Thermodynamics of stable and metastable structures in Fe–C system

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 148–158

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Thermodynamics of stable and metastable structures in Fe–C system Reza Naraghi n, Malin Selleby, John Ågren Department of Materials Science and Engineering, KTH Royal Institute of Technology, Brinellvä gen 23, SE-100 44 Stockholm, Sweden

art ic l e i nf o

a b s t r a c t

Article history: Received 20 February 2014 Received in revised form 13 March 2014 Accepted 14 March 2014 Available online 27 March 2014

The thermodynamic properties and the phase diagram of the Fe–C system are reviewed by means of the CALPHAD method and Gibbs energy functions valid from 0 K upwards are presented. The Fe–C system has been evaluated previously by Gustafson. The information on thermodynamic properties and phase equilibria have now been updated and used as a basis to re-optimize the model parameters. In addition, thermodynamic properties of metastable cementite, Hägg and eta carbides are evaluated on the basis of available experimental data and taking into account the magnetic nature of these carbides. Moreover, a model is proposed for carbon ordering phenomena in martensite. Structural changes during early stages of aging of martensite are described using the proposed model and tempering equilibria with cementite, Hägg, and eta carbides are well reproduced. It should also been mentioned that the present description represents experimental data on the equilibrium with the liquid better than Gustafson's thermodynamic description. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Atomic ordering CALPHAD Martensite Carbides Spinodal decomposition

1. Introduction The binary Fe–C system is of special importance for iron and steel industry and it has been experimentally investigated in great detail, especially on the iron rich side. For the same reason several thermodynamic assessments of the stable and metastable systems have been proposed over the years [1–4]. Gustafson's [4] evaluation of the Fe–C system, using the CALPHAD method, is widely used and included in most CALPHAD databases. In addition to the ordinary bcc, fcc and liquid phases, numerous iron carbides have been reported in this system. None of them is stable at any temperature, but they may be stabilized by impurity elements. θ-Fe3 C (Cementite), χ-Fe5 C2 (Hägg or chi) and ε=η-Fe2 C (epsilon/eta) carbides have thus been observed under metastable equilibrium conditions. Only cementite with a constant heat capacity was included in Gustafson's [4] evaluation and it was recently reassessed by Hallstedt et al. [5]. The formation and decomposition of martensite is of particular interest in connection with hardening and tempering of steels. Since early 1920s, X-ray diffraction measurements have shown that the lattice of freshly formed high-carbon martensite is tetragonaly distorted in contrast to the bcc structure of ferrite [6]. Tetragonality of martensite is usually associated with ordering of interstitial carbon atoms [7]. Tempering of martensite has also been extensively studied. It is widely known that some reactions

n

Corresponding author. Tel.: þ 46 8 790 8325. E-mail addresses: [email protected], [email protected] (R. Naraghi).

http://dx.doi.org/10.1016/j.calphad.2014.03.004 0364-5916/& 2014 Elsevier Ltd. All rights reserved.

occur below conventional tempering temperature of 373 to 873 K, including clustering of carbon atoms in c-oriented (tetragonality axis) octahedral sites (A1), and formation of a modulated tweedlike microstructure consisting of low and high-carbon regions (A2) [8–12]. Compared to the extensive experimental information on early stages of tempering, the quantitative mathematical treatment on carbon ordering in martensite and its relation to early stages of tempering is sparse to our knowledge. The present work was initiated by such needs for a theoretical analysis of the stability of martensite and formation of metastable carbides during tempering. A new thermodynamic description of carbon ordering in martensite, in addition to new descriptions of cementite, Hägg and eta carbides are presented and the binary stable and metastable systems are assessed from 0 K upwards, including the liquid phase.

2. Thermodynamic models 2.1. Pure elements The SGTE [13] data for pure elements is usually not very accurate at low temperatures and below 300 K it should not be used at all. In order to successfully describe phase transformations around room temperature or comparing the energy of formation of the carbides derived from thermodynamic evaluations with that derived from ab-initio calculations, it is thus necessary to extend the Gibbs energy of Fe and C below this limit down to 0 K. Chen

R. Naraghi et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 148–158

and Sundman [14] have performed a comprehensive evaluation of the thermodynamic properties and lattice stability of pure iron based on the Einstein model for the heat capacity from 0 K upwards. The liquid and amorphous phases have been treated as one phase with the generalized two-state model in their work. Their description of iron is accepted and used in this paper. Hallstedt et al. [5] have kept the SGTE Gibbs energy expressions for graphite and added experimentally based polynomials for two lower temperature intervals. Vřešt'ál et al. [15] have also presented a method for the extension of SGTE Gibbs energy expression for graphite and other pure elements to 0 K on the basis of the Einstein model without using the experimental heat capacity data at low temperatures. In this work it was decided to use the Einstein model for the temperature dependence of the heat capacity to extend the SGTE Gibbs energy expressions for graphite and diamond and use the available experimental heat capacity data from close to 0 K up to 300 K. In order to have a smooth connection of Gibbs energy expression below 300 K to the valid SGTE function above this limit, the low temperature function is constrained to have the same function value and the value of the first derivative as the SGTE function at 300 K. Similarly, in order to have a continuous heat capacity, the heat capacity is constrained to have the same function value and the value of the first derivative as the SGTE's CP at the limiting temperature. The heat capacity of graphite and diamond is represented by
2 ΘE eΘE =T C P ¼ 3R þ aT þ bT 4 þ cT 2 ð1Þ Θ T ðe E =T 1Þ2 where ΘE is the Einstein temperature, and the first term stands for the contribution from the harmonic lattice vibrations. The second term consists of contributions from electronic excitations and loworder anharmonic corrections (dilatational and explicitly anharmonic). The third term originates from the high-order anharmonic lattice vibrations. It was also found necessary to add a cT 2 term in order to have a smooth continuation of CP to that of the SGTE description at the limiting temperature. From the expression for CP, the Gibbs energy at 1 bar can be derived as 
 3 ΘE a b c G ¼ E0 þ RΘE þ 3RT ln 1  exp   T2  T5  T3 ð2Þ 2 2 20 6 T where E0 is the energy without lattice vibrations at 0 K, and the second term is the zero-point energy due to the lattice vibrations [16,17]. 2.2. The binary system All binary phase models used in this work are based on the compound energy formalism [18].

