I. INTRODUCTION

I.1.  Summary
We propose to create a Cooperative Research Team (CRT) focused on the study of "Microstructural Effects on the Mechanics of Materials" as part of the Computational Materials Science Network (CMSN).  Our goal is to bring together a diverse set of researchers, each with their own approaches and skills, to develop a hierarchically structured, integrated approach towards materials modeling across all the inherent physical length and time scales relevant to microstructural effects in materials mechanics. To focus the efforts of the Team, we will investigate the interplay between dislocation and grain microstructures in polycrystal plasticity. Our specific goal is to elucidate the fundamental dislocation and grain-boundary processes thought to be responsible for the crossover in the well-known Hall-Petch effect, from "normal" behavior at larger grain sizes to the "inverse" behavior for grain sizes less than typically 20 nm in grain size. Insights gained from this study will naturally lead to a better understanding and predictive capability for related, more complicated deformation processes controlled by the interplay between dislocation and grain-boundary processes that are critical in many applications of microstructurally designed materials. By focusing the efforts of a variety of researchers with broad scientific and computational expertise on the same problem, perhaps the most important outcome of this team effort will not only be the development of a conceptual framework enabling the bridging of length and time scales in materials modeling but also the emergence of new scientific ideas and more predictive models in this important area of materials science.

I.2.  Overview
Within the materials science and mechanics communities it is widely recognized that the key ingredient needed for the development of a predictive model for the mechanical behavior of polycrystalline materials has to come from a deeper understanding of the dynamical interplay between dislocation and grain microstructures. The underlying microstructural length and time scales are critical because in functional materials the spatial arrangement of the dislocations and grain boundaries in the microstructure may be controlled and optimized via sophisticated materials synthesis and processing techniques. The resulting microstructurally designed materials are important not only as light-weight components with superior mechanical performance but also in high-temperature applications.  For example, the combination of low density with high operating temperature gives silicon-based fiber reinforced structural ceramics considerable potential for aerospace applications and as engine components.  Light-weight aluminum alloys are a key component of creating commercially viable, highly fuel-efficient and pollution-free automobiles.  Thermal-barrier and hard and corrosion-resistant coatings can greatly increase the operating temperature of turbine engines or extend the operating life of cutting tools and engine components.  Microstructural control can also facilitate the processing of inherently brittle materials.  For example, the development of superplastic ceramics and alloys is revolutionizing the processing of inherently brittle materials and the manufacturing of components.

One of the greatest challenges in realistically simulating the mechanical behavior of polycrystalline materials is the development of a unified predictive modeling approach that incorporates all the relevant length and time scales of the problem.  These length and time scales range from the level of the atoms and their electronic-bonding environments, via the various length scales associated with the dynamical behavior of interacting dislocation and grain microstructures (the "mesoscale"), all the way up to the continuum level. What makes this task particularly challenging is that the type of materials information needed as input or for the validation of simulations at the mesoscale is inherently difficult to obtain from experiments alone. On the other hand, given the rapidly expanding computational resources that have become available in recent years (a pace anticipated to accelerate considerably with the resources that will become available through the Scientific Simulation Initiative (SSI)), the time has come to tackle microstructure both from the bottom up and from the top down, i.e., in a coordinated team approach that uses a combination of atomistic, mesoscopic and continuum modeling techniques to expose materials behavior at the critical, microstructural length and time scales.

It is the purpose of this CMSN effort to assemble a team to develop a coordinated, multifaceted simulation approach that fully incorporates the physics of the material at all relevant levels and, most importantly, to elucidate materials behavior at the microstructural length scales. For example, incorporation of physical parameters associated with dislocation nucleation and migration into a finite-element or Monte-Carlo computational platform will greatly increase the predictive capability of these simulations, permitting phenomena such as strain hardening, crack nucleation and dislocation emission from cracks, etc. to be predictively modeled. At the level of the grain microstructure, physical input parameters involve, for example, grain-boundary energy, mobility, sliding resistance and diffusion, permitting phenomena such as grain growth, grain-boundary diffusion creep, texture formation, etc. to be modeled quantitatively.

By merging atomic-level materials input into mesoscale simulations and by using insights on microstructural processes and phenomena thus obtained to predict the overall, macroscopic materials response at the continuum level, our coordinated team approach will be used to expose the complex interplay between dislocation and grain-boundary processes in the deformation behavior of polycrystalline materials. So as to focus our effort on a conceptually relatively "clean" phenomenon and yet tackle a technologically important problem, at the workshop held at Argonne on April 22, 1999 we have chosen the crossover from normal to inverse Hall-Petch behavior as our team thrust. As described further in Sec. 2.1, the famous Hall-Petch relation predicts that yield strength or hardness of polycrystalline materials increases with decreasing grain size. Recently, however, studies of nanocrystalline materials have reported a crossover from this "normal" to an "inverse" Hall-Petch behavior, i.e., a decrease of strength with decreasing grain size, beyond a critical grain size of typically about 10-25 nm. This crossover is thought to arise from a change in the underlying deformation mechanism, from one governed by the grain-size dependent dislocation mean free path to one governed by a grain-sliding or other grain-boundary mediated process.

Our investigation of the fundamental dislocation and grain-boundary mechanisms responsible for this crossover in the Hall-Petch effect will also provide insights into related, more complicated deformation processes controlled by the interplay between dislocation and grain-boundary processes that are critical in many applications of microstructurally designed materials, such as superplastic deformation, high-temperature diffusion creep and intergranular cracking.

