A new formalism for use in atomistic simulations to calculate the full local elastic-constant tensor in terms of local stresses and strains is presented. Results of simulations on a high-angle (001) twist grain boundary are illustrated, using both a Lennard–Jones potential for Cu and an embedded-atom potential for Au. The two conceptionally rather different potentials show similar anomalies in all elastic constants, confined to within a few lattice planes of the grain boundary, with an especially dramatic reduction in the resistance to shear parallel to the grain-boundary plane. It is found that the primary cause of the anomalies is the atomic disorder near the grain boundary, as evidenced by the slice-by-slice radial distribution functions for the inhomogeneous interface system.

Atomistic simulations show that high-energy grain boundaries in nanocrystalline copper and nanocrystalline silicon are highly disordered. In the case of silicon the structures of the grain boundaries are essentially indistinguishable from that of bulk amorphous silicon. Based on a free-energy argument, we suggest that below a critical grain size nanocrystalline materials should be unstable with respect to the amorphous phase.

The high-temperature equilibrated atomic structures and energies of large-unit-cell grain boundaries (GB’s) in diamond and silicon are determined by means of Monte-Carlo simulations using Tersoff’s potentials for the two materials. Silicon provides a relatively simple basis for understanding GB structural disorder in a purely sp3 bonded material against which the greater bond stiffness in diamond combined with its ability to change hybridization in a defected environment from sp3 to sp2 can be elucidated. We find that due to the purely sp3-type bonding in Si, even in highly disordered, high-energy GB’s at least 80% of the atoms are fourfold coordinated in a rather dense confined amorphous structure. By contrast, in diamond even relatively small bond distortions exact a considerable price in energy that favors a change to sp2-type local bonding; these competing effects translate into considerably more ordered diamond GB’s; however, at the price of as many as 80% of the atoms being only threefold coordinated. Structural disorder in the Si GB’s is therefore partially replaced by coordination disorder in the diamond GB’s. In spite of these large fractions of three-coordinated GB carbon atoms, however, the three-coordinated atoms are rather poorly connected amongst themselves, thus likely preventing any type of graphite-like electrical conduction through the GB’s.

The growth kinetics of melting nucleated at a high-angle twist boundary in silicon are investigated using molecular dynamics. Melting is found to be a two-stage process. In the first stage order is lost within a single plane at the interface and the density of the solid increases to that of the liquid. In the second stage the atomic coordination changes and an isotropic liquid is formed.

We have recently developed a novel molecular-dynamics simulation method to grow polycrystals from a melt containing randomly oriented crystalline seeds. The resulting microstructures contain only randomly oriented (i.e., high-energy) grain boundaries. We find that these grain boundaries, which are highly constrained by their close proximity to grain junctions, are highly disordered in fcc metals and amorphous in silicon. From simulations of infinitely extended high-energy grain boundaries in bicrystals, we find that such highly disordered and amorphous grain boundaries are actually the thermodynamic ground state; by contrast, low-energy grain boundaries are crystalline. High-energy grain boundaries in diamond, however, are structurally ordered at the expense of a significant amount of graphite-like bonding. We show that these complex grain boundary structures have important effects on properties including grain boundary diffusion (fcc metals and silicon), grain boundary diffusion creep (silicon) and grain boundary electrical activity and strength (diamond). The implications for engineering materials with prescribed properties are discussed.

Lattice statics and lattice dyanamics computer simulation methods are applied to investigate the elastic properties of grain boundaries in fcc metals. The elastic constants in the GB reigon are found to differ significantly from their bulk ideal-crystal values. Due to the broken symmetry in the GB region a pronounced anisotropy of the elastic properties is observed. Whereas Young’s moduli are found to increase in the GB region, the shear moduli and Poisson’s ratios generally decrease. The magnitude of these effects is shown to depend sensitively on the GB structure.

Atomistic computer simulations are used to systematically investigate the role of interfacial disorder on the elastic behavior of composition-modulated superlattices of fcc metals, represented by simple Lennard–Jones potentials. The structures, energies, and average elastic properties of four types of superlattices with various degrees of interfacial disorder are computed as a function of the modulation wavelength along [001]. The four superlattice types studied include perfectly coherent, incoherent, and two types derived from these by introducing relative twists about [001] between alternating layers. A 20% lattice-parameter mismatch between the two modulating materials is assumed. Results are compared with our earlier work on unsupported thin films, grain-boundary superlattices, and incoherent superlattices with a 10% lattice-parameter mismatch. The degree of structural disorder at the interfaces is found to correlate well with the magnitude of the elastic anomalies, which cannot be accounted for by anisotropic lattice-parameter changes alone. The grain-boundary superlattices studied earlier are found to provide a good model limit for the elastic behavior of interfacially disordered dissimilar-material superlattices.

