Thermodynamic Study on Solubility Products of Ti₄C₂S₂ in Fe Using First-Principles Calculations

Abstract

This study elucidates the solubility product of Ti₄C₂S₂ in steels by means of first-principles calculations and thermodynamic analysis. For this purpose, the Gibbs formation energy of Ti₄C₂S₂ was calculated theoretically by considering the effect of lattice vibration and thermal expansion. In addition, the Gibbs energies of the bcc and fcc phases in the Fe–Ti–C ternary system were also obtained using the cluster expansion and cluster variation method. Although some experimental data were considered as required, those results were evaluated as the calculation of phase diagrams (CALPHAD)-type thermodynamic parameters through fitting to the sublattice model. By using those thermodynamic functions, an approximate expression of the solubility product for Ti₄C₂S₂ was derived. The result agrees with an experimental result measured in a relatively large temperature range. Furthermore, the formation behavior of precipitates in typical interstitial-free steels was discussed, incorporating an earlier thermodynamic analysis on the Fe–Ti–S ternary system. The results show that NiAs-type TiS was the main precipitate at higher temperatures and that Ti₄C₂S₂ was the main precipitate at lower temperatures.

1. Introduction

Ti-stabilized interstitial-free (IF) steels have been widely used in automotive industries because of their good formability and drawability. It is well known for these steels that inclusions of S enhance the precipitation of several sulfides and carbosulfide, such as NiAs-type TiS, NaCl-type MnS, and Ti₄C₂S₂; hence, controlling the behavior of these precipitates plays a key role in determining the mechanical properties of the steels.^{1,2,3,4,5,6,7,8,9,10,11,12)} However, because of experimental difficulty in detecting small concentration of C and S in IF steels, the thermodynamic properties of these precipitates have remained unclear. In particular, discrepancies among the reported thermodynamic stabilities for the Ti₄C₂S₂^{3,4,6,7,8,9,10,12)} have been found according to a variety of chemical compositions, temperatures, and experimental methods.

Thermodynamic calculations based on the calculation of phase diagrams (CALPHAD) method have been performed to evaluate the thermodynamic properties of materials.^11,13) However, the thermodynamic parameters optimized in these calculations were obtained only using the experimental values, such as solubility products and phase equilibria, and the calculation accuracy greatly depends on the reliability of experimental values. Recently, first-principles calculations combined with a statistical thermodynamics technique have been applied to evaluate the thermodynamic stability of sulfides in IF steels at finite temperatures.¹⁴⁾ The values calculated by the method are introduced to the CALPHAD approach to estimate the thermodynamic properties in the metastable region. Then, the thermodynamic stability of Ti₄C₂S₂ in a wide temperature range is expected to be evaluated in a more-accurate manner without any limitation for conditions. Based on these backgrounds, the present study attempts to calculate the Gibbs energy of Ti₄C₂S₂ using first-principles calculations and investigates the thermodynamic stability in a wider temperature range. Furthermore, this study investigates the precipitation behavior of several sulfides in IF steel based on the theoretical thermodynamic parameters.

2. Computational Procedures

The first-principles calculations based on density functional theory were performed to evaluate thermodynamic functions at 0 K in this work. The details of this method and the expression of the thermodynamic model using in the CALPHAD approach are noted in this section.

2.1. Calculation of Formation Energies for Compounds

The total energies of the NaCl-type TiC and Ti₄C₂S₂ were calculated using the projector augmented wave method^15,16) implemented in the Vienna Ab-Initio Simulation Package (VASP) code.^17,18) The generalized gradient approximation proposed by Perdew, Burke, and Ernzerhof¹⁹⁾ was applied to treat the exchange and correlation function. The Methfessel–Paxton method of order-one smearing was used, and the width of the smearing was set as 0.2 eV. The plane wave energy cutoff was set as 400 eV, and spin polarization was not included in the calculation. The convergence criteria for energy calculation and structure optimization were set as 10⁻⁵ eV/atom and 0.2 eV/nm, respectively. For both structures, gamma-centered k-point meshes were prepared, and the sizes of the k-point meshes were 7 × 7 × 7 and 7 × 7 × 5 for TiC and Ti₄C₂S₂, respectively. The formation energies of the compounds were computed from the difference between the total energy of each component in its stable state, i.e., hcp-Ti, orthorhombic-S, and graphite. Because of a difficulty in considering the van der Waals interaction for C by means of the first-principles calculation, experimental value for energy difference between diamond structure and graphite structure, 2412 J/mol,²⁰⁾ was added to a calculated energy of C of the diamond structure obtained in the present study. Some calculation results for the van der Waals interaction of orthorhombic-S were reported,²¹⁾ however, discordance not to be overlooked was observed between those calculated values. The largest error was observed in CuS in which the experimental error was reported about 30%. This fact led to the abandonment of this calculation in this study, and we considered that calculation errors up to 15% were inherent in the result of the solubility product Ti₄C₂S₂ containing 25% of S.

2.2. Calculation of Free Energy for fcc and bcc Phases

To clarify the precipitation behavior in IF steel, thermodynamic parameters for matrix fcc-Fe and bcc-Fe phases are necessary. Therefore, the Gibbs free energy of bcc and fcc phases in the Fe–Ti–C ternary system were calculated using the cluster expansion method (CEM)²²⁾ and the cluster variation method (CVM).^23,24) The results were fitted to the compound energy formalism expressed by the two-sublattice model^25,26) as (Fe,Ti)_a(C,Vacancy)_c. The values of a and c are the numbers of sites in each sublattice are a=c=1 for the fcc phase and a=1 and c=3 for bcc phase. The composition area of each ordered structure was defined by four end-member structures, i.e., bcc-Fe, bcc-Ti, FeC₃, and TiC₃ for the bcc phase; and fcc-Fe, fcc-Ti, FeC, and TiC for the fcc phase. Then, the matrix fcc-Fe and the TiC can be described by the same Gibbs free energy surface based on this two-sublattice model. First, for describing the Gibbs energies of these phases, the formation energies of the end-member structures were calculated by the VASP code. Next, there were 890 and 901 ordered structures prepared for the bcc and fcc phases, respectively, to calculate the Gibbs energies of the solid solutions. The convergence criteria for energy calculation and structure optimization were set to be 10⁻⁵ eV/atom and 10⁻⁴ eV/atom, respectively. In addition, the ferromagnetic state of bcc-Fe was adopted as the reference state of formation energy calculation. Structure volume change was merely allowed in the structure optimization, and the k-point meshes were created with 1000 k-points per reciprocal atom. In the calculation of the bcc-ordered structures, spin polarization was considered, but it was not considered in fcc-ordered structures because the magnetic structure of the fcc-Fe is paramagnetic at or above room temperature.