149

of an fcc to bcc lattice, since only one-third of the interstitial positions in the bcc lattice correspond to the interstitial positions in the original fcc. Zener [7] suggested that the observed tetragonality cannot be merely an effect of the diffusionless transformation and the preferred distribution of carbon atoms must be the equilibrium distribution. He used a phenomenological model and predicted a critical point of the order–disorder transition depending on temperature and carbon content of the martensite at which the ordered distribution becomes thermodynamically more stable than the disordered one. Kurdjumov and Khachaturyan later used a more fundamental approach to describe the order–disorder transition using the microscopic elasticity theory (MET) and static concentration waves method [20–22]. As it will be shown in Section 5 in the present report the ordering results in an inflection point in the Gibbs-energy curve at the critical order–disorder transformation concentration at which spinodal decomposition also becomes feasible [23]. In other words, ordering and spinodal decomposition coexist. The ordered martensite is thus not stable and will eventually undergo a spinodal decomposition during aging. The so called Zener-ordering predicts carbon rich regions with a composition close to FeC stoichiometry, 50 at% C (17.7 wt% C). Such a high carbon content modulation has never been observed experimentally. Instead, transmission electron microscopy (TEM) [24], atom-probe field-ion microscopy (APFIM), and more recent three-dimensional atom probe (3DAP) [25] measurements estimate the composition of carbon rich regions to be close to 11 at% C (2.59 wt% C) as in α″-Fe16 C2 (or Fe8 C) compound. Such a structure can be obtained as the result of spinodal decomposition in a secondary α″ type ordering of carbon interstitials within the tetragonal sublattice. It can be shown that the α″-ordered structure can be derived from the Zener-ordered structure by systematic removal of one-eighth of the interstitial atoms. Correspondingly, here we describe martensite with ðFeÞ1 ðC; VaÞa ðC; VaÞb model, where the a and b parameters are 1 and 2 for the Zener-ordered structure and 1/8 and 23/8 for α″-ordered structure, respectively. The chemical part of the Gibbs energy per mole of formula unit of martensite can then be represented by ð1Þ ð2Þ ð3Þ ð1Þ ð2Þ ð3Þ ð1Þ ð2Þ ð3Þ bct bct bct Gbct m ¼ yFe yC yC 1GFe:C:C þ yFe yC yVa 1GFe:C:Va þ yFe yVa yC 1GFe:Va:C ð2Þ ð3Þ ð2Þ ð2Þ ð2Þ ð2Þ bct þ yð1Þ Fe yVa yVa 1GFe:Va:V a þ aRTðyC ln yC þ yVa ln yV a Þ ð3Þ ð3Þ ð3Þ E bct mo bct Gm þ bRTðyð3Þ C ln yC þ yVa ln yVa Þ þ Gm þ

ð3Þ

where yðSÞ denotes the site fraction of component i on sublattice S. i E

Gbct m is the excess Gibbs energy described as

E

ð1Þ ð2Þ ð2Þ bct ð1Þ ð3Þ ð3Þ bct Gbct m ¼ yFe yC yVa LFe:C;V a:n þ yFe yC yV a LFe:n:C;Va

ð4Þ

and ð2Þ ð2Þ ν ν bct Lbct LFe:C;V a:n Fe:C;V a:n ¼ ∑ ðyC  yVa Þ

ν

2.2.1. Austenite, ferrite, and martensite The interstitial solution of carbon in fcc and bcc is modelled with two sublattices, ðFeÞ1 ðC; VaÞa . The first sublattice is occupied by substitutional iron atoms and the second one represents the vacant interstitial octahedral sites which are randomly occupied by carbon atoms. The low number of vacancies, so-called thermal vacancies, on the first sublattice is thus neglected. The number of octahedral sites is the same as the number as substitutional sites in fcc, and they are 3 times as many as the substitutional sites in bcc. Accordingly parameter a is set to 1 for fcc and 3 for bcc. It is known experimentally that the lattice of freshly formed high carbon martensite differs from bcc structure of ferrite and one crystal axis is elongated with respect to the other two, i.e. the structure is tetragonal rather than cubic. According to Bain [19] the tetragonality of freshly formed martensite is due to the restraints imposed by the carbon atoms upon the diffusionless transformation

ð3Þ ð3Þ ν ν bct Lbct LFe:n:C;V a Fe:n:C;Va ¼ ∑ ðyC  yVa Þ

ν

ð5Þ

We have chosen the superscript bct in order to emphasize that it is only the chemical part of the Gibbs energy of martensite that is considered. The total Gibbs energy of martensite has large additional contributions from its high content of defects such as dislocations and twin boundaries and from elastic stresses caused by the large shape changes during the martensitic transformation. These additions will not be dealt with in this paper. The last term in Eq. (3) is the contribution due to magnetic ordering described by the model proposed by Inden [26] and adopted by Hillert and Jarl [27] with a slight change of the parameters involved and the number of terms retained in the truncation [14]: mo

bct Gbct þ 1Þf ðτÞ m ¼ RT lnð〈β 〉

ð6Þ

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R. Naraghi et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 148–158

