Broken Bond Model Research Articles

A model for predicting grain-boundary energy of refractory high-entropy alloys based on local element concentration

Refractory high-entropy alloys (RHEAs) exhibit exceptional high-temperature mechanical properties, and their structural stability is primarily influenced by grain boundaries (GBs). However, it remains challenging to accurately quantify the grain boundary energy (GBE) in RHEAs. Herein, we introduce a quantitative descriptor for point-to-point prediction of GBEs in RHEAs based on local element concentration and valence-electron numbers. Our results demonstrate that the role of the constituent elements on GBEs is primarily determined by their valence-electron numbers in RHEAs. These findings are derived from the cohesive energy and atomic structure of the elements within the framework of the broken-bond model, as well as the differential contributions of s-electrons and d-electrons. The model establishes a structure-property relationship between local element concentration and GBEs, with the prediction consistent with experimental and theoretical results, which is crucial for evaluating the GB stability of RHEAs and facilitating the development of high-performance alloys.

Hook's law scaled broken-bond model for surface energy: From metals to ceramics

Innovative surfaces via lattice engineering are crucial challenges and critical to enhance the mechanical and functional properties of advanced materials. Here, a universal linear scaling rule in terms of key property parameters (KPPs) is proposed to estimate the modifications of local lattice strain (LLS) caused by the surface, in which the bulk modulus (B), the B/G ratio, and the bond energy are utilized to describe the KPPs of the BCC and FCC, the isotropic HCP, and the polarized A2B2O7-type Pyrochlore, individually. The modified broken-bond model by the Hook's law is addressed to predict the elastic-energy-corrected surface energy efficiently and precisely, which is capable to deal with the complex of geometry, chemistry, charge states of clean surface and are verified and validated in 28 metals and ceramics. While the LLS of metals is caused by the improved bond length, the ceramics one is yielded by the reduced one of polarized surface.

A new analytical surface energy model for arbitrary (h k l) planes in BCC and FCC metals

A new analytical surface energy model (Eq. 9) that is free of adjustable parameters has been developed to accurately predict the surface energies of arbitrary (h k l) planes in both BCC and FCC ground-state pure metals at 0 K, using only (100), (110) and (111) surface energies as input. The formulation is established by expanding the original broken-bond model up to the 3rd nearest neighbor shell and considering the surface relaxation effect. In addition, this model consists of only one single formula per crystal structure, overcoming the geometric complexities in previous models in which multiple equations are needed for different surface groups. The analytical model has been validated through the comparison with 4357 independent surface energies. These data include the energies of 517 unique surfaces per metal in Fe, Cr, W, Cu, Ni, and Al (a total of 3102 surface energies) that are calculated through our molecular dynamics simulations, and excellent agreement is obtained. The validation data also include 1255 independent surface energies in literature or online database for up to 46 BCC and FCC metals that were calculated by either empirical potentials or density functional theory. The agreement is still very good for most of the data, demonstrating the robustness of our new model.

Broken bond models, magic-sized clusters, and nucleation theory in nanoparticle synthesis.

Magic clusters are metastable faceted nanoparticles that are thought to be important and, sometimes, observable intermediates in the nucleation of certain faceted crystallites. This work develops a broken bond model for spheres with a face-centered-cubic packing that form tetrahedral magic clusters. With just one bond strength parameter, statistical thermodynamics yield a chemical potential driving force, an interfacial free energy, and free energy vs magic cluster size. These properties exactly correspond to those from a previous model by Mule et al. [J. Am. Chem. Soc. 143, 2037 (2021)]. Interestingly, a Tolman length emerges (for both models) when the interfacial area, density, and volume are treated consistently. To describe the kinetic barriers between magic cluster sizes, Mule et al. invoked an energy parameter to penalize the two-dimensional nucleation and growth of new layers in each facet of the tetrahedra. According to the broken bond model, barriers between magic clusters are insignificant without the additional edge energy penalty. We estimate the overall nucleation rate without predicting the rates of formation for intermediate magic clusters by using the Becker-Döring equations. Our results provide a blueprint for constructing free energy models and rate theories for nucleation via magic clusters starting from only atomic-scale interactions and geometric considerations.

