A Dynamic Atomistic-Continuum Method for the Simulation of Crystalline Materials

Density-functional tight-binding molecular dynamics simulations of SWCNT growth by surface carbon diffusion on an iron clusterYasuhito Ohta a ,Yoshiko Okamoto a ,Stephan Irle a,b,*,Keiji Morokuma a,*a Fukui Institute for Fundamental Chemistry,Kyoto University,34-4Takano Nishihiraki-cho,Sakyo-ku Kyoto,Kyoto 606-8103,Japan bInstitute for Advanced Research and Department of Chemistry,Nagoya University,Nagoya 464-8602,JapanA R T I C L E I N F O Article history:Received 19November 2008Accepted 2January 2009Available online 8January 2009A B S T R A C TIron-catalyzed SWCNT growth by carbon diffusion starting from a carbon cap has been demonstrated in density-functional tight-binding molecular dynamics simulations.A C 40(5,5)SWCNT cap attached to an Fe 38cluster was employed as initial model system.After 40carbon atoms were supplied onto the iron surface for 20ps,dynamics were continued for 160ps without supply of further carbon feedstock.Growth of the SWCNT sidewall is mainly due to surface-diffusion of short carbon chains,and to a lesser degree due to sub-surface diffusion.Newly created rings consist only of pentagons and hexagons,while heptagons are infrequent and short-lived,which seems to be caused by the slower,more ordered sidewall growth due to the diffusion process.Ó2009Elsevier Ltd.All rights reserved.1.IntroductionSingle-walled carbon nanotubes (SWCNTs)[1]are known to typically grow from metal or metal carbide particles,and understanding the growth mechanisms is a major goal in carbon nanotube research [2].Recent in situ environmental transmission electron microscopy (ETEM)experiments have successfully captured remarkable snapshots of SWCNTs dur-ing growth activity,where a carbon cap structure was first formed on an iron–carbide nanoparticle and then evolved into a rod-shaped structure [3].The experiments have clearly demonstrated that metal-catalyzed SWCNT growth is initi-ated by the formation of a carbon cap structure on the sur-face of a catalytic nanoparticle.However,atomistic details of the growth reaction mechanism are still not fully understood.Computer simulations of metal-catalyzed SWCNT growth are very challenging.Several groups have employed classical empirical molecular dynamics (MD)simulations [4,5]mainly based on Brenner’s reactive empirical bond order (REBO)po-tential [9].In such simulations,neither the effect of p -conju-gation nor the change in reactivity on the metal surface due to interactions with the metal d bands is included.Gavillet et al.have used a first-principle MD method based on the Car-Parrinello (CPMD)approach [6],especially focusing on reaction dynamics between feedstock carbons diffusing on a metal surface with a carbonaceous cap structure attached.In their model system,a (6,6)armchair carbon cap structure was attached to an hcp Co metal slab,and carbon atoms were distributed on the metal surface around the cap as feedstock.Starting with this model,they performed molecular dynamics simulations to observe incorporation of carbon atoms into the0008-6223/$-see front matter Ó2009Elsevier Ltd.All rights reserved.doi:10.1016/j.carbon.2009.01.003*Corresponding authors:Address:Fukui Institute for Fundamental Chemistry,Kyoto University,34-4Takano Nishihiraki-cho,Sakyo-ku Kyoto,Kyoto 606-8103,Japan.Fax:+81537886151/757814757.E-mail addresses:sirle@iar.nagoya-u.ac.jp (S.Irle),morokuma@fukui.kyoto-u.ac.jp (K.Morokuma).C A R B O N47(2009)1270–1275a v a i l ab l e a t w w w.sc i e n c ed i re c t.c o mjour nal h omepage:www.el sevie r.co m/lo cate/car bonrim of the carbon cap through surface diffusion on the metal slab.Over the course of15ps,five carbon atoms became incorporated into the carbon honeycomb network.However, their initial geometry was neither obtained by geometry optimization nor by thermal annealing.Thus the system is expected to be initially very unstable,inviting unrealistic drastic movement of feedstock carbon atoms.Since the sub-sequent trajectory was very short and the size of the unit cell used was inadequate to realistically describe diffusion pro-cesses,it is quite difficult to obtain essential key reactions during SWCNT growth from these simulations.With the exception of the work by Raty et al.which also uses CPMD simulations but employs an unrealistic carbon nanodia-mond/Fe particle initial geometry[7],we are not aware of any other quantum chemical MD simulations of SWCNT growth by carbon surface diffusion.In the present study,we demonstrate growth of a SWCNT sidewall through surface/sub-surface diffusion of carbons to-wards a SWCNT cap structure on an iron cluster using quan-tum chemical MD method based on the density functional tight-binding(DFTB)method[10,11]combined with afinite electronic temperature approach[12–14].The DFTB method