cracks and atoms**chaos.ph.utexas.edu/manuscripts/1067546500.pdfexperiment seem as remote as a...

Cracks and Atoms**

By Dominic Holland* and Michael Marder

1. Introduction

Many materials scientists and engineers are, with somejustification, suspicious of theoretical and numerical studiesascending from the atomic scale on the mechanical responseof materials. On the one hand, there is a reluctance to believethat the invisible atomic scale is important for macroscopicmechanical deformation. Out of sight, out of mind. On theother hand, many large scale computer simulations thatproduce brightly colored pictures with gobs of toy atoms, andsometimes even impressive statistics on processing effi-ciency, seem simply to avoid questions on how to comparecomputation with either theory or experiment. For in fact, acalculation involving ten billion atoms, necessarily withquestionable effective atomic interactions, would exceed thepowers of the world's largest computers, and yet describeonly a cube of matter no more than half a micrometer alongeach side. And even when computers become large enoughto store and manipulate the coordinates of this manyparticles, it will not be possible to follow their behavior formuch more than a nanosecond, thus making comparison withexperiment seem as remote as a manned flight to Pluto.

These simple observations lie behind the dominance ofcontinuum mechanics in most studies of mechanical behav-ior of materials. Obviously, so the argument goes, it is anenormous waste of effort to calculate the motion of everyatom when all information of interest is contained incontinuous fields that are most sensibly studied by othermeans. Hence the feeling, widely held but seldom expressed,that ªreal materials are not made of atomsº.

The point of this article is to show that this feeling is wrong.Materials constantly betray their atomic underpinnings.When this happens, it should come as no surprise that thecontinuum theory breaks down, since it requires a great dealof cleverness indeed to apply continuum elastic theory tophenomena that are neither continuous nor elastic. We willdiscuss properties of materials for which atomic features areessential to even a qualitative understanding, and show howto design studies at the atomic scale in an efficient manner,studies which permit direct comparison with experiment.

The mechanical response of materials is an enormous andvaried subject. We will therefore focus on one particular casethat makes it possible to examine the relationship betweenatomic and macroscopic scales in detail: the process of brittlefracture.

Fracture is important because it determines the ultimatestrength of a wide range of materials. Fracture fundamentallyhas to do with the severing of inter-atomic bonds: this pointstheoretical investigations toward atomic-scale studies. Asgem-cutters know, cracks tend to run along crystal planes,showing that the process is sensitive to atomic detail.Nevertheless, most fracture research is carried out in thecontext of continuum elasticity through an elegant frame-work that bypasses most of the questions arising at theatomic scale. Our aim is to identify the questions that thecontinuum approach cannot address, and to show how acombination of theoretical insight and numerical computa-tion can be employed to answer them. The ability to comparedirectly with experiment will then provide a strong test of thecorrectness of the underlying interatomic potentials used insimulations.

2. Continuum Elastic Fracture Mechanics

Drop a metal spoon on the floor and it may only bend alittle. Drop a glass and it will shatter. Why? Continuummechanics cannot resolve this matter, but it has a great dealto say about the rapid fracture taking place when somethingbreaks.

In particular, using continuum elastic fracture mechanicsone can derive, with some effort, an equation of motion for astraight, clean, dynamic crack. Let l(t) be the length of adynamic crack in a two-dimensional large plate under tensileloading of strength s0 applied only to the crack faces. LetG(u) be the amount of mechanical strain energy required forthe crack to advance by unit area while traveling at velocity u;G is called the fracture energy. The equation of motion isshown in Equation 1,[1±7] where n is Poisson's ratio, E isYoung's modulus, and cR is the Rayleigh wave speed (thespeed of sound on a free surface, about 0.9 times the shearwave speed). Equation 1 relates the stress fields near thecrack tip with the total flow of elastic energy in the entiremedium to the crack tip energy sink (assumed to be the onlymechanical dissipative region in the system). The effects ofbulk material properties are incorporated in the elasticconstants (E and n), and the resistance of the material tocrack extension is represented by G.

Adv. Mater. 1999, 11, No. 10 Ó WILEY-VCH Verlag GmbH, D-69469 Weinheim, 1999 0935-9648/99/1007-0793 $ 17.50+.50/0 793

±

[*] Dr. D. Holland, Dr. M. MarderCenter for Nonlinear Dynamics and Department of PhysicsThe University of Texas at AustinAustin, TX 78712 (USA)

[**] This work was supported by the National Science Foundation (DMR-9531187), the Texas Advanced Computing Center, and the NationalPartnership for Advanced Computational Infrastructure. Thanks to JensHauch and Harry Swinney.

794 Ó WILEY-VCH Verlag GmbH, D-69469 Weinheim, 1999 0935-9648/99/1007-0794 $ 17.50+.50/0 Adv. Mater. 1999, 11, No. 10

(1)

Continuum elastic fracture mechanics by itself makes noprediction about how the crack velocity should vary as thefracture energy changes. The fact of the matter is that there isan unknown functional relationship in Equation 1: G(u) isunknown (u, l, and s0 are interrelated). Equation 1 does saythat the Rayleigh wave speed is an upper bound on thepossibilities for the crack speed u (otherwise if u exceeded cR,the energy required for unit area crack extension would benegative); but there is no requirement that u should approachcR as the energy available for fracture increases. If G isbounded, however, then u must approach cR, since l on theright hand side of Equation 1 can become arbitrarily large.Intuitively, one would not expect the energy required toadvance the crack by unit area at some speed u (<cR) tobecome unboundedly large. From a first approximation, anadvancing crack involves only the severing of bonds, whichwould lead to a constant G(u); but even allowing for thepossibility that sound waves also will be generated by theadvancing tip, one would not expect these waves to becapable of carrying off ever-increasing amounts of energy.

The form of G(u) cannot be settled using continuum elasticfracture mechanics, for to do so would require a detailedknowledge of what goes on in the process zone, the regionnear the crack tip where the material is furthest from

equilibrium. To make progress, one must either assume aform for G(u), estimate it numerically, or measure itexperimentally. For dynamic cracks, continuum elasticfracture mechanics then predicts the crack velocity giventhe loading s0. The simplest assumption is that G(u) is aconstant, and taking this to be true, Equation 1 is easily put inthe form[5,8] shown in Equation 2, where l0 is the crack lengthat which the crack first receives enough energy to beginmoving.

(2)

The results of Equation 2, alas, are not in accord withexperiment. The equation predicts that the speed of a crackshould approach the Rayleigh wave speed as the cracklengthens. Experimentally, however, the limiting speed isusually between 20 and 80 % of this value.[8±12]

We will present a detailed description of the nature ofthese limiting speeds in later sections, but a sketch of themain idea is as follows: As a crack approaches the Rayleighwave speed, continuum theory predicts that the elastic regionabout the crack tip must store greater and greater amounts ofelastic energy in the form of rapidly varying stress and strainfields. Because real materials are not prone to handling anever-increasing energy flux in a controlled manner, thecontinuum description of a single straight crack must break

D. Holland, M. Marder/Cracks and Atoms

Dominic Holland is Irish, and received a B.A. and an M.Sc. in theoretical physics from TheUniversity of Dublin, Trinity College. He completed his Ph.D. in physics from the University ofTexas at Austin in the winter of 1998, and joined the Scientific Computing Group at the San DiegoSupercomputer Center in the spring of 1999. His graduate research was on atomic-scaleinvestigations of brittle fracture, using high performance computing; patterns in acoustic andmagnetic signals from solids; and numerical modeling of patterns in granular media. Theseremain active research interests of his, as well as applying hybrid parallel computing methods inmodeling physical systems. In his spare time, he likes to walk among the hills of the desertsouthwest of the USA.

