Y, Li F, Zhang RQ, Lee ST (2005) Strain energy and electronic structures of silicon carbide nanotubes: density functional calculations. Phys Rev B 71(8):085312 35. Reihani A, Soleimani A, Kargar S, Sundararaghavan V, Ramazani A (2018) Graphyne nanotubes: materials with ultralow phonon mean free path and strong optical phonon scattering for thermoelectric applications. J Phys Chem C 122(39):22688–22698 36. Coluci VR, Braga SF, Legoas SB, Galvao DS, Baughman RH (2003) Families of carbon nanotubes: Graphyne-based nanotubes. Phys Rev B 68(3):035430

Chapter 6

Thermal Properties of Carbon Nanothreads

6.1 Introduction Carbon-based nanostructures have been demonstrated with outstanding thermal conductivity with respect to other nanomaterials, and they have been considered as promising candidates for thermal connection and temperature management. Carbon nanothreads that are one-dimensional sp3 -hybridized carbon structures could be one of the best candidates for thermal management of electronic devices. However, the thermal conductivity of carbon nanothreads has not been explored through experiments yet due to experimental inaccessibility. Theoretical prediction becomes an alternate approach to characterize the thermal transport characteristics of carbon nanothreads.

6.2 Computational Methodology 6.2.1 Direct Method for Thermal Conductivity Non-equilibrium molecular dynamics (NEMD) is one of the most widely used approaches to characterize the thermal conductivity of materials at the nanoscale. Different from the traditional MD method, the NEMD deals with the physical system at non-equilibrium states and is adept in studying the time courses of the transport process by controlling the non-equilibrium environmental variables, such as pressure, temperature, and energy. To calculate the thermal conductivity using NEMD, a constant heat flux J heat along one direction is imposed into the system. The schematic representation of the direct method is shown in Fig. 6.1. The presence of heat bath and cold bath will create a constant heat flux when the system reaches a stable state. This method is also called “direct method” [1]. The thermal conductivity κ is determined from Fourier’s law: © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. M. Liew et al., Carbon Nanothreads Materials, Materials Horizons: From Nature to Nanomaterials, https://doi.org/10.1007/978-981-19-0912-2_6

157

158

6 Thermal Properties of Carbon Nanothreads

Fig. 6.1 Schematic representation of three-dimensional periodic system for studying thermal conductivity using direct method

Jheat = −κT /Z ,

(6.1)

where T/Zisthetemperaturegradient along Z direction, Z is the distance between heat bath and cold bath. The heat flux J heat could be computed by accumulated energy E at exchanged interval t: Jheat = E/(2 × A × t),

(6.2)

where A is the cross-section area of the unit cell, and the factor 2 in Eq. 6.3 is due to the heat flux transferring in two directions. The addition/extraction of energy E in heat/cold bath can be achieved by exchanging the velocity [2], for instance, the velocity of the slowest atom in heat bath is exchanged with the velocity of the fastest atom in cold bath. The simulation box is divided into N slabs with identical size along the direction of heat flux, as seen in Fig. 6.2. The kinetic temperature T k in slab k can be calculated as:

Fig. 6.2 Subdividing the periodic simulation box into slabs. Slab 0 is the “cool” slab, slab N/2 the “hot” slab. Kinetic energy is artificially transferred from the cool to the hot slab and then flows back by thermal conduction. Reprinted (adapted) with permission from [2]

6.2 Computational Methodology

159

Tk =

nk 1  m i vi2 , 3n k kb i∈k

(6.3)

where the sum extends over the n k atoms in slab k with masses mi and velocities vi ; kb is the Boltzmann constant. In Fig. 6.2 the heat bath locates at the slab N/2, and the cold bath locates at the slab 0. In the beginning, the temperature difference between the heat bath and cold bath will be very small. As the velocity exchange proceeds, the temperatures of heat bath and cold bath will increase and decrease, respectively. A temperature gradient between the heat bath and cold bath will be generated, resulting in a net heat flux from the heat bath to cold bath. After the system reaches a stable state, the accumulated energy E will be balanced by the heat flux in the opposite direction. The heat flux will induce a stable temperature gradient, which is dependent on the thermal conductivity κ of materials. The higher κ leads to a smaller temperature gradient. Based on Eqs. 6.1–6.3, the thermal conductivity can be further calculated as:  mh 2 mc 2 ν − 2 νc 2 h , (6.4) κ=− 2t A∂ T /∂z where subscript h and c denote the atom in the hot bath and cold bath, respectively. Alternatively, the addition/extraction of energy E in the heat/cold bath can be achieved by rescaling the velocities [3]. To avoid an artificial drift of the kinetic energy, the total momentum during energy exchange should be conserved. For an atom i in the heat/cold bath, the velocity in each exchange is modified by: vi = vG + α(vi − vG ),  α=

1±

E , E cR

(6.5)

(6.6)

where vG is the velocity of the center of mass of the atoms in the heat/cold bath, and α is the scaling factor determining the amount of exchange energy E, and the relative kinetic energy E cR is calculated as: E cR =

1 1 m i vi2 − m i vG2 . 2 i 2 i

(6.7)

In addition, the addition/extraction of energy E in the heat/cold bath can be also realized by coupling the heat bath and cold bath with two separate thermostats. For instance, the temperature of heat bath and cold bath is set as T + T /2 and T − T /2, respectively. Upon reaching a stable state, the temperature gradient along

160

6 Thermal Properties of Carbon Nanothreads

Z direction is T /Z . The energy in heat/cold bath will increase linearly with time, and the slope of the linear line will be the heat flux. The direct method is suitable for thermal conductivity simulation of nanostructure, which has a large thermal conductivity (above the Debye temperature). The thermal conductivity predicted by the direct method monotonically decreases with increasing temperature, which disagrees with the available experimental findings [4]. The disagreement can be attributed to the fact that the quantum effects, which play important roles below the Debye temperature, are neglected in the classical MD simulation. The Debye temperature T D of a material is given by: TD =

ω D , kB

(6.8)

where  is the reduced Planck constant, k B is the Boltzmann constant, ω D is Debye frequency, which refers to the highest phonon frequency in the Debye model. For the case that quantum characteristics cannot be ignored, it is necessary to make quantum corrections to the MD calculations of temperature and thermal conductivity [5]. The temperature correction is made by equating the total energies of the classical and quantum systems. The energy equality can be expressed as [5]: N ,3n 

 EC

κ,ν

K ν

 =

N ,3n  κ,ν

 EQ

 K , ν

(6.9)

where the sums are overall  phonon modes denoted by N wave vectorsK , and 3n K dispersion branchesν, E is the energy of a phonon mode, and the superscripts ν C and Q denote the classic and quantum. Based on two assumptions (1) equipartition of the classical energy and subtracting the three translational degrees of freedom and (2) the quantum energy to be harmonic, Eq. 6.9 can be further expressed as [5]: ⎡

3(N n − 1)k B T C



K     ω N ,3n ⎢  ν K K ⎢ = fQ + ⎢ω ν ν ⎣ 2 κ,ν

⎤ ⎥ ⎥ ⎥, ⎦

(6.10)

  K where  is the Planck constant divided by 2π, ω is the mode-dependent ν   K frequency, f Q is the equilibrium occupation number. The temperature after ν quantum correction T Q can be determined by computing T C in Eq. 6.10 for a series

6.2 Computational Methodology

161

of T Q values and then interpolating. The correction of thermal conductivity is determined by equating the heat fluxes obtained from the Fourier law in the classical and quantum systems [5]: κ Q = κC

dT C . dT Q

(6.11)

Another issue for the direct method is the finite size effects when the length of system along direction of energy flux is not significantly longer than the phonon mean-free path [1]. The thermal conductivity will be limited by the length of system, which is due to the phonon scattering that occurs at the interfaces with the heat and cold baths. This phenomenon is also called Casimir limit. It is important to determine the effective mean-free path leff when l eff ~ l ∞ , where l∞ is the mean-free path for an infinite system. By adding the inverse mean-free path, the effective mean-free path leff is given by: 1 leff

=

1 l∞

+

s . Lz

(6.12)

Here, the factor of s is determined by the average distance L z /s as phonons travel along the length of the simulation cell from the heat bath to the cold bath. For a thermal transport system in Fig. 6.2, the average distance between the heat bath and the cold bath is L z /2 (s = 2). Equation 6.12 assumes that there is no anharmonic phonon–phonon scattering in the region between the heat bath and the cold bath (ballistic transport). The thermal conductivity κ of an infinite system can be obtained by extrapolating the linear line 1/κ vs. 1/L z to 1/L z = 0. It should be noticed that the linear extrapolation approach will be inaccurate when the minimum system size used in the direct method is not sufficiently larger than the largest mean-free paths. In such circumstance, the thermal conductivity will be underestimated. The minimum system size can be determined by comparing the thermal conductivity k predicted by using the linear extrapolation approach and the maximum thermal conductivity max that can be accurately predicted: κ∞ max κ∞ =

 1/2 1/2 Lk B C11 + 2C44 18a 3 ρ 1/2

,

(6.13)

where ρ, C 11 , and C 44 are the bulk density, elastic constants, respectively, L is the length of the system, a is the lattice constant. If the thermal conductivity predicted max , the length of the by using the linear extrapolation approach is larger than the κ∞ max system L needs to be increased until κ ≤ κ∞ .

162

6 Thermal Properties of Carbon Nanothreads

6.2.2 Green–Kubo Method for Thermal Conductivity The Green–Kubo method uses equilibrium MD (EMD) simulations to determine the thermal conductivity κ based on the fluctuation–dissipation theorem. The EMD simulation counts on the equilibrium heat current autocorrelation function to the thermal conductivity via the Green–Kubo expression [1]: κμν (τm ) =

τm 

1

k B

T2

 Jμ (τ )Jν (0) dτ,

(6.14)

0

where is the system volume, k B is the Boltzmann constant, T is the system temperature, and Jμ and Jν are the heat fluxes in the μ and ν directions, respectively, and the angular brackets are heat current autocorrelation function. The EMD simulation is performed for discrete time step t, and hence Eq. 6.14 can be further expressed as: κμν (τ M ) =

M N −m  t  −1 (N − m) Jμ (m + n)Jν (n),

k B T 2 m=1 n=1

(6.15)

where τ M is given by Mt, and Jμ (m + n) is the mth component of the heat flux at timestep m + n. The total simulation steps N mush be considerably larger than the integration steps M to ensure good statistical averaging. The heat flux J is calculated as: J=

d  ri (t)εi (t), dt i

(6.16)

where ri (t) and εi (t) are the coordinate and site energy of atom i at time t, respectively. For site energy εi (t) can be divided into: εi =

1 1   m i vi2 + u 2 ri j . 2 2 j

(6.17)

Here, the first term on the right side denotes the kinetic energy, and the last term denotes the potential energy due to pairwise interactions u 2 (r ). By calculating the time derivative of Eq. 6.16, the heat flux can be further expressed as: J (t) =

 i

v j εi +

 1   ri j Fi j · vi , 2 i j,i= j

(6.18)

6.2 Computational Methodology

163

where Fi j is the force on atom i due to its neighbor j from the pair potential. Equation 6.17 assumes that the potential energy is evenly divided between two atoms i and j. However, this assumption will be invalid for the Sillinger–Weber potential, which considers both pairwise and three-body interactions. The potential energy for the Sillinger–Weber potential is given by: E pot =

 1   1  u 2 ri j + u 3 ri , r j , rk . 2 j 6 i jk

(6.19)

Here, the first term on the right side denotes the pairwise potential energy as introduced above, and the last term denotes the three-body potential energy by assuming that the potential energy is evenly divided between three atoms i, j, and k. Similar to Eq. 6.18, the heat flux using the Sillinger–Weber potential can be expressed as: J (t) =



v j εi +

i

 1    1   ri j + rik Fi jk · vi , ri j Fi j · vi + 2 i j,i= j 6 i jk

(6.20)

where Fi jk is the force on atom i due to its neighbors j and k from the three-body potential, and it is given by:   Fi jk = −i u 3 ri , r j , rk .

(6.21)

Another definition of three-body interaction is to assign the entire energy to atom i at the vertex of the trio ijk, which leads to a different expression of local site energy: εi =

 1 1   1   m i vi2 + u 2 ri j + ε h ri j , r jk , jik , 2 2 j 2 jk

(6.22)

where is θ jik is the angle between ri j and r jk . Similar to Eq. 6.18, the heat flux can be expressed as: J (t) =

 i

vi εi +

    1   ri j Fi j · vi + ri j F j (i jk) · v j , 2 i j,i= j i jk

(6.23)

  F j (i jk) = −ε j h 3 ri j , r jk , θ jik .

(6.24)

Note that the energy partition rule in two- and three-body potentials does not play a significant effect on the simulation results because the variety of temperature gradient is much larger than the interatomic distances [6]. A typical heat current autocorrelation function is provided in Fig. 6.3. The heat current autocorrelation function Jμ (τ )Jν (0) is normalized by the initial value at t = 0. A fast decrease of normalized heat current autocorrelation function is shown when t < 0.1 ps, followed by a much slower decay lasting ∼100 ps. The normalized

164

6 Thermal Properties of Carbon Nanothreads

Fig. 6.3 Representative HCACFs for Si (Sillinger–Weber potential) normalized by their value at t = 0 and plotted on a logarithmic y-axis. The dashed lines are fits using Eq. (6.5). For t ≥ tsignal ≈ 52 ps, the magnitude of the HCACF is comparable to the noise level calculated from the cross-correlation function. Reprinted (adapted) with permission from [7]

heat current autocorrelation function is dominated by noise when t > tsignal , whose magnitude W can be estimated as the root-mean-square of the heat current crosscorrelation functions:   2  2 Jx (t)Jy (0) + Jy (t)Jz (0) + Jz (t)Jx (0)2 . (6.25) W = 3 Because the cross-correlations equal to zero in the thermodynamic limit, W comprises noise only. Therefore, the upper limit of the integral τm in Eq. 6.14 should be larger than the t signal to obtain a convergent value of thermal conductivity. Apart from the direct numerical integration using Eq. 6.14, a fitted exponential function is also capable to characterize the heat current autocorrelation function Jμ (τ )Jν (0). For instance, the sum of three exponential decays is suggested [7]: J (t) · J (0) =

3 

Ai e−t/τi .

(6.26)

i=1

Here, Ai and τi are the fitting parameters. When fitting the parameters by fitting the data up to t = t signal , the calculated thermal conductivity by the direct numerical integration is very close to that by the fitted exponential function. The disadvantage for Green–Kubo method is it needs a very long simulation time for accurate statistical averaging to obtain reliable results by integration (see Eq. 6.15). To reduce the computational cost, Fast Fourier transformations can be applied to the heat current and take the limit ω → 0 of the expression [1]: κμν (ω, T ) =

1 Jμ (ω)Jv∗ (ω),

k B T 2

(6.27)

6.2 Computational Methodology

165

where Jμ (ω) is the frequency spectrum of the μth-component heat current, and Jv∗ (ω) is the complex conjugate of Jv (ω).

6.2.3 Model for Interfacial Thermal Resistance Interfacial thermal resistance, also known as thermal boundary resistance or Kapitza resistance [8], is used to occur at the interface when heat flows between two materials with different vibrational properties. When heat flux transverses the interface, phonon scattering at the interface will cause a temperature discontinuity. The interfacial thermal resistance can be modeled by using the direct method and thermal system presented in Fig. 6.2. According to the Fourier’s law, the interfacial thermal resistance R is given by: R=

T , J

(6.28)

where T is the temperature drop at the interface, J is the heat flux. Low interfacial thermal resistance is required for fast heat dissipation in microelectronic semiconductor devices. Generally, the interfacial thermal resistance could be reduced by enhancing the interfacial interaction energy per area and reducing the mismatch in vibrational density of states between two dissimilar materials.

6.2.4 Effect of Force Field on Thermal Conductivity The interatomic potential plays the most significant role in determining the thermal conductivity in MD simulation. So far lots of force fields have been applied to model the thermal conductivity of carbon-based nanomaterials, including the original Tersoff [9], optimized Tersoff [10], Brenner-II [11], and AIREBO [12] potentials. The thermal conductivity of a (10, 10) CNT predicted by the four potentials is shown in Fig. 6.4 [13]. It can be seen that the thermal conductivity of CNT is strongly dependent on the employ potential. The optimized Tersoff predicts the highest value of thermal conductivity, followed by the original Tersoff, Brenner-II, and AIREBO. The low values of thermal conductivity predicted by the Brenner-II and AIREBO potentials are attributed to: (1) low velocities of acoustic phonons obtained from phonon dispersion relations for CNT and strong anharmonicity of Brenner-II potential resulting in high phonon–phonon scattering rates [10]; (2) the van der Waals interactions added to the Brenner-II potential in AIREBO may further increase the phonon scattering rate, reading to a reduction of thermal conductivity [13]. The optimized Tersoff potential is expected to provide the most reliable value of thermal conductivity of CNT due to: (1) it provides the most accurate phonon properties of

166

6 Thermal Properties of Carbon Nanothreads

Fig. 6.4 a Schematic representation of the thermal system used in NEMD simulations. b Thermal conductivity, k, predicted in NEMD simulations performed with four different potentials for identical CNTs under identical computational conditions. Reprinted (adapted) with permission from [13]

CNT [14]; (2) it can correctly describe the linear temperature dependence of the G peak compared to the experimental values [14].

