Kinetics of Diffusion Induced Recrystallization in the Cu(Al) System

Abstract

Diffusion induced recrystallization (DIR) occurs in Cu of Cu-clad Al (CA) wire during heating at temperatures around 200°C. The kinetics of DIR in the CA wire was experimentally observed in the temperature range of 210–270°C. In this temperature range, CA wires were isothermally annealed for various periods of 3 h to 960 h. Owing to annealing, layers of the α₂, γ₁, δ, η₂ and θ phases form at the original Cu/Al interface, and the DIR region alloyed with Al is produced in the Cu phase from the Cu/α₂ interface. The thickness l of the DIR region monotonically increases with increasing annealing time t. The DIR region grows faster at higher annealing temperatures. The new extended-model proposed in a previous study was used to analyze theoretically the experimental result. According to the analysis, l is proportional to t in the early stages but to a power function of t in the late stages. This means that the DIR growth is governed by the interface reaction at the moving boundary in the early stages but by the boundary diffusion across the DIR region in the late stages. Although the shortest annealing time of 3 h at 240–270°C is located in the early stages, most of the annealing times belong to the intermediate stages. Consequently, under the present experimental conditions, both the interface reaction and the boundary diffusion mainly contribute to the rate controlling process of DIR. On the other hand, the interface reaction is the rate controlling process for the shortest annealing time.

Fig. 3 The mean thickness l of the DIR region versus the annealing time t shown as open rhombuses, squares and circles for T = 483, 513 and 543 K (210, 240 and 270°C), respectively.

1. Introduction

In the automobile industry, fuel efficiency is one of the most important issues for each country. A large number of electric wires are used in the automobile. Thus, to improve the fuel efficiency, the total weight of the electric wires should be reduced. For this purpose, Cu wires are gradually being replaced with Al wires.^1–3) On the other hand, combination of Cu and Al is another method to reduce the total wire weight. Hence, in the automobile industry, the Cu-clad Al (CA) wire is widely utilized for electronic parts as a conductive material exhibiting both the lightness of Al and the good connectivity of Cu. Here, the CA wire is an Al wire coated with Cu. The CA wire is usually manufactured by solid-state bonding due to wire drawing process.

The CA wire used in the automobile is heated at temperatures of about 100–200°C (373–473 K) near the motor. Owing to heating at such temperatures, a layer of intermetallic compounds (IMC’s) will form at the Cu/Al interface and grow gradually. Since the IMC layer is brittle and electrically resistant, the growth of the IMC layer deteriorates mechanical and electrical properties of the CA wire. Thus, information on the growth behavior of the IMC layer during heating is essentially important to assure the reliability of the product.

The growth of the IMC layer was experimentally observed by Gueydan et al.⁴⁾ at temperatures of 300–400°C (573–673 K) for times of 2–48 h (7.2–172.8 ks). Their observation indicates that the IMC layer is composed of the γ₁ (Cu₉Al₄), η₂ (CuAl) and θ (CuAl₂) phases and the total thickness of the IMC layer increases in proportion to the square root of the annealing time. Such a relationship is called a parabolic relationship. The parabolic relationship shows that the growth of the IMC layer is controlled by volume diffusion.⁵⁾

On the other hand, the present authors⁶⁾ experimentally examined the growth behavior of the IMC layer at temperatures of 150–270°C (423–543 K). At these temperatures, CA wires were isothermally annealed for various times of 12–960 h (43.2 ks to 3.456 Ms). According to the experimental result,⁶⁾ the IMC layer consists of the α₂ (Cu₃Al), γ₁ (Cu₉Al₄), δ (Cu₃Al₂), η₂ (CuAl) and θ (CuAl₂) phases and the total thickness of the IMC layer is proportional to a power function of the annealing time. However, unlike the observation by Gueydan et al.,⁴⁾ the exponent of the power function is 0.23–0.44. The exponent smaller than 0.5 deduces that boundary diffusion as well as volume diffusion contributes to the layer growth. Furthermore, the Cu solid-solution phase alloyed with Al was recognized between the α₂ phase and the Cu specimen. This alloyed region was produced by diffusion induced recrystallization (DIR). DIR is the phenomenon that new fine grains with discontinuously different solute concentrations form behind moving grain boundaries owing to recrystallization combined with diffusion of solute atoms along the moving and stationary boundaries surrounding the fine grains.⁷⁾ In various binary alloy systems, DIR occurs at temperatures where volume diffusion is frozen out but boundary diffusion occurs practically. The occurrence of DIR in the Cu(Al) system was reported also by Gueydan et al.,⁴⁾ Vandenberg et al.⁸⁾ and den Broeder et al.⁹⁾ Here, the notation A(B) indicates that a solute B diffuses into either a pure metal A or a binary A–B alloy of the A-rich single-phase according to convention.