Molecular dynamics simulation and thermodynamics calculation on surface segregation in Ni-Cu nano-films under stress

The surface segregation of Cu atoms in a Ni-Cu system was investigated using molecular dynamics simulations. Thermodynamic calculations were performed to verify the results of the molecular dynamics simulations. For the thermodynamic calculations, a model for evaluating the influence of stress on surface segregation was developed using the modified Darken model in combination with the broken-bond model. Using molecular dynamics simulations, it was found that the enrichment of Cu atoms occurred for a free-standing Ni-10 at.% Cu film consisting of 20 layers. Simultaneously, the stress distribution across the Ni-Cu thin film is obtained. The thermodynamic calculation results show that the influence of stress on the surface segregation cannot be ignored because of the considerable surface stress. Surface tension stress promotes the surface segregation of copper in Cu-Ni alloys due to the larger lattice parameter of copper than nickel, which leads to the reduction of surface strain energy. When the thickness is greater than 31 nm (or the number of layers exceeds 89), the size effect disappears, i.e., the surface concentration doesn’t increase with the increase of thickness. The calculation results obtained by the Bragg-William equation used for the surface segregation in equilibrium are in good agreement with the thermodynamic calculation and molecular dynamics simulation results.

Computational analysis of heterogeneous nucleation and precipitation in AA6005 Al-alloy during continuous cooling DSC experiments

A physically-based model for the computational evaluation of differential scanning calorimetry (DSC) curves during continuous cooling of Al alloys from solution annealing temperature is developed. The model is particularly suitable for predicting the nucleation and growth of quench induced precipitates at heterogeneous nucleation sites, such as grain boundaries, the stress field of dislocations or particles like primary phases or dispersoids. The thermokinetic model incorporates an extended formulation of the precipitate nucleation barrier and considers β-Mg2Si and B’-Al4Mg8Si7 phase precipitates. These are the dominating, experimentally observed quench induced phases in an aluminium AA6005 alloy within a wide range of cooling rates. The model is implemented in the thermokinetic software MatCalc, which simulates the precipitation processes and the evolution of excess specific heat capacities during the heat treatments. The Generalized Broken Bond model, incorporating the effects of interface curvature and diffuse interfaces, is used for interface energy prediction. Comparison of experiment and simulation demonstrates that the impact of heterogeneous nucleation sites must be adequately considered in the nucleation energy expression to obtain accurate predictions of the precipitate evolution, particularly for precipitation at low and very low undercooling as present during continuous cooling, particularly at slower cooling rates.

Effects of composition and magnetism on interfacial energy in Cu-Co alloys

The composition and magnetic dependent interfacial energy in Cu-Co immiscible alloys is investigated within a coherent interface model using ab initio calculations. We translate the composition dependence of the interfacial energy to the temperature dependence considering the variations of the equilibrium compositions of precipitate and matrix with respect to temperature. The obtained results are in reasonable agreement with those obtained by experiments and thermodynamic calculations. Reviewing the experimental methods for determining the interfacial energy based on kinetic models for precipitate nucleation and coarsening, as well the thermodynamic models based on broken-bond models, we point out that the temperature effect on the interfacial energy in the above models is primarily due to the composition change of the interface. The present work emphasizes the effort to understand the meaning of the speciously same quantity in different methods.

Controlling anisotropy in 2D microscopic models of growth

The quantitative knowledge of interface anisotropy in lattice models is a major issue, both for the parametrization of continuum interface models, and for the analysis of experimental observations. In this paper, we focus on the anisotropy of line tension and stiffness, which plays a major role both in equilibrium shapes and fluctuations, and in the selection of nonequilibrium growth patterns. We consider a 2D Ising Hamiltonian on a square lattice with first and second-nearest-neighbor interactions. The surface stiffness and line tension are calculated by means of a broken-bond model for arbitrary orientations. The analysis of the interface energy allows us to determine the conditions under which stiffness anisotropy is minimal. These results are supported by a quantitative comparison with kinetic Monte Carlo simulations, based on the coupling of a field of mobile atoms to a condensed phase. Furthermore, we introduce a generic smoothing parameter which allows one to mimic the finite resolution of experimental microscopy techniques. Our results provide a method to fine-tune the interface energy in models of nanoscale non-equilibrium processes, where anisotropy and fluctuations combine and give rise to non-trivial morphologies.

