Thermodynamics and its prediction and CALPHAD modeling: Review,state of the art,and perspectives

Thermodynamics is a science concerning the state of a system, whether it is stable, metastable, or unstable, when interacting with its surroundings. The combined law of thermodynamics derived by Gibbs about 150 years ago laid the foundation of thermodynamics. In Gibbs combined law, the entropy production due to internal processes was not included, and the 2^nd law was thus practically removed from the Gibbs combined law, so it is only applicable to systems under equilibrium, thus commonly termed as equilibrium or Gibbs thermodynamics. Gibbs further derived the classical statistical thermodynamics in terms of the probability of configurations in a system in the later 1800's and early 1900's. With the quantum mechanics (QM) developed in 1920's, the QM-based statistical thermodynamics was established and connected to classical statistical thermodynamics at the classical limit as shown by Landau in the 1940's. In 1960's the development of density functional theory (DFT) by Kohn and co-workers enabled the QM prediction of properties of the ground state of a system. On the other hand, the entropy production due to internal processes in non-equilibrium systems was studied separately by Onsager in 1930's and Prigogine and co-workers in the 1950's. In 1960's to 1970's the digitization of thermodynamics was developed by Kaufman in the framework of the CALculation of PHAse Diagrams (CALPHAD) modeling of individual phases with internal degrees of freedom. CALPHAD modeling of thermodynamics and atomic transport properties has enabled computational design of complex materials in the last 50 years. Our recently termed zentropy theory integrates DFT and statistical mechanics through the replacement of the internal energy of each individual configuration by its DFT-predicted free energy. The zentropy theory is capable of accurately predicting the free energy of individual phases, transition temperatures and properties of magnetic and ferroelectric materials with free energies of individual configurations solely from DFT-based calculations and without fitting parameters, and is being tested for other phenomena including superconductivity, quantum criticality, and black holes. Those predictions include the singularity at critical points with divergence of physical properties, negative thermal expansion, and the strongly correlated physics. Those individual configurations may thus be considered as the genomic building blocks of individual phases in the spirit of the materials genome®. This has the potential to shift the paradigm of CALPHAD modeling from being heavily dependent on experimental inputs to becoming fully predictive with inputs solely from DFT-based calculations and machine learning models built on those calculations and existing experimental data through newly developed and future open-source tools. Furthermore, through the combined law of thermodynamics including the internal entropy production, it is shown that the kinetic coefficient matrix of independent internal processes is diagonal with respect to the conjugate potentials in the combined law, and the cross phenomena that the phenomenological Onsager flux and reciprocal relationships are due to the dependence of the conjugate potential of a molar quantity on nonconjugate molar quantities and other potentials, which can be predicted by the zentropy theory and CALPHAD modeling.     

A generalized computational interface for combined thermodynamic and kinetic modeling

The computational interface developed by Huang et al. (2008) [Z. Huang, P.P. Conway, R.C. Thomson, A.T. Dinsdale, J.A.J. Robinson, CALPHAD 32 (2008) 129-134] has been extended and generalized in different programming and modeling environments, which includes C, Fortran, Python and Java besides MATLAB and COMSOL Multiphysics. The generalized computational interface can be used to integrate various software packages for materials and process modeling into one programming platform, within which complicated modeling processes beyond the capability of these software packages can be achieved, such as combined thermodynamic and kinetic modeling, microstructural morphology evolution modeling for systems with arbitrary geometries and microstructure-based property prediction. The interface is applicable to all software packages that provide a dynamic-link library or DLL and the incorporation of Thermo-Calc and MTDATA are introduced in this paper. Several application examples utilizing the thermodynamic data of a Cu-Sn binary alloy system and an Fe-Cr-C ternary system are presented. In addition, modeling of solidification, using both a phase field and a phase field crystal models with the finite element method, are conducted within the integrated platform.     

