First-principles prediction of yield stress for basal slip in Mg–Al alloys

The finite-temperature yield stress of Mg–Al alloys undergoing basal slip is investigated using a recently developed parameter-free solute strengthening model. The model takes input from first-principles calculations of the dislocation/solute interaction energy and evaluates the solute strengthening due to fluctuations in solute concentration, taking into account the correlation of these fluctuations as a function of dislocation roughening. Due to the wide partial separation of the Mg basal edge dislocation, a smaller roughening is required to decorrelate the solute fluctuations in the partials compared to that required to decorrelate the fluctuations in the “far field”. As a consequence, the dislocation has two stable configurations in the random field of solutes, corresponding to “short-range” and “long-range” solute interactions. The configuration of the “short-range” interactions control the strength at low temperatures or high stress, whereas the “long-range” interactions control the strength at higher temperatures or lower stresses. Predictions of the model are in very good agreement with experiments over a wide range of solute concentrations and temperature. In particular, the model naturally predicts the “plateau stress” observed at high temperatures, which is attributable to the “long-range” solute interactions.     

Solute strengthening from first principles and application to aluminum alloys

Alloys containing substitutional solutes exhibit strengthening due to favorable solute fluctuations within the alloy that hinder dislocation motion. Here, a quantitative, parameter-free model to predict the flow stress as a function of temperature and strain rate of such alloys is presented. The model builds on analytic concepts developed by Labusch but introduces key innovations rectifying shortcomings of previous models. To accurately describe the solute/dislocation interaction energies in and around the dislocation core, density functional theory and a flexible-boundary-condition method are used. The model then predicts the zero temperature flow stress, the energy barrier for dislocation motion, and thus the finite temperature flow stresses. The model is used to predict the flow stresses of various Al alloys. Excellent results are obtained for Al–Mg and Al–Mn. Al–Fe with ppm levels of Fe is not predicted well but, using experimental results for Fe, results for the quasi-binary Al–Cr–(Fe) and Al–Cu–(Fe) alloys agree well with experiments. The model is also consistent with the “stress equivalency” postulate of Basinski. This parameter-free model using first-principles input thus provides a basis for achieving the long-sought goal of computational design of alloys, within the context of solute-strengthening mechanisms.     

Prediction of thermal cross-slip stress in magnesium alloys from a geometric interaction model

We develop a geometry-based model from first-principles data for the interaction of solutes with a prismatic screw dislocation core, and predict the thermally activated cross-slip stress above room temperature in Mg alloys. Electronic structure methods provide data for the change in prismatic stacking fault energy for different possible fault configurations for 29 different solutes. The direct solute-dislocation interaction energies for solutes that produce stable prismatic screw dislocation cores (K, Na, Sc and Ca) is correlated with stacking fault misfits. This geometric interaction model produces similar prediction errors for all 29 solutes. The model predicts alloys with cross-slip stresses lower than pure Mg for three previously considered solutes (K, Na and Sc) and three new solutes (Ca, Y and Zr). The model also qualitatively confirms the experimental observation that Mg-Li alloys have lower cross-slip stress than pure Mg. In particular, low concentrations of Y are predicted to significantly decrease the cross-slip stress in Mg.     

Solute–vacancy binding in aluminum

Previous efforts to understand solute–vacancy binding in aluminum alloys have been hampered by a scarcity of reliable, quantitative experimental measurements. Here, we report a large database of solute–vacancy binding energies determined from first-principles density functional calculations. The calculated binding energies agree well with accurate measurements where available, and provide an accurate predictor of solute–vacancy binding in other systems. We find: (i) some common solutes in commercial Al alloys (e.g., Cu and Mg) possess either very weak (Cu), or even repulsive (Mg), binding energies. Hence, we assert that some previously reported large binding energies for these solutes are erroneous. (ii) Large binding energies are found for Sn, Cd and In, confirming the proposed mechanism for the reduced natural aging in Al–Cu alloys containing microalloying additions of these solutes. (iii) In addition, we predict that similar reduction in natural aging should occur with additions of Si, Ge and Au. (iv) Even larger binding energies are found for other solutes (e.g., Pb, Bi, Sr, Ba), but these solutes possess essentially no solubility in Al. (v) We have explored the physical effects controlling solute–vacancy binding in Al. We find that there is a strong correlation between binding energy and solute size, with larger solute atoms possessing a stronger binding with vacancies. (vi) Most transition-metal 3d solutes do not bind strongly with vacancies, and some are even energetically strongly repelled from vacancies, particularly for the early 3d solutes, Ti and V.     

