Nano Mechanics and Materials

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1 Nano Mechanics and Materials Theory, Multiscale Methods and Applications Wing Kam Liu Northwestern University, Illinois, USA Eduard G. Karpov Northwestern University, Illinois, USA Harold S. Park Vanderbilt University, Tennessee, USA

3 Nano Mechanics and Materials

5 Nano Mechanics and Materials Theory, Multiscale Methods and Applications Wing Kam Liu Northwestern University, Illinois, USA Eduard G. Karpov Northwestern University, Illinois, USA Harold S. Park Vanderbilt University, Tennessee, USA

6 Copyright 2006 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) (for orders and customer service enquiries): Visit our Home Page on All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or ed to permreq@wiley.co.uk, or faxed to (+44) This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA , USA Wiley-VCH Verlag GmbH, Boschstr. 12, D Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: ISBN-10: Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.

7 To Our Families

9 Contents Preface xi 1 Introduction Potential of Nanoscale Engineering Motivation for Multiple Scale Modeling Educational Approach Classical Molecular Dynamics Mechanics of a System of Particles Generalized Coordinates Mechanical Forces and Potential Energy Lagrange Equations of Motion Integrals of Motion and Symmetric Fields Newtonian Equations Examples Molecular Forces External Fields Pair-Wise Interaction Multibody Interaction Exercises Molecular Dynamics Applications Lattice Mechanics Elements of Lattice Symmetries Bravais Lattices Basic Symmetry Principles Crystallographic Directions and Planes Equation of Motion of a Regular Lattice Unit Cell and the Associate Substructure Lattice Lagrangian and Equations of Motion Examples Transforms Fourier Transform Laplace Transform Discrete Fourier Transform

10 viii CONTENTS 3.4 Standing Waves in Lattices Normal Modes and Dispersion Branches Examples Green s Function Methods Solution for a Unit Pulse Free Lattice with Initial Perturbations Solution for Arbitrary Dynamic Loads General Inhomogeneous Solution Boundary Value Problems and the Time History Kernel Examples Quasi-Static Approximation Equilibrium State Equation Quasi-Static Green s Function Multiscale Boundary Conditions Methods of Thermodynamics and Statistical Mechanics Basic Results of the Thermodynamic Method State Equations Energy Conservation Principle Entropy and the Second Law of Thermodynamics Nernst s Postulate Thermodynamic Potentials Statistics of Multiparticle Systems in Thermodynamic Equilibrium Hamiltonian Formulation Statistical Description of Multiparticle Systems Microcanonical Ensemble Canonical Ensemble Maxwell BoltzmannDistribution Thermal Properties of Periodic Lattices Numerical Heat Bath Techniques Berendsen Thermostat Nosé Hoover Heat Bath Phonon Method for Solid Solid Interfaces Introduction to Multiple Scale Modeling MAAD Coarse-Grained Molecular Dynamics Quasi-Continuum Method CADD Bridging Domain Introduction to Bridging Scale Bridging Scale Fundamentals Multiscale Equations of Motion Removing Fine Scale Degrees of Freedom in Coarse Scale Region Relationship of Lattice Mechanics to Finite Elements Linearized MD Equation of Motion

11 CONTENTS Elimination of Fine Scale Degrees of Freedom Commentary on Reduced Multiscale Formulation Elimination of Fine Scale Degrees of Freedom: 3D Generalization Numerical Implementation of Impedance Force Numerical Implementation of Coupling Force Discussion on the Damping Kernel Technique Programming Algorithm for Time History Kernel Cauchy Born Rule Virtual Atom Cluster Method Motivations and General Formulation General Idea of the VAC Model Three-Way Concurrent Coupling with QM Method Tight-Binding Method for Carbon Systems Coupling with the VAC Model Staggered Time Integration Algorithm MD Update FE Update Summary of Bridging Scale Equations Discussion on the Bridging Scale Method Bridging Scale Numerical Examples Comments on Time History Kernel D Bridging Scale Numerical Examples Lennard-Jones Numerical Examples Comparison of VAC Method and Cauchy Born Rule TruncationofTimeHistoryKernel D/3D Bridging Scale Numerical Examples Two-Dimensional Wave Propagation Dynamic Crack Propagation in Two Dimensions Dynamic Crack Propagation in Three Dimensions VirtualAtomClusterNumericalExamples Bending of Carbon Nanotubes VAC Coupling with Tight Binding Non-Nearest Neighbor MD Boundary Condition Introduction Theoretical Formulation in 3D Force Boundary Condition: 1D Illustration Displacement Boundary Condition: 1D Illustration Comparison to Nearest Neighbors Formulation Advantages of Displacement Formulation Numerical Examples: 1D Wave Propagation Time-History Kernels for FCC Gold Conclusion for the Bridging Scale Method Bridging Scale Perspectives ix

