Solid Solution High-entropy Alloys Research Articles

Achieving ultrahigh strength and ductility via high-density nanoprecipitates triggering multiple deformation mechanisms in a dual-aging high-entropy alloy with precold deformation

How to achieve high-entropy alloys (HEAs) with ultrahigh strength and ductility is a challenging issue. Precipitation strengthening is one of the methods to significantly enhance strength, but unfortunately, ductility will be lost. To overcome the strength–ductility trade-off, the strategy of this study is to induce the formation of high-density nanoprecipitates through dual aging (DA), triggering multiple deformation mechanisms, to obtain HEAs with ultrahigh strength and ductility. First, the effect of precold deformation on precipitation behavior was studied using Ni35(CoFe)55V5Nb5 (at.%) HEA as the object. The results reveal that the activation energy of recrystallization is 112.2 kJ/mol. As the precold-deformation amount increases from 15 % to 65 %, the activation energy of precipitation gradually decreases from 178.8 to 159.7 kJ/mol. The precipitation time shortens, the size of the nanoprecipitate decreases, and the density increases. Subsequently, the thermal treatment parameters were optimized, and the DA process was customized based on the effect of precold deformation on precipitation behavior. High-density L12 nanoprecipitates (∼3.21 × 1025 m−3) were induced in the 65 % precold-deformed HEA, which led to the simultaneous formation of twins and stacking fault (SF) networks during deformation. The yield strength (YS), ultimate tensile strength, and ductility of the DA-HEA are ∼2.0 GPa, ∼2.2 GPa, and ∼12.3 %, respectively. Compared with the solid solution HEA, the YS of the DA-HEA increased by 1,657 MPa, possessing an astonishing increase of ∼440 %. The high YS stems from the precipitation strengthening contributed by the L12 nanoprecipitates and the dislocation strengthening contributed by precold deformation. The synergistically enhanced ductility stems from the high strain-hardening ability under the dual support of twinning-induced plasticity and SF-induced plasticity.

Efficient property-oriented optimization of magnetic high-entropy metallic glasses via a multi-stage design strategy

High-entropy metallic glasses (HE-MGs) have drawn much attention as promising multifunctional materials combining the excellent soft magnetic properties of traditional metallic glasses and impressive mechanical properties, thermal stability, corrosion resistance, etc., of solid solution high entropy alloys. Property optimization of HE-MGs is of great significance for promoting their engineering applications. However, various constituent elements and the high chemical complexity make the possible alloying composition space extremely massive, which is very challenging for the rational design of HE-MGs. In this work, we proposed a multi-stage optimization strategy based on machine learning (ML) to accelerate the rational design of magnetic HE-MGs with desired properties. The huge composition search space was significantly narrowed by the ML-based phase prediction model and constraints from user preferences. Utility functions based on the exploitation and exploration strategy were designed to find the global optimization solutions, i.e., alloying compositions. Experiments were conducted as concept validation, and new Fe-Co-Ni-Si-B HE-MGs with balanced saturation magnetic flux density and mixing entropy were developed.

Machine learning prediction of hardness in solid solution high entropy alloys

The mechanical properties of high entropy alloys (HEAs) are enhanced by solid solution strengthening (SSS) mechanism, which is of great importance for the design of HEAs. The compositional design space of HEAs is extensive, enabling the attainment of varying levels of hardness by combining different SSS elements and phases. Consequently, the selection of appropriate alloy compositions and their proportions to achieve the desired hardness has emerged as a present research challenge in the field. Against this backdrop, achieving accurate predictions of the hardness of HEAs has become a focal point of scholarly investigations. Machine learning (ML) adeptly captures the nonlinear relationships between alloy hardness and various characteristic features, and is widely regarded as a pivotal approach in the realm of hardness prediction for contemporary HEAs. A complete ML-based workflow for hardness prediction of HEAs is presented in this paper. Five ML models were trained using the selected feature descriptors. Eventually, a Random Forest model was obtained, which demonstrated a predictive correlation coefficient of 0.91 on the independent test dataset. Furthermore, the Shapley Additive exPlanations (SHAP) theory is employed to elucidate the influence of empirical parameters on the hardness of HEAs.

Ordering-Dependent Hydrogen Evolution and Oxygen Reduction Electrocatalysis of High-Entropy Intermetallic Pt4 FeCoCuNi.

