MASARYKOVA UNIVERZITA přírodovědecká fakulta Ústav chemie Habilitační práce Brno 2015 Jana Pavlů MASARYKOVA UNIVERZITA přírodovědecká fakulta Ústav chemie Ab initio and semiempirical modelling of intermetallic phases Habilitační práce Jana Pavlu Brno 2015 Bibliografický záznam Autor: Název práce: Akademický rok: Počet stran: Klíčová slova: Mgr. Jana Pavlů, Ph.D. Přírodovědecká fakulta, Masarykova univerzita Ústav chemie Ab initio and semiempirical modelling of intermetallic phases 2015/2016 43+95 ab-initio výpočty; semiempirické modelování metodou CALPHAD; stabilita fází; termodynamické vlastnosti; magnetické vlastnosti; intermetalické fáze; sigma fáze; Lavesovy fáze Bibliographic entry Author: Title of Thesis: Academic Year: Number of Pages: Keywords: Mgr. Jana Pavlu, Ph.D. Faculty of Science, Masaryk University Department of Chemistry Ab initio and semiempirical modelling of intermetallic phases 2015/2016 43+95 ab-initio calculations; semiempirical modelling by CALPHAD method; phase stability; thermodynamic properties; magnetic properties; intermetallic phases; sigma phase; Laves phases Abstract This work summarises the results obtained during the study of physical and chemical properties (crystallographic structure, magnetism, lattice stability, Gibbs energy, enthalpy) of solid state phases. In particular, the intermetallic structures with practical impact were investigated, among them sigma phases found in superaustenitic steels and influencing mechanical properties of alloys; Laves phases in systems considered as possible hyper-conducting or hydrogen-storage materials, etc. For such phases, the detailed description providing information about their behaviour in complex systems under various conditions such as composition and temperature is desired. This description is not only a collection of data but it also includes thermodynamic polynomials applicable for predictions of phase equilibria in high-order systems. In this survey, different methods had to be applied depending on the scale of studied issue. To study the relations between electronic structure and crystallographic, magnetic and energetic properties, the DFT calculations based on Kohn-Sham equations were used. These calculations are working on nano-scale level. Nevertheless, sometimes such a detailed attitude was not necessary. In these cases, the macro-scale modelling is more effective as it provides the complex description of complicated systems. One of the macro-scale methods is the semiempirical CALPHAD (Computer Coupling of Phase Diagrams and Thermochemistry) approach assessing the data from both experiments and theoretical sources and providing the thermodynamic properties and phase diagrams. The results obtained were published in several scientific papers which are listed in the List of author's publications where the specification of author's contribution is provided. In principle, the ab initio modelling should start from characterisation and study of properties of known structures and than it can proceed to hypothetical structures or experimentally inaccessible phases. This two steps usually go together especially in case where the ab initio results are intended to be employed in the subsequent thermodynamic modelling and phase diagram calculations. The publications [VI, XIII] deal with experimentally well defined intermetallic phases such as PdBi, PdBi2, FePd, FePd3, FePt, FePt3 and provide comparison of the experimentally found energies of formation with theoretical ones. This type of publications, where one type of variable is studied by various approaches, provide valuable data. In case of disagreement between results from different methods, the weak aspects of approaches can be revealed and improved. In addition to the energies of formation, the information about the crystallographic arrangement and possible magnetic ordering [XIII] are provided. On the same principle, the studies of Fe-based C14 Laves phases [XVI] and sigma phases in Cr-Fe and Cr-Co [X] and Ni-Fe [VII] systems are based. In addition, they also include characterisation of structural configurations that are experimentally inaccessible. In case of sigma phase study [X], the detailed overview of magnetic behaviour and its influence on lattice stability is provided over the whole composition region. Not only pure ab initio calculations can be challenging but also their combinations with thermodynamic CALPHAD modelling can bring interesting results. The interplay of these methods was for the first time applied on the sigma phase in Cr-Fe [I] and Co-Cr [II] binary system. The implementation of the ab initio results into the thermodynamic description of the sigma phase using the two-sublattice model enabled us to perform the phase diagram calculations using the parameters having the physical meaning. This approach can be of course used for many intermetallic phases [IV,IX,XI] and high-order systems [III]. This procedure is now widely used in the CALPHAD community. As the thermodynamic modelling is very efficient tool for phase equilibria predictions, its combination with ab initio calculations pushes its applicability to less and less experimentally explored regions. One of them is the modelling at low temperatures, where phase transformations also occur and interesting physical (superconductivity) phenomena take place. The extension of the ordinary used SGTE (Scientific Group Thermodata Europe) unary data to 0 K temperatures was developed [XIV] and its application to the intermetallic phases was demonstrated [XVII]. Acknowledgement My work presented in this thesis could not be realised without the help of prof. Jan Vřešťál (MU Brno, diploma supervisor, Ph.D. supervisor) and prof. Mojmír Sob (MU Brno, group leader), who are acknowledged for long term cooperation, many encouraging scientific discussions and for a critical reading of this thesis. I would like also to thank my family, namely my parents Ludmila and Ladislav Houserovi, for outstanding support. Declaration I hereby confirm that I have written the habilitation thesis independently, that I have not used other sources than the ones mentioned and that I have not submitted the habilitation thesis elsewhere. Brno 10m September 2015 Jana Pavlů Contents Introduction ............................................................... 1 1 Structures studied....................................................... 4 2 Ab initio calculations.................................................... 6 2.1 Theory and methodology .................................. 6 2.1.1 DFT (Density Functional Theory) ........................ 7 2.1.2 Exchange-correlation potentials.......................... 8 2.1.3 Calculations for periodic solids.......................... 9 2.1.4 Methodology of performed calculations .................... 10 2.2 Results and discussion.................................... 14 2.2.1 Crystal structure .................................... 14 2.2.2 Energies of formation................................. 17 2.2.3 Magnetic properties.................................. 18 2.2.4 Mechanical properties ................................ 20 3 CALPHAD modelling................................................... 21 3.1 Theory and methodology .................................. 21 3.1.1 Sigma phase modelling................................ 23 3.1.2 Modelling at low temperatures........................... 24 3.2 Results and discussion.................................... 26 3.2.1 Binary systems..................................... 26 3.2.2 Ternary systems..................................... 28 3.2.3 Thermodynamic modelling at low temperatures............... 28 Conclusions and main results ............................................... 31 4.1 Ab initio calculations..................................... 31 4.2 CALPHAD modelling .................................... 32 List of author's publications and the specification of author's contribution .... 35 Bibliography............................................................... 39 Appendix I................................................................. 44 Appendix II................................................................ 52 Appendix III............................................................... 62 Appendix IV............................................................... 66 Appendix VI............................................................... 73 Appendix VII.............................................................. 77 Appendix IX............................................................... 83 Appendix X................................................................ 89 Appendix XI............................................................... 98 Appendix XIII ............................................................. 105 Appendix XIV ............................................................. 112 Appendix XVI ............................................................. 124 Appendix XVII............................................................. 131 Introduction There is no doubt that materials and their development are essential for the evolution of the society we live in. During the last century, the achievements in electronic science, computer technology etc. have opened up enormous possibilities for progress in materials science leading to research and development of more sophisticated materials. Most, if not all, of properties of solids can be described by theoretical approaches working on different scale. It is only the question of priority which method is used whether nano-scale ab initio (first-principles) electronic structure calculations or CALPHAD (Computer Coupling of Phase Diagrams and Thermochemistry) modelling, which works on macroscopic level, or even some other approaches. Anyway, both of the above mentioned methods can significantly contribute to understanding and prediction of the physical and chemical properties of materials. At present, the possibilities are almost unlimited and the problem of choice of suitable method to study particular problem makes high demands on author's experience. In principle, all properties of material are directly related to the behaviour of electrons that constitute inter-atomic bonds in solids. The bridge to the macroscopic properties such as e.g. total energy, crystallographic arrangement, magnetic ordering, mechanical properties etc. is constituted by the rules of quantum mechanics implemented to ab initio (first-principles) approaches [1-4]. The theory of electronic structure is not only helpful in understanding and interpreting experiments, but it also becomes a predictive tool in the physics and chemistry of condensed matter. The advantage of methods based on electronic structure calculations consists in their physical transparency and independence on experimental data and fitting parameters as input values. On the other hand, some price for this independence has to be paid in the form of high computational demands on software, hardware and time. For application of ab initio approach in the research, fast development of computing facilities, numeric methods and their increasing accessibility (via networks and workstations) in recent decades has been crucial. As an illustration of the detailed ab initio study of physical and chemical properties of intermetallic phases, the paper [X] can be mentioned here. In this work, the comprehensive study of magnetism of sigma phases in Fe-Cr and Co-Cr binary system and its influence on phase stability is presented. In spite of the fruitfulness of the ab initio methods, some disadvantages related to them should be mentioned here. The first one is that these approaches require many approximations (e.g. adiabatic (Born-Oppenheimer) approximation, approximations of functional of exchange-correlation energy etc.) which result in lowering the accuracy of calculation. The other disadvantage is that the ab initio calculations are performed for 0 K temperature which disables their direct usage for phase equilibria studies at higher -1- Introduction_2 temperature. In this case, the material science turns towards a macro-scale method omitting the role of electrons - the CALPHAD approach [5]. This widely used semiempirical method is based on the laws of thermodynamics and uses the Gibbs energies of phases as building stones for description of system. From this reason, the knowledge of the Gibbs energy dependence on composition and temperature for all structures occurring in the system (stable and even metastable) is crucial. Unfortunately, this information is for metastable phases experimentally inaccessible. Nevertheless, the lack of proper data can be bridged by ab initio calculations of lattice stability [6]. In the field of phase equilibria calculations, the main advantage of the ab initio methods is the ability to deal with systems far from equilibrium or with metastable or hypothetical states providing their total energies of formation with respect to the reference states at T = 0 K [IX, XVII, 7, 8], consequently applied in the CALPHAD modelling. Such approach can put the thermodynamic data describing the metastable states on the sound physical basis. One of the first applications of this combined approach was presented in our work [I] in 2002 and we have continued with further studies of complex phases such as sigma and Laves phase etc. [Ill, IV, IX, XVII]. This approach is now being used by many research groups [7-10] as the combination of both above mentioned different-scale methods becomes very useful tool for description of multi-component systems with complex intermetallic phases where experiments seldom provide satisfactory set of thermodynamic or phase data. In some cases, not only the question of thermodynamic stability should be treated in phase modelling but also the mechanical stability becomes crucial. This topic can be studied via the analysis of the elastic constants [XVII, 11] or, more completely, via the phonon spectra [XVII, 12]. From the point of basic science, the CALPHAD method is currently being developed by extending the theoretical background into the fields which have not been covered yet. In [XIV], a method for the extension of SGTE (Scientific Group Thermodata Europe) Gibbs energy expressions for pure elements [13] to zero Kelvin temperature was presented. It is based on the Einstein formula for the temperature dependence of heat capacity extended to provide the temperature dependence of the Gibbs energies below the limiting temperature of validity of SGTE unary data. The application of this method to low temperature modelling of intermetallic phases was presented in [XVII]. At present, the knowledge of phase diagrams (or their sections) and relevant thermodynamic properties is crucial in a design of modern materials. However, to obtain the full understanding of material behaviour and to make reliable predictions, it is crucial to combine the theoretical methods on one side and experiments on the other because the quality of theoretical predictions of course increases when more information about the system is available. The metallic systems studied in this thesis were chosen not only because of scientific reasons looking for understanding of physical and chemical background of material behaviour but also according to their applicability in material engineering. The stress was laid on the systems found in superaustenitic steels containing the sigma phase (Cr-Fe [I,X], Co-Cr [II,X], Cr-Fe-Ni [III], Co-Mo [IV], Fe-Mo [IV], Cr-Fe-W [VII]) and binary systems with Laves phases (Cr-Zr [IX], Cr-Hf [XI], Cr-Ti [XI], V-Zr [XVII]) which have become candidates for some functional as well as structural applications, e.g. hydrogen Introduction 3 storage materials [14], superconductors [15], and materials with a high strength up to high temperatures [16]. The list of systems and phases studied is not complete here. Further examples can be found in List of author's publications. Chapter 1 Structures studied As this thesis concerns the intermetallic phases, the characterisation of those which form the nub of this work is provided here. Sigma phase was first observed by Bain [17] in Cr-Fe system in 1923 and, at present, about 50 binary transition-metal systems exhibiting this phase are known [18], e.g. Fe-Mo, Co-Mo or Fe-V. The sigma phase has the space group No. 136 (PA2/mnm) and its repeat cell contains thirty atoms accommodated in five crystalographically inequivalent sublattices (2a, 4f, 8i, 8V and 8j) [19-21], see Figure 1.1. If these sublattices are occupied by studied constituents in various succession, 32 different configurations are formed. The sigma phase is very crucial in material science and technology because its properties are very disadvantageous. It is brittle and therefore it can cause a strong degradation of material (crack nucleation sites). It develops in heat affected zones of welded super-austenitic stainless steels [22] and it was concluded that it is formed after longer ageing times in the temperature range of 500-1100 °C. It is also known that high concentrations of Cr and Mo promote precipitation of this phase. From the thermodynamic point of view, the sigma phase is very stable. Figure 1.1: The structure of the sigma phase [XIX]. The brightness of atoms in sublattices is increasing in the order of positions 2a, 4f, 8i, 8V and 8j. Laves phases can be found in metallic systems in three polytypes: cubic C15 (prototype MgCu2, space group 227, Fd3m), hexagonal C14 (prototype MgZn2, space group 194, P63/mmc) and hexagonal C36 (prototype MgNi2, space group 194 P63/mmc). All three -4- Chapter 1. Structures studied 5 structures are shown in Figure 1.2. In this thesis, our attention was mainly drawn to the C14 and C15 Laves phases. Laves phases have a significant influence on mechanical properties of modern high-Cr steels. They precipitate mostly on ferrite subgrain boundaries and on prior austenite grain boundaries. It has been found that the presence of silicon in the steels accelerates the precipitation of Laves phase and that the phase itself than contains significant amount of Si [23]. Figure 1.2: The structures of Laves phases [XIX]. Bright and dark spheres correspond to the A and B atoms in the general formula A2B. Chapter 2 Ab initio calculations This part of thesis addresses problems concerning the properties of particular phase such as equilibrium structure parameters (lattice parameters, angles, atomic positions), magnetic ordering, total energies and mechanical properties. In the case of experimentally accessible phases, the above mentioned values can be compared with experimental data and can form a strong background for methods using these data as input values. The mutual interactions between phases can be also investigated for example by means of lattice stabilities, grain boundary energies, etc., however, these values are usually obtained from the post-processing of ab initio results than from multi-phase ab initio calculations. Accordingly, a range of DFT (Density Functional Theory) quantum chemistry approaches is employed from the LMTO-ASA (Linear Muffin-Tin Orbitals method within the Atomic Sphere Approximation) method, FLAPW (Full-potential - Linear Augmented Plane Wave) method to pseudopotential approach within the LDA (Local Density Approximation) and GGA (Generalised Gradient Approximation) employed for the exchange-correlation term. 2.1 Theory and methodology The properties of material depend on the electronic structure which is described by the wave functions calculated from the Schrodinger equation where {Ra} are positions of atomic nuclei, *F is wave function, E energy and Hamiltonian H is defined as ^,{Ra} = -EV' + E^{Ra}(^)+ £ T-^-T- (2-2) Here, r, and ry are positions of electrons. The first term in Equation (2.2) stands for the kinetic energy of electrons, the second one is the potential acting on the electron i coming from surrounding nuclei and the last term describes the interaction between two electrons. The Ve {Ra} (rj) potential is defined as ^{Ra}(r;-) = -2£-^, (2.3) a lr — Ka| -6- Chapter 2. Ab initio calculations 7 where Za are the proton numbers of studied elements. To evaluate the total energy, the Rydberg atomic units with h = 1, 2me = 1 and Ke2 = 2 are often used, where h = h/lit, h stands for the Planck constant, me and e are the electron mass and charge, respectively, and K is Coulomb law constant (K = 1/(4tt£o), where Eq is permittivity of vacuum). As there are approximately 1023 interacting particles in one mole of a real solid, it is impossible to solve the Schrodinger equation for such a huge number of objects. From this reason various approximations has to be employed resulting in lowering the accuracy of results. 2.1.1 DFT (Density Functional Theory) Ab initio results commented in this thesis are based on the DFT [1-4], which simplifies the many-particle problem. It is based on two theorems published in 1964 by Hohenberg and Kohn [24] showing the elegant reduction of many-electron problem. The first theorem (existence theorem) introduces the density of electrons p (r) whose a non-degenerate ground state defines fully the Hamiltonian of the whole system (by means of determination of external potential). From this Hamiltonian, it is possible to determine all the basic properties of the studied material (e.g. lattice constants, total energy, etc.). Thus, all the characteristics of the system in the ground state may be treated as functionals of one function - electron density p (r). The existence theorem induces a huge decrease in number of degrees of freedom as the electron density is a function of sole three variables. According to the second theorem (variational principle), the total energy of a system of electrons E [p] reaches its minimum for electron density of the ground state. On the base of DFT, the electron density is being changed until the minimum of total energy is obtained, regardless to the number of particles in the system. Based on these theorems, it can be stated that the equilibrium ground-state electron density corresponds to the minimum of total energy and vice versa. Furthermore, the formulation of the above mentioned theorems yielded the introduction of the Kohn-Sham equations [25] Hs\in (r) = emu (r) = [-V2 + Veff (r)] Wi (r) , (2.4) which are again a single-particle equations. These equations describe the behaviour of an electron moving in the field evoked by the other electrons and nuclei. Hs is one-electron Kohn-Sham Hamiltonian. i/a, are one-electron wave functions that are solutions of Kohn-Sham equation, are eigenenergies of one-electron states and Veff is the effective potential which is, in general, nonlocal. Veff defined as Veff (r) = Vext (r) + VH(r) + Vxc(r) (2.5) can be constructed on the basis of electron density and includes important effects of exchange and correlation. The particular potentials - external Vext (r), Hartree Vh (r) and exchange-correlation Vxc (r) potential are characterised as follows: • External potential describes the effect of nuclei and external fields on electron. Chapter 2. Ab initio calculations 8 • Hartree potential corresponds to the classical repulsion of electron with other electrons. • Exchange-correlation potential V M-^eM. (27) Vxc[V)~ 5p(r) ( ° is functional of so-called exchange-correlation energy Exc and its exact form is not known, because its determination is equivalent to the solution of many-electron problem. It contains the non-classical part of the electron-electron interaction and the difference between the kinetic energy of interacting and non-interacting electron system [1]. For this term various approximations has to be used. The one-particle density p (r) used in previous equations is defined as the sum over the occupied one-electron energy states of Af-electron system p(r) = f>;(r)|2 . (2.8) i=l The total energy of the system may be calculated according to following formula E = E £i + J J P{rr)_Pr/([/)d3rd3r/ - J Vxc (r) p (r) d3r + Exc [p] . (2.9) 2.1.2 Exchange-correlation potentials The most often used methods for determination of exchange-correlation energy are LDA (Local Density Approximation), LSDA (Local Spin Density Approximation) and GGA (Generalised Gradient Approximation). The LDA defines the exchange-correlation energy Exc [p] as Exc[p] = Jp(v)exc[p(v)]d3v7 (2.10) where exc [p (r)] is the exchange-correlation energy per particle in a homogeneous system of density p. Similarly, the exchange-correlation energy in LSDA is Exc[pi,pt] = |p(r)e,c[p;(r),pt(r)]d3r. (2.11) The most frequently employed approximations are due to Hedin and Lundqvist [26], von Barth and Hedin [27], Janak [28], Ceperley and Alder [29] as parametrised by Perdew and Zunger [30], Vosko, Wilk and Nusair [31] and Perdew and Wang [32]. As here defined £xc corresponds to a homogeneous electron gas, the application of L(S)DA is limited to the systems with slowly varying electron density. In the case of strong gradients, e.g. due to Chapter 2. Ab initio calculations 9 the directional bonding, these approximations are less successful. For example, they fail in reproduction of the ground state of iron. From this reason, it was necessary to include the magnitude of gradient of the electron density into the exchange-correlation energy evaluation. This was done by the GGA where the term exc [p (r)] in Equation (2.10) is substituted by the term exc [p (r), Vp (r)]. 2.1.3 Calculations for periodic solids Equations (2.4), (2.5) and (2.8) are solved self-consistently until the electron density p (r) and potential Veff (r) correspond to each other within certain limits. This process is significantly simplified by the idea of periodicity of crystal structure, which is characterised by the translation vector T in crystal lattice. In the periodic systems, the effective potential has to obey the periodicity condition Veff (r + T) = Veff (r). This condition results in the Bloch theorem, according to which the solution of Equation (2.4) can be expressed as yk(r)=eikW(r) • (2.12) Here, k is the reciprocal lattice vector and is a periodic function with the same period as the crystal lattice. Based on this assumptions, it is sufficient to find the wave function i/Ak (r) in the primitive cell and the region of k-vectors is constrained to a primitive cell in the reciprocal space, i.e. to the first Brillouin zone [33,34]. If we suppose that the electron interaction between the atoms in the solid is quite weak (i.e. the electrons are mainly localised in the vicinity of atoms) then the wave functions may be written as a linear combination of orbitals localised at the positions of nuclei. When solving the Kohn-Sham equation (2.4), the one-electron wave functions are expanded into a series Ynk (r) = £ c*>kXrk (r) • (2.13) i The index n is a counting index (band index), Cijf± are expansion coefficients and Xik (r) are the basis functions (orbitals) that satisfy the Bloch condition - Equation (2.12). For expansion coefficients Cijf± we obtain £[//ý--£nkOý-Jcí>k = 0, (2.14) j where Hij = (Xik\Hs\Xjk) = í XftMxIkMd'r (2.15) •J í^, are matrix elements of Hamiltonian and Oij = (XiklXjk) = JnXMXji(r)d3r (2-16) are overlap integrals with the volume of the unit cell CI. The energies are determined by the secular equation Chapter 2. Ab initio calculations 10 det [Hij - enkOij] = 0 . (2.17) The Bloch theorem enables us to calculate the electronic wave functions and corresponding electron energies by effective block-diagonalisation of the Hamiltonian matrix, with each block (corresponding to a particular k) having a manageable size. The methods used for the electronic structure calculations differ in the type of basis functions %i which has to be chosen carefully with respect to the problem solved. We can use the plane waves or their modifications (Plane Wave - PW, Orthogonalised Plane Wave - OPW, Augmented Plane Wave - APW); Linear Combination of Atomic (LCAO), Gaussian (LCGO) and Augmented Slater-Type (LASTO) Orbitals; Augmented Spherical Waves (ASW), Muffin-Tin Orbitals (MTO), Linear Muffin-Tin Orbitals (LMTO), etc. The Green function of Kohn-Sham equation is used in Korringa-Kohn-Rostoker (KKR) method, alternatively called Green Function (GF) method. The detailed information about basis functions can be found in many publications [35-37] . The pseudopotential approach [38] is also widely used. This method modifies the potential close to the nucleus (i.e. in the region of electron shell with the lowest energy) to narrow down the basis set. 2.1.4 Methodology of performed calculations The primary reason for the execution of ab initio calculations was to provide the input data for the CALPHAD modelling - so called lattice stabilities. The lattice stabilities are the energies of formation of particular phase which has to be calculated with respect to exactly defined reference states. In the articles listed in List of author's publications, the reference states are structures of the pure constituents that are stable at a temperature T of 298 K and pressure p of 1 bar, such as FM (ferromagnetic) hep (hexagonal close packed) Co, FM bec (body centered cubic) Fe, AFM (antiferromagnetic) bec Cr, NM (non-magnetic) bec Mo, etc. These phases are denoted SER (Standard Element Reference) states in this thesis. From this reason, not only intermetallic phases but also SER structures had to be included in the ab initio calculations. At the beginning of any study, the suitable method has to be chosen. In the case of study of phases with the same symmetry, the LMTO-ASA method [4,39,40] implemented in the code by Krier et al. [41] can be used. This code was employed for calculations of various sigma phase configurations in: Cr-Fe and Cr-Co binary system [X], Fe-Ni binary system [III]; and for structure relaxations of sigma phases of pure elements, e.g. Cr, Fe, Co and Mo [I, II, IV, X]. Here, the exchange-correlation energy was evaluated within the GGA [42]. The s-p-d basis with the / states incorporated by the down-folding procedure and with the combined-correction term included [39,41] was used. This is apparently the best performance the LMTO-ASA method may provide. For all computational methods the optimum technical parameters had to be found to get the required precision of total energy of phase. In case of LMTO-ASA, these parameters were: the number of k-points in the whole Brillouin zone and the sphere radii, which define the size of non-overlapping Muffin-Tin spheres with the spherically symmetric potential. Outside the spheres, the potential was constant. The partitioning of the unit cell into atomic spheres is shown in Figure 2.1. Chapter 2. Ab initio calculations 11 Figure 2.1: Partitioning of the unit cell into atomic spheres (I) and interstitial region (II). Nevertheless, the LMTO-ASA method does not provide reliable structural energy differences for structures of different symmetry, although the total energy differences calculated by this method for the same crystallographic structures are considered to be quite reliable [43^45]. To calculate the energy differences between the structures of different symmetry, the FLAPW and pseudopotential methods were employed. The FLAPW method [38] implemented in the WIEN97 / WIEN2k code [46] / [47] was used in all works related to the sigma phases [I-IV,VII,X] employing the GGA [42] for the exchange-correlation term. This method is considered to be one of the most reliable methods. At the beginning of FLAPW calculations, the optimisations of technical parameters (the RMT (Radius Muffin-Tin) parameter and number of k-points) were performed. The optimum values are provided in relevant publications. The last ab initio approach employed in this thesis is the pseudopotential method [38] incorporated in the VASP (Vienna Ab initio Simulation Package) [48,49] and combined with the PAW-PBE (Projector Augmented Wave-Perdew-Burke-Ernzerhof) pseudopotential [50-52] (i.e. the GGA was employed for the exchange-correlation energy). This method was employed in the study of magnetism of sigma phases in Fe-Cr and Co-Cr binary system [X]; and in the investigations of Laves phases in the following systems: V-Zr [XVII], Cr-Zr [IX], Cr-Hf and Cr-Ti both [XI] and in Fe-based systems [XVI]. Furthermore, the studies of the intermetallies PdBi, PdBi2 [VI] and NiTi, FePd, FePd3, FePt, FePt3 [XIII] used this method. The optimised technical parameters for this approach are: the cut-off energy restricting the number of plane waves in the basis set and the number of k-points. Except for the above mentioned optimised technical parameters, there are further parameters employed in the discussed calculations which influenced the obtained results. Their detailed descriptions are provided in user guides / manuals of particular codes. As the properties of phases studied are directly related to their crystal structure, the optimisation of crystallographic arrangement with respect to total energy had to be performed. In case of LMTO and FLAPW approaches, this optimisation was rather demanding. It was performed by alternating minimisation of total energy as a function of lattice parameter a (unit cell volume V) at a constant c/a ratio and minimisation of total energy as a function of the c/a ratio at the constant parameter amtn (Vmin) from the previous optimisation. These two steps were repeated until the change of total energy was small enough (lower than 0.1 mRy/atom). In this way, the equilibrium energies of studied Chapter 2. Ab initio calculations 12 phases were found. In comparison with LMTO and FLAPW codes used, the structure optimisation by VASP is more comfortable as this code automatically calculates the forces and the stress tensor, which are used to search directions to the equilibrium positions of atoms. Using the structure optimisation, the equilibrium structure parameters (lattice constants, angles, atomic positions) corresponding to the minimum energy were obtained. To find the equilibrium crystal structure and energy of phases studied, their magnetic arrangement had to be also taken into account as it significantly influences the stability of particular phases as it had been found in the case of iron [53]. When the spin polarised calculations are performed [X,XIII], the detailed information about the magnetic moments of individual atoms is obtained, which is usually not accessible by experimental methods. After the evaluation of equilibrium total energies, the molar total energies of formation AEfm can be calculated. The expression for AEfm between two phases of pure constituent is very simple AEp/m = EPh-EsmER, (2.18) where E^R (Ej^1) stands for the molar total energy per atom of SER state (studied phase). The molar energy of formation of the intermetallic phase (int) is calculated with respect to the weighted average of the total energies of SER states of pure constituents as *E% = ET - [xEsmER^ + (1 - x)E^] . (2.19) Here, the subscripts 1 and 2 following the name of structure denote different pure constituents and x is the molar fraction of constituent 1. It is also possible to combine the results of two ab initio methods for the evaluation of the energy of formation of intermetallics. This approach was used for analysis of energetics of various configurations of sigma phases [X]. In this case, it is necessary to use the following equation AE?ni=AEij> +AE{?> = = {Em ~ [xEm + (1 — x)Em] }lMTO or FLAPW or pseudopotential + + ixEm + (1 - x)Em ~ [xE^Rl + (1 _ x)E^R2]}FhAPW or pseudopotential • (2.20) The AE°m in Equation (2.20) consists of two parts: (i) the energy difference of alloy sigma phase with respect to weighted average of total energies of pure constituents in the sigma phase structure, both calculated by means of the LMTO, FLAPW or pseudopotential method (the LMTO method may be used here as the systems considered have the same type of structure), and (ii) the energy difference of weighted average of total energies of pure constituents in the sigma phase and SER states, both calculated by means of the FLAPW or pseudopotential method (here a more reliable, but also more time consuming method had to be used as the structures involved have different types of symmetry). Both energy differences (i) and (ii) may be considered as quite reliable, as the total energies used for their determination were obtained by the same method on equal footing. The mechanical stability can be also evaluated when the mechanical properties (bulk moduli [XIV] and elastic constants [XVII]) are calculated from the total energy depen- Chapter 2. Ab initio calculations 13 dencies on structure deformation. For example, when the dependencies of total energy on volume are expressed in polynomial form of the third order (y = ax3 + bx2 + cx + d), the bulk modulus (B) can be calculated from its second derivative as B = Vmin(6aVmin + 2b) . (2.21) To judge the mechanical stability, the elastic constant has to be calculated and the elastic stability criteria has to be fulfilled. For cubic phase, the elastic stability criteria are as follows: Cn > 0; C44 > 0; Cn > |Ci2|; and (Cn +2Ci2) > 0, where Cn, Ci2 and C44 are elastic constants. The mechanical stability can be also evaluated on the base of phonon spectra [XVII] where no negative branches can occur. Chapter 2. Ab initio calculations 14 2.2 Results and discussion As mentioned in section 2.1.4 Methodology of performed calculations, the results obtained during the ab initio studies are rather complex as they form a logically integrated set of data related to studied system. However, for greater clarity, they are divided into particular sections in this thesis. The results obtained are demonstrated on chosen exemplary systems: Cr-Fe [I,X] for sigma phases and V-Zr [XVII] for Laves phases. The citations of analogous results are provided and if it is needed the details on studies of further phases are provided [VI, XIII]. 2.2.1 Crystal structure iiv,vi,vn,ix-xi,xiii,xiv,xvi,xvii The equilibrium crystallographic data were obtained for all phases commented in this work and were listed in tables in corresponding publications. In the case of SER states, the results are summarised in Table 2.1. The results correspond very well to experimental findings and the deviations from experimental volume A%exp ranges from -5.12 % for NM bcc V to 5.02 % for NM fee Pd. However, the deviation for most structures is within ±3%, which is generally acceptable Structure Method a (nm) c/a Vat (nm3.103) Ref. a (nm) c/a Vat (nm3.103) Ref. A%exp FM hep Co FLAPW 0.2446 1.6025 10.1496 [II] 0.2506 1.6237 11.0650 [18] -8.27 FLAPW 0.2498 1.6194 10.9342 [IV, X] -1.18 PP 0.2492 1.6190 10.8435 [X] -2.00 AFM bcc Cr FLAPW 0.2866 1 11.7743 [UI, X] 0.2879 1 11.9281 [54] -1.29 PP 0.2855 1 11.6327 [IX-XI,XVI] 0.2879 1 11.9281 [54] -2.48 FM bcc Fe FLAPW 0.2865 1 11.7603 [I,IV,X] 0.2858 1 11.6669 [54] 0.80 0.2866 1 11.7709 [55] -0.09 PP 0.2836 1 11.4025 [X, XIII, XVI] 0.2858 1 11.6669 [54] -2.27 0.2866 1 11.7709 [55] -3.13 NM hep Hf PP 0.3195 1.5786 22.2901 [XI] 0.3230 1.5851 23.1300 [18] -3.63 NM bcc Mo FLAPW 0.3160 1 15.7762 [IV] 0.3145 1 15.5553 [18] 1.42 PP 0.3149 1 15.6174 [XVI] 0.40 FM fee Ni PP 0.3523 1 10.9287 [XIII] 0.3520 1 10.9036 [56] 0.23 NM fee Pd PP 0.3954 1 15.4543 [XIII] 0.3890 1 14.7160 [18] 5.02 NM fee Pt PP 0.3977 1 15.7280 [XIII] 0.3923 1 15.0937 [18] 4.20 NM diam. Si PP 0.5469 1 20.4501 [XVI] 0.5431 1 20.0227 [18] 2.13 NM hep Ti PP 0.2924 1.5818 17.1210 [XI, XIII] 0.2950 1.5866 17.6442 [18] -2.97 NM bcc Ta PP 0.3309 1 18.1159 [XVI] 0.3302 1 17.9996 [18] 0.65 NM bcc V PP 0.2978 1 13.2092 [XVII] 0.3031 1 13.9215 [18] -5.12 NM bcc W PP 0.3171 1 15.9360 [XVI] 0.3165 1 15.8492 [18] 0.55 NM hep Zr PP 0.3236 1.5977 23.4332 [IX, XVII] 0.3232 1.5930 23.2838 [54] 0.64 Table 2.1: Structural properties of SER states, a and c are the lattice parameters, Val is the volume per atom and A%exp is the deviation of ab initio results from experimental data in % of experimental value, PP stands for pseudopotential. The high deviation of A%exp = -8.27 in the third row is caused by old set of calculation parameters for FM hep Co. Chapter 2. Ab initio calculations 15 error. The example of FLAPW calculation of energy dependence on volume for FM bcc Fe and AFM bcc Cr is shown in Figure 2.2. Similar investigation was done for hypothetical sigma phases of pure constituents Fe and Cr [I]. In Figure 2.3(a), there are depicted two energy dependencies on volume calculated by LMTO and FLAPW method. In Figure 2.3(b) [I], the curves of energy dependence on volume calculated by FLAPW approach for both Fe and Cr are shown. Figure 2.2: Volume dependence of total energy of AFM bcc Cr (■) and FM bcc Fe (♦) calculated by FLAPW method [I]. The volume corresponds to two-atom unit cell. 2000 2100 Volume [at. units 3 ] 1900 2100 2300 Volume [at.unit3 ] (a) Pure Fe, tion. FLAPW (□) and LMTO (♦) optimisa- (b) Final FLAPW optimisation for pure Cr (□) and Fe (0) sigma phase at constant cja ratio; c/ape = 0.5174, c/cicr = 0.5237 [I]. Full symbols represent the crossing points with previous optimisation of total energy vs. cja ratio. Figure 2.3: Volume dependence of total energy of sigma phases (30 atoms). Chapter 2. Ab initio calculations 16 In the case of complex phases, not only lattice parameters but also the atomic positions were optimised. The easiest way how to perform such an optimisation is to employ the pseudopotential VASP code with its automatic relaxation. The equilibrium atomic positions of sigma phases of pure constituents [X] are listed in Table 2.2, columns 6-8. The internal parameters describing the atomic positions in chosen experimentally studied sigma phases are given in the same table in columns 3-5. The equilibrium data given in Table 2.2 reveal only very small scatter and the fully relaxed parameters describing the positions of atoms correspond well to those determined experimentally for alloy sigma phases. The equilibrium lattice parameters of Co, Cr and Fe sigma phases calculated by LMTO, FLAPW and pseudopotential approaches are summarised in Table 2.3 [X]. In this case, LMTO method results in the highest values of the lattice parameter a and atomic volume Vat, the medium values are provided by the VASP code and finally the lowest numbers are obtained from the WIEN97 calculations. The scatter of the values is reasonably small, in units of percent. From this point of view, the all methods used can be considered as equivalent. Analogously the structural parameters of binary sigma phases can be obtained [X]. Subl. Par am. Cr-Fe a Cr-Co b Co-Mo c Co Cr Fe Ref. [19] Ref. [20] Ref. [21] [X] 4f x 0.3986 0.3984 0.3973 0.4019 0.3982 0.4030 8i x 0.4635 0.4627 0.4635 0.4613 0.4671 0.4572 y 0.1312 0.1291 0.1283 0.1332 0.1285 0.1315 8V x 0.7399 0.7404 0.7450 0.7346 0.7434 0.7366 y 0.0661 0.0654 0.0670 0.0669 0.0594 0.0660 8j x 0.1827 0.1826 0.1820 0.1812 0.1877 0.1821 z 0.2520 0.2500 0.2500 0.2507 0.2553 0.2503 Table 2.2: Experimental (columns 3-5) and by VASP calculated (columns 6-8) equilibrium values of internal structure parameters of NM sigma phases [X]. a xcr = 0.495, T = 923 K; b xcr = 0.564 and c xq0 = 0.4, T = 1673 K. The symbols xcr and xq0 represent the molar fraction of Cr and Co, respectively. The exact atomic positions can be calculated from these parameters using simple relations corresponding to the given sublattice and particular space group. Elem. LMTO WIEN97 VASP a (au) c/a Vat (au3) a (au) c/a Vat (au3) a (au) c/a Vat (au3) Co 16.1116 0.5161 71.9496 15.8602 0.5197 69.1126 15.9252 0.5289 71.2044 Cr 16.6677 0.5216 80.5088 16.3792 0.5237 76.7078 16.5267 0.5214 78.4528 Fe 16.0465 0.5180 71.3427 15.5987 0.5174 65.4592 15.9325 0.5210 70.2374 Table 2.3: Equilibrium lattice parameters and atomic volumes of NM sigma phase of pure constituents calculated by LMTO, WIEN97 and VASP codes [X]. Chapter 2. Ab initio calculations 17 The V-Zr system [XVII] was chosen as an example for modelling of Laves phases. The dependence of structural parameters on composition can be demonstrated for example on C14 Laves phase, see Figure 2.4, where the occupation of the 6h and 4f sublattice is changing. 1200 0.333 0.667 (a) a lattice parameter (♦), c lattice parameter (■) 0.333 0.667 (b) c/a (♦), volume per atom Vat (■) Figure 2.4: Composition dependence of structural parameters of C14 Laves phase in V-Zr system [XVII]. The open symbols correspond to experimental values [18]. It was found that the volume per atom Val decreases with increasing molar fraction of vanadium. Similarly, the composition dependencies of structural parameters were studied in Cr-Zr [IX], Cr-Hf and Cr-Ti [XI] and Fe-X (X = Si, Cr, Mo, W, Ta) [XVI] system. In all the above mentioned systems, the dependence of volume on composition is linear. There are some deviations from linear dependencies of lattice parameters on composition in case of CrgXg and Q^Xio configurations in systems where X = Hf, Ti. Their values of a are higher and values of c are lower than the values from linear approximation. These deviations compensate in volume calculation. 