Phase Field Dislocation Dynamics Research Articles

Phase field dislocation dynamics modeling of shearing modes in Ni2(Cr,Mo,W)-containing HAYNES® 244® Superalloy

Density Functional Theory (DFT) calculations can determine planar defect energies and slip pathways that are present in ordered intermetallic systems used to strengthen Ni-based superalloys, but analytical models used to evaluate the strengthening effects of these phases often involve significant simplifying assumptions. Instead, Phase Field Dislocation Dynamics (PFDD) is a useful modeling tool that incorporates dislocation interactions with precipitates and slip pathways informed by DFT to determine how precipitate shearing might occur under applied stresses with improved accuracy over previous models. In this work, we apply PFDD to study precipitate shearing in HAYNES® 244®, a high strength Ni-based superalloy strengthened through a novel Ni2(Cr, Mo, W) phase that has a low symmetry Body Centered Orthorhombic (BCO) crystal structure which complicates analysis of slip pathways. Through our modeling, we show the formation of and evolution of extended dislocations in the matrix and in the precipitates, the interaction of dislocations with the precipitates, and the formation of planar faults in the precipitate. A key aspect of incorporating the DFT determined slip pathway is the influence of the unstable fault energy and the asymmetry of the energy pathway on the strengthening aspect of the precipitate. The resulting critical strengths are compared to analytical models. The size, orientation, particle distance, and calculated slip pathway for the different variants in this system are all shown to have an important effect on the critical stress to shear these precipitates.

Role of diffusing interstitials on dislocation glide in refractory body centered cubic metals

In this work, we employ a phase field dislocation dynamics technique to simulate dislocation motion in body centered cubic refractory metals with diffusing interstitials. Two distinct systems are treated, Nb with O interstitials and W with H interstitials, to consider both relatively small and large atomic size interstitials. Simulations without and with driving stress are designed to investigate the role of interstitial type and mobility on the glide of edge- and screw-character dislocations. The simulations reveal the various short- and long-range dislocation-interstitial interactions that can take place and their dependency on interstitial type, site occupation, stress state, and mobility of the interstitials relative to dislocations. We show that while interstitial O increases the breakaway stress for both screw and edge dislocations in Nb, interstitial H in low H concentrations makes screw dislocations easier and the edge dislocations harder to move. The simulations find that screw dislocation glide is enhanced by the presence of interstitials in both systems. Edge dislocation glide is enhanced in W–H and inhibited in Nb–O.

A combined kinetic Monte Carlo and phase field approach to model thermally activated dislocation motion

In this work, a kinetic Monte Carlo (KMC) scheme is integrated into a phase field dislocation dynamics (PFDD) model to account for thermal activation. While the implementation is general and could be applied to many different thermally activated processes, we apply the model to simulate a dislocation transmitting through an interface. Two cases are analyzed: a dislocation transmitting through a strong interface, and a dislocation transmitting through a more realistic grain boundary in tungsten. Results and trends for both cases are discussed in detail, and ultimately show that the combined KMC-PFDD algorithm is effective in capturing thermally activated dislocation processes.

Phase-field modeling of dislocation–interstitial interactions

The mechanical behavior of body-centered cubic (BCC) materials can be dramatically affected by the presence of interstitial solute atoms. We present a new phase-field dislocation dynamics formulation to include the diffusion of interstitials. Short-range interactions are accounted for by a concentration-dependent lattice energy, and long-range interactions are accounted for by modifications to the elastic energy. The interstitial diffusion law introduces gradients that require methods for minimizing Gibbs oscillations, which is done via a modified Green’s function. The formulation is general to any solute-solvent system and is applied here to Nb-O as a model system, whose interstitial parameters are obtained from ab initio calculations. The effect of O on the core structures of Nb edge and screw dislocations is calculated. The diffusion of O to form interstitial atmospheres around dislocation cores is simulated, as well as the critical stresses required for dislocations to break away or cross slip from these atmospheres. Future applications of the method to simulate complex interstitial embrittlement mechanisms are discussed.

