Diffusivities and atomic mobilities in bcc Ti-Zr-Nb alloys

Calphad 64 (2019) 160–174

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Diffusivities and atomic mobilities in bcc Ti-Zr-Nb alloys a

a

b

a

a,⁎

c

Weimin Bai , Yueyan Tian , Guanglong Xu , Zhijie Yang , Libin Liu , Patrick J. Masset , ⁎ Ligang Zhanga, a b c

T

School of Materials Science and Engineering, Central South University, Changsha 410083, China Tech Institute for Advanced Materials & School of Materials Science and Engineering, Nanjing Tech University, Nanjing, Jiangsu 211800, China Technallium Engineering and Consulting (TEC), Fliederweg 6, D-92449 Steinberg am See, Germany

ARTICLE INFO

ABSTRACT

Keywords: bcc Ti-Zr-Nb alloys Diffusivity Atomic mobility CALPHAD modeling Whittle and Green method

Diffusion couples were prepared and annealed at 1373 K for 72 h and 1473 K for 48 h, respectively. The interdiffusion coefficients at the intersection composition were obtained using the Whittle and Green method and the ternary trace diffusion coefficients using Hall method. The experimental diffusion coefficients of Ti and Nb in bcc Ti-Zr-Nb system were assessed to develop an atomic mobility database. The calculated diffusion coefficients and composition profiles show good agreement with the experimental data.

1. Introduction Titanium and its alloys are widely used in biomedical applications due to their low elastic modulus which is in addition comparable to bone as well as their high specific strength and high corrosion resistance in human body fluids [1–6]. β-type(with bcc structure) and near β-Ti alloys comprising non-toxic and non-allergic elements have been extensively investigated in the past decades to produce materials exhibiting a low Young’s modulus combined with good mechanical properties [7–10]. Song et al. [11] suggested that Zr, Nb, Ta and Mo additives are the most suitable alloying elements in β-type bio-compatible Ti-based alloys which process an increased strength and a reduced elastic modulus according to electronic structural calculations. Recently, low-modulus biomedical β-Ti alloys have been developed, for example, Ti–29Nb–13Ta–4.6Zr (TNTZ) [12], Ti–24Nb–4Zr–7.9Sn [13] and Ti–8Mo–4Nb–2Zr [14]. By measuring the diffusion coefficients in single bcc phase alloys, combining the CALPHAD method, the atomic mobility database for the β-Ti alloys can be established and then the diffusion information at full temperature and composition range can be obtained by extrapolation. The diffusion paths and phase fractions during homogenization and precipitation [15,16] were precisely predicted with the help of thermodynamic and kinetic data by means of the Thermo-Calc [17] and DICTRA software [18,19]. The microstructure evolution during the heat treatment was not only statistically explained via the classical nucleation and growth model including accurate diffusivity data [20], but also represented by the phase field modeling allying with diffusion kinetic

⁎

database of multi- component and multi-phase systems [15]. The present work aims at studying the diffusion behaviors of Zr and Nb in β-Ti alloys and to develop an atomic mobility database for the bcc phase in Ti-Zr-Nb system using the CALPHAD method [17]. 2. Literature review Self-diffusion, impurity diffusion, tracer diffusion coefficients as well as inter-diffusion coefficients regarding Ti-Zr-Nb system were determined using several experiments or techniques as summarized in Table 1. By assessing the available experimental data of self-diffusion, impurity diffusion, tracer diffusion and inter-diffusion coefficients, Liu et al. optimized the atomic mobilities of bcc Ti-Zr [21], Ti-Nb [22] and Nb-Zr [23] alloys using the CALPHAD technique. Comprehensive comparisons between the calculated and experimental data show that the mobility parameters can be satisfactorily reproduced most of the experimental data. In this work, the binary atomic mobilities were chosen to be the boundary system for their reliability and self-consistency. Thermodynamic modeling of boundary binary systems were the same as Liu’s work, obtained from works of Turchanin et al. [61], Zhang et al. [62], and Guillermet [63]. Studies indicate that the bcc miscibility gap in the Nb-Zr binary system extends into the ternary system, and no ternary compound forms. The corresponding ternary system is composed of solution phases only. Thermodynamic modeling of this ternary system can be carried out using a simple ternary extrapolation, based

Corresponding authors. E-mail addresses: [email protected] (L. Liu), [email protected] (L. Zhang).

https://doi.org/10.1016/j.calphad.2018.12.003 Received 22 August 2018; Received in revised form 26 November 2018; Accepted 2 December 2018 0364-5916/ © 2018 Elsevier Ltd. All rights reserved.

