A thermodynamic re-assessment of Al–V toward an assessment of the ternary Al–Ti–V system

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 51 (2015) 75–88

Contents lists available at ScienceDirect

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad

A thermodynamic re-assessment of Al–V toward an assessment of the ternary Al–Ti–V system Bonnie Lindahl a,n, Xuan L. Liu b, Zi-Kui Liu b, Malin Selleby a a b

KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 30 March 2015 Received in revised form 7 July 2015 Accepted 17 July 2015

Titanium alloys are highly sought after due to their excellent mechanical properties. One of the most commonly used Ti alloys is Ti–6Al–4V, which contains 6% Al and 4% V by weight. Despite the popularity of this alloy, no thermodynamic description of the ternary Al–Ti–V system has been published in the open literature. In this work an assessment procedure of the ternary Al–Ti–V system was initiated based on the binary descriptions by Witusiewitcz et al. (J. Alloys Compds. 465 (2008) 64–77 [1]) for (Al-Ti), Gong et al. (Int. J. Mater. Res. 95 (2004) 978–986 [2]) for (Al–V) and Saunders (COST 507, 2 (1998) 297– 298 [3]) for (Ti–V). When combining the three binary systems and looking at the extrapolated ternary isothermal sections, it was found that there was a very large miscibility gap in the bcc phase. The origin of this miscibility gap was mainly the Al-V system and therefore it was decided to reassess this system. The Al–V system was reassessed according to available experimental data along with the enthalpies of formation of all compounds as well as the enthalpies of mixing for all terminal phases obtained by firstprinciples calculations based on the density functional theory. For the Al8V5 phase there are two different sets of data for the enthalpies of formation. These two sets are investigated in this work and it is found that the set not used by Gong et al. in their assessment of the Al–V binary system gives better extrapolations. The final description produced improved extrapolated ternary isothermal sections. & 2015 Elsevier Ltd. All rights reserved.

Keywords: CALPHAD Al–V Al–Ti–V Thermodynamic modeling

1. Introduction Titanium alloys are known for their excellent mechanical properties, combining high tensile strength with high toughness even at elevated temperatures. One of the most common titanium alloys is the Ti–6Al–4 V alloy, containing 4% Al and 6% V by weight. Despite the technical importance of these alloys the thermodynamic properties of the Al–Ti–V system have never been assessed. The most recent binary descriptions available from Witusiewicz et al. [1] (Al–Ti), Gong et al. [2] (Al–V) and Saunders [3] (Ti–V) were chosen as a starting point for this ternary assessment and the corresponding phase diagrams are shown in Fig. 1. All of the chosen descriptions seemed to fit the available binary experimental data well. However, when combining the three binary assessments and extrapolating into the ternary, the deviation from ternary experimental information was large. According to experimental data the solvus of bcc, originating from pure V should extend quite far toward the Al–Ti side [4,5], in the extrapolation using the binary descriptions by [1–3] however, a very n

Corresponding author. E-mail addresses: [email protected], [email protected] (B. Lindahl).

http://dx.doi.org/10.1016/j.calphad.2015.07.002 0364-5916/& 2015 Elsevier Ltd. All rights reserved.

narrow extension of the bcc phase field is observed, as shown in Fig. 2. When examining the extrapolations more closely it was found that there is a very large metastable miscibility gap in bcc that extends into the stable regions making it very hard, if at all possible, to fit the solvus of the bcc phase using reasonable model parameters. When looking at the metastable binary systems (Fig. 3) it can be seen that there is a metastable miscibility gap in both the Al–V and the Ti–V systems. The miscibility gap in the Ti– V system is to be expected since the interaction in the bcc phase is positive [6]. In the Al–V system however, no such investigation has been made and the extrapolation into the higher order system suggests that it needs to be removed. In order to improve the extrapolations to the ternary Al–Ti–V system it is necessary to use a different thermodynamic description for the Al–V binary system. The only description of the Al–V system preceding the one by Witusiewicz [1] was by Saunders [7]. Since this assessment was published new experimental data has been presented [8] that drastically changed some of the invariant temperatures in the Al–V system. Due to the lack of an adequate description of the Al–V binary system it has been re-assessed in this work. We have here chosen to show the entire process of a thermodynamic assessment, involving an assessment of all available data,

76

B. Lindahl et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 51 (2015) 75–88

1.0

2200

0.9

2000

0.8 0.7

Temperature [K]

1800

0.6

1600

0.5

1400

0.4 0.3

1200

0.2

1000

0.1

800

0 0

600

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

400 1.0

200 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.9

Mole fraction Al

0.8 0.7 0.6

2200

0.5

2000

0.4

Temperature [K]

1800

0.3 0.2

1600

0.1

1400

0 0

1200

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1000 Fig. 2. Isothermal section at 873 K of Al–Ti–V extrapolated from binaries. (a) shows the stable diagram together with experimental information on the solvus of bcc from Ahmed and Flower [4] [○] and Maeda [5] [ ]. (b) shows the metastable miscibility gap in bcc.

800 600 400 200 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mole fraction V 2200

optimization, development of new data and final optimization. In the Calphad community it is not common to show the entire process but we hope that doing this will be helpful for those who are in the beginning of their Calphad studies. All percentages in this work will be given by mole of atom if nothing else is stated.

2000

2. Review of available data

Temperature [K]

1800

The Al–V system contains 8 known stable phases. The liquid phase, Al (fcc), dissolves a small amount of V, V (bcc, A2) which dissolves large amounts of Al and 5 intermetallic compounds. All of the phases can be seen in Table 2. Hcp is not stable in the Al–V system but is an important phase in the ternary Al–Ti–V system, therefore it has also been included in this work. There have also been discussions in the literature whether there is a low-temperature phase forming from bcc (V) [9,10] this will be discussed further later on.

1600 1400 1200 1000 800 600 400

2.1. Review of experimental data

200 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mole fraction Ti Fig. 1. Binary phase diagrams used to initially model the Al–Ti–V system: (a) Al–Ti by [1]; (b) Al–V by [2]; and (c) Ti–V by [3].

The experimental information published before 1989 was reviewed by Murray [17] who found that the system had been studied by several authors [14–16,18–27]. Roth [26] and Varich et al. [27] measured the solubility of V in Al. The former by lattice

B. Lindahl et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 51 (2015) 75–88

77

Table 1 Predicted enthalpies and entropies of formation (J/mol-atom) of Al21V2, Al45V7, Al23V4, Al3V, and Al8V5 (end-members) at 298 K.

1500

Phase

Δf H (J/mol-atom)

Δf S (J/mol-atom)

Source


6.297
2.134

Present work (DFT) Present work (Calphad) DFT [11] Calphad [2] Present work (DFT) Present work (Calphad) DFT [11] Calphad [2] Present work (DFT) Present work (Calphad) DFT [11] Calphad [2] Present work (DFT) Present work (Calphad) DFT [11] DFT [12] (a) Calorimetry [13] Calorimetry [14] Calorimetry [15] Calphad [2] Present work (Calphad) Calorimetry [13] Calorimetry [14] Calorimetry [15] EMF [16] Calphad [2]

Al–V intermetallic compounds

Temperature [K]

1200

Al21V2 298 K 0K 298 K Al45V7 298 K 0K 298 K Al23V4 298 K 0K 298 K Al3V 298 K 0K 0K 298 K

900

600

300 0

Al

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mole fraction V

V

598 K 706 K 298 K

1500


9.029
3.394


8.315
3.683


10.583
4.524

1.937

End-members for Al8V5: (Al)6(Al,V)2(Al,V)3(V)2 Al:Al:Al:V
12.65
4.477 Al:V:Al:V
19.39
4.915 Al:Al:V:V
18.19
6.823 Al:V:V:V
18.18
8.397

1200

Temperature [K]

598 K 298 K Al8V5 298 K


8.40
10.73
8.91
8.3
13.94
16.50
14.65
12.9
15.42
18.03
16.04
14.2
26.91
28.11
27.32
28.91
27.8
27.2
27.2
23.9
22.93
34
23.4
22.6
44.6
35

Present Present Present Present

work work work work

(DFT) (DFT) (DFT) (DFT)

(a) Calculated using FP LAPW

900 Table 2 Information about phases in the Al–V system modeled in this work.

600

300 0

V

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mole fraction Ti

Ti

Fig. 3. Metastable miscibility gaps in the (a) Al–V and (b) Ti–V systems.