6.2.5 Generalized Debye-Peierls/Allen-Feldman Model for Thermal Conductivity In 1959, Callaway proposed an approximate solution of the Peierls Boltzmann equation within the relaxation time approximation invoking a Debye description of solids, which reproduced well the thermal conductivity κ vs. temperature dependence of germanium for low temperatures [14]. Holland extended the Callaway’s model by differentiating longitudinal acoustic (LA) and transverse acoustic (TA) phonons [15]. Based on the Callaway–Holland model, the thermal conductivity κ in the Z direction could be calculated by summing the contributing of the phonon mode branches m and phonon wave vectors q: κZ =

        v q m · Z )2 τ q m C ph q m , m

(6.29)

6.2 Computational Methodology

167

where v(q) = dω/dq is the group velocity, τ (q) is the mode-specific scattering time, and C ph (q) is the mode-specific heat capacity. The thermal conductivity for isotropic solid with frequency-dependent phonon density g(ω) can be further expressed as: κ=





cos2 θ

 v(ω)2 τ (ω, T )C ph (ω, T )g(ω)dω,

m

1 = v(ω) (ω, T )C ph (ω, T )g(ω)dω, d m

(6.30)

where θ is the angle between wave vector q and a temperature gradient T , d is the dimensionality, and cos2 θ  = 1/d is the geometric factor originates from summing over modes propagating in all directions, τ (ω, T ) is the scattering rates, and (ω, T ) = v(ω)τ (ω, T ) is the mean free paths. The heat capacity is determined as C ph = k B x 2 e x (e x − 1)2 , where x = ω/k B T ,  is the reduced Plank constant, and k B is Boltzmann constant. In generally, the phonon mode branches m can be divided into linear dispersion m = l and parabolic dispersion m = p. The density of states g(ω), group velocity v, and cutoff frequency ωc for modes with linear dispersion and parabolic dispersion are provided in Table 6.1. Based on Eq. 6.30 and Table 6.1, the contribution of different dispersion modes to thermal conductivity is given by: sd k B κ = d (2π )d =

sd k B d (2π )d





sd k B κp = d (2π )d

kB T 

kB T 



kB T 

d

xc v

1−d 0

d

xc v2−d 0

 d+1 2 a

1−d 2

(x, T )x d+1 e x d x. (e x − 1)2

τ (x, T )x d+1 e x d x. (e x − 1)2

xc 0

(6.31)

(6.32)

d+3

(x, T )x 2 e x d x. (e x − 1)2

(6.33)

Table 6.1 Density of states g(ω), group velocity v, and cutoff frequency ωc for modes with linear dispersion and parabolic dispersion.sd is the surface area of a d-dimensional sphere with unit radius: sd = (2, 2π, 4π ) for d = (1, 2, 3). n is the average atomic spacing. Reprinted (adapted) with permission from [16]. Copyright {2016} American Physical Society Dispersion

g(ω)

V

ωc

ω = vq(linear)

sd ωd−1 (2π v)d

V

1/d 2π v( nd sd )

ω = aq 2 (parabolic)

sd 2(2π )d

d

d

a − 2 ω 2 −1

√ 2 aω

2/d (2π )2 a( nd sd )

168

6 Thermal Properties of Carbon Nanothreads

sd k B κp = d (2π )d



 d+1 2

kB T 

a

xc

1−d 2

0

d+3

(x, T )x 2 e x d x. (e x − 1)2

(6.34)

Zhu and Ertekin [16] employed Cahill’s approach [17], which enables more delocalized vibrations by using a larger mean-free path = π v/ω equal to half the modal wavelength λ for the random walk step in Eqs 6.31 and 6.33, and generalized the Cahill’s model to arbitrary dimensions: κam

κ am p



d−1

xc

x d ex d x. (e x − 1)2

(6.35)

 d xc 1+ d x k B T 2 1− d x 2e sd k B 2 = a d x. d (2π )d−1  (e x − 1)2

(6.36)

sd k B =π d (2π )d

kB T 

v

2−d 0

0

They categorized the heat carriers-vibrons according to their degree of localization based on Allen-Feldman theory [18]. The vibrons consist of extendons and locons. The locons are high frequency, highly localized energy carriers that contribute negligibly to thermal conductivity. The extendons, which contribute the most to the thermal conductivity, are more delocalized energy carriers and can be further divided into propagons and diffusons. Propagons are the lowest frequency that transport heat in a manner reminiscent of typical phonons. Diffusons transport heat via diffusive random walk steps. The boundary between propagons and diffusons is approximated by a smooth sigmoid function σ (x) = (1 + e−α(x−xξ ) )−1 , where α and xξ = ω/k B T control the steepness and location of the boundary, respectively. The thermal conductivity can hence be expressed as: κ =

κ =

⎧ ⎪ s ⎪ ⎨ dd ⎪ ⎪ ⎩

κp =

κ +



kB (2π)d

π

⎧ ⎪ s ⎪ ⎨ 2 dd ⎪ ⎪ ⎩



 k B T d 

0

 k B T  d+2 2

2π sdd



κp.

(6.37)

p

⎫ x d+1 x ⎪ e ⎪ v2−d (1 − σ (x))τ (x, T ) (ex x −1) 2 d x. ( pr opagons) ⎬

sd k B (k B T )d−1 v2−d d (2π)d

kB (2π)d



a

2−d 2

x 0

d x

e σ (x) (exx −1) 2 d x. (di f f usions)

x p

(1 − σ (x))τ (x, T ) x

0

kB ( k BT (2π)x −1)2

d

)2 a

2−d 2

x p 0

σ (x)

d+2 x 2 ex (e x −1)2

4+d

e

2x

⎪ ⎪ ⎭

(6.38)

⎫ ⎪ d x. ( pr opagons) ⎪ ⎬

d x. (di f f usions)

⎪ ⎪ ⎭ (6.39)

Here, x and x p are the cutoff xc for the linear dispersion and parabolic dispersion, respectively.

6.2 Computational Methodology

169

For a one-dimensional carbon nanothread, there are four acoustic branches (LA, TA1 , TA2 , TW) exhibiting the usual linear dispersion, and there is no contribution from parabolic dispersion. TA1 and TA2 are two transverse acoustic phonons, which are unique to rolled system, and TW is the twist phonon. By approximating all modes as linear in wave vector, the thermal conductivity is calculated as: 

κ=

κ .

(6.40)

=L A,T A1 ,T A2 ,T W

κ =

⎫ ⎧ x k 2B T ⎪ ⎪ x 2 ex ⎪ ⎬ ⎨ π v (1 − σ (x))τ (x, T ) (ex −1)2 d x. ( pr opagons, 1D) ⎪ ⎪ ⎪ ⎩

0

k B v

x 0

x

σ (x) (exxe−1)2 d x. (di f f usions, 1D)

⎪ ⎪ ⎭

(6.41)

6.3 Thermal Conductivity of Carbon Nanothreads 6.3.1 Phonon and Localization Zhu and Ertekin [18] performed a real-space vibrational analysis to establish the characteristics of vibrational transport in the pristine sp3 -(3, 0) configuration (termed as crystalline) and sp3 -(3,0) configuration with a random distribution of Stone–Wales (SW) defects (termed as amorphous). The vibrational modes were assumed to be √ expressed by the simple form u iα,λ = (1/ m i )εiα,λ exp(iωλ t). The eigenfrequencies and eigenvectors could be calculated by solving the lattice dynamical equations: ωλ2 εiα,λ =



iα, jβ ε jβ,λ .

(6.42)

∂2V 1 iα, jβ = √ , m i m j ∂u iα u jβ

(6.43)

jβ

Here the matrix elements are:

where V is the potential energy, λ is a particular vibrational mode, u iα is the displacement of atom i along α axis, m i is the mass of atom i, ω is the modal eigenfrequency, and εiα the corresponding eigenvector component. By calculating the participation ratio pλ that measures the fraction of atoms participate in the vibration, the degree of spatial localization of a vibrational mode λ can be obtained:

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6 Thermal Properties of Carbon Nanothreads

 2   1 ∗ =N εiα,λ εiα,λ . pλ α i

(6.44)

The participation ratio of vibrational modes for crystalline and amorphous carbon nanothreads are shown in Fig. 6.5a. It is shown that pλ varies from 0.6 to 0.1 for crystalline configuration. However, the presence of SW defects reduces the pλ to 0.4~0.6 for most frequency. For frequencies larger than 40 THz, the pλ further drops below 0.4, which is near the Debye vibrational cutoff frequency (~45 THz). The reductions of pλ are common characteristics of disorder-induced localization limited to both vibrational mode and waves (electronic, spin, etc.) in amorphous configuration. The distinct spatial spans can be witnessed in the both localized and delocalized vibrational modes in Fig. 6.5b. For amorphous structure, the direction of the atomic vibrations of a given mode λ will be less wavelike and more randomized. The modal polarization spheres are Fig. 6.5 a Participation ratio of vibrational modes in both crystalline and amorphous carbon nanothreads, the SW defects are at a density of 20%; b Visual comparisons between modes of different frequency, the arrows indicate the vibration direction of each atom. Reprinted (adapted) with permission from [18]. Copyright {2016} American Chemical Society

6.3 Thermal Conductivity of Carbon Nanothreads

171

able to capture the loss of polarization and loss of the definition of the wave vector [19, 20], which can provide more insights into the vibrational modes. It shows the direction of the atomic vibrations of a given mode. The polarization sphere is a plot of the normalized direction of vibration of all atoms i of the eigenvector εiα,λ for mode λ. In general, the projection eiα,λ of the modal eigenvector εiα,λ onto unit spheres is given by: εiα,λ  eiα,λ =   ∗ α εiα,λ εiα,λ

(6.46)

The crystalline structure will have phonon modes with well-defined polarization. For instance, the polarization sphere for a LA mode will compose of dots marked only at ± êz , one dot for each atom. In contrast, the amorphous structure will have lost polarization. For instance, the atomic displacement will be unrelated and uniformly distributed over the sphere. The modal polarization spheres for all modes in a specific frequency range for the crystalline and amorphous carbon nanothreads oriented along the êz direction are shown in Fig. 6.6. The six large spheres in the xy plane indicate the TW modes, and the equatorial rings in the xy plane refer to TA modes. The large spheres at ± êz direction correspond to the LA modes. The three equatorial rings that are uniformly distributed through the ± êz poles are helical modes, including vibration both in longitudinal and transverse directions. It can be seen that the crystalline structure exhibits high symmetries in all modes. For instance, the TW modes have hexagonal symmetry about êz . For the amorphous structure with 20% SW defects, a well-defined polarization along the êz direction is observed for the lowest frequency mode smaller than 0.45 THz, whose characteristics are similar to the longitudinal phonons in nature. The vibrational characteristics with frequency smaller than 3.2 THz are similar to the twist and/or transverse modes, as can be seen from the several new modes involving atoms vibrating in or near the xy plane. With the increase of frequency, the atomic

Fig. 6.6 Polarization spheres for both crystalline and amorphous carbon nanothreads. The size of subspheres denotes the number of atoms oscillating in each direction. In the amorphous one, the lowest frequency modes maintain well-defined longitudinal and twist-like polarizations. Reprinted (adapted) with permission from [18]. Copyright {2016} American Chemical Society

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vibrations turn to randomized and homogeneously spaced over the sphere’s surface, indicating a loss of high-frequency wave character. The modal diffusivity is another metric, which could characterize the modes in amorphous structure. Based on the Allen-Feldman model, the thermal conductivity is given by: κ AF =

1  cλ Dλ , V λ

(6.47)

where V is the volume of the system, cλ is the heat capacity of mode λ, and Dλ is the modal diffusivity expressed as: Dλ =

  πV2  |Sλ,μ |2 δ ωλ − ωμ , 32 ωλ2 λ=μ

(6.48)

where Sλ,μ is the heat current operator in the harmonic approximation [20]. Normally, two transitions can be observed in the modal diffusivity curve in disordered system. A sudden drop at high frequency is called the mobility edge, which is relative to the transition from diffusons (delocalized carriers) to locons (localized carriers). A mild drop at low frequency is called the Ioffe-Regel transition, which indicates the boundary between propagons and diffusons. The modal diffusivity of carbon nanothreads with 20% SW defects is shown in Fig. 6.7. It can be seen the Ioffe-Regel transition occurs at the frequency of ~0.45 THz. Such transition is consistent with the polarization spheres, which also shows a transition in polarization from longitudinal to transverse and twist-like around 0.45 THz. The mobility edge occurs at the frequency of ~47.5 THz, which is close to the vibrational Debye frequency cutoff ∼45 THz. The frequency of the mobility edge corresponds well with the results Fig. 6.7 Modal diffusivity of amorphous carbon nanothreads. Two transitions are observed, including the Ioffe-Regel edges occurring at ~0.45 THz and the mobility edges occurring at ~47.5 THz. Reprinted (adapted) with permission from [18]. Copyright {2016} American Chemical Society

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from the participation ratio pλ in Fig. 6.5, which shows a drop at the frequency of ~40 THz.

6.3.2 Thermal Conductivity Zhu and Ertekin [18] predicted the thermal conductivities of carbon nanothreads using the Green–Kubo approach and generalized Debye-Peierls/Allen-Feldman model. The AIREBO potential was employed to describe the interatomic interaction. Three types of carbon nanothread are concerned, including a sp3 -(3, 0) configuration without Stone–Wales defects and two sp3 -(3, 0) configurations with a random distribution of SW defects at densities of 20 and 50%. The effects of temperature on the thermal conductivities of different carbon nanothreads are provided in Fig. 6.8. Fig. 6.8 a Thermal conductivity κ versus temperature T obtained from EMD approach for a sp3 -(3, 0) configuration without Stone–Wales defects and two sp3 -(3, 0) configurations with a random distribution of SW defects at densities of 20% and 50%. b Thermal conductivity κ versus temperature T obtained from generalized model. The insets show the individual contributions of heat carriers’ propagons (involving TW, LA, and TA) and diffusons. Reprinted (adapted) with permission from [16]. Copyright {2016} American Physical Society

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Zhu and Ertekin [16] predicted the thermal conductivities of carbon nanothreads using the Green–Kubo approach and generalized Debye-Peierls/Allen-Feldman model. The AIREBO potential was employed to describe the interatomic interaction. Three types of carbon nanothread are concerned, including a sp3 -(3, 0) configuration without Stone–Wales defects and two sp3 -(3, 0) configurations with a random distribution of SW defects at densities of 20 and 50%. The effects of temperature on the thermal conductivities of different carbon nanothreads are provided in Fig. 6.8. According to the prediction from Green–Kubo method, the carbon nanothreads without SW defects have a thermal conductivity of ~1000 W/mk at 300 K in Fig. 6.8a. The existence of SW defects will significantly reduce the thermal conductivity. The thermal conductivity of carbon nanothreads with a random distribution of SW defects at a density of 20% is reduced by a factor of 5 at 300 K with respect to the nondefective carbon nanothreads. Besides, a larger concentration of SW defects leads to a smaller thermal conductivity. These results agree well with the thermal conductivities predicted by the generalized Debye-Peierls/Allen-Feldman model in Fig. 6.8b. The thermal conductivity of defective carbon naonthreads decreases with increasing temperature due to the more intense phonon–phonon scattering. The contributions of different vibrational modes to thermal conductivity are also characterized in the inset of Fig. 6.8b. The LA and TW modes are found to be the predominant vibrational modes contributing to thermal conductivity. For the heat carriers’ diffusons and propagons, it is shown that the latter contribute significantly the thermal conductivity, while the former exhibit relatively small contribution. The negligible contribution of diffusons to thermal conductivity will lead to a steady value of κ throughout the full temperature regime. By convention, the κ will be suppressed at low temperatures before diffusons are activated. The suppression will be alleviated with the activation of diffusons at high temperature, which could help recover the κ and reduce the degree of suppression 10~100. However, due to the negligible contribution of diffusons in carbon nanothreads, the degree of suppression would be stable. Zhang et al. [21] performed NEMD simulations to calculate the thermal conductivities of carbon nanothreads. The interatomic interactions are described by the AIREBO potential. Different contents of SW defects are introduced into the sp3 -(3, 0) carbon nanothread, and the poly-benzene rings are connected by SW transformations, as seen in Fig. 6.9a. A DNT unit cell with n poly-benzene rings between two adjacent SW transformations is denoted by DNT-n. It is shown in Fig. 6.9b that increasing the number of poly-benzene rings (reducing SW defects) leads to higher stability. The temperature profile for DNT-55 with a length of ~24 nm was studied. An obvious temperature jump (δT ) is observed at the region with a SW defect, which is attributed to the interfacial thermal resistance induced by the SW transformation. The thermal conductivity of DNT-55 is predicted as ~35.6 ±4.7 W/mK, which is about half of the value of (3, 0) CNT (~63.1 ±6.3 W/mK) [21]. The vibration density of states (DOS) of the (3, 0) CNT, the poly-benzene rings, and the SW transformation region of DNT-8 were also studied. Compared to the CNT, the DNT-8 has a larger suppression of phonon modes at frequencies DNT-H/PMMA > DNT-Z/PMMA, which correspond well with the strengthening efficiency in Fig. 7.10a. Therefore, the DNT-C has the strongest mechanical interlocking effect with the PMMA matrix, while the DNT-Z has the worse relatively. In conclusion, changing the structural morphology of carbon nanothreads will affect the reinforcing efficiency by changing both the non-covalent interaction energy and mechanical interlocking effect. The load transfer capacity of DNT-C/PMMA interface and CNT/PMMA interface are characterized by performing the pull-out test, as shown in Fig. 7.12. Figure 7.13a shows the evolution of the non-covalent interaction energy between reinforcements and matrix as a function of pull-out displacement. It can be seen that the non-covalent