Michael Marder grew up in Champaign, Illinois, and was educated at Cornell University, beingawarded degrees in physics and mathematics. He then went to the University of California atSanta Barbara, where he obtained a Ph.D. in theoretical condensed matter physics under JamesLanger on the subject of phase transformations. After a postdoctoral position at the James FranckInstitute at the University of Chicago, he moved to the University of Texas at Austin, where he is amember of the Center for Nonlinear Dynamics. He is involved in a wide range of investigationsbut specializes in the mechanics of solids, particularly the fracture of brittle materials, and carriesout experiments as well as theoretical and numerical investigations. He has just completed agraduate text, Condensed Matter Physics, that emphasizes materials issues more than has beencustomary in books of this type.

down: the region away from the crack tip becomes so highlydistorted that it is able to spawn new cracks away from theline of the main one. Precisely when and why the new cracksemerge depends upon small-scale details not accessible fromthe continuum point of view.

3. Microscopic Models

The most complete description of dynamic fracture isprovided by lattice theory. It is built around lattice models inone and two dimensions. The models are made from particlesat lattice sites connected to nearest neighbors with Hookesprings that snap after undergoing a small extension. The firstsuch models were studied in the early 1970s by RobbThomson and coworkers,[13] who uncovered a property theycalled lattice trapping, which is the inability of a crack toadvance in a crystal even though, from an energy argumentalone, it should do so. Then in the early 1980s, LeonidSlepyan took things mathematically much further, studyingdynamic cracks in an infinite square lattice, which he thoughtof as modeling inhomogeneous solids such as concrete.However, it is more natural to consider the lattice points tobe atoms, and long thin lattice strips have been studiedextensively with this point of view in the background. Thesestrip lattice models are completely analytically solvable, andhave the following four properties: 1. When the fractureenergy is moderate, i.e., for moderate strains, dynamic cracksare attracted to steady states. In other words, they want to bewell-behaved; 2. When in steady state, there is an intimateconnection between the speed of the crack and the waveproperties of the medium; 3. There is a large minimumvelocity, below which a crack will not travel, i.e., there is avelocity gap or forbidden band of velocities. This is related tothe lattice trapping earlier found by Thomson; 4. When thefracture energy exceeds some threshold, the crack willbecome unstable, it will no longer be in a steady state, and itsdynamics rapidly become very complex.

The last property, instability, is well-established experi-mentally for brittle amorphous materials, where it is clearlyresponsible for the inability of cracks to reach the Rayleighwave speed.[14±16] But the other three properties are difficultto observe experimentally, and have not conclusively beenobserved in the crystals, where they are most confidentlypredicted. These four properties together define an idealbrittle universality class.

It is only proper to ask whether the lattice theory offracture has much bearing on reality; in particular, does thereexist in nature an ideal brittle universality class of materials?After all, there does not exist any two-dimensional materialwith snapping Hooke springs between nearest neighboratoms. On the other hand, many fracture phenomena aredifficult to handle in an analytically tractable way. It is clearlya formidable task to predict what will happen when a largenumber of atoms are driven far from equilibrium; but this isprecisely what happens when a crack moves, at speeds up to a

few kilometers per second, bifurcating and changing direc-tion, or when dislocation loops nucleate, propagate, andentangle. That is, it is difficult to get a theoretical handle onmuch of reality. So between theory (or lack of it) andexperiment, then, is a large, unexplored territory. This iswhere realistic large-scale molecular dynamics simulationscan be a powerful investigative tool.

How realistic is realistic? Although it depends on what ªisºis, the answer is not very. This important matter is discussedmore fully below. How large is large? Again, not very. Butdoes it matter? On a large supercomputer, large and superbeing datable modifiers, one can follow on the order of tenmillion particles for ten nanoseconds, with reasonablysophisticated interactions between the particles, in threedimensions. There are fifteen or so orders of magnitude to goto reach the macroscopic scale studied in the laboratory.Molecular dynamics simulations are not going to get there ina brute-force way. It must be borne in mind that it isimportant to be able to compare with experiment: simula-tions can be extraordinarily complex, and are necessarilycrude approximations of reality. Thus, when one is makingclaims to realism, one had better be able to compare withexperiment, the arbiter at the end of the day, and not get lostin a vast virtual phase space. Toward this end, with moleculardynamics modeling of dynamic fracture, it is imperative thatsimulations be designed around a scaling argument that willtake one through the fifteen or so orders of magnitude fromvirus-sized systems up to the macroscopic scale, and do so in amanner that is well-founded at the atomic level.

4. The Scaling Argument

The scaling argument and the thin strip geometry go handin hand. What one needs control over, after all, is the energyflux to the crack tip. For a long crack moving at steadyvelocity in a strip whose upper and lower boundaries are heldrigid, one can deduce the energy consumed by the fracture ina trivial way. Consider the crack moving in the thin stripdepicted in Figure 1. Far to the right of the crack tip thematerial is under tension, and it stores an elastic energy

per area A (Eq. 3).

(3)

Far to the left of the crack, the material is completelyrelaxed, and therefore if the boundaries of the system arerigid as the crack proceeds in a steady fashion, the energydissipated by crack motion per unit area crack advance mustthen be given by Equation 4.

(4)

The reason for this assertion is that when the crack movesahead in a steady fashion the stress and strain fields around itdo not change. As shown in Figure 1, the fields translate in




the direction of crack motion, and the change in energycomes from the transformation of stressed material ahead ofthe crack to the unstressed material behind.

Fig. 1. The upper surface of a strip of height L and thickness w is rigidlydisplaced upwards by distance d. A crack is cut through the center of the strip,and relieves all stresses in its wake. When the crack moves distance dl from (A)to (B), the net effect is to transfer length dl of strained material into a length dlof unstrained material.

This conclusion rests upon symmetry, and does not evendemand that strains ahead of the crack tip to be so small thatlinear mechanics be applicable. Steady velocity u

®and a

corresponding energy flux G are achieved in the long timelimit, the natural scale in experiments as well as in analyticalcalculations, where the crack tip reaches dynamic equilib-rium with waves reflecting from top and bottom boundaries.That is, the return of acoustic waves from the systemboundaries actually simplifies the task of understandingenergy balance in the thin strip geometry. According tofracture mechanics, the relationship between energy flowingto a crack tip and crack velocity is, for a given latticedirection, universal. Having obtained the relation in a strip,one knows it for any of the vast range of geometries to whichfracture mechanics is applicable, such as a long crack in alarge plate.[17]

In order to relate samples of different size to one another,let Gc be the Griffith energy density;[5,18] that is, twice thecrack surface energy density, a lower bound on the energyper unit area required for a perfectly efficient crack topropagate along a certain plane. One can then define adimensionless measure of loading D as in Equation 5, whereG is the fracture energy density already introduced, i.e., theelastic strain energy stored per unit area (in the fractureplane) ahead of the crack.