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Table 1 Summary of the experimental determined diffusion coefficients in subsystems regarding Ti-Zr-Nb system. Diffusion coefficients

System

Experimental data References

Self-diffusion coefficients

Ti Zr Nb Zr in Ti Nb in Ti Ti in Zr Nb in Zr Ti in Nb Zr in Nb Ti and Zr in bcc Ti-Zr alloys Ti and Nb in Ti-Nb alloys Nb and Zr in Nb-Zr alloys Ti-Zr system Ti-Nb system Nb-Zr system

[24–27] [27–33] [34–38] [39,40] [26,41] [42–44] [29,33,45–47] [48,49] [49,50] [51] [26,41,52] [33,46,53] [42–44,54,55] [45,49,55–58] [59,60]

Impurity diffusion coefficients

Tracer diffusion coefficients Inter-diffusion coefficients

Yi = Table 2 Composition of Ti-Zr-Nb diffusion couples. Diffusion couples 1373 K M1 M2 M3 M4 M5 N1 N2 N3 1473 K O1 O2 O3 O4 O5 P1 P2 P3

ci ci+

ci ci

(1)

Then the Fick’s second low can be expressed as: Actual composition (Atomic percent)

1 dz 2t dYZr

Ti–4.77Zr/Ti–4.73Nb Ti–9.71Zr/Ti–4.84Nb Ti–9.71Zr/Ti–9.82Nb Ti–14.86Zr/Ti–9.72Nb Ti–14.68Zr/Ti–14.78Nb Ti/Ti–4.90Zr–14.68Nb Ti/Ti–9.87Zr–9.57Nb Ti/Ti–14.55Zr–4.87Nb

(1

YZr )

z

YZr dz + YZr

+ z

(1

YZr ) dz

dc Ti Ti = D˜ZrZr + D˜ZrNb Nb dcZr

1 dz 2t dYNb

(1

YNb)

(2) z

YNb dz + YNb

+ z

(1

dc Ti Ti = D˜NbNb + D˜NbZr Zr dcNb

Ti–4.67Zr/Ti–4.75Nb Ti–10.04Zr/Ti–4.70Nb Ti–10.04Zr/Ti–9.35Nb Ti–15.05Zr/Ti–9.46Nb Ti–14.53Zr/Ti–14.81Nb Ti/Ti–5.02Zr–14.80Nb Ti/Ti–9.54Zr–9.64Nb Ti/Ti–14.58Zr–4.88Nb

YNb) dz (3)

By solving this set of four equations from one pair of diffusion Ti couples whose diffusion paths cross at one point, the diffusivities D˜NbNb , Ti Ti Ti ˜ ˜ ˜ DNbZr , DZrNb , DZrZr at that point can be obtained. Note that the molar volume was taken to be constant for the lack of reliable data on the composition-dependent molar volume in bcc Ti-Zr-Nb alloys. Errors introduced by this approximation are believed to be within the accuracy of the results obtained via the Whittle-Green method [65]. In practice, the original experimental profiles should be subjected to smooth by one of algorithms [67,68]: moving average smoothing, Savitzky-Golay smoothing method and PCHIP interpolation, etc. However, errors may be brought in during the smoothing operation. To avoid the errors from fitting or smoothing, a robust error function expansion (ERFEX) was put forward to represent the experimental profiles in a highly accurate analytical form [69,70]:

on the Nb-Ti, Nb-Zr, and Ti-Zr binary systems. Recently, Chen [64] measured ternary inter-diffusivities in bcc Tirich Ti-Nb-Zr alloys at 1273 K using diffusion couple and Matano-Kirkaldy method. A set of mobility parameters of Ti, Nb and Zr in the ternary Ti-Nb-Zr system were assessed based on his experimental data. In this work, inter-diffusion and impurity diffusion coefficients in the Ti-Zr-Nb alloys at 1373 K and 1473 K were determined and taken into account when assessing the temperature-dependent.

X (r ) =

[ai erf(bi z + ci ) + di] i

(4)

where X (r ) is the effective alloying element content at location z ; a , b, c and d are the fitting parameters.

3. Model 3.1. Extraction of inter-diffusion coefficients

3.2. Extraction of impurity diffusion coefficients

In this work, the Whittle and Green (W-G) method [65] was utilized to eliminate the step of locating the Matano plan in traditional MatanoKirkaldy method [66] and the error it may introduce. Whittle and Green introduced a variable change expressing the mass balance in the vicinity of the Matano plane:

The impurity diffusivities of Zr in Ti-Nb and Nb in Ti-Zr alloys were extracted from the M1-M5 profiles by applying the analytical Hall method [71]. Similar to its binary prototype, the profiles were first transformed to a plot of μ vs λ, in which erfµ = 2Y 1 and = z / t .

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Fig. 1. Calculated phase diagrams of (a) Ti-Zr [61], (b) Ti-Nb [62], (c) Nb-Zr [63] binary systems.

Fig. 2. Diffusion Paths in the Gibbs isothermal section of the Ti-Zr-Nb ternary phase diagram: (a) 1373 K and (b) 1473 K.

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Fig. 3. SEM backscattered electron image of the diffusion couples: (a) M1 annealed at 1373 K for 72 h. (b) O1 annealed at 1473 K for 48 h. n

Jk =

Lki i=1

µi = z

n

Lki i=1

µi c i ci z

(7)

where Lki is the phenomenological factor; µi is the chemical potential of specie k ; z is the distance. The diffusion flux is related with thermodynamic driving force of all the species, i.e., the chemical potential gradient in a substitutional solution containing n components. When the inter-diffusion coefficient Dkjn is introduced, it can be expressed as: n 1

Dkjn

Jk = j =1

cj z

(8)

The summation is made over n-1 dependent composition variants of solutes, and the n independent one is treated as solvent. So the interdiffusion coefficient Dkjn can be expressed as [17]:

Dkjn =

Fig. 4. The ERFEX representation of the composition profiles of the M3-N1 couple annealed in 1373 K for 72 h.