In the formalism of CEM, the formation energy of a structure is represented as:   

$$E=\sum {}_{\alpha }{J}_{\alpha }{\xi }_{\alpha },$$(1)

where E, J_α, and ξ_α are the formation energy of the ordered structure, effective cluster interaction for cluster α, and cluster correlation function, respectively. The cluster correlation function is defined by the occupation operator for sites in the cluster and is uniquely determined for each ordered structure because the atomic configuration of an ordered structure is uniquely determined. As E is expressed as a function of the cluster correlation function ξ_α, the values of J_α are determined by the inverse matrix calculation as follows:   

$$J_{\alpha }={\left\{{\xi }_{\alpha }\right\}}^{-1}E.$$(2)

Lastly, the free energy at finite temperatures F was calculated by CVM according to Eq. (3):   

$$F=\sum {}_{\alpha }{J}_{\alpha }{\xi }_{\alpha }-T\sum {}_{\alpha }{\gamma }_{\alpha }{S}_{\alpha }.$$(3)

where S_α is the configurational entropy term, and γ_α is the Kikuchi-Barker coefficient, representing the contribution of entropy from cluster α. In CVM, the free energy of the phase at an arbitrary finite temperature is computed by determining ξ that minimizes free energy by the variational method. In this study, the calculation code developed by Sluiter et al.^27,28) was applied to CEM and CVM.

2.3. Calculation of the Free Energy of Compounds Based on Vibrational Analysis

To consider the vibrational effect on the free energy of compound phase at finite temperatures, finite displacement method^29,30) and quasiharmonic approximation were applied to Ti₄C₂S₂, NaCl-type TiC, NaCl-type FeC, bcc-ordered structure TiC₃, and FeC₃.

The Helmholtz free energy with lattice vibration contribution F(T) can be described as follows:   

$$F\left(T\right)=E_{0K}+k_{B}T{\sum }_q{\sum }_{j}ln\left\{2sinh\left[\frac{\hslash {\omega }_j(q)}{2k_{B}T}\right]\right\},$$(4)

where E_0K, k_B, $\hslash$, ω_j(q) are the total energy at 0 K, Boltzmann constant, reduced Planck constant, and phonon frequency that is decided by the wave vector q and the band index j, respectively. Within the formalism of quasiharmonic approximation, phonon frequency is evaluated from different volumes to introduce the thermal expansion effect that is derived from the anharmonicity of lattice vibration. Then, Eq. (4) can be rewritten as Eq. (5):   

$$F\left(T,V\right)=E_{0K}\left(V\right)+k_{B}T{\sum }_q{\sum }_{j}ln\left\{2sinh\left[\frac{\hslash {\omega }_j(q,V)}{2k_{B}T}\right]\right\}.$$(5)

Lastly, Gibbs free energy G(T,p) and heat capacity at constant pressure C_p(T) is given by:   

$$G\left(T,p\right)=\underset{V}{min}\left[F\left(T,V\right)+pV\right],$$(6)

  

$$C_p\left(T\right)=-T\frac{{\partial }^{2}G\left(T,p\right)}{\partial T^2}.$$(7)

To evaluate the value of $\underset{V}{min}\left[F\left(T,V\right)+pV\right]$, the F(V) curve at each temperature were fitted using the Vinet equation of state³¹⁾ given in Eq. (8):   

$$\begin{array}{l}F\left(V\right)=E_0+{\displaystyle \frac{2B_0{V}_0}{{\left(B_0^{\prime }-1\right)}^2}}\{2-\left[5+3{\left({\displaystyle \frac{V}{V_0}}\right)}^{{\displaystyle \frac{1}{3}}}\left(B_0^{\prime }-1\right)-3B_0^{\prime }\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\times exp\left[-{\displaystyle \frac{3}{2}}\left(B_0^{\prime }-1\right)\left[{\left({\displaystyle \frac{V}{V_0}}\right)}^{{\displaystyle \frac{1}{3}}}-1\right]\right]\},\end{array}$$(8)

where E₀, B₀, $B_0^{\prime }$ are the formation energy at equilibrium volume V₀, bulk modulus at V₀, the first derivative of bulk modulus with pressure, respectively.

The above calculations were performed using phonopy code^32,33,34) combined with VASP. Then, 2 × 2 × 2 supercell structures were prepared, and the sizes of the k meshes were 4 × 4 × 4 for NaCl-type and bcc-ordered structures and 4 × 4 × 2 for Ti₄C₂S₂.

2.4. Thermodynamic Modeling

2.4.1. Ti₄C₂S₂ Phase

In the present study, the Ti₄C₂S₂ phase was treated as being a stoichiometric compound. The molar Gibbs free energy of Ti₄C₂S₂, $G_{\text{Ti}\mathrm{\hspace{0.17em} }\text{:}\mathrm{\hspace{0.17em} }\text{C}\mathrm{\hspace{0.17em} }\text{:}\mathrm{\hspace{0.17em} }\text{S}}^{\text{Ti4C2S2}}$, was expressed as follows:   

$$\begin{array}{l}{G}_{\text{Ti : C : S}}^{\text{Ti4C2S2}}-0.5G_{\text{Ti}}^{\text{hcp}}-0.25G_{\text{C}}^{\text{graphite}}-0.25G_{\text{S}}^{\text{orthorhombic}}=\\ A+BT+CTlnT+D{T}^2+E{T}^3+F{T}^4,\end{array}$$(9)

where $G_{\text{Ti}}^{\text{hcp}}$, $G_{\text{C}}^{\text{graphite}}$, and $G_{\text{S}}^{\text{orthorhombic}}$ denote the Gibbs free energy of each pure element in its stable structure. These parameters are generally called the lattice stability parameters and are used to describe the pure element i as shown in Eq. (10):   

$$\begin{array}{l}{}^{\circ }G_{\mathrm{i}}^{\text{phase}}-{}^{\circ }H_{\mathrm{i}}^{\text{ref}}=\\ A+BT+CTlnT+D{T}^2+E{T}^3+F{T}^7+I{T}^{-1}+J{T}^{-9}.\end{array}$$(10)

where ${}^{\circ }H_{\mathrm{i}}^{\text{ref}}$ denotes the molar enthalpy of the pure element i in its stable state at T = 25°C and A to J are coefficients. The lattice stability parameters were taken from those evaluated by Scientific Group Thermodata Europe.³⁵⁾

2.4.2. bcc Phase

The two-sublattice model (Fe,S,Ti)₁(C,Vacancy)₃ was applied to describe the Gibbs free energy for the bcc phase. The molar Gibbs free energy of the bcc phase for formula unit of the two-sublattice is expressed as follows:   