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Fig. 7.11 a Atomic structure of DNT-C-reinforced (atoms in the backbone are colored blue) PMMA composite; b Atomic structure of DNT-H-reinforced (atoms in the backbone are colored orange) PMMA composite; c Atomic structure of DNT-Z-reinforced (atoms in the backbone are colored pink) PMMA composite; d Effects of nanothread’s morphology on the non-covalent interaction energy of PMMA composites; e Effects of nanothread’s morphology integrated RDF. Reprinted (adapted) with permission from [25]

Fig. 7.12 Snapshots of atomic structures as a function of pull-out displacement. Reprinted (adapted) with permission from [25]

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Fig. 7.13 a Evolution of the non-covalent interaction energy between CNT/DNT-C and polymer matrix as a function of pull-out displacement. Two-dimensional density distribution of polymer atoms as a function of pull-out displacement with b CNT; c DNT-C; d Evolution of the noncovalent interaction energy between different nanothreads and polymer matrix. Reprinted (adapted) with permission from [25]

interaction energy increases with the increase of pull-out displacement and then saturates at a value when the carbon nanothread is fully pulled out (pull-out displacement > 50 Å). The total change of non-covalent interaction energy in the DNT-C/PMMA composite is around 147 kcal/mol. This value is slightly larger than the change of non-covalent interaction energy in CNT/PMMA composite. A slight increase of pull-out energy after pull-out is due to the relaxation of PMMA chains [20]. The movement of atoms as a function of pull-out displacement is tracked by plotting the density distribution analysis in Fig. 7.13b and c. It can be identified that there is a tracking effect between the DNT-C and polymer chains. Polymer chains adjacent to

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the DNT-C reinforcement are dragged to heal up the empty spaces created by the moving reinforcement, resulting in a decrease of potential energy and formation of shearing surface. The shearing surface has been explained by the locking effect of reinforcement [30]. Figure 7.13d shows the evolution of non-covalent interaction energy between different nanothreads and polymer matrix. The total change of non-covalent interaction energy of DNT-C/PMMA, DNT-H/PMMA, and DNT-Z/PMMA composites are 147 kcal/mol, 126 kcal/mol, and 31 kcal/mol, respectively, and the corresponding ISS can be obtained as 59 MPa, 53 MPa, and 13 MPa using Eq. (7.68). During the calculation, the diameters of DNT-H and DNT-Z are taken as 4.44 Å and 5.75 Å, respectively. The DNT-C which is an sp3 -(3,0) carbon nanothread has the largest load transfer capacity in PMMA composites. The trends of ISS correspond well with the strengthening efficiency of different carbon nanothreads in Fig. 7.10a.

7.3.3 Glass Transition Temperature of PMMA Composites The effects of carbon nanothreads on the glass transition temperature (T g ) of PMMA composites were investigated via MD simulation [31]. T g is an important factor reflecting the stability of the polymer matrix at a higher temperature. When temperature increases above T g , the PMMA composites will change from a glassy state to a rubbery state. A higher T g is a favorite for PMMA composites to ensure workability and mechanical properties. The COMPASS force field [22] is applied to describe the bonded and non-bonded interactions in PMMA and carbon nanothreads. Three kinds of carbon nanothreads in Fig. 7.5 are studied. A DNT reinforcement is built and located along the diagonal direction of a unit box. The atomic structures of PMMA matrix and carbon nanothreads reinforced PMMA composites are provided in Fig. 7.14. The T g of the polymer composites can be obtained by a stepwise cooling process. It should be noticed that the cooling rate will affect the value of T g . The density–temperature curves for the PMMA matrix and PMMA composites are shown in Fig. 7.15. The T g can be obtained as the intersection point of two fitted temperature–density lines. The glass transition phenomenon is reflected by the change of slope in density–temperature curves. The temperature–density line with a larger slope corresponds to the glassy state, and the line with a smaller slope is the rubbery state. A transition from a rubbery state to a glassy state is observed in the PMMA matrix as the temperature decreases. The calculated T g for pure PMMA matrix is about 424 K, which is a bit higher than that of 380 K reported in the experiment [32]. The lower T g in MD simulations is attributed to the presence of an external phase in experiments, resulting in the formation of polymer chain aggregates [33]. The calculated T g for PMMA composites is 458 K (DNT-C), 440 K (DNT-H), and 424 K (DNT-Z), respectively. The highest T g is obtained in the DNT-C/PMMA composites, which is 34 K higher than that of bulk PMMA. However, the DNT-Z is found to have limited effects on the T g of the PMMA composite.

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Fig. 7.14 Atomistic models for testing glass transition temperature of a PMMA matrix; b DNT/PMMA composite. Reprinted (adapted) with permission from [31]

The evolution of free volume in PMMA composites during glass transition is presented by the Connolly surface method [34]. The free volume is defined as the unoccupied space when a Connolly probe goes through the vdW surface of each atom. Figure 7.16 shows the distribution of free volume in PMMA matrix and DNTC/PMMA composite at temperatures of 600 K (rubbery state), temperatures around the T g value (glass transition state), and 280 K (glassy state). For the PMMA matrix, well-distributed cavities have already existed at the glassy state. When the structure transforms from a glassy state to a rubbery state, these cavities start to grow and coalesce, while the free volume increases with the increased temperature. The increase in free volume offers more empty positions for the directional and rotational motions of PMMA chains, leading to larger flexibility of PMMA chains at high temperatures. For DNT-C/PMMA composite, the free volume in DNT-C/PMMA composite is less than that in the PMMA matrix in the glassy state, which suggests that the DNT-C is capable to hinder the PMMA chains to move and rotate in the composites during thermal loading. However, cavities growth and coalescence occur to the interface at the rubbery state, resulting in interfacial debonding. The debonded DNTC reinforcement should lead to fewer constraints for the directional and rotational motions of PMMA chains, and thus causes degradation of non-covalent interactions and enhancement of flexibility of chains at high temperatures. The effects of temperature on Young’s modulus of PMMA matrix and DNTC/PMMA composite are analyzed in Fig. 7.17. Results show that Young’s modulus at a glassy state is always larger than that at a rubbery state. At a glassy state, there is a nearly negative linear relationship between Young’s modulus and temperature. At a rubbery state, the slope of reduction of Young’s modulus becomes smaller. In

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Fig. 7.15 a Density–temperature curves for PMMA matrix. Density–temperature curves for PMMA composites with addition of b DNT-C; c DNT-H; d DNT-Z. The values of T g are provided in the figure. Reprinted (adapted) with permission from [31]

Fig. 7.16 Distribution of free volumes (marked in blue regions) in PMMA matrix at temperatures of a 280 K; b 400 K (near T g value); c 420 K (near T g value); d 600 K; Distribution of free volumes in PMMA composite with the addition of DNT-C at temperatures of e 280 K; f 440 K (near T g value); g 460 K (near T g value); h 600 K. Reprinted (adapted) with permission from [31]

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Fig. 7.17 Effects of temperature on Young’s modulus of a PMMA matrix; b DNT/PMMA composite. Reprinted (adapted) with permission from [31]

comparison with the PMMA matrix, a larger drop of Young’s modulus is witnessed in the DNT-C PMMA composite during glass transition. The enhancement of DNTC on Young’s modulus is insignificant at a rubbery state, which is attributed to the degradation of interfacial interaction, which is explained by the degradation of noncovalent interaction at the interface. Based on the findings from Figs. 7.16 and 7.17, conclusions can be made that the degradation of non-covalent interfacial interaction is the predominant reason leading to the lower reinforcing efficiency of reinforcement in polymer composite at elevated temperatures. It is envisioned that improving the interfacial interaction of reinforcement by adding functional groups and cross-links will be useful to maintain the mechanical properties of polymer composites during glass transition. Figure 7.18a shows the change of non-covalent energy in PMMA composite with increasing temperature. After fitting the non-covalent energy with two linear lines, it is shown that there is a breaking point at 440 K, which is close to that of 458 K for the DNT-C/PMMA composite. This finding suggests that the noncovalent interactions play critical roles in affecting the mobility of PMMA chains during glass transition. The evolution of non-bond energy in different components PMMA matrix and interfacial region are further shown in Fig. 7.18b, c. The non-bond energy E I in the interfacial region is calculated as E I = E com − E matri x − E D N T .

(7.72)

Here, E com is the total potential energy of PMMA composite, E matri x is the individual potential energy of the PMMA matrix, and E D N T is the individual potential energy of DNT-C reinforcement. Similarly, two breaking points of 438 K and 450 K are observed for the matrix and interface components, respectively. The 438 K can be considered as the T g contributed by the PMMA chains which have no interactions with the DNT-C, and the 450 K can be considered as the T g contributed by chains

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Fig. 7.18 Non-covalent energy–temperature curves for a DNT-C/PMMA composite; and its b matrix component; c Interface component. Reprinted (adapted) with permission from [31]

adjacent to the reinforcement. Therefore, the chain mobility in a bulk matrix is faster than that in the interfacial region. The effects of different carbon nanothreads on the T g can be explained by the changes of non-covalent interfacial interactions [25]. The non-covalent interfacial energy between DNT-C and PMMA is the highest, which can best inhibit the mobility of the PMMA chains, contributing to the best improvement of T g of DNT/PMMA composite. The effects of cross-link density on the T g have also been explored. The CH2 CH2 is introduced as a cross-linker between the backbone of DNT-C and PMMA chains. The density for the PMMA composites with various percentages of cross-links at different temperatures is shown in Fig. 7.19. Here, the 6% cross-linkers of DNT means 6% of carbon atoms in the DNT are connected to the PMMA chains. Results show that with the increase of cross-link density, the T g increases from 458 to 496 K. The T g of DNT-C/PMMA composite with 24% cross-linkers density is ~70 K higher than that of bulk PMMA. The distribution of free volume in DNT-C/PMMA composite by adding 24% cross-linkers density at various temperatures is shown in Fig. 7.20. Compared to

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Fig. 7.19 Density–temperature curves for PMMA composites reinforced with carbon nanothread by adding a 6% cross-linkers; b 12% cross-linkers; c 18% cross-linkers; d 24% cross-linkers. Reprinted (adapted) with permission from [31]

Fig. 7.20 Distribution of free volume in PMMA composites reinforced with carbon nanothread by adding 24% cross-linkers at temperatures of a 280 K; b 480 K (near T g value); c 500 K (near T g value); d 600 K. Reprinted (adapted) with permission from [31]

Fig. 7.16b, it can be found that adding cross-linkers can impede the growth of cavities at the interface, and the interfacial debonding does not occur at high temperatures. Figure 7.21 compares the mean square displacement (MSD) g(t) of oxygen atoms on PMMA chains in PMMA composites. Two temperatures 300 and 600 K are studied. The g(t) is calculated as g(t) = [ri (t) − ri (0)]2 ,

(7.73)

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Fig. 7.21 MSD curves of oxygen atoms on PMMA chains in PMMA composites with variations in temperature and cross-linker. Reprinted (adapted) with permission from [31]

where ri (t) are the location of atom i at time t. The bracket implies the ensemble average by averaging over all trajectories. Results show slower displacements of oxygen atoms in the polymer composite with cross-linkers, which indicates interfacial cross-linking is efficient to restrain the movement of polymer chains. The chain constraints are explained by the isotropic mobility of chains at the interface. For DNT-C/PMMA composite without crosslink, the PMMA chains are restrained in the transverse direction of reinforcement, and they are easier to move along the longitudinal direction of reinforcement, which is called anisotropic mobility. For DNT-C/PMMA composite with cross-link, the interfacial cross-links can also provide constraints in the longitudinal direction of reinforcement, which is called isotropic mobility. Therefore, increasing the amounts of cross-linker shall further enhance the T g of DNT-C/PMMA composite. Note that higher cross-link density will not be practical due to the severe entanglement of PMMA chains near the DNT-C. This brings difficulty in connecting the DNT-C reinforcement and polymer chains. The DNT/PMMA composites are compared with other reported PMMA composites regarding the T g in Fig. 7.22. It can be concluded that carbon nanothread with cross-linkers has the highest efficiency in improving the T g of PMMA composite.

7.3.4 Tensile Properties of Epoxy Composites The influences of spatial distribution and interfacial cross-link on the mechanical properties of carbon nanothreads-reinforced epoxy composite (DNTs/epoxy composite) were investigated via MD simulation [35]. The consistent valence force field (CVFF) is applied to express the covalent and non-covalent interactions in DNTs/epoxy composite. The DNTs/epoxy composite composes sp3 -(3,0) carbon nanothread, EPON-862, and curing agent DETDA. Regarding the interfacial crosslink, the ends of the carbon nanothread are connected with six DETDA molecules. For

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Fig. 7.22 Comparisons of PMMA composites by adding different nanofillers regarding the T g . Reprinted (adapted) with permission from [31]

epoxy nanocomposites, two kinds of dispersion of carbon nanothreads are considered, including uniform dispersion and aggregation. Epoxy composite where n carbon nanothreads are uniformly dispersed in the epoxy is denoted as Dn -DNTs/epoxy composite. Epoxy composite where n carbon nanothreads are aggregated in the epoxy is denoted as An-DNTs/epoxy composite. All the bulk epoxy and epoxy composites have the same cross-linking degree of 63%. The stress–strain curves of bulk epoxy and epoxy composites with various nanothread’s dispersion under tension are shown in Fig. 7.23. Results show that the stress level of epoxy composites is larger than that of bulk epoxy, indicating that adding carbon nanothreads effectively enhances the mechanical properties of epoxy. The excellent mechanical performance of the epoxy composite is attributed to the

Fig. 7.23 Tensile stress–strain curves of bulk epoxy and epoxy nanocomposites with various nanothread’s dispersion. Reprinted (adapted) with permission from [35]

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outstanding elastic modulus of carbon nanothread, as well as the vdW interactions and mechanical interlocking at the interface. The elastic modulus of epoxy composites is larger than that of bulk epoxy. The elastic modulus of A7-DNTs/epoxy, which has a mass fraction of carbon nanothread of 6.966%, is 8.73 GPa, which is 33% larger than that of 6.56 GPa of bulk cross-linked epoxy. With the increase of mass ratio, the elastic modulus of epoxy composites increases gradually. The elastic modulus of A7-DNTs/epoxy increases from 6.643 to 8.488 GPa when the mass fraction of carbon nanothread increases from 3.109 to 6.966%. The A-DNTs/epoxy composites have a larger elastic modulus than the D-DNTs/epoxy composites. Generally, the aggregation will reduce the surface area of the nanofillers, leading to the degradation of mechanical properties of the nanocomposite. It is found that when the cross-linking degree is beyond 60%, a large molecule will be formed quickly to build up the three-dimensional network, contributing to the enhancement of mechanical properties, such as elastic modulus and yield stress [36]. However, the uniformly dispersed DNT-C reinforcements act as a barrier to impede the formation of a three-dimensional network and thus result in a degradation of the mechanical properties. On the contrary, the aggregated DNT-C reinforcements have a lower degree of impedance to the buildup of a large molecule. The structural morphology of carbon nanothreads in the epoxy composites is provided in Fig. 7.24. Results show that the well-dispersed reinforcements have a curved appearance because of the low bending stiffness of individual ones, which is disadvantageous to the load transfer between carbon nanothread and epoxy chains. The aggregated carbon nanothreads maintain a straight appearance due to the relatively strong bending stiffness of the bundle, allowing better load-transfer efficiency (Fig. 7.25).

