Reviews in computational chemistry, volume 27

Here an atomic version of the elasticity theory for isotropic,homogeneous materials is established and the need is highlighted for includingthree-body interactions in force fields for a

Reviews in

Computational Chemistry

Volume 27

Reviews in

Computational Chemistry 27

Edited by

Kenny B Lipkowitz

Editor Emeritus

Donald B Boyd

Kenny B Lipkowitz

Office of Naval Research

875 North Randolph Street

Arlington, VA 22203-1995 U.S.A.

kenny.lipkowitz@navy.mil

Donald B Boyd Department of Chemistry and Chemical Biology

Indiana University-Purdue University at Indianapolis

402 North Blackford Street Indianapolis, Indiana 46202-3274 U.S.A.

dboyd@iupui.edu Copyright © 2011 by John Wiley & Sons, Inc All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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 as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,

MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com Requests

to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online

at http://www.wiley.com/go/permission.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.

Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic formats For more information about Wiley products, visit our web site at www.wiley.com.

Library of Congress Cataloging-in-Publication Data:

ISBN: 978-0470-58714-0

ISSN: 1069-3599

Printed in Singapore

10 9 8 7 6 5 4 3 2 1

Computational chemistry transcends traditional barriers separating chemistry,physics, and mathematics It is, de facto, a product of the “Computer Age,”but the impetus for its success really lies in the hands of scientists who needed

to better understand how Nature works Chemists in particular were able toadopt computational methodology quickly, in part because there were insti-tutions like the Quantum Chemistry Program Exchange disseminating soft-ware free of charge and websites like the Computational Chemistry List-serve making available a variety of services, but also because of books like

Reviews in Computational Chemistry providing tutorials and reviews,

espe-cially for nontheorists and novice molecular modelers By and large, tional chemistry moved from the domain of the theorist to that of the benchchemist, and it has moved from the realm of chemistry to other disciplines,most notably in the biological sciences where biologists are now adopting amolecular view of living systems

computa-Since this book series began, we sold more than 20,000 books coveringmyriad topics of interest to chemists Those topics were written by mathemati-cians, chemists, computer scientists, engineers, and physicists, and they cover awide swath of computing in science, engineering, and technology One area ofresearch where chemists are under-represented in terms of theory and simula-tion, however, is in multiscale modeling The scales typically involved here arethose in, say, a molecular dynamics protein folding study where picosecondsare required for assessing molecular vibrations but milliseconds are needed tounderstand segmental relaxation, and length scales in materials science whereangstrom-level views are needed to account for bond making and bond break-ing, but micron-level and larger views are required for predicting certain bulkbehavior

For some researchers, multiscale modeling means harnessing huge puting resources at places like Los Alamos National Laboratory where multi-million atom systems can be treated; for others it means extending simulationtimes for as long as possible But throwing “brute force” at a problem hasits limitations, and accordingly, more reasonable and more elegant approaches

com-to solving multiscale problems are needed Many advances in this regard havecome to fruition and are being used by some chemists An especially well-writtentutorial on the topic of multiscale modeling appeared in Volume 26 of this

v

vi Preface

book series; for the novice or uninformed reader, it is a chapter that is wellworth reading because it describes what “multiscale modeling” means, what iscurrently being done, and what still needs to be accomplished in this area oftheory/computation

Many companies rely heavily on simulations of mechanical properties ofmaterials for engineering purposes The mathematical basis for this (mechanics)rests on a continuum treatment of the material That method fails when thegranularity of the system is small (at the molecular level), so something specialneeds to be done to include the small length scales of atoms and molecules.This is true for modeling micro-cracks in bulk materials as an example, but it iseven more pressing for modeling the mechanical behavior of modern materialscomposed of (or incorporating) nanoparticles, which are now being preparedand evaluated for many uses

There is a movement afoot to couple continuum mechanics with atomisticmodels What is most needed in this area of analysis is ensuring that the correctatomistic information is fed back to the continuum mechanics model A con-certed effort is now being made by scientists and engineers to unify modeling in

a way where atomistic information is used, either sequentially or concurrently,with finite element methods employed in the area of mechanics To understandthe stress-strain relationships in polymers, composites, ceramics, and metals, forexample, requires model input at the atomic level and requires treating largevolumes of space incorporating millions of atoms My opinion is that chemistsare missing a golden opportunity, in terms of funding opportunities at agencieslike the National Science Foundation (NSF), U.S Department of Energy, andthe various U.S Department of Defense agencies, but also in terms of contribut-ing their considerable wealth of knowledge about chemical systems toward thisendeavor The following facts validate my opinion First, the number of publi-cations on the topic of multiscale modeling is increasing as depicted in Figure 1.This plot was obtained by searching SciFinder for “multiscale modeling” and

“multiscale simulation.” Omitted are search terms like “multiscale analysis,”

“multiscale approach,” and the like The use of multiscale modeling far exceedsthe relatively small number of publications indicated in this figure, however, be-cause many multiscale modelers work in defense agencies or in industry wherepublication is not de rigueur or it is outright forbidden

Second, the majority of these publications (∼33%) emanated from ments in engineering schools—most notably from mechanical, chemical, civil,aerospace, and bioengineering departments Approximately 8% were published

depart-by researchers in chemistry departments, 8% from physicists, and only 5% frommaterials science departments Industrial organizations like Toyota, Motorola,Samsung, 3-M, and software companies contributed ∼4%, whereas nationallaboratories, worldwide, contributed 14% as one might expect Approximately25% of the publications came from other departments like mathematics, frommixed departments, or could not otherwise be clearly identified Interestingly,less than 1% came from mechanics departments and only 2% came from metal-lurgy departments This assessment does not include the many papers published

Figure 1 Number of multiscale publications between 1998 and 2008.

under the moniker QM/MM and other such publications where small and largescales are being examined simultaneously; it includes only those papers that ex-plicitly refer to their studies as being multiscale in scope The point I am making

is that now is an excellent time for chemists to begin working in a developing

field of computing

With this theme of multiscale modeling, Stefano Giordano, AlessandroMattoni, and Luciano Colombo present in Chapter 1 a tutorial on how tomodel brittle fracture, as found in myriad materials we use everyday, includingmetals, ceramics, and composites The authors begin their tutorial by providing

an overview of continuum elasticity theory, introducing the ideas of stress andstrain, and then providing the constitutive equations for their relationship Thegoverning equations of elasticity and the constitutive equation of an elastic ma-terial is described before the authors focus on the microscopic (i.e., atomistictheory of elasticity) Here an atomic version of the elasticity theory for isotropic,homogeneous materials is established and the need is highlighted for includingthree-body interactions in force fields for a formal agreement with continuumelasticity theory; interatomic potentials for solid mechanics and atomic-scalestress are then described rigorously The authors consider linear elastic me-chanics by first examining stress concentration, the Griffith energy criterion (anenergy balance criterion), and then different modes of crack formation in twoand three dimensions The elastic behavior of multifractured solids is broughtforward before a review of atomistic simulations in the literature is given The

viii Preface

chapter terminates with a detailed look at atomistic simulation of cubic siliconcarbide because it is the prototype of an ideally brittle material up to extremevalues of strain, strain rate, and temperature and because of its relevance intechnology Because the need exists to understand the mechanical properties ofnanoparticles that are becoming so prevalent nowadays, relying on mechanicalphenomena at a length scale where matter is treated as a continuum is not ten-able; this tutorial brings the reader up to speed in the area of mechanics, pointsout potential pitfalls to avoid, and reviews the literature of brittle fracture in arigorous, albeit straightforward, manner