II.  ISSUES

II.1.  Scientific Issues
How polycrystalline materials deform under stress is important not only scientifically, for the understanding of plastic flow, but also technologically.  The mean grain size in polycrystalline materials, d, plays a critical role in mechanical properties as it limits the free path of moving dislocations. The piling up of moving dislocations against the grain boundaries gives rise to a strengthening of the material with decreasing grain size; this is known as the Hall-Petch (HP) effect, according to which hardness or strength increases as s ~ d-1/2 (Fig. 1; T.G. Nieh and J. Wadsworth, Scripta Met. 25, 955 (1991)).

The dislocation pile-up explanation of the Hall-Petch effect involves two key assumptions.  First, the stress to pass a dislocation through a boundary must exceed the stress necessary to move a dislocation in an infinite, homogeneous medium.  Second, slip must be mostly planar, so that once a dislocation is emitted from a source, the dislocation remains on the original slip plane (or at least on a nearby parallel plane).  These two conditions lead to the formation of dislocation pile-ups at the grain boundaries and a simple scaling analysis yields the form of the Hall-Petch relation. Microscopy studies, however, often fail to observe the proposed pile-ups in materials that do display the Hall-Petch effect and the validity of the above explanation must be questioned.

Fig. 1. Schematic diagram showing hardness or strength as a function of grain size.  The strength of a material reaches a maximum at a critical grain size, dc, at which the distance between the dislocations piling up against the grain boundaries becomes comparable to the grain size (from T. G. Nieh and J. Wadsworth, Scripta Met. 25, 955 (1991)).
 

The increase predicted by the HP relation obviously has its limitations because the strength cannot exceed the theoretical strength of the material. As the grain size decreases, the distance between the dislocations in the pile-up eventually becomes comparable to d, and individual grains will not be able to support more than one dislocation, a point at which the HP relation ceases to be valid. Also, assuming highly disordered, practically amorphous grain boundaries in nanocrystalline materials (P. Keblinski et al., Acta Mater. 45, 987 (1997)), in the limit of zero grain size the material is essentially amorphous, with a strength that is smaller than the theoretical strength of the crystal, and the grain-boundary strengthening effect will have completely disappeared (see Fig. 1).
The real question is therefore at what critical grain size, dc, the material reaches its maximum strength ("the strongest size"; S. Yip, Nature391, 532 (1998)) and what the softening mechanism is by which the material subsequently looses this maximum strength. According to Nieh and Wadsworth (1991), dc is given by the distance between the dislocations piling up against the grain boundaries. Typical values for dc are of the order of 10-25 nm depending, among other factors, on the elastic moduli of the material and the Burgers vectors of the dislocations in the pile-up. These values of dc are in the range of grain sizes that can be reached and exceeded by the largest atomic-level simulations possible today (see Sec. II.2)

By contrast with the widely accepted dislocation pile-up explanation of the d-1/2 strengthening behavior (for d>dc), the physical process or combination of processes responsible for the inverse Hall-Petch behavior (for d<dc; see Fig. 1) are not clear at all, and a number of mechanisms have been proposed. Among them are

ï reduction of the effectiveness of dislocation sources under the severe microstructural constraints associated with a very small grain size;
ï suppression of dislocation pile-ups due to the presence of a large concentration of grain boundaries;
ï dislocation motion through multiple grains, i.e., grain boundaries that are "transparent" to dislocations;
ï grain-boundary-sliding  induced  deformation;
ï enhanced diffusional creep in the grain boundaries.

In common to all these mechanisms is a critical role played by the interaction between dislocations and grain boundaries, particularly the detailed mechanisms by which a grain boundary can either absorb dislocations or permit slip across the boundary while, perhaps, modifying its own atomic structure and that of the dislocation. For example, one can easily envision that an amorphous grain boundary might be able to absorb a dislocation without significantly changing its atomic structure, particularly in the presence of grain-boundary diffusion at elevated temperatures. By contrast, dislocation absorption by a crystalline boundary is usually accompanied by the formation of a grain-boundary step or ledge; yet some (which?) grain boundaries might be more or less transparent for dislocations.

One could also envision that with decreasing grain size, processes involving grain sliding are becoming energetically more and more feasible, with a potentially strong effect on dislocation slip in the vicinity of the interfaces. In fact, recent atomic-level simulations by Jacobsen et al. (Nature 391, 561 (1998)) suggest that grain sliding is responsible for the decrease in yield strength with decreasing grain size for d<dc (Fig. 1). However, because of their limitation to zero temperature, in these simulations grain sliding is driven solely by the high applied stress, by contrast with the thermally activated grain-sliding processes thought to be critical in enabling superplastic flow. Another limitation of these simulations is the rather small grain size that has prohibited the verification that normal HP behavior, indeed, exists at the larger grain sizes (d>dc). More realistic, finite-temperature simulations that cover both the normal and inverse HP regime are clearly needed (see Sec. III.2.C)

The above discussion of the physics of the Hall-Petch effect illustrates the richness of phenomena involving the interaction between dislocations and grain boundaries and their effects on plastic deformation that can be elucidated by an in-depth study of the crossover from normal to inverse Hall-Petch behavior.

II.2.  Computational  Approach and Challenges
There are four computational approaches that will be linked through the work of this Cooperative Research Team.  Each of these has needs for both conceptual and algorithmic advances.  Atomistic simulations will be used not only to look at individual defect properties, but also the properties of carefully selected model ensembles of dislocations and grain boundaries.  Discrete dislocation simulations will focus on the development and dynamics of dislocation microstructures, both with and without nearby grain boundaries.  Microstructural simulations of grain growth, recrystallization, etc. will incorporate not only materials input of interfacial properties but also consider the effects of dislocation structures. Finally, continuum finite-element analysis will use new non-local approaches to compare directly with simulations at the smaller scales.  There are a number of challenges in each of these computational areas.  Whereas conceptual challenges involve the development of new concepts and methodologies linking length and time scales, algorithmic challenges include, for example, the development of novel graphic visualization tools, methods for handling large data sets, parallelization and the treatment of long-range forces, etc.