Fig. 7.24 Atomic structures of a D5-DNTs/epoxy composite, and b A5-DNTs/epoxy composite without deformation. Reprinted (adapted) with permission from [35]

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Fig. 7.25 Tensile stress–strain curves of bulk epoxy and epoxy nanocomposites with unreactive and reactive functional groups. Reprinted (adapted) with permission from [35]

7.3.5 Tensile Properties of PE Composites with Curved Carbon Nanothreads The influences of loading environment on the mechanical properties of carbon nanothreads-reinforced (NTHs) PE composite were investigated via MD simulation [37]. In MD simulations, adaptive intermolecular reactive empirical bond order (AIREBO) potential [38] is employed to express the interactions in carbon nanothreads. The PCFF [21] is employed to describe the bonded interactions in PE. The non-bonded interactions are expressed by the COMPASS force field [22]. Three kinds of carbon nanothreads are considered. The atomic structures of three kinds of carbon nanothreads are provided in Fig. 7.26a. The sp3 -(3,0) carbon nanothread is denoted as NTU-1, the carbon nanothread which has a nomenclature of 143,652 (see Fig. 3.11) is denoted as NTU-2, and the carbon nanothread which has a nomenclature of 134,562 (see Fig. 3.11) is denoted as NTU-3. Two kinds of composite models are considered. As seen in Fig. 7.26b, model 1 has a single nanothread whose length is the same as the unit cell, and model 2 has a single nanothread with a shorter length. The stress–strain curves of PE matrix and PE composites under tension are shown in Fig. 7.27a. Different mechanical behaviors for the PE matrix and PE composites (model 1) can be found. In terms of the PE matrix, the stress increases gradually with the increased strain and saturates at a plateau value. For the PE composite, the stress first rises to a higher level and then drops down suddenly, and finally saturates at the same plateau value with the PE matrix. The maximum stress of the PE matrix is ~152 MPa, smaller than that of PE composites. The maximum stress is 549 MPa for the NTH-1/PE composite, 582 MPa for the NTH-7/PE composite, and

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Fig. 7.26 Atomic structures of the PE nanocomposite. a Atomic structures of three carbon nanothreads viewed along the axial and radial directions; b Atomic structures of two nanocomposite models with a variation in carbon nanothread’s length viewed along the cross-section. The corresponding atomic potentials are shown. Reprinted (adapted) with permission from [37]. Copyright {2021} American Chemical Society

Fig. 7.27 Tensile properties of bulk PE and PE composites with individual carbon nanothread. a Stress–strain curves of bulk PE and PE composites deformed along the length direction of the carbon nanothread. M1 and M2 denote the model 1 and model 2; b Strain energy–strain curves of the total composite (model 1), and structural components polymer and NTH-1; b Strain energy–strain curves of the total composite (model 2), and structural components polymer and NTH-1. Reprinted (adapted) with permission from [37]. Copyright {2021} American Chemical Society

389 MPa for the NTH-13/PE nanocomposite. In addition to the maximum stress, the PE composites also have a higher Young’s modulus. The Young’s modulus is 7.66, 6.57, and 3.85 GPa for NTH-1/PE, NTH-7/PE, and NTH-13/PE composites, respectively, which is much larger than that of 2.5 GPa of PE matrix. However, for the PE composite in model 2, a similar mechanical response with the PE matrix is identified, which is explained by the insignificant load transfer between the shorter carbon nanothread and the PE matrix. In other words, shorter carbon nanothread cannot coordinate the tensile deformation of the PE matrix.

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Previous experimental results at room temperature have reported a range of 0.5– 1.5 GPa in Young’s modulus [39], and 25.2–106 MPa in maximum stress [40–42], which are lower than the simulation results. This is attributed to the lower temperature (100 K) studied in the MD simulations. At room temperature, the PE matrix is at a rubbery state, and hence lower mechanical properties are expected. It has been examined that similar mechanical properties can be obtained by increasing the temperature to 300 K [37]. The strain energy evolution of different components in NTH-1/PE composite (model 1) in Fig. 7.27b indicates that the NTH-1 reinforcement absorbs more strain energy than the PE matrix during deformation. The strain energy per atom in NTH-1 reinforcement at 8% strain is about 1.6786 kcal/mol, which is about 100 times the strain energy per atom in the PE matrix. For the strain energy evolution in NTH-1/PE composite (model 2), Fig. 7.27b shows negligible absorption of strain energy in the NTH-1 reinforcement, confirming that shorter carbon nanothread cannot deform with PE matrix simultaneously. The atomic structures of NTH-1 reinforcement in models 1 and 2 at various tensile strains are provided in Fig. 7.28a, b, respectively. For model 1 structure, failure occurs to the reinforcement at a strain of 10%, which corresponds to the sudden drop of stress–strain curve in Fig. 7.28a. For model 2 structure, significant deformation is not observed in the reinforcement with increased strain, demonstrating that the reinforcement does not cooperate with the deformation of the PE matrix. The evolution of interfacial interaction energy E int in Fig. 7.28c shows that the value of E int will decrease if the NTH-1 reinforcement cannot deform with PE matrix simultaneously, and the value will maintain if simultaneous deformation between

Fig. 7.28 Models of the PE composites with carbon nanothread NTH-1. a The carbon nanothread is periodic along the tensile direction, and snapshots of simultaneous deformation with unit cell and fracture are shown; b The carbon nanothread is non-periodic along the tensile direction, and snapshots of carbon nanothread with increasing strain are shown; c Evolution of non-covalent interaction energy with strain; d Integrated RDF values of composite with a variation of strain. Reprinted (adapted) with permission from [37]. Copyright {2021} American Chemical Society

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reinforcement and PE matrix occurs. A similar phenomenon is also found in the evolution of integrated RDF value g(r) in Fig. 7.28d. The mechanical properties of the PE matrix and PE composites (model 2) reinforced with four evenly distributed and vertically aligned carbon nanothreads are shown in Fig. 7.29. The representative atomic structures of NTH-1/PE composite are provided in Fig. 7.29a. The RDF curves in Fig. 7.29b show that there are only weak vdW interactions between carbon nanothread and PE matrix, as evidenced by the absence of peak value in the profile. The stress–strain curves of PE matrix and PE composites under tension along the z-axis are plotted in Fig. 7.29c. The Young’s modulus is 3.15 GPa for the NTH1/PE composite, 3.0 GPa for the NTH-7/PE composite, and 3.52 GPa for the NTH13/PE nanocomposite. The maximum stress is 175, 170, and 196 GPa for NTH1/PE, NTH-7/PE, and NTH-13/PE composites, respectively. The NTH-13 has the best reinforcing efficiency in both Young’s modulus and maximum stress, which is attributed to the strongest mechanical interlocking brought by its spiral morphology. The stress–strain curves of PE matrix and PE composites under tension along the lateral direction are plotted in Fig. 7.29d. The Young’s modulus is 3.07 GPa for the NTH-1/PE composite, 2.98 GPa for the NTH-7/PE composite, and 2.74 GPa for the NTH-13/PE nanocomposite. The maximum stress is 184, 176, and 168 GPa for NTH-1/PE, NTH-7/PE, and NTH-13/PE composites, respectively. The NTH-1 has the best reinforcing efficiency in both Young’s modulus and ultimate stress.

Fig. 7.29 a Models of the PE composites with four carbon nanothreads aligned in the matrix. b Normalized RDF curves for PE composites with different carbon nanothreads; c Stress–strain curves of bulk PE and PE composites deformed along the length direction of the carbon nanothread; d Stress–strain curves of bulk PE and PE composites deformed along the lateral direction of the carbon nanothread. Reprinted (adapted) with permission from [37]. Copyright {2021} American Chemical Society

7.3 Mechanical Properties of Polymer Nanocomposites

235

The influences of functionalization on the mechanical properties of NTH-1/PE composite (model 1) under tension are shown in Fig. 7.30. Functional groups –CH3 are introduced to the NTH-1 reinforcement by random replacement of the hydrogen atoms. Figure 7.30a and b show that functionalization will degrade the mechanical properties of NTH-1/PE composite. The mechanical deterioration is attributed to the additional cavities introduced at the interface, which lowers the interfacial interaction energy as well as load transfer capacity. The worse interfacial interaction of functionalized NTH-1/PE composite is demonstrated by the RDF curves in Fig. 7.30c, and the RDF curves for the composite with functionalized nanothread are lower than those without functionalization. The stress–strain curves of PE matrix and PE composites under compression are shown in Fig. 7.31. The compressive modulus is calculated as 7.85, 6.14, and 3.17 Fig. 7.30 a Stress–strain curves of bulk PE and PE composites with functionalized carbon nanothread deformed along the a length direction (z-axis) and b lateral direction (x-axis). Reprinted (adapted) with permission from [37]. Copyright {2021} American Chemical Society

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Fig. 7.31 Compressive properties of bulk PE and PE composites with individual carbon nanothread. a Stress–strain curves of bulk PE and PE composites deformed along the length direction of the carbon nanothread. M1 and M2 denote model 1 and model 2; b Stress–strain curves of bulk PE and PE composites deformed along the lateral direction; c Strain energy–strain curves of the total composite (Model 2), and structural components of polymer and NTH-1. Reprinted (adapted) with permission from [37]. Copyright {2021} American Chemical Society

GPa for NTH-1/PE, NTH-7/PE, and NTH-13/PE composites, respectively, which is much larger than that of 2.23 GPa of PE matrix. The carbon nanothreads can absorb a large amount of strain energy during compression in model 1. However, for the PE composite in model 2, the mechanical improvement is smaller. Similarly, the maximum compressive stress of PE composites is much larger than that of PE matrix, and the reinforcing efficiency of carbon nanothread becomes lower in model 2 due to its inconsiderable adsorption of strain energy during compression. The atomic structures of NTH-1 reinforcement in models 1 and 2 at various compressive strains are provided in Fig. 7.32a and b, respectively. For model 1 structure, buckling occurs to the reinforcement at a compressive strain of −5%, and failure occurs to the reinforcement at a strain of −19%, causing a big fluctuation of potential energy E NTH-1 in Fig. 7.32c). For model 2 structure, buckling is not observed in the reinforcement with increased compressive strain, demonstrating that the reinforcement does not cooperate with the deformation of the PE matrix. The evolution of interfacial interaction energy E int in Fig. 7.32c shows that the value of E int will decrease if the NTH-1 reinforcement experiences buckling, while the value will maintain if the buckling phenomenon does not occur. The evolution of integrated RDF value g(r) in Fig. 7.32d shows a sudden drop when the carbon nanothread starts to buckle, indicating that some PE chains are losing intimate contacts with nanothread at the interface, thereby leading to a reduction of E int . A conclusion can be drawn that buckling is an important index in identifying the simultaneous deformation of carbon nanothread with PE matrix. The effects of functionalized carbon nanothread on the mechanical properties of PE composite (model 1) under compression are shown in Fig. 7.33. It is shown that functionalization does not affect significantly the mechanical properties of the NTH1/PE composite. Only a slight improvement of mechanical properties is observed for the compression along the length direction, while the stress–strain curves are almost consistent for the compression along the lateral direction.

7.3 Mechanical Properties of Polymer Nanocomposites

237

Fig. 7.32 a The carbon nanothread is periodic along the tensile direction, snapshots of simultaneous deformation with unit cell and buckling are shown; b The carbon nanothread is non-periodic along the tensile direction, snapshots of carbon nanothread with increasing strain are shown; c Evolution of non-covalent interaction energy with strain; d Integrated RDF values of composite with a variation of strain. Reprinted (adapted) with permission from [37]. Copyright {2021} American Chemical Society

Fig. 7.33 Stress–strain curves of bulk PE and PE composites with pristine carbon nanothread compressed along the a length direction (z-axis) and b lateral direction (x-axis). Stress–strain curves of bulk PE and PE composites with functionalized carbon nanothread compressed along the c length direction (z-axis) and d lateral direction (x-axis). Reprinted (adapted) with permission from [37]. Copyright {2021} American Chemical Society

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7.4 Thermal Properties of Polymer Nanocomposites The effects of carbon nanothread on the thermal conductivity of PE composite were investigated via MD simulation [43]. In MD simulations, the AIREBO potential [38] is applied to characterize the interactions in carbon nanothread. The PCFF [21] is applied to describe the interactions in PE and interfacial interactions between PE and carbon nanothread. The supercell of PE composite is composed of vertically aligned carbon nanothreads and PE chains. The carbon nanothread can be considered as a hydrogenated (3,0) carbon nanotube with a single Stone–Wales (SW) defect and is denoted by NTH. The thermal conductivity κ A is calculated using Eq. 6.1. The thermal conductivity of PE matrix is calculated as 0.04 W m−1 K−1 . The effects of volume fraction of carbon nanothreads on the thermal conductivity are analyzed in Fig. 7.34. Figure 7.34a provides the temperature profile of the PE composite with two carbon nanothreads evenly distributed in a matrix at the nonequilibrium stable state. The linear shape of the profile indicates that a constant heat flux has been created between the heat bath and cold bath. The thermal conductivities of the PE composites with various volume fractions of carbon nanothreads are shown in Fig. 7.34b. The thermal conductivity tends to increase linearly with the volume fraction. At a volume fraction of 23%, the calculated thermal conductivity of PE composite is 1.88 W m−1 K−1 , which is more than three Fig. 7.34 a Temperature-location profile (black dots) of PE composite with the addition of a single carbon nanothread. The data are plotted by a red solid line to find the temperature gradient. b Effects of volume fraction on the thermal conductivity (orange dots) of the PE nanocomposite. The data are plotted by a blue solid line using the Nan model [44]. Reprinted (adapted) with permission from [43]

7.4 Thermal Properties of Polymer Nanocomposites

239

times that of the PE matrix. The linear-like relationship can be explained by the Nan model [44]: κf

 f γ κp κe =1+ , κp 3 γ+P

(7.74)

where κe ,κ f , and κ p are the effective conductivity of PE composite, carbon nanothread, and PE matrix, respectively,  f and γ are the volume fraction and aspect ratio of the carbon nanothread, respectively, P can be expressed in terms of interfacial thermal resistance Rk : P=

2Rk κ p κ f . d κp

(7.75)

Here d is the diameter of carbon nanothread. According to the linear-like relationship, the effective thermal conductivity of PE composite with 100% volume fraction of carbon nanothread, namely, carbon nanothread bundle structure without PE matrix, can be obtained as 6.88 W m−1 K−1 . The effects of spatial distribution of carbon nanothread on the thermal conductivity of PE composite are analyzed in Fig. 7.35. The volume fraction of carbon nanothread is consistent at 17%, and six distribution patterns are provided in Fig. 7.35a. To

Fig. 7.35 a Models for the PE composites with carbon nanothreads aligned in different manners; b Effects of relative contact area on the thermal conductivity of PE composites; c Atomistic structure of a bundle of carbon nanothreads viewed along the axial and radial directions; d A CNT with chirality of (10,10) encasing a carbon nanothread; e Effects of interfacial interaction between CNT wall and carbon nanothread on the accumulated average heat transfer of carbon nanothread. The symbols H and C indicate the H and C virtual cylindrical walls, and O1 and O2 indicate two oxygen atoms in the carboxy group as shown in the inset. Reprinted (adapted) with permission from [43]

240 Table 7.2 Detail parameters in the 9-6 Lennard-Jones potential to modify the vdW interactions between the CNT and the enclosed carbon nanothread. Reprinted (adapted) with permission from [43]

7 Carbon Nanothreads-Reinforced Polymer Nanocomposites Type

ε (kcal mol−1 )

σ (Å)

H

0.020

2.995

C

0.054

4.010

a

0.240

3.420

O2 a

0.267

3.300

O1

a

a

O1 and O2 represent the two oxygen atoms in the carboxy group, see the inset of Fig. 7.35e.

characterize the aggregation degree of carbon nanothread, a relative contact area is defined as ηS = S i /S 0 , where S 0 is the total contact area in distribution pattern I in Fig. 7.35a and S i is the total contact area in distribution pattern i. Figure 7.35b shows that the relative contact area decreases from pattern I to VI, and the thermal conductivity decreases accordingly. This finding is explained by the change of interfacial interaction energy E int with the relative contact area. The E int of the interface in PE composite will be smaller than that between adjacent carbon nanothreads. Example is given to a bundle of seven carbon nanothreads with a length of ~24 nm in Fig. 7.35c. The interfacial interaction energy is given as E int = E mp − E 0p , where E 0p and E mp are the potential energy of the initial bundle structure and final structure when the middle nanothread is completely pulled out. The E int of the bundle structure is larger than that of the interface in PE composite. The effects of interfacial interaction on the thermal conductivity are examined in a model where an individual carbon nanothread is enclosed by a (10,10) carbon nanotube (CNT), as seen in Fig. 7.35d. Different degrees of interfacial interaction are created by modifying the parameters in the 9-6 Lennard-Jones potential. The detailed parameters are provided in Table 7.2. The degree of interfacial interaction follows the order of type H > C > O1 > O2 based on the value of ε in the LennardJones potential. Figure 7.35e shows that the accumulated average heat flux (J ave ) in the carbon nanothread from the heat bath to the cold bath decreases with the increased interfacial interactions, indicating that a stronger interfacial interaction between PE matrix and carbon nanothread will suppress the enhancing efficiency of carbon nanothread in thermal conductivity. The effects of functional group (–CH2 CH3 ) on the thermal conductivity of PE composite are analyzed in Fig. 7.36. The percentage of functional group is defined as the ratio between the number of functional groups and the initial number of hydrogen atoms. Results show that the thermal conductivity decreases with increased content of the functional group, so is the average heat flux. The degradation of thermal conductivity is attributed to the additional cavities created by the functional group, weakening the interfacial interaction between functionalized carbon nanothread and PE matrix. The influences of spatial dispersion on the thermal conductivity of PE composite are analyzed in Fig. 7.37. The thermal conductivity is not significantly affected by the volume fraction of randomly distributed carbon nanothread. Compared with the uniformly distributed case, a reduction of thermal conductivity is observed for the

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241

Fig. 7.36 a Model of the PE composite with three aligned carbon nanothreads functionalized by ethyl groups. b Effects of functionalization percentage on the thermal conductivity of PE composites. Values of thermal conductivity are expressed in a boxplot style, with the median values expressed by circle points, and the 25th and 75th percentiles expressed by the bottom and top edges of the box. The symbol “+” indicates the unconsidered outliers. Reprinted (adapted) with permission from [43]

Fig. 7.37 a Model of the PE composite with carbon nanothreads randomly distributed in the matrix; b Effects of volume fraction on the thermal conductivity of the PE nanocomposite. The orange dots indicate the cases of random distribution, and the red dots indicate the vertically aligned cases. Reprinted (adapted) with permission from [43]

randomly distributed case due to the less effective heat transfer channel. Besides, the randomly distributed carbon nanothreads could introduce additional interfacial thermal resistance within the nanothreads, which suppressed the enhancing efficiency of carbon nanothread in thermal conductivity.

