Thermodynamic analysis of the Fe–Ti–P ternary system by incorporating first-principles calculations into the CALPHAD approach

Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 147–158 www.elsevier.com/locate/calphad

Thermodynamic analysis of the Fe–Ti–P ternary system by incorporating first-principles calculations into the CALPHAD approach Hiroshi Ohtani a,∗ , Naoko Hanaya b , Mitsuhiro Hasebe a , Shin-ichi Teraoka c , Masayuki Abe c a Department of Materials Science and Engineering, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan b Graduate School, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan c Yawata R&D Lab., Nippon Steel Co., Tobata-ku, Kitakyushu 804-8501, Japan

Received 1 August 2005; received in revised form 16 September 2005; accepted 21 September 2005 Available online 12 October 2005

Abstract A thermodynamic analysis of the Fe–Ti–P ternary system has been performed by combining first-principles calculations with the CALPHAD approach. Because of the lack of experimental information available, the enthalpies of formation of the Fe–P and Ti–P based binary phosphides were evaluated using the Full Potential Linearized Augmented Plane Wave method, and the estimated values were introduced into a CALPHADtype thermodynamic analysis. Applying this procedure, the phase diagrams of the Fe–P and Ti–P binary systems, whose contents are uncertain so far, were calculated with a high degree of probability. The thermodynamic properties of orthorhombic anti-PbCl2 -type FeTiP were obtained following the same procedure. The calculated phase diagrams were in good accordance with previous experimental results. The ternary phosphide, FeTiP, was in equilibrium with most of the phases in the ternary system, and was dominant in the liquidus surface projection. c 2005 Elsevier Ltd. All rights reserved.  Keywords: Phase diagram; Thermodynamic analysis; First-principles calculation; Phosphide; IF steels

1. Introduction Titanium-stabilized ultra-low carbon steels are generally referred to as interstitial-free (IF) steels, and they are widely used in automobile manufacturing, because of their good formability and drawability. The addition of Ti plays a crucial role in controlling the desired properties of these steels by removal of the interstitials from the solid solution and by forming very fine precipitates. Irrespective of these advantages, a strain ageing or retardation of recrystallization sometimes appears in Ti-containing IF steels due to the presence of P, which is added as solid-solution strengthening element. This is attributed to the exhaustion of Ti during the formation of the ternary phosphide, FeTiP, and, therefore, it is essential to understand the precipitation behavior of this phosphide during the annealing stage, using the information contained in phase diagrams. Because of the high volatilization of P, the Fe–Ti–P ternary phase diagram has been investigated only for the Fe corner of ∗ Corresponding author. Tel.: +81 93 884 3359; fax: +81 93 884 3359.

E-mail address: [email protected] (H. Ohtani). c 2005 Elsevier Ltd. All rights reserved. 0364-5916/$ - see front matter  doi:10.1016/j.calphad.2005.09.006

this system, using thermal analysis, metallography and X-ray diffraction [1]. Based on this work, an outline of the projected liquidus surface, as well as some isothermal and vertical sections, was revealed. Thus, a thermodynamic analysis of the Fe–FeP–Ti system may be possible by applying a conventional CALPHAD-type analysis. Clarification of the thermodynamic properties and the phase equilibrium in this Fe-rich region may fulfill a practical purpose. On the other hand, phase diagrams that are difficult to determine experimentally have been constituted using the thermodynamic quantities evaluated by first-principles calculations in the CALPHAD approach [2,3]. Estimating unknown phase equilibria based on the established theoretical values should play an important role, especially in developing new materials. Thus, we attempted to calculate the thermodynamic properties of various phosphides using a first-principles band energetic calculation method. The objective of this study was to clarify the phase equilibria of the Fe–Ti–P ternary system over the entire composition range, by incorporating first-principles calculations into a CALPHAD method.

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This section discusses the computational procedure used for the first-principles calculations, and the expression of the Gibbs free energy.

where ◦ H iref denotes the molar enthalpy of the pure element i in its stable state at T = 25 ◦ C. The parameter L Li,j denotes the interaction energy between i and j in the liquid phase, and has a compositional dependency following the Redlich–Kister polynomial as

2.1. First-principles energetic calculations

L Li,j = 0 L Li,j + 1 L Li,j (x i − x j ) + 2 L Li,j (x i − x j )2

2. Computational procedure

The formation energies of the various phosphide phases were calculated using the Full Potential Linearized Augmented Plane Wave (FLAPW) method. The FLAPW method, as embodied in the WIEN2k software package [4], is one of the most accurate schemes for electronic calculations, and allows for precise calculations of the total energies in a solid. The FLAPW method uses a scheme to solve many-electron problems based on the local spin density approximation (LSDA) approach. In this framework, the unit cell is divided into two regions: non-overlapping atomic spheres and an interstitial region. Inside the atomic spheres, the wave functions of the valence states are expanded by a linear combination of the radial functions and spherical harmonics, and a plane wave expansion is used in the interstitial region. Because the LSDA approach includes an approximation for both the exchange and correlation energies, it has been recently expanded on by adding gradient terms for the electron density to the exchange–correlation energy. This has led to the generalized gradient approximation (GGA) method suggested by Perdew et al. [5], and we used this improved method over the conventional LSDA approach. Muffin-tin radii of 2.0 au (0.106 nm) for Fe and Ti were assumed in our computations, and a muffin-tin radius of 1.9 au (0.1005 nm) was assumed for P. The value of RKmax was fixed at 9.0, which almost corresponds to the 20 Ry (270 eV) cutoff energy used in our work. Spin-polarized electronic structure calculations were carried out on the FeTiP phase. 2.2. Thermodynamic modeling of the solution phases A description of the Gibbs energy for each phase appearing in the Fe–Ti–P ternary system is presented in this section. 2.2.1. Liquid (L), bcc (α), fcc (γ ), and hcp (α-Ti) solid solutions The regular solution approximation was applied to the liquid phase. The molar Gibbs energy, G Lm , was calculated using the following equation: G Lm = x Fe ◦ G LFe + x P ◦ G LP + x Ti ◦ G LTi + RT (x Fe ln x Fe +x P ln x P + x Ti ln x Ti) + x Fe x P L LFe,P + x Fe x Ti L LFe,Ti +x P x Ti L LP,Ti + x Fe x P x Ti L LFe,P,Ti

