1 Introduction

Needless to say, iron (Fe) is an element of great interests for academic studies and of immense importance for industrial applications. The fact that Fe adopts the bcc structure at the ground state, transits to the fcc structure as temperature increases to 1185 K, and then returns to the bcc structure above 1667 K is a scientifically fascinating phenomenon, which we all know now is due to the intricate play of the magnetism of Fe. In order to accurately predict the structural transition temperature, enthalpy, and entropy, a good thermodynamic description of all the structures or phases is necessary, i.e. the temperature dependent Gibbs energy of each phase needs to be well expressed. The Gibbs energy of a structure can be constructed by considering various physical contributions. For a substance exhibiting magnetic behaviour, the magnetic contribution to the Gibbs energy needs to be properly described and knowledge about the magnetic moment and magnetic transition temperature is the prerequisite.

Fe has the highest magnetic moment among the transition metals and has been applied in areas requiring high magnetisation.[1] The magnetic properties of Fe have therefore been extensively studied.[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] At ambient pressure, fcc Fe is only stable at high temperatures, from 1185 to 1667 K. In order to experimentally study magnetism of fcc Fe at the ground state, it needs to be artificially stabilised and different methods to do this will be discussed.

Modelling the thermodynamic properties with more physical meaning was first put forward at the 1995 Ringberg workshop,[20,21,22,23] where more physically-based models were recommended for both solids and liquid. Over the past years, the models have been tested and developed including choices and treatments of model parameters.[24,25,26,27] Alongside this, a number of thermodynamic descriptions of unary and binary systems have been assessed for the next generation of Calphad descriptions.[27,28,29,30,31,32,33,34] The first attempt to use the suggested models was for describing the thermodynamic data of Fe.[35] However, that thermodynamic description was assessed at the very early stages of developing the new generation of Calphad descriptions. Therefore, the recent suggestions were not considered in that assessment.

Considering the updated magnetic properties of fcc Fe and the updated models for the third generation of Calphad, the thermodynamic descriptions of fcc, hcp and liquid Fe were here re-assessed. The revised thermodynamic models were used and the new treatments of the model parameters were considered. Moreover, the new extrapolation method was used for the solid phases, i.e. bcc, fcc and hcp.

2 Literature Review

2.1 Previous Assessment

The thermodynamic description of pure Fe accepted in the SGTE compilation[36] was given by Fernández Guillermet and Gustafson.[37] In that work, they assessed the thermodynamic properties of the bcc, fcc, hcp and liquid phases of pure Fe and analysed the (P,T) phase diagram of Fe. Later, Chen and Sundman[35] provided a thermodynamic description including bcc, fcc and liquid of pure Fe for the third generation of Calphad descriptions. In their assessment, the physically-based models suggested at the 1995 Ringberg workshop[20,21] were used to model the thermodynamic properties. Bigdeli et al.[38] adopted the equilibrium temperatures of hcp/fcc and hcp/liquid assessed by Fernández Guillermet and Gustafson when modelling the hcp phase and used the models presented by Chen and Sundman to assess the thermodynamic data of hcp Fe. However, as already mentioned, the models have been further developed since then including new treatment of the model parameters. We therefore re-assessed the descriptions of fcc, hcp and liquid, and applied the new extrapolation method suggested by He et al.[27] to the solid phases.

2.2 Selected Datasets

There have been a tremendous number of studies on the thermodynamic properties of pure Fe. Comprehensive reviews on these data can be found in the work by Fernández Guillermet and Gustafson[37] and by Chen and Sundman[35] and will not be repeated here. Only the data which were considered for the present assessment will be discussed.

2.2.1 Selected Data for fcc

Desai[39] recommended the values for the enthalpy and entropy of the transformation between bcc and fcc at 1185 and 1667 K, listed in Table 1 and accepted in the present work. Besides, the heat capacity data of fcc Fe measured by Wallace et al.[40] and Cezairliyan and McClure[41] were recommended by Desai and thus accepted in the present work. These heat capacity data are comparable and coincide almost exactly with the calculated values using the description by Chen and Sundman. Anderson and Hultgren[42] measured the enthalpy of fcc and adopted the experimental heat capacity data from Pallister et al.[43] and Awbery et al.[44] to obtain the functions for enthalpy, entropy and heat capacity for fcc. Orr and Chipman[45] recommended the Gibbs energy, enthalpy and entropy of the transformation between bcc and fcc at their transition temperatures. Their recommended data were based on the measured enthalpy and functions provided by Anderson and Hultgren,[42] and experimental measurements by Wallace et al. [40] and Dench.[46] We need to point out that Orr and Chipman[45] tabulated the enthalpy data beyond the temperature range where fcc is stable by extrapolating the heat capacity data. We therefore decided to exclude these tabulated data and only adopt the data recommended by Orr and Chipman[45] within the temperature range where fcc is stable for optimizing the model parameters whereas the other data were used only for comparison.