2.2.2 Energies of formation iiv,vi,vn,ix-xi,xiii,xiv,xvi,xvii The energies of formation of studied structures (even of hypothetical phases consisting of pure constituents or phases with occupation of sublattices which has not been observed experimentally) can be evaluated according to the Equation (2.18), (2.19) or (2.20). The values obtained for intermetallic phases were compared with experimental data if available (Table 2.4). Sometimes, the ab initio methods can successfully reproduce the experimental values as it was found in case of Fe2Ta C14 Laves phase, NiTi, FePt and FePt3. But there are phases, for which the agreement is worse, e.g. Fe2W, G"2Hf and FePd3. This disagreement between theory and experiments is caused by different temperatures at which the experiments (room temperature or higher) and calculations (0 K) are performed. In case of sigma phase [X], it was found that some atoms preferentially occupy certain sublattices which significantly influences the stability of particular configurations of this phase. Fe and Co (in binary systems with Cr) prefer the 8z'sublattice and Cr the 8i and 8j sublattices. The same findings concerning the site preferences in Cr-Fe system were reported by Korzhavyi [36] and were confirmed by experimental results [20]. Chapter 2. Ab initio calculations 18 Phase a pint f,abinitio (kJ/mol of atoms) Ref. kt-fint f', experiment (kJ/mol of atoms) Ref. Fe2Ta - C14 Laves phase -18.61 [XVI] -19.27 [57] Fe2W - C14 Laves phase 0.68 [XVI] -7.61±3.14 [58] Cr2Hf - C15 Laves phase -10.38 [XI] -4.8±4.3 [59] NiTi -33.1 [XIII] -32.7±1.0 [XIII] FePt -23.1 [XIII] -23.0±1.9 [XIII] FePt3 -19.2 [XIII] -20.7 ± 2.3 [XIII] FePd3 -10.0 [XIII] -16.0 ±2.7 [XIII] Fe-Cr - sigma phase xCr = 0.4, NM 10.23 [X] l.T [60] XCr = 0.4, FM 6.89 [X] xCr = 0.533, NM 8.01 [X] 6.5b [61] xCr = 0.533, FM 7.90 [X] Cr-Co - sigma phase xCr = 0.533, NM 11.50 [X] 9.3T [62] xCr = 0.533, FM 9.98 [X] xCr = 0.6, NM 9.40 [X] (-2.9;5.1)rf [61] xCr = 0.6, FM 9.04 [X] (2.64;6.77f [63] Table 2.4: Ab initio calculated energies of formation and experimental enthalpies of formation of intermetallic phases. a xCr = 0.45, b xCr = 0.45, c xCr = 0.6, d xCr = 0.45 - 0.63 ande xCr = 0.57 - 0.61. 2.2.3 Magnetic properties x'xm Some elements such as Cr, Fe, Co and Ni have tendencies to magnetic ordering in pure state, which was confirmed by experiments. In this case, their magnetic arrangement has to be reproduced by ab initio calculations. The overview of magnetic properties of studied elements in SER states is given in Table 2.5. When intermetallic phases contain elements listed above, there is a high probability that the magnetism will play an important role in their behaviour. Sometimes, it is confirmed by experiment but sometimes the magnetic ordering is important only at very low temperatures which has to be taken into account when performing ab initio calculations at 0 K. The detailed study of magnetic properties for all 32 configurations of sigma phases in Fe-Cr and Co-Cr binary system was performed [X]. It turns out, for example, that elemental iron in the sigma phase structure exhibits different magnetic moments at different sublattices. At the 4f, 8i and 8j sublattices, the local magnetic moment of Fe atoms equals to 2.29 /is/atom, 2.00 /is/atom and 1.87 /is/atom, respectively, the highest magnetic moment being at the sublattice 4f possessing the highest coordination number. Magnetic moments of iron in 8i' and 2a sublattices with the lowest coordination numbers are substantially lower: 1.22 /is/Worn and 1.10 /is/'atom. In case of elemental Co, the magnetic moment decreases from 1.70 /is/atom, 1.67 jis/atom, 1.59 /is/atom to 1.54 /is/atom which corresponds to the sublattices 4f, Chapter 2. Ab initio calculations 19 Structure Method M (Ms) Ref. l^exp (Ms) Ref. (GPa) Ref. Bexp [64] (GPa) FM hep Co FLAPW 1.74 * 1.72 [56] 225.08 * 191.4 PP 1.56 * 210.94 * AFM bec Cr FLAPW 1.07 * 0.59 [54] 193.32 * 190.1 PP 1.08 * 186 [XVI] FM bec Fe FLAPW 2.28 * 2.22 [56] 167.79 * 168.3 PP 2.18 [XIII] 2.12 [54] 194 [XVI] NM hep Hf PP 112.68 * 109 NM bec Mo FLAPW 264.09 * 272.5 PP 271 [XVI] FM fee Ni PP 0.6 [XIII] 0.61 [56] 196.97 * 186 NM fee Pd PP 223.54 * 180.8 NM fee Pt PP 249.33 * 278.3 NM diam. Si PP 90 [XVI] 98.8 NM hep Ti PP 117.46 * 105.1 NM bec Ta PP 201 [XVI] 200 NM bec V PP 188.5 [XVII] 161.9 NM bec W PP 315 [XVI] 323.2 NM hep Zr PP 97.6 [XVII] 83.3 Table 2.5: Magnetic and mechanical properties of SER states, ji is the magnetic moment, B bulk modulus and PP stands for pseudopotential. * this work. (8i, 8j), 8V and 2a, respectively. Again, the atomic magnetic moment decreases with decreasing coordination number. On the other hand, the sigma phase of elemental Cr is nonmagnetic because the magnetic moments found are very close to zero. However, it does not mean at all that the chromium atoms are nonmagnetic through the whole composition region. Their magnetic moment calculated by VASP reaches even -1.21 \XBJatom (-1.29 11b/atom) in CrCoCoCoCo (CrFeFeFeFe) configuration. In binary sigma phases we can see that the atomic magnetic moments of all three constituents (i.e. Fe, Co, Cr) mostly decrease with increasing molar fraction of chromium. Similarly as in elemental Fe and Co, the highest values of magnetic moments are found at the 4f sublattice with the highest coordination number. In Cr-Fe system, the Cr atoms exhibit very often antiferromagnetic behaviour with respect to Fe atoms, i.e. they have the opposite orientation of magnetic moments. This fact is fully manifested at 8i, 8V and 8j sublattices with some exceptions for higher chromium concentrations. The antiparallel arrangement of magnetic moments is also found at the 2a and 4f sublattices, again with some exceptions. In sporadic cases the antiferromagnetic arrangement occurs at Fe atoms in the sublattice 2a for FeCrFeCrFe configuration and in 8V for FeFeCrFeCr and CrCrCrFeCr. In Cr-Co system Cr atoms also exhibit antiferromagnetic behaviour with respect to Co atoms. However, with increasing molar fraction of Cr atoms, we observe increasing amount of Cr atoms that behave in the ferromagnetic way with respect to Co atoms. In contrast to the Cr-Fe system the occurrence of this arrangement is not fully connected with particular sublattices, but it is Chapter 2. Ab initio calculations 20 most frequent at the 2a and 4f sublattices. Similarly, the magnetism had to be included in study of NiTi, FePd, FePd3, FePt and FePt3 phases [XIII]. The FePd, FePd3, FePt and FePt3 structures are ferromagnetic whereas the NiTi intermetallics in both the cubic and the monoclinic arrangement are nonmagnetic. The comparison of found and experimental magnetic moments (Table 4C in [XIII]) in FM bcc Fe, FM fee Ni, FM FePd, FM FePd3 and FM FePt provides an excellent agreement. In the case of FePt3, the AFM arrangement of the structures is reported [65]. Nevertheless, the magnetic moments found in the literature agree very well with the calculated ones. 2.2.4 Mechanical properties AVi>AVii The values of bulk moduli of SER states were presented in [XVI, XVII] and are listed in Table 2.5. It shows that the deviations from experiments are higher than those found for structure parameters but they are usually within ±20 GPa, which is acceptable. This limit was exceeded for FM hep Co, FM bcc Fe, NM fee Pd, NM fee Pt and NM bcc V. To judge the mechanical stability of phase, the elastic constants has to be calculated. This was done for cubic C15 V2Zr Laves phase where three elastic constants Cn = 162.33 GPa, C44 = 6.60 GPa and C12 = 136.62 GPa were evaluated (Table 4 in [XVII]). This phase was declared to be mechanically stable as the elastic stability criteria: Cn > 0; C44 > 0; Cn > |C12|; and (Cn +2Ci2) > 0 were fulfilled. Additionally the stability of C15 V2Zr Laves phase was confirmed by phonon spectra calculations [XVII]. Chapter 3 CALPHAD modelling Nowadays, more sophisticated and complex materials with excellent properties are required in material engineering and technologies. These advanced materials often consist of more than 10 elements and many of them are used in extreme conditions. Nevertheless, the tendency to reach thermodynamic equilibrium, either stable or metastable, after long-term exploitation, is characteristic for most of such materials. Some alloying elements can significantly improve desired properties, but they can also introduce some unexpected features (e.g. precipitation of unwanted or new phases, brittleness, degradation processes, etc.) in long term run that can outweigh their positive influence [66]. In all such cases the deep knowledge of relevant phase diagrams (or their sections) and thermodynamic properties is crucial for prediction of structural and material development towards the equilibrium or metastable state. The powerful combination of ab initio electronic structure calculations, semiempirical thermodynamic approach using the CALPHAD method [5, 67] and carefully selected experimental investigations is employed to model phase diagrams of complex materials and to construct a consistent thermodynamic database for these systems [I-IV, VII, IX, XI, XVII]. Furthermore, the extension of the thermodynamic modelling down to 0 K temperature is presented [XIV, XVII]. 3.1 Theory and methodology The CALPHAD method [5,67] is based on the modelling of the Gibbs energies of all phases possibly existing in the system, followed by the minimisation of total Gibbs energy of the system. The Gibbs energies of relevant phases are obtained by assessing pre-defined polynomials to the experimental phase equilibrium data (the positions of phase boundaries, compositions of phases in equilibrium, etc.) and known thermodynamic quantities (e.g. heat capacities, activities). It means that CALPHAD method is dependent on certain amount of robust experimental or ab initio data for simpler systems, especially for binary and ternary ones. The molar Gibbs energy of the whole system is defined as the sum of molar Gibbs energies of all included phases G^, multiplied by their molar fraction -21- Chapter 3. CALPHAD modelling 22 gtot = ^xfgf, (3.1) / where & = Y^yt °g{ + gid + ge + gmag + gpres. (3.2) i The molar Gibbs energy of phase g? contains the sum of molar Gibbs energies of pure constituents i in the phase / multiplied by their lattice fractions (j^yt °g{^J, the terms describing ideal (gld) and nonideal (ge) mixing and, when needed, some special terms such as magnetic (ygmag) or pressure (gpres) contributions. For a binary system (A-B), the terms describing the mixing may be evaluated by relatively simple formulas gid = RT (yA In yA + yB In yB) (3.3) and ge = yAyB (l°(T) +L1 (T) (yAyB)+L2(T) (yA-yB)2 + ...) , (3.4) where L°, L1 and L2 are the expansion coefficients of the Redlich-Kister polynomial [68], T is temperature and R is the universal gas constant. The temperature dependence of L-parameters is given by an equation of the following type L0orlor2- =a + bT + cTlnT . (3.5) In the CALPHAD modelling, various models for °g{ can be used [67]. Here, more details are provided on a sublattice model as this approach was used for intermetallic phases studied in papers commented in this thesis. In general, the number of sublattices in sublattice model can change according to the crystallography of phase and the needs of modelling but, for simplification, the presented description is limited to two sublattices. In the two-sublattice model, the Gibbs energy of the reference state is rref,f _vlv2 0rf ,,,1,,2xvn To describe the phase equilibria at low temperatures (i.e. below the temperature limit Tum used for the SGTE Gibbs energy expressions for pure elements [13]), it is necessary to find polynomials which • obey the thermodynamic laws at low temperatures and • have the same function value and the value of the first derivative at 7},m as the corresponding SGTE Gibbs energy expressions [13]. Furthermore, the values of SGTE polynomials of Gibbs energy above Tum should be left unchanged, because they are based on experiments and are widely used. In Ref. [XIV], the SGTE polynomials are extended below 7},m using the Einstein formula for the temperature dependence of the heat capacity. In this first step, magnetic and pressure contributions to the Gibbs energy and the temperature and concentration dependence of the Einstein (7b) and Debye (7b) temperature are not considered. According to [73], the heat capacity of pure nonmagnetic elements at low temperatures can be represented by equation Cpjow = 3Ar(^j +aT + bT2 + cT2, (3.12) where T is the temperature in K and A = ^i- The first term in Equation (3.12) represents the contribution of the harmonic lattice vibrations. The second term consists of contributions from electronic excitations and low-order anharmonic corrections (dilatational and explicitly anharmonic) and the parameter a can be related to a non-thermodynamic information, e.g., electron density of states at Fermi level. The third term corresponds to the high-order anharmonic lattice vibrations and it is seldom that one can find experimental information to validate the parameter b. Parameter c is added for smooth continuation of Cp through the Tum. The low temperature Gibbs energy related to the SER states is evaluated as [73] Glow (T) =E0 + 3-RTE + 3RT In (l - - ^T2 - ^T5 - °-T3 , (3.13) Chapter 3. CALPHAD modelling 25 where Eq is the total energy of a nonmagnetic structure of an element at 0 K relative to the SER state and the second term is the energy of zero-point lattice vibrations [74,75]. The condition for smooth connecting of the extended G[ow(T) function to the G(T) in SGTE data [13] at contact temperature 7},m, (usually, but not always 298.15 K) means that function values and values of first derivative of both functions have to be equal at Tum. Similarly, the condition for a smooth connection of heat capacity function below {Cp,iow(T)) and above (Cp(T)) Tum has to be fulfilled. Based on these four conditions, four equations for Giow(Tiim), ^^(7)™), Cp^ow{TUm) and ^%^(7),m) including Eq, a, b and c parameter were obtained and solved. On the left side of these equations, there are expressions for low-temperature polynomials based on Equations (3.12) and (3.13)) for Tum. Here, the Einstein temperature is related to the Debye temperature To as Te = 0.777b [75]. On the right side of equations, there are expressions for the same variables (at 7},m), however, expressed from polynomials valid above Tum and provided in [13]. More details on this approach and values of Eo,a,b and c parameters for particular elements are provided in [XIV]. Chapter 3. CALPHAD modelling 26 3.2 Results and discussion 3.2.1 Binary systems UI,iv,ix,xi,xvii The modelling of sigma phase is represented here by the Cr-Fe binary system [I]. The Gibbs energies and enthalpies for phases found in this system [I] are shown in Figure 3.1. Here, the differences between the results obtained using the old three-sublattice [72] and new (11) two-sublattice [76] model for description of sigma phase is shown. It is obvious that the values based on the (11) two-sublattice model run through the whole composition region, which is convenient in modelling of more-components systems. Figure 3.1: Concentration dependence of Gibbs energy (1000 K) and enthalpy in Cr-Fe system [I]. (1) liquid, (2) bcc phase, (3) fee phase, (4) sigma phase (new two-sublattice model), (4a) sigma phase (three-sublattice model). The phase diagram calculated using the new (1 1) two-sublattice model [76] (Figure 3.2) yields better agreement with experimental data than that obtained by means of an older three-sublattice model [72]. The thermodynamic modelling of sigma phases based on the same theoretical approach as described for Cr-Fe binary system [I] was performed in Cr-Co [II] and both Co-Mo and Fe-Mo [IV] binary systems. The results of modelling in ternary systems are described in Section 3.2.2 Ternary systems. In case of Laves phases, mostly, the two-sublattice model (A,B)2(A,B) with four end-members was employed [IX, XI, XVII] and Cr-Zr binary system [IX] was chosen as an example. The Gibbs energy of all three Laves phases (C14, C15 and C36) in Cr-Zr system was modelled with the help of ab initio calculated total energy differences, presented in Table 3 in Ref. [IX]. The C14 and C36 Laves phases were also modelled by three-sublattice model (A,B)4(A,B)gA2 employing the total energy differences for the CrgZrg and C^Zrio configuration. Chapter 3. CALPHAD modelling 27 Figure 3.2: Phase diagrams of Cr-Fe system [I]. Thick line: calculated by the new two-sublattice model (for data see Table 4 in Ref. [I]), thin line: calculated by three-sublattice model (data from [72]), stars: experimental data [77]. In both cases, the Gibbs energy is obtained from Equations (3.7) and (3.8) where the entropy term, containing also the vibration contribution to the enthalpy, is adjusted to the experimental data. The L-parameters describing the excess Gibbs energy GE of non-ideal mixing in Equation (3.4) are obtained in the same way. The thermodynamic parameters for all other phases (liquid, hep, bec, fee) are based on unary data from [13]. The calculated phase diagram is presented in Figure 3.3, where the equilibria with all three Laves phases are denoted. It was shown that ab initio calculated structural energy differences fit well the two-sublattice model of C15 and three-sublattice model of C14 and Figure 3.3: Phase diagram of Cr-Zr system with experimental data [IX]. Chapter 3. CALPHAD modelling 28 C36 Laves phases. Moreover, a substantially smaller number of adjustable parameters was necessary for thermodynamic description of Laves phases than in previous attempts in literature. By analogy, the thermodynamic modelling and phase equilibria calculations with Laves phases were performed in Cr-Hf [XI] and Cr-Ti [XI] system. 3.2.2 Ternary systems ni'vn As mentioned in Section 3.2.1 Binary systems, the modelling of sigma phases using the (11) two-sublattice model had been performed in several binary systems and it war extended to more-components systems, i.e. to ternary system Fe-Ni-Cr [III] and Cr-Fe-W [VII]. The calculated phase diagrams of Cr-Fe-Ni system at various temperatures are presented in Figure 3.4 and it was found that the (11) two-sublattice model of the sigma phase can be also used for a reasonable description of phase equilibria with sigma phase in the Fe-Cr-W system, although the sigma phase is not stable in two binary subsystems, i.e. in Fe-W and Cr-W. (a) T = 1073 K; experimental data: triangles for (b) T = 1173 K; experimental data: triangles for bcc/fcc [79], squares for fcc/sigma [80]. bcc/fcc [81], squares for fcc/sigma [80], circles for fee [82] . Figure 3.4: Calculated phase diagram of Fe-Ni-Cr system [III]. Full lines: the two-sublattice model of sigma phase, dashed lines: the three-sublattice model [72] using data from [78]. 3.2.3 Thermodynamic modelling at low temperatures AiV>AVii The temperature dependencies of G and Cp for pure elements at low temperatures were modelled and the results for chosen elements are demonstrated, in Figure 3.5. These figures show that the knowledge of the Debye (Einstein) temperature makes it possible to obtain a realistic extension of Gibbs energy function as the temperature falls towards 0 K. The thermodynamic modelling at low temperatures presented for pure elements [XIV] can be also applied to complicated intermetallic phases such as C15 Laves phase in V-Zr Chapter 3. CALPHAD modelling 29 G of Li-bcc_a2, Na-bcc_a2, K-bcc_a2, Rb-bcc_a2. Cs-bcc_a2 Cp of Li-bcc_a2, Na-bcc_a2, K-bcc_a2, Rb-bcc_a2, Cs-bcc_a2 Temperature / K Temperature / K (a) bcc Li, Na, K, Rb, Cs. G of Be-hcp_a3. Mg-hcp_a3, Ca-fcc_al, Sr-fcc_al, Ba-bcc_a2 Cp of Be-hcp_a3, Mg-hcp_a3. Ca-fcc_al, Sr-fcc_al, Ba-bcc_a2 Temperature / K Temperature / K (b) hep Be, Mg; fee Ca, Sr; bcc Ba. Figure 3.5: Temperature dependence of Gibbs energy and heat capacity for extended and SGTE functions [XIV]. system [XVII]. In this system the thermodynamic data above room temperature were taken from literature [10] and improved for C15 Laves phase and a hep phase. The new extension of expression of Gibbs energy of C15 Laves phase and rhombohedral phase to zero Kelvin compatible with Gibbs energy expressions above 298.15 K [10] and based on respective values of Debye temperatures [83-85] was provided. The phase diagram including the phase equilibria at low temperatures is presented in Figure 3.6. The shape of corresponding molar Gibbs energy G(T) functions for V2Zr C15 Laves phase and for V2Zr rhombohedral phase in the temperature region 0-400 K is designed in Figure 3.7. The work [XVII] shows that the methodology of calculation of unary data [XIV] at temperatures below 298.15 K is transferable to more complicated structures. Chapter 3. CALPHAD modelling 30 2400 0.2 0.4 0.6 0.8 x(Zr) Figure 3.6: Phase diagram of V-Zr binary system [XVII] compared with experimental data: stars [86], crosses [87], square [88] and triangle [89]. -5 -10 -15 o -20 E 2 S -25 -30 -35 -40 -45 \ G(T) C15 extrapolated from [10] 1-1-1-1- \/G(T)C15 G(T) RHOMBo'^^*^^^ - G(T) RHOMBO G(T)C15 fromnoixX^ _I_1_1_ 50 100 150 200 Temperature / K 250 300 350 400 Figure 3.7: Temperature dependence of the molar Gibbs energy of V2Zr C15 Laves phase and of V2Zr rhombohedral phase [XVII]. Blue curve represents the Gibbs energy of the V2Zr C15 Laves phase according to [10] including its extrapolation below 298.15 K, the red / green curve show the extension of the Gibbs energy [XIV] of the V2Zr C15 Laves phase / V2Zr rhombohedral phase based on our new model to zero Kelvin temperature. Conclusions and main results In this thesis, the results of ab initio electronic structure calculations and the CALPHAD modelling were summarised. It was shown that the combination of both approaches provides a better physical insight into the construction of phase diagrams [I-IV,VII,IX-XI,XVII]. The advantage of implementation of ab initio calculated lattice stabilities into CALPHAD modelling is especially crucial when the system studied contains complex phases such as sigma and Laves phases. The above mentioned theoretical methods are very important in both basic materials science and practical applications in industrial laboratories to plan effectively their experimental program from the point of view of economy and time. 4.1 Ab initio calculations Our first-principles calculations provide the basic information about the properties of metallic phases. They validate the efficiency of methods applied when the experimental data are available for comparison. On the other hand, they also provide new experimentally inaccessible data, e.g. magnetic moments of particular atoms [X,XIII], preferential occupation of sublattices by particular elements in complex phases [III, VII, X], and energies of formation of metastable phases [I-IV,VII,IX-XI,XVII]. The ab initio data obtained may be considered as the first step to determination of both thermodynamic and mechanical stability of various intermetallic phases influencing the properties of materials. Detailed conclusions about ab initio calculations The structural, magnetic and mechanical properties of SER states of pure constituents were over-viewed and compared with experimental data [I,II,IV,VIII-XI,XIII,XVI,XVII] providing good agreement. The calculated total energies were subsequently used to evaluate the lattice stabilities of intermetallic phases. The detailed ab initio study was accomplished for sigma phases in Cr-Fe and Cr-Co binary system [X] using the LMTO-ASA, FLAPW and pseudopotential method. The energies of formation and magnetic ordering were discussed and compared with experimental data. The inclusion of magnetic ordering stabilises the sigma phase in these systems and results in the shift of stability region towards the configurations with a higher concentration of iron or cobalt. We predicted that Cr-Fe and Cr-Co sigma phases are magnetically ordered at 0 K, which corresponds to experimental findings. Our study reveals that the largest part of magnetisation is carried by the iron or cobalt atoms and that the chromium -31- Conclusions_32 atoms do not contribute to this effect very much although they induce a decrease of the total magnetic moment by their AFM behaviour in some sublattices. The magnetic moments depend on the kind and position of the atom. The influence of the sublattice on the magnetic moment is the same for both Fe and Co atoms and their magnetic moments decrease from the sublattices 4f, Si, 8j, Si' to 2a. The magnetic moment of Cr in the sigma phase structure is close to zero. However, it increases with the increasing number of Fe or Co atoms in the structure. Combining the LMTO-ASA and FLAPW method, the concentration dependence of the energy of formation of sigma phase in Fe-Ni [III], Ni-Cr [III], Cr-W [VII] and Fe-W [VII] systems was calculated and compared with the Gibbs energies of various phases in Fe-Ni and Ni-Cr system and with Gibbs energy of sigma phase in Cr-W and Fe-W system. Energies of formation of the sigma phase in Fe-Ni, Fe-W and Cr-W binary systems are mainly positive and they are in agreement with supposed metastability in mentioned systems. With the help of ab initio electronic structure calculations, the relations between the electronic structure, size of the atoms and the thermodynamic as well as structural properties of C14 Laves phases in Fe-X (X = Si, Cr, Mo, W, Ta) binary systems were understood [XVI]. It was found that the structure parameters and energies of formations strongly depend on the molar fraction of iron and that the calculated equilibrium parameters correspond very well to the experimental values. Our calculations reveal that the C14 Laves phase is unstable at zero temperature in both Fe-Mo and Fe-W system which is in contradiction with experiments. However, the absolute value of the total energy of formation is very low. We suppose that the instability at higher temperatures is suppressed by the entropy effects. The Fe2Ta C14 Laves phase is stable at low temperatures. In Cr-Zr [IX], Cr-Hf [XI], Cr-Ti [XI] and V-Zr [XVII] system, the ab initio calculated structural parameters and energies of formation of Laves phases correspond reasonably well to both experimental data where available and to previous theoretical results. The total energies of formation and magnetism were also studied in NiTi, FePd, FePd3, FePt and FePt3 [XIII]. The ab initio calculated values were compared with the experimental enthalpies of formation and with values calculated using the semi-empirical model of Miedema and co-workers [90]. It was shown that the energies of formation corresponding to the equilibrium arrangement can significantly contribute to the analysis of the energetics of intermetallic phases in spite of the fact that they are calculated at 0 K. Ab initio calculated energies of formation of PdBi and PdBi2 intermetallics were successfully used as a basis for a thermodynamic assessment of the phase diagram of the Bi-Pd system [VI]. Except for the lattice stabilities, the so called mechanical stabilities can be obtained from ab initio calculations of elastic constants and phonon spectra. This was demonstrated in case of V2Zr C15 Laves phase [XVII]. 4.2 CALPHAD modelling We were among the first groups in the world applying the combination of CALPHAD method and ab initio techniques in phase diagram modelling of systems with complex intermetallic phases [I, II]. We implemented this approach into the new (11) two-sublattice model for sigma phase [I-IV,VII] defining the Gibbs energy of this phase in the whole Conclusions 33 composition region which is crucial for predictions of phase equilibria in multicomponent systems. The extensive thermodynamic modelling of phase equilibria with Laves phases was successfully accomplished [IX, XI, XVII]. Recently, we suggested a method for the extension of SGTE data for pure elements to zero Kelvin temperature [XIV]. This approach was extended to intermetallic phases [XVII] which is important for modelling of phase equilibria in multicomponent systems in materials under extreme conditions. Detailed conclusions about CALPHAD modelling A new approach to calculation of phase equilibria with sigma phase using the ab initio calculated total energies of formation of sigma phase of pure constituents was proposed [1,11]. The procedure was successfully tested on four binary systems: Cr-Fe [I], Cr-Co [II], Co-Mo [IV] and Fe-Mo [IV]; and it was extended to two ternary systems: Cr-Fe-Ni [III] and Cr-Fe-W [VII]. The calculated phase equilibria involving the sigma phase agree well with the experimental data. In Cr-Fe [I] and Cr-Co [II] binary system, the phase diagrams constructed using the new (1 1) two-sublattice model [I] yielded better agreement with experimental data than diagrams obtained by means of an older three-sublattice model [72]. It was found that the (1 1) two-sublattice model of the sigma phase can be also used for a reasonable description of phase equilibria in the Fe-Cr-W system [VII], although the sigma phase is not stable in two binary subsystems, i.e. in Fe-W and Cr-W systems. The approach based on combination of ab initio and CALPHAD modelling provides a reasonable physical interpretation of the individual terms in the mathematical expression for the Gibbs energy difference between the sigma phase and SER state of the pure constituents. The proposed procedure enables us to predict the stability region of sigma phase in metallic materials. The ab initio calculated energies of formation also form a solid base for the thermodynamic modelling and phase diagram calculations in systems containing Laves phases. The total energy differences fit well the two-sublattice model of C15 and three-sublattice model of C14 and C36 Laves phase in Cr-Zr [IX], Cr-Hf [XI] and Cr-Ti [XI] systems. The ab initio analysis of relative stability of Laves phase structures confirms the sequence of decreasing stability C15-C36-C14 and it can be fully utilised in the thermodynamic modelling of those phases except for Cr-Hf system, which lacks phase equilibrium data for C36 structure. Phase diagrams in the above mentioned binary systems were calculated and compared with experimental data. The extension of the SGTE data to low temperatures was proposed [XIV]. This extension is based on the knowledge of the Debye (Einstein) temperature and the Einstein function, which can be used to evaluate the heat capacity, Gibbs energy and their first derivatives with respect to temperature. The extension method maintains the SGTE unary data above the limiting temperature Tum and forces the low temperature extension to have the same function value and the value of the first derivative at Tum as the corresponding SGTE polynomial has. The temperature dependencies of Gibbs energies and heat capacities of pure elements were presented and some problems were pointed out [XIV]. It was shown that the methodology of extension of the SGTE data to temperatures below 298.15 K is transferable to more complicated structures such as V2Zr rhombohedral and C15 Laves Conclusions_34 phase [XVII]. The phase equilibria with these phases at low temperatures were evaluated and corresponding phase diagram was constructed. List of author's publications and the specification of author's contribution International refereed journals and one chapter in monography are listed below. The papers I-IV, VI, VII, IX-XI, XIII, XIV, XVI, XVII present the research summarised in this thesis. Scientific papers in journals [I] Houserova J., Friák M., Šob M., Vřešťál J.: Ab Initio Calculations of Lattice Stability of Sigma-phase and Phase Diagram in the Cr-Fe System. Computational Materials Science 25 (2002) 562-569. DOI: 10.1016/s0927-0256(02)00335-x Author's contribution (ca. 40%): All ab initio calculations (including analysis and interpretation), related figures and tables, the preparation of manuscript. The motivation for the project, CALPHAD modelling and finalisation of the manuscript is due to prof. Vřešťál. The help with technical problems and consultations about run of DFT codes were provided by M. Friák. Discussions about DTF approaches and the critical reading of the manuscript is due to prof. Sob. [II] Houserova J., Vřešťál J., Friák M., Šob M.: Phase Diagram Calculation in Co-Cr System Using Ab Initio Determined Lattice Instability of Sigma Phase. CALPHAD 26 (2002) 513-522. DOI: 10.1016/s0364-5916(02)80004-9 Author's contribution (ca. 40%): All ab initio calculations (including analysis and interpretation), related figures and tables, the preparation of manuscript. The motivation for the project, CALPHAD modelling and finalisation of the manuscript is due to prof. Vřešťál. The help with technical problems and consultations about run of DFT codes were provided by M. Friák. Discussions about DTF approaches and the critical reading of the manuscript is due to prof. Sob. [III] Chvátalova K., Houserova J., Šob M., Vřešťál J.: First-principles Calculations of Energetics of Sigma Phase Formation and Thermodynamic Modelling in Fe-Ni-Cr System. Journal of Alloys and Compounds 378 (2004) 71-74. DOI: 10.1016/j.jallcom.2003.10.071 Author's contribution (ca. 30%): Choice of ab initio methods, part of ab initio calculations (including analysis and interpretation), related figures and tables, supervision of the student's (K. Chvátalova) calculations, the preparation of part of manuscript. Part of ab initio calculations (including analysis and interpretation), related figures and tables, the preparation of part of manuscript by K. Chvátalova. The motivation for the project and CALPHAD modelling is due to prof. Vřešťál. -35- List of author's publications and the specification of author's contribution 36 Discussions about DTF approaches and the critical reading of the manuscript is due to prof. Sob. [IV] Houserova J., Vřešťál J., Šob M.: Phase Diagram Calculations in the Co-Mo and Fe-Mo Systems Using First-principles Results for the Sigma Phase. CALPHAD 29 (2005) 133-139. DOI: 10.1016/j.calphad.2005.06.002 Author's contribution (ca. 60%): All ab initio calculations (including analysis and interpretation), figures and tables, the whole text, but corrected by co-authors. The motivation for the project and work on CALPHAD modelling was shared with prof. Vřešťál. Discussions and the critical reading of the manuscript is due to prof. Šob. [V] Vogtenhuber D., Houserova J., Wolf W., Podloucky R., Pfeiler W., Puschl W.: Comparative Ab Initio Study of Point Defect Energies and Atom Migration Profiles in the Ll(2)-ordered Intermetallic Compounds Ni^Al, Ni^Ga, Pt^Ga, Pt^In. Diffusion in Materials: DIMAT 2004, PTS 1 and 2; Book Series: Defect and Diffusion Forum Series Vol. 237-240 (2005) 133-138. Author's contribution (ca. 15%): Part of ab initio calculations (including analysis and interpretation). Introduction into the problem by prof. R. Podloucky is gratefully acknowledged. [VI] Vřešťál J., J. Pinkas, Kroupa A., Houserova J. , Scott A., Watson A.: Assessment of the Thermodynamic Properties and Phase Diagram of the Bi-Pd System. CALPHAD 30 (2006) 14-17. DOI: 10.1016/j.calphad.2005.12.001 Author's contribution (ca. 20%): Ab initio calculations (including analysis and interpretation), the text concerning this part of study. [VII] Chvátalova K., Vřešťál J., Houserova J., Šob M.: First-principles Calculations of Energetics of Sigma Phase Formation and Thermodynamic Modelling in the Cr-Fe-WSystem. Material Science and Engineering A 462 (2007) 153-158. DOI: 10.1016/j.msea.2006.02.474 Author's contribution (ca. 30%): Choice of ab initio methods, some of ab initio calculations, supervision of the student's (K. Chvátalova) calculations, the correction of part of manuscript. [VIII] Pavlů J., Vřešťál J., Šob M.: Re-modeling of Laves Phases in the Cr-Nb and Cr-Ta Systems Using First-principles Results. CALPHAD 33 (2009) 179-186. DOI: 10.1016/j.calphad.2008.04.006 Author's contribution (ca. 45%): Shared work with prof. Vřešťál. Critical reading of the manuscript is due to prof. Šob. [IX] Pavlů J., Vřešťál J., Šob M.: Stability of Laves Phases in the Cr-Zr System. CALPHAD 33 (2009) 382-387. DOI: 10.1016/j.calphad.2008.11.003 Author's contribution (ca. 45%): Shared work with prof. Vřešťál. Critical reading of the manuscript is due to prof. Šob. [X] Pavlů J., Vřešťál J., Šob M.: Ab Initio Study of Formation Energy and Magnetism of Sigma Phase in Cr-Fe and Cr-Co Systems. Intermetallics 18 (2010) 212-220. DOI: 10.1016/j.intermet.2009.07.018 Author's contribution (ca. 90%): The motivation for the project, the choice of the List of author's publications and the specification of author's contribution 37 methods, all ab initio calculations (including analysis and interpretation), figures and tables, the whole manuscript text. Critical reading of the manuscript is due to prof. Šob and prof. Vřešťál. [XI] Pavlů J., Vřešťál J., Šob M.: Thermodynamic Modeling of Laves phases in the Cr-Hf and Cr-Ti Systems: Reassessment Using First-principles Results. CALPHAD 34 (2010) 215-221. DOI: 10.1016/j.calphad.2010.03.003 Author's contribution (ca. 45%): Shared work with prof. Vřešťál. Critical reading of the manuscript is due to prof. Sob. [XII] Pavlů J., Vřešťál J., Chen X.-Q., Rogl P.: Thermodynamic Modeling of Laves Phases in the Ta-V System: Reassessment Using First-principles Results. CALPHAD 35 (2011) 103-108. DOI: 10.1016/j.calphad.2010.12.002 Author's contribution (ca. 40%): Ab initio calculations (including analysis and interpretation), the text concerning this part of study. [XIII] Meschel S.V, Pavlů J., Nash P.: The Thermochemical Behavior of Some Binary Shape Memory Alloys by High Temperature Direct Synthesis Calorimetry. Journal of Alloys and Compounds 509 (2011) 5256-5262. DOI: 10.1016/j.jallcom.2011.01.152 Author's contribution (ca. 30%): Ab initio calculations (including analysis and interpretation), the text concerning this part of study. [XIV] Vřešťál J., Strof J., Pavlů J.: Extension of SGTE Data for Pure Elements to Zero Kelvin Temperature - A Case Study. CALPHAD 37 (2012) 37-48. DOI: 10.1016/j.calphad. 2012.01.003 Author's contribution (ca. 15%): Participation on thermodynamic modelling and on critical reading. The motivation for the project is due to prof. Vřešťál. [XV] Legut D., Pavlů J.: Electronic Structure and Elasticity of Z-phases in Cr-Nb-V-N System. Journal of Physics: Condensed Matter 24 (2012) 195502. DOI: 10.1088/0953-8984/24/19/195502 Author's contribution (ca. 20%): Thermodynamic analysis, cooperation on the preparation of manuscript, critical reading. [XVI] Pavlů J., Šob M.: Ab Initio Study of C14 Laves Phases in Fe-based Systems. Journal of Mining and Metallurgy, Section B: Metallurgy 48 (3) B (2012) 395-401. DOI: 10.2298/jmmbl20704050p Author's contribution (ca. 90%): The motivation for the project, the choice of the methods, all ab initio calculations (including analysis and interpretation), figures and tables, the whole manuscript text. Critical reading of the manuscript is due to prof. Sob. [XVII] Strof J., Pavlů J., Wdowik U. D., Buršík J., Šob M., Vřešťál J.: Laves Phases in the V-Zr System Below Room Temperature: Stability Analysis Using Ab Initio Results and Phase Diagram. CALPHAD 44 (2014) 62-69. DOI: 10.1016/j.calphad.2013.08.003 Author's contribution (ca. 25%): Ab initio calculations (including analysis and interpretation), the text concerning this part of study, motivation for the phonon spectra calculations, cooperation on finalisation of manuscript. List of author's publications and the specification of author's contribution 38 [XVIII] Vfest'al J., Houserova J., Sob M.: Energetics and Phase Diagrams ofFe-Cr and Co-Cr Systems from First Principles. Journal of Mining and Metallurgy 38 (3-4) B (2002) 205-211. Author's contribution (ca. 40%): Ab initio calculations (including analysis and interpretation), the corresponding part of manuscript. The motivation for the project, CALPHAD modelling and finalisation of the manuscript is due to prof. Vfest'al. Discussions about DTF approaches and the critical reading of the manuscript is due to prof. Sob. Summarizing paper of publications [I] and [II]. Chapter in monography [XIX] Sob M., Kroupa A., Pavlü J., Vfest'al J.: Application of Ab Initio Electronic Structure Calculations in Construction of Phase Diagrams of Metallic Systems with Complex Phases. Solid State Phenomena: Solid Phase Transformations II 150 (2009) 1-28. DOI: 10.4028/www.scientific.net/SSP 150.1 Author's contribution (ca. 20%): Ab initio calculations (including analysis and interpretation), the corresponding part of manuscript. Survey paper on combination of ab initio and CALPHAD technique. Bibliography [1] Parr R.G., Yang W.: Density-Functional Theory of Atoms and Molecules. Oxford University Press, New York, USA, 1989. [2] Martin R.M.: Electronic Structure: Basic Theory and Practical Methods. Cambridge University Press, Cambridge, England, 2004. [3] Taylor Ph.L., Heinonen O.: A Quantum Approach to Condensed Matter Physics. Cambridge University Press, Cambridge, England, 2002. [4] Turek I., Drchal V., Kudrnovsky J., Sob M., Weinberger P.: Electronic Structure of Disordered Alloys, Surfaces and Interfaces. Kluwer Academic Publishers, Boston, USA, 1997. [5] Saunders N., Miodownik A.P: CALPHAD (A Comprehensive Guide). Elsevier, London, England, 1998. [6] Grimvall G., Magyari-Koepe B., Ozolins V. et al.: Rev. Mod. Phys. 84 (2012) 945. [7] Wolverton C: Acta Mater. 50 (2002) 2187. [8] Abe T, Chen Y, Yamabe-Mitarai Y, Numakura H.: CALPHAD 32 (2008) 353. [9] Kissavos A., Shallcross S., Meded V., Kaufman L., Abrikosov I.A.: CALPHAD 29 (2005) 17. [10] Zhao X.S., Yuan G.H., Yao M.Y, Yue Q., Shen J.Y: CALPHAD 36 (2012) 163. [11] Karki B.B., Ackland G.J., Crain J.: J. Phys.: Condens. Matter 9 (1997) 8579. [12] Neuhaus T, Likos C.N.: J. Phys. Condens. Matter 23 (2011) 234112. [13] Dinsdale A.T., CALPHAD 15 (1991) 317. [14] Akiba E., Iba H.: Intermetallics 6 (1998) 461. [15] Inoue K., Tachikawa K.: Appl. Phys. Lett. 18 (1971) 235. [16] Stein E, Palm M., Sauthof G.: Intermetallics 12 (2004) 713. [17] Bain E.C.: Chem. and Met. Eng. 28 (1923) 23. -39- Bibliography_40 [18] Villars P., Calvert L.D.: Pearson's Handbook of Crystallographic Data for Inter-metallic Phases. ASM International, Materials Park, Ohio, USA, 1991. [19] Yakel H.L.: Acta Crystallogr. B39 (1983) 20. [20] Dickins G.J., Douglas A.M.B., Taylor W.H.: Acta Crystallogr. 9 (1956) 297. [21] Forsyth J.B., d'Alte da Viega L.M.: Acta Crystallogr. 16 (1963) 509. [22] Heino S., Knutson-Wedel E.M., Kars son B.: Materials Science and Technology 15 (1999)101. [23] Lundin L., Hättestrand M., Andren H.-O.: Parsons 2000 Advanced Materials for 21st Century Turbines and Power Plants, edited by Strang A. et al., The Institute of Materials, London, 2000, GB, p. 603. [24] Hohenberg P., Kohn W.: Phys. Rev. 136 (1964) B864. [25] Kohn W., Sham L.J.: Phys. Rev. 140 (1965) Al 133. [26] Hedin L., Lundquist B.I.: J. Phys. C: Solid St. Phys. 4 (1971) 2064. [27] von Barth U., Hedin L.: J. Phys. C: Solid St. Phys. 5 (1972) 1629. [28] Janak J.F: Solid State Commun. 25 (1978) 53. [29] Ceperley D.M., Alder B.J.: Phys. Rev. Lett. 45 (1980) 566. [30] Perdew J., Zunger A.: Phys. Rev. B 23 (1981) 5048. [31] Vosko S.H., Wilk L., Nusair M.: Can. J. Phys. 58 (1980) 1200. [32] Perdew J.P, Wang Y: Phys. Rev. B 45 (1992) 13244. [33] Ziman J.M.: Principles of the Theory of Solids. Cambridge University Press, London, England, 1972. [34] Vainstein B.K., Fridkin V.M., Indenbom V.L.: Structure of Crystals. Springer-Verlag, Berlin-Heidelberg, Germany, 1995. [35] Yip S.: Handbook of Materials Modeling. Springer, Dordrecht-Berlin-Heidelberg-New York, USA, 2005. [36] Moriarty JA., Benedict L.X., Glosli J.N., Hood R.Q., Orlikowski DA., Patel M.V, Söderlind P., Streitz EH., Tang M., Yang L.H.: J. Mater. Res. 21 (2006) 563. [37] Guo Z.X. (ed.): Multiscale Materials Modelling: Fundamentals and Applications. Woodhead Publishing Ltd., Cambridge, England, and CRC Press LLC, Boca Raton, USA, 2007. [38] Singh D.: Planewaves, Pseudopotentials and the LAPW Method. Kluwer, Boston, USA, 1994. Bibliography_41 [39] Andersen O.K.: Phys. Rev. B 12 (1975) 3060. [40] Skriver H.L.: The LMTO Method. Springer-Verlag, Berlin, Germany, 1984. [41] Krier G., Jepsen O., Burkhardt A., Andersen O.K.: Computer code TB-LMTO-ASA version 4.6. Max-Planck-Institut für Festkörperforschung, Stuttgart, Germany, 1994. [42] Perdew J.P, Chevary J.A., Vosko S.H., Jackson K.A., Pederson M.R., Singh D.J., Fiolhais C, Phys. Rev. B 46 (1992) 6671. [43] Havránková J., Vřešťál J., Wang L.G., Sob M.: Phys. Rev. B 63 (2001) 174104. [44] Andersen O.K., Methfessel M., Rodriguez CO., Blöchl P., Polatoglou H.M.: In Atomistic Simulation of Materials: Beyond Pair Potentials, edited by Vitek V, Srolovitz D.J., Plenum, New York-London, 1989, p. 1. [45] Tank R.W., Arcangeli C, Krier G., Andersen O.K., Jepsen O.: In Properties of Complex Inorganic Solids, edited by Gonis A., Meike A., Turchi, P.E.A., Plenum, New York-London, 1997, p. 233. [46] Blaha P., Schwarz K., Luitz J.: Computer code WIEN97. Vienna University of Technology, Vienna, 1997; improved and updated Unix version of the original copyrighted WIEN code, which was published by Blaha P., Schwarz K., Sorantin P., Trickey S.B.: Comput. Phys. Commun. 59 (1990) 399. [47] Blaha P., Schwarz K., Madsen G., Kvasnicka D., Luitz J.: WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties. Vienna University of Technology, Vienna, 2001. [48] Kresse G., Furthmüller J.: Comput. Mater. Sei. 6 (1996) 15. [49] Kresse G., Furthmüller J.: Phys. Rev. B 54 (1996) 11169. [50] Blöchl P.E.: Phys. Rev. B 50 (1994) 17953. [51] Kresse G., Joubert J.: Phys. Rev. B 59 (1999) 1758. [52] Perdew J.P, Burke K., Ernzerhof M.: Phys. Rev. Lett. 77 (1996) 3865. [53] ŠobM.: In Multiscale Materials Modelling: Fundamentals and Applications, edited by Guo Z.X., Woodhead Publishing Ltd., Cambridge, England, and CRC Press LLC, Boca Raton, USA, 2007. [54] Moroni E.G., Jarlborg T.: Phys. Rev. B 47 (1993) 3255. [55] Polcarová M., Goodwod K., Bak-Misiuk J., Kadečková S., Bradler J.: Phys. Stat. Sol. (a) 106 (1988) 17. [56] Moroni E.G., Kresse G., Hafner J., Furthmüller J.: Phys. Rev. B 56 (1997) 15629. [57] Gale W.F., Totemeier T.C. (eds.): Smithells Metals Reference Book. Elsevier Butterworth-Heinemann, Oxford, GB, 2004. Bibliography_42 [58] Rezukhina T.N., Kashina T.A.: The Journal of Chemical Thermodynamics 8 (1976) 519. [59] Meschel S.V., Chen X.Q., Kleppa O.J., Nash P.: CALPHAD 33 (2009) 55. [60] Dench W.A.: Trans. Faraday Soc. 59 (1963) 1279. [61] Zubkov A.A., Mogutnov B.M., Shaposhnikov N.G.: Dokl. Akad. Nauk SSSR 311 (1990) 388; Dokl. Phys. Chem. 311 (1990) 239. [62] Bell H.B., Hajra J.P, Putland FH., Spencer P.J.: Met. Sei. Journal 7 (1973) 185. [63] Downie D.B., Arlasan F.: J. Chem. Thermodynamics 15 (1983) 654. [64] Kittel C: Introduction to Solid State Physics. John Wiley and Sons, New York, USA, 1976. [65] Buschow K.H.J, (ed.): Handbook of Magnetic Materials. Elsevier Sei. B.V, Amsterdam, the Netherlands, 2001. [66] Nath B., Metcalfe E., Haid J.: Microstructural Development and Stability in New High Strength Steels for Thick Section Applications at up to 620° C. In Microstructural Development and Stability in High Cr Ferritic Power Plant Steels, edited by Strang A., Gooch D.J., TSM, Cambridge, UK, 1997, 123. [67] Lukas H.L., Fries S.G., Sundman B.: Computational Thermodynamics. Cambridge University Press, Cambridge, UK, 2007. [68] Redlich O., Kister A.T.: Indust. Eng. Chem. 40 (1948) 345. [69] Allibert C, Bernard C, Effenberg G., Nüssler H.