Atomistic simulations have traditionally been limited in both length and time scales. Recent advances in computational power through the use of parallel computing has greatly increased the size of systems that can be studied, such that simulations with ten million atoms are not that rare nowadays.  This great increase in system size enables the study of many phenomena for the first time, including interacting defects (dislocation / dislocation, dislocation / grain boundary, etc.), direct simulations of grain growth, grain-boundary sliding, migration and diffusion, etc. In the context of our main thrust, these advances will enable us to simulate directly, at the atomic level, the entire range of grain sizes relevant in the crossover from normal to inverse Hall-Petch behavior (see Fig. 1), and thus to identify and quantitatively capture the changes in the underlying dislocation and grain-boundary processes.  While system size has increased to be able to model systems observable in the laboratory, there is often still a large mismatch in time scales.  In normal molecular dynamics, the time steps are femtoseconds and a nanosecond simulation is long.  Recent advances in time-accelerated molecular dynamics (Voter, Phys. Rev. Lett. 78, 3908 (1997)), however, offer the possibility of simulations in the microsecond to millisecond time scales, at least for some processes. Atomistic simulations can thus be used to model many important phenomena directly even in the time domain, or at least to provide important constitutive relations for mesoscale simulations.

Discrete dislocation methods for the simulation of dislocation microstructures are much less well developed than atomistic simulations and thus present many challenges, ranging from developing more robust representations of dislocation loops in three dimensions to creating better rules for short-range dislocation / dislocation and dislocation / boundary interactions. Some of these require input from other levels of modeling (e.g., atomistics). Also, since dislocations are curvilinear defects, whose topology is fixed by the underlying crystal lattice, the dislocation response and deformation behavior are also highly dependent on the crystal structure.  Moreover, while the interactions between dislocations are well known at long range, at least in the limit of linear elasticity, these interactions are tensorial with the added complication of being very long-ranged (the forces go as the inverse of the distance).  Increasing the complexity is the fact that the number of dislocations in a material is not fixed as dislocations are generated and annihilated in response to external and internal stresses.  These "reactions" between dislocations are also restricted by topology.  Finally, the dislocations form complex structures, the dynamics of which in large part determines the overall deformation response.  The analysis of those structures is in itself a major challenge, and much work needs to be done to develop good analysis techniques of local structures to use for comparison with experiment.

Despite these challenges, much progress has been made in developing discrete simulation methods for dislocations in both two and three dimensions.  There have been a number of alternate approaches proposed by members of the CRT and we will cross compare these methods with our existing discrete dislocation models.  We will employ a new continuum dislocation dynamics model that is based on continuous dislocation density "functions" and will extend that model to the multi-grain geometry.  All the features included in the discrete dislocation dynamics models are represented in this approach.  However, the dislocations are represented by continuous distributions in a phase space consisting of the spatial position vector, the velocity vector, and an angular coordinate measured in each slip system in terms of which the unit tangent to the dislocation line is defined.  A set of kinetic equations is developed which describes the evolution of the dislocations on all slip systems.  We will also examine a hybrid method that couples dislocations into finite-element calculations.

Mesoscale simulations of microstructural evolution are somewhat more advanced than dislocation simulations in that they have been very successful at reproducing the static and dynamic properties of ideal microstructures.  There are a number of methods in use, from the Potts model to phase fields to recently-proposed finite-element methods.  For example, lattice-based Monte-Carlo and cellular-automaton methods accurately capture the steady-state grain size distribution and grain-growth kinetics in pure polycrystals with uniform interface properties. The simplicity of these methods allows them to model systems with tens of thousands of grains, providing a statistically significant sampling of grain sizes. However, the introduction of a realistic sampling of non-uniform interface properties (e.g., multiple grain-boundary types), and the treatment of more complex microstructures (e.g., containing defects), magnifies the computational requirements manifold. In addition, conventional lattice-based models of microstructural evolution rely on Monte-Carlo time scales that cannot be translated into real, physical units (e.g., seconds). Models of interface migration with real time scales do exist, e.g., vertex and front tracking schemes, but these methods are much more computationally demanding than the lattice-based methods and have only been applied to small systems with ideal microstructures. All of these methods therefore require development of more efficient parallel algorithms. Cross comparison of results from these varying approaches is critical.

Finally, the major computational challenge for simulations at the continuum scale is the development of adaptive mesh refinement to capture regions of localized slip during the evolution of patterning.  There are also questions about the order of interpolation necessary to obtain accurate (convergent) solutions when second gradients enter the constitutive response.  The major theoretical challenge is how to rigorously incorporate both atomistic and discrete dislocation descriptions into macroscopic (continuum) models.  Specifically, we will develop finite-element simulations using non-local crystal constitutive relations to study lattice curvature near grain boundaries, localized patterned flow, and the interplay between dislocation dominated and grain-boundary dominated plasticity in the behavior of nanocrystalline materials.

All of these methods will benefit from advances in computational technology and resources.  However, what is truly needed to advance this area of research are conceptual advances.  The overall goal is, of course, to "link the length and time scales".  There are many fundamental issues that are critical to that goal that are unclear.  For example, how do complex structures lead to (relatively) simple behavior (e.g., Hall-Petch)?  How do we find the correct method to average quantities at one scale for use at larger scales?  How do we analyze complex structures?  We have no answers for these questions at this time; these are areas of active research by members of this CRT and others, and the hope is that by linking these disparate efforts, we can make faster progress than we would without the CRT and the Network.