12 x CONTENTS 9 Multiscale Methods for Material Design Multiresolution Continuum Analysis Generalized Stress and Deformation Measures Interaction between Scales Multiscale Materials Modeling Multiscale Constitutive Modeling of Steels Methodology and Approach First-Principles Calculation Hierarchical Unit Cell and Constitutive Model Laboratory Specimen Scale: Simulation and Results Bio-Inspired Materials Mechanisms of Self-Healing in Materials Shape-Memory Composites Multiscale Continuum Modeling of SMA Composites Issues of Modeling and Simulation Summary and Future Research Directions Bio Nano Interface Introduction Immersed Finite Element Method Formulation Computational Algorithm of IFEM Vascular Flow and Blood Rheology Heart Model Flexible Valve Viscous Fluid Interaction Angioplasty Stent Monocyte Deposition Platelet Adhesion and Blood Clotting RBCAggregationandInteraction Electrohydrodynamic Coupling Maxwell Equations Electro-manipulation Rotation of CNTs Induced by Electroosmotic Flow CNT/DNA Assembly Simulation Cell Migration and Cell Substrate Adhesion Conclusions Appendix A Kernel Matrices for EAM Potential 297 Bibliography 301 Index 315

13 Preface Within the past decade, the emphasis of scientific research worldwide has shifted to the study of the behavior of materials at the atomic scale of matter. The proliferation of scientists and engineers studying matter at this length scale has led to the coining of the phrase nanotechnology. This term can generally be taken to imply the investigation and technological utilization of the properties of matter at length scales of one thousand nanometers or smaller. Generally, a few thousand atoms will exist in the space of thousand nanometers. As engineers typically study the mechanical properties of materials, the corresponding emphasis of research in the engineering community has been on nano mechanics. The term nano mechanics is typically associated with the study and characterization of the mechanical behavior of individual atoms, atomic-scale systems and structures in response to various types of forces and loading conditions. The specific nature of nano mechanics research generally varies depending on the discipline of the engineer; the topic of interest can involve the atomic-scale effect of fracture and wear on material performance, mechanical properties of nanocomposites, atomic-scale flow and locomotion of individual biological cells. Regardless of the interest of the particular scientist or engineer, what is universally agreed upon is the overall potential that nanotechnology, and particularly nano mechanics, has for the betterment of our society, including the sectors of private industry, national defense and homeland security. An emphasis on nanoscale entities will make our manufacturing technologies and infrastructure more sustainable in terms of reduced energy usage and environmental pollution. Recent advances made by the research community in this topic have stimulated ever-broader research activities in science and engineering that are devoted to their development and applications. Many areas of research are rapidly advancing owing to the combined efforts of science and engineering. In mechanics and materials, we are particularly excited with the progress in research and education that can be achieved by combining engineering and basic sciences through modeling and simulation together with experimentation. Owing to the combination of constantly increasing computational power and the increased knowledge and understanding of material behavior, multiple scale modeling methods have recently emerged as the tool of choice to link the mechanical behavior of materials from the smallest scale of atoms to the largest scale of structures. Multiple scale methods offer the best hope for bridging the traditional gap that exists between experimental approach, the theoretical approach and computational modeling for studying and understanding the behavior of materials. Owing to the central role that multiple scale methodology appears poised to play in the computational mechanics and materials science in the foreseeable future, this book aims to summarize the past and the current developments in multiple scale modeling to