D � ��G=Gc

p(5)

Analytical solutions for the ideal brittle solid show that therelationship between D and crack velocity becomes inde-pendent of the height of the strip (number of planes stackedvertically) for surprisingly small strips;[19] a strip 80 atomshigh has, for all practical purposes, reached the infinite limit,Figure 2. Guidance for conducting computationally expen-sive, moderately realistic, molecular dynamics simulationscan be had from this simple result: the very rapidconvergence of the main quantity of physical interest allows

one to obtain physically meaningful results from simulationsthat are considerably smaller than many that are currentlybeing carried out.[20±24] This approach can also be viewed asan alternative to methods that join together atoms andcontinua.[25±29]

Fig. 2. Two-dimensional triangular thin strip lattice under tensile loading:Crack velocity u, scaled by the sound speed c =H3/2, versus the driving strainD,Equation 5, for strips 80 and 160 atoms high, with non-central forces betweenatoms one-tenth the strength of the central forces (the non-central forces helpstabilize the lattice). That the results are independent of the system height forsuch small heights suggests that molecular dynamics simulations do not haveto be enormous.

If computational resources were infinite, one would stillneed to properly choose the sample geometry in order tocompare and inform experiment and theory. With limitedcomputational resources, choice of sample geometry andscale are of paramount importance: when performingmolecular dynamics simulations of fracture, one should notask how large a system one can work with, but rather what isthe smallest system that will give results scaleable to themacroscopic level. A good scaling argument can transform aªgrand challengeº problem into something that's betterfocused and less of a challenge. Size does matter: a smallerspatial scale allows one to follow the evolution of systems forlonger times; and with a strip geometry one can then fullyanalyze steady state behavior and connect theory andexperiment.

5. Molecular Dynamics

The molecular dynamics method applies to any system ofparticles with some prescribed inter-particle potential. Itconsists of integrating Newton's equations of motion for allparticles in lock step over a series of time steps, the size of thechosen step being small enough to give converged dynamics.Time step integration for all results reported here wasachieved using the Verlet algorithm.[30±32]

Lattice theory makes predictions that are hard to observein experiment, and one of the reasons for doing computersimulations is to relate the two. The simulations to bedescribed later were carried out for silicon. Silicon is


extremely brittle, and high-quality macroscopic single crystalwafers are cheap. It is therefore an excellent candidate forlaboratory fracture experiments. Silicon is also of greattechnological importance and, as a result, is one of the moststudied materials. One measure of this is that there are overthirty effective (actually rather ineffective) interatomicpotentials for silicon in the literature.[33,34] Furthermore,transmission electron micrographs of cracks in silicon wafersreveal atomically sharp crack tips.[35,36] Silicon therefore is anobvious candidate for molecular dynamics investigations ofdynamic fracture, and an appropriate setting for testinglattice theory.

5.1. Interatomic Potentials

The equation underlying materials physics is not in doubt.It is Schrödinger's equation. This equation can be solvedanalytically for the hydrogen atom, numerically for thehelium atom, and with reliable approximate methods ofquantum chemistry for small molecules. For any solid ofinterest in the study of materials, all hope of controlledapproximations must be abandoned, and the Schrödingerequation is brutally reduced to tractable form in a way that isrefined by comparison with experiment. Quantitativemethods that employ such approximations from the begin-ning are called ab initio.

Despite their origin, ab initio methods are the mostreliable techniques available for numerical treatment ofmaterials. However, they are restricted to perfect crystalswhere the unit cell is not much bigger than 1000 atoms, andthese atoms can be followed for only a few tens ofpicoseconds. The main motivation, then, for constructingempirical or effective ªclassicalº interatomic potentials isspeed of computation and the ability to work with relativelylarge numbers of particles. With effective potentials, it ispossible to follow ~107 atoms for a few tens of nanoseconds.This difference in computational scales becomes importantin the modeling of processes that require a minimumof ~105 atoms to capture just some of the complex under-lying physics; processes involving fracture, dislocation loops,grain boundaries, or amorphous-to-crystal transitions, forexample.

Another motivation for constructing effective interatomicpotentials is that they make the complex physics of what arefundamentally quantum mechanical phenomena more phys-ically intuitive, so that one may interpret the results ofatomistic simulations in terms of the simple principles ofchemical bonding.

5.2. Realistic Potentials for Silicon?

Solid silicon is composed of covalently bonded atoms andhas the open diamond crystal structure. If only two-bodypotentials operated among the atoms, one would expect the

crystal to collapse in on itself to form a close-packedstructure, thereby reducing its energy. Covalent systems,however, are characterized by restoring forces between pairsof contiguous interatomic bonds. That is, pairs of bonds withan atom in common want to maintain a preferred anglebetween them. These extra forces are what stabilize the opendiamond structure in silicon, carbon, grey-tin, and germa-nium, for example. The lowest-order way of capturing thisproperty with effective interatomic potentials is to go beyondbinary bonding and include a three-body term in the system'sHamiltonian.

The Stillinger±Weber (SW) (two- and three-body) poten-tial[37] has proved to be very popular and durable in theliterature. It gives excellent elastic properties, and captureswell the nonlinear physics involved in heating and melting. Itis therefore a reasonable starting point for conductingmolecular dynamics fracture simulations in silicon. Un-fortunately, the potential will not yield fracture along theexperimentally preferred fracture planes (111) and (110). Atlow or moderate strains, what happens is that two disloca-tions open up at the crack tip, blunting it and preventing itfrom advancing (Fig. 3). One can play around with giving atransverse opening velocity to a select few atoms around thetip. But to no avail. The crack simply will not crack. At veryhigh strains, the crack tip region melts.

Fig. 3. Crack tip blunting in SW silicon: two dislocations open up at the tip,preventing it from advancing. The crack is pointing in the direction [110] in theplane (110), and the system is loaded with a strain parameter D = 1.6.

SW does give a type of fracture along the (100) plane whichis quite rough on the atomic scale (Fig. 4). Abraham et al. callthis brittle fracture.[23] There is as yet little consensus on aprecise definition of brittleness. The experimental results ofLawn and Hockey[35,36] for fracture along (111) in siliconshow, however, that it is possible to have atomically sharpfracture, i.e., where the newly created fracture surfaces areatomically flat. The experimental evidence for fracture along(100), on the other hand, is scant and inconclusive,[38] andthat SW yields fracture along this plane, albeit in a rough




manner, might even be yet another indication of thepotential's shortcomings.

It is possible to get cracks going in an ideal brittle manneralong (111) and (110) with the SW potential by increasing therestoring forces between pairs of bonds; i.e., by increasing thestability of the tetrahedra in the diamond lattice, and thusmaking the crystal more brittle. This can be done by scaling aparameter, l, the coupling constant in the three-body term.Originally, Stillinger and Weber set l = 21 (dimensionless).However, by doubling this, one can obtain fast brittlefracture. With a fast crack running, if one quasistaticallydecreases l, the crack arrests well before one reaches l = 21(Fig. 5). Although l = 42 gives fracture phenomenology inreasonable accord with experiment, it has the adverse effectof raising the melt temperature above 3500 K, whereasexperimentally the melt happens at 1685 K. The Youngmoduli also get shifted. Moduli results from tensile tests onsmall samples, for three lattice directions, are given inTable 1 for both the original and modified SW potentials,along with the corresponding experimental values.

Table 1. Elastic constants of silicon, comparing SW, modified SW potentials,and experiment.