allows to carry out electronic structure calculations that are at least two orders of magnitude faster than thefirst-princi-ples density functional theory(DFT),while providing remark-ably similar energetics and structures[15].Using DFTB/MD, we have successfully performed Fe-catalyzed continued SWCNT growth simulations where the reactions of carbon feedstock atoms at the metal–carbon interface of a SWCNT–Fe cluster[8]and the temperature dependence of the growth rate of such SWCNT amplification[16]was investigated,as well as the rapid growth mechanism of a SWCNT from a C20 corannulene cap fragment[15].The simulation time required to accomplish growth by less than10A˚tube length was typi-cally around45ps.Although these rapid growth simulations provided mean-ingful information to understand the reactivity of the nano-tube–Fe interface,gas-phase C atoms were continuously supplied around the nanotube–Fe interface during growth simulations at a rapid pace.This purposeful targeting of the interface area obviously favors the simulations on events occurring at the nanotube–Fe interface,while disfavoring the occurence of events taking place on the metal particle. In addition,the highly exothermic reactions between C atoms and the nanotube–Fe interface instantaneously led to local heating,causing high mobility of atoms in the reaction area. To the contrary,in a‘‘real’’nanotube–Fe cluster,the cross-sec-tion of the metal surface is much larger than that of the nano-tube–metal interface.Therefore,the majority of feedstock carbons shouldfirst experience the reaction with the metal surface before contributing to the continued SWCNT growth. For this reason,we employ in the present simulations a differ-ent carbon supply procedure and molecular model,in order to focus on SWCNT growth by surface diffusion of carbons on an Fe cluster.From trajectories obtained we have observed key reactions leading to SWCNT root growth,which have never before been seen in quantum chemical MD simulations.We will discuss the incorporation mechanism of carbon atoms into the rim of the cap in detail,after introducing methods and model putational methodologyFig.1shows the initial structure of our model system,which is composed of a cap-shaped carbon fragment on an iron cluster.The carbon cap is a fraction of a(5,5)SWCNT,ob-tained by removing a section of Ih C60and‘‘cutting’’it down to C40,while the iron cluster part consists of38iron atoms possessing face-centered cubic(fcc)arrangement in analogy with the c-iron,a stable phase in bulk iron system between 1184K and1665K[17].The diameter of the iron cluster is 7A˚,which is relatively small compared to those normally observed in experiments;the small size was chosen due to computational requirements.In similar studies,we have previously performed growth simulations of SWCNT using an open-ended nanotube frag-ment[8,16].These studies were motivated by the experimen-tal study of amplification of a nanotube seed on Fe catalyst [18].The simulations using a nanotube fragment demon-strated the significance of high reactivity at the nanotube–Fe interface,which can easily incorporate gas-phase carbons into the carbon sidewall,allowing for efficient(=rapid)growth of the SWCNT.However,in the previous high temperature simulations,the carbon cluster was exposed to attack by highly reactive carbon atoms from the gas phase,resulting in significantly enhanced heating of the carbon superstruc-ture.The associated large-amplitude vibrations caused partial destruction and reconstruction of the SWCNT fragment.In the present cap model,only less-reactive,surface-adsorbed carbon atoms can react with the SWCNT cap,and the present simulations are therefore more realistic for the study of con-tinued SWCNT growth.Electronic structure energy and gradient calculations dur-ing the present quantum chemical MD simulations are based on self-consistent charge density functional tight-binding (SCC-DFTB)method[11,19]with the use of electronic temper-ature of10,000K[8].The electronic temperature allows the occupancy of each molecular orbital to change smoothly from 2to0depending on its energy,and effectively incorporates the open-shell nature of the system due near-degeneracy of iron d orbitals as well as carbon dangling bonds.The equation of motion is integrated using the Velocity Verlet algorithm with a time step of1fs,and nuclear temperature was con-trolled at1500K using Nose–Hoover chain thermostat[20].The initial carbon–metal cluster shown in Fig.1wasfirst equilibrated at1500K for10ps,and then10configurations la-beled alphabetically A–J were randomly extractedbetweenFig.1–The Fe38C40cluster for the simulation of SWCNT growth via carbon diffusion.C A R B O N47(2009)1270–127512715ps and10ps.To these initial structures,a gas-phase feed-stock carbon atom was supplied around the iron surface