Disordered solid-solution high-entropy alloys have attracted wide research attention as robust electrocatalysts. In comparison, ordered high-entropy intermetallics have been hardly explored and the effects of the degree of chemical ordering on catalytic activity remain unknown. In this study, a series of multicomponent intermetallic Pt4 FeCoCuNi nanoparticles with tunable ordering degrees is fabricated. The transformation mechanism of the multicomponent nanoparticles from disordered structure into ordered structure is revealed at the single-particle level, and it agrees with macroscopic analysis by selected-area electron diffraction and X-ray diffraction. The electrocatalytic performance of Pt4 FeCoCuNi nanoparticles correlates well with their crystal structure and electronic structure. It is found that increasing the degree of ordering promotes electrocatalytic performance. The highly ordered Pt4 FeCoCuNi achieves the highest mass activities toward both acidic oxygen reduction reaction (ORR) and alkaline hydrogen evolution reaction (HER) which are 18.9-fold and 5.6-fold higher than those of commercial Pt/C, respectively. The experiment also shows that this catalyst demonstrates better long-term stability than both partially ordered and disordered Pt4 FeCoCuNi as well as Pt/C when subject to both HER and ORR. This ordering-dependent structure-property relationship provides insight into the rational design of catalysts and stimulates the exploration of many other multicomponent intermetallic alloys.

PySSpredict: A python-based solid-solution strength prediction toolkit for complex concentrated alloys

The emergence of solid solution high entropy alloys (HEAs) and complex concentrated alloys (CCAs) offers opportunities to design novel alloys with tailored strength and ductility. The growing community of integrated-computational materials engineering (ICME) can benefit from implementing state-of-the-art solid-solution strengthening models to alloy design practices. This paper introduces pySSpredict, an open-source python-based toolkit that automates high-throughput calculations of solid-solution strengths of CCAs and thermodynamic properties. We present the functions of the pySSpredict code: (1) automating high-throughput calculations of strength for CCAs, (2) managing the data of thermodynamic calculations from databases or other software, and (3) visualizing and filtering the data to identify candidate alloys. The toolkit implements the latest theoretical edge dislocation model for face-centered cubic (FCC), and edge/screw dislocation models for body-centered cubic (BCC) alloys. The pySSpredict code is hosted on GitHub and deployed on nanoHUB for demonstrations.

Simple Approach to Model the Strength of Solid-Solution High Entropy Alloys in Co-Cr-Fe-Mn-Ni System

Simple Approach to Model the Strength of Solid-Solution High Entropy Alloys in Co-Cr-Fe-Mn-Ni System

The Evolution of Intermetallic Compounds in High-Entropy Alloys: From the Secondary Phase to the Main Phase

High-performance structural materials are critical to the development of transportation, energy, and aerospace. In recent years, newly developed high-entropy alloys with a single-phase solid-solution structure have attracted wide attention from researchers due to their excellent properties. However, this new material also has inevitable shortcomings, such as brittleness at ambient temperature and thermodynamic instability at high temperature. Efforts have been made to introduce a small number of intermetallic compounds into single-phase solid-solution high-entropy alloys as a secondary phase to their enhance properties. Various studies have suggested that the performance of high-entropy alloys can be improved by introducing more intermetallic compounds. At that point, researchers designed an intermetallic compound-strengthened high-entropy alloy, which introduced a massive intermetallic compound as a coherent strengthening phase to further strengthen the matrix of the high-entropy alloy. Inspired from this, Fantao obtained a new alloy—high-entropy intermetallics—by introducing different alloying elements to multi-principalize the material in a previous study. This new alloy treats the intermetallic compound as the main phase and has advantages of both structural and functional materials. It is expected to become a new generation of high-performance amphibious high-entropy materials across the field of structure and function. In this review, we first demonstrate the inevitability of intermetallic compounds in high-entropy alloys and explain the importance of intermetallic compounds in improving the properties of high-entropy alloys. Secondly, we introduce two new high-entropy alloys mainly from the aspects of composition design, structure, underlying mechanism, and performance. Lastly, the high-entropy materials containing intermetallic compound phases are summarized, which lays a theoretical foundation for the development of new advanced materials.