Solute strengthening of both mobile and forest dislocations: The origin of dynamic strain aging in fcc metals

A full rate-dependent constitutive theory for dynamic strain aging is developed based on two key ideas. The first idea is that both solute strengthening and forest strengthening must exist and must exhibit aging phenomena. The second idea is that a single physical aging mechanism, cross-core diffusion within a dislocation core, controls the aging of both the solute and forest strengthening mechanisms. All the material parameters in the model, apart from forest dislocation density evolution parameters, are derivable from atomistic-scale studies so that the theory contains essentially no adjustable parameters. The model predicts the steady-state stress/strain/strain-rate/temperature/concentration dependent material response for a variety of Al–Mg alloys, including negative strain-rate sensitivity, in qualitative and quantitative agreement with available experiments. The model also reveals the origin of non-additivity of solute and forest strengthening, and explains observed non-standard transient stress behavior in strain-rate jump tests.     

A simulation study of the shape of β′ precipitates in Mg–Y and Mg–Gd alloys

The metastable β′ phase is a key strengthening precipitate phase in a range of Mg–RE (RE: rare-earth elements) based alloys. The morphology of the β′ precipitates changes from a faceted and nearly equiaxed shape in Mg–Y alloys to a truncated lenticular shape in Mg–Gd alloys. In this work, we study effects of interfacial energy and coherency elastic strain energy on the morphology of β′ precipitates in binary Mg–Y and Mg–Gd alloys using a combination of first-principles calculations and phase-field simulations. Without any free-fitting parameters and using the first-principles calculations, CALPHAD databases and experimental characterizations as model inputs (lattice parameters of the β′ phase, elastic constants and chemical free energy of Mg matrix and interfacial energies of the coherent β′/Mg matrix interfaces), the phase-field simulations predict equilibrium shapes of β′ precipitates of different sizes that agree well with experimental observations. Factors causing the difference in the equilibrium shape of β′-Mg₇Y and β′-Mg₇Gd precipitates are identified, and possible approaches to increase the aspect ratio of the β′ precipitates and thus to enhance the strength of Mg–RE alloys are discussed.     

Solute and dislocation junction interactions

In this study, the role of solute segregation on the strength and the evolution behavior of dislocation junctions is studied by utilizing kinetic Monte Carlo and three-dimensional dislocation dynamics simulations. The different solute concentrations and the character of the junctions are all included in the simulations in an effort to make a parametric investigation. The results indicate that the solutes have a profound effect on the strength of the junctions. Solute segregation can lead to both strengthening and weakening behavior, depending upon the evolution of the dislocation junctions. The local solute concentration seems to be the more relevant parameter to characterizing the solute and dislocation interactions, due to the short-range stress field of solutes; and its bounds are set by the unconstrained volume dilatation.     

Do substitutional solutes soften or embrittle cracktips in α-Fe?

We employ atomistic modeling to examine the influence substitutional solutes have on the activated process of nucleating dislocations from cracktips in α-Fe. We demonstrate a strong correspondence between the energy landscape of the solute during an idealized block-like sliding process and the influence on the activation energy barrier for dislocation emission near a idealized cracktip. Using a series of solute potentials developed with input from ab initio calculations, these models predict that Cu and Ni both lower the activation energy barriers for emission, while solutes such as Mo and W raise the barriers. For solutes that raise or lower the emission barriers, the implications on the overall nucleation frequency turns out to be very different.     