every0.5ps for20ps(up to a total of40carbon atoms)with the assumption that atomic carbons are produced experi-mentally either by graphite vaporization or by decomposition of feedstock carbon precursors such as methane on the cata-lytic iron cluster.The velocity vectors of the incident carbon atoms were directed to the center of mass of the iron cluster with an incident energy of0.13eV,which is equivalent to the kinetic energy of1500K.During the20ps carbon supply sim-ulations,about90%of the gas-phase carbon atoms became adsorbed on the surface of the iron cluster,while the rest of them was incorporated into the rim of the carbon cap to form short polyyne chains such as C2and C3without forming any polygonal rings.The carbon atoms on the iron surface exhib-ited Brownian-like motion and reacted with each other to form polyyne chains.After20ps,the carbon supply was stopped and the simulation time‘‘clock’’in each trajectory was reset to zero for starting the cap growth simulations through diffusion of carbons on the iron cluster.The total dif-fusion simulation time is160ps,and height of the cap frag-ment was monitored by calculating the distance between the center of mass of the iron cluster,G Fe and the most distant cap carbon from G Fe.Due to the different equilibration times, the overall total simulation time ranges from185ps to190ps.3.Results and discussionFig.2a shows snapshots of the10trajectories after20ps car-bon supply simulation,and(b)after additional160ps diffu-sion simulation.In most of thefinal structures,the cap-shaped carbon fragments are found to extend their sp2 hybridized network at the rim of the cap by forming new hexagons and pentagons(indicated by purple atom color in Fig.2b)with a ratio close to1:1.We have also observed that no seven-membered rings were formed in the present simula-tions.This is contrastive to the previous carbon supply simu-lations near the nanotube–metal interface[8,16].In the previous simulations,a considerable number of seven-mem-bered rings was produced in the carbon sidewall by rapid hexagon-to-heptagon transformation,induced by the reac-tion of the gas-phase C atoms and hexagonal rings[16],whileFig.2–Snapshots of10trajectories(a)after20ps carbon supply simulations,and(b)after additional160ps carbon-diffusion simulations at T=1500K.Pink spheres highlight newly formed polygonal carbon rings.(For interpretation of the references to colour in thisfigure legend,the reader is referred to the web version of this article.)1272C A R B O N47(2009)1270–1275in the present simulations,only chemisorbed feedstock car-bons reacted with the rim of the carbon cap by surface/sub-surface diffusion.The difference between the previous and present simulations highlights that surface-adsorbed carbons contribute to the construction of carbon sidewall producing a smaller number of defects,indicating that the rim of the car-bon cap is a key area for chirality-specific root growth of a SWCNT.We note that the carbon network surfaces containing newly formed polygonal rings at the rim of the cap are almostparallel to the nanotube axis in each of the 10trajectories,clearly demonstrating thereby carbon sidewall construction based on SWCNT root growth on the iron cluster.Fig.3shows details of the growth process of the carbon-cap observed in trajectory I and an average over all trajectories in Fig.3c dur-ing the diffusion simulations.As seen for trajectory I in Fig.3a and b,intermittent hexagon formation occured repeatedly be-tween t =0and 40ps,and the cap height accordingly in-creases,leading to cap lift-off from the iron cluster.In other trajectories,polygon formation takes place also beyond the first 40ps,with an average rate of only 0.00625/ps =1/160ps for hexagon formation (compare Fig.3c.We expect that fur-ther hexagonal ring formation will occur in all trajectories when longer simulations are performed.During the carbon diffusion simulations,the formation of polygonal rings at the rim of the cap occurred via reactions between the polyyne chains dangling at the rim of the carbon cap and diffusive carbons on the iron surface.Fig.4depicts a typical ring formation process during cap-growth on the iron cluster.In steps (a)to (b),a C 2molecule diffuses on the sur-face of the iron cluster and reacts with a polyyne chain dan-gling from the rim of the carbon cap.In steps (c)to (d),the extended polyyne chain migrates on the iron surface and occasionally approaches an edge carbon at the rim of the car-bon cap to close a polygonal ring such as pentagon and hexa-gon.Once the polygonal ring is formed,it hardly breaks because of the structural stability of the sp 2hybridized net-work,leading to the construction of carbon sidewall neces-sary for SWCNT growth.The importance of polyyne chains at the rim of the cap is consistent with our previous SWCNT growth simulations [8,16].During the simulations,no polyyne chains penetrated into the inside of the iron cluster,but several atomiccarbonsFig.3–Growth process of the carbon cap observed in the trajectory I at T =1500K and average over all trajectories.