Concurrent Atomistic-continuum Research Articles

Bridging length and time scales in predictive simulations of thermo-mechanical processes

Abstract This work introduces a theoretical formulation and develops numerical methods for finite element implementation of the formulation so as to extend the Concurrent Atomistic-Continuum (CAC) method for modeling and simulation of finite-temperature materials processes. With significantly reduced degrees of freedom, the CAC simulations are shown to reproduce the results of atomically resolved molecular dynamics simulations for phonon density of states, velocity distributions, equilibrium temperature field of the underlying atomistic model, and also the density, type, and structure of dislocations formed during the kinetic processes of heteroepitaxy. This work also demonstrates the need of a mesoscale tool for simulations of heteroepitaxy, as well as the unique advantage of the CAC method in simulation of the defect formation processes during heteroepitaxy.

Effect of a micro-scale dislocation pileup on the atomic-scale multi-variant phase transformation and twinning

In this paper, we perform concurrent atomistic–continuum (CAC) simulations to assess the contribution of the internal stress induced by the microscale dislocation pileup at an atomically structured interface to the atomic-scale phase transformations (PTs), reverse PTs, and twinning. The main novelty of this work is to unify the atomistic description of the interface and the coarse-grained (CG) description of the lagging dislocations away from the interface within one single framework. Our major findings are: (a) the interface dynamically responds to a pileup by forming steps/ledges, the height of which is proportional to the number of dislocations arriving at the interface; (b) the pileup-induced internal stress concentration profile follows neither the classical Eshelby model nor the super-dislocation model alone, but a combination of them; (c) when the pre-sheared sample is compressed, a direct square-to-hexagonal PT occurs ahead of the pileup tip and eventually grows into a wedge shape. The two variants of the hexagonal phases form a twin with respect to each other; (d) upon a further increase of the loading, part of the newly formed hexagonal phase transforms back to the square phase. The square product phase resulting from this reverse PT forms a twin with respect to the initial square phase. All phase boundaries (PBs) and twin boundaries (TBs) are stationary and correspond to zero thermodynamic Eshelby driving forces; and (e) the microscale dislocation pileup-induced internal shear stress and the structural change at the atomic-scale interface reduces the stress required for initiating a PT by a factor of 5.5, comparing with that in the sample containing no dislocations. This work is the first characterization of the behavior of PTs/twinning resulting from the reaction between a microscale dislocation slip and an atomically structured interface. The gained knowledge will advance our understanding of how the multi-phase material behaves in many complex physical processes, such as the synthesis of multi-phase high-entropy alloys or superhard ceramics under high-pressure torsion, deep mantle earthquakes in geophysics, and so on, which all involve dislocation slip, PTs, twinning, and their interactions across from the atomistic to the microscale and beyond.

Dislocation formation in the heteroepitaxial growth of PbSe/PbTe systems

The paper presents a multiscale study of the kinetic processes of the heteroepitaxial growth of the PbSe/PbTe(111) and PbTe/PbSe(001) systems, using the Concurrent Atomistic-Continuum (CAC) method as the simulation tool. The CAC simulations have reproduced the Stranski–Krastanov growth mode and the layer-by-layer growth mode of the two systems, respectively; the pyramid-shaped island morphology of the PbSe epilayer on PbTe(111), the square-like misfit dislocation networks within the PbTe/PbSe(001) interface, and the critical thickness for the PbTe/PbSe(001) system at which coherent interfaces transit to semi-coherent interfaces with the formation of misfit dislocations, all in good agreement with experimental observations. Four types of misfit dislocations are found to form during the growth of the two PbTe/PbSe heterosystems, and hexagonal-like misfit dislocation networks are observed within the PbSe/PbTe(111) interfaces. The growth processes, including the formation of misfit dislocations, have been visualized. Dislocation half-loops have been observed to nucleate from the epilayer surfaces. These half-loops extend towards the interface by climb or glide motions, interact with other half-loops, and form misfit dislocation networks at the interfaces and threading dislocations extending from interfaces to epilayer surfaces. The dominant types of misfit dislocations in both systems are found to be those with Burgers vectors parallel to the interfaces, whereas the misfit dislocations with Burgers vectors inclined to the interface have a low likelihood of generation and tend to annihilate. The size of the substrate is demonstrated to have a significant effect on the formation, evolution, and distribution of dislocations on the growth of PbSe on PbTe(111).

