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Twisted 2D materials form complex moiré structures that spontaneously reduce symmetry through picoscale deformation within a mesoscale lattice. We show twisted 2D materials contain a torsional displacement field comprised of three transverse periodic lattice distortions (PLD). The torsional PLD amplitude provides a single order parameter that concisely describes the structural complexity of twisted bilayer moirés. Moreover, the structure and amplitude of a torsional periodic lattice distortion is quantifiable using rudimentary electron diffraction methods sensitive to reciprocal space. In twisted bilayer graphene, the torsional PLD begins to form at angles below 3.89° and the amplitude reaches 8 pm around the magic angle of 1. 1°. At extremely low twist angles (e.g. below 0.25°) the amplitude increases and additional PLD harmonics arise to expand Bernal stacked domains separated by well defined solitonic boundaries. The torsional distortion field in twisted bilayer graphene is analytically described and has an upper bound of 22.6 pm. Similar torsional distortions are observed in twisted WS 2 , CrI 3 , and WSe 2 /MoSe 2 . In twisted 2D materials, spontaneous lattice reconstructions mean that twist angle alone provides an incomplete description. Here, using electron diffraction, the authors show that the displacement field in twisted bilayer graphene can be described as a superposition of three periodic lattice distortion (PLD) waves with wavevectors oriented at 120° from each other, forming a “torsional\" PLD.

Machine learning interatomic potentials (IPs) can provide accuracy close to that of first-principles methods, such as density functional theory (DFT), at a fraction of the computational cost. This greatly extends the scope of accurate molecular simulations, providing opportunities for quantitative design of materials and devices on scales hitherto unreachable by DFT methods. However, machine learning IPs have a basic limitation in that they lack a physical model for the phenomena being predicted and therefore have unknown accuracy when extrapolating outside their training set. In this paper, we propose a class of Dropout Uncertainty Neural Network (DUNN) potentials that provide rigorous uncertainty estimates that can be understood from both Bayesian and frequentist statistics perspectives. As an example, we develop a DUNN potential for carbon and show how it can be used to predict uncertainty for static and dynamical properties, including stress and phonon dispersion in graphene. We demonstrate two approaches to propagate uncertainty in the potential energy and atomic forces to predicted properties. In addition, we show that DUNN uncertainty estimates can be used to detect configurations outside the training set, and in some cases, can serve as a predictor for the accuracy of a calculation.

Nanostructures are technological devices constructed on a nanometer length scale more than a thousand times thinner than a human hair. Due to the unique properties of matter at this scale, such devices offer great potential for creating novel materials and behaviors that can be leveraged to benefit mankind. This paper addresses a particular challenge involved in the design of nanostructures—their stochastic or apparently random response to external loading. This is because fundamentally the function that relates the energy of a nanostructure to the arrangement of its atoms is extremely nonconvex, with each minimum corresponding to a possible equilibrium state that may be visited as the system responds to loading. Traditional atomistic simulation techniques are not capable of systematically addressing this complexity. Instead, we construct an equilibrium map (EM) for the nanostructure, analogous to a phase diagram for bulk materials, which fully characterizes its response. Using the EM, definitive predictions can be made in limiting cases and the spectrum of responses at any desired loading rate can be obtained. The latter is important because standard atomistic methods are fundamentally limited, by computational feasibility, to simulations of loading rates that are many orders of magnitude faster than reality. In contrast, the EM-based approach makes possible the direct simulation of nanostructure experiments. We demonstrate the method’s capabilities and its surprisingly complex results for the case of a nanoslab of nickel under compression.

Control of the interlayer twist angle in two-dimensional van der Waals (vdW) heterostructures enables one to engineer a quasiperiodic moiré superlattice of tunable length scale1–8. In twisted bilayer graphene, the simple moiré superlattice band description suggests that the electronic bandwidth can be tuned to be comparable to the vdW interlayer interaction at a ‘magic angle’9, exhibiting strongly correlated behaviour. However, the vdW interlayer interaction can also cause significant structural reconstruction at the interface by favouring interlayer commensurability, which competes with the intralayer lattice distortion10–16. Here we report atomic-scale reconstruction in twisted bilayer graphene and its effect on the electronic structure. We find a gradual transition from an incommensurate moiré structure to an array of commensurate domains with soliton boundaries as we decrease the twist angle across the characteristic crossover angle, θc ≈ 1°. In the solitonic regime (θ < θc) where the atomic and electronic reconstruction become significant, a simple moiré band description breaks down and the secondary Dirac bands appear. On applying a transverse electric field, we observe electronic transport along the network of one-dimensional topological channels that surround the alternating triangular gapped domains. Atomic and electronic reconstruction at the vdW interface provide a new pathway to engineer the system with continuous tunability.An investigation of the structural and transport properties of bilayer graphene as a function of the twist angle between the layers reveals atomic-scale reconstruction for twist angles smaller than a critical value.