$$\begin{array}{l}{G}_{\text{m}}^{\text{bcc}}=y_{\text{Fe}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{}^{\circ }G_{\text{Fe : C}}^{\text{bcc}}+y_{\text{S}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{}^{\circ }G_{\text{S : C}}^{\text{bcc}}+y_{\text{Ti}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{}^{\circ }G_{\text{Ti : C}}^{\text{bcc}}\\ \hspace{1em}\hspace{1em}+y_{\text{Fe}}^{\text{(1)}}{y}_{\text{Va}}^{\text{(2)}}{}^{\circ }G_{\text{Fe : Va}}^{\text{bcc}}+y_{\text{S}}^{\text{(1)}}{y}_{\text{Va}}^{\text{(2)}}{}^{\circ }G_{\text{S : Va}}^{\text{bcc}}+y_{\text{Ti}}^{\text{(1)}}{y}_{\text{Va}}^{\text{(2)}}{}^{\circ }G_{\text{Ti : Va}}^{\text{bcc}}\\ \hspace{1em}\hspace{1em}+RT\{\left(y_{\text{Fe}}^{\text{(1)}}ln{y}_{\text{Fe}}^{\text{(1)}}+y_{\text{S}}^{\text{(1)}}ln{y}_{\text{S}}^{\text{(1)}}+y_{\text{Ti}}^{\text{(1)}}ln{y}_{\text{Ti}}^{\text{(1)}}\right)\\ \hspace{1em}\hspace{1em}+3\left(y_{\text{C}}^{\text{(2)}}ln{y}_{\text{C}}^{\text{(2)}}+y_{\text{Va}}^{\text{(2)}}ln{y}_{\text{Va}}^{\text{(2)}}\right)\}\\ \hspace{1em}\hspace{1em}+y_{\text{Fe}}^{\text{(1)}}{y}_{\text{S}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{L}_{\text{Fe,S : C}}^{\text{bcc}}+y_{\text{Fe}}^{\text{(1)}}{y}_{\text{S}}^{\text{(1)}}{y}_{\text{Va}}^{\text{(2)}}{L}_{\text{Fe,S : Va}}^{\text{bcc}}\\ \hspace{1em}\hspace{1em}+y_{\text{Fe}}^{\text{(1)}}{y}_{\text{Ti}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{L}_{\text{Fe,Ti : C}}^{\text{bcc}}+y_{\text{Fe}}^{\text{(1)}}{y}_{\text{Ti}}^{\text{(1)}}{y}_{\text{Va}}^{\text{(2)}}{L}_{\text{Fe,Ti : Va}}^{\text{bcc}}\\ \hspace{1em}\hspace{1em}+y_{\text{S}}^{\text{(1)}}{y}_{\text{Ti}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{L}_{\text{S,Ti : C}}^{\text{bcc}}+y_{\text{S}}^{\text{(1)}}{y}_{\text{Ti}}^{\text{(1)}}{y}_{\text{Va}}^{\text{(2)}}{L}_{\text{S,Ti : Va}}^{\text{bcc}}\\ \hspace{1em}\hspace{1em}+y_{\text{Fe}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{y}_{\text{Va}}^{\text{(2)}}{L}_{\text{Fe : C,Va}}^{\text{bcc}}+y_{\text{S}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{y}_{\text{Va}}^{\text{(2)}}{L}_{\text{S : C,Va}}^{\text{bcc}}\\ \hspace{1em}\hspace{1em}+y_{\text{Ti}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{y}_{\text{Va}}^{\text{(2)}}{L}_{\text{Ti : C,Va}}^{\text{bcc}},\end{array}$$(11)

where $y_{\text{i}}^{(n)}$ is the site fraction of i on sublattice n, ${}^{\circ }G_{\text{i : k}}^{\text{bcc}}$ is the Gibbs free energy of the structure where sublattice 1 is occupied by element i, and sublattice 2 is occupied by element k. $L_{\text{i,j : k}}^{\text{bcc}}$ is the interaction parameter between elements i and j when sublattice 2 is occupied by element k, and the compositional dependency of $L_{\text{i,j : k}}^{\text{bcc}}$ is expressed as follows:   

$$L_{\text{i,j : k}}^{\text{bcc}}={}^{0}L_{\text{i,j : k}}^{\text{bcc}}+{}^{1}L_{\text{i,j : k}}^{\text{bcc}}{\left(y_{\text{i}}^{(1)}-y_{\text{j}}^{(1)}\right)}^1+{}^{2}L_{\text{i,j : k}}^{\text{bcc}}{\left(y_{\text{i}}^{(1)}-y_{\text{j}}^{(1)}\right)}^2.$$(12)

2.4.3. fcc and TiC Phases

Because of the similarity in crystal structures between the fcc solid solution and the NaCl-type TiC phase, the Gibbs free energy for these phases is described by the two-sublattice model which has a formula of (Fe,S,Ti)₁(C,Vacancy)_1. Then, the molar Gibbs free energy of the fcc phase for formula unit of the two-sublattice is expressed as follows:   

$$\begin{array}{l}{G}_{\text{m}}^{\text{fcc}}=y_{\text{Fe}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{}^{\circ }G_{\text{Fe : C}}^{\text{fcc}}+y_{\text{S}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{}^{\circ }G_{\text{S : C}}^{\text{fcc}}+y_{\text{Ti}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{}^{\circ }G_{\text{Ti : C}}^{\text{fcc}}\\ \hspace{1em}\hspace{1em}+y_{\text{Fe}}^{\text{(1)}}{y}_{\text{Va}}^{\text{(2)}}{}^{\circ }G_{\text{Fe : Va}}^{\text{fcc}}+y_{\text{S}}^{\text{(1)}}{y}_{\text{Va}}^{\text{(2)}}{}^{\circ }G_{\text{S : Va}}^{\text{fcc}}+y_{\text{Ti}}^{\text{(1)}}{y}_{\text{Va}}^{\text{(2)}}{}^{\circ }G_{\text{Ti : Va}}^{\text{fcc}}\\ \hspace{1em}\hspace{1em}+RT\{\left(y_{\text{Fe}}^{\text{(1)}}ln{y}_{\text{Fe}}^{\text{(1)}}+y_{\text{S}}^{\text{(1)}}ln{y}_{\text{S}}^{\text{(1)}}+y_{\text{Ti}}^{\text{(1)}}ln{y}_{\text{Ti}}^{\text{(1)}}\right)\\ \hspace{1em}\hspace{1em}+\left(y_{\text{C}}^{\text{(2)}}ln{y}_{\text{C}}^{\text{(2)}}+y_{\text{Va}}^{\text{(2)}}ln{y}_{\text{Va}}^{\text{(2)}}\right)\}\\ \hspace{1em}\hspace{1em}+y_{\text{Fe}}^{\text{(1)}}{y}_{\text{S}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{L}_{\text{Fe,S : C}}^{\text{fcc}}+y_{\text{Fe}}^{\text{(1)}}{y}_{\text{S}}^{\text{(1)}}{y}_{\text{Va}}^{\text{(2)}}{L}_{\text{Fe,S : Va}}^{\text{fcc}}\\ \hspace{1em}\hspace{1em}+y_{\text{Fe}}^{\text{(1)}}{y}_{\text{Ti}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{L}_{\text{Fe,Ti : C}}^{\text{fcc}}+y_{\text{Fe}}^{\text{(1)}}{y}_{\text{Ti}}^{\text{(1)}}{y}_{\text{Va}}^{\text{(2)}}{L}_{\text{Fe,Ti : Va}}^{\text{fcc}}\\ \hspace{1em}\hspace{1em}+y_{\text{S}}^{\text{(1)}}{y}_{\text{Ti}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{L}_{\text{S,Ti : C}}^{\text{fcc}}+y_{\text{S}}^{\text{(1)}}{y}_{\text{Ti}}^{\text{(1)}}{y}_{\text{Va}}^{\text{(2)}}{L}_{\text{S,Ti : Va}}^{\text{fcc}}\\ \hspace{1em}\hspace{1em}+y_{\text{Fe}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{y}_{\text{Va}}^{\text{(2)}}{L}_{\text{Fe : C,Va}}^{\text{fcc}}+y_{\text{S}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{y}_{\text{Va}}^{\text{(2)}}{L}_{\text{S : C,Va}}^{\text{fcc}}\\ \hspace{1em}\hspace{1em}+y_{\text{Ti}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{y}_{\text{Va}}^{\text{(2)}}{L}_{\text{Ti : C,Va}}^{\text{fcc}}+y_{\text{Fe}}^{\text{(1)}}{y}_{\text{S}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{y}_{\text{Va}}^{\text{(2)}}{L}_{\text{Fe,S : C,Va}}^{\text{fcc}}\\ \hspace{1em}\hspace{1em}+y_{\text{Fe}}^{\text{(1)}}{y}_{\text{Ti}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{y}_{\text{Va}}^{\text{(2)}}{L}_{\text{Fe,Ti : C,Va}}^{\text{fcc}}+y_{\text{S}}^{\text{(1)}}{y}_{\text{Ti}}^{\text{(1)}}{y}_{\text{C}}^{\text{(2)}}{y}_{\text{Va}}^{\text{(2)}}{L}_{\text{S,Ti : C,Va}}^{\text{fcc}},\end{array}$$(13)