-D., Spencer P.J.: CALPHAD 5 (1981)227. [70] Ansara I., Chart T.G., Fernandez Guillermet A., Hayes F.H., Kattner U.R., Pettifor D.G., Saunders N., Zang K.: CALPHAD 21 (1997) 171. [71] HillertM.: CALPHAD 22 (1998) 127. [72] Anderson J.-O., Sundman B.: CALPHAD 11 (1987) 83. [73] Chen Q., Sundman B.: J. Phase Equilib. 22 (2001) 631. [74] Born M., Huang K.: Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford, UK, 1954. [75] Gimvall G.: Thermophysteal Properties of Materials, Elsevier Science, North-Holland, Amsterdam, the Netherlands, 1999. [76] Vfest'äl J.: Arch. Metall. 46 (2001) 9. [77] Cook A.J., Jones EW.: J. Iron Steel Inst. 148 (1943) 217 and 223. Bibliography 43 [78] Hillen M., Qiu C: Met. Trans. 22A (1991) 2187. [79] Rees W.P., Burns B.D., Cook A.J.: J. Iron Steel Inst. 162 (1949) 325. [80] Sopousek J., Kruml T.: Scripta Materialia 35 (1996) 689. [81 ] Hasebe M., Nishizawa T.: Analysis and Synthesis of Phase Diagrams of the Fe-Cr-Ni, Fe-Cu-Mn and Fe-CuNi Systems. In Applications of Phase Diagrams in Metallurgy and Ceramics: Proceedings of a Workshop Held at the National Bureau of Standards, Gaithersburgh, MD, 1997, p. 911. [82] Schultz J.W., Merrick H.F.: Met. Trans. 3 (1972) 2479. [83] Keiber H., Geibel C, Renker B., Rietschel H., Schmidt H., Wühl H., Stewart G.R.: Phys. Rev. B 30 (1984) 2542. [84] Geibel C, Goldacker W., Keiber H., Oestreich V., Rietschel H., Wühl H.: Phys. Rev. B 30(1984) 6363. [85] Zhang X., Chen L., Ma M., Zhu Y, Zhang S., Liu R.: J. Appl. Phys. 109 (2011) 113523. [86] Smith J.F.: In Binary Alloy Phase Diagrams, edited by Massalski T.B., 2nd ed., vol. 3, ASM International, Materials Park, OH, 1990, p. 3528. [87] Williams J.T.: Trans. AIME 203 (1955) 345. [88] Rapp O., Benediktsson G.: Phys. Lett. A 74 (1979) 449. [89] MonctonD.E.: Solid State Commun. 13 (1973) 1779. [90] de Boer ER., Boom R., Mattens W.C.M., Miedema A.R., Niessen A.K.: Cohesion in Metals. Transition Metal Alloys, Elsevier Sei. Publ., The Netherlands, 1988. Appendix I: Houserova J. et al., Comp. Mater. Sei. 25 (2002) 562. 44 ELSEVIER Computational Materials Science 25 (2002) 562-569 COMPUTATIONAL MATERIALS SCIENCE www.elsevier.com/locate/commatsci Ab initio calculations of lattice stability of sigma-phase and phase diagram in the Cr-Fe system Jana Houserova a'*, Martin Friák b, Mojmír Šob b, Jan Vřešťál a a Faculty of Science, Institute of Theoretical and Physical Chemistry, Masaryk University, Kotlářská 2, CZ-611 37 Brno, Czech Republic b Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Žižkova 22, CZ-616 62 Brno, Czech Republic Abstract Total energy of pure metals in the sigma-phase structure and in the standard element reference (SER) structure were calculated by full-potential linear augmented plane waves method in the general gradient approximation at the equilibrium volume of all phases. Relaxation of lattice parameters of sigma-phase and SER structure were performed. The difference of total energy of sigma-phase and of standard element phase for pure constituents (A°Ef~SER) was used in a new two-sublattice model of sigma-phase, which was subsequently employed for calculation of phase diagram. Entropy term of Gibbs energy of elements in sigma-phase structure and excess Gibbs energy of mixing of sigma-phase have still to be adjusted to the experimental phase equilibrium data. This procedure was tested on the Fe-Cr system. © 2002 Elsevier Science B.V. All rights reserved. 1. Introduction Phase equilibria calculations performed by calculations of phase diagrams (CALPHAD) method are based on the axiom that complete Gibbs energy vs. composition curves can be constructed for all the structures exhibited by the elements right across the whole alloy system. This involves the extrapolation of G(x) curves of many phases into regions where they are either metastable or unstable and, in particular, the relative Gibbs energy for various crystal structures of the pure elements of the system must therefore be established [1]. Utilising the results from ab initio electronic Corresponding author. Tel.: +420-5-41129316. E-mail address: houser@chemi.muni.cz (J. Houserova). structure calculations may be very useful for describing thermodynamic properties of complicated phases in the systems exhibiting slow changes of Gibbs energy with temperature T and concentration x (molar fraction), such as in sigma-phase, u-phase, Laves phase etc. The sigma-phase was first described by Bain [2] in the Cr-Fe system in 1923. At present, Villars et al. [3] report on about 110 different intermetallic phases with sigma-phase structure. This structure (space group no. 136, P42/mnm) contains 30 atoms in the repeat cell distributed into five crystallo-graphically inequivalent sublattices (2a, 4f, 8i, 8i' and 8j). This structure is very brittle and stable and its inconvenient properties cause very strong degradation of materials (crack nucleation sites). In practice, it also develops in heat affected zones of welded superaustenitic stainless steels [4]; it was 0927-0256/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S0927-0256(02)00335-X Reprinted from Computational Materials Science, 25, Houserova J., Friák M., Šob M., Vřešťál J., Ab Initio Calculations of Lattice Stability of Sigma-Phase and Phase Diagram in the Cr-Fe System, 562-569, Copyright (2002), with permission from Elsevier. Appendix I: Houserova J. et al., Comp. Mater. Sei. 25 (2002) 562. 45 J. Houserova et al. I Computational Materials Science 25 (2002) 562-569 concluded there that it formed after longer ageing times in the temperature range of 500-1100 °C and that its composition was 55 wt.% of Fe, 22 wt.% of Cr, 11 wt.% of Mo and 5 wt.% of Ni. It is also known that high concentrations of Cr and Mo promote precipitation of sigma-phase. Therefore, it is very important to have more information about its region of stability. The ab initio (first-principles) electronic structure calculations are able to reproduce the difference of total energy between the standard element reference (SER) structures and the sigma-phase. Calculated results may constitute a basis for a new approach to the determination of phase equilibria and to the prediction of phase diagrams containing more complicated phases. The procedure of choosing equilibrium volume (corresponding to the minimum on the energy-volume curve) as the state of reference overcome the uncertainty connected with the use of experimental atomic volume of sigma-phases for total energy calculation of pure components in sigma-phase, although this procedure was successfully employed in [5]. The aim of this paper is to test the approach mentioned above on the Cr-Fe system. A new model for sigma-phase description [6] using ab initio calculations results is employed and verified. It turns out that for description of energetics of sigma-phase and for construction of phase diagram only structural energy differences A°Ef~SER for pure constituents are needed. Therefore, no ab initio calculations for sigma-phase systems have to be performed. 2. Calculations 2.1. Calculations of phase diagrams The CALPHAD in this paper is based on finding the minimum of the total Gibbs energy of the system at a constant pressure and temperature respecting the mass conservation law. Such a calculation is often performed by the CALPHAD method [1]. This method uses the structural Gibbs energy difference (the difference between the Gibbs energy of the phase in question and the Gibbs energy of SER) for various struc- 563 tures of pure elements, as e.g. bec, fee, hep, Laves phase, sigma-phase etc. These Gibbs energy differences are functions of pressure, temperature and volume. The total Gibbs energy of the system is given by Gto' = ^M/Gf, (1) f i.e. it is equal to the sum of Gibbs energies of all phases (Gf) multiplied by their volume fraction (wf). The Gibbs energy of a phase of certain composition is obtained by Gf = X^'X + G'd + GB + G"12* + • • •, (2) where yt is lattice fraction of the component i (the sum of lattice fractions in each lattice (sublattice) is equal to 1), °Cri is the Gibbs energy of pure element in the phase f, the term Gld describes the Gibbs energy of ideal mixing, GE is the excess Gibbs energy describing real mixing and Gmag is the magnetic contribution to the Gibbs energy. The Gibbs energy of a certain phase G^ is used as an input for phase diagram calculations. It is quite easy to obtain these Gibbs energies for less complicated structures, for example for the fee or bec structures, because they could be determined experimentally or by extrapolation to the pure components, and are summarised in various databases [7]. But it is not the case of the sigma-phase. At the beginning of sigma-phase studies, the Gibbs energy of bec phase was used instead of Gibbs energy of sigma-phase. Later on, the estimations were done using extrapolation of experimental data [8]. Now, one often applies the model of a substitutional structure (B)8(A)4(A,B)i8 or (B)io(A)4(A,B)i6. Such modelling is performed using the assumption that the atoms are ordered in two or more sublattices [9-11]. The problem consists in the dilemma into which sublattice each element goes and, further, how to reduce number of sublattices in order to restrict the number of model parameters. For a binary A-B sigma-phase (A being an element of the VIth group of the periodical table or lower, B being an element of the VIIth group or higher), a guideline for reducing the number of parameters was proposed as follows [10]: Reprinted from Computational Materials Science, 25, Houserova J., Friák M., Sob M., Vřešťál J., Ab Initio Calculations of Lattice Stability of Sigma-Phase and Phase Diagram in the Cr-Fe System, 562-569, Copyright (2002), with permission from Elsevier. Appendix I: Houserova J. et al., Comp. Mater. Sei. 25 (2002) 562. 46 J. Houserova et al. I Computational Materials Science 25 (2002) 562-569 564 (i) Combine all sublattices with the same coordination number (CN) and similar point symmetry into one. (ii) If more than one remain, combine the two with the highest CN into one. (iii) Arrange the reduced set of sublattices in the order of increasing CN. (iv) B elements will go preferentially into the first sublattice but it may dissolve some A. (v) The next sublattice will be preferentially filled with A but it may dissolve some B. (vi) If there is a third sublattice, it will be reserved for A. In the Cr(A)-Fe(B) system the CN of sublattices are following [9]: 2a (CN = 12), 4f (CN = 15), 8i (CN = 12), 8i' (CN = 14) and 8j (CN = 14). The first and third sublattice and the fourth and fifth sublattice are combined according to the point (i) in order to obtain this preliminary formula 16(gi'+8j) :4(4f): 10(2a+8i) • Than the sublattices are arranged in the order of increasing CN according to (iii) and are occupied by atoms in order to satisfy the points (iv)-(vi) getting 10(AB): 16(AB):4(A). At the end of the procedure of reducing of model parameters it is assumed [9] that the occupation of the second sublattice by A atoms is negligible (10(B):16(AB):4(A)) and the sublattice with mixed occupation is moved to the end of formula. Such a way, this procedure yields the formula (B)io(A)4(A,B)i6. The expression for Gibbs energy of sigma-phase having Fe atoms in the first sub-lattice and Cr atoms in the second and third sub-lattices is °G£e:Cr:Cr = 10°G^CC + 4°Ggcc + 16°Ggcc + C°T(T). (3) We can obtain a similar equation for the Gibbs energy of sigma-phase having Fe atoms in the first and the third sublattices and Cr atoms in the second sublattice: °GFe:Cr:Fe = 10°C + 4°Ggcc + 16°G^CC + C%e(T). (4) Here °Gti is Gibbs energy of pure component i (Cr or Fe) in phase f; hbec, hfee are symbols of hypothetical paramagnetic (non-spin-polarised) bec and fee phases, and Cf(T) is a temperature-dependent adjustable parameter. This parameter is defined by Cf{T)=Ai+BiTi, (5) where At, Bt are constants that can be adjusted to the phase equilibrium data. It is obvious from Eqs. (3) and (4) that it is not possible to describe the Gibbs energy of sigma-phase close to the pure elements regions using this model. The Gibbs energy of sigma-phase is described here as empirical combination of Gibbs energies of some absolutely different structures. The above mentioned procedure only enables us to estimate the lattice stability of metastable phase by means of known Gibbs energies of stable phases of pure constituents with the same CN. Using this approach, we are not able to express the Gibbs energy of pure constituents in sigma-phase structure, and Gibbs energy of sigma-phase has to be adjusted to phase equilibrium data. The physical background of this procedure is, therefore, questionable. Further, it is known from X-ray studies [12] that the mixing of the constituents takes place in all sublattices, which is not respected by the proposed approach. Electronic structure calculations could bring a substantial improvement of that model. Namely, knowledge of a correct value of the total energy difference between the sigma-phase and the SER-phase of pure constituents from first principles enables us to build up the Gibbs energy of the sigma-phase of pure elements on a physically correct energetic basis, and only the entropy term must be adjusted to phase equilibrium data. Using this idea and the knowledge that the sigma-phase does not behave like rigid stoichiometric phase (1 1), that means that the sublattices in sigma-phase are not occupied exclusively by one kind of atoms (mixing is possible), we have proposed a new physical (two-sublattice) model (1 1). This is a model of a solid solution, as e.g. fee or bec [1]. The label (11) for sigma-phase means that this solution phase contains two sublattices, each of Reprinted from Computational Materials Science, 25, Houserova J., Friák M., Sob M., Vřešťál J., Ab Initio Calculations of Lattice Stability of Sigma-Phase and Phase Diagram in the Cr-Fe System, 562-569, Copyright (2002), with permission from Elsevier. Appendix I: Houserova J. et al., Comp. Mater. Sei. 25 (2002) 562. 47 J. Houserova et al. I Computational Materials Science 25 (2002) 562-569 them having one lattice site (the one atom only could be placed here). In this thermodynamic description the five sub-lattices found in the X-ray experiments are reduced to two. This is possible because the mixing occurs in all five sublattices as mentioned above. This reduction give us the possibility to describe the sigma-phase in the whole composition region. In this procedure the results of first-principles calculations have crucial importance yielding the correct energetic basis of the model. The Gibbs energy of sigma-phase in binary system Cr-Fe of a certain composition in two-sublattice solution model may be expressed by GCr,Fe = yFe°Gpe +JCr°Ggr + G^Fe + G^Pe, (6) where y is lattice fraction of a component and °Gf is Gibbs energy of hypothetical sigma-phase that contains only one pure element. These energies are defined as °G,a = G,SER + A°E°-SER - S°T, (7) where °GfER is Gibbs energy of pure element in SER state, A°E7~SER expresses the total energy difference of hypothetical sigma-phase and standard state of a pure metal; this difference may be obtained from ab initio electronic structure calculations. Further, S? is the entropy term in the Gibbs energy. It is a constant adjustable to the experimental data. G^pe is the Gibbs energy of ideal mixing of metals in sigma-phase and may be expressed as Gc^Fe = RT(yCr ^ J& + J^Fe ln^e), (8) where R is the gas constant. GE'rCTFe is excess Gibbs energy describing real mixing in sigma-phase. We may write this energy as C=W*e[i°m +L1(T)(ycr -yve)+L2(T) X(jcr-J*e)2], (9) L°(T) =D + ET, L\T)=F + JT and L\T) =K + LT, (10) where L°(T), Ll(T), L2(T) are interaction parameters and D, E, F, J, K, L are fitting parameters adjusted to the phase data. 565 On the basis of presented equations the new two-sublattice model (1 1): (i) is able to describe the Gibbs energy and enthalpy dependencies vs. composition in the whole concentration region, (ii) has a solid physical background, (iii) yields a very simple description of sigma-phase based on ab initio calculated A°Ef~SER term for pure constituents. This model is used in the present paper for CALPHAD of Cr-Fe system. The Gibbs energies of phases needed for construction of phase diagram (bec, fee, liquid) were taken from recent assessments of thermodynamic data [7,11,13] and calculations of phase equilibria were performed by means of THERMO-CALC programme [14]. 2.2. Calculations of total energy difference (A°E^ER ) The structural energy differences between the SER structure and sigma-phase structure of pure constituents were calculated by means of the full-potential linearized augmented plane waves (FLAPW) method incorporated in the WIEN97 code [15] using the generalized gradient approximation [16] for the exchange-correlation term. In all cases, the minima of the total energy as a function of lattice parameters were found, as described in more detail below. By extensive testing, we have found that the changes of positions of atoms in the repeat cell of sigma-phase (within the limits found in literature) do not have a great effect on the total energy (the maximum change in energy was AE = 2 mRy/ atom and, in average, we had AE = 0.5 mRy/ atom). Therefore, we were able to keep the internal parameters constant during the calculations. Using various sigma-phases containing Fe (e.g. Fe-Cr, Fe-Mo etc.) and employing their crystal structure for calculating the total energy of hypothetical Fe sigma-phase, we have chosen that structure (i.e. that set of internal parameters) which exhibited the lowest total energy [17]. The same procedure was applied for hypothetical Cr sigma-phase; the Reprinted from Computational Materials Science, 25, Houserova J., Friák M., Sob M., Vřešťál J., Ab Initio Calculations of Lattice Stability of Sigma-Phase and Phase Diagram in the Cr-Fe System, 562-569, Copyright (2002), with permission from Elsevier. Appendix I: Houserova J. et al., Comp. Mater. Sei. 25 (2002) 562. 48 J. Houserova et al. I Computational Materials Science 25 (2002) 562-569 566 lowest total energy was obtained for the structure given in [18]. In the case of sigma-phase, preliminary optimisation of unit cell volume and c/a ratio was performed using the LMTO-ASA method [19]. Then the optimisations of RMT (muffin-tin radius) and of number of ^-points were done. The final FLAPW optimisation was done in the following way: at first the optimisation of volume at the constant c/a ratio was performed. The second step was the calculation of the dependence of the total energy vs. c/a ratio when the volume of the repeat cell (Vmm from the previous step) was kept constant. These two steps were repeated until the total energy converged to its minimum. Concerning the SER-phase (ferromagnetic bec Fe and antiferromagnetic bec Cr), the preliminary optimisation by LMTO-ASA method was not performed and the RMT parameter was used the same as in the sigma-phase calculation. We have used 2 atoms in the unit cell and found the equilibrium lattice constants corresponding to the minimum of total energy as a function of volume. Then the total energy difference per atom, 3. Results and discussion The total energy of sigma-phase of Cr and Fe as a function of the cell volume in the last step of optimisation described above is shown in Fig. 1. The full symbols represent the crossing points with the previous optimisation curves of c/a ratio at constant volumes. Because the total energies at the crossing points differ less than e = 0.2 mRy/atom, we could stop the optimisation at this level. It turns out that three optimisation steps are sufficient to obtain the equilibrium lattice parameters of sigma-phase with the accuracy needed. The calculated equilibrium values of lattice parameters and cell volumes are given in Table 1. The volume dependence of total energy of ferromagnetic Fe and antiferromagnetic Cr in bec structure is presented in Fig. 2 and the values of equilibrium lattice parameters are given in Table 2. C3 O o o V) -it ID r I + -555 556 -557 -558 559 -560 1 1 1 1 SIGMA - 1Fe / , i - \ / — \ i 1 * 1 i i a + B 1700 1900 2100 2300 2500 Volume [at.unit3 ] Fig. 1. Final FLAPW optimisation of the cell volume of sigma-phase (30 atoms) of pure Cr (□) and Fe (0) at constant c/a ratio (c/aFe = 0.5174, c/acr = 0.5237). Full symbols represent the crossing points with previous optimisation of total energy vs. c/a ratio. Reprinted from Computational Materials Science, 25, Houserova J., Friák M., Sob M., Vřešťál J., Ab Initio Calculations of Lattice Stability of Sigma-Phase and Phase Diagram in the Cr-Fe System, 562-569, Copyright (2002), with permission from Elsevier. Appendix I: Houserova J. et al., Comp. Mater. Sei. 25 (2002) 562. 49 J. Houserova et al. I Computational Materials Science 25 (2002) 562-569 567 Table 1 Values of equilibrium FLAPW lattice parameters (calculated) and cell volumes (30 atoms) for hypothetical sigma-phase of end members in system Cr-Fe Parameter Cr Fe Volume (a.u.3) 2301.38 1963.76 a (a.u.) 16.3792 15.5987 c (a.u.) 8.5783 8.0707 The deviation of the calculated equilibrium lattice constant from the experimental value is -0.43% for Cr and 0.27% for Fe. The total energies of equilibrium hypothetical sigma-phase and the SER state of Fe and Cr as well as their differences are given in Table 3 together with the values obtained for the experimental volumes of Cr-Fe sigma-phase [5]. It turns out that in case of Cr, the volume optimisation has somewhat larger effect than in case of Fe. Recently, we have attempted to apply the results of first-principles calculations to determine the phase diagram. In [6], the first-principles structural energy differences obtained on the basis of extrapolation of experimental volume of sigma-phase to the pure components [5] were used. For Table 2 Values of experimental and equilibrium FLAPW lattice parameters for SER-phase of antiferromagnetic Cr and ferromagnetic Fe Source Cr (ab initio) Cr [22] Fe (ab initio) Fe [22] a (a.u.) 5.41653 5.44 5.41438 5.40 Table 3 Values of equilibrium total energies per atom for sigma-phase and SER-phase of Cr and Fe and their differences Variable (Ry/atom) Cr (ab initio) Fe (ab initio) Total energy per atom of -2101.7603 -2545.5597 cr-phase Total energy per atom of -2101.7832 -2545.5927 SER Total energy difference per 0.0229 0.0330 atom (ct-SER) Total energy difference per 0.0154 0.0309 atom (ct-SER) [5] the other phases, thermodynamic description of Cr-Fe system, based on high temperature vapour pressure measurements results published in [20], was adopted. In spite of approximations used, -570 CS E BCC Cr a Volume [at.unit3 ] Fig. 2. The volume dependence of total energy of antiferromagnetic Cr and ferromagnetic Fe. The volume corresponds to two-atom unit cell. Reprinted from Computational Materials Science, 25, Houserova J., Friák M., Sob M., Vřešťál J., Ab Initio Calculations of Lattice Stability of Sigma-Phase and Phase Diagram in the Cr-Fe System, 562-569, Copyright (2002), with permission from Elsevier. Appendix I: Houserova J. et al., Comp. Mater. Sei. 25 (2002) 562. 50 568 /. Houserova et al. I Computational Materials Science 25 (2002) 562-569 calculated phase equilibria with sigma-phase were reproduced. The procedure described in this work is based on the first-principles total energy calculations at equilibrium atomic volume and on reliably assessed low-temperature thermodynamic data [11,13]. It represents a new approach to the calculations of phase equilibria in systems containing the sigma-phase. The temperature dependence of excess Gibbs energy of sigma-phase (entropy term) has still to be adjusted to phase equilibrium data, following the traditional CALPHAD method. Comparison of phase diagrams with sigma-phase in Cr-Fe system calculated by three-sublattice model [11] and by new two-sublattice model is shown in Fig. 3. The values of adjustable parameters employed in the CALPHAD are given in Table 4. The agreement of the phase diagram calculated by means of a new two-sublattice model employing equilibrium total energy values with experimentally determined phase equilibrium values [21] is better than for the case of three-sublattice model [11] or for the case of the two-sublattice model using the total energy values of pure constituents determined at the experimental atomic volume of the Cr-Fe sigma-phase [6]. Table 4 Values of parameters used in the Cr-Fe phase diagram calculation: parameters of description of cr-phase S H Parameter Cr Fe S" +0.7 +0.7 L° -133950 Ü +31000 I? -127000 Fig. 3. Comparison of phase diagrams of Cr-Fe. Thick line: calculated by the new two-sublattice model (for data see Table 4), thin line: calculated by three-sublattice model (data from [11]), stars: experimental data [21]. Fig. 4. Concentration dependence of Gibbs energy of phases at 1000 K in Cr-Fe system: (1) liquid phase, (2) bec phase, (3) fee phase, (4) sigma-phase (new two-sublattice model), (4a) sigma-phase (three sublattice model). Composition dependencies of Gibbs energy (at 1000 K) and enthalpy calculated using new two-sublattice model are shown in Figs. 4 and 5. Here we also show the differences between the results obtained using the old (three-sublattices [11]) and new (two-sublattices [6]) model of description of sigma-phase. It may be seen that the new two-sublattice model yields the values of Gibbs energy and enthalpy of phases in the whole composition region; the old model gives these quantities only in a limited range of concentrations. 4. Conclusions The results of ab initio calculations of total energy of sigma-phase and SER-phase of pure Reprinted from Computational Materials Science, 25, Houserova J., Friák M., Sob M., Vřešťál J., Ab Initio Calculations of Lattice Stability of Sigma-Phase and Phase Diagram in the Cr-Fe System, 562-569, Copyright (2002), with permission from Elsevier. Appendix I: Houserova J. et al., Comp. Mater. Sei. 25 (2002) 562. 51 J. Houserova et al. I Computational Materials Science 25 (2002) 562-569 0.0 0.2 0.4 0.6 0.8 1.0 Fe XCr Cr Fig. 5. Concentration dependence of enthalpy of phases in Cr-Fe system: (1) liquid phase, (2) bec phase, (3) fee phase, (4) sigma-phase (new two-sublattice model), (4a) sigma-phase (three-sublattice model). constituents performed by FLAPW method were used in a new model of calculation of phase equilibria in systems containing the sigma-phase. The procedure was tested on the Cr-Fe system. The phase diagram calculated using the new two-sublattice model (1 1) [6] yields better agreement with experimental data than that obtained by means of an older three-sublattice model [11]. The proposed procedure has a solid physical background and enables us to predict the stability region of sigma-phase in metallic materials. Acknowledgements This research was supported by the COST Action P3 (Simulation of physical phenomena in technological applications, Project nos. COST OC P3.90 and P3.10), by the Grant Agency of the Czech Republic (Project no. 106/99/1178) and by the Grant Agency of the Academy of Sciences of 569 the Czech Republic (Project no. A1010817). The use of computer facilities at the MetaCenter of Masaryk University, Brno, is acknowledged. References [1] N. Saunders, P. Miodównik, CALPHAD—A Comprehensive Guide, Elsevier, London, 1998. [2] E.C. Bain, Chem. Met. Eng. 28 (1923) 23. [3] P. Villars, L.D. Calvert, Pearson's Handbook of Crystal-lographic Data for Intermetallic Phases, ASM International, Materials park, OH, 1991. [4] S. Heino, E.M. Knutson-Wedel, B. Karsson, Mater. Sei. Technol. 15 (1999) 101-108. [5] J. Havránková, J. Vřešťál, L.G. Wang, M. Šob, Phys. Rev. B 63 (2001) 174101. [6] J. Vřešťál, Arch. Metall. 46 (3) (2001) 9. [7] A.T. Dinsdale, Calphad 15 (1991) 317. [8] C. Allibert, C. Bernard, G. Effenberg, H.-D. Nüssler, P.J. Spencer, Calphad 5 (1981) 227. [9] I. Ansara, T.G. Chart, A. Fernandez Guillermet, F.H. Hayes, U.R. Kattner, D.G. Pettifor, N. Saunders, K. Zang, Calphad 21 (1997) 171. [10] M. Hillert, Calphad 22 (1998) 127. [11] J.-O. Anderson, B. Sundman, Calphad 11 (1987) 83. [12] S.H. Algie, E.O. Hall, Acta Crystallographica 20 (1966) 142, cited from Ref. [7]. [13] B.-J. Lee, Calphad 17 (1993) 251. [14] B. Sundman, THERMO-CALC, version L, Royal Institute of Technology, Stockholm, 1997. [15] P. Blaha, K. Schwarz, J. Luitz, WIEN97, Vienna University of Technology, 1997; improved and updated Unix version of the original copyrighted WIEN code, which was published by P. Blaha, K. Schwarz, P. Sorantin, S.B. Trickey, Comput. Phys. Com-mun. 59 (1990) 399. [16] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. [17] H.Y. Yakel, Acta Crystallographica B 39 (1983) 20. [18] G.J. Dickins, Audrey, M.B. Douglas, W.H. Taylor, Acta Crystallographica 9 (1956) 297. [19] G. Krier, O. Jepsen, A. Burkhardt, O.K. Andersen, computer code TB-LMTO-ASA version 4.6, Max-Planck- Institut für Festkörperforschung, Stuttgart, 1994. [20] J. Vřešťál, J. Tomiska, P. Brož, Ber. Bunsenges. Phys. Chem. 98 (1994) 1601. [21] A.J. Cook, F.W. Jones, J. Iron Steel Inst., London 148 (1943) 217, and 223. [22] E.G. Moroni, T. Jarlborg, Phys. Rev. B 47 (1993) 3255. Reprinted from Computational Materials Science, 25, Houserova J., Friák M., Šob M., Vřešťál J., Ab Initio Calculations of Lattice Stability of Sigma-Phase and Phase Diagram in the Cr-Fe System, 562-569, Copyright (2002), with permission from Elsevier. Appendix II: Houserova J. et ed., CALPHAD 26 (2002) 513. 52 Calphad, Vol. 26, No. 4, pp. 513-522, 2002 Pergamon © 2003 Elsevier Science Ltd. All rights reserved 0364-5916/02/$ - see front matter Pll: S0364-5916(03)00003-8 Phase Diagram Calculation in Co-Cr System Using Ab Initio Determined Lattice Instability of Sigma Phase Jana Houserova1, Jan V eš ál1, Martin Friák2 and Mojmír Šob2 'institute of Theoretical and Physical Chemistry, Faculty of Science, Masaryk University, Kotlá ská 2, CZ-611 37 Brno, Czech Republic institute of Physics of Materials, Academy of Sciences of the Czech Republic, i kova 22, CZ-616 62 Brno, Czech Republic Corresponding author's e-mail: vrestal@chemi.muni.cz (Received May 28, 2002) Abstract The calculations of phase equilibria in the Co-Cr system were performed using the CALPHAD method on the basis of a new two-sublattices model of sigma phase. This model enables us to utilise the results of ab initio calculations of total energy differences between the sigma phase structure and the Standard Element Reference (SER) structures of pure metal at the relaxed lattice parameters (A°E'J~SER). Total energies were calculated by Full-Potential Linear Augmented Plane Waves (FLAPW) method in the General Gradient Approximation (GGA). The entropy contribution to the Gibbs energy of the pure elements in the sigma phase structure, and the excess Gibbs energy of mixing of the sigma phase were adjusted to the experimental phase equilibrium data. © 2003 Elsevier Science Ltd. All rights reserved. Introduction The sigma phase was first observed by Bain [1] in Cr-Fe system in 1923. However, it obtained its name a little bit later. Since its discovery, more than 40 different sigma phases have been found in various binary systems of transition metals (e.g. Co-Cr, Fe-Mo, Ni-V) [2]. Many material engineers were interested in the prediction of regions of stability of the sigma phase because it has very disadvantageous properties. It is well known that the sigma phase is extremely brittle and therefore it increases the probability of the formation of crack nucleation sites. On the other hand, it is very stable from the thermodynamic point of view. Therefore, the knowledge of the regions of stability of sigma phase is very desirable in the Co-Cr system. Experimental data on the sigma phase in the Co-Cr system were published in [3-10]. For the analysis of the region of stability of the sigma phase in the Co-Cr system by the CALPHAD method [11] it is necessary to know the dependence of the Gibbs energy on composition for all structures occurring in the system at all concentrations, even where these structures are metastable or unstable. The relative Gibbs energies of such hypothetical structures (that are unstable or metastable in some composition range) in the pure state are needed for this method. This gap may be bridged by ab initio electronic-structure calculations of lattice stability in the case of complex non-stoichiometric phase such as sigma phase, u phase, Laves phase etc. These calculations enable us to get some idea about the thermodynamic properties of those structures in such regions where the experimental data are unavailable (e.g. for the hypothetical sigma phase of the pure component). The only input for such calculations is the atomic numbers and concentrations of the constituents and the crystallographic data of the underlying structures. In the case of the hypothetical sigma phase for the pure elements, the uncertainty connected with the use of experimental lattice parameters of binary sigma phase is overcome by choosing the 513 Reprinted from CALPHAD, 26, Houserovä J., Vfest'äl J., Friäk M., Sob M., Phase Diagram Calculation in Co-Cr System Using Ab Initio Determined Lattice Instability of Sigma Phase, 513-522, Copyright (2002), with permission from Elsevier. Appendix II: Houserovä J. et ed., CALPHAD 26 (2002) 513. 53 514 J. HOUSEROVA et al. equilibrium state of that phase, corresponding to the minimum of the total energy as a function of the lattice parameters. The total energy differences between the sigma phase and the SER structure of the pure constituents can be implemented into the new model of description of the sigma-phase [12]. The aim of this paper is to verify this approach in the case of the Co-Cr system. Calculations Calculation of Phase Diagrams The behaviour of the system is defined by the minimum of the total Gibbs energy Glot at given conditions of pressure and temperature. Gtot is equal to the sum of the Gibbs energies of all phases (G1) multiplied by their volume fraction (wf), i.e. G,0,= XwfGf. (1) f The G1 is again a function of the thermodynamic conditions considered above, and it is defined as Gf.= Z y i ° Gf + °id + °E + G™8 + G"^' (2) i where y: is the lattice fraction of the component i, °G\ is the Gibbs energy of a pure constituent i in the phase f, the term Gld describes the Gibbs energy of ideal mixing, GE is the excess Gibbs energy describing real mixing, GmaB is the magnetic contribution and Gpres is the pressure contribution to the Gibbs energy. The °Gf constitute the main input information for phase diagram calculations based on the CALPHAD approach. These values are easily available for many structures of the pure elements, e.g. bec, fee, hep, because they are usually quite well measurable. They are summarised in various databases, e.g. [13]. However, there is a problem in the case of the sigma phase because it is much more complicated, it is unstable for the pure constituents and its region of stability is too narrow for reasonable extrapolation. This experimentally unsolvable problem can be treated using ab initio calculations. The ab initio approach allows us to replace methods of construction of the Gibbs energy for the sigma phase that use the combination of Gibbs energies of bec and fee structures according to similarity in coordination number [14,15], or that estimate it using extrapolation of experimental data [16]. The model of a substitutional structure (B)8(A)4(A,B)i8 or (B)io(A)4(A,B)i6 is the one of the most widely used methods in recent time [14,15] but it has some disadvantages. The first step of description of the sigma phase in this model is the reduction of number of sublattices. It is known from X-ray experiments that the sigma phase (space group No. 136, P42/mnm) contains 30 atoms in the unit cell distributed into five crystallographically inequivalent sublattices (2a, 4f, 8i, 8i' and 8j) [17], too many for "a reasonable CALPHAD description. Therefore, the number of sublattices should be reduced [14,15,18]. In [14], some general rules for such a reduction in the number of sublattices and for their occupation were formulated: (i) All sublattices with the same coordination number (CN) and similar point symmetry are combined into one. (ii) If more than one sublattice remains, combine the two with the highest CN into one. (iii) The reduced set of sublattices must be arranged in the order of increasing CN. (iv) B elements will go preferentially into the first sublattice but it may dissolve some A. (v) The next sublattice will be preferentially filled with A but it may dissolve some B. (vi) If there is a third sublattice, it will be reserved for A. If A is an element of the VIth group of the Periodical Table or lower, and B is an element of the VIIth group or higher, then, in the case of a binary sigma phase, the CN of individual sublattices in Co(A)-Cr(B) are, by analogy to [18], as follows: 2a (CN=12), 4f (CN=15), 8i (CN=12), 8i' (CN=14) and 8j (CN=14). The first and third sublattice and the fourth and fifth sublattice are combined according to the point (i) above to obtain the preliminary formula 16(8i'+8j) 4(4f) 10 (2a+8i). Then the sublattices are arranged in the order of increasing CN and are occupied by atoms in order to satisfy the points (iv) - (vi) getting 10(A,B):16(A,B):4(A). At the end of the procedure for reducing the number of model parameters, it is assumed [18] that the occupation of the second sublattice by A atoms is negligible, i.e. we have 10(B):16(A,B):4(A), and the sublattice with mixed occupation is moved to the end of the formula. So the resulting formula is 10(B) 4(A) 16(A,B) or, alternatively, (B)10(A)4(A,B)16. Reprinted from CALPHAD, 26, Houserovä J., Vfest'al J., Friäk M., Sob M., Phase Diagram Calculation in Co-Cr System Using Ab Initio Determined Lattice Instability of Sigma Phase, 513-522, Copyright (2002), with permission from Elsevier. Appendix II: Houserovä J. et ed., CALPHAD 26 (2002) 513. 54 PHASE DIAGRAM CALCULATION IN Co-Cr SYSTEM 515 The Gibbs energy of the sigma phase depends on occupation of the third sublattice, where the mixing occurs, and therefore following two equations are useful: "GcaCrCr = 10°G£f + 4°Ggcc + 16°G^C + C£r(T). (3) °G£a&co = 10°G£C + 4°Gg" + 16°G£r + Q0(T). (4) where0Gf is the Gibbs energy of a pure constituent i (Cr or Co) in the phase f, hbcc' hfcc are symbols of hypothetical paramagnetic (non-spin-polarised) bec and fee phases, and C"(T) is a temperature-dependent adjustable parameter that is defined by C°(T)= A, + BjTj, (5) with Aj, Bj being the constants that can be adjusted to phase equilibrium data. The problem of this model is that it does not capture the main idea of the CALPHAD method, i.e. it is not based on the lattice stability of the pure constituents. Therefore, it is not able to describe the thermodynamic properties of the system in the regions close to the pure constituents (the knowledge of the Gibbs energy in those regions is indispensable for computing ternary and higher-order systems). Nevertheless, the most serious objection from the physical point of the view is that the Gibbs energy of the sigma phase is described as an empirical combination of Gibbs energies of simple structures that are not related to the sigma phase. Our new physical two-sublattices model describes the thermodynamic properties of the sigma phase in the whole composition range [12]. This is a model of solid solution, similar for example to the fee or bec one [11]. Here the sigma phase is described as a two-sublattices structure where each sublattice contains one atomic site per unit cell. The model is capable to use such a strong reduction in the number of sublattices (from five to two) because it is known from X-ray experiments [19] that the sigma phase behaves more or less like a solution, i.e. there is a mixing of atoms in all five sublattices, which was not respected by the previous model. Site occupancies of the binary sigma phases were critically reviewed by Ansara et al. [18]. Experimental evidence for fully preferentially occupied lattice sites was found in less than one half of studied systems. In particular, for Co-Cr system, Algie and Hall [19] reported mixing of atoms in all 5 inequivalent lattice sites. Within the two-sublattices solution model, the Gibbs energy of the sigma phase in binary Co-Cr system is given by G°crCo = yco0G£o + yCr°G£r + Ggjk + GE£0, (6) where y( is the lattice fraction of a component i and °Gfis the Gibbs energy of a hypothetical (unstable) sigma phase that contains only one pure constituent i. Such energies of pure sigma phases are defined as °Gf = GfER+ A°E?-SER- SfT, (7) where °GfER is the Gibbs energy of a pure constituent in the SER state and A°E?~SER denotes the total energy difference between hypothetical sigma phase and the standard state of that constituent. This total energy difference may be obtained from ab initio electronic-structure calculations. Further, S° is the entropy term in the Gibbs energy. It is a constant adjustable to the experimental data. G'^0 is the Gibbs eneTgy of the ideal mixing of metals in the sigma phase and may be expressed as Gg£o = RT (y& In yCr + yCo In yCo), (8) where R is the gas constant. Gq£~0 is the excess Gibbs energy describing real mixing in the sigma phase. It was expressed by frequently used Redlich-Kister equation Gc£o = yoyco(L° (T) + L1 (T) (yCr - Vcc) + L2 (T) (yCr - yCo)2), (9) L° (T) = D + ET, L1 (T) = F + JT and L1 (T) = K + LT, (10) where L°(T),L1(T), L2(T) are interaction parameters and D,E,F,J,K,L are fitting parameters adjusted to the experimental phase equilibrium data. Utilization of ab initio calculations in the determination of the A°Er~SBR term constitutes a considerable improvement of the CALPHAD approach to the phase diagrams. It gives us the opportunity to determine the Reprinted from CALPHAD, 26, Houserovä J., Vfest'äl J., Friäk M., Sob M., Phase Diagram Calculation in Co-Cr System Using Ab Initio Determined Lattice Instability of Sigma Phase, 513-522, Copyright (2002), with permission from Elsevier. Appendix II: Houserovä J. et ed., CALPHAD 26 (2002) 513. 55 516 J. HOUSEROVA et al. Gibbs energy of the sigma phase for the pure constituents on a physically correct energy basis. Let us note, however, that we are not able to incorporate the contribution from the zero-point motion. It is not excluded that, when evaluating the structural energy differences, this contribution may play quite an important role. There is only one disadvantage in this model. We are not able to calculate the entropy of the system and, therefore, it is still adjusted to phase equilibrium data. This model was already successfully used in the case of Cr-Fe system [20] and the Co-Cr system is modelled in the present paper using existing assessments for remaning phases. Calculation Of Total Energy Differences The total energies of both structures (SER and sigma phase) at their equilibrium volumes were calculated within the Full-Potential Linearized Augmented Plane Waves (FLAPW) method incorporated in the WIEN97 code [21] using the Generalised Gradient Approximation (GGA) [22] for the exchange-correlation term. At the beginning of our calculations, the internal parameters (i.e. the positions of all atoms in the unit cell) of the hypothetical sigma phase for pure cobalt were chosen. The total energies of the mentioned structure were calculated at constant lattice parameters [23] using various internal parameters of binary sigma-phases that contained cobalt (e.g. Co-Cr, Co-Mo etc.). We have chosen that set of internal parameters, which exhibited the lowest total energy [24]. The same procedure was applied for the choice of the internal parameters of a hypothetical Cr in the sigma phase structure. The lowest total energy of Cr sigma phase was found at the lattice parameters given in [25] and internal parameters given in [23]. In the following calculations, the internal parameters were kept constant because we have found that their optimisation does not have any significant influence on the total energy of the sigma phase. The changes in total energy caused by changing the internal parameters within the limits found in the literature did not exceed the value of 2 mRy/atom and, on average; they amounted to 0.5 mRy/atom. Now, to estimate the equilibrium values of the lattice constants of the sigma-phase, we performed auxiliary calculations with the Linear Muffin-Tin Orbital method in the Atomic Sphere Approximation (LMTO-ASA) [26]. Then we continued the FLAPW calculations. The optimal RMT (muffin-tin radius) and number of k-points needed for the calculations were chosen. The RMT parameters used in this work are 1.97 a.u. in the case of cobalt and 2.1 a.u. in the case of chromium. Concerning the number of k-points in the irreducible zone found by preliminary optimisation, the used values are 42 in the case of cobalt and 36 in the case of chromium. The process of optimisation of the lattice parameters is very simple. It is based on repeating two steps until the change in the total energy is small enough. These steps are the optimisation of the volume at constant c/a ratio (constant shape of unit cell) and optimisation of the c/a ratio at constant volume (Vmm from the previous step). The calculations for the SER-phases (ferromagnetic hep Co and antiferromagnetic bec Cr) were not so time-consuming because the LMTO-ASA calculations had not been performed and the RMT parameters had been taken from the sigma phase calculations. The k-points convergence tests resulted in using 320 k-points for cobalt and 120 k-points for chromium in all following calculations. The optimisation of the unit cells with two atoms had the same theoretical basis as in the sigma phase calculations but the optimisation of the c/a ratio was not employed in the case of the cubic structure, and therefore the calculations of the bec structure were finished already after the first optimisation step. Results And Discussion The calculated total energies for the pure constituents in both structures were used for evaluating the lattice stability of the sigma phase characterized by the total energy difference A0Ef~SER = 0 e° - 0 EfER . The profiles of the total energy as a function of volume in the case of both constituents in the sigma phase arrangement obtained from the last step of optimisation are shown in Fig. 1. The crossing points with previous optimisation curves are represented by full symbols. The difference in total energies obtained in the Reprinted from CALPHAD, 26, Houserovä J., Vfest'äl J., Friäk M., Sob M., Phase Diagram Calculation in Co-Cr System Using Ab Initio Determined Lattice Instability of Sigma Phase, 513-522, Copyright (2002), with permission from Elsevier. Appendix II: Houserovä J. et ed., CALPHAD 26 (2002) 513. 