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7.5 Coarse-Grained Study of Polymer Nanocomposites The effects of carbon nanothread on the failure behaviors of PMMA composite at mesoscale were investigated via CG simulation [45]. In CG simulations, a structurebased approach is adopted to derive the potential parameters from the all-atom model. The interatomic interaction of the carbon nanothread and PMMA in the all-atom model are described by the COMPASS force field [22]. Figure 7.38a shows the influences of weight ratio (wt.%) on the mechanical properties and failure behaviors of PMMA composite under tension. An obvious improvement on Young’s modulus and maximum stress is observed with the addition of carbon nanothread. It is shown in Fig. 7.38b that cavitation occurs inside the PMMA composite when the weight ratio is 0.5%. Further increase of weight ratio to 5% leads to crazing phenomenon inside the PMMA composite. The cavitation here refers to the formation of growth of cavities which are identified by zero density in the 3D density map. The crazing refers to forming fibrils that are observed from the configuration in the crack region. Figure 7.39 shows the cavity evolution and deformation pattern during deformation. It is shown that interfacial debonding is a crucial factor influencing the resistance of cavitation. As seen in Fig. 7.39a, interfacial debonding occurs before

Fig. 7.38 Effects of carbon nanothread concentration on the tensile properties of PMMA composites. a Stress–strain curves of PMMA composites with the addition of different weight ratios of carbon nanothread. Inset shows the mechanical responses at small strains. b Snapshots of the formation process of cavities and fibrils in PMMA composites with 0.5 and 5 wt% carbon nanothreads. Reprinted (adapted) with permission from [45]

7.5 Coarse-Grained Study of Polymer Nanocomposites

243

Fig. 7.39 a Three-dimensional density distribution of PMMA atoms in composite prior to forming cavities. The slippage of carbon nanothread inside the composite can be identified by the green stripes. Initial damage occurs to the interface by debonding of carbon nanothread; b Evolution of cavity inside the composite with a variation in the weight ratio of carbon nanothread; c Snapshots of bridging and network-bridging in PMMA composites with 0.5 and 5% carbon nanothreads. d Snapshots of crazes formed in PMMA composites during deformation. The corresponding atomistic structure of crazes is shown in the inset. The Connolly surfaces at the carbon nanothreadPMMA interface are plotted in green to show the non-covalent interactions. Reprinted (adapted) with permission from [45]

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the formation of cavities inside the PMMA matrix. During the debonding process, a large amount of energy is dissipated through vdW interaction and mechanical interlock, which delay the absorption of strain energy in the transformation site with low local modulus, resulting in a better cavitation resistance and larger yield strain. With the increased amount of carbon nanothread, an interlocking network will also be formed, and the interfacial debonding will be more intensive. The interlocking network is responsible for the rapid nucleation of cavities to spatial homogenization of cavities, as seen in Fig. 7.39b, c. For PMMA composite without forming an interlocking network of carbon nanothread (0.5 wt.%), cavities quickly form and grow in the vulnerable areas with low local modulus, and crack propagation occurs because of the enlargement and fusion of cavities. In contrast, the formation and growth of the cavity are uniformly confined in the network and impeded from fusion in PMMA composite with an interlocking network (5 wt.%). The confinement of cavitation is due to the presence of carbon nanothreads which act as barriers and force the cavities to develop in a curved path, thereby allowing better blocking of cavity coalescence. During the debonding process, fibrils will be formed on the surface of the carbon nanothread due to the ultrahigh-specific surface area of the latter. The formation of nanofibrils is shown in Fig. 39d. After reaching the yield strain, the interlocking network near the cavitation areas unlock slowly and transform into bridges. As the strain increases, the carbon nanothreads will accommodate the deformation of the PMMA matrix, and they can orientate toward the tensile direction while absorbing PMMA chains on the surface. Finally, the PMMA chains on the carbon nanothreads dissociate during debonding and turn into long fibrils, resulting in a crazing phenomenon. Generally, the crazing is strongly affected by the length and flexibility of polymer chains. The failure behavior will transform from cavitation to crazing if the length of polymer chains is longer than the critical entanglement length. In the current work, the number of monomers in the PMMA chain is 10, and the length should be shorter than the critical entanglement length. Therefore, crazing is not witnessed in the bulk PMMA. The influences of strain rate on the mechanical properties of PMMA composite with 5 wt.% carbon nanothread are shown in Fig. 7.40a, b. A strain rate sensitivity is observed for the PMMA matrix and PMMA composite, and Young’s modulus and maximum stress and residual stress increase with the increased strain rate. The concentration of PMMA atoms as a function of coordinate along the tensile direction is plotted in Fig. 7.40c, d. At the strain rate of 0.00005/ps, no PMMA atoms are detected in the cavitation region at a strain of 0.45, indicating cracks have propagated through the cross-section completely. At the strain rate of 0.001/ps, a small amount of PMMA atoms are identified in the cavitation region at a strain of 0.45, implying the existence of fibrils across the fracture plane. These findings demonstrate that increasing strain rate is efficient to promote the occurrence of crazing phenomenon. The rate-dependent crazing can be explained by the various degrees of disentanglement. Short PMMA chains with lower rigidity in the cavitation region can slide on the surface of carbon nanothread, leading to disentanglement and a larger cavity volume. The chain disentanglement is unfavorable for a steady and successive growth of fibrils. It is shown in Fig. 7.41a that the nucleation rate of cavity (slope of

7.5 Coarse-Grained Study of Polymer Nanocomposites

245

Fig. 7.40 a Tensile stress–strain curves of PMMA matrix at different strain rates; b Tensile stress– strain curves of PMMA composite with 5 wt.% carbon nanothreads at different strain rates. Concentration profile of polymer atoms along the tensile direction at the strain rate of c 0.00005; d 0.001. Reprinted (adapted) with permission from [45]

the curve) is 30% slower at the strain rate of 0.001/ps than that at 0.00005/ps. The cavity volume decreases with increasing strain rate, indicating a slower disentanglement. In Fig. 7.41b, the cavity distribution in PMMA composite also demonstrates that at higher strain rates cracks have more preferential initiating locations, which can delay the fusion of cavities. Besides, it is shown in Fig. 7.41d that local alignment of carbon nanothread is more preferred at higher strain rates due to the simultaneous deformation of carbon nanothread with PMMA matrix, which is beneficial for the absorption of the PMMA chains and forming fibrils. The effects of spatial distribution on the mechanical properties of PMMA composite with 5 wt.% carbon nanothread are shown in Figs. 7.42. The spatial

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7 Carbon Nanothreads-Reinforced Polymer Nanocomposites

Fig. 7.41 a Evolution of volume of cavity as a function of strain rate. The total deformation is divided into three stages, including the interfacial damage, cavitation, and crazing stage. At the interfacial damage stage, only debonding of carbon nanothread occurs. The cavitation occurs at the cavitation stage, and the volume increases nonlinearly. The crazing occurs at the crazing stage, and the volume increases linearly. b Distribution of cavities inside the composites with 5 wt.% carbon nanothread. The gray areas represent the positions of the cavity. c Snapshots of crazing phenomenon at different strain rates, and the crazes are observed in green zones. d Reorientation of carbon nanothreads at different strain rates. Carbon nanothreads are more capable to reorient at higher strain rates. Reprinted (adapted) with permission from [45]

distribution of carbon nanothread is shown in Figs. 7.42a. In addition to the random distribution, there are also partial agglomeration, agglomeration, and full agglomeration, and the agglomeration degree increases gradually. The effects of agglomeration degree on the tensile mechanical properties are shown in Figs. 7.42b. A lower residual stress is obtained at lower agglomeration degree, as a result of the heterogeneity in the composite and stress concentration caused by the agglomerated carbon nanothreads [46]. The concentration of PMMA atoms as a function of coordinate along the tensile direction is plotted in Figs. 7.42c and d. At full agglomeration condition, no PMMA

7.5 Coarse-Grained Study of Polymer Nanocomposites

247

Fig. 7.42 a Model of the PMMA composite with carbon nanothreads distributed in different agglomeration degrees; b Tensile stress–strain curves of PMMA composites with carbon nanothreads distributed in different agglomeration degrees. Concentration profile of polymer atoms along the tensile direction at c full agglomeration case; d random distribution case. Reprinted (adapted) with permission from [45]

atoms are detected in the cavitation region at a strain of 0.45, indicating cracks have propagated through the cross-section completely. At random distribution condition, a small amount of PMMA atoms are identified in the cavitation region at a strain of 0.45, implying the existence of fibrils. These findings indicate that agglomeration is disadvantageous for the formation of fibrils.

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7 Carbon Nanothreads-Reinforced Polymer Nanocomposites

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Chapter 8

Arrangements of Carbon Nanothreads

8.1 Introduction Nowadays, the bottom-up strategy is receiving extensive research interest in the domain of science. For example, the properties of a macro-system can be designed or tuned by controlling the size and arrangement of the integrated nanoscale components. Carbon nanothreads are promising building blocks for microelectromechanical devices due to their excellent mechanical, thermal, and electrical properties. However, due to the single-point defect during pressure-induced polymerization, it is challenging to fabricate a high-dimensional carbon nanothread framework with long-range order. Recently, both experimental and theoretical studies found that structural arrangement and expanding the precursor library offer new avenues to achieve this goal. There is some high-dimensional carbon nanothreads framework with long-range order such as carbon nanothreads rods, carbon nanothreads forests, carbon nanothreads nanomeshes and nanoforms, and carbon nanothreads cubanes. Carbon nanothreads rods are one-dimensional structures with covalent bonding of polycyclic aromatic hydrocarbon molecules. Carbon nanothreads forests can be regarded as parallel arrangements of individual carbon nanothread. Carbon nanothreads nanomeshes and nanoforms are two-dimensional and three-dimensional structures with covalent interconnections of aligned threads, respectively. Carbon nanothreads cubanes are three-dimensional structures which have been synthesized via solid-state diradical polymerization of cubane. This chapter aims to introduce the structural morphology of these supra-nanostructures and predict their properties based on computational modeling. To facilitate the design of high-dimensional carbon nanothread supra-nanostructures, theoretical guidance is given by introducing the existing carbon nanotube framework from both experimental and theoretical insights.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 K. M. Liew et al., Carbon Nanothreads Materials, Materials Horizons: From Nature to Nanomaterials, https://doi.org/10.1007/978-981-19-0912-2_8

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8.2 Carbon Nanothread Rods Using polycyclic aromatic hydrocarbon (PAH) molecule as a precursor, the carbon nanothread rods can be formed by interconnecting each adjacent pair of rings [1]. According to the number of rings and arrangements of PAH molecules, the carbon nanothread rods can be denoted as “NX”, where N is the number of aromatic rings, and X = “A, B, C, …” is the ring arrangements. Note that the ring number for the PAH molecule could range from 2 to ∞, and hence there will be numberless amounts of carbon nanothread rods. When the ring number approaches ∞, the carbon nanothread rods could be regarded as diamond nanowires. This chapter introduces the carbon nanothread rods created from the PAH molecules ranging from 2 to 7 rings. Besides, there are three types of ring arrangements of PAH molecules, including linear, angular, and clustered. As introduced in Chap. 3, different connecting patterns for two adjacent aromatic rings will generate a variety of nanothread configurations. It would be feasible to extend the connecting patterns from benzene molecules to PAH molecules. Figure 8.1 shows examples of possible configurations of carbon nanothread rods created from 2A PAH molecules. The blue dots in the aromatic ring indicate the carbon atoms bonded to equivalent atoms in the ring above, and red dots denote those bonded to atoms of the ring above in a different relative position, indicated by the red arrow. The 2A-Tube structure is built by following the connecting pattern of sp3 -(3,0) carbon nanothread in Fig. 3.11. The 2A-Linear, 2A-Asymmetric, and 2A-Symmetric structures are constructed by following the connecting pattern of Polymer I carbon nanothread in Fig. 3.11. The 2A-Linear structure means the carbon nanothread rod is built from the linear PAH molecule (denoted by A) with linear configuration (denoted by

Fig. 8.1 Connecting patterns and configuration of carbon nanothread rods, taking naphthalene molecules (2A) as an example: a Tube, b Linear, c Asymmetric, and d Symmetric. Two different lateral views are shown for each atomic configuration. The dots and arrows are used to describe the connecting patterns. Reprinted (adapted) with permission from [1]

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Linear). The 2A-Asymmetric and 2A-Symmetric structures differ from each other in symmetry. Note that other connecting patterns could be also applied to the PAH molecules, resulting in different structural stability. The binding energies of carbon nanothread rods as a function of ring number are obtained by density functional theory (DFT) calculations. The GGA/PerdewBurke-Ernzerhof (PBE) exchange–correlation functional is applied during the DFT calculation. The lower the binding energy, the better stability is the structure. Structures with positive binding energy will be less favorable in experimental fabrication. The benzene-derived carbon nanothreads with sp3 -(3,0) and Polymer I configurations are taken as comparative examples. The binding energy of sp3 -(3,0) nanothread is lower than the Polymer I nanothread. It is found that the binding energies of carbon nanothread rods with linear configuration are always positive. The same observation goes to the carbon nanothread rods built from angular and clustered PAH molecules. The carbon nanothread rods built from linear PAH molecules have better stability, and the symmetric and asymmetric carbon nanothread rods created from linear PAH molecules are more stable than their tube counterpart. With the increase of ring number, the binding energy of carbon nanothread rods based on linear PAH molecules decreases except for the linear configuration, while the binding energy of angular and clustered structures increases. Compared to the benzene-derived carbon nanothreads, the asymmetric and symmetric carbon nanothread rods based on linear PAH molecules are more stable and have a higher likelihood of synthesis. The mechanical properties of carbon nanothread rods are investigated by MD simulations [1]. The interactions between C atoms and H atoms are described by the adaptive intermolecular reactive bond order (AIREBO) potential [2]. Figure 8.2 shows the stress–strain curves of different carbon nanothread rods under tension at a temperature of 300 K, and a brittle behavior is observed. The Tube configuration exhibit better mechanical properties than the Asymmetric, Symmetric, and Linear configurations for a given precursor molecule. Among the Tube configuration, the structure 7 has the highest Young’s modulus and ultimate stress. The Linear configuration has the weakest ultimate stress due to its cross-section area of fracture having fewer C–C bonds with respect to other configurations. The Tube and Linear configurations have the lowest and highest fracture strains, respectively. The dependence of mechanical properties on the size of precursor molecules is attributed to the homogeneity of the structure and variations in C–C bonds. The characteristics of C–C bonds are similar throughout the Tube and Linear configurations. For Asymmetric and Symmetric configurations, however, the exterior bonds at the surface experience larger strain than the interior bonds, leading to weaker strength. As the size of the precursor molecule increases, the ratio between exterior and interior bonds decreases, which enhances the mechanical strength. The bending modulus of various carbon nanothread rods is shown in Fig. 8.3. An increase in the size of a precursor molecule leads to enhanced bending modulus. There is a linear-like relationship between the bending modulus and the number of aromatic rings for the carbon nanothread rods based on linear PAH molecules. For a given number of aromatic rings, the structures based on linear PAH molecules present the lowest bending modulus. Besides, the bending modulus is also dependent

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Fig. 8.2 Tensile stress–strain curves at 300 K, for a Tube, b Asymmetric, c Symmetric, and d Linear carbon nanothread rods. Reprinted (adapted) with permission from [1]

Fig. 8.3 Bending modulus of PAH-based carbon nanothread rods as a function of the number of aromatic rings in the precursor molecule. Reprinted (adapted) with permission from [1]

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on the clustered size or diameter of the rings. The larger the clustered size, the high would be the bending modulus due to the larger effective diameter of the carbon nanothread rods. The Tube configurations present the highest bending moduli than other configurations for a given precursor molecule, which is similar to the findings in carbon nanothreads that Tube (3,0) nanothread has the higher bending rigidity than other nanothreads [3]. The calculated bending modulus of carbon nanothread rods can reach up to 2750 eV Å at room temperature (for 7-Tube configuration), which is significantly stiffer than conventional benzene-derived carbon nanothreads [3].