Effect of a Long-Range Dislocation Pileup on the Atomic-Scale Hydrogen Diffusion near a Grain Boundary in Plastically Deformed bcc Iron

In this paper, we present concurrent atomistic-continuum (CAC) simulations of the hydrogen (H) diffusion along a grain boundary (GB), nearby which a large population of dislocations are piled up, in a plastically deformed bi-crystalline bcc iron sample. With the microscale dislocation slip and the atomic structure evolution at the GB being simultaneously retained, our main findings are: (i) the accumulation of tens of dislocations near the H-charged GB can induce a local internal stress as high as 3 GPa; (ii) the more dislocations piled up at the GB, the slower the H diffusion ahead of the slip–GB intersection; and (iii) H atoms diffuse fast behind the pileup tip, get trapped within the GB, and diffuse slowly ahead of the pileup tip. The CAC simulation-predicted local H diffusivity, Dpileup−tip, and local stresses, σ, are correlated with each other. We then consolidate such correlations into a mechanics model by considering the dislocation pileup as an Eshelby inclusion. These findings will provide researchers with opportunities to: (a) characterize the interplay between plasticity, H diffusion, and crack initiation underlying H-induced cracking (HIC); (b) develop mechanism-based constitutive rules to be used in diffusion–plasticity coupling models for understanding the interplay between mechanical and mass transport in materials at the continuum level; and (c) connect the atomistic deformation physics of polycrystalline materials with their performance in aqueous environments, which is currently difficult to achieve in experiments.

Multiscale computational and experimental analysis of slip-GB reactions: In situ high-resolution electron backscattered diffraction and concurrent atomistic-continuum simulations

In this paper, in situ high-resolution electron backscattered diffraction (EBSD) is combined with concurrent atomistic-continuum (CAC) simulations to study the interactions between dislocation-mediated slip and grain boundaries (GBs) in Ni. It is found that the local stress associated with slip-GB intersections first increases upon the pileup of dislocations, then remains high even after the nucleation of dislocations in the neighboring grain, only relaxing after the nucleated dislocations propagate away from the GB due to more incoming dislocations participating in the pileup. The local stress relaxation is accompanied by an atomic-scale GB structure reconfiguration, which affects not only the subsequent dislocation transmission, but also the configuration of those dislocations away from the GB. These findings demonstrate the importance of incorporating local stress history at higher length scale models, such as crystal plasticity finite element.

Investigating shock wave propagation, evolution, and anisotropy using a moving window concurrent atomistic–continuum framework

Despite their success in microscale modeling of materials, atomistic methods are still limited by short time scales, small domain sizes, and high strain rates. Multiscale formulations can capture the continuum-level response of solids over longer runtimes, but using such schemes to model highly dynamic, nonlinear phenomena is very challenging and an active area of research. In this work, we develop novel techniques within the concurrent atomistic–continuum (CAC) multiscale framework to simulate shock wave propagation through a two-dimensional, single-crystal lattice. The technique is described in detail, and two moving window methods are incorporated to track the shock front through the domain and thus prevent spurious wave reflections at the atomistic–continuum interfaces. We compare our simulation results to analytical models as well as previous atomistic and CAC data and discuss the apparent effects of lattice orientation on the shock response of two materials. We then use the moving window techniques to perform parametric studies which analyze the shock front’s structure. Finally, we compare the efficiency of our model to molecular dynamics simulations. This work showcases the framework’s capability for simulating dynamic shock evolution over long runtimes and opens the door to more complex studies involving shock propagation through composites and alloys.

Transmitting multiple high-frequency phonons across length scales using the concurrent atomistic–continuum method

Coupled atomistic–continuum methods can describe large domains and model dynamic material behavior for a much lower computational cost than traditional atomistic techniques. However, these multiscale frameworks suffer from wave reflections at the atomistic–continuum interfaces due to the numerical discrepancy between the fine-scaled and coarse-scaled models. Such reflections are non-physical and may lead to unfavorable outcomes such as artificial heating in the atomistic region. In this work, we develop a technique to allow the full spectrum of phonons to be incorporated into the coarse-scaled regions of a periodic concurrent atomistic–continuum (CAC) framework. This scheme tracks phonons generated at various time steps and thus allows multiple high-frequency wave packets to travel between the atomistic and continuum regions. Simulations performed with this method demonstrate the ability of the technique to preserve the coherency of waves with a range of wavevectors as they propagate through the domain. This work has applications for systems with defined boundary conditions and may be extended to more complex problems involving waves randomly nucleated from an impact within a multiscale framework.