where $y_{\text{i}}^{(n)}$, ${}^{\circ }G_{\text{i : k}}^{\text{fcc}}$ and $L_{\text{i,j : k}}^{\text{fcc}}$ have the same meanings as described in Eqs. (11) and (12). $L_{\text{i,j : k,l}}^{\text{fcc}}$ is the interaction parameter when the mixing of elements is allowed on both sublattices simultaneously. The compositional dependency of $L_{\text{i,j : k,l}}^{\text{fcc}}$ is expressed as follows:   

$$\begin{array}{l}{L}_{\text{i, j : k, l}}^{\text{fcc}}=\\ {}^{0}L_{\text{i, j : k, l}}^{\text{fcc}}+{}^{1}L_{\text{i, j : k, l}}^{\text{fcc}}\left(y_{\text{i}}^{(1)}-y_{\text{j}}^{(1)}\right)+{}^{2}L_{\text{i, j : k, l}}^{\text{fcc}}\left(y_{\text{k}}^{(2)}-y_{\text{l}}^{(2)}\right).\end{array}$$(14)

In addition to these models, the contribution to the Gibbs free energy because of magnetic ordering was considered by adding the formalism suggested by Inden³⁶⁾ and Hillert and Jarl³⁷⁾ to the nonmagnetic term of Gibbs free energy. Thermodynamic analysis and phase diagram calculation were performed using the Thermo-Calc software package.³⁸⁾

3. Results and Discussion

3.1. Formation Energies and Heat Capacities of TiC and Ti₄C₂S₂

The formation energies of TiC and Ti₄C₂S₂ were evaluated using the first-principles calculations. The calculated results and the lattice parameters are listed in Table 1 with the corresponding experimental results. The reference states of Ti, C, and S for these formation energies were set to hcp-Ti, graphite, and orthorhombic-S, respectively. Experimental formation energies of TiC were measured by combustion calorimetry,³⁹⁾ direct synthesis calorimetry,⁴⁰⁾ Knudsen cell method,⁴¹⁾ and solute-solvent drop calorimetry.⁴²⁾ In contrast, the formation energies of Ti₄C₂S₂ have never been evaluated directly in an experiment. Then, the calculated results by first-principles calculation^43,44,45) were compared with the present study in Table 1, which shows that the calculated values in this work show reasonable agreement with those by the previous studies.

Table 1. The calculated thermodynamic properties of NaCl–TiC and Ti₄C₂S₂. The reference states of Ti, C, and S are hcp-Ti, graphite, and orthorhombic-S, respectively.

Compound	Space group	Calculated lattice parameter, nm	Calculated formation energy, kJ/mol of atom	Experimental39,40,41,42) and calculated43,44,45) results, kJ/mol of atom	Ref.
TiC	Fm3m	a=0.433 α=90°	−82.4	−91.7±0.8	39)
				−91.6±9.6	40)
				−95.2	41)
				−92.9±8.9	42)
Ti4C2S2	P63/mmc	a=b=0.320	−122.1	−117.4	43)
		c=1.126		−130.5	44,45)
		α=β=90°, γ=120°

Calculations of the heat capacities for TiC and Ti₄C₂S₂ were performed by the finite displacement method, and the results are given in Figs. 1 and 2. Figure 1 shows the calculated isobaric heat capacity of TiC, which is in good agreement with the experimental data⁴⁶⁾ and evaluated value by Frisk.⁴⁷⁾ Figure 2 shows the isobaric heat capacity of Ti₄C₂S₂. The solid line denotes the calculated value using optimized parameters in the framework of the CALPHAD method, and the result corresponds well with the results by the finite displacement method.

Fig. 1.

The calculated isobaric heat capacity of NaCl-type TiC. The result was compared with the thermodynamic data⁴⁶⁾ and thermodynamic analysis.⁴⁷⁾

Fig. 2.

The isobaric heat capacity of Ti₄C₂S₂ calculated by the finite displacement method. The solid line represents the result based on the present thermodynamic analysis.

3.2. Evaluation of Thermodynamic Properties for the fcc and bcc Phases in the Fe–Ti–S–C System

To calculate the solubility products of Ti₄C₂S₂ as well as TiC and TiS in Fe, thermodynamic properties for the fcc and bcc phases in the Fe–Ti–S–C system were evaluated. To accomplish this objective, the Fe–Ti, Ti–C, Fe–C binary systems and the Fe–Ti–C ternary system were thermodynamically reassessed based on the first-principles calculations. Because of the low concentration of C and S in steels, the Fe–C–S and Ti–C–S ternary systems were excluded from the analysis. Thermodynamic parameters for the Fe–S and Ti–S binary systems and the Fe–Ti–S ternary system were obtained from previous studies.^14,48)

3.2.1. Fe–Ti, Ti–C, and Fe–C Binary Systems

The Fe–Ti binary system is composed of the fcc, bcc, hcp, liquid, Fe₂Ti, and FeTi phases. Figure 3 shows a comparison of the Gibbs free energies for the fcc and bcc phases calculated by CVM with the values from a previous study.⁴⁹⁾ Some discrepancies between our CVM calculation and the thermodynamic analysis are found. However, their parameters⁴⁹⁾ were adopted in the Fe–Ti–S ternary system¹⁴⁾ that was accepted in the present work, then in spite of the slight difference between them, the result was used by this analysis. The thermodynamic parameters in the fcc and bcc phases are shown in Table 2, and the calculated Fe–Ti binary phase diagram is presented in Fig. 4.

Fig. 3.

The Gibbs free energy of the (a) fcc and (b) bcc phases calculated in the Fe–Ti binary system. The reference states are (a) fcc-Fe and fcc-Ti, and (b) bcc-Fe and bcc-Ti, respectively. The black lines denote the results of thermodynamic analysis reported by Kumar et al.⁴⁹⁾

Table 2. Thermodynamic parameters of bcc, fcc, and Ti₄C₂S₂ phases for binary and ternary systems in this study.