Solute Segregation to Grain Boundaries in Al: A First-Principles Evaluation

Solute-induced grain boundary (GB) strengthening is effective in improving the toughness and tensile strength of polycrystalline alloys. In this work, GB segregation behaviors of solute elements in Al alloys and their potential effects on GB binding have been systematically investigated from first-principles energetics. The low-energy Σ3(111) and Σ11(113) are immune to vacancy segregation, while high-energy Al GBs, such as Σ13(320), Σ9(221), Σ5(210), and Σ5(310), can attract both vacancies and solutes. Under-sized elements (Ni, Fe, Co, Cu) and similar-sized elements (Si, Zn, Ag, and Ti) prefer interstitial or vacancy sites at the GB interface, while over-sized elements (Mg, Zr, Sc, Er) tend to substitute Al or vacancy-neighboring sites at the GB interface. Segregated vacancies weaken GBs. Under-sized Ni, Co, Cu, similar-sized Ti, and over-sized Zr, Er, can directly enhance Al GBs, while similar-sized Ag and over-sized Mg reduce the GB binding strength. Solute strengthening or weakening effects tend to be always mitigated, more or less, by GB-segregated vacancies.     

A unified constitutive model for multiphase precipitation and multi-stage creep ageing behavior of Al-Li-S4 alloy

A unified constitutive model is presented to predict the recently observed “multi-stage” creep behavior of Al-Li-S4 alloy. The corresponding microstructural variables related to the yield strength and creep deformation of the alloy during the creep ageing process, including dislocations and multiple precipitates, have been characterized in detail by X-ray diffraction (XRD) and transmission electron microscopy (TEM). For the yield strength, the model considers the multiphase strengthening behavior of the alloy based on strengthening mechanisms, which includes shearable T₁ precipitate strengthening, non-shearable T₁ precipitate strengthening and θ′ precipitate strengthening. Based on creep deformation mechanism, the “multi-stage” creep behavior of the alloy is predicted by introducing the effects of interacting microstructural variables, including the radius of multiple precipitates, dislocation density and solute concentration, into the creep stress-strain model. It is concluded that the results calculated by the model are in a good agreement with the experimental data, which validates the proposed model.     

Solute–vacancy binding of the rare earths in magnesium from first principles

The thermodynamic and kinetic properties of Mg–rare earth (RE) alloys are not widely known, despite increased research in recent years into their mechanical strengthening mechanisms. To aid in the development of new alloys and strengthening models, the solute–vacancy binding energies for all 17 RE elements in hexagonal close-packed (hcp) Mg are calculated by density functional theory (DFT) in several solute/vacancy configurations. As the REs can be considerably larger than Mg, special care is taken to ensure convergence with respect to supercell size, and the binding of other large, non-RE solutes is calculated in the large supercells employed in the current work. It is found that the light and heavy lanthanides have favorable and unfavorable nearest-neighbor solute–vacancy binding, respectively. The binding energies exhibit a strong linear correlation to the magnitude of the solute’s displacement towards the vacancy, suggesting that local lattice relaxation is a significant contribution to the binding energy. There are four notable exceptions to this trend: Na, Sb, Pb and Bi. Explicit calculation of the lattice relaxation energy due to the presence of the solute–vacancy pair confirms that local relaxation is an important component of the binding energy, particularly for the larger solutes. Finally, empirical predictions of the dilute mixing energy disagree with those predicted by DFT. New experimental measurements are necessary to resolve the discrepancies, as there are no published thermodynamic data of the REs in hcp Mg.     

Effect of interstitial carbon on the mechanical properties of electrodeposited bulk nanocrystalline Ni