8.3 Carbon Nanothread Forests By arranging multiple carbon nanothreads along one direction, the carbon nanothread forests can be formed [4]. A seven-strand bundle is given as an example here. Two types of carbon nanothreads, including the polytwistane (143,652) and the stiffchiral-3 (136,425), are considered as bundle components. The carbon nanothreads in the bundle structure are named A, B, C, E, G, and F in clockwise order, and the central nanothread is named as D. Meanwhile, numbers 0 and 1 are used to represent carbon nanothread itself and its mirror image is seen from a cross-sectional view. As seen in Fig. 8.4a, the carbon nanothread forests can be notated as (0_0_0_1_0_0_0), and the atomic configurations based on polytwistane and stiff-chiral-3 carbon nanothreads are shown in Fig. 8.4b. Based on the symmetry, the carbon nanothread forests can be further classified into symmetric structure and asymmetric structure. There are totally 28 kinds of carbon nanothread forests, and their labeling schemes are shown in Fig. 8.5. The torsional properties of carbon nanothread bundles are investigated by MD simulations [4]. The interactions between C atoms and H atoms are described by the adaptive intermolecular reactive bond order (AIREBO) potential [2]. A temperature of 1 K is selected in order to reduce the effect of thermal fluctuation. The torsional energy E t is given by

Fig. 8.4 Carbon nanothread bundle. a Illustration of the labeling scheme for the bundle structure and the (0_0_0_1_0_0_0) bundle from b polytwistane and c stiff-chiral-3 carbon nanothreads. Reprinted (adapted) with permission from [4]. Copyright (2021) American Chemical Society

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Fig. 8.5 Illustration of the labeling scheme for all the bundle structures. Reprinted (adapted) with permission from [4]. Copyright (2021) American Chemical Society

E t = G I p ϕ 2 /2L 0

(8.1)

where G and I p are the shear modulus and polar moment of inertia, ϕ and L 0 are twist angle and initial length, respectively. The twist rate (E) is expressed as ε = ϕ/L 0 . The gravimetric deformation energy is given by J = E t /m, where m is the mass. The torsion rigidity (G I p ) is determined as the slope of the fitting the energy density curve. The effects of loading direction on the torsional properties of individual carbon nanothread are shown in Fig. 8.6. Results show that the polytwistane nanothread has a relatively higher torsional loading limit in an anticlockwise direction than that in the clockwise direction. Here the torsional loading limit is defined as the maximum twist

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Fig. 8.6 Gravimetric energy density as a function of the twist rate for polytwistane and stiff-chiral3 carbon nanothreads in different twisting directions. The symbols + and − represent the clockwise and anticlockwise torsion, respectively. Reprinted (adapted) with permission from [4]. Copyright (2021) American Chemical Society

rate prior to bond breaking. For the stiff-chiral-3 nanothread, the torsional loading limit in the anticlockwise direction (~1.92 rad/nm) is much higher than that in the clockwise direction (~1.47 rad/nm). The calculation results of the torsional loading limit/elastic limit are summarized in Table 8.1. The maximum gravimetric energy density, which characterizes the mechanical energy storage capability, is shown in Table 8.1. A higher maximum gravimetric energy density is observed in both individual carbo nanothread in the anticlockwise direction. Especially, the polytwistane nanothread possesses a much higher energy density (2.04 MJ/kg) in the anticlockwise direction, which is about 80% higher than that in the clockwise direction, indicating an obvious anisotropic energy storage capability. In addition, it is also shown that the polytwistane nanothread exhibits higher torsional rigidity than the stiff-chiral-3 irrespective of the twisting direction. The highest torsional rigidity (61.44 eV·Å) is found in the polytwistane nanothread in the anticlockwise direction. Table 8.1 Torsional properties of individual carbon nanothread. Reprinted (adapted) with permission from [4]. Copyright (2021) American Chemical Society Structure

Max gravimetric density Elastic limit (rad/nm) Torsional rigidity (eV·Å) (MJ/kg)

Polytwistane (+)

1.16

1.44

56.13

Polytwistane (−)

2.04

1.49

61.44

Stiff-chiral-3 (+)

1.09

1.47

48.53

Stiff-chiral-3 (−) 1.41

1.92

42.23

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Fig. 8.7 Atomic configurations showing the torsional deformation process of a polytwistane and b stiff-chiral-3 carbon nanothreads. Atoms are colored according to atomic stress. Positive and negative angles represent the clockwise and anticlockwise torsion. Reprinted (adapted) with permission from [4]. Copyright (2021) American Chemical Society

The atomistic configurations of polytwistane and stiff-chiral-3 nanothreads during torsional deformation are shown in Fig. 8.7. Due to the difference in morphology, von Mises stress distribution shows different stress distribution status under torsion. For the polytwistane one, it is observed that stress concentration in the two ends of the structure becomes more intensive with the increased twist rate. In comparison, better stress distribution is found in the stiff-chiral-3 one. Torsional-induced buckling phenomenon is observed in the polytwistane nanothread under clockwise torsion and stiff-chiral-3 nanothread under anticlockwise torsion. Besides, the C–C bonds in both nanothreads undergo compression and tension during torsional deformation. Note that the size effect exists in the torsional properties of individual carbon nanothread, and the torsional properties tend to converge when the length is larger than 18 nm. The maximum energy density and elastic limit under torsion for carbon nanothread bundle based on polytwistane nanothread are shown in Fig. 8.8. The bundle structures 1_1_1_1_1_1_1, 1_1_1_0_1_1_1, 0_0_1_0_0_1_1, and 0_0_0_1_1_1_1 have the highest maximum energy density of ~1.02 MJ/kg, and the 0_0_1_0_0_1_1 and 0_0_0_1_1_1_1 also have the largest elastic limit of ~0.565 rad/nm. Compared

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Fig. 8.8 Maximum energy density and the elastic limit under torsion for different bundle structures constructed from polytwistane. Reprinted (adapted) with permission from [4]. Copyright (2021) American Chemical Society

to the individual carbon nanothread, the carbon nanothread bundle possesses a lower maximum energy density and elastic limit, which is explained by the more intensive stress concentration of the constituent carbon nanothread. There is a strong correlation between the torsional properties and enantiomer ratio, as seen from the pairwise data relationship in Fig. 8.9. The Pearson’s correlation coefficients (PCCs) with values ranging from −1 to +1 are used to assess the linear relationship. A value of +1 indicates a positive linear relationship, 0 indicates no linear relationship, and −1 indicates a negative linear relationship. A strong positive linear relationship is found between the maximum energy density versus the elastic limit, the maximum energy density versus the torsional rigidity, and the torsional rigidity versus the enantiomer ratio. Here the enantiomer ratio is defined as the ratio of the number of enantiomers to the number of strands in the carbon nanothread bundle. The largest torsional rigidity is found in the 1_1_1_1_1_1_1 bundle structure with the enantiomer ratio equal to 1, while the lowest rigidity is found in the 0_0_0_0_0_0_0 bundle structure with the enantiomer ratio equal to 0. The maximum energy density and elastic limit under torsion for carbon nanothread bundle based on stiff-chiral-3 nanothread are investigated. The torsional properties are dependent on the arrangement of nanothread. The bundle structure 0_0_1_0_0_1_1 has the maximum elastic limit of ~0.59 rad/nm, and the bundle structure 0_0_0_0_1_1_1 has the lowest elastic limit of 0.43 rad/nm and the maximum energy density of 0.32 MJ/kg. There is a strong correlation between the torsional rigidity and enantiomer ratio, as seen from the pairwise data relationship in Fig. 8.10. In contrast to the polytwistane-based bundle, there is no obvious relationship between the maximum energy density and the enantiomer ratio.

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Fig. 8.9 Pairwise data relationship between the maximum energy density, elastic limit, torsional rigidity, and the enantiomer ratio of polytwistane-based seven-strand bundles. Reprinted (adapted) with permission from [4]. Copyright (2021) American Chemical Society

8.4 Carbon Nanothread Nanomeshes and Nanoforms By removing the hydrogen atoms on the surface of two perpendicular carbon nanothreads, the carbon nanothread framework can be formed by creating sp3 C–C bonds between one end of the carbon nanothread and the dehydronated surface of another carbon nanothread [5]. This section introduces two kinds of carbon nanothreads as structural components for the nanomeshes and nanoforms, including the sp3 -(3,0) and Polymer I carbon nanothreads, and their atomic configurations have been introduced in Fig. 3.11. The processes of forming carbon nanothread nanomeshes are depicted in Fig. 8.11. Based on the possible configurations and bonding patterns in Fig. 8.11b, two kinds of carbon nanothread nanomeshes could be formed by connecting the DNT-A part to the DNT-C part, and the ultimate structure with a planar surface (see

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Fig. 8.10 Pairwise data relationship between the maximum energy density, elastic limit, torsional rigidity, and the enantiomer ratio of stiff-chiral-3 C_NTH-based seven-strand bundles. Here, PCC refers to the Pearson correlation coefficient. Reprinted (adapted) with permission from [4]. Copyright (2021) American Chemical Society

Fig. 8.12a) is denoted as DNM-ACP (n,m), the ultimate structure with a corrugated surface (see Fig. 8.12b) is denoted as DNM-ACC (n,m). The indices (n,m) in the notation are the number of carbon nanothread unit cells along the primary and secondary directions. The primary direction (PD) aligns along the larger continuous domains, and the secondary direction (SD) aligns along the smaller discontinuous domains. The third kind of carbon nanothread nanomesh denoted as DNM- DCP (n,m) can be formed by connecting DNT-D part to the DNT-C part, as shown in Fig. 8.12c. The stress–strain curves for DNM-ACP (1,1), DNM-ACC (1,1), and DNMDCP (1,1) under uniaxial tension are shown in Fig. 8.12d–f. Similar to the carbon nanothreads, the carbon nanothread nanomeshes exhibit a brittle behavior. It can be seen that the mechanical properties are dependent on the tensile direction, due to the

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Fig. 8.11 a Atomic configurations of a benzene molecule and a typical 1-D diamond nanothread. b Four possible carbon nanothread configurations with different orientations of stacked benzene molecules and bonding patterns. c Sequence of configurations showing the formation of sp3 C–C bonds between two individual carbon nanothreads, through (i) dehydrogenation of the surface of a DNT-A and (ii) bonding of the dehydronated surface to the end of a DNT-C. Reprinted (adapted) with permission from [5]

different arrangement of C–C bonds in each direction. In other words, the mechanical properties are governed by the carbon nanothread parts aligned in the tensile direction. Previous studies have proved that DNT-A has the largest Young’s modulus and ultimate stress among the four configurations, and the DNT-D has the smallest Young’s modulus and ultimate stress but best flexibility [6]. Both DNM-ACP (1,1), DNM-ACC (1,1), and DNM-DCP (1,1) configurations have similar ultimate stress in Fig. 8.12. The ultimate stress is obtained in DNM-ACP (1,1) and DNM- DCP (1,1) along the secondary direction and DNM-ACC (1,1) along the primary direction. The DNM-ACP (1,1) has a lower fracture strain in both tensile directions due to the lower flexibility of DNT-A and DNT-C. The largest fracture

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Fig. 8.12 a–c Atomic structure of representative carbon nanothread nanomeshes (top and lateral views), with primary and secondary directions defined by the axis depicted in a. d–f Stress–strain curves for the corresponding (a–c) configurations. Reprinted (adapted) with permission from [5]

strain (0.27– 0.32) is found in DNM-ACC (1,1) along the secondary direction due to its corrugated surface allowing out-of-plane C– C rotation. For DNM-DCP (1,1), it also exhibits high flexibility (strains of 0.25–0.3) before fracture along the primary direction, which is attributed to the small Young’s modulus of the DNT-D component. The indices (n,m) which define the number of carbon nanothread unit cells along the primary and secondary directions affect the mechanical properties of carbon nanothread nanomeshes. An increase of n (number of unit cells along the primary direction) reduces Young’s modulus and ultimate stress in the secondary tensile direction, and an increase of m (number of unit cells along the secondary direction) has the same effect. The degradation mechanism of mechanical properties is illustrated such that as the number of unit cells perpendicular to the tensile direction increases, the transverse area increases. For a given force, smaller stress is obtained with a larger transverse area. However, the change of indices (n,m) will not effectively influence the fracture strain, because the fracture strain is governed by the arrangement of C–C bonds in the cross-section area of the fracture. Obviously, the arrangement of C–C bonds in the cross-section area of fracture will not change with the indices (n,m).

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The carbon nanothread nanoforms can be built by connecting a DNT-A-like continuous large component with two small DNT-C-like small components in a hexagonal manner. The proposed carbon nanothread nanoforms are denoted as DNFAC(n,m,k) with indices n, m, k defining the number of unit cells along the primary direction (PD), secondary direction (SD), and tertiary direction (TD). A representative atomic structure of carbon nanothread nanoform is given in Fig. 8.13a. The carbon nanothread nanoforms have extremely low density. For instance, the density of DNF-AC(2,2,2) is about 0.218 g/cm3 , which is much smaller than that of 3.5 g /cm3 of the diamond.

Fig. 8.13 a Atomic structure of DNF-AC(1,1,1). b Stress–strain curves along primary direction (PD), secondary direction (SD), and tertiary direction (TD), as defined in the axis depicted in a. c Pore size distribution plots. d and e Evolution of the atomic structure upon application of uniaxial tensile strain along the SD and TD directions, respectively (thicker arrows define the direction of applied strain). Atoms are colored according to the local von Mises stress × atomic volume. Reprinted (adapted) with permission from [5]

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The stress–strain curves for DNF-AC(1,1,1) under uniaxial tension are shown in Fig. 8.13b. Similar to the carbon nanothread nanomeshes, the carbon nanothread nanoforms exhibit a brittle behavior. It can be seen that the mechanical properties are dependent on the tensile direction. The DNF-AC(1,1,1) structure has the largest Young’s modulus in the primary direction (characterized by DNT-A-like large component) and the highest flexibility in the secondary direction (characterized by DNT-C-like small component). The DNF-AC(1,1,1) has a specific strength of 1.36 × 107 N · m/kg, which is in the same order of magnitude as individual carbon nanothreads [6]. Similar to the carbon nanothread nanomeshes, changing the indices (n,m,k) will affect the mechanical properties of carbon nanothread nanoforms. Simultaneous increases of n, m, k reduce Young’s modulus and the ultimate stress in all tensile directions but enhance the fracture strain. The pore size distribution in Fig. 8.13c shows that the pore diameter is tunable by changing the indices (n,m,k), indicating that the carbon nanothread nanomeshes are promising molecule sieves. The deformation processes of DNF-AC(1,1,1) at various strains along the secondary and tertiary direction are shown in Fig. 8.13d and e, respectively. At low strains, stress is uniformly distributed to the C–C bonds through the structure, and the hexagonal shape is maintained. With the increase of strain, the hexagonal structure loses its stability and flattens to a nearly rectangular structure, accompanied by stress concentration at the carbon nanothread components aligned to the tensile direction. The flatten process enables further deformation of the structure Finally, the C–C bonds in the vicinity of one junction break, and the release of large amounts of strain energy leads to continuous breakage of C–C bonds in the neighboring junctions, resulting in a brittle fracture.

8.5 Carbon Nanothread Cubanes Experimental studies have demonstrated that crystalline with three-dimensional order can be synthesized via high-pressure, solid-state diradical polymerization of cubane [7]. The experimental details are introduced in Chap. 2. In this chapter, the reaction pathways toward a three-dimensional carbon nanothread framework from cubane are elucidated through ab initio MD simulations. We start by introducing the polymerization of cubane molecules to form a one-dimensional cubane-based nanothread. First, multiple cubane molecules are placed in the simulation cell at a hydrostatic pressure of 30 GPa and at a temperature of 500 K under NPT ensemble. In this situation, the cubane molecules maintain stability due to the van der Waals interaction, and no reaction between cubane molecules occurs. The simulation cell is then relaxed with uniaxial stress of 30 GPa along < 111> direction (equivalent stack directions), after which polymerization of cubane molecules appears. The applied uniaxial stress is expected to potentially differentiate nominally equivalent molecular stacks and guide the polymerization along the crystallographic axis. The polymerization process from MD simulations is shown in Fig. 8.14. At 80 fs, the polymerization

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Fig. 8.14 Hexagonal cubane precursor crystal viewed from a the [111] zone axis and b the [010] axis, showing diffraction planes and their interplanar spacings at 24.9 GPa with the orientation and simulated diffraction spots (inset). c Molecular dynamics simulation shows that the diradical polymerization of cubane occurs spontaneously under uniaxial stress of 30 GPa. In the simulation, the reaction propagates along three equivalent crystallographic directions, which are [111], [211], and [121]. Reprinted (adapted) with permission from [7]. Copyright (2020) American Chemical Society

initiates and the adjacent cubane molecules start to form interconnection. At 2.5 ps, a one-dimensional cubane-based nanothread has been formed. With the increase of cubane molecules, linear cubane-based nanothread arrays could also be formed, as seen in Fig. 8.14a and b. These nanothread arrays could serve as intermediates for the formation of three-dimensional carbon nanothread cubane through interthread reaction. The reaction pathways are presented in Fig. 8.15. Nanoindentation results in Fig. 8.16 show that such carbon nanothread cubane crystals have an average hardness near 11 GPa, which is a bit larger than the fused quartz.

