Teilprojekt A3: Thermodynamik

Hallstedt Ph.D. (Institut für Werkstoffanwendungen im Maschinenbau, RWTH Aachen University)

A3 Thermodynamik
A3 Thermodynamik
Ein wesentlicher Grundstein für modellierungsbasierte Ansätze ist eine möglichst genaue und zuverlässige thermodynamische Beschreibung des Legierungs­systems. Eine solche Beschreibung ermöglicht die Berechnung von Phasengleichgewichten und verschiedenen thermodynamischen Daten und wird als Grundlage für die Simulation von Erstarrung, Ausscheidungsprozessen und anderen Phasen­umwandlungen benötigt. Desweiteren werden die thermodynamischen Daten benutzt, um Stapelfehler­energien, Anfangstemperaturen für martensitische Umwandlungen (Ms), Néel- und Curie-Temperaturen zu berechnen.

 

 

 

 

 

 

 

 

 

 

A3 Thermodynamik
A3 Thermodynamik
In der ersten Förderperiode haben wir das Basissystem Fe–Mn–C thermodynamisch in großer Detailtiefe studiert. Dabei haben ab initio-Berechnungen aus anderen Teilprojekten wichtige Erkenntnisse geliefert. In der zweiten Förderperiode haben wir das Element Al hinzugenommen und eine Datenbank für das System Fe–Mn–Al–C aufgebaut. Wir haben ebenfalls Ms für die Bildung von ε-Martensit und Stapelfehlerenergien berechnet und neue Erkenntnisse in Bezug auf Verformungsmechanismen gewonnen. Arbeiten mit einer (sehr komplexen) Modellierung der κ-Phase als Ordnungszustand des Austenits wurden angefangen.

 

Das Kernziel in der dritten Förderperiode ist es, die thermodynamische Grundlage zu liefern, die für die Behandlung (real und simulativ) von mehrphasigen hoch-Mn- und mittel-Mn-haltigen Stählen benötigt wird. Insbesondere wollen wir dadurch die Entwicklung von neuen austenitischen Stählen mit hohen κ-Phasen-Anteilen und ferritisch-austenitischen Mittel-Mn-Stählen wie sie in Cloud II (Engineering of interfaces and precipitates) vorgesehen sind, unterstützen. Weiter wollen wir die Arbeiten über den Zusammenhang zwischen Stapelfehlerenergien (Austenitstabilität) und Verformungsmechanismen im Rahmen von Cloud I (Strain hardening engineering) fortführen.

 

Die thermodynamische Modellierung der κ-Phase und die damit zusammenhängende Modellierung der Systeme Fe–Al–C, Mn–Al–C und Fe–Mn–Al–C stellen im Arbeitsprogramm einen Themenschwerpunkt dar. Aus diesen Arbeiten (zusammen mit Arbeiten außerhalb des SFB) wird eine thermodynamische Datenbank für die Elemente Fe, Mn, Al, Si, V, Nb, Ti, C, N und eventuell auch Cr und Ni aufgebaut. Diese Datenbank wird dazu benutzt, Phasengleichgewichte zwischen Austenit und κ-Phase und zwischen Austenit und Ferrit zu berechnen, sowie als Grundlage für die Berechnung von Stapelfehlerenergien und Martensit-Start-Temperaturen zu dienen. Die Datenbank wird auch als Grundlage für Simulationen von Phasenumwandlungen benutzt. Für die Austenitbildung in Mittel-Mn-Stählen werden Diffusionssimulationen durchgeführt.

Results

The ternary Fe-Mn-C system forms the basis for high-Mn TRIP and TWIP steels. Therefore an accurate thermodynamic description for this system is essential as a foundation for further modeling-based approaches. With such a description the calculation of phase equilibria and different thermodynamic properties is possible. It is also necessary as a basis for the simulation of solidification, precipitation processes and other phase transformations.
The major aim of this part-project is to develop a precise and reliable thermodynamic description for the Fe-Mn-C system. The accuracy and reliability is achieved by including ab initio data to an increased extent. The developed thermodynamic description will be employed to calculate phase equilibria, T0, stacking fault energies, and temperatures for martensitic transformations as well as to partially simulate carbide precipitation.