Solid solution strengthening by carbon and sulfur in bulk nanocrystalline Ni was studied by electrodeposition and first-principles calculations. Bulk nanocrystalline Ni with a carbon content of 30–1600 ppm and a sulfur content of 140–1200 ppm was prepared using a sulfamate bath with different complexing agents and gloss agents. The hardness values of the bulk nanocrystalline Ni were scattered as the grain size decreased to ～12 nm with increasing carbon and sulfur content. It was found that the scatter could be explained by considering the effect of impurities such as solute atoms on the hardness of electrodeposited Ni, in addition to the Hall–Petch relationship. Thus, to determine the structure of Ni–C and Ni–S solid solutions and estimate the contribution of impurities to hardness, the enthalpy of solution and misfit strain were calculated by first-principles calculations. The results indicate that carbon exists as an interstitial solute atom in the Ni matrix, producing large misfit strains, and sulfur exists as a substitutional solute atom, inducing no significant changes. A model of solid solution strengthening due to interstitial solute atoms was developed by considering the interaction between mobile dislocations and solute atoms. This study has effectively divided the observed solid solution effect from the grain refinement effect in electrodeposited nanocrystalline Ni. The results of this study point to the origin of high-strength electrodeposited bulk nanocrystalline Ni.     

Magnetism-induced solid solution effects in intermetallic Ni60Al40

In this study, experimental work on solute effects in an intermetallic alloy, NiAl, has been complemented by first-principles quantum mechanical calculations to investigate the effect of iron and cobalt solutes on lattice parameter and hardening. Ternary additions of iron and cobalt with similar atomic sizes were added to replace nickel in NiAl containing 40% Al. Cobalt solutes did not affect the lattice parameter or the hardening behavior of NiAl alloys. Iron solutes, on the other hand, substantially expanded the lattice, resulting in unusual solid solution softening. These results could not be explained from a consideration of the size mismatch based on the Goldschmidt radii for iron and cobalt atoms. From the contrasting behavior of the iron and cobalt solutes, magnetic interactions induced by iron atoms located on the aluminum sublattice have been identified as a new factor responsible for the unusual large lattice dilation and resultant solid solution softening in these intermetallic alloys.     

Strengthening mechanisms based on reinforcement distribution uniformity for particle reinforced aluminum matrix composites

A modified mixed strengthening model was proposed for describing the yield strength of particle reinforced aluminum matrix composites. The strengthening mechanisms of the composites were analyzed based on the microstructures and compression mechanical properties. The distribution uniformity of reinforcements and cooperation relationship among dislocation mechanisms were considered in the modified mixed strengthening model by introducing a distribution uniformity factor u and a cooperation coefficient f_c, respectively. The results show that the modified mixed strengthening model can accurately describe the yield strengths of Al₃Ti/2024 Al composites with a relative deviation less than 1.2%, which is much more accurate than other strengthening models. The modified mixed model can also be used to predict the yield strength of Al₃Ti/2024 Al composites with different fractions of reinforcements.     

Effect of magnesium on dispersoid strengthening of Al—Mn—Mg—Si (3xxx) alloys

The effects of magnesium addition on the dispersoid precipitation as well as mechanical properties of 3xxx alloys were investigated. The microstructures in as-cast and heat-treated conditions were evaluated by optical microscopy and transmission electron microscopy. The results reveal that Mg has a strong influence on the distribution and volume fraction of dispersoids during precipitation heat treatment. The microhardness and yield strength at ambient temperature increase with increasing Mg content. The solid solution and dispersoid strengthening mechanisms of materials after heat treatment are quantitatively analyzed. Dispersoid strengthening for the alloys is the predominant strengthening mechanism after precipitation heat treatment. An analytical model is introduced to predict the evolution of ambient-temperature yield strength.     

Effect of alloying elements on the elastic properties of Mg from first-principles calculations

The influence of different alloying elements on the lattice parameters and elastic properties of Mg solid solution has been studied using first-principles calculations within the generalized gradient approximation. The solute atoms employed herein are Al, Ba, Ca, Cu, Ge, K, Li, Ni, Pb, Si, Y and Zn. A supercell consisting of 35 atoms of Mg and one solute atom is used in the current calculations. A good agreement between calculated and available experimental data is obtained. Lattice parameters of Mg–X alloys are found to be dependent on the atomic radii of the solute atoms. A correlation between the bulk modulus of Mg–X alloys and the nearest-neighbor distance between Mg and X is shown. Addition of solute atoms belonging to the s-block and p-block of the periodic table results in a lower bulk moduli than d-block elements. A strong dependence of the elastic modulus of Mg–X alloys on the elastic properties of the solute atoms is also observed. Using the bulk modulus/shear modulus ratio (B/G), the change in the ductility of Mg due to the addition of the solute atom is briefly described. Linear regression coefficients for the elastic constants of each of the alloys are obtained as a tool for predicting the trend in the elastic properties of Mg as a function of concentration of the solute atoms.     