Phase and model	System	Thermodynamic parameters, J/mol	Temperature, K	Ref.
bcc (Fe,S,Ti)1(C,Vacancy)3	Fe–S	${}^{0}L_{\text{Fe,S}}^{\text{bcc}}=-119\mathrm{\hspace{0.17em} }675-18.7201T$	298.15<T<6000	48)
	Fe–Ti	${}^{0}L_{\text{Fe,Ti}}^{\text{bcc}}=-57\mathrm{\hspace{0.17em} }943+14.95T$	298.15<T<6000	49)
		${}^{1}L_{\text{Fe,Ti}}^{\text{bcc}}=-6\mathrm{\hspace{0.17em} }059$	298.15<T<6000
	S–Ti	${}^{0}L_{\text{S,Ti}}^{\text{bcc}}=-450\mathrm{\hspace{0.17em} }000$	298.15<T<6000	14)
		${}^{1}L_{\text{S,Ti}}^{\text{bcc}}=-80\mathrm{\hspace{0.17em} }000$	298.15<T<6000
		${}^{2}L_{\text{S,Ti}}^{\text{bcc}}=+200\mathrm{\hspace{0.17em} }000$	298.15<T<6000
	Fe–C	$\begin{array}{l}{}^{\circ }G_{\text{Fe:C}}^{\text{bcc}}-{}^{\circ }G_{\text{Fe:Va}}^{\text{bcc}}-3{}^{\circ }G_{\text{C}}^{\text{graphite}}=+856\mathrm{\hspace{0.17em} }440-109.464T\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+14.253TlnT+0.014T^2\end{array}$	298.15<T<6000	This work
		${}^{0}L_{\text{Fe:C,Va}}^{\text{bcc}}=-735\mathrm{\hspace{0.17em} }989.212+52.973T$	298.15<T<6000
		${}^{1}L_{\text{Fe:C,Va}}^{\text{bcc}}=-5\mathrm{\hspace{0.17em} }196.62+9.385T$	298.15<T<6000
		${}^{2}L_{\text{Fe:C,Va}}^{\text{bcc}}=+214\mathrm{\hspace{0.17em} }766.13-179.487T$	298.15<T<6000
		$T{c}_{\text{Fe:C}}^{\text{bcc}}=+1\mathrm{\hspace{0.17em} }043,{\beta }_{\text{Fe:C}}^{\text{bcc}}=+2.22$	298.15<T<6000	53)
	Ti–C	$\begin{array}{l}{}^{\circ }G_{\text{Ti:C}}^{\text{bcc}}-{}^{\circ }G_{\text{Ti:Va}}^{\text{hcp}}-3{}^{\circ }G_{\text{C}}^{\text{graphite}}=+1\mathrm{\hspace{0.17em} }363\mathrm{\hspace{0.17em} }737.492-109.464T\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+14.253TlnT+0.014T^2\end{array}$	298.15<T<6000	This work
		${}^{0}L_{\text{Ti:C,Va}}^{\text{bcc}}=-578\mathrm{\hspace{0.17em} }438.2067+26.235T$	298.15<T<6000
		${}^{1}L_{\text{Ti:C,Va}}^{\text{bcc}}=+150\mathrm{\hspace{0.17em} }985.682$	298.15<T<6000
		${}^{2}L_{\text{Ti:C,Va}}^{\text{bcc}}=-1\mathrm{\hspace{0.17em} }208\mathrm{\hspace{0.17em} }242.145+36.725T$	298.15<T<6000
fcc (Fe,S,Ti)1(C,Vacancy)1	Fe–S	${}^{0}L_{\text{Fe,S}}^{\text{fcc}}=-108\mathrm{\hspace{0.17em} }733-18T$	298.15<T<6000	48)
	Fe–Ti	${}^{0}L_{\text{Fe,Ti}}^{\text{fcc}}=-50\mathrm{\hspace{0.17em} }304+5.49T$	298.15<T<6000	49)
	Ti–S	${}^{0}L_{\text{S,Ti}}^{\text{fcc}}=-410\mathrm{\hspace{0.17em} }000$	298.15<T<6000	14)
		${}^{1}L_{\text{S,Ti}}^{\text{fcc}}=-100\mathrm{\hspace{0.17em} }000$	298.15<T<6000
		${}^{2}L_{\text{S,Ti}}^{\text{fcc}}=+30\mathrm{\hspace{0.17em} }000$	298.15<T<6000
	Fe–C	$\begin{array}{l}{}^{\circ }G_{\text{Fe:C}}^{\text{fcc}}-{}^{\circ }G_{\text{Fe:Va}}^{\text{fcc}}-{}^{\circ }G_{\text{C}}^{\text{graphite}}=+109\mathrm{\hspace{0.17em} }651+8.2817T\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-1.1365TlnT+4.36\times {10}^{-3}{T}^2\end{array}$	298.15<T<6000	This work
		${}^{0}L_{\text{Fe:C,Va}}^{\text{fcc}}=-107\mathrm{\hspace{0.17em} }953.968-19.196T$	298.15<T<6000
		${}^{1}L_{\text{Fe:C,Va}}^{\text{fcc}}=-108\mathrm{\hspace{0.17em} }062.024+51.399T$	298.15<T<6000
		${}^{2}L_{\text{Fe:C,Va}}^{\text{fcc}}=-70\mathrm{\hspace{0.17em} }710.717+52.69T$	298.15<T<6000
	Ti–C	$\begin{array}{l}{}^{\circ }G_{\text{Ti:C}}^{\text{fcc}}-{}^{\circ }H_{\text{Ti}}^{\text{hcp}}-{}^{\circ }H_{\text{C}}^{\text{graphite}}=-194\mathrm{\hspace{0.17em} }244+264.40T\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-42.248TlnT-4.4167\times {10}^{-3}{T}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\text{31}\mathrm{\hspace{0.17em} }{\text{263T}}^{-\text{1}}+6.96\times {10}^9{T}^{-3}\end{array}$	298.15<T<6000	47)
		${}^{0}L_{\text{Ti:C,Va}}^{\text{fcc}}=-41\mathrm{\hspace{0.17em} }909.757-17.397T$	298.15<T<6000	This work
		${}^{1}L_{\text{Ti:C,Va}}^{\text{fcc}}=-70\mathrm{\hspace{0.17em} }919.290-5.365T$	298.15<T<6000
		${}^{2}L_{\text{Ti:C,Va}}^{\text{fcc}}=-62\mathrm{\hspace{0.17em} }335.120+29.625T$	298.15<T<6000
	Fe–Ti–C	${}^{0}L_{\text{Fe,Ti:C}}^{\text{fcc}}=+18\mathrm{\hspace{0.17em} }735.894+6.956T$	298.15<T<6000	This work
		${}^{1}L_{\text{Fe,Ti:C}}^{\text{fcc}}=-2\mathrm{\hspace{0.17em} }878.682+3.161T$	298.15<T<6000
		${}^{0}L_{\text{Fe,Ti:C,Va}}^{\text{fcc}}=-41\mathrm{\hspace{0.17em} }167.538+137.133T$	298.15<T<6000
		${}^{1}L_{\text{Fe,Ti:C,Va}}^{\text{fcc}}=+294\mathrm{\hspace{0.17em} }560.13-24.886T$	298.15<T<6000
		${}^{2}L_{\text{Fe,Ti:C,Va}}^{\text{fcc}}=-294\mathrm{\hspace{0.17em} }560.13+24.886T$	298.15<T<6000
Ti4C2S2 (Ti)4(C)2(S)2	Ti–C–S	$\begin{array}{l}{}^{\circ }G_{\text{Ti:C:S}}^{{\text{Ti}}_4{\text{C}}_2{\text{S}}_2}-4{}^{\circ }G_{\text{Ti}}^{\text{hcp}}-2{}^{\circ }G_{\text{C}}^{\text{Graphite}}-2{}^{\circ }G_{\text{S}}^{\text{Orthorhombic}}\\ =-1\mathrm{\hspace{0.17em} }013\mathrm{\hspace{0.17em} }379.126+25.794T-8.074TlnT\\ +1.490\times {10}^{-2}{T}^2+2.310\times {10}^{-6}{T}^3-4.986\times {10}^{-10}{T}^4\end{array}$	298.15<T<6000	This work

Fig. 4.