Microstructure formation mechanism and corrosion behavior of FeCrCuTiV two-phase high entropy alloy prepared by different processes

The microstructure and corrosion resistance of FeCrCuTiV high entropy alloy prepared by vacuum arc melting and laser melting deposition were studied. The microstructure and phase composition of FeCrCuTiV high entropy alloy were characterized by SEM, XRD, EBSD, etc. It is found that both of them are composed of the FCC Cu-rich phase, BCC Fe-rich phase and, BCC V-rich phase. Comparing with FeCrCuTiV high entropy alloy prepared by vacuum arc melting, the grain size of FeCrCuTiV high entropy alloy prepared by the laser melting deposition reduces from 29.48 µm to 1.85 µm. Meanwhile, the Cu-rich phase volume fraction decreases from 44.7 % to 31.5 %. Two reasons contribute to this phenomenon: On the one hand, the higher cooling rate of high entropy alloy makes the grain growth limited and the grain size reduced. On the other hand, the higher cooling rate reduces the volume fraction of intergranular segregation Cu-rich phase formed under non-uniform solidification due to the lack of long-range migration of Cu element, which makes the solidification state of FeCrCuTiV high entropy alloy prepared by laser melting deposition deviate from the ideal thermodynamic equilibrium. In addition, the corrosion resistance test of FeCrCuTiV high entropy alloy prepared by two processes indicates: FeCrCuTiV high entropy alloy prepared by laser melting deposition shows better corrosion resistance under the electrochemical corrosion of salt solution. This result is caused by the reduction of the Cu-rich phase, which weakens its corrosion effect as the anode of galvanic cell. Laser melting deposition technology provides new ideas and methods for controlling the phase structure of complex solid solution high entropy alloys to improve their physical properties.

Understanding chemical short-range ordering/demixing coupled with lattice distortion in solid solution high entropy alloys

Chemical short-range ordering (CSRO) or demixing in solid solution high entropy alloys (HEAs) is a fundamental issue yet to be fully understood. In this work, we first developed a generalized quasi-chemical solid solution model that enables quantitative computation of the local chemical ordering or demixing in solid solution HEAs. After that, we performed synchrotron diffraction experiments, extensive Reverse Monte Carlo (RMC) simulations, and first principles calculations on the CoCrFeNi model alloy to study the development of local chemical environments after long time thermal annealing. The outcome of the combined research demonstrates that the development of local chemical ordering or demixing in CoCrFeNi is not only affected by the heat of mixing between dislike atoms but also coupled with local lattice distortion.

A universal configurational entropy metric for high-entropy materials

In recent years, high-entropy materials have become increasingly complex, outgrowing simple solid-solution based calculations of configurational entropy, and with that, the previously established metrics for high-, medium-, and low-entropy. Those previous metrics are derived from metals-specific approximations of the entropy of fusion and the internal energy of a monatomic gas, and thus, do not apply to other materials. A new entropy metric is proposed that is universally applicable for crystalline materials, from simple solid-solution high-entropy alloys to high-entropy materials with complex crystal structures and multiple sublattices. Additionally, the methods for calculating configurational entropy in complex crystal structures are discussed.

Strength and Plasticity of Cast Solid-Soluble High-Entropy Alloys

The strength properties of cast solid-solution high-entropy alloys based on bcc and fcc lattices in the temperature range from -70 to 900°C have been investigated. Melting ingots weighing up to 200 g were carried out in a vacuum-arc furnace MIFI-9 by melting the mixture in an atmosphere of purified argon with a non-consumable tungsten electrode on a water-cooled copper hearth. The X-ray phase analysis and scanning electron microscopy in combination with INCA X-ray microanalyzer were used. Hardness (HIT) and reduced elastic modulus (Er) were determined using automatic microindentation on a Micron-gamma device using the Berkovich pyramid with a 2 N load in accordance with ISO14577-1:2015. High-temperature indentation of the Vickers pyramid was carried out at a 9.8 N load in vacuum. Thehigh-entropy composition of Nb1.5Cr1.25MoV0.75Ta0.5, which is a solid solution based on the bcc lattice, is characterized by high values of hardness up to 900°C. The yield stress and plasticity of solid-solution alloys are determined depending on the temperatures of the compression tests and the type of crystal lattice. For the high-entropy composition TiZrHfVNbTa based on the bcc lattice, the yield points are almost twice as high as those of the CrMnFeCoNi composition based on the fcc lattice. To create high-entropy alloys with improved characteristics of hot hardness, it is necessary to take into account such factors as the melting point, enthalpy of mixing, and dimensional mismatch in the lattice. It was found that alloys based on bcc lattice at positive values of the enthalpy of mixing are characterized by high characteristics of plasticity and deformability, as well as normalized hardness HIT/Er and compressive yield strength at room temperature. This can be attributed to the cluster structure and distortion of the crystal lattice.