(1)

where ◦ G Li denotes the molar Gibbs energy of element i in the liquid state. This quantity is called the lattice stability parameter, and is described by the formula ◦

G Li − ◦ H iref = A + BT + C T ln T + DT 2 + E T 3 +F T 7 + I T −1 + J T −9

(2)

+ · · · + n L Li,j (x i − x j )n

(3)

where n

L Li,j = a + bT + cT ln T + dT 2 + · · · .

(4)

The term L LFe,P,Ti is the ternary interaction parameter between elements Fe, P, and Ti. The compositional dependency of the interaction parameters is expressed as L LFe,P,Ti = x Fe 0 L LFe,P,Ti + x P 1 L LFe,P,Ti + x Ti 2 L LFe,P,Ti .

(5)

The bcc (α), fcc (γ ), and hcp (α-Ti) solid solutions, which exhibit a range of non-stoichiometric alloys, were modeled using the same regular solution approximation. The contribution to the Gibbs free energy due to the magnetic ordering was added to the nonmagnetic part of the free energy [6,7]. 2.2.2. Fe2 Ti Laves phase The C14 Laves phase, Fe2 Ti, shows a wide compositional range. To account for the homogeneity range, Kumar et al.[8] used a three-sublattice    model [9] denoted by Fe y (1) Va y (1) Fe y (2) Ti y (2) Fe y (3)=1 , and this thermodyFe

Va

2

Fe

Ti

4

Fe

6

namic description was used in our study. The Gibbs energy per mole of formula unit of this phase is given by   (1) (2) ◦ Fe2 Ti (2) ◦ Fe2 Ti Fe2 Ti Gm yFe = yFe G Fe:Fe:Fe +yTi G Fe:Ti:Fe   (1) (2) Fe2 Ti (2) Fe2 Ti +yTi ◦ G Va:Ti:Fe + yVa yFe ◦ G Va:Fe:Fe    (1) (1) (1) (1) + RT 2 yFe ln yFe + yVa ln yVa   (2) (2) (2) (2) + 4 yFe ln yFe + yTi ln yTi   (1) (1) (2) Fe2 Ti (2) Fe2 Ti + yTi L Fe,Va:Ti:Fe + yFe yVa yFe L Fe,Va:Fe:Fe   (2) (2) (1) Fe2 Ti (1) Fe2 Ti + yFe yTi yFe L Fe:Fe,Ti:Fe + yVa L Va:Fe,Ti:Fe (1) (1) (2) (2) Fe2 Ti + yFe yVa yFe yTi L Fe,Va:Fe,Ti:Fe

(6)

Fe2 Ti where ◦ G i:j:k represents the Gibbs energy of a hypothetical compound, i2 j4 k6 , and terms relative to the same stoichiometry Fe2 Ti are identical. The parameter L i,j:k:l , for example, denotes the interaction energies between unlike atoms on the first sublattice, varying with composition as a polynomial expansion as   (1) (1) 1 Fe2 Ti 0 Fe2 Ti 2 Ti L Fe L i,j:k:l . (7) i,j:k:l = L i,j:k:l + yi − yj

A similar compositional dependency was introduced in Fe2 Ti 2 Ti L Fe i:j,k:l , and L Fe,Va:Fe,Ti:Fe was assumed to be independent of composition.

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2.2.3. (Fe, Ti)3 P, (Fe, Ti)2 P, Ti5 P3 and TiP phases A slight solubility of Ti in the (Fe, Ti)3 P and (Fe, Ti)2 P phases has been reported from experiments [10]. Therefore, in the present study, the Gibbs energies of these phases were expressed using a two-sublattice model, in which the Fe atoms were substituted   with Ti  atoms in the first sublattice, i.e.,  P y (2) =1 . Fe y (1) Ti y (1) Fe

Ti

v1

v2

P

Ti5 P3 and TiP have some homogeneity range [11], and the Gibbs   energies were modeled using the formula Ti y (1) =1 . Here, w1 and w2 denote the P y (2) Ti y (2) Ti

w1

P

Ti

w2

number of sites of the sublattice in parentheses, and w1 = 0.61 and w2 = 0.39, and w1 = w2 = 1 were assigned for Ti5 P3 and TiP, respectively. 2.2.4. Stoichiometric compounds The binary compound phases with zero homogeneity ranges (i.e., FeP, FeP2 , FeP4 , Ti3 P, Ti1.7 P, Ti4 P3 , and TiP2 ) were treated as being stoichiometric compounds. 2.2.5. FeTiP phase Vogel and Giessen [1] found this phase to be essentially stoichiometric. However, a small homogeneity range was found at 800 ◦ C in the Fe3 P–FeTiP pseudobinary system [10]. Therefore,   the three-sublattice    model denoted by  Fe y (2) Ti y (2) P y (3) =1 Fe y (1) Ti y (1) was applied to this Fe

phase.