Resonant interaction between phonons and PbTe/PbSe (001) misfit dislocation networks

This work aims at a quantitative and mechanistic understanding of the dynamic process of the phonon-dislocation interaction in PbTe/PbSe (001) heterostructures using the Concurrent Atomistic-Continuum (CAC) method as the simulation tool. The misfit dislocation network and the atomic-scale dislocation core structure obtained in the simulations are found to agree reasonably well with the experimental observations of the PbTe/PbSe (001) interface. Through visualizing the dynamic interaction between phonons and dislocations, as well as quantifying the dislocation vibration amplitude, the phonon energy transmission, and the thermal resistance of the misfit interfaces, this work has illustrated and quantified two mechanisms for phonon-dislocation interaction: (1) phonon scattering by the strain field of dislocations, and (2) phonon scattering by dislocations that vibrate via the local modes of a dislocation network; the latter, leads to resonant phonon-dislocation interaction, which is manifested as local maxima of out-of-phase vibration of the atoms on the two sides of the slip plane, leading to local minima of the energy transmission in the heterostructure that contains one interface. The local vibrational modes are found to be excited only by shear stress induced by transverse phonons. Among various resonant modes, the one with the lowest frequency has the strongest effect. This work has also demonstrated the collective motion of dislocations under ultrafast phonon pulses. In addition, the dynamic properties of the misfit dislocation network localized within one interface are found to be significantly altered by the presence of misfit dislocations at other interfaces, thus further confirming the cooperative dynamic nature of the motion of dislocations and phonons.

A parallel algorithm for the concurrent atomistic-continuum methodology

In this work we present a parallel algorithm for the Concurrent Atomistic Continuum (CAC) formulation that can be integrated into existing molecular dynamics codes. The CAC methodology is briefly introduced and its parallel implementation in LAMMPS is detailed and then demonstrated through benchmarks that compare CAC simulation results with corresponding all-MD (molecular dynamics) results. The parallel efficiency of the algorithm is demonstrated when simulating systems represented by both atoms and finite elements. The verification benchmarks include dynamic crack propagation and branching in a Si single crystal, wave propagation and scattering in a Si phononic crystal, and phonon transport through the phase interface in a PbTe/PbSe heteroepitaxial system. In each of these benchmarks the CAC algorithm is shown to be in good agreement with MD-only models. This parallel CAC algorithm thus offers one of the first scalable multiscale material simulation methodologies that relies solely on atomic-interaction models.

Coarse-grained atomistic modeling of dislocations and generalized crystal plasticity

Recent developments in generalized continuum modeling methods ranging from coarse-grained atomistics to micromorphic theory offer potential to make more intimate physical contact with dislocation field problems framed at length scales on the order of microns. We explore a range of discrete dynamical and continuum mechanics approaches to crystal plasticity that are relevant to modeling behavior of populations of dislocations. Predictive atomistic and coarse-grained atomistic models are limited in terms of length and time scales that can be accessed; examples of the latter are discussed in terms of interactions of multiple dislocations in heterogeneous systems. Generalized continuum models alleviate restrictions to a significant extent in modeling larger scales of dislocation configurations and reactions, and are useful to consider effects of dislocation configuration on strength at characteristic length scales of sub-micron and above; these models require a combination of bottomup models and top-down experimental information to inform parameters and model form. The concurrent atomistic-continuum (CAC) method is extended to model complex multicomponent alloy systems using an average atom approach. Examples of CAC are presented, along with potential to assist in informing parameters of a recently developed micropolar crystal plasticity model based on a set of sub-micron dislocation field problems. Prospects for further developments are discussed.

