Chapter 5 Thermodynamic Models for Solution and Compound Phases 5.1. INTRODUCTION Thermodynamic modelling of solution phases lies at the core of the CALPHAD method. Only rarely do calculations involve purely stoichiometric compounds. The calculation of a complex system which may have literally 100 different stoichiometric substances usually has a phase such as the gas which is a mixture of many components, and in a complex metallic system with 10 or 11 alloying elements it is not unusual for all of the phases to involve solubility of the various elements. Solution phases will be defined here as any phase in which there is solubility of more than one component and within this chapter are broken down to four types: (1) random substitutional, (2) sublattice, (3) ionic and (4) aqueous. Others types of solution phase, such as exist in polymers or complex organic systems, can also be modelled, but these four represent the major types which are currently available in CALPHAD software programmes. For all solution phases the Gibbs energy is given by the general formula G = Go + G%? + G%x (5.1) where G° is the contribution of the pure components of the phase to the Gibbs energy, G^- is the ideal mixing contribution and is the contribution due to non-ideal interactions between the components, also known as the Gibbs excess energy of mixing. The random substitutional and sublattice models are intimately related, as a phase with random occupation of atoms on all sites can technically be considered as a phase containing a 'single sublattice* and mathematically the same general equations used in both cases. However, it is useful to separate the two types as the sublattice models implicitly define some internal, spatial substructure and give rise to site occupations which define stoichiometric compounds. Also, and very importantly, G^f1 and (S^ are governed by site occupation of the components in the various sublattices rather than the global concentration of the components themselves. This chapter will begin, very briefly, with the thermodynamic representation of Gibbs energy for stoichiometric compounds before concentrating on the situation when mixing occurs in a phase. 91 92 N. Saunders and A. P. Miodownik 5.2. STOICHIOMETRIC COMPOUNDS The integral Gibbs energy, G[t,p\, of a pure species or stoichiometric compound is given simply by the equation G[T, p] — H[t, p]-TS[tip] (5.2) where H\t,p) and S{t,f] are the enthalpy and entropy as a function of temperature and pressure. Thermodynamic information is usually held in databases using some polynomial function for the Gibbs energy which, for the case of the Scientific Group Thermodata Europe (Ansara and Sundman 1987), is of the form (Dinsdale 1991) Gm[7] - i£er = a + bT + cT\n(T) + f2^- (5-3) 2 The left-hand-side of the equation is defined as the Gibbs energy relative to a standard element reference state (SER) where H^er is the enthalpy of the element or substance in its defined reference state at 298.15 K, a, b, c and dn are coefficients and n represents a set of integers, typically taking the values of 2, 3 and -1. From Eq. (5.3), further thermodynamic properties can be obtained as discussed in Chapter 6. S = -b - c - c ln(T) - ^TmU""-1 H = a - cT - - 1)4^ Cp = -c-^(n-lR^-1. Equation (5.3) is given with respect to temperature but other terms associated with pressure and magnetism can be added. The representation of such effects is described more completely in Chapters 6 and 8, S3. RANDOM SUBSTITUTIONAL MODELS Random substitutional models are used for phases such as the gas phase or simple metallic liquid and solid solutions where components can mix on any spatial position which is available to the phase. For example, in a simple body-centred cubic phase any of the components could occupy any of the atomic sites which define the cubic structure as shown below (Fig. 5.1). In a gas or liquid phase the crystallographic structure is lost, but otherwise positional occupation of the various components relies on random substitution rather than any preferential occupation of site by any particular component. 93 Figure 5.1. Simple body-centred cubic structure with random occupation of atoms on all sites. 