Coarse-Graining and Forecasting Atomic Material Simulations with Descriptors.

Atomic simulations of materials require significant resources to generate, store, and analyze. Here, descriptor functions are proposed as a general, metric latent space for atomic structures, ideal for use in large-scale simulations. Descriptors can regress a broad range of properties, including character-dependent dislocation densities, stress states, or radial distribution functions. A vector autoregressive model can generate trajectories over yield points, resample from new initial conditions and forecast trajectory futures. A forecast confidence, essential for practical application, is derived by propagating forecasts through the Mahalanobis outlier distance, providing a powerful tool to assess coarse-grained models. Application to nanoparticles and yielding of nanoscale dislocation networks confirms low uncertainty forecasts are accurate and resampling allows for the propagation of smooth property distributions. Yielding is associated with a collapse in the intrinsic dimension of the descriptor manifold, which is discussed in relation to the yield surface.

Analysing ill-conditioned Markov chains.

Discrete state Markov chains in discrete or continuous time are widely used to model phenomena in the social, physical and life sciences. In many cases, the model can feature a large state space, with extreme differences between the fastest and slowest transition timescales. Analysis of such ill-conditioned models is often intractable with finite precision linear algebra techniques. In this contribution, we propose a solution to this problem, namely partial graph transformation, to iteratively eliminate and renormalize states, producing a low-rank Markov chain from an ill-conditioned initial model. We show that the error induced by this procedure can be minimized by retaining both the renormalized nodes that represent metastable superbasins, and those through which reactive pathways concentrate, i.e. the dividing surface in the discrete state space. This procedure typically returns a much lower rank model, where trajectories can be efficiently generated with kinetic path sampling. We apply this approach to an ill-conditioned Markov chain for a model multi-community system, measuring the accuracy by direct comparison with trajectories and transition statistics. This article is part of a discussion meeting issue 'Supercomputing simulations of advanced materials'.

Capabilities and limits of autoencoders for extracting collective variables in atomistic materials science.

Free energy calculations in materials science are routinely hindered by the need to provide reaction coordinates that can meaningfully partition atomic configuration space, a prerequisite for most enhanced sampling approaches. Recent studies on molecular systems have highlighted the possibility of constructing appropriate collective variables directly from atomic motions through deep learning techniques. Here we extend this class of approaches to condensed matter problems, for which we encode the finite temperature collective variable by an iterative procedure starting from 0 K features of the energy landscape i.e. activation events or migration mechanisms given by a minimum - saddle point - minimum sequence. We employ the autoencoder neural networks in order to build a scalar collective variable for use with the adaptive biasing force method. Particular attention is given to design choices required for application to crystalline systems with defects, including the filtering of thermal motions which otherwise dominate the autoencoder input. The machine-learning workflow is tested on body-centered cubic iron and its common defects, such as small vacancy or self-interstitial clusters and screw dislocations. For localized defects, excellent collective variables as well as derivatives, necessary for free energy sampling, are systematically obtained. However, the approach has a limited accuracy when dealing with reaction coordinates that include atomic displacements of a magnitude comparable to thermal motions, e.g. the ones produced by the long-range elastic field of dislocations. We then combine the extraction of collective variables by autoencoders with an adaptive biasing force free energy method based on Bayesian inference. Using a vacancy migration as an example, we demonstrate the performance of coupling these two approaches for simultaneous discovery of reaction coordinates and free energy sampling in systems with localized defects.

Quantum de-trapping and transport of heavy defects in tungsten.

The diffusion of defects in crystalline materials1 controls macroscopic behaviour of a wide range of processes, including alloying, precipitation, phase transformation and creep2. In real materials, intrinsic defects are unavoidably bound to static trapping centres such as impurity atoms, meaning that their diffusion is dominated by de-trapping processes. It is generally believed that de-trapping occurs only by thermal activation. Here, we report the direct observation of the quantum de-trapping of defects below around one-third of the Debye temperature. We successfully monitored the de-trapping and migration of self-interstitial atom clusters, strongly trapped by impurity atoms in tungsten, by triggering de-trapping out of equilibrium at cryogenic temperatures, using high-energy electron irradiation and in situ transmission electron microscopy. The quantum-assisted de-trapping leads to low-temperature diffusion rates orders of magnitude higher than a naive classical estimate suggests. Our analysis shows that this phenomenon is generic to any crystalline material.