Another approach for treating systems on the mesoscopic scale is to ploy dissipative particle dynamics (DPD), which is a coarse graining methodthat implements simplified potentials as well as grouping of atoms into a singleparticle In Chapter 2, Igor V Pivkin, Bruce Caswell, and George Em Karni-adakis describe how interacting clusters of molecules, subject to soft repulsions,are simulated under Lagrange conditions The authors begin with a basic math-ematical formulation and then highlight that, unlike the steep repulsion of aLennard–Jones potential, which increases to infinity as the separation distance,

em-r, approaches zero and imposes constraints on the maximum time step that

can be used to integrate the equations of motion, DPD numerically uses a soft,conservative potential obviating that problem The authors compare and con-trast the potentials used in traditional molecular dynamics (MD) simulationswith that of DPD keeping the mathematical rigor but with easy-to-follow ex-planations The thermostat used in DPD along with integration algorithms andboundary conditions are likewise described in a pedagogical manner With thatformal background, the authors then introduce extensions of the DPD method,including DPD with energy conservation, the fluid particle model, DPD fortwo-phase flows, and other extensions The final part of the chapter focuses

on applications of DPD, highlighting the simplicity of modeling complex ids Emphasized are polymer solutions and polymer melts, binary mixtures ofimmiscible liquids like oil-in-water and water-in-oil emulsions, as well as am-phiphilic systems constituting micelles, lipid bilayers, and vesicles The authorsend the chapter with an extreme example of multiscale modeling involvingdeformable red blood cells under flow resistance in capillaries

flu-For those of us who use atomistic MD simulation methods in chemistry,physics, or biology, we encounter rare, yet important, transitions between long-lived stable states These transitions might involve physical or chemical trans-formations and can be explored with classic potential functions or by quantum-based techniques In Chapter 3, Peter G Bolhuis and Christoph Dellago provide

an in-depth tutorial on the statistical mechanics of trajectories for studying rareevent kinetics After a brief introduction, the authors begin with transition statetheory (TST) Using mathematics in tandem with easy-to-follow figures that il-lustrate the concepts, the authors focus on statistical mechanical definitions,rate constants, TST, and variational TST before introducing us to reactive fluxmethods Here the Bennett–Chandler procedure is described in great detail as isthe effective positive flux and the Ruiz–Montero–Frenkel–Brey method Then,

transition path sampling is described, again, with simple cartoon-like figuresfor clarity of complex problems In this section, the authors illuminate pathprobability, order parameter, and sampling the path ensemble Also coveredare the shooting move, sampling efficiency, aimless shooting, and stochasticdynamics shooting, along with an explanation of which shooting algorithm touse The ensuing section of the tutorial covers the computation of rates withpath sampling Included here are the correlation function approach, transitioninterface sampling, partial path sampling, replica exchange, forward flux sam-pling, milestoning, and discrete path sampling Minimizing the action comprisesthe penultimate section of the tutorial Here the nudged elastic band method

is described along with action-based sampling and the string method The thors provide insights about how to identify the mechanism under investigationfrom the computed path ensemble in the final section of the tutorial Because

au-so many modelers are interested in topics beyond simple structure prediction, aneed exists for methods that can be implemented to compute low-probability,rare events; this chapter provides the detailed mathematics of those methods.Micro-electro-mechanical systems (MEMS) are used extensively in manydevices such as radar, disk drives, telecommunication equipment, and the like.Metal contacts that are repetitively opened and closed lead to degradation ofthose materials, and it is imperative that we understand the events leading tothis degradation so that better products can be engineered, especially as weminiaturize such machinery down to the nanoscale The metal surfaces makingcontact are not atomically smooth; instead, they have relatively rough surfaceswith thin metal asperities through which electrical current flows The resistance

of the electrons is a consequence of inelastic interactions between the electronsand the phonons, which in turn leads to Ohmic (Joule) heating As the temper-ature increases from this resistive heating, the ability of an electron to movethrough a wire decreases How one can model such systems is the focus ofChapter 4 where Douglas L Irving provides a tutorial on multiscale modeling

of metal/metal electrical contact conductance The author begins by describingfactors that influence contact resistance Surface roughness and local heatingare paramount in this regard, as are intermixing between different materialsused in the contacts and the dimensions of the contacting asperities He thenintroduces the computational methodology needed to model those influencingfactors, highlighting the fact that modeling metal/metal interfaces is inherently

a multiscale problem Atomistic methods like density functional theory, tightbinding methods, and potential energy functions are described For the treat-ment of systems containing hundreds of thousands to millions of metal atoms,the embedded atom method (EAM) and variations thereof are described Thecoupling of atomistic details to finite element and finite difference techniquesused in the area of mechanics is then described using simple mathematics gearedfor the novice Applications of these hybrid multiscale techniques are then de-scribed with several case studies that focus on electric conduction throughmetallic nanowires and then on the deformations of metals in contact withcompressive stresses This journey into the realm of metallurgy is enlightening,

ma-up to date on advances in the field of atomistic simulations of lipid bilayers Theybegin this tutorial/review by first addressing methodologies used for membranesimulation A focus is placed on force fields (especially those developed and pa-rameterized for lipid materials), the selection of appropriate statistical ensem-bles for simulations, force field validation, and Monte Carlo (MC) simulationmethods where the configuration-biased MC algorithm is described Selectingsuitable experiments with which to compare simulation results is also described.The second part of the chapter uses all of these ideas to show how one can carryout atomistic simulations of lipid bilayers; the authors cleverly disguise their tu-torial by examining four different microscopic level models proposed for choles-terol/phospholipid interactions that can produce liquid-ordered raft domains.

Of special note for the novice modeler is the explanation of the balance betweenenergetics and entropy; for the more experienced modeler, the complexities, util-ity, and pitfalls to avoid when using the isomolar semi-grand canonical ensemble

in MC simulations of bilayers consisting of more than one type of phospholipids

is especially important reading Although much is being done computationally

to characterize phase diagrams of ternary systems, the authors provide insightsabout what must be done next in this exciting area of theory

In 1952, David Bohm presented an interpretation of quantum mechanics(QM) that differs in profound ways from the standard way we think of quan-tal systems During the last decade there has been great interest in Bohm’sinterpretation and, in particular, in its potential to generate computationaltools for solving the time-dependent Schrödinger equation In Chapter 6, So-phya Garashchuk, Vitaly Rassolov and Oleg Prezhdo describe the semiclassicalmethodologies that are inspired by the Bohmian formulation of quantum me-chanics and that are designed to represent the complex dynamics of chemicalsystems The authors introduce the Madelung de Broglie–Bohm formalism bydrawing analogy with classical mechanics and explicitly highlighting the non-classical features of the Bohmian mechanics The nonclassical contributions tothe momentum, energy, and force are then described The fundamental proper-ties of the Bohmian quantum mechanics are discussed, including the conserva-tion and normalization of the QM probability, the computation of the QM ex-pectation values, properties of stationary states, and behavior at nodes Severalways to obtain the classical limit within the Bohmian formalism are considered.Then, mixed quantum/classical dynamics based on the Bohmian formalism isderived and illustrated with an example involving a light and a heavy particle