The calculated Fe–Ti binary phase diagram based on the thermodynamic analysis by Kumar et al.⁴⁹⁾

The Ti–C binary system is composed of fcc (which is identical to NaCl-type TiC), bcc, hcp, liquid, and graphite phases. The calculated Gibbs free energies for the fcc and bcc phases in the Ti–C binary system are shown in Fig. 5. The symbols in these figures represent the calculated results using CVM, and the solid lines show the Gibbs energy evaluated by fitting the CVM values using Eqs. (11) and (13). The solid lines in Fig. 5 are shown for comparison with the results from CVM. From this comparison in Fig. 5(a), thermodynamic parameters for fcc phase evaluated in this study agree with the result of CVM. In contrast, we observe a slight difference which originates from influence of short-range ordering in the CVM calculation in Fig. 5(b). The calculated value of the CVM decreases solubility of C in bcc-Ti, resulting in deviation from the experimental values. Therefore, the experimental solubilities of C in bcc-Ti^50,51) were considered simultaneously with the evaluated free energy based on the CVM calculation in this thermodynamic analysis, and the free energy in conformity to the experimental values was adopted in this study. The Ti–C binary phase diagram calculated using these parameters is compared with that by the previous study⁴⁷⁾ in Fig. 6.

Fig. 5.

The calculated Gibbs free energies of the (a) fcc and (b) bcc phases in the Ti–C binary system. The solid lines denote the results of the thermodynamic analysis in the present study.

Fig. 6.

The calculated Ti–C binary phase diagram in (a) this study compared with a (b) previous study.⁴⁷⁾

In the Fe–C binary system, the fcc, bcc, liquid, and graphite phases constitute the equilibria. The calculated Gibbs free energies for the fcc and bcc phases in the binary system are shown in Fig. 7. The symbols in these figures represent the results of CVM calculation, and the solid lines denote the calculated values in the present study. To check the validity of the optimized thermodynamic parameters, the calculated activity of C was compared with the experimental results. Figure 8 shows the comparison for the activity of C in the (a) fcc and (b) bcc phases with the experimental values.^52,53,54) The calculated activities are consistent with the experimental results in both phases, and the calculated Fe–C binary phase diagram is shown in Fig. 9 and compared with the previous result.⁵⁵⁾

Fig. 7.

The Gibbs free energies for the (a) fcc and (b)(c) bcc phases in the Fe–C binary system calculated using CVM. The solid lines denote the results of the thermodynamic analysis in this study.

Fig. 8.

The comparison of the activities of C in (a) fcc and (b) bcc phases in the Fe–C binary system between the present calculation and experiment values.^52,53,54)

Fig. 9.

The calculated Fe–C binary phase diagram (a) in this study compared with a (b) previous study.⁵⁵⁾

3.2.2. Fe–Ti–C Ternary and Fe–Ti–S–C Quaternary Systems

The ternary fcc phase in the Fe–Ti–C system was thermodynamically analyzed in the present study. Figures 10(a) and 10(b) show the cross-sections of the Gibbs free energy surface along the lines connecting Fe to TiC, and TiC to FeC, respectively. The symbols in this figure represent the results of CVM calculation and the solid lines show the evaluated values by the thermodynamic analysis. The Gibbs energy curves across Fe to TiC are upwardly convex, indicating a two-phase separation between the fcc-Fe and NaCl-type TiC phase. In Fig. 10(a), there exists some discrepancy between the results of thermodynamic analysis and CVM at lower C content, which may be a due to effect of short-range ordering in CVM calculation. When the result of CVM is just used for phase diagram calculation, the solubility of TiC will be evaluated smaller than that obtained in the present analysis. This results in the estrangement from the experimental values. Figure 11 shows a comparison for the activity of C in the Fe–Ti–C ternary system.⁵⁶⁾ The calculated activities agree well with the experimental results. In Fig. 12, the calculated solubilities of TiC in fcc-Fe are shown with some experimental results.^57,58) The good accordance between the results shows that the thermodynamic properties evaluated by this study describe this ternary fcc phase well. In the case of using CVM results faithfully, it is expected that the solubility of TiC in fcc-Fe is calculated smaller because of the instability of CVM results. Then in the present thermodynamic analysis, experimental solubilities of TiC in fcc-Fe^57,58) were also considered simultaneously with the evaluated free energy based on the CVM calculation. For the thermodynamic parameters, except for the bcc, fcc, and Ti₄C₂S₂ phases in the Fe–Ti–S–C quaternary system, the results from the previous works on the Fe–S,⁴⁸⁾ Fe–Ti,⁴⁹⁾ Ti–C,⁴⁷⁾ Fe–C,⁵⁵⁾ Fe–Ti–C,⁵⁹⁾ Ti–S and Fe–Ti–S¹⁴⁾ were adopted. Finally, the thermodynamic parameters for bcc, fcc, and Ti₄C₂S₂ phases are listed in Table 2.

Fig. 10.

The Gibbs free energy curves calculated in the cross sections of (a) Fe–TiC and (b) TiC–FeC in the fcc phase of the Fe–Ti–C ternary system.

Fig. 11.

Comparison of the calculated activities of C in austenitic Fe–Ti alloys with the experimental values.⁵⁶⁾

Fig. 12.

Comparison of the solubility of TiC in fcc-Fe between the present calculation and the experiments.^57,58)

3.3. Solubility Product of Ti₄C₂S₂ in fcc-Fe

In this section, an approximate expression for the solubility product of Ti₄C₂S₂ is derived. When the Ti₄C₂S₂ phase is treated as a stoichiometric compound, the equilibrium between Ti₄C₂S₂ with fcc-Fe can be expressed by the following equation:⁶⁰⁾   

$${}^{\circ }G_{{\text{Ti}}_4{\text{C}}_2{\text{S}}_2}=4{\mu }_{\text{Ti}}^{\text{fcc}}+2{\mu }_{\text{S}}^{\text{fcc}}+2{\mu }_{\text{C}}^{\text{fcc}},$$(15)

where ${}^{\circ }G_{{\text{Ti}}_4{\text{C}}_2{\text{S}}_2}$ is the Gibbs energy of Ti₄C₂S₂ per formula unit and ${\mu }_i^{\text{fcc}}$ is the chemical potential of element i in the fcc-Fe phase. Considering the lower contents of Ti, S, C in the fcc-Fe phase, each chemical potential can be abbreviated as follows:   

$${\mu }_{\text{Ti}}^{\text{fcc}}={}^{\circ }G_{\text{Ti:Va}}^{\text{fcc}}+RTln{y}_{\text{Ti}}^{\text{fcc}}+L_{\text{Fe,Ti:Va}}^{\text{fcc}},$$(16)

  

$${\mu }_{\text{S}}^{\text{fcc}}={}^{\circ }G_{\text{S:Va}}^{\text{fcc}}+RTln{y}_{\text{S}}^{\text{fcc}}+L_{\text{Fe,S:Va}}^{\text{fcc}},$$(17)

  

$${\mu }_{\text{C}}^{\text{fcc}}={}^{\circ }G_{\text{Fe:C}}^{\text{fcc}}-{}^{\circ }G_{\text{Fe:Va}}^{\text{fcc}}+RTln{y}_{\text{C}}^{\text{fcc}}+L_{\text{Fe:C,Va}}^{\text{fcc}}.$$(18)

