126 N. Saunders and A, P. Miodownik Pelton, A. D. and Blander, M. (1984) in Second International Symposium on Metallurgical Slags & Fluxes, eds Fine, A. H. and Gaskell, D. R. (Met. Soc. AIME, New York), p. 281. Pelton, A. D. (1988) CALPHAD, 12, 127. Pelton, A. D. and Blander, M. (1986a) Met. Trans. B, I7B, 805. Pelton, A. D. and Blander, M. (1986b) in Computer Modelling of Phase Diagrams, ed. Bennett, L. H. (TMS, Warrendale), p. 19. Pelton, A. D. and Blander, M. (1988) CALPHAD, 12, 97. Pettifor, D. G. (1995) private communication. Pitzer, K. S. (1973) J. Phys. Chem., 77, 268. Pitzer, K. S. (1975) J. Soln. Chem., 4, 249. Pitzer, K. S. and Brewer, L. (1961) Thermodynamics; 2nd Edition (McGraw-Hill, New York), Pitzer, K. S. and Kim, J. J. (1974) J. Am. Chem. Soc, 96, 5701. Pitzer, K. S. and Mayorga, G. (1973) J. Phys. Chem., 77, 2300. Predel, B. and Oehme, G. (1974) Z. Metallkde, 65, 509. Richardson, F. D. (1956) Trans. Farad. Soc., 52, 1312. Sastri, P. and Lahiri, A. K. (1985) Met. Trans. B, I6B, 325. Sastri, P. and Lahiri, A. K. (1986) Met. Trans. B, 17B, 105. Saunders, N. (1989) Z. Metallkde., 80, 894. Saunders, N. (1996) CALPHAD, 4, 491. Selleby, M. (1996) unpublished research. Sharma, R. C. and Chang, Y. A. (1979) Met. Trans. B., 10B, 103. Sommer, F. (1977) CALPHAD, 2, 319. Sommer, F. (1980) Z Metallkde, 71, 120. Sommer, F. (1982) Z Metallkde., 73, 72 and 77. Sundman, B. (1985) private communication. Sundman, B. (1994) unpublished research. Sundman, B. (1996) private communication. Sundman, B. and Mohri, T. (1990) Z Metallkde, 81, 251. Sundman, B. and Ägren, J. (1981) J. Phys. Chem. Solids, 42, 297. Taylor, I R. and Dinsdale, A. T. (1990) CALPHAD, 14, 71. Temkin, M. (1945) Acta Phys. Chim. USSR, 20, 411. Tomiska, J, (1980) CALPHAD, 4, 63. Toop, G. W. (1965) Trans. Met. Soc. Aime, 233, 855. Toop, G. W. and Samis, C. S. (1962) Can. Met Quart., 1, 129. Villars, P. and Calvert, L. D, (1991) Pearson's Handbook of Crystaltographic Data for Intermetallic Phases; 2nd Edition (ASM International, Materials Park, OH). Wagner, C. (1951) Thermodynamics of Alloys, Addison-Wesley, Reading, Mass. Yokokawa, T. and Niwa, T. (1969) Trans. Japan Inst. Met., 10, 1. Zemaitis, J. F. (1980) in Themodynamics of Aqueous Systems with Industrial Applications, ed. Newman, S. A. (American Chemical Society), p. 227. Chapter 6 Phase Stabilities 6.1. Introduction 129 6.2. Thermochemical Estimations 129 6.2.1 General Procedure for Allotropic Elements 129 6.2.2 General Procedure for Non-Allotropic Elements 132 6.2.2.1 The Van Laar Technique for Estimating Melting Points 134 6.2.2.2 The Estimation of Metastable Entropies of Melting 135 6.2.2.3 Determination of Transformation Enthalpies in Binary Systems 139 6.2.2.4 Utilisation of Stacking Fault Energies 141 6.2.3 Summary of the Current Status of Thermochemical Estimates 141 6.3. Ab Initio Electron Energy Calculations 142 6.3.1 Comparison Between FP and TC Lattice Stabilities 144 6.3.2 Reconciliation of the Difference Between FP and TC Lattice Stabilities for Some of the Transition Metals 148 6.4. The Behaviour of Magnetic Elements 153 6.4.1 Fe 153 6.4.2 Co 158 6.4.3 Ni 159 6.4.4 Mn 159 6.5. The Effect of Pressure 160 6.5.1 Basic Addition of a PAV Term 160 6.5.2 Making the Volume a Function of T and P 161 6.5.3 Effect of Competing States 162 6.6. Determination of Interaction Coefficients for Alloys and Stability of Counter-Phases 165 6.6.1 The Prediction of Liquid and Solid Solution Parameters 166 6.6.1.1 Empirical and Semi-Empirical Approaches 166 6.6.1.2 Ab Initio Electron Energy Calculations 168 6.6.2 The Prediction of Thermodynamic Properties for Compounds 168 6.6.2.1 The Concept of Counter-Phases 168 6.6.2.2 Structure Maps 1?0 6.6.2.3 The Miedema Model and Other Semi-Empirical Methods 170 6.6.2.4 Ab Initio Electron Energy Calculations 171 Chapter 6 Phase Stabilities 6.1. INTRODUCTION The CALPHAD approach is based on the axiom that complete Gibbs energy versus 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. By convention these are referred to as lattice stabilities and the Gibbs energy differences between all the various potential crystal structures in which an element can exist need to be characterised as a function of temperature, pressure and volume. The basic question is how to perform extrapolations so as to obtain a consistent set of values, taking into account various complications such as the potential presence of mechanical instability. Additional complications arise for elements which have a magnetic component in their Gibbs energy, as this gives rise to a markedly non-linear contribution with temperature. This chapter will concern itself with various aspects of these problems and also how to estimate the thermodynamic properties of metastable solid solutions and compound phases, where similar problems arise when it is impossible to obtain data by experimental methods. 6.2. THERMOCHEMICAL ESTIMATIONS 6.2.1 General procedure for allotropic elements Allotropic elements, which exhibit different crystal forms within accessible temperatures and pressures, allow the Gibbs energies to be measured for at least some of the possible alternative structures. Such data, although of necessity restricted to particular regimes of temperature and pressure, were therefore used as the platform for all the initial efforts in this field. The elements Mn (Weiss and Tauer 1958), Ti and Zr (Kaufman 1959a) and Fe (Kaufman et ah 1963) thus provided the basis for obtaining the first lattice stabilities for phase-diagram calculations. However, in order to make the CALPHAD approach universally applicable, a knowledge of the relative stability of the most frequently occurring crystal structures is also required for elements which do not manifest allotropy under normally accessible conditions 129 130 N. Saunders and A. P. Miodownik of temperature and pressure. Kaufman's conviction that a reliable set of basic lattice stabilities could be assembled was undoubtedly the key concept for the whole CALPHAD approach. As with all physical phenomena, an agreed reference state has to be established for each element or component. In order to have a firm foundation, this has generally been taken as the crystal structure in which that element exists at standard temperature and pressure. This ensures that thermochemical (TC) lattice stabilities are firmly anchored to the available experimental evidence. Although the liquid phase might be considered a common denominator, this raises many problems because the structure of liquids is difficult to define; it is certainly not as constant as popularly imagined. It is therefore best to anchor the framework for lattice stabilities in the solid state. Figure 6.1(a) shows the relative position of the experimentally determined Gibbs energy curves for Ti, which includes segments that refer to the stable low-temperature c.p.h. a-phase, the high temperature b.c.c. 0-phase and the liquid phase. Figure 6.1(b) shows curves where properties have been extrapolated to cover regions where, although the phases are metastable, properties can be estimated by extrapolation from alloys which stabilise one or other of the phases in binary systems. It is clearly desirable to see if the total curve can be de-convoluted into parts that can be identified with a specific physical property so that trends can be established for the many cases where data for metastable structures are not experimentally accessible. In principle, the TC lattice stability of an element in a specified crystal structure /? relative to the standard state a can be comprehensively expressed as follows (Kaufman and Bernstein 1970): = H£$ + f(ea, BPl T) + GaE?J f(7a, lp, T) + G£h> V 7) + GZ$ f(Ea, EP) + Gr0 f(Va, V* P). (6.1) Here H%2(f is the enthalpy of transformation at 0 K, f($, 7, A, E) denote functions for the components associated with Debye temperature (#), electronic specific heat (7), lambda transitions such as magnetism (A) and contributions (E) related to multiple electronic states, sljl..., while f(P, Va, Vp) refers to the function which generates the pressure contribution. It may of course be unnecessary to consider all these terms and the equation is much simplified in the absence of magnetism and multiple electronic states. In the case of Ti, it is possible to deduce values of the Debye temperature and the electronic specific heat for each structure; the pressure term is also available and lambda transitions do not seem to be present. Kaufman and Bernstein (1970) therefore used Eq. (6.2), which yields the results shown in Fig. 6.1(c). CALPHAD—A Comprehensive Guide r- 6 "o e 4 >-) O 2 I O" (a) Free energy of afCPH Ti) at ~ used as reference state all temperatures \ 1 t 1 Ö 4 E O ca-['"t (b) \ Nx T0(a-p) \ TM(a) 0 400 800 1200 1600 2000 T (*K) 0 400 800 1200 1600 2000 T(*K) o 4 pa-liq (c) \ _ a 1 1 1 1 o 0 400 800 1200 1600 2000 T(»K) 0 -2 -4 _ Ga-Iiq (d) \ qH —f.C.C, \\ - \ \ go-<0 a i i i i 0 400 800 1200 1600 2000 T(°K) Figure 6.1. Gibbs free energy curves for Ti (a) a, 0 and liquid segments corresponding to the stable regions of each phase; (b) extrapolated extensions into metastable regions; ■ References are listed on pp. 173-178. 132 N. Saunders and A. P. Miodownik The right-hand side can be separated into five parts. The first part is the enthalpy at 0 K, the second represents the zero point energy, the third is the Debye energy term, the fourth is an approximation for the Cv - Cv correction while the last part arises from the difference in electronic specific heats. 6.2.2 General procedure for non-allotropic elements With the exception of a few allotropic elements, the necessary input parameters to Eqs (6.1) or (6.2) are not available to establish the lattice stabilities of metastable structures. Therefore an alternative solution has to be found in order to achieve the desired goal. This has evolved into a standard format where the reference or ground state Gibbs energy is expressed in the form of general polynomials which reproduce assessed experimental Cp data as closely as possible. An example of such a standard formula is given below (Dinsdale 1991): Gm[T] - H™R = a + bT + cT\n{T) + J24» (6.3) The left-hand term is defined as the Gibbs energy relative to the standard element reference (SER) state where is the enthalpy of the element or substance in its defined reference state at 298.15 K, a, bt c and are coefficients and « represents a set of integers, typically taking the values of 2, 3 and -1. From Eq. (6.3) further thermodynamic properties of interest can be obtained. (6.3a) (6.3b) (6.3c) 5 = _i> _ c _ eh (T) - Y, ndn T""1 Jř=sa-rf,-£(n-l)dBTn Other phases are then characterised relative to this ground state, using the best approximation to Eq. (6.1) that is appropriate to the available data. For instance, if the electronic specific heats are reasonably similar, there are no lambda transitions and T»6D, then the entropy difference between two phases can be expressed just as a function of the difference in their Debye temperatures (Domb 1958): loge 9ß (6.4) Combined with the 0 K enthalpy difference, the Gibbs energy of metastable phases can then be obtained by adding terms of the form (A-BT) to the reference state value or, more specifically: _ jja->P _ TSa->0 (6,5) CALPHAD—A Comprehensive Guide 133 where Ha^ and Sa^ are taken as constant with respect to temperature. A linear model as shown in Fig. 6.1(b) is therefore a reasonable approximation at temperatures above 6D. If the Debye temperatures are close enough, then the linear model will also give a reasonable description of the lattice stabilities below the Debye temperature (Miodownik 1986) and can be used to estimate a value of , since this will be equal to the enthalpy of transformation measured at high temperature. Such linear expressions form the basis of many listed metastable lattice stability values but, in the longer term, it is desirable to return to a mode of representation for the unary elements which re-introduces as many physically definable parameters as possible. This is currently being pursued (Chase et al. 1995) by using either the Debye or Einstein equation where , 3 r$D ebye 9íí or and x = hv/ET, or Cm instein (e* -1)" MT J [e^/r-l]2 (6.6) (6.7) Empirical fitting coefficients can be added (ai(i = 1,2...)) so that the Cp is given by Cfit = Cnebye, Einstein + ^ {extrapolated to Tm = 0) and the Lindemann constant [CL) for various crystal structures (from Achar and Miodownik 1974) Structure of solid phase Lindemann constant (Cl) Function of Cz. (10'4Cl) Entropy of fusion (S') b.c.c. (A2) 118 1.4 1.6 f.c.c. (Al) 138 1.9 1.9 c.p.h. (A3) 169 2.8 2.3 Bi (A7) 197 3.9 4.0 Diamond (A4) 215 4.6 4.6 paper on Ti and Zr (Kaufman 1959a), but the treatment did not take into account the inherent mechanical instability of the high temperature phase. More recent work has shown that, in the case of these elements, suitable anharmonic contributions can stabilise the b.c.c. phase in Ti and Zr and that the vibrational entropy contribution accounts for 70% of Sa_^ at the allotropic transformation temperature (Ho and Harman 1990, Perry et al. 1991). As, in this case, the mechanical instability is only marginal the treatment of Kaufman (1959a) may be considered a good approximation. However, when there is a much more marked instability, as in the case for tungsten (Fernandez Guillermet et al. 1995, Einarsdotter et al. 1997) any vibrational entropies calculated from Debye temperatures may be totally inappropriate. < 6.2.2.3 Determination of transformation enthalpies in binary systems. Just as consistent values of Tm for elements can be obtained by back-extrapolation from binary systems, so it is possible to obtain values of Ha^ by extrapolating the enthalpy of mixing vs composition in an alloy system where the phase has a reasonable range of existence. The archetypal use of this technique was the derivation of the lattice stability of f.c.c. Cr from the measured thermodynamic properties of the Ni-based f.c.c. solid solution (7) in the Ni-Cr system (Kaufman 1972). If it is assumed that the f.c.c. phase is a regular solution, the following expression can be obtained: RTIn J = G%r^fcc- + A4; (6.12) where the left-hand side refers to the activity coefficient of Cr in the Ni solid solution, 7, the right-hand side contains the 'lattice stability' value for f.c.c. Cr while ft is the regular solution interaction parameter. Plotting a^Jxct vs then leads to a straight line with an intercept equal to the Gibbs energy difference between the f.c.c. and b.c.c. forms of Cr, at the temperature where measurements were made (Fig. 6.5), while the slope of the line yields the associated regular solution interaction parameter. The lattice stability and the interaction parameter are conjugate quantities and, therefore, if a different magnitude References are listed on pp. 173-178. 140 N. Saunders and A. F. Miodownik 12000 sooo 4000 OCJ o 10500 + 0.63T J mo!" (estimated) -4000 -8000 -120O0 -20000 J/mol a Exptl data from 1448 K • Exptl data from 1550 K i i_l Cr 0.2 0.4 0.6 0.8 tt t t t 1 t Cr0.7 0.5 0.4 0.3 0.2 0.1 Ni Ni Figure 6.5. Plot of the activity coefficient of f.c.c. Cr (relative to pure b.c.c. Cr) vs the square of the atomic fraction of Ni to establish the lattice stability of f ex. Cr (adapted from Kaufrnan 1972). for the lattice stability is adopted, the results can only be fitted by using a more complex solution model (e.g. sub-regular). In the case of Ni-Cr, the use of a. regular solution model leads to a lattice stability for f.c.c. Cr which coincides closely with the value obtained independently through Iiquidus extrapolations (Saunders et al 1988). However, on the basis of calculations of lattice stabilities from spectroscopic data, Brewer (1967, 1979) has consistently maintained that interaction coefficients can change drastically with composition, and that extrapolated lattice stabilities obtained with simple models should therefore be considered as only 'effective' values. While this may indeed be true when mechanical instability occurs, many of the assumptions which underlie Brewer's methodology are questionable. A core principle of the spectroscopic approach is the derivation of 'promotion energies' which require the definition of both ground and excited levels. Assumptions concerning the relevant excited state have always been strongly coloured by adherence to the empirical views of Engel (1964) and Brewer (1967). By definition, the choice References are listed on pp. 173-178. of atomic ground states ignores all band-structure effects, and calculations by Moruzzi and Marcus (1988a) for the 3D transition elements highlight the omitted factors. It is interesting to note that there have been progressive changes in underlying assumptions over the years (Brewer 1975, Kouvetakis and Brewer 1993) and the lattice stabilities obtained by this route are now much closer to those currently produced by the more conventional TC method. 