The electrical conductivity of Cu is decreased by alloying with Al. Furthermore, DIR takes place much faster than the corresponding volume diffusion. As a result, the occurrence of DIR quickly deteriorates the electrical property of the CA wire. Consequently, like the growth behavior of the IMC layer, information on the kinetics of DIR is also important to ensure the reliability. Unfortunately, however, such information is not available. In the present study, the kinetics of DIR in the Cu(Al) system was experimentally examined at temperatures of 210–270°C (483–543 K) using CA wires in a metallographical manner. The experimental result was quantitatively analyzed using a mathematical model. The rate-controlling process of DIR was discussed on the basis of the experimental and analytical results.

2. Experimental

According to the experimental procedures reported in a previous study,⁶⁾ CA wires with Cu and Al volume fractions of roughly 0.15 and 0.85, respectively, were prepared by a wire drawing technique. Here, the diameter was 10 mm and 1.5 mm for the initial CA rod and the final CA wire, respectively. Owing to such wire drawing, Cu and Al were bonded strongly. The CA wires were isothermally annealed in the temperature range of 210–270°C (483–543 K). The annealing time was 192–960 h (0.6912–3.456 Ms) at 210°C (483 K) and 3–192 h (10.8–691.2 ks) at 240 and 270°C (513 and 543 K). Cross-sections of each annealed CA wire were mechanically polished using # 180–1000 emery papers and diamond paste with size of 1 µm. The mechanically polished cross-section was chemically etched with a solution containing ammonia and hydrogen peroxide as the main components. The microstructure of the chemically etched cross-section was observed mainly by scanning electron microscopy (SEM). Transmission electron microscopy (TEM) was partially used for the microstructure observation. A focus ion beam (FIB) technique was utilized to prepare TEM specimens. The chemical composition of each phase in the TEM specimen was measured by energy dispersive spectrometry (EDS). For the EDS measurement, a calibration curve was prepared using various binary Cu–Al alloys with different Al concentrations.

3. Results and Discussion

3.1 Microstructure

Bright filed (BF) images of TEM specimens for the CA wires annealed at T = 513 K (240°C) for t = 10.8 ks (3 h) and at T = 543 K (270°C) for t = 345.6 ks (96 h) are shown in Fig. 1(a) and 1(b), respectively, where the Cu/Al interfaces are located edge on. Here, T is the annealing temperature measured in K, and t is the annealing time measured in s. In Fig. 1, the area with black contrast on the right-hand side is the Cu specimen, and that with white contrast on the left-hand side is the Al specimen. As can be seen, various layers with different gray contrasts are observed between the Cu and Al specimens. For identification of the gray layers, EDS measurements were carried out along the direction normal to the original Cu/Al interface. The result for Fig. 1(b) is shown in Fig. 2. In this figure, the vertical axis indicates the mol fraction y of Al, and the horizontal axis represents the distance x. The concentration profiles along lines A and B in Fig. 1(b) are shown in Fig. 2(a) and 2(b), respectively. According to the result in Fig. 2(a), three of the gray layers are the α₂ (Cu₃Al), δ (Cu₃Al₂) and θ (CuAl₂) phases from the Cu side to the Al side. The γ₁ (Cu₉Al₄) and η₂ (CuAl) phases as well as the α₂, δ and θ phases are recognized for t > 345.6 ks (96 h). In addition to these five compounds, the ζ₂ (Cu₄Al₃) phase is the stable compound at the annealing temperatures of T = 483–543 K (210–270°C) in the binary Al–Cu system.¹⁰⁾ However, the ζ₂ phase was not detected in the present study. On the other hand, in Fig. 2(b), the Cu solid-solution phase alloyed with Al is clearly observed between the Cu specimen and the α₂ phase. This alloyed region is produced by diffusion induced recrystallization (DIR). Hereafter, the region produced by DIR is called the DIR region. In Fig. 2(b), the distance x is measured from the interface between the DIR region and the α₂ phase. Along the Cu/α₂ interface, the DIR region is continuously located in Fig. 1(b) but rather isolated in Fig. 1(a). The continuous DIR region may be produced by connection of neighboring isolated DIR regions during annealing. DIR in the Cu(Al) system was observed also by Gueydan et al.,⁴⁾ Vandenberg et al.⁸⁾ and den Broeder et al.⁹⁾ However, the growth behavior of the DIR region was not reported in their studies.

