2022 Volume 63 Issue 12 Pages 1597-1606
Interdiffusion in Ti-rich β solid solutions in Ti–Al–Zr alloys was investigated at 1473 K. In the β Ti–Al–Zr alloys, the main interdiffusion coefficients (i.e., 
and 
, respectively) and the cross interdiffusion coefficients, (i.e., 
and 
, respectively) have positive values, as well as a slight concentration dependence. The values of 
are larger than those of 
. In Ti–Al–X (= V, Cr, Fe, Co, Zr) alloys, the values of the main coefficients are 
at 1473 K. Repulsive interactions occur between Al and X (= V, Cr, Fe, Co, Zr) atoms in the Ti–Al–X alloys, because the ratio values of the cross coefficients to the main ones are positive in sign. On the other hand, the interactions between Ti (solvent) and X (= V, Cr, Fe, Co, Zr) (or Al) atoms are attractive in the present alloys because the ratio of the converted interdiffusion coefficients is negative values.
This Paper was Originally Published in Japanese in J. JILM 71 (2021) 539–548.
Various types of titanium alloys have been used as the aerospace materials, automobile parts, and biomaterials etc., because of their high specific strength, heat resistance, and excellent corrosion resistance. In recent years, the titanium alloys have been also used for the manufacture of a turbine blades and various implants with Additive Manufacturing (AM), so these alloys have attracting much attention. Based on this background, until now, many titanium-based α, α + β, and β practical alloys have been investigated and developed. In particular, many studies have focused on the strength of materials and the metallography. In the future, the application of computer simulations is expected to improve the understanding of phenomena and materials development in the titanium alloys.1–4)
Most practical β-titanium alloys contain β-stabilizing elements such as vanadium, chromium, niobium, and iron etc., although the neutral elements such as zirconium and tin, and only practical α-stabilizing elements of aluminum up to approximately 5 mass%Al, can be present substitutional solute elements.5–7) The mechanical properties of Ti alloys are closely related to their materials microstructures, and the development of the optimal microstructures is important for improving the performance and properties of these alloys. Alloy microstructures are formed by working processes, thermomechanical treatments, heat treatments, etc. Especially, in the thermomechanical treatments or heat treatments, the microstructures are developed primarily by recovery, recrystallization, grain size growth, transformation, and precipitation, which are controlled by the diffusion phenomena in alloys. Moreover, in a heat-resistant of TiAl–Zr alloy, the oxidation resistance can be improved by adding the aluminum and a small amount of zirconium (approximately 1 at%).8) In the phenomena of high-temperature oxidation, the diffusion of constituent elements plays an important role.9) Therefore, the knowledge of the diffusion mechanism and the diffusion coefficients of aluminum and zirconium elements in binary and ternary Ti alloys is indispensable for understanding the some industrial processes and scientific diffusional phenomena during heat treatment and high-temperature oxidation of basic β-titanium alloys containing aluminum and zirconium.10,11)
Many experimental studies on diffusion in binary titanium-based alloys containing α- or β-stabilizing elements have been performed.10–15) Other than investigation of the diffusion of titanium-based ternary alloys in a Ti–V–Zr system by Brunsch et al.,16) a Ti–Al–Cr system,17) a Ti–Al–V system,18) a Ti–Al–Co system19) and a Ti–Al–Fe system,20) the number of studies on the ternary diffusion is limited. Therefore, the accumulation of the data from diffusion studies is desired for multi-component titanium alloys.
As a series of diffusion studies of multi-component titanium alloys, the purposes of the present work are as follows: (a) to determine, using the Matano–Kirkaldy method,21–23) the binary and ternary interdiffusion coefficients in the β solid solutions of the ternary Ti–Al–Zr alloys at 1473 K and the impurity diffusion coefficients of Zr in Ti–Al alloys at 1473 K, and (b) to estimate the thermodynamic interactions between solute and solute atoms (and solvent-solute atoms) in β Ti–Al–Zr solid solutions at 1473 K.
The experimental method described below involves a Ti–Al–Zr alloy system, and many experiments using similar methods have been carried out in the other Ti–Al–X (X = V, Cr, Fe, Co) ternary alloy systems.17–20)
Eight types of binary alloy ingots and four types of the ternary alloy ingots were prepared with pure metals of 99.9 mass% sponge Ti, 99.99 mass% Al, and 99.9 mass% Zr through an Ar arc melting. The terminal compositions of the binary and ternary alloys of the Ti–Al–Zr system are listed in Table 1. The compositions of the binary and ternary alloys were considered to be in the β solid solutions of the Ti–Al–Zr system at 1473 K, according to the Ti–Al, Ti–Zr and Ti–Al–Zr phase diagrams.24)
The alloy bars were cut from the mother ingots and sealed into quartz capsules with Ar gas at approximately 20 kPa. They were annealed at 1473 K for 2 h for homogenization, and then quenched into ice water. These homogenized alloy bars were cut into alloy plates of 10 × 10 × 3 mm3 in size, after which the surfaces of the alloy plates were polished by SiC papers and 1 µm alumina powder. Immediately after polishing, ternary and binary diffusion couples were assembled from these polished plates using stainless steel clamps in 10 combinations of alloys (Table 1). Then, these assembled diffusion couples for ternary and binary diffusion couples were sealed into a quartz capsule with Ar gas at approximately 20 kPa and a small amount of sponge titanium as an oxygen getter. Diffusion couples and stainless steel were separated by tungsten thin foil of 100 µm thickness to avoid reaction. They were annealed for diffusion treatment at 1473 K for 28.8 ks, and then quenched in ice water after the annealing. The annealed diffusion couples were mounted in synthetic resin and cut in half parallel to the diffusion direction to expose sections that had suffered no oxidation or evaporation of elements. Sections of the diffusion couples were metallographically polished. The characteristic X-ray intensities of Al and Zr parallel to the diffusion direction were measured on the polished surface of these diffusion couples by a JEOL JXA-8900 electron microanalyzer (EPMA), and they were converted to solute concentration profiles of Al and Zr by correcting for atomic number, absorption, and fluorescence effects using the bulk alloy compositions at the ends of the couples as standards.25,26)
2.2 Diffusion coefficientsThe binary interdiffusion coefficients $\tilde{D}_{(\textit{Ti}\text{–}\textit{Al})}$ in Ti–Al alloys and $\tilde{D}_{(\textit{Ti}\text{–}\textit{Zr})}$ in Ti–Zr alloys were determined from the concentration profiles in the binary diffusion couples J1 and J2 (Table 1) using Matano method21) with the following equations.