Phase name Other names

Strukturbericht Space group

Pearson symbol

Liquid fcc

Al

A1

Fm3m

cF4

bcc

V

A2

(Al,V) (Al,V) (Al,V)

A3

Im 3m P63 /mmc

cI2

hcp

hP2

(Al,V)

Al8V5

D82

I43m

cI52

Al3V

D022

I4/ mmm P63 /mmc C2/m

tI8

(Al)6(Al,V)2(Al,V)3 (V)2 (Al)3(V)1

hP54

(Al)23(V)4

mC104 cF176

(Al)45(V)7 (Al)21(V)2

Al23V4

parameter measurements and the latter by paramagnetic susceptibility measurements. They reported the maximum solubility to be 0.2% (620 °C) and 0.3%. The liquidus of Al-rich alloys was studied by Carlson et al. [20], Elliott [21], Bailey et al. [18], Gebhardt and Joseph [23] and Eremenko et al. [22]. The peritectic melting temperatures of compounds were also studied by several authors and the data are shown in Table 3. Rostoker and Yamamoto [25] investigated the maximum solubility of Al in V by means of X-ray diffraction. Carlson et al. [20] also investigated the liquidus of high V alloys using optical pyrometry. In 2000 Richter and Ipser [8] re-investigated the phase diagram between 0 and 50% V. They found that the peritectic melting of the Al8V5 and Al3V phases occurs 262 °C and 90 °C respectively below the previously accepted temperatures. The thermochemical data available for this system are quite scarce. Batalin et al. [19] studied the activity of V

model

Al45V7 Al21V2

Al7V Al10V

Fd3m

Table 3 Melting temperatures of compounds in the Al–V system. Phase

fcc

Al21V2

Melting T (°C)

660 661.8 661.9 665

Al45V7

Al23V4

Al3V

Al8V5

685

735

1360 [25]

1670 [20]

670 727 690

>850 736

730

1270

[18] [21] 1408 [8]

78

B. Lindahl et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 51 (2015) 75–88

in Al-rich liquids. Samokhval et al. [16] estimated the partial Gibbs energies of Al in bcc (V) from EMF measurements. Johnson et al. [24] measured the activities of Al between 12 and 76% Al at 1000 °C. Enthalpies of formation have been studied by Kubaschewski and Heymer [15], Neckel and Nowotny [14] and Meschel and Kleppa [13]. Kubachewski and Heymer [15] studied the enthalpies of formation of alloys containing 40–75% Al using calorimetry. Neckel and Nowotny [14] and Meschel and Kleppa [13] studied the enthalpies of formation of Al3V and Al8V5. All enthalpy data have been summarized in Table 1.

corresponds to the second sublattice being filled with Al atoms and the third with V. The enthalpies at 298 K for the different endmembers are calculated using Density Functional Theory (DFT) and used as starting values for the optimization. The molar Gibbs energy of the Al8V5 phase is described by Al8 V5 Al8 V5 ″ GmAl8 V5 = y″Al yAl ‴ °G Al : Al : Al : V + yAl yV‴°G Al : Al : V : V

Al8 V5 ° Al8 V5 ″ + yV″ y‴ Al G Al : V : Al : V + yV yV‴°G Al : V : V : V

2.2. Review of first principles data

″ + yV″ ln yV″ ) + 3RT (yAl + 2RT (y″Al ln yAl ‴ ln yAl ‴ + yV‴ ln yV‴)

The thermodynamic properties of the Al–V system have only been studied via first principles methods previously by Boulechfar et al. and Wang et al. [12,11]. These values are shown together with experimental data in Table 1. Elastic constants of phases in the Al–V system were studied by Shang et al., Wang et al. and Jahnátek et al. [28,11,29]. Shang et al. and Wang et al. [28,11] also studied the lattice parameters which are shown in Table 5.

The thermodynamic modeling in this work was performed according to the Calphad method [30] in which the Gibbs energy for each phase is described by a mathematical function of temperature, pressure and constitution. The functions contain adjustable model parameters that are optimized to reproduce experimental data found in the literature. The mathematical expression varies depending on which model is chosen. The liquid, fcc, bcc and hcp phases were modeled as substitutional solutions yielding the following molar Gibbs energy expression:

∑ xi oGiθ + RT ∑ xi ln xi + ∑ xi xj Ii, j i

i

″ yAl + yAl ‴ yV‴ IAl : Al : Al, V : V + yV″ yAl ‴ yV‴ IAl : V : Al, V : V ″ yV″ yAl + yAl ‴ yV‴ IAl : Al, V : Al, V : V

(5)

where yi(s) is the mole fraction of i on sublattice s, also denoted site

3. Thermodynamic models

θ Gm =

″ yV″ yV‴ IAl : Al, V : V : V + yAl ″ yV″ yAl + yAl ‴ IAl : Al, V : Al : V

i>j

(1)

The first term represents the mechanical mixing of the endmembers, here pure elements, the second term represents the contribution to the Gibbs energy due to ideal entropy of mixing and the third term represents the excess Gibbs energy where Ii, j describes the deviation from ideal interaction between i and j with a Redlich–Kister polynomial as follows:

fraction, °GiAl: j8: kV:5l are the different end-member compounds, and I are the interaction parameters. Elements on different sublattices are separated by colons and elements interacting on the same sublattice are separated by a comma. The interaction parameters can be expanded using Redlich–Kichter polynomials in the same way as in Eqs. (2) and (3). Although there are no known stable ordered structures based on the bcc lattice in the Al–V system, a four sublattice model (Al,V)1/4(Al,V)1/4 (Al,V)1/4(Al,V)1/4 has been implemented in order to improve the extrapolations of ordered structures in higher order systems. If for instance Fe is added to this system both the B2 and L21 ordered structures will become stable. Using a four sublattice model both these structures can be described. Using the four sublattice model it was also possible to investigate whether the phase that might form at low temperatures is an ordered bcc phase. The ordering is described with the partitioning model, which means that the ordering is described as an addition to the Gibbs energy of the disordered phase so that the molar Gibbs energy will be described as

n

Ii, j =

∑ nLi, j (xi − xj )n

(2)

0

where n

L i, j = A + BT + CT ln T

(3)

and A, B and C are model parameters to be optimized to fit experimental information. It is also possible to add a fourth term to Eq. (1) describing the magnetic properties of the system. The compound phases in the Al-rich part of the diagram i.e. Al21V2, Al45V7, Al23V4 and Al3V are all modeled as stoichiometric compounds. Due to lack of experimental data the so-called Neumann–Kopp approximation is used i.e. the heat capacity (Cp) will be the weighted average of the Cp of the pure elements compound Gm

=

xA GApure

+

xB GBpure

+ D + ET

(4)

Again, D and E are the model parameters to be optimized. For the Al8V5 phase, the model described by Gong et al. [2] is adopted. This model is derived from crystallographic data [31] and yields (Al)6(Al,V)2(Al,V)3(V)2. In this model Al can occupy the first three sublattices and V the last three which means that mixing occurs on the second and third sublattices. The ideal stoichiometry

bcc dis ord Gm = Gm + ΔGm

(6)

where Gmdis is the contribution to the Gibbs energy based on ord is the additional Gibbs composition expressed by Eq. (1) and ΔGm energy due to ordering i.e. configuration ord ord ord ΔGm = Gm (yi ) − Gm (yi = xi ) ord Gm =

(7)

∑ yi(1) y(j2) yk(3) yl(4) °Gijkl + RT ∑ ∑ yi(s) ln yi(s) i

+

s

∑ ∑ ∑ yi(s) yj(s) Ii, j s

i

j

i

(8)

The values for the different end-members (°Gijkl ) are evaluated from DFT calculations since no experimental evidence for ordered structures based on the bcc lattice exists in this system. Another possible low temperature phase that has been reported in the literature is an A15 phase (Al3V) [9,10,32]. The phase is modeled as a two sublattice phase (Al,V)1(Al,V)3. And the molar Gibbs energy is described by

B. Lindahl et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 51 (2015) 75–88

Table 4 Parameter values in initial optimization in J/mol-atom, the difference between the optimization procedure is explained in the text.

