Stephen M. Foiles

The embedded-atom method: a review of theory and applications

Abstract The embedded-atom method is a semi-empirical method for performing calculations of defects in metals. The EAM incorporates a picture of metallic bonding, for which there is some fundamental basis. The limitations of the EAM are fairly well characterized: it works best for purely metallic systems with no directional bonding; it does not treat covalency or significant charge transfer; and it does not handle Fermi-surface effects. The main physical property incorporated in the EAM is the moderation of bond strength by other bonds (coordination-dependent bond strength). Within these constraints, the EAM provides a very useful and robust means of calculating approximate structure and energetics, from which many interesting properties of metals can be obtained. We believe that atomistic calculations will continue to play an important role in the development of materials theory. Where the EAM can be useful, there is a tremendous number of interesting projects that have yet to be carried out. The understanding of mechanical properties on an atomistic level has only just begun. For materials where the EAM is not expected to work well, there are recent developments which may allow calculations similar to those presented here. We have mentioned already the problem of treating directional bonding in semiconductors and elements from the transition series. One approach which promises to be useful for treating directional bonding is reviewed by Carlsson [70]; the interested reader is encouraged to start there.

Atomistic simulations of elastic deformation and dislocation nucleation during nanoindentation

Nanoindentation experiments have shown that microstructural inhomogeneities across the surface of gold thin films lead to position-dependent nanoindentation behavior [Phys. Rev. B (2002), to be submitted]. The rationale for such behavior was based on the availability of dislocation sources at the grain boundary for initiating plasticity. In order to verify or refute this theory, a computational approach has been pursued. Here, a simulation study of the initial stages of indentation using the embedded atom method (EAM) is presented. First, the principles of the EAM are given, and a comparison is made between atomistic simulations and continuum models for elastic deformation. Then, the mechanism of dislocation nucleation in single crystalline gold is analyzed, and the effects of elastic anisotropy are considered. Finally, a systematic study of the indentation response in the proximity of a high angle, high sigma (low symmetry) grain boundary is presented; indentation behavior is simulated for varying indenter positions relative to the boundary. The results indicate that high angle grain boundaries are a ready source of dislocations in indentation-induced deformation.

How Grain Growth Stops: A Mechanism for Grain-Growth Stagnation in Pure Materials

Taking the Rough with the Smooth Even with extensive annealing at high temperatures, most polycrystalline materials will not become a perfect single crystal, which would represent the thermodynamically preferred state. The stability of the polycrystalline state has been attributed to the presence of impurities that accumulate at the grain boundaries, but even very pure materials show grain growth stagnation. Using simulations, Holm and Foiles (p. 1138) show that grain boundaries can be classified as “rough” and “smooth.” Rough boundaries move continuously with well-defined activation energies, while the smooth boundaries have low mobility and move in a jerky, stepwise manner. With heating, a boundary can change from smooth to rough, but the transition temperature can vary by hundreds of degrees from one grain boundary to the next. These smooth, low-mobility boundaries thus pin the polycrystalline structure, even in the absence of impurities. Smooth, low-mobility grain boundaries limit crystallite growth when a pure metal is heated. The thermodynamic equilibrium state of crystalline materials is a single crystal; however, polycrystalline grain growth almost always stops before this state is reached. Although typically attributed to solute drag, grain-growth stagnation occurs, even in high-purity materials. Recent studies indicate that grain boundaries undergo thermal roughening associated with an abrupt mobility change, so that at typical annealing temperatures, polycrystals will contain both smooth (slow) and rough (fast) boundaries. Mesoscale grain-growth models, validated by large-scale polycrystalline molecular dynamics simulations, show that even small fractions of smooth, slow boundaries can stop grain growth. We conclude that grain-boundary roughening provides an alternate stagnation mechanism that applies even to high-purity materials.

Ordered surface phases of Au on Cu

Calculations using the embedded atom method show that Au forms ordered surface layers on the low index faces of Cu which exist in equilibrium with a bulk containing dilute amounts of Au. These surfaces have a mixed Au-Cu surface plane rather than Au adatoms on a Cu surface. The (100) and (110) ordered surfaces contain 1/2 monolayer of Au arranged in c(2×2) structures, though the ordering on the (110) surface is poor in the direction normal to the close packed rows. The (111) surface has 1/3 monolayer of Au in a (√3 ×√3)R30° structure. The surface layers of all three faces are found to be rippled with the Au atoms 0.13 to 0.21 A above the Cu atoms. The results for the (100) surface are in agreement with LEED results for the structure found for Au deposited on Cu.

Kinetic phase field parameters for the Cu–Ni system derived from atomistic computations

In the phase field model of binary solidification the mobility terms which appear in the governing rate equations can be estimated from the liquid diffusion coefficients of the pure elements and the velocity of the solid-liquid interface as a function of undercooling. Molecular dynamics simulations utilizing embedded atom potentials have been employed to compute the liquid diffusivities for pure Cu and Ni in the vicinity of their melting points. In both cases the diffusion coefficient is found to vary linearly with temperature and the results are in good agreement with experimental values which are available for Cu. The simulations were also employed to obtain the boundary velocities in three different low index growth directions. The results for Cu and Ni were found to be very similar, with the slope of the velocity-undercooling curve at small undercoolings varying in the range 45--18 cm/s/K. Anisotropy in the growth behavior was observed with V{sub 100} > V{sub 110} > V{sub 111}. The solid-liquid interface velocities were found to be a factor of 4--5 less than the theoretical upper limit derived previously.