At this point, the Bohmian representation is used as a tool to couple the tum and classical subsystems The quantum subsystem can be evolved by either

quan-Bohmian or traditional techniques The quantum/classical formulation startswith the Ehrenfest approximation, which is the most straightforward and com-mon quantum/classical approach The Bohmian formulation of the Ehrenfestapproach is used to derive an alternative quantum/classical coupling schemethat resolves the so-called quantum backreaction problem, also known as thetrajectory branching problem The partial hydrodynamic moment approach tocoupling classical and quantum systems is outlined The hydrodynamic mo-ments provide a connection between the Bohmian and phase-space descrip-tions of quantum mechanics The penultimate section of this tutorial describesapproaches based on independent Bohmian trajectories It includes the deriva-tive propagation method, the stability approach, and the Bohmian trajectorieswith complex action Truncation of these hierarchies at the second order re-veals connection to other semiclassical methods Next, the focus shifts towardBohmian dynamics with globally approximated quantum potentials Separatesubsections are devoted to the global energy-conserving approximation for thenonclassical momentum, approximations on subspaces and spatial domains,and nonadiabatic dynamics Each approach is first introduced at the formaltheoretical level, and then, it is illustrated by an example The final sectiondeals with computational issues, including numerical stability, error cancella-tion, dynamics linearization, and long-time behavior The numerical problemsare motivated and illustrated by considering specific quantum phenomena, such

as zero-point energy and tunneling The review concludes with a summary ofthe semiclassical and quantum/classical approaches inspired by the Bohmianformulation of quantum mechanics The three appendices prove the quantumdensity conservation, introduce quantum trajectories in arbitrary coordinates,and explain optimization of simulation parameters in many dimensions.The final chapter by Dr Donald B Boyd is an overview of career opportu-nities in computational chemistry It was written in part to examine this aspect ofour history in computational chemistry but also as an aid for students and theiradvisors who are now deciding whether they should enter this particular work-force In addition to presenting trends in employment, the author provides data

on the types of computational chemistry expertise that have been most helpfulfor securing employment After an introduction, Dr Boyd describes how, in theearly days (1960s–1970s), computational scientists had meager support andpoor equipment with which to work; moreover, there was abundant skepticism

in those days that computing could become a credible partner with experiment.Those hard-fought efforts in computational chemistry allowed it to stand on

an equal footing with experiment, and accordingly, there was a commensuratespate of hiring in that field Dr Boyd provides a dataset of jobs available from

1983 to 2008 and then provides a detailed assessment of the kinds of jobsthey were (e.g., tenure-track positions, nontenured academic staff positions,positions at software or hardware companies, and other such positions) Hefurther elaborates on the specific type of expertise employers were seeking atdifferent periods in time, tabulating for us the rankings of desired skill sets like

xii Preface

“working with databases,” “library design,” and “QSAR” as well as of morebroadly defined skills like “molecular modeling” and “computational chem-istry.” The author dissects all of his data in an interesting way, showing theebbs and flows of employment over time and weaving his story into the fabric

of social and economic changes that occurred over the years, especially in thepharmaceutical companies

Finally, an appendix is provided by the editor that lists the names ande-mail addresses of∼2500 people who regularly publish in the field of compu-tational molecular science (their postal addresses are available from the editorupon request) Those people are not called computational chemists, althoughmany are Instead, they are referred to as computational molecular scientists,and as you will note, many are physicists, biologists, engineers, mathematicians,materials scientists, and so on What they all have in common, however, is thatthey either develop or use computing tools to understand how nature works atthe atomic/molecular level

Because computational molecular science is so important in today’s oratory setting, we know that many experimentalists want to use the theoriesand the associated software developed by computational scientists for theirown needs The theoretical underpinnings and philosophical approaches used

lab-by theorists and software developers can sometimes be buried in terse ematics or hidden in other ways from the view of a traditional, black-box-using bench chemist who has little time to become truly proficient as a theorist.Yet, those experimentalists want very much to use computational tools to ra-tionalize their results or, in some instances, to make predictions about whatnext to do along their research trajectory Because of this need, we started

math-the Reviews in Computational Chemistry book series that, in hindsight, could

just as well have been called “Tutorials in Computational Chemistry.”

Because the emphasis of the material covered in this book series is directedtoward the novice bench chemist wanting to learn about a particular method

to solve their problems (or for that matter the veteran computational chemistneeding to learn a new technique with a modicum of effort), we have againasked our authors to provide a tutorial on the topic being reviewed As before,they have risen to the occasion and prepared pedagogically driven chapters withthe novice in mind

Note that our publisher now makes our most recent volumes available

in an online form through Wiley InterScience; please consult the Web (http://www.interscience.wiley.com/onlinebooks) or contact reference@wiley.com forthe latest information For readers who appreciate the permanence and conve-nience of bound books, these will, of course, continue

I thank the authors of this and previous volumes for their excellentchapters

Kenny B Lipkowitz,Washington, DCApril 2009

1 Brittle Fracture: From Elasticity Theory to Atomistic Simulations 1

Stefano Giordano, Alessandro Mattoni, and Luciano Colombo

Governing Equations of Elasticity and Border Conditions 18

xiii

xiv Contents

Igor V Pivkin, Bruce Caswell, and George Em Karniadakis

Peter G Bolhuis and Christoph Dellago

Rare Event Kinetics from Transition State Theory 113

Reactive Flux Methods 128

Enhancement of Sampling by Parallel

Transition Path Theory and the String Method 193Identifying the Mechanism from the Path Ensemble 196

Transition State Ensemble and Committor

xvi Contents

4 Understanding Metal/Metal Electrical Contact Conductance

Calculating Conductance of Nanoscale Asperities 230

Multiscale Methods Applied to Metal/Metal Contacts 241

5 Molecular Detailed Simulations of Lipid Bilayers 253

Max L Berkowitz and James T Kindt

Detailed Simulations of Bilayers Containing Lipid Mixtures 266

Sophya Garashchuk, Vitaly Rassolov, and Oleg Prezhdo

The Classical Limit of the Schr ¨odinger Equation and

the Semiclassical Regime of Bohmian Trajectories 297Using Quantum Trajectories in Dynamics

Mean-Field Ehrenfest Quantum-Classical Dynamics 301Quantum-Classical Coupling via Bohmian Particles 302Numerical Illustration of the Bohmian

Properties of the Bohmian Quantum-Classical

Hybrid Bohmian Quantum-Classical Phase–Space Dynamics 311

The Bohmian Trajectory Stability Approach Calculation

of Energy Eigenvalues by Imaginary Time Propagation 314

Dynamics with the Globally Approximated Quantum

Global Energy-Conserving Approximation of

Approximation on Subspaces or Spatial Domains 324

Stabilization of Dynamics by Balancing

7 Prospects for Career Opportunities in Computational Chemistry 369

Donald B Boyd

Max L Berkowitz, Department of Chemistry, Venable and Kenan Laboratories,

The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3290U.S.A (Electronic mail: maxb@unc.edu)

Peter Bolhuis, Computational Physics and Chemistry, van’t Hoff Institute for

Molecular Sciences, University of Amsterdam, Nieuwe Achtergracht 166, 1018

WV Amsterdam, The Netherlands (Electronic mail: bolhuis@science.uva.nl)

Donald B Boyd, Department of Chemistry and Chemical Biology, Indiana

University-Purdue University at Indianapolis, 402 North Blackford Street, dianapolis, IN 46202-3274 U.S.A (Electronic mail: dboyd@iupui.edu)

In-Bruce Caswell, Division of Applied Mathematics, Brown University, 182

George Street, Providence, RI 02912 U.S.A (Electronic mail: caswell@dam.brown.edu)

Luciano Colombo, Department of Physics of the University of Cagliari and

CNR-IOM (SLACS Unit), Cittadella Universitaria, I-09042 Monserrato (Ca),Italy (Electronic mail: luciano.colombo@dsf.unica.it)