Kaufman et al.[47] gave a comprehensive analysis of the thermodynamic and magnetic properties of fcc and bcc considering different contributions, such as magnetism and lattice vibrations. A conclusion was drawn that the electronic heat capacity coefficient and the high-temperature entropy-Debye temperature of fcc are approximately equal to those of bcc. Grimvall[48] conducted a similar analysis and gave the following relations for the electronic heat capacity coefficients and the high-temperature entropy-Debye temperatures between fcc and bcc: \(\gamma_{\rm el}^{\rm fcc}\approx \gamma _{\rm el}^{\rm bcc}\) and \(\theta _D^{\rm fcc}/\theta _D^{\rm bcc}=0.99\). These relations were taken into account when assessing the electronic heat capacity coefficient and the Einstein temperature of fcc.

Table 1 Comparison between the assessed and the experimental enthalpy and entropy for bcc and fcc phase transformations
Full size table

To experimentally investigate the ground state magnetic properties of fcc Fe, artificial stabilisation is needed. Traditionally, there are two common methods to do this: 1) precipitate fcc Fe in a Cu matrix; 2) grow fcc Fe thin films on a Cu substrate. In addition to these two experimental methods, a third method based on density functional theory (DFT) calculations has recently been successfully applied to theoretically predict the magnetic properties of different structures of Fe. In the following, these three methods will be discussed in order to justify the selected magnetic properties of fcc and hcp Fe in the present assessment work.

Method 1) has been used in a number of investigations[2,3,4,5] to obtain the magnetic properties of fcc Fe from fcc Fe precipitates in a Cu matrix. The Néel temperature was given as 67 K and the magnetic moment was reported to be 0.7 \(\mu _B\). These magnetic properties were adopted by Fernández Guillermet and Gustafson[37] and accepted in the SGTE compilation[36] and also by Chen and Sundman.[35] In these studies it was pointed out that the lattice constants of the fcc Fe precipitates and the Cu matrix are different, which causes coherence stress. This stress can be lowered by repositioning the atoms. Consequently, the lattice constant of the crystal structure is lowered, which will affect the magnetic spin. The stress also depends on the size of the precipitates and is lowered as the size increases. Moreover, as the precipitate grows, the magnetic moment increases. In order to minimize the incurred strain energy, the coherency between the precipitates of fcc Fe and the Cu matrix will gradually be broken as the precipitate size increases. Eventually, a martensitic transition occurs as the precipitate continues to grow and reach the upper limit size, i.e. the bcc structure forms.[4, 49] This raises the question whether the magnetic properties of the finely-dispersed precipitates can represent the properties of bulk fcc Fe. Later, Tsunoda[50] attempted to keep the fcc structure non-distorted for larger precipitates by adding small amounts of Co. He claimed that fcc Fe\(_{100-x}\)Co\(_x\) precipitates are independent of the particle size except for extremely small particles (d\( \approx 50\) nm) and considered this as an indication that the magnetism of fcc Fe\(_{100-x}\)Co\(_x\) precipitates is almost like that of bulk fcc Fe. The experimental results showed that fcc Fe has a non-collinear helical spin structure, which was later confirmed by a theoretical study.[51] The reported Néel temperature was 50 K with a magnetic moment of 0.5 \(\mu _B\).[50] However, the effects on the magnetic properties due to alloying elements are complex and may not be ignored.

Method 2), i.e. growing thin films of fcc Fe on a Cu substrate, was used in several studies.[6,7,8,9,10,11] Among these studies, a general conclusion was that the magnetic state of fcc Fe thin films changes with increasing thickness of the films. A ferromagnetic high-spin (FM-HS) state was observed with thinner layers while a ferromagnetic low-spin (FM-LS) or antiferromagnetic (AFM) states were observed in thicker layers with undistorted fcc structure. Moreover, the magnetic moment depends on the obtained lattice constant of fcc Fe in the thin films. The magnetic properties obtained by investigating the grown thin films were contradictory. The paramagnetic (PM) state was found by Macedo and Keune[6] at 295 K while the AFM state was found below 80 K. Shinjo and Keune[11] suggested the Néel temperature to be  70 K. Li et al.[7] detected the magnetism of fcc Fe as a function of both thin film thickness and the substrate temperature during the growth of the film. They reported that bulk fcc Fe is in an AFM state with a Néel temperature of  200 K. This result was obtained by measurements on 6-11 monolayers of fcc Fe grown at room temperature. They also suggested that the magnetic moment of fcc Fe may be considerably larger than 0.7 \(\mu _B\) obtained from the fcc Fe precipitates in a Cu matrix. The contradictory results obtained from fcc Fe thin films can be attributed to the various substrate temperatures during the growth of the films.