6.2.2.4 Utilisation of Stacking-Fault Energies. Experimental values of stacking-fault energies (SFE) offer a method of providing energy differences between stable and metastable close-packed structures. A rigorous relationship involves modelling the interface between the fault and the matrix, but a good working formula can be established by assuming that this interfacial energy term is constant for a given class of alloys (Miodownik 1978c). A = 2(G^c-c-,c-p-h- + cr) where G f.c.c.—>c.p.h. _ 221 ATI/3 [p/M}2/3G^c-^- (6.13) (6.14) ^f-cc-cph. |s tjje qjoos energy difference/unit area between the f.c.c. and c.p.h. phases in mJ m-2, a is the energy of the dislocation interface, p is the density in g cm-3, Mthe molecular weight in grammes and f?^c<:-'cPh' the Gibbs energy in J mol-1. Eqs (6.13) and (6.14) were originally used to predict the SFE of a wide range of stainless steels, but they have also been used, in reverse, to estimate values Qf Qf.c.c.-.c.p.h. 0f some f c c eiemeirits (Saunders et al. 1988) where they provide values which are in excellent agreement (Miodownik 1992) with those obtained by FP methods (Crampin et al. 1990, Xu et al 1991). Although this method is essentially restricted to a particular sub-set of lattice stabilities, it nevertheless provides an additional experimental input, especially in cases where it is not possible to access the metastable phase by other methods. It is therefore disappointing that there are no experimental values of the SFE available for Ru or Os, which could provide confirmation of glass transition) onto existing Gibbs energy expressions so as to avoid changes in well-substantiated high-temperature data (Agren et al 1995). If high-temperature data were also to change this would inevitably require major re-calcuiation of many systems. (4) The increasing availability of electron energy calculations for lattice stabilities has produced alternative values for enthalpy differences between allotropes at 0 K. which do not rely on the various TC assumptions and extrapolations. Such calculations can also provide values for other properties such as the Debye temperature for metastable structures, and this in turn may allow the development of more physically appropriate non-linear models to describe low-temperature Gibbs energy curves. 6.3. AB INITIO ELECTRON ENERGY CALCULATIONS Ideally, the 0 K values of the relative enthalpy for various crystal structures can be References are listed on pp. 173-178. CALPHAD—A Comprehensive Guide 143 obtained by ab initio (first-principles) electron energy calculations, using merely atomic numbers and the desired atomic geometry as input. However, such methods did not have sufficient accuracy at the time lattice stabilities were first brought into use. An apocryphal analogy is that to obtain lattice stabilities by calculating the difference in total energy of two crystal structures, is like determining the weight of a ship's captain by first weighing the ship empty and then weighing the ship with the captain on the bridge! Even as late as 1971 some methods were still unable to predict the correct ground state for elements such as Zn, while the scatter obtained from various calculation routes was far too high for ab initio phase stabilities to be introduced into a TC database (Kaufman 1972). However, owing to a combination of improved modelling and the availability of more powerful computers, results have become increasingly more consistent (Pettifor 1977, Skriver 1985, Watson et al. 1986, Paxton et al. 1990, Asada and Terakura 1993). In many cases, the values obtained from first principles (FP) have confirmed the values obtained by the thermochemica! (TC) methods outlined in the previous section. The convergence of FP and TC values for the 0 K enthalpy is of great potential benefit to both the CALPHAD and physics communities for the following reasons. > A, 1 vDes|iite the success of phase-diagram calculations, there is still a considerable reluctance by sections of the scientific community to accept that TC lattice stabi-"ties represent a real physical entity as distinct from an operational convenience. This inevitably creates doubts concerning the ultimate validity of the calculations. It is therefore important to verify that TC lattice stabilities, largely derived by extrapolation, can be verified by ab initio calculations and placed on a sound physical basis. 2. From the physicist's point of view, agreement with the CALPHAD figures for metastable allotropes represents one of the few ways of assessing the validity of their technique which is based on the principle that, once a theoretical model is developed, there are no further adjustable input parameters other than the atomic number and a fixed geometry for relative atomic positions. However, just as a number of different extrapolative techniques have been used in the TC approach, so a variety of assumptions have been used to solve the Schrodinger equation for a complex ensemble of atoms, combining different levels of accuracy with solutions that can be attained on a realistic time scale. A selection is given in Table 6.2. As many methods were developed in parallel, the order in which they are listed should not be considered important. The most significant assumptions made in these various electron energy calculations are indicated in the various acronyms listed in Table 6.2. These can be permutated in many combinations and a proper comparison of these methods is beyond the scope of the present article. Excellent review articles (Pettifor 1977, Turchi and Sluiter 1993, de Fontaine 1996) are available if further detail is required. Other references of particular interest are those which compare the results N. Saunders and A. P. Miodownik Table 6.2. Selection of various methods used to produce first-principles (FP) values for the relative stability at 0 K of different crystal structures EPM Empirical pseudo-potential method Heine and Weaire (1971) CPA Coherent potential approximation Ehrenreich and Schwartz (1976) GPM Generalised perturbation method Ducastelle and Gautier (1976) FRO Frozen core approximation Yin (1982) GPT Generalised pseudo-potential theory Moriarty and McMahan (1982) CFT „ Concentration-functional theory Gyorffy and Stocks (1983) LMTO Linear muffin tin orbital Skriver (1983) FLAPW Full potential linearised augmented plane wave Jansen and Freeman (1984) ASA Atomic spherical wave approximation Skriver (1985) LASTO Linear augmented slater type orbital Watson et al. (1986) "'life LSDA Local spin density approximation Moruzzi et al. (1986) ECM Embedded cluster method Gonis et al. (1987) GGA Generalised gradient approach Asada and Terakura (1993) BOP Bond order approximation Pettifor et al. (1995) 4 of making different combinations for specific groups of elements, see for example Moriarty and McMahan (1982), Fernando et al. (1990) and Asada and Terakura (1993). 6.3.1 Comparison between FP and TC lattice stabilities Despite the variety of assumptions that have been used, some general trends for the resultant lattice stabilities have been obtained for various crystal structures across the periodic table. The mean values of such (FP) lattice stabilities can therefore be compared with the equivalent values determined by thermochemical (TC) methods. Such a comparison shows the following important features (Miodownik 1986, Watson et al. 1986, Saunders et al. 1988, Miodownik 1992): (1) In the main there is reasonable agreement for the sign and the magnitude of the lattice stabilities for elements whose bonding is dominated by sp electrons (Table 6.3) (see also Figs 6 and 7 of Saunders et al. 1988). In the case of elements such as Na, Ca and Sr, predictions have even included a reasonable estimate of their transition temperatures. Good agreement is also obtained for Group IVB elements such as Ge and Si (Table 6.4). The agreement for these elements is particularly striking because the values obtained by TC methods entailed large extrapolations, and for the case of Si(f.c.c) a virtual, negative melting point is implied. At first sight such a prediction could be considered problematical. However, it can be interpreted in terms of the amorphous state being more stable at low temperatures than the competing f.c.c. crystalline configuration, and it does not contravene the third law as it is either metastable or unstable compared to the stable A4 structure. In the case of W and Mo there is now experimental evidence that supports this viewpoint (Chen and Liu 1997). CALPHAD—A Comprehensive Guide 145 Table 6.3. Comparison of lattice stabilities (kJ mol'1) obtained by TC and FP routes for elements whose bonding is dominated by sp electrons Element Method b.c.c. —* c.p.h. f.c.c. —* cp.h. Reference Al Pseudopotentials -4.6 +5.5 Moriarty and McMahan (1982) TC -4.6 +5.5 Kaufman and Bernstein (1970) Saunders et al. (1988) Mg Pseudopotentials -2.7 -0.8 Moriarty and McMahan (1982) LMTO -1.2 -0.9 Moriarty and McMahan (1982) TC -3.1 -2.6 Saunders et al. (1988) Zn Pseudopotentials -7.5 -1.7 Moriarty and McMahan (1982) LAPW -9.2 -1.6 Singh and Papaconstantopoulos (1990) TC -2.9 -3.0 Dinsdale (1991) Be Pseudopotentials -8.1 -7.2 Lam et al. (1984) LMTO -6.4 -5.9 Skriver (1982) TC -6.8 -6.3 Saunders et al. (1988) Table 6.4. Comparison of lattice stabilities for Si and Ge (kJ mol-1) obtained by TC and FP routes for transformations from A4 to Al (f.c.c), A2 (b.c.c), A3 (cp.h.) and A5 Q3-Sn) Structure Element Method Al A2 A3 , A5 Reference Si TC 51 47 49 Saunders et at. (1988) TC 50 42 51 Kaufman and Bernstein (1970) FP 55 51 53 26 Yin (1982) FP Goodwin et al. (1989) Ge TC 36 34 35 Saunders et al. (1988) TC 36 28 Miodownik (1972b) FP 44 42 43 24 Yin (1982) FP Goodwin etal. (1989) (2) A sinusoidal variation of the 0 K energy difference between b.c.c. and close-packed structures is predicted across the transition metal series, in agreement with that obtained by TC methods (Saunders et al. 1988). For the most part magnitudes are in reasonable agreement, but for some elements FP lattice stabilities are as much as 3-10 times larger than those obtained by any TC methods (Fig. 6.6). (3) FP methods inherently lead to a marked sinusoidal variation of Hc•ec--,cP-h-across the periodic table (Pettifor 1977) and for Group V and VI elements, electron energy calculations predict Jf-c-t-,c-Pi 0f opposite sign to those obtained by TC methods. It is worth noting, however, that a sinusoidal variation is reproduced by one of the more recent TC estimates (Saunders et al. 1988) although displaced on the energy axis (Fig. 6.7). Some reconciliation can be achieved by considering the effect of changing References are listed on pp. 173-178. 146 N. Saunders and A. P. Miodownik 40 -40 -80 80 r- -80 Figure 6.6. Variation of the enthalpy difference between f.c.c. and b.c.c. structures obtained by various TC and FP routes. • Saunders et ai (1988); A Kaufinan and Bernstein (1970); □ Skriver (1985). CALPHAD—A Comprehensive Guide o B t u -10 Ti Mn \ Fe •—' Co -20 L Co Ni Cu 20 I W -10 -20 Figure 6.7. Variation of the enthalpy difference between f.c.c. and c.p.h. structures obtained by various TC and FP routes. • Saunders et at. (1988); A Kaufinan and Bernstein (1970); □ Skriver (1985). References are listed on pp. 173-178. 148 N. Saunders and A. P. Miodownik the da ratio (Fernando et al. 1990), as some of the hexagonal phases seem to be most stable at c/a ratios which depart substantially from the ideal close-packed value (~1.63). This factor is of significance in the case of elements like Cd and Zn where c/a ~ 1.8 and it is clearly not appropriate to equate the Gibbs energies of the ideal and non-ideal hexagonal forms (Singh and Papaconstanto-poulus 1990). Making this kind of adjustment could reverse the sign of jrf.c.c-w.p.h. for ■pfo ^oes not reaijy make much impact on the situation for Mo, Ta and W. Likewise a change in volume within reasonable limits does not resolve the issue (Fernando et al. 1990). (4) In the past, electron energy calculations have failed dramatically for magnetic elements since spin polarisation was not included. However, this can now be taken into account quite extensively (Moruzzi and Marcus 1988b, 1990, Asada and Terakura 1993) and calculations can reproduce the correct ground states for the magnetic elements. 6.3.2 Reconciliation of the difference between FP and TC lattice stabilities for some of the transition metals Several approaches have been made in order to remove the outstanding conflicts between FP and TC lattice stabilities. Niessen et al. (1983) proposed a set of compromise values, which are also listed in de Boer et al. (1988), but these do not constitute a solution to this problem. These authors merely combined the two sets of values in relation to an arbitrary reference state in order to refine their predictions for heats of formation (see Section 6.6.1.1). The accuracy with which a phase diagram can be fitted with competing values of phase stabilities appeared, at one time, to be a fairly obvious route to discover whether the TC or FP values were closer to reality. Tso et al. (1989) calculated the Ni-Cr phase diagram using a series of different values for the lattice stability of f.c.c. Cr. They were able to reasonably reproduce the phase diagram with energy differences close to those proposed by electron energy calculations. However, they could only do so by introducing a compensating change in the Gibbs energy of mixing for the f.c.c. phase, which had to become large and negative in sign. It is difficult to accept such a proposal for the enthalpy of mixing for f.c.c. alloys when the liquid exhibits almost ideal behaviour and the b.c.c. phase has mainly positive interactions. In addition, calculations of mixing energy by using ^-band electron models (Colinet et al. 1985, Pasturel et al. 1985) supports values much closer to those already in use by CALPHAD practitioners. A series of publications by various authors used a similar process to establish which lattice stability gave the best fit for particular phase diagrams (Anderson et al. 1987, Fernandez Guillermet and Hillert 1988, Fernandez Guillermet and Huang 1988, Kaufman 1993, Fernandez Guillermet and Gustafson 1985). Au-V formed a particularly useful test vehicle (Fernandez Guillermet and Huang 1988, Kaufman References are listed on pp. 173-178. CALPHAD—A Comprehensive Guide 149 2500 i4 1500 1000 500 00 H(b-c-c-)-H(f-c-c) = 5 kJ/mol DS = -1.7 J/mol/K b.c.c. / / f.c.c. Rfcs=^_ / Experimental Values / 4 UQ/UQ+BCC / O BCC/LIQ+BCC / » LIQ/LIQ+FCC / * FCC/L1Q+FCC 1 f FCC/BCC+FCC .....t ■— J * BCC/BCC+FCC 0.2 0.4 0.6 0.8 1.0 Mole fraction Au 0 0.2 0.4 0.6 0.8 1.0 Mole fraction Au H(b-cc)-H(f-c-c) = 15 kJ/mol DS = -1.7 J/mol/K Sad 2S00 2000 a. S 0.2 0.4 0.6 0.8 Mole fraction Au _ 700 500 "— * * i H(b-"-)-H('co-) = 25 kJ/mol DS = -1.7 J/mol/K b.c.c. j / 1 • ' f.c.c. * i ' it 1 11 ii // ....... 1.0 s s a 300 100 0.2 0.4 0.6 0.8 Mole fraction Au AS (A) = -2.4 AS (B) = -1.7 AS (C) = -0.8 1.0 5 10 15 20 25 Enthalpy difference (Id mol-1) Figure 6.8. (a-d) The effect of varying the f.c.c.-b.c.c. V lattice stability on the matching of experimental and calculated phase boundaries in the Au-V system (after Fernandez Guillermet and Huang 1988) and (e) the effect of varying ASb on the minimum error sum, giving a preferred value of 7.5 kj mol-1. 1993) as V was one of the elements which showed a serious discrepancy (Fig. 6.8). Using a regular solution model and restricting the range of Sft such an analysis (Fernandez Guillermet and Huang 1988) suggested that the optimum values of 150 N. Saunders and A, P. Miodownik £b.c.c.