MICROSTRUCTURAL features in metals profoundly affect what happens on a larger scale, particularly when systems are pushed to their limits. For example, jet aircraft turbine blades can fail if small concentrations of particular impurities cluster at the boundaries between the individual crystalline grains of the metal. Even in the absence of impurities, the strength and plastic deformation of a metal are controlled by extended crystal defects called dislocations. Such features also affect the performance of nuclear weapon systems, where materials are pushed to extremes of temperatures and pressures.
Modeling macroscopic mechanical properties such as strength and failure at different length scales-that is, multiscale modeling-is of major interest at Lawrence Livermore because of its relevance to the Department of Energy's Stockpile Stewardship Program. With the cessation of underground nuclear testing, weapons scientists must be able to predict with confidence the properties of materials in stockpiled warheads and their effect on weapons performance.
This need to predict performance has put a high premium on understanding materials behavior. In this respect, the mechanical properties of nuclear weapon materials are uniquely complex. Unlike thermodynamic properties, such as the equation of state, which are fully determined at the atomic length scale, mechanical properties are inherently multiscale, depending on phenomena at all length scales. Thus, multiscale modeling is a huge scientific challenge as well as a critical necessity for successful stockpile stewardship. To meet these demands, Livermore's multiscale-modeling effort involves some 25 researchers from a variety of disciplines (theoretical and weapons physics, engineering, and chemistry and materials science) as well as many outside collaborators, including the Massachusetts Institute of Technology, Stanford University, the University of California at Los Angeles, the University of Illinois, Brown University, Yale University, Carnegie-Mellon University, and IBM.

Making Connections among Scales
In the multiscale program, scientists are creating and validating computer models to predict and explain the mechanical properties of metals at dimensions ranging from a fraction of a nanometer to meters. The focus is on three major length scales-the atomic scale (nanometers), the microscale (micrometers), and the mesoscale (millimeters and above) (Figure 1). What sets this effort apart from previous ones is that fundamental physical and mathematical principles are rigorously applied to the modeling at each scale, and data are then passed to the next scale up. In the past, such efforts were hampered by the lack of computational power needed to simultaneously model the individual and collective behavior of a large number of atoms and defects. Now, by combining a multiscale-modeling strategy with spectacular advances in computational technology, scientists are shedding light on the fundamental mechanisms that determine how materials deform and fail.
John Moriarty, a leading physicist in Livermore's multiscale-modeling effort, notes, "In the days of underground weapons testing, hydrodynamic computer codes relied on purely phenomenological models of mechanical properties based upon limited experimental data obtained at or near ambient conditions. In the multiscale-modeling program, we are developing a predictive capability based on first principles. That is, the predictions we make at the everyday macroscopic level will be based on fundamental quantities and rules derived from the atomic scale and microscale."
The program is currently focused on the prototype problem of strength and plastic deformation in body-centered-cubic (bcc) metals, such as molybdenum and tantalum (Figure 2). These metals are of special interest because of their physical and structural similarity to stockpile materials. Tantalum, in particular, is predicted to remain a bcc metal to extremely high pressures. In addition, notes Moriarty, the thermodynamic and mechanical properties of metals such as tantalum are of long-standing interest to both the high-pressure and materials physics communities.
Over the years, the materials scientists have accumulated substantial data on the yield strength and other mechanical properties of bcc metals at or near ambient pressure, and more detailed and accurate data are being obtained as part of the multiscale program. Theoretical corroboration has been lacking, however, as has information on tantalum's mechanical properties at high pressures. Rigorous mathematical answers are difficult to come by because mechanical properties depend on phenomena at all length scales. The advent of the Department of Energy's Accelerated Strategic Computing Initiative means that the computational power is now available to bridge the different length scales and accurately model the mechanical properties of tantalum and other metals from first principles.