The light alloy Calphad databases PanAl and PanMg

Thermodynamic databases for multi-component aluminum alloys, PanAl, and magnesium alloys, PanMg, are reviewed and applications are highlighted. Precipitation simulations by combining thermodynamic and mobility databases for Al and Mg alloys with the PanPrecipitation module of Pandat are also demonstrated. These simulations can serve as virtual experiments to understand the effects of alloy composition and heat treatment condition on the target properties therefore provide guidance for the design of real experiments, save time and reduce cost. For PanAl the focus in this work is on 7xxx alloy and high throughput calculation (HTC) to understand/predict the effects of major and minor alloying elements on the selected properties of an alloy. For PanMg applications are exemplified in a wider spectrum of Mg alloy design and related processing parameters and the usefulness of the CALPHAD modeling tool in Mg technology is demonstrated.     

PANDAT software with PanEngine, PanOptimizer and PanPrecipitation for multi-component phase diagram calculation and materials property simulation

The newly enhanced PANDAT, integrating PanEngine, PanOptimizer and PanPrecipitation, bridges thermodynamic calculation, property optimization, and kinetic simulation of multi-component systems based on CALPHAD (CALculation of PHAse Diagram) approach. This software package, in combination with thermodynamic/kinetic/thermo-physical databases, provides an integrated workspace for phase diagram calculation and materials property simulation of multi-component systems. The simulation results, which include thermodynamic, kinetic, thermo-physical properties, and microstructure related information, are critically needed in materials design, in the selection of parameters for fabrication steps such as heat treatment, prediction of performance, and failure analysis. In addition to the functionalities provided by PANDAT as a stand-alone program, its calculation/optimization engines (PanEngine, PanOptimizer and PanPrecipitation) are built as shared libraries and enable their integration with broader applications in the field of Materials Science and Engineering.     

An overview on phase equilibria and thermodynamic modeling in multicomponent Al alloys: Focusing on the Al-Cu-Fe-Mg-Mn-Ni-Si-Zn system

Knowledge of thermodynamics and phase diagram is a prerequisite for understanding many scientific and technological disciplines. To establish a reliable thermodynamic database, an integrated approach of key experiments and thermodynamic modeling, supplemented with first-principles calculations, can be utilized. In this paper, first investigations of phase diagram and thermodynamics of technologically important Al alloys (focusing on the Al-Cu-Fe-Mg-Mn-Ni-Si-Zn system, which covers the major elements in most commercial Al alloys) is reviewed with an emphasis on the need of the integrated approach. Second, the major experimental methods (X-ray diffraction, metallography, electron probe microanalysis, differential thermal analysis, diffusion couple method, and calorimetry), which are widely employed to provide phase diagram and thermodynamic data, are briefly described. Third, the basics of the first-principles calculations and CALPHAD are presented focusing on the integration of these two computational approaches. Case study for the representative Al-Fe-Ni ternary system is then demonstrated, followed by a thermodynamic modeling of the quinary Al-Fe-Mg-Mn-Si system and a brief summary to our recent activities on investigations of phase equilibria in multicomponent Al alloys.     

Towards high-throughput microstructure simulation in compositionally complex alloys via machine learning

The coupling of computational thermodynamics and kinetics has been the central research theme in Integrated Computational Material Engineering (ICME). Two major bottlenecks in implementing this coupling and performing efficient ICME-guided high-throughput multi-component industrial alloys discovery or process parameters optimization, are slow responses in kinetic calculations to a given set of compositions and processing conditions and the quality of a large amount of calculated thermodynamic data. Here, we employ machine learning techniques to eliminate them, including (1) intelligent corrupt data detection and re-interpolation (i.e. data purge/cleaning) to a big tabulated thermodynamic dataset based on an unsupervised learning algorithm and (2) parameterization via artificial neural networks of the purged big thermodynamic dataset into a non-linear equation consisting of base functions and parameterization coefficients. The two techniques enable the efficient linkage of high-quality data with a previously developed microstructure model. This proposed approach not only improves the model performance by eliminating the interference of the corrupt data and stability due to the boundedness and continuity of the obtained non-linear equation but also dramatically reduces the running time and demand for computer physical memory simultaneously. The high computational robustness, efficiency, and accuracy, which are prerequisites for high-throughput computing, are verified by a series of case studies on multi-component aluminum, steel, and high-entropy alloys. The proposed data purge and parameterization methods are expected to apply to various microstructure simulation approaches or to bridging the multi-scale simulation where handling a large amount of input data is required. It is concluded that machine learning is a valuable tool in fueling the development of ICME and high throughput materials simulations.     