(a)Time-variation of total number of pentagons and hexagons for trajectory I.(b)SWCNT height as function of time for trajectory I.(c)Time-variation of total number of pentagons and hexagons averaged over all 10diffusion trajectories.C A R B O N47(2009)1270–12751273exhibited sub-surface diffusion into the iron cluster.We ob-served that these sub-surface atomic carbons also contrib-uted to sidewall construction by reacting with the rim of the carbon cap.One of remarkable incorporation processes involving sub-surface carbon atom is shown in Fig.5.In step (a),a carbon atom penetrates into the metal cluster,exhibit-ing Brownian‘‘hopping’’on metal atom sites.In step(b),the carbon atom occasionally emerges on the metal surface en-closed underneath the cap.In step(c),the emerged carbon atom reacts with one of the cap edge carbons to form a r C–C bond.In steps(d)to(e),the newly formed C–C bond dangles at the rim of the cap and then reacts with the neighbor poly-yne chain to close a polygonal ring such as hexagon and pen-tagon.In this atomic carbon incorporation process,the fluctuation of iron atoms played two important roles.One is frequent C–Fe bond breaking at the rim of the carbon cap. This bond breaking increases the opportunity for the diffusive carbons to become incorporated into the rim of the carbon cap.Another role is promoting hopping motion of diffusive carbons.In the high temperature environment,the amplitude of iron–iron distance is increased with large deformation of the metal cluster.These two aspects promote carbon atom migration into openings between iron atoms,and increases the frequency of reactions between the diffusive carbons and the rim of the carbon cap.4.SummaryIn summary,we have demonstrated single-walled carbon nanotube(SWCNT)growth starting from a C40carbon cap fragment through surface/sub-surface diffusion of carbons on an iron cluster using a quantum chemical molecular dynamics(QM/MD)method based on the density-functional tight-binding(SCC-DFTB)method with afinite electronic tem-perature approach.In carbon diffusion simulations of up to 160ps length,we have found that relatively short polyyne chains such as C2,C3,and C4are incorporated into the cap fragment through surface diffusion on the iron cluster,while carbon atoms are also incorporated via surface as well as sub-surface diffusion.In the case of sub-surface diffusion,carbon atoms emerging from the iron cluster underneath the carbon cap contribute to the extension of the growing SWCNT side-wall.During the diffusion growth,only pentagons and hexa-gons were created in contrast to previous simulations[16], where carbon atom was supplied near the Fe/C interface lead-ing to creation of an equal amount of pentagons and hepta-gons with smaller to equal numbers of hexagon rings.The present simulations mark thefirst time that a SWCNT frag-ment was extended significantly during QM/MD simulations of carbon diffusion on a metal nanoparticle.AcknowledgementsThis work was in part supported by a CREST(Core Research for Evolutional Science and Technology)grant in the Area of High Performance Computing for Multi-scale and Multi-phys-ics Phenomena from the Japan Science and Technology Agency(JST).One of the authors(S.I.)also acknowledges sup-port by the Program for Improvement of Research Environ-ment for Young Researchers from Special Coordination Funds for Promoting Science and Technology(SCF)commis-sioned by the Ministry of Education,Culture,Sports,Science and Technology(MEXT)of Japan.The simulations were per-formed in part using the computer resources at Academic Center for Computing and Media Studies(ACCMS)at Kyoto University.Appendix A.Supplementary dataInitial Cartesian coordinates of the Fe38C40complex,AVI-mo-vie of hexagon formation through surface diffusion of poly-yne chains,AVI-movie of incorporation reaction of cap-caged carbon atom into the rim of the carbon cap fragment. 