1 2k1 1+ exp(µ2 ) × Y (x ) 4h12

(5)

D˜ (x ) =

1 1 4h 22

(6)

Y (x )]

ik

xk ) x i Mi

µi

µi

xj

xn

(9)

As suggested by Andersson and Ågren [17] and later modified by Jönsson [72], the atomic mobility Mi of species i can be expressed as:

Mi = Mi0 exp

Qi 1 ×mg RT RT

= exp

RT ln Mi0 RT

Qi

1 mg × RT (10)

D˜ (x ) =

exp(µ2) × [1

( i

Fitting the plot with a linear equation µ = h + k , the parameters h and k of Zr and Nb were determined at the left and right ends of the transformed profile. The impurity diffusion coefficients were derived using Eqs. (5) and (6).

2k2

n

where R is the gas constant; T is the temperature; is the frequency factor; Qi is the activation energy; mg is the correction factor taking into account the ferromagnetic contribution. Since there is no ferromagnetic transition reported, mg was taken Gi as equal to 1. Provided that Gi = RT ln Mi0 Qi , Eq. (10) can be written as:

Mi0

where x is the terminal composition.

Mi = exp

3.3. Atomic mobility and diffusivity

Gi RT

1 RT

(11)

For a solid solution phase, the parameter Gi is assumed to be dependent of the composition, and can be expressed using the RedlichKister polynomial form as follow:

According to the Fick-Onsager law, the diffusion flux Jk of specie k , can be written as:

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Table 3 Interdiffusion coefficients in the bcc Ti-Zr-Nb alloys at 1373 K and 1473 K. Temp. (K)

Diffusion Couple

1373

M1-N1 M1-N2 M1-N3 M2-N1 M2-N2 M2-N3 M3-N1 M3-N2 M3-N3 M4-N1 M4-N2 M4-N3 M5-N1 M5-N2 M5-N3 O1-P1 O1-P2 O1-P3 O2-P1 O2-P2 O2-P3 O3-P1 O3-P2 O3-P3 O4-P1 O4-P2 O4-P3 O5-P1 O5-P2 O5-P3

1473

x p Gi p +

Gi = p

q>p v>q

Zr

Nb

Ti D˜ZrZr

1.579 2.742 3.531 1.734 3.662 5.665 2.421 4.739 6.717 2.486 5.586 8.857 3.365 6.861 10.000 1.524 2.597 3.365 1.707 3.589 5.701 2.43 4.724 6.881 2.552 5.489 8.913 3.438 6.723 9.993

3.71 1.849 0.721 4.504 3.286 1.783 8.516 5.243 2.364 9.07 6.636 3.437 12.944 8.153 3.934 3.605 1.898 0.81 4.399 3.386 1.951 8.144 5.243 2.508 8.725 6.461 3.466 12.795 8.091 3.932

6.22 7.26 7.82 6.35 6.67 7.11 4.19 5.49 6.76 4.31 5.06 6.17 2.77 4.28 5.86 15.12 15.85 19.50 13.74 13.80 16.93 10.63 11.67 16.84 11.57 12.02 16.48 8.14 9.94 15.72

(r ) G p, q (x p i

q>p

[xs + (1

xp

xq

x v )/3]

(s ) G p, q, v i

s = p, q, v

(12) where denotes the solid solution phase; x p is the mole fraction of species p ; Gip is the value Gi of species i in pure species p ; (r ) Gi p, q and (s) Gip, q, v are the binary and ternary interaction parameters. Assuming that the mono-vacancy exchange process is the dominate diffusion mechanism, and neglecting correlation factors, the tracer diffusion of element i , Di*, is directly related to the mobility Mi by the Einstein relation [17,72]:

Di* = RTMi

13

m2s 1)

Ti D˜ZrNb

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.05 0.07 0.10 0.12 0.04 0.14 0.07 0.07 0.16 0.17 0.03 0.03 0.01 0.01 0.07 0.58 0.40 0.53 0.37 0.07 0.20 0.12 0.13 0.10 0.28 0.13 0.07 0.06 0.08 0.09

0.18 0.35 0.45 0.08 0.16 0.62 0.09 0.26 0.61 0.03 0.20 0.61 0.11 0.49 0.80 − 0.01 − 0.94 1.35 0.18 − 0.46 4.15 0.31 0.01 3.46 0.08 − 0.66 2.93 0.39 0.35 3.99

Ti D˜NbZr

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.01 0.05 0.17 0.03 0.02 0.17 0.03 0.04 0.16 0.04 0.04 0.23 0.01 0.01 0.06 0.14 0.30 1.07 0.02 0.04 0.16 0.08 0.08 0.05 0.11 0.09 0.05 0.04 0.08 0.14

0.41 0.06 0.06 − 0.22 − 0.29 − 0.08 − 0.47 − 0.24 − 0.03 − 0.49 − 0.24 − 0.06 − 0.33 − 0.05 − 0.12 − 0.33 − 0.96 0.31 0.03 − 0.59 0.43 − 0.13 − 0.83 0.40 − 0.81 − 1.10 0.03 − 0.58 − 0.76 0.09

Ti D˜NbNb

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.10 0.03 0.03 0.06 0.04 0.04 0.01 0.02 0.01 0.24 0.11 0.01 0.02 0.03 0.05 0.12 0.03 0.04 0.24 0.05 0.03 0.31 0.08 0.08 0.17 0.08 0.05 0.10 0.03 0.10