56 PHASE DIAGRAM CALCULATION IN Co-Cr SYSTEM 517 last two steps was smaller than 0.2 mRy/at, and therefore we could stop the optimisation at this level. The lattice parameters of the unit cell corresponding to the minimum of total energy are listed in the Table 1. Table 1. FLAPW calculated equilibrium volumes per atom (a.u.3) and lattice parameters (a.u.) of the thirty-atom unit cell of hypothetical sigma phase of end members Cr and Co. Parameter Cr Co Volume 76.7127 69.1070 a 16.3792 15.8602 c 8.5783 8.2419 The optimised total energy profiles for the SER structures (the first one for antiferromagnetic Cr in the bec structure and the third one in the case of ferromagnetic Co in the hep structure) are presented in Fig.2. The optimised lattice parameters together with the experimental ones are listed in Table 2. Table 2. Values of experimental and FLAPW equilibrium lattice parameters (a.u.) for the SER phase of antiferromagnetic Cr and ferromagnetic Co. Lattice parameter Cr Co ab-initio experiment [30] ab-initio experiment [2] a 5.41653 5.44 4.62149 4.74 c 5.41653 5.44 7.40598 7.69 The deviation of the calculated equilibrium lattice constant from the experimental values is -0.43% for Cr and -2.41% for the a lattice parameter and -3.68% for the c lattice parameter in the case of Co. The calculated total energies for the optimised structures and their differences are summarised in Table 3. Table 3. Ab initio calculated values of equilibrium total energies per atom (Ry/atom) for sigma phase and SER phase of Cr and Co and their differences. Variable Cr Co Total energy per atom of a-phase Total energy per atom of SER Total energy difference per atom (a-SER) -2 101.7603 -2 786.8954 -2 101.7832 -2 786.9190 0.0229 0.0236 It is well known that the energy difference between the bec and fee structures and fee and hep structures of Cr and Co, predicted by first principles methods [27], are substantially larger than those estimated by the CALPHAD approach. So, it is not surprising that the energy differences between the bec and sigma-phase Cr or hep and sigma-phase Co are so large. These total energy differences were used in the new two-sublattices model presented in [12] for phase diagram calculations. The temperature dependence of the excess Gibbs energy of the sigma phase (entropy term) had still to be adjusted to phase equilibrium data, following the traditional CALPHAD method. The calculations of phase diagram and thermodynamic values were performed by means of the THERMO-CALC program [28]. The recent assessment [29] gave us the Gibbs energies of all phases (bec, fee, hep, and liquid) that exist in the Co-Cr system using data for the pure constituents from [13]. The final calculated phase diagram is given in Fig.3 (full lines). The dashed lines represent the phase diagram calculated by the old model [29]. There is an important improvement in the position of the line that describes the equilibrium between the sigma phase, the paramagnetic hep, and the ferromagnetic hep phases at the Co-rich side. Our calculated position corresponds better to that reported in [30] which is approximately 610 K. The Reprinted from CALPHAD, 26, Houserovä J., Vfest'äl J., Friäk M., Sob M., Phase Diagram Calculation in Co-Cr System Using Ab Initio Determined Lattice Instability of Sigma Phase, 513-522, Copyright (2002), with permission from Elsevier. Appendix II: Houserovä J. et ed., CALPHAD 26 (2002) 513. 57 518 J. HOUSEROVA era/. phase diagram calculated using the new model is in good agreement with experimental data given in [3-10]. The values of the adjustable parameters used in this calculation are summarised in Table 4. Table 4. Values of the adjustable parameters for the sigma phase (eqs. (7), (9), and (10)) used in the calculation of the phase diagram of Co-Cr. The values of E, J, and L in eq. (10) were set to zero. PARAMETER Cr Co S° + 0.7 +0.75 L° -115 950 L' + 10 800 L2 - 95 000 The composition dependences of the Gibbs energy and the enthalpy were calculated for both models at 1200 K and they are shown in Figs. 4 and 5. We may see that the lines 5 and 5a obtained using the new two-sublattices model and the three-sublattices model, respectively, are quite different. Regrettably, available experimental values of enthalpy of formation provide no possibility to prefer one of them (see Fig. 5). It is worth noting that the description obtained from the old model is constrained to a limited range of concentrations while the two-sublattices model [12] provides these values in the whole range of composition. Conclusions The ab initio calculations of lattice stability for various structures performed by FLAPW method provide a possibility for improving phase diagram calculations. The results of ab initio calculations may be utilised in a new two-sublattices model [12] that yields a better agreement with experimental data than the old three-sublattices model [15]. In the present paper, the procedure was tested on the Co-Cr system. Our approach has a solid physical background and enables us to predict the region of stability of the sigma phase in metallic materials. Acknowledgements This research was supported by the Grant Agency of the Czech Republic (Project No. 106/02/0877). The use of computer facilities at the MetaCenter of Masaryk University, Brno, is acknowledged. References 1. E. C. Bain, Chem. and Met. Eng., 28 (1923) 23. 2. P. Villars and L.D.Calvert, Pearson's Handbook of Crystallographic Data for Intermetallic Phases, ASM International, Materials Park, OH, 1991. 3. C.A. Allibert, C. Bernard, N. Valigant and M. Dombre, J. Less-Common Met., 59 (1978) 221-228. 4. F. Wever and U. Hashimoto, Mitt. Kaiser-Wilhelm-Inst. Eisenforsch., 11 (1929) 293-308. 5. Y. Matsungana, Kinzoku-no-Kenkyu, 8 (1931) 549-561. 6. A.G. Metcalfe, Trans. A1ME, 197 (1953) 357-364. 7. A. Chiba, Ph.D. thesis, Tohoku Univ. Sendai, Japan, 1971. 8. Z. Jin, Scand. J. Metall, 10 (1981) 279-287. 9. M. Hasebe, K. Oikawa and T. Nishizawa, J. Jpn. Inst. Met, 46 (1982) 557-583. 10. J.W. Smits, S.B. Luitjens and F.J.A. den Broeder, J. Appl. Phys., 55 (1984) 2260-2262. 11. N. Saunders and P. Miodownik, CALPHAD - A Comprehensive Guide, Elsevier, London, 1998. 12. J. V es al, Archives of Metallurgy, 46 (2001) 239-247. 13. A.T. Dinsdale, Calphad, 15 (1991) 317-425. Reprinted from CALPHAD, 26, Houserovä J., Vfest'äl J., Friäk M., Sob M., Phase Diagram Calculation in Co-Cr System Using Ab Initio Determined Lattice Instability of Sigma Phase, 513-522, Copyright (2002), with permission from Elsevier. Appendix II: Houserovä J. et ed., CALPHAD 26 (2002) 513. 58 PHASE DIAGRAM CALCULATION IN Co-Cr SYSTEM 519 14. M. Hillen, Calphad, 22 (1998) 127-133. 15. J.-O. Anderson and B. Sundman, Calphad, 11 (1987) 83. 16. C. Allibert, C. Bernard, G. Effenberg, H.-D. Nüssler and P.J Spencer, Calphad, 5 (1981) 227. 17. J.L.C. Daams, P. Villars and J.H.N. van Vucht, Atlas of Crystal Structure Types for Intermetallic Phases, vols. 1-4, ASM International, 1991. 18. I. Ansara, T.G. Chart, A. Fernandez Guillermet, F.H. Hayes, U.R. Kattner, D.G. Pettifor, N. Saunders and K. Zang, Calphad, 21 (1997) 171-218. 19. S.H. Algie and E.O. Hall, Acta Crystallographica, 20 (1966) 142. 20. J. Houserovä, M.Friak, M. Sob and J. V es äl, Computational Materials Science, in print. 21. P. Blaha, K. Schwarz and J. Luitz, WIEN97, Vienna University of Technology 1997 (improved and updated Unix version of the original copyrighted WIEN code, which was published by P. Blaha, K. Schwarz, P. Sorantin and S.B. Trickey in Comput. Phys. Commun. 59 (1990) 399). 22. J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh and C. Fiolhais, Phys. Rev. B, 46(1992) 6671. 23. G.J. Dickins, Audrey M.B.Douglas and W.H.Taylor, Acta Crystallographica, 9 (1956) 297. 24. J.B. Forsyth, and d'Alte da Viega, Acta Crystallographica, 16 (1963) 509-512. 25. H.L. Yakel, Acta Crystallographica B, 39 (1983) 20-28. 26. G. Krier, O. Jepsen, A. Burkhardt and O.K. Andersen, Computer code TB-LMTO-ASA version 4.6, Max-Planck-Institut für Festkörperforschung, Stuttgart, 1994. 27. H.L. Skriver, Phys.Rev.B, 31 (1985) 1909. 28. B. Sundman, THERMO-CALC, version L, Royal Institute of Technology, Stockholm, 1997. 29. A. Kusoffski and Bo Jansson, Calphad, 21 (1997) 321-333. 30. G. W. Qin, K. Okinawa, T. Ikeshoji, R. Kainuma and K. Ishida, Journal of Magnetism and Magnetic Materials, 234 (2001) L1-L5. 31. H.B. Bell, J.p. Hajra,F.H. Putland and P.J. Spencer, Mat.Sei Journal, 7 (1973) 185. 32. D.B. Downie and F.Arslan, J.Chem. Thermodynamics, 15 (1983) 654. 33. A.A. Zubkov, B.M. Mogutnov and N.O. Shaposhnikov, Dokl. Akad. Nauk SSSR, 31 (1990) 388. [Dokl. Phys. Chem., 311, (1990) 239.] Reprinted from CALPHAD, 26, Houserovä J., Vfest'äl J., Friäk M., Sob M., Phase Diagram Calculation in Co-Cr System Using Ab Initio Determined Lattice Instability of Sigma Phase, 513-522, Copyright (2002), with permission from Elsevier. Appendix II: Houserova J. et ed., CALPHAD 26 (2002) 513. 59 520 J. HOUSEROVA etal. -554 r SIGMA -i (S tí B -555- -556: m g] h CN + O U -5571- -558^ o Co Cr 3 sa -559 í -560 60 65 70 75 80 Volume [a.u.3/atom] 85 Fig. 1. Final FLAPW optimisation of the volume per atom of the sigma phase (30 atoms/cell) for pure Cr (squares) and Co (diamonds) at constant c/a ratio (c/ac0 - 0.5237 c/aCr= 0.5197, respectively.). Full symbols represent the crossing points with previous optimisation profile of total energy (per atom) vs. c/a ratio. -570 r já o er, cr, + o U ** -5751-- -580 -585- -590 -595 Cr - BCC Co -HCP m r* S h + Kí Kí O O Dl 65 70 75 80 85 90 Volume [a.u. /atom] Fig. 2. The volume dependence of the total energy (per atom) of antiferromagnetic Cr and ferromagnetic Co. Reprinted from CALPHAD, 26, Houserovä J., Vfest'äl J., Friäk M., Sob M., Phase Diagram Calculation in Co-Cr System Using Ab Initio Determined Lattice Instability of Sigma Phase, 513-522, Copyright (2002), with permission from Elsevier. Appendix II: Houserova J. et al, CALPHAD 26 (2002) 513. 60 PHASE DIAGRAM CALCULATION IN Co-Cr SYSTEM 521 0) i_ -I—> fü i_ Cü CL £ cu 2200 2000 Fig. 3. Comparison of phase diagrams of Co-Cr. Full line: calculated by the new two-sublattices model (this work, for the adjustable parameters see Table 4), dashed line: calculated by the three-sublattices model [28], the experimental data: © [3], V [4], • [5], + [6], [7], 0 [8], * [9], [10]. The dotted line represents the concentration dependence of the Curie temperature [29], letter f denotes the ferromagnetic phase and p is the paramagnetic phase. Reprinted from CALPHAD, 26, Houserova J.. Vřešťál J., Friák M., Šob M.. Phase Diagram Calculation in Co-Cr System Using Ab Initio Determined Lattice Instability of Sigma Phase. 513-522, Copyright (2002), with permission from Elsevier. Appendix II: Houserova J. et ed., CALPHAD 26 (2002) 513_ .61 522 J. HOUSEROVA era/. -15 ~i-1-r~ Fig. 4. Concentration dependence of the Gibbs energy for the Co-Cr system at 1200 K. 1: liquid phase, 2: bec phase, 3: hep phase, 4: fee phase, 5: sigma phase (new two-sublattices model), 5a: sigma phase (three-sub lattices model). I Fig. 5. Same as for Fig. 4 but for the concentration dependence of the enthalpy in the Co-Cr system at 1200 K. Available experimental data are denoted as + [31], A [32], 0 [33]. Reprinted from CALPHAD, 26, Houserovä J., Vfest'al J., Friäk M., Sob M., Phase Diagram Calculation in Co-Cr System Using Ab Initio Determined Lattice Instability of Sigma Phase, 513-522, Copyright (2002), with permission from Elsevier. Appendix III: Chvátalova K. et al, J. Alloy. Compd. 378 (2004) 71. 62 Journal of ALLOYS ELSEVIER Journal of Alloys and Compounds 378 (2004) 71-74 www.elsevier.com/locate/jallcom First-principles calculations of energetics of sigma phase formation and thermodynamic modelling in Fe-Ni-Cr system K. Chvátalova3'*, J. Houserováb, M. Šobb, J. Vřešťál3 a Faculty of Science, Institute of Physical Chemistry, Masaryk University, Kotlářská 2, 611 37 Bmo, Czech Republic b Institute of Physics of Materials, Academy of Sciences of Czech Republic, Žižkova 22, 616 62 Bmo, Czech Republic Received 1 September 2003; accepted 14 October 2003 Abstract A combination of full-potential linear augmented plane-wave (FLAPW) electronic structure method (for elemental constituents) with simpler linear muffin-tin orbital method in the atomic-sphere approximation (LMTO-ASA method, for mixtures) was employed for nickel systems (Ni-Fe and Ni-Cr) to calculate the energy of formation of sigma phase with respect to standard element reference (SER) structures of pure constituents. In all cases, the optimisation of the equilibrium volume of sigma phase and of the SER structures was performed. Calculated energies of formation of sigma phase with respect to SER state of pure constituents in Ni-Cr and Ni-Fe systems were compared with the results of thermodynamic modelling. Phase diagram of ternary Fe-Ni-Cr system determined with the help of ab initio calculated sigma phase parameters in pure elements are in a good agreement with experimental data. Enthalpies of formation of sigma phase in Ni-Cr and in Fe-Ni systems obtained by thermodynamic modelling agree reasonably well with the energy of formation in these systems, calculated ab initio. © 2004 Elsevier B.V. All rights reserved. Keywords: Transition metals; Sigma phase; First-principles calculations 1. Introduction The brittle sigma-phase (5 inequivalent lattice sites, 30 atoms per repeat cell, with large concentration range of stability, similarly as in solutions) was first described by Bain [1] in Fe-Cr system. Recently, Villars et al. [2] have reported 110 intermetallic phases with stable sigma-phase structure. Energy of formation of sigma phase can be calculated ab initio for pure constituents and for alloys even in those systems where a stable sigma phase does not exist. In this case, experimental verification of calculated values is not possible. On the other hand, thermodynamic modelling in ternary systems containing sigma phase yields the optimised values of enthalpy of formation of sigma phase, which can be compared with calculated energies of formation. Therefore, the thermodynamic modelling of sigma phase in Fe-Ni and * Corresponding author. Tel.: +420-541-129-316; fax: +420-541-211-214. E-mail address: chvatalova@email.cz (K. Chvátalova). 0925-8388/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2003.10.071 Cr-Ni systems may be performed by CALPHAD (CALcu-lation of PHAse Diagrams) method based on values of thermodynamic functions of phases where ab initio calculated energy differences for sigma- and standard element reference (SER) phases for pure constituents are used. The ternary system Fe-Ni-Cr, where the sigma phase is stable in the Fe-Cr basis with addition up to 15 wt.% Ni, enables us to perform this modelling as a boundary condition. Modelled phase diagrams in both binaries and also in ternary Fe-Ni-Cr system will be confronted with experimental data and the energies of formation of (hypothetical) sigma phases calculated ab initio in Ni-Cr and Fe-Ni systems will be compared with thermodynamically optimised values of enthalpy of formation. 2. The model of sigma-phase Knowledge of the value of the total energy difference between the sigma phase and SER phase of end-members calculated from first-principles (we use the Full Potential Reprinted from Journal of Alloys and Compounds, 378, Chvátalova K., Houserova J., Sob M., Vřešťál J., First-principles Calculations of Energetics of Sigma Phase Formation and Thermodynamic Modelling in Fe-Ni-Cr System, 71-74, Copyright (2004), with permission from Elsevier. Appendix III: Chvátalova K. et al, J. Alloy. Compd. 378 (2004) 71. 63 72 K. Chvátalova et al./Journal of Alloys and Compounds 378 (2004) 71-74 Augmented Plane wave (FLAPW) method, the WIEN97 code [3]) makes it possible to determine the lattice stability of pure constituents (Gibbs energy differences) on physically correct energetic basis. Our model of sigma phase stems from the two-sublattice model, which is similar to a model of solid solution phase [4], and the structural energy differences A£tot ,slgma_SER between the sigma and SER phase of pure constituents are obtained from first-principles electronic structure calculations. The Gibbs energy of the sigma-phase may be then expressed by means of the following relations: TS. ideal i y^U i + Cr , (1) where o sigma _ o^SER _|_ sigma-SER _ yg_vib,sigma-SER Vdeal = -iffi,-(x,-mx,-) (2) (3) (4) where L-parameters can be temperature dependent. The entropy term (Eq. (2)) and the excess term (Eq. (4)) have to be adjusted to phase equilibrium data. Fig. 1. Calculated phase diagram of Fe-Ni-Cr system at 1073 K according to the present model of sigma-phase (full lines), compared with the phase diagram calculated by the three-sublattice model [8] (dashed lines), using data from [9], and with experimental data: triangles for bcc/fcc from [10], squares for fcc/sigma from [11]. 3. Results and discussion First-principles total energy calculations (FLAPW method was employed, the energies were optimised at relaxed equilibrium volume) yield the following structural energy differences between the sigma and SER phases at 0 K for Fe, Cr and Ni [3]: A i? sigma-i A J7 „ sigma-i A i? sigma-5 A-Etot.Ni B (this work). -.A33301mo\-1 : 30070 J mop1 = 31950 J mol : 33.0mRyat_1, : 22.9mRyat_1, = 24.3mRyar1 Total energies of the SER phases were calculated in the corresponding spin-polarized state (ferromagnetic Fe and Ni, antiferromagnetic Cr). The values of A5,vib>sigma"SER for Fe, Cr and Ni were adjusted to phase diagram data; they were found to be 0.7JKT1 mol-1 in all three elements studied (for Fe and Cr, see also [3]). The excess term GE was expressed by Redlich-Kister polynomial (4), and the parameters L were also adjusted to phase equilibrium data. The resulting values (in Jmol-1) amount to (Lo-:Fe from [5]): 0.4 0.6 1 Ni Fig. 2. Calculated phase diagram of Fe-Ni-Cr system at 1173 K according to the present model of sigma-phase (full lines), compared with the phase diagram calculated by the three sublattice model [8] (dashed lines), using data from [9] and with experimental data: triangles for bcc/fcc from [12], squares for FCC/sigma from [11], and circles for FCC from [13]. °LCr:Fe = -133950 lLCrFe = 31000 2LCr:Fe = -127000 °LCr:Ni = -104000 ^CnNi = -14000 2LCr:Ni = 80000 °ÍFe:Ni = "7000 ^L-Fe-m = -40000 2LFe:Ni = -50000 Reprinted from Journal of Alloys and Compounds, 378, Chvátalova K., Houserova J., Sob M., Vřešťál J., First-principles Calculations of Energetics of Sigma Phase Formation and Thermodynamic Modelling in Fe-Ni-Cr System, 71-74, Copyright (2004), with permission from Elsevier. Appendix III: Chvátalova K. et al, J. Alloy. Compd. 378 (2004) 7h .64 K. Chvátalova et al./Journal of Alloys and Compounds 378 (2004)71-74 73 Fig. 3. Gibbs energy of phases in the Fe-Ni system at 1000 K: (1) liquid, (2) bcc, (3) fee, (4) sigma, present model. Thermodynamic optimisation. No Gibbs energy of sigma phase according to [8,9]. Fig. 5. Gibbs energy of phases in the Ni-Cr system at 1000 K: (1) liquid, (2) bcc, (3) fee, (4) sigma, present model, (4a) sigma according to [8,9]. Thermodynamic optimisation. Further, we performed first-principles calculations of total energies both of pure constituents and mixtures in the sigma-phase at OK by means of a simpler LMTO-ASA method [6]. In this way, we obtained the concentration dependence of the total energy £tot of sigma-phase in various systems. Combining LMTO-ASA and FLAPW calculations, we obtained the energy of formation (Figs. 4 and 6). Detailed description of this procedure may be found in [7], where we applied it for Fe-Cr and Co-Cr systems using unrelaxed volumes of the sigma phase. As it follows from Figs. 1 and 2, the phase diagram calculation of Fe-Ni-Cr system at 1073 and 1173 K with the help of the sigma-phase description by means of the present model and ab initio results yields good agreement with ex- Fig. 4. Comparison of optimized enthalpy of formation (full line, thermodynamic calculations) and ab initio calculated energy of formation (diamonds, ab initio calculations) of sigma-phase in the Fe-Ni system with respect to energies of sigma phases of both constituents. The straight line denoted by SER shows the weighted average of total energies of SER states of pure constituents. Fig. 6. Comparison of optimized enthalpy of formation (full line, thermodynamic calculations) and ab initio calculated energy of formation (diamonds, ab initio calculations) of sigma-phase in the Ni-Cr system with respect to energies of sigma phases of both constituents. The straight line denoted by SER shows the weighted average of total energies of SER states of pure constituents. Reprinted from Journal of Alloys and Compounds, 378, Chvátalova K., Houserova J., Sob M., Vřešťál J., First-principles Calculations of Energetics of Sigma Phase Formation and Thermodynamic Modelling in Fe-Ni-Cr System, 71-74, Copyright (2004), with permission from Elsevier. Appendix III: Chvátalova K. et al, J. Alloy. Compd. 378 (2004) 71. 65 K. Chvátalova et al./Journal of Alloys and Compounds 378 (2004) 71-74 74 perimental data and also with calculations based on the three-sublattice model [8]. Optimised thermodynamic functions describing the Fe-Ni and Ni-Cr systems are shown in Figs. 3-6. Figs. 3 and 5 display the Gibbs energies of various phases in Fe-Ni and Ni-Cr systems and Figs. 4 and 6 compare the optimised enthalpies of formation with the energies of formation with respect to energies of sigma phases of both constituents. The diamonds represent the values of energies of formation of binary sigma phases with various occupancies of sublattices [7]. Similarity in calculated energies of formation in Fe-Cr [5] and Ni-Cr systems is a strong support for the existence of sigma-phase in Ni-Cr system reported in [14] 30 years ago. From the results presented in our article, it may be concluded that ab initio calculated energy differences between the sigma and SER structures for pure constituents can be successfully employed in the CALPHAD method. Acknowledgements This research was supported by the Grant Agency of the Czech Republic (Projects Nos. 106/02/0877 and 106/03/P002). First-principles electronic structure calculations were performed using the facilities of Metacentrum of Masaryk University Brno and phase equilibria were determined by using Thermocalc software. References [1] E.C. Bain, Chem. Met. Eng. 28 (1923) 23. [2] P. Villars, L.D. Calvert, Pearson's Handbook of Crystallographic Data for Intermetallic Phases, ASM International, Materials Park, Ohio, USA, 1991. [3] P. Blaha, K. Schwarz, L. Luitz, WIEN97, Vienna University of Technology, 1997. [4] J. Vřešťál, Arch. Metall. 46 (2001) 9. [5] J. Houserova, J. Vřešťál, M. Friák, M. Šob, Comp. Mater. Sei. 25 (2002) 562-569. [6] G. Krier, O. Jepsen, A. Burkhardt, O.J. Andersen, Computer Code TB-LMTO-ASA, ver. 4.6, Max-Planck-Institute für Festkörperforschung Stuttgart, 1994. [7] J. Havránková, J. Vřešťál, M. Friák, M. Šob, Phys. Rev. B 63 (2001) 174101. [8] O.J. Anderson, B. Sundman, Calphad 11 (1987) 83-92. [9] M. Hillen, C. Qiu, Met. Trans. 22A (1991) 2187-2198. [10] W.P Rees, B.D. Bums, A.J. Cook, J. Iron Steel Inst. 162 (1949) 325. [11] J. Sopoušek, T. Kruml, Scripta Materialia 35 (1996) 689. [12] M. Hasebe, T. Nishizawa, in Workshop NBS. 1997. Gaithersburgh, MD, p. 911. [13] J.W. Schultz, H.F Merrick, Met. Trans. 3 (1972) 2479. [14] N. Yukawa, M. Hida, T. Imura, M. Kawamura, Y. Miyuno, Met. Trans. 3 (1972) 887-8953. Reprinted from Journal of Alloys and Compounds, 378, Chvátalova K., Houserova J., Šob M., Vřešťál J., First-principles Calculations of Energetics of Sigma Phase Formation and Thermodynamic Modelling in Fe-Ni-Cr System, 71-74, Copyright (2004), with permission from Elsevier. Appendix IV: Houserova J. et al, CALPHAD 29 (2005) 133. 66 Available online at www.sciencedirect.com [directs äCIE NCE^, ELSEVIER 7/ Computer Coupling of Phase Diagrams and Thermochemistry 29 (2005) 133-139 www.elsevier.com/locate/calphad Phase diagram calculations in the Co-Mo and Fe-Mo systems using first-principles results for the sigma phase -b,a Jana Houserováa'*, Jan Vřeštál , Mojmír Sob ^Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Žižkova 22, CZ-616 62 Brno, Czech Republic ^Department of Theoretical and Physical Chemistry, Faculty of Science, Masaryk University, Kotlářská 2, CZ-611 37 Brno, Czech Republic Received 16 February 2005; received in revised form 3 June 2005; accepted 8 June 2005 Available online 28 June 2005 Abstract We calculate the phase diagrams of Co-Mo and Fe-Mo systems by means of a combination of ab initio electronic structure calculations and the CALPHAD approach. Ab initio calculations of the total energy differences between the sigma phase and standard element reference (SER) structures for pure constituents are accomplished by means of the full-potential linearized augmented plane wave (FLAPW) method in the generalized gradient approximation (GGA). During these calculations the structure relaxation is performed. The results obtained are employed as a part of the Gibbs energy describing the thermodynamics of the sigma phase. Subsequently, the thermodynamic data sets describing the sigma phases in the Co-Mo and Fe-Mo systems are optimized using the CALPHAD approach by employing both the experimental and thermodynamic data extended by ab initio calculations. Description of all other phases is taken from the literature. © 2005 Elsevier Ltd. All rights reserved. Keywords: Ab initio calculation; Sigma phase; Phase diagram 1. Introduction Molybdenum plays a controversial role in steel development. On one hand, it improves steel corrosion resistivity and, on the other hand, it increases the probability of formation of the unwelcome sigma phase. A reliable thermodynamic description of the Co-Mo and Fe-Mo systems is therefore needed to assess the regions of stability of various phases and may be used for further modelling of ternary systems containing Fe and Mo, which constitute a basis of some corrosion-resistant materials. Information about phase equilibria and thermodynamic properties in the above-mentioned systems is of fundamental importance in the design of materials for applications under extreme conditions. The possibility of predicting phase equilibria provides a powerful tool for evaluating the properties and behaviour of materials. Such predictions can be performed by the CALPHAD (CALculation of PHAse * Corresponding author. Tel.: +420 532290 461; fax: +420 541218 657. E-mail address: houserova@ipm.cz (J. Houserova). 0364-5916/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2005.06.002 Diagram) approach [1], which allows one to calculate the phase diagrams using thermodynamic functions of both pure constituents and all phases formed from these constituents. The sigma phase is an intermetallic compound that can precipitate e.g. at grain boundaries as a secondary phase. It is extremely brittle and, therefore, it increases the probability of formation of crack nucleation sites. It also decreases the resistance of materials to various corrosive media. This phase can be found in many transition-metal systems (e.g. Fe-Cr, Co-Cr, Fe-Mo, Ni-V) [2]. It is well known that molybdenum stabilizes the sigma phase in iron-based alloys. A full understanding of the behaviour of the sigma phase relies on the knowledge of the phase equilibria. This is primarily limited by a correct thermodynamic description of binary systems where the sigma phase can be treated using various models. The three-sublattice model [3-8] is one of the most frequently used ones. The crystallographic structure of the sigma phase belongs to the space group No. 136 and is represented by 30 atoms in the unit cell. These atoms are accommodated in five in-equivalent sublattices. The above-mentioned three-sublattice Reprinted from CALPHAD, 29, Houserova J., Vřešťál J., Sob M., Phase Diagram Calculations in the Co-Mo and Fe-Mo Systems Using First-principles Results for the Sigma Phase, 133-139, Copyright (2005), with permission from Elsevier. Appendix IV: Houserova J. et al, CALPHAD 29 (2005) 133. 67 J. Houserova et al. / Computer Coupling of Phase Diagrams and Thermochemistry 29 (2005) 133-139 134 model (which derives from the regular solution model [1] and the generalized multiple sublattice model [9]) is a substantial simplification of the sigma phase description, which is based on merging the sublattices with the same (or similar) coordination number to get a lower number of sublattices than found in the structural analysis. The Gibbs energy of the sigma phase modelled in this way is defined as a sum of the Gibbs energies of the bcc and fee structures in the ratio given by simplified formula plus a correlation term, which is fitted to phase equilibrium and thermodynamic data. We have shown that another approach resulting in a two-sublattice model is sufficient [10-12]. This model describes the sigma phase as a solid solution with two sublattices (each having one site). In this case the Gibbs energy of the sigma phase is based on the Gibbs energies of the sigma phases of the pure constituents. The mixing of the elements involved occurs in the first sublattice and it is described by the sum of the Gibbs energies of the sigma phases of pure constituents weighted by their molar fractions. The second sublattice can be used for modelling interstitial phases and in the simplest cases, it is occupied by vacancies. The assessment of stability of any structure (characterized by its energy of formation) is usually based on the standard element reference (SER) states, i.e. structures of the pure constituents that are stable at a temperature T of 298 K, and pressure p of 1 bar, such as hep ferromagnetic Co, bcc ferromagnetic Fe and bcc non-magnetic Mo. This approach is very convenient because the thermodynamic functions describing these structures have been summarized in a published worldwide-accepted database [13] and verified in many thermodynamic assessments. Total energy differences between the sigma phase and the SER structure constitute a part of the Gibbs energy differences, and can be reliably calculated from first principles, e.g. by the full-potential linearized augmented plane waves (FLAPW) method [14] that provides high-precision solutions to Kohn-Sham equations as well as total energies for solid state structures. The purpose of this work is to improve the thermodynamic modelling of the more complicated structures, such as the sigma phase, by employing the results of ab initio electronic structure calculations. We show that our model based on a combination of ab initio results and CALPHAD treatment provides a reliable approach for evaluating phase equilibria that involve such structures. The paper is organized as follows. After the introduction, we briefly characterize our thermodynamic model in Section 2. Section 3 describes the ab initio calculations of the total energy differences, which are used in Section 4 for predicting the phase diagrams. There we provide the values of the fitted parameters and discuss our results. Section 5 concludes the paper. 2. Thermodynamic model Thermodynamic modelling allows one to describe arbitrary systems under various conditions (i.e. of pressure, temperature or composition) and enables us to predict the behaviour of systems which are technologically important for the improvement of properties and manufacturing of current materials as well as for the development of new ones. In contrast to experiments, this treatment not only gives us an overview of the properties of actually existing structures but also is capable of characterizing metastable or even unstable configurations. The main variable used in such modelling is the molar Gibbs energy of the whole system Gtot, which is defined as the sum of molar Gibbs energies G-f of all included phases / multiplied by their molar fractions %f: Gtot = J2xfGf, (1) where Gf = J2 y- °G{ + Gid + GE + Gmag + Gpres. (2) As shown in Eq. (2), the molar Gibbs energy G-f of phase / can be further divided in several terms, i.e. into the sum yt°Gf) of molar Gibbs energies °Gf of pure constituents i in the phase / multiplied by molar fractions yi of the corresponding element in a given structure, terms describing ideal (Gld) and non-ideal (GE) mixing and, when needed, some special terms such as the magnetic (Gmag) or pressure (Gpres) contribution. The last two terms are omitted in the case of the sigma phase in our work because we are modelling a solid metallic structure that is not very much influenced by pressure in the low-pressure region and this structure is supposed to be non-magnetic over the whole composition interval. We cannot exclude the possibility that the sigma phases of pure constituents could prefer some magnetic ordering. But the choice of the non-magnetic sigma phase as the reference state for the Gibbs energy in construction of the phase diagram could be justified by the fact that no magnetic sigma phase has been found in any system. Further, employing this nonmagnetic reference state does not significantly influence thermodynamic modelling of the sigma phase in regions where this structure is stable. The terms describing mixing are evaluated using relatively simple formulas as follows: G;dy = RTiy.lny,-+yJlnyJ), (3) Gfj =ylyJ(L0(T) + L1(T)(yl-yJ) + L2(T)(yl-yJ)2), (4) where L°, L1 and L2 are the expansion coefficients of the Redlich-Kister polynomial [15] and T stands for temperature. The molar Gibbs energies of the pure components in more complicated phases are obtained in some cases with substantial difficulty, in contrast to the Gibbs energies of simple structures, such as bcc, fee, hep, which are summarized in various databases, e.g., Ref. [13]. Structures such as the Laves or sigma phase do not exist in the pure elemental state and therefore their Gibbs energies are experimentally inaccessible. In these situations ab initio Reprinted from CALPHAD, 29, Houserova J., Vřešťál J., Sob M., Phase Diagram Calculations in the Co-Mo and Fe-Mo Systems Using First-principles Results for the Sigma Phase, 133-139, Copyright (2005), with permission from Elsevier. Appendix IV: Houserova J. et al, CALPHAD 29 (2005) 133. 68 J. Houserova et al. / Computer Coupling of Phase Diagrams and Thermochemistry 29 (2005) 133-139 135 Table 1 FLAPW equilibrium and experimental lattice parameters of the SER structures Structure This work Experiment a (a.u.) cja Vat (a.u.3) a (a.u.) cja Vat (a.u.3) Ref. FM hep Co 4.7211 1.6194 73.7888 4.7357 1.6237 74.6722 [2] FM bec Fe 5.4144 1 79.3626 5.40 5.4160 1 78.7320 79.4339 [18] [19] NM bec Mo 5.9714 1 106.4619 5.9434 1 104.9723 [2] FM stands for ferromagnetic and NM for non-magnetic states; Vat denotes the volume per atom. methods can help us to get deeper insight into the thermodynamics of such structures. The Gibbs energy of the sigma phase of the pure constituents can be written as O^sigrr OgSER OgSER _j_ ^ 0^ sigma—SER _ 0 osigma—SER T, (5) where H is enthalpy and S entropy. The enthalpy of the sigma phase expressed with respect to the SER state, A o#sigma-SER can be replaced by the ab initio calculated total energy difference between the sigma phase and the SER state, A o^sigma-SER^ under the assumption that these two differences are equal. This is exactly fulfilled at zero temperature. We also suppose that A O/zsigma-SER an(j A 0£sigrr ' do not vary significantly with increasing temperature. This assumption may be verified on the basis of a combination of • the Kirchhoff equation (A//(7i) = AH(T0) + jj ACpdT, where AH(TQ) and AH(T\) are the reaction enthalpies at Tq and 7), respectively, and the difference between the heat capacities at constant pressure is ACP = Clp°dacts - qfactants) and • the Neumann-Kopp rule (Cp(AnBm) = nCp(A) + mCp(B)), which is reasonably valid for metallic systems. This combination results in ACP = Cp products r* reactants Cp{AnBm) - nCp{A) + mCp{B) = 0 and, subsequently, AH(Tt) = AH(T0) + OdT = AH(T0). Therefore, we can write Ogsigma _ OgSER . OgSER . 0/^ sigma—SER ■ A uGslgl 0 rrsigma— SER ■ A°£slgl AUS: 0 rsigma—SER T. (6) The entropy term A°5'igma_SER may be adjusted to experimental data (see Section 4). Expression (6) can then be employed in phase diagram calculations according to Eq. (2). This procedure has been successfully used for the calculation of phase diagrams in other systems containing the sigma phase, such as Fe-Cr [11] and Co-Cr [12]. In this paper the Co-Mo and Fe-Mo systems are presented as further examples. 3. Ab initio calculations of total energies We used the FLAPW method [14] incorporated into the WIEN97 code [16] to calculate the total energy differences between the sigma phase and the SER structure similarly to our previous studies [11,12]. The exchange-correlation energy was evaluated in the generalized gradient approximation (GGA) [17]. We used constant muffin-tin radii (RMT parameters) during all our calculations (1.97 a.u. for Co, 1.96 a.u. for Fe and 2.30 a.u. for Mo). The optimum numbers of /c-points in the irreducible part of the Brillouin zone providing a sufficient convergence of the total energies of the SER states were found to be 288 for Co and 120 for Fe and Mo. We decided to relax all structures to get reliable reference states for evaluating the energy differences. The relaxation is based on the alternating optimization of the volume and c/a ratio. In the case of the cubic structures only volume relaxation is sufficient to get the lowest-energy state. The resulting optimum lattice parameters for the SER states are listed in Table 1. The same approach was applied to equilibrium calculations for the sigma phase. Since the sigma phase of the pure constituents does not exist we have used the structure parameters from the binary phases. For this purpose, we have utilized the lattice parameters a and c from such binary sigma phases that involve elements with similar atomic radii to the element chosen, i.e. for Co from the Co-Cr sigma phase [20], for Fe from the Fe-Cr sigma phase [21] and for Mo from the Mo-Mn sigma phase [22]. There are some other parameters besides lattice ones describing the structure of the sigma phase—so-called internal parameters, which describe the positions of atoms within the unit cell. We have examined their values from various binary sigma phases (e.g. the internal parameters for Cr-Fe, Fe-Mo etc. in the case of Fe) and, employing the linear muffin-tin orbital method in the atomic sphere approximation (LMTO-ASA) [23-25] within the GGA [17], we looked for those values that provided the lowest energies. These were acquired using the internal structure parameters of Fe-Cr [21] for Fe and of Co-Mo [26] for Co and Mo. The values of the internal structure parameters obtained in this way were kept constant in all subsequent calculations. Afterwards we employed the FLAPW method again for the optimization of lattice parameters a and c in the same Reprinted from CALPHAD, 29, Houserova J., Vřešťál J., Sob M., Phase Diagram Calculations in the Co-Mo and Fe-Mo Systems Using First-principles Results for the Sigma Phase, 133-139, Copyright (2005), with permission from Elsevier. Appendix IV: Houserova J. et al, CALPHAD 29 (2005) 133. 69 J. Houserova et al. / Computer Coupling of Phase Diagrams and Thermochemistry 29 (2005) 133-139 136 way as was described for the SER structures. Performing convergence tests, we have found that 42 ^-points in the irreducible part of the Brillouin zone were sufficient for all three constituents. As the sigma phases existing in nature are not spin-polarized, the calculations for pure constituents were also performed for a non-spin-polarized state. The resulting lattice parameters are summarized in Table 2. It may be shown that the differences in atomic volumes between the SER and sigma phase related to the equilibrium atomic volume of the SER state are comparable with the differences between the values obtained for the bcc, fee, and hep structures [27]. Table 2 FLAPW equilibrium lattice parameters of the sigma phase of pure constituents Element in the This work sigma phase structure a (a.u.) cja Vat (a.u.3) Co 15.8602 0.5197 69.1069 Fe 15.5987 0.5174 65.4586 Mo 18.1870 0.5241 105.1003 This procedure provided the total energies of the sigma phases of pure constituents at the equilibrium volumes and the justification for the statement that the sigma phase of pure constituents is stable with respect to the volume changes and tetragonal deformation. Unfortunately there is no absolute guarantee of the dynamic stability of this phase in the pure state. On the other hand, even if the sigma phase structure of the pure constituent were dynamically unstable, the total energy difference between this structure and SER phase at zero temperature has a well-defined value [28,29], which can be used as an effective value in the procedure of adjusting the thermodynamic parameters in the phase equilibrium data calculations. Hence we evaluated the total energy differences between the sigma phase and SER state (Table 3). Table 3 Ab initio calculated total energy differences between the sigma and SER phases Element Co Fe Mo O^sigma-SER (mRyatom-1) 45.58 33.03 17.77 i (kJmol-1) 59.84 43.36 23.33 The present values for Co are somewhat different in comparison with those in [12] due to better approximations used in this work. The physical conclusions in paper [12] are not influenced by this difference. 4. Phase diagram calculations The thermochemical basis of the CALPHAD method relies explicitly on the assumption that the composition of the equilibrium phase arises as a result of minimization of the Gibbs energy in a closed system at constant external conditions (temperature and pressure). For modelling the sigma phase we employed the two-sublattice model [11,12] because it has a good physical background, it enables us to describe the sigma phase over the whole composition range and it should be able to provide a substantial improvement of the thermodynamic modelling of more complicated systems. The sigma phase is treated here as a solid solution (see Eq. (1)) and its Gibbs energy in the pure state is defined by Eq. (6) where the entropy term is adjusted to the experimental data. The L parameters describing the excess Gibbs energy GE of non-ideal mixing in Eq. (4) are obtained in the same way. All these parameters (A O^sigma-SER an(j L) ^ Usted in TaWes 4 and 5 Table 4 The adjusted entropy of the transition from the SER state to the sigma phase Element Co Fe [11] Mo ^ 0^,sigma—SER (jr'mor1) 0.75 0.70 0.75 Table 5 The adjusted L parameters describing the non-ideal mixing Parameter/System Co-Mo Fe-Mo L° -171 500 + 25.0 *T -119 000 + 5.1 *T Ll (J mol-1) 35 000 -50000 L2 75 000 -108400 The complete thermodynamic description of the sigma phase in the two-sublattice model is given in Appendix. The thermodynamic parameters for all other phases in Co-Mo and Fe-Mo systems, namely of the liquid, fee, hep, EPS, Mu, bcc, Laves phase and R phase, were taken from [6] and [30], respectively. The calculated phase diagrams are presented in Fig. 1. It is obvious that our approach provides a very good description of the stability range of Co-Mo and Fe-Mo sigma phases with quality comparable to that of the earlier three-sublattice model ([6] and [30], respectively). Moreover, our model employs a transparent approach to the modelling of the sigma phase over the whole composition region that is based on ab initio calculations. It can be thought of as a reasonable starting point in approaching otherwise inaccessible regions of unstable sigma phases. In contrast, the three-sublattice model describes the sigma phase as a combination of the Gibbs energies of SER states, which has no relevance in the composition range close to the pure constituents. Therefore we expect the two-sublattice model to provide a better understanding when studying multicomponent systems. Our calculated phase equilibria also agree very well with the experimental phase equilibrium data [3,5,31-33] for both systems. It is not the purpose of this paper to study those regions of the phase diagram not containing the sigma phase because they cannot Reprinted from CALPHAD, 29, Houserova J., Vřešťál J., Sob M., Phase Diagram Calculations in the Co-Mo and Fe-Mo Systems Using First-principles Results for the Sigma Phase, 133-139, Copyright (2005), with permission from Elsevier. Appendix IV: Houserova J. et al, CALPHAD 29 (2005) 133. 