Ti

1

Fe

Ti

1

P

1

3. Results and discussion 3.1. Calculation of the thermodynamic properties of the phosphide phases The enthalpy of formation of the various phosphide phases in the binary and ternary systems was evaluated using firstprinciples calculations. Information on crystallographic data were obtained from reference [12]. The calculated results are listed in Table 1, together with the equilibrium lattice constants. The calculated values denote the formation enthalpies based on bcc Fe, hcp Ti, and black P. The reference material for P in the Scientific Group Thermodata Europe (SGTE) database [13] is white or red P; however, neither the crystal structure nor the internal atomic positions of these materials are available. Therefore, in this work, the total energy of pure P was calculated using the crystallographic data of black P (space group = Cmca) [12]. The thermodynamic properties of the binary phosphides have not been evaluated experimentally. However, Spencer and Kubaschewski [14] derived the enthalpy of formation of Fe3 P from bcc Fe and white P at room temperature to be −41 kJ/mol based on heat capacity data and high-temperature calorimetric data. The corresponding value obtained using first-principles calculations is −48.4 kJ/mol. If all the discrepancy is attributed to the difference in stability between white and black P, then white P is more stable than black P by about 22.4 kJ/mol at most. However, this value is not applicable at all times, because

149

the evaluated enthalpy of formation for Fe2 P based on firstprinciples calculations (−54.56 kJ/mol) is in close agreement with the estimated result (−53.3 kJ/mol) [14]. Black P is inherently an allotropic modification of white P. However, white P has a monoclinic structure composed of tetrahedra, whereas black P has an orthorhombic structure. Thus, the phase stability of these two structures should be different. Working against such a prior expectation is that the energy difference between the black P and metastable fcc structures is about 0.01604 Ry(= 21.06 kJ/mol) according to first-principles calculations, which is almost equivalent to the phase stability between white P and the fcc structure advocated by the SGTE database. This result supports the assumption that the phase stabilities of black and white P are not much different, but this might be difficult to confirm. Thus, in the present work, an error of about 10% was considered for the absolute values of the enthalpy of formation derived using first-principles calculations. The thermodynamic property of Ti1.7 P is not shown in Table 1, as no crystallographic data were available. Therefore, the thermodynamic quantity of this structure was evaluated using thermodynamic analysis. Ti2 P was reported to exhibit two types of crystal structure: hexagonal and trigonal [11]. The former structure corresponds to the same structure as ¯ P 62m Fe2 P, while the detail of the latter structure is not ¯ clear. Thus, in the present study, the hexagonal P 62m Ti2 P is assumed to be stable in the Ti–P binary system, and the thermodynamic quantity is listed in the table. Rundqvist and Nawapong [15] investigated FeZrP in detail using single-crystal X-ray diffraction. They showed that FeZrP crystallized in an orthorhombic anti-PbCl2 -type structure (space group = Pnma), and the internal atomic coordinates for this phosphide are known. Ivanov et al. [16] recently investigated FeTiP found in a meteorite using in situ synchrotron X-ray diffraction, and determined the internal atomic coordinates. Their results almost corresponded to those found for FeZrP [15]. However, the formation enthalpy of FeTiP calculated using the internal position data was slightly more unstable than that obtained based on the results for FeZrP. Therefore, the formation enthalpy of FeTiP in the ground state was evaluated using the internal atomic coordinates of Rundqvist and Nawapong, as listed in Table 1. According to the spin-polarized calculations, the magnetic moment of FeTiP was 0.85 μB . 3.2. The electronic structure and phase stability of the FeTiP phase Fig. 1(a)–(d) show the total density of states (DOS) and the angular-momentum-resolved density of states (p-DOS) for each element of the FeTiP phase, respectively. The term E f denotes the Fermi energy, and no electrons occupy the electronic states above this energy level. The Fermi level is located at the origin in Fig. 1(a)–(d). Gupta et al. [17] studied the electronic structure of FeTiP using self-consistent linear muffin-tin orbitals (LMTOs) in the atomic sphere approximation (ASA). The shape of the density of states for FeTiP in Fig. 1(a) is in good agreement with the results of Gupta et al., and their explanation of the electronic

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Table 1 The calculated thermodynamic parameters for phosphides System

Phase

Status in phase diagrams

Space group

Calculated lattice parameter (nm)

Observed lattice parameter (nm)

Fe3 P in (Fe, Ti)3 P

stable

I 4¯

a = 0.9074 c = 0.4444

a = 0.9107 c = 0.4460

−48.4

−41.0 [14]

Fe2 P in (Fe, Ti)2 P

stable

¯ P 62m

a = 0.5723 c = 0.3372 γ = 120◦

a = 0.5865 c = 0.3456 γ = 120◦

−54.6

−53.3 [14]

Fe2 P in FeTiP

metastable

Pnma

a = 0.5782 b = 0.3467 c = 0.6638

– – –

−49.2

–

FeP

stable

Pna21

a = 0.5157 b = 0.5751 c = 0.3077

a = 0.5193 b = 0.5792 c = 0.3099

−82.5

–

FeP2

stable

Pnnm

a = 0.4988 b = 0.5674 c = 0.2731

a = 0.4973 b = 0.5657 c = 0.2723

−95.0

–

a = 0.4616 b = 1.3670 c = 0.7002 γ = 101.48◦

−87.1

–

Fe–P

Ti–P

Fe–Ti–P

Calculated enthalpy of formation (kJ/mol)

Observed enthalpy of formation (kJ/mol)