Christoph Dellago, Faculty of Physics, University of Vienna, Boltzmanngasse

5, 1090 Vienna, Austria (Electronic mail: christoph.dellago@univie.ac.at)

Sophya Garashchuk, Department of Chemistry and Biochemistry, University

of South Carolina, 631 Sumter Street, Columbia, SC 29208 U.S.A (Electronicmail: sgarashc@chem.sc.edu)

Stefano Giordano, Department of Physics of the University of Cagliari and

CNR-IOM (SLACS Unit), Cittadella Universitaria, I-09042 Monserrato (Ca),Italy (Electronic mail: stefano.giordano@dsf.unica.it)

xix

xx Contributors

Douglas L Irving, Department of Materials Science and Engineering, North

Carolina State University, Campus Box 7907, Raleigh, NC 27695-7907 U.S.A.(Electronic mail: dlirving@ncsu.edu)

James Kindt, Department of Chemistry, Emory University, 1515 Dickey Drive,

Atlanta, GA 30322 U.S.A (Electronic mail: jkindt@emory.edu)

George Em Karniadakis, Division of Applied Mathematics, Brown

Uni-versity, 182 George Street, Providence, RI 02912 U.S.A (Electronic mail:gk@dam.brown.edu)

Alessandro Mattoni, Department of Physics of the University of Cagliari and

CNR-IOM (SLACS Unit), Cittadella Universitaria, I-09042 Monserrato (Ca),Italy (Electronic mail: alessandro.mattoni@dsf.unica.it)

Igor V Pivkin, Department of Materials Science and Engineering,

Mas-sachusetts Institute of Technology, 77 MasMas-sachusetts Avenue, 8-139, bridge, MA 02139 U.S.A (Electronic mail: piv@mit.edu)

Cam-Oleg Prezhdo, Department of Chemistry, University of Rochester, Rochester

NY 14627 U.S.A (Electronic mail: prezhdo@chem.rochester.edu)

Vitaly Rassolov, Department of Chemistry and Biochemistry, University of

South Carolina, 631 Sumter Street, Columbia, SC 29208 U.S.A (Electronicmail: rassolov@chem.sc.edu)

Contributors to

Previous Volumes

Volume 1 (1990)

David Feller and Ernest R Davidson, Basis Sets for Ab Initio Molecular Orbital

Calculations and Intermolecular Interactions

James J P Stewart, Semiempirical Molecular Orbital Methods.

Clifford E Dykstra, Joseph D Augspurger, Bernard Kirtman, and David J Malik, Properties of Molecules by Direct Calculation.

Ernest L Plummer, The Application of Quantitative Design Strategies in

Pesti-cide Design

Peter C Jurs, Chemometrics and Multivariate Analysis in Analytical Chemistry Yvonne C Martin, Mark G Bures, and Peter Willett, Searching Databases of

Three-Dimensional Structures

Paul G Mezey, Molecular Surfaces.

Terry P Lybrand, Computer Simulation of Biomolecular Systems Using

Molec-ular Dynamics and Free Energy Perturbation Methods

Donald B Boyd, Aspects of Molecular Modeling.

Donald B Boyd, Successes of Computer-Assisted Molecular Design.

Ernest R Davidson, Perspectives on Ab Initio Calculations.

xxi

xxii Contributors to Previous Volumes

Volume 2 (1991)

Andrew R Leach, A Survey of Methods for Searching the Conformational

Space of Small and Medium-Sized Molecules

John M Troyer and Fred E Cohen, Simplified Models for Understanding and

Predicting Protein Structure

J Phillip Bowen and Norman L Allinger, Molecular Mechanics: The Art and

Science of Parameterization

Uri Dinur and Arnold T Hagler, New Approaches to Empirical Force Fields Steve Scheiner, Calculating the Properties of Hydrogen Bonds by Ab Initio

Methods

Donald E Williams, Net Atomic Charge and Multipole Models for the Ab

Initio Molecular Electric Potential

Peter Politzer and Jane S Murray, Molecular Electrostatic Potentials and

Chem-ical Reactivity

Michael C Zerner, Semiempirical Molecular Orbital Methods.

Lowell H Hall and Lemont B Kier, The Molecular Connectivity Chi Indexes

and Kappa Shape Indexes in Structure-Property Modeling

I B Bersuker and A S Dimoglo, The Electron-Topological Approach to the

QSAR Problem

Donald B Boyd, The Computational Chemistry Literature.

Volume 3 (1992)

Tamar Schlick, Optimization Methods in Computational Chemistry.

Harold A Scheraga, Predicting Three-Dimensional Structures of Oligopeptides Andrew E Torda and Wilfred F van Gunsteren, Molecular Modeling Using

NMR Data

David F V Lewis, Computer-Assisted Methods in the Evaluation of Chemical

Toxicity

Volume 4 (1993)

Jerzy Cioslowski, Ab Initio Calculations on Large Molecules: Methodology and

Applications

Michael L McKee and Michael Page, Computing Reaction Pathways on

Molec-ular Potential Energy Surfaces

Robert M Whitnell and Kent R Wilson, Computational Molecular Dynamics

of Chemical Reactions in Solution

Roger L DeKock, Jeffry D Madura, Frank Rioux, and Joseph Casanova,

Com-putational Chemistry in the Undergraduate Curriculum

Volume 5 (1994)

John D Bolcer and Robert B Hermann, The Development of Computational

Chemistry in the United States

Rodney J Bartlett and John F Stanton, Applications of Post-Hartree–Fock

static Calculations and Brownian Dynamics Simulations

K V Damodaran and Kenneth M Merz Jr., Computer Simulation of Lipid

Systems

Jeffrey M Blaney and J Scott Dixon, Distance Geometry in Molecular

Mod-eling

Lisa M Balbes, S Wayne Mascarella, and Donald B Boyd, A Perspective of

Modern Methods in Computer-Aided Drug Design

Volume 6 (1995)

Christopher J Cramer and Donald G Truhlar, Continuum Solvation Models:

Classical and Quantum Mechanical Implementations

xxiv Contributors to Previous Volumes

Clark R Landis, Daniel M Root, and Thomas Cleveland, Molecular

Mechan-ics Force Fields for Modeling Inorganic and Organometallic Compounds

Vassilios Galiatsatos, Computational Methods for Modeling Polymers: An

In-troduction

Rick A Kendall, Robert J Harrison, Rik J Littlefield, and Martyn F Guest,

High Performance Computing in Computational Chemistry: Methods and chines

Ma-Donald B Boyd, Molecular Modeling Software in Use: Publication Trends Eiji Osawa and Kenny B Lipkowitz, Appendix: Published Force Field Param-

Jiali Gao, Methods and Applications of Combined Quantum Mechanical and

Molecular Mechanical Potentials

Libero J Bartolotti and Ken Flurchick, An Introduction to Density Functional

Zdenek Slanina, Shyi-Long Lee, and Chin-hui Yu, Computations in Treating

Fullerenes and Carbon Aggregates

Gernot Frenking, Iris Antes, Marlis Böhme, Stefan Dapprich, Andreas W Ehlers, Volker Jonas, Arndt Neuhaus, Michael Otto, Ralf Stegmann, Achim Veldkamp, and Sergei F Vyboishchikov, Pseudopotential Calculations of Tran-

sition Metal Compounds: Scope and Limitations

Thomas R Cundari, Michael T Benson, M Leigh Lutz, and Shaun O merer, Effective Core Potential Approaches to the Chemistry of the Heavier

Som-Elements

Jan Almlöf and Odd Gropen, Relativistic Effects in Chemistry.