G m = y′Al y″Al °GAl + y′Al yV″ °G AlV3 + yV′ y″Al °G Al3 V + yV′ yV″ °GV ′ + yV′ ln yV′ + 3 (y″Al ln y″Al + yV″ ln yV″ )) + RT (y′Al ln yAl

79

(9)

The Gibbs energies for the end-members are calculated from DFT and inserted directly. Ideal mixing within the sublattices is assumed since no other data are available.

Parameter

Optimization #1

0 liq


48 700þ 0.64T

L

L


95 800þ 17T

0 bcc

L


130 530 þ 34.62T 20 640

L


61 100


65 770þ 4.46T

Al

V2

fcc − xAl °G Al − xV °GVbcc


9540 þ 1.32T


9655 þ1.35T

Al

V7

fcc − xAl °G Al − xV °GVbcc


14 770 þ2.24T


14 920 þ 2.23T


16 260 þ 2.52T


16 440þ 2.56T


27 430 þ 4.83T


27 710þ 4.87T


25 000 þ0.39T


33 990 þ5.55T

0 fcc

L

Gm 21

In the initial optimization only the data available in the literature were used. During the optimization procedure, several different combinations of experiments and model parameters were tried in order to get the best fit and the best extrapolations, this process is explained in the following. Most of the data produced for this system is consistent. However there are two sets of data that are in direct contradiction to each other. The first set of data suggests that Al3V is the phase that has the most negative enthalpy of formation [15,14] and the other suggests that the enthalpy of formation continues to decrease with increasing vanadium content until it reaches a minimum for Al8V5 [13,16]. Therefore two different optimizations were done one for each set of data. The procedures were similar and are explained below. The first step in both optimizations was to fit the enthalpy of formation data available in order to get the right topology in the phase diagram. Since no experimental values were available the enthalpies of formation calculated by first principles by Wang et al. [11] were used to optimize the temperature independent model parameters for the high Al compounds i.e. Al21V2, Al45V7 and Al23V4. The enthalpy of formation for Al3V has been measured several times with similar results [13–15] and the temperature independent model parameter was optimized to fit these values. Since there is so little information about the Al8V5 phase it is approximated to be stoichiometric in the initial step. This is where the two different sets of data start to differ. In the first optimization the enthalpy information from Neckel and Nowotny [14] is used to fit the temperature independent model parameter for Al8V5. The information from Kubaschewski and Heymer [15] and Johnson et al. [24] is used to fit the temperature independent part of the regular solution parameter for the bcc phase (0L bcc ). In the second optimization, the enthalpy of formation from Meschel and Kleppa [13] was used to fit the temperature independent model parameter for Al8V5. The temperature independent part of the regular solution parameter for bcc (0L bcc ) was optimized to fit the data on partial Gibbs energy from Samokhval et al. [16]. A regular solution parameter for the liquid phase (0L liq ) was optimized to fit activity data from Batalin et al. [19] in both optimizations. After this, the temperature independent model parameters for the compounds were allowed to vary to make all compounds stable at room temperature. The regular solution model parameter for fcc ( 0L fcc ) is then optimized to fit the solubility of V in fcc according to Roth [26]. The temperature dependent model parameters of the compounds were then optimized to fit the melting temperatures of the compounds according to Richter and Ipser [8]. In order to fit the liquidus, especially in the vanadium rich part, temperature dependence was added to the bcc and liquid phases. In the second optimization it was necessary to add a subregular parameter to both the bcc and liquid phases ( 0L bcc , 0L liq ). Table 4 shows the values of the parameters in the initial optimization, as can be seen, the second optimization resulted in more and larger parameters for the bcc and liquid phases. The produced phase diagrams as well as extrapolations to the ternary Al–Ti–V system are shown in Figs. 4 and 5. Descriptions of


69 140þ 9.63T 12 700

1 liq

1 bcc

4. Initial optimization

Optimization #2

Gm 45

Al V Gm 23 4

−

fcc xAl °G Al

−

xV °GVbcc

Al V

fcc Gm 3 − xAl °G Al − xV °GVbcc Al V5

Gm 8

fcc − xAl °G Al − xV °GVbcc

the Al–Ti and Ti–V systems were, as previously taken from Witusiewicz et al. [1] and Saunders [3] respectively. The phase diagrams produced are very similar but there are some small differences in the shape of the liquidus and the solvus for bcc. More parameters for the bcc and liquid phases were needed to get a reasonable fit to the experimental data in the second optimization. The extrapolations are also similar, the largest difference lies in the solubility of vanadium in the two compounds AlTi and AlTi3 originating from the Al–Ti system. The solubilities of vanadium in AlTi and AlTi3 were larger in the first optimization process although still not as large as suggested by experimental data. At this point it was not possible to deduct which of the values for the enthalpy of formation of Al8V5 was closest to the truth, therefore it was decided to investigate this using DFT.

5. DFT study of the Al–V system The Vienna ab initio Simulation Package (VASP) [33] v. 5.3.2 has been employed for all DFT calculations. The projector augmentedwave method (PAW) was used to account for electron–ion interactions in order to increase computational efficiency [34,35]. Electron exchange and correlation effects were described using the generalized gradient approximation (GGA) as implemented by Perdew, Burke, and Ernzerhof (PBE) [36]. Reciprocal k-meshes used for fcc-Al/V, hcp-Al/V, bcc-Al/V, Al21V2, Al23V4, Al45V7 and Al8V5 primitive cells are 2  23  23, 35  35  17, 27  27  27, 6  6  6, 8  8  3, 19  19  23, and 8  8  8, respectively. A plane-wave basis cutoff energy of 400 eV was consistently used, as recommended by the VASP manual [37]. Smearing at the Fermi level is introduced by implementing the Methfessel–Paxton method [38]. After relaxations, a final calculation using the tetrahedron method with Blöchl corrections [34] was applied to ensure an accurate total energy static calculation. Finite temperature predictions were performed by calculating the Helmholtz energy, F (V , T ), of phases in the Al–V system as a function of volume (V ) and temperature (T ) based on the DFT and can be expressed as [28,39]

F (V , T ) = E0K (V ) + FVib (V , T ) + FT − el (V , T )

(10)

where E0K (V ) is the static contribution at 0 K without zero-point vibrational energy. FVib (V , T ) and FT − el (V , T ) represent the temperature-dependent phonon-vibrational and thermal-electronic contributions, respectively. At ambient pressures used in the modeling, the Helmholtz free energy is taken to be equivalent to the Gibbs free energy for solid state systems. It is important to also

80

B. Lindahl et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 51 (2015) 75–88

1.0 0.9 0.8

2500

0.7 0.6 0.5 0.4

Temperature [K]

2000

0.3 0.2 0.1 0

1500

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole fraction Ti 1.0 0.9

1000

0.8 0.7 0.6 0.5

500

0.4 0.3

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mole fraction V

0.2 0.1 0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole fraction Ti

Fig. 4. Binary phase diagram and ternary extrapolations using parameters evaluated during optimization #1.

include the thermal-electronic entropic contribution due to the metallic nature of all species, Al and V, with finite electronic density at the Fermi level. Fermi–Dirac statistics [28] was used to

account for thermal-electronic contribution based on the calculated electronic density of states. Equilibrium E0K (V ), volume, bulk modulus, and its pressure derivative is obtained using a four 1.0 0.9 0.8

2500

0.7 0.6 0.5 0.4

Temperature [K]

2000

0.3 0.2 0.1 0

1500

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole fraction Ti 1.0 0.9

1000

0.8 0.7 0.6 0.5

500

0.4 0.3

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mole fraction V

0.2 0.1 0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole fraction Ti

Fig. 5. Binary phase diagram and ternary extrapolations using parameters evaluated during optimization #2.