Interaction of extended dislocations with nanovoid clusters

Voids of nanoscale dimensions in irradiated metals can act as obstacles to dislocation motion and cause strengthening. In this work, nanovoid strengthening and the influences of void size, void spacing and material properties, such as stacking fault energies, on dislocation bypass mechanisms are investigated using Phase Field Dislocation Dynamics, a three-dimensional mesoscale model that predicts the minimum energy pathway taken by discrete dislocations. A broad range of face centered cubic metals (copper, nickel, silver, rhodium, and platinum) and nanovoid sizes and spacings are treated, altogether spanning void size–to–dislocation stacking fault width ratios from less than unity to ten. Material γ-surfaces, calculated from ab initio methods, are input directly into the formulation. The analysis reveals that the critical bypass stress scales linearly with the linear void fraction, effective isotropic shear modulus, and ratio of the intrinsic to unstable stacking fault energies. With only a few exceptions, the critical stress is controlled by the stress required for the leading partial to impinge the voids (to move within range of the attractive image stress field of the void). When the void diameter is nearly an order of magnitude greater than the stacking fault width, the mechanism determining critical strength shifts to the stress for the dislocation to breakaway after partially cutting the void. This situation corresponds to that treated by line tension models and is realized here for Pt, with a sub-nanometer stacking fault width.

Multi-scale investigation of short-range order and dislocation glide in MoNbTi and TaNbTi multi-principal element alloys

Refractory multi-principal element alloys (RMPEAs) are promising materials for high-temperature structural applications. Here, we investigate the role of short-range ordering (SRO) on dislocation glide in the MoNbTi and TaNbTi RMPEAs using a multi-scale modeling approach. Monte carlo/molecular dynamics simulations with a moment tensor potential show that MoNbTi exhibits a much greater degree of SRO than TaNbTi and the local composition has a direct effect on the unstable stacking fault energies (USFEs). From mesoscale phase-field dislocation dynamics simulations, we find that increasing SRO leads to higher mean USFEs and stress required for dislocation glide. The gliding dislocations experience significant hardening due to pinning and depinning caused by random compositional fluctuations, with higher SRO decreasing the degree of USFE dispersion and hence, amount of hardening. Finally, we show how the morphology of an expanding dislocation loop is affected by the applied stress.

Temperature dependent phase field dislocation dynamics model

In this work, we present a three-dimensional phase field-based formulation for simulating the temperature-dependent motion of discrete dislocations in crystals. For demonstration, this thermal phase field dislocation dynamics method is applied to study pyramidal-type dislocations in three exemplar hexagonal close packed materials, Mg, Ti, and Zr. Pyramidal-type dislocations are well known for temperature sensitivity and influence on the yield strength of these metals. Calculations include the predictions of activation stress, dislocation velocity, and dislocation nucleation over temperature ranges from 0 K to up to half the melting temperatures of these metals. We show that pyramidal slip is glissile, but the activation stress and stacking fault widths are asymmetric with respect to forward/backward glide. The stress to initiate glide reduces logarithmically with homologous temperature. The role of temperature in Frank-Read source operation is shown to decrease the size and time of first loop formation. The analysis identifies a master inverse relationship for the dislocation loop nucleation rate with homologous temperature followed by all three metals.

Dislocation dynamics in heterogeneous nanostructured materials

Crystalline materials can be strengthened by introducing dissimilar phases that impede dislocation glide. At the same time, the changes in microstructure and chemistry usually make the materials less ductile. One way to circumvent the strength–ductility dilemma is to take advantage of heterogeneous nanophases which simultaneously serve as dislocation barriers and sources. Owing to their superior mechanical properties, heterogeneous nanostructured materials (HNMs) have attracted a lot of attention worldwide. Nevertheless, it has been difficult to characterize dislocation dynamics in HNMs using classical continuum models, mainly due to the challenges in describing the elastic and plastic heterogeneity among the phases. In this work, we advance a phase-field dislocation dynamics (PFDD) model to treat multi-phase materials, consisting of phases differing in composition, structural order, and size in the same system. We then apply the advanced PFDD model to exploring two important but divergent materials design problems in HNMs: dislocation/obstacle interactions and dislocation/interface interactions. Results show that the interactions between a dislocation and distribution of obstacles varying in structure and composition cannot be understood by simply interpolating from their individual interactions with a dislocation. It is also found that materials containing interfaces with nanoscale thicknesses and compositional gradients have a much higher dislocation bypass stress than those with sharp interfaces, providing an explanation for the simultaneous high strength and toughness of thick interface-containing nanolaminates as observed in recent experiments.