Donald B Chesnut, The Ab Initio Computation of Nuclear Magnetic

Reso-nance Chemical Shielding

Volume 9 (1996)

James R Damewood, Jr., Peptide Mimetic Design with the Aid of

Computa-tional Chemistry

T P Straatsma, Free Energy by Molecular Simulation.

Robert J Woods, The Application of Molecular Modeling Techniques to the

Determination of Oligosaccharide Solution Conformations

Ingrid Pettersson and Tommy Liljefors, Molecular Mechanics Calculated

Conformational Energies of Organic Molecules: A Comparison of ForceFields

Gustavo A Arteca, Molecular Shape Descriptors.

Volume 10 (1997)

Richard Judson, Genetic Algorithms and Their Use in Chemistry.

Eric C Martin, David C Spellmeyer, Roger E Critchlow Jr., and Jeffrey

M Blaney, Does Combinatorial Chemistry Obviate Computer-Aided Drug

Design?

Robert Q Topper, Visualizing Molecular Phase Space: Nonstatistical Effects

in Reaction Dynamics

xxvi Contributors to Previous Volumes

Raima Larter and Kenneth Showalter, Computational Studies in Nonlinear

Dynamics

Stephen J Smith and Brian T Sutcliffe, The Development of Computational

Chemistry in the United Kingdom

Volume 11 (1997)

Mark A Murcko, Recent Advances in Ligand Design Methods.

David E Clark, Christopher W Murray, and Jin Li, Current Issues in De Novo

Molecular Design

Tudor I Oprea and Chris L Waller, Theoretical and Practical Aspects of

Three-Dimensional Quantitative Structure–Activity Relationships

Giovanni Greco, Ettore Novellino, and Yvonne Connolly Martin, Approaches

to Three-Dimensional Quantitative Structure–Activity Relationships

Pierre-Alain Carrupt, Bernard Testa, and Patrick Gaillard, Computational

Ap-proaches to Lipophilicity: Methods and Applications

Ganesan Ravishanker, Pascal Auffinger, David R Langley, Bhyravabhotla yaram, Matthew A Young, and David L Beveridge, Treatment of Counterions

Ja-in Computer Simulations of DNA

Donald B Boyd, Appendix: Compendium of Software and Internet Tools for

Computational Chemistry

Volume 12 (1998)

Hagai Meirovitch, Calculation of the Free Energy and the Entropy of

Macro-molecular Systems by Computer Simulation

Ramzi Kutteh and T P Straatsma, Molecular Dynamics with General

Holo-nomic Constraints and Application to Internal Coordinate Constraints

John C Shelley and Daniel R Bérard, Computer Simulation of Water

Ph-ysisorption at Metal–Water Interfaces

Donald W Brenner, Olga A Shenderova, and Denis A Areshkin,

Quantum-Based Analytic Interatomic Forces and Materials Simulation

Henry A Kurtz and Douglas S Dudis, Quantum Mechanical Methods for

Predicting Nonlinear Optical Properties

Chung F Wong, Tom Thacher, and Herschel Rabitz, Sensitivity Analysis in

Thomas Bally and Weston Thatcher Borden, Calculations on Open-Shell

Molecules: A Beginner’s Guide

Neil R Kestner and Jaime E Combariza, Basis Set Superposition Errors: Theory

and Practice

James B Anderson, Quantum Monte Carlo: Atoms, Molecules, Clusters,

Liq-uids, and Solids

Anders Wallqvist and Raymond D Mountain, Molecular Models of Water:

Derivation and Description

James M Briggs and Jan Antosiewicz, Simulation of pH-dependent Properties

of Proteins Using Mesoscopic Models

Harold E Helson, Structure Diagram Generation.

Volume 14 (2000)

Michelle Miller Francl and Lisa Emily Chirlian, The Pluses and Minuses of

Mapping Atomic Charges to Electrostatic Potentials

T Daniel Crawford and Henry F Schaefer III, An Introduction to Coupled

Cluster Theory for Computational Chemists

Bastiaan van de Graaf, Swie Lan Njo, and Konstantin S Smirnov, Introduction

to Zeolite Modeling

Sarah L Price, Toward More Accurate Model Intermolecular Potentials For

Organic Molecules

xxviii Contributors to Previous Volumes

Christopher J Mundy, Sundaram Balasubramanian, Ken Bagchi, Mark E Tuckerman, Glenn J Martyna, and Michael L Klein, Nonequilibrium Molec-

ular Dynamics

Donald B Boyd and Kenny B Lipkowitz, History of the Gordon Research

Conferences on Computational Chemistry

Mehran Jalaie and Kenny B Lipkowitz, Appendix: Published Force Field

Pa-rameters for Molecular Mechanics, Molecular Dynamics, and Monte CarloSimulations

Volume 15 (2000)

F Matthias Bickelhaupt and Evert Jan Baerends, Kohn-Sham Density

Func-tional Theory: Predicting and Understanding Chemistry

Michael A Robb, Marco Garavelli, Massimo Olivucci, and Fernando Bernardi,

A Computational Strategy for Organic Photochemistry

Larry A Curtiss, Paul C Redfern, and David J Frurip, Theoretical Methods

for Computing Enthalpies of Formation of Gaseous Compounds

Russell J Boyd, The Development of Computational Chemistry in Canada.

Volume 16 (2000)

Richard A Lewis, Stephen D Pickett, and David E Clark, Computer-Aided

Molecular Diversity Analysis and Combinatorial Library Design

Keith L Peterson, Artificial Neural Networks and Their Use in Chemistry Jörg-Rüdiger Hill, Clive M Freeman, and Lalitha Subramanian, Use of Force

Fields in Materials Modeling

M Rami Reddy, Mark D Erion, and Atul Agarwal, Free Energy Calculations:

Use and Limitations in Predicting Ligand Binding Affinities

Volume 17 (2001)

Ingo Muegge and Matthias Rarey, Small Molecule Docking and Scoring Lutz P Ehrlich and Rebecca C Wade, Protein-Protein Docking.

Christel M Marian, Spin-Orbit Coupling in Molecules.

Lemont B Kier, Chao-Kun Cheng, and Paul G Seybold, Cellular Automata

Models of Aqueous Solution Systems

Kenny B Lipkowitz and Donald B Boyd, Appendix: Books Published on the

Topics of Computational Chemistry

Steven W Rick and Steven J Stuart, Potentials and Algorithms for

Incorporat-ing Polarizability in Computer Simulations

Dmitry V Matyushov and Gregory A Voth, New Developments in the

Theo-retical Description of Charge-Transfer Reactions in Condensed Phases

George R Famini and Leland Y Wilson, Linear Free Energy Relationships

Using Quantum Mechanical Descriptors

Sigrid D Peyerimhoff, The Development of Computational Chemistry in

Ger-many

Donald B Boyd and Kenny B Lipkowitz, Appendix: Examination of the

Em-ployment Environment for Computational Chemistry

Volume 19 (2003)

Robert Q Topper, David, L Freeman, Denise Bergin, and Keirnan R LaMarche, Computational Techniques and Strategies for Monte Carlo Ther-

modynamic Calculations, with Applications to Nanoclusters

David E Smith and Anthony D J Haymet, Computing Hydrophobicity Lipeng Sun and William L Hase, Born-Oppenheimer Direct Dynamics Classical

Trajectory Simulations

Gene Lamm, The Poisson-Boltzmann Equation.

xxx Contributors to Previous Volumes

Volume 20 (2004)

Sason Shaik and Philippe C Hiberty, Valence Bond Theory: Its History,

Fun-damentals and Applications A Primer

Nikita Matsunaga and Shiro Koseki, Modeling of Spin Forbidden Reactions Stefan Grimme, Calculation of the Electronic Spectra of Large Molecules Raymond Kapral, Simulating Chemical Waves and Patterns.