70 J. Houserova et al. / Computer Coupling of Phase Diagrams and Thermochemistry 29 (2005) 133-139 137 (a) 3000 2000 1000 (b) 2800 2400 .2000 1600 1200 t Fig. 1. Calculated phase diagrams of Co-Mo (a) and Fe-Mo (b). Thick dashed lines: the two-sublattice model (thermodynamic data are given in Tables 3-5), thin full lines: the three-sublattice model (thermodynamic data from [6] and [30]), experimental data according to [5] (A) and [31] (□) for Co-Mo and [32] (A), [33] (□) and [3] (+) for Fe-Mo. The dashed line in the left-hand part of the Co-Mo phase diagram divides the region of the fee phase into the ferromagnetic (lower) and paramagnetic (upper) parts. be influenced by changes in the sigma phase modelling and, therefore, the corresponding experimental data are not presented in Fig. 1. The comparison between the Gibbs energy and enthalpy of the sigma phase calculated within both models and those for other phases (described using the data from [6,30]) is shown in Figs. 2 and 3. From these figures, we conclude that the two-sublattice model provides the proper composition dependence for both thermodynamic functions (i.e. G-f and Hf) over the whole composition range in contrast to the three-sublattice model (see lines 8 and 8a in those figures). Fig. 2. Concentration dependence of the molar Gibbs energy Gf for various phases in the Co-Mo (a) and Fe-Mo (b) systems at 1700 K: 1—liquid, 2—fee, 3—bec, 4—R, 5—Mu, 6—hep, 7—EPS, 8—sigma (two-sublattice model), 8a—sigma (three-sublattice model). This is a great advantage of our method, which will enable us to extend the calculations to multicomponent systems. 5. Conclusions Our approach provides a reasonable physical interpretation of the individual terms in the mathematical expression for the Gibbs energy difference between the sigma phase and SER state of the pure constituents. Total energy differences are reliably evaluated by the ab initio methods, and can be incorporated into the two-sublattice model [11,12] which consequently provides a transparent and effective way for constructing phase diagrams. This model can be used Reprinted from CALPHAD, 29, Houserova J., Vřešťál J., Sob M., Phase Diagram Calculations in the Co-Mo and Fe-Mo Systems Using First-principles Results for the Sigma Phase, 133-139, Copyright (2005), with permission from Elsevier. Appendix IV: Houserova J. et al, CALPHAD 29 (2005) 133. 71 J. Houserova et al. / Computer Coupling of Phase Diagrams and Thermochemistry 29 (2005) 133-139 (a) 120 100 O E 2 x "i-1-r i 1 r (b) 100 ~ 80 Ö E 40 1 1 1 1 1 1 1 1 1— 1 , -5 1 . SEŘ~ -j- 0.2 0.4 0.6 0.8 Fig. 3. Concentration dependence of the molar enthalpy for various phases in the Co-Mo (a) and Fe-Mo (b) systems at 1700 K. The labelling of the lines is the same as in Fig. 2. The thin dashed straight line denoted as SER shows the weighted average of the molar enthalpy of the pure constituents in the SER states. The arrows indicate our predictions of molar enthalpies of formation of the alloy sigma phases with respect to the weighted average of molar enthalpies of SER states of pure constituents. because the inequivalent lattices are not exclusively occupied by one component [7]. Recently, we have shown that the two-sublattice model can reproduce well the phase equilibria in a multicomponent system [34]. This paper illustrates the use of such model in the case of the Co-Mo and Fe-Mo binary systems. It was shown that the calculated equilibria involving the sigma phase agree well with the experimental data ([5,31] for Co-Mo and [3, 32,33] for Fe-Mo). Unlike the previously considered three-sublattice model, the two-sublattice model employed in this study provides a description of the molar Gibbs energy G-f and the molar enthalpy of the sigma phase over the whole composition range. Unfortunately, there are no experimental data which could prove or disprove our results concerning the enthalpy of the sigma phase in Mo-based systems. For the Cr-based systems, we have shown that the calculated enthalpies obtained from the two-sublattice model agree quite well with the experiments [12]. Acknowledgements This research was supported by the Grant Agency of the Czech Republic (Projects Nos. 106/03/P002 and 106/02/0877), by Grant Agency of the Academy of Sciences of the Czech Republic (Project No. S2041105) and by the Research Projects AV0Z20410507 and MSM0021622410. The use of the computer facilities at the MetaCentre of the Masaryk University in Brno is acknowledged. Phase equilibria were calculated by means of the Thermocalc code. Appendix Summary of sigma phase parameters: For 298.15 < T < 6000: two sublattices, sites 1:1. Constituents: Co, Mo, Fe:Va: 0,^-FM-hcp 0/^sigma Opsigma_ 0/^FM—bcc 'Fe:Va ^Fe n Opsigma _ OpNM—bcc wMo:Va uMo Or sigma Co,Mo:Va " 1 j sigma -171500-+35 000. - 59 840- 0.75 *T. 43 360- 0.7 *7\ f- 23 300- 0.75 *7\ ■25*7". Co,Mo:Va 2^CrMo:Va=+75000 Or sigma Fe,Mo:Va " 1 j sigma LFe,Mo:Va " 2 j sigma ^Fe,Mo:Va " -119000- -50000. -108400. 5.1 * T. References [1] N. Saunders, A.P. Miodownik, CALPHAD (Calculation of Phase Diagrams): A Comprehensive Guide, in: Pergamon Materials Series, vol. 1, Elsevier Science Ltd, Oxford, UK, 1998. [2] P. Villars, L.D. Calvert, Pearson's Handbook of Crystallographic Data for Intermefallic Phases, ASM International, Materials Park, OH, USA, 1991. [3] A.F. Guillermet, CALPHAD 6 (1982) 127. [4] J.O. Andersson, B. Sundman, CALPHAD 11 (1987) 83. [5] A.V. Davydov, U.R. Kattner, J. Phase Equilib. 20 (1999) 5. [6] A.V. Davydov, U.R. Kattner, J. Phase Equilib. 24 (2003) 209. [7] I. Ansara, T.G. Chart, A.F. Guillermet, F.H. Hayes, U.R. Kattner, D.G. Pettifor, N. Saunders, K Zeng, CALPHAD 21 (1997) 171. [8] M. Hillen, CALPHAD 22 (1998) 127. [9] B. Sundman, J. Agren, J. Phys. Chem. Solids 42 (1981) 297. [10] J. Vfestal, Arch. Metall. 46 (3) (2001) 9. Reprinted from CALPHAD, 29, Houserova J., Vřešťál J., Sob M., Phase Diagram Calculations in the Co-Mo and Fe-Mo Systems Using First-principles Results for the Sigma Phase, 133-139, Copyright (2005), with permission from Elsevier. Appendix IV: Houserova J. et al, CALPHAD 29 (2005) 133. 72 J. Houserova et al. / Computer Coupling of Phase Diagrams and Thermochemistry 29 (2005) 133-139 [11] J. Houserova, M. Friák, M. Šob, J. Vřeštil, Comput. Mater. Sei. 25 (2002) 562. [12] J. Houserova, J. Vřeštil, M. Frilk, M. Šob, CALPHAD 26 (2002) 513. [13] A.T. Dinsdale, CALPHAD 15 (1991) 317. [14] D. Singh, Planewaves, Pseudopotentials and the LAPW Method, Kluwer, Boston, 1994. [15] O. Redlich, A.T. Kister, Ind. Eng. Chem. 40 (1948) 345. [16] P. Blaha, K. Schwarz, J. Luitz, Computer code WIEN97, Vienna University of Technology, 1997 (improved and updated Unix version of the original copyrighted WIEN code, which was published by P. Blaha, K. Schwarz, P. Sorantin, S.B. Trickey, Comput. Phys. Commun. 59 (1990) 399). [17] J.P. Perdew, JA. Chevary, S.H. Vosko, KA. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. [18] E.G. Moroni, T. Jarlborg, Phys. Rev. B 47 (1993) 3255. [19] M. Polcarova, K. Goodwod, J. Bak-Misiuk, S. Kadeckova, J. Bradler, Phys. Status Solidi a 106 (1988) 17. [20] G.J. Dickins, A.M.B. Douglas, W.H. Taylor, Acta Crystallogr. 9 (1956) 297. 139 [21] H.L. Yakel, Acta Crystallogr. B 39 (1983) 20. [22] B.F. Decker, R.M. Waterstrat, J.S. Kasper, J. Met., Trans. ALME 6 (1954) 1406. [23] O.K. Andersen, Phys. Rev. B 12 (1975) 3060. [24] H.L. Skriver, The LMTO Method, Springer Verlag, Berlin, 1984. [25] G. Krier, O. Jepsen, A. Burkhardt, O.K. Andersen, Computer code TB-LMTO-ASA version 4.6, Max-Planck-Institut für Festkörperforschung, Stuttgart, 1994. [26] J.B. Forsyth, d'Alte da Viega, Acta Crystallogr. 16 (1963) 509. [27] Y. Wang, S. Curtarolo, C. Jiang, R. Arroyave, T. Wang, G. Ceder, L.-Q. Chen, Z.-K. Liu, CALPHAD 28 (2004) 79. [28] G. Grimvall, Ber. Bunsenges. Phys. Chem. 102 (1998) 1083. [29] G. Grimvall, CALPHAD and Alloy Thermodynamics, TMS, Warrendale, PA, USA, 2002, p. 81. [30] J.-O. Andersson, CALPHAD 12 (1988) 9. [31] CP Heijwegen, G.D. Rieck, J. Less-Common Met. 34 (1974) 309. [32] P. Gustafson, Z. Metallkde. 79 (1988) 388. [33] CP Heijwegen, G.D. Rieck, J. Less-Common Met. 37 (1974) 115. [34] K. Chvátalova, J. Houserovi, M. Sob, J. Vřeštil, J. Alloys Compounds 378 (2004) 71. Reprinted from CALPHAD, 29, Houserova J., Vřešťál J., Šob M., Phase Diagram Calculations in the Co-Mo and Fe-Mo Systems Using First-principles Results for the Sigma Phase, 133-139, Copyright (2005), with permission from Elsevier. Appendix VI: Vřešťál J. et al, CALPHAD 30 (2006) 14. 73 Available online at www.sciencedirect.com SCIEI :NCE (^)Dlrt ECT" ELSEVIER Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 14-17 www.elsevier.com/locate/calphad Assessment of the thermodynamic properties and phase diagram of the Bi-Pd system J. Vřešťála-*, J. Pinkasa, A. Watsonb, A. Scottb, J. Houserovác, A. Kroupa0 ^-Masaryk University, Faculty of Science, Department of Physical Chemistry, Kotlářská 2, 611 37 Brno, Czech Republic k University of Leeds, Institute for Materials Research, S.P.E.M.E., Leeds, UK c Institute of Physics of Materials, Academy of Sciences of Czech Republic, Žižkova 22, 616 62 Brno, Czech Republic Received 11 October 2005; received in revised form 2 December 2005; accepted 2 December 2005 Available online 29 December 2005 Abstract New experimental DTA, XRD, SEM, solution calorimetry and ab initio calculated results are used together with data from the literature to assess the thermodynamic parameters of the Bi-Pd system, necessary for calculation of the phase diagram by the CALPHAD method. Samples were prepared with compositions of 51 and 82 at.% Pd, and their homogeneity checked by SEM/EDX before being employed in DTA studies. High temperature solution calorimetry was used for the determination of the enthalpies of mixing of liquid Pd and Bi, in the range from 0 up to 50 at.% Pd at 1028 K. XRD determination of the structures of equilibrated samples after quenching to room temperature provided important information for the evaluation of structure. Ab initio electronic structure calculations provided information on the energetics of the intermetallic phases of the system. The calculated phase diagram was compared with new phase equilibrium data and with the data presented in the literature. © 2005 Elsevier Ltd. All rights reserved. 1. Introduction The use of bismuth is not only important as a solvent of uranium in nuclear metallurgy, but also as a component in lead-free solder materials. In the study of the interaction of such solder materials with substrates, which often contain palladium, knowledge of the Bi-Pd phase diagram is of interest. Phase diagram studies have been carried out by Zhuravlev and Zhdanov [1], Zhuravlev [2] (without experimental details) and Brasier and Hume-Rothery [3]. The published liquidus data were confirmed by Schweitzer and Weeks [4] by the method of analysis after equilibrating annealing. The third paper of these [3] is a rich source of accurate phase equilibrium data, obtained by cooling curve analysis (2-3 °C/min, silica stirrer, argon atmosphere, recalibrated thermocouple, high purity metals), suitable for use in the assessment. New information on the solubility of bismuth in palladium at lower temperatures has been provided by Oberndorff [5], reporting no solubility of Bi in Pd at 200 °C and 215 °C following diffusion couple studies. For prediction of phase equilibria by * Corresponding author. Tel.: +420 549498134; fax: +420 541211214. E-mail address: vrestal@chemi.muni.cz (J. Vfest'al). 0364-5916/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2005.12.001 the CALPHAD (CALculation of PHAse Diagram) method, the thermodynamic properties of the phases are required. Unfortunately, no thermodynamic information is available for this system. Therefore, the aim of this work is to perform a thermodynamic assessment of the Bi-Pd system using new thermodynamic information gained by calorimetric experiments and first-principles calculations. Complementary SEM, DTA and XRD measurements have been performed to confirm the results. 2. Experimental 2.1. Calorimetry A high temperature SETARAM HT1500 calorimeter was used for the measurement of the heat of dissolution of palladium in liquid bismuth at 1028 K by the drop technique; the apparatus has been described elsewhere [6]. Solid palladium (Alfa Aesar, 99.99%) was dropped from room temperature into a liquid bismuth (Alfa Aesar, 99.99%) bath held at 1028 K in an alumina crucible under a pure argon atmosphere. The heat flow from the reaction crucible to the reference crucible Reprinted from CALPHAD, 30, Vřešťál J., J. Pinkas, Kroupa A., Houserova J. , Scott A., Watson A., Assessment of the Thermodynamic Properties and Phase Diagram of the Bi-Pd System, 14-17, Copyright (2006), with permission from Elsevier. Appendix VI: Vřešťál J. et al, CALPHAD 30 (2006) 14. 74 J. Vřešťál et al. / Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 14-17 0 0.2 0.4 0.6 0.8 Fig. 1. Integral enthalpy of mixing of liquid Bi-Pd alloys (liquid standard state). Results of three independent runs. 1—Least squares fit of experimental results by a polynomial of second order. 2—Result of global optimization of calorimetric and phase equilibrium results, expressed by a polynomial of second order. was recorded with respect to time via a thermopile. The total heat effect was then calculated by integration of the thermogram, the graphical representation of the output of the thermopile with respect to time. By subtracting appropriate amounts for the enthalpy of fusion and enthalpy increment of palladium (Hpd,io23-Hpd,298) the partial enthalpy of mixing can be calculated. By making successive palladium drops and summing the partial enthalpies, the integral enthalpy of mixing curve can be derived. The calorimeter was calibrated by dropping a bismuth sample immediately before the palladium drops and by dropping an alumina sample at the end of the run. The resulting integral enthalpies of mixing of Bi-Pd liquid alloys from three independent runs are shown in the Fig. 1, where the approximation of experiments to a smooth curve is also drawn. The value of the enthalpy of mixing at the composition xpd = 0.5 is determined to be AH^iFA = —27 kJ mol-1, approximating the concentration dependence of AH^iFA by a quadratic polynomial. The reproducibility of calorimetric measurements can be estimated to be ± 1 kJ mol~1 considering Fig. 1, where the results of three independent runs are displayed, and approximated by a polynomial of second order by the least squares method. On the other hand, global optimization of the calorimetric experiment with phase equilibrium data [1^1] expressed by a polynomial of second order for the thermodynamic parameters of the liquid phase, Fig. 1, lead to the small differences from the measured values of its enthalpy of mixing which may be regarded as acceptable. 2.2. DTA measurements ofliquidus temperature Liquidus temperatures were taken from the literature [1^1], but two samples of Bi-Pd alloys with jrpd = 0.51 and with 15 ;tpd = 0.82 were prepared for verification of the published results. A sample with xpd = 0.51 was prepared from the elements of the same purity as given above. Appropriate amounts of bismuth and palladium were melted together in an evacuated silica capsule, in a flame. After solidification, the sample was annealed at 450 °C for 2 h followed by quenching into water. The homogeneity of the sample was checked by SEM, and two phases were found: an orthorhombic PdBi intermetallic and very small amount of a second phase, both of which were confirmed by XRD analysis (the Pd5Bi3 phase, which was expected considering the phase diagram, was not confirmed unambiguously, as its crystallographic structure is not reliably known). Part of the sample was used in the determination of the melting temperature of the PdBi intermetallic compound by differential thermal analysis (DTA). The DTA was performed under an argon atmosphere, using a Derivatograph (MOM Budapest) instrument, employing a heating rate of 5 °C/min and alumina as the reference material. The melting temperature of PdBi was found to be 624 °C and the eutectic temperature on the palladium side was 578 °C. A similar procedure was followed for a sample with xpd = 0.82. SEM/EDS analysis of this sample revealed a two-phase structure with compositions 0.77 (Pd3Bi) and 0.83 of palladium (FCC), respectively. This structure corresponds to that at the crystallization temperature and therefore represents a frozen equilibrium. Air cooling of the sample does not provide a slow enough cooling rate to give the true equilibrium at low temperature. Corresponding DTA analysis revealed endothermic peaks at 939 °C and at 1081 °C, which correspond to the peritectic temperature and to the temperature of full melting of the sample. These are in agreement with values in the literature [ 1^1] and were used in the optimization procedure. 3. Theoretical calculations 3.1. Ab initio calculated total energies of formation ofPdBii and PdBi We have performed ab initio calculations of the total energies of PdBi (room temperature structure) and PdBi2 (high temperature structure) intermetallic s and of Bi and Pd in their experimentally observed structures (i.e., RHOMBOHEDRAL A7 Bi and FCC Al Pd), which represent the standard element reference (SER) states. The structural information for the PdBi and PdBi2 intermetallics was taken from [7] and [8], respectively. Spin polarization was not included in our calculations as all phases studied are paramagnetic at ambient temperature. The FLAPW (Full Potential Augmented Plane Wave) method implemented in the WIEN 97 code [9] using the generalized gradient approximation (GGA) [10] for the exchange-correlation term was employed to evaluate the total energy difference between PdBi and the SER-phases at 0 K at equilibrium volume Reprinted from CALPHAD, 30, Vřešťál J., J. Pinkas, Kroupa A., Houserova J. , Scott A., Watson A., Assessment of the Thermodynamic Properties and Phase Diagram of the Bi-Pd System, 14-17, Copyright (2006), with permission from Elsevier. Appendix VI: Vřešťál J. et al, CALPHAD 30 (2006) 14. 75 J. Vřešťál et al. / Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 14-17 16 (i.e., at the minima of the energy/volume curves). The calculated energy of formation of PdBi is: AEvfiBi = -34.13 kJ moP1 = -26.00 mRy at"1. This result was checked using the pseudopotential Vienna Ab initio Simulation Package (VASP) code [11,12] by using the Projector Augmented Wave (PAW) pseudopotential for the Local Density Approximation (LDA). These calculations yielded results similar to the FLAPW case: AEJdBi = -33.16 kJ mol"1 = -25.26 mRy at"1. For PdBi2, the pseudopotential codes VASP and CASTEP [13] were employed, with the results (at equilibrium structure parameters): A£PdBi2 = _23.17 ij molr = -17.65 mRy at"1 (VASP) A£EdB'2 = -27.0 kJ mol"1 = -20.6 mRy at"1 (CASTEP). For thermodynamic considerations, the mean values AE^dB' = —33.6 kJ mol"1 and A£EdB'2 = —25.1 kJ mol"1 were used, because we cannot prefer any one of the computational methods over the rest. The structure of the Pd3Bi intermetallic compounds is very complex (Bi4PbPdi5 structure type—16 atoms in the repeat cell with 6 inequivalent lattice sites) [14]. Ab initio calculations of total energy of this structure were not performed owing to this complexity. 3.2. Thermodynamic assessment The main aim of this work is to derive a set of thermodynamic model parameters, describing the Gibbs energy of the phases in the system, which then can be used for phase diagram calculation by the CALPHAD method. This is based on finding the minimum total Gibbs energy of the closed system at constant temperature and pressure, yielding the composition and the amount of phases in equilibrium. The total molar Gibbs energy (Gtot) is equal to the sum of the molar Gibbs energies of all phases (fif) multiplied by their molar fractions (xf), i.e., gtot = J2xfGf- (!) / g? is defined as gf = ^y,°G/' + Gid + GE + Gma8 + ..., (2) where yi is the molar fraction of the component (' in phase /, °g{ is the molar Gibbs energy of pure component in the phase /, the terms G'd and GE stand for the molar Gibbs energy of ideal and excess mixing, and Gmag is the magnetic contribution to the Gibbs energy. The excess Gibbs energy GE can be expressed as an empirical Redlich Kister polynomial [15]: Gfj = yiJj Cl{j + ll(j (yi - yj) + 2l(j (yi - yj)2). (3) 2000 1600 400 0 II 1 f^r—w— ** / >7 / i11 1 1 1 0 0.2 0.4 0.6 0.8 XPJ Fig. 2. Phase diagram of the Bi-Pd system calculated using thermodynamic parameters (see Appendix) determined in this work (solid lines) compared with experimental phase equilibrium data (> [1], V [2], A [3], 0 [4], □ [5], # this work). whereas Gld can be calculated according the theoretical relationship: Gid = -RTj2(yt Inyt). (4) The Gibbs energies of phases, g?, are used as an input for the phase diagram calculations. To obtain parameters describing the Gibbs energies of the individual phases, various models for the excess Gibbs energy description are proposed. For the liquid phase, a simple subregular model was used (3), and for the solid phases, the sublattice model and compound energy formalism (CEF) [16] were applied. Thermo-calc software [17], and its module PARROT for optimization of model parameters by the least squares method, were employed to give the calculated phase boundaries and thermodynamic properties which gave the best fit with experimental data within their experimental uncertainties. The resulting values of these parameters are presented in the Appendix. The phase diagram calculated using these parameters is shown in the Fig. 2. The experimental data from the literature and those determined in the present work are also given. The homogeneity range of Pd5Bi3 is not described experimentally with any reliability; therefore, it is denoted in [3] by dotted lines. X-ray studies by Zhuravlev [18] suggested that the phase has a structure of the AsNi type, but data describing the mixing of the atoms are missing. Therefore, a simple two sublattice model was used (Bi,Pd:Va) and parameters were determined to best reproduce the dotted line region. Ab initio calculations were not performed. 4. Discussion The value of the enthalpy of mixing in the liquid phase at the composition x-pi = 0.5 determined by drop calorimetry Reprinted from CALPHAD, 30, Vřešťál J., J. Pinkas, Kroupa A., Houserova J. , Scott A., Watson A., Assessment of the Thermodynamic Properties and Phase Diagram of the Bi-Pd System, 14-17, Copyright (2006), with permission from Elsevier. Appendix VI: Vřešťál J. et al, CALPHAD 30 (2006) 14. 76 ]. Vřešťál et al. / Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 14-17 (AHM = —27 kJ moU1) is in fair agreement with value estimated by Miedema's model, AHM = — 21 kJ mol-1, published in [19], and it was used as the fixed experimental thermodynamic quantity in this assessment. In the optimisation of the thermodynamic parameters, the values of the energy of formation of the PdBi and PdBi2 intermetallics were not optimized. These values were used to check the reliability of the thermodynamic parameters resulting from the optimisation of the Bi-Pd system. Calculated values from first principles at 0 K represent a better choice for the temperature independent term than any constant value used in many cases previously. The optimisation of the temperature dependent parameters of the Gibbs energy of these compounds was used for tuning the assessment to the experimental phase equilibrium data. Liquidus data published in the literature [1-4] are in mutual agreement, so all of these data were used in the assessment. Phase equilibrium data for Pd5Bi3 were taken from [3], similarly the data for the peritectic temperature of Pd3Bi which were verified experimentally in this work. Data for the solubility of Bi in Pd at 200 °C and 215 °C, published in [5], complement the data set. It is evident from Fig. 2 that the phase diagram calculated using the parameters in the Appendix is in very good agreement with the experimentally determined phase equilibrium data. 5. Conclusions The experimental thermodynamic enthalpies of mixing of liquid Bi and Pd measured by drop calorimetry, and ab initio calculated energies of formation of PdBi and PdBi2 intermetallics were successfully used as a basis for a thermodynamic assessment of the phase diagram of the Bi-Pd system. New phase equilibrium data were determined by DTA, SEM and XRD methods, and were used together with phase equilibrium data from the literature in this thermodynamic assessment. Calculated thermodynamic parameters are presented that enable reliable calculation of the phase diagram of Bi-Pd system. Acknowledgements This research was supported by the Ministry of Education of Czech Republic (Projects Nos. OC 531.001, OC 531.002, AB0Z 20410507 and MSM 0021622410), EPSRC (UK) Platform Grant (GR/R95798) and COST Action 531 (STSM). First-principles electronic structure calculations were performed using the facilities of Metacentrum of Masaryk University Brno and phase equilibria were calculated by employing Thermo-calc software. Appendix Values of thermodynamic parameters for the Bi-Pd system (in Joules) for the temperature interval 298^000 K (Eqs. (2) and (3)): o,FCC ^Bi.PiVa "L —78 000 + IT; = -107 000 + 247; rLiq JBi,Pd 1 ,Liq Bi,] °G™'d2 = 0.334 * GHSERPD --25 100 + 2.5 * T; 0.666 * GHSERBI orPd3Bi °Bi:Pd = 0.5 * GHSERPD + 0.5 * GHSERBI -33 600 + 7.65 *T; -- 0.75 * GHSERPD + 0.25 * GHSERBI -22 300 + 0.4 *r; °G™J = GHSERBI + 31 000; orPd5Bi3 0 , Pd5Bi3 ^Bi.PiVa 1 , Pd5Bi3 GPďVa = GHSERPD + 9000; = -179000 + 27.3 * T; o ^KHÜMBÜ °Pd:Va o,RHOMBO ^Bi.Pd^a ■ : GHSERPD + 4000 -32 000 Symbols GHSERBI and GHSERPD in the expression for Gibbs energy of the intermetallic phases represent the Gibbs energy of the SER state for Bi (RHOMBOHEDRAL A7) and Pd (FCC Al), respectively. They were taken from [20]. L-parameters are related to the Redlich-Kister polynomial (Eq. (3)), and G-parameters express Gibbs energy of phases (Eq. (2)). Elements occupying the same sublattice are separated by commas; different sublattices are separated by a colon. References [1] N.N. Zhuravlev, G.S. Zhdanov, Zhur. Exper. i Teor. Fiz. 25 (1953) 485. [2] N.N. Zhuravlev, Zhur. Exper. i Teor. Fiz. 32 (1957) 1305. [3] J. Brasier, W. Hume-Rothery, J. Less-Common Met. 1 (1959) 157. [4] D.G. Schweitzer, J.R. Weeks, Trans. ASM 54 (1961) 185. [5] P. Oberndorff, Ph.D. Theses, TU Eindhoven, 2001. [6] A. Watson, FH. Hayes, J. Alloys Compounds 220 (1995) 94. [7] Y.C. Bhatt, K. Schubert, J.Less-Common Met. 64 (1979) 17. [8] N.N. Zhuravlev, Soviet Phys.-JETP, translated from Zhur. Exper. i Teor. Fiz. 5 (1957) 1064. [9] P. Blaha, K. Schwarz, J. Luitz, WIEN97, Technical University of Vienna 1997—improved and updated Unix version of the original copyrighted Wien-code, published by P. Blaha, K. Schwarz, P. Sorantin, S.B. Trickey, Comput. Phys. Commun. 59 (1990) 399. [10] J.R Perdew, JA. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. [11] G. Kresse, J. FurthmUller, Comput. Mater. Sei. 6 (1996) 15. [12] G. Kresse, J. FurthmUller, Phys. Rev. B 54 (1996) 11169. [13] M.D. Segall, P.L.D. Lindan, M.J. Probert, C.J. Pickard, P.J Hasnip, S.J. Clark, M.C. Payne, J. Phys.: Condens. Mater. 14 (11) (2002) 2717. [14] K. Schubert, S. Bhan, T.K. Biswas, K. Frank, P.K. Panday, Naturwissenschaften 55(1968) 542. [15] O. Redlich, A.T. Kister, Ind. Eng. Chem. (2) (1948) 345. [16] N. Saunders, P. Miodownik, CALPHAD-A Comprehensive Guide, Elsevier, London, 1998. [17] B. Sundman, Thermo-calc, version L, Royal Institute of Technology, Stockholm, 1997. [18] N.N. Zhuravlev, Soviet physics-crystallography, translated from Kristal- lograflya 3 (1958) 506. [19] A.K. Niesen, FR. de Boer, R. Boom, RF. de Chatel, W.C.M. Mattend, A.R. Miedema, CALPHAD 7 (1983) 401. [20] A.T. Dinsdale, CALPHAD 15 (1991) 317. Reprinted from CALPHAD, 30, Vřešťál J., J. Pinkas, Kroupa A., Houserova J. , Scott A., Watson A., Assessment of the Thermodynamic Properties and Phase Diagram of the Bi-Pd System, 14-17, Copyright (2006), with permission from Elsevier. Appendix VII: Chvátalova et al. Mat. Sci. Eng. A 462 (2007) 153. 77 ELSEVIER Materials Science and Engineering A 462 (2007) 153-158 MATERIALS SCIENCE & ENGINEERING www.elsevier.com/locate/msea First-principles calculations of energetics of sigma phase formation and thermodynamic modelling in the Cr-Fe-W system K. Chvátalova3*, J. VřešťáP, J. Houserováb, M. Šobab a Masaryk University, Faculty of Science, Department of Theoretical and Physical Chemistry, Kotlářská 2, CZ-611 37 Brno, Czech Republic b Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Žižkova 22, CZ-616 62 Brno, Czech Republic Received 30 August 2005; received in revised form 2 February 2006; accepted 2 February 2006 Abstract Ab initio total energies of the alloy sigma phase in Cr-W and Fe-W systems were calculated by means of the linear muffin-tin orbital method in the atomic sphere approximation. Total energies of the pure constituents in the sigma phase and standard element reference (SER) states were evaluated using the full-potential linear augmented plane wave method. The combination of these results was used for the prediction of the energy of formation of the sigma phase with respect to both sigma phase of pure constituents and SER states at equilibrium structure parameters. We have also provided a novel physical approach to the thermodynamic modelling of the sigma phase and a reliable construction of the phase diagram of the Cr-Fe-W system, using ab initio calculated total energy differences in the calculation of phase diagram (CALPHAD) approach. © 2006 Elsevier B.V. All rights reserved. Keywords: Thermodynamic modelling; CALPHAD approach; Sigma phase; First-principles calculations; Chromium; Iron; Tungsten 1. Introduction The sigma phase (space group 136, P42/mnm) has a relatively complicated structure. It contains 30 atoms in the repeat cell, which are distributed among five crystallographically inequiva-lent sublattices (2a, 4f, 8i, 8i' and 8j). The sigma phase is very brittle and stable and its presence causes strong degradation of materials. Therefore, the reliable and accurate prediction of the regions of stability of this structure is desirable. The full-potential electronic structure calculations can reliably determine the total energy difference between the standard element reference (SER) states and the sigma phase. This enables us to model thermodynamic functions of the sigma phase not only in systems where it really exists but also even in systems, where it is not stable at all. The purpose of this work is to perform a thermodynamic modelling of the sigma phase in the Fe-W and Cr-W systems (where this structure is not stable at 1 bar at any temperature) and, using the calculation of phase diagram (CALPHAD) approach [1], to construct the phase diagram of the Cr-Fe-W system, which contains the sigma phase regions. * Corresponding author. Tel.: +420 541 129 316; fax: +420 541 211 214. E-mail address: chvatalova@email.cz (K. Chvátalova). 0921-5093/$ - see front matter © 2006 Elsevier B.V. All rights reserved, doi: 10.1016/j .msea.2006.02.474 2. Thermodynamic modelling Phase equilibrium calculations performed by the CALPHAD method [1] are based on finding the minimum of the total Gibbs energy of the system at constant pressure, temperature and composition. The total molar Gibbs energy (Gtot) is equal to the sum of the molar Gibbs energies of all phases (G^) multiplied by their molar fractions (x?), i.e. Gtot = J2xf(}f- / The Cf is defined as G/ = 53>i0G,/ + Gid + GE + Gma8 + ---, (2) where y; is the molar fraction of the component (' in phase /, °g{ is the molar Gibbs energy of pure constituent in the phase /, the terms Gld and GE stand for the molar Gibbs energy of ideal and real mixing of components in the phase / and Gmag is the magnetic contribution to the Gibbs energy. The Gibbs energies Cf are used as input values for phase diagram calculations. To overcome the difficulty caused by extrapolations of thermodynamic functions of the sigma phase from the concentra- Reprinted from Material Science and Engineering A, 462, Chvátalova K., Vřešťál J., Houserova J., Sob M., First-principles Calculations of Energetics of Sigma Phase Formation and Thermodynamic Modelling in the Cr-Fe-W System, 153-158, Copyright (2007), with permission from Elsevier. Appendix VII: Chvátalova et al. Mat. Sci. Eng. A 462 (2007) 153. 78 K. Chvátalova et al. / Materials Science and Engineering A 462 (2007) 153-158 154 tion region of its real existence to regions close to the pure constituents, a three-sublattice model for the sigma phase was proposed by Anderson and Sundman [2]. This empirical model reduces five sigma phase sublattices to three and contains adjustable parameters in the expression for the Gibbs energy, which are determined by fitting to all available experimental data. This procedure was summarised in the guideline for reducing the number of sublattices in modelling of intermetallic phases by the compound energy formalism (CEF) [3]. One of the most significant shortcomings of the CEF is the restriction of the entropy of mixing to that of an ideal random mixture. This assumption is contained also in the paper of Pratt and Jones [4], who used a pair-wise interaction model for the description of the sigma phase. By the restriction to randomness, the short-range order (SRO) is ignored and a part of the contribution of SRO to Gibbs energy is hidden in the "excess term" after the adjustment of parameters to experimental data. A better description of SRO can be achieved by the cluster variation method (CVM) for con-figurational entropy and by the cluster expansion method (CEM) for the internal energy. The application of these approaches in the CALPHAD treatment is promising, but not commonly used yet [5]. Using the three-sublattice model, the Gibbs energy of the sigma phase is described as an empirical combination of Gibbs energies of some absolutely different structures, i.e. bcc, fee, etc. In this way, we are not able to express the Gibbs energy of pure constituents in the sigma phase structure and the Gibbs energy of the sigma phase of the corresponding binary system has to be adjusted to phase equilibrium data. The physical background of this procedure is, therefore, questionable. Further, it is known from X-ray studies [6] that the mixing of the constituents takes place in all sublattices, which is not respected by the above mentioned approach. 2.1. Two-sublattice model of sigma phase Recently, we have extended the application of ab initio calculations of the total energies of complex phases to calculations of phase diagrams [7-12]. Here, the energy differences between the SER and the sigma phase structure of pure constituents were calculated by the full-potential augmented plane wave (FLAPW) method using the WIEN 97 code [13]. This approach enables us to utilize full physical information about the sigma phase (based on X-ray results) for its thermodynamic description. Knowledge of the above mentioned total energy differences from the first-principles calculations allows us to define the Gibbs energy difference between the sigma phase and SER structures (lattice stability) on physically correct energetic basis with only the entropy term adjusted to phase equilibrium data. Using this idea and the fact that the sigma phase does not behave like a strictly ordered structure (mixing is possible), we have employed a new physical two-sublattice [8] model for its thermodynamic modelling. The molar Gibbs energy of the sigma phase in this model is based on Eq. (2) and is defined as: Gsigma = ^;y.OGsigma_:r5id + G,Ei (3) where O^sigma _ O^SER _|_ ^^.sigma-SER _ j ^O^sigma-SER ^ Sid = -fll>ln;y;, (5) and z^E /0 i sigma , 1 r sigma. , , 2 r sigma. *2\ Gij = ytyji Ltj + Lt6j (yi-yj)+ Lu) (yi - yj) ). (7) Here R is the gas constant, Tin the temperature and A£-'8ma SER denotes the total energy difference between the hypothetical sigma phase and the standard state of a given constituent. Further, S"1 expresses the entropy of ideal mixing in the sigma phase. The parameters L can be temperature-dependent and ternary correction term Gter is defined by: n 1 7 sigma | „ 2, sigma , „ 3 , sigma^ iQ\ Gter = ytyjykiyi L B +yj L B + yt L B ). (8) This procedure gives a physical meaning to parameters from the mathematical expression for the Gibbs energy difference between the sigma phase and SER states of pure constituents. So, the knowledge of AS-'81"3 SER can substantially simplify the physical modelling of the thermodynamic properties of the sigma phase (and other relevant intermetallic phases) in the CALPHAD method. This idea is further illustrated in this work, where the calculations of the phase equilibrium in the Cr-Fe-W system are performed by means of the THERMO-CALC programme [14]. 3. Results and discussion We performed ab initio calculations of the total energy of Cr, Fe and W in their experimentally observed structure (i.e. bcc) and in the sigma phase structure using the FLAPW method implemented in the WIEN 97 code [13] at the equilibrium lattice parameters, i.e. at the volume corresponding to the minimum total energy. The exchange-correlation term was evaluated within the generalized gradient approximation (GGA) [15]. Subsequently, we evaluated the total energy differences between the sigma phase and SER state at 0 K for all studied elements as follows: A£sigma-SER = 30.07kJmor1 = 22.9mRyat_1, fromref. [7], A£sigma-SER = 43 33kTmorl = 33.0mRyar\ fromref. [7], A£sigma-SER = ,44 62kjmol-l = 110.1 mRy at"1, this work. Reprinted from Material Science and Engineering A, 462, Chvátalova K., Vřešťál J., Houserova J., Sob M., First-principles Calculations of Energetics of Sigma Phase Formation and Thermodynamic Modelling in the Cr-Fe-W System, 153-158, Copyright (2007), with permission from Elsevier. Appendix VII: Chvátalova et al. Mat. Sci. Eng. A 462 (2007) 153. 79 K. Chvátalova et al. / Materials Science and Engineering A 462 (2007) 153-158 The magnetic ordering of the SER phases (ferromagnetic bcc Fe, antiferromagnetic bcc Cr and nonmagnetic bcc W) was taken into account. The A£^,gma SER difference is very large (in comparison with, e.g. A£jjJ[80ma SER = 17.77 mRyar1 [12]), which may be indication of a very high degree of unstability of the W sigma phase and this issue should be subject of further investigation. The values of A°5Js'8ma SER were adjusted only to phase equilibrium data, supposing the sigma phase structure to be metastable. In the case of an unstable structure, where entropy is not defined, the value of A°5Js'8ma adjusted to experimental phase data could be regarded as an "effective" value. In general, the A°5Js'8ma SER values (Eq. (4)) providing the phase diagram exhibiting the best agreement with experiments are obtained by optimisation of particular binary systems containing the sigma phase and, if necessary, by a re-optimization using the ternary phase equilibrium data. But in our case, the sigma phase is not stable either in Fe-W or in Cr-W system, so the value of A°5^8ma SER could not be optimised to experimental binary data and it was set to the same value as for Cr and Fe taken from ref. [7]. Values used for A°51s'8ma SER(in J Kr1 mol-1) are: A Sc" =0.7, fromref. [7], A Sf ° =0.7, fromref. [7], . n 0sigma—SER ~ ^, . . A Sw =0.7, this work. The last quantity needed for the evaluation of the 0(^lgma in Eq. (4) is °GfER term and its values for particular constituents were taken from ref. [16]. The G? was expressed by the Redlich-Kister polynomial, Eq. (7), taking into account the SRO. Values of GE in the binary systems Fe-W and Cr-W could be adjusted only to phase equilibrium in ternary Cr-Fe-W system by means of the ternary term defined by Eq. (8). Therefore, we set the starting values of L-parameters in the Fe-W sigma phase to such values that provided Gpf™,^ not too much higher than the values of Gibbs energy of other present phases. Further, we performed optimisation of L-parameters of the sigma phase for Fe-W and Cr-W (including ternary ones) with respect to ternary experimental phase equilibrium data [17]. Resulting values of L-parameters (in J mol-1) are: °LCr:Fe = -133950 °LCr:W = 225661 - 335.883L + 0 ^CnFe = 31000 ^cnw = -999061 + 1083.025L 2Lcr.Fe = -127000 2LCr:W=0 155 To be able to describe the unstability of the sigma phase in whole temperature region, the expression for L-parameters of the Cr-W and Fe-W system are quadratic in temperature. The values of thermodynamic functions of other phases in Cr-Fe-W system were taken from ref. [17]. This paper also presents a lot of reliable experimental phase equilibrium data (obtained by diffusion couple technique and precisely evaluated by SEM-EDX method) describing phase equilibria in the studied system at six temperatures in range between 1173 and 1673 K. These experiments were applied in our thermodynamic modelling which used the two-sublattice model for the sigma phase. The same two-sublattice model was employed also for fee and bcc phases (including magnetic contribution). For other intermetallic phases the proper models [17] were used. The phase diagrams of the Cr-Fe-W system at 1173 and 1673 K, calculated using the two-sublattice model, are given in Fig. l(a and b). They are in reasonable agreement with experimental data taken from ref. [17]. The composition dependencies of the Gibbs energies were calculated for Cr-W and Fe-W systems at 1473 K. They are shown in Fig. 2(a and b). We can see that lines 6 and 7 in Fig. 2(b) obtained using the two- and three-sublattice model [2], respectively, yield different description of metastability of the sigma phase in the Fe-W system, which cannot be resolved experimentally. It is also evident that the two-sublattice model yields the values of Gibbs energy of the sigma phase in the whole composition region, whereas the old three-sublattice model gives this quantity only in a limited range of concentrations. In addition to our FLAPW results for pure constituents, we have also performed first-principles calculations of total energies of both pure elements and alloy sigma phase by means of the linear muffin-tin orbital method in the atomic sphere approximation (LMTO-AS A), using the code by Krier et al. [ 18]. In this way, we obtained the concentration dependence of the total energy L^'8™ in Fe-W and Cr-W system. As the LMTO-ASA method does not provide reliable structural energy differences for structures of different symmetry, we applied the values of A£-'8ma SER calculated by means of the FLAPW method to obtain the energy of formation of the sigma phases with respect to SER states in the whole composition range (Fig. 3(a and b)). These results suggest that the sigma phase is unstable in the whole concentration range both in the Cr-W and Fe-W systems. Fig. 3 (a and b) also displays the comparison of optimised enthalpy of .06114L2 - 0.1964L2 °LFe:W = -390770 + 16.146L + 0.0589L2 °LCr:Fe:W = 986778 - 829.86L ^Fe-w = 103470 - 10.1357 - 0.012L2 1 LCr.Fe:W = 49184.5 - 44.595L 2LFe:W = 63257 + 43.749L - 0.0091L2 2LCr:Fe:W = 197355 - 172.03L, where Lcr:Fe were taken from ref. [7]. Reprinted from Material Science and Engineering A, 462, Chvátalova K., Vřešťál J., Houserova J., Sob M., First-principles Calculations of Energetics of Sigma Phase Formation and Thermodynamic Modelling in the Cr-Fe-W System, 153-158, Copyright (2007), with permission from Elsevier. Appendix VII: Chvátalova et al. Mat. Sci. Eng. A 462 (2007) 153. 80 156 (a) K. Chvátalova et al. / Materials Science and Engineering A 462 (2007) 153-158 Fig. 1. (a) Calculated phase diagram of Fe-Cr-W system at 1173 K according to the two-sublattice model of sigma phase (full lines), compared with experimental equilibrium data with sigma phase from ref. [17]: crosses for equilibrium sigma/bcc, triangles for sigma/chi, squares for bcc/sigma/chi. The greek letters olx,x, k 4 x)Ea (1 + (1 -x)ERS-AES(">. I x£a> + (1 - x)Ea* J FLAPW or VASP different types of symmetry). Both energy differences (i) and (ii) may be considered as quite reliable, as the total energies used for their determination were obtained by the same method on equal footing. If we apply the same method (FLAPW or VASP) for all calculations, the subscripts concerning the method used may be omitted and the formula is simplified to AE* = Ea - [x£RS> + (1 - x)ERSA . The subscripts 1 and 2 following the name of structure denote different pure constituents. We can see that the A£f consists of two parts: (i) the energy difference of alloy sigma phase with respect to weighted average of total energies of pure constituents in the sigma phase structure, both calculated by means of the LMTO, FLAPW (WIEN97) and pseudopotential (VASP) method (the LMTO method may be used here as the systems considered have the same type of structure), and (ii) the energy difference of weighted average of total energies of pure constituents in the sigma phase and RS (Table 4), both calculated by means of the FLAPW (WIEN2k) or pseudopotential (VASP) method (here a more reliable, but also more time consuming method had to be used as the structures involved have The results of these calculations are illustrated in Fig. 3 and the corresponding values of total energies are listed in Tables 4 and 5. From Table 5, we can see that all methods provide slightly different values of the energy of formation of the sigma phase with respect to RS of pure constituents. The differences between our results might be caused by different level of relaxation included. Let us note that all values of total energies used in this paper correspond to the equilibrium lattice volumes. The choice of the equilibrium volumes for the reference states eliminates the uncertainties connected with the use of experimental volumes of binary sigma phases for total energy calculation of hypothetical (i.e. unstable) sigma phase structure of pure constituents as it was used in Ref. [12]. ° • B a o a * • 8 9 ;^>f.f«c, ■ -eF.CtFtF«Cr OUT -2 -4 C -6 H -8 |-10 < -14 -16 -18 -20 : ■ a ctQQCdCi' □ B ■ jj ^ac^fc^ -22 1— 0.0 Fig. 2. Energies of formation of the sigma phase with various occupancies of sublattices calculated by the LMTO-ASA WIEN97 and VASP codes: (a) Cr-Fe, (b) Cr-Co. Hypothetical sigma phase structures of pure constituents are taken as reference states. The squares correspond to the nonmagnetic LMTO results from this work. Open (full) diamonds show nonmagnetic (magnetic) results obtained by the WIEN97 code. The dashed (dotted) line and open (full) circles show nonmagnetic (magnetic) results obtained using the VASP code. The occupation of the sublattices at the open circles on the dashed line is given in the order 2a, 4f, 8i, 8i', and 8j. Reprinted from Intermetallics, 18, Pavlů J., Vřešťál J., Sob M., Ab Initio Study of Formation Energy and Magnetism of Sigma Phase in Cr-Fe and Cr-Co Systems, 212-220, Copyright (2010), with permission from Elsevier. Appendix X: Pavlů J. et al., Intermetallics 18 (2010) 212. 