| \begin{equation} \tilde{D}_{(\textit{Ti–Al})} = -\frac{1}{2t}\int_{C_{\textit{Al}}^{(-\infty)}}^{C_{\textit{Al}}}xdC_{\textit{Al}} \end{equation} | (1) |
| \begin{equation} \tilde{D}_{(\textit{Ti–Zr})} = -\frac{1}{2t}\int_{C_{\textit{Zr}}^{(-\infty)}}^{C_{\textit{Zr}}}xdC_{\textit{Zr}} \end{equation} | (2) |
| \begin{equation} \int_{C_{1}^{(-\infty)}}^{C_{1}}xdC_{1} = -2t\left(\tilde{D}_{11}^{3}\frac{\partial C_{1}}{\partial x} + \tilde{D}_{12}^{3}\frac{\partial C_{2}}{\partial x}\right) \end{equation} | (3) |
| \begin{equation} \int_{C_{2}^{(-\infty)}}^{C_{1}}xdC_{2} = -2t\left(\tilde{D}_{21}^{3}\frac{\partial C_{1}}{\partial x} + \tilde{D}_{22}^{3}\frac{\partial C_{2}}{\partial x}\right) \end{equation} | (4) |
| \begin{equation} \int_{C_{i}^{(-\infty)}}^{C_{i}^{(+\infty)}}xdC_{i} = 0\ (i = 1,2) \end{equation} | (5) |
Diffusion couple (Ti–Al/Ti–Zr) (e.g., R3 in Table 1, as described in Fig. 1(b)) is designated as diffusion couple A, and the diffusion couple (Ti/Ti–Al–Zr) (Z4 in Table 1, Fig. 1(a)) is designated as diffusion couple B, the integral terms (areas) on the left-hand side of eq. (3) are designated as S1A and S1B, respectively, and in the common compositions (C1, C2) of the components 1 and 2 (2 = Al) in the same diffusion couples A and B, the partial differential terms (gradients) on the right side in eq. (3) are designated G1A, G2A, G1B, and G2B. The main and cross-interdiffusion coefficients of component 1 are given by eqs. (6) and (7), respectively.
| \begin{equation} \tilde{D}_{11}^{3} = \left(-\frac{1}{2t}\right)\frac{S_{1}^{A}G_{2}^{B} - S_{1}^{B}G_{2}^{A}}{G_{1}^{A}G_{2}^{B} - G_{1}^{B}G_{2}^{A}} \end{equation} | (6) |
| \begin{equation} \tilde{D}_{12}^{3} = \left(-\frac{1}{2t}\right)\frac{S_{1}^{B}G_{1}^{A} - S_{1}^{B}G_{1}^{B}}{G_{1}^{A}G_{2}^{B} - G_{1}^{B}G_{2}^{A}} \end{equation} | (7) |
Typical concentration profiles in diffusion couples (a) Z4 and (b) R3 at 1473 K for 28.8 ks.
Because the error in the diffusion (annealing) time t is extremely small, it can be neglected. The including error δD in the diffusion coefficients in eqs. (6) and (7) arises from the errors in the experimental areas and gradients of the concentration profiles at the common compositions (C1, C2). When the errors of the areas and gradients are designated as δS and δG, respectively, the relative errors of the direct and cross-interdiffusion coefficients of the component 1 are expressed by eqs. (8) and (9), respectively.
| \begin{equation} \frac{\delta\tilde{D}_{11}^{3}}{|\tilde{D}_{11}^{3}|} = \frac{\delta(S_{1}^{A}G_{2}^{B} - S_{1}^{B}G_{2}^{A})}{|S_{1}^{A}G_{2}^{B} - S_{1}^{B}G_{2}^{A}|} + \frac{\delta(G_{1}^{A}G_{2}^{B} - G_{1}^{B}G_{2}^{A})}{|G_{1}^{A}G_{2}^{B} - G_{1}^{B}G_{2}^{A}|} \end{equation} | (8) |
| \begin{equation} \frac{\delta \tilde{D}_{12}^{3}}{|\tilde{D}_{12}^{3}|} = \frac{\delta(S_{1}^{B}G_{1}^{A} - S_{1}^{A}G_{1}^{B})}{|S_{1}^{B}G_{1}^{A} - S_{1}^{A}G_{1}^{B}|} + \frac{\delta(G_{1}^{A}G_{2}^{B} - G_{1}^{B}G_{2}^{A})}{|G_{1}^{A}G_{2}^{B} - G_{1}^{B}G_{2}^{A}|} \end{equation} | (9) |
The relative errors of the first term on the right-hand side of eqs. (8) and (9) are different relative errors, whereas those of the second term have the same relative error. Regarding the relative errors of the cross-interdiffusion coefficients in eq. (9), the value of the numerator, i.e., ($S_{1}^{B}G_{1}^{A} - S_{1}^{A}G_{1}^{B}$), becomes smaller because the values of S1BG1A and S1AG1B have the same signs, and the absolute values of $|S_{1}^{B}G_{1}^{A} - S_{1}^{A}G_{1}^{B}|$ are equal to those of $|S_{1}^{B}G_{1}^{A}| - |S_{1}^{A}G_{1}^{B}|$ and $|S_{1}^{A}G_{1}^{B}| - |S_{1}^{B}G_{1}^{A}|$. When the difference between the absolute values of $|S_{1}^{B}G_{1}^{A}|$ and $|S_{1}^{A}G_{1}^{B}|$ is obtained with similar absolute values in eq. (9), the cancellation of significant digits arises from a decrease in significant digits. In particular, when $|S_{1}^{B}G_{1}^{A}|$ and $|S_{1}^{A}G_{1}^{B}|$ have the similar values in eq. (9), it is easy to be introduced the cancellation of significant digits in the cross coefficients. In the diffusion couples A and B with extremely closed the terminal composition, (i.e., at a composition of intersection points with small intersection angles in the diffusion paths), the evaluated cross coefficients at the composition of the intersection point correspond to this type. In this work, to prevent this type of error as much as possible, we have designed the following the combination of diffusion couples with the terminal compositions.19) That is, in the Ti–Al–Zr system phase diagram, their couples correspond to the couples forming diffusion paths with angles close to a right angle at the composition of the intersection point of the diffusion paths in the phase diagram (see in Fig. 1).