2.08 2.44 2.68 2.00 2.34 2.99 1.25 2.00 2.82 1.20 1.87 2.73 0.93 1.63 2.85 5.85 6.38 7.93 5.43 5.48 7.86 3.89 5.02 7.84 3.85 4.95 8.33 3.02 4.59 7.98

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.02 0.04 0.05 0.03 0.04 0.05 0.01 0.01 0.01 0.03 0.11 0.12 0.02 0.02 0.04 0.02 0.02 0.05 0.13 0.06 0.08 0.05 0.03 0.14 0.04 0.11 0.09 0.04 0.06 0.14

Fig. 1). The melting process was repeated six times to attain complete melted and homogeneous alloys. The ingots were annealed at 1473 K for 12 h to obtain alloys with average grain size larger than several millimeters such that the effect of grain boundary diffusion can be ignored. The annealed ingots were cut into rectangular solids of 10 × 10 × 5 mm size using wire-electrode cutting. Subsequently, surfaces of the blocks were polished to mirror-like quality. The well-contacted diffusion couples were assembled with appropriate pairs of blocks under vacuum at 1173 K for 4 h. The diffusion couples were sealed into quartz capsules filled with pure argon and annealed at 1373 K for 72 h, followed by quenching in ice water. A same set of diffusion couples were made and annealed at 1473 K for 48 h. The diffusion couples were mounted, ground, and polished by standard metallographic techniques. The microstructure of diffusion zone was observed by scanning electron microscopy (SEM) and the composition profiles were obtained using electron microprobe analysis (EPMA, JEOL JAX-8230). The diffusion paths were presented in Fig. 2.

x q )r

r = 0,1,2,...

x p xq x v p

Interdiffusion coefficients ( ×10

x p xq p

+

Intersection composition (mole%)

(13)

4. Experiment

5. Results and discussions

16 binary and ternary Ti-based alloys were prepared from 99.99 wt % Ti, 99.99 wt% Zr and 99.99 wt% Nb by arc melting in electric arc furnace under an argon atmosphere. The nominal compositions of the alloys are listed in Table 2. All the alloy compositions were selected in the bcc solid solution area of Ti-Zr-Nb system at 1373 K and 1473 K according to accepted binary phase diagrams [61–63] (illustrated in

5.1. Diffusion behaviors of the bcc Ti-Zr-Nb ternary alloys Fig. 3 shows the SEM backscattered electron image of diffusion zone of the diffusion couples of M1 annealed at 1373 K for 72 h and O1 annealed at 1473 K for 48 h. As shown, no second phase and obvious

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Ti Ti Ti Ti Fig. 5. The iation of ternary inter-diffusion coefficients with the compositions: (a) D˜ZrZr with Zr, (b) D˜NbNb with Zr, (c) D˜ZrZr with Nb, and (d) D˜NbNb with Nb at 1373 K.

Table 4 Impurity diffusion coefficients of Zr in Ti-Nb and Nb in Ti-Zr alloys at 1373 K and 1473 K. Temperature/K

Composition

1373

D* Zr (Ti

4.78Nb )

D* Zr (Ti

9.78Nb )

1473

D* Zr (Ti

14.78Nb)

D* Zr (Ti

4.73Nb )

D* Zr (Ti

9.40Nb )

D* Zr (Ti

14.81Nb)

Impurity diffusion coefficients ( ×10

13

m2s 1)

5.675 3.359 2.073 12.229 9.144 5.961

Kirkendal void were observed in the diffusion zone. Fig. 4 shows a couple of composition profiles (M3-N1) used for solving the inter-diffusion coefficients at the intersection composition. Both the EPMA experimental profiles and the corresponding fitted curves were appropriately smooth and show a conventional S-shape. The experimental composition-distance curve indicates that the penetration depths of Zr were fairly larger than that of Nb in both in M3 and N1 diffusion couples. It reveals that Zr diffuses faster than Nb in bcc Ti-Zr-Nb alloys,

Composition

D* Nb (Ti

4.77Zr )

D* Nb (Ti

9.71Zr )

D* Nb (Ti

14.76Zr )

D* Nb (Ti

4.70Zr )

D* Nb (Ti

10.02Zr )

D* Nb (Ti

14.74Zr )

Impurity diffusion coefficients ( ×10

13

m2s 1)

3.129 3.187 3.323 7.867 9.161 13.122

as shown in Fig. 4. It is can be seen that there was a slight drift of the Matano plane, and that was the reason why the Whittle and Green method was applied. The diffusion paths of 16 diffusion couples at 1373 K and 1473 K were plotted on the Ti-Zr-Nb ternary isotherms in Fig. 2. All the paths were bended and S-shaped, showing the difference of diffusion rates between Zr and Nb (Fig. 4).