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Lec 2 (2/8/02) Survey of Simulation TechniquesThe Multiscale Materials Modeling ConceptDuring the past fifteen or so years, materials research has gained increasingly wide-spread recognition in the scientific community for its relevance to technological innovation and competitiveness. The advent of another powerful driver -high performance computing -has given rise, essentially in parallel, to intense interest in the computational approach to advanced materials research. The term multiscale materials modeling (MMM) has now taken on the meaning of theory and simulation of materials properties and behavior across length and time scales from the atomistic to the macroscopic. Although sometimes the importance of experiments is not explicitly acknowledged, it is nevertheless understood, at least by the more informed practitioners of materials simulation that selected experimental information will be indispensable to successful modeling, for providing appropriate database to determine the theoretical parameters and for validating the critical model assumptions.The relevance of multiscale modeling is fundamentally predicated on the belief that such analysis and prediction will bring about better understanding and control of materials microstructures, which in turn are essential in the application areas of processing, performance evaluation, and ultimately the design of new materials. Because physics, chemistry, and various fields of engineering are all disciplines important to materials modeling, this emerging field has broad appeal to students with a diverse range of backgrounds. For basically the same reason, there is considerable impetus for different groups of research investigators at universities and research labs to team up to tackle grand-challenge type problems that lie beyond the expertise and capabilities of any single group.As materials research expands in both breadth and depth, molecular engineering of materials becomes more of a reality. This is the long-held dream of every materials scientist and engineer where new materials can be created with such benefits as enhanced performance, extended service life, and acceptable environmental impact, not to mention cost reduction. Even though computer-aided materials design has yet to achieve the success of computer-aided molecular (drug) design, there is impressive progress being made, especially in the area of functional materials for microelectronics, optical and magnetic applications. In contrast, for structural materials the complexities of mechanical, thermal, chemical (alloying, corrosion, etc.) phenomena continue to pose formidable challenges to reliable and predictive modeling. As a result, it is blieved that the most promising approach to understanding and control of these phenomena is to effectively combine several simulation techniques, each one being suited for a particular length and time scale.One can think of MMM as an open environment in which materials phenomena are described over a continuous distribution of length and time scales. The spirit of MMM is outward looking, intuitively logical, and flexible that it invites the investigtor to come up with his own interpretation. Yet despite the different ways of saying what is MMM, no general concensus on how to define the underlying concept has been established. Much like a good painting, it can have different meanings for different people, and in this way it maintains a certain freshness and contemporary appeal.Simulation Techniques in Multiscale Materials ModelingReturning to Fig. 1-1, shown at the last lecture, one sees that in materials processing the length scales span the range from 1 angstrom to about 1 m. The 10 decades go from nanotechnologies at the short end to system components and structures we normally associate with the macroscopic world. The order in which the topics themselves are arranged is interesting in that it begins with atomistic modeling which is one of the 4 characteristic length scales in MMM (see below) but not the smallest length scale. Several entries in this listing could be relevant to what we will be discussing later in the term, such as molecular dynamics/diffusion, phase equilibria/transformation, elasticity-plasticity, thermodynamics, thermomechanics, and deformation processing. Just these alone already give one the impression that materials processing involves a very wide range of physical phenomena, and this is only a partial list. We can regard this viewgraph as a reminder of the vastness of the material modeling landscape and the need to have a sense of direction in one's own research.In many fields of scientific inquiry, it is commonly recognized that a single physical phenomenon can be examined at several levels or length (time) scales. For example, the complicated motion of a ocean wave as it washes onto a beach can be observed by seeing it in real life, or it can be visualized (in our minds eye) in terms of the individual movements of the water molecules which make up the wave. Depending on the scale of interest, the relevant dynamics requires quite different ways of analysis -in this case, continuum fluid dynamics to describe waves breaking on a beach and discrete-particle molecular dynamics to describe the atomic motions of the molecules.Fig. 2-1For many materials problems one can identify four distinct length scales where different aspects of a physical phenomenon can be analyzed. As shown in Fig. 2-1 these four regions maybe referred to as