where 〈β〉bct is the mean magnetic moment per mole of formula unit, τ ¼ T=T C , and TC is the Curie temperature. 
 τ1 1 þ 0:63570895  1 f ðτÞ ¼ 1  0:38438376 p p
3  τ τ9 τ15 τ21 þ þ þ =D for τ r1 6 135 600 1617 
 7  τ τ  21 τ  35 τ  49 f ðτ Þ ¼  þ þ þ =D for τ 4 1 21 630 2975 8232
 1 ð7Þ D ¼ 0:33471979 þ 0:49649686  1 p The parameter p is a structural factor defined as the fraction of the total ordering enthalpy which is due to short-range order and is absorbed above the Curie temperature. For fcc metals p is given as 0.25 and for bcc metals as 0.37. Due the similarities of the bcc and bct structures, the parameter p is selected as 0.37 for bct as well. The order independent contribution to the coefficients of the bct model can be calculated by converting the two-sublattice bcc ðFeÞ1 ðC; VaÞ3 to three sublattices, ðFeÞ1 ðC; VaÞa ðC; VaÞb . In order to favor the stability of the ordered state in certain temperature or composition ranges, it is also necessary to impose some constraints on the coefficients of the Gibbs energy applied to this phase. It is sufficient to require that the Gibbs energy should ð3Þ always have a minimum at yð2Þ C ¼ yC when the disordered state is stable [28]. The following relations for the coefficients of the Gibbs energy expression of the bct can thus be derived mathematically when limiting the model to regular interaction terms: bcc 1Gbct Fe:Va:V a ¼ 1GFe:V a bcc 1Gbct Fe:C:C ¼ 1GFe:C a bcc b bcc ab bcc L 1Gbct  ðc þ dTÞ Fe:C:V a ¼ 1GFe:C þ 1GFe:Va þ 3 3 9 Fe:C;V a a bcc b bcc ab bcc L  ðc þ dTÞ 1Gbct Fe:Va:C ¼ 1GFe:V a þ 1GFe:C þ 3 3 9 Fe:C;V a  a 2 Lbct Lbcc Fe:C;V a:n ¼ Fe:C;V a þ ðc þ dTÞ 3
2 b Lbct Lbcc Fe:n:C;Va ¼ Fe:C;V a þ ðc þdTÞ 3

ð8Þ

The composition and ordering dependence of the Curie temperature and the Bohr magneton number can be described in the same way as for the Gibbs energy. However, since the effect of composition and ordering of carbon on the magnetic properties of iron is not experimentally well established, the magnetic contributions to the Gibbs energy of the ordered and disordered states are treated as independent of composition and ordering. 2.2.2. Cementite Cementite has an orthorhombic crystal structure with a ferromagnetic transition around 483 K [29]. In Gustafson's [4] evaluation of the binary Fe–C system, cementite is described by a constant heat capacity without the magnetic transition, mainly due to the lack of reliable experimental data and weak temperature dependence of the heat capacity at high temperatures. Hallstedt et al. [5] re-evaluated the thermodynamic properties of cementite including the magnetic transition and presented Gibbs energy functions valid from 0 K upwards in terms of conventional polynomials in three temperature intervals. It is noteworthy that the compositions of the metastable carbides are somewhat uncertain; even cementite which has long been regarded as stoichiometric may exist over a narrow composition range deviating from the stoichiometric Fe3 C composition. As the variation is quite small cementite is here described with a stoichiometric model ðFeÞ3 ðCÞ1 . It was also decided to use the Einstein model for the heat capacity of cementite. Following the Chen and Sundman [14], the heat

capacity of cementite per mole of atoms is represented by
2 ΘE eΘE =T C P ¼ 3R þaT þbT 4 þ mag C P Θ T ðe E =T  1Þ2

ð9Þ

where the last term denotes the contribution from the magnetic transformation. From the expression of heat capacity, the Gibbs energy per mole of formula unit at 1 bar can be derived as 
   3 ΘE a b Gθm ¼ 4 E0 þ RΘE þ 3RT ln 1  exp   T2  T5 2 2 20 T þ mdoð1Þ Gθm þ mo Gθm Gθm ¼ f m Gθm þ mdoð1Þ Gθm þ mo Gθm where ing: mdoð1Þ

mdoð1Þ

ð10Þ ð11Þ

θ

Gm is the total Gibbs energy for magnetic disorder-

  TC Gθm ¼ R lnð〈β 〉θ þ 1Þ T  0:38438376 pD

ð12Þ

and mo Gθm is the contribution due to magnetic ordering given in Eq. (6). f m Gθm is the Gibbs energy of the ferromagnetic ground state. Due to the similarities of orthorhombic and fcc structures, p is selected as 0.25 for cementite as well, TC is set to 485 K and 〈β 〉θ is fitted to the experimental heat capacity data. 2.2.3. Hägg and eta carbides Formation of other iron carbides besides cementite has an important dual role in catalytic chemistry and in metallurgy. Formation of these carbides during carburization experiments and in Fischer–Tropsch synthesis with iron catalyst was investigated more than 80 years ago [30–32]. The same carbides can also be obtained from tempered martensite [33]. The first transition carbide that appears at lower temperatures was originally identified as hexagonal ε-carbide [33], but later work showed that it is in fact the orthorhombic η-Fe2 C [34]. Lately it was shown that η-Fe2 C is slightly more stable than ε and that hexagonal structure can relax to orthorhombic [35]. ε=η-Fe2 C carbide also appears to have ferromagnetic properties for which Cohn and Hofer [36] gave a Curie temperature around 653 K. The η carbide is thus modeled in this work with a stoichiometric model ðFeÞ2 ðCÞ1 . In high carbon martensite, another transition carbide appears as the precursor for the formation of cementite. At first this carbide was denoted as Fe2 C which was later called Hägg carbide [37]. In 1962 Senateur et al. [38] determined the monoclinic structure of Hägg carbide by X-ray diffraction analysis. Fruchart et al. [39] confirmed the similarity of its crystal structure with Mn5 C2 carbide which led to the conclusion that the composition of Hägg carbide should be Fe5 C2 rather than Fe2 C [40]. Chemical analysis by Naumann and Langenscheid [41] with carbon content of 28.6–30.2 at% C further confirmed this conclusion. Hägg carbide also appears to be ferromagnetic with a Curie temperature around 520 K [36]. A stoichiometric ðFeÞ5 ðCÞ2 model was used to describe the Hägg carbide in this work. A summary of crystal structures, Curie temperatures and averaged saturation magnetic moment per iron atom of the carbides from the literature [29,34–36,42–46] is given in Table 1. Due to the lack of experimental data, independent modeling of the heat capacity of these carbides was not possible. The frequently used Neumann–Kopp rule cannot be used for iron carbides due to the very different character of bcc Fe and the carbides. Instead it was decided to give the reference to cementite according to the following reactions χ

ð13Þ

η

ð14Þ

fm

C P ¼ 53 f m C θP þ 13 C gr P

fm

C P ¼ 23 f m C θP þ 13 C gr P

R. Naraghi et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 148–158

151

Table 1 Summary of crystal structures, Curie temperatures and averaged saturation magnetic moment per iron atom of the carbides at 0 K. Phase