The zero-temperature energies and equilibrium volume expansions of point-defect free asymmetrical grain boundaries (GBs) involving the four densest planes in fcc bicrystals (with some higher-index plane on the other side of the interface) have been determined using an embedded-atom-method potential fitted to Au. It is found that the two asymmetrical tilt GBs at the endpoints of the related GB energy vs. twist misorientation curves give rise to pronounced energy cusps. As for symmetrical GBs, a practically linear relationship between the GB energy and equilibrium volume expansion is observed. The volume expansion and the destruction of the perfect-crystal stacking at the GB are shown to cause a pronounced local decrease in the resistance towards shear parallel to the GB plane.

It is suggested that the “supermodulus effect” observed for composition-modulated strained-layer superlattices may arise from the presence of the structurally disordered solid interfaces and not necessarily from electronic-structure effects. The latter are excluded by investigating the elastic properties of a so-called grain-boundary superlattice in which chemically identical materials are joined to form a three-dimensional superlattice. Both an embedded-atom-method and Lennard-Jones potential are employed in our zero-temperature atomistic calculations of the elastic constants and moduli of such a superlattice. They yield qualitatively similar results which, for large modulation wavelengths, can be represented by a mean-field model in which the interfacial regions are characterized by a set of effective elastic constants which are different from those of the bulk regions. The appearance of a maximum in the biaxial modulus and a minimum in the shear modulus is shown to arise from the interaction between interfaces. It is also shown that such extreme anomalies appear only in the moduli but not in the elastic constants of the grain-boundary superlattice.

Traditionally, due to the impurities inevitably present in real materials that limit the grain-boundary (GB) mobility, the processes of grain growth and GB diffusion are thought to involve similar activation barriers. However, recent molecular dynamics (MD) simulations of GB migration in bicrystals have suggested that, in pure materials, GB diffusion and GB migration involve distinct mechanisms and, hence, different activation barriers. Here we report MD simulations of grain growth in an impurity-free model nanocrystalline palladium microstructure containing only high-energy GBs. By contrast with the bicrystal simulations, we observe virtually identical activation energies for grain growth and GB diffusion. We discuss several mechanisms that might be responsible for this difference between the bicrystal and polycrystal results, including accommodation of the curvature-driven GB migration by the elimination of GB area and GB excess free volume and a possible finite mobility of the triple junctions. The qualitative agreement of our observations with the general experimental findings is remarkable given the absence of any impurities in our model system. This suggests that a fundamental, intrinsic process may be responsible for the similar activation energies for GB migration and GB diffusion in polycrystals. In our model system this process seems to involve the accommodation of the curvature-driven GB migration by GB diffusion to eliminate the GB area and related GB free volume at the triple junctions.

The zero-temperature energies and equilibrium volume expansions of point-defect-free grain boundaries (GBs) on the three densest planes of cubic-diamond silicon have been determined using the Stillinger-Weber potential. It is found that the energy of GBs on the second-densest plane are two to three times higher than the energy of those on the first-densest plane, and GBs on the third-densest planes have even higher energies. By examining the radial and angular distribution functions of a GB on each plane, it is shown that the energy of a GB is determined by the disorder in both bond lengths and bond angles.

We review the results of recent molecular-dynamics simulations of the structure and deformation behavior of nanocrystalline materials, i.e., polycrystalline materials with a grain size of typically less than about 100 nm. These simulations have now become large enough and sophisticated enough that they are beginning to cover the entire range of grain sizes over which the experimentally suggested transition from a dislocation-based deformation mechanism to one involving GB processes takes place. Their atomic-level resolution provides novel insights into the intricate interplay between the dislocation and GB processes responsible for this crossover. These simulations also reveal how and why this crossover in the dominant mechanism leads to a transition in the mechanical behavior. However, in spite of these early successes, these simulations are inherently limited to rather idealized model microstructures and extremely high deformation rates. We therefore address the critical question as to the degree to which they begin to capture the experimentally observed, albeit controversial, deformation behavior of real nanocrystalline materials. (Supplementary material to this article, in the form of color graphs of some of the figures and several deformation-simulation movies, can be viewed at http://phillpot.mse.ufl.edu/review.html.)

Mesoscale simulations of grain-boundary (GB) diffusion creep in which GB migration-induced static grain growth is suppressed were carried out based on the variational principle of dissipated power. Assuming that the boundaries exhibit no sliding resistance in response to shear stress, the variation of the normal-stress distribution and the diffusive fluxes along the grain boundaries during Coble creep were analysed. The effects of microstructural inhomogeneity, including topological and physical inhomogeneity (i.e. distributions in the grain sizes and GB diffusivities) were investigated. We find that because of the lack of GB migration as an accommodation process to relax the stress concentrations in the microstructure, a topologically inhomogeneous microstructure becomes unphysical at high strains (of typically between 50-100%). Consistent with earlier simulations by Pan and Cocks (1993 Comput. Mater Sci. 1 95), we find that even in the absence of static grain growth an inhomogeneous microstructure exhibits dynamic grain growth induced by grain-switching induced grain-disappearance events. Our simulations also reveal that in a topologically inhomogeneous microstructure, the diffusive fluxes along any given GB can be in the same direction at both delimiting triple points; i.e. qualitatively different from a homogeneous (i.e. regular hexagonal) microstructure in which, according to Spingarn and Nix (1978 Acta Metall. 26 1389), these fluxes oppose each other.