Fig. 1

Bright field (BF) TEM images of the isothermally annealed CA wires: (a) T = 513 K (240°C) with t = 10.8 ks (3 h) and (b) T = 543 K (270°C) with t = 345.6 ks (96 h).

Fig. 2

The mol fraction y of Al versus the distance x measured by EDS along lines shown in Fig. 1(b): (a) line A and (b) line B.

3.2 Penetration depth of Al in Cu

As shown in Fig. 2(b), the concentration profile of Al across the DIR region was measured by EDS. According to the result in Fig. 2(b), the DIR region grows to a thickness of about 1 µm due to annealing at T = 543 K (270°C) for t = 345.6 ks (96 h). Furthermore, the mol fraction y of Al discontinuously varies across the boundary between the DIR region and the untransformed Cu phase. On the other hand, in the DIR region, the concentration of Al gradually decreases from y = 0.129 to y = 0.044 with increasing distance from x = 0.1 µm to x = 1.1 µm. As previously mentioned, x is measured from the interface between the α₂ phase and the DIR region.

Let us consider the interdiffusion of Al and Cu due to volume diffusion in the Cu phase with the interdiffusion coefficient D and the molar volume V_m. If both D and V_m are constant and independent of the composition at a constant annealing temperature, y is expressed as an explicit function of x and t by the following equation for this interdiffusion:¹¹⁾   

\begin{equation} y = y_{0} + (y_{\text{c}} - y_{0})\left\{1 - \text{erf}\left(\frac{x}{2\sqrt{Dt}} \right) \right\}. \end{equation} (1)

Here, y₀ is the initial concentration of Al in the Cu phase, and y_c is the concentration of Al in the Cu phase at the Cu/α₂ interface. In eq. (1), x is measured from the Cu/α₂ interface and positive towards the Cu phase.

When y, y₀ and y_c are much smaller than unity, D is close to the tracer diffusion coefficient $D_{\text{Al}}^{\text{Cu}}$ of Al in Cu for y = 0.^12,13) The dependence of $D_{\text{Al}}^{\text{Cu}}$ on T is expressed by the equation   

\begin{equation} D_{\text{Al}}^{\text{Cu}} = D_{\text{Al0}}^{\text{Cu}} \exp \left(- \frac{Q}{RT} \right). \end{equation} (2)

Here, $D_{\text{Al0}}^{\text{Cu}}$ is the pre-exponential factor, Q is the activation enthalpy, and R is the gas constant. The following values are reported for $D_{\text{Al}}^{\text{Cu}}$: $D_{\text{Al0}}^{\text{Cu}} = 1.31 \times 10^{ - 5}$ m²/s and Q = 185 kJ/mol.¹⁴⁾ Using these parameters, y was calculated as a function of x at T = 543 K (270°C) for t = 345.6 ks (96 h). The result is shown as a solid curve in Fig. 2(b). The solid curve indicates very steep slope almost along the DIR/α₂ interface. According to eq. (1), the penetration depth d of Al in the Cu phase is roughly estimated as follows:   

\begin{equation} d = 2\sqrt{Dt}. \end{equation} (3)

From eq. (3), d = 5.4 nm is obtained at T = 543 K (270°C) for t = 345.6 ks (96 h). On the other hand, the thickness of the DIR region in Fig. 2(b) is greater than 1 µm. Thus, the thickness is almost two hundred times greater than the penetration depth. As a consequence, alloying of Cu with Al takes place considerably fast due to DIR under the present annealing conditions.