By substituting Eqs. (16), (17), and (18) in Eq. (15), the formation energy of Ti₄C₂S₂ phase ${}^{\circ }G_{{\text{Ti}}_4{\text{C}}_2{\text{S}}_2}$ can be expressed as Eq. (19):   

$$\begin{array}{l}\mathrm{\Delta }{G}_{{\text{Ti}}_4{\text{C}}_2{\text{S}}_2}\equiv {}^{\circ }G_{{\text{Ti}}_4{\text{C}}_2{\text{S}}_2}-4{}^{\circ }G_{\text{Ti}}^{\text{hcp}}-2{}^{\circ }G_{\text{S}}^{\text{orthorhombic}}-2{}^{\circ }G_{\text{C}}^{\text{graphite}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}=4({}^{\circ }G_{\text{Ti}}^{\text{fcc}}-{}^{\circ }G_{\text{Ti}}^{\text{hcp}})+2({}^{\circ }G_{\text{S}}^{\text{fcc}}-{}^{\circ }G_{\text{S}}^{\text{orthorhombic}})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}+2\left({}^{0}G_{\text{Fe:C}}^{\text{fcc}}-{}^{0}G_{\text{Fe:Va}}^{\text{fcc}}-{}^{0}G_{\text{C}}^{\text{graphite}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}+\text{4}{L}_{\text{Fe,Ti:Va}}^{\text{fcc}}+2L_{\text{Fe,S:Va}}^{\text{fcc}}+2L_{\text{Fe:C,Va}}^{\text{fcc}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}+4RTln{y}_{\text{Ti}}^{\text{fcc}}{(y_{\text{S}}^{\text{fcc}})}^{0.5}{(y_{\text{C}}^{\text{fcc}})}^{0.5}.\end{array}$$(19)

By applying the thermodynamic parameters listed in Table 2 and neglecting the composition dependency for interaction parameter L, Eq. (20) can be obtained:   

$$\begin{array}{l}RTln{y}_{\text{Ti}}^{\text{fcc}}{(y_{\text{S}}^{\text{fcc}})}^{0.5}{(y_{\text{C}}^{\text{fcc}})}^{0.5}\\ =-226\mathrm{\hspace{0.17em} }698.69+14.87T-1.45TlnT\\ +1.54\times {10}^{-3}{T}^2+5.77\times {10}^{-7}{T}^3-1.25\times {10}^{-10}{T}^4.\end{array}$$(20)

The term $ln{y}_{\text{Ti}}^{\text{fcc}}{(y_{\text{S}}^{\text{fcc}})}^{0.5}{(y_{\text{C}}^{\text{fcc}})}^{0.5}$ in Eq. (20) can be approximated as follows:   

$$\begin{array}{l}ln{y}_{\text{Ti}}^{\text{fcc}}{(y_{\text{S}}^{\text{fcc}})}^{0.5}{(y_{\text{C}}^{\text{fcc}})}^{0.5}\\ \cong ln{x}_{\text{Ti}}^{\text{fcc}}{(x_{\text{S}}^{\text{fcc}})}^{0.5}{(x_{\text{C}}^{\text{fcc}})}^{0.5}\\ \cong ln\left({\displaystyle \frac{M_{\text{Fe}}}{M_{\text{Ti}}}}\cdot w_{\text{Ti}}^{\text{fcc}}\right){\left({\displaystyle \frac{M_{\text{Fe}}}{M_{\text{S}}}}\cdot w_{\text{S}}^{\text{fcc}}\right)}^{0.5}{\left({\displaystyle \frac{M_{\text{Fe}}}{M_{\text{C}}}}\cdot w_{\text{C}}^{\text{fcc}}\right)}^{0.5}\\ =ln\left({\displaystyle \frac{M_{\text{Fe}}}{M_{\text{Ti}}}}\cdot {\displaystyle \frac{[\text{mass\%Ti}]}{100}}\right){\left({\displaystyle \frac{M_{\text{Fe}}}{M_{\text{S}}}}\cdot {\displaystyle \frac{[\text{mass\%S}]}{100}}\right)}^{0.5}{\left({\displaystyle \frac{M_{\text{Fe}}}{M_{\text{C}}}}\cdot {\displaystyle \frac{[\text{mass\%C}]}{100}}\right)}^{0.5}\\ =2.303\times \left\{log[\text{mass\%Ti}]{[\text{mass\%S}]}^{0.5}{[\text{mass\%C}]}^{0.5}+\text{log}\left({\displaystyle \frac{M_{\text{Fe}}^{\text{2}}}{\text{10}\mathrm{\hspace{0.17em} }\text{000}{M}_{\text{Ti}}{M}_{\text{S}}^{\text{0}\text{.5}}{M}_{\text{C}}^{\text{0}\text{.5}}}}\right)\right\},\end{array}$$(21)

where x_i is the mole fraction of element i, w_i is the mass fraction of element i, M_i is the molar mass of element i and [mass%i] is the mass percent of i in the matrix respectively.

Therefore, the equation for the solubility product can be described as Eq. (22):   

$$\begin{array}{l}log[\text{mass\%Ti}]{[\text{mass\%S}]}^{0.5}{[\text{mass\%C}]}^{0.5}\\ ={\displaystyle \frac{-98\mathrm{\hspace{0.17em} }436.25+6.46T-0.63TlnT+6.70\times {10}^{-4}{T}^2+2.51\times {10}^{-7}{T}^3-5.41\times {10}^{-11}{T}^4}{RT}}\\ +\text{3}\text{.479}\text{.}\end{array}$$(22)

The solubility product of Ti₄C₂S₂ calculated by using Eq. (22) is listed in Table 3. Figure 13 shows the calculated solubility product of Ti₄C₂S₂ in the fcc phase, which is compared to some experimental values.^4,6,9,10) The filled squares are the solubility product listed in Table 3. In the solubility lines determined by experiments, the portion given by the bold line shows the measured experimental region. In general, the solubility product changes linearly with the reciprocal temperature, at least in a small temperature range. That is one of the reasons why the temperature dependency of the solubility product significantly differs outside the region of each measurement. Regarding the observation by Mitsui et al.,¹⁰⁾ a wider temperature range was covered compared with that observed in other measurements.^4,6,9) The solubility product obtained in the present study coincides well with their results.

Table 3. The solubility product of Ti₄C₂S₂ calculated using Eq. (22).

Temperature, T / K	$\frac{10\mathrm{\hspace{0.17em} }000}{T}$	Solubility product log[mass%Ti][mass%S]0.5[mass%C]0.5
1250	8	−5.62
1428.5714	7	−4.43
1666.6667	6	−3.22
2000	5	−2.01

Fig. 13.

The calculated variation of solubility products with reciprocal temperature. Some experimental results^4,6,9,10) are shown for comparison. The portion given by bold line shows the measured experimental region.

3.4. Calculation of Precipitation Behavior in IF Steels

Figure 14 presents the calculated amount of precipitates in Fe–Ti–S–C quaternary system and Fig. 14(a) shows the results of Fe–0.0010%C–0.02%Ti–0.025%S alloy varying with temperature. The main precipitate in austenite is NiAs-type TiS in a higher temperature range, while the Ti₄C₂S₂ phase is formed with descending temperature. For the thermodynamic properties of the NiAs-type TiS, estimated values by Hirata et al.¹⁴⁾ were accepted in the present study as listed in Table 4.