Molecular-dynamics simulations have been used to synthesize and characterize a fully dense, three-dimensional, relaxed nanocrystalline material with an average grain size of 4.3 nm by crystallization from the melt. In spite of some striking similarities, the atomic structures of the highly constrained grain boundaries in this material are considerably less ordered and their energies are less anisotropic than those well-known from bicrystal studies. Although this GB phase is neither bulk-glass like nor can it be described in terms of structural models derived from bicrystal studies, the concept of a cement-like phase connecting the nanometer-sized grains, reminiscent of Rosenhains`s amorphous-cement model, describes the observations well.

Classical shell-model potentials for describing the complex ferroelectric behaviour of barium titanate and strontium titanate are developed and used to simulate BaxSr1−x TiO3 solid solutions. The temperature versus composition phase diagram is very well described and the local behaviour of the structure and polarization is analysed. It is shown that the ferroelectric properties of the solid solution can be understood in terms of the effects of average density and the local chemical environment. The experimentally observed static dielectric peak broadening around Tc at low x is reproduced in the simulation and seems to be related to the average volume rather than to the local chemical environment.

Molecular-dynamics simulations are used to elucidate the effects of grain growth on grain-boundary diffusion creep and grain-boundary sliding during high-temperature deformation of a nanocrystalline Pd model microstructure. The initial microstructure consists of a 25-grain polycrystal with an average grain size of about 15 nm and a columnar grain shape. Prior to the onset of significant grain growth, the deformation proceeds via the mechanism of Coble creep accompanied by grain-boundary sliding. While grain growth is generally known to decrease the creep rate due to the increase of the average grain size, the results obtained in this study reveal an enhanced creep rate at the onset of the grain growth, when rapid grain-boundary migration occurs. The enhanced creep rate is shown to arise from topological changes during the initial growth phases, which enhance both the stress-induced grain-boundary diffusive fluxes and grain-boundary sliding. Dislocations generated as a result of grain-rotation-induced grain coalescence and grain-boundary decomposition in the vicinity of certain triple junctions also contribute to the deformation.

A deterministic approach for meso-scale modeling of grain growth in stressed polycrystalline materials based on the principle of virtual power is presented. The variational equation is formulated based on the power balance of the system associated with grain boundary surface tension and curvature, rate of strain energy stored in each grain, strain energy density jump across the grain boundaries, and external work rate. The numerical solution of stressed grain growth variational equation requires discretization of grain interiors and grain boundaries. This cannot be effectively modeled by Lagrangian, Eulerian, or Arbitrary Lagrangian Eulerian finite element method if grain boundary migration (moving interfaces) and topological changes of grain boundary geometry are considered. This paper presents a double-grid method to resolve the above mentioned difficulty. In this approach, the material grid points carry material kinematic variables, whereas the grain boundary grid points carry grain boundary kinematic variables. The material domain is discretized by a moving least squares reproducing kernel approximation with strain discontinuity enrichment across the grain boundaries. The grain boundaries, on the other hand, are discretized by the standard finite elements. An interface enrichment function to accurately capture strain jump conditions across the grain boundaries is introduced. A reproducing kernel approximation that includes the periodicity of the unit cell in the construction of reproducing kernel shape function for material velocity is also presented. This proposed double-grid method allows modeling of arbitrary evolution of grain boundaries without remeshing.

Molecular-dynamics simulations have recently been used to elucidate the transition with decreasing grain size from a dislocation-based to a grain-boundary-based deformation mechanism in nanocrystalline f.c.c. metals. This transition in the deformation mechanism results in a maximum yield strength at a grain size (the 'strongest size') that depends strongly on the stacking-fault energy, the elastic properties of the metal, and the magnitude of the applied stress. Here, by exploring the role of the stacking-fault energy in this crossover, we elucidate how the size of the extended dislocations nucleated from the grain boundaries affects the mechanical behaviour. Building on the fundamental physics of deformation as exposed by these simulations, we propose a two-dimensional stress-grain size deformation-mechanism map for the mechanical behaviour of nanocrystalline f.c.c. metals at low temperature. The map captures this transition in both the deformation mechanism and the related mechanical behaviour with decreasing grain size, as well as its dependence on the stacking-fault energy, the elastic properties of the material, and the applied stress level.