The Calphad method

Fig. 1: Schematic representation of the Calphad method.
Fig. 1: Schematic representation of the Calphad method.
An equilibrium phase diagram is usually a diagram with axes for temperature and composition of a chemical system. It shows the regions where substances or solutions (i.e. phases) are stable and regions where two or more of them coexist. Phase diagrams are a very powerful tool for predicting the state of a system under different conditions and were initially a graphical method to rationalise experimental information on states of equilibrium. The CALPHAD (Computer Coupling of Phase Diagrams and Thermochemistry) approach is based on the fact that a phase diagram is a manifestation of the equilibrium thermodynamic properties of the system, which are the sum of the properties of the individual phases. It is thus possible to calculate a phase diagram by first assessing the thermodynamic properties of all the phases in a system. In the CALPHAD method, schematically presented in Fig. 1, all available thermodynamic and phase equilibrium data for the system are evaluated simultaneously to obtain one set of model equations for the Gibbs energies of all phases as functions of temperature and composition. Some of these expressions contain adjustable coefficients that are often referred to as model parameters. The optimal values for the model parameters providing the best match between available experimental values and the calculated quantities are usually obtained by a weighted non-linear least square minimisation procedure. This methodology is known as thermodynamic optimisation. From these equations, the thermodynamic properties and the phase diagrams can be calculated. Traditionally Calphad evaluations are based on experimental phase diagram and thermodynamic data. However, ab initio data are increasingly used as complementary data, in particular for properties which are difficult or impossible to measure experimentally.

 

 

Fig. 2: Heat capacity of cementite (Fe3C).
Fig. 2: Heat capacity of cementite (Fe3C).
An equilibrium phase diagram is usually a diagram with axes for temperature and composition of a chemical system. It shows the regions where substances or solutions (i.e. phases) are stable and regions where two or more of them coexist. Phase diagrams are a very powerful tool for predicting the state of a system under different conditions and were initially a graphical method to rationalise experimental information on states of equilibrium. The CALPHAD (Computer Coupling of Phase Diagrams and Thermochemistry) approach is based on the fact that a phase diagram is a manifestation of the equilibrium thermodynamic properties of the system, which are the sum of the properties of the individual phases. It is thus possible to calculate a phase diagram by first assessing the thermodynamic properties of all the phases in a system. In the CALPHAD method, schematically presented in Fig. 1, all available thermodynamic and phase equilibrium data for the system are evaluated simultaneously to obtain one set of model equations for the Gibbs energies of all phases as functions of temperature and composition. Some of these expressions contain adjustable coefficients that are often referred to as model parameters. The optimal values for the model parameters providing the best match between available experimental values and the calculated quantities are usually obtained by a weighted non-linear least square minimisation procedure. This methodology is known as thermodynamic optimisation. From these equations, the thermodynamic properties and the phase diagrams can be calculated. Traditionally Calphad evaluations are based on experimental phase diagram and thermodynamic data. However, ab initio data are increasingly used as complementary data, in particular for properties which are difficult or impossible to measure experimentally.
Fig. 1: Schematic representation of the Calphad method.
Fig. 3: The calculated Mn-C phase diagram.
An equilibrium phase diagram is usually a diagram with axes for temperature and composition of a chemical system. It shows the regions where substances or solutions (i.e. phases) are stable and regions where two or more of them coexist. Phase diagrams are a very powerful tool for predicting the state of a system under different conditions and were initially a graphical method to rationalise experimental information on states of equilibrium. The CALPHAD (Computer Coupling of Phase Diagrams and Thermochemistry) approach is based on the fact that a phase diagram is a manifestation of the equilibrium thermodynamic properties of the system, which are the sum of the properties of the individual phases. It is thus possible to calculate a phase diagram by first assessing the thermodynamic properties of all the phases in a system. In the CALPHAD method, schematically presented in Fig. 1, all available thermodynamic and phase equilibrium data for the system are evaluated simultaneously to obtain one set of model equations for the Gibbs energies of all phases as functions of temperature and composition. Some of these expressions contain adjustable coefficients that are often referred to as model parameters. The optimal values for the model parameters providing the best match between available experimental values and the calculated quantities are usually obtained by a weighted non-linear least square minimisation procedure. This methodology is known as thermodynamic optimisation. From these equations, the thermodynamic properties and the phase diagrams can be calculated. Traditionally Calphad evaluations are based on experimental phase diagram and thermodynamic data. However, ab initio data are increasingly used as complementary data, in particular for properties which are difficult or impossible to measure experimentally.

Evaluation of the Fe-Mn-C system

Both Mn and C stabilise the fcc (austenite or γ) phase and the phase diagram is characterised by an extended fcc region. The carbon solubility in fcc is high and almost independent of the Mn content, but decreases with decreasing temperature. The Fe-Mn-C system contains several carbides (M3C, M23C6, M5C2 and M7C3) where Fe and Mn can extensively replace each other. For practical purposes (in high-Mn steels) probably only M3C (cementite) is of interest. M3C can be expected to form during extended holding at relatively low temperature (e.g. 600-700 °C) for steels with relatively high carbon content.
The present evaluation of the Fe-Mn-C system is based on previous evaluations of the Fe-Mn (W. Huang, Calphad, 13 (1989) 243-52), Fe-C (P. Gustafson, Scand. J. Metall., 14 (1985) 259-67) and Mn-C (own evaluation, see above) systems. In the Fe-Mn system, the description of the hcp phase was slightly modified to better reproduce data on the fcc/hcp martensitic transformation and in the Fe-C the description of Fe3C was modified as described above. Based on current ab initio data from part project A1, the stability of the metastable carbides Fe5C2 and Fe7C3 were changed considerably compared to previous evaluations. Calculated isothermal sections are shown in Figs. 4 and 5 and the calculated liquidus surface is shown in Fig. 6. The corresponding invariant equilibria are listed in Table 1. In order to calculate stacking fault energies (SFE) it is important that the fcc/hcp martensitic transformation is well reproduced. This is shown as function of carbon content in Fig. 7, where the so called T0 line is calculated. The T0 line shows where the Gibbs energies of the fcc and the hcp phases are equal.