Pearson symbol

Space group

Strukturbericht designation

Tc (K)

β ðμB =FeÞ

θ-Fe3 C

oP16 [42]

Pnma [42]

D011 [42]

χ-Fe5 C2

mC28 [45]

ε-Fe2 C η-Fe2 C

hPn [46] oP6 [34]

C2=c [45] C12=c1 [44] P63 22 [46] Pnnm [34]

480( 7 20) [43] 483 [29] 520( 73) [36]

1.86 1.80 1.69 1.72 1.60

653 [36]

mdoð1Þ χ Gχm ¼ a þ 53 f m Gθm þ 13 Ggr Gm þ mo Gχm mþ

ð15Þ

mdoð1Þ η Gm þ mo Gηm Gηm ¼ b þ 23 f m Gθm þ 13 Ggr mþ

ð16Þ

Due to the similarity of the crystal structures of χ-Fe5 C2 and η-Fe2 C with cementite, parameter p is selected as 0.25 for these carbides as well. TC is set to 520 K and 663 K for χ and η respectively. Recent ab-initio calculations show that the averaged local magnetic moment per iron atom only slightly decreases with increasing C concentrations [35,44]. Assuming that iron atoms in χ and η have the same average local magnetic moment as in θ χ η cementite ðβFe  βFe  β Fe Þ, we can estimate the mean magnetic moment per formula unit for χ and η from that of cementite, θ, by the following relations: 〈β 〉χ  ð〈β〉θ þ 1Þ5=3 1

ð17Þ

〈β 〉η  ð〈β〉θ þ1Þ2=3  1

ð18Þ

[35] [44] [35] [44] [35]

where P is expressed in atmospheres, and C is dissolved carbon. If graphite is chosen as the reference state for carbon, the constants of the equations above become identical with the constants of the equations CO2 ðgÞ þ CðgrÞ ¼ 2 COðgÞ

ð25Þ

2 H2 ðgÞ þ CðgrÞ ¼ CH4 ðgÞ

ð26Þ

and the carbon activity can be written as aC ¼ r 20 =K 25

ð27Þ

aC ¼ r 21 =K 26

ð28Þ

where aC is given relative to graphite. Bradley et al. [47] made a detailed analysis of different values of reactions constants used in the literature and recommended the data given in JANAF thermochemical tables [48]. Accordingly all the activities in the present evaluation are recalculated using the equilibrium constants given in those tables.

where we have used θ

ð19Þ

3.1. Carbon ordering and early stages of aging of martensite

χ

ð20Þ

η

ð21Þ

Olson and Cohen [10] and Krauss [11] divided the tempering of martensite into seven stages. Tempering reactions in martensite and associated temperatures from the literature [9,11,33,49–52] are summarized in Table 2. Structure changes at, below or above room temperature are designated as A1, A2 and A3 stages of aging corresponding to clustering of carbon atoms, spinodal decomposition, and ordering respectively and carbide forming stages at higher temperatures are designated with T1, T2, T3, T4 of tempering. Due to the scatter in experimental data, an accurate determination of the lowest temperature for formation of transition carbides seems to be difficult. However, temperature intervals reported by Nagakura et al. [9] are in reasonable agreement with most of the other observations and were used in the present evaluation. The value of the critical point of order–disorder transition in martensite at room temperature has been given theoretically with quite scattered results, i.e. 0.22 wt% C [7], 0.64 wt% C [53], 0.23 wt% C [54], and 0.18 wt% C [21,55]. Khachaturyan has obtained a critical C concentration of 0.03 wt% C at room temperature by taking into account the direct contact chemical repulsion of nearest-neighbor C–C pairs [22]. This value is considerably lower than the previous approximations. Udyansky et al. [56] suggested that this discrepancy may be due to the insufficient description of the C–C chemical interactions and studied the stability limits using a combination of MET and atomistic potentials. Using this approach a critical C concentration of 0.16 wt% C at room temperature has been determined which is consistent with [21,55]. The experimental results show discrepancy as well due to the difficulty of detecting the low tetragonality and the self-tempering effect of low-carbon martensite. Ren and Wang [57] suggested that A1 and A2 stages of aging are both spinodal decomposition, but to different extents. The minimum carbon concentration to form A1, A2 at room temperature is given as 0.2 wt% C by Nagakura et al. [9]. An experimental critical

mo θ Sm

¼ R lnð〈β 〉θ þ 1Þ ¼ 3R lnðβ Fe þ 1Þ

mo χ Sm

¼ R lnð〈β 〉χ þ 1Þ ¼ 5R lnðβ Fe þ 1Þ

mo η Sm

¼ R lnð〈β 〉η þ 1Þ ¼ 2R lnðβ Fe þ 1Þ

In general, 〈β〉 is obtained for a mole of substitutional atoms and if one would like to consider another formula unit, e.g. with b substitutional atoms, 〈β〉 cannot be used. The new 〈β 〉 has to be calculated according to 〈β 〉new ¼ ð〈β 〉 þ1Þb=a  1:

ð22Þ

3. Experimental information Gustafson [4] carefully selected a set of binary data which itself was to a considerable extent based on the classic evaluation by Chipman [2]. His selection will be used in the present work. The reference temperature scale varies among reported experiments data. Although in comparison with other experimental uncertainties such as unknown impurities in alloys, cooling/heating rate effects and uncertainty in chemical compositions of specimens, the temperature scale has a minor effect, for the sake of consistency, all experimental temperatures used in the present evaluation are corrected to the international temperature scale of 1990 (ITS-90). Another uncertainty arises in calculation of activities from gas equilibria. Activities of carbon in fcc, bcc and liquid iron have been determined by measuring the gas constituents in equilibrium with the solid or liquid phase according to CO2 ðgÞ þ C ¼ 2 COðgÞ

r 20 ¼

2 H2 ðgÞ þ C ¼ CH4 ðgÞ

r 21 ¼

P 2CO P CO2 P CH4 P 2H2

ð23Þ ð24Þ

152

R. Naraghi et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 148–158

carbon concentration of the cubic to tetragonal transition at room temperature is also reported by Xiao et al. [58]. Their value of 0.18 wt % C is in good agreement with Nagakura et al.'s observations [9] and theoretical estimations of [21,55,56]. Therefore, this point was used in the present evaluation.