An evaluation of the Mn–Ga system: Phase diagram,crystal structure,magnetism, and thermodynamic properties

The Ni–Mn-Ga alloy system is attractive due to its functional properties with potentials in various applications. However, the fundamental alloy thermodynamics of the binary Mn–Ga system is still lack of investigation. Therefore, a comprehensive evaluation of the Mn–Ga is performed in this work. Different versions of Mn–Ga phase diagrams available in the literature are reviewed. A new version of the Mn–Ga phase diagram is recommended along with possible invariant reactions. The crystal structure, magnetic transition, and thermochemical properties of intermetallic compounds are reviewed by considering available experimental and modeling such as ab initio calculations. In fact, more experimental information on the Mn-rich side is required in order to perform CALPHAD thermodynamic modeling for a reliable database. Further experiments are recommended to study the high-temperature phase equilibria between liquid, (γMn), (δMn) and Mn₂Ga(h), phase reactions between Mn₈Ga₅ and Mn₇Ga₆, and invariant reactions involving the MnGa phase. Nevertheless, the summarized information on phase equilibria, phase diagram, crystallography, magnetic transition temperature, magnetic moment, heat capacity, and enthalpy of formation can support the future thermodynamic investigation of the Mn–Ga system, which is critical for the materials design and discovery of Ni–Mn-Ga alloys.     

Genomic materials design: CALculation of PHAse Dynamics

The CALPHAD system of fundamental phase-level databases, now known as the Materials Genome, has enabled a mature technology of computational materials design and qualification that has already met the acceleration goals of the national Materials Genome Initiative. As first commercialized by QuesTek Innovations, the methodology combines efficient genomic-level parametric design of new material composition and process specifications with multidisciplinary simulation-based forecasting of manufacturing variation, integrating efficient uncertainty management. Recent projects demonstrated under the multi-institutional CHiMaD Design Center notably include novel alloys designed specifically for additive manufacturing. With the proven success of the CALPHAD-based Materials Genome technology, current university research emphasizes new methodologies for affordable accelerated expansion of more accurate CALPHAD databases. Rapid adoption of these new capabilities by US apex corporations has compressed the materials design and development cycle to under 2 years, enabling a new “materials concurrency” integrated into a new level of concurrent engineering supporting an unprecedented level of manufacturing innovation.     

Interface between quantum-mechanical-based approaches,experiments, and CALPHAD methodology

The increased application of quantum-mechanical-based methodologies to the study of alloy stability has required a re-assessment of the field. The focus is mainly on inorganic materials in the solid state. In a first part, after a brief overview of the so-called ab initio methods with their approximations, constraints, and limitations, recommendations are made for a good usage of first-principles codes with a set of qualifiers. Examples are given to illustrate the power and the limitations of ab initio codes. However, despite the “success” of these methodologies, thermodynamics of complex multi-component alloys, as used in engineering applications, requires a more versatile approach presently afforded within CALPHAD. Hence, in a second part, the links that presently exist between ab initio methodologies, experiments, and the CALPHAD approach are examined with illustrations. Finally, the issues of dynamical instability and of the role of lattice vibrations that still constitute the subject of ample discussions within the CALPHAD community are revisited in the light of our current knowledge with a set of recommendations.     

ICME guided design of heat-treatable Zn-modified Al–Mg alloys

Currently, there is a tendency towards breaking through conventional 5000 series aluminum alloys framework and designing heat-treatable Al–Mg based alloys with the improved mechanical properties. Here, based on a small amount of experimental work and our previously developed Integrated Computational Materials Engineering (ICME) framework, the authors systematically and efficiently optimize the Zn content and two-step aging treatment process in the heat-treatable Al–Mg–Zn alloys. It is found that the addition of 3.0 wt.% Zn can lower the nucleation activation energy, promote the precipitation of the T phase, and thus enhance and accelerate the age-hardening response of Al–Mg–Zn alloys. Different from the single-step aging process, the pre-aging treatment leads to a large number of fine and dispersive T precipitates due to the precipitation of numerous, stable and dispersive precursors. Further, we optimized two-stage aging treatment for the Al-5.1Mg-3.0Zn alloy which enables the peak-aged yield strength to increase by 21.1% as compared to that in the single-step aging process. This research presents an efficient strategy to design new-generation aluminum alloys via combining the key experimental data and an ICME framework.     

