Preface

# Computational Thermodynamics: Recent developments and future potential and prospects

Article first published online: 14 JAN 2014

DOI: 10.1002/pssb.201470107

© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Additional Information

#### How to Cite

Hickel, T., Kattner, U. R. and Fries, S. G. (2014), Computational Thermodynamics: Recent developments and future potential and prospects. Phys. Status Solidi B, 251: 9–13. doi: 10.1002/pssb.201470107

#### Publication History

- Issue published online: 14 JAN 2014
- Article first published online: 14 JAN 2014

- Abstract
- Article
- References
- Cited By

### 1 Motivation

- Top of page
- 1 Motivation
- 2 CALPHAD method
- 3 SGTE unary database
- 4 First-principles approaches
- 5 The Ringberg unary workshop 2013
- 6 Synergies and outcomes
- 7 Summary
- Acknowledgement
- References

Modern technology is often closely connected to the computer-based development of novel, tailor-made alloys with complex composition and competing structural phases. State-of-the-art goals for material scientists are, for example, the adaptive change of crystal structures under mechanical load, the stabilization and size control of precipitates, and the development of alloys with special magnetic, thermal or electrical properties. In most of these cases, knowledge of temperature dependent phase stabilities and the evolution of thermodynamic quantities during phase transformations is required. Therefore, thermodynamic computer simulations, along with extensive materials databases, are becoming increasingly important for scientists and materials producers.

The accuracy of thermodynamic predictions is a key issue for computational development of new alloys. The reliability of the thermodynamic databases, therefore, needs to be of the highest possible level. Since all materials are built from the elements in the periodic table, it is evident that a detailed understanding of unaries is required for subsequent accurate modelling of multi-component systems. In order to work systematically towards new and improved materials, it is critical that different research and development activities use the same set of recommended data for these unaries, so that the results can be meaningfully compared. For the desired generalization to multi-component systems, it is additionally important that all unaries are described in a comparable way, i.e., using a similar set of functions.

### 2 CALPHAD method

- Top of page
- 1 Motivation
- 2 CALPHAD method
- 3 SGTE unary database
- 4 First-principles approaches
- 5 The Ringberg unary workshop 2013
- 6 Synergies and outcomes
- 7 Summary
- Acknowledgement
- References

An example of the success of such a strategy is documented by the SGTE (Scientific Group Thermodata Europe) unary database, which is part of the Calphad (CALculation of PHAse Diagrams) method [1] for anticipating the properties of stable and metastable phase equilibria in multi-component systems. The underlying concept is that of “lattice stabilities”, which has been introduced by Kaufman [2] and which was originally mostly applied to relative stabilities, enthalpy and entropy of transformation. In an empirical fashion mathematical models are employed to describe these properties mainly in terms of polynomials as function of temperature, pressure and for multi-component systems also as function of composition. The parameters of these models for systems with usually less than four components are determined using available experimental data. For the description of a multi-component system the mathematical models of the unaries, binaries and ternaries are combined into databases and the results from Gibbs energy minimisation calculations obtained from such a database are then validated by comparison with experimental data of systems with more than three components.

The Calphad method is currently the only method that can be used efficiently with the required accuracy for practical applications in multi-component and multi-phase systems. As a consequenc, the method has become an everyday tool in many industries, including some of the largest (and conceived to be) most conservative, such as steelmaking [3]. For example, the method is successfully used to simulate the thermodynamic stability of precipitate phases, which are of central importance for the mechanical properties of steels [4].

An additional advantage of the Calphad methodology is the fact that its application is not only limited to thermodynamics, but can be extended to the description of any phase based property [5]. For example, for diffusion simulations thermodynamics is coupled with the description of mobilities to obtain diffusion coefficients [6]. For the description of the composition dependence of mobilities the same formalisms are used as for thermodynamics. Furthermore, Calphad software has been coupled with simulation tools such as microstructure evolution software [7].

This wide applicability is one of the reasons why the Calphad method was identified in the report of the U.S. National Research Council on Integrated Computational Materials Engineering (ICME) [8] as the most important tool for ICME in order to enable the development of widespread capabilities in the future. The core role of Calphad in ICME makes it an important component of the Materials Genome Initiative [9] by providing data needed for the development of new, advanced materials.