5.3,1 Simple mixtures 53.1.1 Dilute solutions. There are a number of areas in materials processing where low levels of alloying are important, for example in refining and some age-hardening processes. In such cases it is possible to deal with solution phases by dilute solution models (Wagner 1951, Lupis and Elliott 1966). These have the advantage that there is a substantial experimental literature which deals with the thermodynamics of impurity additions, particularly for established materials such as ferrous and copper-based alloys. However, because of their fundamental limitations in handling concentrated solutions they will only be discussed briefly. In a highly dilute solution, the solute activity (a;) is found to closely match a linear function of its concentration (xi). In its simplest form the activity is written as CH = ffxi (5.4) where 7? is the value of the activity coefficient of i at infinite dilution. This is known as Henry's law. Equation (5.5) may be rewritten in terms of partial free energies as Gi = G*s + KT\ogeXi (5.5) where Gfs has a constant value in the Herman concentration range and is obtained directly from 7°. The expression can also be modified to take into account interactions between the solute elements and Eq. (5.5) becomes Gi = G?* + KT loge Xi + RTJ24^ (5-6) where Xi is the concentration of solute i and is an interaction parameter taking into account the effect of mixing of component i and j in the solvent. There are References are listed on pp. 124-126. 94 N. Saunders and A. P. Miodownik various other treatments of dilute solutions, mainly associated with how secondary interactions are considered (Foo and Lupis 1973, Kirkaldy et al. 1978, Bhadeshia 1981, Enomoto and Aaronson 1985). Recently a more sophisticated dilute solution treatment consistent with the Gibbs-Duhem integration in more concentrated solutions has been put forward by Bale and Pelton (1990). This allows the solute range handled by dilute solution models to be expanded significantly. Unfortunately, the dilute solution model is limited in its applicability to concentrated solutions. This causes problems for alloys such as Ni-based superalloys, high alloy steels, etc., and systems where elements partition strongly to the liquid and where solidification processes involve a high level of segregation. It is also not possible to combine dilute solution databases which have been assessed for different solvents. The solution to this problem is to use models which are applicable over the whole concentration range, some of which are described below. 5.3.1.2 Ideal solutions. An ideal substitutional solution is characterised by the random distribution of components on a lattice with an interchange energy equal to zero. The entropy due to configuration is easily calculated and is related to the proability of interchange of the components. The configurational entropy 5conf is given by S0"* = ft loge Wp (5.7) where ft is Boltzmann's constant and Wp is the number of configurations in which the components can be arranged for a given state. For a multicomponent system Wp is equal to the number of permutations given by ATI WP==— (5.8) where AT = J> (5.9) n; is the number of components of i and N is the total number of components in the system. For one mole of components, JV is equal to Avogadro's number. From Stirling's formula, S™"* is now equal to i The ideal molar entropy of mixing is then given by (5.10) (5.11) CALPHAD—A Comprehensive Guide 95 where a* is the mole fraction of component i. With the assumption that the interchange energy is zero it is possible to substitute the atoms without changing the energy of that state and G*g* is then given by (~tideal _ moidea! _ jxrV^ i Wax---^mbc — Xi l°Se xi (5.12) where R is the gas constant. The Gibbs energy of an ideal solution phase will then be G» - 5>iG? + AT £ xt log, Xi (5.13) with Gf defining the Gibbs energy of the phase containing the pure component i. For the case of gases, ideal mixing is often assumed and this assumption can often be quite reasonable. However, in condensed phases there is always some interaction between components. 5.3.1.3 Non-ideal solutions: regular and non-regular solution models. The regular solution model is the simplest of the non-ideal models and basically considers that the magnitude and sign of interactions between the components in a phase are independent of composition. Assuming the total energy of the solution (Eb) arises from only nearest-neighbours bond energies in a system A — B then - uAA Eaa + ojbb Ebb + vab Bab (5.14) where uAA, wBB, uAB, EAA, EBB and EA8 are the number of bonds and energies associated with the formation of different bond types AA, BB and AB. If there are JV atoms in solution and the co-ordination number for nearest neighbours of the crystal structure is z, the number of bond types being formed in a random solution is uAA=-Nzx2A uab = NzxAxB (5.15a) (5.15b) (5.15c) where xA and xB are the mole fractions of A and B. Substituting Eqs (5 15 fa-cYi into Eq. (5.14) gives " V " Nz, e°~~2~ (xa eaa + xb ebb + xa xb$Eab - Eaa - Ebb))■ (5.16) If the reference states are taken as pure A and B then Eq. (5.16) reverts to References are listed on pp. 124-126. 