Optimal dimensionality reduction of Markov chains using graph transformation.

Markov chains can accurately model the state-to-state dynamics of a wide range of complex systems, but the underlying transition matrix is ill-conditioned when the dynamics feature a separation of timescales. Graph transformation (GT) provides a numerically stable method to compute exact mean first passage times (MFPTs) between states, which are the usual dynamical observables in continuous-time Markov chains (CTMCs). Here, we generalize the GT algorithm to discrete-time Markov chains (DTMCs), which are commonly estimated from simulation data, for example, in the Markov state model approach. We then consider the dimensionality reduction of CTMCs and DTMCs, which aids model interpretation and facilitates more expensive computations, including sampling of pathways. We perform a detailed numerical analysis of existing methods to compute the optimal reduced CTMC, given a partitioning of the network into metastable communities (macrostates) of nodes (microstates). We show that approaches based on linear algebra encounter numerical problems that arise from the requisite metastability. We propose an alternative approach using GT to compute the matrix of intermicrostate MFPTs in the original Markov chain, from which a matrix of weighted intermacrostate MFPTs can be obtained. We also propose an approximation to the weighted-MFPT matrix in the strongly metastable limit. Inversion of the weighted-MFPT matrix, which is better conditioned than the matrices that must be inverted in alternative dimensionality reduction schemes, then yields the optimal reduced Markov chain. The superior numerical stability of the GT approach therefore enables us to realize optimal Markovian coarse-graining of systems with rare event dynamics.

Rare events and first passage time statistics from the energy landscape.

We analyze the probability distribution of rare first passage times corresponding to transitions between product and reactant states in a kinetic transition network. The mean first passage times and the corresponding rate constants are analyzed in detail for two model landscapes and the double funnel landscape corresponding to an atomic cluster. Evaluation schemes based on eigendecomposition and kinetic path sampling, which both allow access to the first passage time distribution, are benchmarked against mean first passage times calculated using graph transformation. Numerical precision issues severely limit the useful temperature range for eigendecomposition, but kinetic path sampling is capable of extending the first passage time analysis to lower temperatures, where the kinetics of interest constitute rare events. We then investigate the influence of free energy based state regrouping schemes for the underlying network. Alternative formulations of the effective transition rates for a given regrouping are compared in detail to determine their numerical stability and capability to reproduce the true kinetics, including recent coarse-graining approaches that preserve occupancy cross correlation functions. We find that appropriate regrouping of states under the simplest local equilibrium approximation can provide reduced transition networks with useful accuracy at somewhat lower temperatures. Finally, a method is provided to systematically interpolate between the local equilibrium approximation and exact intergroup dynamics. Spectral analysis is applied to each grouping of states, employing a moment-based mode selection criterion to produce a reduced state space, which does not require any spectral gap to exist, but reduces to gap-based coarse graining as a special case. Implementations of the developed methods are freely available online.

Unsupervised Calculation of Free Energy Barriers in Large Crystalline Systems.

The calculation of free energy differences for thermally activated mechanisms in the solid state are routinely hindered by the inability to define a set of collective variable functions that accurately describe the mechanism under study. Even when possible, the requirement of descriptors for each mechanism under study prevents implementation of free energy calculations in the growing range of automated material simulation schemes. We provide a solution, deriving a path-based, exact expression for free energy differences in the solid state which does not require a converged reaction pathway, collective variable functions, Gram matrix evaluations, or probability flux-based estimators. The generality and efficiency of our method is demonstrated on a complex transformation of C15 interstitial defects in iron and double kink nucleation on a screw dislocation in tungsten, the latter system consisting of more than 120 000 atoms. Both cases exhibit significant anharmonicity under experimentally relevant temperatures.

Compact A15 Frank-Kasper nano-phases at the origin of dislocation loops in face-centred cubic metals.