An atomistic-to-microscale computational analysis of the dislocation pileup-induced local stresses near an interface in plastically deformed two-phase materials

Taking the two-phase material as a model system, here we perform atomistic-to-microscale computational analysis on how the dislocations pileup is formed at a buried interface through two-dimensional concurrent atomistic-continuum simulations. One novelty here is a simultaneous resolution of the μm-level dislocation slip, the pileup-induced stress complexity, and the atomic-level interface structure evolution all in one single model. Our main findings are: (i) the internal stresses induced by a pileup spans a range up to hundreds of nanometers when tens of dislocations participate the pileup; (ii) the resulting stress concentration decays as a function of the distance, r, away from the pileup tip, but deviates from the Eshelby model-based 1/r0.5, where the interface was assumed to be rigid without allowing any local structure reconstruction; and (iii) the stress intensity factor at a pileup tip is linearly proportional to the dislocation density nearby the interface only when a few dislocations are involved in the pileup, but will suddenly ”upper bend” to a very high level when tens of or more dislocations arrive at the interface. The gained knowledge can be used to understand how the local stresses may dictate the plastic flow-induced phase transformations, twinning, or cracking in heterogeneous materials such as polycrystalline steel, Ti-, Mg-, high entropy alloys, fcc/bcc, fcc/hcp, and bcc/hcp composites, containing a high density of interfaces.

Moving window techniques to model shock wave propagation using the concurrent atomistic–continuum method

Atomistic methods have successfully modeled different aspects of shock wave propagation in materials over the past several decades, but they suffer from limitations which restrict the total runtime and system size. Multiscale methods have been able to increase the length and time scales that can be modeled but employing such schemes to simulate wave propagation and evolution through engineering-scale domains is an active area of research. In this work, we develop two distinct moving window approaches within a Concurrent Atomistic–Continuum (CAC) framework to model shock wave propagation through a one-dimensional monatomic chain. In the first method, the entire CAC system travels with the shock in a conveyor fashion and maintains the shock front in the middle of the overall domain. In the second method, the atomistic region follows the shock by the simultaneous coarsening and refinement of the continuum regions. The CAC and moving window frameworks are verified through dispersion relation studies and phonon wave packet tests. We achieve good agreement between the simulated shock velocities and the values obtained from theory with the conveyor technique, while the coarsen-refine technique allows us to follow the propagating wave front through a large-scale domain. This work showcases the ability of the CAC method to accurately simulate propagating shocks and also demonstrates how a moving window technique can be used in a multiscale framework to study highly nonlinear, transient phenomena.

Multiscale Concurrent Atomistic-Continuum (CAC) modeling of multicomponent alloys

Strengthening in complex multicomponent systems such as solid solution alloys is controlled primarily by the dynamic interactions between dislocation lines and heterogeneously distributed solute species. Modeling of extended defect length scales in such multicomponent systems becomes prohibitively expensive, motivating the development of reduced order approaches. This work explores the application of the Concurrent Atomistic-Continuum (CAC) method to model dislocation mobility in random alloys at extended length scales. By employing recently developed average-atom interatomic potentials, the average “bulk” material response in coarse-grained regions interacts with true random solute species in the atomistic-scale domain. We demonstrate that spurious stresses in domain resolution transition regions are eliminated entirely due to the CAC formulation. Simultaneously, the key details of local stress fluctuation due to randomness in the dislocation core region are captured, and fluctuating stress smoothly decays to the long-range dislocation stress field response. Dislocation mobility calculations, for line lengths over 400 nm, are computed as a function of alloy composition in the model FeNiCr system and compared to full molecular dynamics (MD). The results capture the composition-dependent trends, while reducing degrees of freedom by nearly 40%. This approach can be readily extended to any system described by an EAM potential and facilitates the study of large-scale defect dynamics in complex solute environments to support computational alloy design.

Crystal Plasticity Phase-Field Model with Crack Tip Enhancement Through a Concurrent Atomistic-Continuum Model