96 n. Saunders ana a. r. Mioaownw Nz EAA — Ebb) (5.17) where is the enthalpy of mixing. If the bond energies are temperature dependent there will also be an excess entropy of mixing leading to the well-known regular solution model for the Gibbs excess energy of mixing xAxBQ (5.18) Gxs mix where £1 is now a temperature-dependent interaction parameter. When generalised and added to Eq. (5.13) this gives i i i j>i However, it has been realised for a long time that the assumption of composition-independent interactions was too simplistic. This led to the development of the sub-regular solution model, where interaction energies are considered to change linearly with composition. The following expression for G*^ is then obtained as (Kaufman and Bernstein 1970) (5.20) Taking this process further, more complex composition dependencies to Ci can be considered and it is straightforward to show that a general formula in terms of a power series should provide the capability to account for most types of composition dependence (Tomiska 1980). The most common method is based on the Redlich-Kister equation and Eq. (5.19) is expanded to become Gm = £ Xi G° + AT $> log, xt + £ £ Xi xj £ f% - xtf (5.21) i i i j>i v where ftV- is a binary interaction parameter dependent on the value of v. The above equation for G^ becomes regular when v = 0 and sub-regular when v=l. In practice the value for v does not usually rise above 2. If it is found necessary to do so, it is probable that an incorrect model has been chosen to represent the phase. Equation (5.21) assumes ternary interactions are small in comparison to those which arise from the binary terms. This may not always be the case and where evidence for higher-order interactions is evident these can be taken into account by a further term of the type Gy* = xt Xj xk iy*, where Liik is an excess ternary interaction parameter. There is little evidence for the need for interaction terms of any higher order than this and prediction of the thermodynamic properties of substitutional solution phases in multi-component alloys is usually based on an assessment of binary and ternary terms. Various other polynomial expressions for the excess term have been considered, see for example the reviews by Ansara (1979) and Hillert (1980). All are, however, based on predicting the properties of References are listed on pp. 124-126. the higher-order system from the properties of the lower-component systems. Equation (5.21) is normally used in metallic systems for substitutional phases such as liquid, b.c.c, f.c.c, etc. It can also be used to a limited extent for ceramic systems and useful predictions can be found in the case of quasi-binary and quasi-ternary oxide systems (Kaufman and Nesor 1978). However, for phases such as interstitial solutions, ordered intermetallics, ceramic compounds, slags, ionic liquids and aqueous solutions, simple substitutional models are generally not adequate and more appropriate models will be discussed in Sections 5.4 and 5.5. J 5,3.1.4 The extrapolation of the Gibbs excess energy to multi-component systems. As mentioned in the previous section, most methods of extrapolating the thermodynamic properties of alloys into multi-component systems are based on the summation of the binary and ternary excess parameters. The formulae for doing this are based on various geometrical weightings of the mole fractions (Hillert 1980). To demonstrate this principle, emphasis will be given to three of the main methods which have been proposed. For simplicity the binary systems will be represented as sub-regular solutions. Muggianu's equation The predominant method at the present time uses the equation developed by Muggianu et at. (1975). In this circumstance the excess energy in a multi-component system, as given by the last term of Eq. (5.21), expands for a sub-regular solution to G; mix --xAxB{L°AB + LlAB(xA - xB)} + xBxc{L%c + LBC(xB - xc)} + xA xc{LAC + l\c{xA - xc)}. (5.22) Kohler's equation In the Kohler equation (Kohler 1960) G*^ is described by ---—U'ab + lU*^ XA + Xb XA+XB\ A" Aa\xA + Xbj + (xB + xc) + (xA + xc) xb + xc xb 2 xa xc xc f 3 + xc | rO , rl Ixb ~Xc^ nbc + ^BC XB + XC) xA + xc xA + xc[ AC a0\xa+xCjl (5.23) Although different in form to the Muggianu equation, it becomes identical in a binary system as the various terms (xj + xj) terms become equal to 1. 