The (1 × 2) missing-row phase of Au(110): energetics determined from an extended embedded atom method

The genesis of order in the missing-row phase of the Au(110) surface and the disordering of that phase have been investigated via examination of the underlying static energetics. We determined the total cohesive energy of various clusters of Au “adatoms” on an Au(110) surface and that of various extended defects on the surface using the embedded atom method with an extension to improve the accuracy of treatment in regions where charge density gradients are large. From the energetics of the clusters we have extracted lattice-gas “adatom-interaction” parameters and studied the resulting disordering phase transition using transfer-matrix scaling. We find Tc = 670 K, in reasonable agreement with experiment. (Experimental Tcs reported range from 650 to 770 K.) From our study of the energy of steps on the (1 × 1) surface and roughening excitations of the reconstructed surface two important results emerge: the energy of steps whose edges are parallel to the rows is very small suggesting the latter play an important role in initiating ordering; certain defects which roughen the (1 × 2) surface are also found to be low in energy, suggesting that the disordering transition includes intrinsic roughening.

Spectral neighbor analysis method for automated generation of quantum-accurate interatomic potentials

We present a new interatomic potential for solids and liquids called Spectral Neighbor Analysis Potential (SNAP). The SNAP potential has a very general form and uses machine-learning techniques to reproduce the energies, forces, and stress tensors of a large set of small configurations of atoms, which are obtained using high-accuracy quantum electronic structure (QM) calculations. The local environment of each atom is characterized by a set of bispectrum components of the local neighbor density projected onto a basis of hyperspherical harmonics in four dimensions. The bispectrum components are the same bond-orientational order parameters employed by the GAP potential 1]. The SNAP potential, unlike GAP, assumes a linear relationship between atom energy and bispectrum components. The linear SNAP coefficients are determined using weighted least-squares linear regression against the full QM training set. This allows the SNAP potential to be fit in a robust, automated manner to large QM data sets using many bispectrum components. The calculation of the bispectrum components and the SNAP potential are implemented in the LAMMPS parallel molecular dynamics code. We demonstrate that a previously unnoticed symmetry property can be exploited to reduce the computational cost of the force calculations by more than one order of magnitude. We present results for a SNAP potential for tantalum, showing that it accurately reproduces a range of commonly calculated properties of both the crystalline solid and the liquid phases. In addition, unlike simpler existing potentials, SNAP correctly predicts the energy barrier for screw dislocation migration in BCC tantalum.

A dislocation-based description of grain boundary dissociation: application to a 90° 〈110〉 tilt boundary in gold

Abstract High resolution transmission electron microscopy (HRTEM) observations and atomistic simulations of {1 1 1}/{121} facets in a gold 90° tilt boundary show the presence of a ∼10 A wide layer with stacking faults distributed one to every three close-packed planes. This interfacial reconstruction, which forms the rhombohedral 9R stacking arrangement, is similar to that found previously for near−Σ=3{112} boundaries in low stacking fault energy metals. We discuss a general approach for partitioning grain boundary orientation into a set of Shockley partial dislocations and then apply this description to the {1 1 1}/{121} interface. The analysis explains both the distribution of faults and the geometry of the local plane bending and shows, further, that the 9R stacking occurs in both the Σ=3{112} and {1 1 1}/{121} interfaces due to the similar ratios of 30° and 90° Shockleys in both cases. Finally, we discuss limitations of this description, in particular concerning a 1/8[ 1 01] relaxation that is predicted by the atomistic simulations.

Multi-scale modeling of polycrystal plasticity: a workshop report

Abstract The workshop on multi-scale modeling of polycrystal plasticity was held on April 9–11, 1997 at the Institute for Mechanics and Materials at the University of California, San Diego in La Jolla, CA. This workshop addressed length-scale issues associated with developing a predictive capability in the modeling of the plastic deformation of polycrystals by the incorporation of more physically based information in the models. The goals of the workshop were to: (1) establish a model system that is well suited to the multi-scale modeling methodology; (2) explore a set of discrete simulation methods at the continuum-scale, meso-scale, micro-scale, and atomic-scale; and (3) identify critical links connecting the length scales which will allow information to be passed among scales and allow the end goal of predictive models at the continuum scale. This paper presents the technical summary of the topics covered by the speakers at the workshop and a discussion of critical issues at each length scale.

Al(f.c.c.):Al3Sc(L12) interphase boundary energy calculations

gave g(100)<g(110)<g(111) with values of 32.5, 51.3 and 66.3 mJ/m 2 , respectively. LTE calculations of the excess grand potential of the (100) interface predicted a nearly temperature independent interfacial energy below 400 K that decreased modestly above 400 K. Monte Carlo (MC) simulations produced a compositional diAuseness of about 4 atomic layers separating the two bulk phases. Because the spatial extent of this region is very similar to the classically determined critical nucleus dimensions extracted from nucleation rate data, it is concluded that critical nuclei of Al3Sc are most likely of nonclassical design at high undercooling. # 1998 Acta Metallurgica Inc.