93 216 Pavlů et al. / Intermetallics 18 (2010) 212-220 Table 3 WIEN97 and VASP equilibrium lattice parameters of RS structures, compared with experiment. The WIEN97 parameters were also used in the WIEN2k calculations. Element WIEN97 VASP Experiment a(au) cla Vu (au3) a(au) cla Vu (au3) a(au) cla Vu (au3) Co (FM hep) Cr (AFM bec) Fe (FM bec) 4.7211 5.4165 5.4144 1.6194 1 1 73.7878 79.4559 79.3635 4.7084 5.3947 5.3589 1.6190 1 1 73.1758 78.5004 76.9479 4.7357 5.44 5.40 1.6237 1 1 74.6702 80.4946 78.7320 Ref.[2] Ref.[34] Ref.[34] 3. Results and discussion Fig. 1(a) and (b) display the LMTO total energies of formation of the sigma phases at various compositions. The open squares represent the total energies of all 32 possible occupations of sublattices in sigma phase structure calculated at equilibrium volumes. Let us note that for each studied Cr concentration belonging to the interval (0.267;0.733> we encounter three different configurations according to the occupation of sublattices 8i, 8i' and 8j. This corresponds to three different energies at each of these concentrations in Figs. 1-3 and in Table 6. These three configurations occurring at the same composition are called triplets in our article. The full squares in Fig. 1(a) and 1(b) correspond to the boundaries that approximately show the region of experimentally determined existence of sigma phase. The solid U-shaped line connects the points with the lowest energies and therefore the energy of formation may be related to this line. The descriptions of occupation of sublattices of the most stable configurations (lying on the line) are also given. The triangles represent the results of calculations based on experimental lattice parameters [12] and the dashed U-shaped line has the same meaning for calculations based on experimental parameters as the full one for the equilibrium results. The points in parentheses correspond to the states which are unstable with respect to line connecting the energies of the most stable configurations. The solid arrows show the energies of formation of sigma phase related to the weighted average of total energy of pure sigma phases calculated at equilibrium volume and the dashed ones correspond to the experimental lattice parameters [12]. The values of total energy of formation of alloy sigma phase calculated at equilibrium volume are somewhat smaller for both Cr-Fe and Cr-Co system in comparison with the results obtained at experimental volume and constant internal structure parameters [12]. The decrease in energy at the positions designated by full squares caused by LMTO relaxation is about 1.5-2 mRy/atom in the Cr-Fe system and 1 mRy/atom in the case of Cr-Co system. This effect could be expected because the calculations using the same experimental volume of an alloy sigma phase of certain composition in the whole interval of concentrations provide correct results only for the experimental compositions (no significant decrease in total energy) but they yield higher (non-equilibrium) values of total Table 4 Energies of formation of NM sigma phases of pure constituents with respect to RS states. Element RS AF°-RS = A£f(ii) (kj/mol) WIEN2ka WIEN2kb VASP Ref. [12] Ref. [37] Ref. [16] Co hep 40.567 37.457 34.265 34.4 6.07c - Cr bec 15.513 14.096 13.394 15.4 13.20 13.3 Fe bec 32.617 26.512 25.777 30.9 17.60 38.2 a In WIEN2k calculations, the equilibrium lattice parameters obtained by WIEN97 were used. b In WIEN2k calculations, the equilibrium lattice parameters obtained by VASP code were used. c This value was obtained using the energy difference £™-£™p= 1.97 kj/mol [39], as the value of energy of formation of NM sigma phase in Co obtained in Ref. [37] was related to FM fee Co. energy for pure element phases and configurations close to them. Their energies fall down during the relaxation and cause a decrease of the total energy of formation related to the sigma phase of pure constituents. From these figures and on the basis of our LMTO calculations, we can say that some atoms prefer to occupy certain sublattices, i.e. Fe and Co (in binary systems with Cr) prefer the sublattice 8i' and Cr the sublattices 8i and 8j. The same findings concerning the site preferences in Cr-Fe were reported by Korzhavyi [36] and were confirmed by experimental results [20]. This explains why for compositions xq-> 0.5 only one configuration in the triplet is favoured and the remaining two have higher and similar energies (the 8i' sublattice can be occupied by Fe or Co only once). On the other hand, in case of xcr < 0.5 two configurations are more stabilised and one of them exhibits a higher energy (the 8i' sublattice can be occupied by Fe or Co in two cases). These tendencies may be clearly seen from Table 6, where we present the total energies calculated by the VASP code. Fig. 2(a) and (b) summarise all results of sigma phase calculations obtained in this study. It is obvious that all methods employed, i.e. LMTO-AS A, FLAPW (WIEN97, WIEN2k) and pseudo-potentials (VASP) independently exhibit the same trends. In the case of nonmagnetic configurations (open symbols) the shapes of the U-curves are quite similar. This reveals that in Cr-Fe system the sigma phase of the highest stability lies close to the composition xFe = 0.333 (in the NM case) or at xFe = 0.6 (in the FM case). On the other hand, in the Cr-Co system the sigma phase of the highest stability lies close to the composition xc0 = 0.6 (in the NM case). For FM configurations at xc0 > 0.667 the trend of the U-curve is significantly changed, indicating a strong stabilisation of sigma phase by magnetic ordering. All methods employed also usually predict the same relative stability of configurations at the same composition. In general we can say that accommodation of Fe or Co atoms to sublattices 2a and 8i' has also a stabilising effect. Furthermore the influence of the magnetic order is obvious here. The non-polarised states (WIEN97 and VASP) have higher energies than the polarised ones. Therefore, according to our WIEN97 and VASP calculations, the sigma phase in the Cr-Fe and Cr-Co system should be spin-polarised in most cases at zero temperature. Stabilisation effects of magnetic ordering were not found in the following configurations: CrCrCrCrCr (elemental Cr in the sigma phase structure), FeCrCrCrCr, CrCrCrFeCr and CrCrCoCrCr by VASP and CrCrCrCrCr with CrFeCrFeCr by WIEN97. However these configurations either contain too much Cr or their energies of NM and FM state are too close. The magnetic order in sigma phases has already been found by experiments in Cr-Fe system [14,15]. Our stabilisation energies induced by magnetisation of the most stable configurations at given composition in Cr-Fe system are —2.36 kj/mol at xcr = 0.4 in FeCrCrFeFe and -0.26 kj/mol at xCr = 0.533 in FeFeCrFeCr (WIEN97) and -3.33kJ/mol at xCr = 0.4 in FeCrFeFeCr and -0.11 kj/mol at xCr = 0.533 in FeFeCrFeCr (VASP). These values are in rough agreement with —6.36 kj/mol published by A.L. Udovskii for the FeCrFeFeCr configuration (xq. = 0.40) [17]. However, the effect of magnetic ordering in some unstable configurations is even higher, e.g. for the FeFeFeCrCr configuration Reprinted from Intermetallics, 18, Pavlů J., Vřešťál J., Sob M., Ab Initio Study of Formation Energy and Magnetism of Sigma Phase in Cr-Fe and Cr-Co Systems, 212-220, Copyright (2010), with permission from Elsevier. Appendix X: Pavlů J. et al., Intermetallics 18 (2010) 212. 94 J. Pavlů et al. / Intermetallics 18 (2010) 212-220 Table 5 Energies of formation of sigma phase with respect to RS states, comparison with experiments. System A£f(kJ/mol) NM LMTO NM LMTO VASP Ref. [12] Exp. + WIEN2k + VASP NM FM NM Cr-Fe 0.400 16.16 11.21 10.23 6.89 20.28 7.7a 0.533 13.66 9.34 8.01 7.90 16.78 6.5" Cr-Co 0.533 15.22 11.15 11.50 9.98 17.56 9.37c 0.600 12.47 8.68 9.40 9.04 15.32 (-2.9; 5.1)d (2.64; 6.77)e * xa = 0.45, Ref. [8]. b xCr = 0.45, Ref. [11]. c xCr = 0.6, Ref. [9]. d xCr = 0.45-0.63, Ref. [11]. e xCr = 0.57-0.61, Ref. [10]. at xq- = 0.533 the magnetic stabilisation energy amounts to -5.38 kj/mol (VASP). Similar stabilising effects were found in the Cr-Co system, where the configuration CoCoCrCoCr at xq. = 0.533 is stabilised by -1.52 kj/mol and CrCoCrCoCr at xCr = 0.6 by -0.36 kj/mol (VASP). From analysis of total energies it follows that magnetism does not influence the order of configurations in triplets at given composition very much. However, it does change the shape of the U-curve. The region of the most stable configurations is shifted towards higher concentrations of iron or cobalt. The stability in the chromium rich part is not influenced by the magnetic effects. 217 The above mentioned stabilisation by magnetic effect is enhanced with increasing number of Fe (Co) atoms in the system and it can even reach values of —10.67 kj/mol (VASP) for pure Fe sigma phase and —27.03 kj/mol (VASP) for pure Co sigma phase. This may indicate that only the iron or cobalt atoms carry the magnetization. In both systems it was found that the stabilisation effect in triplets increases with average magnetic moment but the most stable configurations are not those with the highest average magnetic moment. On the contrary, they are usually the least stable ones (see Table 6). Very significant results obtained by the ab initio analysis of magnetism in the studied systems deal with distribution of the magnetic moments in dependence on the kind and positions of atoms (Table 6). It turns out, for example, that elemental iron in the sigma phase structure exhibits quite different magnetic moments at different sublattices. At the 4f, 8i and 8j sublattices, the local magnetic moment of Fe atoms equals to 2.29 /iirj/atom, 2.00 jiiB/atom and 1.87 jiiB/atom, respectively, the highest magnetic moment being at the sublattice 4f with the highest coordination number. Magnetic moments of iron in 8i' and 2a sublattices with the lowest coordination numbers are substantially lower: 1.22 jiiB/atom and 1.10 /iB/ atom (see the first line of Table 6). In case of elemental Co, the magnetic moment decreases from 1.70 jiiB/atom, 1.67 jiiB/atom, 1.59 jiiB/atom to 1.54 jiiB/atom, which corresponds to the sublattices 4f, (8i, 8j), 8i' and 2a, respectively. Again, the atomic magnetic moment decreases with decreasing coordination number. On the other hand, the sigma phase of elemental Cr is nonmagnetic because the magnetic moments found Table 6 Magnetic moments per atom (VASP) at particular sublattices with corresponding energies of formation of the FM configuration (Af/Cl)) related to the energies of elemental NM sigma phases {Ea>NM, E°2"M) as reference states. The numbers in parentheses give the coordination number of atom in the corresponding sublattice. Bold-face numbers mark the sublattices occupied by Cr. Ji denotes average magnetic moment per atom. Cr-Fe Cr-Co Magnetic moment (fts/atom) A£f0> (kj/mol) Magnetic moment (ftB/atom) AEm (kj/mol) 2a (12) 4f (15) 8i (14) 8i' (12) 8j (14) 2a (12) 4f (15) 8i (14) 8i' (12) 8j (14) 0 1.10 2.29 2.00 1.22 1.87 1.75 -8.130 1.54 1.70 1.67 1.59 1.67 1.64 -20.587 0.067 -1.29 2.14 2.11 1.34 1.84 1.63 -8.721 -1.21 1.49 1.69 1.38 1.54 1.34 -16.993 0.133 0.47 -0.92 1.98 1.27 1.76 1.26 -9.527 1.11 -0.55 1.47 1.46 1.60 1.20 -15.378 0.2 -0.69 -0.68 1.99 1.14 1.72 1.17 -8.560 0.01 -0.33 1.52 1.27 1.49 1.09 -13.456 0.267 1.49 2.05 -0.83 0.82 1.78 0.85 -10.132 1.55 1.32 -0.10 0.95 1.36 0.86 -12.424 0.94 2.04 1.75 -1.10 1.78 1.00 -8.847 0.57 1.44 0.86 -0.44 1.14 0.64 -10.224 1.03 2.05 1.68 1.06 -0.61 0.92 -10.113 1.10 1.53 1.47 1.17 -0.28 0.90 -13.203 0.333 -0.84 1.93 -0.64 0.83 1.76 0.73 -10.277 -0.70 1.12 0.00 0.56 1.21 0.58 -11.497 0.08 1.79 1.78 -0.88 1.75 0.96 -6.714 0.08 1.18 0.88 -0.22 1.11 0.63 -8.308 -0.29 1.22 1.50 0.99 -0.16 0.76 -12.074 0.4 0.88 0.30 -0.28 0.53 1.66 0.62 -8.930 1.20 0.41 -0.35 0.47 1.12 0.46 -10.340 -0.19 -0.37 1.56 -0.74 1.66 0.61 -7.385 0.36 0.61 0.63 -0.16 1.17 0.54 -9.203 0.87 -0.44 1.70 1.01 -0.37 0.63 -10.612 -0.06 -0.58 1.18 0.75 0.06 0.45 -12.674 0.467 -0.45 0.41 -0.30 0.51 1.63 0.52 -8.570 0.01 0.29 -0.30 -0.01 0.96 0.21 -9.940 0.25 -0.10 1.65 -0.65 1.68 0.73 -5.312 0.10 0.43 0.71 -0.18 1.16 0.52 -7.398 0.06 -0.22 1.69 0.94 -0.33 0.59 -8.105 0.43 -0.13 1.26 0.86 0.06 0.59 -10.405 0.533 0.56 2.06 1.52 -0.33 -0.13 0.60 -6.217 0.21 1.14 0.15 -0.01 -0.11 0.17 -7.592 0.35 1.24 -0.16 -0.03 -0.07 0.12 -8.586 0.99 1.03 0.35 0.60 0.01 0.46 -10.018 1.01 1.81 -0.13 -0.29 1.60 0.63 -4.114 0.67 1.14 -0.03 0.02 0.86 0.42 -4.799 0.6 0.03 1.95 1.52 -0.24 -0.10 0.58 -4.102 0.09 0.99 0.12 0.08 -0.02 0.18 -5.193 -0.08 1.56 -0.20 0.25 -0.05 0.21 -7.645 -0.40 0.50 0.36 0.37 0.02 0.24 -9.678 -0.25 1.64 -0.15 -0.11 1.56 0.55 -2.129 0.09 0.88 -0.01 0.04 0.85 0.36 -3.129 0.667 0.27 0.21 1.50 -0.39 -0.01 0.35 -5.049 0.45 0.59 0.20 0.05 -0.04 0.17 -6.722 0.62 0.04 -0.03 0.17 -0.06 0.07 -9.173 0.04 -0.29 0.37 0.40 -0.02 0.17 -10.873 0.07 0.27 -0.04 -0.19 1.53 0.39 -3.350 0.33 0.37 -0.04 0.01 0.93 0.31 -3.904 0.733 0.04 -0.06 1.44 -0.31 -0.05 0.28 -3.014 -0.03 0.22 0.24 -0.05 -0.02 0.08 -4.476 0.01 0.00 0.00 -0.04 0.00 -0.01 -7.434 0.06 0.18 0.27 0.40 -0.01 0.21 -9.052 -0.12 0.33 -0.08 -0.10 1.51 0.40 -1.566 -0.04 0.10 -0.06 0.02 0.60 0.16 -1.980 0.8 0.60 1.80 -0.11 0.16 -0.01 0.29 -0.872 0.45 1.04 0.00 0.08 0.01 0.19 -1.093 0.867 0.08 1.73 -0.11 0.18 0.05 0.27 1.200 0.22 0.83 0.01 0.08 0.00 0.15 1.413 0.933 0.00 0.00 0.00 0.00 0.00 0.00 -1.561 0.18 0.16 -0.03 0.10 0.03 0.06 -1.808 1 0.00 0.00 0.00 0.00 0.00 0.00 0.000 0.00 0.00 0.00 0.00 0.00 0.00 0.000 Reprinted from Intermetallics, 18, Pavlů J., Vřešťál J., Sob M., Ab Initio Study of Formation Energy and Magnetism of Sigma Phase in Cr-Fe and Cr-Co Systems, 212-220, Copyright (2010), with permission from Elsevier. Appendix X: Pavlů J. et al., Intermetallics 18 (2010) 212. 95 J. Pavlů et al / Intermetallics IS (2010) 212-220 218 are very close to zero. However, it does not mean at all that the chromium atoms are nonmagnetic through the whole composition region. Their magnetic moment calculated by VASP reaches even —1.21 jiiR/atom (—1.29 jiiB/atom) in CrCoCoCoCo (CrFeFeFeFe) - see the second line of Table 6. In binary sigma phases we can see that the atomic magnetic moments of all three constituents (i.e. Fe, Co, Cr) mostly decrease with increasing molar fraction of chromium. Similarly as in elemental Fe and Co, the highest values of magnetic moments are found at the 4f sublattice with the highest coordination number. In Cr-Fe system, the Cr atoms exhibit very often antiferromag-netic behaviour with respect to Fe atoms, i.e., they have the opposite orientation of magnetic moments. This fact is fully manifested at 8i, 8i' and 8j sublattices with some exceptions for higher chromium concentrations. The antiparallel arrangement of magnetic moments is also found at the 2a and 4f sublattices, again with some exceptions. In sporadic cases the antiferromagnetic arrangement occurs at Fe atoms in the sublattice 2a for FeCrFeCrFe configuration and in 8i' for FeFeCrFeCr and CrCrCrFeCr. In Cr-Co system Cr atoms also exhibit antiferromagnetic behaviour with respect to Co atoms. However, with increasing molar fraction of Cr atoms, we observe increasing amount of Cr atoms that behave in the ferromagnetic way with respect to Co atoms. In contrast to the Cr-Fe system the occurrence of this arrangement is not fully connected with particular sublattices, but it is most frequent at the 2a and 4f sublattices. The WIEN97 (VASP) calculated average magnetic moment per atom of the most stable configurations in Cr-Fe extends from 0.26 jiiB to 0.10 fig (0.63 fiB to 0.12 fiB) for xCr = 0.400-0.533. The corresponding values of the WIEN97 (VASP) calculated average magnetic moment per Fe atom of the most stable configurations in Cr-Fe extends from 0.65 fiB to 0.21 fiB (1.30 jiiB-0.39 fiB) for xCr = 0.400-0.533, which is comparable with experiments. For example, Cieslak et al. [ 14] reported the average magnetic moment per Fe atom in the range of 0.287(3) jiiB-0.142(l) jiiB for the composition range xcr = 0.450-0.496, where the average Curie temperature 7c ranges from 38.9 K to 8.3 K. The magnetism of the sigma phase at low temperatures was also confirmed by Read et al. [38]. Furthermore Korzhavyi [36] claims that magnetic moments are vanishing on Fe atoms occupying icosahedral sites (sublattices 2a, 8i') and they are high, cca 2 jiiB, on Fe atoms occupying the sites with a high coordination numbers (sublattices 4f, 8i, 8j). Similar trend s were confirmed by our calculations. The magnetic moment of the Fe in icosahedral sites is significantly lower but not vanishing. The VASP calculated average magnetic moment per atom for the most stable configurations in Cr-Co extends from 0.46 jiiB to 0.24 jiiB for xcr = 0.533-0.600 (see Table 6). Fig. 3(a) and (b) demonstrate our approach to the calculations of the total energies of formation of sigma phase related to the total energies of RS structures. The triangles and squares have the same meaning as in Fig. 1(a) and (b) whereas the lower lines represent the linear combination of total energies of RS structures based on equilibrium lattice parameters (full line - WIEN2k results based on WIEN97 structure data, long-dashed line - VASP) or on the results calculated for the experimental lattice parameters [12] (dashed line). The zero of total energy is constituted by the weighted average of total energies of sigma phases of pure constituents similarly as in Fig. 1(a) and (b). The solid arrows show the energies of formation of sigma phase calculated at equilibrium lattice parameters and the dashed ones correspond to the experimental lattice parameters [12]. The lattice parameters corresponding to the minimum of total energies of RS are listed in Table 3. Fig. 3(a) and (b) demonstrate that the total energy differences for the pure constituents (A£f'"' = £0_RS, Table 4) calculated by various methods are close. The results obtained are even quite close to the differences based on the experimental volumes. Let us note that for elemental Cr, our values of energy of formation agree very well with the previous calculations [12,16,37]. In case of Fe our calculated numbers are between the values calculated in Ref. [37] and Ref. [16]. Co sigma phase was not treated in Ref. [16] and the value from Ref. [37] is related to another reference state. Trying to get this value on equal footing, we arrived to 6.07 kj/mol which is surprisingly low in comparison with other available values. The equilibrium energies of formation of the sigma phase related to the weighted average of total energy of pure RS structures in the range of composition that corresponds to the experimental one (arrows in Fig. 3(a) and (b)) are summarised in Table 5 and in Fig. 4(a) and (b) together with the results of calculations performed at the experimental volumes [12] and the experimental values measured calorimetrically by various authors [8-11]. We can see that the energies of formation obtained by various methods reveal some scatter. Concerning the nonmagnetic arrangement, the values of the energies of formation of the sigma phase in the Cr-Fe system at both compositions decrease in series 0,0 0.2 0,4 0.6 0,8 1,0 0.0 0,2 0.4 0.6 0.8 1.0 Fig. 3. Total energies of formation of sigma phase with respect to RS states: (a) Cr-Fe, (b) Cr-Co. The full (dashed) line and squares (triangles) correspond to the LMTO + WIEN2k (LMTO +WIEN97) results from this work (from Ref. [12]), based on the WIEN97 equilibrium (experimental) structure data. Long-dashed line shows the energy differences between the RS and the sigma phase of pure constituents calculated by the VASP code. The arrows mark the energies of formation of the most stable configurations with respect to the RS of pure constituents close to the border of the experimental region of existence of the sigma phase. Similar arrows for the VASP calculations can be also constructed but they were omitted for better clarity of the figure. Reprinted from Intermetallics, 18, Pavlů J., Vřešťál J., Sob M., Ab Initio Study of Formation Energy and Magnetism of Sigma Phase in Cr-Fe and Cr-Co Systems, 212-220, Copyright (2010), with permission from Elsevier. Appendix X: Pavlů J. et al., Intermetallics 18 (2010) 212. 96 J. Pavlů et al. / Intermetallics 18 (2010) 212-220 1 1 1 1 1 1 1 Hiwinkůví (200]), _ ab-initio calculations I based on CHptrimenta] -volumes — t | I.MTO + WLEN2k - ■ LMTO + VASP VASP NM Dench (l«3).i40QK reaction. c.iEor inn-try Zubkov (1990), 304K ~~ reaction -1- dissolution CAiorimrtry VASPFM a i , i 1 0-4 0-6 - 10 1 Havránková (2001). A ab-initio calculations based i on experimental volumes \ q LMTO + WIENZk Bell (1973), dni., 1473K Downic (1983), j J Ml.. 473K VASPNM LMTO + VASP VASPFM -5 0.0 Zubkov (1990), sol.. 304K _i_I_i_L_ 0.2 0-4 I 06 Fig. 4. Calculated energies of formation of sigma phase, compared with experimentally determined enthalpies of formation: (a) Cr-Fe, (b) Cr-Co. The RS structures correspond to the zero line. The full (dashed) arrows and open squares (full triangles) correspond to the LMTO + WIEN2k (LMT0 +WIEN97) results from this work (from Ref. [12]), both based on the WIEN97 lattice parameters. Similar arrows for the VASP calculations and experiments were omitted for better clarity. Open and full circles and full diamonds describe the results of the NM VASP, FM VASP and LMTO + VASP analysis. Experiments done by Dench [8], Bell [9], Downie [10], and Zubkov [11 ] are marked by asterisks, crosses, open triangles and full squares, respectively. LMTO + WIEN97 (based on experimental lattice parameters) [12], LMTO + WIEN2k, LMTO + VASP, and VASP. The same statement is valid also for the Cr-Co system except of the fact that the VASP numbers are above the LMTO + VASP ones. This tendency fully reflects the level of structure relaxation, i.e. the more relaxed sigma phase the lower energies of formation are obtained. Furthermore the results provided by LMTO + VASP and VASP calculations correspond to the experimental data very well. From Fig. 4(a) and (b), we can see that inclusion of magnetic ordering improves the overall agreement with experiments. On the other hand, it is also necessary to take into account the reliability of the experimental data in these comparisons, and Fig. 4(b) shows that the scatter of measured values is indeed very large. The accuracies given for the calorimetrie measurements usually refer to their physical performance. Errors of chemical nature may often be much more serious (e.g. impurities, incomplete reactions, side reactions). So inspecting the results of Zubkov [11], whose results suggest decrease in heat of formation with Co concentration, the experimental error has to be assessed from results given in their Table 1, where the values of —AF*5501 for the sigma phase that contains more than 45 at% of Co were obtained by extrapolation of experimental data. It seems to be questionable to believe in decrease of heat of formation with Co composition on the basis of these results. The comparison of heat of formation measured experimentally with the calculated values is performed under the assumption that AE(0 K) = AH(T), which need not be valid exactly. Further we should take into account that AH(T) contains the PV term and vibrational / CpdT term for both sigma and RS phase which are not included in AE(0 K). It may elucidate the difference of calculated and measured energies of formation. The difference between the experimental values of heat of formation in Cr-Co system for C00.4&0.6 of Bell [9] (9.37 kj/mol) and Zubkov [11] (2.1 kj/mol) is more than 400%. So, the higher values of heat of formation calculated by the ab initio methods in comparison with the measured ones are not too bad in the case of Cr-Co system. parameters of experimentally found sigma phase for calculations of total energy of hypothetical configurations of pure constituents. Our study also provides the analysis of the influence of magnetism on the stability of the Cr-Fe and Cr-Co sigma phase. The results of ab initio calculations of total energies of sigma and RS phases of pure constituents (obtained by FLAPW method) combined with total energies of sigma phases for different compositions and for pure elements calculated by LMTO-ASA and FLAPW method were employed to study the stability of sigma phase in Cr-Fe and Cr-Co systems. Furthermore the additional calculations performed by the pseudopotential code VASP and the analysis of the effect of magnetism were carried out. The influence of relaxation on lattice parameters of all structures was also included in this work. It turns out that the more structural relaxations are used the lower energies of formation are obtained. The lowest energies of formation were acquired by the VASP code with the full relaxation and the obtained energies are fully comparable with experimental data. The preferential occupation of the sub-lattices was analysed. Inclusion of magnetic ordering in the Cr-Fe and Cr-Co sigma phase calculations causes the decrease of energy of formation and results in the shift of stability region towards the configurations with a higher concentration of iron or cobalt. We predict that Cr-Fe and Cr-Co sigma phases are magnetically ordered at 0 K, which corresponds to experimental findings. Our study reveals that the largest part of magnetization is carried by the iron or cobalt atoms and that the chromium atoms do not contribute to this effect very much although they induce a decrease of the total magnetic moment by their antiferromagnetie behaviour at some sublattiees. The magnetic moments depend on the kind and position of the atom and agree satisfactorily with available experimental data. The influence of the sublattiee on the magnetic moment is the same for both Fe and Co atoms and their magnetic moments decrease from the sublattiees 4f, 8i, 8j, 8i' to 2a. The magnetic moment of Cr in the sigma phase structure is close to zero. However it increases with the increasing number of Fe or Co atoms. 4. Conclusions The new achievement of the present paper consists in calculations of total energies of sigma phase and RS at equilibrium volume. This removes the uncertainty connected with the use of lattice Acknowledgments This research was supported by the Ministry of Education of the Czech Republic (COST Project No. OC 164). Reprinted from Intermetallics, 18, Pavlů J., Vřešťál J., Sob M., Ab Initio Study of Formation Energy and Magnetism of Sigma Phase in Cr-Fe and Cr-Co Systems, 212-220, Copyright (2010), with permission from Elsevier. Appendix X: Pavlů J. et al., Intermetallics 18 (2010) 212. 97 220 J. Pavlů et al. / Intermetallics 18 (2010) 212-220 References [1] Bain EC. Chem and Met Eng 1923;2S:23. [2] Villars P, Calvert LD. Pearson's handbook of crystallographic data for inter- metalic phases. Materials Park (OH): ASM International; 1991. [3] Cook AJ, Jones FW. J Iron Steel Inst 1943:148:217. London:1943:148:223. [4] Yakel HL Acta Crystallogr 1983:B39:20. [5] Dickins GJ, Douglas AMB, Taylor WH. Acta Crystallogr 1956:9:297. [6] Forsyth JB, d'Alte da Viega. Acta Crystallogr 1963:16:509. [7] Heino S, Knutson-Wedel EM, Karlsson B. Mater Sei Technol 1999:15:101. [8] Dench WA. Trans Faraday Soc 1963:59:1279. [9] Bell HB, Hajra JR Putland FH, Spencer PJ. Met Sei Journal 1973:7:185. [10] Downie DB, Arlasan F. J Chem Thermodyn 1983:15:654. [11] Zubkov AA, Mogutnov BM, Shaposhnikov NG. Dokl Akad Nauk SSSR 1990:311(2):388: Dokl Phys Chem 1990:311:239. [12] Havránková J, Vřeštil J, Wang LG, Šob M. Phys Rev B 2001:63:174104. [13] Sluiter MHF, Esfarjani K, Kawazoe Y. Phys Rev Lett 1995:75:3142. [14] Cieslak ], Reissner M, Steiner W, Dubiel SM. ] Magn Magn Mater 2004:534:272. [15] Cieslak ], Reissner M, Steiner W, Dubiel SM. Physica Status Solidi (A) 2008:205:1794. [16] Korzhavyi PA, Sundman B, Selleby M, ]ohansson B. Mater Res Soc Symp Proc 2005;842:S4.10.1. [17] Udovskii AL, SCTE2008 (16th International conference on solid compounds of transition elements in Dresden) Abstract Book, 2008. Poster No.146-3, p. 139. [ 18] Kabliman EA, Mirzoev AA, Udovskii AL Fizika Metallov i Metallovedenie, to be published. [19] Ackland GJ. Phys Rev B 2009:79:094202. [20] Joubert J- M. Prog Mater Sei 2008:53:528. [21] Ansara I, Chart TG, Fernandez Guillermet A, Hayes FH, Kattner UR, Pettifor DC, et al. Calphad 1997:21:171. [22] Andersen OK. Phys Rev B 1975:12:3060: Skriver HL The LMTO method. Berlin: Springer Verlag; 1984. Turek I, Drchal V, Kudrnovsky J, Sob M, Weinberger P. Electronic structure of disordered alloys, surfaces and interfaces. Boston-London-Dordrecht: Kluwer; 1997. [23] Krier G, Jepsen O, Burkhardt A, Andersen OK. Computer code TB-LMTO-ASA version 4.6. Stuttgart: Max-Planck-Institut für Festkörperforschung: 1994. [24] Perdew JR Chevary JA, Vosko SH, Jackson KA, Pederson MR, Singh DJ, et al. Phys Rev B 1992:46:6671. [25] Singh D. Planewaves, pseudopotentials and the LAPW method. Boston: Kluwer; 1994. [26] Blaha R Schwarz K, Luitz J. Computer code WIEN97. Vienna: Vienna University of Technology; 1997; improved and updated Unix version of the original copyrighted WIEN code, which was published by Blaha R Schwarz K, Sorantin R Trickey SB. Comput Phys Commun 1990;59:399. [27] Kresse G, Furthmüller J. Comput Mat Sei 1996;6:15. [28] Kresse G, Furthmüller J. Phys Rev B 1996:54:11169. [29] Blöchl P. Phys Rev B 1994:50:17953. [30] Kresse G, Joubert J. Phys Rev B 1999:59:1758. [31] Perdew JR Burke K, Ernzerhof M. Phys Rev Lett 1996:77:3865. [32] Andersen OK, Methfessel M, Rodriguez CO, Blöchl R Polatoglou HM. In: Vitek V, Srolovitz DJ, editors. Atomistic simulation of materials: beyond pair potentials. New York-London: Plenum; 1989. p. 1. [33] Tank RW, Arcangeli C, Krier G, Andersen OK, Jepsen O. In: Gonis A, Meike A, Turchi PEA, editors. Properties of complex inorganic solids. New York-London: Plenum; 1997. p. 233. [34] Moroni EG, Jarlborg T. Phys Rev B 1993:47:3255. [35] Blaha R Schwarz K, Madsen GKH, Kvasnicka D, Luitz J. WIEN2k, An augmented plane wave plus local Orbitals program for calculating crystal properties. Vienna: Vienna University of Technology; 2001. [36] Korzhavyi PA. ADIS2006 (Ab initio description of iron and steel), February 19-24, 2006. Ringberg Castle, Germany. Invited talk: Theoretical design of hard materials. [37] Sluiter MHF. Calphad 2006:30:357. [38] Read DA, Thomas EH, Forsythe JB. J Phys Chem Solids 1968:29:1569. [39] Moroni EG, Kresse G, Hafner J, Furthmüller J. Phys Rev B 1997:56:15629. Reprinted from Intermetallics, 18, Pavlů J., Vřešťál J., Šob M., Ab Initio Study of Formation Energy and Magnetism of Sigma Phase in Cr-Fe and Cr-Co Systems, 212-220, Copyright (2010), with permission from Elsevier. Appendix XI: Pavlů J. et ed., CALPHAD 34 (2010) 215. 98 CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 215-221 Contents lists available at ScienceDirect ELSEVIER CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad Thermodynamic modeling of Laves phases in the Cr-Hf and Cr-Ti systems: Reassessment using first-principles results J. Pavlu *, J. Vfesfal, M. Sob Department of Chemistry, Faculty of Science, Masaryk University, Brno, Czech Republic Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Brno, Czech Republic ARTICLE INFO Article history: Received 14January 2010 Received in revised form 8 March 2010 Accepted 9 March 2010 Available online 30 March 2010 Keywords: Ab initio calculations Laves phases Chromium-hafnium system Chromium-titanium system Phase diagrams ABSTRACT The Cr-Hf and Cr-Ti belong to interesting systems exhibiting the existence of all polytypes of Laves phases, i.e. lower-temperature cubic C15 and higher-temperature hexagonal C14 and C36, although in the Cr-Hf phase diagram only C14 and C15 phases occur. Comparison of total energies of these structures calculated from first principles with the total energy of the ideal mixture of elemental constituents reveals the relative stability of Laves phases in these systems. The effect of magnetic order in the Laves phases is also briefly discussed. The calculated total energies of formation of all the three polytypes are employed in two- and three-sublattice models to revise the thermodynamic description of both the systems published recently. New remodeled Gibbs energies of Laves phases require less fitting parameters than those obtained in previous treatments and corresponding phase diagrams provide an excellent agreement with the experimental phase data found in the literature. The proposed procedure allows us to compare the optimised heat capacity differences with those determined experimentally or theoretically and to use them in phase diagram calculations. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction The Laves phases consisting of chromium and an element from the IV group of the Periodic Table have been systematically investigated because they are part of structural materials at temperatures up to 1200 °C e.g. Nb-silicide based in situ composites [ 1,2]. It has been widely accepted that Cr improves oxidation resistance and Hf enhances both oxidation resistance and strength of these materials. However, additions of new elements may introduce some detrimental phases to the composites which deteriorate the properties of the materials [3]. Therefore, reliable thermodynamic description of stability of phases in both Cr-Hf and Cr-Ti systems is needed for successful materials design. The Cr-Hf and Cr-Ti systems have been critically reviewed by Yang et al. [3] and Zhuang et al. [4], respectively. Here the sublattice model [5,6] successfully described the thermodynamic behaviour and homogeneity ranges of solid solutions and different types of Cr2Hf and Cr2Ti Laves phases. In both systems, 6 equilibrium phases have been reported: liquid, HCP, BCC, C14, C15 and C36 Laves phases. In the Cr-Hf system, the Laves phase structure transforms as a function of temperature from the Corresponding author at: Department of Chemistry, Faculty of Science, Masaryk University, Brno, Czech Republic. Tel.: +420 549493742: fax: +420 549491453. F-mail address: houserova@chemi.muni.cz (J. Pavlu). 0364-5916/$ - see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2010.03.003 lower-temperature cubic C15 polytype to the hexagonal C14 polytype at higher temperatures, avoiding C36 structure [7]. The Cr-Ti system was studied in more detail. It was found that the Laves phase structures transform with increasing temperature from the cubic C15 polytype to the hexagonal C36 polytype at intermediate temperatures and to the hexagonal C14 polytype at the highest temperatures. The C15-C36 transformation temperatures are reported to increase strongly with increasing Ti content from 804 °C at 33.5 at.% Ti to 1223 °C at 37.3 at.% Ti, whereas the transformation temperature from C36 to C14 seems to be nearly independent of composition at 1270 ± 1 °C. However, as described in detail in [4], there is contradicting information on the existence of C36 structure in the literature, connected with its metastability. C36 is an intermediate structure occurring in transformation from C14 to C15 during cooling and the transformation from C36 to C15 is very sluggish. Therefore, the phase diagram of Cr-Ti is still far from being well established. Existing experimental studies [4,8] conclude that the stoichiometric compositions are not incorporated in any of the phase fields of the Laves phases and show that all structures of Cr2Ti are stable only with an excess of Ti atoms. During the last decade or so, structural energy differences (lattice stabilities) calculated from first principles have been employed in the construction of phase diagrams of systems containing complex phases (see e.g. [9-12]); a recent review may be found in [13,14]. In the present paper, the ab initio calculated total energies of formation of two polytypes in the Cr-Hf system and of all the Reprinted from CALPHAD, 34, Pavlu J., Vfesfal J., Sob M., Thermodynamic Modeling of Laves phases in the Cr-Hf and Cr-Ti Systems: Reassessment Using First-principles Results, 215-221, Copyright (2010), with permission from Elsevier. Appendix XI: Pavlu J. et al, CALPHAD 34 (2010) 215. 99 J. Pavlu etal./ CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 215-221 216 three polytypes of Laves phases in the Cr-Ti system are successfully employed in the two- and three-sublattice models to revise the thermodynamic descriptions published in [3,4]. It turns out that total energies of individual phases from the ab initio calculations strongly reduce the number of fitting parameters needed in construction of phase diagrams in Cr-Ti and Cr-Hf systems and, in this way, bring more physics into CALPHAD method. 2. Thermodynamic modeling Using thermodynamic modeling, behaviour of any system under various conditions (i.e. pressure, temperature or composition) can be described. This approach allows us to predict properties and behaviour of systems, which are technologically important for the improvement of current materials or for development of new ones. Contrary to experiments, this treatment provides not only the overview of properties of really existing structures but it is also capable to characterise metastable or even unstable configurations. The main quantity used in such modeling is the molar Gibbs energy of the whole system, which is defined as the sum of molar Gibbs energies of all included phases Cf multiplied by their molar fractions x!: Gtot = Ef>/Gf, (1) where G! = EyfC* + Gid + GE + Gmls + GVIl:s. (2) The molar Gibbs energy of phase (/ contains the sum of molar Gibbs energies of pure constituents i in the phase / multiplied by their lattice fractions 27;y;°G{, the terms describing ideal (G'd) and non-ideal (GE) mixing and, when needed, some special terms as magnetic (Gmag) or pressure (Gpres) contribution. For a binary system (A-B), the terms describing mixing may be evaluated by relatively simple formulas as follows: Gicl =RT(yAlnyA+yBlnyB) (3) GE = yAyB(l°(T) +i1(r)(yA -yB) +I2(T)(yA -yB)2 + ■ ■ ■) (4) where 1°, I1 and I2 are the expansion coefficients of the Redlich-Kister polynomial [ 15 ] and T is temperature. Temperature dependence of L-parameters is given by an equation of type L> = a, + bjT 4-c/rTnT, (5) where a,, b, and c, are constants. For liquid phase, we use a model (A, B)i, for BCC and Laves phases, we employ the (A, B)i(Va)3 and (A, B)2(A, B), or (A, B)4(A, B)6A2 models [5,6], respectively. Similar approach has already been used in systems Cr-Nb, Cr-Ta and Cr-Zr [11,12]. Ideal mixing in Laves phases is then described by expression Gld = RT[(yA lnyA + yB lnyB) + (y2 lny2 + y2 lny2)] (6) where the superscripts are related to sublattices 1 and 2. In the case of two-sublattice model of Laves phase [5,6], the sum of molar Gibbs energies of pure constituents i in the phase / multiplied by their lattice fractions 27;y;°G{ is substituted by equation cref = y]A "Caa + yAy2 °Ga-.b + yBy2 "gba + yBy2 °cB:B (7) where Gibbs energies of all "end-members" (°G;:J) have to be calculated, as mentioned above. For the sake of thermodynamic modeling with the help of the two-sublattice model, some of the experimentally found sublattices of Laves phases have to be merged, e.g. the 2a and 6h sublattice occupied by Cr atoms are joined to form one sublattice in thermodynamic model of the C14 phase. Similarly the 4e and 4f sublattices for X atoms (X = Hf, Ti) and 4f, 6g, and 6h sublattices for Cr atoms in C36 structure are also merged into two sublattices. The difference in Gibbs energies of the pure constituents in the Laves (L) phase and in the Standard Element Reference (SER) state (i.e. antiferromagnetic (AFM) BCC Cr and nonmagnetic (NM) HCP Ti and Hf) is given by equation ^O^I-SER _ Oj~I _ O^SER _ ^O^L-SER _ ^-^O^I-SER ^gj where H is enthalpy and S entropy. At T = 0, we may write that A0HISER(T = 0) = A0£ISER(T = 0), i.e. the difference in enthalpies is equal to the difference of total energies, because the total energy at 0 K is calculated at equilibrium volume, where it holds (311/3 V)T = T(dp/dT)v-p = 0. The total energy differences have been calculated ab initio at equilibrium volume in the present paper. The difference in enthalpies, A°HtSER at finite temperature, is then obtained as (Kirchhoff law) A0Ht-SER = A0£t-SER + J ACI-SERd^ (g) In the region without phase transformation, entropy can be expressed as A0S1-SER = J (ACI-SER/r)dr (1Qj which, substituted in Eq. (8), after integration (supposing ACp~SER = const.) yields A0Gt-SER = A0£t-SER + (ACI-SER)r _ (ACI-SER)r ,n j {u] This equation may be successfully employed in the phase diagram calculations. Usually, A°EI_SER is calculated ab initio, ACE'SER is optimised, as we show below. However, also the value of ACp~SER may be taken from calculations (e.g. from phonon spectra) or from measurements (e.g. from the analysis of Einstein or Debye functions). In this way, we could further reduce the number of fitting parameters needed for description of phase diagrams. 3. Ab initio calculations of total energies Our electronic structure calculations were performed within the Density Functional Theory (DFT). We have used the pseudopo-tential method [16] incorporated into the Vienna Ab initio Simulation Package (VASP) code [17,18] combined with the Projector Augmented Wave-Perdew-Burke-Ernzerhof (PAW-PBE) pseu-dopotential [19-21] (i.e. we have employed the generalised gradient approximation for the exchange-correlation energy), and calculated the total energies of all three Laves phase structures (C14, C15 and C36) as well as the total energy differences between the Laves phases and the SER structures. The cut-off energy restricting the number of plane waves in the basis set was 295 eV, 232 eV and 286 eV for Cr, Ti and Hf, respectively, for both pure constituents and the Laves phases. Spin polarisation was not included in our calculations, except when noted. Reason for this is the fact that all Laves phases found in the systems studied are paramagnetic at ambient temperatures. Nevertheless, calculations for spin-polarised Cr2Cr Laves phase (C15) confirmed the influence of magnetism also on the stability of this structure at 0 K. Preliminary calculations of A2A, A2B, B2A, B2B (A = Cr, B = Ti, Hf) configurations of C14, C15 and C36 Laves phase structures needed for two-sublattice model and also calculations of A6B6 and A2B]0 configurations for C14 and C36 structures needed for three-sublattice model were accomplished using the experimentally found lattice parameters for Cr2Ti and Cr2Hf configurations published in [22] except for the C14 and C15 (C36) Laves phase of pure chromium, which was studied using the parameters Reprinted from CALPHAD, 34, Pavlu J., Vfestal J., Sob M., Thermodynamic Modeling of Laves phases in the Cr-Hf and Cr-Ti Systems: Reassessment Using First-principles Results, 215-221, Copyright (2010), with permission from Elsevier. Appendix XI: Pavlu J. et al, CALPHAD 34 (2010) 215. 100 J. Pavlu etal.j CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 215-221 217 Table 1 Calculated and experimental lattice parameters of the SER structures. Symbols a and c stand for lattice constants, Vat is the atomic volume and A shows the relative difference between the calculated and experimental atomic volume. SER Structure o(pm) c (pm) c/a Vat (107 pm3) AFM BCC Cr Exp. [23] 287.87 287.87 1.0000 1.19281 -2.48 Relax. 285.48 285.48 1.0000 1.16327 NM HCP Hf Exp. [22] 323 512 1.5851 2.31300 -3.63 Relax. 