It is generally considered that the elementary building blocks of defects in face-centred cubic (fcc) metals, e.g., interstitial dumbbells, coalesce directly into ever larger 2D dislocation loops, implying a continuous coarsening process. Here, we reveal that, prior to the formation of dislocation loops, interstitial atoms in fcc metals cluster into compact 3D inclusions of A15 Frank-Kasper phase. After reaching the critical size, A15 nano-phase inclusions act as a source of prismatic or faulted dislocation loops, dependent on the energy landscape of the host material. Using cutting-edge atomistic simulations we demonstrate this scenario in Al, Cu, and Ni. Our results explain the enigmatic 3D cluster structures observed in experiments combining diffuse X-ray scattering and resistivity recovery. Formation of compact nano-phase inclusions in fcc structure, along with previous observations in bcc structure, suggests that the fundamental mechanisms of interstitial defect formation are more complex than historically assumed and require a general revision. Interstitial-mediated formation of compact 3D precipitates can be a generic phenomenon, which should be further explored in systems with different crystallographic lattices.

Defining, Calculating, and Converging Observables of a Kinetic Transition Network.

Kinetic transition networks (KTNs) of local minima and transition states are able to capture the dynamics of numerous systems in chemistry, biology, and materials science. However, extracting observables is numerically challenging for large networks and generally will be sensitive to additional computational discovery. To have any measure of convergence for observables, these sensitivities must be regularly calculated. We present a matrix formulation of the discrete path sampling framework for KTNs, deriving expressions for branching probabilities, transition rates, and waiting times. Using the concept of the quasi-stationary distribution, a clear hierarchy of expressions for network observables is established, from exact results to steady-state approximations. We use these results in combination with the graph transformation method to derive the sensitivity, with respect to perturbations of the known KTN, giving explicit terms for the pairwise sensitivity and discussing the pathwise sensitivity. These results provide guidelines for converging the network, with respect to additional sampling, focusing on the estimates obtained for the overall rate coefficients between product and reactant states. We demonstrate this procedure for transitions in the double-funnel landscape of the 38-atom Lennard-Jones cluster.

Femtosecond quantification of void evolution during rapid material failure.

Understanding high-velocity impact, and the subsequent high strain rate material deformation and potential catastrophic failure, is of critical importance across a range of scientific and engineering disciplines that include astrophysics, materials science, and aerospace engineering. The deformation and failure mechanisms are not thoroughly understood, given the challenges of experimentally quantifying material evolution at extremely short time scales. Here, copper foils are rapidly strained via picosecond laser ablation and probed in situ with femtosecond x-ray free electron (XFEL) pulses. Small-angle x-ray scattering (SAXS) monitors the void distribution evolution, while wide-angle scattering (WAXS) simultaneously determines the strain evolution. The ability to quantifiably characterize the nanoscale during high strain rate failure with ultrafast SAXS, complementing WAXS, represents a broadening in the range of science that can be performed with XFEL. It is shown that ultimate failure occurs via void nucleation, growth, and coalescence, and the data agree well with molecular dynamics simulations.

Fast, vacancy-free climb of prismatic dislocation loops in bcc metals.

Vacancy-mediated climb models cannot account for the fast, direct coalescence of dislocation loops seen experimentally. An alternative mechanism, self climb, allows prismatic dislocation loops to move away from their glide surface via pipe diffusion around the loop perimeter, independent of any vacancy atmosphere. Despite the known importance of self climb, theoretical models require a typically unknown activation energy, hindering implementation in materials modeling. Here, extensive molecular statics calculations of pipe diffusion processes around irregular prismatic loops are used to map the energy landscape for self climb in iron and tungsten, finding a simple, material independent energy model after normalizing by the vacancy migration barrier. Kinetic Monte Carlo simulations yield a self climb activation energy of 2 (2.5) times the vacancy migration barrier for 1/2〈111〉 (〈100〉) dislocation loops. Dislocation dynamics simulations allowing self climb and glide show quantitative agreement with transmission electron microscopy observations of climbing prismatic loops in iron and tungsten, confirming that this novel form of vacancy-free climb is many orders of magnitude faster than what is predicted by traditional climb models. Self climb significantly influences the coarsening rate of defect networks, with important implications for post-irradiation annealing.