->f.c.c. were majkedly lower than those obtained by electron energy calculations (Fernandez Guillermet and Huang 1988) and also significantly lower than proposed by Saunders et al. (1988). The permitted range of lattice stability values can be increased by changing these constraints, in particular letting S! vary as shown in Fig. 6.4(a), but any values still exclude FP predictions. This impasse was eventually resolved by taking into consideration the calculated elastic constants of metastable structures in addition to their energy difference at 0 K. Craievich et al. (1994), Craievich and Sanchez (1995) and Guillermet et al. (1995), using independent calculations, have suggested that the difference between TC and ab initio predictions may be associated with mechanical instabilities in the metastable phase. This point had been raised earlier by Pettifor (1988) and has the following consequence as reported by Saunders et al. (1988): "...the analogy with titanium and zirconium is useful. In these elements there is a reported lattice softening in the b.c.c. ätiotrope near the c.p.h. —> b.c.c. transformation temperature. This instability is reflected by a substantial Cp difference between the c.p.h. and b.c.c. structures which effectively destabilises the b.c.c. phase with decreasing temperature. High-temperature extrapolations which ignore this, then overestimate the low temperature stability of the b.c.c. phase. Äs ab initio calculations give values for heats of transformation at 0 K, comparison with high-temperature phase diagram and thermodynamic extrapolations will produce different results." Furthermore, such instabilities will extend into the alloy system up to a critical composition and must therefore be taken into account by any effective solution-phase modelling. In the case of Ni-Cr, it is predicted that mechanical instability, as defined by a negative value of c' = l/2(cn— cu), will occur between 60 and 70 at%Cr (Craievich and Sanchez 1995), so beyond this composition the f.c.c. phase cannot be considered as a competing phase. While this concept of mechanical instability offers a potential explanation for the large discrepancies between FP and TC lattice stabilities for some elements, the calculations of Craeivich et al. (1994) showed that such instabilities also occur in many other transition elements where, in fact, FP and TC values show relatively little disagreement. The key issue is therefore a need to distinguish between 'permissible' and 'non-permissible' mechanical instability. Using the value of the elastic constant C as a measure of mechanical instability, Craievich and Sanchez (1995) have found that the difference between the calculated elastic constant C for f.c.c. and b.c.c. structures of the transition elements is directly proportional to the FP value of -fcc- (Fig. 6.9(a)). The position of Ti and Zr is again important in this context. While the b.c.c. phase in these elements has long been known to indicate mechanical instability at 0 K, detailed calculations for Ti (Perry 1991) and Zr (Ho and Harmon 1990) show that it is stabilised at high temperatures by additional entropy contributions arising from low values of the elastic constants (soft modes) in specific crystal directions. This concept had already been raised in a qualitative way by Zener (1967), but the References are listed on pp. 173-178. (a) 4000 o B ^ 1000 0 1 3000 - Os 2000 Ru ■«• to the associated elastic instability as measured by the elastic constant C (Private Communication from Craievich and Sanchez 1995). (b) The difference between TC and FP lattice stabilities plotted vs the absolute value of the calculated FP lattice stability. When combined with Fig. 6.8 this suggests a critical value of the elastic instability that cannot be compensated for by thermal contributions. 152 N. Saunders and A. P. Miodownik CALPHAD—A Comprehensive Guide 153 key issue is to find out whether there is a maximum value to the additional entropy that is available to counterbalance specific values of mechanical instability at 0 K. This would be consistent with the observation that major discrepancies between FP and TC lattice stabilities only seem to arise when the mechanical instability rises beyond a critical value (Fig. 6.9(b)). This is also consistent with the findings of (Fernandez Guillermet et at. 1995), who could only reproduce the phase diagram for W-Pt by invoking an empirical and highly anomalous value of 5fcc-,bcc-which would also have to exhibit a strong temperature dependence to avoid the appearance of f.c.c. W as a stable phase below the melting point. The effect of incorporating a more complex entropy function has also been examined by Chang et al. (1995) using the Ni-Cr system as a test vehicle. Here an anomalous entropy contribution was simulated by incorporating a dual Debye temperature function, linked to the critical composition at which the elastic constants had been calculated to change sign in this system. The resulting lattice stability for the f.c.c. phase nicely shows how this approach rationalises the extrapolated liquidus derived using the van Laar method and values that would be consistent with a FP approach. Figure 6.10 shows that the phase diagram in the stable region does not suffer inaccuracies by using high FP values as happened in the earlier attempts on Au-V (Fig. 6.8). It should, however, be emphasised that the equations used by Chang et al. (1995) to define the Gibbs energy of f.c.c. Cr are highly empirical and require a whole new set of adjustable parameters, such as a critical temperature which defines the onset of mechanical instability. This poses an > 2500 2000 — Liq / JCC 1500 ~~ J 1000 FCC / 1 500 / / / / 0 1 1 I i I 0.2 0.4 0.6 0.8 1.0 MOLE FRACTION Cr Figure 6.10. Comparison of the extrapolated liquidus/solidus lines relating to the f.c.c. phase in Ni-Cr alloys, derived by (a) using the van Laar method and (b) the trajectory obtained using the modified FP approach used by Chang et al. (1995). exciting challenge to provide a sounder basis for a treatment that can encompass low-temperature mechanical instability. The present situation concerning FP and TC lattice stabilities can probably be described as follows. 1. Thermochemical methods generate lattice stabilities based on high-temperature equilibria that yield self-consistent multi-component phase-diagram calculations. However, as they are largely obtained by extrapolation, this means that in some cases they should only be treated as effective lattice stabilities. Particular difficulties may occur in relation to the liquid —► glass transition and instances of mechanical instability. 2. By contrast, electron energy calculations have the inherent capability of yielding accurate values for many metastable structures at 0 K but have little or no capability of predicting the temperature dependence of the Gibbs energy, especially in cases where mechanical instabilities are involved. Although the two methodologies are complementary, attempts at producing Gibbs energy curves which combine the two approaches are currently still empirical and would, in practice, be very difficult to incorporate in a general CALPHAD calculation. A more fundamental treatment of various entropy contributions is required to achieve proper integration. Until this is realised in practice, the use of TC lattice stability values as well established operational parameters, valid at high temperatures and for most CALPHAD purposes, is likely to continue for the foreseeable future. 6.4. THE BEHAVIOUR OF MAGNETIC ELEMENTS The treatment outlined so far has not included the magnetic Gibbs energy contributions in Eq. (6.1) because the ground state of the majority of elements is paramagnetic. This, however, is certainly not the case for some key 3d transition elements, which exhibit various forms of magnetic behaviour not only in the ground state but also in one or more allotropes. Fe is a classic example and since ferrous metallurgy has been a major driving force in the development of phase diagram calculations, one of the first steps was to establish the magnitude of the magnetic component on the Gibbs energy. The basic factors that control this are detailed in Chapter 8, but it is worthwhile here to review some of the implications for specific elements, with particular emphasis on Fe. 6.4.1 Fe Considerable information is available on the magnetic parameters associated with three different crystal structures of Fe which are b.c.c. and f.c.c. at ambient pressures and c.p.h. which is observed at high pressures. Table 6.5 gives the corresponding values of the maximum enthalpy and entropy contributions due to References are listed on pp. 173-178. 154 N. Saunders and A. P. Miodownik Table 6.5. Maximum values of the magnetic enthalpy and entropy for various allotropes of Fe, Co and Ni based on data and methodology drawn from Miodownik (1977) and additional data from de Fontaine et al. (1995). Structures in brackets correspond to metastable forms that have not been observed in the TP diagram of the element Element Structure H™* (kJmor1) $""* (J mol"' K~l) Fe b.c.c. 9.13 9.72 f.c.c. 0.32 4.41 c.p.h. 0 0 Co (b.c.c.) 11.17 8.56 f.c.c. 10.69 8.56 c.p.h. 9.66 8.26 Ni (b.cc.) 1.91 3.37 f.c.c. 2.27 4.01 (c.p.h.) 0.94 3.48 magnetism, and 5^ respectively, and Fig. 6.11(a) shows how this affects the total Gibbs energy for the three structures in Fe. At high temperatures Fe behaves like many other allotropic elements, solidifying in a b.c.c. lattice, (fi-Fe). As the temperature is reduced, there is a transition to the f.c.c. 7-Fe phase which has both a lower energy and a lower entropy. However because of C™8 the b.c.c. phase reappears at lower temperature as a-Fe. While the Curie temperature of a-Fe, 770°C, is actually 140°C below the a/7 transformation temperature at 910°C, there is already sufficient short-range magnetic order to cause the transformation. A consequence of this unique behaviour is that the value of Gbc'c~*f cc- remains exceedingly small in the region between the two transition points and never exceeds 50-60 J mol-1 (see Fig. 6.11(b)). This means that small changes in Gibbs energy, due to alloying, will substantially alter the topography of the 7-phase region in many Fe-base systems. If the alloying element also significantly affects the Curie temperature, the proximity of the latter to the a/7 transus will mean that this boundary is disproportionately altered in comparison to the 7/5 transus. This latter point can be shown to account for the otherwise puzzling asymmetric effects of alloying additions on the two transition points (Zener 1955, Miodownik 1977, Miodownik 1978b). Because it is the basis of so many important commercial systems, the allotropy of Fe has been re-examined at frequent intervals. There is relatively more experimental information available for a-Fe, as this is the ground state, and relatively little controversy about characterising this phase. There is also little difference between the various proposals for Gbcc~*u<:- in the temperature range where 7-Fe is stable, because of the need to reproduce the two well-known transformation temperatures Ta^ and T*^ and their associated enthalpies of transformation (Fig. 6.11(b)). There are, however, different points of view regarding the extrapolation of the Gibbs energy of the 7-phase to low temperatures, which relate to different weightings given to various specific heat measurements, assumptions made References are listed on pp. 173-178. S -5 -10 -15 -30 (a) Iron V- Liquid reference state Paramagnetic \ "s. b.c.c. ' — f,c-c-^^"^-^ b.c.c. ~ AjN^^pf.c.c. — -20 -25 1000K Temperature 100 (50) (100) (ISO) --*--.—»--- 4 Q • • ■ ■ ■ Darker? and Smith • Pepperhoff — Expt (1951) Calc (1982) . * Weiss and Tauer □ Kaufman et al Calc (1956) 1 1 Calc (1963) 1 1 1,200 1,400 1,600 1,800 Temperature Figure «.11. General overview of the relative stability of the f.c.c. and b.c.c. phases in pure Fe (a) over the whole temperature range (Miodownik 1978b) and which were based on the specific heat measurements of Austin (1932). 156 JV. Saunders ana A. f. Mioaowmit 5,000 6,000 H oo 4,000 \- o 2,000 r- 200 400 600 800 Temperature (K) 1,200 Figure 6.12. Effect of different models on the Gibbs energy difference between the f.c.c. and b.cc. phases in pure iron. Data (O) of Darken and Smith (1951), (A) Weiss and Tauer (1956), (•) Kaufman et al. (1963), (B) Orr and Chipman (1967), (A) Agren (1979), (°) Bendick and Pepperhoff (1982), (») Fernandez Guillermet and Gustafson (1985), (■) Chang et at, (1985). Relevant data extracted from activity measurements in the Fe-C phase were later included in the assessment by Zener (1946) and Darken and Smith (1951). All this work followed a traditional approach solely based on thermodynamic data derived under equilibrium conditions. By contrast, Kaufman and Cohen (1958) showed that the data of Johannson (1937) were more consistent with information derived from Fe-based martensite transformations than the interpretation used by Darken and Smith (1951). Together with the work of Svechnikov and Lesnik (1956) this was a notable attempt to combine thermodynamic information derived from low-temperature metastable transformations with those from more traditional sources. Another such departure was the attempt by Weiss and Tauer (1956) to decon-volute the global value of Ghcctec' into magnetic, vibrational and electronic components. This represented the first attempt to obtain a physical explanation for the overall effect. An even more comprehensive approach by Kaufman et al. (1963) led to the further inclusion of two competing magnetic states for 7-Fe. Since the computer programmes available at that time could not handle such a sophisticated approach, the relevant values of Ghcc~>ic-C- were converted into polynomial form and subsequently used for the calculations of key Fe-base diagrams such as Fe-Ni, Fe-Co and Fe-Cr (Kaufman and Nesor 1973). Subsequent reassessments now took divergent routes. Agren (1979) used thermodynamic data largely drawn from Orr and Chipman (1967) and re-characterised the magnetic component of the a-phase with the Hillert and Jarl model (see Chapter 8). The concept of two competing states in the 7-phase was abandoned as Orr and 13/ Chipman (1967) favoured a high value for #b-«:--f ■<=■<;. and a suitable fit could be obtained without invoking the added complications of this model. The Agren treatment has been further refined in the most recent assessment of the T-P properties of Fe made by Fernandez Guillermet and Gustafson (1985). By contrast, both Miodownik (1978b) and Bendick and Pepperhoff (1982) pursued further evidence for the existence of two gamma states from other physical properties and built alternative assessments for Fe around this concept. Although attention has tended to concentrate on the equilibrium between a- and 7-Fe, the hexagonal variant, e, has also to be considered. This can only be accessed experimentally in pure Fe when the b.cc. phase is destabilised by pressure or alloying additions. The extrapolated properties of £-Fe are consistent with it being paramagnetic, or very weakly anti-ferromagnetic, and so magnetism will provide a negligible contribution to its stability. Nevertheless, the extrapolation of SFE versus composition plots confirms that the e-Fe becomes more stable than the 7-Fe at low temperatures, despite the magnetic contributions in the 7-phase (see Fig. 8.11 of Chapter 8). Hasegawa and Pettifor (1983) developed a model based on spin-fluctuation theory which accounts for the T-P diagram of Fe without invoking any differences in vibrational entropy or multiple magnetic states, but their theory predicts that both close-packed states should exhibit a temperature-induced local moment. The f.c.c. phase then behaves in a way which is rather similar to that predicted by the phenomenological two-7-state model (Kaufman et al. 1963). The prediction of a high-temperature moment for the e-phase is, however, at variance with the assumptions or predictions of most other workers who consider that the £-phase has a zero or negligibly low moment. Various other models have been proposed which break down the total Gibbs energy in different ways, for instance by placing a different emphasis on the contribution of vibrational and electronic terms (Grimwall 1974). Both the Pettifor-Hasegewa and the Grimwall models can account for the some of the necessary qualitative features exhibited by the allotropy of Fe, but the intrinsic assumptions of these two approaches are incompatible with each other. The various approaches were again reviewed by Kaufman (1991) who quoted further evidence for the two-garruna-state concept, but virtually all current databases incorporate the data generated by Fernandez Guillermet and Gustafson (1985) (Fig. 6.12). This has certainly been well validated in phase-diagram calculations, and it is hard to see how it could be improved at elevated temperatures. However, the differences that appear at low temperatures may be important in relation to metastable equilibria such as martensite transformations which, since the early work of Kaufman and Cohen (1956, 1958), have tended to be excluded from any optimisation procedure. Certainly there seems to be no doubt about the existence of many competing combinations of crystal structure and magnetic moments (Asada and Terakura 1993) (Fig. 6.13) which suggest that there must be room for an improved model. References are listed on pp. 173-178. 158 N. Saunders and A. P. Miodownik (a) -2546.61 .63 2 .65 .67 Fe by OGA \ ™-.-fl-..o- -2'' - b.c.c. --- h.c.p. ..... f.c.c. A * FM o . AF O • NM -2318.20 — b.c.c. -- h.c.p. ----f.c.c. A FM D AF O NM Fieure 6.13. Enthalpy vs volume plots for f.c.c, b.c.c. and c.p.h. for (a) Fe and ft) Mn, incorporating the effect of assuming different magnetic interactions A ferromagnetic, □ anti-ferromagnetic, o non magnetic (from Asada and Terakura 1993). 6.4.2 Co . t ,, . Table 6.5 shows that Coc*h- has a high value of so it is tempting to say that, as with Fe, the low-temperature allotrope forms are due to strong stabilisation by References are listed on pp. 173-178. ^ / k^,t i mi/ - magnetic forces. However this is not the case, as Cofcc- is also ferromagnetic with very similar values of /? and Tc. The magnetic contributions virtually cancel each other out in this case, although the f.c.c./c.p.h. transformation temperature will clearly be sensitive to small changes in magnetic parameters on alloying. The magnetic properties of metastable b.c.c. Co were derived indirectly by Inden (1975) as part of a calculation involving ordering in b.c.c. Fe-Co alloys. Interestingly, these values are substantially confirmed by more recent FP calculations. Table 8.2 and Fig. 8.11 of Chapter 8 can be consulted for further details. 