319.48 504.34 1.5787 2.22896 NM HCPTi Exp. [22] 295.04 468.10 1.5866 1.76438 -2.97 Relax 292.39 462.49 1.5818 1.71204 Table 2 Equilibrium structural parameters of the C14, C15 and C36 Laves phases found in this work. Symbols a and c stand for lattice constants, z, x andy are the internal structural parameters and Vat is the atomic volume. Structure C14 C15 "(pm) c (pm) Vat (107 pm3) 4f-z 6h-x 6h-y "(pm) Vat(107 pm3) Exp. Cr2Hf [22] 505.6 821 1.51463 _ _ _ 714 1.51664 Cr2Cr 464.09 766.32 1.19117 0.0464 0.8289 0.6578 658.04 1.18726 Cr2Hf 503.14 805.61 1.47179 0.0604 0.8315 0.6631 706.55 1.46966 Cr6Hf6 626.68 634.46 1.79822 0.1223 0.8256 0.6513 - - CrHf2 552.85 865.58 1.90930 0.0627 0.8253 0.6505 777.45 1.95797 Cr2Hf,„ 602.11 783.44 2.04974 0.0702 0.8325 0.6650 - - Hf2Hf 569.53 950.50 2.22500 0.0540 0.8267 0.6535 815.72 2.26154 Structure C36 a(pm) c(pm) Vat(107 pm3) 4e-z 4f-z 4f-z 6h-x 6h-y Exp. Cr2Hf [22] 506.4 1647 1.52405 - - - - - Cr2Cr 465.11 1523.11 1.18893 0.1003 0.8485 0.1236 0.1623 0.3247 Cr2Hf 500.90 1624.27 1.47056 0.0945 0.8439 0.1234 0.1647 0.3294 Cr6Hf6 625.94 1275.76 1.80365 0.0638 0.8140 0.1323 0.1568 0.3137 CrHf2 561.55 1771.11 2.01529 0.0600 0.8585 0.1266 0.1603 0.3207 Cr2Hf,„ 601.38 1569.56 2.04832 0.0895 0.8394 0.1250 0.1652 0.3305 Hf2Hf 569.26 1915.81 2.24022 0.0999 0.8487 0.1302 0.1589 0.3177 Structure C14 C15 a(pm) c (pm) Vat (107 pm3) 4f-z 6h-x 6h-y a(pm) Vat(107 pm3) Exp. Cr2Ti [22] 493.22 800.53 1.40543 0.0625 0.833 0.666 6.9324 1.38816 Cr2Cr 464.09 766.32 1.19117 0.0464 0.8289 0.6578 658.04 1.18726 Cr2Ti 486.19 778.39 1.32789 0.0605 0.8299 0.6599 682.86 1.32676 Cr6Ti6 569.68 631.96 1.48011 0.1111 0.8259 0.6518 - - CrTi2 513.21 786.29 1.49463 0.0669 0.8237 0.6474 715.00 1.52301 Cr2Ti10 563.73 706.72 1.62080 0.0754 0.8289 0.6579 - - Ti2Ti 517.22 894.65 1.72722 0.0523 0.8263 0.6527 749.63 1.75520 Structure C36 a(pm) c (pm) Vat (107 pm3) 4e-z 4f-z 4f-z 6h-x 6h-y Exp. Cr2Ti [22] 493.22 1601.10 1.40546 0.094 0.844 0.125 0.167 0.334 Cr2Cr 465.11 1523.11 1.18893 0.1003 0.8485 0.1236 0.1623 0.3247 Cr2Ti 484.27 1568.36 1.32720 0.0947 0.8437 0.1236 0.1633 0.3266 Cr6Ti6 562.73 1290.70 1.47483 0.0729 0.8203 0.1286 0.1583 0.3167 CrTi2 512.43 1587.44 1.50412 0.0908 0.8450 0.1288 0.1556 0.3112 Cr2Ti10 563.17 1417.17 1.62188 0.0871 0.8374 0.1264 0.1615 0.3229 Ti2Ti 506.90 1882.28 1.74524 0.1004 0.8496 0.1276 0.1630 0.3259 of corresponding Cr2Ta (Cr2Zr) phases, also given in [22]. The structural parameters for the SER states of Ti and Hf were taken from [22] and of Cr from [23]. First, we performed convergence tests of total energies with respect to the number of k-points. The range of optimum values for the C14 Laves phases extends from a grid of 15 x 15 x 11 points (Cr2Cr, Ti2Ti), 17 x 17 x 15 points (Cr2Ti) and 19 x 19 x 13 points (Cr6Ti6 and Cr2Tii0) towards 19 x 19 x 15 points (CrHf2), 21 x21 x 15 points (Cr2Hf, Cr2Hf10), 21 x 21 x 17 points (Hf2Hf, CrTi2) and 23 x 23 x 15 points (Cr6Hf6). A similar interval was obtained for theC15 Laves phases which goes from a grid of 15 x 15 x 15 points (Cr2Cr) and 19 x 19 x 19 points (Cr2Ti) towards 21 x21 x21 points (Hf2Hf, Cr2Hf, CrHf2 and CrTi2) and 23 x 23 x 23 points (Ti2Ti). A smaller range was obtained for the C36 Laves phases which spreads from a grid of 15 x 15 x 13 points (Cr2Cr) and 17 x 17 x 13 points (Hf2Hf, Cr2Hf, Cr2Ti, CrTi2) towards 19 x 19 x 15 points (Ti2Ti, Cr6Hf6, CrHf2, Cr2Hf,0, Cr6Ti6 and Cr2Ti,0). In the case of SER structures, we used a grid of 13x13x13 points for AFM BCC Cr, of 19 x 19 x 15 points for NM HCP Ti and of 21 x 21 x 13 points forNM HCP Hf. After these test calculations, each structure was fully relaxed which yielded the minimum total energy and the equilibrium structural parameters. The results obtained are summarised in the following subsections. As the Cr SER structure and the C15 Laves phase are cubic, only volume relaxation is sufficient to get their lowest energy state. The calculated optimum lattice parameters for the SER structures are listed in Table 1 and those for the C14, C15 and C36 structures in Table 2. Spin polarisation was included in AFM BCC Cr calculations. As for Laves phases, we performed a pilot study in [24] for Reprinted from CALPHAD, 34, Pavlu J., Vfestal J., Sob M., Thermodynamic Modeling of Laves phases in the Cr-Hf and Cr-Ti Systems: Reassessment Using First-principles Results, 215-221, Copyright (2010), with permission from Elsevier. Appendix XI: Pavlu J. et al, CALPHAD 34 (2010) 215. 101 218 J. Pavlu eta!./ CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 215-221 Table 3 Total energy differences, A°£L"SER, between the Laves phases of various types and the weighted average of the SER states (antiferromagnetic BCC Cr, nonmagnetic HCP Hf and Ti) calculated in this work, rounded, and compared with the calculated [25-28] and experimental [29] values available in literature. Ah initio results published in [25,26,28] were calculated using the generalised gradient approximation. All values are given in kj/mol of atoms (1 eV atom"1 — 96 485 kj mol"1 atom"1). The values marked by an asterisk (*) were obtained with the help of the energy difference £Fcc — £hcp from Ref. [26], as the value of energy of formation of Laves phase obtained in Ref. [25] was related to the FCC structure. (Let us note that in Ref. [25], nonmagnetic BCC structure is employed as the SER structure for Cr.) Composition a0£L-ser (kj m0|- 1 atom-1) Cr2Cr Cr2Hf Cr6Hf6 CrHf2 Cr2Hf,„ Hf2Hf C14 This work Refs. 28.63 27.8 [25] -8.70 -9.9 [28] 47.25 96.63 19.33 32.10 30.92« [25] This work 27.29 -10.38 - 103.46 - 38.00 C15 Refs. 26.4 [25] -13 [27] -11.6 [28] -4.8 ±4.3 [29] - - - 35.82« [25] C36 This work Ref. 27.75 -9.68 -10.8 [28] 49.77 97.02 19.57 34.14 Composition ao£l-ser (kj mor 1 atom-1) Cr2Cr Cr2Ti Cr6Ti6 CrTi2 Cr2Ti10 Ti2Ti C14 This work Refs. 28.63 27.8 [25] -8.47 -10.2 [28] 35.46 51.39 12.59 27.20 24.71« [25] C15 This work Refs. 7.29 26.4 [25] -10.16 -11 [27] -12.0 [28] - 57.27 - 32.26 25.81«[25] C36 This work Ref. 27.75 -9.51 -11.0 [28] 37.06 53.53 13.47 28.36 hypothetical Cr2Cr Laves phase with a C15 structure. It turns out that the spin polarised Cr2Cr in the C15 structure is more stable than the nonmagnetic one by 31.2 meV/at(3.01 kjmol-1 atom-1) which means that the magnetism of Laves phases should be considered at 0 K. However, at ambient temperatures, Laves phases Cr2Hf and Cr2Ti are paramagnetic. Therefore, in the present paper, calculations of total energy differences for comparison of their relative stability were performed for nonmagnetic states only. It is a major approximation which appears to work reasonably well in this system. Experimental and calculated structural parameters of all CI4, C15 and C36 arrangements in Cr2Hf structure agree quite well. The calculated equilibrium atomic volume ranges from 96.49% for C36 and 96.90% Vexp forC15 to 97.17% V^p for CI 4, where V^p is the experimental atomic volume. A little worse agreement was found for Cr2Ti where calculated equilibrium atomic volume ranges from 94.43% Vexp for C36 and 94.48% Vexp for C14 to 95.58% Vexp for C15. The total energy differences a°Ei-ser between the Laves phases and SER states are then given in Table 3, where they are compared with the results of other authors [25-29]. It is obvious that total energy of C36 structure of pure constituents lies between the values for C14 and C15 structures. However, for pure Cr occupation the stability of Laves phase structures increases from C14 over C36 to C15. This order is opposite to that found for pure Hf and Ti Laves phases where the stability increases from C15 over C36 to C14 configuration. This difference is supposed to be caused by different atomic size of elements studied. The stability of particular Cr2X binary Laves phase configuration decreases in the sequence C15, C36 and C14 which is the same as in the case of pure Cr. According to our calculations, the C15 structure should be the most stable configuration in both Cr2Hf and Cr2Ti system (Table 3). Stability of C14 and C36 structures at higher temperatures is facilitated by the effect of entropy. As for CrHf2 and CrTi2 configurations, the absolute values of calculated structural energy differences are nearly an order of magnitude higher than those of Cr2Hf and Cr2Ti. This is a quantitative confirmation of the fact that the Laves phase structures CrHf2 and CrTi2 are energetically very disadvantageous configurations. Experimental evidence of existence of all three structure types of Laves phases in Cr2Hf and Cr2Ti can be found in [22]. However in Cr-Hf system, only phase equilibria with C14 and C15 structures are reported [3,7]. Quantitative conclusions for improvement of phase diagrams in mentioned systems are not yet available. 4. Calculations of phase diagrams The thermodynamic basis of the CALPHAD method relies explicitly on the assumption that the equilibrium phase composition arises as a result of minimisation of Gibbs energy in closed system at constant external conditions (temperature and pressure). For modeling of C14 and C15 Laves phases in the Cr-Hf system we employ the two-sublattice model [5,6] as it was used in [3]. We describe these structures as the ordered solid solution phases (see Eq. (7)) with four "end-members". Their Gibbs energy is modeled with the help of ab initio total energy differences, presented in Table 3, where the C36 structure data are included for possible future reference. The Gibbs energy of the Laves phases is obtained from Eq. (11) where the heat capacity term is adjusted to the experimental data. In addition the C14 Laves phase structure was modeled also using three-sublattice model inspired by crystallography. For modeling of C15 Laves phases in the Cr-Ti system we employ the two-sublattice model [5,6], and for C14 and C36 structure the three-sublattice model, as it was used in [4]. Their Gibbs energy is modeled with the help of ab initio total energy differences, presented in Table 3. In addition, the modeling of C14 and C36 structure by using the two-sublattice model is performed for possible future use. Results of this procedure are presented in Tables 4 and 5. Let us note here that the values in Table 3 are given in kj/mol of atoms, whereas the values given in Tables 4 and 5 are given in J/mol of compound (formula unit). That is why the quantities transferred from Table 3 to Tables 4 and 5 are 3000 times higher (in the case of two-sublattice model) or 12 000 times higher (in the case of three-sublattice model). Non-rounded values are given here. Reprinted from CALPHAD, 34, Pavlu J., Vfesfal J., Sob M., Thermodynamic Modeling of Laves phases in the Cr-Hf and Cr-Ti Systems: Reassessment Using First-principles Results, 215-221, Copyright (2010), with permission from Elsevier. Appendix XI: Pavlü J. et ed., CALPHAD 34 (2010) 215. 102 }. Pavlu eta!./ CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 215-221 219 Table 4 Optimised thermodynamic parameters describing the CI 4, CI 5 and C36 Laves phases in Cr-Hf system. Ab initio calculated values shown in boldface were taken from Table 3 and were kept fixed during the CALPHAD optimisation. Values of standard Gibbs energies are taken from [30]. Structure Parameters G (Eq. (1)) and L (Eq. (5)) in J mol 1 of formula unit G(Cr:Cr) = 85 900 + 3 * GHSERCR G(Cr:Hf) = -26095 + 0.67 * T - 0.67 * T * ln(T) + 2 * GHSERCR + GHSERHF CI 4 two-subl. G(Hf:Cr) = 289901 + 2 * GHSERHF + GHSERCR G(Hf:Hf) = 96 312 + 3 * GHSERHF L(Cr, Hf:Hf; 0) = -3500 G(Cr:Cr:Cr) = 343 598 + 12 * GHSERCR G(Cr:Hf:Cr) = -104379 + 2.67*T-2.67*T*ln(T) + 8*GHSERCR+4*GHSERHF CI 4 three-subl. G(Cr:Cr:Hf) = 566979 + 6 * GHSERHF + 6 * GHSERCR G(Cr:Hf:Hf) = 231 990 + 10 * GHSERHF + 2 * GHSERCR I(Cr:Hf:Cr, Hf; 0) = -10 000 G(Cr:Cr) = 81877 + 3 * GHSERCR G(Cr:Hf) = -31130 + 0.17 * T-0.17 *T* ln(T) + 2 * GHSERCR + GHSERHF C15 G(Hf:Cr) = 310380 + 2 * GHSERHF + GHSERCR G(Hf:Hf) = 114000 + 3 * GHSERHF L(Cr, Hf:Hf; 0) = -24000 G(Cr:Cr:Cr) = 333 020 + 12 * GHSERCR G(Cr:Hf:Cr) = -116208+1.59*T-1.59*T*ln(T) + 8*GHSERCR+4*GHSERHF C36 prediction G(Cr:Cr:Hf) = 597 242 + 6 * GHSERHF + 6 * GHSERCR G(Cr:Hf:Hf) = 234 827 + 10 * GHSERHF + 2 * GHSERCR I(Cr:Hf:Cr, Hf; 0) = -15 500 Table 5 Optimised thermodynamic parameters describing the C14, C15 and C36 Laves phases in Cr-Ti system. Ab initio calculated values shown in boldface were taken from Table 3 and were kept fixed during the CALPHAD optimisation. Values of standard Gibbs energies are taken from [30]. Structure Parameters G (Eq. (1)) and L (Eq. (5)) inj mol 1 of formula unit G(Cr:Cr) = 85 900 + 3 * GHSERCR G(Cr:Ti) = -25 401 - 0.788 * T + 0.788 * T * ln(T) + 2 * GHSERCR + GHSERTI CI 4 two-subl. G(Ti:Cr) = 154157 + GHSERCR + 2 * GHSERTI G(Ti:Ti) = 81592 + 3 * GHSERTI L(Cr, Ti:Ti; 0) = -24 000 G(Cr:Cr:Cr) = 343 598 + 12 * GHSERCR G(Cr:Ti:Cr) = -101 605-3.157*T+3.157*T*ln(T) + 8*GHSERCR+4*GHSERTI CI 4 three-subl. G(Cr:Cr:Ti) - 425 552 + 6 * GHSERTI + 6 * GHSERCR G(Cr:Ti:Ti) = 151040 + 10 * GHSERTI + 2 * GHSERCR I(Cr:Ti:Cr, Ti; 0) - -20 000 G(Cr:Cr) = 81877 + 3 * GHSERCR G(Cr:Ti) = -30486 - 1.414 * T + 1.414 * T * ln(T) + 2 * GHSERCR + GHSERTI C15 G(Ti:Cr) - 171806 + 2 * GHSERTI + GHSERCR G(Ti:Ti) - 96780 + 3 * GHSERTI L(Cr, Ti:Ti; 0) - -57 450 G(Cr:Cr) = 83255 + 3 * GHSERCR G(Cr:Ti) - -28 534 - 1.107 * T + 1.107 * T * ln(T) + 2 * GHSERCR + GHSERTI C36 two-subl. G(Ti:Cr) = 160581 + GHSERCR + 2 * GHSERTI G(Ti:Ti) - 85 066 + 3 * GHSERTI L(Cr, Ti:Ti; 0) - -28 000 G(Cr:Cr:Cr) = 333 020 + 12 * GHSERCR G(Cr:Ti:Cr) - -114 139 - 4.435*T+4.435*T*ln(T) + 8*GHSERCR+4*GHSERTI C36 three-subl. G(Cr:Cr:Ti) - 444725 + 6 * GHSERCR + 6 * GHSERTI G(Cr:Ti:Ti) = 161657 + 10 * GHSERTI + 2 * GHSERCR I(Cr:Ti:Cr, Ti; 0) - -31 000 Table 6 Thermodynamic parameters for equilibrium phases (liquid, HCP and BCC) in the Cr-Hf and Cr-Ti system. Phase Parameter value (Eq. (4)) in J mol ' 1 per formula unit for Cr-Hf system [3] Parameter value (Eq. (4)) inj mol 1 per formula unit for Cr-Ti system [4] L° - 42847.5 - 12 *7 BCC I1 = 12 064 1° = 11 824 I1 = 5012 L° = 43 774 + 0.64729 * T HU L1 = 15 000 1° = 25 500 .. ., L° = -30 000 + 8 * T -this work LiqUld I1 — 3800 — this work 1° = -992 I1 = 1811 The I-parameters describing the excess Gibbs energy GE of from [30]. For the sake of completeness, they are presented in non-ideal mixing in Eqs. (4), (5) are obtained by fitting to the Table 6. experimental data and are listed in Tables 4 and 5, too. The For remodeling of liquid phase in Cr-Hf system, Miedema's thermodynamic parameters for all other phases (liquid, BCC and guess of enthalpy of formation [27] and Tanaka's rule [31] for the HCP) in both systems were taken from [3,4] based on unary data enthalpy to entropy ratio in liquid phase were applied. Reprinted from CALPHAD, 34, Pavlii J., Vfestal J., Sob M., Thermodynamic Modeling of Laves phases in the Cr-Hf and Cr-Ti Systems: Reassessment Using First-principles Results, 215-221, Copyright (2010), with permission from Elsevier. Appendix XI: Pavlu J. et al, CALPHAD 34 (2010) 215. 103 J. Pavlu etal./ CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 215-221 Fig. la. Phase diagram of the Cr-Hf system calculated in this work in comparison with experimental data. Here three- (two-)sublattice model for C14 (C15) phase was used. Parameters for BCC and HCP phases are from [3]. Experimental points were taken from Refs. [32] (squares) and [33] (triangles). Phase boundaries agree very well with [7]. The optimisation of parameters of Gibbs energies of Laves phases is, of course, not the standard CALPHAD optimisation as the ab initio calculated values shown in boldface in Tables 4 and 5 are kept fixed. Nevertheless this procedure gives us the possibility to employ values with clear physical meaning in our assessment. The calculated phase diagrams of Cr-Hf and Cr-Ti system are presented in Figs. 1 and 2, where they are compared with existing experimental data. In spite of the above mentioned ab initio calculated intermediate lattice stability of C36 Laves phase in Cr-Hf system, the modeling of its thermodynamic properties could not be performed yet because of lack of experimental phase equilibrium data for the Cr-Hf system. The heat capacity and interaction parameters of C36 Laves phase in Cr-Hf system were calculated on the basis of the analogical values in Cr-Ti system for C14 and C36 structures, exploiting similarity of the electronic structure of Hf and Ti. The relative differences of the values of heat capacity and interaction parameter of those structures in both systems were retained. From Figs. 1 and 2 it is evident that the use of total energy differences calculated ab initio into the two- or three-sublattice model of Laves phases describes the phase diagrams (experimental points) in both Cr-Hf and Cr-Ti systems very well. Let us note that in the present treatment only two optimised parameters are needed for a complete thermodynamic description of any Laves phase: (1) the value of the ACp~SER describing the temperature dependence of Gibbs energy of real Laves phases (Cr2Hf, Cr2Ti), connected with difference of heat capacity of Laves phases and SER states by Eqs. (9)-(ll); it may be, in principle, calculated from Einstein or Debye functions or from phonon spectra, and (2) the value of the first I-coefficient of Redlich-Kister polynomial (Eq. (4)), which is obtained as a fitting parameter to experimental phase equilibrium data (here we neglect its temperature dependence as well as the higher-order terms). Its physical meaning is, in principle, connected with the binary interaction of atoms in the structure. On the other hand, analyses employed in [3,4] require 4 and 6 fitting parameters, respectively, for each Laves phase. In addition to that, our approach provides a physical meaning to some coefficients in temperature dependence of Gibbs function connected with the differences in heat capacities (see Eq. (11)), which can further reduce the number of fitting parameters needed. 1900 1800 1700 U !> 1600 Z) 2 g. 1500 E tu 1400-1 1300 1200 A 0.30 0.32 0.34 0.36 0.38 0.40 A x(Hf) Fig. lb. Detail of phase diagram of the Cr-Hf system. Three- (two-)sublattice model (marked by full (dashed) lines) for C14 phase is combined with two-sublattice model for C15 phase in both cases. Experimental points were taken from Refs. [32] (squares) and [33] (triangles). 2000 1800 1600 o 1400- 1200- per Tem 1000- 600 400 Y C14' f C36 ' I ' C15 '' I HCP/ *Ref 34 2Ref 35 „Ref 36 0Ref 37 aRef 38 + Ref 39 ARef 40 YRef 41 ARef 42 vRef 43 0Ref 44 0.2 0.4 0.6 x(Ti) 0.8 1.0 Fig. 2a. Phase diagram of the Cr-Ti system calculated in this work in comparison with existing experimental data [34-44]. Parameters for liquid, BCC and HCP phases are from [4]. Three-sublattice model was employed for C14 and C36 and two-sublattice model for C15 phase. It seems that physically based energy part of Gibbs energy surface of Laves phases (i.e. that one obtained from ab initio calculations) does not require such a strong entropy correction in the present model. This situation is a little bit surprising, as the ab initio calculated parameters for Cr2Cr, Hf2Hf and Ti2Ti energies of formation have more positive values than the guessed ones [3,4], which is, however, in agreement with [25]. We can see from Tables 4 and 5 that the stability of polytypes at 0 K is the highest for C15 structure and decreases in the sequence C15-C36-C14. On the other hand, as it may be expected, the vibrational entropy contribution (heat capacity contribution) increases in the same order when comparing the same models and values per atoms. When comparing phase equilibria with Laves phases in Figs, lb and 2b we can see that both our models (two- and three-sublattice) based on ab initio results provide a very good description of experimental phase data. Let us note, however, that the phase equilibria presented in [3,4] reproduce phase diagram satisfactorily as well, Reprinted from CALPHAD, 34, Pavlu J., Vfesfal J., Sob M., Thermodynamic Modeling of Laves phases in the Cr-Hf and Cr-Ti Systems: Reassessment Using First-principles Results, 215-221, Copyright (2010), with permission from Elsevier. Appendix XI: Pavlü J. et al, CALPHAD 34 (2010) 215_ .104 }. Pavlu eta!./ CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 215-221 1100 BCC C14 , m □ n □ □ LTJ i □ □ □ 7C36 J f C15 / / Re14 A(Laves + BCC) Re18 c(Laves + BCC) «.C14 ZC36 T(C36 + C15) * CI5 0-34 0.36 xT,in ■ 2cTu, a = ^-(Tlim)-DER-4bTlk dT Eo = C (Tum) - -RTe - 3KT,im In (l - e-*'7) a 2 b 5 c 3 + "Tlim + — "Tiim + — T|im 2 lu b (7) (8) (9) (10) where Tiim is the mentioned contact temperature (usually, but not always, 298.15 K), for which the extension of unary data is connected to the SGTE unary data [1]. G(Tnm), dG/dT(Tnm), Cp(Tnm) and dCp/dr(Tiim) represent function values and values of first derivative of respective SGTE polynomials at Tiim. TE is the Einstein temperature. (Expressions (1 — e TE/Tlir -l£/T„„ ) and ln(l■ ) can be evaluated in some softwares, e.g. Thermocalc, by using of FUNCTION statements in advance.) 3. A case study The Einstein function used in Eq. (1), requires a value for the Einstein temperature. The Einstein temperature is related to the Debye temperature as TE = 0J7TD [5]. The Debye temperature of interest is the low temperature limit of the Debye temperature, required for the purpose of extending the unary Gibbs energy data. This low temperature limit of Debye temperature for an ideal solid, TD(—3), can be obtained from either low temperature elastic constants (theoretically) or from heat capacity data (experimentally). TD(—3) means Debye temperature from the (—3) moment of phonon frequencies [6]. At temperatures T < To, anharmonic effects are still rather small, and therefore the low temperature limit of the Debye temperature describes well the behavior of the heat capacity of elements in the low temperature region. The values of Debye temperature, TD(—3), were taken from the publications [6,7], for elements in their thermodynamically stable structures. Some values of TD for unstable structures of elements are added from literature [2] for Fe, from [6] for Ti, Zr and Hf, and from [8] for V on the empirical basis described below. The Einstein temperatures TE were obtained through use of the equation TE = 0.777/D [5]. The values of parameters a, b, c and E0 in Eq. (2), representing the extension of the Gibbs energy of SGTE data [ 1 ] to zero Kelvin for this case are given in Table 1. For dynamically unstable structures (e.g. V-hcp, Ti, Zr, Hf-bcc), it is impossible to calculate TD(—3). In this case the value of TD(0) can be calculated as the high temperature limit of TD. The exact relation between TD(—3) and TD(0) cannot be found theoretically, but in [6] the empirical relation TD(-3) = TD(0)/0.94 was published for Ti, Zr, and Hf. Data found in this empirical way are also included in the Table 1. An excel file for the calculation of parameters a, b, c, and E0 is attached as supplementary material for future use when further Debye temperatures are available. Data in the form of G-HSER are added in Appendix together with the graphical pictures of extension of G(T) and Cp curves to zero Kelvin temperature for all elements in Table 1. It is obvious, that the simple extrapolation of G(T) values from [ 1 ] below the contact temperature Tiim fails close to zero Kelvin because of the term with negative exponent at T. Using the Einstein formula in the expression for G(T) cures this problem and the smooth link to the SGTE data at the temperature Tiim makes it possible to use the existing SGTE data without change. For low values of Debye temperature, the continuation of Cp-curve in some cases (e.g. Ba, Cd, Pb, and in smaller extent in Sr, Zn, Sb, La, Co, Hf-bcc, Zr-bcc, Fe-bcc) exhibits extremum in continuation from SGTE values to the Einstein function values. Reason for it may be in anharmonicity contribution to Cp or more probably in the fact that some SGTE-data close to (but above) Tiim are not fully consistent with the low temperature information represented by the Debye temperature. Such a behavior may indicate that Tiim should be set to higher temperatures (in the case of Ba shift of Tiim to about 350 K cure this problem, Fig. 2b), but in some cases (Cd, Pb) the shift of Tiim to the melting temperature of the respective element is not enough for removing the extremum (Fig. 15b, Fig. 17b). Anyway, Gibbs energy continues fluently to zero Kelvin also in such cases. Reprinted from CALPHAD, 37, Vřešťál J., Strof J., Pavlů J., Extension of SGTE Data for Pure Elements to Zero Kelvin Temperature - A Case Study, 37-48, Copyright (2012), with permission from Elsevier. Appendix XIV: Vřešťál J. et al, CALPHAD 37 (2012) 37. 115 40 J. VfeSt'al et al. j CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 37-48 4. Conclusion Acknowledgement Financial support of the grants GACR P108/10/1908, MSMT of Knowledge of the Debye (Einstein) temperature at low MSM0021622410 and of the Project "CEITEC MU" (CZ.1.05/1.1.00/ temperatures makes it possible to obtain a realistic extension 02.0068) from the European Regional Development Fund is of Gibbs energy function as the temperature falls towards 0 K. gratefully acknowledged. For the stable structures, the low temperature limit of the Debye temperature, TD(-3), can be calculated on the basis of Appendix A elastic constants or experimentally by measurement of the heat ^ , , . , „„ . , , , „ , Table Al contains extended Gibbs energy data (Eq. (2)) in the capacity in the low temperature region. For dynamically unstable form 0f G-HSER for elements listed in Table 1. structures, the high temperature limit of Debye temperature TD(0) (ln some softwares, the argument of logarithmic function can be calculated. The relation between TD(0) and TD(—3) cannot should be calculated first). be found theoretically, but an empirical relation can be validated Figs. 1-18 illustrate connection of G(T) and Cp(T) of extended and values of TD(-3) can be than used for the extension of SGTE functions with SGTE functions. Red color: extended functions, blue unary data to zero Kelvin temperature [2,6,8]. color: SGTE functions. LÍ-BCCA2 (0 < T < 200.00) -7962.63 + (3/2) (8.31451)(264.88) + 3(8.31451)Tln(l - exp(-264.88/T)) - (0.0112590/2)T2 - (-1.53294E - 04/6)T3 - (2.37900E - OaßO)!6 Na-BCC_A2 (0 < T < 200.00) -8016.02 + (3/2) (8.31451) (121.66) + 3(8.31451)Tln(l - exp(-121.66/T)) - (0.0257557/2)T2 - (-1.50434E - 04/6)T3 - (1.66252E - 09/20)1= K-BCCA2 (0 < T < 200.00) -7988.74 + (3/2) (8.31451) (70.07) + 3(8.31451)Tln(l - exp(-70.07/T)) - (0.0194388/2)T2 - (-1.17296E - 04/6)T3 - (1.81586E - OaßO)!6 Rb-BCC_A2 (0 < T < 200.00) -8042.92 + (3/2) (8.31451) (43.12) + 3(8.31451)Tln(l - exp(-43.12/T)) - (0.0120858/2)T2 - (-4.12642E - 05/6)T3 - (1.12973E - OaßO)!6 CS-BCCA2 (0 < T < 200.00) -8041.85 + (3/2) (8.31451) (29.26) + 3(8.31451)Tln(l - exp(-29.26/T)) - (-0.0127130/2)!2 - (1.81210E - 04/6)T3 - (-1.14058E - 09/20)1= Be-HCP_A3 (0 < T < 298.15) -15659.3 + (3/2) (8.31451) (1108.8) + 3(8.31451)Tln(l - exp(-1108.8/T)) - (0.00610649/2)T2 - (1.49314E - 04/6)T3 - (-9.49706E - WßO)!6 Mg-HCP_A3 (0 < T < 298.15) -8833.11 + (3/2) (8.31451) (308) + 3(8.31451)1 ln(l - exp(-308/T)) - (0.0539769/2)T2 - (-2.30295E - 04/6)T3 - (8.10174E - 10/20)T5 Ca-FCC_A1 (0 < T < 298.15) -8060.32 + (3/2) (8.31451) (177.1) + 3(8.31451)Tln(l - exp(-177.1/T)) - (0.0169877/2)T2 - (-4.22890E - 05/6)T3 - (5.21547E - ll/^O)!5 Sr-FCC_A1 (0 < T < 298.15) -8186.46 + (3/2) (8.31451) (113.2) + 3(8.31451)Tln(l - exp(-113.2/T)) - (0.0457474/2)!2 - (-1.94534E - 04/6)T3 - (7.33730E - WßO)!6 Ba-BCC_A2 (0 < T < 298.15) -7911.40 + (3/2) (8.31451) (84.7) + 3(8.3145 l)Tln(l - exp(-84.7/7)) - (0.0684640/2)T2 - (-4.52605E - 04/6)T3 - (2.92889E - OaßO)!6 Sc-HCP_A3 (0 < T < 298.15) -8706.90 + (3/2) (8.31451) (277.2) + 3(8.31451)Tln(l - exp(-277.2/T)) - (0.0469648/2)T2 - (-1.83312E - 04/6)T3 - (5.81278E - 10/20)1= Y-HCPJV3 (0 < T < 298.15) -8431.21 + (3/2) (8.31451) (184.8) + 3(8.31451)Tln(l - exp(-184.8/T)) - (0.0482034/2)T2 - (-1.89282E - 04/6)T3 - (6.10041E - WßO)!6 La-DHCP(0 < T< 298.15) -8252.86 + (3/2) (8.31451) (109.34) + 3(8.31451)Tln(l - exp(-109.34/T)) - (0.0446360/2)T2 - (-1.70763E - 04/6)T3 - (5.46438E - 10/20)T5 TÍ-HCPJV3 (0 < T < 298.15) -8861.83 + (3/2) (8.31451) (323.4) + 3(8.31451)Tln(l - exp(-323.4/T)) - (0.0441439/2)T2 - (-1.74763E - 04/6)T3 - (6.18793E - 10/20)1= TÍ-BCCA2 (0 < T < 298.15) -1800.32 + (3/2) (8.31451) (220.22) + 3(8.31451)Tln(l - exp(-220.22/T)) - (0.0210120/2)T2 - (-5.16686E - 05/6)T3 - (5.66515E - 11/20)T5 Ti-FCCAl (0 < T < 298.15) -2091.78 + (3/2) (8.31451) (255.64) + 3(8.31451)Tln(l - exp(-255.64/T)) - (0.00378822/2)T2 - (9.79921E - 07/6)T3 - (5.88120E- ll/^O)!5 Zr-HCP_A3 (0 < T < 130.00) -8313.16 + (3/2) (8.31451) (224.07) + 3(8.31451)Tln(l - exp(-224.07/T)) - (0.0806209/2)T2 - (-7.48881E - 04/6)T3 - (1.31270E - 08/20)T5 Zr-BCC_A3 (0 < T < 298.15) -674.409 + (3/2) (8.31451) (147.07) + 3(8.31451)Tln(l - exp(-147.07/T)) - (0.0328312/2)!2 - (-1.48985E - 04/6)T3 - (5.42043E - 10/20)1= Zr-FCC_A1 (0 < T < 298.15) 177.008 + (3/2) (8.31451) (144.76) + 3(8.31451)Tln(l - exp(-144.76/T)) - (-0.0346112/2)T2 - (1.90319E - 04/6)T3 - (-6.41923E - 10/20)T5 HÍ-HCPJV3 (0 < T < 298.15) -8421.33 + (3/2) (8.31451) (194.04) + 3(8.31451)Tln(l - exp(-194.04/T)) - (0.0557452/2)T2 - (-2.42998E - 04/6)T3 - (8.33639E - 10/20)T5 Hf-BCC_A2 (0 < T < 298.15) 4357.63 + (3/2) (8.31451) (126.28) + 3(8.31451)Tln(l - exp(-126.28/T)) - (0.0386224/2)!2 - (-1.73357E - 04/6)T3 - (6.09732E - 10/20)1= Hf-FCC_A1 (0 < T < 298.15) 1905.99 + (3/2) (8.31451) (158.62) + 3(8.31451)Tln(l - exp(-158.62/T)) - (0.0342399/2)T2 - (-1.44210E - 04/6)T3 - (4.98051E - 10/20)T5 V-BCCA2 (0 < T < 298.15) -8475.16 + (3/2) (8.31451) (292.6) + 3(8.31451)Tln(l - exp(-292.6/T)) - (0.0327470/2)T2 - (-1.41840E - 04/6)T3 - (5.26993E - WßO)!6 V-FCCAl (0 < T < 298.15) (continued on nextpage) Reprinted from CALPHAD, 37, Vřešťál J., Strof J., Pavlů J., Extension of SGTE Data for Pure Elements to Zero Kelvin Temperature - A Case Study, 37-48, Copyright (2012), with permission from Elsevier. Appendix XIV: Vřešťál J. et al, CALPHAD 37 (2012) 37_116 J. Vřešťál etat/ CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 37-48 Table A. 1 (continued) -997.557 + (3/2) (8.31451) (306.46) + 3(8.31451)Tln(l - exp(-306.46/T)) - (0.0255088/2)T2 - (-9.96915E - 05/6)T3 - (3.48182E - 10/20)75 V-HCP_A3 (0 < T < 298.15) -4580.80 + (3/2) (8.31451) (318.78) + 3(8.31451)Tln(l - exp(-318.78/T)) - (0.0258721/2)T2 - (-9.62630E - 05/6)T3 - (3.1635E - 10/20)T5 Nb-BCC_A2 (0 < T < 298.15) -7944.08 + (3/2) (8.31451) (211.75) + 3(8.31451)Tln(l - exp(-211.75/T)) - (0.0196347/2)T2 - (-1.01055E - 04/6)T3 - (4.61047E - 10/20)T5 Ta-BCC_A2 (0 < T < 298.15) -8101.73 + (3/2) (8.31451) (184.8) + 3(8.31451)Tln(l - exp(-184.8/T)) - (0.0295502/2)T2 - (-1.21167E - 04/6)T3 - (3.93215E - ÍO/20)t Cr-BCC_A2 (0 < T < 298.15) -9933.46 + (3/2) (8.31451) (485.1) + 3(8.31451)Tln(l - exp(-485.1/T)) - (0.0488572/2)T2 - (-1.60734E - 04/6)T3 - (3.86670E - i0ß0)t MO-BCCA2 (0 < T < 298.15) -8861.12 + (3/2) (8.31451) (346.5) + 3(8.31451)Tln(l - exp(-346.5/T)) - (0.0428376/2)!2 - (-1.78282E - 04/6)T3 - (5.86744E - i0ß0)t W-BCCA2 (0 < T < 298.15) -8790.52 + (3/2) (8.31451) (308) + 3(8.31451)Tln(l - exp(-308/T)) - (0.0562411/2)T2 - (-2.47579E - 04/6)T3 - (8.30265E - 10/20)T5 Mn-BCC_A12 (0 < T < 298.15) -8968.11 + (3/2) (8.31451) (315.7) + 3(8.31451)Tln(l - exp(-315.7/T)) - (0.0477805/2)T2 - (-1.79805E - 04/6)T3 - (6.66697E - i0ß0)t Tc-HCP_A3 (0 < T < 298.15) -9735.06 + (3/2) (8.31451) (388.08) + 3(8.31451)Tln(l - exp(-388.08/T)) - (0.0942685/2)T2 - (-4.00134E - 04/6)T3 - (1.31947E - 09/20)75 Re-HCP_A3 (0 < T < 298.15) -9466.06 + (3/2) (8.31451) (331.1) + 3(8.31451)Tln(l - exp(-331.1/T)) - (0.0939376/2)T2 - (-3.93606E - 04/6)T3 - (1.23562E - mßOjT3 Fe-BCC_A2 (0 < T < 298.15) 22.9525 + (3/2) (8.31451) (308.77) + 3(8.31451)Tln(l - exp(-308.77/T)) - (0.0963983/2)T2 - (-4.61144E - 04/6)T3 - (1.75273E - m/20)t Fe-FCC_A1 (0 < T < 298.15) -1141.91 + (3/2) (8.31451) (308.77) + 3(8.31451)Tln(l - exp(-308.77/T)) - (0.0822187/2)T2 - (-3.70372E - 04/6)T3 - (1.36384E - 09/20)75 Ru-HCP_A3 (0 < T < 298.15) -10149.2 + (3/2) (8.31451) (462) + 3(8.31451)Tln(l - exp(-462/T)) - (0.0826785/2)T2 - (-3.20209E - 04/6)T3 - (9.22441E - 10/20)T5 Os-HCP_A3 (0 < T < 298.15) -9711.29 + (3/2) (8.31451) (385) + 3(8.31451)Tln(l - exp(-385/T)) - (0.0857969/2)T2 - (-3.38392E - 04/6)T3 - (9.43881E - 10/20)T5 Co-HCP_A3 (0 < T < 298.15) -1070.33 + (3/2) (8.31451) (342.65) + 3(8.31451)Tln(l - exp(-342.65/T)) - (0.109565/2)T2 - (-4.92418E - 04/6)T3 - (1.73121E - 09/20)T5 Rh-FCC_A1 (0 < T < 298.15) -9434.27 + (3/2) (8.31451) (369.6) + 3(8.31451)Tln(l - exp(-369.6/T)) - (0.0729270/2)T2 - (-2.97118E - 04/6)T3 - (9.59375E - lO^OJT5 Ir-FCC_A1 (0 < T < 298.15) -9287.76 + (3/2) (8.31451) (323.4) + 3(8.31451)Tln(l - exp(-323.4/T)) - (0.0816946/2)T2 - (-3.36578E - 04/6)T3 - (1.01455E - m/20)t Ni-FCCAl (0 < T < 298.15) -7481.20 + (3/2) (8.31451) (346.5) + 3(8.31451)Tln(l - exp(-346.5/T)) - (0.0734882/2)T2 - (-2.98323E - 04/6)T3 - (9.20651E - ÍO/20)t Pd-FCC_A1 (0 < T < 298.15) -8108.69 + (3/2) (8.31451) (210.98) + 3(8.31451)Tln(l - exp(-210.98/T)) - (0.0287734/2)!2 - (-1.35483E - 04/6)T3 - (5.98583E - 10/20)75 Pt-FCC_A1 (0 < T < 298.15) -8152.49 + (3/2) (8.31451) (184.8) + 3(8.31451)Tln(l - exp(-184.8/T)) - (0.0279310 -/2)T2 - (-1.03994E - 04/6)T3 - (3.31714E - 10/20)T5 Cu-FCCAl (0 < T < 298.15) -8342.85 + (3/2) (8.31451) (264.11) + 3(8.31451)Tln(l - exp(-264.11/T)) - (0.0328876/2)T2 - (-1.45771E - 04/6)T3 - (5.34699E - 10/20)75 Ag-FCC_A1 (0 < T < 298.15) -8035.52 + (3/2) (8.31451) (173.25) + 3(8.31451)Tln(l - exp(-173.25/T)) - (0.0251270/2)T2 - (-9.84096E - 05/6)T3 - (3.03702E - 10/20)75 Au-FCCAl (0 < T < 298.15) -7702.16 + (3/2) (8.31451) (127.05) + 3(8.31451)Tln(l - exp(-127.05/T)) - (0.00399439/2)T2 - (-1.14985E - 06/6)T3 - (-3.03981E - 11/20)T5 Zn-HCP_A3 (0 < T < 298.15) -8961.08 + (3/2) (8.31451) (251.79) + 3(8.31451)Tln(l - exp(-251.79/T)) - (0.0919485/2)T2 - (-4.17994E - 04/6)T3 - (1.47137E - 09/20)75 Cd-HCP_A3 (0 < T < 298.15) -8609.56 + (3/2) (8.31451) (160.93) + 3(8.31451)Tln(l - exp(-160.93/T)) - (0.0917643/2)T2 - (-4.38435E - 04/6)T3 - (1.67170E - 09/20)75 Hg-RHOMBCLA10 (0 < T < 200) -10050.4 + (3/2) (8.31451) (55.363) + 3(8.31451)Tln(l - exp(-55.363/T)) - (-0.0437813/2)!2 - (3.77809E - 04/6)T3 - (-2.41620E - 09/20)T5 B-BETA_RHOMBO_B (0 < T < 298.15) -16255.1 + (3/2) (8.31451) (1216.6) + 3(8.31451)Tln(l - exp(-1216.6/T)) - (0.0121461/2)!2 - (2.41989E - 06)T3 - (-1.85030E - i2/20)t Al-FCCAl (0 < T < 298.15) -8676.15 + (3/2) (8.31451) (329.56) + 3(8.31451)Tln(l - exp(-329.56/T)) - (0.0306683/2)T2 - (-1.20157E - 04/6)T3 - (4.14808E - 10/20)75 Ga-ORTHORHOMBIC (0 < T < 298.15) -8798.22 + (3/2) (8.31451) (246.4) + 3(8.31451)Tln(l - exp(-246.4/T)) - (0.0844341/2)!2 - (-4.13421E - 04/6)T3 - (1.78765E - mßOjT3 In-TETRAGONAL_A6 (0 < T < 298.15) -7672.15 + (3/2) (8.31451) (83.16) + 3(8.31451)1 ln(l - exp(-83.16/T)) - (-0.00241390/2)T2 - (3.54384E - 05/6)T3 - (-3.97574E - UßOjt T1-HCP_A3 (0 < T < 200) -7440.98 + (3/2) (8.31451) (60.445) + 3(8.31451)Tln(l - exp(-60.445/T)) - (0.00150820/2)T2 - (-9.03568E - 05/6)T3 - (1.89647E - mßO)t C-HEXAGONAL_A9 (graphite) (0 < T < 298.15) -13723.16 + (3/2) (8.31451) (1717.1) + 3(8.31451)Tln(l - exp(-1717.1/T)) - (0.362613/2)!2 - (-1.68581E - 03/6)T3 - (6.03075E - 09/20)T5 Si-DIAMOND_A4 (0 < T < 298.15) -9397.17 + (3/2) (8.31451) (496.65) + 3(8.31451)Tln(l - exp(-496.65/T)) - (0.0344888/2)T2 - (-1.72822E - 04/6)T3 - (6.55998E - 10/20)75 (continued on next page) Reprinted from CALPHAD, 37, Vřešťál J., Strof J., Pavlů J., Extension of SGTE Data for Pure Elements to Zero Kelvin Temperature - A Case Study, 37-48, Copyright (2012), with permission from Elsevier. Appendix XIV: Vřešťál J. et al, CALPHAD 37 (2012) 37. 117 42 J. VfeSt'al et al. j CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 37-48 Table A. 1 (continued) Ge-DIAM0ND_A4 (0 < T < 298.15) -8350.18 + (3/2) (8.31451) (287.98) + 3(8.31451)Tln(l - exp(-287.98/T)) - (0.0393973/2)T2 - (-2.04327E - 04/6)T3 - (8.44279E - 10/20)T5 Sn-BCTA5 (0 < T < 100) -8217.96 + (3/2) (8.31451) (154) + 3(8.31451)Tln(l - exp(-154/T)) - (0.165492/2)T2 - (-2.13202E - 03/6)T3 - (6.48250E - OSßO)!6 Pb-FCC_A1 (0 < T < 298.15) -8198.50 + (3/2) (8.31451) (80.85) + 3(8.31451)Tln(l AS-RHOMBCLA7 (0 < T < 298.15) -8008.48 + (3/2) (8.31451) (217.1) + 3(8.31451)Tln(l Sb- RHOMBCLA7 (0 < T < 298.15) -8079.96 + (3/2) (8.31451) (162.5) + 3(8.31451)Tln(l Bi- RHOMBCLA7 (0 < T < 298.15) -7725.07 + (3/2) (8.31451) (91.63) + 3(8.31451)Tln(l - exp(-80.85/T)) - (0.0533286/2)T2 - (- - exp(-217.1/T)) - (0.0151794/2)T2 - (- - exp(-162.5/T)) - (0.0420057/2)T2 - (- - exp(-91.63/T)) - (0.0163926/2)T2 - (- 2.34318E - 04/6)T3 - (8.82599E - W/20)P 5.85733E - 05/6)T3 - (1.87854E - W/20)P -2.01951E - 04/6)T3 - (8.00203E - WßO)!6 -7.17683E - 05/6)T3 - (2.86831E - WßO)!6 Temperature / K Temperature / K Fig. 1. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for elements Li, Na, K, Rb, Cs (Color online). G of Be-hcp_a3, Mg-hcp_a3, Ca-fcc_al, Sr-fcc_al, Ba-bcc_a2 ^ Cp of Be-hcp_a3, Mg-hcp_a3, Ca-fcc_al, Sr-fcc_al, Ba-bcc_a2 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 Temperature / K Temperature / K Fig. 2. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for elements Be, Mg, Ca, Sr, Ba (Color online). G of Sc-hcp_a3, Y-hcp_a3, La-DHCP , Cp of Sc-hcp_a3, Y-hcp_a3, La-DHCP a 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 Temperature / K Temperature / K Fig. 3. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for elements La, Y, Sc (Color online). Reprinted from CALPHAD, 37, Vřešťál J., Strof J., Pavlů J., Extension of SGTE Data for Pure Elements to Zero Kelvin Temperature - A Case Study, 37-48, Copyright (2012), with permission from Elsevier. Appendix XIV: Vřešťál J. et al, CALPHAD 37 (2012) 37. 118 }. Vřešťál etal.j CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 37-48 43 a Temperature / K Temperature / K Fig. 4. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for element Ti in hep, bec and fee structures (Color online). G of Hf-fcc_al, Hf-bcc_a2, Hf-hcp_a3 , Cp of Hf-fcc_al, Hf-bcc_a2, Hf-hcp_a3 a 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 Temperature / K Temperature / K Fig. 6. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for element Hf in hep, bec and fee structures (Color online). Reprinted from CALPHAD, 37, Vřešťál J., Strof J., Pavlů J., Extension of SGTE Data for Pure Elements to Zero Kelvin Temperature - A Case Study, 37-48, Copyright (2012), with permission from Elsevier. Appendix XIV: Vřešťál J. et al, CALPHAD 37 (2012) 37. 119 J. Vřeíťál et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 37-48 a 5000 r G of V-bcc_A2, V-hcp_a3, V-fcc_al Cp of V-bcc_A2, V-hcp_a3, V-fcc_al 0 50 100 150 200 250 300 350 400 450 500 Temperature / K 0 50 100 150 200 250 300 350 400 450 500 Temperature / K Fig. 7. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for element V in hep, bec and fee structures (Color online). a Temperature / K Temperature / K Fig. 8. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for elements V, Nb, Ta (Color online). G of Cr-bcc_a2, Mo-bcc_a2, W-bcc_a2 Cp of Cr-bcc_a2, Mo-bcc_a2, W-bcc_a2 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 Temperature / K Temperature / K Fig. 9. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for elements Cr, Mo, W (Color online). Reprinted from CALPHAD, 37, Vřešťál J., Strof J., Pavlů J., Extension of SGTE Data for Pure Elements to Zero Kelvin Temperature - A Case Study, 37-48, Copyright (2012), with permission from Elsevier. Appendix XIV: Vřešťál J. et al, CALPHAD 37 (2012) 37. 120 }. Vřešťál etal.j CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 37-48 45 a G of Mn-bcc_al2, Tc-hcp_a3, Re-hcp_a3 CP of Mn-bcc_al2, Tc-hcp_a3, Re-hcp_a3 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 Temperature / K Temperature / K Fig. 10. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for elements Mn, Tc, Re (Color online). Fig. 11. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for elements Ru, Os, and, Fe in bcc and fee structures (Color online). a 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 Temperature / K Temperature / K Fig. 12. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for elements Co, Ir, Rh (Color online). Reprinted from CALPHAD, 37, Vřešťál J., Strof J., Pavlů J., Extension of SGTE Data for Pure Elements to Zero Kelvin Temperature - A Case Study, 37-48, Copyright (2012), with permission from Elsevier. Appendix XIV: Vřešťál J. et al, CALPHAD 37 (2012) 37. 46 J. Vřešťál et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 37-48 a G of Ni-fcc_al, Pd-fcc_al, Pt-fcc_al b CP of Ni-fcc_al, Pd-fcc_al, Pt-fcc_al 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 Temperature / K Temperature / K Fig. 13. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for elements Ni, Pd, Pt (Color online). a G of Cu-fcc_al, Ag-fcc_al, Au-fcc_al ^ Cp of Cu-fcc_al, Ag-fcc_al, Au-fcc_al 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 Temperature / K Temperature / K Fig. 14. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for elements Cu, Ag, Au (Color online). 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 Temperature / K Temperature / K Fig. 15. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for elements Zn, Cd, Hg (Color online). Reprinted from CALPHAD, 37, Vřešťál J., Strof J., Pavlů J., Extension of SGTE Data for Pure Elements to Zero Kelvin Temperature - A Case Study, 37-48, Copyright (2012), with permission from Elsevier. Appendix XIV: Vřešťál J. et al, CALPHAD 37 (2012) 37. }. Vřešťál etal.j CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 37-48 O 50 100 150 200 250 300 350 400 450 500 O 50 100 150 200 250 300 350 400 450 500 Temperature / K Temperature / K Fig. 16. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for elements B, Al, Ga, In, Tl (Color online). G of C-hex_a9(graphite), Si-diamond_a4, Ge-diamond_a4, Sn-bct_a5, Pb-fcc_al Cp of C-hex_a9(graphite), Si-diamond_a4, Ge-diamond_a4, Sn-bct_a5, Pb-fcc_al Fig. 17. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for elements C, Si, Ge, Sn, Pb (Color online). q G of As-rhombo-a7, Sb-rhombo-a7, Bi-rhombo-a7 ]^ Cp of As-rhombo-a7, Sb-rhombo-a7, Bi-rhombo-a7 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 Temperature / K Temperature / K Fig. 18. Temperature dependence of Gibbs energy (a) and heat capacity (b) for extended and SGTE functions for elements As, Sb, Bi (Color online). Reprinted from CALPHAD, 37, Vřešťál J., Strof J., Pavlů J., Extension of SGTE Data for Pure Elements to Zero Kelvin Temperature - A Case Study, 37-48, Copyright (2012), with permission from Elsevier. Appendix XIV: Vřešťál J. et al, CALPHAD 37 (2012) 37. 123 48 J. Vřešťál et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 37 (2012) 37-48 Appendix B. Supplementary data Supplementary material related to this article can be found online at doi:10.1016/j.calphad.2012.01.003. References [1] A.T. Dinsdale, CALPHAD 15 (1991)317. [2] Q. Chen, B. Sundman, J. Phase Equilib. 22 (2001) 631. [3] M.W. Chase, I. Ansara, A. Dinsdale, G. Eriksson, G. Grimvall, L. Hoglund, H. Yokokawa, CALPHAD 19(1995)437. [4] M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford, UK, 1954. [5] G. Gimvall, Thermophysical Properties of Materials, Elsevier Science, North-Holland, Amsterdam, 1999. [6] Q. Chen, B. Sundman, Acta mater. 49 (2001) 947. [7] C Kittel, Introduction to Solid State Physics, seventh ed. John Wiley and Sons, Inc. New York, 1996. [8] X. Li, Y. Xie, Y. Nie, H. Peng, Chinese Sei. Bull. 52 (22) (2007) 3041. [9] L Fast, J.M. Wills, B. Johansson, O. Eriksson, Phys. Rev. B 51 (1995) 17431. [10] H.R. Schober, RH. Dederichs, in: K.-H. Hellwege, J.L Olsen (Eds.), Lan-dolt-Bernstein Numerical Data and Functional Relationships In Science and Technology New Series, III/l3a, Springer, Berlin, 1981. [11] G.V.Tsagareishvili.J. Less-Common. Met. 75 (1980) 141. Reprinted from CALPHAD, 37, Vřešťál J., Strof J., Pavlů J., Extension of SGTE Data for Pure Elements to Zero Kelvin Temperature - A Case Study, 37-48, Copyright (2012), with permission from Elsevier. Appendix XVI: Pavlů J. et al, J. Min. Met. B48(3)B (2012) 395. 124 J. Min. Metall. Sect. B-Metall. 48 (3) B (2012) 395 - 401 Journal of Mining and Metallurgy, Section B: Metallurgy AB INITIO STUDY OF C14 LAVES PHASES IN Fe-BASED SYSTEMS J. Pavlů"'" *, M. Šoba"c "Central European Institute of Technology, CEITEC MU, Masaryk University, Brno, Czech Republic b Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Brno, Czech Republic c Department of Chemistry, Faculty of Science, Masaryk University, Brno, Czech Republic (Received 04 July 2012; accepted 30 September 2012) Abstract Structural properties and energetics of Fe-based C14 Laves phases at various compositions (i.e. Fe^Fe, Fe^C, X^Fe, X^C, where X stands for Si, Cr, Mo, W, To) were investigated using the pseudopotential VASP (Vienna Ab initio Simulation Package) code employing the PAW-PBE (Projector Augmented Wave - Perdew-Burke-Ernzerhof) pseudopotentials. Full relaxation was performed for all structures studied including the reference states of elemental constituents and the equilibrium structure parameters as well as bulk moduli were found. The structure parameters of experimentally found structures were very well reproduced by our calculations. It was also found that the lattice parameters and volumes of the unit cell decrease with increasing molar fraction of iron. Thermodynamic analysis shows that the Fe^C configurations of Laves phases are more stable than the X^Fe ones. Some of the X^Fe configurations are even unstable with respect to the weighted average of the Laves phases of elemental constituents. Our calculations predict the stability ofFefTa. On the other hand, Fe^Mo and Fe2W are slightly unstable (3.19 and 0.68 kJ.mol', respectively) and hypothetical structures Fe2Cr and Fe2Si are found unstable as well. Keywords: C14 Laves phases; Fe-based systems; Lattice stability; Ab-initio calculations 1. Introduction C14 Laves phase represents a complex intermetallic structure, which can be found in many systems (e.g. Cr-Ta, Cr-Zr, Fe-Mo, Fe-W, Fe-Ta, Ta-V). If present it significantly influences material characteristics. The Fe-based systems are very important for the development of ferritic steels and Laves phases are considered to be promising strengthening phases under certain conditions. In binary systems, these phases can be usually found as stoichiometric but in higher systems they can exhibit some solubility of other elements (Si, Cr). For example, Si has been found to play an important role in the stability of this phase and, because of lack of experimental data concerning the influence of Si on Laves phase in simpler subsystems, a theoretical study of this effect is highly desirable. In principle, basic properties of any phase are determined by its electronic structure. The purpose of this paper is to study, from first principles, the stability of C14 Laves phases in various binary systems consisting of Fe and some of the following elements: Si, Cr, Mo, Ta or W. The paper is organised as follows: After the Introduction, we describe our ab initio calculations in Section 2. Section 3 presents the structural and * Corresponding author: houserova@chemi.muni.cz DOI:10.2298/JMMB120704050P energetic analysis and discussion of the results. Section 4 concludes the paper. 