A more recent, and also much more sophisticated,potential for silicon is the environment-dependent inter-atomic potential, EDIP, by Bazant et al.[34,39±41] It has muchfunctional similarity with the SW potential, but its noveltyand greater complexity lie in its sophisticated environmentdependence. Silicon is tetravalent, and can form a number ofcovalent bonding configurations, corresponding to different

bonding orbital hybridizations. Different hybridizations kickin for different coordination numbers. The method of EDIPis to provide an accurate potential for each of the differentpreferred integer coordinations, and to have smooth inter-polation between these configurations; sp2 hybrids aroundthree-fold coordinated atoms in a hexagonal plane (thetheoretical graphitic structure), sp3 hybrids with four-foldcoordination arranged in tetrahedra (the diamond struc-ture), six-fold coordination for simple cubic, and eight-foldcoordination for bcc, for example. The potential wasdeveloped for bulk materialsÐexplicitly excluding asym-metric distributions of neighborsÐwhich occur for under-coordinated structures like surfaces and small clusters. Ascalar representation of the environment, the coordinationnumber, will therefore suffice; it varies continuously ashybridizations change. It is reasonable to assume thatchanging hybridization is important at an advancing cracktip. Unfortunately, EDIP is even more resistant to dynamicfracture than SW. Again, dislocations will open up at the seedcrack tip, blunting it and keeping it in arrest.

Fig. 5. Crack velocity profile a) along (110) [110], and b) along (211) [111] inmodified SW silicon: l scales the strength of angular forces between pairs ofbonds that stabilize the diamond lattice structure. Stillinger and Weber's valuefor l is 21. However, leaving all other parameters unchanged, cracks will notpropagate unless one uses a larger value of l. D specifies the loading.

As with SW, EDIP also captures reasonably well thenonlinear physics of heating and melting. But both potentialsfail rather dramatically when applied to the nonlinearprocesses of stretching and rupturing interatomic bondsinvolved in fracture. A key to understanding why comes fromdensity functional theory, or rather an empirical fit to DFTcohesive energy curves, viz. the universal energy relation ofRose et al.,[42] which characterizes materials strained far from


Fig. 4. A two-dimensional projection of rough cracking along the (100) planein SW silicon. The system is loaded with a strain parameter D = 1.7,corresponding to a fracture energy density of almost three times the Griffithenergy density. The average crack speed is 1.9 km/s.

equilibrium. The universal energy relation (Eq. 6) gives thecohesive energy as a function of a uniformly strained lattice,where s = (1 ± a/a0)c, with a being the scaled nearest neighboratomic separation and a0 the equilibrium atomic separation(2.35 � in Si). b = 4.64 and c = 4.88 for silicon. E(s) is in eV.

E(s) = b[exp(s)] (±1 + s + 0.05s3) (6)

This equation gives remarkably good fits (with appropriatevalues of b and c) to experimental data and density functionalcalculations for all materials it has been applied to,accurately modeling moderate uniform compression andalso moderate uniform expansion (where experimental dataare lacking). Figure 6 (upper) is a comparison plot of theuniversal energy relation applied to silicon, Equation 6,showing an interaction cutoff at ^5.5 �, and the cohesiveenergy curves for SWand EDIP, the latter two having cutoffsless than the second-nearest neighbor distance, <3.84 �.Figure 6 (lower) shows the corresponding restoring forces.The salient point is that the nearest neighbor potentials aretoo short in range, and must rise from the bottom of thecohesive well and go to zero rapidly as the system is dilated.This results in a large gradient before cutoff, which meansthere is an unreasonably large force of attraction beforerupture. This is more pronounced in the shorter-rangedEDIP. These large forces inhibit crack propagation.

5.3. Remarks on Potentials for Silicon

Most of the potentials for silicon in the literature have acutoff restricted to nearest neighbors,[33] examples of which,along with SW and EDIP, are the Dodson potential[43] andthe three Tersoff potentials.[44±46] Two longer range potentialsare those due to Biswas and Hamann,[47] and Pearson, Takai,Halicioglu, and Tiller (PTHT),[48] with interactions up to thethird (cutoff at 5 �) and seventh (cutoff at 7.3 �) shells,respectively. The elastic properties of these longer rangepotentials are very poorÐthe bottoms of the wells are toonarrow and steep. These potentials are also too steep for allcompression strain values. The short range potentials, on theother hand, describe compression well, and expansion up to anearest neighbor separation r^ 2.8 �. The PTHT potentialgives a bulk cohesive energy for diamond of ±5.45 eV,whereas SW and EDIP give ±4.34 and ±4.65 eV, respectively,in close agreement with density functional theory. TheBiswas±Hamann potential and Tersoff's third potential[46]

overestimate the melt temperature, giving 2900 K and3000 K, respectively (not too far from the modified SWpotential!). None of the potentials is able to model thevarious (1 ´ 1, 2 ´ 1, 7 ´ 7) reconstructions of the (111)surface.

The potential by Bolding and Andersen[49] has over 30adjustable parameters, fit to a wide range of structures, and isclaimed to describe accurately bulk phases, defects, surfaces,and small clusters.

Fig. 6. Cohesive energies and restoring-forces for SW, EDIP, and the universalenergy relation (UER) representations of silicon. a0 is the equilibrium nearestneighbor distance.

The potential of Chelikowsky and Phillips,[50] like EDIP,depends in a complex way on the local atomic environment.The environment description involves a ªdangling bondvectorº which is a weighted average over displacementvectors to neighbors. It vanishes for bulk equilibrium, and islarge for asymmetric neighborhoods, and thus might bebetter suited than EDIP to describing under-coordinatedstructures like surfaces, defects, and clusters. The back-bonding mechanism employed in the Chelikowski±Phillipspotential, which alters the bonding of under-coordinatedatoms, may play an important role at the crack tip.

With so many potentials for silicon, it is unfortunate thatnone is particularly superior and that all do so poorly inreproducing the basic cohesive energy curves. There is noclear choice for fracture simulations which do, it should benoted, demand more of a potential than tests not involvingbond rupture. Faced nevertheless with making a choice, weproceeded with an arbitrary one, whose consequences wenow describe.

6. Fracture Simulations

In this section, we will describe molecular dynamicssimulations of fracture in silicon. The scaling argument ofSection 4 will be used in determining the appropriate sizeand geometry of the sample, the boundary conditions, andalso the rate at which the sample is strained, i.e., the rate atwhich the fracture energy G changes in time. All of the




results presented are from extensive computational runsperformed using the modified SW potential as describedabove in Section 5.2, i.e., where the restoring force betweenpairs of contiguous bonds, the three-body term, is doubled instrength: l = 42 is used instead of l = 21.

Simulations were run to investigate the fracture propertiesof silicon for a number of crystal planes, and for differentdirections in those planes. Different strain rates were used,but these were generally quasistatic.[51,52] The temperaturedependence of crack properties was also investigated. Andfinally, room temperature fracture runs were carried out anddirectly compared with experiment.

The main physical quantity of interest in what follows isD(u) (or G(u)). To determine u, the crack tip was located andrecorded every few time steps. A simple opening criterionwas used. A sample of tip location versus time is in Figure 7.Each bond breaking event is clearly distinguished. Velocitycan then be measured as accurately as possible.

Fig. 7. Accurate crack tip tracking in silicon; each bond-breaking event isclearly distinguished. The origin has been set for convenience.