4. Parameter evaluation The optimization was performed with the selected set of data, with each set given a weight corresponding to their experimental uncertainty. It was found that the same number of parameters as in Gustafson's evaluation were enough for describing all available experimental data with one exception. A temperature dependent interaction parameter 1 Lliquid Fe;C in liquid was used in order to obtain a better fit to graphite solubility in liquid, eutectic temperature and composition. The solubility of carbon in ferrite is very small and consequently it 0 bcc is practically very difficult to optimize 1Gbcc Fe:C and LFe:C;Va separately. In the assessment by Gustafson [4] no interaction term was initially evaluated. This choice would lead to reappearance of bcc phase at high temperatures. In order to avoid this problem a fixed 0 bcc LFe:C;Va ¼ −190 T was used. Later, Franke [59] made another choice ð0 Lbcc Fe:C;Va ¼ 150000−290TÞ to prevent bcc becoming more stable than

the liquid at higher temperatures. However this choice will lead to a miscibility gap in bcc at low temperatures. In the present evaluation it was decided to keep Franke's choice of 0 Lbcc Fe:C;V a parameter and optimize 1Gbcc Fe:C with the activity data in

α-ferrite, equilibrium with

austenite, and δ-ferrite equilibrium with austenite and liquid. The cementite CP coefficients in Eq. (9) and 〈β〉θ are fitted using the heat capacity data from Naeser [60], Seltz et al. [61] and Andes [62]. The solubility of cementite in fcc by Smith [63] in the range 1027 to 1285 K and metastable eutectic temperature 1421 K by Chicco and Thorpe [64] give tight conditions to determine the remaining parameter in Gibbs energy of cementite, i.e. the E0 parameter in Eq. (10).

The heat capacity parameters of χ and η carbides were obtained using the method described in Section 2.2.3. In addition, the tempering reactions in martensite according to Nagakura et al. [9] (i.e. Fe5 C2 þ Fe ¼ 2Fe3 C at 720 K and 2Fe2 C þ Fe ¼ Fe5 C2 at 470 K) give enough constraints to determine the remaining parameters in Eqs. (15) and (16), i.e. a and b parameters. The only available experimental point of cubic to tetragonal transition by Xiao et al. [58] is sufficient to evaluate the ordering parameter at room temperature. As it was discussed above, the ordering of carbon atoms results in spinodal decomposition and formation of a modulated tweed-like microstructure consisting of low and high-carbon regions (A2). However, it is found in this work that a temperature independent ordering parameter would lead to a miscibility gap up to very high temperatures. Whereas, experimentally it is observed that the modulated structure disappears at about 370 K and η-Fe2 C begins to form (T1) [9], see Table 2. This condition together with the critical C concentration at room temperature enables us to obtain a temperature dependent ordering parameter. The parameter values obtained from the optimization are summarized in Table 3.

5. Results and discussion The heat capacities of carbon and diamond calculated with the present set of parameters are presented in Fig. 1 in comparison with the experimental data [65,66]. The experimental heat capacity data are well reproduced by the present description. The calculated stable and metastable phase diagrams from the present evaluation are presented in Fig. 2 together with the stable and metastable phase diagrams calculated from Gustafson's [4] set of parameters. Calculated carbon activities in the fcc are shown in Fig. 3 together with the calculated values from experimental data on gas equilibria [67,68]. The present evaluation is in good agreement with the experimental data. In comparison, Gustafson's evaluation predicts slightly lower carbon solubility at higher temperatures. Fig. 4 gives a comparison between the calculated carbon activities in bcc and the experimental data calculated from the gas equilibria [69,70]. In comparison to Gustafson's evaluation, the present evaluation predicts slightly higher carbon

Table 2 Tempering reactions in steel. Reaction and symbol (if designated)

Temperature range (K)

Comments

Clustering of carbon atoms on octahedral sites of martensite (A1); segregation of carbon atoms to dislocations and boundaries Modulated clusters of carbon atoms on (102) martensite planes (A2)

233–373 [11] 298–373 [49]

Associated with diffuse spikes around fundamental electron diffraction spots of martensite

o 370 [9] 293–373 [11] 333–423 [49] 333–353 [11]

Identified by satellite spots around electron diffraction spots of martensite

Long period ordered phase with ordered carbon atoms arranged (A3) Precipitation of (ε/η) transition carbides as aligned 2 nm diameter particles (T1)

Transformation of retained austenite to ferrite and cementite (T2) Precipitation of χ transition carbides Formation of α and θ-cementite; eventual development of well spheroidized carbides in a matrix of equiaxed ferrite grains (T3)

Formation of alloy carbides in Cr, Mo, V, and W containing steels. The mix and composition of the carbides may change significantly with time (T4) Segregation and co-segregation of impurity and substitutional alloying elements

370–470 [9] 373–473 [11] 473–513 [33] 423–498 [49] 363 o [50] 350–430 [51] 353–453 [52] 473–623 [11] 470–720 [9] 513–743 [33] 720o [9] 743o [33] 523–973 [11] 470–550 [51] 498–548 [49] 773–973 [11]

623–823 [11]

Identified by superstructure spots in electron diffraction patterns Recent work identifies carbides as η-Fe2 C (orthorhombic,); earlier studies identified the carbides as ε (hexagonal)

Associated with tempered martensite embrittlement in low and medium carbon steels

Appears to be initiated by χ formation in high carbon Fe–C alloys

The alloy carbides produce secondary hardening and pronounced retardation of softening during tempering or long time service exposure around 773 K Responsible for temper embrittlement

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153

Table 3 Thermodynamic parameters (in SI units and with R ¼8.31448). Graphite ðCÞ1 3 2 1Ggra T  2:1662 10  5 T 3 þ 3:0953 10  11 T 5 þ GEINð2219:2Þ C ¼  28725  1:786 10