FeP4

stable

P21 /c

a = 0.4618 b = 1.3667 c = 0.7000 γ = 101.48◦

Ti3 P

stable

P42 /n

a = 1.0022 c = 0.5018

a = 0.9959 c = 0.4987

−99.2

–

Ti3 P in (Fe, Ti)3 P

metastable

I 4¯

a = 0.9781 c = 0.4790

– –

−94.8

–

Ti2 P in (Fe, Ti)2 P

stable

¯ P 62m

a = 0.6402 c = 0.3773 γ = 120◦

– – –

−118.0

–

Ti2 P in FeTiP

metastable

Pnma

a = 0.6399 b = 0.3837 c = 0.7347

– – –

−106.4

–

Ti5 P3

stable

P63 /mcm

a = 0.7232 c = 0.5088 γ = 120◦

a = 0.7234 c = 0.5090 γ = 120◦

−132.4

–

Ti4 P3

stable

¯ I 43d

a = 0.7477

a = 0.7430

−134.9

–

a = 0.3499 c = 1.1700 γ = 120◦

−147.4

−132.6 [11]

TiP

stable

P63 /mmc

a = 0.3498 c = 1.1697 γ = 120◦

TiP2

stable

Pnma

a = 0.6240 b = 0.3378 c = 0.8337

a = 0.6181 b = 0.3346 c = 0.8258

−117.6

–

FeTiP

stable

Pnma

a = 0.6025 b = 0.3613 c = 0.6918

a = 0.6007 b = 0.3602 c = 0.6897

−123.0

–

structure of this compound agrees well with our results. That is, the DOS associated with the 3s states of P is located more than 10 eV below the Fermi level. The DOS for the p-states of P and the d-states of Fe and Ti are in the same energy range, located just below E f , and a significant hybridization between these elements is expected. Fe has more electrons in its 3d-state than Ti has, and thus the d levels of Fe are located at energies below the Fermi level, whereas the d-electrons of Ti are predominantly located at energy levels above E f . This results in the formation of a pseudogap in the metal d-band complex. For the Pnma space group, which the FeTiP structure exhibits, the Fe, Ti, and P atoms occupy three 4c internal positions

of Wyckoff-letter multiplicity, respectively. Therefore, the unlike metal atoms were replaced with the same type of atoms at the 4c sites, and electronic band energy calculations for hypothetical anti-PbCl2 -type Fe2 P and Ti2 P (space group = Pnma) were performed. The calculated formation energies for these structures form the parameters required for defining the Gibbs energy of the FeTiP phase, and the values are listed in Table 1. Fig. 2 shows the total density of states of the anti-PbCl2 -type Fe2 P. The two distributions of the DOS diagrams of FeTiP in Fig. 1(a) and Fe2 P in Fig. 2 are similar. However, one can see a difference between these two distributions, in that the Fermi level is located close to the pseudogap for the FeTiP structure,

H. Ohtani et al. / Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 147–158

151

Fig. 1. (a) The total density of states, and the angular-momentum-resolved density of states of the FeTiP structure for (b) Fe, (c) Ti, and (d) P.

analysis. Therefore, the entropy terms of the formation energies were estimated by considering the phase equilibria in the ternary system. More specifically, the thermodynamic parameters of the phosphides in the Fe–P system and those of the liquid phase were evaluated initially from the experimental phase boundaries in the Fe-rich portion of the Fe–Ti–P system. Next, the Ti–P binary system was analyzed by introducing the assumption that FeTiP was mainly in equilibrium with the Ti–Pbased binary phosphides in the ternary system. The details are summarized in this section.

Fig. 2. The total density of states for a hypothetical anti-PbCl2 type Fe2 P.

but for Fe2 P, the Fermi level is located in the hybridization band of the Fe d-states and the P p-states. This indicates that, from an energetic point of view, the FeTiP structure forms a stable structure by dissolving Ti in Fe2 P. 3.3. Thermodynamic analysis A brief outline of our thermodynamic analysis strategy will be presented in this section. Most of the descriptions of the lattice stability parameters for each pure element were obtained from the SGTE data [13,18], and are shown in Table 2. The enthalpies of formation of the phosphides appearing in these binary systems were evaluated using the first-principles calculations listed in Table 1. The Fe–P and Ti–P phase diagrams could be obtained using such enthalpy values estimated by first-principles calculations. However, these results proved not to be satisfactory in our exploratory

3.3.1. The Fe–Ti binary system The Fe–Ti binary system is composed of the liquid (L), bcc (α), fcc (γ ), Fe2 Ti, FeTi, and hcp (α-Ti) phases. A thermodynamic analysis of this binary system has been carried out by Kumar et al. [8], and these results were used in our study. The adopted thermodynamic description is shown in Table 3, and the calculated Fe–Ti binary phase diagram is shown in Fig. 3. 3.3.2. The Fe–P binary system The Fe–P binary system is composed of the liquid (L), bcc (α), fcc (γ ), Ni3 P-type tetragonal Fe3 P ((Fe, Ti)3 P phase), hexagonal Fe2 P ((Fe, Ti)2 P phase), orthorhombic FeP, FeS2 type orthorhombic FeP2 , monoclinic FeP4 , and P (red and white) phases. However, the P-rich portion (P > 50 at.%) is undetermined as seen in Fig. 4(a), because of the high volatilization of P. A critical assessment of the Fe–P binary system has been performed by Okamoto [19]. Gustafson [20] presented a thermodynamic description for the Fe-rich region of the binary system, which enabled the calculation of the phase diagram

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Table 2 Lattice stability parameters for Fe, P, and Ti Element Phase

L

α

Lattice stability parameters (J/mol)

Temperature (◦ C)

Ref.