Method 3), DFT calculations, was used in.[12,13,14,15] The most recent theoretical study on the magnetism of fcc Fe was performed by Haglöf et al.[52]. They calculated the magnetism of fcc Fe in non-collinear spin spiral (SS), ferromagnetic (FM), single-layered antiferromagnetic (AFM-I) and double-layered antiferromagnetic (AFM-D) structures. Besides, the AFM-D collinear structure was also accounted for in their calculations for comparison with the previous studies by Marsman and Hafner,[12] and Sjöstedt and Nordström.[13] The energy they obtained for the SS structure is in good agreement with those obtained in,[12, 13] and the energy for collinear AFM-D structure agrees with that obtained by Sjöstedt and Nordström.[13] Furthermore, Haglöf et al.[52] found that the energy of the AFM-D structure is slightly lower than that of the SS structure, which was thus concluded to be the magnetic ground state for fcc Fe with a Néel temperature of 192 K and a magnetic moment of 1.77 \(\mu _B\).

As discussed above, the precipitates of fcc Fe in a Cu matrix tend to distort the crystalline and magnetic structures. Besides, it has been shown that the magnetism changes with increasing particle size. One cannot use the magnetic properties obtained from the small precipitates to represent those of bulk fcc Fe. Even though alloying the Cu matrix with Co can suppress the structural transitions, it decreases the lattice constant of the precipitates while increases the lattice mismatch between the precipitates and the matrix. As a consequence, the Néel temperature is reduced. Moreover, the alloying effects on the magnetic properties are complex, e.g. as Marsman and Hafner discussed,[12] a different observation was found when alloying the Cu matrix with Co from alloying it with Au. Even though the mismatch between the precipitates and the matrix is increased by alloying Au, the Néel temperature increases instead of decreasing. The magnetic properties of fcc Fe thin films strongly depend on the thickness of the layers and the growth temperature of the Cu matrix. It also has a strong tendency to form a distorted crystal structure. Theoretical calculations can provide a definite minimum energy and a definite axial ratio, which can govern the corresponding symmetries of the magnetic and crystal structures. Therefore, we decided to accept the magnetic properties obtained by theoretical calculations. The non-collinear AFM-D structure was not considered by Marsman and Hafner or by Sjöstedt and Nordström but it was later found to be the stable magnetic ground state for fcc Fe by Haglöf et al. As already noted, the results for SS and collinear FM structures reported Haglöf et al. agree well with those obtained by Marsman and Hafner and Sjöstedt and Nordström, which confirms the soundness of the calculation setup used by Haglöf et al. Thus, we decided to accept the results on magnetism provided by Haglöf et al.

2.2.2 Selected Data for hcp

Hcp Fe is metastable at ambient pressure. The methods to obtain information on hcp Fe are based on extrapolations from binary systems or from high-pressure experimental measurements. Fernández Guillermet and Gustafson[37] analysed existing studies on the vibrational contributions to the Gibbs energy and assumed that the electronic contribution and the heat capacity of hcp Fe equals that of fcc Fe. Based on these assumptions, they obtained the description of hcp Fe. The enthalpy part of the lattice stability between hcp and fcc was given as −2243.38 J/mol while the difference in entropy was given as −4.309 J/(mol K). These values were adopted to model the thermodynamic data of hcp Fe in the present work.

Fernández Guillermet and Gustafson[37] treated hcp as a non-magnetic phase while Bigdeli et al.[38] took into account the contribution from magnetism in their assessment. Bigdeli et al. adopted the magnetic properties given by Ohno,[53] which were extrapolated from three binary systems containing Fe, i.e. Fe-Ru, Fe-Os and Fe-Mn. Ohno[53] concluded that hcp is antiferromagnetic with \(T_N=100\) K and \(\beta =0.1\) \(\mu _B\). It is worth noting that in the work by Ohno the lattice parameters and the c/a ratios of hcp Fe extrapolated from these three binary systems are different. Besides, the Néel temperatures extrapolated from Fe-Ru and Fe-Os show close agreement while the Néel temperature extrapolated from Fe-Mn is considerably higher. The Néel temperature extrapolated from Fe-Ru and Fe-Os was accepted by Ohno, \(T_N=100\) K. However, there was no explicit explanation given by Ohno why the Néel temperature extrapolated from Fe-Mn is significantly higher than the one extrapolated from the other two systems. Besides, the magnetic interactions between the alloying elements, Ru, Os and Mn, and Fe were not considered when extrapolating the Néel temperature. In addition, Ohno used neutron diffraction to detect the magnetic moment of Fe\(_{0.8}\)Ru\(_{0.2}\). However, no detectable scattering was observed. As the lower limit to observe the detectable scattering is 0.2 \(\mu _B\), they estimated the magnetic moment of Fe\(_{0.8}\)Ru\(_{0.2}\) to be 0.1 \(\mu _B\), which they also assumed to be the magnetic moment of hcp Fe. It is uncertain how the alloying elements affect the crystal symmetry and the magnetic properties of hcp Fe. As mentioned above, DFT calculations can be performed on a rigid lattice with definite minimum energy, which governs the crystal structure and are a better solution compared to experiments to investigate such a metastable phase. Furthermore, the calculation setup by Haglöf et al.[52] was confirmed to be correct by calculations on the stable phases and their calculated magnetic moment for hcp Fe shows close agreement with the one given by Bigdeli et al.[38] (\(\beta =1.15\) \(\mu _B\)). Thus, we decided to adopt the calculated magnetic properties of hcp Fe by Haglöf et al., i.e. an AFM magnetic state with \(T_N=310\) K and \(\beta =0.98\) \(\mu _B\).