B. Lindahl et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 51 (2015) 75–88

Table 5 Lattice parameters of Al, V, Al21V2, Al45V7, Al23V4, Al3V, and Al8V5 (end-members). The calculated values at 0 K are compared to previous experiments and calculations at various temperatures. Bolded structures are stable at finite temperatures. Phase

a (Å)

Al (fcc)

4.041 4.048 4.050 3.234 3.244 3.811 3.810 3.002 2.992 3.030

Al (bcc) V (fcc) V (bcc)

Al–V intermetallic compounds Al21V2 14.456 14.449 14.586 11.105 Al45V7 11.097 11.088 7.680 Al23V4 7.672 7.692 Al3V 3.769 3.764 3.778

b (Å)

c (Å)

Source Present work DFT [39] (a) Exp [54] Present work DFT [39] (a) Present work DFT [39] (a) Present work DFT [39] (a) Exp [54]

7.611 7.604 7.621

Present work DFT [11] Exp [55] Present work DFT[39] Exp [56] Present work DFT [11] Exp [57] Present work DFT [11] Exp [58]

11.084 11.079 11.081 17.039 17.027 17.040 8.313 8.307 8.324

End-members for Al8 V5:(Al)6(Al,V)2(Al,V) 3(V)2 Al:Al:Al:V 9.343 Al:V:Al:V 9.295 Al:Al:V:V 9.165 9.234 Al:V:V:V 9.137

Present work Present work Present work Exp [59] Present work

(a) Calculated using PAW-GGA PW91

parameter Birch–Murnaghan (BM4) equation of state (EOS) fitting, which is recommended for metallic systems [28], as shown below [40,28]

E0K (V ) = a + bV −2/3 + cV −4/3 + dV −2

(11)

In the present work, calculations were performed for five volumes in the fitting of parameters a, b, c, and d. The atomic supercell used in the fitting was initially relaxed with respect to ionic positions, cell shape and cell size to obtain a volume close to the equilibrium lattice structure. After this relaxation, 5 additional volumes with up to 6% difference in lattice parameter in expansion and compression were constructed around the equilibrium volume. These additional supercells with fixed volumes were relaxed with respect to ionic positions and cell shape. Finally, energy (E) and pressure (P) vs. volume (V) relationships were determined by fitting energies and pressures to the BM4 equations. The Debye– Grüneisen model was chosen in the present work with the benefit of both accuracy and efficiency [41] over the phonon supercell method. In the Debye–Grüneisen approximation, the vibrational contribution to the Helmholtz energy is described as [28]

FVib (V , T ) =

⎡ Θ (V ) ⎤ 9 kB ΘD (V ) − kB T {D ⎢ D ⎣ T ⎥⎦ 8 +3 ln ⎡⎣ 1 − e−ΘD (V ) / T ⎤⎦ }

(12)

In the above expression, ΘD (V ) is the non-0 K Debye temperature, kB Boltzmann's constant, T the temperature, and D [ the Debye function, expressed as D (x ) = temperature, ΘD (V ), can be written as

3 x3

x z 3 dz

∫0

ez−1

ΘD (V ) ], T

. The Debye

⎛ B ⎞1/2 ⎛ V ⎞γ ΘD = sAV01/6 ⎜ 0 ⎟ ⎜ 0 ⎟ ⎝ M⎠ ⎝ V ⎠

81

(13)

where s is the Debye scaling factor [42], γ the Grüneisen parameter, and A a constant equal to (6π 2)1/3ℏ/kB with the equilibrium volume V0 given in Å3, the bulk modulus B0 in GPa, and the atomic mass M in grams. The differences in transverse and longitudinal phonon modes [43] in an isotropic medium produce a scaling factor that scales the Debye temperature that better fits experimental measurements. Moruzzi et al. [42] fit a scaling factor of 0.617 for non-magnetic bcc metals based on experimental longitudinal and shear moduli. However, it has been shown that the scaling factor is highly dependent on the crystal structure [44,45] and can be better estimated from elastic constant calculations using DFT. Elastic constant calculations are implemented in this work for fcc-Al, bcc-V, and all Al–V intermetallics using the method proposed by Shang et al. [46]. Thereafter, the Voigt– Reuss–Hill (VRH) approximation for an isotropic medium is applied to the elastic constants in order to obtain a Poisson ratio and scaling factor [47]. The Grüneisen parameter, which accounts for anharmonicity in the system, can be expressed as γ = (1 + B0′ ) /2 − x [42], where B0′ is the first derivative of the bulk modulus B0 with respect to pressure. Expressing γ as shown above captures the linearity of γ with respect to the derivative of B0 [48] and can represent the high and low temperature assumptions, respectively [42,44,28]. By choosing x ¼2/3, x¼ 0, or x ¼1, one arrives at the Slater (low), VZ and DM (high) limits [49] for the Grüneisen parameter. The Slater limit, x ¼2/3, is used in this work. Special quasirandom structures are used to predict solution mixing in the bcc, fcc, and hcp phases. This is especially important for the thermodynamic properties of the bcc solution phase which has a large extension near pure V. Binary 16-atom SQS-bcc, SQS-fcc, SQS-hcp supercells developed by Wolverton [50], Jiang et al. [51], and Shin et al. [52] are used in the study. Relaxations of SQS cells are performed carefully to retain cell symmetry and atomic coordinate environments [53]. First, the supercells are relaxed with respect to cell volume only. Cell volume and shape are then allowed to relax simultaneously. If convergence is reached, cell volume, shape and ionic positions are all relaxed concurrently. To address symmetry concerns, radial distribution functions (RDF) of relaxed supercells are compared with the ideal bcc, fcc, and hcp structures after each relaxation step [53]. Only structures that retain their original coordination environments and have the lowest energies at any specific composition are used in the analysis. Pure element calculations are listed together with previous calculations and experiments in Table 5. Lattice parameters for Al and V in bcc and fcc structures are compared to previous DFT investigations by Wang et al. [39]; experimental measurements [54] for the stable structures of Al and V are also shown. For the pure elements, excellent agreement with previous studies is seen in all cases. Al–V intermetallic lattice parameters are shown to be in good agreement with experimental [55–59] and calculated results [11], as seen in Table 5. Calculated scaling factors for all elements and intermetallics are used in the Debye–Grüneisen model to predict accurate finite temperature formation enthalpies and entropies, as seen in Table 1. The results from this work at 298 K are compared to previous calculations and are shown to be in excellent agreement [12,11]. Unfortunately, no entropy data exists in the literature to compare to. SQS calculation results are shown in Table 6. No solution enthalpies are available for comparison. It is seen that bcc phase is the most stable solution phase near the V-rich region of the phase diagram. Solution phase calculations for fcc and hcp are also performed as they are crucial to extrapolations in the ternary system Al–Ti–V. The fcc phase is shown to be more stable on the Al-rich side.