◦GL − ◦Gα Fe Fe

= 12 040.17 − 6.55843T − 3.6751551 × 10−21 T 7

25 < T < 1537.85

[13]

◦GL − ◦ H α Fe Fe

= −10 839.7 + 291.302T − 46T ln(T )

1537.85 < T < 5726.85

◦Gα − ◦ H α Fe Fe

= 1225.7 + 124.134T − 23.5143T ln(T ) − 0.00439752T 2 − 5.8927 × 10−8 T 3 + 77359T −1 = −25 383.581 + 299.31255T − 46T ln(T ) + 2.29603 × 1031 T −9

25 < T < 1537.85

α = 1043, β α = 2.22 T cFe Fe

Fe γ

γ

◦ G γ − ◦ G α-Ti Ti Ti

α-Ti

◦ G α-Ti − ◦ H α-Ti Ti Ti

1537.85 < T < 5726.85

25 < T < 1537.85

[13]

1537.85 < T < 5726.85

= 4134.494 + 126.7062T − 23.9933T ln(T ) − 0.004777975T 2 + 1.06716 × 10−7 T 3 + 72636T −1 = 4382.601 + 126.0791T − 23.9887T ln(T ) − 0.0042033T 2 − 9.0876 × 10−8 T 3 + 42680T −1 = 13 103.253 + 60.0676T − 14.9466T ln(T ) − 0.0081465T 2 + 2.02715 × 10−7 T 3 − 1477 660T −1 = 369 519.198 − 2553.9505T + 342.059267T ln(T ) − 0.163409355T 2 + 1.2457117 × 10−5 T 3 − 67 034 516T −1 = −19 887.006 + 298.8087T − 46.29T ln(T ) = −1272.064 + 134.78618T − 25.5768T ln(T ) − 6.63845 × 10−4 T 2 − 2.78803 × 10−7 T 3 + 7208T −1 = 6667.385 + 105.438379T − 22.3771T ln(T ) + 0.00121707T 2 − 8.4534 × 10−7 T 3 − 2002 750T −1 = 26 483.26 − 182.354471T + 19.0900905T ln(T ) − 0.02200832T 2 + 1.228863 × 10−6 T 3 + 1400 501T −1 = 6000 − 0.1T

25 < T < 5726.85

626.85 < T < 881.85 881.85 < T < 1026.85 1026.85 < T < 1667.85 1667.85 < T < 3726.85 25 < T < 881.85 881.85 < T < 1667.85 1667.85 < T < 3726.85 25 < T < 3726.85

TiP Ti5 P3

◦ G Ti5 P3 − ◦ G α-Ti = 30 000 + 6T Ti:Ti Ti

25 < T < 5726.85

◦ G L − ◦ H White P P

α

◦ G α − ◦ G Red P P ◦ G γ − ◦ G Red P P

Red

◦ G Red − ◦ H White P P

[13]

25 < T < 626.85

25 < T < 626.85

L

[8]

25 < T < 5726.85

= −8059.921 + 133.687208T − 23.9933T ln(T ) − 0.004777975T 2 + 1.06716 × 10−7 T 3 + 72636T −1 = −7811.815 + 133.060068T − 23.9887T ln(T ) − 0.0042033T 2 − 9.0876 × 10−8 T 3 + 42680T −1 = 908.837 + 67.048538T − 14.9466T ln(T ) − 0.0081465T 2 + 2.02715 × 10−7 T 3 − 1477 660T −1 = −124 526.786 + 638.878871T − 87.2182461T ln(T ) + 0.008204849T 2 − 3.04747 × 10−7 T 3 + 36 699 805T −1 ◦ G TiP − 2 ◦ G α-Ti = 50T Ti Ti:Ti

γ

P

[13]

= −3957.199 + 5.24951T + 4.9251 × 1030 T −9

Fe2 Ti

α

25 < T < 1537.85

= −3705.78 + 12.591T − 1.15T ln(T ) + 6.4 × 10−4 T 2

◦ G Fe2 Ti − 8 ◦ G γ −4 ◦ G α = 69869 Fe:Fe:Fe Fe Fe ◦ G Fe2 Ti ◦ G γ −4 ◦ G α = 60724 − 6 Va:Fe:Fe Fe Fe

◦ G α − ◦ H α-Ti Ti Ti

[18]

[18]

◦ G α-Ti − ◦ G α Fe Fe

◦ G L − ◦ H α-Ti Ti Ti

25 < T < 5726.85

25 < T < 5726.85

α-Ti

L

Ti

◦Gγ − ◦Gα = −1462.4 + 8.282T − 1.15T ln(T ) + 6.4 × 10−4 T 2 Fe Fe ◦Gγ − ◦ H α = −27098.266 + 300.25256T − 46T ln(T ) + 2.78854 × 1031 T −9 Fe Fe γ γ T cFe = −201, βFe = −2.1

1537.85 < T < 5726.85

626.85 < T < 881.85 881.85 < T < 1667.85 1667.85 < T < 3726.85 25 < T < 5726.85

present work

= −26 316.111 + 434.943931T − 70.7440584T ln(T ) − 0.002898936T 2 + 3.9049341 × 10−5 T 3 + 1141 147T −1 = −7232.449 + 133.304873T − 26.326T ln(T )

−23.15 < T < 44.15

= 44769 − 13.26T

25 < T < 5726.85

= 37 656 − 13.26T

25 < T < 5726.85

= −25 976.559 + 148.685002T − 25.55T ln(T ) + 0.0034121T 2 − 2.418867 × 10−6 T 3 + 160 095T −1 = −21 723.721 + 77.684737T − 14.368T ln(T ) − 0.00957685T 2 + 3.93917 × 10−7 T 3 − 141 375T −1

−23.15 < T < 226.85

[13]

44.15 < T < 2726.85

226.85 < T < 579.2 (continued on next page)

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153

Table 2 (continued) Element Phase

Temperature (◦ C)

Lattice stability parameters (J/mol) = −119 408.413 + 1026.04262T − 149.449556T ln(T ) + 0.067272364T 2 − 6.651929 × 10−6 T 3 + 12 495 943T −1 = −24 524.119 + 153.852181T − 26.326T ln(T )

White

= −43 821.799 + 1026.70689T − 178.426T ln(T ) + 0.290708T 2 − 1.04022667 × 10−4 T 3 + 1632695T −1 ◦ G White − ◦ H White = −9587.448 + 152.354487T − 28.7335301T ln(T ) P P + 0.001715669T 2 − 2.2829 × 10−7 T 3 + 172 966T −1 = −8093.075 + 135.889831T − 26.326T ln(T )

Ref.