Random Generation of Lattice Structures with Short-Range Order

Thermodynamically driven short-range order (SRO) is known to exist in multi-component alloys and can influence their mechanical properties. Atomistic and mesoscale simulations are frequently used to study the effects of SRO, but introducing SRO into the lattice usually requires costly Monte Carlo simulations. Here, we develop a novel method and open-source code to create lattices with specified SRO parameters that can be used as inputs to simulations to study the interaction of SRO with dislocations or other deformation mechanisms. The method, order through informed swapping (OTIS), uses a statistical smart swapping procedure to create lattices with known SRO parameters from an initially random lattice. We illustrate the use of OTIS on ternary and quinary equimolar alloys to create both BCC and FCC lattices with hundreds of thousands of atoms. The computation time and size scaling are discussed. As an example application of the method, the generated SRO lattices are incorporated into a phase-field dislocation dynamics to study dislocation loop growth.

Phase field modeling of dislocations and obstacles in InSb

We present a phase-field dislocation dynamics (PFDD) model informed by first-principle calculations to elucidate the competitive dislocation nucleation and propagation between the glide and shuffle sets in InSb diamond cubic crystal. The calculations are directly informed with generalized stacking fault energy curves on the (111) slip plane for both the “glide set,” with the smaller interplanar spacing, and the “shuffle set,” with the larger interplanar spacing. The formulation also includes elastic anisotropy and the gradient term associated with the dislocation core. The PFDD calculations show that under no stress the equilibrium structure of screw glide set dislocations dissociates into Shockley partials, while those of the shuffle set dislocations do not dissociate, remaining compact. The calculated dislocation core widths of these InSb dislocations agree well with the measured values for other semiconductor materials, such as Si and GaN. We find that a shuffle set dislocation emits from a dislocation source at an applied stress about three times smaller than that needed to emit leading and trailing partials successively on the glide set plane. Once the partial dislocations in the glide set are emitted, they propagate faster than the shuffle set perfect dislocation at the same stress level.

Thick interface size effect on dislocation transmission in nanolaminates

Recent experimental studies have reported that thick interfaces in nanolaminates can lead to greater strengths than conventionally sharp interfaces without sacrificing deformability. Using a multi-phase phase-field dislocation dynamics model, dislocation transmission across a compositionally graded, nanoscale thick interface is investigated. Thicker interfaces over a finite range are found to lead to greater resistance to transmission. The limit interface thickness at which the peak resistance is reached, and the strengthening capacity of the interface are greater when the dislocation is dissociated, as in a face-centered cubic lattice, than when it is compact, as in a body-centered cubic lattice. The composition transitions within the interface are treated with multiple sublayers, and it is found that the interface transmission barrier is as strong as its most resistance composition.

Recent progress in the phase-field dislocation dynamics method

The phase-field dislocation dynamics (PFDD) method, originated in 2002, is a continuum dislocation model that uses order parameters to describe dislocation slips in crystalline materials. In the past two decades, and especially since it was last reviewed in 2016, PFDD was advanced significantly in terms of the mathematical formulation, numerical implementation, and applicability. The main purpose of this short review is to summarize recent progress made to improve the energy functional formulation and numerical techniques of PFDD as well as its recent applications. Some recommendations for future work to further extend the PFDD method are presented.

Simultaneous High-Strength and Deformable Nanolaminates With Thick Biphase Interfaces.

Two-phase nanolaminates are known for their high strength, yet they suffer from loss of ductility. Here, we show that broadening heterophase interfaces into "3D interfaces" as thick as the individual layers breaks this strength-ductility trade-off. In this work, we use micropillar compression and transmission electron microscopy to examine the processes underlying this breakthrough mechanical performance. The analysis shows that the 3D interfaces stifle flow instability via shear band formation through their interaction with dislocation pileups. To explain this observation, we use phase field dislocation dynamics (PFDD) simulations to study the interaction between a pileup and a 3D interface. Results show that when dislocation pileups fall below a characteristic size relative to the 3D interface thickness, transmission across interfaces becomes significantly frustrated. Our work demonstrates that 3D interfaces attenuate pileup-induced stress concentrations, preventing shear localization and offering an alternative way to enhanced mechanical performance.