98 N. Saunders and A. P. Miodownik Toop's equation Both the Muggianu and Kohler equations can be considered symmetrical as they treat the components in the same way and do not differentiate between them. This is true for another method suggested by Colinet (1967) which is derived differently than either the Kohler or Muggianu equations but which can be reduced to the Muggianu equation (Hillert 1980). Toop's equation (1965) is essentially different in that it considers one of the binary systems does not behave in the same way as the others and the extrapolation is based essentially on two of the binaries having identical mole-fraction products with the mole-fraction products of the third being different. For a sub-regular solution Toop's equation breaks down to (Hillert 1980) Gm\x = x* xb{L°ab + L\B{xA -xb-xc)} + xa xc{Lac + L\c(xA-xc-xB)} + xBxc< LBC + L^ xc H--;-Xa XB + xc v (5.24) Again this reduces to the normal sub-regular form in the binaries. The geometrical representation of the Muggianu, Kohler and Toop equations is best summarised using the figures below. In the Muggianu extrapolation it can be seen that the line from the ternary alloy composition to the edge binaries forms a right-angle to the binary. This has the consequence that, when the alloy composition is dilute in two of the components, the interaction parameter of these two components will approach regular behaviour because the term (xi — Xj) becomes small. This is the case even if the ratio x\ : Xj differs substantially from unity. This is not important for systems where the interaction terms higher than sub-regular are small in comparison to the regular term. However, for systems such as Cr-Ni where the sub-regular part predominates, any composition dependence of Cr-Ni interactions in dilute solutions with other elements will be ignored. This could have significant consequences in the prediction of activity coefficients in low-alloy steels for example. Kohler's equation allows for the composition dependence of the binaries and in these circumstances Muggianu Kohler Toop Figure 5.2. Geometrical constructions of (a) Muggianu, (b) Kohler and (c) Toop models. CALPHAD—A Comprehensive Guide 99 may be preferable. In practice, the phase boundaries calculated by either the Muggianu and Kohler extrapolations seem to provide comparable results (Ansara et al. 1978), but it was noted that the choice of extrapolation method should receive more attention when exact knowledge of partial quantities such as activity coefficients is more critical. The Toop equation is not suitable for metallic systems but may be appropriate for some ionic liquid systems. However, it should be used with care in all cases as the extrapolation is dependent on which binary is chosen to behave differently, and it is possible to obtain three different answers depending on this choice. 5.4. SUBLATTICE MODELS 5.4J Introduction A sublattice phase can be envisaged as being composed of interlocking sublattices (Fig. 5.3) on which the various components can mix. It is usually crystalline in nature but the model can also be extended to consider ionic liquids where mixing on particular 'ionic sublattices' is considered. The model is phenomenological in nature and does not define any crystal structure within its general mathematical formulation. It is possible to define internal parameter relationships which reflect structure with respect to different crystal types, but such conditions must be externally formulated and imposed on the model. Equally special relationships apply if the model is to be used to simulate order-disorder transformations. Sublattice modelling is now one of the most predominant methods used to describe solution and compound phases. It is flexible and can account for a variety of different phase types ranging from interstitial phases such as austenite and ferrite in steels to intermetallic phases, such as sigma and Laves, which have wide homogeneity ranges. Furthermore, it can be extended to simulate Bragg-Williams-Gorsky ordering in metallic systems and be adapted to account for 0--1--O' o-o- Figure 5.3. Simple body-centred cubic structure with preferential occupation of atoms in the body-centre and corner positions. References are listed on pp. 124-126. 100 N. Saunders and A. P. Miodownik ionic liquids. This section will begin by giving a detailed description of the multiple sublattice model of Sundman and Agren (1981) and give some specific examples of usage. 5.4.2 The generalised multiple sublattice model (Sundman and Agren 1981) 5.4.2.1 Definition of site fractions. The multiple sublattice model is an extension of earlier treatments of the two-sublattice models of Hillert and Steffansson (1970), Harvig (1971) and Hillert and Waldenstrom (1977). It allows for the use of many sublattices and concentration dependent interaction terms on these sublattices. To work with sublattice models it is first necessary to define what are known as site fractions, yf. These are basically the fractional site occupation of each of the components on the various sublattices where * ni Vi = — (5.25) is the number of atoms of component i on sublattice $ and Ns is total number of sites on the