2.2.3 Selected Data for Liquid

The enthalpy of melting accepted in the present work was provided by[54,55,56,57] and their data are comparable. Additionally, Desai[39] assessed the thermodynamic properties from[55,56,57,58] for liquid Fe at the melting temperature. The recommended values given by Desai are listed in Table 2, and were adopted in the present work. Chen and Sundman[35] reviewed the different studies on the high-temperature entropy-Debye temperatures of the ideal amorphous phase of Fe and adopted the value 343 K. Based on the relation between the Einstein temperature and high-temperature entropy-Debye temperature,[59] they obtained 245 K as the Einstein temperature of ideal amorphous Fe. This value was accepted in the present work.

Table 2 Calculated thermodynamic properties of liquid compared to the experimental data and previous Calphad descriptions
Full size table

Chen and Sundman[35] comprehensively analysed different measurements and estimations on the magnetic properties of amorphous Fe. They concluded that it is appropriate to assume \(T_C=200\) K and \(\beta =1.7\) \(\mu _B\) for amorphous Fe. As shown in their assessment, their magnetic description of amorphous Fe can give a reasonable representation of the contribution to the Gibbs energy. Thus, their selected magnetic properties of amorphous Fe were adopted in the present work.

3 Thermodynamic Models

He and Selleby[60] provided a comprehensive review of the thermodynamic models of non-magnetic elements for the third generation Calphad and this will not be repeated here. Instead, the focus will be given to the discussion on how to describe the magnetic contribution to the thermodynamic properties.

3.1 Modelling the fcc Phase

To model the thermodynamic properties of a solid crystalline phase like the fcc phase, different contributions are considered and described individually.[20, 22, 35] The expression of the heat capacity for magnetic crystalline solid phases is given as follows:

$$\begin{aligned} C_P=3R \left( \frac{\theta _E}{T}\right) ^2\frac{e^{\theta _E/T}}{(e^{\theta _E/T}-1)^2}+aT+bT^n+C_P^{\rm mag} \end{aligned}$$
(1)

where the first term is the contribution from the harmonic lattice vibrations described using the Einstein model; the second term consists of the contributions from the electronic excitations, low-order anharmonic lattice vibrations and the correction from \(C_V\) to \(C_P\) associated with the variation of volume; the third term is the contribution from high-order anharmonic lattice vibrations; the fourth term is the contribution due to magnetism. n can be selected among 2, 3, and 4. Moreover, based on the shape of the heat capacity curve, more \(T^n\) terms can be added.

Following the suggestion by Inden,[61] the hypothetical paramagnetic ground state needs to be used as the reference state in considering the composition dependence of the magnetic contribution in binary and higher order systems for the purpose of avoiding unrealistic high temperature miscibility gaps. With this consideration and by integration, the Gibbs energy expression can be obtained from Eq 1 and given by Chen and Sundman[35] as follows:

$$\begin{aligned} \begin{aligned} G=&\,E_0+\frac{3}{2}R\theta _E+3RT\ln \left[1-\exp \left(- \frac{\theta _E}{T}\right)\right]-\frac{a}{2}T^2-\frac{b}{n(n+1)}T^{n+1}\\&+G^{\rm mdo}(\infty )-\int _\infty ^T\left(\int _\infty ^T\frac{C_P^{\rm mag}}{T}dT\right)dT \end{aligned} \end{aligned}$$
(2)

where \(E_0\) denotes the cohesive energy (total energy excluding the vibrational contribution) at 0 K; \(\theta _E\) is the Einstein temperature. \(E_0\), \(\theta _E\), a and b are evaluated by fitting to the experimental data. Moreover, \(E_0\) can be obtained by ab initio calculations while \(\theta _E\) can be estimated using the methods introduced in the work by He et al.[27]. The first five terms together are the Gibbs energy of a ferromagnetic/antiferromagnetic state, which can be denoted as \(G^{\rm FM}\). The sixth term denotes the total Gibbs energy for magnetic disordering from 0 K to infinity. The first six terms together are thus the Gibbs energy of a phase in a hypothetical paramagnetic state, which can be denoted as \(G^{\rm PM}\). The seventh term represents the magnetic contribution with the paramagnetic state as the reference state, i.e. the Gibbs energy for the magnetic ordering, which can be denoted as \(G^{\rm mo}\) and described using the model first presented by Inden,[62] later simplified by Hillert and Jarl[63] using Maclaurin expansion (the IHJ model) and recently extended by Chen and Sundman[35] by adding one more term to the expansion (the IHJ-CS model). It is worth pointing out that parameters evaluated with one magnetic model cannot be directly used in the other magnetic model.