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Ti Ti Ti Ti Fig. 6. 3D plot of the ternary inter-diffusion coefficients (a) D˜ZrZr , (b) D˜ZrNb , (c) D˜NbZr and (d) D˜NbNb in the bcc Ti-Zr-Nb alloys at 1373 K, together with the impurity * (Ti Nb) and DNb * (Ti Zr ) , and binary diffusion coefficients obtained from the literature [21,23]. diffusion coefficients DZr

5.2. Inter-diffusion and impurity diffusion coefficients at 1373 K and 1473 K

coefficients are larger than the cross ones. It is to mention, that based on the formula calculation error from solving Fick’s second law to extract the diffusion coefficients, the accuracy of the cross-diffusion coefficients is much lower than the main ones. Ti Ti The variation of D˜ZrZr and D˜NbNb with the composition of Zr and Nb at 1373 K were presented in Fig. 5. From Fig. 5(a) and (b), one can Ti Ti observe that D˜ZrZr and D˜NbNb decrease with the increase of concentration of Zr. However, the decreasing rate is weighted by the Nb contents in the alloys (the three colors correspond to inter-diffusion coefficients extracted from three different diffusion couples, N1, N2 and N3). It appears that high x(Nb)/x(Zr) ratio slows down the diffusion process. Whereas in Fig. 5(b), the diffusion coefficients show an increasing trend Ti Ti (green spots) when the ratio is big enough. D˜ZrZr and D˜NbNb decrease rapidly with the increasing of the concentration of Nb in Fig. 5(c) and (d). It seems that these diffusion coefficients did not affected by the change of x(Zr)/x(Nb) ratios. That is because that the influence of Nb Ti Ti on D˜ZrZr and D˜NbNb is much greater than Zr. The impurity diffusion coefficients of Zr in Ti-Nb binary alloys and Nb in Ti-Zr alloys are listed in Table 4. They were extracted using the Hall method [71] applied to the profiles M1-M5 and O1-O5. Variation rules of the impurity diffusion coefficients from the table were easily * (Ti xNb) decreases with the increase of x; (2) DNb * (Ti xZr ) observed: (1) DZr increases with the increase of x; (3) they all increase with the increase of the temperature.

As shown in Fig. 2, the diffusion paths of 8 diffusion couples have 15 intersections. The inter-diffusion coefficients extracted at the 15 intersecting compositions at 1373 K and 1473 K are summarized in Table 3. All the values were satisfied the thermodynamic constraints derived by Kirkaldy [73]: Ti Ti D˜ZrZr + D˜NbNb > 0

(14)

Ti Ti D˜ZrZr × D˜NbNb

(15)

2 Ti Ti (D˜ZrZr + D˜NbNb)

Ti Ti D˜ZrNb × D˜NbZr > 0 Ti Ti 4(D˜ZrZr × D˜NbNb

Ti Ti D˜ZrNb × D˜NbZr )

0

(16)

Note that the standard deviations were determined from four independent calculations upon two independent measurements. Ti As expected, the main diffusion coefficients are positive and D˜ZrZr Ti has a larger value than D˜NbNb at each single intersecting composition

which is fully consistent with the phenomenon observed in Figs. 2 and 4, whereas the cross coefficients are much smaller (in most case in magnitude) and scattered. Generally, the main diffusion coefficients, Ti Ti D˜ZrZr and D˜NbNb stand for the primary diffusion rate of Zr or Nb under its Ti Ti own concentration gradient, D˜ZrNb and D˜NbZr , capture the interaction between the two alloying elements. It is typical that the main diffusion

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Ti

Ti

Ti

Ti

Fig. 7. The variation of ternary interdiffusion coefficients with the compositions: (a) D˜ZrZr with Zr, (b) D˜NbNb with Zr, (c) D˜ZrZr with Nb, and (d) D˜NbNb with Nb, and 3D Ti Ti Ti Ti plot of the ternary interdiffusion coefficients (e) D˜ZrZr , (f) D˜ZrNb , (g) D˜NbZr and (h) D˜NbNb in the bcc Ti-Zr-Nb alloys at 1473 K.

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Table 5 Atomic mobilities for the bcc phase of the Ti-Zr-Nb ternary system. Mobility Mobility of Nb MQ(BCC_A2,NB,NB: VA;0) MQ(BCC_A2,NB,TI: VA;0) MQ(BCC_A2,NB,ZR: VA;0) MQ(BCC_A2,NB,NB,TI: VA;0) MQ(BCC_A2,NB,NB,ZR: VA;0) MQ(BCC_A2,NB,NB,ZR: VA;1) MQ(BCC_A2,NB,TI,ZR: VA;0) MQ(BCC_A2,NB,NB,TI,ZR: VA;0) Mobility of Ti MQ(BCC_A2,TI,NB: VA;0) MQ(BCC_A2,TI,TI: VA;0) MQ(BCC_A2,TI,ZR: VA;0) MQ(BCC_A2,TI,NB,TI: VA;0) MQ(BCC_A2,TI,TI,ZR: VA;0) MQ(BCC_A2,TI,TI,ZR: VA;1) Mobility of Zr MQ(BCC_A2,ZR,NB: VA;0) MQ(BCC_A2,ZR,TI: VA;0) MQ(BCC_A2,ZR,ZR: VA;0) MQ(BCC_A2,ZR,NB,ZR: VA;0) MQ(BCC_A2,ZR,NB,ZR: VA;1) MQ(BCC_A2,ZR,TI,ZR: VA;0) MQ(BCC_A2,ZR,TI,ZR: VA;1) MQ(BCC_A2,ZR,NB,TI: VA;0) MQ(BCC_A2,ZR,NB,TI: VA;1) MQ(BCC_A2,ZR,NB,TI,ZR: VA;0) MQ(BCC_A2,ZR,NB,TI,ZR: VA;1) MQ(BCC_A2,ZR,NB,TI,ZR: VA;2)