sublattice. This can be generalised to include vacancies, which are important to consider in interstitial phases, such that Eq. (5.25) becomes Vi (5.26) and nya is the number of vacancies on sublattice s. Mole fractions are directly related to site fractions by the following relationship 3ji (5.27) 5.4.2.2 Ideal entropy of mixing. The ideal entropy of mixing is made up of the configurational contributions by components mixing on each of the sublattices. The number of permutations which are possible, assuming ideal interchanges within each sublattice, is given by the following equation WP = J[ n (5.28) and the molar Gibbs ideal mixing energy is References are listed on pp. 124-126. CALPHAD—A Comprehensive Guide s i where yf includes contribution from vacancies. 101 (5.29) 5.4.2.3 Gibbs energy reference state. The Gibbs energy reference state is effectively defined by the 'end members' generated when only the pure components exist on the sublattice. Envisage a sublattice phase with the following formula {A, B\(C, D)1. It is possible for four points of 'complete occupation' to exist where pure A exists on sublattice I and either pure B or C on sublattice 2 or conversely pure B exists on sublattice 1 with either pure JS or C on sublattice 2. The composition space of the phase can then be considered in Fig. 5.4 below as consisting of four compounds, the so-called 'end members', at the corner of each square. The composition of the phase is then encompassed in the space between the four 'end member' compounds (Fig. 5.4(a)) and the reference energy surface will look like Fig. 5.4(b). AD BO t Id **~gbd °bc / / AC 1 BC Figure 5.4. (a) Composition space encompassed by the system (A, j3)j(C,jD)i and (b) the reference energy surface described by equation (5.30) after Hillert and Staffanson (1980). The surface in Fig. 5.4(b) can be represented by the equation G£ = yA yc G°AC + yB yc G%c + yA yo G°AD + yB yD G°BD. (5.30) Equation (5.30) holds for the simple case of a phase with the formula {A, 5)j(C, D\. But for more complex phases the function for the Gibbs reference energy surface may be generalised by arranging the site fractions in a (I + c) matrix if there are / sublattices and c components. y\ y\ y\ y\ vl vl vl • • • • • 2/i • • • • id 102 JV. Saunders and A. P. Miodownik CALPHAD—A Comprehensive Guide 103 Each row represents a sublattice and each column a component. The information contained in the subscript and superscript can be represented by a component array f 7, which defines one component for each sublattice. Using this notation a more generalised formula can be obtained (Sundman and Agren 1981) Ctf^PiOO-GJ (5.31) i where G° represent the Gibbs energy of the compound defined by 7 and Pi{Y) i represents the corresponding product of site fractions from the Y matrix. 5.4.2.4 Gtbbs excess energy of mixing. The method for describing the Gibbs I excess energy can, again, be best shown by using a two-sublattice system (A, S)i(C, ■ D)i before generalising to a multi-component system. In this alloy A-C, A-D, B-C and B-D interactions are controlled by the Gibbs energy of the compounds AC, BC, < AD and BD. Mixing on the sublattices controls A-B and C-D interactions and the | simplest form of interaction is a regular solution format such that = Va Vb L%b* + Vc Vd L%c,d (5-32) \ i where LAB., and l£.CD denote regular solution parameters for mixing on the j sublattices irrespective of site occupation of the other sublattice. A sub-regular model can be introduced by making the interactions compositionally dependent on the site occupation in the other sublattice. + vcvdva la:c,d + Vc Vd Vl L%Cfi (5'33) Finally some site fraction dependence to these parameters can be added such that ^a,b:c = AvbVc^lX&c{va V -Vb) (5.34a) ^a,b-.d = VaVbVdJ2L*,*d{Va V -Vb) (5.34b) = vavIvdY,l*c,d(Vc V -Vd) (5.34c) ^b-.c.d = VBVcvV£f^b!cfi (vc -Vd) (5.34d) It is clear that this can be expanded to any number of sublattices and components and Eq. (5.33) can be generalised using the notation of Eq. (5.31) 71 i 71 refers to a component array of the first order, where one sublattice contains two components, but the remaining sublattices are occupied by only one component with the summation being taken over all different 71. The type of component array that was introduced in Eq. (5.31) can then be denoted 10 and referred to as a component of the zeroth order. Equation (5,35) is general for the case of regular solutions but can be further extended to include interactions of a higher order, as in Eq. (5.34), by introducing the appropriate arrays IZ but with a further restriction that component array must not contain any component more than once in each sublattice. In this way the excess Gibbs energy can be written as