In the present work, we followed the suggestion by He et al.[27] to use a single Gibbs energy expression, i.e. Eq 2, to describe the Gibbs energy of the solid phases for the entire temperature range. As a consequence, the solid phases may be re-stabilised at high temperatures. Thus, the Equal-Entropy Criterion (EEC) suggested by Sundman et al.[64] must be applied to exclude any solid phase with higher entropy than the liquid phase.

Earlier theoretical studies show that there are two magnetic states in fcc Fe varying with volume or alloying elements, one with a lower magnetic moment and volume while the other with a higher magnetic moment and volume.[10, 47, 49, 65,66,67,68] Furthermore, the thermal excitation between these two states gives an additional contribution to the thermodynamic properties of fcc Fe, e.g. the entropy, making it stable again at 1185 K. To model this additional thermodynamic properties, a so-called two state model was used.[35, 37, 47, 65, 69] In addition to the terms in the Gibbs energy expression given in Eq 2, the contribution to the Gibbs energy due to the thermal excitation between the two magnetic states was added, which is expressed using the following equation:

$$\begin{aligned} G^{\rm 2st}=-RT\ln \left[1+\frac{g_2}{g_1}\exp (-\frac{\Delta E^{\rm 2st}}{RT})\right] \end{aligned}$$
(3)

where \(\Delta E^{\rm 2st}\) is the energy gap between the two magnetic states. The term \(g_2/g_1\) denotes the degeneracy ratio of the two magnetic states. We need to point out that the two-state model can well represent the experimental data but it does not provide any information on the physical nature of these two magnetic states, e.g. electronic structure, magnetism, atomic volume or the thermally-induced transitions.[49] Besides, it describes how Gibbs energy changes with temperature instead of volume.

Some recent studies[17,18,19] qualitatively suggested a different picture of the magnetism of fcc Fe and fcc in Fe-Ni alloys. The randomness of the magnetic configuration increases and the magnetisation is decreasing with increasing temperatures. Besides, a large number of magnetic states depend sensitively on volume. The volume change is induced by changing the temperatures or by adding alloying elements to pure Fe, these two ways of changing the volume can both result in a change of the magnetic state. There is a transition region from the low-spin state to the high-spin state upon the variation in volume in fcc Fe. In this transition region, the total energies of many magnetic states have very close values. Moreover, alloying elements, for instance Ni, can shift the equilibrium volume towards lower volumes.

As discussed above, there are more than two magnetic states in fcc Fe changing from one to the other with variations in volume or in alloying elements. Furthermore, this additional contribution due to the thermal excitation between two magnetic states[47, 65] for stabilising fcc Fe at 1185 K was estimated on the basis of the magnetic properties obtained from precipitates in a Cu matrix.[3] As discussed in Section 2.2.1, these magnetic properties may not be reliable. If we calculate the entropy of fcc Fe at 1185 K using the magnetic properties adopted in the present work, there is no additional entropy needed in order to stabilise fcc. To conclude, accurate magnetic properties at 0 K are critical to describe the thermodynamic properties at ambient pressure. Once the correct magnetic properties are used, the IHJ-CS model is capable to describe the decreasing magnetisation from 0 K to finite temperature until the magnetic transition temperature. For future work, molar volume and magnetisation coupling needs to be added in order to describe various magnetic states with reduced magnetisation at given temperature, pressure and composition when modelling e.g the Invar effect. Thus, we decided to use Eq 2 to describe all the contributions to the Gibbs energy of the fcc phase.

3.2 Modelling the hcp Phase

Hcp Fe is a metastable allotrope. In the second generation of Calphad descriptions, lattice stabilities were used to model such metastable allotropes, i.e.

$$\begin{aligned} ^{o}{G}_i^{\rm meta}-\,^{o}G_i^{\rm stab}=\Delta {H}-T\Delta {S} \end{aligned}$$
(4)

which gives a non-zero entropy, \(\Delta {S}\), at 0 K and violates the third law of thermodynamics.

Considering that the difference in high-temperature entropy between two phases, here stable and metastable allotropes, due to different harmonic vibrations goes towards a constant value at high temperatures,[59] Fernández Guillermet and Huang[70] derived the relations of harmonic vibrational entropies and Debye temperatures between different allotropes in Mn using the Debye model. Based on a similar consideration, Dinsdale et al.[26] later derived a relation to estimate the Einstein temperature of a metastable allotrope from that of the stable one to model the contribution from the entropy difference in lattice stability.

$$\begin{aligned} \theta _E^{\rm meta}=\theta _E^{\rm stab}\exp ((S_i^{\rm stab}-S_i^{\rm meta})/3R)=\theta _E^{\rm stab}\exp (-\Delta {S}/3R) \end{aligned}$$
(5)

which means that the contribution from a constant entropy difference can be expressed by the Gibbs energy using the Einstein model with a different Einstein temperature from that of the stable allotrope. If there is information available for the metastable allotrope, such as melting temperature and transition temperature from the stable allotrope, the values of \(\Delta {H}\) and \(\Delta {S}\) are suggested to be evaluated incorporating the third generation descriptions of the liquid phase and the stable allotrope. Otherwise the values can be taken directly from the lattice stability in the SGTE compilation.