In addition, the impurity diffusion coefficients of Zr in Ti–Al alloys were determined from the concentration profiles of ternary diffusion couples (R1, R2, R3, and R4) using the Hall’s method.27,28) This Hall’s method enables the estimation of accurate diffusion coefficients near the concentration extremes, where the diffusion element is very dilute. The application of the Hall’s method to the analysis of these impurity diffusion coefficients has already been described elsewhere.29)
Figures 1(a) and (b) show the concentration profiles of Al and Zr elements, respectively, in the Z4 and R3 diffusion couples annealed at 1473 K for 28.8 ks with respect to distance x from the Matano interface. The penetration depth of Zr is larger than that of Al, as shown in Fig. 1(a). This indicates that Zr element diffuses faster than Al element in Ti–Al–Zr alloys. The diffusion paths for all ternary diffusion couples at 1473 K are shown in the phase diagram triangles of the Ti–Al–Zr system in Fig. 2. The terminal compositions of the diffusion couples Z1 to R4 are also shown by closed squares (■) in Fig. 2. Diffusion paths are considered to exist in the solid solution of the β (bcc) phase in the Ti–Al–Zr alloy at 1473 K. As shown in Fig. 2, the diffusion paths, consisting of the eight lines, are drawn, and the four interdiffusion coefficients are evaluated for the 16 compositions at the intersections of the diffusion paths using eqs. (3), (4), and (5). Their paths bend markedly and exhibit S-shaped curves. The S-shaped curves of the diffusion paths are principally attributed to the large difference between the diffusion rates of Al and Zr. The initial direction of the diffusion path tends to be along the line of constant composition of the more slowly diffusing component (i.e., Al), and the degree of curvature increases the difference in the rate of diffusion of the two components (i.e., Al and Zr).
Diffusion paths for the diffusion couples annealed at 1473 K for 28.8 ks.
Table 2 lists the four interdiffusion coefficients, namely, $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}$, $\tilde{D}_{\textit{AlZr}}^{\textit{Ti}}$, $\tilde{D}_{\textit{ZrZr}}^{\textit{Ti}}$, and $\tilde{D}_{\textit{ZrAl}}^{\textit{Ti}}$ ($\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}$, $\tilde{D}_{\textit{ZrZr}}^{\textit{Ti}}$; main coefficients; $\tilde{D}_{\textit{AlZr}}^{\textit{Ti}}$, $\tilde{D}_{\textit{ZrAl}}^{\textit{Ti}}$: cross coefficients), at the intersection of the diffusion paths with respect to the combination of the eight diffusion couples (Table 1). For example, in Table 2, the combination of the diffusion couple Z1 (Ti/Ti–3.5Al–13.5Zr) and diffusion couple R1 (Ti–3.5Al/Ti–3.5Zr) is designated as Z1-R1. The four ternary interdiffusion coefficients of the Z1-R1 couples in Table 2 are the coefficients at the intersection composition of diffusion paths (CAl = 0.3 at%, CZr = 2.8 at%; see in Fig. 2 and Table 2). In addition, Table 2 lists the binary interdiffusion coefficients, namely, $\tilde{D}_{(\textit{Ti}\text{–}\textit{Al})}$, and $\tilde{D}_{(\textit{Ti}\text{–}\textit{Zr})}$, which are obtained from the diffusion profiles in the binary diffusion couples of Ti/Ti–Al alloy (J1) and Ti/Ti–Zr alloy (J2) using the Matano method outlined in eqs. (1) and (2).
In Figs. 3(a)–(d), the 16 compositions in Table 2 are plotted by the symbol (
) on the intersections of the diffusion paths at 1473 K on the ternary Ti–Al–Zr diagram triangles, and the seven compositions of the binary Ti–Al and Ti–Zr alloys are also plotted on the Ti–Al and Ti–Zr sides of the same diagram triangles. The values of the ternary interdiffusion coefficients $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}$, $\tilde{D}_{\textit{AlZr}}^{\textit{Ti}}$, $\tilde{D}_{\textit{ZrZr}}^{\textit{Ti}}$, and $\tilde{D}_{\textit{ZrAl}}^{\textit{Ti}}$ are clearly shown in the vicinity of the symbol () in Figs. 3(a)–(d), and the values of the binary interdiffusion coefficients $\tilde{D}_{(\textit{Ti}\text{–}\textit{Al})}$ and $\tilde{D}_{(\textit{Ti}\text{–}\textit{Zr})}$ are also shown in the vicinity of the symbol () on the Ti–Al and Ti–Zr sides in Figs. 3(a) and (c). In the present alloys, all the main coefficients $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}$ and $\tilde{D}_{\textit{ZrZr}}^{\textit{Ti}}$ are positive values. Conversely, as shown in Figs. 3(b) and 3(d), the cross $\tilde{D}_{\textit{AlZr}}^{\textit{Ti}}$, and $\tilde{D}_{\textit{ZrAl}}^{\textit{Ti}}$ values have both positive and negative values; most number of $\tilde{D}_{\textit{AlZr}}^{\textit{Ti}}$ values and approximately half of the $\tilde{D}_{\textit{ZrAl}}^{\textit{Ti}}$ values were positive.
Main and cross interdiffusion coefficients (a) $\tilde{D}_{\text{AlAl}}^{\text{Ti}}$, (b) $\tilde{D}_{\text{AlZr}}^{\text{Ti}}$, (c) $\tilde{D}_{\text{ZrZr}}^{\text{Ti}}$ and (d) $\tilde{D}_{\text{ZrAl}}^{\text{Ti}}$ in ternary Ti–Al–Zr alloys at 1473 K.
As highlighted by Heyward and Goldstein, and by Roper and Whittle,30,31) the cross coefficients in the vicinity of the terminal compositions of diffusion couples inevitably include a large analysis error induced from small values of the $[\partial C_{k}/\partial x]$ and $[\int x dC_{i}]$ factors in eqs. (3) and (4) near the terminal compositions. Judging from the values of $\tilde{D}_{\textit{AlZr}}^{\textit{Ti}}$ and $\tilde{D}_{\textit{ZrAl}}^{\textit{Ti}}$ considering the negative values of cross coefficients near the terminal compositions, it can be recognized that both cross coefficients $\tilde{D}_{\textit{AlZr}}^{\textit{Ti}}$ and $\tilde{D}_{\textit{ZrAl}}^{\textit{Ti}}$ should be intrinsically positive.