14 xii PREFACE provide a coherent starting point from which interested scientists and engineers can begin their journey into this vast and rapidly expanding subject. We hope that this book is one of the first systematic works aimed at providing knowledge about fundamental concepts behind nanoscale mechanics and materials and the relevant applications. The book contains both published and previously unpublished material and is aimed at nanoscale engineers, designers, materials scientists and interested students and researchers. A salient feature of this book is that it is also intended to be used as an educational tool. The major reason is to synthesize the state of the art in multiple scale modeling techniques into the classroom such that the crucial tools being made available today are passed onto the next generation of scientists and engineers. Thus, the materials in this book which were previously used for courses at Northwestern University and the National Science Foundation (NSF) Summer Institute on Nano Mechanics and Materials have been coherently combined with Powerpoint lecture notes and selected computer codes (available online at to make the material presented readily accessible for those researchers who are interested in joining and contributing to the field of multiple scale modeling and analysis. Along with the review of basic theoretical concepts, they present the solutions and dynamic visualization of numerous practical problems, ranging from simple one-dimensional systems to state-of-the-art applications. The solutions of the simple illustrative problems are augmented by Matlab and Mathematica codes which serve to highlight the numerical implementation of the theoretical approaches presented in this book. There are many other novel and unique aspects to this book. As mentioned above, the integration of teaching and research is one of the key features. The material contains detailed expositions on all the topics that are necessary to fully comprehend multiple scale analysis. As such, the book is logically divided into three parts. The first part consists of Chapters 2 4, which cover the theoretical basis needed to understand the behavior of multiparticle atomistic systems. The second part consists of Chapters 5 8, and introduces multiple scale methods. In particular, the bridging scale concurrent approach, which is based on the theoretical considerations provided in the first part of the book, is given special attention here. The third part comprising Chapters 9 10 is devoted to contemporary applications in the area of nanostructured and bioinspired materials, biofluidics and cell mechanics. Chapter 1 contains an introduction, and emphasizes the need for multiple scale simulations by presenting case studies from different scientific disciplines, including materials design and biofluidics. Chapter 2 introduces the notion of Lagrangian dynamics description of systems of interacting particles, including nonconservative equations of motion, multibody interatomic potentials and arbitrary molecular shapes. Chapter 3 details the extension of the Lagrangian method to spatially periodic lattice structures; it reviews the relevant symmetry concepts, and derives the basic response solutions for a general three-dimensional lattice in semianalytical forms that are important in nanoscale engineering applications. Chapter 4 gives a systematic, though condensed, exposition on contemporary approaches that allow an averaged macroscopic characterization of multiparticle systems in thermodynamic equilibrium; these include the methods of thermodynamic potentials, statistical averaging, microcanonical and canonical ensemble theories. Chapter 5 provides an overview of multiple scale modeling. As such, previously developed multiple scale methods are reviewed and analyzed, and capabilities that are needed in

15 PREFACE multiple scale modeling are discussed and provided as a basis for the remaining chapters. Chapter 6 introduces the bridging scale concurrent method, which couples atomistic and continuum scale models; here, connections are made between the bridging scale, particle dynamics and lattice mechanics concepts introduced in Chapters 2 3. Numerical validation of the bridging scale approach is given in Chapter 7. The numerical examples in one, two and three dimensions highlight the applicability of the bridging scale to highly nonlinear physical phenomena, including the fracture and subsequent failure of materials. The recent extension of the bridging scale to incorporate quantum mechanical information into the coupling of length scales framework is also described in this chapter. Chapter 8 provides an extension of the MD impedance force such that it can be utilized with long-ranged interatomic potentials; this extension is crucial as most realistic interatomic potentials incorporate non-nearest neighbor bonding. This chapter concludes the section on multiple scale modeling with comments on future research directions. Chapter 9 highlights applications of multiple scale methods in crucial areas of physical interest. In the realm of solids, the topics covered are the hierarchical and concurrent design of realistic materials, including novel steel and metallic alloys, shape memory composites and self-healing materials. Lastly, Chapter 10 emphasizes new research in the area of computational biofluidics, electrohydrodynamics, bioengineering and nano-bio interfacial problems. The topics include electrophoresis multiscale and multiphysics modeling of red blood cell (RBC) aggregation and the effect on blood rheology, capillary flow, cell migration, nanomanipulation and assembly of macromolecules. We would like to thank our colleagues and graduate students for their contributions to this book, in particular, Ted Belytschko, Antonio Bouze, Dmitry Dorofeev, David Farrell, Mark Horstemeyer, Sukky Jun, Hiroshi Kadowaki, Adrian Kopacz, Yaling Liu, Cahal McVeigh, Sergey Medyanik, Dong Qian, Leonid Shilkrot, Shaoqiang Tang, Franck Vernerey and Sulin Zhang. Finally, we would like to thank and acknowledge the following sponsors for their support: the National Science Foundation (NSF), the NSF Summer Institute on Nano Mechanics and Materials, the NSF Integrative Graduate Education and Research Traineeship (IGERT) program, the Army Research Office (ARO), the Office of Naval Research (ONR) CyberSteel 2020 project and the ONR Nanofilament-Based Combined Chemical/Biological Detectors Project. xiii Evanston, Illinois Nashville, Tennessee Wing Kam Liu Eduard G. Karpov Harold S. Park

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