3.3 Growth behavior of DIR region

As mentioned in Section 2, SEM was mainly used for the microstructure observation. From SEM images of the annealed CA wires, the mean thickness l of the DIR region was estimated by the following equation.⁶⁾   

\begin{equation} l = \frac{A}{w} \end{equation} (4)

Here, A and w are the total area and the total length of the DIR region, respectively, on the cross-section. The result is shown in Fig. 3. In this figure, the vertical axis indicates the thickness l, and the horizontal axis represents the annealing time t. Both axes are shown in logarithmic scales, and open rhombuses, squares and circles indicate the results of T = 483, 513 and 543 K (210, 240 and 270°C), respectively. As can be seen in Fig. 3, the thickness l monotonically increases with increasing annealing time t at all the annealing temperatures. The higher the annealing temperature is, the faster the DIR region grows. If we omit the experimental points for the shortest annealing time of t = 10.8 ks (3 h) at T = 513 and 543 K (240 and 270°C), the open symbols for each annealing temperature lie well on a straight line. The reason of this omission will be discussed later on. Thus, l is expressed as a power function of t by the equation   

\begin{equation} l = k\left(\frac{t}{t_{0}} \right)^{n}, \end{equation} (5)

where t₀ is unit time, 1 s, which is adopted to make the argument t/t₀ of the power function dimensionless. The proportionality coefficient k has the same dimension as that of the thickness l, and the exponent n is dimensionless. Using the experimental points excluding the ones with t = 10.8 ks (3 h) in Fig. 3, k and n in eq. (5) were determined by the least-squares method, as indicated by dotted, dashed and solid straight-lines for T = 483, 513 and 543 K (210, 240 and 270°C), respectively. According to these straight-lines, the exponent takes values of n = 0.52, 0.52 and 0.66 for T = 483, 513 and 543 K (210, 240 and 270°C), respectively.

Fig. 3

The mean thickness l of the DIR region versus the annealing time t shown as open rhombuses, squares and circles for T = 483, 513 and 543 K (210, 240 and 270°C), respectively.

The rate-controlling process for the growth of the IMC layer in the CA wire was extensively discussed in a previous study.⁶⁾ If the IMC growth is controlled by volume diffusion, n = 0.5.⁵⁾ In contrast, at low temperatures where volume diffusion is negligible, boundary diffusion can control the IMC growth. In such a case, n is smaller than 0.5 on condition that grain growth occurs in the IMC layer.^15,16) In the case of DIR, however, boundary diffusion is the only way to form the DIR region. Thus, at first glance, we may expect that n is smaller than 0.5 also for DIR. Nevertheless, as shown in Fig. 3, n is slightly greater than 0.5. Therefore, unlike the IMC growth, the value of n cannot be used to estimate the rate-controlling process in a straightforward manner.

3.4 Kinetic analysis

In a previous study,¹⁷⁾ a new extended (NE) model was proposed to describe mathematically the growth rate of the DIR region in the A(B) system. As mentioned earlier, the notation A(B) indicates that a solute B diffuses into either a pure metal A or a binary A–B alloy of the A-rich single-phase according to convention. For DIR in the Cu(Al) system, Cu and Al are elements A and B, respectively. The NE model will be explained briefly below.

If one mol of the DIR region with composition y_e is produced from (y_e − y₀)/(1 − y₀) mol of element B and (1 − y_e)/(1 − y₀) mol of the solid-solution α phase of element A with composition y₀, y_e is expressed as a function of l, s, r, δ, D^b and y_i by the following equation according to the NE model.¹⁷⁾   

\begin{equation} y_{\text{e}} = \frac{y_{0}slr + y_{\text{i}}2\delta D^{\text{b}}}{slr + 2\delta D^{\text{b}}} \end{equation} (6)

Such a reaction is schematically depicted in Fig. 4,¹⁷⁾ where the α and β phases correspond to the untransformed Cu phase and the α₂ phase, respectively, in the Cu(Al) system. In eq. (6), y_i is the initial value of y_e for l ≈ 0 m, s is the diameter of crystal grain in the DIR region, r is the migration rate of the moving boundary of the DIR region, D^b is the boundary diffusion coefficient of element B, and δ is the thickness of the grain boundary. In a manner similar to eq. (2), D^b is expressed as a function of T as follows   

\begin{equation} D^{\text{b}} = D_{0}^{\text{b}} \exp \left(- \frac{Q^{\text{b}}}{RT}\right). \end{equation} (7)