Fig. 14.

Calculated amount of precipitates in (a) Fe–0.0010%C–0.02%Ti–0.025%S alloy varying with temperature, and effect of alloying contents of (b) S and (c) C on the formation behavior of the precipitates.

Table 4. Thermodynamic parameters of NiAs-type TiS phase for binary and ternary systems in this study.^14,48)

Phase and model	System	Thermodynamic parameters, J/mol	Temperature, K	Ref.
NiAs–TiS (Fe,Ti,Vacancy)1(S,Vacancy)1	Fe–S	$\begin{array}{l}{}^{\circ }G_{\text{Fe:S}}^{\text{NiAs-TiS}}-{}^{\circ }G_{\text{Fe:Va}}^{\text{bcc}}-{}^{\circ }G_{\text{S}}^{\text{Orthorhombic}}\\ =-107\mathrm{\hspace{0.17em} }518-18.19T+1.78TlnT\end{array}$	298.15<T<6000	48)
		${}^{\circ }G_{\text{Fe:Va}}^{\text{NiAs-TiS}}-{}^{\circ }G_{\text{Fe:Va}}^{\text{bcc}}=+65\mathrm{\hspace{0.17em} }000$	298.15<T<6000
		${}^{\circ }G_{\text{Va:S}}^{\text{NiAs-TiS}}-{}^{\circ }G_{\text{S}}^{\text{Orthorhombic}}=+258\mathrm{\hspace{0.17em} }600$	298.15<T<6000
		${}^{\circ }G_{\text{Va:Va}}^{\text{NiAs-TiS}}=+1\mathrm{\hspace{0.17em} }000\mathrm{\hspace{0.17em} }000$	298.15<T<6000
		${}^{0}L_{\text{Fe,Va:S}}^{\text{NiAs-TiS}}=-409\mathrm{\hspace{0.17em} }000+10T$	298.15<T<6000
		${}^{1}L_{\text{Fe,Va:S}}^{\text{NiAs-TiS}}=+60\mathrm{\hspace{0.17em} }000+20T$	298.15<T<6000
		${}^{0}L_{\text{Fe:S,Va}}^{\text{NiAs-TiS}}=+100\mathrm{\hspace{0.17em} }000$	298.15<T<6000
		${}^{0}L_{\text{Va:S,Va}}^{\text{NiAs-TiS}}=-409\mathrm{\hspace{0.17em} }000+10T$	298.15<T<6000
		${}^{1}L_{\text{Va:S,Va}}^{\text{NiAs-TiS}}=+60\mathrm{\hspace{0.17em} }000+20T$	298.15<T<6000
		${}^{0}L_{\text{Fe,Va:Va}}^{\text{NiAs-TiS}}=+100\mathrm{\hspace{0.17em} }000$	298.15<T<6000
	Ti–S	$\begin{array}{l}{}^{\circ }G_{\text{Ti:S}}^{\text{NiAs-TiS}}-{}^{\circ }G_{\text{Ti:Va}}^{\text{hcp}}-{}^{\circ }G_{\text{S}}^{\text{Orthorhombic}}\\ =-279\mathrm{\hspace{0.17em} }121.17-14T+0.8TlnT\\ +6.6\times {10}^{-3}{T}^2-4.6\times {10}^{-7}{T}^3\end{array}$	298.15<T<6000	14)
		${}^{\circ }G_{\text{Ti:Va}}^{\text{NiAs-TiS}}-{}^{\circ }G_{\text{Ti:Va}}^{\text{hcp}}=+121\mathrm{\hspace{0.17em} }418+30T$	298.15<T<6000
		${}^{\circ }G_{\text{Va:S}}^{\text{NiAs-TiS}}-{}^{\circ }G_{\text{S}}^{\text{Orthorhombic}}=+258\mathrm{\hspace{0.17em} }600$	298.15<T<6000
		${}^{\circ }G_{\text{Va:Va}}^{\text{NiAs-TiS}}=+1\mathrm{\hspace{0.17em} }000\mathrm{\hspace{0.17em} }000$	298.15<T<6000
		${}^{0}L_{\text{Ti,Va:S}}^{\text{NiAs-TiS}}=-125\mathrm{\hspace{0.17em} }000-45T$	298.15<T<6000
		${}^{0}L_{\text{Ti:S, Va}}^{\text{NiAs-TiS}}=-125\mathrm{\hspace{0.17em} }000-45T$	298.15<T<6000
		${}^{0}L_{\text{Va:S,Va}}^{\text{NiAs-TiS}}=-409\mathrm{\hspace{0.17em} }000+10T$	298.15<T<6000
		${}^{1}L_{\text{Va:S,Va}}^{\text{NiAs-TiS}}=+60\mathrm{\hspace{0.17em} }000+20T$	298.15<T<6000
	Fe–Ti–S	${}^{0}L_{\text{Fe, Ti:S}}^{\text{NiAs-TiS}}=-45\mathrm{\hspace{0.17em} }000-20T$	298.15<T<6000	14)
		${}^{1}L_{\text{Fe, Ti:S}}^{\text{NiAs-TiS}}=-20\mathrm{\hspace{0.17em} }000$	298.15<T<6000

Figures 14(b) and 14(c) show the results for the effect of S and C contents in the alloy on the formation behavior of the precipitates. The increase of the S content leads to an increase of the amount of TiS; however, the formation temperature range of Ti₄C₂S₂ and its total amount of precipitation little change, as shown in Fig. 14(b). In contrast, as seen in Fig. 14(c), the increase of the C content in the alloy scarcely influences the formation behavior of TiS; however, it yields an increase in the amount of Ti₄C₂S₂. This fact confirms that the formation of Ti₄C₂S₂ is mainly influenced by the C levels in IF steels.

4. Conclusion

This work evaluates the solubility and formation behavior of Ti₄C₂S₂ in steel, which have not been clarified because of experimental difficulty, by a combination of the first-principles calculations and the thermodynamic analysis. Based on this result, the formation temperatures for the Ti₄C₂S₂ and other sulfides in the IF steel was calculated, and the formation behavior was discussed. As a result, the following conclusions were obtained.

(1) The free energies of formation for the Ti₄C₂S₂ and NaCl-type TiC were evaluated by first-principles calculations and vibrational analysis. The free energies of bcc and fcc phases in the Fe–Ti–C ternary system were also calculated by the cluster expansion and CVM to clarify the formation behavior of these precipitates in IF steels. Thermodynamic parameters for these phases were evaluated by fitting those results as well as the thermodynamic analysis in which some experimental data were considered to the sublattice model.

(2) By using the evaluated thermodynamic parameters, an approximate expression of the solubility product for Ti₄C₂S₂ was derived. The result agrees well with the experimental result measured in a relatively large temperature range.

(3) The formation behavior of these precipitates in some typical IF steels was calculated by considering the result of the thermodynamic analysis on the Fe–Ti–S ternary system by a previous work. NiAs-type TiS was the main precipitate at higher temperatures, while the Ti₄C₂S₂ phase was formed with descending temperature. The results show that these precipitates are influenced by the contents of alloying elements, and the amount of precipitation of Ti₄C₂S₂ increases as C concentration increases in the steels.

Acknowledgments

This study was supported by JSPS KAKENHI (grant number, 16H02387). The authors also gratefully acknowledge the useful discussions with Dr. Masanori Enoki at Tohoku University.

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