8.6 Carbon Nanotube Superstructure Over the past two decades, high-dimensional CNT superstructures have received great interest in the field of bottom-up design due to their fantastic physical and chemical properties. Since the carbon nanothreads have a similar atomic structure with the CNTs, it is expected that some of the experimental findings in the highdimensional CNT superstructures could be applied to the carbon nanothread and facilitate the design in the future.

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Fig. 8.15 a Scheme of breaking three orthogonal bonds and extending the covalent connectivity into a three-dimensional structure. b Schematic view of the cubane-derived carbon nanothread framework. Reprinted (adapted) with permission from [7]. Copyright (2020) American Chemical Society

Fig. 8.16 a The loading/unloading displacement curves of the crystalline and amorphous component of carbon nanothread cubanes. b The indentation hardness and c the elastic modulus as a function of loads. Reprinted (adapted) with permission from [7]. Copyright (2020) American Chemical Society

8.6.1 Carbon Nanotube Yarn CNT yarn could be synthesized by drawing a thin sheet of CNT forests grown on an iron catalyst-coated substrate by the chemical vapor deposition (CVD) method, as

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Fig. 8.17 SEM of directly spun CNT yarns from the as-grown CNT array. Reprinted (adapted) with permission from [8]

seen in Fig. 8.17. The unplied CNT yarns are then twisted and spun with a variablespeed motor to produce multiply, torque-stabilized yarns [8]. The SEM images of CNT yarns are shown in Fig. 8.18. The twist-spun CNT yarns are promising for torsional artificial muscles and tensile artificial muscles. The twist-spun CNT yarns will rotate when half immersed in an electrolyte, functioning as a torsional artificial muscle in a simple three-electrode electrochemical system [10]. A twist-spun CNT yarn with a diameter of 12 μm can produce a reversible 15,000° rotation at speeds of up to 590 rpm. The generated torque of CNT yarns is about 1.85 N · m/kg, which is quite impressive compared to large commercial electric motors (2.5–6 N · m/kg) [11]. The torsional actuation mechanism is attributed to the yarn volume increase due to electrochemical double-layer charge injection, resulting in simultaneous motions of lengthwise contraction and torsional rotation. Besides, the twist-spun CNT yarn can provide faster rotations of ~11,500 rpm when half infiltrated with volume expanding guest-paraffin wax, and highly reversible torsion of ~2 million times without sign of degradation [12].

Fig. 8.18 a SEM image of dual Archimedean-type twist insertion during spinning from a CNT forest. The inset illustrates a dual Archimedean scroll. Scale bar, 500 μm. b Schematic illustration of muscle configurations and yarn structures for torsional actuation. c SEM images of unfilled CNT yarn (Z chirality) and wax-SEBS-infiltrated CNT yarn (S chirality). The average diameter of S and Z yarn is ~20 μm. The average bias angle is ~25°. Scale bars, 5 μm. Reprinted (adapted) with permission from [9]

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The twist-spun CNT yarns can provide a highly reversible tensile contraction, functioning as a tensional artificial muscle. A twist-spun CNT yarn infiltrated with paraffin wax delivers 7.3% tensile contraction when electrothermally heated from ambient to incandescent temperature (~2560 °C). The CNT yarn could also provide 3% tensile contraction at 1200 cycles/min for more than 1.4 million cycles, during which a power output of 27.9 kW/kg is provided, which is 85 times the peak output of mammalian skeletal muscles (0.323 kW/kg) [13]. The reversible actuation mechanism is attributed to the volume expansion. When the wax volume decreases during cooling, the nanotube-paraffin interfacial energy will be replaced by nanotubeair interfacial energy at the expense of energy, which generates additional force to compress the CNT yarn. When the wax volume increases during heating, the elastic energy stored in the yarn are released gradually, ensuring volume consistency between the molten wax and yarn over the entire actuation circle. The coiled CNT yarn could generate contractile work of 1.36 kJ/kg, which is 29 times the work capacity of natural muscle [14]. In addition to the energy conversion from electricity to contraction and rotation, the twist-spun CNT yarns could also electrochemically convert mechanical energy into electrical energy, functioning as energy harvesters without an external bias voltage. When stretch and release the coiled CNT yarns inside an electrolyte, it can generate 250 W/kg of peak electrical power at a frequency of 30 Hz and up to 41.2 J/kg of electrical energy each cycle [15]. The energy harvesters also have potential applications in harvesting ocean energy. When tested in an ocean environment, the coiled CNT yarns deliver a peak power of ~94 W/kg for 30% stretch and frequencies of 0.25–12 Hz. Combined with the thermally-driven artificial muscles, small temperature fluctuations in the ocean environment can up-twist and stretch the coiled CNT yarns, generating considerable amounts of electrical energy. The energy transformation originates from the reversible electrochemical capacitance change for the coiled CNT yarn during the stretch release process. During the stretching process, the up-twist of CNT yarn will impose internal stress and reduce the surface energy of adjacent CNTs. The reduced surface energy leads to a decrease in electrochemical capacitance, thereby converting the mechanical energy to electrical energy.

8.6.2 Carbon Nanotube Sponge CNT sponges are a randomly arranged three-dimensional framework of interconnected CNTs, featured by extreme lightweight, high porosity, and high elasticity. The CNT sponges can be synthesized by the CVD process using ferrocene and 1,2dichlorobenzene as the catalyst precursor and carbon source, respectively [16]. The macroscopic and microscopic views of the sponges are shown in Fig. 8.19. The bulk density of sponges can be as small as 5–10 mg/cm3 , and the porosity is ~99%. The CNT sponges exhibit outstanding mechanical properties, such as large-strain deformation, super-elasticity, and high fatigue resistance in both air and aqueous

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8 Arrangements of Carbon Nanothreads

Fig. 8.19 a A monolithic sponge with a size of 4 cm × 3 cm × 0.8 cm. b Cross-sectional SEM image of the sponge showing a porous morphology and overlapped CNTs. c TEM image of largecavity, thin-walled CNTs. d Schematic of the sponge containing CNTs. Reprinted (adapted) with permission from [16]

conditions [16]. When compressed in air, the sponges will undergo elastic deformation in the early stage, followed by strain hardening until very high strains up to 80%. When unloading the strain, the sponges could return to the initial status with complete volume recovery, indicating purely elastic deformation. The volume recovery is attributed to the squeezing of inter-tube pores. Due to their high porosity, the sponges can be compressed to more than 95% volume reduction. When compressed in a solvent, the solvents will be driven out of the pores, and the strain hardening phenomenon is not obvious. When unloading the strain, the reversible absorption of the pores with solvents can accelerate the volume recovery of carbon nanotube sponges. Although the CNTs have an ultrahigh thermal conductivity of up to 3000 W · K−1 m−1 at room temperature, the measured thermal conductivity of CNT sponges is less than 0.15 W · K−1 m−1 at temperatures of 200–360 K due to their high porosity [16], comparable to the thermal conductivity of insulating materials of 0.02–1 W · K−1 m−1 . Therefore, the CNT sponges are an excellent low-density thermal insulator. The CNT sponges are excellent supercapacitors for electrochemical energy storage. They can be used as bulk electrodes directly in aqueous electrolytes, exhibiting an electric double-layer capacitor (EDLC) behavior without redox reactions [17]. Specifically, the CNT sponges have highly stable electromechanical

8.6 Carbon Nanotube Superstructure

271

behavior under compression. For instance, they can be compressed to 50% strain during electrochemical testing without property degradation and maintain their specific capacitance under strain as large as 80%. Despite the ultralow specific capacitance of CNT sponges ( k O2 > k N2 ≈ kco . The specific mass transfer rate tends to increase with the kinetic diameter of molecules. Although the CO has a larger kinetic diameter than the N2 , the higher adhesion of CO to the membrane will decrease the specific mass transfer rate.

9.3 Gas Membrane

291

Fig. 9.15 Dependence of effective pore diameter (top) and ideal selectivities for O2 /N2 and H2 /CO with applied strain, for CNM-2,3,4. Reprinted (adapted) with permission from [12]

The effects of applied strain on the gas selectivity of membranes CNM-2, CNM3, and CNM-4 are shown in Fig. 9.15. With the increase of strain, the pore size of the membrane becomes narrow, which leads to an increase in gas selectivity. When the pore size is close to the kinetic diameter of gas molecules, the gas selectivity increases dramatically. The sudden change of gas selectivity is attributed to the intrinsic transport mechanism of molecule sieves [14]. The membrane has a higher selectivity of H2 /CO pair than the O2 /N2 pair for a given applied strain and membrane type due to the larger difference of kinetic diameter between H2 and CO. The highest selectivity for H2 /CO pair is as high as 283 for the CNM-2, which is higher than the reported values of conventional gas membranes. There used to be an inverse relationship between gas selectivity and gas permeability. It is meaningful to design the membrane with large gas permeability while maintaining high selectivity. The permeances Pi of CNM-2 strained at the highest selectivity are calculated as 1.0–1.7 × 107 GPU for H2 (1 GPU = 3.35 × 10−10 mol /(m2 s Pa)), 0.1–2.0 × 105 GPU for CO, 0.9–2.0 × 106 GPU for O2 , and 0.4– 2.3 × 105 GPU for N2 . The permeances Pi of CNM-4 strained at the highest selectivity are calculated as 1.8–4.4 × 107 GPU for H2 (1 GPU = 3.35 × 10−10 mol /(m2 s Pa)), 1.3 × 106 –1.1 × 107 GPU for CO and N2 . These permeances are higher than the reported values of conventional membranes, and they are in the same order of magnitude as other carbon-based nanomaterials such as porous graphene [15]. The ideal selectivity of Si O2/N2 as a function of the permeability of O2 is shown in Fig. 9.16. The continuous straight line denotes the upper bound relationship defined by Robeson for polymeric membranes, following a power-law relationship Pei = k(Pi /Pj )n , with n = − 5.666 and k = 1.396 × 106 Barrers for i = O2 and j = N2 [16]. The upper bound can be a criterion to assess the efficiency of membranes regarding the ideal selectivity and permeability. It can be seen that the CNM-2 has a high efficiency than the upper bound for strains ranging from 0.022 to ~ 0.036. For

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Fig. 9.16 Ideal selectivity (O2 /N2 ) versus permeability of O2 (Robeson-like plot) for membranes CNM-n (n = 2,3,4) at strains that provide the optimal selectivity (within 0.022–0.036, 0.074–0.086, and 0.077–0.083 for n = 2, 3, and 4, respectively). The continuous straight line denotes the upper bound defined by Robeson for polymeric membranes. Reprinted (adapted) with permission from [12]

CNM-4, although the permeability is higher than the CNM-2, the selectivity is lower, and it falls below the upper bound, which is comparable to those of conventional polymeric membrane [16]. The CNM-3 has a good combination of both selectivity and permeability. The real selectivity of different membranes for different pairs of gases is studied by simulating the gas permeation. It is found that the membrane efficiently separates the O2 from the N2 . Besides, there is a discrepancy between the ideal and real selectivity, which is due to the interaction between species and possible sorption/diffusion competition, as well as other possible effects of the diffusing gases on the membrane pore structure [14, 17]. For the equimolar gas mixtures, the specific mass transfer rate ki of CO and O2 in is 10–30% larger than the pure gas, and the ki of H2 and N2 is 15–30% smaller. The difference of ki between the gas mixture and pure gas is due to the varying degrees of molecule adsorption at the membrane surface. The CO molecules prefer to absorb on the surface of the membrane, which leads to an increase in the CO concentration gradient and subsequent flux across the membrane. The CO molecules near the surface could inhibit the passage of H2 molecules, leading to a smaller ki . The same phenomenon is observed in the O2 /N2 mixture, where the preferred absorption of the O2 molecules at the membrane surface inhibit the passage of N2 molecules.

9.4 Water Membrane The two-dimensional nanoporous nanomeshes based on carbon nanothreads are promising membranes for water desalination [18]. FB-NEMD simulations are conducted to investigate the salt rejection and water permeability of carbon

9.4 Water Membrane

293

nanothread-based membranes (CNM) whose construction methods have been introduced in Sect. 9.3. The atomic configurations of CNM and desalination system are shown in Fig. 9.17a. As seen in Fig. 9.17b, seawater with a NaCl concentration of ~35 g/L and freshwater are placed on the left and right sides of the membrane, respectively. The effective pore area allowing the passage of water molecules is

Fig. 9.17 a Atomic structure of CNMs. b Atomic configuration of water desalination system with higher pressure (P1 ) applied to the saltwater on the left side and lower pressure (P2 = 1 atm) applied to the pure water on the right side. c Pore structure of CNM-3 under various applied strains along with maps of water density inside the pore. Reprinted (adapted) with permission from [18]

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9 Technologically Relevant Applications

geometrically represented in the form of probability maps in Fig. 9.17c for CNM-3 at various strains. The pore becomes narrower with increased strain. Besides, it is also shown in the probability maps that water molecules prefer to move to the C–H bonds. In contrast to conventional two-dimensional membranes such as graphyne, the geometric packing of water molecules induced by the CNM is not obvious, which suggests that permeation occurs with insignificant entropic constraints in regard to the change of the pore size. The intermolecular interactions in the simulation system between C and H atoms are described by 12-6 Lennard-Jones (LJ) and Coulombic potentials using the CHARMM general force field [19]. The interactions between water molecules, sodium, and chloride are described by the TIP3P model [20]. Shake algorithm is applied to the water model to reduce computational cost. Two rigid pistons are placed at two sides of the membrane, and a pressure difference (30, 40, and 50 MPa) is created to push the water molecules through the membrane. Note that the pressures used in the FB-NEMD simulations are much higher than those used in reverse osmosis. If the computed flow rate scales linearly with the applied pressure, the permeability is expected to be valid for realistic situations. The early stage at which water flux increases linearly is considered for the calculation of water permeability. The water permeability is calculated as the slope of volumetric water flow (determined by the slope of the number of water molecules that crossed the membranes (N m ) × time (t) curves) normalized by the total membrane area and the applied pressure, with unit L·cm−2 day−1 ·MPa−1 . The salt rejection R is determined by the ratio between the salinity of the permeate solution C and the original feed concentration C 0 , which is calculated as R = 1−C/C 0 . The permeate solution C can be approximated by the ratio between salt flux and water flux across the membrane. The water flux in the linear regime across the CNMs-3,4,5 as a function of pressure is shown in Fig. 9.18. It can be seen that the water flow rate increases linearly with increased pressure. Besides, a decrease of water flow rate with the applied strain causes a reduction of effective pore area (see Fig. 9.17c). For zero strain and pressure at 30 MPa, the water flow rate follows an order of CNM-3 < CNM-4 < CNM-5 due to an increase of pores size. The density profiles of water molecules along the direction perpendicular to the membrane plane are plotted in Fig. 9.19. The two highest peaks in the curves indicate two adsorption layers on each side of the membrane, and the left peak is higher due to the higher pressure on the left side. The density of water molecules in the position of the membrane is lower due to the confinement effect of the nanopore. The highest peak value follows an order of CNM-2 > CNM-3 > CNM-4, indicating that the number of water molecules near the surface of the membrane increases with decreased pore size, due to the fact that CNM-2 with smaller pore size has a larger surface area for water adsorption. There is also a shift in the position of the highest peak. The position shift is attributed to the hydrophobic nature of the membrane. CNM-4 with a larger pore size allows more water molecules confined in the pore, which induces a high concentration of hydrogen bond and hence reduces the hydrophobicity of the membrane.