6.1. Design of Simulations

In the simulations depicted in Figure 8, the crack runsalong x, exposing either (111) or (110) planes. Three separateboundary conditions are employed:l x±y planes: Two layers of atoms at the top and bottom of

the strip are held rigid during the simulation. By pullingthem apart, elastic energy of any desired amount can bestored ahead of the crack tip. Sometimes the distancebetween top and bottom layers is held fixed during thewhole simulation, while other times it is increased ordecreased quasistatically.

l y±z planes: To model a crack in an infinitely long strip,cutting and pasting on a conveyor belt is used. First,whenever the crack tip approaches within 150 � of theright-hand boundary, new crystal is pasted onto the right-hand side, and broken crystal cut from the left. Second, toprevent elastic waves from informing the crack tip that itlives in a strip of finite extent along x, there are energyabsorbing regions 20 � thick at both the left and right

endsÐthough this is not important for the region aheadof the crack tip, which remains practically undisturbeduntil the crack becomes unstable.

l x±z planes: These boundaries are periodic. This choiceenables one to describe correctly the flux of energy to thecrack tip in a macroscopic sample, at least while the crackis stable. Nakamura and Parks[53] have shown that in amacroscopic plate of thickness d, at distances from thecrack tip much smaller than d, the appropriate elasticsolutions are found to be those with such periodicboundary conditions.

The initial equilibrium sample is 614 ´ 20 ´ 153 �3. Thelength, 614 � along the x-axis, is sufficient so that the cracktip will not feel the effect of the damped head and tailboundaries, and also that the waves emanating from thecrack tip can be characterized well and analyzed. Thethickness, 20 � along the y-axis, is three unit cells deep, whichis more than generous for this periodic axis. The primarydimension informed by the scaling argument, the height, is153 � along the z-axis, a value which, like the thickness, errson the generous side. It is sufficiently large to give scalingbehavior for all values of G up to instability, i.e., where thesteady state analysis breaks down. A system twice as largegives identical results, although after a longer propagationtime and a much longer central processing unit (CPU) time.

With the above dimensions, the number of atoms involvedat any stage of the simulation is approximately 94 000,although after numerous cuts and pastes at left and right


Fig. 8. Visualizations of the simulations, showing a stable steady-state crack at6.24 % strain, top, and an unstable crack at 8.97 % strain, bottom, along (111).Animations can be seen at http://chaos.ph.utexas.edu/~holland/Crack/.

hand boundaries tens of millions of atoms are cumulativelyinvolved. This number is small enough to enable one tofollow crack motion for the times (~10 ns) needed for thecracks to proceed through a succession of steady statesbetween arrest and instability.

The top and bottom layers of atoms are pulled apart so thatthe load on the system that drives the crack is given by thestrain parameter D defined in Equation 5. The two mostexplored fracture planes were (110) and (111), for which theGriffith energy densities are Gc = 3.3 J/m2 and Gc = 2.7 J/m2,respectively.

To initiate the simulation, the two x±y boundaries arepulled apart so that D in Equation 5 is 1.6; a narrow seedcrack is inserted running half the sample length; an initialvelocity is given to a few atoms near the crack tip; and it is letrip, i.e., Newtonian mechanics takes over. The time step usedis 4 fs. There are about 30 time steps in the smallest period ofvibration in the system, giving very good energy conserva-tion. Decreasing the time step by a factor of ten shows nochange in the steady state dynamics. At very high strains,there is much more energetic particle motion, and a time stepof ~0.4 fs is required.

6.2. Steady States

Steadystatesaremostdirectlymanifestedbythepropertiesof the waves emanating from the crack tip and running alongthe newly created crack surfaces. To investigate the existenceand character of steady states, it is desirable to be able tomonitor the motions of an arbitrary number of particles alongthe fracture plane. In a steady state, it might be that the atomsat every equivalent lattice site along the crack line behave inthe same way, or perhaps only every second or third atom, etc.This can be checked by monitoring atoms at successiveequivalent lattice sites along the crack line.

Particles are found up ahead of the crack and tracked asthe crack advances, hits them, and on until they reach the tailwhere they get removed by the advancing conveyor beltmechanism, Figure 9. For silicon, the only steady statesfound were those where a particle at every successive latticesite along the fracture plane behaved in the same way.

7. Results of Zero Kelvin Calculations in Silicon

Questions to be answered: (1) Are there loads D wherecracks are attracted to steady states? (2) Do cracks emitphonons at the predicted frequencies? (3) Do cracks refuseto travel below a minimum velocity u1 > 0?, and (4) Do theygo unstable above an upper load Dc? The answer to allquestions is yes.

Figure 9 shows the time history of two different atoms onan upper (111) fracture surface for D = 1.6 after approxi-mately 50 ps. The atoms are not yet behaving identically. Thisis because the advancing crack tip has not yet reached

equilibrium with the waves it sends out that reflect from theupper and lower rigid boundaries. Figure 10, on the otherhand, shows atomic motions after the crack has beentraveling for over 0.24 ns. As anticipated by the theory ofideal brittle fracture, the crack has reached steady state withvelocity u = 3460 m/s, which means that the verticaldisplacement z

R! of an atom originally at crystal location R

!

is related to the vertical displacement zR!� na

! of an atom nlattice spacings a

!º ax to the right by Equation 7.

zR!� na!(t + na/u) = z

R!(t) (7)

Fig. 10. Same as Figure 9, except that here the crack has been traveling forover ~0.24 ns. That the two overlapped curves are almost completelyindistinguishable shows the crack has reached steady state, according toEquation 7, and is emitting phonons in accord with Equation 9.

For a range of loads D, Equation 7 applies for any pair ofatoms, whatever their separation along the crack surface. Inorder to obtain the perfect periodicity shown in Figure 10,the crack was allowed to run first for 60 000 time steps so as tocome into equilibrium with the waves it sends towards topand bottom boundaries.

The longer and/or higher the system the longer it will taketo reach a steady state. The 240 ps required to reach thesteady state depicted in Figure 10 is about an order ofmagnitude longer than the duration of most large-scalemolecular dynamics simulations of fracture. Using a reducedspatial system size, as validated by the scaling argument, iswhat makes this possible. This becomes crucially important



Fig. 9. After ~50 ps at D =1.6, the crack has not yet reached a steady state:height z of two atoms lying on the crack line as a function of time, showingpassage of a crack on a (111) plane. The second atom lies 184 � along x relativeto the first, and is displaced backwards by 5.32 ps in time.


when proceeding through a sequence of steady states, whichrequires quasistatic changes in the loading,D or G. To obtaina full set of results, like Figure 12, a crack actually will traveltens of micrometers, or for times on the order of a tenth of amicrosecond. To achieve this, one needs not only efficientcode and a high performance computer, but also a physicallymotivated smallest computational cell: a minimum thin stripon a conveyor belt.

Under steady state conditions as described by Equation 7,the radiation far from the crack tip obeys Equation 8, so thatthe crack excites all surface phonons whose frequency o(k

!)

and wave number k!

in the extended zone scheme obey theCherenkov condition (Eq. 9), where k

!¢ is restricted to the

first Brillouin zone and K!

is a reciprocal lattice vector, so thatcrack velocity u! equals phonon phase velocity.

exp�i k!��R!� na!� ÿ io� k

!��t � na=u��

exp�i k!�R!ÿ io� k

!)t] (8)

o ( k!� � u!� k!� o�k ¢� � u

!�� k!¢�K!