6.4.3 Ni Ni does not exhibit any allotropy with respect to either temperature or pressure, which implies that its f.c.c. structure must be significantly more stable than the nearest competitors. This is consistent with values for (jf-«-c--*«-p-h- derived from SFE measurements and changes in SFE with alloying also suggest that there is a magnetic component in Gf-<^--,<:-pJ\ Since the magnetic enthalpy is much smaller than for Fe or Co, this in turn implies that that Nicph- is paramagnetic or has a low Tc (Miodownik 1978a). However, FP calculations imply a value of j3 not much lower than for Nif cc-, which would indicate that Ni behaves much like Co. Further experimental information on the magnetic properties of Nic,p,h-, possibly from thin films, would clearly be desirable. 6.4.4 Mn Experimentally Mn solidifies as the b.c.c. 6-Ma phase, which then transforms to f.c.c. 7-Mn on cooling. This is entirely analogous to the behaviour of Fe. However, it then undergoes two further low-temperature transitions to /?-Mn and a-Mn which are complex crystal structures with large unit cells. De-convoluting the experimental data into separate contributions is difficult. Three of the phases are anti-ferromagnetic with widely differing values of the saturation magnetisation at 0 K, and there are also significant differences in the Debye temperatures and electronic specific heat. It is therefore difficult to make a consistent characterisation which matches all these observations. By concentrating on matching phase transition temperatures and Debye temperature data, Fernandez Guillermet and Huang (1990) were forced to use effective magnetisation values which could be as low as \ of those experimentally observed. This makes it impossible to estimate the real role of any magnetic factors in their treatment. By contrast, in an earlier assessment, Weiss and Tauer (1958) decided to use the measured magnetisation values as a primary input over the whole temperature range, but then had to depart from the experimentally observed Debye temperatures. Interestingly, their treatment led to the conclusion that the G curves for /?-Mn and 7-Mn would also intersect twice so that the behaviour of Fe could no longer be 160 N. Saunders and A. P. MiodowniK considered unique. However in this scenario it is 7-Mn that has the largest magnetic component, while the stability of /3-Mn is dictated by a larger electronic specific heat. The finding by Gazzara et al. (1964) that the value of the saturation magnetisation for a-Mn (and other anti ferromagnetic materials) can be temperature dependent may have to be taken into account in order to finally reconcile what are otherwise different interpretations of the same data. The relative Gibbs energy curves for b.c.c, f.c.c. and c.p.h. Mn as calculated by Asada and Terakura (1993) are shown in Fig. 6.13(b). 6.5. THE EFFECT OF PRESSURE 6.5.1 Basic addition of a PAV term It is commonplace to assume a form of the Gibbs energy function which excludes the pressure variable for solid-state phase transformations, as the magnitude of the PAV term is small at atmospheric pressures. This is of course not the case in geological systems, or if laboratory experiments are deliberately geared to high-pressure environments. Klement and Jayaraman (1966) provide a good review of the data available at the time when some of the earliest CALPHAD-type calculations were made (Kaufman and Bemstein 1970, Kaufman 1974). Much work was also carried out on specific alloy systems such as Fe-C (Hilliard 1963) and the Tl-In system (Meyerhoff and Smith 1963). The extra pressure term to be added to the Gibbs energy can be expressed as Jo (6.15) where is the change in molar volume due to the transformation of a to /?. Although this can be considered as constant to a first approximation, this will no longer be true at high pressures, and several empirical descriptions have been developed, depending on the pressure range in question. Bridgman (1931) used a simple second-power polynomial to define the effect of pressures up to 3 GPa: Vm = K0 + ViP + V2JP2 (6.16) where Vo,a,2 are temperature dependent and have to be. determined by experiment. If the entropy, enthalpy and volume differences between these phases are assumed independent of temperature and pressure is not excessive, then Eq. (6.15) will obviously reduce to: Qa->0 pyt*->ß (6.17) where P and are given in GPa and m3 respectively. This simple treatment also leads to the familiar Clapeyron equation where the slope of the temperature vs References are listed on pp. 173-178. Table 6.6. Linear Gibbs energy equations for WBW taken (Kaufman and Bemstein 1970) Values an? I ?" elements that include a PKSS*™ te™ ---_">■ vaiues are applicable only at temperatures >300 K Element Transformation Ca Sr Be Yb Ba Pb C f.c.c. ->b.c.c. f.c.c.-.b.c.c. c.p.h.->b.c.c. f.c.c—tb.c.c. f.c.c.-tcp.h. f.c.c.-»c.p.h. A4->Graphite AH (J mol"1) 243 837 4602 3180 4351 2510 1255 -TAS +PAV (5 = J mol"'; T=K) (p = GPa; V = m«) -0.33T -0.96T -3.01 T -2.97T +3.35T -0.42T -4.77T +0.01P -0.1 OP -0.18P -0.80P -0.74P -0.1SP +1.76P Note: Although current values of AH and A 0 (6.18) Some examples of equations based on adding a simole P<\V i™™ ^ *1. a « 6.5.2 Making the volume a function ofT and P These earlier treatments have now been superseded by a more general approach, where the molar volume of each phase as a function of temperature and pressure is expressed as a function of the compressibility x (Fernandez Guillermet et al. 1985). V*>p = f0(l + nPXyi/n exp (ao T + \ a, T2) (6.19) where Vo is an empirically fitted parameter with the dimensions of volume and ao and ai are parameters obtained from fitting the experimental lattice parameter and dilatometric data such that (6.20) ^rifexit' ?emiai expansivity at zero Pressure as a function of temperature. X * also expressed as a function of temperature using the polynomial X = Xo + XiT+X2T2. (6.21) reduces t0 the simp,er—— 162 xV. Saunders and A. P. Miodownik VT'P = ■ yT,P~0 r TTJ ,.- (6.21) (l+nPXT>p=0fn where V£'P=0 and xT'P=0 respectively, the molar volume and isothermal compressibility at zero pressure. Gm as a function of both temperature and pressure can then be obtained by adding the following expression to Eq. (6.3): GY - = a + bT + Crioge(T) + £dTn + f V™PdP (6-22) Jo where Jo V^dP = Vo{(l + nPxf-^-l) V JL-(6, 23) Available experimental data for various solid-liquid and solid-solid state transformations has been successfully fitted for C, Mo and W (Fernandez Guillermet et al. 1985) and Fe (Fernandez Guillermet and Gustafson 1985) using the above expression. It gives better results than Eq. (6.16) at high pressures and also has the advantage that Eq. (6.19) can be inverted to give an expression for Pr>v">. This in turn allows an explicit function to be written for the Helmholtz energy. Another approach that has been considered recently is the Rose equation of state (Rosen and Grimwall 1983). This however requires the additional input the expansion coefficient and the pressure derivative of the bulk modulus. With the increasing availability of such quantities from first-principles calculations, this may well become the basis for future formulations (Burton et al. 1995). Depending on the data available, Eqs (6.17)-(6.23) reproduce experimental pressure effects with considerable accuracy in many cases. In particular, Eq. (6.18) can be used to confirm entropy data derived using more conventional techniques and can also provide data for metastable allotropes. Ti again provides a leading example, as pressure experiments revealed that the w-phase, previously only detected as a metastable product on quenching certain Ti alloys, could be stabilised under pressure (Fig. 6.14). Extrapolation of the transus line yields the metastable allotropic transformation temperature at which the /3-phase would transform to ui in the absence of the a-phase, while the slope of the transus lines can be used to extract a value for the relevant entropy via Eq. (6.18). 6.5.3 Effect of competing states All of the above approaches assume that missing pressure data can be estimated from consistent periodic trends and that individual phases do not exhibit anomalous behaviour. This is reasonable when the alternative crystal structures are each associated with specific energy minima, but there can be some important exceptions ^ALfttAU—A ^ompreHemm^mme^"- xc 800» \— 600° h O o f- 400° h 200° }- PressuTe 120 Kbars Figure 6.14. T-P diagram for Ti showing experimental data from Bundy (1965b) and Jayaraman (1966) and the extrapolation of the (3-v transus to yield the metastable at atmospheric pressure (from Vanderpuye and Miodownik 1970). to this general rule. For example, in magnetic systems it is possible to obtain markedly different combinations of exchange and correlation forces in the same crystal structure (Fig. 6.13) although the two structures can differ in energy by only a small amount (Asada and Terakura 1993). The possibility of Schottky excitations between alternative states can then produce an anomalous changes in volume with temperature and pressure. Although it is always possible to handle such a situation by a choice of suitable coefficients in Eq. (6.23), an alternative treatment is to explicitly determine the effect on the Gibbs energy of having a combination of states. This was attempted for 7-Fe using the following additional relationships (Kaufman et al. 1963, Blackburn et al. 1965): JT.P (6.24) (6.25) References are listed on pp. 173-178. 164 JV. Saunders and A. P. Miodownik i^ALfnAu—a isomprenensive ijuiae tfJT72 = Ep=o2 + PV*^ (6-26) | 0$p = lOOPI^1 + RT\n{l - P). (6-27) \ Here V71, V72 and Z71, p* refer to the molar volume and fractions of each state, j jE^1"*72 is the difference in energy between the two states and G71-*72 is the extra j increment of Gibbs energy resulting from the existence of the two states. Including j this term, the 7/5 temperature-pressure transus for pure iron was then quantitatively J predicted before this was verified experimentally by Bundy (1965a) (Fig. 6.15). The \ fit is clearly not as good as that which can be obtained by Fernandez Guillermet and \ Gustafson (1985) using Eqs (6.3) and (6.23), but on the other hand these authors j included the experimental pressure data in their optimisation and did not make any | specific provision for the possibility of multiple states. Which treatment is to be -j preferred depends on whether priority is to be given to the development of simple I universally applicable algorithms or to more physically realistic models. ^ > 1 \ 1 V: 0 50 100 150 P (kbar) Figure 6.15. Comparison of experimental T-P diagram for Fe (Bundy 1965) * with calculated phase boundaries from Blackburn et al. (1965) and Fernandez Guillermet and Gustafson (1985) o. Data derived from pressure experiments on semi-conducting elements by Klement and Jayamaran (1966) and Minomura (1974) have also been useful in obtaining confirmation that the entropies associated with transitions in Si, Ge and Sn form a consistent pattern, supporting the concept that each crystallographic transformation tends to have a characteristic associated entropy change (Miodownik 1972a, 1972b). Similarly, extrapolations from pressure data on alloys can be used to obtain estimates of lattice stabilities at P = 0, which can then be compared with estimates obtained by other routes, such as SFE measurements. Calculation of the critical pressure required to cause a phase transformation at 0 K can also be obtained from first-principle calculations. Assuming the various phases exhibit normal elastic behaviour the tangency rule can be applied to energy vs volume plots to yield values for the critical pressure that would generate a phase transformation: -Pent = -£y • (6.28) However, if there are large variations in the elastic constants with pressure, this can seriously affect Perit by as much as an order of magnitude (Lam et al. 1984). 6.6. DETERMINATION OF INTERACTION COEFFICIENTS FOR ALLOYS AND STABILITY OF COUNTER-PHASES The determination of individual binary equilibrium diagrams usually only involves the characterisation of a limited number of phases, and it is possible to obtain some experimental thermodynamic data on each of these phases. However, when handling multi-component systems or/and metastable conditions there is a need to characterise the Gibbs energy of many phases, some of which may be metastable over much of the composition space. This requires a methodology for characterising a large range of metastable solutions and compounds which, by definition are difficult, if not impossible, to access experimentally. The available methods involve various levels of compromise between simplicity and accuracy and can be categorised by the choice of atomic properties used in the process. At one end of the spectrum are first-principles methods where the only input requirements are the atomic numbers Za, Zb, ... the relevant mole fractions and a specified crystal structure. This is a simple extension to the methods used to determine the lattice stability of the elements themselves. Having specified the atomic numbers, and some specific approximation for the interaction of the relevant wave functions, there is no need for any further specification of attractive and repulsive terms. Other properties, such as the equilibrium atomic volumes, elastic moduli and charge transfer, result automatically from the global minimisation of References are listed on pp. 173-178. 166 N. Saunders ana A. f. mivuuww.*. the energy for the total assembly of electrons and ions (Zunger 1980a, 1980b, Pettifor 1992). At the other end of the spectrum are more empirical methods which utilise secondary properties which reflect more conventional terms, such as the atomic volume, electronegativity or some function of the enthalpy of evaporation (Darken and Gurry 1953, Hildebrand and Scott 1956, Mott 1968, Pauling 1960, Kaufman and Bernstein 1970). In many cases the desire for simplicity has led to allocation of a set of constant parameters to each atomic species, even though such properties are known to vary substantially with the geometric or chemical environment in which they are placed. This is probably adequate if such parameters are to be fed into a correspondingly simple regular solution model or line compound model, but will not be sufficient if there is a need for sub-regular solution parameters or if substantial directional bonding is involved. Many treatments have been modified by feedback to give better agreement with experimental trends. In some cases this has been achieved by introducing additional terms and in others by merely altering the numerical values that were originally tied rigidly to a measured property (Miedema 1976, Niessen et al. 1983). 6.6.1 The prediction of liquid and solid solution parameters 6.61.1 Empirical and semi-empirical approaches. The problem of making theoretical estimates for the interaction coefficients for the liquid phase has been tackled in different ways by various authors. Kaufman and Bernstein (1970) considered that the liquid state would exhibit the lowest repulsive forces of all the states of condensed matter and that a description of the interaction parameters for the liquid state would be the best basis for the prediction of interaction parameters for various solid phases. Their expression for a liquid interaction parameter (!) follows Mott (1968) and starts with the simple sum of an attractive term, e0, and a repulsive term, ep, L = e0 + e. (6.29) Kaufman and Bernstein (1970) derived e0 via the electronegativity values, & and i-r + Q(An J/3)2 - R\. (6.36) Here nw3 is an electron density based on the volume of Wigner-Seitz atomic cells, Vm, and it is assumed that differences in electron density Anws between different species of atoms will always lead to local perturbations that give rise to a positive energy contribution. On the other hand, * is a chemical potential based on the macroscopic work function , so that differences, at the cell surfaces between different species of atoms lead to an attractive term, analogous to the gain in energy from the formation of an electric dipole layer. P, Q and R are empirically derived proportionality constants. P and Q were initially taken to be adjustable parameters, but turned out to be almost universal constants for a wide variety of atomic combinations, especially amongst the transition metals. The parameter R was introduced as an arbitrary way of adjusting for the presence of atoms with p References are listed on pp. 173-178. 