III.  PROJECT  PLAN

III.1.  Objectives
Our investigation of the crossover from normal to inverse Hall-Petch behavior enables us to elucidate a number of important processes and mechanisms involving the interaction between dislocations and grain boundaries and their effects on plastic deformation in general. We distinguish between the underlying elementary mechanisms and their interplay in the collective deformation behavior.

Elementary dislocation / grain-boundary interaction mechanisms include (see Sec. III.2.A):

ï Dislocation / grain-boundary interactions;
ï Operation of dislocation sources in confined geometries;
ï Kinetics and mechanisms of grain-boundary sliding, diffusion and migration;
ï Effects of solute interactions.

Collective  mechanisms include (see Sec. III.2.B):

ï Evolution of lattice curvature near grain boundaries;
ï Collective dislocation behavior;
ï Grain dynamics in the presence of dislocation microstructures;
ï Constitutive models for (poly)crystalline plasticity.

Our CRT effort culminates in:

ï a direct, atomistic and mesoscale simulation study of the crossover in the Hall-Petch effect (see Sec. III.2.C). Comparison of the results of this investigation with the insights gained on the underlying elementary and collective dislocation / grain-boundary processes will enable us to identify which of these are predominantly responsible for the crossover effect.

III.2.  Project Description
A.  Elementary Mechanisms
Elucidation of the crossover between dislocation-mediated and grain-boundary-mediated plastic flow requires a fundamental understanding of the interactions between grain boundaries and dislocations.  Investigation of these interactions involves the four subthrusts listed above.  In the following each of these is described in some detail.

A.1.  Dislocation / Grain-Boundary Interactions
Partners: Bulatov, Chrzan, Cleri, Ghoniem, Henager, Morris, Phillpot, Swaminarayan, Quong, Wolf and Yip
Using a combination of atomistic and mesoscopic techniques, this work will explore the fundamental mechanisms of transmission of lattice dislocations through grain boundaries and their absorption by and repulsion from grain boundaries, including slip transmission through grains.  Atomistic computations will provide input to discrete dislocation simulations of dislocation sources operating in a multi-grain environment to determine the effects of boundary type and grain size on source operation.

At the atomic level we will map out statically the energetics and stress distributions as the two types of defects approach each other. The mechanistic aspects of their interaction will be investigated by direct, dynamical simulation. By coupling these atomic-level insights to appropriate mesoscale model, key elementary mechanisms and processes involved in dislocation / grain-boundary interactions will be elucidated.

One means of checking atomistic calculations against mesoscale models is to calculate the same property within two different frameworks. For example, the behavior of low-angle grain boundaries can be explored entirely within a continuum theory of dislocation dynamics.  The results of such calculations can then be compared with the results of the atomistic simulations as a means of advancing our understanding.

Given the size and number of simulations that need to be performed, it is critical that we optimize our approach. There are two possible routes to increase the number of simulations conducted during the duration of this project: (a) using massively parallel processing architectures to bring the runtimes down and (b) finding new methods using fewer atoms to calculate the properties of interest.  To address point (a), the CRT includes experts in large-scale simulation. Towards (b) we will develop "evolving" boundary conditions for dislocation intersection, dislocation mobility, and dislocation / grain-boundary interaction.

A.2.  Kinetics and mechanisms of grain-boundary sliding, diffusion and migration
Partners: Biner, Bulatov, Cleri, Henager, Morris, Phillpot, Wolf and Yip
Grain boundary sliding, diffusion and migration are high-temperature processes that are known to be intricately coupled to the underlying grain-boundary structure. On the one hand, understanding grain boundary structure is a prerequisite for elucidating the mechanisms and activation energies for these processes; on the other hand, these processes can be used to probe the underlying grain-boundary structure. This structure can be expected to have a significant effect also on how dislocations interact with the grain boundary. Determining high-temperature grain-boundary structure via, and together with, structure-sensitive properties therefore represents an important step towards elucidating the mechanism by which dislocations and grain boundaries interact.

Within this subtask we will perform systematic molecular dynamics simulations of the high-temperature structure, diffusion and sliding behavior of bicrystalline grain boundaries as a function of the type of grain boundary defined by the five macroscopic grain-boundary degrees of freedom. Of particular interest are the underlying atomic-level mechanisms involved in these processes in dislocation (i.e., low-angle or vicinal) boundaries by comparison with high-angle, high-energy grain boundaries. Because of fundamental differences in the underlying grain boundary geometry, a detailed comparison of the behaviors of tilt vs. twist and symmetric vs. asymmetric boundaries will also be performed.

The members of the proposed CRT bring together considerable experience in atomistic modeling of the high-temperature structure and properties of grain boundaries. The older work clearly needs to be updated with more sophisticated potential models, boundary conditions, and methods of inducing grain boundary motion. Besides bicrystals, we will also study nanocrystalline microstructures in which grain-boundary sliding competes with intragranular deformation in responding to an applied load.  One approach to be pursued is the development of a Peierls-Nabarro model of grain-boundary sliding. Obtaining reliable atomic-level "misfit potentials" for grain-boundary sliding is clearly one of the outstanding challenges.

A.3. Effects of solute interactions on dislocation mobility and grain-boundary sliding, diffusion and migration
Partners: Chrzan, Henager, Horstemeyer, Liu, Swaminarayan and Warren
Using a combination of atomistic and mesoscopic techniques, this work will explore effects of solutes and point defects on dislocations and grain boundaries, on the core structures of grain boundaries, the boundary diffusivity and grain-sliding energetics and dynamics.  Dislocation-solute, dislocation-vacancy and solute-vacancy binding energies will be computed and used as input for dislocation mobilities and boundary-migration rates in kinetic Monte-Carlo models.  "Evolving" boundary conditions for dislocation intersection, dislocation mobility and dislocation / grain-boundary interaction studies based on the dislocation treadmill will be developed.  Using this method the effects of solutes and vacancies on these dislocation interactions will be determined. The effects so elucidated will be incorporated into continuum dislocation-dynamics models as a means of exploring their importance to the overall dynamics of the dislocation (e.g., investigations of whether or not a nonlinear velocity law leads to instabilities in the dislocation dynamics).