82

B. Lindahl et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 51 (2015) 75–88

Table 6 Calculated bcc, fcc, and hcp solution phase mixing enthalpies at 0 K using dilute and SQS methods. Stable pure phases for Al (fcc) and V (bcc) are taken as reference states. xV

Pure 32-atom dilute 27-atom dilute 16-atom dilute 16-atom SQS 16-atom SQS 16-atom SQS 16-atom dilute 27-atom dilute 32-atom dilute Pure

0.00 9.39 0.03 0.04 8.00 0.06 0.25
3.30 0.50
11.30 0.75
10.45 0.94 0.96
4.06 0.97 1.00 0.00

bcc (kJ/molatom)

fcc (kJ/molatom)

hcp (kJ/molatom)

2100

0.00
0.85

3.31

1800


0.67
8.51
8.55 3.89 14.10


7.39
8.45 0.57

19.46 23.47

Temperature [K]

Structure

2400

1200 900

26.02

600

Table 7 Parameters obtained after final optimization. Parameter

1500

Value (J/mol-atom)

300 0

Liquid 0 liq


78 540 þ 10.07 T

1 liq

17 594

L

L

Mole fraction V

bcc
79 106þ0.654T

1 bcc

6645

bcc − ord °G Al V

67 772 − 1.246T

L

L

3

− ord °GBbcc 2

63 285 − 2.373T

− ord °GBbcc 32 bcc °G AlV− ord 3

54 368 − 2.373

33 040 − 4.688T

fcc
64 732

0 fcc

L

hcp
68 596

0 hcp

L

Al21V2 Al

V2

fcc − xAl °G Al − xV °GVbcc


10 729þ 2.13T

V7

fcc − xAl °G Al − xV °GVbcc


16 503 þ3.39T

Gm 21 Al45V7 Al

Gm 45

−

fcc xAl °G Al

−

xV °GVbcc


18 028 þ 3.68T

1000 950 900 850 800 750 700

600 0

Al3V Al V Gm 3

1050

650

Al23V4 Al V Gm 23 4

1100

Temperature [K]

0 bcc

fcc − xAl °G Al − xV °GVbcc


28 114þ 4.52T

Al8V5 Al V

fcc GAl :8Al5: Al : V − xAl °G Al − xV °GVbcc


12 854

fcc GAl :8V :5Al : V − xAl °G Al − xV °GVbcc


19 309

Al V fcc GAl :8Al5: V : V − xAl °G Al − xV °GVbcc Al8 V5 fcc GAl : V : V : V − xAl °G Al − xV °GVbcc

22 996 − 1.87T

Al V

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


18 258

6. Final optimization According to the DFT-study, the enthalpy of formation of Al8V5 was
18.19 kJ/mol-atom which is closer to the value from Kubaschewski and Heymer (
22.6) [15] and Neckel and Kleppa (
23.4) than the value from Meschel and Kleppa (
34) [13] and Samokhval et al. (
44.6) [16], therefore Optimization #1 from the initial optimization step is used as the starting point for the final optimization. According to the DFT calculations in this work, the Al8V5 phase would not be stable at room temperature. Therefore

0.05

0.10

0.15

0.20

0.25

0.30

Mole fraction V Fig. 6. Phase diagram after final optimization together with experimental data from [8] (◯) , [26] (▵) and [20] (□).

the enthalpies of formation are kept at the same values as were reached in the initial optimization. In order to get some solubility in the Al8V5 phase, the enthalpies of formation of the metastable end-members are set to the values calculated by DFT in this work. In Table 6, the calculated values for fcc, bcc and hcp are shown. The values for the pure elements relative to their stable state are also known as lattice stabilities. In the case of Al, the values for the metastable phases bcc and hcp are quite similar to the accepted Calphad values of 10.083 kJ and 5.481 kJ respectively. In the case of V however, the values differ quite drastically. The Calphad value for fcc V is 7.5 kJ as opposed to 23.47 kJ and for hcp the Calphad value is 4 kJ compared to the DFT value of 26.02 kJ. Because of these large differences on the vanadium side it is impossible to use the values on enthalpy of mixing directly. It is however possible to use the trends, such as shape of the curve and stability range of the

B. Lindahl et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 51 (2015) 75–88

1.0

15

0.009

T = 1807 K

0.008

0.9

Enthalpy of formation [kJ/mol]

0.007 0.8

0.006 0.005

Activity of V

0.7

0.004 0.003

0.6

0.002 0.001

0.5

-Batalinet al. 1985

0 0

0.4

83

0.02

0.04

0.06

0.08 0.10

0.3 0.2

10 5 0 -5 -10 -15

0.1

-20

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mole fraction V

Mole fraction V 30

16 T = 1273.15K

25

Enthalpy of formation [kJ/mol]

14

12

log

10

8

6

4

20 15 10 5 0 -5 -10

2

-15

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mole fraction V

Mole fraction Al Fig. 7. (a) Calculated activity of the stable state at 1807 K compared to experimental values from [19] and (b) activity data from [24] compared to the same calculated quantity at 1273 K.

phase compared to the other phases. The bcc phase shows some slight asymmetry in the enthalpy of mixing, therefore a subregular parameter is added and the bcc phase is re-optimized against the data from Carlson et al. [20], Johnson et al. [24], Kubaschewski and Heymer [15] and Richter and Ipser [8]. A subregular parameter is also added to the liquid phase in order to keep the shape of the phase diagram. Temperature dependence was added to the fcc phase and the model parameters were fitted to the solubility data by Roth [26], the melting temperature determined by Richter and Ipser [8] and the composition where the enthalpies of formation of fcc and bcc are equal according to the present DFT calculations. Since there is no information about metastable hcp in this system the model

Fig. 8. Enthalpy of formation of bcc (◯) , fcc (▵) and hcp (□), (a) at 298 K according to the current description and (b) calculated with DFT at 0 K.

parameters are optimize to have the same enthalpy of formation as fcc at equiatomic composition as is suggested by the DFT calculations. After this a final round of optimization is then performed including all model parameters and experimental data.

7. Results and discussion A self-consistent set of parameters have been obtained to reproduce the Al–V system. All the binary parameters are shown in Table 7. Fig. 6 shows the phase diagram produced by the parameters assessed in this work along with the experimental phase diagram data used for optimization. The liquidus is reproduced nicely as

84

B. Lindahl et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 51 (2015) 75–88

Enthalpyof formation [kJ/mole]

0

-Kubaschewskiand Heymer1960 -Meschelet al. 1993

-5

-Neckelet al.