A computational experiment on deducing phase diagrams from spatial thermodynamic data using machine learning techniques

Derivation and discovery of physical dynamics inherent in big data is one of the most major purposes of machine learning (ML) in the field of modern natural science. In the materials science, phase diagrams are often called as “road maps” to perfectly understand the conditions for phase formation and/or transformation in any material system caused by the associated thermodynamics. In this paper, we report a numerical experiment investigating whether the underlying thermodynamics can be derived from the big data constructed of local spatial composition and phase distribution data along with the help of ML. The artificial data analysed have been created assuming a steel composition based on the calculation phase diagram (CALPHAD) thermodynamics combined with the order-statistics-based sampling model. The hypothetical procedures of data acquisition assumed in this numerical experiment are as follows; (i) obtaining local analysis data on the composition and phase distribution in the same observation area using instruments such as electron probe micro analyser (EPMA) and electron backscattering diffraction (EBSD), and (ii) training the classification model based on a ML algorithm with compositional data as input and the phase data as output. The accuracies of the reconstructed phase diagrams have been estimated for three ML algorithms, i.e. support vector machine (SVM), random forest, and multilayer perceptron (MLP). The phase diagrams predicted using SVM and MLP are found to be adequately consistent with those of the CALPHAD method. We have also investigated the regression performance of the continuous data involved in the CALPHAD thermodynamics, such as the phase fractions of body-centred cubic, face-centred cubic, and cementite phases. Compared with the ML algorithms, the CALPHAD method is found to show superior predictive performance since it is based on the sophisticated physical model.     

On the Theories and Numerics of Continuum Models for Adaptation Processes in Biological Tissues

Computational continuum mechanics have been used for a long time to deal with the mechanics of materials. During the last decades researches have been using many of the theoretical models and numerical approaches of classical materials to deal with biological tissue which, in many senses, are a much more sophisticated material. We aim to review the last achievements of continuum models and numerical approaches on adaptation processes in biological tissues. In this review, we are looking, in particular, at growth in terms of changes of density and/or volume as, e.g., in collagen remodeling, wound healing, arterial thickening, etc. Furthermore, we point out some of the most relevant limitations of the current state-of-the-art in terms of these well established computational continuum models. In connection with these limitations, we will finish by discussing the trend lines of future work in the field of modeling biological adaptation, focusing on the computational approaches and mechanics that could overcome the current drawbacks. We would also like to attract the attention of all those researchers in classical materials (metal, alloys, composites, etc), to point out how similar the continuum and computational models between our fields are. We hope we can motivate them for getting their expertize in this challenging field of research.     

Induced rational parametrisations of special curves

Rational parametrisation of curves and surfaces is a defacto industry standard for computer graphics and shape representation. While it is well known that rational parametrisations of the conic sections exist, and their explicit forms are used in geometric modeling (Hoschek and Lasser, 1993), other classical curves for which rational parametrisations may be determined have received less attention in the literature. This paper presents an elegant method for the explicit computation of rational parametrisations for many 'special' or famous curves. The technique 'induces' a parametrisation of a 'higher' or 'target' curve from a 'primitive' or 'source' curve for which rational parametrisations are known - e.g., the straight line and the conic sections. A number of examples and modeling applications are given.     

An assessment of Fe-Nb-B melts using the two-state liquid model

Amorphous alloys of the (Fe-Co-Ni)-(Cr-Mo-Nb)-B system are promising materials to supply the demands for higher wear resistance components in the petrochemical industry. Since the development of the CALPHAD method, the development of new metallic alloys has been accompanied by thermodynamic modelling and calculations. The prediction of the formation of amorphous alloys requires special care with the modeling of the liquid and or an amorphous phase. As a initial stage in the more complex system, the basic Fe-Nb-B ternary system was selected. In order to predict the stability and tendencies of transformations of these amorphous alloys, the Fe-Nb-B system was reassessed using Ågren's two-state model to describe the liquid phase. The results of the present assessment show very good agreement with the recently reported stable phase diagram. Furthermore, the use of the two-state model for the liquid is more accurate and physically consistent when evaluating transformations from supercooled liquid, as shown it the present work.     

