Session G11: Invited Session: Concurrent Multiple Length-Scale Modelling

Quantum Mechanics Based Multiscale Modeling of Materials

We present two quantum mechanics based multiscale approaches that can simulate extended defects in metals accurately and efficiently. The first approach (QCDFT) can treat multimillion atoms effectively via density functional theory (DFT). The method is an extension of the original quasicontinuum approach with DFT as its sole energetic formulation. The second method (QM/MM) has to do with quantum mechanics/molecular mechanics coupling based on the constrained density functional theory, which provides an exact framework for a self-consistent quantum mechanical embedding. Several important materials problems will be addressed using the multiscale modeling approaches, including hydrogen-assisted cracking in Al, magnetism-controlled dislocation properties in Fe and Si pipe diffusion along Al dislocation core.   

\textit{Ab initio} prediction of environmental embrittlement at a crack tip in aluminum

This talk reports on our \textit{ab initio} predictions of environmental embrittlement in aluminum. We have used an atomistic-continuum multiscale framework to simulate the behavior of a loaded crack tip in the presence of oxygen and hydrogen. The multiscale simulations and subsequent analysis suggest that electronegative surface impurities can inhibit dislocation nucleation from a loaded crack tip, thus raising the likelihood for incremental brittle crack growth to occur during near-threshold fatigue. The metal-impurity bonding characteristics have been analyzed using a Bader charge transfer approximation, and the effect of this bond on the theoretical slip distribution has been investigated using a continuum Peierls model. The Peierls model, which is a function of the position dependent stacking fault energy along the slip plane, was used to estimate the effects of several common environmental impurities.   

Concurrent multiscale modeling of amorphous materials

An approach to multiscale modeling of amorphous materials is presented whereby atomistic scale domains coexist with continuum-like domains. The atomistic domains faithfully predict severe deformation while the continuum domains allow the computation to scale up the size of the model without incurring excessive computational costs associated with fully atomistic models and without the introduction of spurious forces across the boundary of atomistic and continuum-like domains. The material domain is firstly constructed as a tessellation of Amorphous Cells (AC). For regions of small deformation, the number of degrees of freedom is then reduced by computing the displacements of only the vertices of the ACs instead of the atoms within. This is achieved by determining, a priori, the atomistic displacements within such Pseudo Amorphous Cells associated with orthogonal deformation modes of the cell. Simulations of nanoscale polymer tribology using full molecular mechanics computation and our multiscale approach give almost identical prediction of indentation force and the strain contours of the polymer. We further demonstrate the capability of performing adaptive simulations during which domains that were discretized into cells revert to full atomistic domains when their strain attain a predetermined threshold.   

Coarse-graining molecular dynamics models using an extended Galerkin method

I will present a systematic approach to coarse-grain molecular dynamics models for solids. The coarse-grained models are derived by Galerkin projection to a sequence of Krylov subspaces. On the coarsest space, the model corresponds to a finite element discretization of the continuum elasto-dynamics model. On the other hand, the projection to the finest space yields the full molecular dynamics description. The models in between serve as a smooth transition between the two scales. We start with a molecular dynamics (MD) model, $m_i\ddot{\mathbf x}_i= -\frac{\partial V}{\partial \mathbf x_i}$. First, let $Y_0$ be the approximation space for the continuum model. By projecting the MD model onto the subspace, we obtain a coarse-grained model, $ M \ddot{\mathbf q} = F(\mathbf q)$. Using the Cauchy-Born approximation, this model can be shown to coincide with the finite element representation of the continuum elastodynamics model. This model has limited accuracy near lattice defects. One natural idea is to switch to the MD model in regions surround local defect. As a result, one creates an interface between the continuum and atomistic description, where coupling conditions are needed. Direct coupling methods may involve enforcing constraints or mixing the energy or forces. Such an approach may suffer from large phonon reflections at the interface, and introduce large modeling error. In order to seamlessly couple this model to MD, we successively expand the approximation space to the Krylov spaces, $ K_\ell = Y_0 + A Y_0 + \cdots + A^\ell Y_0$. Here $A$ is the force constant matrix, computed from the atomistic model. Due to the translational invariance, only a smaller number of such matrices need to be computed. By projecting the MD model onto this new subspace, we obtain an extended system, $M \ddot{\mathbf q} = F_0(\mathbf q, \xi_1, \cdots, \xi_{\ell}), \ddot{\xi}_1= F_1(\mathbf q, \xi_1, \cdots, \xi_{\ell}), \cdots \cdots, \ddot{\xi}_{\ell}= F_\ell(\mathbf q, \xi_1, \cdots, \xi_{\ell}).$ The additional variables $\xi_j$ represent the coefficients in the extended approximation space. Using this systematic approach, one can build a hierarchy of models with increasing accuracy, each of which is a well-posed model. At the top of the hierarchy is the continuum model, represented on a finite element mesh. Then, on the same mesh, we obtain higher order approximations of MD. In the limit $\ell \to N$, the full MD description is recovered.   

Seamless bridging of quantum-mechanics with mechanics and electronic structure calculations at macroscopic scales

Defects play a crucial role in influencing the macroscopic properties of solids---examples include the role of dislocations in plastic deformation, dopants in semiconductor properties, and domain walls in ferroelectric properties. These defects are present in very small concentrations (few parts per million), yet, produce a significant macroscopic effect on the materials behavior through the long-ranged elastic and electrostatic fields they generate. The strength and nature of these fields, as well as other critical aspects of the defect-core are all determined by the electronic structure of the material at the quantum-mechanical length-scale. Hence, there is a wide range of {\it interacting} length-scales, from {\it electronic structure to continuum}, that need to be resolved to accurately describe defects in materials and their influence on the macroscopic properties of materials. This has remained a significant challenge in multi-scale modeling, and a solution to this problem holds the key for predictive modeling of complex materials systems. In an attempt to address the aforementioned challenge, this talk presents the development of a {\it seamless} multi-scale scheme to perform electronic structure calculations at macroscopic scales. The key ideas involved in its development are (i) a real-space variational formulation of electronic structure theories, (ii) a nested finite-element discretization of the formulation, and (iii) a systematic means of adaptive coarse-graining retaining full resolution where necessary, and coarsening elsewhere with no patches, assumptions or structure. This multi-scale scheme has enabled, for the first time, calculations of the electronic structure of multi-million atom systems using orbital-free density-functional theory, thus, paving the way for an accurate electronic structure study of defects in materials. The accuracy of the method and the physical insights it offers into the behavior of defects in materials is highlighted through studies on vacancies and dislocations. Current efforts towards extending this multi-scale method to Kohn-Sham density functional theory will also be presented.