579.2 < T < 1226.85 1226.85 < T < 2726.85 −23.15 < T < 44.15 44.15 < T < 726.85 726.85 < T < 2726.85

Fig. 3. The calculated binary phase diagram of the Fe–Ti system.

for compositions up to 50 at.% P. However, it was necessary to correct his thermodynamic parameters, as the results led to the unexpected formation of a miscibility gap in the liquid phase. In our study, the temperature dependencies of the formation energy for (Fe, Ti)3 P, (Fe, Ti)2 P, and FeP were determined initially from the phase boundary data of the ternary system. Next, the free energy of the liquid phase was determined from the experimental phase boundaries of the binary alloy, using the thermodynamic description of the above phosphides. Then, the formation energy of FeP2 was estimated by assuming this phase is in equilibrium with FeTiP and FeP at 800 ◦ C. For FeP4 , the phase equilibrium was not clear, and so a melting point almost similar to that for FeP2 was assumed, and the formation energy was evaluated. The evaluated thermodynamic parameters are shown in Table 3. The calculated Fe–P binary phase diagram is shown in Fig. 4(b), and is compared with the assessed diagram as illustrated in Fig. 4(a) [19]. 3.3.3. The Ti–P binary system As regards the Ti–P binary system, a critical assessment by Murray [11] has been reported, but no thermodynamic analysis was performed. According to Murray’s assessment, the Ti–P binary system is composed of the liquid (L), hcp (α-Ti) and bcc (α) solid solutions, tetragonal Ti3 P,

Fig. 4. (a) The assessed phase diagram for Fe–P, and (b) the calculated binary phase diagram.

Ti2 P((Fe, Ti)2 P phase), Ti1.7 P, hexagonal Ti5 P3 , Ti4 P3 , AsTitype hexagonal TiP, Al2 Cu-type tetragonal TiP2 , and P (red and white). The assessed phase diagram is shown in Fig. 5(a). Homogenous Ti5 P3 occurs at a composition of ∼39 at.% P. The phase diagram was constructed in the range 0–55 at.% P because the gaseous phase was included in the phase equilibria.

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Table 3 Optimized thermodynamic parameters for the binary and ternary systems System Fe–Ti

Phase L

α γ FeTi Fe2 Ti α-Ti Fe–P

L α γ (Fe, Ti)3 P (Fe, Ti)2 P FeP FeP2 FeP4 FeTiP

Ti–P L α Ti3 P (Fe, Ti)2 P Ti1.7 P Ti5 P3

Thermodynamic parameters (J/mol)

Ref.

0 LL Fe,Ti 1 LL Fe,Ti 0 Lα Fe,Ti 1 Lα Fe,Ti

[8]

= −67 589 + 9.809T = −4731 = −57 943 + 14.954T = −6059

α T cFe,Ti = 637.79

0 Lγ Fe,Ti = −50 304 + 5.487T ◦ G FeTi − ◦ G α − ◦ G α-Ti = −53 650 + 7.495T Fe:Ti Fe Ti ◦ G Fe2 Ti − 8 ◦ G γ −4 ◦ G α = −429 782 + 120.825T Fe Fe:Ti:Fe Ti ◦ G Fe2 Ti − 6 ◦ G γ −4 ◦ G α = −356 573 + 109.065T Fe Va:Ti:Fe Ti 0 L α-Ti = 15 132 − 8.668T Fe,Ti 0 LL Fe,P = −266 000 + 41.3T 2 LL Fe,P = 96 900 − 40.84T 0 Lα Fe,P = −200 300 + 9.0T 0 Lγ Fe,P = −198 500 + 19.24T ◦ G (Fe,Ti)3 P − 3 ◦ G α − ◦ G Red = −193 600 + 9.75T Fe:P Fe P ◦ G (Fe,Ti)2 P − 2 ◦ G α − ◦ G Red = −184 546 + 8.8T Fe:P P Fe ◦ G FeP − ◦ G α − ◦ G Red = −165 000 + 18.84T Fe:P Fe P ◦ G FeP2 − ◦ G α − 2 ◦ G Red = −282 880 + 45T Fe:P Fe P ◦ G FeP4 − ◦ G α − 4 ◦ G Red = −435 500 + 80T Fe:P Fe P ◦ G FeTiP − 2 ◦ G α − ◦ G Red = −147 600 Fe:Fe:P P Fe 0 LL = −276 200 + 50.25T P,Ti 1 L L = −12 950 + 17.57T P,Ti 2 L L = −11 100 + 18T P,Ti 0 L α = −170 000 P,Ti ◦ G Ti3 P − 3 ◦ G α-Ti − ◦ G Red = −396 800 + 90T P Ti Ti:P ◦ G (Fe,Ti)2 P − 2 ◦ G α-Ti − ◦ G Red = −363 800 + 78T P Ti:P Ti ◦ G Ti1.7 P − 0.63 ◦ G α-Ti −0.37 ◦ G Red = −130 000 + 27.7T P Ti:P Ti ◦ G Ti5 P3 − 0.61 ◦ G α-Ti −0.39 ◦ G Red = −133 000 + 28T Ti:P Ti P Ti5 P3 L Ti:P,Ti = −2080 − 20.8T