1Ggra C ¼  17; 368:441 þ 170:73 T  24:3 T LnðTÞ  4:723 10  4 T 2 þ 2; 562; 600 T  1  2:643 108 T  2 þ 1:2 1010 T  3 Diamond ðCÞ1 3 2 1Gdia T  1:3059 10  5 T 3 þ þ 1:9750 10  11 T 5 þ GEINð1571:63Þ C ¼  18; 646 þ 1:807 10

1Gdia C ¼  16; 359:441 þ 175:61 T  24:31 T LnðTÞ  4:723 10  4 T 2 þ 2; 698; 000 T  1  2:61 108 T  2 þ 1:11 1010 T  3

0.01o T o 298.15 298.15 o T o6000.00

0.01o T o 298.15 298.15 o T o6000.00

Bcc ðFeÞ1 ðC; VaÞ3 0.01o T o 6000.00

1Gbcc Fe:C ¼ 162; 505 þ 184:465 T 0 bcc LFe:C;Va

¼ 150; 000  290 T

T C ¼ 1043; 〈β〉bcc ¼ 2:22; p ¼ 0:37 Zener-ordered bct ðFeÞ1 ðC; VaÞ1 ðC; VaÞ2 0.01o T o 6000.00

bcc 1Gbct Fe:C:C ¼ 1GFe:C bcc bcc bcc 1Gbct Fe:C:Va ¼ 1=31GFe:C þ 2=31GFe:V a þ 2=9LFe:C;Va  ð 1; 146; 502  2911:105 TÞ bcc bcc bcc 1Gbct Fe:Va:C ¼ 1=31GFe:Va þ 2=31GFe:C þ 2=9LFe:C;Va  ð1; 146; 502  2911:105 TÞ 0 bct LFe:C;Va:n 0 bct LFe:n:C;Va

¼ 1=9Lbcc Fe:C;Va þ ð1; 146; 502  2911:105 TÞ ¼ 4=9Lbcc Fe:C;Va þ ð1; 146; 502  2911:105 TÞ

T C ¼ 1043; 〈β〉bct ¼ 2:22; p ¼ 0:37 α″-ordered bct ðFeÞ1 ðC; VaÞ1=8 ðC; VaÞ23=8 0.01o T o 6000.00

bcc 1Gbct Fe:C:C ¼ 1GFe:C bcc bcc bcc 1Gbct Fe:C:Va ¼ 1=241GFe:C þ 23=241GFe:V a þ 23=576LFe:C;V a  ð28; 797  52:983 TÞ bcc bcc bcc 1Gbct Fe:Va:C ¼ 1=241GFe:Va þ 23=241GFe:C þ 23=576LFe:C;V a  ð28; 797  52:983 TÞ 0 bct LFe:C;Va:n 0 bct LFe:n:C;Va

¼ 1=576Lbcc Fe:C;Va þ ð28; 797  52:983 TÞ ¼ 529=576Lbcc Fe:C;Va þ ð28; 797  52:983 TÞ

T C ¼ 1043; 〈β〉bct ¼ 2:22; p ¼ 0:37 Fcc ðFeÞ1 ðC; VaÞ1 cc 1GfFe:C ¼ 72; 729  15:212 T 0 f cc LFe:C;Va

0.01o T o 6000.00

¼  30; 532

T N ¼ 67; 〈β〉f cc ¼ 0:7; p ¼ 0:25 Liquid–amorphous ðFe; CÞ1 0 liquid LFe;C 1 liquid LFe;C 2 liquid LFe;C

¼  102; 958 þ 18:236 T

0.01o T o 6000.00

¼ 87; 869  34:089 T ¼ 111; 085  50:533 T

Cementite ðFeÞ3 ðCÞ1 3 2 1Gcem T þ GEINð320:91Þ þ GMDOð1Þ þ GMO Fe:C ¼  1900  2:808 10 T C ¼ 485; 〈β〉θ ¼ 2:44; p ¼ 0:25

Hägg ðFeÞ5 ðCÞ2 5 fm θ 1GM5 Gm þ 13 Ggra Fe:C ¼ 3 m þ GMDOð1Þ þ GMO þ 2554 T C ¼ 520; 〈β〉χ ¼ 6:839; p ¼ 0:25

0.01o T o 6000.00

0.01o T o 6000.00

Eta ðFeÞ2 ðCÞ1 2 fm θ 1GM2 Gm þ 13 Ggra Fe:C ¼ 3 m þ GMDOð1Þ þ GMO þ 3904:8 T C ¼ 663; 〈β〉η ¼ 1:278; p ¼ 0:25

0.01o T o 6000.00

Auxiliary functions GEINðΘE Þ ¼ 1:5RΘE þ 3RT ln ½1  expð ΘE =TÞ h i TC GMDOð1Þ ¼  R lnð〈β〉 þ 1Þ T  0:38438376pD   D ¼ 0:33471979 þ 0:49649686 1p  1 GMO ¼ RTh lnð〈β〉 þ 1Þf ðτÞ   3 i 1 τ9 τ15 τ21 f ðτÞ ¼ 1 0:38438376τ p þ 0:63570895 1p  1 τ6 þ 135 þ 600 þ 1617 =D h  7  i  21 τ  35 τ  49 =D þ 8232 f ðτÞ ¼  τ21 þ τ630 þ 2975

solubility in ferrite at low temperatures as shown in Fig. 5 compared with experimental data [70–73]. Fig. 6 gives a comparison between the experimental data [63,68,71,74–76] and calculated eutectoid part of the stable and metastable phase diagram. The calculated graphite and cementite solubilities are in good agreement with experimental data. The calculated g/(g þa) boundary, i.e. the A3 line, is somewhat lower compared to Gustafson's evaluation, and in better agreement with the experimental data from Mehl and Wells [74]. The eutectoid temperature is remained

τ ¼ T=T C τr1 τ41

unchanged at 1011 K. The metastable eutectoid temperature, 1001 K, is one degree higher than Gustafson's estimation (1000 K) but in good agreement with the value of 1000.8 70.4 K reported by Smith and Darken [77]. Fig. 7 shows the peritectic part of the phase diagram. The calculated peritectic part is essentially unchanged compared to Gustafson's evaluation and in reasonable agreement with the experimental data [1,64,68,78,79]. The calculated austenite/liquid phase boundaries and the graphite solubility in liquid are shown in Fig. 8 in comparison with experimental data [1,64,68,78–85].