Transitions in the morphology and critical stresses of gliding dislocations in multiprincipal element alloys

Refractory multiprincipal element alloys (MPEAs) are promising material candidates for high-temperature, high-strength applications. However, their deformation mechanisms, particularly at the dislocation scale, are unusual compared to those of conventional alloys, making it challenging to understand the origin of their high strength. Here, using an atomistically informed phase-field dislocation dynamics model, we study transitions in the morphology and critical stresses of long gliding screw dislocations over extended distances in a refractory MPEA. The model MPEA crystal accounts for the atomic-scale fluctuations in chemical composition across the glide planes via spatially correlated lattice energies and for the differences in glide resistances between screw and edge dislocations of unit length. We show that the dislocation moves in a stop-start motion, alternating between wavy morphology in free flight and nearly recovered straight screw orientation in full arrest. The periods of wavy glide are due to variable kink-pair formation and migration rates along the length, where portions with higher rates glide more quickly. The critical stress to initiate motion corresponds to the stress required to form and migrate a kink pair at the weakest region along the length of the dislocation. Heterogeneity in lattice energy leads to variability in the local stress-strain response and to a strain hardening-like response, in which the critical stress to reactivate glide increases with glide distance. Statistical assessment of hundreds of realizations of dislocations indicates that the amount of hardening directly scales with the dispersion in underlying lattice energy.

Phase-field modeling of the interactions between an edge dislocation and an array of obstacles

Obstacles, such as voids and precipitates, are prevalent in crystalline materials. They strengthen crystals by serving as barriers to dislocation glide. In this work, we develop a phase-field dislocation dynamics (PFDD) technique for investigating the interactions between dislocations and second-phase obstacles, which can be either voids or precipitates. The PFDD technique is constructed to account for elastic heterogeneity, elastic anisotropy, dissociation of the dislocation, and dislocation transmission across bicrystalline interfaces. Within the framework, we present a model for “pseudo-voids”, which are voids shearable by dislocations, in contrast to unphysical, unshearable voids in conventional phase-field dislocation formulations. We employ the PFDD technique to investigate the in-plane interactions between an edge dislocation and an array of nano-scale obstacles with different spacings. In this application, the interactions take place in glide planes of either a face-centered cubic (FCC) Cu or a body-centered cubic (BCC) Nb matrix, while the precipitates have a Cu1−xNbx composition, with x varying from 0.1 to 0.9. Our atomistic simulations find that the alloy precipitates can have an FCC, an amorphous, or a BCC phase, depending on the compositional ratio between Cu and Nb, i.e., value of x. Among all types of obstacles, the critical stresses for dislocation bypass are the highest for unshearable amorphous precipitates, followed by shearable crystalline precipitates, and then the pseudo-voids.

Dislocation transmission across Σ3{112} incoherent twin boundary: a combined atomistic and phase-field study

Grain boundaries (GBs) in polycrystalline materials act as impediments to dislocation motion and result in strengthening. Understanding slip transmission through GBs, specifically twin boundaries, is essential to understand the plastic deformation behavior of polycrystalline fcc materials. In this study the interaction between a glide dislocation and Σ3{112} incoherent twin boundary (ITB) in copper is investigated using a combined atomistic and mesoscale approach. The material parameters and structure of the GB in the mesoscale phase field dislocation dynamics (PFDD) model are informed from Molecular Statics (MS) simulations. The structural unit of the ITB consists of an array of three partial dislocations. The interaction between a glide dislocation impinging on each of the GB partial dislocations is investigated using both PFDD and Molecular Dynamics (MD) with two boundary conditions. Transmission planes predicted by both PFDD and MD (NVT) are in agreement, and show that not all transmission events are direct. Critical transmission stresses predicted by PFDD are in the range of 276 MPa to 1380 MPa, while MD predictions are in the range from 100 MPa to 700 MPa. The PFDD and MD predictions of slip transmission are explained using dislocation theory based on isotropic linear elasticity.

Non-orthogonal computational grids for studying dislocation motion in phase field approaches

In this work, new non-orthogonal computational grids are implemented into a phase field model called Phase Field Dislocation Dynamics (PFDD). We demonstrate that the new non-orthogonal grid can accommodate multiple slip planes in either the face centered cubic (FCC) or body centered cubic (BCC) crystallographic systems. We show that they avoid numerical errors induced when modeling glide on inclined slip planes in an orthogonal grid. The Gibbs effect that arises in the orthogonal or rotated orthogonal grids is substantially diminished when a non-orthogonal grid is employed. A few test cases demonstrate the effectiveness of using non-orthogonal grids in solving systems with multiple non-planar slip systems.