Costel Sârbu and Horia Pop, Fuzzy Soft-Computing Methods and Their

Ap-plications in Chemistry

Sean Ekins and Peter Swaan, Development of Computational Models for

En-zymes, Transporters, Channels and Receptors Relevant to ADME/Tox

Volume 21 (2005)

Roberto Dovesi, Bartolomeo Civalleri, Roberto Orlando, Carla Roetti, and Victor R Saunders, Ab Initio Quantum Simulation in Solid State Chemistry Patrick Bultinck, Xavier Gironés, and Ramon Carbó-Dorca, Molecular Quan-

tum Similarity: Theory and Applications

Jean-Loup Faulon, Donald P Visco, Jr., and Diana Roe, Enumerating

Molecules

David J Livingstone and David W Salt, Variable Selection- Spoilt for Choice Nathan A Baker, Biomolecular Applications of Poisson-Boltzmann Methods Baltazar Aguda, Georghe Craciun, and Rengul Cetin-Atalay, Data Sources and

Computational Approaches for Generating Models of Gene Regulatory works

Net-Volume 22 (2006)

Patrice Koehl, Protein Structure Classification.

Emilio Esposito, Dror Tobi, and Jeffry Madura, Comparative Protein

Modeling

Joan-Emma Shea, Miriam Friedel, and Andrij Baumketner, Simulations of

Pro-tein Folding

Marco Saraniti, Shela Aboud, and Robert Eisenberg, The Simulation of Ionic

Charge Transport in Biological Ion Channels: An Introduction to NumericalMethods

C Matthew Sundling, Nagamani Sukumar, Hongmei Zhang, Curt Breneman, and Mark Embrechts, Wavelets in Chemistry and Chemoinformatics.

Volume 23 (2007)

Christian Ochsenfeld, Jörg Kussmann, and Daniel S Lambrecht, Linear Scaling

Methods in Quantum Chemistry

Spiridoula Matsika, Conical Intersections in Molecular Systems.

Antonio Fernandez-Ramos, Benjamin A Ellingson, Bruce C Garrett, and ald G Truhlar, Variational Transition State Theory with Multidimensional Tun-

Don-neling

Roland Faller, Coarse-Grain Modelling of Polymers.

Jeffrey W Godden and Jürgen Bajorath, Analysis of Chemical Information

Content Using Shannon Entropy

Ovidiu Ivanciuc, Applications of Support Vector Machines in Chemistry Donald B Boyd, How Computational Chemistry Became Important in the

Pharmaceutical Industry

Volume 24 (2007)

Martin Schoen, and Sabine H L Klapp, Nanoconfined Fluids Soft Matter

Between Two and Three Dimensions

Volume 25 (2007)

Wolfgang Paul, Determining the Glass Transition in Polymer Melts.

Nicholas J Mosey and Martin H Müser, Atomistic Modeling of Friction.

xxxii Contributors to Previous Volumes

Jeetain Mittal, William P Krekelberg, Jeffrey R Errington, and Thomas M Truskett, Computing Free Volume, Structured Order, and Entropy of Liquids

and Glasses

Laurence E Fried, The Reactivity of Energetic Materials at Extreme Conditions Julio A Alonso, Magnetic Properties of Atomic Clusters of the Transition Ele-

ments

Laura Gagliardi, Transition Metal- and Actinide-Containing Systems Studied

with Multiconfigurational Quantum Chemical Methods

Hua Guo, Recursive Solutions to Large Eigenproblems in Molecular

Spec-troscopy and Reaction Dynamics

Hugh Cartwright, Development and Uses of Artificial Intelligence in Chemistry.

Volume 26 (2009)

C David Sherrill, Computations of Noncovalent  Interactions.

Gregory S Tschumper, Reliable Electronic Structure Computations for Weak

Noncovalent Interactions in Clusters

Peter Elliott, Filip Furche, and Kieron Burke, Excited States from

Time-Dependent Density Functional Theory

Thomas Vojta, Computing Quantum Phase Transitions.

Thomas L Beck, Real-Space Multigrid Methods in Computational Chemistry Francesca Tavazza, Lyle E Levine, and Anne M Chaka, Hybrid Methods for

Atomic-Level Simulations Spanning Multi-Length Scales in the Solid State

Alfredo E Cárdenas and Eric Bath, Extending the Time Scale in Atomically

Detailed Simulations

Edward J Maginn, Atomistic Simulation of Ionic Liquids.

Brittle Fracture: From Elasticity Theory

to Atomistic Simulations

Stefano Giordano, Alessandro Mattoni,

and Luciano Colombo

Department of Physics of the University of Cagliari and

CNR-IOM (SLACS Unit), Cittadella Universitaria, I-09042 Monserrato (Ca), Italy

INTRODUCTION

Understanding the mechanical properties of materials with theory tionally has been done by using continuum methods, ranging from elastic theory(in both linear and nonlinear regimes), to plastic theory, and to fracture me-chanics The computational counterpart of continuum modeling is represented

tradi-by finite element analysis Continuum theories have been extremely ful, as proved by the tremendous achievements reached in structural design ofbuildings, ships, bridges, air-/space crafts, nuclear reactors, and so on Overallthis represents the core of theoretical and computational solid mechanics

success-In the last 20 years or so, the technological rush toward nano-sized systemshas forced researchers to investigate mechanical phenomena at a length scale

in which matter no longer can be considered as a continuum This is the case,for instance, of investigating the crack-related features in a material displayingelastic or structural complexity (or, equivalently, inhomogeneity or disorder)

at the nanoscale This problem of atomic-scale granularity immediately seems

to be prohibitive for (standard) solid mechanics To better elaborate on this

Reviews in Computational Chemistry, Volume 27

edited by Kenny B Lipkowitz Copyright © 2011 John Wiley & Sons, Inc.

1

2 Brittle Fracture: From Elasticity Theory to Atomistic Simulations

concept, let us focus on the case of a crack propagating into a nano-compositematerial, where occasionally it faces phase boundaries between the matrix andthe fiber There are serious conceptual limitations in applying solid mechanics

in such situations because continuum stress and strain fields are mathematicallysingular at vanishing distances from the crack tip This, of course, prevents anymeaningful application of continuum mechanics over a region in the near vicin-ity of the crack tip (i.e., at the length scale where a direct interaction between thecrack and the phase boundary indeed occurs) Computational limitations exist

as well for the same problem As a matter of fact, the total numerical workload

of the continuum analysis based on finite elements could become prohibitivelylarge because of the extreme refinement of the numerical mesh that is required

to take into account the inhomogeneity displayed at the nanoscale Such amesh refinement, therefore, would be stopped at a larger length scale, repre-senting a (possibly bad) coarse grain picture of the actual elastic or structuraldisorder