Continuum mechanics and thermodynamics are foundational theories of many fields of science and engineering. This book presents a fresh perspective on these fundamental topics, connecting micro- and nanoscopic theories and emphasizing topics relevant to understanding solid-state thermo-mechanical behavior. Providing clear, in-depth coverage, the book gives a self-contained treatment of topics directly related to nonlinear materials modeling. It starts with vectors and tensors, finite deformation kinematics, the fundamental balance and conservation laws, and classical thermodynamics. It then discusses the principles of constitutive theory and examples of constitutive models, presents a foundational treatment of energy principles and stability theory, and concludes with example closed-form solutions and the essentials of finite elements. Together with its companion book, Modeling Materials, (Cambridge University Press, 2011), this work presents the fundamentals of multiscale materials modeling for graduate students and researchers in physics, materials science, chemistry and engineering.

Molecular dynamics simulations are used to obtain mode I and mode II fracture energies and cohesive laws for bulk epoxy and interfaces formed between epoxy and single-layer graphene (SLG), multilayer graphene (MLG), and multilayer graphene oxide (MLGO). The elastic moduli and ultimate tensile and shear strengths of epoxy–graphene interfaces are calculated from uniaxial tension and simple shear loadings. The results show that Young’s modulus and the ultimate tensile strength increase relative to bulk epoxy, whereas the shear modulus and ultimate shear strength are reduced. Failure of epoxy–graphene interfaces in tension occurs due to the formation of voids in the epoxy. Failure in shear is due to tangential slipping at the interface. Under mixed-mode conditions, the shear modulus and shear strength decrease with increasing tensile load. The critical energy release rate G c for the studied epoxy–SLG/MLG/MLGO systems is obtained using a continuum fracture mechanics approach and is found to be significantly lower than for bulk epoxy. All of the results are combined to define mode I and II cohesive laws for bulk epoxy and epoxy–SLG/MLG/MLGO interfaces that can be used in theoretical models and numerical methods, such as finite elements, that employ cohesive zones.

A simulation can stand its ground against an experiment only if its prediction uncertainty is known. The unknown accuracy of interatomic potentials (IPs) is a major source of prediction uncertainty, severely limiting the use of large-scale classical atomistic simulations in a wide range of scientific and engineering applications. Here we explore covariance between predictions of metal plasticity, from 178 large-scale (~10 8 atoms) molecular dynamics (MD) simulations, and a variety of indicator properties computed at small-scales (≤10 2 atoms). All simulations use the same 178 IPs. In a manner similar to statistical studies in public health, we analyze correlations of strength with indicators, identify the best predictor properties, and build a cross-scale “strength-on-predictors” regression model. This model is then used to estimate regression error over the statistical pool of IPs. Small-scale predictors found to be highly covariant with strength are computed using expensive quantum-accurate calculations and used to predict flow strength, within the statistical error bounds established in our study.

We derive and validate a partition function for low-dimensional systems interacting with a heat bath, addressing the general issue of thermodynamic modeling of nanoscale systems. In contrast to bulk systems in the canonical (NVT) ensemble where the partition function is solely determined by the Hamiltonian of the system and the temperature of the heat bath, our formulation demonstrates that accounting for the interactions with the heat bath is essential for describing the statistical mechanics of low-dimensional materials. To validate our theoretical findings, we develop a molecular dynamics (MD) algorithm for directly modeling the heat bath as a gas reservoir. We first validate our approach using a 1D harmonic oscillator, calculating its length distribution through explicit numerical integration and confirming these results with MD simulations. We then extend our method to investigate the out-of-plane fluctuations of a 2D graphene monolayer immersed in a gas at finite temperature and pressure. Comparisons with conventional NVT ensemble simulations controlled by a thermostat reveal that environmental interactions significantly influence the properties of the 2D material system.

Monolayer and multilayer graphene are promising materials for applications such as electronic devices, sensors, energy generation and storage, and medicine. In order to perform large-scale atomistic simulations of the mechanical and thermal behavior of graphene-based devices, accurate interatomic potentials are required. Here, we present a new interatomic potential for multilayer graphene structures referred to as \"hNN--Gr\\(_x\\).\" This hybrid potential employs a neural network to describe short-range interactions and a theoretically-motivated analytical term to model long-range dispersion. The potential is trained against a large dataset of monolayer graphene, bilayer graphene, and graphite configurations obtained from ab initio total-energy calculations based on density functional theory (DFT). The potential provides accurate energy and forces for both intralayer and interlayer interactions, correctly reproducing DFT results for structural, energetic, and elastic properties such as the equilibrium layer spacing, interlayer binding energy, elastic moduli, and phonon dispersions to which it was not fit. The potential is used to study the effect of vacancies on thermal conductivity in monolayer graphene and interlayer friction in bilayer graphene. The potential is available through the OpenKIM interatomic potential repository at https://openkim.org.