) (9)

Thus, the Cherenkov condition is equivalent to demandingthat a propagating wave obey Equation 7, so Figure 10 alsoshows that Equation 9 is satisfied. A close-up snapshot of asteady-state crack in silicon is in Figure 11.

Fig. 11. Close-up of the steady-state crack tip in Figure 8, traveling at about3 km/s.

In order to find fracture speed u as a function of the loadingparameter D, the separation between the x±y boundaries isdecreased adiabatically while allowing the simulation to run.The crack tip is precisely located every second time step,showing clearly every single bond-breaking event, (Fig. 7).In order to estimate just how slow the strain rate must be toachieve the adiabatic limit, numerical simulations of theanalytically solvable models were carried out (Fig. 2), andthe results compared with analytical results, leading to thecriterion that the dimensionless strain rate ehz/c should bemuch less than one, where e is the strain rate, c = 5500 m/s is a

sound speed, and hz = 153 � is the height of the sample alongz. For the simulations described here, low rates e < 100 ms±1,or ehz/c < 10±4 were employed. Prior molecular dynamicssimulations of fracture have been carried out with strain ratesof order >104 times greater, for which steady states areunattainable and the crack very rapidly goes unstable. Inlaboratory experiments,[11] it is possible to have ehz/c ~ 10±8.

7.1. Along (110)

The relation between velocity u! and load D for cracksexposing (110) and traveling along [110] is shown inFigure 12. The crack velocity smoothly decreases as Ddecreases, until at u = 2256 m/s and D = 1.258, the crackabruptly comes to a halt. Raising D again, the crack does notbegin to move until D = 1.366, a value that is sensitive toresidual vibrations in the crystal, but the rising curve thenperfectly overlaps the descending one. Crack speed con-tinues to rise smoothly until u = 3586 m/s, D » 2.2, at whichpoint steady state motion becomes unstable. When a crackbecomes unstable, complicated phenomena such as forma-tion of small branches, emission of dislocations, and changesin the plane of propagation can occur, and intermittencywhere the crack makes repeated attempts at branching.

Fig. 12. Relation between crack speed u and loadingD for crack along (110) at0 K. As D descends, velocity drops abruptly to zero at a lower critical value,and as D ascends resumption of crack motion is hysteretic. For convergencecheck, system size was doubled along x and z, and u(D) measured. Cf. Figure 2.

7.2. Along (111): Crackons

The relation between velocity u! and load D for cracksalong (111) and traveling along [011] is shown in Figure 13.For 1.44 <D< 2.2, the crack has stable steady states, and forD> 2.2 it goes unstable in a similar manner to cracks along(110). However, for 1.175 < D < 1.44 the dynamics of thecrack exhibit a number of interesting features that have notbeen seen previously, and for which there is not yet acomplete theoretical description. There is a variety ofdifferent dynamic states available for each value of D, where


the crack travels at different speeds. Each of these statescorresponds to a plateau in u(D); D can change by as much asone fifth of the amount needed to go from arrest to instabilityand the crack velocity does not alter within numericalresolution. When the crack finally decides to accelerate outof the plateau, it may jump by over 1 km/s and reach an upperplateau to within a few m/s. On cyclical loading the sameplateaus are always reached. All of these transitions arehysteretic, as depicted in Figure 13. The different states emitnoticeably different phonons; on a given plateau, the phononfrequencies appear fixed and their amplitude changes, whilebetween plateaus the frequencies change in accord withEquation 9. This is crackon behavior. All these phenomenaare easily disguised if strain rates are too high. Resolving allthe fine structure visible in Figure 13 required e. < 8 ms±1, ore.hz/c < 10±5.

7.3. Other Lattice Directions

A crack running along (111) but in a direction [2 1 1],perpendicular to that shown in Figure 13, was also simulated.Its u(D) relation is in Figure 14. It has less plateau structure, alarger velocity gap, and higher lattice trapping. Since crackvelocity and phonon phase velocity are directly related, andsound velocities depend on the direction of propagation, itfollows that different u(D) relations will result for a singlefracture plane, with well-defined Griffith point, but with thecracks traveling in different directions in the plane.

Some crystal planes are not fracture planes at all, and someplanes barely allow crack propagation. Figure 15 shows asnapshot of two such situations. In Figure 15a, a seed crack isoriented along (211) [11 1] (same as (2 1 1) [1 1 1]) at D = 1.6.However, the crack immediately takes off in a diagonal along(111) [2 1 1]. This can be guessed from looking at a ball-and-stick model of silicon.

Figure 15b shows a crack reluctantly traveling along (001)[100] at D = 1.2. This plane will only permit unstable crackpropagation in a narrow window around D = 1.2. The (100)plane is the plane of propagation only in an average sense, asthe crack keeps jumping about.

8. Implementing Temperature

The cracking crystal is maintained at a particulartemperature by having a strip region running the height of



Fig. 13. Relation between crack speed u and loadingD for cracking along (111)[011] at 0 K. Dotted lines indicate forbidden velocities. The lower figure showscrackonsfor1.18<D<1.44:thecrackisabletoexposemanydifferentstateslyingalong many hysteresis loops. Ideal steady states are unstable aboveD^2.2.

Fig. 14. Crack velocity profile for Si (111) [21 1] at 0 K. Dashed lines indicateforbidden velocities. The crack is unstable above D^ 1.9.

Fig. 15. In a) a seed crack is aligned along (211) [11 1] atD = 1.6 but a dynamiccrack takes off along (111) [2 1 1]. In b) a crack loaded atD= 1.2 will propagateonly in an unstable manner along (100) [100].


the system far ahead of the crack act as a heat bath at thedesired temperature. In the heat bath, the fluctuation±dissipation theorem[54] is used to set the temperature. That is,at each time step, the atoms are given random forces or kickswithin a certain amplitude and their motion undergoes acorresponding damping. The square amplitude of therandom force áF2ñ is given by Equation 10, where b is theapplied damping, áp2ñ is the average momentum squared of aparticle at a time step, and Dt is the size of the time step.Random forces are correlated over a time step, anduncorrelated from time step to time step.

áF2ñ = 2b áp2ñ / Dt (10)

Taking into account Equation 11, where m is the mass ofan atom, k is Boltzmann's constant, and T is the temperature,the amplitude of the random force is given by Equation 12.*

p2

2m

+� 3kT

2(11)

FA ��6bmkT=Dt

p(12)

Note that if there are different atomic species in the crystalwith different masses, there will be a different random forceamplitude for each. The random forces and damping areimplemented at the start of each time step. For eachdimension (x, y, and z), the random forces come from asquare distribution of amplitude FA centered on zero. Thecentral limit theorem[55] shows that the actual shape of therandom distribution is unimportant; one will still end up witha Boltzmann distribution of particle velocities correspondingto the desired temperature. Thus, for example, at the start ofsome time step, particle i will have its force along the x-axisinitialized to fix, see Equation 13, where pix is the particle'spresent x-momentum, and w is a random number, ±1 £w £ 1.

fix = ±bpix + wFA (b > 0) (13)

9. Crack Behavior at Non-Zero Temperaturesin Silicon

9.1. Vanishing Velocity Gap

Temperature implies energy fluctuations in time, so that ifthe crack gets trapped due to an energy fluctuation thatreduces the fracture energy, further kinetic fluctuations maysubsequently enable the crack to move on, giving rise to thepossibility of creep. This suggests that the strength of latticetrapping should be a function of temperature. A series ofsimulations at 50 K intervals were carried out to investigatethe effect of temperature on lattice trapping along (111)[011]. The results are shown in Figure 16. The latticetrapping/forbidden band of velocities hysteresis loop nar-

rows while the lowest velocity before arrest remains the sameas temperature increases, and eventually the velocity gapvanishes above about 200 K. It might therefore be necessaryto go to liquid nitrogen temperatures in order to observelattice trapping and a velocity gap. Lattice trapping has beenpostulated since the early 1970s,[13] but has never been seen inexperiment. Careful fracture experiments have not yet beencarried out at low temperatures.