168 N. Saunders and A. P. Miodownik electrons and takes on values that increase regularly with the valency of the B-group element. In contrast to the Kaufman model (Kaufman and Bernstein 1970), Eq. (6.36) contains no parameters which refer to crystal structure and therefore the heats of solution of f.c.c, b.c.c. and c.p.h. structures are predicted to be, a priori, identical. The same conclusion can also be expected for liquid solutions, and indeed the suggested parameters for liquid and solid solutions are very similar. The constants P and Q are practically the same, although it was necessary to introduce different values of R. 6.6.1.2 Ab initio electron energy calculations. While there are many methods for predicting the relative stability of ordered structures, electron energy calculations for predicting interaction parameters for the liquid is still a major problem. Calculations of the heat of solution for disordered solutions falls in an intermediate category. It is a surprising fact that it is at present still impossible to calculate the melting point, or the entropy and enthalpy of fusion of the elements to any reasonable degree of accuracy, which poses a major challenge to a full calculation of any phase diagram by FP methods. One method to overcome this problem for liquids is to assume that it will exhibit sro parameters based on the predominant solid state structure^) in the phase diagram (Pasturel et al. 1992). Pasturel et al. (1985) and Colinet et al. (1985) have used a tight-binding model which considers the moments of the density of states. The model provides an estimate for the d electron transfer between the elements in an alloy/compound by using the concept of partial density of states. The model is, however, simplified so that neither atomic position or crystal structure is calculated in detail. The results can therefore be considered to provide a good general prediction for disordered phases but is independent of crystal structure. The reason why this works reasonably well is because the magnitude of the fundamental electronic rf-band effect for many such alloys is substantially larger than the effect arising from differences in crystal structure. More accurate results probably require the incorporation of directional bonding (Pettifor et at. 1995) which can lead to highly composition-dependent interaction parameters. 6.6.2 The prediction of thermodynamic properties for compounds 6.6.2.1 The concept of counter-phases. When a stable compound penetrates from a binary into a ternary system, it may extend right across the system or exhibit only limited solubility for the third element. In the latter case, any characterisation also requires thermodynamic parameters to be available for the equivalent metastable compound in one of the other binaries. These are known as counter-phases. Figure 6.16 shows an isothermal section across the Fe-Mc—B system (Pan 1992) which involves such extensions for the binary borides. In the absence of any other guide- I CALPHAD—A Comprehensive Guide 169 Fe Mo-B (Mo) 0-2 0.4 0.6 0.8 MOLE FRACTION MO Mo (a) (b) Counlerphases FeB Coumerphases Pseado-iernary Compound COMPOSITION MoB Fe3B2 COMPOSITION Figure 6.16. Counter phases in the Fe-Mo-B system: (a) the section FeB-MoB and (b) the section Fe3B2-Mo3B2 (from Pan 1992 and Miodownik 1994). lines it is necessary to use empirical methods, which have been well documented in Kaufman and Bernstein (1970), and constitute a useful fitting procedure. Accurate results can be obtained if there are some prior phase-boundary data for the ternary system which can be used to validate the assumed thermodynamics of the counter-phase. However, problems arise when there is little or no experimental information and when the extension of the compound into the ternary is small. Further, when multiple sublattice modelling is used (see Chapter 5) it may be necessary to input thermodynamic properties for compounds which are compositionally far away from the area of interest in the CALPHAD calculation. It is therefore desirable to have a method of predicting the thermodynamic properties of as many counter-phases as possible. References are listed on pp. 173-178. 170 6.6.2.2 Structure maps. Attempts have been made to chart the occurrence of particular structures in relation to familiar parameters such as size, electronegativity and electron concentration (Darken and Gurry 1953, Villars et al. 1989). Some of these schemes have only limited applicability to particular areas of the periodic table, but more recently there has been greater success through structure maps which are based on the concept of a Mendeleev number (Pettifor 1985a). These have been extended to ternary compounds (Pettifor 1985b) and give considerable insight into the choice of structures which may be significant competitors in a given multicomponent situation. Calculations which give more detailed energies of various competing structures in particular regions of the map, such as for the Laves phases (Ohta and Pettifor 1989) are now emerging with increased frequency. 6.6.2.3 The Miedema model and other semi-empirical methods. The Miedema model was originally devised as a tool for merely predicting the sign of the heat of solution at the equiatomic composition. Therefore Eq. (6.36) does not contain any concentration-dependent terms (Miedema 1973, Miedema et al. 1973). However, the treatment was extended in subsequent publications, and modifications were made to include ordering contributions and asymmetric effects (Miedema et al. 1975, de Boer et al. 1988). Additional functions /(c^, csB) and g(cA, cb, VmA, K>i.b) were kept very simple and did not include additional parameters other than (Vm), which had already been used to determine (nws). AH(cA, cB) = f(csA, csB) g (cA, cB, VmA, VmB) -P(A^)2+Q(Anw//3) -H (6-37) * where f(cfA, Cg) = c^c^(l + 8(c^Cg)2) and c^, etc., are surface concentrations. These arise from the concept that in their 'macroscopic atom' model the enthalpy effects are generated at the common interface of dissimilar atomic cells. Such an expression contains only a limited degree of asymmetry and essentially use a very primitive ordering model that does not even consider nearest-neighbour interactions but assumes that #0id/#disord is virtually a constant. Since there is no provision for any crystallographic parameters it is impossible to make any predictions about the relative stability compounds with different crystal structures at the same stoichio-metry. A quantum-mechanical rationale of the Miedema approach has been published by Pettifor (1979a), outlining its strengths and weaknesses. Machlin (1974, 1977) developed a semi-empirical treatment which used a constant set of nearest-neighbour interactions and was one of the earliest semi-empirical attempts to obtain the relative enthalpies of formation between different crystal structures. This successfully predicted the correct ground states in a substantial number of cases, but the treatment was generally restricted to transition metal combinations and a limited number of crystal structures. References are listed on pp. 173-178. 6.6.2.4 Ab initio electron energy calculations. Some of the earliest electron energy calculations were made for transition metals where it was noted that trends in cohesive energy could be approximated to a very reasonable degree by considering only the (/-band electrons (Friedel 1969, Ducastelle and Cyrot-Lackman 1970). Various means of representing the density of states were then applied in an attempt to quantify this. Van der Rest et al. (1975) utilised the coherent potential approximation (CPA) method with off-diagonal disorder to compute the density of states curve. This method was employed by Gautier et al. (1975) and Ehrenreich and Schwartz (1976) for extensive calculations of transition metal alloys. Later work by Pettifor (1978, 1979b), Varma (1979) and Watson and Bennett (1981) extended the rf-band concept further and comparison with experimental enthalpies of formation for equi-atomic compounds were quite reasonable, although there is a clear tendency for the predictions to be too exothermic. They are also limited because they are insensitive to crystal structure and, subsequently, give no predictions for meta-stable compounds. Pseudo-potential calculations are also a method by which heats of formation can be predicted (Hafner 1989, 1992). These work better for sp-bonded compounds such as the alkali metal Laves phases (Hafner 1987) where nearly free electron theory is well suited, but they are not so applicable to rf-band calculations which are necessary for transition metal alloys. Electron energy calculations utilise methods which solve the Schrodinger equation and start with the specification of the lattice space group taking into account a wide variety of interatomic forces. It is therefore possible to make calculations for a large group of ordered structures and calculate the difference in enthalpy that arise from any specified change in crystallography at a given composition for any combination of elements (Pettifor 1985a, Finnis et al. 1988). Such calculations incorporate the effects of changes in band width, the centre of gravity of the bands, various forms of hybridisation and can also include directional bonding (Pettifor 1989, Phillips and Carlsson 1990, Pettifor et al. 1995). Although there are inherent approximations in any electron energy calculation, it is significant that this route now has a high success rate in predicting the correct ground state amongst a large range of competing structures (Gautier 1989, Sluiter and Turchi 1989, Nguyen-Manh et al. 1995). In order to shorten the calculation time these are often limited to those structures thought most likely to occur. This can potentially lead to the omission of other contenders but algorithms have recently been developed which can markedly extend the number of structures sampled and remove such arbitrary limitations (Lu et al. 1991). Electron energy calculations now offer a coherent explanation of trends observed both across and down the periodic table and the grouping and overlaps observed in structure maps. Of particular importance are the marked changes that occur on moving to elements of higher atomic number, which means that some of the earlier assumptions concerning similarities of behaviour for compounds of the 3d, 4d, and 5d elements (Kaufman and Bernstein 1970) have had to be revised. Quantum 172 JV. Saunders and A. P. Miodownik mechanical effects also lead to significantly greater sinusoidal variations in stability with atomic number and hence also to markedly greater asymmetry m the vanation of enthalpy of formation of similar structures in a given binary system. 6.7. SUMMARY Despite a sometimes turbulent history a series of standard and effective lattice stability values are now readily available for the majority of elements. Most of the remaining areas of controversy concern structures which seem to be elastically unstable or where the magnetic structure is uncertain. The relative stability of different compounds seems to be taking a similar route, but clearly involves a much wider range of structures. The availability of many different calculation routes has led to numerous comparisons being made between the various calculated results and whatever experimental information is available for the stable phases (Kaufman 1986, Bimie et al. 1988, Colinet et al 1988, Watson et al. 1988, de Boer et al 1988, Aldinger et al. 1995). Figure 6.17 shows that there is often good general agreement for a given class of structures so that it is now possible to make a better estimate for the enthalpies of unknown metastable compounds. The success of earlier empirical schemes in predicting the sign and magnitude of the enthalpy of formation can in part be attributed to making predictions for AB compounds. This can hide asymmetrical effects which are more apparent when other stoichiometrics are considered. The empirical and semi-empirical schemes do Z o S a a a, u. o w X -75 Sc Ti V Cr Mt» Fe Co Nl V Z, Nb Mo Tc Ru Rh Pd La Hf Ta W Rc Os Figure 6.17. Comparison of enthalpies of formation for AB TUanmm Alloys Mictions of Miedema (de Boer et al. 1988), W^redactions of Wateon and Bennett (1984) and C=predictions of Colinet et al. (1985). F.gure from Aldinger et al. (1995). Ir Pt CALPHAD—A Comprehensive ^^^^""^ 173 not have the capacity to handle a wide range of bonding characteristics without the introduction of many additional parameters with arbitrary numerical values. This much diminishes their predictive value as far as defining the relative stability of counter-phases is concerned. The more sophisticated electron energy calculations are now undoubtedly a better guide for the relative enthalpy of a much wider range of structures and should be increasingly used as input data in any CALPHAD optimisation. However, such calculations cannot predict the relative entropies of the competing structures and this is still an area where an empirical approach is required for the present. REFERENCES Achar, B. S. and Miodownik, A. P. (1974) Report on Phase Equilibria in Alloys, Report to Science Research Council (UK) under Contract B/RG/19175. Ägren, J. (1979) Met. Trans., 10A, 1847. Agren, J., Cheynet, B. Clavaguera-Mora, M. T., Hack, K., Hertz, J., Sommer, F. and Kattner, U. (1995) CALPHAD, 19,449. Aldinger, F., Fernandez Guillermet, A., lorich, V. S., Kaufman, L., Oates, W. A., Ohtani, H., Rand, M. and Schalin, M. (1955) CALPHAD, 19, 555. Anderson, J. O., Fernandez Guillermet, A. and Gustafson, P. (1987) CALPHAD, 11, 365. Ardell, A. J. (1963) Acta. Met., 11, 591. 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V. and Weiss, R. J. W^^J Klement, W. and Java» A. (1966) Progress m Sohd State Chemistry, 3,289. Kmetko, E. A. and Hill, H. H. (1976) J. Phys. F, 6, 1025. Kouvetakis, J. and Brewer, L. (1993) J. Phase Equilibria, 14, 563. 8.7.2 Thermodynamic Consequences of Multiple States 8.8. Changes in Phase Equilibria Directly Attributable to Gma« 8.9. Interaction with External Magnetic Fields References 247 248 253 256 Chapter 8 The Role of Magnetic Gibbs Energy 8.1. INTRODUCTION In paramagnetic materials there is no polarisation of electron spins and therefore it is unnecessary to consider a magnetic contribution to the Gibbs energy if this condition is taken to be the standard state. However in ferromagnetic, anti-ferromagnetic and ferri-magnetic materials, there is also competition between different spin arrangements and the enthalpy of certain transition metals, rare earths and their associated alloys and compounds are lowered by specific forms of spin polarisation. This includes the technologically important elements Fe, Ni and Co and their alloys and the effect can be of sufficient magnitude to have a major effect on phase transformation. There are many different forms of spin polarisation. The ferromagnetism exhibited by b.c.c. Fe is probably the best-known variant. Here the coupling between atoms favours parallel spin configurations between nearest neighbours and the critical ordering temperature is known as the Curie temperature (Tc). In other materials the exchange forces favour anti-parallel spins between nearest neighbours. If there is only one species of atoms this is known as anti-ferromagnetism and the critical ordering temperature is known as the Neel temperature (Tn). If the anti-parallel coupling involves more than one species of atoms with different values of spin, there will be a net spin in one particular direction. Such a material, e.g., Fe304, is considered to be ferri-magnetic and again behaves like a ferromagnetic with a Curie temperature. Many other subtle differences can occur, such as periodic changes in the crystallographic direction in which the spins are orientated, but these details are rarely considered in thermodynamic terms. In the treatments which follow, it is sufficient to note that the quantum number s = +V2 or -V2 defines the spin-up or spin-down direction for individual electrons, while the magnitude of any magnetic effects depends on the number of electrons that are being polarised. The unit magnetic moment associated with spin V2 is equal to a Bohr magneton (jis) and the number of magnetic electrons/atom is defined by the symbol /?, so that /? = 2s (Bozorth 1956). Unlike chemical ordering, which can only occur in alloys, magnetic ordering can occur in unary systems and the magnetic Gibbs energy turns out to be sufficiently large to cause fundamental changes in structure. For example, the high-temperature 229 230 N. Saunders and A. P. Miodownik CALPHAD—A Comprehensive Guide 231 form of solid Fe is 6 which has a b.c.c. structure and transforms to 7-Fe with the more close-packed f.c.c. structure at 1394°C. Both of these phases are paramagnetic at these elevated temperatures, but the onset of ferromagnetism in the b.c.c, phase causes this phase to reappear at 912°C as a-Fe. This makes Fe a unique element in that the high-temperature allotrope reappears at low temperatures, and it should be noted that the thermodynamic effect due to short-range magnetic order is already sufficient to cause the re-appearance of the b.c.c. form; a-Fe only becomes ferromagnetic below its Curie temperature at 770°C. At 0 K the magnetic enthalpy («9000 J mol-1) is an order of magnitude larger than was involved in the 8 - 7 high temperature transformation («900 Jmol-1). The change in Gibbs energy with temperature of Fe, with respect to the paramagnetic form 5-Fe, is shown in Fig. 8.1(a). This can be correlated directly with the value of the magnetic specific heat (Fig. 8.1(b)) (Nishizawa et al. 1979), which in turn is related to a corresponding change in the saturation magnetisation, /?, (Fig. 8.1(c)) (Nishizawa 1978). The critical magnetic ordering temperature, the Curie temperature (Tc), is defined either by the peak in C^6 or by the maximum rate of change in the variation of /? with T. The situation is made more complex on alloying as both Tc and /? vary with composition. It has already been mentioned in the previous chapter that the Ising model, which underlies the formalism used for most chemical ordering treatments, was originally used to describe magnetic transitions. It is, however, technically only valid if the various magnetic states correspond to multiples of the unit magnetic spin, s. Early attempts to describe the associated thermodynamic functions within formal ordering theory were by Saito et al. (1959), Arita (1978) and Moran-Lopez and Falicov (1979). More recent examples of the combination of chemical and magnetic ordering, as applied to b.c.c. Fe-base alloys, have been given by Inden and Pitsch (1991) and Kosakai and Miyazaki (1994). The BWG treatment appears to be a reasonable assumption in such systems because the saturation magnetisation of Fe (2.2 hb) is close to the integral value /3 — 2 which is equivalent to s = 1. However, the concentration dependence of the saturation magnetisation, and the important role played by magnetic short-range order, makes it difficult to use such a treatment in the general case and it has been necessary to devise alternative methodologies to describe the magnetic Gibbs energy of alloy systems. 