New approach to model the yield strength of body-centered cubic solid solution refractory high-entropy alloys

A simple fitting approach is used for modeling the compressive yield strength of body centered cubic (bcc) solid solution high entropy alloys in Al-Hf-Nb-Mo-Ta-Ti-V-Zr system. It is proposed that the yield strength could be modeled by a polynomial where the experimental data can be used for finding the polynomial coefficients. The results show that the proposed polynomial could model the yield strength of solid solution alloys relatively well. The developed polynomial is used for predicting the strength of RHEAs in Hf-Mo-Nb-Ta-Ti-V-Zr system. It is observed that the yield strength of alloys within this system increases with the additions of Mo and Zr and decreases with the addition of Ti. Furthermore, the model predicts that the yield strength increases with increasing the value of parameters valence electron concentration (VEC) and atomic size difference (ASD). Although the developed polynomial does not consider the mechanisms involved in the strengthening of alloys, it can be considered as a straightforward method for assessing the strength of solid solution RHEAs.

Theories for predicting simple solid solution high-entropy alloys: Classification, accuracy, and important factors impacting accuracy

This viewpoint paper reviews theories for predicting simple solid solution high-entropy alloys (HEAs). The models are first classified into two categories based on their core strategies, and representative models under each category are reviewed. One hundred HEAs are then selected from the literature and used to benchmark the accuracies of the models. The accuracies are compared, and factors that impact accuracy are pointed out and discussed. Some factors have global effects, while others affect specific groups of alloys. Important future directions to better assess these theories and to improve their accuracies are also proposed.

A thermodynamic study on phase formation and thermal stability of AlSiTaTiZr high-entropy alloy thin films

Abstract The vast, unexplored chemical realm of multi-principal element alloys (MPEAs) has encouraged researchers to design and synthetize advanced materials based on refractory metals, metalloids, and deoxidizer elements, that are expected to follow single-phase, solid solution high-entropy alloys (HEAs)’ footsteps. In this paper, we have theoretically and experimentally studied the AlSiTaTiZr alloy in the thin film form, due to potential applications as a high-temperature oxidation-resistant coating. The MPEA film, synthetized by radio-frequency magnetron sputtering (RFMS), shows a considerable glass-forming ability. However, the metallic glass structure transits to several intermetallic compound phases by post-thermal annealing at ∼873 K, confirming that both thermodynamics and kinetics determine the formation of phases in sputtered MPEAs. Moreover, a strong tendency of Si and Al to form compounds with transition metals was revealed by X-ray diffraction (XRD), X-ray photoelectron spectroscopy (XPS), and thermodynamic investigations. Special focus is given to the role played by both entropy and enthalpy on the phase evolution, demonstrated by thermodynamic calculations using the regular solution model and the CALculation of PHAse Diagrams (CALPHAD) method. This methodology is presented as a useful approach for starting deeper thermodynamic investigations in the thin film literature of MPEAs and HEAs, currently focused on the ultimate properties of such alloys.

Consideration of kinetics on intermetallics formation in solid-solution high entropy alloys

There are overwhelming experimental observations indicating intermetallic (IM) phase formed in the intermediate temperatures in high entropy alloys. In this study, we proposed a model to show how kinetics could intervene the thermodynamic determination of phase formation. The model offers a good explanation for the prevalent formation of IM formation in the intermediate temperature range in high entropy alloys available in the literature. To further demonstrate the kinetic effect, we selected the equiatomic CrMnFeCoNi alloy (i.e., Cantor Alloy) as a surrogate material and employed differential scanning calorimetry (DSC) at different heating rates to investigate IM formation in this alloy. In this case, we found the presence of four IM compounds, BCC-Cr, L10-NiMn, B2-FeCo, and Cr-rich σ phases, in the temperature ranging from about 200 to 900°C, above which the alloy was a complete solid solution. From the DSC measurements, we were also able to build a quasi-equilibrium TTT diagram for Cantor Alloy. It was the first time that such a TTT diagram was established for a high entropy alloy.