3.3 Modelling the Liquid-Amorphous Phase

The two-state model has been suggested to describe the liquid-amorphous phase.[24, 71] This model is based on the assumption that all the atoms of the liquid-amorphous phase are either in the translational state or vibrational state (amorphous-like state). The Gibbs energy is formulated as follows:

$$\begin{aligned} G_i^{\rm liq-am}= {^{o}G_{i}^{\rm liq-am}}-RT\ln \left[1+\exp (-\frac{\Delta {G_d}}{RT})\right] \end{aligned}$$
(6)

where \(^{o}G_{i}^{\rm liq-am}\) represents the Gibbs energy of the ideal amorphous phase; \(\Delta G_d\) denotes the difference in Gibbs energy between the two states.

The Gibbs energy of the ideal amorphous phase was suggested to be described in a similar way as that of a crystalline solid phase[24, 71]:

$$\begin{aligned} ^{o}G_{i}^{\rm liq-am}&=\frac{3}{2}R\theta _E^{\rm liq-am}+3RT\ln [1-\exp (- \frac{\theta _E^{\rm liq-am}}{T})]\\ &\quad +a-\frac{1}{2}\gamma _{\rm el}^{\rm liq-am}T^2\\&\quad+G^{\rm mdo}(\infty )+G^{\rm mo} \end{aligned}$$
(7)

where \(\theta _E^{\rm liq-am}\) is the Einstein temperature of the ideal amorphous phase. \(\theta _E^{\rm liq-am}\) can be scaled to that of the stable crystalline solid phase if there are not enough heat capacity data for the liquid-amorphous phase. The ratio of the Einstein temperatures between the ideal amorphous phase and the crystalline solid phase has been discussed by Grimvall.[59] \(\gamma _{\rm el}^{\rm liq-am}\) is the electronic heat capacity coefficient and it is close to that of the stable crystalline solid phase at the melting temperature for most metals, exceptions are Cr, Mo and W.[59] If there is not enough information on \(\gamma _{\rm el}^{\rm liq-am}\), one can fix it to that of the crystalline solid phase at the melting point. \(G^{\rm mdo}(\infty )\) denotes the total Gibbs energy contribution of the magnetic disordering effect to infinitely high temperatures; \(G^{\rm mo}\) represents the Gibbs energy due to the magnetic ordering effect.

Ågren[71] suggested a general expression for describing \(\Delta {G_d}\) as follows:

$$\begin{aligned} \Delta {G_d}=A-BT+CT\ln T+\cdots \end{aligned}$$
(8)

where A, B and C are the model parameters which can be evaluated by fitting to the experimental or theoretical data. The communal entropy, R,[72] or the entropy of melting can be used as a starting value for optimizing parameter B. When evaluating the parameters, it is useful to check the entropy from \(\Delta {G_d}\) and make sure the value is reasonable. If the experimental information is not enough, one can fix parameter B to the communal entropy or the entropy of melting.

4 Optimization Procedures

4.1 Parameter Optimisation for fcc

Equation 2 was used to describe the Gibbs energy of the fcc phase. The selected data for fcc Fe as discussed in Section 2.2.1 were fitted by optimizing the model parameters. During the optimization, we made sure to fit the properties at the transition temperatures. The relations of the electronic heat capacity coefficients and the high-temperature entropy-Debye temperatures between fcc and bcc suggested in[47, 48] were taken into account to make sure that the assessed Einstein temperature and the electronic heat capacity coefficient of fcc are close to those of bcc.

4.2 Parameter Optimisation for hcp

Hcp Fe was treated as a magnetic phase in the present work instead of non-magnetic as done by Fernández Guillermet and Gustafson.[37] Besides, we adopted different magnetic properties for fcc Fe. Thus, the difference in Gibbs energy between hcp and fcc due to magnetism needs to be re-evaluated in order to calculate the lattice stability of hcp for the present thermodynamic description.