8.1. J Polynomial representation of magnetic Gibbs energy While acknowledging that the 'Gibbs energy anomaly' was associated with magnetism, early attempts to characterise the behaviour of Fe, did not derive C"*8 explicitly from ferromagnetic parameters such as /?. Instead, G""* was derived either by the direct graphical integration of C^* or simply incorporated into the overall Gibbs energy difference between the 7- and a-phases in Fe by means of a global polynomial expression such as used by Kaufman and Nesor (1973) References are listed on pp. 256-258. t = T/Tc 1.0 1 ^tn! I l(c? w 1 \ P 1 1 t 400 800 1200 Temp / K 1600 l^J'1' S^Wm betWeen (a) ^S"*"* Gibbs energy, (b) magnetic specific heat (from Ntshizawa 1992) and (c) change of satmnon magnetisTtion for pure Fe (from Nishizawa 1978). 300 < T < 1100Ä" (P^a = 6109 - 3.462T - 0.747217 x 10~2r2 + 0.5125 x HT5T3. (8.1) Different temperature regions then required different sets of coefficients and, in specific systems, the coefficients had to be modified when used to describe the low-temperature martensitic transformation (Kaufman and Cohen 1956). It soon became evident that it was preferable to work with functions that properly reflected the physical origin of the extra Gibbs energy. This was also important when moving from pure Fe to increasingly complex steels, as specific heat measurements are not often as generally available as measurements of or Tc. 8.1.2 Consideration of the best reference state The first step is to consider the methods which can be used to represent the change of magnetisation with temperature, and how this can be translated into the corresponding change of G"1*8. By analogy with configurational ordering, it would be reasonable to choose the ground state at 0 K as the fundamental reference point. This view was taken by Zener (1955) who may be considered the first author to try and obtain an explicit description of Graag. However, many important phase changes are associated with the high-temperature behaviour, especially the region where short-range magnetic order is present. Starting at 0 K therefore presents certain problems. Firstly, an accurate representation of the temperature dependence of the magnetic parameters is required, and this can be highly non-linear. Secondly, anomalies can arise if there is a rapid variation of the magnetic ordering temperature with composition (Inden 1991). This is particularly true if Tc reaches zero in the middle of a system. Finally, it is necessary to devise a system which can take care of mixtures of magnetic and non-magnetic components. In contrast to most treatments of configurational ordering, the high-temperature paramagnetic state has therefore been adopted as the best reference state. Objections to using this state on the grounds that it cannot be retained below Tc are unwarranted, as the situation can be considered analogous to using the liquid phase as a reference state at high temperatures. 8.1.3 Magnitude of the short-range magnetic order component The magnetic specific heat shows a marked degree of short-range magnetic order above Tc which decreases asymptotically to zero when T»TC. Figure 8.2 illustrates how the equivalent values for Ni are obtained by careful subtraction of other components from the total specific heat (Hofmann et al. 1956). The importance of standardising this procedure has been emphasised by de Fontaine et al. (1995). The fraction () of the total magnetic enthalpy retained above Tc is clearly always an important quantity and, by analysing experimental results, Inden (1976) obtained a References are listed on pp. 256-258. Experimental Cp (NSel) Experimental Cp (Sykes & Wilkinson) Subtracted magnetic Cp From Neel From Sykes &. Wilkinson Extrapolated 100 200 300 400 500 600 700 800 900 1000 Temperature °K 1200 1400 Figure 8.2. Extraction of magnetic specific heat from the total specific heat for pure Ni (from Hoffman et al. 1956). value of 4> - 0.27 for both f.c.c. Ni and Co while the larger value of = 0.43 was found for b.c.c. Fe. These values were subsequently,, assumed equally valid for all other f.c.c. and b.c.c. phases (Hillert and Jarl 1978, Inden 1981a). While this is a reasonable assumption for steels, it may not hold for atoms with larger magnetic moments (de Fontaine et al. 1995). This has been confirmed by a CVM treatment applied to Gd (Schon and Inden 1996). In order to incorporate this result into a semi-empirical treatment suitable for multi-component systems it may be worthwhile to revert to an earlier suggestion (Paskin 1957) that is a function of s and the nearest-neighbour co-ordination number, (z), *-£ks- (8-2) 8.2. DERIVATION OF THE MAGNETIC ENTROPY While it is important to partition the long-range and short-range magnetic components correctly, the maximum entropy contribution due to magnetism, 5™*s, is an equally crucial factor. 8.2.1 Theoretical value for the maximum magnetic entropy Despite the intrinsic complexity of magnetic phenomena, there is general agreement 232 300 < T < lluuA Airlines (atzKs), OS 774 Belgrade, = DJLUy - CS.40üi — Xt.lZCZtt X 1U "T' + 0.5125 x 10-5T3. Vienna, 320 (8.1) Different temperature regions then required different sets of coefficients and, in specific systems, the coefficients had to be modified when used to describe the low-temperature martensitic transformation (Kaufman and Cohen 1956). It soon became evident that it was preferable to work with functions that properly reflected the physical origin of the extra Gibbs energy. This was also important when moving from pure Fe to increasingly complex steels, as specific heat measurements are not often as generally available as measurements of (3 or Tc. 8.1.2 Consideration of the best reference state The first step is to consider the methods which can be used to represent the change of magnetisation with temperature, and how this can be translated into the corresponding change of G™88. By analogy with configurational ordering, it would be reasonable to choose the ground state at 0 K as the fundamental reference point. This view was taken by Zener (1955) who may be considered the first author to try and obtain an explicit description of Gmag. However, many important phase changes are associated with the high-temperature behaviour, especially the region where short-range magnetic order is present. Starting at 0 K therefore presents certain problems. Firstly, an accurate representation of the temperature dependence of the magnetic parameters is required, and this can be highly non-linear. Secondly, anomalies can arise if there is a rapid variation of the magnetic ordering temperature with composition (Inden 1991). This is particularly true if Tc reaches zero in the middle of a system. Finally, it is necessary to devise a system which can take care of mixtures of magnetic and non-magnetic components. In contrast to most treatments of configurational ordering, the high-temperature paramagnetic state has therefore been adopted as the best reference state. Objections to using this state on the grounds that it cannot be retained below Tc are unwarranted, as the situation can be considered analogous to using the liquid phase as a reference state at high temperatures. 8.1.3 Magnitude of the short-range magnetic order component The magnetic specific heat shows a marked degree of short-range magnetic order above Tc which decreases asymptotically to zero when T »TC. Figure 8.2 illustrates how the equivalent values for Ni are obtained by careful subtraction of other components from the total specific heat (Hofinann et al. 1956). The importance of standardising this procedure has been emphasised by de Fontaine et al. (1995). The fraction (0) of the total magnetic enthalpy retained above Tc is clearly always an important quantity and, by analysing experimental results, Inden (1976) obtained a 0 100 200 300 400 500 600 700 800 900 1000 Temperature °K 1200 1400 Figure 8.2. Extraction of magnetic specific heat from the total specific heat for pure Ni (from Hoffman et al. 1956). value of 4> = 0.27 for both f.c.c. Ni and Co while the larger value of $ = 0.43 was found for b.c.c. Fe. These values were subsequently assumed equally valid for all other f.c.c. and b.c.c. phases (Hillen and Jarl 1978, Inden 1981a). While this is a reasonable assumption for steels, it may not hold for atoms with larger magnetic moments (de Fontaine et al. 1995). This has been confirmed by a CVM treatment applied to Gd (Schon and Inden 1996). In order to incorporate this result into a semi-empirical treatment suitable for multi-component systems it may be worthwhile to revert to an earlier suggestion (Paskin 1957) that ^ is a function of s and the nearest-neighbour co-ordination number, (z), *-£ry <8'2) 8.2. DERIVATION OF THE MAGNETIC ENTROPY While it is important to partition the long-range and short-range magnetic components correctly, the maximum entropy contribution due to magnetism, S££g, is an equally crucial factor. 8.2,1 Theoretical value for the maximum magnetic entropy Despite the intrinsic complexity of magnetic phenomena, there is general agreement References are listed on pp. 256-258. that the maximum entropy generated by the de-coupling of spins between atoms can be expressed by: S%* = Rloge(2s + l). (8-3) Technically, the values of s are theoretically restricted to integral multiples of s and one should consider the sum of each contributing species, but these values are not always available. 8.2.2 Empirical value for the maximum magnetic entropy The saturation magnetisation 0, in (iB per atom, of many materials often corresponds to non-integral values of s. This is due to contributions other than s being involved, for example polarised conduction electrons. It is, therefore, general practice to substitute the experimental value of the saturation magnetisation at 0 K, 0o, for 2s in Eq. (8.3) which leads to (Miodownik 1977) S^ = Rloge(0o + l). (8-4) 8.2.3 Explicit variation in entropy with magnetic spin number and temperature The magnetic entropy may be rigorously specified if the BWG mean-field approximation is combined with specific values of the magnetic spin. The expression for the magnetic entropy corresponding to s = \, as a function of the degree of order % is identical to that already given in Eq. (7.2) of the previous chapter on configurational ordering ^BWG -NkB (8.5) However, the permutation of permissible spins rapidly becomes more complicated with higher values of s. The equivalent expression for s = 1, as derived by Semenovskaya (1974) and used by Inden (1975, 1981), is given by 5^=1 = - NkB [loge(8 - 6t? + 2^4 -3rf) - (1 - rj) log,(2(l - r,)) - (1 + 77) loge(7/+ V4-3772)]. (8.6) Expressions corresponding to s = 3.2 and s = 2 are listed in Inden (1981). S3. DERIVATION OF MAGNETIC ENTHALPY, H™*0 8.3.1 Classical derivation The simplest case is to start with a situation where all the atoms have the same spin. References are listed on pp. 256-258. The general expression for the BWG (mean field) approximation then gives (8.7) This allows for a summation for magnetic interactions in successive (&-th) neighbouring shells but, unless J values can be derived from first-principles calculations, there is generally insufficient experimental data to allow anything other than a single nearest-neighbour interaction parameter to be used. This, then, must be taken to incorporate any other longer-range effects, and application to real alloy systems is also technically restricted by having to use multiples of s = 1/2. Much therefore depends on whether this realistically simulates experimental values of 0, which fortunately seem to be the case in Fe-rich, Co-rich and Ni-rich alloys. However, this is less secure at higher solute concentrations, especially if there are transitions between different forms of magnetism in the system (see Fig. 8.4). An equation which includes the effect of adding a magnetic term to the enthalpy of an ordered system has already been given in Section 7.3.2.3 of Chapter 7, but no details were given of how to determine the value of the magnetic interaction energy For a binary alloy A^BX this involves the introduction of three magnetic interaction parameters JAl\, TgB and J^B which describe the strength of the coupling between nearest-neighbour atoms whose (magnetic) electrons have parallel spins as compared to anti-parallel spins. The magnetic interchange energy MAB is then obtained by combining these parameters to yield M(1) - J1 mAB — JAA (1) , r(D JBB 2J AB' (8.8) This is entirely analogous to the treatment of the chemical interaction energies V£, treated in Chapter 7. Negative values of Jtj correspond to ferromagnetism while positive values correspond to anti-ferromagnetism. Positive values of Mitj imply that the strength of the ferro-magnetic coupling between unlike atoms is stronger than between like atoms. Most treatments do not consider magnetic interactions beyond the nearest-neighbour coordination shell. The most convenient way of determining the interchange energies is from experimental values of the critical ordering temperature, which is designated the Curie temperature for ferromagnetic ordering The Curie temperature for alloys with two magnetic components having the same spin (1/2 or 1) can be written as follows: f SKa I a(l — x) + ßx (1-x) t(1) t(1) ^ + x^-x(l k k ■x) (8.9) 236 Saunders and A. P. MiodawniK In this equation, \i is the magnetic analogue to the x factor used in the chemical ordering case to compensate for the higher critical temperatures always generated by the BWG formalism (see Chapter 7). Comparing the results from the BWG treatment with those of other more sophisticated treatments leads to a value of H = 0.8 (Inden 1975) and the long-range ordering parameter 77 takes values between 0 and 1 as in the case of chemical ordering. In the magnetic case r\ equals the ratio of the mean spin value per atom at temperature T to the maximum value of 5 (at 0 K). Ks is a parameter which varies with the magnitude of the magnetic spin s such that K, = 1 for 3 = 1/2 and Ks = 2/3 for s = 1. The coefficients a and 0 can be set either to 0 or 1 to cover various combinations of magnetic and non-magnetic components. 