3.3 Comparison of impurity diffusion coefficients in Ti–Al alloys and interdiffusion coefficients in the present alloysThe limiting value of $\tilde{D}_{ii}^{k}$ on the j-k side is equal to the impurity diffusion coefficient of component i (i.e., $D_{i(j - k)}^{*}$), in a binary j-k alloy, according to the relationship presented by Shuck and Tool.32)
| \begin{equation} \lim_{C_{i} \to 0}\tilde{D}_{ii}^{k} = D_{i(j - k)}^{*} \end{equation} | (10) |
In Fig. 3(c), the impurity diffusion coefficients $D_{\textit{Zr}(\textit{Ti}\text{–}\textit{Al})}^{*}$ of Zr in Ti–Al alloys at 1473 K are plotted on the Ti–Al side. The dependence on the concentration of Al for $D_{\textit{Zr}(\textit{Ti}\text{–}\textit{Al})}^{*}$ is also shown in Fig. 4 with the reported impurity diffusion coefficients of V,18) Cr,17) Fe,20) and Co19) in Ti–Al alloys at 1473 K ($D_{x(\textit{Ti}\text{–}\textit{Al})}^{*}$ (X = Co, Fe, Zr, Cr, V)). The dependence on the concentration of Al for $D_{V(\textit{Ti}\text{–}\textit{Al})}^{*}$ is vague because there are only two data points. However, those of the $D_{\textit{Zr}(\textit{Ti}\text{–}\textit{Al})}^{*}$ and $D_{\textit{Cr}(\textit{Ti}\text{–}\textit{Al})}^{*}$ are small in the concentration range of 3 at% Al to approximately 12 at% Al. The values of $D_{\textit{Co}(\textit{Ti}\text{–}\textit{Al})}^{*}$ and $D_{\textit{Fe}(\textit{Ti}\text{–}\textit{Al})}^{*}$ are more or less constant up to approximately 7 at% Al, and their values decrease slightly as the concentration approaches 14 at% Al.
Relation between impurity diffusion coefficients $D_{\text{X(Ti–Al)}}^{\text{*}}$ (X = Co, Fe, Zr, Cr, V) in Ti–Al alloys and Al concentration at 1473 K.
Figure 5 shows the relationship between the impurity diffusion coefficient $D_{x(\textit{Ti}\text{–}\textit{Al})}^{*}$ of X (= V, Cr, Fe, Co, or Zr) in the Ti–Al alloy and the metallic radius of each solute atom rx. The impurity diffusion coefficient of each solute atom is shown by the average value of the data obtained in the present work and those reported in previous work. For comparison, Fig. 6 shows the relationship between the impurity diffusion coefficient $D_{i(\textit{Ti})}^{*}$ at 1473 K for various elements i in Ti(β) and the each metallic radius.14) Figure 6, also shows the impurity diffusion coefficients $D_{\textit{Al}(\textit{Ti})}^{*}$ of Al in β-Ti at 1473 K, and the value of $D_{\textit{Al}(\textit{Ti})}^{*}$ (= 8.4 × 10−13 m2/s; symbol in Fig. 6) is calculated at 1473 K using eq. (6) in literature.17) In addition, the impurity diffusion coefficient $D_{\textit{Zr}(\textit{Ti})}^{*}$ (= 2.0 × 10−12; symbol 
in Fig. 6) of Zr in β-Ti at the same temperature is shown, and the value of $D_{\textit{Zr}(\textit{Ti})}^{*}$ can be obtained by extrapolation to 0 at% Al using the data for $D_{\textit{Zr}(\textit{Ti}\text{–}\textit{Al})}^{*}$ in Fig. 4 of the present work. The value of $D_{{\textit{Zr}(\textit{Ti})}}^{*}$ in the present work is similar to that given in the literature.14)
Relation between impurity diffusion coefficient ($D_{\text{x}}{}^{*}{}_{(\text{Ti–Al})}$) of X (= V, Cr, Fe, Co or Zr) in Ti–Al alloy and metallic radius of each solute atom at 1473 K.
Relation between impurity diffusion coefficient Di(x)* of various element in Ti (β) and each metallic radius.
Figures 7(a) and (b) show a comparison of the average values of (a) the main interdiffusion coefficients $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}$ and (b) the main diffusion interdiffusion coefficients $\tilde{D}_{\textit{XX}}^{\textit{Ti}}$ in Ti–Al–X (X = V,18) Cr,17) Fe,20) Co,19) and Zr) alloys at 1473 K. The value of $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}$ in each system alloy increased slightly, from 0.68 × 10−12 m2/s (Ti–Al–V system) to 1.3 × 10−12 m2/s (Ti–Al–Co system) in the order V, Cr, Zr, Fe, and Co. Conversely, the trend for $\tilde{D}_{\textit{XX}}^{\textit{Ti}}$ in Fig. 7(b) is similar to that of the impurity diffusion coefficients $D_{x(\textit{Ti}\text{–}\textit{Al})}^{*}$ in Fig. 4, and the values of $\tilde{D}_{\textit{XX}}^{\textit{Ti}}$ indicate the difference in the rate of diffusion for each alloy system. That is, the order is $\tilde{D}_{\textit{VV}}^{\textit{Ti}} < \tilde{D}_{\textit{CrCr}}^{\textit{Ti}} < \tilde{D}_{\textit{ZrZr}}^{\textit{Ti}} < \tilde{D}_{\textit{FeFe}}^{\textit{Ti}} < \tilde{D}_{\textit{CoCo}}^{\textit{Ti}}$, and their values increase by approximately 50 times, from 0.41 × 10−12 m2/s (Ti–Al–V system) to 19 × 10−12 m2/s (Ti–Al–Co system).
Comparison of (a) main interdiffusion coefficients $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}$ and (b) main interdiffusion coefficients $\tilde{D}_{\textit{XX}}^{\textit{Ti}}$ in Ti–Al–X (X = V, Cr, Fe, Co, Zr) alloys at 1473 K.