Machine-learning informed prediction of high-entropy solid solution formation: Beyond the Hume-Rothery rules

The empirical rules for the prediction of solid solution formation proposed so far in the literature usually have very compromised predictability. Some rules with seemingly good predictability were, however, tested using small data sets. Based on an unprecedented large dataset containing 1252 multicomponent alloys, machine-learning methods showed that the formation of solid solutions can be very accurately predicted (93%). The machine-learning results help identify the most important features, such as molar volume, bulk modulus, and melting temperature. As such a new thermodynamics-based rule was developed to predict solid–solution alloys. The new rule is nonetheless slightly less accurate (73%) but has roots in the physical nature of the problem. The new rule is employed to predict solid solutions existing in the three blocks, each of which consists of 9 elements. The predictions encompass face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal closest packed (HCP) structures in a high throughput manner. The validity of the prediction is further confirmed by CALculations of PHAse Diagram (CALPHAD) calculations with high consistency (94%). Since the new thermodynamics-based rule employs only elemental properties, applicability in screening for solid solution high-entropy alloys is straightforward and efficient.

Passivation Phenomena in Single Phase High Entropy Alloys: The Evolution of Oxide Composition in Chloride

High entropy alloys (HEAs), sometimes known more broadly as compositionally complex alloys (CCAs) are emerging materials that have recently attracted great interest due to their potentially attractive properties. This class of alloys deviates from traditional alloys by containing five or more alloying elements which results in a high entropy of mixing promoting the formation of a solid solution alloy [1-3]. Initial electrochemical testing and characterization of HEAs revealed excellent passivity and resistance to local corrosion[3-6]. For some HEAs, a non-stoichiometric solid solution oxide forms during fast passivation. In this case, all alloying elements may be oxidized if the passivation potential is far from equilibrium. Also, some elements will dissolve preferentially at the oxide/electrolyte interface leading to an enrichment of other alloying elements in the passivating oxide[4]. The focus of this study is to understand the evolution of the electrochemically grown oxide formed on solid solution HEAs: Ni38Fe20Cr21Ru13Mo6W2 and Ni38Fe20Cr22Mn10Co10 within a slightly acidic Cl- environment. Results are compared to a conventional Ni-20Cr binary alloy. This study attempts to establish the fate of the alloying elements with respect to the outer and inner layers of the oxide, whether enriched or depleted at the oxide metal interface, or preferentially dissolved into solution. The E-log(i) behavior was first characterized on the HEA within a chloride environment enabling the identification of the passive current density, pitting potential, and repassivation potential. Passive films were electrochemically grown on alloy surfaces utilizing a potential hold for times ranging from 102 to 105 seconds within the passive region. In-situ analysis was conducted using AESEC, AC as well as DC electrochemistry methods to monitor passive current density, oxide thickness, and cation dissolution into solution. Ex-situ analysis was utilized to characterize the passive film and consisted of the following: 3D-atom probe tomography and X-ray photoelectron spectroscopy. Results are compared against theories such as non-equilibrium solute capture compared to thermodynamic expectations as well as enrichment models such as those proposed by Grimal and Marcus, Kirchheim, and others[7-9]. Acknowledgements This work was supported as part of the Center of Performance and Design of Nuclear Waste Forms and Containers, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award # DE-SC0016584 References [1] Y. Qiu, M.A. Gibson, H.L. Fraser, N. Birbilis, Corrosion characteristics of high entropy alloys, Materials Science and Technology, 31 (2015) 1235-1243. [2] Y. Qiu, S. Thomas, M.A. Gibson, H.L. Fraser, N. Birbilis, Corrosion of high entropy alloys, npj Materials Degradation, 1 (2017) 15. [3] M.H. Tsai, J.W. Yeh, High-Entropy Alloys: A Critical Review, Materials Research Letters, 2 (2014) 107-123. [4] K.F. Quiambao, S.J. McDonnell, D.K. Schreiber, A.Y. Gerard, K.M. Freedy, P. Lu, J.E. Saal, G.S. Frankel, J.R. Scully, Passivation of a corrosion resistant high entropy alloy in non-oxidizing sulfate solutions, Acta Materialia, 164 (2019) 362-376. [5] Tianshu Li, Orion J. Swanson, G.S. Frankel, Angela Y. Gerard, Pin Lu, James E. Saal, J.R. Scully, Localized corrosion behavior of a single-phase non-equimolar high entropy alloy, Electrochimica Acta, 306 (2019) 71-84. [6] P. Lu, J.E. Saal, G.B. Olson, T.S. Li, O.J. Swanson, G.S. Frankel, A.Y. Gerard, K.F. Quiambao, J.R. Scully, Computational materials design of a corrosion resistant high entropy alloy for harsh environments, Scripta Mater, 153 (2018) 19-22. [7] J.E. Castle, K. Asami, A more general method for ranking the enrichment of alloying elements in passivation films, Surface and Interface Analysis, 36 (2004) 220-224. [8] R. Kirchheim, B. Heine, H. Fischmeister, S. Hofmann, H. Knote, U. Stolz, The passivity of iron-chromium alloys, Corrosion Science, 29 (1989) 899-917. [9] P. Marcus, J.M. Grimal, The anodic dissolution and passivation of NiCrFe alloys studied by ESCA, Corrosion Science, 33 (1992) 805-814.

