Jonathan A. Zimmerman

Calculation of stress in atomistic simulation

Atomistic simulation is a useful method for studying material science phenomena. Examination of the state of a simulated material and the determination of its mechanical properties is accomplished by inspecting the stress field within the material. However, stress is inherently a continuum concept and has been proven difficult to define in a physically reasonable manner at the atomic scale. In this paper, an expression for continuum mechanical stress in atomistic systems derived by Hardy is compared with the expression for atomic stress taken from the virial theorem. Hardys stress expression is evaluated at a fixed spatial point and uses a localization function to dictate how nearby atoms contribute to the stress at that point; thereby performing a local spatial averaging. For systems subjected to deformation, finite temperature, or both, the Hardy description of stress as a function of increasing characteristic volume displays a quicker convergence to values expected from continuum theory than volume averages of the local virial stress. Results are presented on extending Hardys spatial averaging technique to include temporal averaging for finite temperature systems. Finally, the behaviour of Hardys expression near a free surface is examined, and is found to be consistent with the mechanical definition for stress.

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.

Generalized stacking fault energies for embedded atom FCC metals

Atomistic calculations for the 112 -generalized stacking fault (GSF) energy curve are performed for various embedded atom models of FCC metals. Models include those by Voter and Chen; Angelo, Moody and Baskes; Oh and Johnson; Mishin and Farkas; and Ercolessi and Adams. The resulting curves show similar characteristics but vary in their agreement with the experimental estimates of the intrinsic stacking fault energy, sf , and with density functional theory (DFT) calculations of the GSF curve. These curves are used to obtain estimates of the unstable stacking fault energy, us , a quantity used in a criterion for dislocation nucleation. Curves for nickel and copper models show the theoretically expected skewed sinusoidal shape; however, several of the aluminium models produce an irregularly shaped GSF curve. Copper and aluminium values for us are underestimates of calculations from DFT, although some of the nickel models produce a value matching one of the available DFT results. Values for sf are either fitted to, or underestimate, the measured results. For use in simulations, the authors recommend using the Voter and Chen potential for copper, and either the Angelo, Moody and Baskes potential or the Voter and Chen potential for nickel. None of the potentials model aluminium well, indicating the need for a more-advanced empirical potential.

Dislocation emission around nanoindentations on a (001) fcc metal surface studied by scanning tunneling microscopy and atomistic simulations.

We present a combined study by scanning tunneling microscopy and atomistic simulations of the emission of dissociated dislocation loops by nanoindentation on a (001) fcc surface. The latter consist of two stacking-fault ribbons bounded by Shockley partials and a stair-rod dislocation. These dissociated loops, which intersect the surface, are shown to originate from loops of interstitial character emitted along the <110> directions and are usually located at hundreds of angstroms away from the indentation point. Simulations reproduce the nucleation and glide of these dislocation loops.

Coupled atomistic-continuum simulations using arbitrary overlapping domains

We present a formulation for coupling atomistic and continuum simulation methods for application to both quasistatic and dynamic analyses. In our formulation, a coarse-scale continuum discretization is assumed to cover all parts of the computational domain with atomistic crystals introduced only in regions of interest. The geometry of the discretization and crystal are allowed to overlap arbitrarily. Our approach uses interpolation and projection operators to link the kinematics of each region, which are then used to formulate a system potential energy from which we derive coupled expressions for the forces acting in each region. A hyperelastic constitutive formulation is used to compute the stress response of the defect-free continuum with constitutive properties derived from the Cauchy-Born rule. A correction to the Cauchy-Born rule is introduced in the overlap region to minimize fictitious boundary effects. Features of our approach will be demonstrated with simulations in one, two and three dimensions.