Fig. 4: Calculated isothermal section at 600 °C in the Fe-Mn-C system.
Fig. 4: Calculated isothermal section at 600 °C in the Fe-Mn-C system.

 

Fig. 5: Calculated isothermal section at 1100 °C in the Fe-Mn-C system.
Fig. 5: Calculated isothermal section at 1100 °C in the Fe-Mn-C system.

 

Fig. 6: Calculated liquidus projection of the Fe-Mn-C system.
Fig. 6: Calculated liquidus projection of the Fe-Mn-C system.

 

Fig. 7: Fcc/hcp martensitic transformation temperatures (Ms and As) and T0 line as function of carbon content at 17 mass-% Mn.
Fig. 7: Fcc/hcp martensitic transformation temperatures (Ms and As) and T0 line as function of carbon content at 17 mass-% Mn.

 

Tabelle 1

 

Calculation of stacking fault energies (SFE)

Since a stacking fault can be viewed as a two atomic layers thin slice of hcp within the fcc crystal, it seems possible that the SFE could be related to the difference in Gibbs energy between the hcp and fcc phases. This idea has been applied successfully by e.g. Allain et al. (S. Allain, J.-P. Chateau, O. Bouaziz, S. Migot, N. Guelton, Mater. Sci. Eng., A 387-389 (2004) 158-62) and Cotes et al. (S.M. Cotes, A. Fernández Guillermet, M. Sade, Metall. Mater. Trans. A, 35A (2004) 83-91), using the following simple equation:

A3 Formel1.1
 

where ρ is the number of atoms per m2 in one atomic layer, ΔGγ→ε is the difference in Gibbs energy between the hcp and fcc phase, and ζ is the interface energy (between fcc and hcp). The number of atoms, ρ, is given by:

A3 Formel2

 

where a is the lattice parameter and N is Avogadros number. Cotes et al. also added and analysed a strain energy contribution to the SFE, but, in the case of Fe-Mn alloys, found it to be small (about 2 mJ/m2). The interface energy, ζ, is typically selected in the range 5 to 15 mJ/m2 (for Fe-Mn based alloys).
The major contribution to the composition and temperature dependence of the SFE comes from ΔGγ→ε. The Gibbs energies of the fcc and hcp phases can be taken from Calphad evaluations of the appropriate systems and ΔGγ→ε calculated by taking the difference. However, it should be noted that the hcp phase is not a stable phase in the Fe-Mn-C (or other steel systems) and its properties cannot be directly measured. But the hcp phase can form martensitically (as ε-martensite) from the fcc phase in the range 15 to 30 mass-% Mn and data from this transformation are used to model the Gibbs energy of the hcp phase (Fig. 8). Thus, the ΔGγ→ε from Calphad evaluations is closely related to the formation of ε-martensite, which is the same kind of martensite which is formed by transformation induced plasticity (TRIP). Below 15% Mn, α-martensite forms instead of ε-martensite and above 30% Mn a magnetic transition in the fcc phase increases the stability of the fcc phase so that ε-martensite does not form. The Gibbs energies of the fcc and hcp phases at 25 °C are shown in Fig. 9 and the resulting SFE is shown in Fig. 10. Note that these calculations were made without carbon. Carbon strongly increases the SFE. The increase is linear with about 5 mJ/m2 per 0.1 mass-% C.

Fig. 8: Experimental data on the fcc/hcp martensitic transformation in the Fe-Mn system (from Cotes et al.).
Fig. 8: Experimental data on the fcc/hcp martensitic transformation in the Fe-Mn system (from Cotes et al.).

 

Fig. 9: Calculated Gibbs energies for the fcc and hcp phases in the Fe-Mn system at 25 °C.
Fig. 9: Calculated Gibbs energies for the fcc and hcp phases in the Fe-Mn system at 25 °C.

 

Fig. 10: Calculated SFE for Fe-Mn (without carbon) at 25 °C.
Fig. 10: Calculated SFE for Fe-Mn (without carbon) at 25 °C.

 

Publications

1. B. Hallstedt, D. Djurovic, J. von Appen, R. Dronskowski, A. Dick, F. Körmann, T. Hickel, J. Neugebauer, Calphad, 34, 129-33(2010).
2. D. Djurovic, B. Hallstedt, J. von Appen, R. Dronskowski, Calphad, accepted for publication, May 2010.