A new feeling within the computational materials science community isthat a completely different approach (other than continuum modeling) really

is needed for predicting mechanical properties at the nanoscale Such a novelapproach is based on a direct atomistic description of relevant phenomena,and therefore, it has been named atomistic (or atomic-scale) modeling Thekey idea of atomistic modeling is to look at a solid body under mechanicalload as being an assembly of atoms interacting through direct coupling; theircollective response to loading eventually will drive the overall mechanical re-sponse (for the above discussed case, such a collective response will drive thepropagation of the crack) Because the material is now resolved atomistically,there is no ambiguity in representing its actual nanostructure, displaying inprinciple any combination (at any possible relative distance) of cracks, phaseboundaries, or whatever kind of elastic inclusion In other words, atomisticmodeling naturally operates at the length scale, which falls out-of-reach ofcontinuum theories Furthermore, because the response is represented by thecollective displacement of atoms, the mechanical behavior is governed by theselected interatomic potentials, which in turn, are derived from a fundamen-tal analysis of chemical bonding between atoms In other words, no guess isneeded any longer about the constitutive equations for the mechanical behavior(i.e., the actual stress–strain relation for the investigated material) To clarifythis conceptual breakthrough, it is useful to turn back to the crack-inclusioninteraction problem; the mathematical singularity of stress and strain fields atthe crack tip is removed naturally when mapping the problem onto a discreteatom-resolved lattice The elementary step for crack advancement, in fact, isrepresented by a bond breaking event, whereas the corresponding strain fieldsimply is computed by the prediction of the new atomic coordinates (just afterthe bond snaps) Similarly, the local stress is computed on each displaced atom

so that no singular behavior ever is reached In this respect, atomistic modelingcould be viewed at as a first-principles mechanical theory

The present chapter mainly is intended as a tutorial introduction to brittlefracture Although the emphasis is on atomistic simulations, a detailed (but,hopefully, gentle) introduction to the continuum elasticity theory and to fracturemechanics is offered as well We believe that basic mechanical concepts likestrain, stress, and border conditions—which are central to this topic—moreeffectively are introduced and discussed within a continuum framework Thisallows us to develop such concepts at the needed degree of rigorous formalism,

as is actually done in the “Essential Continuum Elasticity Theory” Section

In the “Microscopic Theory of Elasticity” Section, we introduce the scopic theory of elasticity, in which the atomic (discrete) structure of materialsexplicitly is taken into account as the main underlying constitutive hypothesis

micro-By making use of simple two-dimensional model systems, we develop the mostfundamental features of the microscopic description of elasticity This will deter-mine the minimal degree of complexity that any interatomic force model mustdisplay to describe correctly essential elasticity We then will describe moderninteraction potentials and outline their most recent applications Another veryimportant topic discussed in this section is the atomic-scale formulation of thestress; here we develop the formalism under the most general assumptions, andprovide practical recipes for any two-body or many-body potential Establish-ing a clean and complete theory for atomic-level stress tensor, which today isstill a matter of investigation, is a crucial part of this section

The “Linear Elastic Fracture Mechanics” Section is devoted to presentingthe basics of brittle fracture, starting from the energy balance criterion devel-oped by Griffith Here, we also discuss the typical border conditions reflectingthe kind of loading that can be applied to a cracked solid The importance ofthis issue often is underestimated in typical atomistic simulations, which there-fore, sometimes do not correspond—even if technically correct—to any realisticsituation We also present some of our recent continuum results obtained formulticracked systems Finally, the section is completed with a qualitative intro-duction to the atomistic view of fracture

This review concludes with a section titled “Atomistic Investigation onBrittle Fracture” in which we discuss extensively our investigations on brittlefracture in silicon carbide Several topics are developed, all of them being un-derpinned by the same concept: Atomistic simulations are both consistent withstandard fracture mechanics (when referred to a situation that can be treatedequally well by two different approaches), and they provide a valuable source

of hints for developing improved continuum models Our main message is that

by means of molecular dynamics simulations, it is indeed possible to developatomically informed mesoscopic models that enlarge the range of validity ofcontinuum theory down to the nanoscale

Fracture mechanics is a beautiful example of how a natural science hasdeveloped over the years It is, therefore, instructive to consider its historicalevolution The attempt to formulate a microscopic (i.e., atomistic or molecular)theory of elasticity has been addressed largely in the scientific literature since

4 Brittle Fracture: From Elasticity Theory to Atomistic Simulations

the first approaches to model the mechanical behavior of elastic bodies Duringthe nineteenth century, different approaches have been followed Fresnel1andNavier2published in 1820 and 1821, respectively, very similar results based on

the so-called corpuscular approach They systematically adopted the Lagrange

“M´ecanique analytique,” describing the motion of an elastic solid decomposedinto a given collection of point masses interacting by means of distance-varyingelastic forces This approach did not consider the modern concept of stressbecause the forces were transmitted at the molecular level only Although thismicroscopic description of fundamental interactions is qualitatively consistentwith modern solid state physics,3–5the model by Fresnel and Navier (as well astheir actual understanding of microscopic material physics) was too rudimentaland, therefore, resulted in being insufficient for developing either a consistent

or a predictive theory An alternative methodology was followed by assumingthe mass distribution within a solid body to be continuous throughout its vol-ume; in 1822, Cauchy6introduced the continuum approach to study the elastic

properties of solid bodies Cauchy obtained the equilibrium equations exactly

in the same form in which they appear in modern textbooks; in particular, hedefined a tensorial pressure (stress), and he proved that the stress tensor di-vergence is zero (at equilibrium and in absence of volumetric external forces).Moreover, in 1828, Cauchy7 introduced the linear constitutive relations (theHooke law established in 1678) defining two different elastic constants needed

to model isotropic media

Despite several efforts, the problem of reconciling the opposing puscular and continuum approaches remained an intriguing challenge formany years.8The simplest atomistic models—including only central two-bodyinteractions—describe the mechanical behavior of any material by means of

cor-a single elcor-astic constcor-ant, cor-a sort of sccor-alcor-ar stiffness At vcor-aricor-ance, the continuum

approach predicts, in the isotropic case, the need for two independent andmaterial-specific parameters So, the basic question is as follows: Do we needjust one modulus or actually two elastic moduli to describe elastic isotropicmedia properly?

The first robust attempt to address this problem was given by Voigt.9According to his model, the regular structure of a crystal suggests that, when

a molecule (or atom) is added to the lattice, an ad hoc couple of forces act

on the molecule to set its correct orientation within the crystal In modernterminology, we can say that such a molecular torque corresponds to an effectivemany-body interaction that is at work among the elementary constituents of thelattice (either atoms or molecules) By considering both the central forces andthe three-body interactions (i.e., the simplest effective molecular torques), Voigtobtained the general equations of elasticity theory for isotropic solids containingtwo independent constants, as predicted by the continuum approach and as isconsistent with experimental knowledge In conclusion, three-body forces andangle-dependent forces must be considered to reproduce the correct behavior

of a solid elastic body

The modern theory of elasticity is concerned with the mechanics of formable media, which completely recover their original shape and give up allwork expended in the deformation after the applied deforming forces are re-moved The development of the theory of elasticity was based on the concept of

de-a continuous medium, which ende-ables one to ignore its de-atomic structure de-and todescribe macroscopic phenomena by the methods of continuum mechanics.10Within the framework of elasticity theory, the so-called fracture mechanics hasbeen introduced, which deals with the failure of a given body or structure due

as a result of the propagation of cracks or fractures.11

The fundamental science underlying fracture is rich, spanning fromphysics and chemistry at the atomic scale to micromechanics of materials and

to continuum mechanics of structures on the large scale Most real materials,when loaded with some stresses, can exhibit internal cuts in their microstruc-ture, called cracks or fractures, which cause degradation of the mechanicalproperties or complete breaking (failure) Thus, it is observed that fracture is

a significant problem in the industrialized world and that a theoretical andpractical basis for design against fracture is needed Fracture mechanics dealsessentially with the following problems Given a structure with a preexistingcrack or crack-like flaw, we must determine what loads can be tolerated bythe structure for any given crack size or configuration Similarly, considering