Kirkaldy et al.33) presented the relationship between the ratio of $\tilde{D}_{ij}^{3}/\tilde{D}_{ii}^{3}$ and Wagner’s interaction parameter, εi(j), in very dilute solutions of the ternary alloys, as follows:
| \begin{equation} \tilde{D}_{ij}^{3}/\tilde{D}_{ii}^{3} = [1 + \varepsilon_{i}{}^{(j)}]N_{i} \end{equation} | (11) |
Relation between $\tilde{D}_{\textit{ZrAl}}^{\textit{Ti}}/\tilde{D}_{\textit{ZrZr}}^{\textit{Ti}}$ and Zr concentration at 1473 K.
As described in the introduction of this paper, the microstructures are formed via various processes, including recovery, recrystallization, grain size growth, transformation, precipitation, and oxidation, which are basic phenomena associated with the diffusion in alloys. Consequently, knowledge of the diffusion in alloys is extremely important with respect to the microstructures of metallic-materials. In the present work, a series of investigations of the ternary titanium-base alloys, as presented in Table 1 was conducted. First, the preparation of the diffusion couples was designed to determine the interdiffusion coefficients in the β-phase region of the binary Ti–Al and Ti–Zr alloys and the ternary Ti–Al–Zr alloys at 1473 K. As an example, the concentration profiles of the diffusion couples Z4 and R3 are shown in Figs. 1(a) and (b), and the diffusion paths are obtained from the all of the concentration profiles of the Z1-R4 diffusion couples, as shown in Fig. 2. The four interdiffusion coefficients in eqs. (3), (4), and (5) are evaluated at the common composition (C1 and C2) of the diffusion paths in two independent diffusion couples in the β-phase region of the binary Ti–Al and Ti–Zr alloys and the ternary Ti–Al–Zr alloys (Table 2 and Figs. 3(a)–(d)). As shown in Fig. 3(a), the values of $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}$ are (0.6∼1.3) × 10−12 m2/s and have a slight concentration dependence. The values of $\tilde{D}_{\textit{ZrZr}}^{\textit{Ti}}$ in Fig. 3(c) are (1.5∼2.1) × 10−12 m2/s and have a slightly lower concentration dependence than that of $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}$. The average values of their coefficients are helpful for comparison between the diffusion rates of Al and Zr, although there are some concentration dependences of the binary and ternary diffusion coefficients, as will be discussed. As listed in Table 2, the average values of $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}$, $\tilde{D}_{\textit{AlZr}}^{\textit{Ti}}$, $\tilde{D}_{\textit{ZrZr}}^{\textit{Ti}}$, and $\tilde{D}_{\textit{ZrAl}}^{\textit{Ti}}$ for the 16 compositions in Fig. 3 are respectively 0.97 × 10−12 m2/s, 0.09 × 10−12 m2/s, 1.9 × 10−12 m2/s and 0.15 × 10−12 m2/s respectively, and the average values of the 16 compositions is Ti–4.3 at%Al–4.2 at%Zr.
However, the negative $\tilde{D}_{\textit{AlZr}}^{\textit{Ti}}$ and $\tilde{D}_{\textit{ZrAl}}^{\textit{Ti}}$ values are eliminated during calculation of their average values because their cross coefficients include the relatively many errors (Section 2.2). The average value of the cross coefficients $\tilde{D}_{\textit{ZrAl}}^{\textit{Ti}}$ is slightly higher than that of $\tilde{D}_{\textit{AlZr}}^{\textit{Ti}}$, and the average value of the main coefficients $\tilde{D}_{\textit{ZrZr}}^{\textit{Ti}}$ is apparently higher than that of $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}(\tilde{D}_{\textit{ZrZr}}^{\textit{Ti}} > \tilde{D}_{\textit{AlAl}}^{\textit{Ti}})$. As shown in Figs. 1(a) and Fig. 2, this is consistent with the facts that the penetration depth of Zr is relatively larger than that of Al (Fig. 1(a)), and the initial direction of the diffusion path tends to be along the line of constant composition of the more slowly diffusing component, i.e., of Al (Fig. 2).
As described later, in Ti–Al–Zr alloys, the attractive interactions exist between Ti (solvent atom) and Zr (solute atom), and the solution enthalpies of Al and Zr in Ti(L) are −119 kJ/mol for Al and −1 kJ/mol for Zr,34) the bond energy of Ti–Zr is smaller than that of Ti–Al. Therefore, the probability of a Zr atom being captured by a Ti atom is smaller than that of an Al atom being captured by a Ti atom, and the Zr atom in comparison with the Al atom can more easily diffuse in the Ti–Al–Zr alloy.
Shuck and Tool32) described the relationship between the ternary main interdiffusion coefficient $\tilde{D}_{ii}^{k}$ and the binary interdiffusion coefficient as follows:
| \begin{equation} \lim_{C_{j} \to 0}\tilde{D}_{ii}^{k} = \tilde{D}_{(i - k)} \end{equation} | (12) |
To compare with the binary diffusion coefficients and the ternary main coefficients in the dilute region of Al or Zr elements in the Ti–Al–Zr alloy, some data regarding the binary $\tilde{D}_{(\textit{Ti}\text{–}\textit{Al})}$ in the Ti–Al alloy and $\tilde{D}_{(\textit{Ti}\text{–}\textit{Zr})}$ in the Ti–Zr alloy are also plotted on the Ti–Al side and the Ti–Zr side on the ternary Ti–Al–Zr diagram triangles in Figs. 3(a) and (c), respectively. In fact, in the Ti–Al–Zr diagram triangle in Fig. 3(a), the ternary values $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}$ in the dilute range of Zr in the Ti–Al–Zr alloy range from approximately 0.6 × 10−12 m2/s to approximately 1.3 × 10−12 m2/s (near the Ti–Al side), so the value of $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}$ shows quite small of concentration dependence. The binary values of $\tilde{D}_{(\textit{Ti}\text{–}\textit{Al})}$ are range from 0.97 × 10−12 m2/s to 1.4 × 10−12 m2/s, and their values increase slightly, so the both diffusion coefficients of $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}$ and $\tilde{D}_{(\textit{Ti}\text{–}\textit{Al})}$ are approximately equal.