electronic structure, atomistic, microstructure, and continuum. Imagine a piece of material, say a crystalline solid. The smallest length scale of interest is about a few angstroms (10-8 cm). On this scale one can deal with the individual electrons of an atom. The appropriate method for modeling relevant processes is called density functional theory or first-principles (ab initio) calculation. Being fully quantum mechanical, it is the most computationally demanding of all the techniques, and as a result it can be applied only to small simulation systems, the current limit being about 300-400 atoms. The system being modeled at the electronic structure level is therefore a collection of atoms in the form of nuclei (or ions) and electrons, see Fig. 2-2.Fig. 2-2Sometimes the electrons are further separated into core and valence electrons. The ions are described classically using Newton's equations of motion, F = ma , while the electrons are governed by the Schroedinger equation, H y = E y . The computationally intensive part of the calculation lies in the solving the Schroedinger equation. Once the wave function y is determined, we then know the electron charge distribution in the system, r = y 2 , which tells us about the bonding between atoms, a very fundamental and useful piece of information.The next scale in Fig. 2-1, spanning hundreds of angstroms, is called atomistic. Here the system is represented a collection of atoms. The appropriate simulation techniques are called molecular dynamics (MD) and Monte Carlo (MC); they are quite well developed and will be discussed in detail in Weeks 3 and 4 respectively. Such methods require the knowledge of an interatomic potential function U(r 1,..., r N ) (recall Lec 1) which are often obtained empirically byfitting a functional form with several parameters to experimental data. By not treating the electrons explicitly, atomistic simulation is computationally much simpler than electronic structure calculations, which means that one can treat a much larger number of particles, the limit being about 109 at the present time. The equations to be solved in the case of molecular dynamics is again Newton's equation of motion, while in the case of Monte Carlo one does not solve any equations of motion but instead sample a certain distribution of positions. With either simulation the essential input is the same, the potential function U. The goodness of a simulation therefore depends on how accurately do we know U for the material of interest. Because the electronic structure effects are ignored, atomistic simulations are not as reliable as ab initio calculations. This is especially true in situations where chemical bonds are broken as in fracture or rearranged as in chemical reactions. On the other hand, one has the advantage of being able to simulate much larger systems over much longer times. This in turn means one can study more physical properties and behavior of the material.The length scale above atomistic is sometimes called the mesoscale, the characteristic length being a micron (104 angstrom). Here the simulation technique commonly in use is the finite-element method (FEM), where the system is represented by a grid of elements, triangles in 2 dimensions, cubes or tetrahedra in 3 dimensions, for example. The equations to be solved describe force balance on each element, the output being the nodal displacements of the specified grid. Typically, the grid is chosen to be fine enough to represent idealized microstructural features of micron size, which makes this method well suited for the simulation of materials microstructure in many applications. Notice, however, for FEM calculation one needs to specify the force acting on an element due to its neighbor, and this kind of mterials property has to be provided either from experiment, if available, or from studies at the atomistic or ab initio level.Beyond the mesoscale one has the macroscale where the material is represented as a continuum described by distributions which vary rather smoothly on the scale of mm and larger. As an example we can imagine the temperature distribution in a rod that is heated at one end. The equation describing this system is the time-dependent heat conduction equation (Fig. 2-2), with k denoting the thermal conductivity of the material. Like the FEM simulation, the materials properties needed for simulation at the continuum level have to be supplied externally. In this case the conductivity is a quantity that one measure experimentally, or calculate by means of atomistic simulation.We can see why it is important to couple the different length scales. For practical design calculations, either continuum or finite-element methods are the most useful. However, these methods require parameter or property specifications that cannot be generated internally, nor