Ab initio calculations will be required to address accurately arbitrary binary or ternary systems. Approximate interaction potentials may be developed using input from CALPHAD calculations.

A.4.  Operation of dislocation sources in confined geometries
Partners: Chrzan, LeSar, Morris, Phillpot and Swaminarayan
As discussed in detail in Sec. II.1, the formation of dislocation pile-ups has long been considered as a possible explanation of the Hall-Petch effect. However, this explanation involves two key assumptions, namely that (i) the stress to pass a dislocation through a boundary must exceed the stress necessary to move a dislocation in an infinite, homogeneous medium and (ii) slip must be mostly planar, so that once a dislocation is emitted from a source, the dislocation remains on the original slip plane (or at least on a nearby parallel plane). The collaborative effort proposed in this Network offers the unique opportunity to study the operation of dislocation sources in confined media.  We envision two approaches to the problem.

First, we will use large-scale molecular-dynamics methods to simulate the operation of a Frank-Read dislocation source as a function of the size of the grain in which it operates.  We will follow the evolution of the generated dislocations until they equilibrate or are annihilated by other dislocations or at surfaces and grain boundaries.  We will validate the simulations by attempting to reproduce the known relationship of the critical stress for the equilibration of a dislocation loop on the elastic properties of the material, the Burgers vector of the dislocation and on the radius of the dislocation loop.

Concomitant simulations of the operation of sources as viewed from a continuum dislocation-dynamics perspective will also be performed.  A major aspect of the continuum work will be to ascertain the relative importance of cross slip (i.e., non-planar slip) to ultimate yielding behavior of the grain.  By comparing with atomic-scale calculations, we will also understand the accuracy of our formalisms for describing dislocation dynamics.

B.  Collective Behavior
A critical aspect in the competition between dislocation dominated and grain-boundary dominated plasticity is the role of the collective behavior of the dislocation and grain-boundary microstructures.  How complex dislocation structures respond to stress and how these dislocation structures interact with grain boundaries are open questions that are critical to understanding the coupling between grain-boundary and dislocation plasticity.  These are, however, quite complex phenomena and a robust theoretical description will require linking many scales of length and time.  The goal of the Network is to couple activities at various laboratories and universities and thus to bring together the disparate activities needed for this task.
The initial focus of this part of this CRT will be on four subtasks.  Here we go through each of them in turn.  We also list the proposed partners and each of the proposed activities.

B.1.  Evolution of lattice curvature near grain boundaries
Partners:  Bassani, Henager, King, LeSar, Phillpot, Rollett, Stlken, Warren, Wolf
Here we shall explore the relationship between grain-boundary character, applied stress, and dislocation substructure, using a combination of direct atomistic simulations, dislocation dynamics in two and three dimensions, and continuum plasticity.  We will relate predictions from this work to experimental observations, in particular to lattice rotations near boundaries.  The goal of this subtask is to develop a better understanding and predictive ability for the evolution of dislocation substructures.

Atomistic simulations will be used in this subtask in two ways.  First, we will calculate the fundamental interactions important to the development of dislocations substructures near boundaries, namely the short-range dislocation / dislocation and dislocation / grain-boundary interactions.  We will also use atomistic simulations to examine dislocation generation at boundaries and the structures that develop with a small number of dislocations.
Dislocation dynamics in two and three dimensions will be used to examine substructure development near boundaries and the local orientational changes that accompany the dislocation substructures.  Critical issues in these simulations are the local dislocation / boundary interactions (to be obtained from atomistics) and the development of a representation of the long-range elastic interactions between dislocations and grain boundaries of different types.  We will compare the various dislocation simulation methods, including the coupled discrete dislocation / finite-element approach.  The latter method may be the best approach for examining the dislocation / boundary structures for large-angle boundaries, where dislocation models of boundaries do not hold.  Orientational changes can be calculated directly from the displacements fields of the dislocations.  The orientational changes will be compared to experimental results (e.g., Hughes (SNL), Adams (CMU)) and to predictions from continuum theory.

Motivated by the fact that the lattice is distorted inelastically only at the core of a dislocation, continuum theory assumes that the lattice distorts only elastically as a crystal undergoes macroscopic elastic-plastic straining.  If straining is non-uniform, which generally is the case near interfaces, the resulting distortion leads to lattice curvature and, in some instances, to the formation of subgrains.  The extent of curvature depends upon the incompatibility of deformation between grains and, therefore, will be sensitive to how the grain boundary itself deforms in order to accommodate that incompatibility.  This problem will be studied using macroscopic (continuum) simulations and compared to results from mesoscopic (discrete dislocation) simulations.  Along with direct experimental observations, these simulations will (i) identify mechanisms of grain-boundary deformation  and (ii) elucidate the interplay between dislocation dominated and grain-boundary dominated plasticity in the behavior of nanocrystalline materials.  Lattice curvature also gives rise to geometrically-necessary dislocations, and these can lead to patterning of flow as discussed below.