We adopted the value, \(\Delta S^{\rm PM}=S^{\rm PM(hcp)}-S^{\rm PM(fcc)}=-4.309\) J/(mol K), obtained by Fernández Guillermet and Gustafson.[37] In the third generation, the paramagnetic contribution to the entropy is expressed as \(S^{\rm PM}=S^{\rm FM}+S^{{\rm mdo}(\infty )}\), we can therefore express the difference in entropy between hcp and fcc as \(\Delta S^{\rm PM}=\Delta S^{\rm FM}+\Delta S^{{\rm mdo}(\infty )}\). The entropy difference, \(\Delta S^{{\rm mdo}(\infty )}\), between hcp and fcc is −2.7916 J/(mol K) which was obtained from the magnetic properties selected in the present work. The entropy difference, \(\Delta S^{\rm FM}\), between hcp and fcc is therefore −1.5174 J/(mol K). Based on the Einstein temperature of fcc assessed in the present work, 304 K, and the entropy difference, \(\Delta S^{\rm FM}=-1.5174\) J/(mol K), the Einstein temperature of hcp was calculated to 323 K. The metastable melting point of hcp, 1337 K, given by Fernández Guillermet and Gustafson[37] was adopted to calculate the enthalpy difference between hcp and fcc in the present work, \(\Delta H^{\rm PM}\). Using the magnetic properties, we determined \(\Delta H^{{\rm mdo}(\infty )}\) to 113.117 J/mol. Thereafter, \(\Delta H^{\rm PM}=-2445.38\) J/mol was obtained. Accordingly, the enthalpy difference between hcp and fcc, \(\Delta H^{\rm FM}=\Delta H^{\rm PM}-\Delta H^{{\rm mdo}(\infty )}\), was calculated to −2558.5 J/mol.

4.3 Parameter Optimisation for Liquid

The two-state model was used to describe the thermodynamic properties of the liquid-amorphous phase. The Einstein temperature was fixed to the value used by Chen and Sundman.[35] There is no available information on the electronic heat capacity coefficient and we thus adopted the approximation proposed by He et al.[27] to use the same value as the solid bcc phase has at the melting temperature. The electronic heat capacity expression provided in the work by Forsblom et al.,[73] \(C_{\rm el}=\gamma T/(1+\lambda _{\rm el-ph})\), was used to obtain the value, 3.51 mJ/(mol K)\(^2\), which was fixed during the optimization. Furthermore, parameter B in Eq 8 was fixed to the entropy of melting. Thereafter, the rest of the model parameters were optimized by fitting to the selected experimental data discussed in Section 2.2.3.

5 Results and Discussions

The third generation Calphad descriptions of fcc, hcp and liquid Fe were re-assessed in the present work. The evaluated thermodynamic parameters are listed in Table 3. The Gibbs energy, enthalpy, entropy and heat capacity of bcc, fcc, hcp and liquid Fe are plotted using the thermodynamic description presented in Table 3 and shown in Fig. 1. As mentioned in Section 3.1, the extrapolation method for solid phases suggested by He et al.[27] was adopted in the present work using one single Gibbs energy expression for the entire temperature range. Therefore, the Gibbs energy expression for the low-temperature range given by Chen and Sundman was adopted for bcc in the present work. With the new extrapolation treatment a drastic increase in heat capacity appears for bcc. Nevertheless, the bcc phase is only re-stabilised at really high temperatures, approximately 3700 K (see Fig. 1a). Moreover, as can be seen in Fig. 1(c), the entropy of bcc is higher than that of liquid above 2900 K and applying the EEC it will be excluded from the equilibrium calculations above that temperature which is lower than the temperature where it is re-stabilised. In the present work, the Equal-Entropy Temperature (EET) was manually calculated in Thermo-Calc.[74] However, the EEC has been implemented in the thermodynamic software OpenCalphad[75] and will be implemented in Thermo-Calc. With this implementation, equilibrium calculations can be done automatically with EEC activated. If the calculation reaches the EET of the solid phase, the software will automatically suspend this solid phase.

Table 3 A third generation Calphad description of pure Fe. Note that all values are given relative to the SER state at 1 bar pressure and in SI units. For details on the new models and their parameters please refer to the Appendix
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Fig. 1
figure 1

The calculated Gibbs energy, enthalpy, entropy and heat capacity of bcc, fcc, hcp and liquid Fe obtained using the description presented in Table 3. The Gibbs energy is calculated taking bcc as reference state while the enthalpy was calculated relative to the SER state

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5.1 The fcc Phase

The thermodynamic properties at the bcc and fcc phase transformation temperatures were calculated using the present description and are listed in Table 1. These calculated results are compared with the selected experimental data and calculated data using the previous Calphad descriptions. The calculated data obtained using the present Calphad description show a good agreement with the selected experimental data. The assessed Einstein temperature of fcc is 304 K which is close to the Einstein temperature of bcc, 309 K. This follows the suggested relation that fcc and bcc have approximately equal high-temperature entropy-Debye temperatures.[47, 48] The heat capacity of fcc is shown in Fig. 2(a), which was calculated using the present description and the descriptions given by Chen and Sundman[35] and Fernández Guillermet and Gustafson[37] respectively. We need to point out that the description by Fernández Guillermet and Gustafson is not valid below 298.15 K. Thus, the heat capacity was only calculated down to 298.15 K using their description. At low temperatures, the one obtained using the present description differs from the one obtained by the description given by Chen and Sundman. The magnetic properties selected in our assessment are different from their selected ones, which contribute to such a deviation at low temperatures. However, as can be seen, the heat capacity calculated using the present description agrees well with the experimental data and the calculation using the description by Chen and Sundman at the temperatures where fcc is stable. The differences in Gibbs energy, enthalpy and entropy between bcc and fcc are shown in Fig. 2. A good agreement is achieved between the calculations and the experiments.