Fig. 16. Velocity gap vanishes near 200 K. Dotted lines indicate forbiddenvelocities. All low-lying velocity states exist at room temperature. Cracking isalong (111) [011].

Two further sets of runs were carried out at 300 K wherethe strain was quasistatically increased up to where thecracks became unstable. The first of these (Fig. 17) is acontinuation of the 300 K segment in Figure 16. Notice thatalthough all low-lying velocity states now exist, there stillexists some dynamic forbidden band of velocities andhysteresis at higher velocities and strain.


Fig. 17. Crack velocity profile with respect to quasistatic loadingD along (111)[011] in silicon at 300 K. All low-lying velocity states exist.

With all low-lying velocity states accessible at roomtemperature, as just mentioned, it might then be possiblefor the crack to creep.[56] Whether a crack can creep forvalues of D all the way down to 1, or as the fracture energydensity approaches the Griffith point, cannot be settled withmolecular dynamics; the time scales required for thermalfluctuations simply are beyond the reach of computers.

Note also in Figure 17 that as D is decreased below 1 thecrack actually heals and travels backwards. This is perfectlyreasonable as there has not been any non-uniform surfacedamage, and no oxide layer has formed.

9.2. Direct Comparison with Experiment

The second full set of room-temperature runs wasperformed along (111) [1 1 2] in order to allow directcomparison with experiment[57] (the zero Kelvin runs forthis lattice direction are in Fig. 14). The results are inFigure 18, and were obtained for a thin strip of size 532´ 15 ´154 �3, periodic along the thin axis. As before, new materialwas added ahead of the crack tip and old material lopped offat the tail every time the crack advanced to within 200 � ofthe forward end of the strip. In this fashion, the crack traveled7 mm during the course of the simulations as G was variedbetween 5 and 14 J/m2. The room-temperature experimentsfor fracture in silicon wafers along (111) [1 1 2] are describedin the literature.[57] These experiments are difficult toperform because of the high Young's modulus and brittlenessof silicon.

The highest experimental and numerical crack velocitiesshown in Figure 18 are reasonably close, but the minimumfracture energies at which a crack propagates differ: 2.3 J/m2

in the experiments and 5.2 J/m2 in the simulations. Since thescale of crack velocities in a material is bounded by soundspeeds,[5] which are given correctly by the SW potential, it isnot surprising that the experimental and computationalcrack velocity scales agree. Furthermore, the potential givesthe correct cohesive energy of silicon (but an inaccuratecohesive energy curve, see Section 5.2), leading to theagreement in numerical and experimental energy scales.However, the nonlinear parts of the potential involved instretching and rupturing bonds play an important role indetermining the actual fracture energies and crack velocities,in particular where the crack arrests and what its highestvelocity will be. The quantitative disagreements shown herepoint to a shortcoming of the nonlinear parts of the potential,which have not received much attention. The modified SWpotential cannot be expected to correctly reproduce experi-mental data. But it highlights the control over brittleness inthe three-body term, and the inadequacy of the potential tailsin the two-body term. These are complicated matters, and yetperhaps only hint at greater difficulties on the road to a betterpotential.

The lowest fracture energy density at which a crackpropagated in the experiments was close to the Griffith energy

densityfora(111)plane,2.2 J/m2.[58]Sinceacrackcannottravelwithlessenergy,theremustbeanarrowrangeoffractureenergyover which the crack velocity rises rapidly from zero to thelowestvaluemeasured,^2 km/s.Thisphenomenonisalsoseenin glass and polymers.[14] Because of the extreme precisionrequiredattheboundaries,experimentswithsiliconarenotyetcapableofsettlingthematterofwhetherthissharpvelocityrisesignals a velocity gap. However, as shown in Figure 16, innumerical silicon, the velocity gap is temperature dependent,vanishing above 200 K.At300 K theredoexist ªsteady statesºat all velocities between 0 and 3 km/s. For glass and Plexiglas,carefully controlled crack arrest experiments rule out theexistence of a velocity gap.[59]

The silicon experiments covered a range of fracture energydensities, 2±16 J/m2, in which the cracks produced verysmooth surfaces. Thus, cracks in silicon can dissipate largeamounts of energy, more than seven times the amountneeded to create a clean cleavage through the whole crystal,without leaving behind any large scale damage on thefracture surfaces. Investigation by atomic force microscopyshows that for low fracture energies the fracture surfaces areflat on the nanometer scale, while at higher energies thesurfaces have pronounced features. These features, however,are smooth on the micrometer scale, and account for heightvariations on the order of 30 nm over an area of 16 mm2. Theroughness gives an area increase of only ~0.1 % above that ofa flat cleaved surface. This extra surface cannot account forthe sevenfold increase in dissipated energy. The simulations,however, indicate that most of the energy can be carried offin lattice vibrations.

10. End Notes

It is possible to perform molecular dynamics simulations offracture in a way that is directly comparable with laboratoryexperiments. A simple scaling argument demonstrates this.Every indication from such simulations is that the predic-



Fig. 18. Experimental and numerical determination of the crack velocity u as afunction of fracture energy G along (111) [1 1 2] at room temperature. The fuzzindicates the spread in simulation velocity data. Experimental data courtesy ofJ. A. Hauch [57].


tions of lattice theory are correct. Molecular dynamicssimulations, however, enable one to work with sophisticatedinterparticle potentials and crystal structures that aredifficult to analyze analytically in the manner of latticetheory. More dynamic possibilities then open up.

Evolution has a long way to go in the zoo of interatomicpotentials. How things will develop for silicon, at least, can bemeasured in part by the strong new test of comparingmolecular dynamics simulations of fracture and laboratoryexperiments. For many years it has been computationallydesirable to use potentials with a short cutoff. Such nearestneighborpotentialscandescribe fairlywellquasi-equilibriumphenomena, like elastic constants and bulk defects, andperhaps even fortuitously or by design give a reasonable melttemperature. What has been demonstrated in this article,however, is that in order to characterize properly non-equilibrium phenomena, better longer-range potentials arerequired, in particular when modeling the severing of bondsinvolved in fracture. Although our discussion has centered onsilicon,themethodologyofcomparingsimulationswithactualexperimentsshouldapplytoabroadrangeofbrittlematerials.For any particular material, then, with a better potential forfracture, one which necessarily will reproduce the full densityfunctional theory cohesive energy curve, and not just thequasi-equilibrium deep-well part, one should have a moretransferable and correct potential than has hitherto existed.