B.2.  Collective dislocation behavior under varying loads
Partners: Baskes, Bassani, Battaile, Biner, Bulatov, Chrzan, Horstemeyer, LeSar and Morris
The goal of this part of the CRT is to bridge the various spatial size scales arising from evolution of dislocation initiation, multiplication, and motion that is reflected in deformation patterning of substructures. We propose to analyze patterning behavior under plastic deformation by employing three length scales of analysis: atomistics, crystal plasticity, and macroscale internal-state-variable theory.

The main goal of the atomistics will be to develop the local rules of dislocation behavior.  Continuum-level definitions of stress and strain tensors as well as kinematic variables that are used to construct crystal plasticity and macroscale formulations have also been developed from atomistic simulations.  These quantities will form the basis of simulations at the meso- and macroscales.

Generally the elastic distortion of the lattice is not compatible with a regular displacement field, and there is evidence that this can strongly influence plastic flow due to the presence of geometrically-necessary dislocations.  Dislocation dynamics in two and three dimensions will be used to examine both the statics and dynamics of this dislocation-microstructure development under varying loading conditions.  We will calculate local stress-strain behavior that arises from these structures and compare those to predictions of continuum theory.

Lattice incompatibility is characterized in terms of the gradient of the elastic distortion field, and this measure can play a natural role in a non-local, gradient-type theory of crystal plasticity.  Recent work at the continuum level has incorporated related ideas of lattice incompatibility in an attempt to model numerous observations of size-dependent plastic phenomena with the general trend that smaller is harder.  This phenomenon is observed particularly when specimens, their geometric features, or their microstructure are roughly below tens of microns (the Hall-Petch effect is a prominent example).  A related size-dependent problem involving localized patterned plastic flow will be studied using non-local continuum simulations.  A simple theory where non-locality only enters the hardening relations leads to multiple-slip hardening equations that have a reaction-diffusion character and, therefore, in principle can lead to patterned flows.  Continuum simulations of patterned flows will be carried out in parallel with discrete dislocation simulations for bicrystals and other multiphase systems.

B.3.  Grain dynamics in the presence of dislocation microstructures
Partners: Battaile, Bulatov, Cleri, LeSar, Phillpot, Stlken, Uras and Wolf
Deformation can dramatically affect the way in which interfaces migrate in a polycrystal.  Pre-existing grain boundaries behave very differently after a material is deformed (e.g., strain-induced boundary migration), and dislocation subcells that form within a grain during or after deformation can grow heterogeneously to dictate the material's crystallographic texture (e.g., recrystallization). In addition, the migration of an interface through a defected material can fundamentally change the material's character (e.g., strain-induced boundary migration). However, little is known about the fundamental processes associated with boundary formation and migration in deformed materials, and thus existing predictive models of microstructural evolution are restricted to ideal strain-free polycrystals. However, most materials processing techniques impart some deformation to the material, and thus the ability to understand and predict microstructural evolution in deformed materials is key to addressing numerous technologically relevant applications.

This subtask is one for which no direct methods exist and we will compare the many approaches represented by the CRT members.  Our first plan for this subtask will be to incorporate realistic interface properties from atomistic (molecular-dynamics and Monte-Carlo) simulations into the microstructural-evolution model to predict development of crystallographic texture and annealed boundary character.  The strategy will be to first validate against prototypical atomistic simulations of grain-boundary evolution. Further tasks will incorporate information from dislocation dynamics and atomistic simulations into microstructural-evolution models to predict effects of strain on microstructural evolution and effects of interface migration on evolution of stored plastic energy.  Finally we will validate against prototypical dislocation dynamics and atomistic simulations of grain-boundary evolution in the presence of dislocations.

As part of this subtask, we will examine alternate approaches to simulating grain microstructures.  One approach will be to define appropriate degrees of freedom for describing grain microstructures.  Based on these, we will develop a mesoscopic approach of "grain dynamics" to simulate the evolution of grain microstructures as a function of temperature and stress. We will also take advantage of the development of a computational formulation based on particle methods, especially, the Reproducing Kernel Particle Method (RKPM), to simulate the mechanics of grains under different loading conditions.  Currently, the finite-element method (FEM) is being applied to mesoscale material simulations.  In contrast to FEM, RKPM does not require an explicit mesh and, consequently, is better suited for cases with large deformations. Effective computational tools, such as RKPM, combined with analytical and experimental techniques will lead to the establishment of a bridge between different material scales by incorporating the findings from mesoscale simulations into macroscale material models.  Specifically, the three-dimensional RKPM code developed for large-deformation continuum problems will be modified to simulate finer-scale phenomena such as grain mechanics at the mesoscale, crystal plasticity, and more realistic macroscale material models.

B.4.  Constitutive models for (poly)crystalline plasticity
Partners: Baskes, Bassani, Battaile, Horstmeyer, Rollett, Stlken and Warren
Recent success in developing new methods for polycrystal plasticity is encouraging.  Comparisons with two-dimensional discrete dislocation simulations are quite good, but there are several central issues requiring input into continuum models from atomistic and discrete dislocation simulations. For example: (i) hardening under multiple slip; (ii) the role of non-glide stresses, particularly in bcc metals and intermetallic compounds; and (iii) effects of lattice curvature and the associated patterning of deformation. There are many open questions, including the role of higher-order stresses and their implication on boundary conditions particularly at interfaces.  Another important issue is how to apportion geometrically-necessary dislocations (i.e., the effects of lattice incompatibility) to individual slip systems.  Of course, a central question is whether the material length scale that must naturally enter non-local constitutive relations is, in fact, a material constant.
The goal is to develop physically-based constitutive models for polycrystal plasticity that realistically capture the effects of internal interfaces on material deformation. We will incorporate (temperature-dependent) dislocation / grain-boundary interaction laws, condensed from dislocation and atomistic simulations, into constitutive polycrystal plasticity equations to capture pile-up effects (e.g., Hall-Petch) during dislocation-dominated plasticity.  We will then incorporate grain-boundary diffusion behavior from atomistic simulations into constitutive equations to address creep during boundary-dominated plasticity.  Incorporating grain-boundary sliding behavior from atomistic simulations into constitutive equations will help us capture superplastic effects during boundary-dominated plasticity.
One part of this subtask will be to attempt to develop a crystal-plasticity formulation that deviates from the classical formulation by including size effects below that of the grains. The kinematics and constitutive assumptions need complete revamping.  As such, we will perform atomistic simulations to determine the development of the kinematic and force variables that are used in crystal-plasticity theory.  Furthermore, we will examine work hardening in terms of the partition of isotropic and kinematic hardening.  This will allow evaluation of the Bauschinger effect, which is very critical in large-strain, non-monotonic loading conditions and in cyclic fatigue.