9.4 Water Membrane

295

Fig. 9.18 Number of water molecules that crossed the membranes as a function of time. Results for CNM-3 are presented for two applied strains (0 and 6%) and three different pressures (30, 40, and 50 MPa). Results for CNM-2 and CNM-4 are presented for 30 MPa and no applied strain. Reprinted (adapted) with permission from [18]

Fig. 9.19 Normalized density profiles of water along the direction perpendicular to the membrane plane. These were computed for membranes at a pressure difference of 50 MPa. Reprinted (adapted) with permission from [18]

The calculated water permeability for membranes as a function of effective pore area is shown in Fig. 9.20a. The water permeability increases linearly with the effective pore area, and it is independent of the pore shape. For instance, the strained CNM-4 and non-strained CNM-3 have different pore shapes, and they have very close effective pore areas and similar water permeability. The salt rejection decreases with the increased effective pore area, as seen in Fig. 9.20b. Finally, the membrane has a maximum water permeability of 66 L·cm−2 day−1 · MPa−1 while maintaining

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Fig. 9.20 a Water permeability and b salt rejection achieved by CNMs under various applied strains, expressed as a function of the effective pore area. Dotted lines correspond to linear fits to the data, and the labels within the plots (%) denote the applied strain. Reprinted (adapted) with permission from [18]

a 100% salt rejection rate. When water molecules cross the membrane, they will reorient themself to achieve a more favorable alignment to cross ellipse-shaped pores of CNMs. Such orientation could be characterized by projecting the H–O–H area of water molecules inside the pores onto the plane of the membrane. The areas of water molecules with different orientations projected on the plane CNM-3 are shown in Fig. 9.21a. When the projected area of a water molecule is smaller in the pore, the energy barriers for permeation will be smaller. Besides, the calculated projected H–O–H area at various strains in Fig. 9.21b shows that the area is always lower than the average projected area in the bulk solution, which indicates a reorientation of water molecules across the membrane. The projected area decreases with increased applied strain due to the narrower pore size. The simulation snapshots in Fig. 9.21c shows that the water molecules reorient themselves across the membrane during permeation. Such reorientation leads to a single-file water file with concerted dipole orientation, which improves the water permeability [21].

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Fig. 9.21 a Areas of water molecules projected on the plane CNM-3 at different strain values. In the left image, the H–O–H plane is parallel to the membrane plane (with a larger projected area) before permeation, while in the right one, the same is rotated with respect to the plane (with a smaller projected area) after permeation. b Plane-projected H–O–H area for molecules inside the pores of CNMs under applied strain. The dotted lines are linear fits to the data. The dashed line corresponds to the average projected area computed for molecules in the bulk solution. c Simulation snapshots of a water molecule crossing CNM-3 at maximum strain (before, during, and after crossing the membrane). Reprinted (adapted) with permission from [18]

9.5 Energy Storage The one-dimensional carbon nanothread bundles based on carbon nanothreads are promising candidates for energy storage with a gravimetric energy density of up to 1.76 MJ/kg due to their excellent mechanical properties and ultralight density [22]. MD simulations are conducted to investigate the contributions from different deformation modes to the energy storage capacity in carbon nanothread bundles. The C–C and C–H interactions are described by the AIREBO potential. The AIREBO potential includes the bonded interaction (bond, angle, and dihedral terms) and nonbonded interactions (vdW), which is suitable to represent the binding energy and elastic properties of carbon materials. To avoid thermal effects on the mechanical properties of the system, a temperature of 1 K is adopted. According to the equilibrium inter-thread distance, different carbon nanothread bundles can be constructed with proper filament number n which allows the close-packing morphology. The atomic configurations of individual carbon nanothread are shown in Fig. 9.22a. Two kinds of carbon nanothreads are selected, which are nanothreadA (135,462) and nanothread-C (134,562), respectively. The torsional strain energy density E t /m (kJ/kg) as a function of the dimensionless torsional strain is depicted

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Fig. 9.22 a Atomic configurations of nanothread-A (left, 135,462) and nanothread-C (right, 134,562). The six integers (some underscored) represent the bonding topology in the structure. Upper panels are side-views, and bottom panels are end-on views. CW refers to the clockwise torsion. b Torsional strain energy density versus the dimensionless torsional strain. c Tensile strain energy density versus the tensile strain. Solid lines in (b), (c) are the corresponding fitting curves based on Hooke’s law. Reprinted (adapted) with permission from [22]

in Fig. 9.22b. The dimensionless torsional strain is calculated from εt = ϕ D0 /l0 , where D0 and l0 are the equivalent diameter and length of carbon nanothread and ϕ is the twist angle. The torsional elastic limit is determined as the maximum dimensionless torsional strain before bond breakage. The gravimetric energy density is determined as the strain energy density at the elastic limit. The gravimetric energy density is calculated as ~884 kJ/kg and ~737 kJ/kg for nanothread-A and nanothreadC, respectively, and the torsional elastic limit is ~0.71 and ~0.51, respectively. These results indicate that the nanothread-A has a larger gravimetric energy density and torsional elastic limit than the nanothread-C. For a (10,10) CNT, the torsional elastic limit is ~0.06, and the gravimetric energy density is one order of magnitude smaller (~65 kJ/kg). The smaller torsional elastic limit of CNT is due to the early onset of the flattening stage. The ultimate torsional limit is ~0.92, which leads to a gravimetric energy density of ~2720 kJ/kg. The tensile strain energy density E s /m (kJ/kg) as a function of the dimensionless tensile strain is depicted in Fig. 9.22c. The tensile strain is calculated from

9.5 Energy Storage

299

εs = (l − l0 )/l0 , where l0 and l are the initial length and current length of carbon nanothread, respectively. Similarly, the tensile elastic limit is determined as the maximum dimensionless tensile strain before bond breakage. The gravimetric energy density is determined as the strain energy density at the elastic limit. The gravimetric energy density is calculated as ~2051 kJ/kg and ~906 kJ/kg for nanothread-A and nanothread-C, respectively, and the tensile elastic limit is ~0.18 and ~0.15, respectively. The gravimetric energy density of carbon nanothreads under tension is obviously larger than that under torsion. For a (10,10) CNT, the tensile elastic limit is ~0.22, and the gravimetric energy density is ~2374 kJ/kg. The maximum gravimetric energy density for CNT before bond breakage is larger (~6810 kJ/kg). The bending strain energy density E b /m (kJ/kg) as a function of the dimensionless bending strain is depicted in Fig. 9.23. The bending strain is calculated from εb = D0 /R, where R is the local radius of the bending curvature. The bending direction along Y-axis is shown in Fig. 9.22. The gravimetric energy density is calculated as ~468 kJ/kg and ~288 kJ/kg for nanothread-A and nanothread-C, respectively, and the bending elastic limit is ~0.34 and ~0.26, respectively. The (10,10) CNT exhibits early buckling during bending, which leads to a smaller bending strain energy density (~49.3 kJ/kg). The maximum gravimetric energy density for CNT having the maximum curvature is ~618.6 kJ/kg. The radial compressive strain energy density E c /m (kJ/kg) as a function of the radial compressive strain is depicted in Fig. 9.24a. The radial compressive strain in the elastic deformation is approximated by εc,i j = 1 − di j /d0 , where di j is the inter-thread distance. The gravimetric energy density is calculated as ~6051 kJ/kg

Fig. 9.23 Bending strain energy density as a function of bending strain. Solid lines are the corresponding fitting curves based on Hooke’s law. Reprinted (adapted) with permission from [22]

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9 Technologically Relevant Applications

Fig. 9.24 Radial compression of nanothread triangular lattice from simulation. a The compressive strain energy density as a function of compressive strain for nanothread-A, nanothread-C, and CNT(10,10). b Nanothread-A lattice compressed from 0 GPa (left panel) to 30 GPa (middle panel) and relaxed to 0 GPa (right panel). c Nanothread-C lattice compressed from 0 GPa (left panel) to 30 GPa (middle panel) and relaxed to 0 GPa (right panel). Atoms in (b), (c) are colored according to the axial virial stress. In each panel, left images show the periodic triangular lattice (in top-view) and right images show a representative individual nanothread (in side-view) in the triangular lattice. Reprinted (adapted) with permission from [22]

and ~3063 kJ/kg for nanothread-A and nanothread-C, respectively, and the compressive elastic limit is ~0.19 and ~0.18, respectively. The deformation snapshots of nanothread-A and nanothread-C are shown in Fig. 9.24b and c. Due to the helical morphology, the nanothread-C undergoes lateral deformation at higher pressure, which causes earlier bond breakage and smaller gravimetric energy density. For (10,10) CNT, there is a sudden change of the compressive strain from ~ 0.017 to 0.04 as the pressure increases from 1.4 GPa to 1.5 GPa. This phenomenon is due to the fact that the flattening occurs around 1.4 GPa, which leads to a dramatic change of the inter-thread distance. At a compressive strain of 0.017 (compressive elastic limit), the gravimetric energy density for CNT is calculated as ~14.1 kJ/kg. The bond breakage of CNT initiates at a strain of ~0.33, with a maximum gravimetric energy density of ~4053 kJ/kg. Figure 9.25a and c shows that the strain energy density of the carbon nanothread bundle increases with larger number of filaments. A parabolic relationship between the strain energy density and torsional strain is clearly observed. The torsional elastic limit and gravimetric energy density, however, decrease with larger number of filaments. The calculated gravimetric energy density for bundle-3 based on nanothreadA is ~991 kJ/kg, and the dimensionless torsional strain limit is ~0.47. In comparison, the calculated gravimetric energy density for bundle-19 is ~370 kJ/kg only, and the corresponding dimensionless torsional strain limit is ~0.14. Similarly, the calculated dimensionless torsional strain limit for bundle-3 (~0.43) based on nanothread-C is larger than that for bundle-19 (~0.14). For (10,10) CNT bundle, the torsional elastic limit is ~0.13, and the gravimetric energy density is ~577 kJ/kg. A deviation of the strain energy density from a parabolic relationship is observed for the CNT bundle, which is caused by the flattening of individual CNT during torsion. The CNT bundle has a smaller torsional angle than the nanothread bundle. The torsional angle at the torsional elastic limit is ~4.92 and 4.71 rad for bundle-19 based on nanothread-A

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301

Fig. 9.25 Mechanical energy storage in bundle structures from simulation. a Strain energy density as a function of torsional strain of nanothread-A bundles and b the corresponding atomic configurations of bundle-19 at different torsional angles. The last column is colored based on axial atomic virial stress and the rest is colored based on the atomic von Mises (VM) stress; c strain energy density as a function of dimensionless torsional strain of nanothread-C bundles; and d the corresponding atomic configurations of bundle-19 at different torsional angles. For clarity, in the last column only one fractured and one connected (stressed) filaments are colored based on axial atomic virial stress; e strain energy density as a function of the dimensionless torsional strain of CNT bundles; and f the corresponding atomic configurations of bundle-19 at different torsional angles. For (b), (d), (f), the atomic configurations show only a middle segment (~6 nm) of the whole sample, and only C atoms are visualized for nanothread bundles. Reprinted (adapted) with permission from [22]

and nanothread-C, respectively, while the torsional angle is only ~1.47 rad for CNT bundle-19. These findings show that the nanothread-A bundle has the highest gravimetric energy density and dimensionless torsional strain limit. The nanothread-C bundle has a similar gravimetric energy density and dimensionless torsional strain limit with the CNT bundle, but the latter has a lower torsional angle.

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The atomic configurations of nanothread bundle-19 at different torsional angles are shown in Fig. 9.25b and d. For bundle-19 based on nanothread-A, buckling is not observed in each nanothread during torsion. Stress concentration occurs on the exterior surface of the bundle, leading to early bond breakage. At the moment of bond breakage, lots of the axial stress at the outer layer is released, and the interfilament is able to maintain the curve profile through vdW interactions. For bundle19 based on nanothread-C, each nanothread coiled together during torsion, resulting in smaller pitch length and buckling. Stress concentration also occurs to the exterior surface of the nanothread-C bundle, and each nanothread filament experiences double-helix stress distribution, e.g., one helix experiencing tensile stress and the other experiencing compressive stress. The nanothread-C bundle can also maintain the curve profile after bond breakage at the outer layer. The flattening occurring to the individual nanothread during torsion is not observed in the bundle. In general, the strain energy density is closely dependent on the strain through E X /m = k X ε2X , where E is the strain energy, m is the mass of the structure, k is the elastic constant with unit MJ/kg, ε is the strain, the subscript X denotes torsion, stretching, bending, and compression. The elastic constant k can be obtained from strain energy density versus strain curves by quadratic fitting functions. The mechanical properties of individual carbon nanothread and CNT are shown in Table 9.2. It can be seen that the CNT possesses a higher elastic constant and gravimetric Table 9.2 Mechanical properties of individual carbon nanothread and (10,10) CNT as derived from MD simulations. Reprinted (adapted) with permission from [22] Deformation mode

Type

Elastic constant k (MJ/kg)

Elastic limit εmax

Gravimetric energy density E/M (kJ/kg)

Torsion

Nanothread-A

1.78

0.71

884

Nanothread-C

2.70

0.51

737

CNT

19.76 (4.74a , 2.78b )

0.06 (0.92)

65 (2720)

Nanothread-A

77.60

0.18

2051

Nanothread-C

34.81

0.15

906

Tension

Bending

Compression

a,b

CNT

183.51

0.12 (0.22)

2374 (6810)

Nanothread-A

5.12

0.34

468

Nanothread-C

6.04

0.28

288

CNT

24.11

0.05 (0.23)

49.3 (618.6)

Nanothread-A

43.74

0.19

6051

Nanothread-C

31.80

0.18

3063

CNT

48.72

0.017 (0.33)

14.1 (4053)

The torsional elastic constants for CNT at the second and third elastic deformation stages. For CNT, the values within the parentheses are the elastic limit and the corresponding gravimetric energy density before the onset of bond breakage, and the values outside the brackets are the elastic limit and the corresponding gravimetric energy density before flattening or buckling

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energy density than carbon nanothread under torsion, tension and bending, which indicates that the CNT has a better energy storage in these deformation circumstances. According to Hook’s law, the strain energy density E tot /M can be expressed as [23]    1 2 2 2 εs,i + kb εb,i + kc εc,i [nkt εt2 + ks j ], n i=1 i=1 i< j n

E tot /M =

n

(9.18)

with

εs,i =

εt = ϕ D0 /l0 ,

(9.19)

1 + (ρi ϕ/l0 )2 − 1,

(9.20)

εb = D0 /R,

(9.21)

εc,i j = 1 − di j /d0 .

(9.22)

Here,ρi is the rotational radius, which is relative to the closest neighbor interthread distance (di j ) through ρi = di j /2sin(π/n) when filament number n is < 7. By assuming that each nanothread has the same equilibrium distance d0 to its closest neighbors, the dimensionless torsional strain (εt ) is equal to the total torsion strain. Therefore, both tensile strain and bending strain can be correlated with the dimensionless torsional strain and compressive strain using the Taylor series. More details about the numerical derivation and ρi for larger n can be referred to reference [22]. Finally, the tensile strain and bending strain can be expressed as εs,i = η2 d02 (1− 2 2 /2D02 , εb,i = ηd0 (1 − εc,i j )εt,i /D0 , respectively, with η = [2sin(π/n 1 )]−1 . εc )2 εt,i The total strain energy density is given as E tot (εt , εc )/M ≈

1 ηd0 2 ks ηd0 4 ) (1 − εc )4 εt4 + n 1 kb ( ) (1 − εc )2 εt4 + n c kc εc2 ] [nkt εt2 + n 1 ( n 4 D0 D0

(9.23)

where n 1 is the number of threads having tension and bending, and n c is the number of pair threads. Given that the system has reached an energy minimization, the torsional strain εt can be expressed by the compressive strain ∂ E tot /∂εc = 0. The predicted strain energy densities based on the theoretical model for bundle-3, -7, and -19 that are comprised of nanothread-A, nanothread-C, and (10,10) CNT are depicted in Fig. 9.26. The predicted results agree with the MD results at low strain (e.g., εt < 10%). Therefore, the theoretical model is available to analyze the contributions from various deformation modes to the energy storage in the linear elastic

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Fig. 9.26 Theoretical predictions of the contributions of different deformation modes. Strain energy density for bundle-3, -7, and -19 that are comprised of a nanothread-A; b nanothread-C; and c CNT(10,10). Reprinted (adapted) with permission from [22]

regime. It can be seen in Fig. 9.26 that torsion and tension are two predominant deformation modes for the energy storage of nanothread-A and nanothread-C bundles. At low strain, torsion is the dominant mode for bundles with small numbers of thread. At high strain, tension is the dominant mode for bundles with large numbers of threads. Although the theoretical model is established for the small deformation, the predicted strain energy density for nanothread bundles still agrees well with the MD results when εt is smaller than the elastic limit. However, the predicted strain energy density at the elastic limit for CNT bundles deviates largely from the MD simulation in Fig. 9.27, which is attributed to the nonlinearities induced by the twistinduced flattening. The energy stored in the twisted nanothread bundle is dominated

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Fig. 9.27 The theoretical predictions of the strain components as a function of the torsional strain for nanothread-A bundles. Reprinted (adapted) with permission from [22]

by the torsion in the early stage, and it is taken up by the tension gradually with the increased number of threads. Due to the small bending constant (kb ), the bending mode makes limited contribution to the energy storage despite the faster increase of bending strain than the tensile strain.

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