8.3.2 Empirical derivation Given that S£g can be described using Eq. (8.4), it may be asked whether a corresponding value for can be obtained by combining S££% with Tc. If magnetic ordering was a first-order transformation with a critical temperature, T*, it would then follow that (Fig. 8.3). tf^ = rsz*. (8.10) "max max A-rino ie a second-order transformation, this has to be However, since magnetic ordering is a second uiuc -1! h . Kaufman Clougherty and Weiss [20] Calculated values _L 1000° 1043° Temperature, "K o -> r, , * «f ™ *fWtive first-order transformation temperature (T*) to ^*J£^t££~» TO (*» Miodown* 1977). modified to take into account the proportion of short- and long-range order on either side of Tc (see Section 8.1.3). Miodownik (1977) assumed that T* « 0.9TC for Fe, Co, Ni and their alloys. It then follows that #^ = 0.9Tcloge(/?o + l). Weiss and Tauer (1958) made the alternative assumption that (8.11) (8.12) Such equations have the advantage of simplicity but comparison with Eq. (8.2) implies that (p is totally independent of s and z. Whichever expression is used, the next step is to incorporate the effect of temperature on the alignment of the magnetic spins. In view of the other simplifications which have already been adopted, this can be achieved (Miodownik 1977, 1978a) by approximating the Brillouin-Langevin formalism for the number of aligned spins {0T) remaining at T < 0.9TC through the following expression: ßT=ß0[l-T6] (8.13) where r defines the ratio (T/Tc). Assuming that the magnetic enthalpy scales as (ff/Po) this can then be combined with an arbitrary power function to account for short-range magnetic order for T > 0.9TC) ßT = /?o(l/2)|2+10{T- 1)1 (8.14) Equation (8.14) leads to a value for = 0.25 at Tc, which is of comparable magnitude to the values assumed by Inden (1981a), although it clearly does not include any dependence on the co-ordination number z. 8.4. DERIVATION OF MAGNETIC GIBBS ENERGY Even if magnetic interaction energies are available to define a magnetic enthalpy, the development of a viable Gibbs energy expression is difficult. As already noted in the preceding section, the expressions for the magnetic entropy in a BWG treatment become increasingly complicated when s > 1 /2 and, indeed, may become inexact if the distribution of individual spins is not known. In practice their application also requires a series of nested Gibbs energy minimisation calculations which become costly in computing time. Also, of course, the BWG approximation does not take into account any short-range order except through empirical corrections. Despite these drawbacks remarkably good results have been obtained in the equi-atomic region of the Fe-Co system (Inden 1977) which are virtually indistinguishable from those obtained by CVM calculations (Colinet 1993). However, this References are listed on pp. 256-258. 4* JO region deals only with A2/B2 equilibria where the BWG method is known to work well. It is otherwise inadequate to deal with f.c.c.-b.c.c. equilibria, notably in pure Fe or Fe-rich alloys, because of the very small differences in the free energies of the fee. and b.c.c. allotropes. In such cases it is vitally important to include short-range magnetic order and find alternative means of denning the magnetic Gibbs energy. 8.4.1 General algorithms for the magnetic Gibbs energy With expressibns for the magnetic enthalpy and entropy it is now possible to build up an algorithm for G1"*5 at any temperature. The maximum magnetic entropy is taken as S™S = SS-4<=T (8-15) In an analogous way the maximum magnetic enthalpy is « = ^-^r=o- (8-16) The magnetic entropy and enthalpy at a given temperature, 5™^ and H™ag, respectively, are S™s,tf^ = /(T,TC,/?) (8.17) while the magnetic Gibbs energy at a given temperature is given by G™*6 = G£rro - G^a. (8.18) 8.4.2 Magnetic Gibbs energy as a direct function of /3 and Tc Combining the expressions for magnetic entropy and enthalpy and assuming that the magnetic enthalpy scales as /3t//3q G£ag = -0.9RTcloM + l)(/3r/A>) - «Tloge(A - f + 1). (8-19) Here the first term represents H™* and the second term the effect of S™8. Eq. (8.19) represents one of the many empirical ways in which Eq. (8.18) can be made to work in practice (Miodownik 1977). However, this expression does not lend itself easily to a derivation of the associated value of Cv nor to an explicit formulation of SAG^/ST. Therefore, most alternative expressions have been based on approximate analytical expressions for the magnetic specific heat. However, all such expressions also incorporate the functions log^/? + 1) and r and are a far cry from the original methods of graphical integration. 8.4.3 Magnetic Gibbs energy as a function of C™9 for ferromagnetic systems 8.4.3.1 The model of Inden. This approach was pioneered by Inden (1976) who developed the following empirical equations: i- f°r r < 1 Cpmag = K^Rlo&il + r3)/loge(l - r3) (8.20a) forr>l Gpmag = X5roiiloge(H-r5)/loge(l~^) (8.20b) where Klro and KSTO are empirically derived coefficients. At first sight these equations appear to differ substantially from Eq. (8.19), but the values of Klr° and are constrained by the need to correctly reproduce both the total entropy and take into account experimental values of 4>. Since the total entropy is an explicit function of /?, the net result is that this treatment also has a mixed dependence on the two key magnetic parameters 0 and Tc. Further, a relationship between A"sro and Klro can be obtained by considering HmAg: Ksr0 = 518/675(Klro + 0.6K3ro) = loge(/J + 1) (8.21) Klr0 = 474/497[(l - fi/W™- (8.22) A dependence on crystallographic parameters is introduced through making a function of z, as already indicated in Section 8.1.3. Introducing fixed values of appropriate to each crystal structure is a useful simplification as it is then possible to uniquely define the values of Klro and Ksro (de Fontaine et al 1995). 8.4.3.2 Model ofHillert and Jarl. In his original treatment, Inden (1976) used a complicated but closed expression for the enthalpy, but had to use a series expansion in order to calculate the entropy. Hillert and Jarl (1978) therefore decided to convert the Cp expression directly through a series expansion which substantially simplifies the overall calculation and leads to a maximum error of only 1-2 J/mol at the Curie temperature of Fe. The equivalent equations to those used by Inden (1976) are given by for r < 1 Cpmag = 2KlroR{Tm +^T3m +|T5m) (8.23a) for r > 1 G™ag = 2ifsro«(r-n +1?-3" + ^T~5n). (8.23b) 8.4.3.3 Alternative Cp models. Chuang et al (1985) have developed alternative exponential functions to describe the Cp curves and applied this to various Fe-base alloys (Chuang et al. 1986) for t < 1 Cp = A"'i?rexp(-p(l - t)) (8.24a) for r > 1 Cp = K"Rrexp(-q(l - r)). (8.24b) These equations have the advantage of greater mathematical simplicity, but it is still necessary to evaluate the constants K'\ K", p and q. References are listed on pp. 256-258. 240 N. Saunders and A. P. Miodownik CALPHAD—A Comprehensive Guide 241 8.4.3.4 Comparison of models for the ferromagnetic Gibbs energy. It is difficult to compare the various treatments described in Sections 8.4.3.1 to 8.4.3.3, because simultaneous changes were made in both the models and input parameters during their formulation. This is a perennial problem if different weightings are attached to magnetic and thermodynamic data (de Fontaine el al. 1995). The suggestion by Inden (1976) that values for m and n in Eqs (18.18a) and (18.18b) should be taken as 3 and 5 respectively has been preserved by Hillert and Jarl (1978). This is still the core assumption as far as 3d elements are concerned in current models. However, it may be necessary to relax this assumption in systems with larger magnetic spin numbers (de Fontaine et al. 1995) 8.4.4 Anti-ferromagnetic and ferri-magnetic systems Providing there is no change in value of 0a with temperature, Eq. (8.4) can also be used to determine the maximum magnetic entropy in anti-ferromagnetic and ferri-magnetic materials (Miodownik 1978a, Smith 1967, Hofmann et al. 1956). A comparison of predicted and experimental magnetic entropies and energies (Table 8.1) indicates that this is a reasonable assumption in most cases. If serious discrepancies remain between theoretical expectations and experimental results, it may be necessary to consider the existence of multiple magnetic states (see Section 8.7). 8.5. THE EFFECT OF ALLOYING ELEMENTS i ■''■111 The treatment of compounds and other end-members of fixed composition does not differ from that of the elements, but the next vital step is to consider the representation of these parameters for solid solutions. j Table 8.1. A comparison of predicted and experimental magnetic entropies and enthalpies. Exp. SSS Calc. (Eq. (8.4)) Exp. ff^f Calc. (Eq. (8.11)) Element (J/moWC) (J/mo!/K) (J/mol) (J/mol) Ni 3.39 3.47 1757 1987 Fe 8.99 9.20 8075 8556 Gd 17.74 17.41 3372 4602 Dy 23.09 23.05 — — Ho 23.56 23.56 — — Mn02 10.87 11.51 816 900 NiF2 9.46 9.12 590 657 Cr203 22.72 23.22 5314 6255 8.5.1 Effect of changes in Te and 0 with composition The earliest derivation of G™ag was made by Zener (1955) who postulated that the effect of an alloying element was proportional to its effect on Tc. With the assumption that $Tc/$% is a constant, it is then possible to write: G%&(T) = G™*{T) - x (g) (8.25) where T = Tc — x{ST/Sx). The Zener method therefore depends totally on the assumption of a linear variation of 6Tc/6x and any associated changes in 0 are reflected in the change of Tc. It can, nevertheless, give reasonable results for dilute solutions and was extensively used by Hillert et al. (1967) in the period before the Inden formulation was more generally adopted in Europe. It, however, remained popular in Japan for the characterisation of Fe alloys, which was its original field of application (Nishizawa et al. 1979, Hasebe et al. 1985). Zener's method does not explicitly take into account the effect of changes in 0 with composition but implicitly assumes that any changes in 0 are reflected in the changes in Tc. Dougherty and Kaufman (1963) considered how changes in 0 with composition might affect the terminal magnetic entropy of f.c.c. Ni-Zn alloys, but this did not lead to a generalised approach which was applicable across the whole system. Miodownik (1977) showed that values of Tc and 0 for alloys could be introduced into the same algorithm developed for the elements, but this was superseded by the advent of the Hillert-Jarl formalism. With the need to handle multi-component calculations it was then a natural step to develop (separate) Redlich-Kister polynomials for the variation of 0 and Tc with composition (Chin et al. 1987). 8.5.2 Systems whose end-members exhibit different forms of magnetism 8.5.2.1 Ferromagnetic to anti-ferromagnetic transition. When end-members exhibit different forms of magnetism there are often significant deviations from linearity in both Tc and 0 and these parameters may become zero in the middle of the system. One example is associated with a change from ferromagnetic to anti-ferromagnetic behaviour which occurs in a number of industrially important systems, such as Fe-Ni (Fig. 8.4(a)) and the related ternary system Fe-Ni-Mn (Fig. 8.4(b)). b.c.c. Fe-Cr, rare-earth and oxide systems can also exhibit similar complexity. In such situations special measures need to be taken to allow a Redlich-Kister-type polynomial to be used to describe Tc and 0. Since ferromagnetism and anti-ferromagnetism are associated with values for Jij of opposite sign, Weiss and Tauer (1956) suggested that a continuous change between these two forms of magnetic ordering could best be described by attaching a negative sign to Neel temperatures. These authors also suggested a semi-empirical References are listed on pp. 256-258. 242 N. Saunders and A. P. Miodownik (a) CALPHAD—A Comprehensive Guide 243 2.0 ■ Asa no *, • Crangle and Hailam P1(X2<0.385) £\ 1000 800 600 400 200 s H (b) Electrons per atom E/A 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 700 600 a j> 500 3 1 400 a, E £ 300 200 100 0 ii 1 r i i i Fe-Ni-_/ 7/ ft /1 ' T= / / Fe-Mn N. Illllllllllllllllllll AF iii! ) ^F+AF^ 1 10 20 30 40 50 60 70 80 90 100 X Figure 8.4. Variation of magnetic parameters T and /? with composition in (a) Fe-Ni (from Chuang et al. 1986), (b) Fe-Ni-Mn alloys (from Ettwig and Pepperhoff 1974). relation between Tcrit and /3 which is useful if one or other of these parameters is not experimentally available. TCTit = 113.5(2? -Z[) In (/? + 1). (8.26) Here the two parameters, Z\ and Zl, refer to the relative number of spins of opposite sign, which allows this equation to be used for both ferromagnetic and anti-ferromagnetic materials. One implication of Eq. (8.26) is that, for a given value of p sad type of magnetism, the critical temperature is expected to be higher for f.c.c. materials than for b.c.c. materials, which accounts for the observation that Tp° > TcFe although /3°° < j3Fe. There are, however, problems when applying this simple empirical equation to anti-ferromagnetic BCC materials such as Cr and to structures with lower symmetry. A more important conclusion of Eq. (8.26) is that, for a given value of 0 in an f.c.c. lattice, the value of Tn derived for an anti-ferromagnetic alloy will be 1/3 of the equivalent value of Tc in the ferromagnetic case. The utility of this approach has been verified by Chin et al. (1987) who, in order to optimise parameters for their Redlich-Kister polynomials for the critical temperatures of various binary alloy systems, converted positive values of Tn to negative values of Tc and were able to reproduce the critical temperature variation arising from the ferromagnetic/anti-ferromagnetic transition. These authors also used the same convention to differentiate ferromagnetic and anti-ferromagnetic values of /3 when adapting the Hillert-Jarl formalism into Thermo-Calc, although this must be considered purely as a mathematical convenience and has no theoretical justification. Another simplification that has been made with respect to the implementation of magnetic algorithms into software packages, such as Thermo-Calc, relates to the composition at which Tc and (3 reach zero. One might expect that this would happen at one particular composition but Figs. 8.4(a) and (b) show that the situation may be much more complicated. In order to be consistent with a simplified magnetic model Chin et al. (1987) chose a set of parameters which effectively smoothed out the differences in the two critical compositions for binary alloys. This however precludes a proper prediction of regions of mixed magnetism, which is important in Invar alloys (Miodownik 1978b) as well as making it impossible to reproduce effects such as shown in Figs. 8.4(a) and (b). 8.5.2.2 Ferromagnetic-paramagnetic transition. In other cases, a zero Tc may be associated with the transition between ferromagnetism and paramagnetism. Even the use of higher-order Redlich-Kister polynomials cannot properly reproduce such a sudden change in slope and some smoothing of the measured variation of Tc and j3 has to be accepted in the vicinity of 0 K. As both enthalpy and entropy contributions scale approximately with Tc, at low temperatures, this will only lead to small errors on the overall Gibbs energy at such compositions. However, under certain circumstances spurious miscibility gaps have been predicted at high temperatures References are listed on pp. 256-258. 244 JV. Saunders and A. P. Miodownik CALPHAD—A Comprehensive Guide 245 in systems that exhibit abrupt changes in Tc with composition (Inden 1981). Care has, therefore, to be taken in deriving the input parameters to the Hillert-Jarl formalism in such cases. The fact remains that such miscibility gaps have never been observed in practice and it may, therefore, be that as Tc approaches zero some other effects, such as itinerant ferromagnetism (Wohlfarth 1974), may occur which essentially smooth out the variation of magnetic entropy. In this instance, the Redlich-Kister formalism may actually be a better approximation to reality than would be indicated by trying to reproduce a Tc or Tn obtained by linear extrapolation of experimental results to 0 K. 8.6. THE ESTIMATION OF MAGNETIC PARAMETERS 8.6.1 Magnetic versus thermochemical approaches to evaluating the magnetic Gibbs energy In principle, the value of G™g should be the same whether it is derived from knowledge of ß or from the magnetic specific heat. Since the latter is derived by subtraction of other major terms from the total specific heat, such an agreement is a useful confirmation that the deconvolution process has been properly conducted. However, this is not always achieved. In some studies, e.g., for Cr (Andersson 1985), Co (Femandez-Guillermet 1987) and Ni (Dinsdale 1991), the listed ß values are 'effective thermochemical moments' which have been derived on the basis of a global thermochemical assessment rather than from magnetic measurements. Small differences will always arise when considering ways of averaging data and truncating series, but one should be cautious of a tendency for ß to become an adjustable parameter. Such discrepancies will hopefully be reduced in the future by the incorporation of better magnetic models (de Fontaine et al. 1995). 