Here, $D_{0}^{\text{b}}$ is the pre-exponential factor, and Q^b is the activation enthalpy. In contrast, the migration rate r is determined by the effective driving force Δ^efG acting on the moving boundary and the mobility M of the moving boundary as follows   

\begin{equation} r = \frac{\mathrm{d}l}{\mathrm{d}t} = M\Delta^{\text{ef}}G, \end{equation} (8)

where Δ^efG is expressed by the following equation.¹⁷⁾   

\begin{align} \Delta^{\text{ef}}G &= (1 - y_{\text{e}})\left\{\frac{RT}{V_{\text{m}}}\left(\ln \frac{1 - y_{\text{nf}}}{1 - y_{\text{e}}} + \frac{y_{0}}{1 - y_{0}}\ln \frac{y_{\text{nf}}}{y_{\text{e}}} \right) \right.\\ &\qquad \qquad \left. - Y\eta^{2}\frac{(y_{\text{nf}} - y_{0})^{2}}{1 - y_{0}}\right\} \end{align} (9)

Here, y_nf is the composition of the penetration zone at the interatomic distance λ from the moving boundary, Y is the biaxial elastic modulus of the untransformed matrix along the plane parallel to the moving boundary, and η is the misfit parameter. For the α phase with elastic isotropy, Y is expressed by the equation   

\begin{equation} Y = \frac{E}{1 - \nu}. \end{equation} (10)

Here, E is the Young’s modulus, and ν is the Poisson’s ratio. The composition y_nf in eq. (9) is approximately described as a function of y_pf, y₀, r and λ by the following equation.¹⁷⁾   

\begin{equation} y_{\text{nf}} = y_{0} + (y_{\text{pf}} - y_{0})\exp \left(- \frac{r\lambda}{D} \right) \end{equation} (11)

Here, D is the diffusion coefficient for volume diffusion of element B in the penetration zone, and y_pf is the composition in the penetration zone at the interface between the untransformed matrix and the moving boundary. When the concentration gradient of element B across the moving boundary is negligible, y_pf is evaluated by the parallel-tangent construction (PTC) method¹⁸⁾ as follows:   

\begin{equation} \ln \frac{y_{\text{pf}}}{1 - y_{\text{pf}}} + \frac{2YV_{\text{m}}\eta^{2}}{RT}(y_{\text{pf}} - y_{0}) = \ln \frac{y_{\text{e}}}{1 - y_{\text{e}}}. \end{equation} (12)

As to the parameters in eqs. (10)–(12), the following values were reported for the Cu(Al) system: E = 129.8 GPa and ν = 0.343;¹⁹⁾ V_m = 7.1106 × 10⁻⁶ m³/mol¹⁴⁾ and η = 0.070160;²⁰⁾ and λ = 0.3 nm.¹⁷⁾ Furthermore, the following parameters were reported for boundary diffusion of Cu in Cu: $\delta D_{0}^{\text{b}} = 1.31 \times 10^{ - 14}$ m³/s and Q^b = 102.1 kJ/mol.^21,22) Unfortunately, however, the corresponding information on boundary diffusion of Al in Cu is not available. As a result, we used these parameters^21,22) for boundary diffusion of Al in Cu. On the other hand, we consider that the composition y_i of the DIR region initially formed at the Cu/α₂ interface coincides with that y_c of the Cu/(Cu + α₂) phase boundary in the binary Cu–Al system. According to a phase diagram in the binary Cu–Al system,¹⁰⁾ y_c = 0.185 and 0.197 at T = 573 and 636 K (300 and 363°C), respectively. However, no reliable information on the phase equilibria is available at T < 573 K (300°C). Hence, assuming the linear dependence of y_c on T, we obtain y_c = 0.168, 0.174 and 0.179 at T = 483, 513 and 543 K (210, 240 and 270°C), respectively.

Fig. 4

Schematic of DIR region in the A(B) system.¹⁷⁾

Combining the value y₀ = 0 with the parameters mentioned above, we numerically calculated the thickness l_i at each experimental annealing time t_i from eqs. (6)–(12). For this calculation, the mobility M was chosen as the fitting parameter to minimize the function f defined as   

\begin{equation} f = \sum_{i = 1}^{p} (l_{i} - l_{i}^{\text{e}})^{2}, \end{equation} (13)