Z>0 IZ and the total energy of the phase including the reference energy, the ideal entropy and excess terms becomes <3m =EPn{Y) ■ G% + RT^2N'^vtlog.ti+Y,EP™V) ' L™- (5-37) 10 a i Z>0 IZ 5.4.3 Applications of the sublattice model 5.4.3.1 Line compounds. These are phases where sublattice occupation is restricted by particular combinations of atomic size, electronegativity, etc., and there is a well-defined stoichiometry with respect to the components. Many examples occur in transition metal borides and silicides, III-V compounds and a number of carbides. Although such phases are considered to be stoichiometric in the relevant binary systems, they can have partial or complete solubility of other components with preferential substitution for one of the binary elements. This can be demonstrated for the case of a compound such as the orthorhombic Cr2B-type boride which exists in a number or refractory metal-boride phase diagrams. Mixing then occurs by substitution on the metal sublattice. For example in the Fe-Cr-B system (Fig. 5.5) Cr2B extends across the diagram at a constant stoichiometry of boron as (Cr,Fe)2B (Villars and Calvert 1985). Also shown is the corresponding extension of the tetragonal Fe2B compound. Such phases are commonly called "line compounds'. Generally, sublattice occupation can be described as (A, B,C...)U(Z)V where A, B and C are the elements which mix on the first sublattice and Z is the element with fixed stoichiometry. The Gibbs energy of such a phase can be written Gm - E v\i V References are listed on pp. 124-126, 104 N. Saunders and A. P. Miodownik At % Cr Figure 5.5. Partial isothermal section for Fe-Cr-B showing stoichiometric behaviour of the (CrJFe^B compound. where i and j denote the elements mixing on sublattice 1 while Z is the stoichiometric element, u is the molar number of sites in sublattice 1 and G°.g denotes the Gibbs energies of the various compounds when sublattice 1 is completely occupied by one of the components mixing on sublattice 1. The excess energy term is associated only with interactions between the components on the first sublattice. Because of the fixed stoichiometry, mole fractions can straightforwardly be expressed as xt = {u/(u + v)) y] and Eq. (5.38) becomes equivalent to the line compound models used by Kaufman and Bernstein (1970). 5.4.3.2 Interstitial phases. These are predominant in steels and ferrous-based alloys, where elements such as C and N occupy the interstitial sites of the ferrite and austenite lattices. In this case the structure of the phase can be considered as consisting of two sublattices, one occupied by substitutional elements, such as Fe, Cr, Ni, Mn, etc., and the other occupied by the interstitial elements, such as C or N, and interstitial vacancies (Va). As the concentration of C, N,..., etc., increases the interstitial vacancies are filled until there is complete occupation. The occupation of the sublattices is shown below as (Fe, Cr.Ni, Mn.;.)u(Va, C,N...)„. In the case of an f.c.c._Al structure, with u = v = 1, the state of complete occupation of interstitial carbon corresponds to a MC carbide with the NaCl lattice. For the c.p.h._A3 structure, with u = 1 and v = 0.5, complete occupation of the interstitial sites by carbon gives the M2C carbide with a hexagonal Fe2N-type structure. For the case of the f.c.c._Al, austenite phase in Cr-Fe-C, the Gibbs energy of the phase is represented by the formula (Cr, Fe)i (C, Va)i and its Gibbs energy given by the following equation References are listed on pp. 124-126. CALPHAD—A Comprehensive Guide + vkycG^c+• (yhr 2/cr + vk ^ vk) + 1- facl0Se^C + yv>geJ/va)) + ^F^Va^^Cr.FsVa^Cr ~ Vf*)") + VCtVc^* (Z Lh:Q,vM - y%S) + ykycvl*. LFe:c,va {vh - star) ■ 105 (5.39) The first two terms represent the Gibbs energy of f.c.c._Al Cr and Fe because the second sublattice consists entirely of unfilled interstitial sites. The next two terms represent the Gibbs energy of the CrC and FeC compounds which are formed when all of the interstitial sites of the f.c.c._Al lattice are filled with C, The second line is the ideal entropy term while the last four lines correspond to the excess term. It can be seen that, when y\ = 0, the model reduces to Eq. (5.21) for the simple substitutional mixing of Cr and Fe. This is particularly useful as thermodynamic assessments of metallic binary systems can be extended to include interstitial elements without the need for changing any of the existing parameters. 5.4.3.3 Complex intermetallic compounds with significant variation in stoichiometry. In many binary systems intermetallic compounds may exhibit significant solubility and there may be more than two crystallographic sublattices to consider. This is the case for compounds such as the