Phase prediction and microstructure of centrifugally cast non-equiatomic Co-Cr-Fe-Mn-Ni(Nb,C) high entropy alloys

The microstructural evolutions and hardness of two non-equiatomic multi elements CoCrFeMnNi and microalloy CoCrFeMnNiNbC high entropy alloys were studied after centrifugal casting. The chemical composition, microstructural changes and hardness were analysed using XRF, Leco-combustion technique, optical microscopy, SEM-WDX, XRD and Vickers hardness testing. Moreover, the thermal behaviour of both alloys was examined by differential thermal analysis technique at heating rate of 10 °C/min up to their melting points. The experimental results confirmed theoretical calculations based on Calphad method and thermodynamic rules, suggesting two solid solution high entropy alloys with FCC-crystal structure after centrifugal casting. Microscopy observations and hardness analysis revealed insignificant microstructural variations along the thickness of both alloys. From the microscopy results, it was also found that the microstructure consisted of a dendritic structure with the segregation of Fe and Mn in dendritic areas, while interdendritic areas were rich in Co, Cr, Ni. Furthermore, SEM-WDX results showed Nb rich nano-scale precipitates at interdendritic areas with spherical and oval shaped morphologies. Differential thermal analysis demonstrated no peak up to melting point, suggesting the high temperature stability of solid solution structure in both alloys.

High temperature deformation in fine grained high entropy alloys

There is considerable interest currently in developing and understanding microstructural evolution, stability and deformation in the new class of highly concentrated, multi-principal element solid solution high entropy alloys. This overview provides a brief description of potential high temperature deformation mechanisms in such fine grained alloys, discusses the significance of diffusion, and compares the available experimental results with appropriate theoretical models. It is shown that diffusion is not substantially slower in such alloys, and that superplastic deformation follows the standard model established for conventional polycrystals. The data also suggest that deformation at higher stresses occurs by dislocation glide creep, followed by a power-law breakdown regime. It is necessary to exercise caution in interpreting spherical nanoindentation creep data, although some data on a nanocrystalline HEA suggest that deformation at room temperature occurs by Coble diffusion creep. Based on the available data and understanding, a deformation mechanism map is developed highlighting the dominance over different regimes of grain size and stress of Coble diffusion creep, superplastic flow, dislocation creep and power-law breakdown.

Elemental segregation in solid-solution high-entropy alloys: Experiments and modeling

Recent experiments showed that elemental segregation occurs in some high entropy alloys (HEAs) although their X-ray diffraction patterns exhibit a nominal single-phase solid solution structure. Based on the 3-D atom probe tomography (APT) results obtained from the FeNiCoCrCu HEA, here we propose a simple thermodynamics model based on the notion that elemental segregation facilitates stabilizing the solid solution phase with a minimized Gibbs free energy. Our model enables distinguishing nominal single-phased solid-solution HEAs with and without elemental segregation. The predictions of our model are found in good agreement with the experimental data reported hitherto in the literature.