Single-mechanism rate theory for dynamic strain aging in fcc metals

A full thermal activation rate theory for dynamic strain aging is developed for the case where a single rate dependent strengthening mechanism controls dislocation motion in a material (e.g. solute diffusion). The analysis shows that negative strain-rate sensitivity (SRS) cannot be obtained within such a framework, a conclusion previously reached by Hähner [Hähner P. Mater Sci Eng A 1996;207:208]. However, the SRS can be greatly reduced over a range of strain rates, making the inverse behavior more accessible by other mechanisms. In addition, the aging mechanism naturally gives rise to an instantaneous positive SRS and stress relaxation behavior under strain-rate jump conditions, putting the concepts advanced by McCormick [McCormick PG. Acta Metall 1988;36:3061; Estrin Y, McCormick PG. Acta Metall Mater 1991;39:2977] on a quantitative footing. The results here set the stage for subsequent work wherein consideration of multiple strengthening mechanisms (solute and forest hardening) operating together can predict negative SRS in quantitative agreement with data in Al–Mg alloys.     

Theory of the nonlinear internal friction associated with bulk diffusion of solute atoms around dislocations in Al–Mg alloys

In this paper the nonlinear (amplitude-dependent) internal friction (P₃ peak) in cold-worked Al–Mg alloys is theoretically studied by solving the bulk diffusion equations of the solute atoms (Mg atoms) under the action of dislocation drag. The results in the case of a constant external stress show that the bow-out distance of the dislocation has an exponential relation with time, which can be well described by an exponential creep function with a Gaussian distribution in τ. With the increasing strain amplitude, the relaxation strength Δ and relaxation time τ decrease, while the distribution parameter increases. Both activation energy H and pre-exponential factor τ₀ deduced from τ through Arrhenius relation are dependent on strain amplitude.     

Non-linear anelasticity due to dislocation–solute atom interactions in solid solutions

A theory for the non-linear, strain-amplitude-dependent internal friction (ADIF) in solid solutions due to dislocation–solute atom interactions has been developed. The suggested model accounts for different modes of dislocation–solute atom interactions:

• solute atoms, distributed in the dislocation glide plane, represent short-range obstacles for the dislocation motion;

• solute atoms, situated away from the dislocation glide plane, create diffuse weak long-range elastic stress fields, also impeding dislocation motion.

Dislocations overcome localised obstacles under the combined action of the applied stress and thermal energy, whereas diffuse long-range obstacles can be surmounted only athermally. Numerical calculations of the ADIF, strain-amplitude-dependent modulus defect, their ratio, and of the fraction of the athermal ADIF component have been performed. The model predicts a complicated multistage behaviour of the parameters of the non-linear anelasticity in the strain amplitude–temperature–solute concentration domain which is in excellent agreement with recent experimental data.     

First Principle and Experimental Study for Site Preferences of Formability Improved Alloying Elements in Mg Crystal

Some alloying elements (Al, Er, Gd, Li, Mn, Sn, Y, Zn) were proved recently by calculations or experiments to improve the formability of Mg alloys, but ignoring their site preference in Mg crystals during the calculated process. A crystallographic model was built via first principle calculations to predict the site preferences of these elements. Regularities between doping elements and site preferences were summarized. Meanwhile, in the basis of the crystallographic model, a series of formulas were deduced combining the diffraction law. It predicted that a crystal plane with abnormal XRD peak intensity of the Mg-based solid solutions, compared to that of the pure Mg, prefers to possess solute atoms. Thus, three single-phase solid solution alloys were then prepared through an original In-situ Solution Treatment, and their XRD patterns were compared. Finally, the experiment further described the site preferences of these solute atoms in Mg crystal, verifying the calculation results.