8.6.2 Values of the saturation magnetisation, 8 It should be emphasised that it is the rule rather than the exception for 8 to change markedly with crystal structure (Table 8.2). It is therefore unwise to assume that various metastable allotropes can be given the same value of 8 as for the stable structure. In some cases values of 3 can be extrapolated from stable or metastable alloys with the requisite crystal structure, but in others this is not possible. A significant development is that it is now possible to include spin polarisation in electron energy calculations (Moruzzi and Marcus 1988, 1990a, b, Asada and Terakura 1995). This allows a calculation of the equilibrium value of 3o to be made in any desired crystal structure. More importantly, such values are in good accord with known values for equilibrium phases (Table 8.2). It has also been shown that magnetic orbital contributions play a relatively minor role (Eriksson et al. 1990), so calculated values of 3 for metastable phases should be reasonably reliable. References are listed on pp. 256-258. Element Source Ferromagnetic moment (pB) Anti-ferromagnetic moment b.c.c f.c.c. (pB) c.p.h. b.c.c. f.c.c. c.p.h. 0 0.4 0.82 0 0 0.008 0.82 0 0 0/0.5 0/3.0 0 0/0.6 0/0.7 0.4 f [7] 0.0+ 0.09 0.62 0 2.0 2.76 2.2S 0.52 2.4/0 0 0) 2.3 0.70 0 0 0.70 0 2.56 1.55 1.21 0 J.75 0.5/0 0.40 0.70 0.57 Cr Chin et al. 1987* Cr Dinsdale 1991* Cr Asada 1993 (FP> 0 Cr Moruzzi 1990 (FP) 0 Cr Weiss 1979 Cr Exp (/J) Cr Exp \CP) Mn Chin et al. 1987* Mn Dinsdale 1991* Mn Asada 1993 (FP) 1.0 Mn Moruzzi 1990 (FP) 0.9 Mn Weiss 1979 Mn Exp (j3) (1) Fe Chin et al. 1987* 2.22 Fe Dinsdale 1991* 2.22 Fe Asada 1993 (FP) 2.32 Fe Moruzzi 1990 (FP) 2.2 Fe Weiss 1979 Fe Exp (0) 2.22 Fe Chuang 1985 (Cy) 2.05 Fe Lytton 1964 (Cp) 1.03 Co Chin et al. 1987* 1.80 Co Dinsdale 1991* 1.35 Co Asada 1993 (FP) 1.80 Co Moruzzi 1990 (FP) 1.70 Co Miodownik 1978 1.70 Co Exp ffl) 2.00 Co Chuang 1985 (Cp) Co Lytton 1964 (C„) Ni Chin et al. 1987* 0.85 Ni Dinsdale 1991* 0.85 Ni Asada 1993 (FP) 0.50 Ni Moruzzi 1990 (FP) 0.30 Ni Weiss 1956 Ni Exp (0) Ni Chuang 1985 (Cp) Ni Lytton 1964 (Cv) Ni Meschter 1981 (Cp) 0.53 0/4 4.5 2.56 2.7 2.6 1.70 1.35 1.70 1.80 1.80 1.80 0.62 0.52 0.66 0.85 0.62 0.25 0.58 1.70 1.35 1.61 1.70 0.89 1.21 0.52 0.60 1.0 0.1 1.2 0.25 In addition to offering a comparison between theoretical and experimental values Table 8.2 also indicates that there may be a significant difference between the 8 values that have been independently obtained by the deconvolution of C measurements and from magnetic measurements. There is also a significant spread between the 8 values obtained by using different methods to extract the magnetic 246 N. Saunders and A. P. MiodowniK specific heat (Lytton 1964, Meschter et al. 1981, Chuang et al 1985). Some of these differences can be rationalised by the occurrence of mixed magnetic states resulting from increasing temperatures, as most of the data in Table 8.2 essentially refers to 0 K. This is almost certainly the case for Cr and Mn (Weiss 1972, 1979, Moruzzi and Marcus 1990a, b). In the case of Co, the c.p.h. form, e>Co, is a close competitor to the f.c.c. form, a-Co, near the Curie temperature. High stacking-fault densities may therefore also lead to unusual effects (Miodownik 1977). In the case of a-Mn, there is documented evidence for a considerable decrease in ft on passing through Tn which could be associated with a transition from an anti-ferromagnetic to a 'non-magnetic state' (Gazzara et al. 1964, Weiss 1979). Nevertheless, a single temperature-independent value of /? seems a useful starting point in the majority of cases, especially when no other data is available to predict values for metastable elements and solutions. It is noted that j3 is very sensitive to atomic volume (Moruzzi and Marcus 1990a,b) which is certainly one of the reasons for the non-linearity observed for the variation of /? even in simple solid solutions. states, 7i and 72, and 51 and 52 are the corresponding degeneracies of the two states. Clearly if AE (the difference in energy between the two states) is large and/or the temperature is low, there is effectively only one state and one magnetic moment. However, as AE becomes smaller there can be changes in the effective magnetic moment, especially in the case of 7-Fe. Here the two states correspond to a ferromagnetic moment of « 2.8 \iB and an anti-ferromagnetic moment of « 0.5 (jlB, which leads to the effective moment being given by 0F* = [2.8/(1 + a) + 0.5a/(l + a)]. (8.28) This contrasts with the assumption made in virtually all other magnetic models that the value of 0 is independent of temperature. Several variants of the Shottky model have been developed by Miodownik (1977, 1978a) to take into account the situation where one of the two states subsequently undergoes magnetic ordering (Fig. 8.5). In such cases it may also be necessary to consider a temperature-dependent AE (Miodownik and Hillert 1980). 8.7. MULTIPLE MAGNETIC STATES It is not generally appreciated that there are many competing forms of magnetic ordering. When one particular magnetic ground state is substantially more stable than other alternatives, the conventional disordering of magnetic spins, as described in the previous sections, is the only scenario which needs to be considered. However, additional excitations at high temperatures can arise if another magnetic configuration with a comparable ground-state energy exists. Such a hypothesis was used to explain a number of anomalous experimental results in the case of metastable phases in various Fe, Ni and Co alloys and particularly in the case of Invar alloys (Weiss 1963, Chikazumi and Matsui 1978). Together with the switch from ferromagnetic to anti-ferromagnetic behaviour (Fig. 8.4) this makes compelling experimental evidence for the co-existence of two magnetic states in Fe-Ni and Fe-Co alloys (Miodownik 1978b). Weiss (1979) later extended this concept to other Zd elements. 8.7.1 Treatments of multiple states The simplest way to describe the equilibrium between various competing states is to use a Shottky model (Weiss 1963) where a = it = ^exp(-A£/iKT). H 9i (8.27) Here a is the ratio of the fraction of atoms, f, corresponding to the two magnetic 8.7.2 Thermodynamic consequences of multiple states One of the consequences of accepting the presence of multiple magnetic states is an additional contribution to the entropy and, therefore, several authors have considered the inclusion of multiple states in their description of low-temperature phase transformations in Fe and its alloys (Kaufman et al. 1963, Miodownik 1970, Bendick and Pepperhoff 1978). However, most authors have, in the end, preferred to describe the magnetic effects in Fe using more conventional temperature-independent values for the magnetic moments of the relevant phases. This is partly linked to the absence of any provision for the necessary formalism in current WEISS HILLERT CHIKAZUMI ?2 PEPPERHOFF •T2 Figure 8.5. Schematic comparison of various two-state models (from Miodownik 1979). References are listed on pp. 256-258. 248 JV. Saunders and A. P. Miodownik CALPHAD—A Comprehensive Guide 249 Energy difference between Two Gamma States Electron-atom ratio Figure 8.6. Variation of SE between two competing magnetic states with electron/atom ratio for some 3d elements (from de Fontaine et al. 1995). ■ Weiss (1963), A Miodownik (1978b), • Bendick et at. (1977), * Bendick and Pepperhoff (1978), + Bendick et at. (1978), □ Moruzzi and Marcus (1990a), A Moroni and Jarlsberg (1990), O Asada and Terakura (1995). software packages for phase-diagram calculations, and also to the fact that, with the exception of Roy and Pettifor (1977), there was little theoretical backing for the concept of multiple states in its early stages of development. It is interesting to note that recent electron energy calculations (Moroni and Jarlsberg 1990, Moruzzi and Marcus 1990a, b, Asada and Terakura 1995) have not only confirmed the necessary energetics for the existence of multiple states, but have also confirmed both the values of the moments (Table 8.2) and the energy gaps which had been previously inferred from experimental evidence (Fig. 8.6). It has therefore been suggested (de Fontaine et al. 1995) that future developments should make provision for the inclusion of multiple states. While having only a marginal effect on AG at high temperatures (Fig. 8.7a) such a model would lead to significant changes in the driving force for low-temperature (martensitic) transformations and, more importantly, should lead to better modelling of many associated changes in physical properties (Bendick and Pepperhoff 1978, Bendick et al. 1977). S.8. CHANGES IN PHASE EQUILIBRIA DIRECTLY ATTRIBUTABLE TO <3MAG The energy of magnetic transformations can be deceptively large, often exceeding that released by ordinary phase transformations. The following effects have been 5 ■ Purely Y2 " \ Terminal mixture of 7, and v2 \ ^ ----^ -v. Purely y. Terminal slope 5. -Rln(a+1) Temperature Figure 8.7. Variation of AG with temperature for a two-state model (from Miodownik 1977). observed in various systems and confirmed by Gibbs energy calculations (Miodownik 1982). A marked change in solid solubility. This occurs at the point where the solubility limit is intersected by the locus of magnetic transformation temperatures (Fig. 8.8). The magnitude of such effects obviously scales with the value of Gmag/dT and are most marked in Fe- and Co-rich alloys (Nishizawa et al. 1979, Hasebe et al. 1985). An interesting recent example is given by the Fe-Cc—Zn system as the latter two elements have opposite effects on the Curie temperature of Fe (Takayama et al. 1995). In some early diagrams, such abrupt changes were inexplicable and deemed to be due to 'experimental error'. Distortion of miscibility gaps. In the case of some miscibility gaps such intersections are accompanied by an even more marked distortion (Fig. 8.9). This is now often called the Nishizawa Horn, due to the extensive work of Nishizawa and coworkers (1979, 1992) on this effect, but it is interesting to note that the effect had previously been noted by Meijering (1963). Here, too, the apparent presence of more than one maximum in a miscibility gap was believed to represent experimental error before it was shown to have a sound theoretical foundation. Continuous transition between first- and second-order transformations. When examined more closely, the Nishizawa Horn represents a situation where there is a continuous transition between a first- and second-order transformation. This remarkable situation is not restricted to systems which exhibit a miscibility gap (Inden 1981a) (Fig. 8.10), and it therefore remains to be seen whether it is possible to maintain a hard and fast distinction between these two types of transformation References are listed on pp. 256-258. 1400 1200 1000 900 800 700 K 7 8 9 10 11 12 13 14 15 Reciprocal Temperature (107t) Figure 8.8. Change in solubility where a solubility transus is intersected by Tc (from Takeyama et al 1995). when faced with such well-documented effects (Hillert 1996). Stabilisation of metastable phases by the magnetic Gibbs energy contribution. The allotropy of Fe represents an important example where, in the 7 region of pure Fe, the Gibbs energy difference between b.c.c. Fe and f.c.c. Fe is small (50-60 V mol). As a corollary, small changes in Gmag have an apparently disproportionate (a) Gibbs Energy Curve (b) Miscibility Gap A B Figure 8.9. The formation of a Nishizawa Horn from the intersection of a magnetic transition and a miscibility gap (from Nishizawa 1992). effect on the topography of the a/7 region in Fe-base alloys. Because of the location of the Curie temperature in pure Fe, the effect of magnetic forces on the A3 is much more pronounced than on the A4 and alloying additions can produce asymmetric effects on the two transitions (Zener 1955, Miodownik 1977, 1978a, b). Interestingly, although the Curie temperature of Co is much higher, the value of ^mag ~c'P'h 's not as affected because the magnetic properties of both phases are similar (Miodownik 1977). However, because gf-cc-c-Ph js also small, the allotropic transition temperature can be substantially reduced (Fig. 8.11). Magnetic effects on metastable transformations. The underlying factor in all the above effects is the magnitude of the ratio (Gmag/Gtotal) and especially its variation with temperature. It follows that there can also be a substantial effect on the driving force for phase transformations, including shear transformations. Thus the martensite start temperature, MJ^"*, in most Fe alloys is dominated by the References are listed on pp. 256-258. 252 1500 N. Saunders and A. P. Miodownik (a) 1000 H a. E 500 H Co 15 P-V al% 5 10 Figure 8.10. Continuous transition between first- and second-order transformations in Co-V alloys: (a) prediction and (b) experiment (from Inden 1985 and 1982). ferromagnetism of the a phase while the M]^e in Fe-Mn alloys is controlled by the anti-ferromagnetism of the 7 phase (Miodownik 1982). Complex interactions can be expected in systems where the critical ordering temperatures for both chemical and magnetic ordering intersect (Inden 1982, Inden 1991, Skinner and Miodownik 1979), Fe-Si being a good example (Fig. 8.12). Magnetic effects on stacking fault energy. As the width of a stacking fault in CALPHAD—A Comprehensive Guide 253 o 1 uw Y \ Mn 0 v. / T / N -1000 -2000 1 1 1 1000 500 1000 1500 \ Co 0 ^-V. -1000 -2000 €mag| T,(Y)\ 1 1 Fe T \ \ 1 —-. 1 1 Tf(Y)Vs 1 500 1000 1500 e Ni Y "j. —''mag ("V. Tc 1 ! T,(Y)V-1 500 1000 1500 Temperature, °K 500 1000 1500 Temperature, °K Figure 8.11. Effect of magnetic G on f.c.c.-c.p.h.-phase transitions in certain 3d elements (from Miodownik 1977). f.c.c. lattices is a function of the Gibbs energy difference between f.c.c. and c.p.h. structures, this will also be affected by any magnetic component in either or both these structures (Fig. 8.11). This has been analysed by both Ishida (1975) and Miodownik (1978c). Although some of the parameters in both papers need to be reexamined in the light of more recent data, it is difficult to account for the observed trends without taking the magnetic Gibbs energy into consideration. 8.9. INTERACTION WITH EXTERNAL MAGNETIC FIELDS AH the effects noted above can be considered to arise from an internal magnetic field. But a Gibbs energy contribution, G^, is also generated if two phases differing in saturation magnetisation by an amount, A J, are placed in an external magnetic field, H, and is given by References are listed on pp. 256-258. 254 N. Saunders and A. P. Miodownik 1700 1500 1300 1100 900 700 1500 j^" B2,pm ^rfS^51 1300 A2,pm 1100 mm!m!tmmntmnnn // DO-^pm mMtf--^L-' B2,fm+D03,pm 900 A2,fm B2,fm - / B2,fm+D03,fni "'«ľli 700 ...1 / DOj.fm A2,fm+D0,,fm 1 't 1 500 Fe 0.05 0.10 0.15 mole fraction 0.20 0.25 Si Figure 8.12. Equilibrium between magnetic and non-magnetic-ordered derivatives of b c c Fe-Si alloys: (a) experimental and (b) calculated values (from Inden 1982). CALPHAD—A Comprehensive Guide = HAJ. 255 (8.24) This term is entirely analogous to the more commonly recognised product PAV and is yet another constituent of the expression for the total Gibbs energy. The effect has been confirmed by many experiments and an external magnetic field can destabilise austenite during martensitic transformations, causing a rise in Ms temperature of approximately 3°K Tesla-1 (Sadovskiy et al. 1961, Fields and Graham 1976). This effect turns out to be directly proportional to the difference in entropy between the two phases (Satyanarayan et al. 1972). A magnetic field can also markedly alter the rate of transformation in isothermal martensite reactions (Peters et al. 1972, Kakeshita et al. 1993). While changes in Ms due to deformation are well known, the presence of high magnetic fields generated by superconductors at cryogenic temperatures can produce similar effects. Diffusion-controlled transformations are also prone to magnetic-field effects as the driving force for transformation is affected. Thus Fe-Co alloys can be converted from 100% austenite to 100% ferrite at elevated temperatures (Fig. 8.13) under the influence of a field generated by a conventional bench magnet (Peters and Miodownik 1973). The expansion of a miscibility gap and the precipitation of the cr phase in Fe-Cr alloys (Chen et al. 1983) are another example of effects that can be induced by an external field. Although a magnetic field is not commonly applied, it has certain advantages in allowing a precise increment of Gibbs energy to:be added instantaneously to a 1270 1260 — 1250 1240 0.40 0.45 0.50 Mole fraction Co. 0.55 Figure 8.13. Effect of an external magnetic field on phase transformations in Fe-Co (from Peters and Miodownik 1973). References are listed on pp. 256-258. 256 iV. Saunders and A. P. Miodownik system simply by means of a field and without any requirement for direct contact with the specimen. It is also a salutary reminder that the Gibbs energy expressions in common use do not contain the complete range of terms implicit in Gibbs original formulation. REFERENCES Andersson, J.-O. (1985) Int. J. Thermophysics, 6, 411, Arita, M. (1978) Acta Metall., 26, 259. Asada, T, and Terakura, K. (1995) in Computer Aided Innovation of New Materials, Vol. I, eds Doyama, M. et al. (Elsevier Science, Amsterdam), p. 169. Bendick, W. and Pepperhoff, W. (1978) J. Phys. F, 8, 2535. Bendick, W., Ettwig, H. H. and Pepperhoff, W. (1978) J. Phys. F, 8, 2525. Bendick, W., Ettwig, H. H., Richter, F. and Pepperhoff, W. (1977) Z Metallkde., 68, 103. Bozorth, R. M. (1956) in Ferromagnetism (Van Nostrand). 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