Ti4 P3

◦ G Ti4 P3 − 4 ◦ G α-Ti −3 ◦ G Red = −963 700 + 206T P Ti:P Ti

TiP

◦ G TiP − ◦ G α-Ti − ◦ G Red = −294 800 + 68T P Ti Ti:P

present work

present work

L TiP Ti:P,Ti = −160 000 − 20T

(Fe, Ti)3 P

◦ G TiP2 − ◦ G α-Ti − 2 ◦ G Red = −352 800 + 84T P Ti:P Ti ◦ G (Fe,Ti)3 P − 3 ◦ G α-Ti − ◦ G Red = −380 000 + 90T P Ti Ti:P ◦ G FeTiP − 2 ◦ G α-Ti − ◦ G Red = −318 000 + 75T P Ti:Ti:P Ti 0 LL Fe,P,Ti = 58 400 1 LL Fe,P,Ti = −876 000 2 LL Fe,P,Ti = −334 300 (Fe,Ti)3 P L Fe,Ti:P = −180 000

(Fe, Ti)2 P

L Fe,Ti:P2 = −100 000 − 40T

FeTiP

◦ G FeTiP − ◦ G α − ◦ G α-Ti − ◦ G Red = −364 500 + 119.64T − 0.19731T 2 + 6.0 × 10−5 T 3 Fe P Fe:Ti:P Ti L FeTiP = −25 000 Fe:Fe,Ti:P

TiP2 (Fe, Ti)3 P FeTiP Fe–Ti–P L

(Fe,Ti) P

present work

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Fe2 P–FeTiP, Fe3 P–FeTiP, Fe–FeTiP, and Fe2 Ti–FeTiP, as well as 7.5% P–Ti and 7.5% Ti–P, were determined. The liquidus surface in the Fe–FeP–Ti region was also investigated experimentally, and the isothermal sections at room temperature and at 800 ◦ C were elucidated [1,10]. Our thermodynamic analysis was performed using the experimental phase diagrams of Vogel and Giessen [1] and Kaneko et al. [10]. No data concerning the thermodynamic properties of the ternary phosphide were available in the literature. Thus, the formation energy of this phase was determined using the results of the first-principles calculations listed in Table 1, as well as the experimental phase boundary data. The optimized thermodynamic parameters are summarized in Table 3. 3.4. Comparison of the calculated phase equilibria with the experimental data

Fig. 5. (a) The assessed phase diagram for Ti–P, and (b) the calculated binary phase diagram.

The binary phase diagram was analyzed according to the following procedure. The experimental results for the ternary phase boundaries [1] indicated that the Ti–P based phosphides did not occur in the equilibria in the Fe-rich region. Then, Ti3 P, (Fe, Ti)2 P, Ti1.7 P, Ti5 P3 , Ti4 P3 , and TiP were assumed to be in equilibrium with FeTiP at 800 ◦ C, and the formation energy of these phosphides was evaluated. Almost the same temperature dependency of the formation energy for TiP was ascribed to TiP2 , because of a lack of experimental information on this phase. The Gibbs energy of the binary liquid phase was estimated according to the phase equilibrium in the Ti-rich region [11]. The evaluated thermodynamic parameters are shown in Table 3, and the calculated Ti–P binary phase diagram is shown in Fig. 5(b). 3.3.4. The Fe–Ti–P ternary system Vogel and Giessen [1] studied the Fe-rich corner of the Fe–Ti–P ternary phase diagram using thermal analysis, metallography, and X-ray diffraction. The pseudobinary sections for

The Fe–Ti–P ternary phase diagrams were calculated using the thermodynamic parameters described in Section 3.3, and the isothermal sections are shown in Figs. 6 and 7 for T = 25 and 800 ◦ C, respectively. The calculated values agree well with the experimental results. In the isothermal section diagram at T = 25 ◦ C from Vogel and Giessen [1], the phase boundaries for FeTiP were depicted as being in equilibrium with α-Ti. However, this result is not confirmed by experiment. On the other hand, FeTiP forms a three-phase equilibrium with FeP and TiP at room temperature in our calculations, based on the assumptions introduced in our thermodynamic analysis. The calculated vertical section diagrams for Fe3 P–FeTiP, Fe–FeTiP, 7.5% P–Ti, and 7.5% Ti–P are shown in Fig. 8. Compared with the experimental phase boundaries determined using thermal analysis [1], the correspondence with the full phase diagram is acceptable. Vogel and Giessen [1] estimated the melting point of FeTiP to be 1850 ◦ C by extrapolation of the liquidus line for FeTiP in the Fe–FeTiP section. However, this type of extrapolation can involve large errors, and so this value was not considered in our thermodynamic analysis. The melting point of FeTiP calculated using our method was 1808 ◦ C. The calculated liquidus projection can be compared to the assessed diagram [1,21] in Fig. 9. The FeTiP phase dominates the primary crystallization region. Some invariant reactions appear in the Fe-rich region, and were calculated as follows. U1 : L + Fe2 Ti ⇔ FeTi + FeTiP at 1229.8 ◦ C, 0.38 mass% P, and 50.76 mass% Ti. U2 : L + FeTiP ⇔ FeP + (Fe, Ti)2 P at 1254.9 ◦ C, 28.86 mass% P, and 3.51 mass% Ti. U3 : L + (Fe, Ti)2 P ⇔ (Fe, Ti)3 P + FeTiP at 1147.1 ◦ C, 12.99 mass% P, and 0.41 mass% Ti. E1 : L ⇔ α + Fe2 Ti + FeTiP at 1272.5 ◦ C, 0.98 mass% P, and 13.73 mass% Ti. E2 : L ⇔ FeTi + Ti3 P + α at 853.6 ◦ C, 4.44 mass% P, and 68.15 mass% Ti. and E3 : L ⇔ α + (Fe, Ti)3 P + FeTiP at 1051.2 ◦ C, 9.77 mass% P, and 0.39 mass% Ti.