where $l_{i}^{\text{e}}$ is the experimental value of l at t = t_i, and p = 3, 5 and 5 at T = 483, 513 and 543 K (210, 240 and 270°C), respectively. From all the open symbols in Fig. 3, M = 8.66 × 10⁻²¹, 5.27 × 10⁻²⁰ and 1.89 × 10⁻¹⁹ m⁴/Js were obtained at T = 483, 513 and 543 K (210, 240 and 270°C), respectively. Using these values of M, l was calculated as a function of t from eqs. (6)–(12). The results of T = 483, 513 and 543 K (210, 240 and 270°C) are shown as solid curves in Fig. 5(a), 5(b) and 5(c), respectively. Furthermore, the open rhombuses, squares and circles in Fig. 3 are represented as open circles in Fig. 5(a), 5(b) and 5(c), respectively. At first glance, for T = 543 K (270°C) in Fig. 5(c), the open circle for the shortest annealing time of t = 10.8 ks (3 h) seems to deviate rather remarkably from the solid curve. As previously mentioned, the minimization for the summation of the differences between l_i and $l_{i}^{\text{e}}$ was conducted using the function f defined by eq. (13). In contrast, in Fig. 5, the vertical axis shows the logarithm of l. Thus, the deviation from the solid curve is visually exaggerated for a small absolute value of l at the shortest annealing time in Fig. 5(c). As can be seen in Fig. 5, the dependence of l on t is satisfactorily reproduced by the calculation at each annealing temperature. Consequently, the NE model is able to explain quantitatively the kinetics of DIR in the Cu(Al) system.

Fig. 5

The annealing time dependencies of the thickness l of the DIR region with solid curves numerically calculated from eqs. (6)–(12) at (a) 483 K (210°C), (b) 513 K (240°C) and (c) 543 K (270°C). The experimental results in Fig. 3 are shown again as open circles at each annealing temperature. The solid curves are extrapolated as dashed lines in the early and late stages.

3.5 Rate-controlling process

The calculation with eqs. (6)–(12) in Fig. 5 was made for annealing times up to t = 10⁸ s to discuss the rate-controlling process of DIR. The limit t = 10⁸ s is more than one order of magnitude greater than the longest experimental annealing time of t = 3.456 Ms (960 h). On the other hand, the dependence of d on t was approximately evaluated from eq. (3). The results of T = 483, 513 and 543 K (210, 240 and 270°C) are shown as dotted lines in Fig. 5(a), 5(b) and 5(c), respectively. As can be seen, l is almost more than two orders of magnitude greater than d at each experimental annealing time. In eq. (5), l is expressed as a power function of t. If the interface reaction at the moving boundary controls the growth of the DIR region, l is linearly in proportion to t and hence n = 1.¹⁷⁾ In Fig. 5, this corresponds to the early stages. For this type of rate-controlling process, we obtain the following equation from eq. (5) with n = 1:   

\begin{equation} l = \frac{k}{t_{0}}t = Kt, \end{equation} (14)

where K is the kinetic coefficient. Since K = k/t₀, the dimension of K is m/s. From the solid curves in Fig. 5, K = 7.49 × 10⁻¹³, 4.99 × 10⁻¹² and 1.95 × 10⁻¹¹ m/s are obtained at T = 483, 513 and 543 K (210, 240 and 270°C), respectively. As the annealing time increases up to t = 10⁸ s, the exponent monotonically decreases from n = 1 to n = 0.517, 0.505 and 0.493 at T = 483, 513 and 543 K (210, 240 and 270°C), respectively. The annealing time t = 10⁸ s homologizes the late stages. As a consequence, the transition stages appear between the early and late stages. Such stages of the rate-controlling process were extensively discussed in a previous study.¹⁷⁾ In Fig. 5, straight dashed lines represent the extrapolations of each solid curve in the early and late stages. At T = 483 K (210°C) in Fig. 5(a), all the experimental annealing times belong to the transition stages. In contrast, at T = 513 and 543 K (240 and 270°C) in Fig. 5(b) and 5(c), respectively, the shortest annealing time of t = 10.8 ks (3 h) lies in the early stages, but the other annealing times are located in the transition stages. Therefore, we conclude that the growth of the DIR region is mainly controlled by both the interface reaction and the boundary diffusion under the present annealing conditions. For the shortest annealing time mentioned above, however, the interface reaction is the rate-controlling process for the DIR growth. As a result, the open square and circle for t = 10.8 ks (3 h) rather remarkably deviate from the dashed and solid lines, respectively, in Fig. 3. This is the reason why these open square and circle were omitted for the evaluation of k and n in eq. (5).