A material frame approach for evaluating continuum variables in atomistic simulations

We present a material frame formulation analogous to the spatial frame formulation developed by Hardy, whereby expressions for continuum mechanical variables such as stress and heat flux are derived from atomic-scale quantities intrinsic to molecular simulation. This formulation is ideally suited for developing an atomistic-to-continuum correspondence for solid mechanics problems. We derive expressions for the first Piola-Kirchhoff (P-K) stress tensor and the material frame heat flux vector directly from the momentum and energy balances using localization functions in a reference configuration. The resulting P-K stress tensor, unlike the Cauchy expression, has no explicit kinetic contribution. The referential heat flux vector likewise lacks the kinetic contribution appearing in its spatial frame counterpart. Using a proof for a special case and molecular dynamics simulations, we show that our P-K stress expression nonetheless represents a full measure of stress that is consistent with both the system virial and the Cauchy stress expression developed by Hardy. We also present an expanded formulation to define continuum variables from micromorphic continuum theory, which is suitable for the analysis of materials represented by directional bonding at the atomic scale.

Reconsideration of Continuum Thermomechanical Quantities in Atomic Scale Simulations

As motivation builds to consider mechanics of nanometer scale objects, it is increasingly advantageous to implement models with finer resolution than standard continuum approaches. For such exercises to prove fruitful, these models must be able to quantify continuum thermomechanical quantities; furthermore, it may be necessary to do so on a sub-system level in order to assess gradients or distributions in a given property. Herein we review the calculation of stress, heat flux, and temperature in atomic scale numerical simulations such as the molecular dynamics method.

An embedded-atom method interatomic potential for Pd-H alloys

Palladium hydrides have important applications. However, the complex Pd–H alloy system presents a formidable challenge to developing accurate computational models. In particular, the separation of a Pd–H system to dilute (α) and concentrated (β) phases is a central phenomenon, but the capability of interatomic potentials to display this phase miscibility gap has been lacking. We have extended an existing palladium embedded-atom method potential to construct a new Pd–H embedded-atom method potential by normalizing the elemental embedding energy and electron density functions. The developed Pd–H potential reasonably well predicts the lattice constants, cohesive energies, and elastic constants for palladium, hydrogen, and PdHx phases with a variety of compositions. It ensures the correct hydrogen interstitial sites within the hydrides and predicts the phase miscibility gap. Preliminary molecular dynamics simulations using this potential show the correct phase stability, hydrogen diffusion mechanism, and mechanical response of the Pd–H system.

Atomistic models of dislocation formation at crystal surface ledges in Si1-xGex/Si(100) heteroepitaxial thin films

Abstract Mechanisms of defect formation near surface ledges of a diamond cubic crystal subjected to compressive strain parallel to the surface are investigated as precursory processes to dislocation nucleation in Si1-xGex/Si(100) heteroepitaxial thin films under surface diffusion conditions. This study is motivated by our preliminary calculations of dislocation formation at surface ledges in a model crystal characterized by the 6–12 Lennard-Jones interatomic potential, and by our controlled annealing experiments on evolution of a Si1-xGex/Si(100) film from an atomically flat, defect-free, surface morphology to an undulating surface morphology with cusp-like surface features and dislocation formation at the cusp valley. When subjecting such films to high temperature anneals, we observed nucleation and growth of three types of dislocations: the 60° glide dislocations, the 90° Lomer-Cottrell dislocations with stair rod Shockley partials and twinned wedge disclinations with twofold ∑9 coincidence boundaries b...

Shear deformation kinematics of bicrystalline grain boundaries in atomistic simulations

The shear deformation behavior of bicrystalline grain boundaries is analyzed using continuum mechanical metrics extracted from atomistic simulations. Calculating these quantities at this length-scale is premised on determining the atomic deformation gradient tensor using interatomic distances. Employing interatomic distance measurements in this manner permits extension of the deformation gradient formulation to estimate important continuum-scale quantities such as lattice curvature and vorticity. These continuum metrics are calculated from atomic deformation fields produced in 2D and thin 3D equilibrium bicrystalline grain boundary structures under shear at 10 K. Results from these simulations show that interface structure strongly influences the resulting accommodation mechanisms under shear and deformation fields produced in the surrounding lattice. Calculating these continuum quantities at the nanoscale lends insight into localized and collective atomic behavior during shear deformation for various mechanisms, and it is shown that different mechanisms lead to differing behavior. Additionally, the results of these calculations can perhaps serve as an intermediary form to inform continuum models seeking to explore larger-scaled grain boundary deformation behavior in 3D, and to evaluate the veracity of continuum models that overlap the nanoscale.