a structure in a given state of load, it is important to predict the creation orthe growth of a crack Moreover, for a given number of cycles of loading in asystem, we are interested in determining when a crack propagates catastroph-ically Finally, we might ask what size crack can be allowed to exist in a givencomponent of a device or engineering structure for it to operate safely.From a historical point of view, the first experiments on fracture mechan-ics were performed by Leonardo da Vinci, who measured the strength of ironwires in terms of their length He found that the strength varied inversely withwire length This result implied that flaws in materials govern the strength Infact, for a longer wire, we have a larger volume of material, and therefore, there

is a higher probability of encountering many flaws Of course, it is a tive result only The first quantitative result connecting mechanical stress andcrack size was found by Griffith in 1920,12and fracture mechanics became ascience-based engineering discipline during World War II For a brief review ofthe history and development of fracture mechanics, see Ref 13

qualita-ESSENTIAL CONTINUUM ELASTICITY THEORY Conceptual Layout

The classical theory of elasticity is based on the approximation of

con-tinuum medium, which consists of replacing the full set of pointlike atomic

masses distributed within a solid body by a continuum distribution of mass

6 Brittle Fracture: From Elasticity Theory to Atomistic Simulations

This approximation is valid when the spatial wavelength of the displacementfield (describing the imposed deformation) is much greater than the interatomicdistance In this case, the crystalline structure is not relevant for determiningthe variation of the shape of the solid body; the continuum macroscopic de-scription is, in fact, sufficient to study its mechanical response The next mostimportant ideas of elasticity theory are the concepts of strain and the stress,both of which are described easily by means of specific mathematical objectscalled tensors.14–16

A deformation relates two configurations (or states) of the material The

initial state is called the reference configuration and usually refers to the initial time; the other is called the current configuration and refers to a following

time (which may be regarded conveniently as the present moment).17,18 Inlinear elasticity, the strains (typically extensions and shears) and the angles ofrotation are considered small.19 In this case, we use the infinitesimal strain

tensor (or small strain tensor), which is the main object introduced to describe

all deformation features.20,21

To calculate the force of interaction between volume elements situated in

an arbitrary closed region (imagined to be isolated within the body) and volumeelements situated outside this region, it was advantageous to introduce the con-cept of the average force of interaction between them This approach provides

us with the definition of the stress tensor, which takes into consideration all

interaction forces among the volume elements of the continuum body.22,23The strain in a given body can be considered the effect of the appliedstress The relationship between the strain tensor and the stress tensor depends

on the material under consideration, and therefore, it is called the

constitu-tive equation.22The empirical Hooke law establishes a linear relation betweenstresses (forces inside the body) and strains (deformations of the body itself)

In its general form, Hooke’s law can describe an arbitrary inhomogeneous andanisotropic behavior of the material under consideration.20However, the mostsimple and important constitutive equation used in elasticity theory applies tomaterials that are homogeneous (the elastic behavior is the same at any point

of the body) and isotropic (the direction of application of the stress is not vant) The linear, homogeneous, and isotropic constitutive equation is obtainedand discussed in the following sections

rele-The Concept of Strain

Let x be the position vector of a volume element within a body in its

reference (equilibrium) configuration, and let X be the position of the samevolume element in the current configuration Both configurations are framedwithin the same cartesian coordinate system (see Figure 1) Because X is afunction of x, we can write the following:

X = f (x) =f1(x) , f2(x) , f3(x) [1]

Figure 1 Reference configuration and current configuration after a deformation.

We observe that the function f, connecting the vector  X to the vector x, is

a vector field Of course, the relation f (x) /= f (y) is verified for any pair of

volume elements with x /= y in the reference configuration This means that f

is a biunivocal vector function, and therefore, the inverse function f−1always

exists We also assume that f and f−1 are differentiable functions Basically,

the vector field f (x) contains all the information about the deformation driving

the solid body from the reference to the current configuration In the theory of

elasticity, the deformation gradient ˆF=F ij , i, j = 1, 2, 3, and

F ij= ∂f i

∂x j

[2]

is introduced The matrix ˆF also is referred to as the Jacobian matrix of the

transformation and has two important properties: (1) It is not singular because

of the invertibility of f (∃ ˆF−1such that ˆF ˆF−1 = ˆF−1ˆF = ˆI); and (2) its nant is always strictly positive (det F > 0).17We can better exploit the concept

determi-of deformation by introducing the displacement fieldu(x) as:

The Jacobian matrix of the displacement ˆJ = {J ij , i, j = 1, 2, 3} (i.e., the

displace-ment gradient), therefore, is calculated as:

8 Brittle Fracture: From Elasticity Theory to Atomistic Simulations

In linear elasticity, the extent of the deformations is assumed small

Al-though this notion is intuitive, it can be formalized by imposing that, for small

deformations, ˆF is very similar to ˆI or, equivalently, that ˆJ is very small fore, we adopt as an operative definition of small deformation the following

Accordingly, we define the (symmetric) infinitesimal strain tensor (or small

strain tensor) as:

 ij =12

prop-of deformation prop-of an elastic body:

r For a pure local rotation (a volume element is rotated but not changed

in shape and size), we have ˆJ = ˆ and, therefore, ˆ = 0 This means that

the small strain tensor does not take into account any local rotation butonly the changes of shape and size (dilatations or compression) of thatelement of volume.22

Let us clarify this fundamental result with pointx inside a volume

ele-ment that is transformed to x + u(x) in the current configuration

Un-der a pure local rotation, we have x + u(x) = ˆRx, where ˆR is a given orthogonal rotation matrix (satisfying ˆR ˆR T = ˆI) We simply obtain

u(x) = ( ˆR − ˆI)x or, equivalently, ˆJ = ˆR − ˆI Because the applied

defor-mation (i.e., the local rotation) is small by hypothesis, we observe that

the difference ˆR − ˆI is small too The product ˆJˆJ T, therefore, will be

negligible, leading to the following expression:

0 ∼= ˆJˆJ T = ( ˆR − ˆI)( ˆR T − ˆI) = ˆR ˆR T − ˆR − ˆR T + ˆI

Therefore, ˆJ = −ˆJ T or, equivalently, ˆJ is a skew-symmetric tensor It follows that ˆJ = ˆ and ˆ = 0 We have verified that a pure rotation cor-

responds to zero strain In addition, we remark that the local rotation

of a volume element within a body cannot be correlated with any

arbi-trary force exerted in that region (the forces are correlated with ˆ and

not with ˆ); for this reason, the infinitesimal strain tensor is the onlyrelevant object for the analysis of the deformation because of appliedloads in elasticity theory

r The infinitesimal strain tensor allows for the determination of the lengthvariation of any vector from the reference to the current configuration

By defining  nas the relative length variation in directionn, it is possible

˛ n1, n2= 2n1× (ˆ n2) [11]

The present result is also useful for giving a direct geometrical

in-terpretation of the components 12, 23, and 13 of the infinitesimalstrain tensor As an example, we take into consideration the com-

ponent 12, and we assume that n1= e1 and n2= e2 The quantity

˛ n1, n2 represents the variation of a right angle lying on the plane

(x1, x2) Because 12= e1× (ˆ e2), we easily obtain the relationship

˛ n1, n2= 212= ∂u1

∂x2 +∂u2

∂x1 In other words, 12 is half the variation

of the right angle formed by the axis x1 and x2 Of course, the same

interpretation is valid for the other components 23and 13