Thermodynamic modelling of the ternary Bi-Ga-Te system for potential application in thermoelectric materials

Thermoelectric materials have drawn widespread attention because they can enable the direct conversion between electric and thermal energy. Over the years, different materials such as skutterudites, clathrates, intermetallic alloys, eutectic alloys, chalcogenides have been explored for Thermoelectric (TE) applications. Amongst the eutectic alloys, the Bi-Ga-Te system exhibits promising potential as a TE material. Accordingly, in this study, we performed the thermodynamic optimization and critical evaluation of binary Bi–Ga, Bi–Te, Ga–Te, and ternary Bi-Ga-Te systems using the CALPHAD method. It is observed that the Ga–Te system shows asymmetric liquid solution properties with strong negative enthalpy of mixing, whereas the Bi–Te liquid exhibits the symmetric regular solution behavior. Moreover, the Bi–Ga liquid solution has a positive enthalpy of mixing. Therefore, Modified Quasichemical Model (MQM) using pair approximation was utilized to describe the diversified thermodynamic properties of liquid solution in sub-binaries by taking into account the Short-Range Ordering (SRO). By merging the binary optimization results with a proper interpolation method, the liquid solution properties and phase diagram information in the Bi-Ga-Te ternary system were also reproduced successfully without any adjustable ternary parameter. Several ternary eutectic compositions were suggested for designing TE alloy with enhanced properties using the developed database.     

Property evaluation in The Expert System for Thermodynamics (“TEST”) web application

Ever since its release, TEST has found use as thermodynamic courseware in many universities around the world. TEST offers web-based analysis tools–java applets called daemons–for property evaluation of working substances, energy, entropy, and exergy analysis of generic open and closed systems, IC engines, gas and vapor power cycles, refrigeration, HVAC, combustion, chemical equilibrium, and gas dynamics. Other modules of TEST include animations, problems, examples, and system navigations that are closely integrated with the daemon module to create a comprehensive analysis tool for engineering thermodynamics. In this paper the methodology of thermodynamic state evaluation by TEST is discussed with several examples.     

Phase equilibria and microstructure development in Mg-rich Mg-Gd-Sr alloys: Experiments and CALPHAD assessment

Mg-Sr alloys are promising to fabricate orthopedic implants. The alloying of rare earth elements such as Gd may improve the comprehensive mechanical properties of Mg-Sr alloys. The information on the phase diagram and the microstructure development are required to design chemical composition and microstructure of Gd alloyed Mg-Sr alloys. The phase equilibria and the microstructure development in Mg-rich Mg-Gd-Sr alloys (Gd, Sr < 30 at. %) are experimentally investigated via phase identification, chemical analysis, and microstructure observation with respect to the annealed ternary alloys. The onset temperatures of liquid formation are measured by differential scanning calorimetry. A thermodynamic database of the Mg-rich Mg–Gd–Sr ternary system is developed for the first time via CALPHAD (CALculation of PHAse Diagram) approach assisted by First-Principles calculations. The thermodynamic calculations with the developed database enable a well reproduction of the experimental findings and the physical-metallurgical understanding of the microstructure formation in solidification and annealing.     

A CALPHAD-type Young's modulus database of Ti-rich Ti–Nb–Zr–Mo system

Accurate Young's modulus is the necessity for the design of biomedical Ti alloys. A combinatorial method of the diffusion couple, nanoindentation, electron probe microanalysis (EPMA), and CALculation of PHAse Diagrams (CALPHAD) techniques has been utilized to construct the Young's modulus database of Ti alloys with various compositions in the present work. Two groups of body-centered cubic (bcc) Ti–Nb–Zr–Mo quaternary diffusion couples annealed at 1273 K for 25 h were experimentally prepared. Subsequently, the composition-dependent mechanical properties in the wide compositional range of Ti-based alloys were obtained by using EPMA and nanoindentation probes. Finally, on the basis of the measured Young's moduli in the present and previous work and the modeling parameters of Young's modulus of Ti–Nb–Zr system, the Young's modulus database of bcc Ti–Nb–Zr–Mo system was established through the CALPHAD approach. The CALPHAD-type database of bcc Ti–Nb–Zr–Mo system can provide the accurate Young's moduli of Ti alloys with wide compositions.