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(a) Experimental.

(b) Calculated.

Fig. 6. Experimental isothermal section diagram of the Fe–Ti–P system at 25 ◦ C [1] and (b) calculated phase diagram.

(a) Experimental.

(b) Calculated.

Fig. 7. Experimental isothermal section diagram of the Fe–Ti–P system at 800 ◦ C [10] and (b) calculated phase diagram.

4. Conclusions The phase equilibria in the Fe–Ti–P ternary system were investigated by incorporating first-principles calculations into the CALPHAD approach, yielding the following results. 1. We attempted to obtain the formation enthalpy of five phosphides in the Fe–P binary system using the Full Potential Linearized Augmented Plane Wave method. The thermodynamic functions were deduced, and a proposed phase diagram of the Fe–P binary system was composed from our analysis of these results and the limited experimental phase boundaries of the Fe–P and Fe–Ti–P systems. Applying the same procedure to the Ti–P binary system, whose content is unclear at present, a highly plausible phase diagram was calculated. 2. The enthalpy of formation of the FeTiP phase was calculated to be about −123 kJ/mol in the ground state. This phosphide is magnetic, with a magnetic moment of 0.85 μB . According to electronic structure calculations, the DOS for the P p-states and the Fe and Ti d-states involves significant hybridization between these elements. The d-levels of Fe

occupy energy levels below the Fermi level, and the Ti delectrons are predominantly located at energy levels above E f . This results in a pseudogap forming in the metal d-band complex. The Fermi level for the FeTiP structure is located close to this pseudogap, and this brings about the stability of FeTiP. 3. The phase diagrams of the Fe–Ti–P ternary system over the entire composition range were constructed from our thermodynamic calculations. The calculated phase boundaries corresponded to the experimental results. We confirmed that FeTiP is in equilibrium with most of the phases appearing in the ternary system, and that the phosphide is dominant in the liquidus surface projection. Therefore, our understanding of the formation behavior of this precipitate can play an important role in the structural control of IF steels, even for low P content steels. Acknowledgements The authors gratefully acknowledge the useful information provided by Prof. Roberto de Avillez at the Pontif´ıcia Universidade Cat´olica, Brazil. One of the authors (H.O.) also

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Fig. 8. Calculated vertical section diagrams for: (a) Fe3 P–FeTiP, (b) Fe–FeTiP, (c) 7.5% P–Ti, and (d) 7.5% Ti–P. Each section contains data from the experimental phase diagrams [1].

(a) Experimental.

(b) Calculated. Fig. 9. A comparison of the calculated liquidus projection and the experimental data.

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sincerely acknowledges the financial support from Nippon Steel Co., Japan. References [1] R. Vogel, B. Giessen, Arch Eisenh¨uttenwes. 30 (1959) 565–576. [2] H. Ohtani, M. Yamano, M. Hasebe, ISIJ International 44 (2004) 1738–1747. [3] H. Ohtani, M. Yamano, M. Hasebe, CALPHAD 28 (2004) 177–190. [4] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luiz, WIEN2k, in: Karlheinz Schwarz (Ed.), An Augmented Plane Wave and Local Orbitals Program for Calculating Crystal Properties, Technische Universit¨at Wien, Austria, ISBN: 3-9501031-1-2, 2001. [5] J.P. Perdew, K. Burke, Y. Wang, Phys. Rev. B 54 (1996) 16533–16539. [6] G. Inden, Proc. CALPHAD V, D¨usseldorf, 1976, III – (4) – 1. [7] M. Hillert, M. Jarl, CALPHAD 2 (1978) 227–238. [8] K.C.H. Kumar, P. Wollants, L. Delaey, CALPHAD 18 (1994) 223–234. [9] M. Hillert, L.-I. Staffansson, Acta Chem. Scand. 24 (1970) 3618–3626. [10] H. Kaneko, T. Nishizawa, K. Tamaki, Nippon Kinzoku Gakkai-Si 29 (1965) 159–165 (in Japanese).

[11] J.L. Murray, Phase Diagram of Binary Titanium Alloys, in: Monograph Series on Alloy Phase Diagrams, ASM International, Metals Park, OH, 1987, pp. 234–236. [12] P. Villars, Pearson’s Handbook, Crystallographic Data for Intermetallic Phases, Desk edition, vol. 2, ASM International, Materials Park, OH, USA, 1997, p. 2609. [13] A.T. Dinsdale, CALPHAD 15 (1991) 317–425. [14] P. Spencer, O. Kubaschewski, Arch Eisenh¨uttenwes. 49 (1978) 225–228. [15] S. Rundqvist, C. Nawapong, Acta Chem. Scand. 24 (1966) 2250–2254. [16] A.V. Ivanov, M.E. Zolensky, A. Saito, K. Ohsumi, S.V. Yang, N.N. Kononkova, T. Mikouchi, Am. Mineral. 85 (2000) 1082–1086. [17] R.P. Gupta, G. Martin, S. Lanteri, P. Maugis, M. Guttmann, Phil. Mag. A 80 (2000) 2393–2403. [18] A.T. Dinsdale, NPL Report DMA(A)195, 1989. [19] H. Okamoto, Phase Diagrams of Binary Iron Alloys, in: Monograph Series on Alloy Phase Diagrams, ASM International, Materials Park, OH, USA, 1993, pp. 310–316. [20] P. Gustafson, Inst. Met. Res. (IM-2549, 1990), The assessed parameters are recorded in the SGTE Solution Database v2, 1985. [21] V. Raghavan, Phase Diagrams of Ternary Iron Alloys, The Indian Institute of Metals, Calcutta, India, 1988, pp. 189–197.