Parameter, J/mole

Reference

-395,598.95–82.03*T -171,237.75–115.83*T -135,119.44–148.59 *T, T ≤ 1600 K -187,191.53–114.17*T, T ≥ 1600 K 107,764.17–14.52*T + 43,010.37 + 74.52*T − 20568.08 − 585,135.83 + 455.48*T − 1,203,673.97

[22,23] [22] [23]

-369,002.77–87.15*T -151,989.95–127.37*T -140,356.54–138.12*T + 86,711.4 + 2.61 *T − 15,826.04 + 62.55 *T 8243.54

[22] [21,22] [21] [22] [21] [21]

-358,612.72–84.43*T -131,670.56–133.36*T -104,624.81–163.15 *T, T ≤ 1573 K -161,543.53–126.1 *T, T ≥ 1573 K + 83,103.91 + 50.78*T 106,181.31 − 12,581.03 + 33.38 *T 2898.6 − 302,263.57 + 301.38*T 90,944.77 3,937,629.21 − 934,150.46 − 2,161,629.83

[23] [21] [21,23]

The diffusion coefficients obtained in this work, together with the binary diffusion coefficients from the reference [21,22] were fitted using the Redlich-Kister polynomial [74], and are illustrated with threedimensional plots in Fig. 6 (1373 K), which shows a complete diffusion picture of the bcc Ti-Zr-Nb phase over a wider range of concentration. When extended to the boundaries, the diffusion coefficients coincide with the boundary data obtained from experiments and literatures. Screening the map of the main inter-diffusion coefficients, Fig. 6(a) and (d), one can observe that when the concentration of diffusion element tends to 0, the limit of inter-diffusion coefficients would tally with the impurity coefficients obtained in this work. When the concentration of gradient element tends to 0, the limits match the binary diffusion coefficient from literature [21,22] well. As for cross inter-diffusion coefficients, when the concentrations of diffusion elements tend to 0, the limits are both 0. Ti Ti The variations of D˜ZrZr and D˜NbNb with the concentration of Zr and the 3D plots of the diffusion coefficients in the bcc Ti-Zr-Nb system at 1473 K are given in Fig. 7. They show similar trends with those obtained at 1373 K. The average values of determined diffusion coefficients are compared with data in binary systems (x(nb) = 0, x (zr) = 0) in Table S1 in Supplementary materials, showing they are Ti of the same order of magnitude. The D˜ZrZr varies from 2.77 × 10−13 −13 2 to 7.82 × 10 m /s at 1373 K, and from 8.14 × 10−13 to Ti −12 2 1.95 × 10 m /s at 1473. The value of D˜NbNb varies from −14 −13 2 9.32 × 10 to 2.99 × 10 m /s at 1373 K, and from 3.02 × 10−13 to 8.33 × 10−13 m2/s at 1473. The ratios of Ti Ti D˜ZrZr (1473K )/ D˜ZrZr (1373K ) varies from 2.07 to 2.93 and

[22] [23] [23] This work This work

[23] [23] [21] [21] This work This work This work This work This work

Ti Ti D˜NbNb (1473K )/D˜NbNb (1373K ) from 2.34 to 3.24. For the impurity * (Ti Nb) (2.07 × 10−13-5.68 × 10−13 at diffusion coefficients, DZr 1373 K and 5.96 × 10−13–1.22 × 10−12 at 1473 K) and * (Ti Zr ) DNb (3.13 × 10−13-3.32 × 10−13 at 1373 K and 7.97 × 10−13–1.31 × 10−12 at 1473 K), the values at 1473 K are 2.15–2.87 and 2.51–3.95 time of those at 1373 K, respectively. The Ti mean value of these diffusion coefficients, D˜ZrZr are 5.76 × 10−13 at

Ti 1373 K and 1.39 × 10−12 at 1473 K, D˜NbNb are 2.12 × 10−13 at −13 * (Ti Nb) are 3.70 × 10−13 DZr 1373 K and 5.89 × 10 at 1473 K, −13 * (Ti Zr ) are DNb at 1373 K and 9.11 × 10 at 1473 K, 3.21 × 10−13 at 1373 K and 1.00 × 10−12 at 1473 K, as listed in Table S1. The ratios between the values at 1473 and 1373 for these four mean data are 2.41, 2.77, 2.46 and 3.13, respectively.

5.3. Atomic mobilities of bcc Ti-Zr-Nb system The diffusion coefficients derived from diffusion couple experiments were used in optimizing process to obtain the atomic mobility parameters for the Ti-Zr-Nb ternary bcc alloys using the DICTRA module embedded in ThermoCalc software [18]. Liu’s works [21–23] were chosen as the boundary binary systems. The assessed results are listed in Table 5. 5.3.1. Calculated diffusion coefficients compare with experimental results The calculated inter-diffusion coefficients at the intersection composition at 1373 K and 1473 K are shown in Fig. 8. They are compared with experimental data given in brackets. The calculated results

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Ti

Fig. 8. Calculated inter-diffusion coefficients of the ternary Ti-Zr-Nb system compared with the experimental measurements (in brackets) in this work: (a) D˜ZrZr , (b) Ti Ti Ti Ti Ti Ti Ti D˜ZrNb , (c) D˜NbZr , and (d) D˜NbNb . at 1373 K and (e) D˜ZrZr , (f) D˜ZrNb , (g) D˜NbZr , and (h) D˜NbNb at 1473 K.