Predicting activation energies for vacancy-mediated diffusion in alloys using a transition-state cluster expansion

Kinetic Monte Carlo models parameterized by first principles calculations are widely used to simulate atomic diffusion. However, accurately predicting the activation energies for diffusion in substitutional alloys remains challenging due to the wide variety of local environments that may exist around the diffusing atom. We address this challenge using a cluster expansion model that explicitly includes a sublattice of sites representing transition states and assess its accuracy in comparison with other models, such as the broken bond model and a model related to Marcus theory, by modeling vacancy-mediated diffusion in Pt-Ni nanoparticles. We find that the prediction error of the cluster expansion is similar to that of other models for small training sets, but with larger training sets the cluster expansion has a significantly lower prediction error than the other models with comparable execution speed. Of the simpler models, the model related to Marcus theory yields predictions of nanoparticle evolution that are most similar to those of the cluster expansion, and a weighted average of the two approaches has the lowest prediction error for activation energies across all training set sizes.

Strongly trapping soluble lithium polysulfides using polar cysteamine groups for highly stable lithium sulfur batteries

Sulfur has become one of the most promising positive electrode materials for lithium sulfur batteries due to its high theoretical capacity and high energy density (2500 Wh kg−1). The use of common nonpolar carbon/sulfur composites has proved to be a good way to improve the performance, but they still cannot efficiently trap highly polar lithium polysulfides due to the weak interactions between nonpolar carbon and polar polysulfides. Herein, we report a new strategy of using polar cysteamine groups to trap polar polysulfides, leading to greatly enhanced capacity of ∼920 mAh g−1 at 1 C with a high Coulombic efficiency of ∼99.1%, and a long cycle life of over 600 cycles with a capacity retention higher than 80%. Importantly, in situ UV/Vis spectroscopy was employed to identify intermediates during cycling, which demonstrates the constructed unique polar cysteamine functionalized carbon nanotubes (CNTs) can greatly reduce the production of polysulfides and suppress the shuttle effect. The broken-bond model of linear polysulfane during cycling was further demonstrated by density functional theory calculations. The present strategy of using polar cysteamine-functionalized CNTs to trap soluble intermediates is promising and has significant potential for the development of highly efficient lithium sulfur batteries.

Atomistic simulations of energies for arbitrary grain boundaries. Part II: Statistical analysis of energies for tilt and twist grain boundaries

The orientation-dependency of grain boundary (GB) energy in Al is investigated using a cutoff sphere molecular dynamics model presented in Part I, which is applicable to arbitrary GBs. A GB energy dataset is generated using the model with deletion of over-close atoms and without rigid body translations in building the candidate configurations for a given GB. The dataset includes 12,062 tilt GBs (TIGBs) and twist GBs (TWGBs) with a broad range of misorientation angle/axis (θ/O) and GB-plane orientation (n). Statistical analyses of the results show that the energies overall increase with θ in the low-θ range and then level off for higher θ, and that the energies for non-coincident site lattice GBs also follow the increasing trend in the low-θ range. The energies of TIGBs decrease with the variation of O from the central regions to the edge and then corners of the stereographic triangle, and the O-dependency of GB energy becomes more severe for TWGBs. The current energy dataset suggests that the n-dependency of GB energy is best correlated with the orientation-dependency of surface energy following the broken-bond model and it does not support the existing perspective of low energies for GBs terminated by low-index or dense planes. For both types of GBs, the energy suffers a stronger influence from θ than n.

The role of surface energy anisotropy in the formation of a stepped relief of polycrystalline W under sputtering with Ar ions

This paper is devoted to the study of the effect of surface energy anisotropy on the tungsten surface relief modification under ion sputtering. In our experiments, the sputtering of textured polycrystalline tungsten with Ar ions resulted in a stepped surface relief formation. The surface after sputtering was analyzed using the electron backscatter diffraction and confocal laser scanning microscopy techniques. The formation of the stepped relief is explained by different surface energies of differently oriented grains (surface energy anisotropy). The surface energies of the three low-index W planes were calculated in the model of broken bonds. It is shown for the first time that the ratio of the sputtering depth of differently oriented grains of W is equal to the ratio of the differences of the surface energies of these grains. In that way, the ratio of the sputtering yields of differently oriented grains of tungsten can be quantified by knowing the surface energy, which depended on the reticular grain density and the number of broken bonds with nearest and next-nearest neighboring atoms on the surface using the broken bond model. It is shown that ion sputtering can be used as an instrument for studying the surface energy of solids.

Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model

In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study analytically an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface corrugation in 1+1 dimensions via a continuum model below the roughening transition temperature. The model invokes a continuum evolution law expressed by a highly degenerate parabolic partial differential equation (PDE) for surface diffusion, which is related to the nonlinear gradient flow of a convex, singular surface free energy with a certain exponential mobility in homoepitaxy. This evolution law is motivated both by an atomistic broken-bond model and a mesoscale model for crystal steps. By constructing an explicit solution to this PDE, we demonstrate the lack of symmetry in the evolution of top and bottom facets in periodic surface profiles. Our explicit, analytical solution is compared to numerical simulations of the continuum law via a regularized surface free energy.

Phase Stability and Anisotropic Sublimation of Cubic Ge–Sb–Te Alloy Observed by In Situ Transmission Electron Microscopy

Phase stability and anisotropic sublimation dynamics of the cubic Ge–Sb–Te alloy have been investigated by in situ transmission electron microscopy (TEM). The starting point of the phase-transition study is an epitaxially aligned Ge1Sb2Te4 grain on a Si(111) substrate. Upon in situ heating, the cubic phase remains stable up to the sublimation point without a transition to the thermodynamically stable trigonal crystal structure which is attributed to Si diffusion into the Ge1Sb2Te4 grain. The sublimation process is made visible with atomic resolution. The anisotropic process leads to the formation of stable {111} facets via kink nucleation on stable steps and subsequent sublimation from those kink sites as predicated by the terrace-step-kink model. Kink nucleation sites are identified and are in accordance with the broken-bond model approach.

Self-learning kinetic Monte Carlo simulations of diffusion in ferromagnetic α-Fe–Si alloys

Diffusion of Si atom and vacancy in the A2-phase of α-Fe–Si alloys in the ferromagnetic state, with and without magnetic order and in various temperature ranges, are studied using AKSOME, an on-lattice self-learning KMC code. Diffusion of the Si atom and the vacancy are studied in the dilute limit and up to 12 at.% Si, respectively, in the temperature range 350–700 K. Local Si neighborhood dependent activation energies for vacancy hops were calculated on-the-fly using a broken-bond model based on pairwise interaction. The migration barrier and prefactor for the Si diffusion in the dilute limit were obtained and found to agree with published data within the limits of uncertainty. Simulations results show that the prefactor and the migration barrier for the Si diffusion are approximately an order of magnitude higher, and a tenth of an electron-volt higher, respectively, in the magnetic disordered state than in the fully ordered state. However, the net result is that magnetic disorder does not have a significant effect on Si diffusivity within the range of parameters studied in this work. Nevertheless, with increasing temperature, the magnetic disorder increases and its effect on the Si diffusivity also increases. In the case of vacancy diffusion, with increasing Si concentration, its diffusion prefactor decreases while the migration barrier more or less remained constant and the effect of magnetic disorder increases with Si concentration. Important vacancy-Si/Fe atom exchange processes and their activation barriers were identified, and the effect of energetics on ordered phase formation in Fe–Si alloys are discussed.

The role of chemistry and bonding in regulating fracture in multiphase transition metal carbides and nitrides

The presence of the zeta phase in the tantalum carbides has been experimentally shown to give rise to very high fracture toughness which has been attributed to its complex microstructure. In this paper, electronic structure density functional theory is used to investigate the role chemistry and bonding play in regulating the fracture. These simulations demonstrate that fracture in the cubic carbide form is preferred along the {100} planes while the closed packed planes are preferred when carbon atoms are depleted from these planes. This is a consequence of the loss of all the primary covalent bonds across these planes, and idea that is rationalized using a broken-bond model. Thus, the fracture paths should follow the closed packed planes when carbon depleted stacking faults are present in the zeta phase, and the {100} planes when they are not. Furthermore, our results demonstrate that these features are not unique to the tantalum carbides and are present in the vanadium carbides, niobium carbides and hafnium nitrides. This suggests that these materials may also have similar high fracture toughness’s if the correct microstructure can be obtained through processing.