-10 -15 DFT (this work)

-20

Al21V2 Al45V7 Al23V4 Al3V Al8V5

-25 -30 -35 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mole fraction V Fig. 9. Calculated enthalpy of formation across the entire composition range compared to experimental and DFT data.

well as the invariant temperatures. The solubility of V in fcc-Al is now reproduced within experimental error. The overall fit to phase diagram data is very good however the solubility of Al in bcc at the invariant reaction Liquid + bcc → Al8 V5 deviates from the experimental value by Richter and Ipser [8]. The fit to Thermochemical data is shown in Figs. 7–9. The activity of V 1.0

relative to the stable phase of each element is shown in Fig. 7a. The fit to the data in the liquid phase by Batalin et al. [19] is good. Fig. 7b shows activity data from Johnson et al. [24]. Fig. 8 compares the enthalpies of mixing for fcc and bcc calculated by the current Calphad assessment and as calculated by DFT in this work. As can be seen the trend between the two agrees very well but it is not possible to compare the values directly since the lattice stability of fcc-V differs greatly between Calphad and DFT. Fig. 9 shows the calculated total enthalpy relative to the stable phases of the pure elements at 298 K compared to available experimental data and DFT calculations in this work. The overall agreement is very good. The discrepancy between measured enthalpies of formation for Al8V5 is quite large. The data from Kubaschewski and Heymer [15] suggests the Al8V5 phase should have an enthalpy of formation less negative than that of the Al3V phase whilst Meschel and Kleppa [13] and Neckel and Nowotny [14] suggest that it should be more negative. According to DFT this phase should not be stable at 0 K at least not with the stoichiometry Al8V5. If the phase is not stable at low temperatures that could be one reason why the results are so different from each other. Depending on which of the enthalpy of formation data is trusted, very different results are achieved. Gong et al. [2] reassessed the system under the assumption that Al8V5 has an enthalpy of formation that is more negative than that of the Al3V phase which produced a set of parameters much larger than the ones in this work and in turn produced inferior extrapolations. Extrapolations into the Al–Ti–V ternary system using the current work together with the assessments by Witusiewitcz et al. [1] (Al–Ti) and Saunders [3] (Ti–V) are shown in Fig. 10 compared to experimental information from Maeda [5] and Ahmed and Flower [4]. The extrapolations are quite good and it would be expected that large improvements could be made with a few ternary

1.0

0.9

1.0

0.9

0.8

0.9

0.8

0.7

0.8

0.7

0.6

0.7

0.6

0.5

0.6

0.5

0.4

0.5

0.4

0.3

0.4

0.3

0.2

0.2

3

0.1

0.3 0.2 3

0.1

0

0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole fraction Ti

3

0.1 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole fraction Ti

1.0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole fraction Ti

1.0

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

8 5

0.4

0.3

0.3

0.2

0.2 3

0.1 0

0.1 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole fraction Ti

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole fraction Ti

Fig. 10. Extrapolations into ternary Al–Ti–V using assessment of Al–V derived in this work. The experimental points are from Ahmed and Flower [4] (◯) and Maeda [5] ( ).

B. Lindahl et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 51 (2015) 75–88

85

6000

T = 600 o C

5000

-4

Temperature [K]

Gibbs energy of formation [kJ/mole]

0

-8 -12 3

-16

4000

3000

2000

-20 1000

-24 0

-28 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole fraction V

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mole fraction Ti Fig. 12. Metastable phase diagram for the AlTi (L10) phase according to the description by [1].

3000 A15 phase is more stable than the disordered bcc phase according to the calculations. In Fig. 11b the phase diagram produced when inserting the calculated enthalpy of formation of the A15 into the database is shown.

2700

2100 8. Future work

1800 1500 3

1200

7 5

45

21

600

23

2

4

900

8

Temperature [K]

2400

300 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole fraction V

Fig. 11. Stability of low temperature phase forming from DFT according to DFT calculations in this work.

parameters. However, according to the experimental information the AlTi phase (L10) is supposed to extend quite far toward the Al– V binary i.e. toward very low Ti-contents. Unfortunately this is not possible using the current description of Al–Ti because the AlTi phase is described as a separate phase in the description by Witusiewitcz [1], as opposed to an ordered configuration of fcc, and therefore it does not extrapolate very well in the metastable regions i.e. very low Ti-contents. The possibility of a low temperature phase forming from the high V bcc phase has been investigated by DFT in this work. It was found that all of the ordered configurations of bcc were less stable than the disordered bcc. However, as can be seen in Fig. 11a the

When trying to optimize the solubility of AlTi (L10) in the ternary Al–Ti–V system it was not possible beyond a certain point. According to the experimental data by Ahmed and Flower [4] the AlTi (L10) phase should extend to as low as 15% Ti. When examining the metastable phase diagram including only the AlTi (L10) phase (Fig. 12), it was found that it exhibits some strange behavior. The ordered phase disorders beyond certain limits in both composition directions and at high temperatures the first order transition becomes second order. The two phase region between ordered and disordered AlTi (L10) is probably one of the reasons for the difficulty in fitting the experimental data. In order to properly describe the Al–Ti–V system the Al–Ti system would need to be reassessed using an order-disorder model for fcc-L10. To fully understand if any other phase forms from the bcc phase at low temperatures an experimental study would need to be performed.

9. Conclusions

 The Al–V system has been thermodynamically assessed and a 

self-consistent set of parameters to calculate the thermodynamic properties of this system has been produced. The assessment is an improvement to previous versions of Al–V. Extrapolations into the ternary Al–Ti–V system as well as the solubility of V in fcc-Al has been improved compared to the assessment by Gong et al. [2]. The work by Richter and Ipser [8] has been taken into account in this work which was published

86

B. Lindahl et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 51 (2015) 75–88

Table 8 Calculated elastic constants (Cij) in Voigt notation and units of (GPa) for Al, V, Al21V2, Al45V7, Al23V4, Al3V, and Al8V5 (end-members). Experimental and previously calculated results are shown as well (temperatures listed if known). Note: 0 K values represent extrapolation from low temperatures. Phase

C11

Al (fcc)

109.6 101.0 114.3 259.4 272.0 238.2

0K V (bcc) 0K

C22

C33

Al–V intermetallic compounds Al21V2 127.5 116 113.7 Al45V7 187.8 185.1 183.1 188.7 184.4 184.3 Al23V4 190.0 199.5 186.6 200.9 Al3V 239.8 266.0 233 258 225.1 262.5

C44

C55

C66

C12

C13

C15

C23

C25

C35

30.8 25.4 31.6 34.6 17.6 46.8

62.5 61.0 61.9 122.6 144.8 122.0

Present work DFT [28] (a) Experiment [60] Present work DFT [28] (a) Experiment [61]

66.3 62 62.3 53.5 54.9 45.6 42.3 102.1 104 100.9

66.9 69 72.9 53.0 50.2 55.1 53.6 135.1 77 97.8

Present work DFT [29] (a) DFT [11]
1.7

71.9 69.7

66.7 59.0 67.7 66.8 129 133.3

End-members for Al8V5: (Al)6(Al,V)2 (Al,V)3(V)2 Al:Al:Al:V 185.6 34.7 Al:V:Al:V 185.2 47.8 Al:Al:V:V 221.9 62.5 Al:V:V:V 237.5 88.0

49.3 49.7

0.1

55.3 58.3


4.7

C46

Source


5.7

Present work DFT [11]

Present work DFT [11] 102.6 47 47.2

52.1

Present work DFT [29] (a) DFT [11]

61.8 73.8 89.4 85.9

Present Present Present Present

work work work work

(a) Calculated using PAW-GGA PW91

Table 9 Various properties of Al, V, Al21V2, Al45V7, Al23V4, Al3V, and Al8V5 (end-members) calculated using predicted elastic constants. Bulk moduli and Poisson ratios estimated using the Voigt–Reuss–Hill averages are represented by B0-VRH and υ-VRH. Previous investigations are shown for comparison. For the listed Debye temperatures, ‘el’ refers to values derived from elastic constants at low temperatures (listed if known). Phase

B0 (GPa) − EOS

B0 (GPa) − VRH

υ − VRH

s (υ )

ΘD (K )

Source

Al (fcc)

78.0 74.3

0K

78.2 74.3 79.4 167.7 181.3 160.7

0.342 0.359 0.335 0.375 0.421 0.356

0.67 0.63 0.68 0.59 0.46 0.63

419 384(el) 430 381 317(el) 402(el)

Present work DFT [28] (a) Experiment [60] Present work DFT [28] (a) Experiment [61]

Al–V intermetallic compounds Al21V2 85.2 86.1 Al45V7 94.8 96.2 Al23V4 96.7 97.9 Al3V 120.0 121.6

86.7 86.7 96.7 97.0 98.5 98.8 128.7 121.8

0.265

0.83

533

0.227

0.90

595

0.251

0.86

566

0.188

0.98

700 582

Present work DFT [11] Present work DFT [11] Present work DFT [11] Present work DFT [11] DFT [12] (b)

490 495 537 578

Present Present Present Present

0K V (bcc)

181.0 182.9

End-members for Al8V5: (Al)6(Al,V)2 (Al,V)3(V)2 Al:Al:Al:V 97.0 Al:V:Al:V 104.2 Al:Al:V:V 128.6 Al:V:V:V 135.2

102.9 110.9 133.5 136.4

0.313 0.301 0.293 0.247

0.73 0.76 0.77 0.87

work work work work

(a) Calculated using PAW-GGA PW91 and (b) calculated using FP-LAPW Note: 0 K values represent extrapolation from low temperatures.

after the assessment by Saunders [7].