The values of M shown in Fig. 5 are plotted against the annealing temperature T as open circles in Fig. 6. In this figure, the vertical axis shows the logarithm of M, and the horizontal axis indicates the reciprocal of T. Since the open circles lie well on a straight line, M is expressed as a function of T by the following equation of the same formula as eqs. (2) and (7):   

\begin{equation} M = M_{0}\exp \left(- \frac{Q_{M}}{RT} \right). \end{equation} (15)

From the open circles in Fig. 6, the pre-exponential factor and the activation enthalpy in eq. (15) were determined to be M₀ = 1.30 × 10⁻⁸ m⁴/Js and Q_M = 112 kJ/mol, respectively, by the least-squares method as shown with a solid line.

Fig. 6

The mobility M for the moving boundary of the DIR region versus the reciprocal of the annealing temperature T shown as open circles with a solid line. A dashed line passing through open squares indicates the annealing temperature dependence of the kinetic coefficient K.

In Fig. 6, the values of K indicated in Fig. 5 are also represented as open squares. Like the open circles for M, the open squares for K are located well on a straight line. Hence, K is described as a function of T by the equation   

\begin{equation} K = K_{0}\exp \left(- \frac{Q_{K}}{RT} \right), \end{equation} (16)

where K₀ = 5.44 m/s and Q_K = 119 kJ/mol. The value Q_K = 119 kJ/mol is rather close to Q^b = 102.1 kJ/mol but much smaller than Q = 185 kJ/mol. For the moving boundary of the DIR region, boundary diffusion occurs not only along the boundary but also across the boundary. The activation enthalpy for the boundary diffusion may not be so dissimilar to each other between along and across the moving boundary. Since Q_K is close to Q^b, the boundary diffusion across the moving boundary may be the most predominant process of the interface reaction. Furthermore, Q_M = 112 kJ/mol is close to Q_K = 119 kJ/mol. This means that the mobility M is correlated to the interface reaction at the moving boundary and thus to the boundary diffusion across the moving boundary.

The open circles with the solid line in Fig. 6 are shown again in Fig. 7. In contrast, the values of M for DIR in the Cu(Ni) system reported in a previous study²³⁾ are represented as open triangles with a dotted line of M₀ = 1.15 × 10⁻⁷ m⁴/Js and Q_M = 168 kJ/mol in Fig. 7. Furthermore, in this figure, the corresponding results for diffusion induced grain-boundary migration (DIGM) in the Cu(Zn) system^24–26) are indicated as different open symbols with a dashed line of M₀ = 1.24 × 10⁻³ m⁴/Js and Q_M = 177 kJ/mol. Here, DIGM is the phenomenon that a region with different composition is left behind a moving boundary due to grain boundary migration combined with the diffusion of solute atoms along the moving boundary. Although M is smaller for DIR in the Cu(Ni) system than for DIGM in the Cu(Zn) system, it is close to each other between DIR in the Cu(Al) system and DIGM in the Cu(Zn) system. As previously mentioned, no reliable information on boundary diffusion of Al in Cu is available. Thus, the values of $\delta D_{0}^{\text{b}}$ and Q^b for boundary diffusion of Cu in Cu^21,22) were adopted in the present study. To discuss the similarity of M between DIR in the Cu(Al) system and DIGM in the Cu(Zn) system, information on boundary diffusion of Al in Cu is important. Such discussion is valid on condition that this information becomes available.

Fig. 7

The values of M in Fig. 6 are shown again as open circles with a solid line. Open triangles indicate the corresponding result of DIR in the Cu(Ni) system,²³⁾ and different open symbols show those of DIGM in the Cu(Zn) system.^24–26)

4. Conclusions

The kinetics of diffusion induced recrystallization (DIR) in the Cu(Al) system was experimentally examined using the Cu-clad Al (CA) wires. The CA wires were isothermally annealed at temperatures of 483–543 K (210–270°C) for various times between 10.8 ks and 3.456 Ms (3 and 960 h). During annealing, the compound layer consisting of the α₂, γ₁, δ, η₂ and θ phases is produced at the original Cu/Al interface in the CA wire. Furthermore, the DIR region alloyed with Al forms in the Cu phase at the Cu/α₂ interface and grows towards the Cu phase. The concentration of Al discontinuously varies across the moving boundary between the DIR region and the Cu phase. The thickness l of the DIR region is expressed as a power function of the annealing time t within t = 43.2 ks (12 h) and t = 3.456 Ms (960 h). On the basis of the experimental result, the rate-controlling process of DIR was theoretically analyzed using the new extended-model proposed in a previous study.¹⁷⁾ According to the analysis, l is linearly in proportion to t in the early stages but to a power function of t in the late stages. Thus, the interface reaction at the moving boundary is the rate-controlling process in the early stages, but the boundary diffusion across the DIR region is that in the late stages. The transition stages appear between the early and late stages, and the experimental annealing times within t = 43.2 ks (12 h) and t = 3.456 Ms (960 h) belong to the transition stages. As a consequence, under the present experimental conditions, both the interface reaction and the boundary diffusion mainly contribute to the rate controlling process of DIR. On the other hand, the shortest annealing time of t = 10.8 ks (3 h) belongs to the early stages, where the interface reaction is the rate-controlling process.