This paper develops a method for physics-based augmentation of the Helmholtz free energy density functionals, used in coupled crystal plasticity phase-field finite element (CP-PF FE) models of fracture in crystalline metallic materials. Specifically, the defect and crack surface energy components are enhanced with terms that mechanistically account for the presence of atomic-scale, crack-tip nucleated dislocations. The additional terms in the free energy representation are motivated and calibrated by energy equivalence between a concurrent atomistic–continuum scale model and the CP-PF FE model. The atomistic domain of the concurrent model incorporates a time-accelerated, molecular dynamics (MD) LAMMPS code, while the continuum domain is modeled by a dislocation-density crystal plasticity FE model. The concurrent model transfers and transforms discrete dislocations in the atomistic domain to dislocation densities in the crystal plasticity domain. Dislocation densities are transported in the continuum domain by solving the advection equation using a particle-based reproducing kernel particle method with collocation. A new form of the defect energy density is proposed by considering the effect of crack-tip nucleated dislocations. Parameters in the augmented defect and surface energies are evaluated by comparing with the concurrent model results. A comparison of crack propagation with and without contributions from the nucleated dislocations demonstrates a significant effect of nucleated dislocations on the evolution of the crack.

Lattice dislocation induced misfit dislocation evolution in semi-coherent {111} bimetal interfaces

The study of dislocation plasticity mediated by semi-coherent interfaces can aid in the design of certain heterostructured materials, such as nanolaminates. The evolution of interface misfit patterns under complex stress fields arising from dislocation pileups can influence local dislocation/interface interactions, including effects of multiple incoming dislocations. This work utilizes the Concurrent Atomistic-Continuum modeling framework to probe the evolution of misfit structures at semi-coherent Ni/Cu and Cu/Ag interfaces impinged by dislocation pileups generated via nanoindentation. A continuum microrotation metric is computed at various stages of the indentation process and used to visualize the evolution of the interface misfit dislocation pattern. The stress state from approaching dislocations induces mixed contraction and expansion of misfit dislocation structures at the interface. A lower number of misfit nodes per unit interface area coincides with greater localized deformation with regard to atoms near misfit nodes for Ni/Cu. The decreased misfit node spacing for Cu/Ag alternatively distributes the restructuring associated with plastic deformation over a larger percentage of atoms at the interface. Interface sliding facilitated by misfit dislocation motion is found to facilitate deformation extending into the bulk lattices centered on misfit nodes. The depth of penetration of those fields is found to be greater for Ni/Cu than for Cu/Ag. The study of dislocation plasticity mediated by semi-coherent interfaces can aid in the design of certain heterostructured materials, such as nanolaminates. The evolution of interface misfit patterns under complex stress fields arising from dislocation pileups can influence local dislocation/interface interactions, including effects of multiple incoming dislocations. This work utilizes the Concurrent Atomistic-Continuum modeling framework to probe the evolution of misfit structures at semi-coherent Ni/Cu and Cu/Ag interfaces impinged by dislocation pileups generated via nanoindentation. A continuum microrotation metric is computed at various stages of the indentation process and used to visualize the evolution of the interface misfit dislocation pattern. The stress state from approaching dislocations induces mixed contraction and expansion of misfit dislocation structures at the interface. A lower number of misfit nodes per unit interface area coincides with greater localized deformation with regard to atoms near misfit nodes for Ni/Cu. The decreased misfit node spacing for Cu/Ag alternatively distributes the restructuring associated with plastic deformation over a larger percentage of atoms at the interface. Interface sliding facilitated by misfit dislocation motion is found to facilitate deformation extending into the bulk lattices centered on misfit nodes. The depth of penetration of those fields is found to be greater for Ni/Cu than for Cu/Ag.

A concurrent atomistic-crystal plasticity multiscale model for crack propagation in crystalline metallic materials

This paper develops a coupled concurrent atomistic–continuum multiscale model for analyzing crack propagation and associated mechanisms in crystalline metallic materials. This modeling framework can be used to develop effective continuum-scale constitutive models for crack evolution in deforming domains characterized by crystal plasticity. The atomistic region is modeled by the molecular dynamics (MD) code LAMMPS, while the continuum region is modeled by a dislocation density crystal plasticity FE model. A novel method is developed to transfer discrete dislocations in the atomistic domains to dislocation densities in the continuum domain. Propagation of dislocation densities in the continuum domain is modeled by the advection equation of a conserved quantity using the reproducing kernel particle method (RKPM) in conjunction with the collocation method. Validation studies are conducted by comparing results of the concurrent model with those by MD for nickel single crystal specimen with an embedded crack. An analytical model of the crack tip nucleated dislocation density evolution is developed for inclusion in crystal plasticity models. The effect of applied strain-rate and temperature on the parameters of this model are studied.