 The thermodynamic properties of the solid phases in this system has been studied by several authors with quite different results. Some of the authors found the energies to be quite a lot more negative than others. In this work it has been shown that the values that are less negative in nature are also predicted by DFT and gives much more reasonable extrapolations into the ternary Al–Ti–V system.

 The DFT calculations in this work showed that it is possible that an A15 phase forms at low temperatures from the vanadium rich bcc.

Acknowledgments Financial support from the European RFCS (Research Fund for

B. Lindahl et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 51 (2015) 75–88

Coal and Steel) project “Precipitation in High Manganese Steels” under the Grant agreement no. RFSR-CT-2010-00018 is greatly acknowledged.

Appendix See Tables 8 and 9.

Appendix A. Supplementary material Supplementary data associated with this paper can be found in the online version at http://dx.doi.org/10.1016/j.calphad.2015.07. 002.

References [1] V. Witusiewicz, A. Bondar, U. Hecht, S. Rex, T. Velikanova, The Al–B–Nb–Ti system III. Thermodynamic re-evaluation of the constituent binary system Al– Ti, J. Alloys Compds. 465 (1-2) (2008) 64–77, http://dx.doi.org/10.1016/j. jallcom.2007.10.061. [2] W. Gong, Y. Du, B. Huang, R. Schmid-Fetzer, C. Zhang, H. Xu, Thermodynamic reassessment of the Al–V system, Int. J. Mater. Res. 95 (11) (2004) 978–986 , URL 〈http://www.ijmr.de/MK018057 〉. [3] N. Saunders, System Ti–V, in: I. Ansara, A. Dinsdale, M. Rand (Eds.), COST 507: Thermochemical Database for Light Metal Alloys, vol. 2, European Communities, Brussels, 1998, pp. 297–298. [4] T. Ahmed, H.M. Flower, Partial isothermal sections of Ti–Al–V ternary diagram, Mater. Sci. Technol. 10 (4) (1994) 272–288, http://dx.doi.org/10.1179/ 026708394790163915, URL 〈http://www.ingentaconnect.com/content/maney/ mst/1994/00000010/00000004/art00002?to ken¼ 005318b781cd83f3383a4b3b25703a2b2f5f3f382 d574967282a726e2d58464340592f3f3b57503a609〉. [5] T. Maeda, Phase equilibrium and beta stability in Ti-rich Ti-V-Al alloys (Ph.D. Thesis), University of London, Imperial College, 1988. [6] E. Rolinski, M. Hoch, C. Oblinger, Determination of thermodynamic interaction parameters in solid V–Ti alloys using the mass spectrometer, Metall. Trans. 2 (1971) 12613 , URL 〈http://link.springer.com/article/10.1007/BF02814902〉. [7] N. Saunders, System AlV, in: I. Ansara, A. Dinsdale, M. Rand (Eds.), COST 507: Thermochemical Database for Light Metal Alloys, vol. 2, European Communities, Brussels, 1998, pp. 95–98. [8] K.W. Richter, H. Ipser, The Al–V phase diagram between 0 and 50 atomic percent vanadium, Z. Metall. 91 (5) (2000) 383–388. [9] H. Holleck, F. Benesovsky, H. Nowotny, Die Kristallstruktur von V3Al und sein Mischungsverhalten mit isotypen Phasen, Mon. Chem. 94 (2) (1963) 477–481 , URL 〈http://link.springer.com/10.1007/BF00900283〉. [10] J.-M. Léger, H. Hall, Pressure and temperature formation of A3B compounds I. Nb3Si and V3Al, J. Less Common Metals 32 (2) (1973) 181–187 , URL 〈http:// www.sciencedirect.com/science/article/pii/0022508873900866 〉. [11] J. Wang, S.-L. Shang, Y. Wang, Z.-G. Mei, Y.-F. Liang, Y. Du, Z.-K. Liu, Firstprinciples calculations of binary Al compounds: enthalpies of formation and elastic properties, Calphad 35 (4) (2011) 562–573 , URL 〈http://www.science direct.com/science/article/pii/S036459161100099X 〉. [12] R. Boulechfar, S. Ghemid, H. Meradji, B. Bouhafs, FP-LAPW investigation of structural, electronic, and thermodynamic properties of Al3V and Al3Ti compounds, Phys. B: Condens. Matter 405 (18) (2010) 4045–4050 , URL 〈http://www.sciencedirect.com/science/article/pii/S0921452610006745 〉. [13] S. Meschel, O. Kleppa, The standard enthalpies of formation of some 3d transition metal aluminides by high-temperature direct synthesis calorimetry, in: J.S. Faulkner, R.G. Jordan (Eds.), Metallic Alloys: Experimental and Theoretical …, 1, Kluwer Academic Publishers, Dordrecht, Boston and London, 1994, pp. 103–112 , URL 〈http://link.springer.com/chapter/10.1007/978-94011-1092-1_12 〉. [14] A. Neckel, H. Nowotny, Thermochemisty of aluminides, in: International Leichtmetalltagung, Leoben, 1968, p. 72. [15] O. Kubaschewski, G. Heymer, Heats of formation of transition-metal aluminides, Trans. Faraday Soc. 56 (1960) 473–478 , URL 〈http://pubs.rsc.org/EN/ content/articlepdf/1960/tf/tf9605600473 〉. [16] V. Samokhval, P. Poleshchuk, A. Vecher, Thermodynamic properties of aluminum–titanium and aluminum–vanadium alloys, Russ. J. Phys. Chem. 45 (8) (1971) 1174–1176 , URL 〈http://www.osti.gov/scitech/biblio/4664564 〉. [17] J.L. Murray, Al–V (aluminum–vanadium), Bull. Alloy Phase Diagr. 10 (4) (1989) 351–357, http://dx.doi.org/10.1007/BF02877591, URL 〈http://www.spring erlink.com/content/7k475240u2604616/〉. [18] D. Bailey, O. Carlson, J. Smith, The aluminium-rich end of the aluminium– vanadium system, Trans. ASM 51 (1959) 1097–1102. [19] G. Batalin, V. Sudavtsova, N. Maryanchik, Thermodynamic properties of liquid binary alloys of the Al–Sc, Al–V and Al–Ti system, Ukr. Khimicheskii Z. 51 (8)