### 3 SGTE unary database

- Top of page
- 1 Motivation
- 2 CALPHAD method
- 3 SGTE unary database
- 4 First-principles approaches
- 5 The Ringberg unary workshop 2013
- 6 Synergies and outcomes
- 7 Summary
- Acknowledgement
- References

The introduction of the SGTE unary database in 1991 by Dinsdale [10] became necessary since over the years a number of descriptions of the unaries were developed to include heat capacities and/or assign values to the metastable structures, resulting in a “Tower of Babel” of lattice stabilities. The formalisms used in the 1991 descriptions took several major steps in generalizing the lattice stability concept such as introducing heat capacities for the elements and a separate model for the contributution due to the ferromagnetic transition for some elements. However, it also included some “interim” solutions to avoid problems that may occur when the functions are used for the extrapolation of data for the solid phase significantly above the melting point, or for the liquid phase significantly below it.

Although thermodynamic modelling with this database in most cases works very well, it is not transparent with respect to the physical phenomena underlying the thermodynamic description and (as a consequence) also has some quantitative and qualitative deficiencies. For example, the formalism resulted in artificial kinks and spikes in heat capacities. Another shortcoming is the limitation to temperatures above 298.15 K because no need was perceived that would require the modelling of phase equilibria at lower temperatures. It was soon realised that these lattice stabilities needed improvements.

The relevance for the high accuracy of unaries can be illustrated by the above mentioned application for precipitation in steels. For example, the element Cr is an important alloying constituent in many steels that are currently considered for power plant applications at high temperatures, since it triggers the formation of carbides such as M**infi*63C, M**infi*7C**infi*3, M**infi*23C**infi*6. During processing and application of the steel products, heat treatments as well as service temperatures lead to carbide growth, dissolution, and coarsening [11]. Since the steels are targeted to last for more than 20 years under high-temperature conditions in service, even relatively small error bars in the thermodynamic and also kinetic data for Cr can lead to increasingly inaccurate long-term predictions of the microstructural evolution and therefore the performance of the material.

### 4 First-principles approaches

- Top of page
- 1 Motivation
- 2 CALPHAD method
- 3 SGTE unary database
- 4 First-principles approaches
- 5 The Ringberg unary workshop 2013
- 6 Synergies and outcomes
- 7 Summary
- Acknowledgement
- References

Since the original development of the SGTE unary database, there have been significant developments in understanding the nature of basic physical relations and the computation of thermodynamics properties. Based on these insights, one can nowadays go beyond some of the assumptions made in these kind of databases. One of the outstanding developments in the last decade is the possibility to let first-principles calculations replace laboratory experiments to a certain extent. As compared to other computational approaches, these quantum-mechanically based methods have the advantage that no empirical assumptions or fittings to experiment are required. Even though the most established first-principles method, density functional theory (DFT), makes assumptions for the exchange-correlation functional of the electrons, it is a decisive advantage that these approximations are independent of a particular material system at hand.

A second advantage of employing first-principles method for databases is the full control over the different physical contributions to the Gibbs energy, in contrast to experimental techniques. While DFT was traditionally only used to determine ground state material constants (e.g., lattice constants, enthalpies of formation) at *T* = 0 K, the prediction of finite temperature properties (e.g., heat capacities, Gibbs energies) of unaries have recently become significantly more reliable [12]. Not only can vibrational properties now be routinely computed in the quasiharmonic approximation, there has also been dramatic progress in considering the full lattice anharmonicity in first principles calculations. Similarly the consideration of magnetic entropy is no longer limited to the high-temperature paramagnetic limit, but recent breakthroughs now allow its accurate calculation in regions of long- and short-range order.

Finally, a third advantage of first-principles calculations is its ability to provide values for quantities that cannot be experimentally determined, such as the enthalpies of some metastable states. The nature of the Calphad method requires lattice stability values for a large number of metastable end-members, i.e. compounds with sublattices completely occupied by atoms or vacancies that normally only appear as defects. Traditionally, such values were derived from extrapolations of the homogeneity range of solution phases. Using DFT, the Gibbs energies of these phases are now directly accessible, although sophisticated techniques are often needed to consider dynamic instabilities.