Climbing the Multiscale-Modeling Ladder
To define the plastic deformation problem in detail, multiscale modelers use a top-down strategy to pose questions and a bottom-up strategy to obtain solutions. Modeling at each length scale helps pose critical questions to be addressed at the next lower scale. To achieve the corresponding solution, appropriate simulations at the atomic scale, for example, provide input at the microscale.
For tantalum, Moriarty and others start with its fundamental atomic properties, using rigorous quantum-mechanical principles and first-principles calculations to develop accurate interatomic force laws that can be applied to atomistic simulations involving many thousands of atoms. From these simulations, they derive the properties of individual dislocations in a perfect crystal and then, with new microscale simulation techniques, look at the behavior of large collections of interacting dislocations at the microscale in a grain-sized crystal. They model the grain interactions in detail with finite-element simulation codes, and from those simulations, they finally construct appropriate models of properties such as yield strength in a macroscopic chunk of tantalum. At each length scale, the models are experimentally tested and validated with available data. Once validated, the models can be used to predict behavior in regimes not achievable in the laboratory.
The challenge at the atomistic level is to learn how individual dislocations move and interact in the presence of an applied stress. Dislocations-which appear as extra or displaced planes of atoms inserted into the regular latticelike structure of a metal crystal-allow otherwise crystalline material to deform plastically without brittle fracture or failure. Figure 3 shows examples of edge and screw dislocations. Edge dislocations resemble an extra sheet of paper slipped part way into a stack of sheets. In a screw dislocation, the atomic planes are twisted like the steps of a spiral staircase.
The energy of a dislocation is stored largely as strain in the surrounding lattice. The important property of a dislocation is its ability to move easily through the lattice, allowing slip to propagate rapidly. (Slip, or the movement of one atomic plane over another, is the primary way that plastic deformation occurs in a solid.) In bcc metals, screw dislocations limit plastic flow because they are much less mobile than edge dislocations, especially at low temperature or under high strain-rate deformation conditions.

Progress So Far
To date, the team has calculated a wide range of deformation and defect properties for tantalum, validated those calculations, and carried them up to extremely high pressures for many of those properties, including bcc elastic constants. At ambient pressure, the elastic constants agree with measured values. Experiments are under way to measure these quantities at high pressure. Atomistic simulations have been used to predict the atomic structure of selected grain boundaries in niobium, molybdenum, and tantalum. The predicted structures in niobium and molybdenum were confirmed by high-resolution electron microscopy (HREM) experiments, and additional experiments on tantalum are in progress.
The team has also used atomistic simulations to study fundamental properties of screw and edge dislocations in molybdenum and tantalum at ambient pressure. The simulations predicted atomic core structures with unique threefold spreading for the screw dislocations; for molybdenum, this spreading was recently confirmed by HREM experiments in Germany. The minimum, or Peierls, stress required to move these screw dislocations has also been studied as a function of the orientation of the applied stress. This minimum stress can be further reduced by forming local excitations called kinks along the dislocation line, and a study of kink energetics leading to dislocation mobility is in progress.

Bridging Length-Scale Worlds
Microscale modeling bridges the atomic and mesoscale worlds. At the microscale, researchers are developing entirely new three-dimensional, dislocation-dynamics (DD) simulation techniques to model single 15-micrometer-long crystals. In these simulations, dislocation structures are resolved, but individual atoms are not, and the basic building blocks are small segments of individual dislocations. In DD simulations, dislocations move and interact according to linear elasticity laws as well as rules established by atomistic simulations and fine-grained DD simulations of small numbers of dislocations. Developing these rules rigorously is one of the most difficult aspects of the multiscale program. A complementary experimental program is examining dislocation microstructures with transmission electron microscopy and providing stress-strain data on well-characterized, high-purity samples.
The DD simulations provide insights and detailed information about the collective behavior of large numbers of interacting dislocations. They also simulate the evolution of a complex dislocation microstructure under an applied stress (Figure 4). Adds Moriarty, "Dislocations and their distribution are an essential part of plastic deformation. But never before has there been such a powerful tool to model this phenomenon." In the multiscale strategy, the goal at the microscale is to provide a full quantitative description of single-crystal plasticity, including the yield stress and stress-strain relationships. With currently available phenomenological input, the DD simulations have provided accurate results for the single-crystal yield stress in tantalum, including its temperature dependence.

The derived laws of single-crystal plasticity will ultimately be used in mesoscale-modeling simulations to predict the deformation of millimeter-sized tantalum polycrystals, that is, multiple single crystals with different orientations packed together into a single specimen. In mesoscale modeling, the individual crystals and their boundaries are resolved, but microstructures and individual dislocations are not. At this scale, scientists are using finite-element simulation codes such as NIKE3D and ALE3D to examining how a system of randomly arranged, computer-generated single crystals-a virtual test sample-deforms in response to an applied stress. The mesoscale-modeling results will finally be used to derive constitutive relations that describe macroscopic plasticity.
The multiscale-modeling program expects to complete its task in about eight more years, linking quantum-based atomistic models all the way up to finite-element-based mesoscopic simulations. When complete, the models will help stockpile stewardship scientists confidently predict the performance of stored weapons and changes that might occur in the stockpile, as well as provide basic information about material behavior of interest to the nation's industrial products manufacturers.
-Ann Parker

Key Words: atomic scale modeling, body-centered-cubic (bcc) crystal structure, dislocation dynamics (DD), edge dislocation, mesoscale modeling, microscale modeling, multiscale modeling, polycrystals, screw dislocation, stockpile stewardship, transmission electron microscopy.

For further information contact John Moriarty (925) 422-9964 (moriarty2@llnl.gov).

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