Si/Ge (111) Semicoherent Interfaces: Responses to an In‐Plane Shear and Interactions with Lattice Dislocations

Concurrent atomistic–continuum simulations are employed to study Si/Ge (111) semicoherent interfaces in terms of their responses to an in‐plane shear and interactions with lattice dislocations. Three types of Si/Ge interfaces, differing in interfacial structures and energy, are considered. Type I interface coincides with the shuffle‐set slip plane and contains a hexagonal network of edge dislocations. Type II and Type III interfaces both coincide with the glide‐set slip plane, yet they contain, respectively, a triangular and a hexagonal network of Shockley partial dislocations. The simulations show that among the three types of interfaces, 1) Type I interface is the least stable subject to an in‐plane shear and 2) Type III interface impedes the gliding of lattice dislocations the most significantly.

Comparative modeling of the disregistry and Peierls stress for dissociated edge and screw dislocations in Al

Many elementary deformation processes in metals involve the motion of dislocations. The planes of glide and specific processes dislocations prefer depend heavily on their atomic core structures. Atomistic simulations are desirable for dislocation modeling but their application to even sub-micron scale problems is in general computationally costly. Accordingly, continuum-based approaches, such as the phase-field microelasticity, phase-field dislocation dynamics (PFDD), generalized Peierls–Nabarro (GPN) models, and the concurrent atomistic–continuum (CAC) method, have attracted increasing attention in the field of dislocation modeling because they well represent both short-range cores interactions and long-range stress fields of dislocations. To better understand their similarities and differences, it is useful to compare these methods in the context of benchmark simulations and predictions. In this paper, we apply the CAC method and different PFDD variants – one of them is equivalent to a GPN model – to simulate an extended (i.e., dissociated) dislocation in Al with initially pure edge or pure screw character in terms of the disregistry. CAC and discrete forms of PFDD are also employed to calculate the Peierls stress. By conducting comprehensive convergence studies, we quantify the dependence of these measures on time/grid resolution and simulation cell size. Several important but often overlooked differences between PFDD/GPN variants are clarified. Our work sheds light on the advantages and limitations of each method, as well as the path towards enabling them to effectively model complex dislocation processes at larger length scales.

Concurrent atomistic-continuum modeling of crystalline materials

In this work, we present a concurrent atomistic-continuum (CAC) method for modeling and simulation of crystalline materials. The CAC formulation extends the Irving-Kirkwood procedure for deriving transport equations and fluxes for homogenized molecular systems to that for polyatomic crystalline materials by employing a concurrent two-level description of the structure and dynamics of crystals. A multiscale representation of conservation laws is formulated, as a direct consequence of Newton's second law, in terms of instantaneous expressions of unit cell-averaged quantities using the mathematical theory of distributions. Finite element (FE) solutions to the conservation equations, as well as fluxes and temperature in the FE representation, are introduced, followed by numerical examples of the atomic-scale structure of interfaces, dynamics of fracture and dislocations, and phonon thermal transport across grain boundaries. In addition to providing a methodology for concurrent multiscale simulation of transport processes under a single theoretical framework, the CAC formulation can also be used to compute fluxes (stress and heat flux) in atomistic and coarse-grained atomistic simulations.

A comparison of different continuum approaches in modeling mixed-type dislocations in Al

Mixed-type dislocations are prevalent in metals and play an important role in their plastic deformation. Key characteristics of mixed-type dislocations cannot simply be extrapolated from those of dislocations with pure edge or pure screw characters. However, mixed-type dislocations traditionally received disproportionately less attention in the modeling and simulation community. In this work, we explore core structures of mixed-type dislocations in Al using three continuum approaches, namely, the phase-field dislocation dynamics (PFDD) method, the atomistic phase-field microelasticity (APFM) method, and the concurrent atomistic-continuum (CAC) method. Results are benchmarked against molecular statics. We advance the PFDD and APFM methods in several aspects such that they can better describe the dislocation core structure. In particular, in these two approaches, the gradient energy coefficients for mixed-type dislocations are determined based on those for pure-type ones using a trigonometric interpolation scheme, which is shown to provide better prediction than a linear interpolation scheme. The dependence of the in-slip-plane spatial numerical resolution in PFDD and CAC is also quantified.