REFERENCES

• 1)   K.  Oba: Fujikura Tech. J. 42 (2013) 77–80.
• 2)   K.  Taguchi,  T.  Shimada,  J.  Yoshimoto,  T.  Kuwabara and  Y.  Akasofu: SEI Tech. Rev. 84 (2017) 125–130.
• 3)   S.  Koch and  H.  Antrekowitsch: BHM Berg. 152 (2007) 62–67.
• 4)   A.  Gueydan,  B.  Domenges and  E.  Hug: Intermetallics 50 (2014) 34–42.
• 5)   M.  Kajihara: Acta Mater. 52 (2004) 1193–1200.
• 6)   T.  Kizaki,  M.  O and  M.  Kajihara: Mater. Trans. 60 (2019) 895–901.
• 7)   M.  Kajihara and  W.  Gust: Acta Metall. Mater. 39 (1991) 2565–2574.
• 8)   J.M.  Vandenberg,  F.J.A.  den Broeder and  R.A.  Hamm: Thin Solid Films 93 (1982) 277–280.
• 9)   F.J.A.  den Broeder,  M.  Klerk,  J.M.  Vandenberg and  R.A.  Hamm: Acta Metall. 31 (1983) 285–291.
• 10)  T.B. Massalski, H. Okamoto, P.R. Subramanian and L. Kacprzak: Binary Alloy Phase Diagrams, Vol. 1, (ASM International, Materials Park, OH, 1990) p. 142.
• 11)  P.G. Shewmon: Diffusion in Solids, (McGraw-Hill, New York, 1963) p. 14.
• 12)   L.S.  Darken: Trans. AIME 175 (1948) 184–201.
• 13)   G.S.  Hartley: Trans. Faraday Soc. 27 (1931) 10–29.
• 14)  Japan Inst. Metals: Metals Data Book, (Maruzen, Tokyo, 1993) p. 21.
• 15)   Y.L.  Corcoran,  A.H.  King,  N.  Lanerolle and  B.  Kim: J. Electron. Mater. 19 (1990) 1177–1183.
• 16)   A.  Furuto and  M.  Kajihara: Mater. Trans. 49 (2008) 294–303.
• 17)   Y.  Yamamoto and  M.  Kajihara: Mater. Trans. 42 (2001) 1763–1770.
• 18)  M. Hillert: Lectures on the Theory of Phase Transformations, 2nd ed., ed. by H.I. Aaronson, (TMS-AIME, Warrendale, PA, 1999) p. 1.
• 19)  S. Nishikawa: Introduction to Metallurgy, (AGNE, Tokyo, 2001) p. 530.
• 20)   N.  Kuwano,  I.  Ogata and  T.  Eguchi: J. Japan Inst. Metals 39 (1975) 609–614.
• 21)   A.  Hässner: Kristall und Technik 9 (1974) 1371–1388.
• 22)  I. Kaur, W. Gust and L. Kozma: Handbook of Grain and Interphase Boundary Diffusion Data, Vol. 1, (Ziegler Press, Stuttgart, 1989) p. 390.
• 23)   Y.  Yamamoto and  M.  Kajihara: J. Electron. Mater. 37 (2008) 1710–1720.
• 24)   Y.  Yamamoto and  M.  Kajihara: Acta Mater. 47 (1999) 1195–1201.
• 25)   Y.  Yamamoto,  M.  Moriyama,  M.  Kajihara and  T.  Mori: Acta Mater. 47 (1999) 1757–1766.
• 26)   N.  Goukon,  T.  Ikeda and  M.  Kajihara: Acta Mater. 48 (2000) 1551–1562.