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Fig. 4. Carbon activities in bcc in comparison with experimental data [69,70]. Fig. 1. Heat capacity of graphite and diamond in comparison with data from JANAF thermochemical tables [48] and experimental data [65,66].

Fig. 5. The solubility of carbon in bcc in comparison with experimental data [70– 73].

Fig. 2. The stable and metastable phase diagrams from the present evaluation in comparison with the stable and metastable phase diagrams from Gustafson's [4] evaluation.

Fig. 6. The eutectoid part of the stable and metastable phase diagram together with experimental data [63,68,71,74–76].

Fig. 3. Carbon activities in fcc in comparison with experimental data [67,68].

It can be seen that the present evaluation is in better agreement with the phase boundary data compared to Gustafson's evaluation. Fig. 9 shows the solubility of graphite in liquid at higher temperatures in comparison with experimental data [80–87]. In contrast to Gustafson's evaluation, the rapid increase of the solubility at higher temperatures is well reproduced by the present evaluation in good

R. Naraghi et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 148–158 agreement with measurements of Cahill et al. [86]. The carbon activities in liquid [88–91] are also well reproduced as shown in Fig. 10. The present description yields an inverse miscibility gap in liquid at temperatures higher than 4000 K in the absence of the gas phase. This phenomena in addition to high temperature appearance of bcc is discussed recently by Shubhank and Kang [92]. The same issue was also discussed by Hallstedt and Djurovic [93]. The description of the gas phase was not included in the present work. However, by considering the the evaporation temperature of iron at 1 bar (  3160 K) and sublimation temperature of graphite (  3972 K) it can be imagined that these phenomena in liquid at temperatures higher than about 4000 K are not of practical interest. The calculated heat capacity of cementite, available experimental data [29,60– 62], and results of previous thermodynamic assessments are presented in Fig. 11. The experimental data of Naeser [60] from 78 to 1023 are in good agreement with results of Seltz et al. [61] in the temperature range of 68 to 298. The Andes data [62] from 102 to 323 K are relatively lower. The heat capacity measured by Umemoto et al. [29] shows a large discrepancy. Their data are in fair agreement up to the Curie temperature, but show a sharp increase above 500 K. Data of Naeser [60] and ab-initio calculations of Dick et al. [94] show a local minimum at about 600 K and a gradual increase at higher temperatures. The latest assessment of cementite's CP by Hallstedt et al. [5] coincides with the experiments of Seltz et al. [61] in the temperature range of 68 to 298 K but it was kept rather constant above 500 K in order to induce minimum change in high temperature equilibria as given by Gustafson [4] in the temperature range 1000 to 1500 K. The result of the present evaluation falls within experimental scatter of various measurements below 350 K [29,60–62] and in excellent agreement with the experimental data of Naeser [60] above 350 K. Fig. 12 shows the Gibbs energy of formation of cementite, Hägg, and eta carbide (per formula unit with 1 mol of carbon) as a function of temperature. Browning

155

Fig. 9. Calculated solubility of graphite in liquid at high temperatures in comparison with experimental [80–87].

Fig. 10. Carbon activities in liquid in comparison with experimental data [88–91].

Fig. 7. The peritectic part of the phase diagram together with experimental data [1,64,68,78,79].

Fig. 11. Heat capacity of Fe3 C in comparison with experimental data [29,60–62], ab-inito calculations [94] and previous assessments [4,5].

Fig. 8. The eutectic part of the stable and metastable phase diagram together with experimental data [1,64,68,78–85].

et al. [95] have measured the equilibrium constants for formation of cementite and Hägg carbide from iron in high carbon activity H2 =CH4 atmosphere. Hägg carbide is denoted as having approximate composition Fe2 C in their work; however, as it was

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discussed in Section 2.2.3, later investigations led to the conclusion that the Hägg carbide should have the composition Fe5 C2 [40]. The formation reactions can thus be written as CH4 ðgÞ þ 3Fe ¼ Fe3 C þ 2 H2 ðgÞ

ð29Þ

CH4 ðgÞ þ 52 Fe ¼ 12 Fe5 C2 þ 2 H2 ðgÞ

ð30Þ

Browning et al. calculated the energies of formation of both carbides on the basis of the ΔC P for reaction (29) reported by Kelley [96] and assuming that reaction (30) has the same ΔC P . Schneider and Inden [97] determined the Gibbs energy of formation of Hägg carbide at 773 K and calculated the Gibbs energy of formation for cementite and Hägg carbide from the gas equilibria constants for Eqs. (29) and (30) from the experiment of Browning et al. [95] and reaction (26) as