### 5 The Ringberg unary workshop 2013

- Top of page
- 1 Motivation
- 2 CALPHAD method
- 3 SGTE unary database
- 4 First-principles approaches
- 5 The Ringberg unary workshop 2013
- 6 Synergies and outcomes
- 7 Summary
- Acknowledgement
- References

Due to these developments the time is now ripe to reconsider the computational thermodynamic modelling of materials, to make use of new insights into physical relations, to implement new methodological approaches and to adjust quantitative values in databases, whenever more accurate data are available. As stated above, the natural starting point for such considerations are the unaries, though any improvement should eventually aim at the application to multi-component materials. To bring these ideas forward and to develop suggestions for the community, a workshop was held at Ringberg castle in Germany, called the “Ringberg unary workshop 2013”.

For this workshop, the format of a previous workshop in 1995 that was organized by SGTE and the Max-Planck-Institut für Metallforschung was adapted. The intention at this time was to develop models for the unaries that reflect the basic physical properties such as lattice vibrations, electronic and magnetic properties etc. [13]. The goal was to find models that reproduce the experimental data, behave reasonably well in the temperature range between 0 K and 6000 K and allow rapid computer calculation of their values. The workshop resulted in the publication of six articles on (i) heat capacity models for crystalline phases [14], (ii) heat capacity of liquid and amorphous phases [15], (iii) estimation of enthalpies for stable and metastable states [16], (iv) lambda transitions [17], (v) estimation of enthalpies and entropies of transition [18], and (vi) periodic system effects [19]. However, the concepts discussed at this workshop still correspond to the semi-empircal Calphad concept of mathematically modelling experimental data. Furthermore, only a few lattice stabilities of the pure elements were subsequently developed using the recommendations from this workshop. The goal of the present workshop was to develop new formalisms to describe the Gibbs energy (or Helmholtz energy) of all relevant phases as a function of temperature and strain tensor (including volume) using fundamental physical understanding of the properties of the unary bulk and to test these descriptions for a selected set of elements. All relevant aspects for thermodynamically modelling unaries were covered by dividing the workshop into five working groups focusing on the topics (see also Fig. 1)

- crystalline phases,
- liquid phases,
- lambda transitions in materials,
- effects of pressure and stress, and
- point defect thermodynamics.

### 6 Synergies and outcomes

- Top of page
- 1 Motivation
- 2 CALPHAD method
- 3 SGTE unary database
- 4 First-principles approaches
- 5 The Ringberg unary workshop 2013
- 6 Synergies and outcomes
- 7 Summary
- Acknowledgement
- References

A broad range of perspectives was ensured by bringing together scientists with background and expertise in Calphad simulations, first-principles calculations, atomistic simulations, experiments, and industrial applications. Correspondingly, a major part of the discussions and one of the biggest achievements of the workshop was to establish joint concepts that reflect the completely different approaches in these communities.

A typical example of a need to bridge the gap between communities is the consideration of free energies of end-member compounds that involve vacancies and, therefore, automatically yield sublattices or even entire crystals that consist of 100% vacant sites. Such a concept is completely pointless in the defect-centric point of view of first-principles calculations, but needs to be considered in the compound-centric formalism of Calphad. The latter requires a more-or-less arbitrary function, whose role is to penalize the vacant phase with sufficiently high Gibbs energy to ensure that it cannot become stable. Such sources of misunderstanding have been resolved by the group focused on point defects. An advantage of the Calphad technique recognized by the first-principles community is the possibility to model defects that eventually become stable compounds, such as Al and Ni replacing each other in different sublattices of the B2 structure [20].

An example of how experiment and first-principle calculations can benefit from one another is the determination of high-temperature anharmonicities in lattice vibrations of crystalline phases. Their experimental determination is possible with neutron-scattering experiments, and the problem can now be addressed with ab initio molecular dynamics simulations (AIMD). New insights can be obtained from the comparison of both approaches. First-principles calculations are particularly suitable for determining the vibrational contribution to the heat capacity at very low temperatures, allowing a systematic test of empirical models, e.g., the Debye model, to efficiently describe the thermodynamics in this regime. This turns out to be a promising strategy for extending Calphad databases down to 0 K, as concluded by the group on crystalline phases [21].