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* (Ti Fig. 9. Calculated impurity diffusion coefficients (a) DZr

Nb)

* (Ti and (b) DNb

Zr )

compared with the experimental data.

Table 6 Calculated ternary interdiffusion coefficients in bcc Ti-Zr-Nb alloys at 1273 K using the present atomic mobilities and experimental results from Ref. [64]. Composition at%

Experimental results in Ref. [64] (10

14

m2s

1)

Calculated results (10

14

m2s

1)

Nb

Zr

Ti D˜NbZr

Ti D˜NbZr

Ti D˜NbZr

Ti D˜NbZr

Ti D˜NbZr

Ti D˜NbZr

Ti D˜NbZr

Ti D˜NbZr

2.19 13.40 4.97 18.48 8.70 24.71

6.01 13.81 4.70 10.10 1.88 4.76

7.08 2.22 5.13 1.62 2.09 0.97

− 1.18 0.24 − 0.13 0.15 − 0.72 − 0.13

− 0.84 1.92 0.33 0.65 − 0.42 − 0.18

24.49 5.93 19.73 3.50 13.12 1.74

6.37 1.76 4.77 0.88 3.14 0.38

− 0.29 0.19 − 0.33 − 0.19 − 0.32 − 0.24

0.79 0.85 0.48 0.42 0.13 0.13

28.29 5.16 18.05 3.85 11.20 2.44

reproduced agree well with the main inter-diffusion coefficients, within the error allowed. In contrast, the cross coefficients show certain fluctuation with a large scatter. Since the calculated and experimental cross coefficient values are lying in the same order of magnitude and are Ti following a same variation tendency with composition (viz. the D˜ZrNb Ti ˜ shows an increasing trend when close to the Zr boundary, and DNbZr are

deviation in diffusion couple C3. This is ascribed to the calculated diffusivity of Zr at 1273 K, which is higher than the experiment. Globally, measured and calculated profiles and diffusion paths show good agreement, which in turn indicates the accuracy of the atomic mobility parameters derived from this work.

negative), the accessed mobilities are feasible to represent the cross coefficients. Moreover, the calculated impurity coefficients compared with experimental results were also presented in Fig. 9, and showed a good agreement. The calculated inter-diffusion coefficients at 1273 K using the present atomic mobilities compared with experimental results from Ref. [64] are listed in Table 6.

6. Conclusion In this work, the diffusion behaviors in bcc Ti-Zr-Nb alloys were investigated at 1373 K and 1473 K, and the ternary inter-diffusion coefficients matrixes and impurity diffusion coefficients were extracted from composition profiles of diffusion couples using Whittle-Green and Hall methods. By analyzing the diffusion coefficients obtained experimentally, the diffusion rate of Zr in bcc Ti-Zr-Nb alloy was found to be much greater than that of Nb. The ternary diffusion coefficients are largely impact by the variation of concentration of Nb. The impurity diffusion coefficients of Zr in Ti-Nb binary alloys decrease as the concentration of Nb increases, while the impurity diffusivities of Nb in TiZr alloys exhibits an opposite trend. The diffusion coefficients obtained from experiments, coupled with the binary mobilities data from the literature, were assessed to develop an atomic mobility database for the bcc phase in Ti-Zr-Nb ternary system. Using the database, inter-diffusion coefficient at the intersection points of diffusion paths were calculated and compared with the data extracted directly from the composition profiles. They showed good agreement. Simulation of the diffusion couples annealed at 1273 K, 1373 K and 1473 K were also proceeded and the result agree with the EPMA profiles very well.

5.3.2. Simulation of composition profiles and diffusion paths The diffusion simulations were set up to model the semi-infinite ternary diffusion couples in experiment. It was conducted using different initial conditions and heat treatment processes in the experiment. The simulated diffusion paths are compared with the experimental curves in Fig. 10. For both temperatures, a good agreement between the experimental and calculated values is observed conforming the hypothesis used for the calculations. The predicted composition profiles of the couples M1 and N1 annealed at 1373 K and O1 and P1 annealed at 1473 K are compared with the experimental data in Fig. 11. The composition profiles and diffusion paths of diffusion couples annealed at 1273 K from Chen’s work [64] were simulated and compared with the experimental results, and are illustrated in Fig. 12. The calculated and experimental profiles exhibit a very close behavior except a slight

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Fig. 10. Simulated diffusion paths for diffusion couples compared with the experimental measurements: (a) 1373 K and (b) 1473 K.

Fig. 11. Simulated composition profiles for diffusion couples (a) M1 at 1373 K, (b) N1 at 1373 K, (c) O1 at 1473 K and (d) P1 at 1473 K, compared with the experimental data obtained in this work.

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Fig. 12. Simulated composition profiles for diffusion couples at 1473 K from Ref. [64] (a) C1, (b) C2, (c) C3, (d) C4 and (e) C5 and their diffusion paths (f), compared with experimental data in literature.