C.  Direct simulation of crossover from normal to  inverse Hall-Petch behavior
Partners: all
Our CRT effort culminates in a direct, atomistic and mesoscale simulation study of the crossover in the Hall-Petch effect. Comparison of the results of this investigation with the insights gained on the underlying elementary and collective dislocation / grain-boundary processes will enable us to identify which of these are predominantly responsible for the crossover effect. In this effort, large-scale molecular-dynamics and mesoscale dislocation and grain-boundary simulations will be performed to quantitatively investigate the crossover between normal and inverse Hall-Petch behavior that takes place in fcc metals in the 10-25 nm grain-size range (see Fig. 1).  This investigation will not only provide a quantitative test of the Hall-Petch relation in its normal (s~d-1/2) and (postulated) inverse forms (s~d1/2) but also expose the underlying mechanisms activated on either side of the maximum in Fig. 1 and in the grain-size region of maximum strength.

To set up initial polycrystalline microstructures for the molecular-dynamics simulations, a recently developed method will be used that populates with atoms a log-normal grain-size distribution obtained from a simple Monte-Carlo type mesoscale simulation of microstructural evolution.  Initially, an existing parallel molecular-dynamics code will be used to simulate quasi-three-dimensional polycrystalline microstructures containing cylindrical grains (corresponding to a columnar microstructure) with grain sizes up to approximately 40 nm.  A simulation of 20 cylindrical grains with  an average grain size of 40 nm will require approximately 2 million atoms, which is well within the reach of current parallel machines. Subsequently, with the computational resources that will become available with the 3 TOps IBM-SP at NERSC, which will be fully operational in 2000, we will perform fully three-dimensional simulations of polycrystals containing approximately 20 grains of up to about 40 nm grain size and requiring up to about 300 million atoms.

The mesoscale simulations will combine dislocation dynamics with mesoscopic descriptions of the grain microstructures.   The mesoscale code will be verified and validated by comparison with the results of carefully chosen benchmark molecular-dynamics simulations. These simulations will also be compared with polycrystal plasticity (continuum) simulations.  The hope is that, by approaching this phenomena from all the points of view represented in the CRT, we will not only develop a deeper understanding of the coupling between dislocation and grain-boundary plasticity but also gain insight into the inherent limitations and strengths of the various approaches. We will thus make progress on an important scientific problem while simultaneously advancing the simulations methods.

III.3.  Computational Issues
Computational resources now available. All of the members have access to reasonable computational resources at the present time, with a multitude of single-processor and multi-processor computers being available to the individual efforts.  Most simulations will be done using those resources.  However, the ultimate goal is to link scales, which will require some use of larger-scale computational resources than are currently available to most of the members.
Required computational resources.  We will need to use large-scale computation for some of the tasks and subtasks proposed for this CRT.  This will require access to massively-parallel computing, such as the 3 TOps IBM-SP at NERSC, which will be fully operational in 2000.  The needed resources will depend on the method and scale of simulation.  For example, on a full 3 TOps computer, one can simulate about 1 billion atoms for 104 time steps in a day, which corresponds roughly to a cube of copper 0.2 µm a side for 0.1 nanoseconds.  Thus the simulations proposed above to directly simulate the crossover in the Hall-Petch effect (Fig. 1) are well within the capabilities of the NERSC computer.  Given that discrete dislocation simulations require about 100 times as many computations per time step than atomistics, a three-dimensional dislocation simulation could then include ten million segments for 104 time steps in a day.  Assuming a typical segment length of a 0.25 µm and a time step of about 10-10 seconds, this calculation would correspond to a cube (given a dislocation density of 1014/m2) about 30 µm a side for about 1 microsecond.  The level of computing at the new NERSC computer will thus enable simulations at a scale where we can directly compare with continuum solutions.  Similar scalings can be done for the other methods to be used in the Network as well.

IV.   PARTNERING
IV.1.  Members (initial)
This initial list of members includes those who responded to the white paper on the CMSN web page.  Most attended an organizational meeting at Argonne National Laboratory on April 22, 1999.  The final configuration of the Cooperative Research Team will depend on interest, funding and other issues.  Each personís membership in a team is given in the task descriptions above.