In addition to extending the temperature range, another important goal of the workshop was to include the dependence on pressure and stress in thermodynamic models. Compositional trends of, for example, elastic constants are accessible experimentally and can be predicted even more extensively with first-principles methods. The description of the chemical trend by Redlich-Kister polynomials, as typically employed for Gibbs energies in Calphad turned out to be insufficient and suggestions for improvements have been made [22]

The comparison of atomistic simulations based on empirical potentials and Calphad assessments turned out to be fruitful in the case of liquid phases. While the heat capacity of a liquid is traditionally often simulated as being constant even in the undercooled regime, the atomistic simulations indicated that short-range order could yield substantial deviations from this assumption. This gave support to a two-state model, suggested by the group to describe liquids including the undercooled regime and solids in the overheated regime [23].

Synergy effects of a different kind have been revealed by the group focused on lambda transitions. The potential of first-principles based simulations to improve or replace empirical Calphad approaches to magnetism was also highlighted in this group. At the same time, similarities in thermodynamics of different lambda transitions (mainly magnetic as well as chemical order-disorder transitions) have been investigated and the coupling of these different degrees of freedom has been addressed. Capturing these kinds of coupling effects in thermodynamic models will remain a highly important, but challenging task for first-principles as well as Calphad for the next years [24].

### 7 Summary

- Top of page
- 1 Motivation
- 2 CALPHAD method
- 3 SGTE unary database
- 4 First-principles approaches
- 5 The Ringberg unary workshop 2013
- 6 Synergies and outcomes
- 7 Summary
- Acknowledgement
- References

All of the insights and findings, including a variety of interesting ideas, have been documented in the five publications [20-24] printed hereafter. Together these papers provide a valuable set of suggestions about, how computational thermodynamics should be done in the future. All have in common that they demonstrate the great potential of first-principles and atomistic simulations in obtaining qualitative insights as well as quantitative values for thermodynamic properties. Placing these suggestions into practice will result in more physical and first-principles based approaches, yielding in the long-term to a higher predictive power of Calphad approaches.

The success of the Ringberg Unary Workshop 2013 is based on the active involvement and excellent contributions of all its participants. In particular, the organizers would like to thank Bo Sundman, Göran Grimvall and Mike Finnis for their great support and valuable advices at all stages of the event. Further, we gratefully acknowledge financial support of the RUB International programme in the scope of the SAPIENS MUNDI project, which is the extention of the SAPIENS project from Interdisciplinary Centre for Advanced Materials Simulation (ICAMS), to international cooperation.

### Acknowledgement

- Top of page
- 1 Motivation
- 2 CALPHAD method
- 3 SGTE unary database
- 4 First-principles approaches
- 5 The Ringberg unary workshop 2013
- 6 Synergies and outcomes
- 7 Summary
- Acknowledgement
- References

Participants at the Ringberg Unary Workshop 2013, at which the articles for the topical section on â€œComputational Thermodynamicsâ€ have been initiated. From left to right: Jorge Munoz, Jörg Koßmann, Olle Hellmann, Mauro Palumbo, Abed Breidi, Pavel Korzhavyi, Bo Sundman, Suzana G. Fries, André Costa e Silva, Göran Grimvall, Marcello Baricco, Ursula R. Kattner, Xiao-Gang Lu, Sergey Dudarev, Jörg Neugebauer, André Schneider, Marcel H.F. Sluiter, Chandler Becker, Gautam Ghosh, Thomas Hammerschmidt, Sergei A. Decterov, Albert Glensk, Igor Abrikosov, John H. Perepezko, Tilmann Hickel, Bengt Hallstedt, Nathalie Dupin, Sergiy Divinski, Wei Xiong, Malin Selleby, Bonnie Lindahl, Lars HÃ¶glund, Qing Chen, John A. Ågren, Georg Paul, Dario Alfè, Gernot Pottlacher, Jutta Rogal, Ikuo Ohnuma, Sergei Schuwalow, Fritz Körmann, Brent Fultz, Mike Finnis, Patrice E.A. Turchi, Michel H.G. Jacobs, and Benjamin Burton.

### References

- Top of page
- 1 Motivation
- 2 CALPHAD method
- 3 SGTE unary database
- 4 First-principles approaches
- 5 The Ringberg unary workshop 2013
- 6 Synergies and outcomes
- 7 Summary
- Acknowledgement
- References

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