ΔGðFe3 CÞ ¼ ΔGð29Þþ ΔGð26Þ

ð31Þ

ΔGð12 Fe5 C2 Þ ¼ ΔGð30Þ þ ΔGð26Þ

ð32Þ

It was found in this work that an outdated equilibrium constant for reaction (26) was used in Schneider and Inden's [97] work and the calculated values were not in agreement with those reported by Browning et al. [95] based on ΔC P for reaction (29). Therefore, the formation energies were recalculated by using the equilibrium constant for reaction (26) given in JANAF thermochemical tables [48] and a good agreement between two sets of data was achieved as shown in Fig. 12. Although these data are not used in present evaluation, a perfect agreement between the recalculated data of Browning et al. [95] for cementite and the present evaluation is found in the temperature range of 600–800 K. Below 600 K data of Browning et al. [95] predicts a lower energy of formation. The intersection of the lines for cementite and Hägg carbide defines the triple point between α/χ/θ below which Hägg carbide is more stable than cementite. Similarly the intersection of the Hägg and eta carbides gives the α/η/χ triple point below which eta carbide is more stable than Hägg carbide. In the present evaluation these points are located at 720 K and 470 K respectively according to the tempering reactions in martensite given by Nagakura et al. [9]. The recalculated data for Hägg carbide show a rather large scatter. A line drawn through the data of Browning et al. [95] would pass through the result of Schneider and Inden [97] at 773 K. However, such a line would intersect the current cementite line at much lower temperature and predict a α/χ/θ triple point around 550 K which is thought to be low compared to Nagakura et al.'s experiment [9]. Energy of formation of cementite, Hägg, and eta carbides (per formula unit with 1 mol of carbon) given by ab-initio calculations [35,44,98–103] at 0 K is also shown in Fig. 12. In general, ab-initio calculations somewhat underestimate the energy of formation compared to the present thermodynamic evaluation. However, the difference is considered to be within the limits of current ab-initio calculations. Moreover, ab-initio calculations also confirm a trend of stability from high to low as η 4 χ 4 θ at 0 K as it is predicted from the present thermodynamic evaluation. Gibbs energy curves for disordered bcc, Zener-ordered, and α″-ordered Fe–C solutions at room temperature with bcc iron and graphite as the reference are shown in Fig. 13. The ordered Gibbs energy curves contains two inflection points, denoted by X 1s and X 2s i.e. spinodal decomposition limits. The X 1s corresponds also to the critical order/disorder transition composition. At compositions lower than this limit the ordered and disordered curves become identical. The miscibility gap boundaries are denoted by X 1e and X 2e as determined by the common-tangent construction. The X 2s and X 2e for Zener-ordered curve will be close to FeC composition or 50 at% C (17.7 wt% C) which falls outside the scale of the graph.

Fig. 12. Gibbs energy of formation of cementite, Hägg and eta carbides per formula unit with 1 mol of C, versus temperature. Together with calculated energies of formation by ab-initio at 0 K [35,44,98–103], and experimental data [95,97].

Fig. 14 gives an overview of the Zener-ordering temperature given by various authors [7,21,22,53–56,104]. The only experimental critical composition of cubic to tetragonal transition [58] at room temperature is plotted as a filled circle. Previous theoretical models of Zener-ordered martensite have some shortcomings. First of all that the equilibrium state of spinodal decomposition consists of a mixture of almost pure bcc iron and the compound FeC and the secondary ordering as the result of longer-range atomic interactions are not treated. Previous estimations also neglect the effect of temperature on the interactions which leads to the stability of spinodal decomposition ground state up to very high temperature. It is possible to destabilize the spinodal decomposition compared to eta carbide formation at temperatures above 370 K [9] by using a temperature dependent ordering parameter, even though such high carbon regions have never been experimentally observed. TEM, APFIM [24], and 3DAP [25] measurements of carbon content of high-carbon regions of the spinodal decomposition structure show about 11 at% C (2.59 wt% C). Sinclair et al.'s molecular dynamics study of ordering of carbon atoms in octahedral sites [105] also shows that there is a minimum in C–C interaction energy in Fe1  x Cx at 11 at% C for the fully ordered structure. The computed separation distance also corresponds exactly to the octahedral positions expected in α″-Fe16 C2 . In contrast to Zener-ordering, α″-type ordering gives a more consistent picture with the experimental and molecular dynamics studies. The predicted α″ order/disorder transformation and the chemical spinodal decomposition curves are shown in Fig. 15. It must be mentioned that above reactions are conditional and a martensitic transformation is required to produce the initial compositional instability. Therefore, for comparison the Ms temperature data [106–111] are also depicted in Fig. 15. Presence of two Ms temperatures at low carbon contents were accepted after Borgenstam and Hillert [112]. Interpretation of atomic changes in martensite is rather difficult due to the complex structural changes and discrepancies about the martensite growth rate

Fig. 13. Gibbs energy curves for disordered, Zener ordered, and α″ ordered Fe–C solutions with bcc iron and graphite as the reference at 300 K.

Fig. 14. Calculated Zener-order/disorder transformation curves from the present evaluation and previous estimations in literature [7,21,22,53–56,104].

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Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.calphad.2014.03. 004. References

Fig. 15. Calculated α″-order/disorder transformation and the chemical spinodal decomposition curves together with the experimental Ms temperatures [106–111]. and the transformation path. It is interesting to note that the predicted order/ disorder curve intersects with Ms temperatures at approximately 1 wt% carbon. (i.e. the same composition as the lath to plate martensite structural change in ordinary quenching experiments). The Ms temperatures associated with plate martensite at low carbon contents are observed as a thermal arrest temperature during ultra-rapid quenching experiments. Theoretically one atomic jump is enough for carbon atoms to go from the ordered to disordered state. On the basis of the present evaluation for compositions below 1 wt% carbon, the disordered cubic state is thermodynamically stable at the Ms temperature. Therefore, with the normal quenching rates it seems possible for carbon atoms to gain the disordered distribution which leads to the cubic structure. By lowering the temperature the ordered state becomes stable which is responsible for the observed tetragonality at room temperature. At ultra-high cooling rates though, there is no time for carbon rearrangement and the carbon atoms may inherit their original positions from the austenite which would lead the tetragonality of martensite. Above 1 wt% carbon, at Ms and below, the tetragonal ordered state is more stable than the cubic.

6. Conclusions Thermodynamic properties of metastable iron carbides, cementite, Hägg and eta carbides were evaluated by taking into account the magnetic properties of these carbides and the experimental data from different stages of tempering of martensite. The proposed model for ordering of carbon atoms in martensite resolves the two main drawbacks of previous models of Zener-ordering, namely formation of a FeC compound as a result of spinodal decomposition and stability of it relative to the transition carbides. According to the new model, the ordered martensite will spinodaly decompose at room temperature to carbon free and carbon rich regions with Fe16 C2 composition, consistent with the experimental observations in literature. The spinodal structure will however disappear when the transition carbides form at higher temperatures. In comparison with the previous evaluation of the binary system by Gustafson [4], reproduction of high temperature solid/liquid phase equilibria was also improved.

Acknowledgments The work was performed within the VINN Excellence Center Hero-m, financed by VINNOVA (Grant number 2012-02892), the Swedish Governmental Agency for Innovation Systems, Swedish industry, and KTH Royal Institute of Technology. The authors gratefully acknowledge Prof. Annika Borgenstam for valuable discussions about the martensitic transformation. The authors also would like to thank Prof. Bo Sundman for helpful comments regarding the thermodynamic models.

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