Conversely, as shown in Fig. 3(c), the $\tilde{D}_{\textit{ZrZr}}^{\textit{Ti}}$ values in the dilute range of Al element are approximately 1.9 × 10−12 m2/s, and their values are similar to those of the binary $\tilde{D}_{(\textit{Ti}\text{–}\textit{Zr})}$, i.e., 1.9 × 10−12 m2/s and 2.0 × 10−12 m2/s. Thus, the ternary main diffusion coefficients are closely linked to the binary diffusion coefficients, and their concentration dependence is similar to that of binary interdiffusion in the dilute region of the element.
Shuck and Tool32) also presented the relationship between the cross coefficient $\tilde{D}_{ij}^{k}$ and the concentration of element I (i.e., Ci) as follows:
| \begin{equation} \lim\limits_{C_{i} \to 0}\tilde{D}_{ij}^{k} = 0 \end{equation} | (13) |
As shown in Fig. 4, referring to the magnitude of the impurity diffusion coefficients $D_{X(\textit{Ti}\text{–}\textit{Al})}^{*}$ of X (= V, Cr, Zr, Fe, Co) in β Ti–Al alloys at 1473 K, the order of their coefficients is $D_{\textit{V}(\textit{Ti}\text{–}\textit{Al})}^{*} < D_{\textit{Cr}(\textit{Ti}\text{–}\textit{Al})}^{*} < D_{\textit{Zr}(\textit{Ti}\text{–}\textit{Al})}^{*} < D_{\textit{Fe}(\textit{Ti}\text{–}\textit{Al})}^{*} < D_{\textit{Co}(\textit{Ti}\text{–}\textit{Al})}^{*}$, regardless of the Al concentration in the β Ti–Al alloys. When comparing the diffusion coefficients of the Co atom with that of V atom, their coefficients of the Co atom are approximately 20 times than those of the V atom. On the other hand, the values for the metallic radius of each atom rx (X = V, Cr, Fe, Co or Zr)10,35–37) are rV = 0.135 nm, rCr = 0.128 nm, rFe = 0.124 nm, rCo = 0.125 nm, and rZr = 0.160 nm, so the order of the rx is rV > rCr > rFe > rCo, with rZr excluded.
Figure 5 shows the relationship between the impurity diffusion coefficients $(D_{\textit{X}(\textit{Ti}\text{–}\textit{Al})}^{*})$ of X (= V, Cr, Zr, Fe, Co) in β Ti–Al alloys and the metallic radius of each atom rx at 1473 K. It is apparent that the average values of the impurity diffusion coefficients decrease with the metallic radius, except for the impurity diffusion coefficient of Zr. As shown in Fig. 5, the $D_{\textit{X}(\textit{Ti}\text{–}\textit{Al})}^{*}$ values are negatively correlated with the rx values, except for $D_{\textit{Zr}(\textit{Ti}\text{–}\textit{Al})}^{*}$. For comparison with Fig. 5, Fig. 6 shows the relationship between the impurity diffusion coefficients of various elements i in β-Ti (bcc) at 1473 K and each metallic radius rx. As shown in Fig. 6, the impurity diffusion coefficient of Zr in β-Ti is larger than that of Al in β-Ti at 1473 K (Fig. 6 (: Al, : Zr)). Further, as shown in Fig. 6, the $D_{\textit{X}(\textit{Ti})}^{*}$ values a negative correlation and steep slope with respect to rx values less than 0.14 nm, and the $D_{\textit{X}(\textit{Ti})}^{*}$ values have a positive slope with respect to rx values greater than 0.14 nm. Conversely, in α-Ti (hcp), which exists in a lower temperature range than that of β-Ti, a similar correlation is observed for the impurity diffusion coefficients of Co, Fe, and Cr in the α-Ti at 1100 K and the metallic radius.10,11) In descending order of the metallic radii rx of solute X in the α-Ti, (i.e., in the order of rV > rCr > rFe > rCo), the impurity diffusion coefficients of V, Cr, Fe, or Co increase in descending order of rx (X = V, Cr, Fe, Co).10,11) The impurity diffusion of Co, Fe, and Ni in α-Ti occurs via an interstitial diffusion mechanism, which is attributed to the atomic size effect. The fast diffusion of Co, Fe, or Ni with comparatively small radii has been discussed in detail.10,11)
As shown in Fig. 5, regarding the impurity diffusion of Co, Fe, Cr, or V in β Ti–Al alloys, the $D_{\textit{X}(\textit{Ti}\text{–}\textit{Al})}^{*}$ values have a negative correlation and a steep slope with respect to rx (X = Co, Fe, Cr, V) in the β Ti–Al alloys. That is, the atomic size effect (i.e., negative correlation) exists in the impurity diffusion of X in the Ti–Al alloys. Consequently, it is the possible that an interstitial diffusion mechanism similar to that of α-Ti exists for diffusion with smaller metallic radii rx in Ti–Al alloys. In addition, as mentioned above in Fig. 4, because the impurity diffusion coefficients $D_{\textit{X}(\textit{Ti}\text{–}\textit{Al})}^{*}$ have little dependence on Al concentration, it can be understood from the analogy that the element Al in the β Ti–Al alloys elicits no major change in the diffusion mechanism of X in the Ti–Al alloys. However, as shown in Fig. 6, Zr has a larger metallic radius than Ti, and its atomic radius is greater than 0.14 nm. The $D_{i(\textit{Ti})}^{*}$ and $D_{\textit{X}(\textit{Ti}\text{–}\textit{Al})}^{*}$ values are positive correlated and have a steep slope with respect to the metallic radius rX (Figs. 5 and 6). In the periodic table, Ti and Zr are homogeneous elements, and the chemical properties of the two elements are very similar. It is inferred that the diffusion of Zr atoms occurs via a diffusion mechanism similar to that of Ti in the β Ti. That is, the diffusion mechanism of Zr atoms in the β-Ti differs from that of V, Cr, Fe, or Cr atom in the β phase; namely, it is said that the diffusion of Zr atoms in the β-Ti occurs via the normal vacancy diffusion mechanism.
In addition, as indicated by eq. (10), the ternary diffusion coefficients $\tilde{D}_{ii}^{k}$ is closely linked to the impurity diffusion coefficients of i in the j-k alloy $D_{i(\textit{Ti}\text{–}\textit{Al})}^{*}$ (e.g., see Fig. 3(c)), and it is expected that these diffusion coefficients have an atomic size effect similar to that of $D_{i(j - k)}^{*}$.
