Finite-Temperature Quasicontinuum: Molecular Dynamics without All the Atoms

Using a combination of statistical mechanics and finite-element interpolation, we develop a coarse-grained (CG) alternative to molecular dynamics (MD) for crystalline solids at constant temperature. The new approach is significantly more efficient than MD and generalizes earlier work on the quasicontinuum method. The method is validated by recovering equilibrium properties of single crystal Ni as a function of temperature. CG dynamical simulations of nanoindentation reveal a strong dependence on temperature of the critical stress to nucleate dislocations under the indenter

Quasicontinuum simulation of fracture at the atomic scale

We study the problem of atomic scale fracture using the recently developed quasicontinuum method in which there is a systematic thinning of the atomic-level degrees of freedom in regions where they are not needed. Fracture is considered in two distinct settings. First, a study is made of cracks in single crystals, and second, we consider a crack advancing towards a grain boundary (GB) in its path. In the investigation of single crystal fracture, we evaluate the competition between simple cleavage and crack-tip dislocation emission. In addition, we examine the ability of analytic models to correctly predict fracture behaviour, and find that the existing analytical treatments are too restrictive in their treatment of nonlinearity near the crack tip. In the study of GB-crack interactions, we have found a number of interesting deformation mechanisms which attend the advance of the crack. These include the migration of the GB, the emission of dislocations from the GB, and deflection of the crack front along the GB itself. In each case, these mechanisms are rationalized on the basis of continuum mechanics arguments

Quasicontinuum Models of Interfacial Structure and Deformation

Microscopic models of the interaction between grain boundaries (GBs) and both dislocations and cracks are of importance in understanding the role of microstructure in altering the mechanical properties of a material. A recently developed mixed atomistic and continuum method is extended to examine the interaction between GBs, dislocations and cracks. These calculations elucidate plausible microscopic mechanisms for these defect interactions and allow for the quantitative evaluation of critical parameters such as the stress to nucleate a dislocation at a step on a GB and the force needed to induce GB migration.Comment: RevTex, 4 pages, 4 figure

Origin of the structural phase transition in Li7La3Zr2O12

Garnet-type Li7La3Zr2O12 (LLZO) is a solid electrolyte material with a low-conductivity tetragonal and a high-conductivity cubic phase. Using density-functional theory and variable cell shape molecular dynamics simulations, we show that the tetragonal phase stability is dependent on a simultaneous ordering of the Li ions on the Li sublattice and a volume-preserving tetragonal distortion that relieves internal structural strain. Supervalent doping introduces vacancies into the Li sublattice, increasing the overall entropy and reducing the free energy gain from ordering, eventually stabilizing the cubic phase. We show that the critical temperature for cubic phase stability is lowered as Li vacancy concentration (dopant level) is raised and that an activated hop of Li ions from one crystallographic site to another always accompanies the transition. By identifying the relevant mechanism and critical concentrations for achieving the high conductivity phase, this work shows how targeted synthesis could be used to improve electrolytic performance

From Electrons to Finite Elements: A Concurrent Multiscale Approach for Metals

We present a multiscale modeling approach that concurrently couples quantum mechanical, classical atomistic and continuum mechanics simulations in a unified fashion for metals. This approach is particular useful for systems where chemical interactions in a small region can affect the macroscopic properties of a material. We discuss how the coupling across different scales can be accomplished efficiently, and we apply the method to multiscale simulations of an edge dislocation in aluminum in the absence and presence of H impurities.Comment: 4 page

Matching Conditions in Atomistic-Continuum Modeling of Materials

A new class of matching condition between the atomistic and continuum regions is presented for the multi-scale modeling of crystals. They ensure the accurate passage of large scale information between the atomistic and continuum regions and at the same time minimize the reflection of phonons at the interface. These matching conditions can be made adaptive if we choose appropriate weight functions. Applications to dislocation dynamics and friction between two-dimensional atomically flat crystal surfaces are described.Comment: 6 pages, 4 figure

Moments of spectral functions: Monte Carlo evaluation and verification

The subject of the present study is the Monte Carlo path-integral evaluation of the moments of spectral functions. Such moments can be computed by formal differentiation of certain estimating functionals that are infinitely-differentiable against time whenever the potential function is arbitrarily smooth. Here, I demonstrate that the numerical differentiation of the estimating functionals can be more successfully implemented by means of pseudospectral methods (e.g., exact differentiation of a Chebyshev polynomial interpolant), which utilize information from the entire interval $(-\beta \hbar / 2, \beta \hbar/2)$. The algorithmic detail that leads to robust numerical approximations is the fact that the path integral action and not the actual estimating functional are interpolated. Although the resulting approximation to the estimating functional is non-linear, the derivatives can be computed from it in a fast and stable way by contour integration in the complex plane, with the help of the Cauchy integral formula (e.g., by Lyness' method). An interesting aspect of the present development is that Hamburger's conditions for a finite sequence of numbers to be a moment sequence provide the necessary and sufficient criteria for the computed data to be compatible with the existence of an inversion algorithm. Finally, the issue of appearance of the sign problem in the computation of moments, albeit in a milder form than for other quantities, is addressed.Comment: 13 pages, 2 figure

The theory and implementation of the quasicontinuum method,

While atomistic simulations have provided great insight into the basic mechanisms of processes like plasticity, diffusion and phase transformations in solids, there is an important limitation to these methods. Specifically, the large number of atoms in any realistic macroscopic structure is typically much too large for direct simulation. Consider that the current benchmark for large-scale fully atomistic simulations is on the order of 10 9 atoms, using massively-paralleled computer facilities with hundreds or thousands of CPUs. This represents 1/10,000 of the number of atoms in a typical grain of aluminum, and 1/1,000,000 of the atoms in a typical micro-electro-mechanical systems (MEMS) device. Further, it is apparent that with such a large number of atoms, substantial regions of a problem of interest are essentially behaving like a continuum. Clearly, while fully atomistic calculations are essential to our understanding of the basic "unit" mechanisms of deformation, they will never replace continuum models altogether. The goal for many researchers, then, has been to develop techniques that retain a largely continuum mechanics framework, but impart on that framework enough atomistic information to be relevant to modeling a problem of interest. In many examples, this means that a certain, relatively small, fraction of a problem require full atomistic detail while the rest can be modeled using the assumptions of continuum mechanics. The quasicontinuum method (QC) has been developed as a framework for such mixed atomistic/continuum modeling. The QC philosophy is to consider the atomistic description as the "exact" model of material behaviour, but at the same time acknowledge that the sheer number of atoms make most problems intractable in a fully atomistic framework. Then, the QC uses continuum assumptions to reduce the degrees of freedom and computational demand without losing atomistic detail in regions where it is required. The purpose of this article is to provide an overview of the theoretical underpinnings of the QC method, and to shed light on practical issues involved in its implementation. The focus of the article will be on the specific implementation of the QC method as put forward in Tadmor et al. (1996a,b); Shenoy et al. (1998b,a). Variations on this implementation, enhancements, and details of specific applications will not be presented. For the interested reader, these additional topics can be found in several QC review article