Finding these corrected potentials has a relevance goingbeyond the particular details of how cracks behave in amaterial; indeed, the niceties of dynamic fracture arethemselves usually of no practical importance whatsoever.More significant is the broad hope that materials of thefuture will be designed from the atoms up on computers. Thishope is more widespread among those who use computersthan among those who make materials. If it is to be realized, itmust be based on physics that is correct and has been tested indifficult cases. Brittle fracture provides a test case that issimple enough to be tractable, yet demanding enough thatattempts at quantitative prediction have so far beendefeated. Ultimately, it will be necessary to crack thisdifficult matter in order to design and uncover the fullpotential of advanced materials.

Received: March 1, 1999Final version: April 26, 1999

±[1] J. D. Eshelby, J. Mech. Phys. Solids 1969, 17, 177.[2] L. Freund, J. Mech. Phys. Solids 1972, 20, 129.[3] L. Freund, J. Mech. Phys. Solids 1973, 21, 47.[4] L. Freund, J. Mech. Phys. Solids 1974, 22, 137.[5] L. Freund, Dynamic Fracture Mechanics, Cambridge University Press,

Cambridge 1990.[6] B. Kostrov, Appl. Math. Mech. 1966, 30, 1241.[7] J. R. Willis, in Elasticity: Mathematical Methods and Applications (Eds:

G. Eason, R. W. Ogden), Halston, New York 1990, pp. 397±409.[8] J. Fineberg, M. Marder, Phys. Rep. 1999, 313, 1.

[9] A. Kobayashi, N. Ohtani, T. Sato, J. Appl. Polym. Sci. 1974, 18, 1625.[10] J. Fineberg, S. Gross, M. Marder, H. Swinney, Phys. Rev. Lett. 1991, 67,

457.[11] J. Fineberg, S. Gross, M. Marder, H. Swinney, Phys. Rev. B 1992, 45, 5146.[12] E. Sharon, S. P. Gross, J. Fineberg, Phys. Rev. Lett. 1995, 74, 5146.[13] R. Thomson, C. Hsieh, V. Rana, J. Appl. Phys. 1971, 42, 3154.[14] M. Marder, S. Gross, J. Mech. Phys. Solids 1995, 43, 1.[15] E. Sharon, S. P. Gross, J. Fineberg, Phys. Rev. Lett. 1996, 76, 2117.[16] E. Sharon, J. Fineberg, unpublished.[17] J. W. Dally, Exp. Mech. 1979, 19, 349.[18] A. Griffith, Mech. Eng. 1920, A221, 163.[19] M. Marder, in FractureÐInstability Dynamics, Scaling, and Ductile/

Brittle Behavior (Eds: R. L. B. Selinger, J. J. Mecholsky, A. E. Carlsson,E. R. Fuller), Materials Research Society, Pittsburgh, PA 1996, pp. 289±296.

[20] P. Gumbsch, S. J. Zhou, B. L. Holian, Phys. Rev. B 1997, 55, 3445.[21] S. J. Zhou, D. M. Beazley, P. S. Lomdahl, B. L. Holian, Phys. Rev. Lett.

1997, 78, 479.[22] F. F. Abraham, IEEE Comput. Sci. Eng. 1997, 4, 66.[23] F. F. Abraham, in Some New Directions in Science on Computers (Eds:

G. Bhanot, S. Chen, P. Seiden), World Scientific, Singapore 1997, pp. 91±113.

[24] R. K. Kalia, A. Nakano, K. Tsuruta, P. Vashishta, Phys. Rev. Lett. 1997,78, 689.

[25] V. B. Shenoy, R. Miller, E. B. Tadmor, R. Phillips, Phys. Rev. Lett. 1998,80, 742.

[26] H. Rafii-Tabar, L. Hua, M. Cross, J. Phys.: Condens. Matter 1998, 10,2375.

[27] E. B. Tadmor, M. Ortiz, R. Phillips, Phil. Mag. A 1996, 73, 1529.[28] T. Honglai, Y. Wei, Acta Mech. Sin. 1994, 10, 237.[29] T. Honglai, Y. Wei, Acta Mech. Sin. 1994, 10, 150.[30] L. Verlet, Phys. Rev. 1967, 159, 98.[31] D. Frenkel, B. Smit, Understanding Molecular Simulation, Academic,

San Diego, CA 1996.[32] M. Allen, D. Tildesley, Computer Simulations of Liquids, Clarendon,

Oxford 1987.[33] H. Balamane, T. Halicioglu, W. A. Tiller, Phys. Rev. B 1992, 46, 2250.[34] M. Z. Bazant, E. Kaxiras, J. Justo, Phys. Rev. B 1997, 56, 8542.[35] B. Lawn, Fracture in Brittle Solids, 2nd ed., Cambridge University Press,

Cambridge 1993.[36] B. Lawn, B. Hockey, S. Wiederhorn, J. Mater. Sci. 1980, 15, 1207.[37] F. H. Stillinger, T. A. Weber, Phys. Rev. B 1985, 31, 5262.[38] C. P. Chen, M. H. Leipold, Am. Ceram. Soc. Bull. 1980, 59, 469.[39] M. Z. Bazant, E. Kaxiras, Phys. Rev. Lett. 1996, 77, 4370.[40] J. Justo, M. Z. Bazant, E. Kaxiras, V. Bulatov, S. Yip, Phys. Rev. B 1998,

58, 2539.[41] E. Kaxiras, M. Z. Bazant, J. F. Justo, Mater. Res. Soc. Proc. 1997, 491,

339.[42] J. H. Rose, J. R. Smith, F. Guinea, J. Ferrante, Phys. Rev. B 1984, 29,

2963.[43] B. Dodson, Phys. Rev. B 1987, 35, 2795.[44] J. Tersoff, Phys. Rev. Lett. 1986, 56, 632.[45] J. Tersoff, Phys. Rev. B 1988, 37, 6991.[46] J. Tersoff, Phys. Rev. B 1988, 38, 9902.[47] R. Biswas, D. Hamann, Phys. Rev. B 1987, 36, 6434.[48] E. Pearson, T. Takai, T. Halicioglu, W. A. Tiller, J. Cryst. Growth 1984,

70, 33.[49] B. Bolding, H. Andersen, Phys. Rev. B 1990, 41, 10 568.[50] J. Chelikowsky, J. Phillips, Phys. Rev. B 1990, 41, 5735.[51] D. Holland, M. Marder, Phys. Rev. Lett. 1998, 80, 746.[52] D. Holland, M. Marder, Phys. Rev. Lett. 1998, 81, 4029.[53] T. Nakamura, D. Parks, J. Appl. Mech. 1988, 55, 805.[54] L. D. Landau, E. M. Lifshitz, Statistical Physics Part 1, Pergamon, New

York 1980, Vol. 5, p. 362.[55] B. Martin, Statistics for Physicists, Academic, London 1971, pp. 47±49.[56] M. Marder, Phys. Rev. E 1996, 54, 3442.[57] J. A. Hauch, D. Holland, M. Marder, H. L. Swinney, Phys. Rev. Lett.

1999, 82, 3823.[58] J. C. H. Spence, Y. M. Huang, O. Sankey, Acta Metall. Mater. 1993, 41,

2815.[59] J. A. Hauch, M. P. Marder, Int. J. Fract. 1998, 90, 133.


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cracks and atoms**chaos.ph.utexas.edu/manuscripts/1067546500.pdfexperiment seem as remote as a...

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