As shown in Fig. 7(a), the $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}$ values in the Ti–Al–X (X = V, Cr, Zr, Fe, Co) alloys at 1473 K increase slightly in the order of X = V, Cr, Zr, Fe, Co, because the diffusion of Al in the alloys, as well as that of Ti, occurs primarily by the vacancy diffusion mechanism. Consequently, regarding $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}}$, it is inferred that a large difference in diffusion rate does not occur in each Ti–Al–X alloy. However, the $\tilde{D}_{\textit{XX}}^{\textit{Ti}}$ (X = V, Cr, Zr, Fe, Co) values in Fig. 7(b) have different diffusion rates for each Ti–Al–X alloy system; the order of the magnitude is $\tilde{D}_{\textit{VV}}^{\textit{Ti}} < \tilde{D}_{\textit{CrCr}}^{\textit{Ti}} < \tilde{D}_{\textit{ZrZr}}^{\textit{Ti}} < \tilde{D}_{\textit{FeFe}}^{\textit{Ti}} < \tilde{D}_{\textit{CoCo}}^{\textit{Ti}}$ at 1473 K. In this manner, the $\tilde{D}_{\textit{XX}}^{\textit{Ti}}$ values in the ternary Ti–Al–X alloys have a tendency similar to that of the impurity diffusion coefficients of X in the β Ti and β Ti–Al alloys. This reveals that the diffusion mechanism of the interdiffusion in the ternary Ti–Al–X alloys is similar to that of the impurity diffusion mechanism in the β Ti and β Ti–Al alloys.
4.3 Thermodynamic interactions among the constituent atomsAs mentioned in the Section 3.4, Wagner’s interaction parameters $\varepsilon_{i}^{(j)}$ are necessary to discuss the thermodynamic interactions among the solute components in the alloys. In general, to experimentally obtain their interaction parameters, it is necessary to perform activity measurement experiments. In the this study, as indicated by eq. (11), the Wagner’s interaction parameter $\varepsilon_{\textit{Zr}}^{(\textit{Al})}$ can be obtained from the ratio of cross coefficient to the main coefficient, i.e., $\tilde{D}_{\textit{ZrAl}}^{\textit{Ti}}/\tilde{D}_{\textit{ZrZr}}^{\textit{Ti}}$ (see in Fig. 8).
Conversely, Tanaka et al.38,39) derived a method for evaluation of the interaction parameter, $\varepsilon_{2}^{(1),L}$ in a dilute liquid phase for a ternary 3-2-1 alloy (3 = solvent; and 2, 1 = solutes) on the based on the free volume theory of Shimoji and Niwa,40) and they proposed eq. (14) to evaluate of $\varepsilon_{j}^{(i),L}$:
| \begin{align} \varepsilon_{j}^{(i),L} &= \{(\partial^{j}G^{Ex}/\partial N_{i}\partial N_{j})_{N_{j \to 0,}N_{i \to 0}}\}kT \\ &= (\eta_{j}^{(i)} - T\sigma_{j}^{(i)})/kT \end{align} | (14) |
Morita and Tanaka43) also derived a simple relationship between the $\varepsilon_{i}^{(j),L}$ in the dilute liquid phase and the $\varepsilon_{i}^{(j),s}$ in the dilute solid phase in a ternary k-i-j alloy (k: solvent, i, j: solutes), in the case where the elastic energies between the constituent i and j solute atoms and that between solvent k and i atoms resulting from the difference in their atomic radii, elastic modulus of constituent atoms and so on are similar to each other (eq. (15)).
| \begin{equation} \varepsilon_{i}^{(j),L} \cong \varepsilon_{i}^{(j),s} \end{equation} | (15) |
Morita and Tanaka43,44) have defined the equilibrium distribution coefficient ($k_{0}^{i,3}$) of solute i in an Fe-i-j ternary system to that ($k_{0}^{i,2}$) in Fe-i binary one, i.e., the “Distribution Interaction Coefficient” ($k_{i}^{(j)}$), in the form of have presented the following equation:
| \begin{equation} \ln k_{i}^{(j)} \equiv \ln \frac{k_{0}^{i,3}}{k_{0}^{i,2}} = (1 - k_{0}^{j,3})\varepsilon_{i}^{(j),L}N_{j}^{L} \end{equation} | (16) |
Therefore, assuming that the relationship in eq. (15) holds in dilute liquid and solid phases in the ternary Ti–Al–Zr alloy, and applying eqs. (14) and (15), the interaction parameters between the constitutional atoms in the β solid solution in Ti–Al–Zr alloys at 1473 K are evaluated to be $\varepsilon_{\textit{Zr}}^{(\textit{Al})} = + 0.07$ and $\varepsilon_{\textit{Al}}^{(\textit{Zr})} = + 0.07$. These evaluated values are positive and similar to the experimental value $\varepsilon_{\textit{Zr}}^{(\textit{Al})} = + 0.1$ determined in this study, and the positive values support the existence of the repulsive interactions between the Al and Zr atoms in the present alloys.
Furthermore, if we assume that eq. (16) holds in the ternary Ti–Al–Zr alloy, because the equilibrium distribution coefficient of solute j (Zr) in the ternary Ti–Al–Zr alloy, $k_{0}^{j,3}$, is less than 1, the value of $(1 - k_{0}^{j,3})$ in eq. (16) has a positive value, so the positive/negative sign in eq. (16) depends only on the sign of $\varepsilon_{i}^{(j),L}$,44) i.e., only on the interaction between i (Al) and j (Zr).
In this study, because Wagner’s interaction parameters $\varepsilon_{\textit{Zr}}^{(\textit{Al})}$ and $\varepsilon_{\textit{Al}}^{(\textit{Zr})}$ are positive values in the ternary Ti–Al–Zr alloys ($\varepsilon_{\textit{Zr}}^{(\textit{Al})} = \varepsilon_{\textit{Al}}^{(\textit{Zr})}$ in dilute solution45)), the Distribution interaction coefficient ($k_{i}^{(j)}$) in eq. (16) increases with the addition of Zr, which has a repulsive interaction with the Al atoms in the ternary alloy. That is, in the present work, the equilibrium distribution coefficients $k_{0}^{i,3}$ in the ternary Ti–Al–Zr alloys increase with the addition of Zr into the Ti–Al alloys. In general, the hot-tearing susceptibility of aluminum and magnesium alloys during casting increases as the equilibrium distribution coefficient of the alloys decreases; that is, the alloys easily crack during casting.46) Conversely, as mentioned previously, because the distribution interaction coefficient $k_{i}^{(j)}$ increases with the addition of Zr into Ti–Al alloys, hot tearing during casting can develop in the ternary Ti–Al–Zr alloys. In fact, it has been reported that the γ-based TiAl alloys with added Zr exhibit superior strength-ductility balance.47) In Ti-based practical alloys, the fact that Zr is an important added element in the Ti-base alloys,48) can be confirmed from the viewpoint of the increase in the distribution interaction coefficient $k_{i}^{(j)}$ and the equilibrium distribution coefficient $k_{0}^{i,3}$ (i.e., from a thermodynamic viewpoints).