87

(1985) 817–819. [20] O. Carlson, D. Kenney, H. Wilhelm, The aluminum–vanadium alloy system, Trans. ASM 47 (1955) 520–537. [21] R. Elliott, Written discussion in the aluminum–vanadium alloys system, Trans. ASM 47 (1955) 538–542. [22] V. Eremenko, Y. Natanzon, V. Titov, Kinetics of dissolution of vanadium in liquid aluminum, Russ. Metall. 5 (1981) 34–37. [23] E. Gebhardt, G. Joseph, On the ternary aluminum–silicon–vanadium system II, Z. Metall. 52 (1961) 310–317. [24] W. Johnson, K. Komarek, E. Miller, Thermodynamic properties of the Cr-Al and V-Al systems, Conract no. 285 (64) 1967 (1967). [25] W. Rostoker, A. Yamamoto, A survey of vanadium binary systems, Trans. ASM 46 (1954) 1136. [26] A. Roth, An investigation of the aluminum–vanadium system, Z. Metall. 32 (1940) 356–359 , URL 〈http://scholar.google.com/scholar? hl ¼ en&btnG ¼ Search&q ¼intitle:An þ investigation þ ofþ the þAluminumVanadium þ System#0〉. [27] N. Varich, L. Burov, K. Kolesnichenko, A. Maksimenko, Highly supersaturated Al–V, Al–Mo and Al–W solid solutions obtained at high rate of cooling, Met. Metallogr. 15 (2) (1963) 111–113. [28] S. Shang, A. Saengdeejing, Z. Mei, D. Kim, H. Zhang, S. Ganeshan, Y. Wang, Z. Liu, First-principles calculations of pure elements: equations of state and elastic stiffness constants, Comput. Mater. Sci. 48 (4) (2010) 813–826, http: //dx.doi.org/10.1016/j.commatsci.2010.03.041. [29] M. Jahnátek, M. Krajčí, J. Hafner, Interatomic bonding, elastic properties, and ideal strength of transition metal aluminides: a case study for Al3(V,Ti), Phys. Rev. B 71 (2) (2005) 024101, http://dx.doi.org/10.1103/PhysRevB.71.024101. [30] U. Kattner, The thermodynamic modeling of multicomponent phase equilibria, JOM 49 (12) (1997) 14–19 , URL 〈http://jomgateway.net/ArticlePage.aspx? DOI¼ 10.1007/s11837-997-0024-5 http://link.springer.com/article/10.1007/ s11837-997-0024-5〉. [31] P. Villars, L. Calvert, Pearson's Handbook of Crystallographic Data for Intermetallic Phases, 2nd Edition, ASM, Materials Park, OH, USA, 1991. [32] B. Huber, K.W. Richter, Observation of the new binary low temperatures compound AlV, J. Alloys Compds. 493 (1–2) (2010) L33–L35 , URL 〈http:// www.sciencedirect.com/science/article/pii/S0925838809027169 〉. [33] G. Kresse, J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54 (16) (1996) 11169–11186. [34] P.E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50 (24) (1994) 17953–17979. [35] G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59 (3) (1999) 1758–1775. [36] J. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77 (18) (1996) 3865–3868. [37] G. Kresse, M. Marsman, J. Furthmüller, VASP: The Guide, Wein, 2013. http:// cms.mpi.univie.ac.at/vasp/vasp.pdf. [38] M. Methfessel, A.T. Paxton, High-precision sampling for Brillouin-zone integration in metals, Phys. Rev. B 40 (6) (1989) 3616–3621. [39] Y. Wang, Z.-K. Liu, L.-Q. Chen, Thermodynamic properties of Al, Ni, NiAl, and Ni3Al from first-principles calculations, Acta Mater. 52 (9) (2004) 2665–2671, http://dx.doi.org/10.1016/j.actamat.2004.02.014. [40] F. Birch, Finite elastic strain of cubic crystals, Phys. Rev. 71 (11) (1947) 809–824. [41] C.L. Zacherl, S.-L. Shang, A. Saengdeejing, Z.-K. Liu, Phase stability and thermodynamic modeling of the Re–Ti system supplemented by first-principles calculations, Calphad 38 (null) (2012) 71–80. [42] V.L. Moruzzi, J.F. Janak, K. Schwarz, Calculated thermal properties of metals, Phys. Rev. B 37 (2) (1988) 790–799, http://dx.doi.org/10.1103/PhysRevB.37.790. [43] G.A. Alers, Use of sound velocity measurements in determining the Debye temperature of solids, Phys. Acoust. 3(Part B) (1965) 1–42. [44] Q. Chen, B. Sundman, Calculation of Debye temperature for crystalline structures—a case study on Ti, Zr, and Hf, Acta Mater. 49 (2001) 947–961. [45] X.-G. Lu, M. Selleby, B. Sundman, Calculations of thermophysical properties of cubic carbides and nitrides using the Debye–Grüneisen model, Acta Mater. 55 (4) (2007) 1215–1226, http://dx.doi.org/10.1016/j.actamat.2006.05.054. [46] S.L. Shang, Y. Wang, Z.-K. Liu, First-principles elastic constants of α- and θAl2O3, Appl. Phys. Lett. 90 (10) (2007) 101909, http://dx.doi.org/10.1063/ 1.2711762. [47] X.L. Liu, B.K. Vanleeuwen, Y. Du, Z.-K. Liu, A Case Study of the Mg–Zn Binary System Using the Debye–Grüneisen Model. [48] N. Vočadlo, G.D. Price, The Grüneisen parameter—computer calculations via lattice dynamics, Phys. Earth Planet. Inter. 82 (3–4) (1994) 261–270, http://dx. doi.org/10.1016/0031-9201(94)90076-0. [49] D. Kim, S. Shang, Z. Liu, Effects of alloying elements on elastic properties of Ni3Al by first-principles calculations, Intermetallics 18 (6) (2010) 1163–1171, http://dx.doi.org/10.1016/j.intermet.2010.02.024. [50] C. Wolverton, Crystal structure and stability of complex precipitate phases in Al–Cu–Mg–(Si) and Al–Zn–Mg alloys, Acta Mater. 49 (16) (2001) 3129–3142, http://dx.doi.org/10.1016/S1359-6454(01)00229-4. [51] C. Jiang, C. Wolverton, J. Sofo, L.-Q. Chen, Z.-K. Liu, First-principles study of binary bcc alloys using special quasirandom structures, Phys. Rev. B 69 (21) (2004) 214202. [52] D. Shin, R. Arróyave, Z.-K. Liu, A. Van de Walle, Thermodynamic properties of binary hcp solution phases from special quasirandom structures, Phys. Rev. B 74 (2) (2006) 24204.

88

B. Lindahl et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 51 (2015) 75–88

[53] A.C. Lieser, C.L. Zacherl, A. Saengdeejing, Z.-K. Liu, L.J. Kecskes, First-principles calculations and thermodynamic re-modeling of the Hf-W system, Calphad 38 (2012) 92–99, http://dx.doi.org/10.1016/j.calphad.2012.04.005. [54] C. Kittel, Introduction to Solid State Physics, Wiley, Hoboken NJ, 1995. [55] P.J. Brown, The structure of α (V–Al), Acta Crystallogr. 10 (2) (1957) 133–135, http://dx.doi.org/10.1107/S0365110X57000389. [56] P.J. Brown, The structure of the intermetallic phase ‘α’ (VAl), Acta Crystallogr. 12 (12) (1959) 995–1002, http://dx.doi.org/10.1107/S0365110X59002821. [57] A.E. Ray, J.F. Smith, A test for electron transfer in V4Al23, Acta Crystallogr. 13

(11) (1960) 876–884, http://dx.doi.org/10.1107/S0365110X60002168. [58] D.J. Kenney, H.A. Wilhelm, O.N. Carlson, Aluminum–vanadium system, Technical Report, June 1953. [59] W.B. Pearson, P.W. Riley, Gamma-brasses with R cells, Acta Crystallogr. Sect. B 33 (1977) 1088–1095. [60] G.N. Kamm, G.a. Alers, Low-temperature elastic moduli of aluminum, J. Appl. Phys. 35 (2) (1964) 327, http://dx.doi.org/10.1063/1.1713309. [61] D.I. Bolef, R.E. Smith, J.G. Miller, Elastic properties of Vanadium, Phys. Rev. B 3 (June (1971)) (1971) 4100.