Mike Baskes, Los Alamos National Laboratory
John Bassani, University of Pennsylvania
Corbett Battaile, Sandia National Laboratories
S. B. Biner, Ames Laboratory
Vasily Bulatov, Lawrence Livermore National Laboratory
Daryl C. Chrzan, University of California, Berkeley and Lawrence Berkeley National Laboratory
Tomas Diaz de la Rubia, Lawrence Livermore National Laboratory
Nasr M. Ghoniem, University of California, Los Angeles
Chuck Henager, Jr. Pacific Northwest National Laboratory
M. F. Horstemeyer, Sandia National Laboratories
Wayne King, Lawrence Livermore National Laboratory
Richard LeSar, Los Alamos National Laboratory
James R. Morris, Ames Laboratory
Simon Phillpot, Argonne National Laboratory
Andrew Quong, Lawrence Livermore National Laboratory
B. Radhakrishnan, Oak Ridge National Laboratory
Tony Rollett, Carnegie Mellon University
James Stolken, Lawrence Livermore National Laboratory
Sriram Swaminarayan, Los Alamos National Laboratory
R. A. Uras, Argonne National Laboratory
A. F. Voter, Los Alamos National Laboratory
James Warren, National Institute for Standards and Technology
Dieter Wolf, Argonne National Laboratory
Sidney Yip, Massachusetts Institute of Technology

IV.2.  Plan
Organization and leads.  The organization of the CRT will be focused around the tasks and subtasks.  Each member will be associated with one or more subtasks (as listed above) and will have responsibilities primarily to the other members of that subtask.  For ease of organization, each subtask will elect a subtask leader who will also serve as representative on an informal CRT council.  The overall leads for the CRT will be Dieter Wolf (ANL) and Richard LeSar (LANL).  It will be the responsibility of the CRT leads to distribute funds, given the general guidance discussed below. The authority to add new members or remove members who do not contribute and to refine the focus of the various tasks and subtasks will belong to the CRT leads, who will work with the subtask leaders, the membership, and DOE headquarters to keep the work on track and productive.

Meetings.  There will be two meetings of the CRT a year.  An informal meeting at which short presentations of results will be given will be held in conjunction with a regular society meeting (MRS, TMS, etc.).  We will also hold a two-day workshop every year where more substantive discussions of the science can take place and plans will be drawn up for the future.  We plan a kickoff meeting shortly after the CRT is officially started.  At that time, the subtask teams will be formalized and the subtask team leaders selected.  After the initial meeting, the subtask teams will be responsible for creating and maintaining a communication network.  The CRT leads will keep track of progress through the subtask leaders.  The CRT leads will also serve as representatives to other research teams in the Computational Materials Science Network.

Budget.  A budget of $290,000 per year is requested for the Cooperative Research Team.  The initial budget breakdown is as follows:
 General travel costs spread over all institutions            $130,000
 Four postdoctoral fellows (cost-shared)                       $160,000
 TOTAL                                                                         $290,000

To facilitate collaborations between the thirteen institutions involved in the initial stages of the CRT, approximately $130,000 will be set aside to cover travel and other costs.  Some portion of these funds will be set aside to cover the cost of long-term student visits to other institutions to aid in the sharing of ideas and methods across the teams.  The exact split of these funds and the method of distributing them will be decided at the kickoff meeting and will be administered by the CRT council and/or CRT leads.  The plan is to partially support up to four postdoctoral fellows with the CRT funds.  If only three postdoctoral fellows are in part funded, then the remaining funds will be used to partially support students.  The decision on the number of postdoctoral fellows/students to be funded, their location, and their tasks will be made at the kickoff meeting.  Every effort will be made to cost-share the postdoctoral fellows and students so that we can support more task areas.  The postdoctoral fellows and students will be expected to spend time at other institutions within the CRT and will serve as the principle means of furthering the collaborative efforts and of transferring information and methods across the teams.  The council and CRT leads will monitor the allocation of the postdoctoral and student support.

IV.3.  Rationale
Our goal is to keep membership as broad as possible, with the constraints that all work must be relevant and focused on the subtasks and that there is limited funding.  Some of the subtasks listed above may be postponed or dropped as the membership and funding become more set.
Throughout this proposal we have identified areas where the collaborations promoted by the Network will lead to enhancements in our ability to model and simulate the mechanics of materials.  These enhancements will take the form of sharing of methods and codes, cross-comparison of more than one method for the same problem, transfer of information across scales of length and time, and so on.  Perhaps the most important feature of the Network will be the focus of a variety of researchers from a broad background in training and expertise on the same problems.  It is through this multidisciplinary teaming that we expect the most positive benefits, with the expectation that not only will new methods be developed and enhanced, but that new ideas will be created as well.
When possible, we will make links with existing programs and facilities, especially on the experimental side.  We will, for example, work with Carnegie-Mellon University's NSF center on Mesoscale Interface Mapping to examine grain boundary structures and energetics.  We will develop ties to the work by Darcy Hughes at Sandia National Laboratories on misorientation distributions in subgrains..  We will also expand our connections with other, non-CRT, modeling efforts, especially with regard to developing new methods and techniques.  Developing ties with industry will be an important goal.

V.  OUTCOME

V.1.  Implications of Success
The ultimate goal of this team effort is to develop a hierarchically structured, integrated approach towards materials modeling across all the inherent physical length and time scales. Success in the development of this approach would have truly revolutionary impact on many types of predictive materials modeling based on fundamental physical principles.

This Network will lead to research that is "greater than the sum of the parts". The proposed tasks and subtasks are at the forefront of materials modeling and simulation.  Much of the work being done by the members of this Team is pioneering and there is no doubt that bringing the members together on a focused set of problems will greatly enhance the impact of their individual efforts.

V.2.  What Success Might Look Like
Success would be the development of a computational and theoretical hierarchy of models and simulation tools that will span the needed length and time scales to accurately describe the effects of microstructure on the mechanical properties of materials.  Ultimately, the demonstrated ability to incorporate local (interfacial and dislocation) materials properties obtained from atomic-level simulations, for example, into finite-element and phase-field simulations, would represent a major conceptual and practical advance in predictive materials model. This would put within reach the development of core industrial modeling tools incorporating physics-based materials information into macroscopic industrial-design tools, such as the DOE sponsored Casting Process Simulator.
 

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