2. Ab initio calculations Our first-principles calculations were performed by the pseudopotential Vienna Ab initio Simulation Package (VASP) code [1,2] using the Projector Augmented Wave - Perdew-Burke-Ernzerhof (PAW-PBE) pseudopotential [3-5]. The exchange-correlation energy was evaluated within the Generalised Gradient Approximation [6]. Spin polarisation was not included in our calculations, except when noted. Reason for this is the fact that all Laves phases found in the above mentioned systems at ambient temperatures are paramagnetic. From the crystallographic point of view, the C14 Laves phase belongs to the hexagonal structures. It has 12 atoms in its unit cell and its symmetry is described by the space group No. 194 [7, 8]. Optimum calculation settings were achieved starting with the experimentally found lattice parameters. The structure parameters for the reference states (RS) (structures of elemental constituents, which are stable at Standard Ambient Temperature and Pressure (SATP), i.e. FM bcc Fe, NM diamond Si and NM bcc Mo, W and Ta) were taken from [8] and of AFM bcc Cr from [9] (Table 1). For Laves phases in Fe-(Cr,Mo,Si), Fe-Ta and Fe-W Dedicated to the memory of Professor Zbigniew Moser Journal of Mining and Metallurgy, Section B: Metallurgy, 48 (3) B, Pavlu J., Sob M., Ab Initio Study ofC14 Laves Phases in Fe-based SystemsJ, 395-401, Copyright (2012), available at http://www.jmmab.com/images/pdf/2012/aisclpfbs-dec-2012-395-401.pdf. Appendix XVI: Pavlů J. et al., J. Min. Met. B48(3)B (2012) 395_ 396 J. Pavlů et al. / JMM 48 (3) B (2012) 395 - 401 125 systems the lattice parameters from Fe2Mo, Fe2Ta and Fe2W [8] (Table 2), respectively, were implemented. Convergence tests of the total energies with respect to number of k-points have shown that the range of optimum values goes from a grid of 11x11x8 points (Fe2Mo, W2Fe and elemental Mo and W in C14 structure) towards 25x25x13 points (Fe2W). In the case of RS, this range spreads from a grid of 9x9x9 points for FM bcc Fe and NM Si in diamond structure to 23x23x23 points for NM bcc Ta and W. 3. Results and discussion 3.1 Structure Calculated structure parameters in comparison with experimental data are shown in Table 1 for RS structures of elemental constituents and in Table 2 for the C14 Laves phases studied. It may be seen that the theoretical results agree well with experimental values. Table 1. Calculated and experimental lattice parameters of the RS structures (AFM stands for antiferromagnetic, FM for ferromagnetic and NM for nonmagnetic state, V denotes the volume per atom). Structure This work (relaxed) Experiment a (a.u.) (a.u.3) B„ (Mbar) a (a.u.) (a.u.3) Ref. AFM bcc Cr 5.3947 78.5015 1.86 5.44 80.4946 [8] FM bcc Fe 5.3589 76.9466 1.94 5.40 5.4160 78.7320 79.4339 [9] [10] NM bcc Mo 5.9513 105.3921 2.71 59.4340 104.9723 [8] NM diam. Si 10.3354 138.0030 0.90 102.6290 135.1209 [8] NM bcc Ta 6.2531 122.2539 2.01 62.3970 121.4672 [8] NM bcc W 5.9915 107.5396 3.15 59.8060 106.9582 [8] Table 2. Calculated and experimental structure parameters and bulk moduli (BJ of the CI4 Laves phases in the Fe-X systems (X= Si, Cr, Mo, Ta, W). Structure a (a.u.) c / a VJsl.u.3) 4/-z 6h-x B0 (Mbar) Fe2Fe 8.5562 1.5976 72.2224 0.0804 0.8260 2.62 Fe2Cr 8.6291 1.5938 73.9032 0.0775 0.8250 2.68 Cr2Fe 8.7111 1.6095 76.7794 0.0621 0.8287 2.38 Cr2Cr 8.7701 1.6512 80.3841 0.0464 0.8289 2.38 Fe2Mo 8.8485 1.6146 80.7275 0.0695 0.8273 2.88 Fe2Mo <"*> [8] 8.9668 1.6299 84.8053 0.063 " 0.83 " --- Mo2Fe 9.6076 1.5838 101.3668 0.0612 0.8280 2.11 Mo2Mo 9.7465 1.6466 110.0248 0.0413 0.8298 2.42 Fe2Si 8.6043 1.6601 76.3173 0.0714 0.8265 2.13 Si2Fe 9.6519 1.3569 88.0536 0.0230 0.8214 1.43 Si2Si 10.5905 1.2334 105.7317 0.0718 0.8158 0.89 Fe2Ta 8.9710 1.6347 85.1753 0.0656 0.8293 2.68 Fe2Ta [8] 9.1841 1.6284 91.0364 0.063 " 0.83 " --- Ta2Fe 10.0163 1.5921 115.4628 0.0329 0.8294 1.58 Ta2Ta 10.3003 1.5922 125.5696 0.0623 0.8292 2 Fe2W 8.8403 1.6219 80.8705 0.0686 0.8279 3.11 Fe2W [8] 8.9522 1.6242 84.0975 0.063 " 0.83 " --- W2Fe 9.7322 1.5896 105.7456 0.0595 0.8291 2.26 W2W 9.8110 1.6515 112.5596 0.0417 0.8301 2.82 3 As the corresponding experimental data were not found in [8], the values of CI 4 MgZn2 Laves phase were used here. Journal of Mining and Metallurgy, Section B: Metallurgy, 48 (3) B, Pavlu J., Sob M., Ab Initio Study ofC14 Laves Phases in Fe-based SystemsJ, 395-401, Copyright (2012), available at http://www.jmmab.com/images/pdf/2012/aisclpfbs-dec-2012-395-401.pdf. Appendix XVI: Pavlů J. et al., J. Min. Met. B48(3)B (2012) 395_ J. Pavlů et al. / JMM 48 (3) B (2012) 395 - 401 .126 397 The deviations in atomic volume are within the range from -3.1 to 2.1 % for RS (from Fe to Si) and from -3.8 to -6.4 % for Laves phases (from Fe2W to Fe2Ta). We have also found unambiguous trends in the geometry of Laves phase configurations (i.e. Fe2Fe, Fe2X, X2Fe, X2X) depending on the amount of Fe atoms in unit cell and on the size (or number of electrons) of X atom. These trends are demonstrated in Figure 1. The most obvious tendency may be observed in behaviour of the lattice parameters (a, c) and atomic volumes (Va) which increase with the concentration of X atoms involved except of the case of c parameter in Fe-Si system. This common trend results from the fact that Fe atoms are ones of the smallest particles studied in this work [11]. The differences in the size between the Fe and X atoms affect the distances between the individual lines in Figure 1 (c) as here the effects of mutual interplay of a and c are "integrated" into the value of atomic volume. We can see there that the values of V, of all structures increase in the series at Fe-Cr, -Si, -Mo, -W and -Ta - individual curves in Figure 1(c) do not intersect. This tendency is similar to trends in empirical atomic radii [11] which give the order Si, W, Fe, Cr, Mo, Ta. What are the reasons for disagreement in case of Si and W? The answer concerning W is simple. As we do not study separate atoms but bulk material our results should be compared rather with other experimental data e.g. covalent radii [12-16], metallic radii [12] or valence shell orbital radii [17]. These data show that Vat of W in bounded state moves toward higher values. The best agreement with our results was found in the case of the valence d-orbital radii in [17] (0.72 Fe, 0.86 Cr, 1.38 Mo, 1.47 W and 1.55 Ta (a.u.)), which increase exactly in the same order as our atomic volumes V,. However it must be taken into at account that there are several ways how to define radius for atoms. The further question is why such small atoms as Si provide a Laves phase with so relatively large Vat. In case of Si, although the atoms themselves are relatively small, covalent bonds can increase the average atomic volume quite a lot (cf. V, = 138 a.u.3 in diamond structure). For metallic at ' constituents, we may conclude that the effective atomic sizes (here valence d-orbital radii) play an important role in determining of the atomic volume of studied configurations. We can also see that even the internal parameters describing positions of atoms in the unit cell depend Journal of Mining and Metallurgy, Section B: Metallurgy, 48 (3) B, Pavhi J., Sob M., Ab Initio Study ofC14 Laves Phases in Fe-based SystemsJ, 395-401, Copyright (2012), available at http://www.jmmab.com/images/pdf/2012/aisclpfbs-dec-2012-395-401.pdf. Appendix XVI: Pavlů J. et al., J. Min. Met. B48(3)B (2012) 395_ 398 J. Pavlů et al. / JMM 48 (3) B (2012) 395 - 401 127 on the molar fraction of Fe in a regular way. (Fig. 2) The internal parameters 6h-x and Af-z describe positions of atoms in 6h sublattice in direction of both x andy axes within the range of 0.8158-0.8301 and in Af sublattice in the direction of the axis z in the interval 0.0230-0.0804, respectively. It is obvious that these parameters strongly depend on the composition of the structure. Except of Fe-Si system, the 6h-x parameter slightly increases with decreasing concentration of Fe atoms. The Af-z parameter, on the other hand, reveals the opposite tendency in most cases. There are some exceptions from this trend in the Fe-Ta and Fe-Si system which may correspond to the irregularities in trends in c/a ratio (Fig. 1(d)). 0.832 Table 3. The average 1" NN distances the 1" NN shells (w) for C14 Fe-X systems (X = Si, Cr, Mo, (r) and the widths of Laves phases in the Ta, W). 0.4 0.6 b) x(Fe) Figure 2. Concentration dependence of the 6h-x and 4f-z internal parameter for C14 Laves phases in the Fe-X systems (X= Si, Cr, Mo, Ta, W). To study the distance in coordination spheres of atoms and kind of the Nearest Neighbours (NN), the values of the 6h-x and Af-z parameters must be considered. In the case of sublattices in Laves phases studied, we found that the 1st NN atoms do not lie in the same distance from the central atom. They form some kind of shells whose widths are listed in Table 3. The interplay of internal and lattice parameters can be seen in Figure 3 where the deviations from the trends observed earlier are not manifested and we get Structure r (a.u.) w (a.u.) 2a 4f 6h 2a 4f 6h No. of atoms in shell 12 16 12 12 16 12 Fe2Fe 2.48 2.67 2.44 0.41 0.41 0.54 Fe2Cr 2.50 2.68 2.46 0.41 0.36 0.52 Cr2Fe 2.50 2.72 2.49 0.40 0.17 0.45 Cr2Cr 2.53 2.76 2.54 0.34 0.50 0.54 Fe2Mo 2.55 2.76 2.53 0.40 0.19 0.48 Mo2Fe 2.75 2.98 2.73 0.46 0.22 0.50 Mo2Mo 2.81 3.07 2.82 0.39 0.67 0.61 Fe2Si 2.51 2.71 2.49 0.35 0.19 0.56 Si2Fe 2.65 2.87 2.63 0.61 0.81 0.66 Si2Si 2.88 3.05 2.82 0.78 0.92 0.61 Fe2Ta 2.59 2.81 2.58 0.40 0.15 0.47 Ta2Fe 2.85 3.12 2.87 0.44 0.84 0.63 Ta2Ta 2.95 3.20 2.94 0.49 0.22 0.52 Fe2W 2.55 2.76 2.53 0.40 0.18 0.49 W2Fe 2.78 3.02 2.77 0.46 0.21 0.50 W2W 2.83 3.09 2.84 0.39 0.70 0.60 smooth concentration dependencies of average 1st NN distances from central atom on composition. We can draw the following conclusions concerning the 1st NN distances in various sublattices: • In each configuration of C14 Laves phase, the lsl NN distance increases in series 6h, 2a and Af except for the Ta2Fe, Mo2Mo and W2W where the 2a sublattice reveals slightly lower average 1st NN distance than the 6h sublattice. • The 2a sublattice has the Coordination Number (CN) 12 and the 1st NN consist of six 6h and six Af atoms in all Laves phase configurations studied. The average distances between the central atom and the lsl NN atom reveal smooth dependence on Fe composition, see Figure 3(a). The width of the 1st NN shell ranges from 0.34 a.u. for Cr2Cr to 0.78 a.u. for Si2Si configuration. • The 1st NN coordination shell of the 6h sublattice (CN=12) consists of two 6h, two 2a, two 6h, four Af and two Af atoms (in the order of increasing distances) and it reveals only tiny changes in its arrangement in the C14 Laves phases studied. There is only some rearrangement between the furthest six (4+2) Af atoms found in the Fe-rich configurations. On the other hand, the X-rich configurations prefer to arrange these six Af atoms in the opposite order, i.e. in subgroups of 2 and 4 atoms. Let us note that there are some anomalies in Si2Fe and Si2Si: - In Si2Fe the couple of Af atoms substitutes the Journal of Mining and Metallurgy, Section B: Metallurgy, 48 (3) B, Pavhi J., Sob M., Ab Initio Study ofC14 Laves Phases in Fe-based SystemsJ, 395-401, Copyright (2012), available at http://www.jmmab.com/images/pdf/2012/aisclpfbs-dec-2012-395-401.pdf. Appendix XVI: Pavlů J. et al., J. Min. Met. B48(3)B (2012) 395_ J. Pavlů et al. / JMM 48 (3) B (2012) 395 - 401 .128 399 first two 6h atoms in the shell i.e. the number of 4fl6h atoms is increased/lowered here by two. This fact can be a consequence of the significant decrease in Af-z parameter which causes that the atoms of the 4f sublattice become to be the nearest surrounding of the 6h atoms. - Si2Si reveals, on the other hand, a shift of the first two 6h nearest atoms to the farthest position of the coordination sphere. However, even in spite of these rearrangements in c) T T ' ' Ja2Ta 1 1 1 I 7S\2S\~~~~--~^_ - Mo2Mo^T~—^3%, \ - ■Cr2Cr - XjX X2Fe i 1 i 1 1 Fb* 1 Fe2Fě 0.2 0.4 0.6 x(Fe) T a2T % 1 1 1 ^-W2W \^ - - CtjCT——— - XjX X2Fe ! Fe2Fe 0.2 0.4 0.6 x(Fe) 0.4 0.6 x(Fe) Figure 3. Concentration dependences of the average 1" NN distances for the 2a, 4f and 6h sublattice for C14 Laves phases in the Fe-Xsystems (X= Si, Cr, Mo, Ta, W). the 1st NN coordination shell of the 6h sublattice we can still see a smooth dependence of the average lsl NN distances on composition (Fig. 3(c)). The width of the 1stNN shell ranges from 0.45 a.u. for Cr2Fe to 0.66 a.u. for Si2Fe configuration. • The 4f sublattice (CN=16) reveals many rearrangements of the order of three 2a, six 6h, three 6h, three 4f and one 4f atom in the 1st NN coordination shell but there are no evident rules here. Again, the concentration dependences of the average NN distances are quite smooth. Using ab initio methods, the values of bulk moduli of RS and Laves phases can be calculated and their values are listed in Tables 1 and 2. The composition dependence of bulk moduli is shown in Figure 4. It can be seen there that bulk moduli of Laves phases in all systems studied exhibit a strong composition dependence. The least influence of molar fraction of Fe can be found in Fe-Cr system which can be caused by similar size of atoms of these elements. Bulk moduli of both binary and pure constituent Laves phases also strongly depend on the kind of X component and their values decrease in the same order as in case of the RS, i.e. from W, Mo, Ta to Si, except for Fe-Cr system, which can be caused by magnetic ordering of Cr and Fe RS. The values of bulk moduli of pure element structures and of the Laves phases are very close in case of NM elements, i.e. bulk moduli of Mo, W, Si and Ta Laves phase are 89.3 %, 89.5 %, 98.9 % and 99.5 % of the RS value, respectively. On the other hand the bulk moduli of NM Cr and Fe Laves phase are much higher than those of magnetic RS (i.e. 128.0 % and 135.1 % of RS value, respectively). 3.5 E 1.5 n 1.0 0.5 . 1 1 1 1 1 + w ; + Mo " Mo2Mo— Ta _ Fe Cr - -Ta2Ta~---_____^ S / "SÍ2Sl^^ - "X2X ! X2Fe l Fe2í Fe2Fe" 1 ~ + Si 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.2 0.8 04 0.6 x (Fe) RS Figure 4. Concentration dependence of the bulk modulus for CI 4 Laves phases in the Fe-X systems (X= Si, Cr, Mo, Ta, W). 3.2 Energetics Total energy of formation per atom may be obtained as Journal of Mining and Metallurgy, Section B: Metallurgy, 48 (3) B, Pavlii J., Sob M., Ab Initio Study ofC14 Laves Phases in Fe-based SystemsJ, 395-401, Copyright (2012), available at http://www.jmmab.com/images/pdf/2012/aisclpfbs-dec-2012-395-401.pdf. Appendix XVI: Pavlů J. et ed., J. Min. Met. B48(3)B (2012) 395_ 400 J. Pavlů et al. / JMM 48 (3) B (2012) 395 - 401 129 have been experimentally found. The comparison in Table 4 shows that our calculated data for C14 Laves phase of elemental constituents are in perfect agreement with ab initio calculated data published in [18]. The agreement with experimental data is worse. This is given by the fact that experiments are not performed at 0 K temperature. 4. Conclusions The purpose of this work was to understand, with the help of ab initio electronic structure calculations, the relations between the electronic structure, size of the atoms and the thermodynamic as well as structural properties of the C14 Laves phases. We demonstrate that the structure parameters and heats of formations strongly depend on the molar fraction of Fe atoms and that the calculated equilibrium parameters correspond very well to the experimental values. Analysis of our results shows that both geometric and electronic factors are important for stabilization of the C14 Laves phase in the iron-based systems. Our ab initio electronic structure calculations reveal that the C14 Laves phase is unstable at zero temperature in both Fe-Mo and Fe-W system which is in contradiction with experiments. However, the absolute value of the total energy of formation is very low. We suppose that the instability at higher temperatures is suppressed by the entropy effects. But, on the other hand, some other C14 Laves phases (e.g. Fe2Ta) are stable even at low temperatures. We have found that the phase stability related to the reference state is significantly influenced by occupations of sublattices. Table 4. Ab initio calculated total energies offormation A£^aves RS (kl. mol<]> of atoms) of various Laves phase configurations with respect to the weighted ratio of the RS compared with literature data. Elem. X Fe2X X2Fe X2X VASP Exp. VASP VASP Calc. Cr 20.22 30.10 28.63 27.8 [18] Mo 3.19 -14.1 [19] 67.38 36.60 36.5 [18] Si 19.91 48.92 56.52 53.6 [18]" Ta -18.61 -6.3±1.9 [20]b 83.70 9.35 9.7 [18] -19.7 [21] -19.25±2.79 [22] -19.27 [23] W 0.68 -7.61±3.14 [24] 82.60 44.40 43.8 [18] Fe 38.27 31.1 [18] 3 This value was obtained using the energy difference Ebcc - Ediamond = 0.53 eV= 51.1 kJ/mol [25], as the value of energy of formation of Laves phase in Si obtained in Ref. [18] was related to bec Si. b The sample showed significant amount of second phases in X-ray diffraction analyses [20]. Journal of Mining and Metallurgy, Section B: Metallurgy, 48 (3) B, Pavht J., Sob M., Ab Initio Study ofC14 Laves Phases in Fe-based SystemsJ, 395-401, Copyright (2012), available at http://www.jmmab.com/images/pdf/2012/aisclpfbs-dec-2012-395-401.pdf. where Eat stands for total energy per atom and xFe denotes the molar fraction of Fe. 120 80 40 Lif < -40 I 1 1 1 1 1 1 1 1 ♦ Cr2Cr - • M ozM o " ■ Sl,Si — a T a2Ta — iL/ + WjW y 2 1 1 1 ,FeJ< i Fe,Fe i 1 2 i 1 i 2 0 0.2 0.8 0.4 0.6 x(Fe) Figure 5. Concentration dependence of AEL*ves-RS for Q14 Laves phases in the Fe-Xsystems (X= Si, Cr, Mo, Ta, W) (fully relaxed results). All equilibrium total energy differences Afi^™8 RS are summarised in Table 4 and Fig. 5 which shows that the only stable configuration is the Fe2Ta one. But there are two other structures Fe2W and Fe2Mo which are only slightly unstable in comparison with the remaining configurations. These three structures revealing the highest tendencies to stability are the only ones among the Fe-X systems studied which Appendix XVI: Pavlů J. et ed., J. Min. Met. B48(3)B (2012) 395_ J. Pavlů et al. / JMM 48 (3) B (2012) 395 - 401 .130 401 Acknowledgements Financial support of the Grant Agency of the Czech Republic (Project No. PI08/10/1908) and of the Project "CEITEC - Central European Institute of Technology" (CZ. 1.05/1.1.00/02.0068) from the European Regional Development Fund are gratefully acknowledged. This research was also supported by the Academy of Sciences of the Czech Republic (Project No. RVO:68081723). The access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum, provided under the programme "Projects of Large Infrastructure for Research, Development, and Innovations" (LM2010005) is appreciated. References [19] J. H. Zhu, C. T. Liu, L. M. Pike, P. K. Liaw. Intermetallics 10 (2002) 579. [20] S. V. Meschel, O. J. Kleppa, Journal of Alloys and Compounds 415 (2006) 143. [21] A. Miedema, R. Boom, F. De Boer, Journal of the Less Common Metals 41 (1975) 283. [22] Y. I. Gerassimov, V. I. Lavrentev, O. von Goldbeck, D. T. Livey, R. Ferro, A. L. Dragoo, Tantalum: Physico-chemical properties of its compounds and alloys, IAEA Review, Special Issue No. 3, IAEA, Vienna, 1972. In O. Kubaschewski and C.B. Alcock, Metallurgical Thermochemistry, 5th ed., Pergamon Press, Oxford 1979. [23] T. C. Totemeier, C. J. Smithells, Smithells Metals Reference Book, Elsevier Butterworth-Heinemann publications, Oxford, GB, 2004. [24] T. N. Rezukhina, T. A. Kashina, The Journal of Chemical Thermodynamics 8 (6) (1976) 519. [25] M. Y. Yin, M. L. Cohen, Phys. Rev. B 26 (1982) 5668. [I] G. Kresse, J. Furthmüller, Comput. Mater. Sei. 6 (1996) 15. [2] G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169. [3] P. E. Blöchl, Phys. Rev. B 50 (1994) 17953. [4] G. Kresse, J. Joubert, Phys. Rev. B 59 (1999) 1758. [5] J. P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [6] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson. M. R. Pederson, D. J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. [7] J. L. C. Daams, P. Villars, J. H. N. van Vucht, Atlas of crystal structure types, ASM International, Materials Park, Ohio, USA, 1991. [8] P. Villars, L. D. Calvert, Pearson's Handbook of Crystallographic Data for Intermetallic Phases, ASM International, Materials Park, Ohio, USA, 1991. [9] E. G. Moroni, T. Jarlborg, Phys. Rev. B 47 (1993) 3255. [10] M. Polcarovä, K. Godwod, J. Bak-Misiuk, S. Kadeckovä, J. Bradler, Phys. Stat. Sol. (a) 106 (1988) 17. [II] J. C. Slater, J. Chem. Phys. 41 (1964) 3199. [12] R. T. Sanderson, Chemical periodicity, Reinhold, New York, USA, 1962. [13] L. E. Sutton (Ed.), Table of interatomic distances and configuration in molecules and ions, Supplement 1956-1959, Special publication No. 18, Chemical Society. London, UK, 1965. [14] J. E. Huheey, E. A. Keiter, R. L. Keiter, Inorganic chemistry: Principles of structure and reactivity, 4th Edition, HarperCollins, New York, USA, 1993. [15] W. W. Porterfield, Inorganic chemistry, a unified approach, Addison Wesley Publishing Co., Reading. Massachusetts, USA, 1984. [16] A. M. James, M. P. Lord, Macmillan's chemical and physical data, Macmillan, London, UK, 1992. [17] J. B. Mann, Atomic structure calculations II. Hartree-Fock wave functions and radial expectation values: hydrogen to lawrencium, LA-3691, Los Alamos Scientific Laboratory, USA, 1968. [18] M. H. F. Sluiter, CALPHAD 30 (2006) 357. Journal of Mining and Metallurgy, Section B: Metallurgy, 48 (3) B, Pavlu J., Sob M., Ab Initio Study ofC14 Laves Phases in Fe-based SystemsJ, 395-401, Copyright (2012), available at http://www.jmmab.com/images/pdf/2012/aisclpfbs-dec-2012-395-401.pdf. Appendix XVII: Strof J. et al, CALPHAD 44 (2014) 62. .131 CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 44 (2014) 62-69 i:--r Contents lists available at ScienceDirect ELSEVIER CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad A CALPHAD a=S. ---- Laves phases in the V-Zr system below room temperature: Stability analysis using ab initio results and phase diagram Jan Strof3, Jana Pavlůbc*. Urszula D. Wdowikd, Jiří Buršíkc, Mojmír Šobbca, Jan Vřešťálbc CrossMark a Department of Chemistry, Faculty of Science, Masaryk University, Brno, Czech Republic b Central European Institute of Technology, CEITEC MU, Masaryk University, Bmo, Czech Republic c Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Bmo, Czech Republic d Institute of Technology, Pedagogical University, Cracow, Poland ARTICLE INFO Available online 27 August 2013 Keywords: Laves phases V-Zr system Ab initio calculations Phase diagram Zero Kelvin ABSTRACT V-Zr is a well known system where a phase transformation from rhombohedral V2Zr structure to cubic C15 Laves phase occurs during heating at about 115 K. Here we provide a thermodynamic description of this phenomenon, supported by ab initio calculations. We utilize our new method of extension of the Scientific Group Thermodata Europe (SGTE) unary thermodynamic database to zero Kelvin and demonstrate that it may be applied also to complicated intermetallic phases. To keep our analysis on equal footing with previous results for higher temperatures, data regarding a recent thermodynamic assessment of the V-Zr system (valid for temperatures above 298.15 K) were reassessed. With the help of ab initio approach, we demonstrate that the ZrV2 rhombohedral phase is not stable at 0 K and transforms to C15 Laves phase. The stability of C15 Laves phase is confirmed by analysis of elastic stability criteria and phonon spectra calculations. © 2013 Elsevier Ltd. All rights reserved. 1. Introduction Topologically close-packed intermetallic compounds such as Laves phases are promising candidates for high-temperature materials due to their interesting mechanical and corrosion resistance features [1] which can be influenced by their possible polymorphism and peculiar magnetic and electrical properties [2,3]. The solubility of hydrogen in Laves phases [4] and superconductivity [5] are also important properties of these compounds. Electronic structure, elastic properties and total energies of C15 Laves phases of V2M type (M=Zr, Hf, or Ta) were studied with the help of ab initio calculations in [6-8] and the relations between electronic structure, elastic moduli and stability were analyzed. It was found that the V2Zr C15 Laves phase compound, which is cubic at room temperature, undergoes a structural transformation to a non-Laves rhombohedral phase at low temperatures [9-11]. The transformation temperature was determined to be 116.7 K by Moncton [11] and 110 K by Rapp [10], who also reported the corresponding latent heat of transformation as 31 J/mol. In general, it is very difficult to explore low-temperature phase diagrams describing such transformations both experimentally and theoretically and it is great challenge to * Corresponding author at: Central European Institute of Technology, CEITEC MU, Masaryk University, Kamenice 753/5, CZ-625 00 Brno, Czech Republic. Tel.: +420 549493742: fax: +420 549492556. E-mail addresses: houserova@chemi.muni.cz, houserova@ipm.cz (J. Pavlu). 0364-5916/$ - see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.Org/10.1016/j.calphad.2013.0S.003 perform thermodynamic modeling in this region. It is the reason for doing this work. The first step towards the low-temperature predictions of thermodynamic functions was done in works [12,13], where equations for Gibbs energy valid at low temperatures and necessary values of Debye temperatures for many elements are given. In our recent work [14], the expressions of Gibbs energies of 52 elements were extended to zero Kelvin on this basis and they may form the base for modeling of phase equilibria by the CALPHAD method at low temperatures. Furthermore, one of the options how to model the Gibbs energy of any intermetallic phase at low temperatures is based on the extension of its Gibbs energy from high-temperature region towards zero Kelvin which also takes into account the value of its Debye temperature (TD). A fundamental prerequisite for the success of this method is the existence of precise expression for the Gibbs energy of studied phase at high temperature i.e. above 298.15 K. In the V-Zr system the data of phase equilibria above room temperature were determined in [15-17] and thermodynamic assessments were published for equilibria above room temperature by Servant [18] and Zhao et al. [19], where the description of Gibbs energy of V2Zr C15 Laves phase was evaluated by the ATAT software [20,21]. In this work, we have taken the results of Zhao [19] as reliable and improved the description of C15 Laves phase and a hexagonal close-packed (HCP_A3) phase above room temperature. However, Reprinted from CALPHAD, 44, Strof J., Pavlů J., Wdowik U. D., Buršík J., Šob M., Vřešťál J., Laves Phases in the V-Zr System Below Room Temperature: Stability Analysis Using Ab Initio Results and Phase Diagram, 62-69, Copyright (2014), with permission from Elsevier. Appendix XVII: Strof J. et al, CALPHAD 44 (2014) 62. 132 J. Strofetal. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 44 (2014) 62-69 63 Rapp [10], Moncton [11], Keiber [22] and Geibel [23] have found rhombohedral phase as stable phase below 113 K, which cannot be drawn in high-temperature phase diagram. Therefore, we have performed the extension of calculation of phase diagram down to zero Kelvin, using the description of unary data below room temperature [14] and the new extension of expression of Gibbs energy of C15 Laves phase and rhombohedral phase to zero Kelvin compatible with Gibbs energy expressions above 298.15 K [ 19] and based on respective values of Debye temperatures [22-24]. In addition to that, our ab initio calculations of elastic constants and phonon spectra bring new findings concerning the stability and behavior of C15 and rhombohedral V2Zr phase. Table 1 Calculated and experimental lattice parameters and bulk moduli (B) of the SER states. Symbols a and c stand for lattice parameters, Vat is the atomic volume and A shows the relative difference between the calculated and experimental atomic volume and bulk modulus. SER state a(pm) cja Vat(107pm3) A% B (GPa) A% NM BCC_A2 V Exp. [32] 303.09 1 1.392 -5.11 161.9 [33] +16.46 This work 297.83 1 1.321 188.5 NM HCP_A3 Zr Exp. [32] 323.18 1.593 1.164 +0.64 83.3 [33] +17.22 This work 323.55 1.598 1.172 97.6 Equilibrium structural parameters and bulk moduli (B) of Laves phases and a non-Laves rhombohedral phase found in this work. Symbols a and c stand for lattice and Vat is the atomic volume; A denotes the relative difference between the calculated and experimental atomic volume. Structure a (pm) cja IVOO'pm3) parameters 5 (GPa) C14 V2V This work V2Zr Exp. [32] This work VZr2 This work Zr2Zr This work 527.70 521.34 557.23 575.87 1.6386 1.6032 1.5929 1.6667 2.007 1.893 2.297 2.652 181.8 1474 91.6 88.6 C15 V2V This work V2Zr Exp. [32] This work VZr2 This work Zr2Zr This work 745.00 732.90 789.31 824.88 1.723 1.640 2.049 2.339 139= 145.2 875 85.3 C36 V2V This work V2Zr This work VZr2 This work Zr2Zr This work 493.30 528.06 558.41 577.14 3.1324 3.0890 3.1993 3.3310 1.567 1.895 2.321 2.668 181.1 146.9 90.0 877 Rhombohedral phase (hexagonal coordinates) V2V This workb V2Zr Exp. [22] This workb VZr2 This work Zr2Zr This work 535.61 518.37 622.01 620.07 2.3267 2.4480 1.8027 2.0429 1.720 1.641 2.087 2.343 180.6 145.5 81.0 85.8 1 Calculated value published in Ref. [24], 5 These values (written in rhombohedral coordinates) correspond to the C15 Laves phase arrangement and were obtained by the full structure relaxation. Reprinted from CALPHAD, 44, Strof J., Pavlů J., Wdowik U. D., Buršík J., Šob M., Vřešťál J., Laves Phases in the V-Zr System Below Room Temperature: Stability Analysis Using Ab Initio Results and Phase Diagram, 62-69, Copyright (2014), with permission from Elsevier. Appendix XVII: Strof J. et al, CALPHAD 44 (2014) 62. 133 J. Strof et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 44 (2014) 62- 2. Ab initio calculations of stability of V2Zr phases The energy of formation of chosen phase at 0 K can be obtained with the help of ab initio electronic-structure calculations, performed within the Density Functional Theory (DFT). We used the pseudopotential method [25] incorporated into the Vienna Ab initio Simulation Package (VASP) code [26,27] combined with the Projector Augmented Wave-Perdew-Burke-Ernzerhof (PAW-PBE) pseudopotential [28-30] (i.e. we employed the Generalized Gradient Approximation (GGA) for the exchange-correlation energy), and we calculated the total energies of the C15 V2Zr Laves phase and a rhombohedral structure as well as the total energy of the Standard Element Reference (SER) states i.e. HCP_A3 Zr and BCC_A2 V, which are the phases stable at pressure of 105 Pa (1 bar) and temperature 298.15 K. In general, Laves phases form the largest group of intermetallic compounds and crystallize in cubic (MgCu2, C15), or hexagonal (MgZn2, C14 and MgNi2, C36) type structures which differ only by a different stacking of the same four-layered structural units [31 ]. Therefore an additional ab initio study of stability of C14, C15 and C36 structures was performed to obtain a sound physical background for possible future thermodynamic modeling. The preliminary calculations of C14 and C15 V2Zr Laves phase modifications and SER states of Zr and V were accomplished using Total energy differences A°E between the Laves phases of various types and the weighted average of the SER states (NM BCCA2 V and NM HCP_A3 Zr) calculated in this work and compared with values available in literature [34,35], The ab initio results published in [34,35] were obtained using the GGA. The energy of the energetically most favorable V2Zr structure is denoted as boldface. All values are given in k]/mol of atoms (1 eV/atom —96.485 k]/mol of atoms). Composition V2Zr Zr2Zr C14 This work C15 This work Ref C36 This work Rhombohedral phase This work 11.23 11.0 [34] 10.42 11.22= 3.32 5.07 4.03 5.07= 77.85 27.35 26.50 [34,35] 71.95 22.80 65.23 24.68 a These values correspond to the C15 Laves phase arrangement and were obtained by the full structure relaxation. The total energy difference between this relaxed structure and C15 Laves phase in V2V (V2Zr) configuration is -0.0109 (-0.0015) kj/mol of atoms. the corresponding experimentally found lattice parameters published in [32]. For the C36 V2Zr Laves phase, we employed the data of the Cr2Zr phase from [32] and for rhombohedral phase, the data from [22]. The cut-off energy restricting the number of plane waves in a basis set was 241 eV for HCP_A3 Zr, Laves phases and rhombohedral structures except for pure V structures (BCC_A2 and V2V - rhombohedral and Laves phase modifications) where the value of 201 eV was used. The convergence tests of total energies with respect to the number of fc-points were also performed. In the case of SER states with 2 atoms in the unit cell, we used a grid of 19 x 19 x 19 points for nonmagnetic (NM) BCCLA2 V and of 21 x21 x 15 points for NM HCP_A3 Zr. The optimum values obtained for the C14 Laves phases are 15x15x13 (V2V), 21x21x13 (V2Zr), 21x21x15 (Zr2Zr) and 21 x 21 x 17 (VZr2) fc-points, for the C15 Laves phases 13 x 13 x 13 (V2V), 19 x 19 x 19 (V2Zr, VZr2) and 21 x 21 x 21 (Zr2Zr) fc-points and for the C36 Laves phases 13 x 13 x 13 (V2V), 17x17x13 (Zr2Zr, V2Zr) and 19x19x13 (VZr2) fc-points. The optima 15 x 15 x 13 (V2V, V2Zr, VZr2) and 17 x 17 x 13 (Zr2Zr) k-points were found in case of rhombohedral phase described using the hexagonal lattice parameters. The spin polarization was not included in our calculations. The reason for this is the fact that all Laves phases found in the systems studied are paramagnetic at ambient temperatures and V2V and V2Zr C15 Laves phase were found NM from ab initio calculations. After the test calculations, each structure was fully relaxed yielding the minimum total energy and the equilibrium structural parameters. As the SER state of vanadium and the CI 5 Laves phases are cubic, only a volume relaxation is sufficient to get their lowest energy state. The calculated optimum lattice parameters and bulk moduli (B) for SER states, C14, C15 and C36 Laves phases and the non-Laves rhombohedral structure are listed in Tables 1 and 2. Experimental and calculated structural parameters of SER states and V2Zr configurations in C15 and C14 Laves phase agree quite well. Concerning the rhombohedral phase, the situation is more complicated. Here, the analyses of Nearest Neighbors (NN) distances and primitive cells were performed and the results obtained were compared with C15 Laves phase equilibrium data. It was found that the V2V and V2Zr configuration of rhombohedral phase transforms to the C15 Laves phase during the relaxation which is confirmed by the identity of their primitive cells. Furthermore, the corresponding total energy differences A0^ S£B between the rhombohedral (C15 Laves) phase and SER states given in Table 3 are almost identical. The same statement is also valid for their bulk moduli B listed in Table 2. On the other hand, the VZr2 and Zr2Zr configurations 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 (J/THz Fig. 1. Density of states of phonons (a) and dispersion relations of phonons (b) in V2Zr C15 Laves phase, calculated by Phonon software [36], Reprinted from CALPHAD, 44, Strof J., Pavlů J., Wdowik U. D., Buršík J., Sob M., Vřešťál J., Laves Phases in the V-Zr System Below Room Temperature: Stability Analysis Using Ab Initio Results and Phase Diagram, 62-69, Copyright (2014), with permission from Elsevier. Appendix XVII: Strof J. et al, CALPHAD 44 (2014) 62. 134 ]. Strof et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 44 (2014) 62-69 Table 5 Reassessed L-parameters of C15 Laves phase and HCP_A3 phase. Phase T(K) Parameter [I mol-1 per formula unit) This work Ref. [19] HCP_A3 298-3000 Di 41,000+l.OT 1,180,172.2+10.0966981 i-V,Zr 8000+2.5T 1,137,938.4+25.731291 V2Zr_C15 298-2000 Dt-V:V,Zr 60,000 -1998.517+40.884161 't-V:V,Zr - 2478.3464+ 5.0971524T 2. WiVZr - 3388.8469+2.0837273T LV,Zr:Zi 25,000 19,133.339-31.6527241 i-V,Zr:Zr - 9002.261+18.7242431 2, LV,Zr:Zi - 10,837.862-9.36713881 O OC & ID *t t< íó íó x(Zr) Fig. 2. Phase diagram of V-Zr system above 298.15 K calculated according to [19] (with HCP_A3 and C15 data reassessed), compared with experimental data: stars from [15], crosses from [16], 30 OC o o: d íó *t a a M K d a of rhombohedral phase do not converge to the C15 Laves phase arrangement during their structural relaxation. Even more, these structures become more stable than the C15 Laves phase. The absolute values of energies of formation of VZr2 configuration of C14, C15 and C36 Laves phase and rhombohedral structure are nearly 21,15,18 and 13 times higher, respectively, than those of V2Zr configuration. This is a quantitative confirmation of the fact that the VZr2 arrangement of the above mentioned structures is energetically very disadvantageous. The stability of the C15 structure at higher temperatures is facilitated by the vibrational energy effect in spite of the positive value of energy of formation of C15 and non-Laves (rhombohedral) structure at 0 K. 3. Elastic constants and phonon spectra of V2Zr C15 Laves phase The phonon spectra of V2Zr C15 Laves phase were calculated using the Phonon software [36]. The behavior of phonon density of states (DOS) is displayed in Fig. la together with the dispersion relations of phonons (Fig. lb). It may be seen that the V2Zr in C15 Laves phase structure is dynamically stable at zero Kelvin which is in agreement with the findings published in [24] where the V2Zr C15 structure is presented as mechanically stable according to the elastic stability Reprinted from CALPHAD, 44, Strof J., Pavlů J., Wdowik U. D., Buršík J., Sob M., Vřešťál J., Laves Phases in the V-Zr System Below Room Temperature: Stability Analysis Using Ab Initio Results and Phase Diagram, 62-69, Copyright (2014), with permission from Elsevier. Appendix XVII: Strof J. et al, CALPHAD 44 (2014) 62. 135 J. Strof et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 44 (2014) 62- criteria: Cu > 0; C44 > 0; Cu > |C12| and (Cu +2C12) > 0. To confirm this finding, the elastic constants of C15 Laves phase were calculated and are listed in Table 4. From this table, we can see that the mechanical stability of the C15 phase predicted in [24] was confirmed by our results listed in last two lines. These results were obtained by calculating the total energy as a function of the shears [37,38] and are in very good agreement with data published in [24]. Furthermore, the elastic constants in cubic systems are related to the phonon and sound velocities as reported in [39]. Hence, our ab initio elastic constants were compared to those obtained from the least-square fits of the appropriate acoustic phonon slopes and applying density of ZrV2 /?=6518kgm~3 determined using the calculated lattice constant (see Table 2). Resulting values are given in the last line of Table 4. The Cu and C44 that are directly obtained from the slopes of the phonon branches differ from the respective elastic constants calculated in the conventional manner by about 14% and 5%, respectively. We note significant discrepancies between the slopes that involve combinations of the elastic constants. Therefore, the C]2 appearing only in the combination of the elastic constants cannot be determined unambiguously. One has to note that estimates of the elastic constants appropriate for zero-sound propagation (long wave-length limit, i.e. q->0 where q denotes the wave-vector) are usually not as precise as those determined from deformation energy or stress-strain relationships and the discrepancies may reach several or more percent (see for example [40]). 1200 1180 1160 1140 - 0 1120 "5 1100 i_ 01 §■ 1080 H H 1060 1040 1020 - 1000 C15 + FJCC C15 + HCP 0.80 0.85 —i— 0.90 x(Zr) 0.95 1.00 Fig. 3. Detail of the V-Zr phase diagram (Fig. 2) in the region of BCC_A2/HCP_A3 transformation of Zr. Stars from [15], crosses from [16], to zero Kelvin. In this temperature region, the Gibbs energy of elemental constituents may be expressed with respect to SER state by equation [14] G°(T) = E0 +§J?rE + 3J?rln(l -e-T£lT)-%T2-±T5-%T3. m 4. Thermodynamic modeling and phase diagram above 298.15 l< In this temperature region, we adopted a recent assessment of thermodynamic parameters of V-Zr system [19] except for the overestimated number of parameters of C15 Laves phase which was reduced in our work. Subsequently, the corresponding parameters for HCP_A3 phase had to be also reoptimized. The obtained data are presented in Table 5 together with data published in [19]. It is obvious that lower values of °Lv.zr and 'iy.zr parameters of HCP_A3 phase are sufficient in our modeling. In addition to it, ]L and 2L parameters of C15 Laves phase are not needed in our approach. The same parameters as in [19] were used for a liquid and a body-centered cubic (BCC_A2) phase. The rhombohedral phase has not been modeled at this stage as it is not stable in this temperature region. The calculated phase diagram is shown in Fig. 2. From Table 6, we can see that the differences in invariant temperatures calculated in [19] and in this work are very small. The detail of the phase diagram in the region of BCC_A2/ HCP_A3 transformation of Zr is presented in Fig. 3. where TE is the Einstein temperature and a, b and c are constants. It is assumed that Gibbs energy of stoichiometric phases can be expressed also in the form of Eq. (1) and it is evaluated to reach a smooth connection at limiting temperature. Namely, the extended Gibbs energy of C15 Laves phase should have the same function value and the same values of the first and the second derivatives as the Gibbs energy polynomial for V2Zr published in [19] at the limiting temperature (rUm), which was chosen as 298.15 K. The resulting system of equations is based on general expressions for Gibbs energy valid for low temperatures published in [14] and was used as follows (parameter c in Eq. (1) is neglected): GC,5(r„m) = +^ +3J?r„mln(l -e-^/^)-^Tlim2-±Tlim = 1135.53119-15.8195494rlim + 1.34559042rln(rlim) (2.1) dGu5(rlim) 3J?ln(l-e-T"5/T|i™)-3R r»5 e-r|-yr,im _b 4 Tfim(l-e-rf,5/r„m) a'"m 4J,im 5. Thermodynamic modeling of V2Zr phases below 298.15 l< -14.47395898 +1.345590421n(Tlim) (2.2) The Compound Energy Formalism (CEF) [41,42] was also employed for thermodynamic modeling of the V-Zr system down Table 6 Temperatures of invariant reactions in the V-Zr system. Reaction T(K) from [19] T(K) [this work] V-BCC_A2+liquid ^V2Zr C15 1573 1569.1 Liquid^V2Zr C15+Zr-BCC_A2 153S 1535.7 Zr-BCC_A2^V2Zr C15+Zr-HCP_A3 1050 1052.9 and rf2Gcl5(r„m)_ 3RT?5e-^'^T™/Tlim2 Tlim (l_e-r?s/rlim)2 1 -a-bTh, ■-1.34559042= (2.3) The Einstein temperature (TE) can be calculated from the Debye temperature (TD) according to the relation [43] rF = 0.77Tn (3) Reprinted from CALPHAD, 44, Strof J., Pavlů J., Wdowik U. D., Buršík J., Sob M., Vřešťál J., Laves Phases in the V-Zr System Below Room Temperature: Stability Analysis Using Ab Initio Results and Phase Diagram, 62-69, Copyright (2014), with permission from Elsevier. Appendix XVII: Strof J. et al, CALPHAD 44 (2014) 62. 136 J. Sfro/er al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 44 (2014) 62-69 -5 -10 3 -20 E 3 3 -25 -40 I . I G(T) C15 extrapolated from [19] I 1 1 1 \/G(T)C15 - ^S^^. GfT) RHOMBO GfT) C15 from [19]X^X 0 50 100 150 200 250 300 350 400 Temperature / K Fig. 4. Temperature dependence of the molar Gibbs energy of V2Zr C15 Laves and of V2Zr rhombohedral phase. Blue curve represents the Gibbs energy of the V2Zr C15 Laves phase according to [19] including its extrapolation below 298.15 K performed in the present paper, the red/green curves show the extension of the Gibbs energy [14] of the V2Zr C15 Laves phase/V2Zr rhombohedral phase based on our new model (this work) to zero Kelvin. Gibbs energy of V2Zr rhombohedral phase above 298.15 is given by Eq. (6). (Our data of V2Zr C15 Laves and rhombohedral phase are listed in Table 7. For elemental constituents, we employed SGTE data [44] above 298.15 Kand data from [14] below 298.15 K.) Table 7 The Gibbs energy data for V2Zr C15 Laves and rhombohedral phase used for the calculation of Fig. 4. V2Zr C15 Laves phase (0 < T< 298.15) -2173.70+2062.8+24.9435Tln(l-exp(-165.4T-'))+0.0597826T2-6.3207 x 10-"T5+2 GHSERV+GHSERZR V2Zr C15 Laves phase (298.15298.15IC) (6) Reprinted from CALPHAD, 44, Strof J., Pavlů J., Wdowik U. D., Buršík J., Šob M., Vřešťál J., Laves Phases in the V-Zr System Below Room Temperature: Stability Analysis Using Ab Initio Results and Phase Diagram, 62-69, Copyright (2014), with permission from Elsevier. Appendix XVII: Strof J. et al, CALPHAD 44 (2014) 62. 137 J. Sfro/er al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 44 (2014) 62- The zirconium Gibbs energy unary data extended to zero Kelvin in the form of GHSER in J/mol connected with corresponding SGTE data [14,44], Zr HCPA3 (0