Elastic interaction-induced anisotropic growth of dislocation loop arrays

The elastic interactions and reactions of dislocations lead to the formation of complex dislocation substructures, which is critical to the strain hardening and fatigue failure. Phase field dislocation dynamics simulations are conducted as a first step to understand the elastic interactions between dislocation loops. When the interloop spacing is small, the elastic interactions with neighboring loops become strong, rendering the edge segments strongly pinned, while allowing for the screw segments to propagate more easily. The interactions are found to result in an anisotropic stress distribution around the dislocation loops, leading to the formation of arrays of long, straight edge dislocations that could act as barriers to subsequent slip. Furthermore, the effect of initial loop size and applied strain rate on the elastic interaction-induced anisotropic pinning effect is investigated and discussed. The results are important for coarse-graining dislocation substructures formation into continuum level models of deformation in crystalline solids.

Phase field dislocation dynamics (PFDD) modeling of non-Schmid behavior in BCC metals informed by atomistic simulations

Body-Centered Cubic (BCC) metals exhibit anomalous mechanical properties such as twinning/anti-twinning and tension/compression asymmetry, which are attributed to asymmetric behavior of screw dislocations. Unlike Face-Centered Cubic (FCC) metals, the Critical Resolved Shear Stress (CRSS) of BCC metals deviates from the Schmid law. We use a mesoscale modeling approach called Phase Field Dislocation Dynamics (PFDD) to understand the critical factors that control this non-Schmid behavior and reproduce atomistic predictions of the CRSS (Peierls stress). All inputs to the PFDD model are obtained from Molecular Statics (MS) simulations. Multiple pathways for modeling the non-Schmid behavior are investigated by incorporating the representative dislocation properties into different energy terms in the PFDD model. One way to understand non-Schmid behavior is to incorporate stress components projected on inclined planes into the external energy term within PFDD. Alternatively, we propose that non-Schmid behavior can also be accounted for by considering the variation of the {110} unstable stacking fault energy and the dislocation core width as a function of the applied tensorial stress field. The CRSS predicted using PFDD modeling is in excellent agreement with MS predictions.

Asymmetric equilibrium core structures of pyramidal-II 〈c+a〉 dislocations in ten hexagonal-close-packed metals

The structures of pyramidal-II $\ensuremath{\langle}c+a\ensuremath{\rangle}$ dislocations, one of the most important defects in structural hexagonal-close-packed (HCP) metals, have not been fully characterized for many of the HCP metals in use today. Here, we employ ab initio informed phase-field dislocation dynamics to determine the minimum energy structure of pyramidal $\left\{\overline{1}\overline{1}22\right\}\ensuremath{\langle}11\overline{2}3\ensuremath{\rangle}$ dislocations in ten HCP metals, including Be, Co, Mg, Re, Ti, Zn, Cd, Hf, Y, and Zr. As input for the simulations, we calculate, using first-principles density functional theory, the $\left\{\overline{1}\overline{1}22\right\}$ generalized stacking fault energy (GSFE) curves for all ten metals. From these calculations, it is found that magnetism in Co is necessary for achieving a local minimum in the GSFE curve. We observe in simulations that edge and screw character dislocations split into two partials separated by a low-energy intrinsic stacking fault. The splitting distance is shown to scale inversely with the local minimum energy normalized by the product of its shear modulus and Burgers vector. Interestingly, some HCP metals exhibit an asymmetric structure, with either unequal partial Burgers vectors or widths, in contrast to the symmetric configuration expected from linear elastic dislocation theory. We explain these structures by properties of the local maxima in their GSFE curves. Metals with larger degrees of elastic anisotropy result in dislocations with larger splitting distances than would be expected under the commonly used assumption of elastic isotropy. These findings on the sizes and asymmetry in the structures of pyramidal-II $\ensuremath{\langle}c+a\ensuremath{\rangle}$ dislocations are fundamental to understanding how these dislocations glide and interact or react with other defects when these metals are mechanically strained.