Fig. 2
figure 2

(a) Comparison between the calculated and experimental data for the heat capacity of fcc Fe. The calculations were obtained using the present Calphad description and previous Calphad descriptions.[35, 37] The datapoints show the experimental data from.[39,40,41] Comparisons between the calculated and experimental data for the Gibbs energy differences (b), enthalpy (c) and entropy (d). The calculations were obtained taking bcc as reference state and using the present and previous Calphad descriptions.[35, 37] The experimental data were from Smith[76] and the tabulated data were taken from Orr and Chipman[45]

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5.2 The hcp Phase

The heat capacity, Gibbs energy, enthalpy and entropy of hcp Fe are plotted and shown in Fig. 3. The calculations were obtained using the descriptions assessed in the present work, and the assessments by Fernández Guillermet and Gustafson[37] and by Bigdeli et al.[38]. The calculations were only performed down to 298.15 K using the description by Fernández Guillermet and Gustafson as their description is not valid below 298.15 K. Different magnetic properties were used in the present description from those used by Bigdeli et al.,[38] which give a different \(\lambda \) peak in the calculated heat capacity curve. Instead of going to a constant value the heat capacity calculated using the present description continues to increase gradually at high temperatures.

Fig. 3
figure 3

(a) Comparison between calculations of heat capacity (a), Gibbs energy (b), entropy (c) and enthalpy (d) of hcp Fe. The calculations were performed relative to the SER state and obtained by the present Calphad description and previous Calphad descriptions[37, 38]

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5.3 The Liquid-Amorphous Phase

Table 2 lists the thermodynamic properties of liquid Fe at the melting point, including the heat capacity, and the enthalpy and entropy of melting. A comparison between the selected experimental data and calculated data using the present and previous Calphad descriptions is also shown in Table 2. The calculated data using the present description shows a good agreement with the selected experimental data. The heat capacity of the liquid phase was plotted using the present description and the descriptions given by Chen and Sundman[35] and Fernández Guillermet and Gustafson[37] respectively (see Fig. 4a). The difference in the heat capacity calculated using the present description and the description provided by Chen and Sundman attributes to the electronic contribution and the Gibbs energy difference between the translational and vibrational atoms. The electronic heat capacity coefficient was optimized in the assessment by Chen and Sundman. However, in the present work, the electronic heat capacity coefficient was fixed to that of the solid bcc phase at the melting temperature following the suggestion by He et al.[27]. This treatment gave a different electronic heat capacity coefficient from the one obtained by Chen and Sundman. The enthalpy of liquid Fe is calculated using the present and previous descriptions given by[35, 37] compared with the experimental data (see Fig. 4b).

Fig. 4
figure 4

(a) Comparison between the calculated heat capacity using the present and previous Calphad descriptions.[35, 37] (b) Comparison between the calculated and experimental enthalpy of liquid Fe. The calculations were performed relative to the SER state and the experimental data are from[54,55,56,57]

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6 Conclusions

The third generation Calphad models have been developed over the past decade, in order to adopt the revised models and the new treatments of model parameters, the Gibbs energies of the fcc, hcp and liquid phases of pure Fe at ambient pressure were re-assessed in the present work. The extrapolation method with only one Gibbs energy expression was used to model the thermodynamic properties of the solid fcc, hcp and bcc phases for the entire temperature range.

  1. 1.

    Fcc: the recently-published magnetic properties of fcc Fe were selected to describe the magnetic effect. The magnetic two-state model is not used in the present work supported by a conclusion that using the IHJ-CS model and the magnetic properties selected in the present work, the decreasing magnetisation at ambient pressure with increasing temperature can be well described. Furthermore, the contribution from magnetism to thermodynamic properties satisfactorily accounts for the selected experimental data.

  2. 2.

    Hcp: the lattice stability of hcp relative to fcc was re-evaluated taking into account the description of fcc re-assessed in the present work. The entropy difference of the lattice stability was used to estimate the Einstein temperature of hcp and the contribution due to the entropy difference to the Gibbs energy was described using the Einstein model in order to obey the third law of thermodynamics. The present description of hcp gives comparable results as the SGTE database.

  3. 3.

    Bcc: the new extrapolation method results in re-stabilising of bcc at around 3700 K which is the lowest re-stabilising temperature among all the solid phases of pure Fe. However, the EEC can be applied to exclude bcc from equilibrium calculations above the temperature where bcc and liquid have equal entropy, which is at approximately 2900 K.

  4. 4.

    Liquid-amorphous: the electronic contribution to the Gibbs energy was re-evaluated following the most recent suggestion. The present description shows a good reproduction of the experimental data.

The present description of pure Fe will be applied to binary systems, e.g. Fe-Ni, to check its validity.