can they provide the electronic structure or atomic-level understanding of mechanisms that are essential for prediction and design. It is only when the different methods are effectively integrated that one can expect materials modeling to give fundamental insight, or reliable predictions on the appropriate length scale. Once this is achieved, then materials modeling can justifiably claim to be a valuable complement to experiment and theory.In Fig. 2-1 we also indicate the regions where nanoscience and technology would operate, in contrast what is previously the microscale. With regard to biological science at the cellular levl, the range spans from the nanoscale up to tens or even hundreds of microns. The presnetlength scale classification has counterparts in time scales. Moreover, one can discuss various physical properties that manifest on the different length-time scales, and finally one can even roughly associate different disciplines with each level of scales. For example, electronic structure would be the domain of quantum chemistry and condensed matter physics, atomistic and microstructural levels would be materials science and engineering, and continuum level would be the domain of engineering.In as much as the advantages of linking simulation and modeling techniques across different length scales are quite apparent, actual implementation of coupling the different levels of description is highly nontrivial. It is fair to say that no single research group thus far has fully succeeded in demonstrating such a capability, so there are still many opportunities for original research. To emphasize this point we return to Fig. 1-3 which describes a research program currently on-going at the Lawrence Livermore National Laboratory with the goal of understanding dislocation dynamics for applications in high strain-rate deformation. The idea is to take a bottoms-up approach of linking atomistic simulation of dislocation interactions and mobility with mesoscale simulation of single crystal and polycrystal plastic deformation to arrive at appropriate constitutive relations which then may be incorporated into design codes at the continuum level. This is presently one of the most comprehensive efforts to solve a critical materials problems. Its success will go a long way in demonstrating the unique power of the MMM approach to materals research.Fig. 2-3Fig. 2-3 shows several examples of FEM simulations in materials processing. One sees (counter clockwise) the grain microstructure in a turbine blade and a block undergoing directional solidification, the flow pattern or velocity plot in a casting where solid and melt zones begin toseparate, and the thermal profile or temperature zones in a directionally chilled continuous casting. The results demonstrate the usefulness of modeling microstructure in materials technology.Image RemovedFig. 2-4Fig. 2-4 is a cover of a journal devoted to computer-aided materials design, showing a composite of simulation results at different length scales. Starting at the bottom one has the charge distributions in crystalline silicon given by electronic-structure method of density functional theory. As we have mentioned, such information cannot be obtained by any of the other methods in Fig. 2-1. Going up on the left side, one has a molecular dynamics simulation of a crack tip in a two-dimensional crystal under uniaxial tension. The crack (dark region), while moving upward, has started to branch out to take on a jagged appearance which is also seen in laboratory measurement. In Fig. 2-5 we see the same simulation in greater details, the individual atoms in the simulation are now visible. The sequence of six snapshots begins on the left from the top down. The two colors denote the two opposing directions of strain. The fact that the strain directions start to mix as the simulation proceeds means that the state of strain is no longer simpleonce the crack moves further into the crystal. This simulation involves about 30 million atoms, quite large by present standards. If one were to show all the particles in some kind of colorFig. 2-5Fig. 2-6coding scheme, as in Fig. 2-6, one would lose the discreteness of the simulation and the system starts to look like a continuum, including the stress waves that are reflected from the fixed borders of the simulation cell (reminiscent of ripples on the surface of a lake). This is an illustration of how one level of simulation can merge into the next level. The impression we would like to leave here is that atomistic simulation can reveal very local details of how the material is tearing apart, a level of microscopic information that experiments at present cannot provide. The challenge is to understand this kind of complexity and use the knowledge to design materials that would bemore resistant to failure.。