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Acknowledgments

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The authors would like to express gratitude to the financial support of the National Natural Science Foundation of China (Grant no. 51671218), the National Key Research and Development Program of China (Grant no. 2016YFB0701301) and the National Key Basic Research Program of China (973Program) (Grant no. 2014CB644000). Data availability statement The raw data required to reproduce these findings are available to download from http://dx.doi.org/10.17632/46wbk2xfcb.1. The processed data required to reproduce these findings are available to download from http://dx.doi.org/10.17632/46wbk2xfcb.1. Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at doi:10.1016/j.calphad.2018.12.003. References [1] A.K. Gogia, High-temperature titanium alloys, Def. Sci. J. 55 (2005) 149–173, https://doi.org/10.14429/dsj.55.1979. [2] M. Niinomi, Recent metallic materials for biomedical applications, Metall. Mater. Trans. A 33 (2002) 477, https://doi.org/10.1007/s11661-002-0109-2. [3] M. Geetha, A.K. Singh, R. Asokamani, A.K. Gogia, Ti based biomaterials, the ultimate choice for orthopaedic implants – a review, Prog. Mater. Sci. 54 (2009) 397–425, https://doi.org/10.1016/j.pmatsci.2008.06.004. [4] D. Wu, L.B. Liu, L.G. Zhang, L.J. Zeng, X. Shi, Investigation of the influence of Cr on the microstructure and properties of Ti6Al4VxCr alloys with a combinatorial approach, J. Mater. Eng. Perform. 26 (2017) 4364–4372, https://doi.org/10.1007/ s11665-017-2822-4. [5] D. Wu, L. Zhang, L. Liu, X. Shi, S. Huang, Y. Jiang, Investigation of the influence of Fe on the microstructure and properties of Ti5553 near-β titanium alloy with combinatorial approach, Int. J. Mater. Res. 108 (2017) 355–363, https://doi.org/ 10.3139/146.111487. [6] W. Zhang, W.K. Pang, V. Sencadas, Z. Guo, Understanding high-energy-density Sn4P3 anodes for potassium-ion batteries, Joule 2 (2018) 1534–1547, https://doi. org/10.1016/j.joule.2018.04.022. [7] D. Kuroda, M. Niinomi, M. Morinaga, Y. Kato, T. Yashiro, Design and mechanical properties of new β type titanium alloys for implant materials, Mater. Sci. Eng. A 243 (1998) 244–249, https://doi.org/10.1016/S0921-5093(97)00808-3. [8] X. Tang, T. Ahmed, H.J. Rack, Phase transformations in Ti-Nb-Ta and Ti-Nb-Ta-Zr alloys, J. Mater. Sci. 35 (2000) 1805–1811, https://doi.org/10.1023/ A:1004792922155. [9] X. Wang, L. Zhang, Z. Guo, Y. Jiang, X. Tao, L. Liu, Study of low-modulus biomedical β Ti–Nb–Zr alloys based on single-crystal elastic constants modeling, J. Mech. Behav. Biomed. Mater. 62 (2016) 310–318, https://doi.org/10.1016/j.jmbbm. 2016.04.040. [10] X.D. Zhang, L.B. Liu, J.-C. Zhao, J.L. Wang, F. Zheng, Z.P. Jin, High-efficiency combinatorial approach as an effective tool for accelerating metallic biomaterials research and discovery, Mater. Sci. Eng. C 39 (2014) 273–280, https://doi.org/10. 1016/j.msec.2014.02.039. [11] Y. Song, D.S. Xu, R. Yang, D. Li, W.T. Wu, Z.X. Guo, Theoretical study of the effects of alloying elements on the strength and modulus of β-type bio-titanium alloys, Mater. Sci. Eng. A 260 (1999) 269–274, https://doi.org/10.1016/S0921-5093(98) 00886-7. [12] M. Tane, S. Akita, T. Nakano, K. Hagihara, Y. Umakoshi, M. Niinomi, H. Nakajima, Peculiar elastic behavior of Ti–Nb–Ta–Zr single crystals, Acta Mater. 56 (2008) 2856–2863, https://doi.org/10.1016/j.actamat.2008.02.017. [13] Y.L. Hao, S.J. Li, S.Y. Sun, C.Y. Zheng, R. Yang, Elastic deformation behaviour of Ti–24Nb–4Zr–7.9Sn for biomedical applications, Acta Biomater. 3 (2007) 277–286, https://doi.org/10.1016/j.actbio.2006.11.002. [14] P.S. Nnamchi, C.S. Obayi, I. Todd, M.W. Rainforth, Mechanical and electrochemical characterisation of new Ti–Mo–Nb–Zr alloys for biomedical applications, J. Mech. Behav. Biomed. Mater. 60 (2016) 68–77, https://doi.org/10.1016/j.jmbbm.2015. 12.023. [15] Q. Chen, N. Ma, K. Wu, Y. Wang, Quantitative phase field modeling of diffusioncontrolled precipitate growth and dissolution in Ti–Al–V, Scr. Mater. 50 (2004) 471–476, https://doi.org/10.1016/j.scriptamat.2003.10.032. [16] T. Helander, J. Ågren, Diffusion in the B2-b.c.c. phase of the Al–Fe–Ni system—application of a phenomenological model, Acta Mater. 47 (1999) 3291–3300, https://doi.org/10.1016/S1359-6454(99)00174-3.

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