Energy for the interface system of (Nb, Mo)C/γ-Fe

The interfacial energies of MC/γ-Fe and formation energies of MC carbides have been investigated using first-principles calculations based on density functional theory (DFT). Results show that the replacement of Nb by Mo in the NbC lattice is unfavorable with respect to the formation energy. However, it reduces the lattice parameter of MC and decreases the $$\sigma_{\text{chemical}}$$ (interfacial chemical energy) of MC/γ-Fe, thus favoring the formation of complex (Nb, Mo)C carbide. The substitution of Nb by Mo at the interface of MC/γ-Fe system promotes the hybridizations of Mo–1NNFe and C–1NNFe (or 2NNFe) (the first or second nearest neighboring Fe atoms), which leads to a decrease in $$\sigma_{\text{chemical}}$$ . The influence of bond energy is estimated using the discrete lattice plane/nearest neighbor broken bond (DLP/NNBB) model. It is found that the reduced $$\sigma_{\text{chemical}}$$ is attributed to the much smaller value of $$e_{{{\rm{Fe-C}}}} - e_{{{\rm{Mo-C}}}}$$ (the difference between Fe–C and Mo–C interactions) compared to $$e_{{{\rm{Fe-C}}}} - e_{{{\rm{Nb-C}}}}$$ (the difference between Fe–C and Nb–C interactions). The results obtained from the analysis of the precipitates in Nb- and Nb–Mo-bearing steels are in a good agreement with the calculations.

Cohesion and coordination effects on transition metal surface energies

Here we explore the accuracy of Stefan equation and broken-bond model semiempirical approaches to obtain surface energies on transition metals. Cohesive factors are accounted for either via the vaporization enthalpies, as proposed in Stefan equation, or via cohesive energies, as employed in the broken-bond model. Coordination effects are considered including the saturation degree, as suggested in Stefan equation, employing Coordination Numbers (CN), or as the ratio of broken bonds, according to the bond-cutting model, considering as well the square root dependency of the bond strength on CN. Further, generalized coordination numbers CN¯ are contemplated as well, exploring a total number of 12 semiempirical formulations on the three most densely packed surfaces of 3d, 4d, and 5d Transition Metals (TMs) displaying face-centered cubic (fcc), body-centered cubic (bcc), or hexagonal close-packed (hcp) crystallographic structures. Estimates are compared to available experimental surface energies obtained extrapolated to zero temperature. Results reveal that Stefan formula cohesive and coordination dependencies are only qualitative suited, but unadvised for quantitative discussion, as surface energies are highly overestimated, favoring in addition the stability of under-coordinated surfaces. Broken-bond cohesion and coordination dependencies are a suited basis for quantitative comparison, where square-root dependencies on CN to account for bond weakening are sensibly worse. An analysis using Wulff shaped averaged surface energies suggests the employment of broken-bond model using CN to gain surface energies for TMs, likely applicable to other metals.

First-principles investigation on the geometries, stabilities and defective properties of fluoride surfaces

From first-principles calculations, we perform a systematic study of the stoichiometric surface morphology of NaF, MgF2 and CaF2 and the associated stability, charge transfer and defective properties. Given the geometries of their low index surfaces, it is found that the surfaces with the lowest surface energies for NaF, MgF2, and CaF2 are (100), (110) and (111) surfaces, respectively. The dependence of surface energies, electrostatic potentials and effective charges on the slab thickness is discussed. Moreover, we demonstrate the broken bond model, which is based on the covalent interactions, is also suitable for ionic fluoride crystals after modification. By setting a fitting parameter k around 0.5, the estimated surface energies are close to the ones by slab modeling for all the 10 surfaces considered in this work.

Embedded-atom study of low-energy equilibrium triple junction structures and energies

We present an atomistic study of the structures and defect energies of triple junctions (TJs) in polycrystalline materials. A new concept to calculate the excess energy of isolated TJs is proposed and applied to a molecular dynamics (MD) study of iron tricrystals. Line energies of bulk TJs (merging three grain boundaries (GBs)) and surface TJs (merging one grain boundary and two surfaces) are found to be very low. In absolute value they amont to only a few 10−10 Jm−1. Remarkably, defined as a correct excess energy relative to the GBs, the bulk TJ energy is determined to be negative in all studied configurations with an average value of −2.8 × 10−10 Jm−1. These quantitative results are in contrast to various experimental attempts, but they fully agree with simple geometric estimates and broken-bond models, which prompts a re-interpretation of reported measurements.