The interdiffusion coefficients in a ternary system can be described in a different form by the choice of the dependent component “solvent”.16,49) For example, the dependent component can be changed from superscript 3 (Ti) to 2 (Al) or 1 (Zr), as follows:
| \begin{equation} \tilde{D}_{11}^{2} = \tilde{D}_{11}^{3} - (V_{1}/V_{2})\tilde{D}_{12}^{3}, \end{equation} | (17) |
| \begin{equation} \tilde{D}_{13}^{2} = -(V_{3}/V_{2})\tilde{D}_{12}^{3}, \end{equation} | (18) |
| \begin{equation} \tilde{D}_{22}^{1} = \tilde{D}_{22}^{3} - (V_{2}/V_{1})\tilde{D}_{21}^{3}, \end{equation} | (19) |
| \begin{equation} \tilde{D}_{23}^{2} = -(V_{3}/V_{1})\tilde{D}_{21}^{3}. \end{equation} | (20) |
| \begin{equation} \tilde{D}_{13}^{2}/\tilde{D}_{11}^{2} = [1 + \varepsilon_{1}^{(3)}]N_{1}, \end{equation} | (21) |
| \begin{equation} \tilde{D}_{23}^{1}/\tilde{D}_{22}^{1} = [1 + \varepsilon_{2}^{(3)}]N_{1}. \end{equation} | (22) |
Figure 9 shows the relationship between the experimental values of $\tilde{D}_{\textit{ZrTi}}^{\textit{Al}}/\tilde{D}_{\textit{ZrZr}}^{\textit{Al}}$ and the Zr concentrations at 1473 K. The values of the partial molar volumes of VZr, VAl, and VTi are obtained from the literature;42) their values are VZr = 14.0 × 10−6 m3/mol, VAl = 99.9 × 10−6 m3/mol, and VTi = 10.58 × 10−6 m3/mol. The values of $\tilde{D}_{\textit{ZrTi}}^{\textit{Al}}/\tilde{D}_{\textit{ZrZr}}^{\textit{Al}}$ in Fig. 9 decrease with increasing Zr concentration in the Ti–Al–Zr alloys, although they scatter around the broken line of $\varepsilon_{\textit{Zr}}^{(\textit{Ti})} = - 2$ at 1473 K. This plot using eq. (21) suggests that $\varepsilon_{\textit{Zr}}^{(\textit{Ti})}$ has an approximate value of −2 in Ti–Al–Zr alloys. In addition, the value of $\varepsilon_{\textit{Al}}^{(\textit{Ti})}$ is estimated to be approximately −1.4 at 1473 K using a method similar to that shown in Fig. 9. Conversely, because the value of the cross coefficient $\tilde{D}_{\textit{AlZr}}^{\textit{Ti}}$ contains a relatively large amount of error, using the average values of the interdiffusion coefficients in the average compositions (= Ti–4.3 at%Al–4.2 at%Zr) as shown in Table 2, i.e., $\tilde{D}_{\textit{AlAl}}^{\textit{Ti}} = 0.97 \times 10^{ - 12}$ m2/s, $\tilde{D}_{\textit{AlZr}}^{\textit{Ti}} = 0.09 \times 10^{ - 12}$ m2/s, $\tilde{D}_{\textit{ZrZr}}^{\textit{Ti}} = 1.9 \times 10^{ - 12}$ m2/s, and $\tilde{D}_{\textit{ZrAl}}^{\textit{Ti}} = 0.15 \times 10^{ - 12}$ m2/s, we obtain the interaction parameter $\varepsilon_{\textit{Al}}^{(\textit{Ti})} = - 1.7$ at 1473 K using eqs. (19), (20), and (22). This value is approximately similar to $\varepsilon_{\textit{Al}}^{(\textit{Ti})} \cong - 1.4$, as mentioned in Fig. 9. The negative values of $\varepsilon_{\textit{Zr}}^{(\textit{Ti})}$ and $\varepsilon_{\textit{Al}}^{(\textit{Ti})}$ suggest that the interactions between the Ti (solvent atom) and Zr (or Al) atoms are attractive in the present alloys.
Relation between $\tilde{D}_{\textit{ZrTi}}^{\textit{Al}}/\tilde{D}_{\textit{ZrZr}}^{\textit{Al}}$ and Zr concentration at 1473 K.
Wagner’s interaction parameters $\varepsilon_{1}^{(2)}$, $\varepsilon_{2}^{(1)}$, $\varepsilon_{1}^{(3)}$, and $\varepsilon_{2}^{(3)}$ in the β phase of Ti–Al–X (= V, Cr, Fe, Co, Zr) alloys at 1473 K are summarized in Table 3. The solute atoms of X are designated as solute 1, the solute atom Al is designated as 2, and the solvent atom Ti is designated as 3. The interaction parameters $\varepsilon_{1}^{(2)}$ and $\varepsilon_{2}^{(1)}$ are positive for the Ti–Al–X alloys. The interactions between Al and X (= V, Cr, Fe, Co, Zr) atoms in the Ti–Al–X alloys are repulsive. The interaction between Ti (solvent) and X (= V, Cr, Fe, Co, Zr) is attractive in the present alloys, because the interaction parameters $\varepsilon_{1}^{(3)}$ and $\varepsilon_{2}^{(3)}$ are negative values.
Ternary interdiffusion experiments in the β-solid solutions of Ti–Al–Zr alloys were performed at 1473 K. The results are summarized as follows:
This research was funded by JSPS KAKENHI, (grant numbers 20K05141) and the Light Metal Educational Foundation, Inc., Japan.
